RACE CAR SUSPENSION OPTIMIZATION

Jacobs School of Engineering University of California San Diego

Abstract

Shock dampers are an important factor in race car set up and they should be tuned to provide optimum handling and maximize tire contact In order to achieve this a shock dynamometer was used to characterize and tune the 4-way adjustable Cane Creek Double Barrel damper according to theoretical calculations from the Quarter Car Model UCSDs Society of Automotive Engineers has been operating without the use of a fully functional dynamometer and with the rise of this project the SAE team took advantage of the opportunity The suspension system for the SAE car was characterized and utilized to develop ideal settings for maximum

performance

Mechanical Engineering Laboratory Course ndash 171B Spring Quarter 2008 Group B7 Neal Bloom

Isaiah Freerkson Norman Molina Vinh Nguyen

TABLE OF CONTENTS -

PAGE

LIST OF TABLES iii

LIST OF FIGURES iv

INTRODUCTION 1

THEORY

Quarter Car Model 2

Shock Damper 3

Matlab Simulink Quarter Car Simulation 3

EXPERIMENTAL SETUP

Shock Dynamomemeter 5

Transducers - Loadcell LDVT LVT 5

Cane Creek Double Barrel Shock Absorber 6

Labview ndash Data Acquisition 6

Matlab ndash Data Processing 7

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration 8

Shock Characterization 8

EXPERIMENTAL RESULTS

Sensor Equipment Calibration 10

Shock Characterization 10

Simulink Model 12

DISCUSSION

Sensor Equipment Calibration 14

Shock Characterization 14

Simulink Model 15

ERROR ANALYSIS 16

CONCLUSION 17

REFERENCES 18

TABLE OF CONTENTS -

APPENDICES

APPENDIX ldquoArdquo ndash Measurement Data amp Sensor Calibration Figures

APPENDIX ldquoBrdquo ndash Matlab Simulink Figures amp Data

APPENDIX ldquoCrdquo ndash Matlab Code

LIST OF TABLES

PAGE

TABLE 1 ndash Damper Tuning Ratios 4

TABLE 2 ndash Summary of Results for Calibration Curves 10

TABLE 3 ndash Minimum Damping Coefficients ndash Measured 12

TABLE 4 ndash Maximum Damping Coefficients ndash Measured 12

TABLE 5 ndash Minimum Damping Coefficients ndash Cane Creek Data 12

TABLE 5 ndash Minimum Damping Coefficients ndash Cane Creek Data 12

INTRODUCTION

PAGE

FIGURE 1 ndash Quarter car model of a suspension system 2

FIGURE 2 ndash Reduced Quarter Car Model 2

FIGURE 3 ndash 4-way Damping Switch 4

FIGURE 4 ndash LVDT Displacement Transducer 5

FIGURE 5 ndash LVT Velocity Transducer 5

FIGURE 6 ndash Cane Creek Double Barrel cutout view 6

FIGURE 7 ndash Can Creek Double Barrel external adjustment nuts 6

FIGURE 8 ndash Virtual Instrument (VI) Block Diagram 7

FIGURE 9 ndash Virtual Instrument Data Acquisition Screen 7

FIGURE 10 ndash Load cell Calibration Curve ndash Load vs Voltage 10

FIGURE 11 ndash Double Barrel Shock Response ndash All Low and All High Adjustments 11

FIGURE 12 ndash Force vs Velocity ndash Low Speed Rebound 12

FIGURE 13 ndash Single Damping Coefficient ndash Car Body Position vs Time 12

FIGURE 14 ndash 4-way damping ndash Car Body Position vs Time 13

INTRODUCTION

Formula SAEreg competition is an annual collegiate competition for SAE student members to design fabricate and compete with small formula-one style racing cars The student organization SAE at the University of California San Diego is planning on competing in the Formula SAE collegiate competition in June of 2008 The Formula SAE competition is highly competitive therefore it is important to have a well set-up race car Having a well set-up race car allows the driver to utilize its full potential as opposed to having to fight a poorly set-up car around the race track Furthermore having a well set-up car allows it to be more predictable around the race track

Part of having a well set-up car is having an optimized suspension system In doing so requires choosing a proper spring and damper for the application A wide range of dampers are available and for the UCSD SAErsquos racecar the Cane Creek Double Barrel was chosen This is a 4-way adjustable damper that supports adjustability for high and low speed damping for bump and rebound For previous years UCSD SAE has taken a more empirical approach to tuning the damper where adjustments were made according to the driverrsquos feedback at the race track The intent of this experiment is to take a more methodical approach to tuning the suspension system an approach that identifies the theory behind the controls of a suspension system along with theoretical results that can be empirically verified

This experiment sponsored by UCSD SAE will investigate optimization of the vehicle dynamics of the SAE race car This involves characterizing frontrear wheel frequencies spring stiffness and the dampedun-damped masses of the car The dampers will be tuned to the theoretical characteristics of the damping system and after this is completed the suspension will be analyzed and tested on the shock dynamometer This process will optimize the performance of the SAE car vehicle dynamics and the benefits will be realized in the FSAE collegiate competition in June of 2008

THEORY

Figure 1 ndash Quarter-car model of a suspension system

mu

ms

ks

kt

bs

xs

xu

u(t)

Quarter Car Model

The quarter car model is a simplification of the carrsquos suspension system consisting of four shock assemblies down to one Although it does not fully represent the physics of the carrsquos suspension system it can be a useful tool for understanding the workings of the suspension1 Figure 1 shows a diagram of the quarter car model The model consists of the following parameters

Applying Newtonrsquos Second Law the governing equations of motion for the sprung mass and unsprung mass are

To further simplify the equations of motion it is assumed that multltmsand ksltltkt The high stiffness of the tire spring rate combined with the lower mass of the unsprung weight limits the movement of mu and reduces equation (2) to zero The equation of motion for the sprung mass is then This equation simply states that the acceleration of the sprung mass is the sum of the spring rate times the movement of the damper and the damping coefficient times the damper velocity The reduced quarter car model (simple spring-mass-damper system) is shown in Figure2 Applying a Laplace transform to equation (3) to transform it from the time-domain to the frequency domain results in equation (4)

ms = sprung mass (weight of the car) (lb) mu = unsprung mass (wheel and suspension

components) (lb) ks = sprung mass spring rate (lbin) kt = tire spring rate (lbin) bs = sprung mass damping (lbsin) u(t) = displacement of the road as a function of

time

( ) ( ) ( )u u s s u s s u t um x b x x k x x k u x= minus + minus + minusampamp amp amp

( ) ( )s s s u s s u sm x b x x k x x= minus + minusampamp amp amp

s s s s s sm x k x b x= minus minusampamp amp

Eq (1) Eq (2)

Eq (3)

2 0s s

s s

b ks s

m m+ sdot + = Eq (4)

Eq (5)

2 22 0n ns sζ ω ω+ sdot sdot + = Figure 2 ndash Reduced Quarter Car model

THEORY

Comparing equation (4) to the ideal second order system equation (5) where ωnis the natural frequency and ζ is the damping ratio It can be seen that

The damping coefficient is chosen by specifying the damping ratio and the natural frequency is the frequency at which the spring mass system oscillates without the damper With the simplifying assumptions it is expected that the actual values will be lower than the theoretical values

Shock Damper

The shock damper is a critical component of the carrsquos suspension system because it helps control the car chassisrsquo response going into and out of turns (pitch and roll) as well as its response to variations in the road (bouncy or bumpy ride) Without a shock damper the spring would cause the suspension system to oscillate over an extended period of time as it compresses stores mechanical energy and then releases it past its neutral state to a point of extension and then it would continue to oscillate over an extended period of time until all of the stored energy has dissipated The shock damper dissipates energy stored in the spring usually in the form of heat In addition to controlling the carrsquos response in turns the damper also plays a vital role in controlling the temperature of the tires1 In automobile racing controlling tire load variation and maintaining an optimum temperature for the tires is very important as it increases the carrsquos mechanical grip The damping ratio varies and is tuned according to desired handling characteristics and track conditions These ratios are specified in Table 1 LS and HS refers to the damper shaft velocity where LS is low-speed and HS is high-speed Table 1- Damper Tuning Ratios1

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio

is 05 If the circuit is smooth choose 07 If the tires need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where

body control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Matlab Simulink Quarter Car Simulation

Simulink is a graphical approach to a Matlab environment it allows the user to simulate a real word environment such as a quarter car system The Simulink model is used to calculate specific values that are required to tune the SAE cars suspension

To begin the model requires specifications of constants (model parametersmeasured values) needed to produce the equations of motion These constants are placed in two summing blocks that formulate the two equations of motion for the sprung and unsprung masses These equations

sn

s

k

mω =2s n sb mω ζ= sdot sdot sdot Eq (6)

Eq (7)

THEORY

are shown below (Eg (8) and Eq (9)) After equations of motion are placed in the form that solves for the two accelerations of the un-sprung mass (mu)and the sprung mass (ms) the equations are then placed through two integrators the first representing the velocity of (mu)and (ms) and the second producing an equation representing the displacement of (mu)and (ms) These values of acceleration velocity and displacement for mu and ms are then fed back into the controlling parts of the equations of motion This feedback loop allows for continuous control of the damping coefficients

It was imperative for the model to simulate real world performance of the double barrel shock absorber In order to assure a real world simulation the damping coefficients for rebound and compression would have to be determined To determine the system was in rebound or compression the use of a detect increase and detect decrease block respectively These two blocks use the displacement feedback which was mention previously to sense the direction of the simulated damper shaft After each displacement block there resides a switch (1) that is turned on to feed the input of the velocity into another set of switches This switch (1) on the rebound and compression is to engage that particular damping regime Once that switch is on it feeds in the velocity of the damper shaft onto an array of another two switches These two switches switch (2) and switch (3) are responsible for delivering the damping coefficients to the driving part of the system These two switches are tuned to be active at different shaft speeds Switch (2) is on only when switch (1) is on switch (3) is activated when the shaft speed reaches a pre determined limit In this system only the specific damping coefficients that are need for a particular regime are activated this allows for a realistic system in a simulated environment This damping coefficient system can is shown in Figure 3

At the end of each simulation there are specific outputs placed in the Matlab workspace environment Once these outputs are in the workspace there can be manipulate to show the response to the inputs This allows the user to determine to performance of the frac14 car model in a simulated environment

Figure 3 ndash 4-way Damping Switch

( ) ( ) 0s s s s u s s u sm x b x x k x x m g+ minus + minus minus =ampamp amp amp

Eq (8) Eq (9)

( ) ( ) ( ) 0u u s u s s u s t u um x b x x k x x k x q m g+ minus + minus + minus minus =ampamp amp amp

EXPERIMENTAL SETUP

Shock Dynamometer

The Aurora V8 dynamometer itself was donated in non-working condition by a racing specialist Wayne Mitchell to help in the project and the optimization of the shock absorber for the race car A shock dynamometer is a machine used to test and measure force velocity and displacement for shock dampers

Transducers - Load Cell LVDT LVT

Looking in depth into the dynamometer there are three main componentssensors that make up the dynamometer These are the Linear Variable Differential Transformer (LVDT) Linear Velocity Transducer (LVT) and the load cell The load cell sensor is an Omega S-type Model that has a range of 0 to 2000 pound force (lbf) and measured in both compression and tension The LVDT (245 Model) and LVT (110 Model) are manufactured by Trans-Tek a specialist in transducers In particular the LVDT sensor measures linear displacement The transducer itself is a core within a cylindrical tube and contains primary and secondary magnetic coils in which change in voltage output is proportional to the movement of the core The LVT sensor on the other hand measures the velocity at which the motor is running The LVT is similar to the LVDT in terms of coils inside the cylindrical tube The difference being that no external excitation voltage is needed a magnet is passed between the coils and results in a DC voltage output proportional to the instantaneous velocity Again an amplifying circuit board and a data acquisition board were needed to read the voltage outputs out of all three sensors Software used for data logging was National Instruments Labview and MATLAB

Figure 4 ndash LVDT Displacement Transducer Figure 5 ndash LVT Velocity Transducer

Cane Creek Double Barrel Shock Absorber2

The shocks that were acquired for the SAE race car was the Cane Creek Double Barrel shock absorber manufactured by Cane Creek using the technology from the European company Ohlins The double barrel design allows the shock to have four independent external adjustments meaning each adjustment contributes independently to shock response The four independent adjustments consist of high and low speed compression and rebound The traditional shock

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

TABLE OF CONTENTS -

PAGE

LIST OF TABLES iii

LIST OF FIGURES iv

INTRODUCTION 1

THEORY

Quarter Car Model 2

Shock Damper 3

Matlab Simulink Quarter Car Simulation 3

EXPERIMENTAL SETUP

Shock Dynamomemeter 5

Transducers - Loadcell LDVT LVT 5

Cane Creek Double Barrel Shock Absorber 6

Labview ndash Data Acquisition 6

Matlab ndash Data Processing 7

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration 8

Shock Characterization 8

EXPERIMENTAL RESULTS

Sensor Equipment Calibration 10

Shock Characterization 10

Simulink Model 12

DISCUSSION

Sensor Equipment Calibration 14

Shock Characterization 14

Simulink Model 15

ERROR ANALYSIS 16

CONCLUSION 17

REFERENCES 18

TABLE OF CONTENTS -

APPENDICES

APPENDIX ldquoArdquo ndash Measurement Data amp Sensor Calibration Figures

APPENDIX ldquoBrdquo ndash Matlab Simulink Figures amp Data

APPENDIX ldquoCrdquo ndash Matlab Code

LIST OF TABLES

PAGE

TABLE 1 ndash Damper Tuning Ratios 4

TABLE 2 ndash Summary of Results for Calibration Curves 10

TABLE 3 ndash Minimum Damping Coefficients ndash Measured 12

TABLE 4 ndash Maximum Damping Coefficients ndash Measured 12

TABLE 5 ndash Minimum Damping Coefficients ndash Cane Creek Data 12

TABLE 5 ndash Minimum Damping Coefficients ndash Cane Creek Data 12

INTRODUCTION

PAGE

FIGURE 1 ndash Quarter car model of a suspension system 2

FIGURE 2 ndash Reduced Quarter Car Model 2

FIGURE 3 ndash 4-way Damping Switch 4

FIGURE 4 ndash LVDT Displacement Transducer 5

FIGURE 5 ndash LVT Velocity Transducer 5

FIGURE 6 ndash Cane Creek Double Barrel cutout view 6

FIGURE 7 ndash Can Creek Double Barrel external adjustment nuts 6

FIGURE 8 ndash Virtual Instrument (VI) Block Diagram 7

FIGURE 9 ndash Virtual Instrument Data Acquisition Screen 7

FIGURE 10 ndash Load cell Calibration Curve ndash Load vs Voltage 10

FIGURE 11 ndash Double Barrel Shock Response ndash All Low and All High Adjustments 11

FIGURE 12 ndash Force vs Velocity ndash Low Speed Rebound 12

FIGURE 13 ndash Single Damping Coefficient ndash Car Body Position vs Time 12

FIGURE 14 ndash 4-way damping ndash Car Body Position vs Time 13

INTRODUCTION

Formula SAEreg competition is an annual collegiate competition for SAE student members to design fabricate and compete with small formula-one style racing cars The student organization SAE at the University of California San Diego is planning on competing in the Formula SAE collegiate competition in June of 2008 The Formula SAE competition is highly competitive therefore it is important to have a well set-up race car Having a well set-up race car allows the driver to utilize its full potential as opposed to having to fight a poorly set-up car around the race track Furthermore having a well set-up car allows it to be more predictable around the race track

Part of having a well set-up car is having an optimized suspension system In doing so requires choosing a proper spring and damper for the application A wide range of dampers are available and for the UCSD SAErsquos racecar the Cane Creek Double Barrel was chosen This is a 4-way adjustable damper that supports adjustability for high and low speed damping for bump and rebound For previous years UCSD SAE has taken a more empirical approach to tuning the damper where adjustments were made according to the driverrsquos feedback at the race track The intent of this experiment is to take a more methodical approach to tuning the suspension system an approach that identifies the theory behind the controls of a suspension system along with theoretical results that can be empirically verified

This experiment sponsored by UCSD SAE will investigate optimization of the vehicle dynamics of the SAE race car This involves characterizing frontrear wheel frequencies spring stiffness and the dampedun-damped masses of the car The dampers will be tuned to the theoretical characteristics of the damping system and after this is completed the suspension will be analyzed and tested on the shock dynamometer This process will optimize the performance of the SAE car vehicle dynamics and the benefits will be realized in the FSAE collegiate competition in June of 2008

THEORY

Figure 1 ndash Quarter-car model of a suspension system

mu

ms

ks

kt

bs

xs

xu

u(t)

Quarter Car Model

The quarter car model is a simplification of the carrsquos suspension system consisting of four shock assemblies down to one Although it does not fully represent the physics of the carrsquos suspension system it can be a useful tool for understanding the workings of the suspension1 Figure 1 shows a diagram of the quarter car model The model consists of the following parameters

Applying Newtonrsquos Second Law the governing equations of motion for the sprung mass and unsprung mass are

To further simplify the equations of motion it is assumed that multltmsand ksltltkt The high stiffness of the tire spring rate combined with the lower mass of the unsprung weight limits the movement of mu and reduces equation (2) to zero The equation of motion for the sprung mass is then This equation simply states that the acceleration of the sprung mass is the sum of the spring rate times the movement of the damper and the damping coefficient times the damper velocity The reduced quarter car model (simple spring-mass-damper system) is shown in Figure2 Applying a Laplace transform to equation (3) to transform it from the time-domain to the frequency domain results in equation (4)

ms = sprung mass (weight of the car) (lb) mu = unsprung mass (wheel and suspension

components) (lb) ks = sprung mass spring rate (lbin) kt = tire spring rate (lbin) bs = sprung mass damping (lbsin) u(t) = displacement of the road as a function of

time

( ) ( ) ( )u u s s u s s u t um x b x x k x x k u x= minus + minus + minusampamp amp amp

( ) ( )s s s u s s u sm x b x x k x x= minus + minusampamp amp amp

s s s s s sm x k x b x= minus minusampamp amp

Eq (1) Eq (2)

Eq (3)

2 0s s

s s

b ks s

m m+ sdot + = Eq (4)

Eq (5)

2 22 0n ns sζ ω ω+ sdot sdot + = Figure 2 ndash Reduced Quarter Car model

THEORY

Comparing equation (4) to the ideal second order system equation (5) where ωnis the natural frequency and ζ is the damping ratio It can be seen that

The damping coefficient is chosen by specifying the damping ratio and the natural frequency is the frequency at which the spring mass system oscillates without the damper With the simplifying assumptions it is expected that the actual values will be lower than the theoretical values

Shock Damper

The shock damper is a critical component of the carrsquos suspension system because it helps control the car chassisrsquo response going into and out of turns (pitch and roll) as well as its response to variations in the road (bouncy or bumpy ride) Without a shock damper the spring would cause the suspension system to oscillate over an extended period of time as it compresses stores mechanical energy and then releases it past its neutral state to a point of extension and then it would continue to oscillate over an extended period of time until all of the stored energy has dissipated The shock damper dissipates energy stored in the spring usually in the form of heat In addition to controlling the carrsquos response in turns the damper also plays a vital role in controlling the temperature of the tires1 In automobile racing controlling tire load variation and maintaining an optimum temperature for the tires is very important as it increases the carrsquos mechanical grip The damping ratio varies and is tuned according to desired handling characteristics and track conditions These ratios are specified in Table 1 LS and HS refers to the damper shaft velocity where LS is low-speed and HS is high-speed Table 1- Damper Tuning Ratios1

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio

is 05 If the circuit is smooth choose 07 If the tires need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where

body control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Matlab Simulink Quarter Car Simulation

Simulink is a graphical approach to a Matlab environment it allows the user to simulate a real word environment such as a quarter car system The Simulink model is used to calculate specific values that are required to tune the SAE cars suspension

To begin the model requires specifications of constants (model parametersmeasured values) needed to produce the equations of motion These constants are placed in two summing blocks that formulate the two equations of motion for the sprung and unsprung masses These equations

sn

s

k

mω =2s n sb mω ζ= sdot sdot sdot Eq (6)

Eq (7)

THEORY

are shown below (Eg (8) and Eq (9)) After equations of motion are placed in the form that solves for the two accelerations of the un-sprung mass (mu)and the sprung mass (ms) the equations are then placed through two integrators the first representing the velocity of (mu)and (ms) and the second producing an equation representing the displacement of (mu)and (ms) These values of acceleration velocity and displacement for mu and ms are then fed back into the controlling parts of the equations of motion This feedback loop allows for continuous control of the damping coefficients

It was imperative for the model to simulate real world performance of the double barrel shock absorber In order to assure a real world simulation the damping coefficients for rebound and compression would have to be determined To determine the system was in rebound or compression the use of a detect increase and detect decrease block respectively These two blocks use the displacement feedback which was mention previously to sense the direction of the simulated damper shaft After each displacement block there resides a switch (1) that is turned on to feed the input of the velocity into another set of switches This switch (1) on the rebound and compression is to engage that particular damping regime Once that switch is on it feeds in the velocity of the damper shaft onto an array of another two switches These two switches switch (2) and switch (3) are responsible for delivering the damping coefficients to the driving part of the system These two switches are tuned to be active at different shaft speeds Switch (2) is on only when switch (1) is on switch (3) is activated when the shaft speed reaches a pre determined limit In this system only the specific damping coefficients that are need for a particular regime are activated this allows for a realistic system in a simulated environment This damping coefficient system can is shown in Figure 3

At the end of each simulation there are specific outputs placed in the Matlab workspace environment Once these outputs are in the workspace there can be manipulate to show the response to the inputs This allows the user to determine to performance of the frac14 car model in a simulated environment

Figure 3 ndash 4-way Damping Switch

( ) ( ) 0s s s s u s s u sm x b x x k x x m g+ minus + minus minus =ampamp amp amp

Eq (8) Eq (9)

( ) ( ) ( ) 0u u s u s s u s t u um x b x x k x x k x q m g+ minus + minus + minus minus =ampamp amp amp

EXPERIMENTAL SETUP

Shock Dynamometer

The Aurora V8 dynamometer itself was donated in non-working condition by a racing specialist Wayne Mitchell to help in the project and the optimization of the shock absorber for the race car A shock dynamometer is a machine used to test and measure force velocity and displacement for shock dampers

Transducers - Load Cell LVDT LVT

Looking in depth into the dynamometer there are three main componentssensors that make up the dynamometer These are the Linear Variable Differential Transformer (LVDT) Linear Velocity Transducer (LVT) and the load cell The load cell sensor is an Omega S-type Model that has a range of 0 to 2000 pound force (lbf) and measured in both compression and tension The LVDT (245 Model) and LVT (110 Model) are manufactured by Trans-Tek a specialist in transducers In particular the LVDT sensor measures linear displacement The transducer itself is a core within a cylindrical tube and contains primary and secondary magnetic coils in which change in voltage output is proportional to the movement of the core The LVT sensor on the other hand measures the velocity at which the motor is running The LVT is similar to the LVDT in terms of coils inside the cylindrical tube The difference being that no external excitation voltage is needed a magnet is passed between the coils and results in a DC voltage output proportional to the instantaneous velocity Again an amplifying circuit board and a data acquisition board were needed to read the voltage outputs out of all three sensors Software used for data logging was National Instruments Labview and MATLAB

Figure 4 ndash LVDT Displacement Transducer Figure 5 ndash LVT Velocity Transducer

Cane Creek Double Barrel Shock Absorber2

The shocks that were acquired for the SAE race car was the Cane Creek Double Barrel shock absorber manufactured by Cane Creek using the technology from the European company Ohlins The double barrel design allows the shock to have four independent external adjustments meaning each adjustment contributes independently to shock response The four independent adjustments consist of high and low speed compression and rebound The traditional shock

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

TABLE OF CONTENTS -

APPENDICES

APPENDIX ldquoArdquo ndash Measurement Data amp Sensor Calibration Figures

APPENDIX ldquoBrdquo ndash Matlab Simulink Figures amp Data

APPENDIX ldquoCrdquo ndash Matlab Code

LIST OF TABLES

PAGE

TABLE 1 ndash Damper Tuning Ratios 4

TABLE 2 ndash Summary of Results for Calibration Curves 10

TABLE 3 ndash Minimum Damping Coefficients ndash Measured 12

TABLE 4 ndash Maximum Damping Coefficients ndash Measured 12

TABLE 5 ndash Minimum Damping Coefficients ndash Cane Creek Data 12

TABLE 5 ndash Minimum Damping Coefficients ndash Cane Creek Data 12

INTRODUCTION

PAGE

FIGURE 1 ndash Quarter car model of a suspension system 2

FIGURE 2 ndash Reduced Quarter Car Model 2

FIGURE 3 ndash 4-way Damping Switch 4

FIGURE 4 ndash LVDT Displacement Transducer 5

FIGURE 5 ndash LVT Velocity Transducer 5

FIGURE 6 ndash Cane Creek Double Barrel cutout view 6

FIGURE 7 ndash Can Creek Double Barrel external adjustment nuts 6

FIGURE 8 ndash Virtual Instrument (VI) Block Diagram 7

FIGURE 9 ndash Virtual Instrument Data Acquisition Screen 7

FIGURE 10 ndash Load cell Calibration Curve ndash Load vs Voltage 10

FIGURE 11 ndash Double Barrel Shock Response ndash All Low and All High Adjustments 11

FIGURE 12 ndash Force vs Velocity ndash Low Speed Rebound 12

FIGURE 13 ndash Single Damping Coefficient ndash Car Body Position vs Time 12

FIGURE 14 ndash 4-way damping ndash Car Body Position vs Time 13

INTRODUCTION

Formula SAEreg competition is an annual collegiate competition for SAE student members to design fabricate and compete with small formula-one style racing cars The student organization SAE at the University of California San Diego is planning on competing in the Formula SAE collegiate competition in June of 2008 The Formula SAE competition is highly competitive therefore it is important to have a well set-up race car Having a well set-up race car allows the driver to utilize its full potential as opposed to having to fight a poorly set-up car around the race track Furthermore having a well set-up car allows it to be more predictable around the race track

Part of having a well set-up car is having an optimized suspension system In doing so requires choosing a proper spring and damper for the application A wide range of dampers are available and for the UCSD SAErsquos racecar the Cane Creek Double Barrel was chosen This is a 4-way adjustable damper that supports adjustability for high and low speed damping for bump and rebound For previous years UCSD SAE has taken a more empirical approach to tuning the damper where adjustments were made according to the driverrsquos feedback at the race track The intent of this experiment is to take a more methodical approach to tuning the suspension system an approach that identifies the theory behind the controls of a suspension system along with theoretical results that can be empirically verified

This experiment sponsored by UCSD SAE will investigate optimization of the vehicle dynamics of the SAE race car This involves characterizing frontrear wheel frequencies spring stiffness and the dampedun-damped masses of the car The dampers will be tuned to the theoretical characteristics of the damping system and after this is completed the suspension will be analyzed and tested on the shock dynamometer This process will optimize the performance of the SAE car vehicle dynamics and the benefits will be realized in the FSAE collegiate competition in June of 2008

THEORY

Figure 1 ndash Quarter-car model of a suspension system

mu

ms

ks

kt

bs

xs

xu

u(t)

Quarter Car Model

The quarter car model is a simplification of the carrsquos suspension system consisting of four shock assemblies down to one Although it does not fully represent the physics of the carrsquos suspension system it can be a useful tool for understanding the workings of the suspension1 Figure 1 shows a diagram of the quarter car model The model consists of the following parameters

Applying Newtonrsquos Second Law the governing equations of motion for the sprung mass and unsprung mass are

To further simplify the equations of motion it is assumed that multltmsand ksltltkt The high stiffness of the tire spring rate combined with the lower mass of the unsprung weight limits the movement of mu and reduces equation (2) to zero The equation of motion for the sprung mass is then This equation simply states that the acceleration of the sprung mass is the sum of the spring rate times the movement of the damper and the damping coefficient times the damper velocity The reduced quarter car model (simple spring-mass-damper system) is shown in Figure2 Applying a Laplace transform to equation (3) to transform it from the time-domain to the frequency domain results in equation (4)

ms = sprung mass (weight of the car) (lb) mu = unsprung mass (wheel and suspension

components) (lb) ks = sprung mass spring rate (lbin) kt = tire spring rate (lbin) bs = sprung mass damping (lbsin) u(t) = displacement of the road as a function of

time

( ) ( ) ( )u u s s u s s u t um x b x x k x x k u x= minus + minus + minusampamp amp amp

( ) ( )s s s u s s u sm x b x x k x x= minus + minusampamp amp amp

s s s s s sm x k x b x= minus minusampamp amp

Eq (1) Eq (2)

Eq (3)

2 0s s

s s

b ks s

m m+ sdot + = Eq (4)

Eq (5)

2 22 0n ns sζ ω ω+ sdot sdot + = Figure 2 ndash Reduced Quarter Car model

THEORY

Comparing equation (4) to the ideal second order system equation (5) where ωnis the natural frequency and ζ is the damping ratio It can be seen that

The damping coefficient is chosen by specifying the damping ratio and the natural frequency is the frequency at which the spring mass system oscillates without the damper With the simplifying assumptions it is expected that the actual values will be lower than the theoretical values

Shock Damper

The shock damper is a critical component of the carrsquos suspension system because it helps control the car chassisrsquo response going into and out of turns (pitch and roll) as well as its response to variations in the road (bouncy or bumpy ride) Without a shock damper the spring would cause the suspension system to oscillate over an extended period of time as it compresses stores mechanical energy and then releases it past its neutral state to a point of extension and then it would continue to oscillate over an extended period of time until all of the stored energy has dissipated The shock damper dissipates energy stored in the spring usually in the form of heat In addition to controlling the carrsquos response in turns the damper also plays a vital role in controlling the temperature of the tires1 In automobile racing controlling tire load variation and maintaining an optimum temperature for the tires is very important as it increases the carrsquos mechanical grip The damping ratio varies and is tuned according to desired handling characteristics and track conditions These ratios are specified in Table 1 LS and HS refers to the damper shaft velocity where LS is low-speed and HS is high-speed Table 1- Damper Tuning Ratios1

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio

is 05 If the circuit is smooth choose 07 If the tires need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where

body control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Matlab Simulink Quarter Car Simulation

Simulink is a graphical approach to a Matlab environment it allows the user to simulate a real word environment such as a quarter car system The Simulink model is used to calculate specific values that are required to tune the SAE cars suspension

To begin the model requires specifications of constants (model parametersmeasured values) needed to produce the equations of motion These constants are placed in two summing blocks that formulate the two equations of motion for the sprung and unsprung masses These equations

sn

s

k

mω =2s n sb mω ζ= sdot sdot sdot Eq (6)

Eq (7)

THEORY

are shown below (Eg (8) and Eq (9)) After equations of motion are placed in the form that solves for the two accelerations of the un-sprung mass (mu)and the sprung mass (ms) the equations are then placed through two integrators the first representing the velocity of (mu)and (ms) and the second producing an equation representing the displacement of (mu)and (ms) These values of acceleration velocity and displacement for mu and ms are then fed back into the controlling parts of the equations of motion This feedback loop allows for continuous control of the damping coefficients

It was imperative for the model to simulate real world performance of the double barrel shock absorber In order to assure a real world simulation the damping coefficients for rebound and compression would have to be determined To determine the system was in rebound or compression the use of a detect increase and detect decrease block respectively These two blocks use the displacement feedback which was mention previously to sense the direction of the simulated damper shaft After each displacement block there resides a switch (1) that is turned on to feed the input of the velocity into another set of switches This switch (1) on the rebound and compression is to engage that particular damping regime Once that switch is on it feeds in the velocity of the damper shaft onto an array of another two switches These two switches switch (2) and switch (3) are responsible for delivering the damping coefficients to the driving part of the system These two switches are tuned to be active at different shaft speeds Switch (2) is on only when switch (1) is on switch (3) is activated when the shaft speed reaches a pre determined limit In this system only the specific damping coefficients that are need for a particular regime are activated this allows for a realistic system in a simulated environment This damping coefficient system can is shown in Figure 3

At the end of each simulation there are specific outputs placed in the Matlab workspace environment Once these outputs are in the workspace there can be manipulate to show the response to the inputs This allows the user to determine to performance of the frac14 car model in a simulated environment

Figure 3 ndash 4-way Damping Switch

( ) ( ) 0s s s s u s s u sm x b x x k x x m g+ minus + minus minus =ampamp amp amp

Eq (8) Eq (9)

( ) ( ) ( ) 0u u s u s s u s t u um x b x x k x x k x q m g+ minus + minus + minus minus =ampamp amp amp

EXPERIMENTAL SETUP

Shock Dynamometer

The Aurora V8 dynamometer itself was donated in non-working condition by a racing specialist Wayne Mitchell to help in the project and the optimization of the shock absorber for the race car A shock dynamometer is a machine used to test and measure force velocity and displacement for shock dampers

Transducers - Load Cell LVDT LVT

Looking in depth into the dynamometer there are three main componentssensors that make up the dynamometer These are the Linear Variable Differential Transformer (LVDT) Linear Velocity Transducer (LVT) and the load cell The load cell sensor is an Omega S-type Model that has a range of 0 to 2000 pound force (lbf) and measured in both compression and tension The LVDT (245 Model) and LVT (110 Model) are manufactured by Trans-Tek a specialist in transducers In particular the LVDT sensor measures linear displacement The transducer itself is a core within a cylindrical tube and contains primary and secondary magnetic coils in which change in voltage output is proportional to the movement of the core The LVT sensor on the other hand measures the velocity at which the motor is running The LVT is similar to the LVDT in terms of coils inside the cylindrical tube The difference being that no external excitation voltage is needed a magnet is passed between the coils and results in a DC voltage output proportional to the instantaneous velocity Again an amplifying circuit board and a data acquisition board were needed to read the voltage outputs out of all three sensors Software used for data logging was National Instruments Labview and MATLAB

Figure 4 ndash LVDT Displacement Transducer Figure 5 ndash LVT Velocity Transducer

Cane Creek Double Barrel Shock Absorber2

The shocks that were acquired for the SAE race car was the Cane Creek Double Barrel shock absorber manufactured by Cane Creek using the technology from the European company Ohlins The double barrel design allows the shock to have four independent external adjustments meaning each adjustment contributes independently to shock response The four independent adjustments consist of high and low speed compression and rebound The traditional shock

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

LIST OF TABLES

PAGE

TABLE 1 ndash Damper Tuning Ratios 4

TABLE 2 ndash Summary of Results for Calibration Curves 10

TABLE 3 ndash Minimum Damping Coefficients ndash Measured 12

TABLE 4 ndash Maximum Damping Coefficients ndash Measured 12

TABLE 5 ndash Minimum Damping Coefficients ndash Cane Creek Data 12

TABLE 5 ndash Minimum Damping Coefficients ndash Cane Creek Data 12

INTRODUCTION

PAGE

FIGURE 1 ndash Quarter car model of a suspension system 2

FIGURE 2 ndash Reduced Quarter Car Model 2

FIGURE 3 ndash 4-way Damping Switch 4

FIGURE 4 ndash LVDT Displacement Transducer 5

FIGURE 5 ndash LVT Velocity Transducer 5

FIGURE 6 ndash Cane Creek Double Barrel cutout view 6

FIGURE 7 ndash Can Creek Double Barrel external adjustment nuts 6

FIGURE 8 ndash Virtual Instrument (VI) Block Diagram 7

FIGURE 9 ndash Virtual Instrument Data Acquisition Screen 7

FIGURE 10 ndash Load cell Calibration Curve ndash Load vs Voltage 10

FIGURE 11 ndash Double Barrel Shock Response ndash All Low and All High Adjustments 11

FIGURE 12 ndash Force vs Velocity ndash Low Speed Rebound 12

FIGURE 13 ndash Single Damping Coefficient ndash Car Body Position vs Time 12

FIGURE 14 ndash 4-way damping ndash Car Body Position vs Time 13

INTRODUCTION

Formula SAEreg competition is an annual collegiate competition for SAE student members to design fabricate and compete with small formula-one style racing cars The student organization SAE at the University of California San Diego is planning on competing in the Formula SAE collegiate competition in June of 2008 The Formula SAE competition is highly competitive therefore it is important to have a well set-up race car Having a well set-up race car allows the driver to utilize its full potential as opposed to having to fight a poorly set-up car around the race track Furthermore having a well set-up car allows it to be more predictable around the race track

Part of having a well set-up car is having an optimized suspension system In doing so requires choosing a proper spring and damper for the application A wide range of dampers are available and for the UCSD SAErsquos racecar the Cane Creek Double Barrel was chosen This is a 4-way adjustable damper that supports adjustability for high and low speed damping for bump and rebound For previous years UCSD SAE has taken a more empirical approach to tuning the damper where adjustments were made according to the driverrsquos feedback at the race track The intent of this experiment is to take a more methodical approach to tuning the suspension system an approach that identifies the theory behind the controls of a suspension system along with theoretical results that can be empirically verified

This experiment sponsored by UCSD SAE will investigate optimization of the vehicle dynamics of the SAE race car This involves characterizing frontrear wheel frequencies spring stiffness and the dampedun-damped masses of the car The dampers will be tuned to the theoretical characteristics of the damping system and after this is completed the suspension will be analyzed and tested on the shock dynamometer This process will optimize the performance of the SAE car vehicle dynamics and the benefits will be realized in the FSAE collegiate competition in June of 2008

THEORY

Figure 1 ndash Quarter-car model of a suspension system

mu

ms

ks

kt

bs

xs

xu

u(t)

Quarter Car Model

The quarter car model is a simplification of the carrsquos suspension system consisting of four shock assemblies down to one Although it does not fully represent the physics of the carrsquos suspension system it can be a useful tool for understanding the workings of the suspension1 Figure 1 shows a diagram of the quarter car model The model consists of the following parameters

Applying Newtonrsquos Second Law the governing equations of motion for the sprung mass and unsprung mass are

To further simplify the equations of motion it is assumed that multltmsand ksltltkt The high stiffness of the tire spring rate combined with the lower mass of the unsprung weight limits the movement of mu and reduces equation (2) to zero The equation of motion for the sprung mass is then This equation simply states that the acceleration of the sprung mass is the sum of the spring rate times the movement of the damper and the damping coefficient times the damper velocity The reduced quarter car model (simple spring-mass-damper system) is shown in Figure2 Applying a Laplace transform to equation (3) to transform it from the time-domain to the frequency domain results in equation (4)

ms = sprung mass (weight of the car) (lb) mu = unsprung mass (wheel and suspension

components) (lb) ks = sprung mass spring rate (lbin) kt = tire spring rate (lbin) bs = sprung mass damping (lbsin) u(t) = displacement of the road as a function of

time

( ) ( ) ( )u u s s u s s u t um x b x x k x x k u x= minus + minus + minusampamp amp amp

( ) ( )s s s u s s u sm x b x x k x x= minus + minusampamp amp amp

s s s s s sm x k x b x= minus minusampamp amp

Eq (1) Eq (2)

Eq (3)

2 0s s

s s

b ks s

m m+ sdot + = Eq (4)

Eq (5)

2 22 0n ns sζ ω ω+ sdot sdot + = Figure 2 ndash Reduced Quarter Car model

THEORY

Comparing equation (4) to the ideal second order system equation (5) where ωnis the natural frequency and ζ is the damping ratio It can be seen that

The damping coefficient is chosen by specifying the damping ratio and the natural frequency is the frequency at which the spring mass system oscillates without the damper With the simplifying assumptions it is expected that the actual values will be lower than the theoretical values

Shock Damper

The shock damper is a critical component of the carrsquos suspension system because it helps control the car chassisrsquo response going into and out of turns (pitch and roll) as well as its response to variations in the road (bouncy or bumpy ride) Without a shock damper the spring would cause the suspension system to oscillate over an extended period of time as it compresses stores mechanical energy and then releases it past its neutral state to a point of extension and then it would continue to oscillate over an extended period of time until all of the stored energy has dissipated The shock damper dissipates energy stored in the spring usually in the form of heat In addition to controlling the carrsquos response in turns the damper also plays a vital role in controlling the temperature of the tires1 In automobile racing controlling tire load variation and maintaining an optimum temperature for the tires is very important as it increases the carrsquos mechanical grip The damping ratio varies and is tuned according to desired handling characteristics and track conditions These ratios are specified in Table 1 LS and HS refers to the damper shaft velocity where LS is low-speed and HS is high-speed Table 1- Damper Tuning Ratios1

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio

is 05 If the circuit is smooth choose 07 If the tires need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where

body control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Matlab Simulink Quarter Car Simulation

Simulink is a graphical approach to a Matlab environment it allows the user to simulate a real word environment such as a quarter car system The Simulink model is used to calculate specific values that are required to tune the SAE cars suspension

To begin the model requires specifications of constants (model parametersmeasured values) needed to produce the equations of motion These constants are placed in two summing blocks that formulate the two equations of motion for the sprung and unsprung masses These equations

sn

s

k

mω =2s n sb mω ζ= sdot sdot sdot Eq (6)

Eq (7)

THEORY

are shown below (Eg (8) and Eq (9)) After equations of motion are placed in the form that solves for the two accelerations of the un-sprung mass (mu)and the sprung mass (ms) the equations are then placed through two integrators the first representing the velocity of (mu)and (ms) and the second producing an equation representing the displacement of (mu)and (ms) These values of acceleration velocity and displacement for mu and ms are then fed back into the controlling parts of the equations of motion This feedback loop allows for continuous control of the damping coefficients

It was imperative for the model to simulate real world performance of the double barrel shock absorber In order to assure a real world simulation the damping coefficients for rebound and compression would have to be determined To determine the system was in rebound or compression the use of a detect increase and detect decrease block respectively These two blocks use the displacement feedback which was mention previously to sense the direction of the simulated damper shaft After each displacement block there resides a switch (1) that is turned on to feed the input of the velocity into another set of switches This switch (1) on the rebound and compression is to engage that particular damping regime Once that switch is on it feeds in the velocity of the damper shaft onto an array of another two switches These two switches switch (2) and switch (3) are responsible for delivering the damping coefficients to the driving part of the system These two switches are tuned to be active at different shaft speeds Switch (2) is on only when switch (1) is on switch (3) is activated when the shaft speed reaches a pre determined limit In this system only the specific damping coefficients that are need for a particular regime are activated this allows for a realistic system in a simulated environment This damping coefficient system can is shown in Figure 3

At the end of each simulation there are specific outputs placed in the Matlab workspace environment Once these outputs are in the workspace there can be manipulate to show the response to the inputs This allows the user to determine to performance of the frac14 car model in a simulated environment

Figure 3 ndash 4-way Damping Switch

( ) ( ) 0s s s s u s s u sm x b x x k x x m g+ minus + minus minus =ampamp amp amp

Eq (8) Eq (9)

( ) ( ) ( ) 0u u s u s s u s t u um x b x x k x x k x q m g+ minus + minus + minus minus =ampamp amp amp

EXPERIMENTAL SETUP

Shock Dynamometer

The Aurora V8 dynamometer itself was donated in non-working condition by a racing specialist Wayne Mitchell to help in the project and the optimization of the shock absorber for the race car A shock dynamometer is a machine used to test and measure force velocity and displacement for shock dampers

Transducers - Load Cell LVDT LVT

Looking in depth into the dynamometer there are three main componentssensors that make up the dynamometer These are the Linear Variable Differential Transformer (LVDT) Linear Velocity Transducer (LVT) and the load cell The load cell sensor is an Omega S-type Model that has a range of 0 to 2000 pound force (lbf) and measured in both compression and tension The LVDT (245 Model) and LVT (110 Model) are manufactured by Trans-Tek a specialist in transducers In particular the LVDT sensor measures linear displacement The transducer itself is a core within a cylindrical tube and contains primary and secondary magnetic coils in which change in voltage output is proportional to the movement of the core The LVT sensor on the other hand measures the velocity at which the motor is running The LVT is similar to the LVDT in terms of coils inside the cylindrical tube The difference being that no external excitation voltage is needed a magnet is passed between the coils and results in a DC voltage output proportional to the instantaneous velocity Again an amplifying circuit board and a data acquisition board were needed to read the voltage outputs out of all three sensors Software used for data logging was National Instruments Labview and MATLAB

Figure 4 ndash LVDT Displacement Transducer Figure 5 ndash LVT Velocity Transducer

Cane Creek Double Barrel Shock Absorber2

The shocks that were acquired for the SAE race car was the Cane Creek Double Barrel shock absorber manufactured by Cane Creek using the technology from the European company Ohlins The double barrel design allows the shock to have four independent external adjustments meaning each adjustment contributes independently to shock response The four independent adjustments consist of high and low speed compression and rebound The traditional shock

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

INTRODUCTION

PAGE

FIGURE 1 ndash Quarter car model of a suspension system 2

FIGURE 2 ndash Reduced Quarter Car Model 2

FIGURE 3 ndash 4-way Damping Switch 4

FIGURE 4 ndash LVDT Displacement Transducer 5

FIGURE 5 ndash LVT Velocity Transducer 5

FIGURE 6 ndash Cane Creek Double Barrel cutout view 6

FIGURE 7 ndash Can Creek Double Barrel external adjustment nuts 6

FIGURE 8 ndash Virtual Instrument (VI) Block Diagram 7

FIGURE 9 ndash Virtual Instrument Data Acquisition Screen 7

FIGURE 10 ndash Load cell Calibration Curve ndash Load vs Voltage 10

FIGURE 11 ndash Double Barrel Shock Response ndash All Low and All High Adjustments 11

FIGURE 12 ndash Force vs Velocity ndash Low Speed Rebound 12

FIGURE 13 ndash Single Damping Coefficient ndash Car Body Position vs Time 12

FIGURE 14 ndash 4-way damping ndash Car Body Position vs Time 13

INTRODUCTION

Formula SAEreg competition is an annual collegiate competition for SAE student members to design fabricate and compete with small formula-one style racing cars The student organization SAE at the University of California San Diego is planning on competing in the Formula SAE collegiate competition in June of 2008 The Formula SAE competition is highly competitive therefore it is important to have a well set-up race car Having a well set-up race car allows the driver to utilize its full potential as opposed to having to fight a poorly set-up car around the race track Furthermore having a well set-up car allows it to be more predictable around the race track

Part of having a well set-up car is having an optimized suspension system In doing so requires choosing a proper spring and damper for the application A wide range of dampers are available and for the UCSD SAErsquos racecar the Cane Creek Double Barrel was chosen This is a 4-way adjustable damper that supports adjustability for high and low speed damping for bump and rebound For previous years UCSD SAE has taken a more empirical approach to tuning the damper where adjustments were made according to the driverrsquos feedback at the race track The intent of this experiment is to take a more methodical approach to tuning the suspension system an approach that identifies the theory behind the controls of a suspension system along with theoretical results that can be empirically verified

This experiment sponsored by UCSD SAE will investigate optimization of the vehicle dynamics of the SAE race car This involves characterizing frontrear wheel frequencies spring stiffness and the dampedun-damped masses of the car The dampers will be tuned to the theoretical characteristics of the damping system and after this is completed the suspension will be analyzed and tested on the shock dynamometer This process will optimize the performance of the SAE car vehicle dynamics and the benefits will be realized in the FSAE collegiate competition in June of 2008

THEORY

Figure 1 ndash Quarter-car model of a suspension system

mu

ms

ks

kt

bs

xs

xu

u(t)

Quarter Car Model

The quarter car model is a simplification of the carrsquos suspension system consisting of four shock assemblies down to one Although it does not fully represent the physics of the carrsquos suspension system it can be a useful tool for understanding the workings of the suspension1 Figure 1 shows a diagram of the quarter car model The model consists of the following parameters

Applying Newtonrsquos Second Law the governing equations of motion for the sprung mass and unsprung mass are

To further simplify the equations of motion it is assumed that multltmsand ksltltkt The high stiffness of the tire spring rate combined with the lower mass of the unsprung weight limits the movement of mu and reduces equation (2) to zero The equation of motion for the sprung mass is then This equation simply states that the acceleration of the sprung mass is the sum of the spring rate times the movement of the damper and the damping coefficient times the damper velocity The reduced quarter car model (simple spring-mass-damper system) is shown in Figure2 Applying a Laplace transform to equation (3) to transform it from the time-domain to the frequency domain results in equation (4)

ms = sprung mass (weight of the car) (lb) mu = unsprung mass (wheel and suspension

components) (lb) ks = sprung mass spring rate (lbin) kt = tire spring rate (lbin) bs = sprung mass damping (lbsin) u(t) = displacement of the road as a function of

time

( ) ( ) ( )u u s s u s s u t um x b x x k x x k u x= minus + minus + minusampamp amp amp

( ) ( )s s s u s s u sm x b x x k x x= minus + minusampamp amp amp

s s s s s sm x k x b x= minus minusampamp amp

Eq (1) Eq (2)

Eq (3)

2 0s s

s s

b ks s

m m+ sdot + = Eq (4)

Eq (5)

2 22 0n ns sζ ω ω+ sdot sdot + = Figure 2 ndash Reduced Quarter Car model

THEORY

Comparing equation (4) to the ideal second order system equation (5) where ωnis the natural frequency and ζ is the damping ratio It can be seen that

The damping coefficient is chosen by specifying the damping ratio and the natural frequency is the frequency at which the spring mass system oscillates without the damper With the simplifying assumptions it is expected that the actual values will be lower than the theoretical values

Shock Damper

The shock damper is a critical component of the carrsquos suspension system because it helps control the car chassisrsquo response going into and out of turns (pitch and roll) as well as its response to variations in the road (bouncy or bumpy ride) Without a shock damper the spring would cause the suspension system to oscillate over an extended period of time as it compresses stores mechanical energy and then releases it past its neutral state to a point of extension and then it would continue to oscillate over an extended period of time until all of the stored energy has dissipated The shock damper dissipates energy stored in the spring usually in the form of heat In addition to controlling the carrsquos response in turns the damper also plays a vital role in controlling the temperature of the tires1 In automobile racing controlling tire load variation and maintaining an optimum temperature for the tires is very important as it increases the carrsquos mechanical grip The damping ratio varies and is tuned according to desired handling characteristics and track conditions These ratios are specified in Table 1 LS and HS refers to the damper shaft velocity where LS is low-speed and HS is high-speed Table 1- Damper Tuning Ratios1

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio

is 05 If the circuit is smooth choose 07 If the tires need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where

body control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Matlab Simulink Quarter Car Simulation

Simulink is a graphical approach to a Matlab environment it allows the user to simulate a real word environment such as a quarter car system The Simulink model is used to calculate specific values that are required to tune the SAE cars suspension

To begin the model requires specifications of constants (model parametersmeasured values) needed to produce the equations of motion These constants are placed in two summing blocks that formulate the two equations of motion for the sprung and unsprung masses These equations

sn

s

k

mω =2s n sb mω ζ= sdot sdot sdot Eq (6)

Eq (7)

THEORY

are shown below (Eg (8) and Eq (9)) After equations of motion are placed in the form that solves for the two accelerations of the un-sprung mass (mu)and the sprung mass (ms) the equations are then placed through two integrators the first representing the velocity of (mu)and (ms) and the second producing an equation representing the displacement of (mu)and (ms) These values of acceleration velocity and displacement for mu and ms are then fed back into the controlling parts of the equations of motion This feedback loop allows for continuous control of the damping coefficients

It was imperative for the model to simulate real world performance of the double barrel shock absorber In order to assure a real world simulation the damping coefficients for rebound and compression would have to be determined To determine the system was in rebound or compression the use of a detect increase and detect decrease block respectively These two blocks use the displacement feedback which was mention previously to sense the direction of the simulated damper shaft After each displacement block there resides a switch (1) that is turned on to feed the input of the velocity into another set of switches This switch (1) on the rebound and compression is to engage that particular damping regime Once that switch is on it feeds in the velocity of the damper shaft onto an array of another two switches These two switches switch (2) and switch (3) are responsible for delivering the damping coefficients to the driving part of the system These two switches are tuned to be active at different shaft speeds Switch (2) is on only when switch (1) is on switch (3) is activated when the shaft speed reaches a pre determined limit In this system only the specific damping coefficients that are need for a particular regime are activated this allows for a realistic system in a simulated environment This damping coefficient system can is shown in Figure 3

At the end of each simulation there are specific outputs placed in the Matlab workspace environment Once these outputs are in the workspace there can be manipulate to show the response to the inputs This allows the user to determine to performance of the frac14 car model in a simulated environment

Figure 3 ndash 4-way Damping Switch

( ) ( ) 0s s s s u s s u sm x b x x k x x m g+ minus + minus minus =ampamp amp amp

Eq (8) Eq (9)

( ) ( ) ( ) 0u u s u s s u s t u um x b x x k x x k x q m g+ minus + minus + minus minus =ampamp amp amp

EXPERIMENTAL SETUP

Shock Dynamometer

The Aurora V8 dynamometer itself was donated in non-working condition by a racing specialist Wayne Mitchell to help in the project and the optimization of the shock absorber for the race car A shock dynamometer is a machine used to test and measure force velocity and displacement for shock dampers

Transducers - Load Cell LVDT LVT

Looking in depth into the dynamometer there are three main componentssensors that make up the dynamometer These are the Linear Variable Differential Transformer (LVDT) Linear Velocity Transducer (LVT) and the load cell The load cell sensor is an Omega S-type Model that has a range of 0 to 2000 pound force (lbf) and measured in both compression and tension The LVDT (245 Model) and LVT (110 Model) are manufactured by Trans-Tek a specialist in transducers In particular the LVDT sensor measures linear displacement The transducer itself is a core within a cylindrical tube and contains primary and secondary magnetic coils in which change in voltage output is proportional to the movement of the core The LVT sensor on the other hand measures the velocity at which the motor is running The LVT is similar to the LVDT in terms of coils inside the cylindrical tube The difference being that no external excitation voltage is needed a magnet is passed between the coils and results in a DC voltage output proportional to the instantaneous velocity Again an amplifying circuit board and a data acquisition board were needed to read the voltage outputs out of all three sensors Software used for data logging was National Instruments Labview and MATLAB

Figure 4 ndash LVDT Displacement Transducer Figure 5 ndash LVT Velocity Transducer

Cane Creek Double Barrel Shock Absorber2

The shocks that were acquired for the SAE race car was the Cane Creek Double Barrel shock absorber manufactured by Cane Creek using the technology from the European company Ohlins The double barrel design allows the shock to have four independent external adjustments meaning each adjustment contributes independently to shock response The four independent adjustments consist of high and low speed compression and rebound The traditional shock

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

INTRODUCTION

Formula SAEreg competition is an annual collegiate competition for SAE student members to design fabricate and compete with small formula-one style racing cars The student organization SAE at the University of California San Diego is planning on competing in the Formula SAE collegiate competition in June of 2008 The Formula SAE competition is highly competitive therefore it is important to have a well set-up race car Having a well set-up race car allows the driver to utilize its full potential as opposed to having to fight a poorly set-up car around the race track Furthermore having a well set-up car allows it to be more predictable around the race track

Part of having a well set-up car is having an optimized suspension system In doing so requires choosing a proper spring and damper for the application A wide range of dampers are available and for the UCSD SAErsquos racecar the Cane Creek Double Barrel was chosen This is a 4-way adjustable damper that supports adjustability for high and low speed damping for bump and rebound For previous years UCSD SAE has taken a more empirical approach to tuning the damper where adjustments were made according to the driverrsquos feedback at the race track The intent of this experiment is to take a more methodical approach to tuning the suspension system an approach that identifies the theory behind the controls of a suspension system along with theoretical results that can be empirically verified

This experiment sponsored by UCSD SAE will investigate optimization of the vehicle dynamics of the SAE race car This involves characterizing frontrear wheel frequencies spring stiffness and the dampedun-damped masses of the car The dampers will be tuned to the theoretical characteristics of the damping system and after this is completed the suspension will be analyzed and tested on the shock dynamometer This process will optimize the performance of the SAE car vehicle dynamics and the benefits will be realized in the FSAE collegiate competition in June of 2008

THEORY

Figure 1 ndash Quarter-car model of a suspension system

mu

ms

ks

kt

bs

xs

xu

u(t)

Quarter Car Model

The quarter car model is a simplification of the carrsquos suspension system consisting of four shock assemblies down to one Although it does not fully represent the physics of the carrsquos suspension system it can be a useful tool for understanding the workings of the suspension1 Figure 1 shows a diagram of the quarter car model The model consists of the following parameters

Applying Newtonrsquos Second Law the governing equations of motion for the sprung mass and unsprung mass are

To further simplify the equations of motion it is assumed that multltmsand ksltltkt The high stiffness of the tire spring rate combined with the lower mass of the unsprung weight limits the movement of mu and reduces equation (2) to zero The equation of motion for the sprung mass is then This equation simply states that the acceleration of the sprung mass is the sum of the spring rate times the movement of the damper and the damping coefficient times the damper velocity The reduced quarter car model (simple spring-mass-damper system) is shown in Figure2 Applying a Laplace transform to equation (3) to transform it from the time-domain to the frequency domain results in equation (4)

ms = sprung mass (weight of the car) (lb) mu = unsprung mass (wheel and suspension

components) (lb) ks = sprung mass spring rate (lbin) kt = tire spring rate (lbin) bs = sprung mass damping (lbsin) u(t) = displacement of the road as a function of

time

( ) ( ) ( )u u s s u s s u t um x b x x k x x k u x= minus + minus + minusampamp amp amp

( ) ( )s s s u s s u sm x b x x k x x= minus + minusampamp amp amp

s s s s s sm x k x b x= minus minusampamp amp

Eq (1) Eq (2)

Eq (3)

2 0s s

s s

b ks s

m m+ sdot + = Eq (4)

Eq (5)

2 22 0n ns sζ ω ω+ sdot sdot + = Figure 2 ndash Reduced Quarter Car model

THEORY

Comparing equation (4) to the ideal second order system equation (5) where ωnis the natural frequency and ζ is the damping ratio It can be seen that

The damping coefficient is chosen by specifying the damping ratio and the natural frequency is the frequency at which the spring mass system oscillates without the damper With the simplifying assumptions it is expected that the actual values will be lower than the theoretical values

Shock Damper

The shock damper is a critical component of the carrsquos suspension system because it helps control the car chassisrsquo response going into and out of turns (pitch and roll) as well as its response to variations in the road (bouncy or bumpy ride) Without a shock damper the spring would cause the suspension system to oscillate over an extended period of time as it compresses stores mechanical energy and then releases it past its neutral state to a point of extension and then it would continue to oscillate over an extended period of time until all of the stored energy has dissipated The shock damper dissipates energy stored in the spring usually in the form of heat In addition to controlling the carrsquos response in turns the damper also plays a vital role in controlling the temperature of the tires1 In automobile racing controlling tire load variation and maintaining an optimum temperature for the tires is very important as it increases the carrsquos mechanical grip The damping ratio varies and is tuned according to desired handling characteristics and track conditions These ratios are specified in Table 1 LS and HS refers to the damper shaft velocity where LS is low-speed and HS is high-speed Table 1- Damper Tuning Ratios1

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio

is 05 If the circuit is smooth choose 07 If the tires need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where

body control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Matlab Simulink Quarter Car Simulation

Simulink is a graphical approach to a Matlab environment it allows the user to simulate a real word environment such as a quarter car system The Simulink model is used to calculate specific values that are required to tune the SAE cars suspension

To begin the model requires specifications of constants (model parametersmeasured values) needed to produce the equations of motion These constants are placed in two summing blocks that formulate the two equations of motion for the sprung and unsprung masses These equations

sn

s

k

mω =2s n sb mω ζ= sdot sdot sdot Eq (6)

Eq (7)

THEORY

are shown below (Eg (8) and Eq (9)) After equations of motion are placed in the form that solves for the two accelerations of the un-sprung mass (mu)and the sprung mass (ms) the equations are then placed through two integrators the first representing the velocity of (mu)and (ms) and the second producing an equation representing the displacement of (mu)and (ms) These values of acceleration velocity and displacement for mu and ms are then fed back into the controlling parts of the equations of motion This feedback loop allows for continuous control of the damping coefficients

It was imperative for the model to simulate real world performance of the double barrel shock absorber In order to assure a real world simulation the damping coefficients for rebound and compression would have to be determined To determine the system was in rebound or compression the use of a detect increase and detect decrease block respectively These two blocks use the displacement feedback which was mention previously to sense the direction of the simulated damper shaft After each displacement block there resides a switch (1) that is turned on to feed the input of the velocity into another set of switches This switch (1) on the rebound and compression is to engage that particular damping regime Once that switch is on it feeds in the velocity of the damper shaft onto an array of another two switches These two switches switch (2) and switch (3) are responsible for delivering the damping coefficients to the driving part of the system These two switches are tuned to be active at different shaft speeds Switch (2) is on only when switch (1) is on switch (3) is activated when the shaft speed reaches a pre determined limit In this system only the specific damping coefficients that are need for a particular regime are activated this allows for a realistic system in a simulated environment This damping coefficient system can is shown in Figure 3

At the end of each simulation there are specific outputs placed in the Matlab workspace environment Once these outputs are in the workspace there can be manipulate to show the response to the inputs This allows the user to determine to performance of the frac14 car model in a simulated environment

Figure 3 ndash 4-way Damping Switch

( ) ( ) 0s s s s u s s u sm x b x x k x x m g+ minus + minus minus =ampamp amp amp

Eq (8) Eq (9)

( ) ( ) ( ) 0u u s u s s u s t u um x b x x k x x k x q m g+ minus + minus + minus minus =ampamp amp amp

EXPERIMENTAL SETUP

Shock Dynamometer

The Aurora V8 dynamometer itself was donated in non-working condition by a racing specialist Wayne Mitchell to help in the project and the optimization of the shock absorber for the race car A shock dynamometer is a machine used to test and measure force velocity and displacement for shock dampers

Transducers - Load Cell LVDT LVT

Looking in depth into the dynamometer there are three main componentssensors that make up the dynamometer These are the Linear Variable Differential Transformer (LVDT) Linear Velocity Transducer (LVT) and the load cell The load cell sensor is an Omega S-type Model that has a range of 0 to 2000 pound force (lbf) and measured in both compression and tension The LVDT (245 Model) and LVT (110 Model) are manufactured by Trans-Tek a specialist in transducers In particular the LVDT sensor measures linear displacement The transducer itself is a core within a cylindrical tube and contains primary and secondary magnetic coils in which change in voltage output is proportional to the movement of the core The LVT sensor on the other hand measures the velocity at which the motor is running The LVT is similar to the LVDT in terms of coils inside the cylindrical tube The difference being that no external excitation voltage is needed a magnet is passed between the coils and results in a DC voltage output proportional to the instantaneous velocity Again an amplifying circuit board and a data acquisition board were needed to read the voltage outputs out of all three sensors Software used for data logging was National Instruments Labview and MATLAB

Figure 4 ndash LVDT Displacement Transducer Figure 5 ndash LVT Velocity Transducer

Cane Creek Double Barrel Shock Absorber2

The shocks that were acquired for the SAE race car was the Cane Creek Double Barrel shock absorber manufactured by Cane Creek using the technology from the European company Ohlins The double barrel design allows the shock to have four independent external adjustments meaning each adjustment contributes independently to shock response The four independent adjustments consist of high and low speed compression and rebound The traditional shock

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

THEORY

Figure 1 ndash Quarter-car model of a suspension system

mu

ms

ks

kt

bs

xs

xu

u(t)

Quarter Car Model

The quarter car model is a simplification of the carrsquos suspension system consisting of four shock assemblies down to one Although it does not fully represent the physics of the carrsquos suspension system it can be a useful tool for understanding the workings of the suspension1 Figure 1 shows a diagram of the quarter car model The model consists of the following parameters

Applying Newtonrsquos Second Law the governing equations of motion for the sprung mass and unsprung mass are

To further simplify the equations of motion it is assumed that multltmsand ksltltkt The high stiffness of the tire spring rate combined with the lower mass of the unsprung weight limits the movement of mu and reduces equation (2) to zero The equation of motion for the sprung mass is then This equation simply states that the acceleration of the sprung mass is the sum of the spring rate times the movement of the damper and the damping coefficient times the damper velocity The reduced quarter car model (simple spring-mass-damper system) is shown in Figure2 Applying a Laplace transform to equation (3) to transform it from the time-domain to the frequency domain results in equation (4)

ms = sprung mass (weight of the car) (lb) mu = unsprung mass (wheel and suspension

components) (lb) ks = sprung mass spring rate (lbin) kt = tire spring rate (lbin) bs = sprung mass damping (lbsin) u(t) = displacement of the road as a function of

time

( ) ( ) ( )u u s s u s s u t um x b x x k x x k u x= minus + minus + minusampamp amp amp

( ) ( )s s s u s s u sm x b x x k x x= minus + minusampamp amp amp

s s s s s sm x k x b x= minus minusampamp amp

Eq (1) Eq (2)

Eq (3)

2 0s s

s s

b ks s

m m+ sdot + = Eq (4)

Eq (5)

2 22 0n ns sζ ω ω+ sdot sdot + = Figure 2 ndash Reduced Quarter Car model

THEORY

Comparing equation (4) to the ideal second order system equation (5) where ωnis the natural frequency and ζ is the damping ratio It can be seen that

The damping coefficient is chosen by specifying the damping ratio and the natural frequency is the frequency at which the spring mass system oscillates without the damper With the simplifying assumptions it is expected that the actual values will be lower than the theoretical values

Shock Damper

The shock damper is a critical component of the carrsquos suspension system because it helps control the car chassisrsquo response going into and out of turns (pitch and roll) as well as its response to variations in the road (bouncy or bumpy ride) Without a shock damper the spring would cause the suspension system to oscillate over an extended period of time as it compresses stores mechanical energy and then releases it past its neutral state to a point of extension and then it would continue to oscillate over an extended period of time until all of the stored energy has dissipated The shock damper dissipates energy stored in the spring usually in the form of heat In addition to controlling the carrsquos response in turns the damper also plays a vital role in controlling the temperature of the tires1 In automobile racing controlling tire load variation and maintaining an optimum temperature for the tires is very important as it increases the carrsquos mechanical grip The damping ratio varies and is tuned according to desired handling characteristics and track conditions These ratios are specified in Table 1 LS and HS refers to the damper shaft velocity where LS is low-speed and HS is high-speed Table 1- Damper Tuning Ratios1

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio

is 05 If the circuit is smooth choose 07 If the tires need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where

body control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Matlab Simulink Quarter Car Simulation

Simulink is a graphical approach to a Matlab environment it allows the user to simulate a real word environment such as a quarter car system The Simulink model is used to calculate specific values that are required to tune the SAE cars suspension

To begin the model requires specifications of constants (model parametersmeasured values) needed to produce the equations of motion These constants are placed in two summing blocks that formulate the two equations of motion for the sprung and unsprung masses These equations

sn

s

k

mω =2s n sb mω ζ= sdot sdot sdot Eq (6)

Eq (7)

THEORY

are shown below (Eg (8) and Eq (9)) After equations of motion are placed in the form that solves for the two accelerations of the un-sprung mass (mu)and the sprung mass (ms) the equations are then placed through two integrators the first representing the velocity of (mu)and (ms) and the second producing an equation representing the displacement of (mu)and (ms) These values of acceleration velocity and displacement for mu and ms are then fed back into the controlling parts of the equations of motion This feedback loop allows for continuous control of the damping coefficients

It was imperative for the model to simulate real world performance of the double barrel shock absorber In order to assure a real world simulation the damping coefficients for rebound and compression would have to be determined To determine the system was in rebound or compression the use of a detect increase and detect decrease block respectively These two blocks use the displacement feedback which was mention previously to sense the direction of the simulated damper shaft After each displacement block there resides a switch (1) that is turned on to feed the input of the velocity into another set of switches This switch (1) on the rebound and compression is to engage that particular damping regime Once that switch is on it feeds in the velocity of the damper shaft onto an array of another two switches These two switches switch (2) and switch (3) are responsible for delivering the damping coefficients to the driving part of the system These two switches are tuned to be active at different shaft speeds Switch (2) is on only when switch (1) is on switch (3) is activated when the shaft speed reaches a pre determined limit In this system only the specific damping coefficients that are need for a particular regime are activated this allows for a realistic system in a simulated environment This damping coefficient system can is shown in Figure 3

At the end of each simulation there are specific outputs placed in the Matlab workspace environment Once these outputs are in the workspace there can be manipulate to show the response to the inputs This allows the user to determine to performance of the frac14 car model in a simulated environment

Figure 3 ndash 4-way Damping Switch

( ) ( ) 0s s s s u s s u sm x b x x k x x m g+ minus + minus minus =ampamp amp amp

Eq (8) Eq (9)

( ) ( ) ( ) 0u u s u s s u s t u um x b x x k x x k x q m g+ minus + minus + minus minus =ampamp amp amp

EXPERIMENTAL SETUP

Shock Dynamometer

The Aurora V8 dynamometer itself was donated in non-working condition by a racing specialist Wayne Mitchell to help in the project and the optimization of the shock absorber for the race car A shock dynamometer is a machine used to test and measure force velocity and displacement for shock dampers

Transducers - Load Cell LVDT LVT

Looking in depth into the dynamometer there are three main componentssensors that make up the dynamometer These are the Linear Variable Differential Transformer (LVDT) Linear Velocity Transducer (LVT) and the load cell The load cell sensor is an Omega S-type Model that has a range of 0 to 2000 pound force (lbf) and measured in both compression and tension The LVDT (245 Model) and LVT (110 Model) are manufactured by Trans-Tek a specialist in transducers In particular the LVDT sensor measures linear displacement The transducer itself is a core within a cylindrical tube and contains primary and secondary magnetic coils in which change in voltage output is proportional to the movement of the core The LVT sensor on the other hand measures the velocity at which the motor is running The LVT is similar to the LVDT in terms of coils inside the cylindrical tube The difference being that no external excitation voltage is needed a magnet is passed between the coils and results in a DC voltage output proportional to the instantaneous velocity Again an amplifying circuit board and a data acquisition board were needed to read the voltage outputs out of all three sensors Software used for data logging was National Instruments Labview and MATLAB

Figure 4 ndash LVDT Displacement Transducer Figure 5 ndash LVT Velocity Transducer

Cane Creek Double Barrel Shock Absorber2

The shocks that were acquired for the SAE race car was the Cane Creek Double Barrel shock absorber manufactured by Cane Creek using the technology from the European company Ohlins The double barrel design allows the shock to have four independent external adjustments meaning each adjustment contributes independently to shock response The four independent adjustments consist of high and low speed compression and rebound The traditional shock

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

THEORY

Comparing equation (4) to the ideal second order system equation (5) where ωnis the natural frequency and ζ is the damping ratio It can be seen that

The damping coefficient is chosen by specifying the damping ratio and the natural frequency is the frequency at which the spring mass system oscillates without the damper With the simplifying assumptions it is expected that the actual values will be lower than the theoretical values

Shock Damper

The shock damper is a critical component of the carrsquos suspension system because it helps control the car chassisrsquo response going into and out of turns (pitch and roll) as well as its response to variations in the road (bouncy or bumpy ride) Without a shock damper the spring would cause the suspension system to oscillate over an extended period of time as it compresses stores mechanical energy and then releases it past its neutral state to a point of extension and then it would continue to oscillate over an extended period of time until all of the stored energy has dissipated The shock damper dissipates energy stored in the spring usually in the form of heat In addition to controlling the carrsquos response in turns the damper also plays a vital role in controlling the temperature of the tires1 In automobile racing controlling tire load variation and maintaining an optimum temperature for the tires is very important as it increases the carrsquos mechanical grip The damping ratio varies and is tuned according to desired handling characteristics and track conditions These ratios are specified in Table 1 LS and HS refers to the damper shaft velocity where LS is low-speed and HS is high-speed Table 1- Damper Tuning Ratios1

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio

is 05 If the circuit is smooth choose 07 If the tires need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where

body control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Matlab Simulink Quarter Car Simulation

Simulink is a graphical approach to a Matlab environment it allows the user to simulate a real word environment such as a quarter car system The Simulink model is used to calculate specific values that are required to tune the SAE cars suspension

To begin the model requires specifications of constants (model parametersmeasured values) needed to produce the equations of motion These constants are placed in two summing blocks that formulate the two equations of motion for the sprung and unsprung masses These equations

sn

s

k

mω =2s n sb mω ζ= sdot sdot sdot Eq (6)

Eq (7)

THEORY

are shown below (Eg (8) and Eq (9)) After equations of motion are placed in the form that solves for the two accelerations of the un-sprung mass (mu)and the sprung mass (ms) the equations are then placed through two integrators the first representing the velocity of (mu)and (ms) and the second producing an equation representing the displacement of (mu)and (ms) These values of acceleration velocity and displacement for mu and ms are then fed back into the controlling parts of the equations of motion This feedback loop allows for continuous control of the damping coefficients

It was imperative for the model to simulate real world performance of the double barrel shock absorber In order to assure a real world simulation the damping coefficients for rebound and compression would have to be determined To determine the system was in rebound or compression the use of a detect increase and detect decrease block respectively These two blocks use the displacement feedback which was mention previously to sense the direction of the simulated damper shaft After each displacement block there resides a switch (1) that is turned on to feed the input of the velocity into another set of switches This switch (1) on the rebound and compression is to engage that particular damping regime Once that switch is on it feeds in the velocity of the damper shaft onto an array of another two switches These two switches switch (2) and switch (3) are responsible for delivering the damping coefficients to the driving part of the system These two switches are tuned to be active at different shaft speeds Switch (2) is on only when switch (1) is on switch (3) is activated when the shaft speed reaches a pre determined limit In this system only the specific damping coefficients that are need for a particular regime are activated this allows for a realistic system in a simulated environment This damping coefficient system can is shown in Figure 3

At the end of each simulation there are specific outputs placed in the Matlab workspace environment Once these outputs are in the workspace there can be manipulate to show the response to the inputs This allows the user to determine to performance of the frac14 car model in a simulated environment

Figure 3 ndash 4-way Damping Switch

( ) ( ) 0s s s s u s s u sm x b x x k x x m g+ minus + minus minus =ampamp amp amp

Eq (8) Eq (9)

( ) ( ) ( ) 0u u s u s s u s t u um x b x x k x x k x q m g+ minus + minus + minus minus =ampamp amp amp

EXPERIMENTAL SETUP

Shock Dynamometer

The Aurora V8 dynamometer itself was donated in non-working condition by a racing specialist Wayne Mitchell to help in the project and the optimization of the shock absorber for the race car A shock dynamometer is a machine used to test and measure force velocity and displacement for shock dampers

Transducers - Load Cell LVDT LVT

Looking in depth into the dynamometer there are three main componentssensors that make up the dynamometer These are the Linear Variable Differential Transformer (LVDT) Linear Velocity Transducer (LVT) and the load cell The load cell sensor is an Omega S-type Model that has a range of 0 to 2000 pound force (lbf) and measured in both compression and tension The LVDT (245 Model) and LVT (110 Model) are manufactured by Trans-Tek a specialist in transducers In particular the LVDT sensor measures linear displacement The transducer itself is a core within a cylindrical tube and contains primary and secondary magnetic coils in which change in voltage output is proportional to the movement of the core The LVT sensor on the other hand measures the velocity at which the motor is running The LVT is similar to the LVDT in terms of coils inside the cylindrical tube The difference being that no external excitation voltage is needed a magnet is passed between the coils and results in a DC voltage output proportional to the instantaneous velocity Again an amplifying circuit board and a data acquisition board were needed to read the voltage outputs out of all three sensors Software used for data logging was National Instruments Labview and MATLAB

Figure 4 ndash LVDT Displacement Transducer Figure 5 ndash LVT Velocity Transducer

Cane Creek Double Barrel Shock Absorber2

The shocks that were acquired for the SAE race car was the Cane Creek Double Barrel shock absorber manufactured by Cane Creek using the technology from the European company Ohlins The double barrel design allows the shock to have four independent external adjustments meaning each adjustment contributes independently to shock response The four independent adjustments consist of high and low speed compression and rebound The traditional shock

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

THEORY

are shown below (Eg (8) and Eq (9)) After equations of motion are placed in the form that solves for the two accelerations of the un-sprung mass (mu)and the sprung mass (ms) the equations are then placed through two integrators the first representing the velocity of (mu)and (ms) and the second producing an equation representing the displacement of (mu)and (ms) These values of acceleration velocity and displacement for mu and ms are then fed back into the controlling parts of the equations of motion This feedback loop allows for continuous control of the damping coefficients

It was imperative for the model to simulate real world performance of the double barrel shock absorber In order to assure a real world simulation the damping coefficients for rebound and compression would have to be determined To determine the system was in rebound or compression the use of a detect increase and detect decrease block respectively These two blocks use the displacement feedback which was mention previously to sense the direction of the simulated damper shaft After each displacement block there resides a switch (1) that is turned on to feed the input of the velocity into another set of switches This switch (1) on the rebound and compression is to engage that particular damping regime Once that switch is on it feeds in the velocity of the damper shaft onto an array of another two switches These two switches switch (2) and switch (3) are responsible for delivering the damping coefficients to the driving part of the system These two switches are tuned to be active at different shaft speeds Switch (2) is on only when switch (1) is on switch (3) is activated when the shaft speed reaches a pre determined limit In this system only the specific damping coefficients that are need for a particular regime are activated this allows for a realistic system in a simulated environment This damping coefficient system can is shown in Figure 3

At the end of each simulation there are specific outputs placed in the Matlab workspace environment Once these outputs are in the workspace there can be manipulate to show the response to the inputs This allows the user to determine to performance of the frac14 car model in a simulated environment

Figure 3 ndash 4-way Damping Switch

( ) ( ) 0s s s s u s s u sm x b x x k x x m g+ minus + minus minus =ampamp amp amp

Eq (8) Eq (9)

( ) ( ) ( ) 0u u s u s s u s t u um x b x x k x x k x q m g+ minus + minus + minus minus =ampamp amp amp

EXPERIMENTAL SETUP

Shock Dynamometer

The Aurora V8 dynamometer itself was donated in non-working condition by a racing specialist Wayne Mitchell to help in the project and the optimization of the shock absorber for the race car A shock dynamometer is a machine used to test and measure force velocity and displacement for shock dampers

Transducers - Load Cell LVDT LVT

Looking in depth into the dynamometer there are three main componentssensors that make up the dynamometer These are the Linear Variable Differential Transformer (LVDT) Linear Velocity Transducer (LVT) and the load cell The load cell sensor is an Omega S-type Model that has a range of 0 to 2000 pound force (lbf) and measured in both compression and tension The LVDT (245 Model) and LVT (110 Model) are manufactured by Trans-Tek a specialist in transducers In particular the LVDT sensor measures linear displacement The transducer itself is a core within a cylindrical tube and contains primary and secondary magnetic coils in which change in voltage output is proportional to the movement of the core The LVT sensor on the other hand measures the velocity at which the motor is running The LVT is similar to the LVDT in terms of coils inside the cylindrical tube The difference being that no external excitation voltage is needed a magnet is passed between the coils and results in a DC voltage output proportional to the instantaneous velocity Again an amplifying circuit board and a data acquisition board were needed to read the voltage outputs out of all three sensors Software used for data logging was National Instruments Labview and MATLAB

Figure 4 ndash LVDT Displacement Transducer Figure 5 ndash LVT Velocity Transducer

Cane Creek Double Barrel Shock Absorber2

The shocks that were acquired for the SAE race car was the Cane Creek Double Barrel shock absorber manufactured by Cane Creek using the technology from the European company Ohlins The double barrel design allows the shock to have four independent external adjustments meaning each adjustment contributes independently to shock response The four independent adjustments consist of high and low speed compression and rebound The traditional shock

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

EXPERIMENTAL SETUP

Shock Dynamometer

The Aurora V8 dynamometer itself was donated in non-working condition by a racing specialist Wayne Mitchell to help in the project and the optimization of the shock absorber for the race car A shock dynamometer is a machine used to test and measure force velocity and displacement for shock dampers

Transducers - Load Cell LVDT LVT

Looking in depth into the dynamometer there are three main componentssensors that make up the dynamometer These are the Linear Variable Differential Transformer (LVDT) Linear Velocity Transducer (LVT) and the load cell The load cell sensor is an Omega S-type Model that has a range of 0 to 2000 pound force (lbf) and measured in both compression and tension The LVDT (245 Model) and LVT (110 Model) are manufactured by Trans-Tek a specialist in transducers In particular the LVDT sensor measures linear displacement The transducer itself is a core within a cylindrical tube and contains primary and secondary magnetic coils in which change in voltage output is proportional to the movement of the core The LVT sensor on the other hand measures the velocity at which the motor is running The LVT is similar to the LVDT in terms of coils inside the cylindrical tube The difference being that no external excitation voltage is needed a magnet is passed between the coils and results in a DC voltage output proportional to the instantaneous velocity Again an amplifying circuit board and a data acquisition board were needed to read the voltage outputs out of all three sensors Software used for data logging was National Instruments Labview and MATLAB

Figure 4 ndash LVDT Displacement Transducer Figure 5 ndash LVT Velocity Transducer

Cane Creek Double Barrel Shock Absorber2

The shocks that were acquired for the SAE race car was the Cane Creek Double Barrel shock absorber manufactured by Cane Creek using the technology from the European company Ohlins The double barrel design allows the shock to have four independent external adjustments meaning each adjustment contributes independently to shock response The four independent adjustments consist of high and low speed compression and rebound The traditional shock

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

EXPERIMENTAL SETUP

absorber has one tube inside that allows the oil to pass through narrow orifice in both compression and rebound With the twin tube technology there are different orifice (poppet valves) for the oil to travel for both low and high speed compression and rebound

Labview ndash Data Acquisition Software

In order to read the data coming off the dynamometer a few intermediate steps were needed First a 14-bit Digital Acquisition Card (DAC) was acquired The DAC is capable of taking positive and negative voltage inputs and then interfacing it to a computer via USB During initial testing it was determined that the DAC board did not have enough resolution to read the voltage output from the load cell therefore an amplifying circuit with a gain of 100 was made National Instrumentsrsquo Labview was used to log the voltage output from the three sensors on the dynamometer In Labview a Virtual Instrument (VI) interface was created for data acquisition Below in Figure 8 and Figure 9shows the VI that was created for data acquisition The voltage reading while the dynamometer was operating was logged at 4800 samples per second There is a low-pass filter that cuts off anything above 40 Hz

Figure 6 ndash Cane Creek Double Barrel cutout view2 (left) Figure 7 ndash Cane Creek Double Barrel external adjustment nuts2 (below)

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

EXPERIMENTAL SETUP

Figure 8- Virtual Instrument (VI) Block Diagram

Figure 9- Virtual Instrument Data Acquisition Screen

MATLAB ndash Data Processing Software

As stated above the Labview VI created data logs at a sample rate of 4800 Hz in a spreadsheet format specified by the user In order to plot recorded data Mathworks MATLAB was used A MATLAB code file was created in order to extract the points from the spreadsheet Commands were used to load the spreadsheet files to MATLAB and extract the data point into vectors The sampled data (in voltage) is used to calculate the force velocity and displacement measured by the transducers See Appendix C for the code

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

EXPERIMENTAL PROCEDURES

Sensor Equipment Calibration

Load Cell

Once the dynamometer was up and running a calibration of all the sensors was needed The first calibration was the load cell and it consisted of using chains and weights The first weight on the load cell was 35 pounds and was hung on using chain links The next four weights included 70 90 125 and 160 pounds This was then graphed versus the voltage output measured through the Labview VI software The resulting data is plotted using Microsoft Excel and then a linear curve fit applied with an equation to characterize the

LVDT

In calibrating the LVDT the cam shaft was first rotated so that the crank arm was at its lowest position The voltage was recorded to determine the offset The cam was then rotated manually to eight different positions and each time the displacement and voltage was recorded using a caliper and the Labview VI The data was then graphed as voltage versus displacement A curve fit was again applied to the plot resulting in an equation

LVT

The cam on the dynamometer contains three different radius positions in which the stroke and maximum velocity of the shock assembly attachment can be adjusted To calibrate the velocity transducer the cam radius setting was set at two different positions (50 75 in) and the voltage reading was recorded for each The maximum voltage for each maximum velocity during the maximum compression and rebound cycle was recorded To set the positions the cam bolt was removed and the cam was rotated The cam bolt is then placed into a different hole depending the desired radius Again it was graphed as voltage versus velocity where a trend line was added to acquire an equation

General Dynamometer Usage

After the calibrations of all three sensors were performed the dynamometer was ready to log data for the shock absorber The data acquisition equipment was first set up according to the setup information in the previous section The speed of the motor was set at its default setting which is the low speed The height of the apparatus was first adjusted according to the maximum displacement of the absorber and then was then placed onto the mounting points of the dynamometer The power supply for the data acquisition board was turned on which was set at a constant 24 volts The Labview VI on a laptop was then turned on as a continuous run and the dynamometer was switched on as well The Labview graph on the laptop displayed live output data of the load cell displacement and velocity transducer After a few seconds the ldquoLog Datardquo button on the Labview VI was pressed and data logging was initiated After approximately 3 seconds the data logging was stopped as well as the VI and the dynamometer The spread sheets of the data was saved in the default Labview format lvm The lvm files were then extracted to MATLAB and the data was then plotted This procedure for data logging was performed several times at different settings to get different results

Shock Characterization

The purpose of characterizing the damper was to gather information on the damperrsquos response in terms of damping coefficient when different adjustments were made to the damperrsquos adjustment

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

EXPERIMENTAL PROCEDURES

ppppp

bolts In attempting to find how the shock reacts to any knob change there were a number of testing procedures As seen in Figure 7 there are four ways to adjust the damperrsquos response First both the rebound and compression high settings were turned to high as well as the low settings for rebound and compression and the response was graphed on a force vs velocity curve Next all four settings were turned to low and the response was graphed Next all were turned to the middle and graphed Then everything was left at middle while one knob was turned one notch and this was graphed Little by little the same knob was turned one notch more each test to get an accurate picture of incremental change along with the already large change graphs from earlier On each of these graphs the slope was found which presented the damping coefficient at every change of the damper settings

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

EXPERIMENTAL RESULTS

Sensor Calibration

To calibrate the Load cell LVDT and LVT sensors quantitative values were measured with corresponding voltage and then plotted in Microsoft Excel For the load cell forces were measured the LVDT displacements and for the LVT the maximum velocities for each given cam radius The data was plotted using Microsoft Excel and then a linear curve fit was applied to each dataset Figure 10 shows the plotted data for the load cell calibration along with a linear curve fit The resulting slope obtained from the curve fit is 48255 lbfvolt with an offset of 5129 lbs The correlation coefficient (R2) for the curve fit to the load cell data calculated by Excel is 9995 The curve fits for the LVDT and LVT resulted in slopes of 1319 inchesvolt and 81719 inchessecvolt Examining the correlation coefficient for each of the equations determined from the linear curve fits indicates that the measured parameters have a very strong linear relationship to the measured voltages The closer the correlation coefficient is to 1 the higher the linear dependence Table 2 summarizes the determined equations for the calibration of each sensor along with the R2 values Figures A-1 and A-2 the curve fits for the LVDT and LVT sensors are located in Appendix A

y = 48255x + 5129

R2 = 09995

-50

0

50

100

150

200

-01 0 01 02 03 04

Voltage

Load

(Lb

s)

Figure 10 ndash Load cell Calibration Curve - Load vs Voltage

From the above calibration curve since the R2 value was high we can trust that our points are linear and therefore can expect accurate results from the following load cell data

Table 2 ndash Summary of Results for Calibration Curves

Sensor Equation for Calibration Curve fit R2

Loadcell y= 48255 (lbfvolt) x + 5129lb 09995

LDVT y= 01319 (involt) x + 4856in 09976

LVT y= 81719 (ins-1)volt x + 3523 ins-1 09988

Comparing the R2 values from the three sensors from our curve our sensors are accurate and will give us truer results during testing

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

EXPERIMENTAL RESULTS

Shock Characterization This graph shows the results of the four-way dampers being tested in two main configurations With all four damper settings set to the minimum damping coefficients the dark blue line shows the force over velocity By taking the slopes of this graph the exact damping coefficients can be found The same can be done with the light blue data which shows the results of testing after the four damper setting were turned to all maximum damping coefficients The purpose of the characterization tests are to find the outer limits of the shock damper this was achieved successfully which can be seen in figure 11

Figure 11 ndash Double Barrel shock response - All Low and All High Adjustment Settings

From figure 11 the main difference between the two colored lines is their slopes The light blue line is steeper and therefore has a higher slope This would be accurate since the light blue represents when the damper was tuned to high damping coefficient The slope of the force vs velocity graph shows damping coefficient and this means a higher slope is a higher damping coefficient and therefore the light blue line has a higher damper coefficient

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

EXPERIMENTAL RESULTS

Figure 12- Force vs Velocity - Low Speed Rebound

Figure 12shows readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the rebound regime The damping coefficient is therefore 1005 lbs ins The tables below show the difference between measured values of damping coefficient and the corresponding theoretical damping coefficients The R2 value confirms the accuracy of the model as a linear fit

Table 3 ndash Minimum Damping Coefficients ndash Measured

Table 4 ndash Maximum Damping Coefficients ndash Measured

Table 5 ndash Minimum Damping Coefficients ndash Cane Creek Data

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 1005 951 568 Rebound High Speed 2617 475 45000

Compression Low Speed 939 1109 Within parameters Compression High Speed 2441 475 41400

Regime Damping Coefficient (lbs ins)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 6697 951 Within parameters Rebound High Speed 1154 475 Within parameters

Compression Low Speed 6633 1109 Within parameters Compression High Speed 1175 475 Within parameters

Regime Damping Coefficients (lbs in s)

Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 100 951 515 Rebound High Speed 286 475 Within parameters

Compression Low Speed 100 1109 Within parameters Compression High Speed 286 475 Within parameters

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

EXPERIMENTAL RESULTS

Table 6 ndash Maximum Damping Coefficients ndash Cane Creek Data Regime Damping Coefficient

(lbs ins) Theoretical Damping Coefficients (lbs in s)

Percent diff ()

Rebound Low Speed 4580 951 Within parameters Rebound High Speed 233 475 Within parameters

Compression Low Speed 4580 1109 Within parameters Compression High Speed 233 475 Within parameters

Tables 3 and 4 show measured and theoretical values for the characterization tests A sample graph of the experiment is shown in figure 12 In table 3 the measured values are the absolute minimum damping coefficient found for the damper Comparing to the calculated theoretical values only the compression low speed could be tuned down to meet the theoretical value The other three values cannot be attained due to testing errors or internal tuning of the shock This is discussed in detail in the discussion Table 4 shows that all the damping coefficients can be achieved with simple tuning Tables 5 and 6 are comparing the experimental data from the manufacturer Cane Creek of the shock to our calculated theoretical damping coefficients The majority of their damping coefficients are within our theoretical calculations This shows the damper is capable of attaining our theoretical values but we were not able to attain them with the same shock during testing This leads us to believe that either the shocks were given retuned when given from the manufacturer or there were errors during the testing

Simulink Model

Figures 13 and 14 are the results from the theoretical Simulink model Figure 13 shows the step response of a typical car body with one constant damping coefficient while the second shows a step response with the four-way damper tested for the race car

Figure 13 ndash Single Damping Coefficient ndash Car Body Position vs Time

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

EXPERIMENTAL RESULTS

Figure 14 ndash 4-way damping ndash Car Body Position vs Time It is evident that the four-way damper displayed in figure 14 cuts down the settling time by 04 seconds compared to figure 13 which is quite significant in racing conditions This was expected because the advantages of the four way damper are that it can change its damping coefficient mid-stroke in both directions as opposed to one damping coefficient of the average car The discussion section will go into more detail on other advantages of the four-way damper found through theoretical modeling and testing

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

DISCUSSION

Sensor Calibration

The purpose of the calibration experiment is to establish a relationship between load velocity and displacement to voltage measurements from the transducers This was accomplished by taking known physical quantities and recording the voltage measurement for each that were sensed by the transducers Manufacture data sheet for the specific models of the transducers used was not available for comparison due to the old age of the dynamometer However the strong linear relationship between the load velocity and displacement to the measured voltage inspires confidence in its use for the experiment

Shock Characterization

The goal in the initial stages of characterizing the shock damper was to find the maximum and minimum limits of damping for the double barrel damper In the process of characterizing the shock dampers accuracy became an issue Initially we started with a data logger system that was reading the data at 12 bits The data proved to be unreadable and we switched to a data logger that would read at 14 bits this was a vast improvement and the data was far superior to data that was taken by the first data logger As a precaution data was procured from double barrel they provided a graph of force versus absolute velocity with this we determined the high and low damping coefficients It was clear from a comparison of the teamrsquos data versus the companies that the high and low damping coefficients we were not in agreement The graph from double barrel contained upper and lower limits that provided a bigger window of adjustment The reason for this could come from two sources The first being the internal adjustment that control the limits of damping could have been removed or increased this would provided more extreme upper and lower limits The other reason for this discrepancy could be the sensitivity of shock dynamometer sensing equipment The sensor that is in question is the load cell this unit may be providing false readings In the future this inconsistency will be resolved by consulting double barrel At the conclusion of this experiment the analysis at was as accurate as it could be without spending an exorbitant amount of money on new sensors if this is even the reason for the discrepancy

The empirical data that was retrieved from the shock damper provided the team with lower upper limits of the shock damper performance It was found that the high speed and the low speed were damping coefficient were nearly identical for compression and rebound The values for the upper and lower limits can be found in tables 3 and 4 in the results section of the report or in figures (blank thru blank) in the appendix From the theoretical calculation for the desired damping coefficients we find that the limits of the damper are insufficient The percent difference from the theoretical to the data retrieved from the dynamometer and the values derive from the graph received from cane creek can be seen in the results section on tables 5 and 6 It can be seen that the theoretical values can be achieve with respect to the cane creek data all except the low speed compression and this only has a percent different of 515

Simulink Model

To give the team a better idea of what would really be happening to the SAE car after we made our theoretical adjustments Simulink was utilized In this simulated environment all the theoretical hand calculations were implemented and the outputs were investigated to see if our model provided improved results There were several different aspects of performance that were felt to be

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

DISCUSSION

important to verify to insure that the cars performance would be optimized Among them was the displacement of the wheel (mu) and body (ms) and the difference between the two To make a comparison we also simulated a content damping scenario to compare the actively damped scenario When looking at these outputs the first thing that was evident was that the displacement of (mu) for the constant damped and four actively damped scenarios were almost exactly the same At first this was perplexing but after a moment it was realized that this was attributed to the tires lack of damping and not the damping of the car through the actual damping units The results of this can be seen in the appendix figures B1 and B2The next important area to look into was the displacement of the car body (ms) In the comparison of these graphs there was a significant improvement in the settling time of the actively damped system over the constantly damped system The improvement is seen in a shortening of settling for the four way damper time by approximately 04 seconds over the constantly damped system These graphs can be verified in the results section in figures 13 and 14 In addition to the body displacement the forces on the damper system can be analyzed it is seen in figure B3 That the main contributor between the spring and the damper is initially the damper it peaks out and tappers off and then the spring continues to give the greatest contribution to the resistance in the system As the forces go down from the initial step input past the origin it can be observed the damper in the only one to contribute This is consistent with the theory of damper force versus spring force

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

CONCLUSION

Error values have been accounted for throughout the experiment and have not been a significant factor in the determination of final values According the specification sheets acquired for similar LVDT and LVT models the maximum error values are 5 and 25 respectively These nominal values of non linearity are a definite plus in the experimentrsquos data acquisition A specification sheet was unavailable for the load cell however due to its accuracy during the calibration procedure it was safe to assume that itrsquos non-linearity is a negligible effect Error propagation of the natural frequency of the system and the damping coefficients were solved using the following equations

Uncertainty

2 22 2 a b

f ff

a bσ σ σpart part = + + part part

Uncertainty of Natural Frequency

( ) ( )2 2

2n nn s s

s s

k mk m

ω ωω part part∆ = ∆ + ∆ part part

The equation therefore becomes ( ) ( )2 2

2 2

3 2

1

22s

n s sss s

kk m

mm kω

∆ = ∆ + ∆

The calculated error for the natural frequency is 04 radss or 32

Damping Coefficient 2 n sb mω ζ= sdot sdot sdot with uncertainty of

( ) ( )2 2

2 2n nn s

n s

b bb m

mω

ω part part∆ = ∆ + part part

The damping ratioζ was specified as an exact value and therefore did not contribute to error

propagation The values of b for low speed and high speed are 1939 lbsin and 969 lbsin respectively The specified values for ζ low speed was 06 and 03 for high speed The equation

thus changed to ( ) ( ) ( ) ( )2 2 2 22 2s n n sb m mζ ω ω ζ∆ = sdot sdot ∆ + sdot sdot ∆ with a propagated error of

1152 (59) for low speed and 576 (59) for high speed Overall the propagated error remained below 10 in both the calculations of the natural frequency and the damping coefficients

Eq (10)

Eq (11)

Eq (12)

Eq (13)

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

CONCLUSION

As opposed to previous years a more theoretical approach was taken to tuning the shock dampers on the SAE race car This involved using the Quarter Car Model to determine the theoretical natural frequency of the system and the damping coefficient with respect to the damping ratio Damping ratio for the desired shock damper characteristics were specified to calculate the theoretical damping coefficients The shock dynamometer was successfully used to characterize the damping capabilities of the Cane Creek Double Barrel shock damper It was found that the theoretical damping coefficients for the desired handling characteristics of the race car are not achievable with the current configuration of the shock damper However according to the manufacturerrsquos test data the damping coefficients can be attained if the valves in the damping mechanism are changed

The calculated theoretical damping coefficients were entered into Simulink and a simulation was performed to observe the response of the race car suspension to a step response The step input simulated a sudden jerk from the steering wheel of the race car or an encounter with a bumpraised curb The Simulink model showed that the theoretical damping provided the desired suspension response For comparison the simulation was ran using a non-adjustable shock damper (one damping coefficient) and the result was that the suspension was less adaptable to the step input which caused the system to oscillate more over a longer period of time before reaching steady state Overall the experiment was successful in theoretically tuning the shock damper for specific desired handling characteristics in the race car The Simulink model confirmed the choice of damping coefficients for the shock damper Not only is this experiment useful in tuning the shock damper but it can also apply to specifying a shock damper for the application as well Although it empirical verification of the theory remains to be seen it is expected that the shock dampers to be well tuned for the race car and at the very least provide an optimum base starting point for tuning the suspension at the race track The only recommendation for the future is having access to or using a newer shock dynamometer with proprietary software that was made for the application

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

REFERENCES

1 Nowlan Danny ldquoFind the sweet spotrdquo Racecar engineering 182 (2008) 52-58

2 Double Barrel Shock Cane Creek 18 April 2008 lthttpwwwcanecreekcomdouble-barrel-shockhtmlgt

3 Rajamani Rajesh Vehicle Dynamics and Control Springer 2006

4 Dixon John C The Shock Absorber Handbook Society of Automotive Engineers 1999

5 Thomson William T and Marie Dahleh Theory of Vibration with Applications Prentice Hall 1998

6 Chen Yi ldquoVehicle Suspension System Modelingrdquo 2006

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

REFERENCES

Region Value Comments

ζ LS-Compression 05 This is body control so if it is a bumpy circuit the ratio is 05 If

the circuit is smooth choose 07 If the tyres need to be worked hard then choose 12

07 12

ζ LS-Rebound 03 This will be dependent on a number of factors Where body

control is not significant use 3 However where body control is significant use 5 to 7

05 07

ζ HS- Compression 03 This is the high pass area of the damping curve The values here should be 3 to 4 ζ HS- Rebound 04

Table A1 Region ζ Value b - Damping Coefficient (lbsin) Error ωn (rads)

LS - Compression 07 1109 065 590LS - Rebound 06 951 056 590HS - Compression 03 475 028 590HS - Rebound 03 475 028 590 ωn= (Ksms)

12

Damping Coefficient Formula b= 2ωnmsζ

195

Table A2

Calibration Curves

Figure A-1 ndash LVDT Calibration Curve ndash Displacement vs Voltage

APPENDIX ldquoArdquo ndashMEASUREMENTS amp SENSOR CALIBRATION FIGURES

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

APPENDIX

Figure A-2 ndash LVT Calibration Curve ndash Velocity vs Voltage

Figure A-3 ndash Low Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 939 lbs ins

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

APPENDIX

Figure A-4 ndash High Speed Rebound for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the rebound regime The damping coefficient is 2617 lbs ins

Figure A-5 ndash High Speed Compression for Lowest Settings These readings were taken when all four settings were turned to low and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of the compression regime The damping coefficient is 2441 lbs ins

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

APPENDIX

Figure A-6 ndash Low Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the low speed range of the compression regime The damping coefficient is 6633 lbs ins

Figure A-7 ndash High Speed Compression for Highest Settings These readings were taken when all four settings were turned to high and displays force vs velocity when the damping coefficient was found from the slope in the high speed range of of the compression regime The damping coefficient is 6637 lbs ins

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

APPENDIX

Figure A-8 ndash Shock Characterization - Force vs Velocity Low speed on lowest settings high speed on highest settings Low speed on highest settings High speed on lowest settings All adjustments on medium The following two figures display wheel hop and tire deflection are propagated by a step input of 1 inch this can be seen as the red line in the graphs When comparing the two graphs it is noticed that there is very little difference the reason being is tire spring rate being constant throughout

Figure B-1 ndashOne Damping Coefficient (Tire Position vs Time Wheel Hop)

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

APPENDIX

Figure B-2 ndash Four Damping coefficients (tire position vs time wheel hop)

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-3 ndash Force on Spring and Damper vs Time

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

APPENDIX

This figure shows the two forces encountered by the spring and the shock during a 1 inch bump situation These two components share the load of the step input

Figure B-4 ndash Velocity vs Time with change in damping coefficients This graph displays the velocity encountered in a step input and the damping coefficients that accompany the different velocities Matlab Code for extracting data from dynamometer and plotting

Load data and then graph clc clear all load testlvm load test_1lvm load test_2lvm load test_3lvm load test_4lvm ===== Extract data from file and store into Vecto rs ===== t=test(1) time vector f=48255test(2) + 5129 Load Cell voltage v=81719test(3) +03523 Velocity Transducer voltage d=test(4) Displacement Transducer voltage t1=test_1(1) time vector f1=48255test_1(2) + 5129 Load Cell voltage v1=81719test_1(3) +03523 Velocity Transducer voltage d1=test_1(4) Displacement Transducer voltage t2=test_2(1) time vector f2=48255test_2(2) + 5129 Load Cell voltage v2=81719test_2(3) +03523 Velocity Transducer voltage d2=test_2(4) Displacement Transducer voltage

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on

APPENDIX

changed radius to 5 and high speed t3=test_3(1) time vector f3=48255test_3(2) + 5129 Load Cell voltage v3=81719test_3(3) +03523 Velocity Transducer voltage d3=test_3(4) Displacement Transducer voltage changed radius 75 highspeed t4=test_4(1) time vector f4=48255test_4(2) + 5129 Load Cell voltage v4=81719test_4(3) +03523 Velocity Transducer voltage d4=test_4(4) Displacement Transducer voltage ================================================== ======== figure(1) plot(tftvtd) hold on plot(t1f1t1v1t1d1) hold on plot(t2f2t2v2t2d2) hold on plot(t3f3t3v3t3d3) hold on plot(t4f4t4v4t4d4) grid on figure(2) plot(abs(v)f b ) hold on plot(abs(v1)f1 c ) legend( All low All high Location SouthWest ) figure(3) plot(abs(v2)f2 r ) hold on plot(abs(v3)f3 m ) hold on plot(abs(v4)f4 y ) legend( low-low high-high Low-high high-low All medium Location SouthWest ) grid on