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Engineers, Part D: Journal of AutomobileProceedings of the Institution of Mechanical

http://pid.sagepub.com/content/218/3/259The online version of this article can be found at:

DOI: 10.1243/095440704322955795

2004 218: 259oceedings of the Institution of Mechanical Engineers, Part D: Journal of Automobile EngineeringJ B Hoyle

truck suspensionModelling the static stiffness and dynamic frequency response characteristics of a leaf spring

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259

Modelling the static stiness and dynamic frequencyresponse characteristics of a leaf spring trucksuspension

J B Hoyle

Alvis Vickers Limited, Hadley Castle Works, Telford, Shropshire TF1 6QW, UK

Abstract: The suspension characteristics of a 10 ton truck with a highly non-linear leaf springsuspension have been investigated. The acceleration transmissibility frequency response characteristics

of the truck were rst investigated. Results showed that whereas the bounce (sprung mass) resonantfrequency was quite close to that expected, the wheel-hop (unsprung mass) resonant frequencies wereroughly twice that expected. As no obvious mechanism for this eect was evident, the static stinesscharacteristics of the suspension were measured in an attempt to understand this phenomenon. Asimple plot of the resulting suspension-generated force against suspension deection showed that two

distinct stiness regimes were present; the rst, which was associated with small suspension deectionsof between 2 and 4 mm, was signicantly greater than the second, which was associated with largersuspension deections.

A state-variable linear 5 degree-of-freedom spring mass damper model of the vehicle was createdwithin MATLAB in an attempt to predict the frequency response characteristics of the vehicle. Thestate-variable model predictions were close to those seen experimentally for the sprung mass atfrequencies below 3 Hz, but were quite incorrect for both the sprung and unsprung masses abovethis frequency. The state-variable model was then used again with increased values of suspensionstiness and results showed that using these higher values led to a more accurate predicted responsefor both the sprung and unsprung masses above 3 Hz, whereas below this frequency their responseswere clearly incorrect. Comparison of experimental and predicted results showed that the vehiclefrequency response was dominated above 3 Hz by the low suspension deectionhigh stiness regime

identied in the static stiness results, whereas below this frequency, the high suspension deectionlow stiness regime tended to dominated the response.

To overcome the problem of the transition from the low to the high stiness regimes when predictingthe frequency response characteristics, a non-linear model was created using MATLAB andSIMULINK. Results showed that the model was able to capture this transition, with resulting fre-quency response curves for both the sprung and unsprung masses being very similar to the experimen-tal results. Further renements to the model were able to account for additional discrepancies between

predictions and experimental results.The non-linear nature of the spring stinesses, resulting in a very useful hysteresis damping eect

at the sprung mass resonant frequency, meant that for most normal road conditions the vehicle wouldride on a near solid suspension. The apparent advantages of the extra damping would therefore beoset by a signicant decrease in ride quality, and this was especially evident in respect to road noise

and vibrationa nding borne out in practice.This paper establishes the principal characteristics of the truck suspension and goes on to describe

the linear state-variable and non-linear models created to simulate the frequency response character-istics of the vehicle suspension.

Keywords: static stiness, dynamic frequency response, leaf spring truck suspension

1 INTRODUCTION that the leaves are similar, the eective stiness of the

suspension can be considered to be the sum of the

stinesses of the individual leaves.Leaf spring suspensions can consist of a single leaf orOne feature of leaf spring suspensions is their rela-multiple leaves in parallel. In the latter case, provided

tively large amount of inherent damping compared to

other forms of suspension design. As a multileaf springThe MS was received on 4 April 2003 and was accepted after revisionfor publication on 11 September 2003. suspension deects, the leaves must slide over one

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260 J B HOYLE

Fig. 1 Vehicle mounted on a four-post rig

another at some point along their length or else they energy was lost into the suspension during cycling due

would act like a solid beam. Due to friction between the to the hysteresis eects of Coulomb friction. Otherleaves, the sliding of the leaves generates friction forces results by the same authors showed that the Coulomb

that resist further deection of the suspension. This friction increased with load and deection and that theresistance to further deection acts like normal damping hysteresis losses were independent of frequency.in as much as it is direction dependent, but because it

tends to be Coulombic in nature (stiction), it does not2 TRUCK LEAF-SPRING SUSPENSIONact like the viscous damping associated with normal

shock absorbers. This form of Coulomb friction is not CHARACTERISTICS

conducive to the creation of good ride characteristics for

the vehicle because it is dicult to control. Both the acceleration transmissibility frequency responseFor designs where a number of similar leaves are used characteristics and the static stiness characteristics of a

in parallel, the leaves will all carry approximately the 10-ton truck with leaf-spring suspension have beensame load. This means that where the leaves are in con- investigated using a four-post rig. This rig, partly showntact, large friction forces will be generated when the sus-

in Fig. 1, consisted of four servo-controlled hydraulicpension deects and the leaves slide over one another. vertical-axis actuators with horizontal plates on the ends,Furthermore, if a certain load has to be applied to the one plate and actuator to support each wheel. The actu-suspension in order to overcome the friction between the ators incorporated displacement and load transducers,leaves, then until this load has been reached the leaves the former for position feedback control and the latterwill act like a solid beam. Implicit in this scenario is the for load feedback control. Instrumentation was includedpossibility that for small deections the locked leaves that enabled frequency sweeps and data capture to bewill have a dierent, and probably higher, stiness than conducted, as was a computer for the analysis of thethe sum of the stinesses of the individual leaves. resulting data.

The static stiness of multileaf spring suspensions has

been measured by various groups [13 ]. The various

authors presented results which showed that, depending 2.1 Dynamic resultson the suspension design, the stiness was highly non-

The bounce (sprung mass) and wheel-hop (unsprunglinear in the fashion described above. Dierent stiness

regimes were evident depending on the amplitude, and mass) acceleration transmissibility frequency responses

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261MODELLING THE STATIC STIFFNESS AND DYNAMIC FREQUENCY RESPONSE CHARACTERISTICS

were derived from the four post rig. Sine wave displace- 2.1.3 Predicted natural frequencies

ments were input to the wheels via the actuators, theirApproximate values for the undamped natural frequen-frequencies being increased in increments while thecies of the sprung and unsprung masses can be predictedamplitude decreased similarly. Accelerometers mountedusing standard techniques [4 ]. These techniques assumeon the wheels, the vehicle body and the actuators enabledthat the vehicle can be modelled as a simple linearthe acceleration transmissibility frequency responses to2 degree-of-freedom (DoF) system. This is a reasonablebe generated.assumption considering that the unsprung natural fre-

quencies were very similar and that the vehicle body2.1.1 Acceleration transmissibility frequency responsesexperienced very little pitch in the primary bounce mode.

For an undamped linear 2 DoF system, the predictedThe results from the transmissibility frequency responsesprung (v

s) and unsprung (v

us) resonant frequencies aretests are shown in Fig. 2a (sprung mass) and Fig. 2b

given by(unsprung masses). Figure 2a shows three curves, these

being the response of the centre of gravity of the vehicle vs=1/2 K

sK

t/[M

s(K

s+K

t)] Hz (1)

body (marked CG), the front of the vehicle bodyv

us=1/2 (K

s+K

t)/M

usHz (2)

(marked F) and of the rear (marked R). Figure 2bwhere M

sis the sprung (vehicle body) mass, M

usis theshows four curves, one curve for each wheel. The two

sum of the two unsprung masses (front and rear axlescurves for the wheels on the front axle (marked F) werewith wheels), K

sis the sum of the front and rear suspen-very similar, as were those for the wheels on the rear

sion stinesses and Kt

is the sum of the front and rearaxle (marked R). tyre stinesses. Substitution of experimentally measured

values into these equations [8210 kg (Mb-M

e), 1300 kg

2.1.2 Sprung and unsprung mass natural frequencies(M

f+M

r), 1.74 MN/m (K

Lowf+K

Lowr) and 4.28 MN/m

(2Kt) respectively (see Table 1)] give predictions for theThe bounce (sprung mass) resonant frequency and corre-

bounce (sprung mass) and wheel-hop (unsprung mass)sponding amplitude ratio of the centre of gravity of thenatural frequencies as 1.95 and 10.8 Hz respectively.vehicle body can be seen (in Fig. 2a) to have beenWhereas the bounce natural frequency prediction was2.44 Hz and 10.24 dB respectively, with the natural fre-close to that measured experimentally on the actualquency of the front of the vehicle being slightly lowervehicle (2.44 Hz), the wheel-hop natural frequency wasand that of the rear being slightly higher. This indicatednot. The latter, at 19 Hz, was roughly twice that pre-that although the vehicle body responded mainly indicted by equation (2). It is in response to these discrep-bounce, there was a slight amount of pitch evident, withancies that further experimental and indeed theoretical

the slightly higher natural frequency and lower ampli- investigations were carried out.tude of the rear reecting the slightly stier suspension

and greater damping (eects of the assistor springs).

The wheel hop (unsprung mass) resonant frequency 2.2 Static stiness resultsand amplitude ratio can be seen (see Fig. 2b) to have

been about 19 Hz and 1 dB respectively, with both front In an attempt to determine the reason for these unusual

frequency response characteristics, the static stinesseswheels being very similar to the rear wheels.

Fig. 2 (a) Experimental sprung mass frequency responses and (b) experimental unsprung mass frequency

responses



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262 J B HOYLE

Table 1 Look-up table of data used in the model

Symbol Vehicle parameter Source Value

Ce

Damping of engine mounts Estimated 2.28 kN s/mC

fand C

rLinearized front shock absorber damping for 5 DoF state-variable Calculated 20 kN s/m

model; value based on manufacturers damper dataIb

Pitch moment of inertia of vehicle body; derived from Mb

(Lfr/2)2 Calculated 23000 kg m2

and rounded to nearest whole thousandK

asrrStiness of assistor spring relaxation and recovery element Estimated 80 kN/m

Kax Stiness of axle stand Estimated 10.0 MN/mK

cA constant dening the velocity input to the friction plates in the

leaf-spring modelK

eStiness of engine mounts (total stiness) Estimated 1.3 MN/m

KHighf

Total locked leaf stiness of front suspension (sum of left and Measured 4.82MN/mright springs)

KHighr

Total locked leaf stiness of rear suspension (sum of left and Measured 5.04M N/mright springs)

KLowf

Total n ominal stiness of front suspension (sum of left and right Measured 850 kN/msprings)

KLowr

Total nominal stiness of rear suspension (sum ofleft and right springs) Mesured 890kN/mK

stStiness of tie-down strop Estimated 5.0 MN/m

Kt

Total stiness of two tyres in parallel (front two tyres being Measured 2.14MN/midentical to the rear two)

Laccelf

Distance from body CG to front body mounted measurement Measured 2.83maccelerometer

Laccelr

Distance from body CG to front body mounted measurement Measured -2.75 maccelerometer

Lax

Distance fro m bo dy CG to ax le stand (fro nt and rear) Me asure d 2.5 7 m an d-1.70 mL

eDistance fro m bo dy CG to en gin e CG a nd moun tings Me asure d 1.7 0 m

Lf

Distance from body CG to front axle Measured 1.574L

frWheel base Measured 3.3 m

Lr

Distance from body CG to rear axle Measured -1.726 mL

stDistance from body CG to tie-down strop (front and rear) Measured 2.83m and-2.72 m

Mb

Body (sprung) mass Measured 8610 kgM

eMass of engine Estimated 400 kg

Mf

Mass of front axle (unsprung mass) Estimated 700 kgM

rMass of rear axle (unsprung mass) Estimated 600 kg

yb

, ybb

, yb

Vertical displacement, velocity and acceleration of vehicle bodyy

e, yb

e, y

eVertical displacement, velocity and acceleration of engine

yf

, ybf

, yf

Vertical displacement, velocity and acceleration of front axle

yr,yb

r, yr Vertical displacement, velocity and acceleration of rear a xleY1

, Yb1

Suspension displacement and velocity input into the leaf-springmodel and assistor-spring model

Y2

Displacement of the friction plates in the spring-modelY

3Displacement of relaxation and recovery element in the

leaf-spring modelY

4Nominal d eection of the rubber assistor spring

Y5

Deection of relaxation and recovery element in the assistor-spring model

mf

Eective coecie nt of fric tion of f ro nt suspe nsion leav es Me asure d 0.0 9m

rEective coecient of fric tion of re ar suspe nsion lea ves Me asure d 0.1 2

Hb

, Hbb

, Hb

Pitch angular displacement, velocity and acceleration of vehicle body

of the suspension were measured. This was achieved by tive to the vehicle body) and Figs 3b and 4b are the time

histories of the suspension-generated load, deection oftying the vehicle chassis down to the actuator bed plateand jacking up the actuators under the wheels. Separate the suspension and the velocity of the suspension deec-

tion (axle velocity relative to vehicle body velocity).displacement transducers were used to measure the sus-

pension deection of the axles relative to the vehicle Figures 3 and 4 are for the front right and rear right

suspensions respectively and are typical of the left sides.body, and the load transducers in the actuator heads

were used to measure the resulting reaction at the tyre-

to-ground contact. Load versus deection curves were 2.2.1 Stiness characteristicsthus generated for the suspension, and all these curves

included a load oset due to the static weight of the The dynamic characteristics of the vehicle are generally

determined by the stiness and damping characteristicsvehicle.

The results that were obtained are shown in Figs 3 of the suspension, and the sprung and unsprung masses.

Measuring the dynamic characteristics of the suspensionand 4, where Figs 3a and 4a are the plots of suspension-

generated load (actually the tyre-to-ground contact is dicult due to the large number of variables involved.

However, measuring the static characteristics is moreforce) against suspension deection (axle movement rela-



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Fig. 3 (a) Experimental front static suspension stiness and (b) experimental front stiness time histories

Fig. 4 (a) Experimental rear static suspension stiness and (b) experimental rear stiness time histories

straightforward and can generate useful information for 2.2.2 Rubber bush relaxation

the creation of models of such suspensions.When the suspension was held at a constant deectionIn the plots of generated load against suspension dis-for the period of time after a change in deection hadplacement (Figs 3a and 4a) for this vehicle, six dierentoccurred, two more regimes were identied (regimes 3regimes were identied, these being numbered 1 to 6.and 6). When the suspension was loaded up and thenThe curves were non-linear and two distinct stinessesheld, the generated load at the tyre-to-ground contactwere evident in the stiness characteristics. Regimes 1dropped o slightly. Similarly, when the suspension wasand 4 were associated with the suspension leaves whensubsequently unloaded and then held, the generated loadthey were in their locked state due to friction, the leavesincreased slightly. This phenomenon was assumed to beacting like a solid beam to produce a stiness that was

due to relaxation and recovery of the numerous rubbermuch higher than the sum of the stinesses of the indi-bushes that were incorporated in the suspension.vidual leaves. Regime 1 was generated at the initiation

of increased suspension deection as the already stati- 2.2.3 Rubber assistor springscally loaded suspension was further loaded, whereas

regime 4 was generated at the initiation of suspension Although nominally identical, for deections greater

than those of static equilibrium, the rear suspension dis-recovery as the greater-than-statically-loaded suspension

was subsequently unloaded. played a considerably higher stiness than did the front,

with also more relaxation and recovery of the suspensionRegimes 2 and 5 were associated with the suspension

leaves when they had broken free from their locked-up load when it was held at a constant deection. This was

due to the rear suspension having rubber assistor springsstate and exhibited stinesses that were close to those

expected from the sum of the stinesses of the individual in parallel with the normal leaf springs. These assistor

springs were tted in such a way that they just contactedleaves. Regime 2 was for the time when the suspension

was loaded up and regime 5 was for the subsequent the axle at the normal ride height of the vehicle, and

thus only assisted the leaf springs for deections greaterunloading of the suspension.



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264 J B HOYLE

than those caused by the static loads in the suspension. loading and unloading parts of the curves respectively,

represent the changes in the suspension loads from aThe relaxation and recovery eects of these assistor

springs can be clearly seen in a comparison of regimes static state that were required to break the suspension

springs free from their locked states. These values were3 and 6 in Figs 3a and 4a.

derived from regimes 1 and 4 in the curves shown in 2.2.4 Suspension stiness values Figs 3a and 4a. These breakaway loads must in some

way be related to the coecient of friction between theThe nominal stiness of the leaf springs was quoted by

leaves and the load carried by the suspension. A valuethe manufacturer as 480 kN/m (120 kN/m per leaf ), that is related to the coecient of friction between thethere being four identical leaf spring assemblies on the

leaves could thus be derived, and is shown in the lastvehicle. If the loading and unloading regimes (regimes 2

row in Table 2.and 5) are considered for the front suspension, the aver-

The coecients of friction derived were low comparedage of these gives a nominal stiness for the suspension

to normal steel to steel friction, but this was due to theof 410 kN/m, considerably less than quoted by the

large mechanical disadvantage under which the frictionmanufacturer.

acted. The higher stiness beam was only created by theThe same could not be measured directly for the rear

inter-leaf friction preventing the leaves sliding relative tosuspension because of the inclusion of the rubber assistor

one another at their outer ends. As the leaves were nom-springs. However, the manufacturers quoted stiness

inally straight and near-horizontal in the staticallyfor the assistor springs was 0.15 MN/m. This value can

loaded condition, the unsupported centre of the leavesbe subtracted from the loading and unloading portions

was able to move signicantly further in the verticalof the rear suspension stiness curves, and the results of direction than could the supported outer ends in thethis exercise are shown in Table 2 (bracketed values

horizontal direction. This meant that even a high inter-being without the eects of the assistor springs). The

leaf coecient of friction preventing the ends of theaverage of the loading and unloading regimes (regimes

leaves from moving horizontally relative to one another2 and 5) can thus be calculated, and at 460 kN/m is

translated into only a small eective coecient of fric-much closer to the manufacturers quoted stiness.

tion preventing the centre of the leaves from moving inThe stiness of the leaf springs in regime 2 (loading)

the vertical direction, and it is this latter value that haswas higher than in regime 5 (unloading). The dierences

been measured directly in the static stiness results andbetween these stiness regimes can be accounted for by

appears in Table 2.the fact that the friction between the leaves was pro-

The stiness of regimes 1 and 4 when the suspensionportional to the load being carried by the spring. As this

leaves were locked together to form a solid beam canload was increased, so too was the load required to over-

also be seen in Table 2. These stinesses were all greatercome the friction between the leaves and unlock them.than those of regimes 2 and 5. Table 3 shows the aver-

Thus the dierence between the curves on the loadaged stinesses for these regimes (averages of regimes 1

against displacement plots at any one suspension deec-and 4 and of 2 and 5, for both front and rear suspen-

tion should have been greater at the higher loads thansions). The values for the stinesses of the rear suspen-

at the lower loads. This eect, resulting in a dierencesion springs in regimes 1 and 4 (bracketed values in

in the slopes (stiness) of the curves, was just aboutTable 2) were obtained, as described above, by sub-

detectable in the front suspension, but was rather moretracting the assistor spring nominal stiness (the value

evident in the rear suspension (compare regimes 2 andof the relatively linear portion below deections of

5 in Table 2).40 mm of the assistor spring) from the measured

The rows marked breakaway force loading andsuspension stinesses.

breakaway force unloading in Table 2, referring to theThese stiness characteristics were further analysed to

give the ratio between these low and high stinessregimes (the average of regimes 1 and 4 compared to theTable 2 Suspension stiness characteristicsaverage of regimes 2 and 5). These are also shown in

Rear rightSpring Front right Rear right bump spring

Stiness (MN/m)Table 3 Consolidated stiness characteristicsRegime 1 1.60 2.30 (2.15) 0.15

Regime 2 0.42 0.67 (0.52) 0.15Front right Rear rightRegime 4 3.00 3.08 (2.93) 0.15

Regime 5 0.40 0.55 (0.40) 0.15 Spring position suspension suspension

Break away force ( KN)Averaged low stiness

Loading 4.0 5.5(regim es 2 and 5) 0.41 MN/m 0.46 MN/m

Unloading 5.5 13.0Averaged high stiness

(regim es 1 and 4) 2.30 MN/m 2.54 MN/mEective coecientof friction 0.09 0.12 Stiness ratio 5.61 5.52



8/21


Table 3, the stiness ratio for the front being very similar MATLAB. The model and its predictions are given in

full in the Appendix.to that of the rear.

It is evident that the suspension exhibited two dierent When using the nominal stinesses (lower values) of

the leaf springs, the linear 5 DoF state-variable modelstiness regimes, which were the nominal stiness

expected from the sum of the individual leaves and a was able to predict the resonant frequency of only the

sprung mass with any reasonable accuracy (1.8 Hz pre-higher stiness that was a result of the leaves locking

together to form a solid beam of some form. The dicted compared to 2.44 Hz). Furthermore, the predicted

sprung mass amplitude ratio (13.5 dB) at this frequencyresulting stiness characteristics would have a denite

energy absorbing capability (damping) once the leaves was greater than that seen experimentally (about 10 dB).

Above this frequency the predictions completely failedhad broken free from their locked-up state. The amount

of energy dissipated into the suspension would be pro- to match the experimental behaviour of either the sprung

or unsprung masses (compare Figs 1a and b with Figsportional to the area of the loop in the load against

deection curves, which would produce a contribution 20a and b in the Appendix respectively).

A much better prediction for the sprung and unsprungto the overall damping in the suspension.

The stiness characteristics of the assistor springs were masses was obtained using the higher stiness values of

the locked-leaf suspension, but again this was only trueobtained from the manufacturer. They were of the

bellow type and acted as a bump spring only in the for frequencies above 3 Hz (compare Figs 2a and b with

Figs 21a and b in the Appendix respectively). Below thisextremes of deection (bellow bound!). During test,

they had been subjected to a sine wave displacement with frequency, the linear model predictions again completely

failed to match the experimental results. The close matchan amplitude of 0.0295 mm and a frequency of 2 Hz, theassistor spring being kept in contact with the actuator between the linear state-variable model and the exper-

imental results above 3 Hz indicated that above this fre-over the whole stroke. The load versus deection charac-

teristics for these assistor springs are shown as the curve quency the response of the vehicle was dominated by the

higher locked-leaf state of the leaf springs, at least fororiginating and terminating at the origin in Fig. 5. This

curve shows that the assistor springs exhibited hysteresis- the amplitudes of excitation used during these tests.

It was clear that the linear model was unable to simu-type behaviour, in that a characteristic loop was evident.

The stiness of the assistor spring was found by taking late the transition from the high deectionlow stiness

regime of the leaf springs to the low deectionhighthe average of the loading and unloading portions of the

loop over the lower half of the deection range. For low stiness regime when making predictions for the fre-

quency responses. It was for this reason that a signi-deections of less than 40 mm, the springs had a stiness

of about 0.15 MN/m. cantly more complex non-linear model was created using

MATLAB and SIMULINK.

4 NON-LINEAR 5 DoF MODEL

The unusual frequency response behaviour of the vehicle

and subsequently identied low and high stiness

regimes in the static stiness measurement results,

together with the failure of the linear state-variable

model to accurately predict the frequency response over

the full frequency range, indicated where the emphasis

was to be placed during the creation of the non-linear

model. There were three component parts to the suspen-

sion that deserved special attention: the leaf springs (sec-

tion 4.1), the rubber assistor springs (section 4.2 ) and

the shock absorbers (viscous dampers) (section 4.3) . TheFig. 5 Assistor spring stiness characteristics

models for these are developed separately, with all

notation and values being given in Table 1 and Fig. 6.

3 LINEAR 5 DoF STATE-VARIABLE MODEL4.1 Leaf-spring model

Various methods by which the non-linear stiness andHaving measured the frequency response of the vehicle

and its static stiness characteristics, an attempt was friction characteristics of leaf-spring suspensions can be

modelled are available [3, 4 ]. The method adopted heremade to predict the frequency responses of the vehicle

using a linear 5 DoF state-variable model within is described below. As there were two relatively clearly



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266 J B HOYLE

Fig. 6 Suspension leaf bush, assistor spring and shock absorber characteristics

delineated stiness regimes, it was anticipated that a sus- ated by the higher stiness element of the suspension,

KHigh

, that would act on the axle and therefore react onpension model that had two springs in series, with some

kind of friction model between the two, might suce, as the vehicle body. The load P was assumed to be non-

zero for all suspension deections other than that of theshown in the right-hand half of Fig. 7. The friction

device would consist of friction plates with a load acting completely unloaded condition (unladen free length of

spring).on them in such a manner as to push them together.

This load, P in Fig. 7, was assumed to be proportional One of the friction plates was xed to the vehicle body.

The friction plate attached between the low and highto the load generated by the nominal stiness element

of the suspension, KLow

, but it would be the load gener- stiness springs slides relative to this body-mounted



10/21


Considering the high and low stiness elements,

Fk=(Y

1-Y

2)K

High(4)

where the deection of the low stiness element, KLow

,

was denoted by Y2

. However, this force was resisted by

the forces generated by the force P acting on the friction

plates and the reaction of the low stiness element. The

sum of the resistive forces, Fresist

, was given byFig. 7 Leaf-spring suspension model

Fresist=-(Y

2K

Low)(mP) (5)

plate only when the high stiness spring had been com- where m was the eective coecient of friction betweenpressed or extended suciently to overcome the friction the friction plates. The force P was dened as

loads generated by the plates and the resistive loads gen-P=-(Y

2K

Low)+P

0(6)

erated by the low stiness spring. Thus for small deec-

tions of the statically preloaded suspension, the stiness where P0

was the preload at the zero suspension loadwould generally be that of the high stiness spring, (the free length of spring). Load P was assumed to bebecause the friction plate would not slide relative to the generated by the low stiness element as it was the grossvehicle body. For larger deections, the load generated (large) deections of the springs that produced the sup-by the high stiness spring would be sucient to over- portive loads that carried the weight of the vehicle, and

come the friction and resistive loads generated by the the high stiness elements were only created by the leavesplates and low stiness spring, and thus force the plates locking together in the static condition when this loadto slide relative to one another. In these circumstances, was exerted by the springs. The resistive forces thereforethe stiness characteristics would be largely governed by becomethe low stiness element of the suspension. The resistive

Fresist=-(1m)K

LowY

2(7)forces produced by the friction plates would be pro-

portional to a load P acting on the plates and the where P0

was assumed to be zero. The direction of thecoecient of friction between the plates, and would friction forces was given by the sign in equation (7),always act to oppose the axle motion relative to the such that the friction forces a lways opposed the directionvehicle body. of deection of the suspension.

The other element of the leaf-spring model was the When the suspension was in the locked state with therelaxation and recovery regimes generated by the use of leaves acting like a solid beam, the forces generated by

rubber bushes in the suspension. This element was mod- the high and low stiness elements of the suspensionelled as a spring in series with a damper/Coulomb fric- were given by equation (4), where Y

2remained constant.

tion element, as shown in the left-hand half of Fig. 7. However, when the leaves became unlocked and slippedThe stiness of the bush and the damping/Coulomb fric- relative to one another, the force generated by thesetion element ( the two in series allowing a degree of relax- springs was still given by equation (4), but the deectionation and recovery in the suspension) are denoted by Y

2did not remain constant. This displacement, Y

2, was

KBush

and DBush

respectively and were modelled in paral- derived by integration of a component of the inputlel with the main suspension stiness element to generate velocity Yb

1.

only the relaxation and recovery eect. These suspension characteristics were implemented in

the model with the use of logic operators.4.1.1 Leaf-spring suspension governing equations

Leaf-spring suspension relaxation and recoveryThe equations governing the force output from the leaf- characteristics. The forces generated by the relaxationspring suspension model are developed below. The

and recovery element in the suspension, Frr

in equa-inputs into the model were the deection and velocitytion (3), were given byacross the suspension, Y

1and Yb

1respectively, which are

derived from the relative motions of the axles and Frr=(Y

1-Y

3)K

Bush(8)

vehicle body.How this spring in series with a damper was

implemented in the model is described below.Leaf-spring suspension non-linear stiness characteristics.

The force on the vehicle generated by the suspension,4.1.2 Model implementationF

susp, was made up of two components, these being the

force generated by the high and low stiness elementsThe model of the leaf springs was split into two dis-

and that generated by the relaxation and recoverytinct parts: that dealing with the non-linear stiness

element, Fk and Frr, respectively. Thus: characteristics and that dealing with the relaxation and

recovery characteristics.Fsusp=F

k+F

rr( 3)



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268 J B HOYLE

Leaf-spring suspension non-linear stiness characteristics. needed as far as the logic was concerned, but inclusion

of these made the implementation of the model muchThe block diagram for the implementation of the non-

linear stiness characteristics is shown in Fig. 8. The more stable.

inputs to the model were the relative deection, Y1

, andLeaf-spring suspension relaxation and recoverythe relative velocity, Yb

1, across the suspension.

characteristics. The block diagram for the implemen-The suspension deection was rst compared to thetation of the relaxation and recovery element of the leafdisplacement of the slider, or friction plate, Y

2, to calcu-

springs is shown in Fig. 9. The input into the model waslate the force in the high stiness spring [equation (4)].the deection, Y

1, across the suspension. This was com-This force was then compared to the resistive forces

pared to the deection of the damper, Y3

, and the result-created in both directions by the friction plates (frictionant deection of the spring, of stiness K

Bush, was usedcoecient ofm) and the low stiness spring.

to nd the force acting on the damper [equation (8)].If the high stiness force was greater than the higherAs this type of model was always in a state of equilibriumlimit, or less than the lower limit, then the logic oper-(because there was no inertial mass associated with it),ator blocks returned a value of 1 ( logical true) tothe force generated by the spring would always equalswitch 1. If the inputs to input 2 of either of the switchthat developed by the damper. Thus the spring force wasblocks exceeded their respective thresholds (both set todivided by the damping in order to derive the velocity0.5), then the switches would pass through their input 1.of the damper. This velocity, Yb

3, was then integrated toIf it did not exceed their thresholds, they would pass

derive the displacement of the damper.through their input 3. If the input threshold (input 2) ofA linear version of this system was tested against the

switch 2 was exceeded by the output from switch 1 (the normal transfer function representation of this type oflatter being the state of the logic operators), it wouldmodel, to demonstrate that it worked. The advantage ofpass through a component of the suspension velocity tothe implementation presented here was that non-linearthe integrator, which in turn would then derive thestinesses and damping could be incorporated in thedeection of the friction plates, Y

2.

model by replacing the linear gain blocks by look-upThe velocity passed to the integrator was not thetables. Indeed, the damping included a Coulomb-likeabsolute velocity across the suspension, but a componentfriction element as well as a viscous damping element,of the velocity in proportion to the respective stinessesas shown in Fig. 7. The viscous damper element wasof the in-series low and high stiness springs. Theneeded to derive a meaningful velocity; i.e. the velocitycomponent was given by the constant K

c, where

had to be in some way proportional to the generated

force. Failure to do this would result in a highly unstableKc=

KHigh

KHigh+

KLow

(9)

model. The velocityforce relationship of the damper/

Coulombic friction element is shown in Fig. 6a.with the velocity being passed to the integrator always

being less in absolute terms than the velocity across the

suspension, Yb1

. The constant labelled preload [P0

in

equation (6)] was set to zero. A non-zero constant here

would create a friction force even when the force created

by the low stiness spring was zero. Switch 1 and the

sign block, the latter returning a 1 or -1 depending

on the sign (direction) of its input velocity (relative veloc-

ities of the axle and vehicle body), were not strictly

Fig. 9 Relaxation and recovery element block diagram

4.2 Rubber assistor springs

As noted previously, the rubber assistor springs were

only included in the rear suspension, and even then only

acted for suspension deections beyond the static deec-

tion of the springs. A spring and damper model was

designed based on the available stiness data and had

approximately the right characteristics over the range of

deections of interest. This model is shown in Fig. 10,

and is based on a spring KAS

in parallel with a damper

Fig. 8 Spring element block diagram DAS

, the characteristics of which are shown in Figs 6b



12/21


and c respectively. The model so far described, that of as that in the leaf-spring relaxation and recovery model,

which was of nding the force on the non-linear dampera spring in parallel with a damper, would not produce

any appreciable relaxation and recovery eects. For this DASrr

due to spring KASrr

and integrating the derived

velocity. The characteristics of spring KAShigh

werelatter eect, a relaxation and recovery component was

included by having a spring KASrr

in series with a damper derived from spring KAS

, in that the spring generated

the same compressive forces as KAS

, but in only oneDASrr

, the characteristics of which are shown in Fig. 6d,

this combination being in parallel with the spring and tenth of the compression distance (eectively making

KAShigh

=10*KAS

) and thereby enabling the desired non-damper described above KASrr

.

The spring element KAS

would largely dictate the linearities of this spring to be exercised during com-

pression of the assistor spring.stiness characteristics of the assistor spring, whereas

the damper element DAS

would largely dictate the4.2.2 Assistor-spring model implementationhysteresis characteristics. For completion, a non-linear

spring of high stiness was included, KAShigh

.The assistor-spring model was implemented as shown in

The assistor springs were manufactured as athe block diagram in Fig. 11. The damping generated by

reinforced rubber moulding and exhibited typicalthe damping element D

ASwas non-linear and was

characteristics for components made from this materialdeection-dependent. Thus the force exerted on the

[5 ]. Other reinforced rubber components, such as auto-damper was divided by the damping at that particular

motive tyres, have been similarly modelled as springsdeection (input U2 divided by input U1 in the oper-

in parallel with dampers [6 ], which is the approachator block), this damping (D

AS) being derived with the

developed here. use of a look-up table with the deection Y4 as the input.The stiness of the spring K

ASwas, in tension, equal in4.2.1 Governing equations

value but opposite in sense to that of compression.

The equations governing the output from the rubber

assistor springs are developed below. The input into the

model was the deection across the suspension, Y1

. This

was compared to the nominal displacement of the

assistor spring, Y4

, to nd the force developed by the

spring, KAShigh

. This was achieved with the aid of a

look-up table, the resulting force, FAS

, being that which

was developed by the assistor spring on the axle and

vehicle body. KAShigh

was non-linear and generated zero

force when it was in tension. This meant that the assistor

spring was not physically attached to the axle and could

come away from it. Fig. 11 Assistor-spring block diagramAs with the relaxation and recovery model in the leaf

spring, all of the points in the model of the assistor

springs would be in equilibrium. The sum of the forces4.3 Shock absorbersdeveloped at point Y

4in the model would therefore be

zero. The forces developed by the springs, KAS

, KAShigh The shock absorbers (viscous dampers) were included in

and KASrr

, were summed, the resulting force being thatthe model as a viscous damper in series with a compliant

developed by the damper DAS

. As with the relaxationbush, and were modelled in a similar manner to that

and recovery element in the leaf-spring model, a look-upshown in Fig. 9. Both the damping and the spring

table was used to derive the damper velocity correspond- elements were derived from look-up tables (see Figs 6eing to the impressed force, this velocity being integratedand 6f ), the damping being a plot of velocity against

to give the nominal assistor spring displacement, Y4

. Theforce, the input to the look-up table being the force and

displacement of the point Y5

was found in the same waythe output being the velocity associated with this force.

The integrator would then derive the displacement of

the damper piston against which the input displacement

(displacement across the suspension) would be com-

pared. The bush force could then be calculated by multi-

plying the bush compression (or extension) by its

stiness. In this fashion, a system could be made to gen-

erate a viscous damping dominated output (and there-

fore a system dominated by the velocity across the

system), even though the input to the model was a

Fig. 10 Assistor-spring model displacement.



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270 J B HOYLE

4.4 Vehicle model and similarly for the front, but excluding the assistor

spring forces Fasf

. The force Rear_susp_F was the staticThe suspension model so far developed was included in load in the suspension due to gravity, the equivalenta relatively simple 5 DoF model of the vehicle, the free- static load also being present in the front suspension.doms being the vertical displacement of the body, engine The front and rear axle equations of motion were givenand front and rear axles, and the pitch displacement of respectively bythe body. Tyre stinesses were measured on the real

Mf

ymf=-F

sf-(y

f-y

gf)K

tvehicle, lumped masses were used to represent the axles

and engine, and a lumped mass with pitch inertia was +Front_tyre_load (13)used to represent the vehicle body. The same types of

Mry

mr=-F

sr-(y

r-y

gr)K

ttests that were carried out on the real vehicle were simu-

lated with the model, to ensure that the model was sub- +Rear_tyre_load (14)jected to as near as possible the same input conditions

The equation of motion for the engine was given byas the real vehicle. Thus the vehicle tie-down (or strop)

Mey

e=F

e(15)and axle stands that were used to measure the static

stinesses were incorporated in the model, and all the wheremeasurements taken from the model were the same as

Fe= [y

e-(y

b+H

bL

e)]K

ethose taken on the real vehicle.

The general arrangement of the model is shown in [ybe-(yb

e+Hb

bL

e)]C

e+Eng_susp_F (16)

Fig. 12, with the vehicle tie-down strop and axle stand

The load Eng_susp_F was the static load in the enginebeing shown at the front. The springs shown for themounts.suspension incorporate the leaf-spring and assistor

The force generated by the axle stand, Fax

, was givenspring models developed earlier. The dampers (conven-

bytional shock absorbers) shown in the model were derived

from look-up tables based on the manufacturers data Fax=-(y

b+H

bL

ax)K

ax(17)

(see section 4.3).with the provision that this force could only be positive;

i.e. it could only push the vehicle upwards. The force4.4.1 Equations of motion

generated by the vehicle tie-down strop, Fst

, was

generated byThe equations of motion for the vehicle model were

developed as follows. The vertical (or bounce) equation Fst=-(y

b+H

bL

st)K

st(18)

of motion for the vehicle body was given by

with the provision that this force could only be negative;Mby

b=F

sf+F

sr+F

ax+F

st+F

e+M

bg (10) i.e. it could only be in tension and pull the vehicle down-

wards. The deection and velocity across the frontand the pitch equation of motion for the vehicle bodysuspension, Susp_disp_F and Susp_vel_F were given bywas given by

Susp_disp_F=yb+H

bL

f-y

f(19)

IbH

b=F

sfL

f+F

srL

r+F

axL

ax+F

stL

st+F

eL

e(11)

Susp_vel_F=ybb+Hb

bL

f-yb

f(20)

where Fsf

and Fsr

were the front and rear suspensionand similarly for the rear suspension, these values beingforces generated by the leaf springs, assistor springs andused as inputs into the suspension models describedshock absorbers, such thatearlier in the paper (sections 4.1, 4.2 and 4.3).

Fsr=F

suspr+F

asr+F

Shockr+Rear_susp_F

=(Fkr+F

rrr)+F

asr+F

Shockr5 THEORETICAL PREDICTIONS

+Rear_susp_F (12)

The results from the theoretical predictions of the sus-

pension characteristics, and of the rubber assistor

springs, are discussed below. All of the data used in the

simulations are presented in Table 1 and Fig. 6.

5.1 Suspension stinesses

The vehicle model was used to simulate the same tests

that the real vehicle had been subjected to. The results

were presented in a similar format to those of the exper-

imental results. The front and rear static suspension

Fig. 12 Vehicle model stiness results are presented in Figs 13 and 14.



14/21


Fig. 13 (a) Theoretical front static suspension stiness and (b) theoretical front stiness time histories

Fig. 14 (a) Theoretical rear static suspension stiness and (b) theoretical rear stiness time histories

Examination of these theoretical predictions showed ( Fig. 5). The theoretical predictions followed the exper-

that the suspension model was capable of simulating the imental curve accurately for most of the deection range;

same basic characteristics as the real suspension. Again, only at very small deections were any discrepancies

six regimes were identied, these being the high and low evident.stiness regions (regimes 1 and 4 and 2 and 5 respect- The predicted eects of the assistor springs on theively) and the relaxation and recovery regions when the suspension stiness characteristics were similar to thosesuspension was held at constant deection after a change of the real suspension. The assistor springs had the eectin deection had occurred (regimes 3 and 6 respectively). of increasing the dierence between the loading and

Small discrepancies between the theoretical predic- unloading regions of the stiness curves, this being duetions and the experimental results were evident, however, to the increase in the degree of relaxation and recovery

particularly in the amount of hysteresis in the force in the suspension. However, as noted above, the staticagainst deection loops. It is thought that these were nature of the actual suspension stiness measurementslikely to be due to the characteristics of the rubber compared to the dynamic measurement data of theassistor springs under the non-dynamic compression test assistor springs meant that the small discrepancies notedconditions and to other variables not included in the above were probably due in large part to incompletemodel (drive line stiness, etc.). data concerning the behaviour of the assistor springs.

The time histories of the suspension load, deection

and velocity also showed very similar characteristics to

those of the real suspension.5.3 Model implications

The model so far developed made predictions that com-5.2 Assistor springs

pared well with the experimental data for a single loading

and unloading cycle. If, however, the suspension modelThe simulation results of the assistor-spring model were

plotted on the same graph as the experimental data were to be loaded and unloaded in a series of steps, then



15/21

272 J B HOYLE

the model would predict that only when a change of followed basically the same straight lines. Only when the

direction of deection changed did the high stinessdirection occurred would the high stiness regime mani-

fest itself. This would be due to the fact that at the end regime manifest itself. The loading and unloading curves

show evidence of relaxation and recovery respectivelyof any one loading or unloading step, the high stiness

spring would be in a state of compression or tension to whenever the deection was held constant at the end of

each step. Furthermore, the eect of the assumed zerothe point where any further change in load in the same

direction would immediately overcome the resistive value of preload [P0

in equation (6)] can be seen as the

converging loading and unloading portions of the loopforces to further deection created by the friction plates

and the low stiness spring. Thus only when a change in Fig. 15a as the load approached zero.

The results from a similar test on the real vehicle areof direction occurred would the high stiness spring rst

have to be loaded or unloaded from one state (com- shown in Fig. 16. The front suspension was loaded up

in six steps from zero, held and then unloaded in vepression or tension) to the other, with the change in load

required to achieve the breakaway force in the new direc- steps. The loading and unloading regions of the load

against deection curve can be seen to have followed ation being twice that generated by the product of the

load acting on the friction plates and the eective straight line in each case, with evidence of relaxation and

recovery being observed at the end of each step. Thecoecient of friction between these plates.

The model was used to simulate the suspension as it high stiness regime only manifested itself when a

change of direction of the suspension deectionwas loaded up in a series of six steps from zero load.

Upon reaching the maximum deection, the suspension occurred. These results are exactly as predicted by the

model. Also evident is the fact that the loading andwas briey held and then unloaded back to zero in aseries of ve steps. The results from this test are shown unloading portions of the loop in Fig. 16a do not con-

verge at zero load. This would be due to the existencein the usual format in Fig. 15. The loading and

unloading portions of the load against deection curve of a small, but nevertheless non-zero, preload P0

existing

Fig. 15 (a) Theoretical step loading static stiness and (b) theoretical step loading stiness time histories

Fig. 16 (a) Experimental step loading static stiness and (b) experimental step loading stiness histories



16/21


between the leaf springs. The ratio between the low and stiness regime at high frequencies during a single fre-

quency sweep, it could reasonably be concluded that thehigh stiness regimes, being on the order of 5.5

(Table 3), did not suggest any simple form of beam non-linear model was able to successfully simulate the

principal dynamic characteristics of the vehicle suspen-bending mechanism for the high stiness regime.

Furthermore, the model also assumed that kinetic fric- sion. Furthermore, it would also provide good evidence

justifying the assumption that dynamic friction wastion (friction during movement) was the same as static

friction in both directions. eectively the same as static friction.

5.4.1 Frequency response predictions

5.4 Dynamic model predictionsThe model thus developed was subjected to a frequency

analysis of a very similar nature to that of the actualThus far, the models created have been able to simulate

the static stiness characteristics of the leaf spring sus- vehicle. A sine wave vertical displacement was input into

the tyres to simulate the experimental hydraulic actu-pension, the hysteresis relaxation and recovery eects of

the suspension after a change in load and deection had ators, the frequency being increased in a number of dis-

crete steps. For each step, a number of cycles were inputtaken place and the stiness and damping eects of the

assistor springs. With the assumption that the dynamic into the vehicle model and the steady state response of

the model recorded. A fast Fourier transform ( FFT)friction eects were the same as the static friction eects,

the acceleration transmissibility frequency responses of algorithm was used to determine the various acceleration

amplitudes of the vehicle response, which were comparedthe sprung and unsprung masses predicted by the model

were compared to those obtained from the actual vehicle. to the acceleration amplitude of the input to determine

the sprung and unsprung acceleration transmissibilityAs noted in section 2.1.3, linear undamped predictions

for the sprung and unsprung natural frequencies ( 1.95 ratios for each frequency. As with the experimental pro-

cedure, the displacement input amplitude was decreasedand 10.8 Hz respectively) were dierent from those mea-

sured experimentally (2.44 and 19 Hz). Furthermore, it at a rate of 2 dB/octave above 2 Hz and the starting

amplitude was 5 mm. The acceleration transmissibilitywas established that although the linear 5 DoF state-

variable model was able to predict the sprung and ratios were then plotted against frequency on a suitable

log graph to enable direct comparison of the experimen-unsprung mass resonant frequencies ( 1.9 Hz using the

low stiness regime and 17 Hz using the high stiness tally derived frequency responses to those predicted by

the model.regime respectively), it could only do this over a limited

range of frequencies, and it completely failed to capture The results of the theoretical frequency analyses are

shown in Fig. 17a (sprung mass) and Fig. 17b (unsprungthe transition between the low and high sti

ness regimeswithin one single frequency sweep. As it was the eects masses). A number of curves are shown in Fig. 17a, three

of these being the response of the vehicle body front,of the friction between the leaves that gave rise to the

static non-linear characteristics of the springs, if the centre of gravity and rear (marked F, CG and R

respectively), and correspond to the three experimentallyvehicle model was able to predict similar dynamic results

to those measured experimentally, and in particular cap- derived curves shown in Fig. 2a. Another curve is shown

in Fig. 17a, which is the response of the engine (markedture the transition from the high deection low stiness

regime at low frequencies to the low deection high Engine).

Fig. 17 (a) Theoretical sprung mass frequency responses and (b ) theoretical unsprung mass frequency

response



17/21

274 J B HOYLE

Comparison of the frequency response predictions in 5.4.2 Eect of rubber bushed shock absorbers

Fig. 17 with those of the actual vehicle in Fig. 2 revealedThe results so far generated, both experimentally andthat the predictions of the non-linear model weretheoretically, were compared in Table 4. They show thatextremely good, far better in fact than the linear dampedthe non-linear suspension model developed in this paper5 DoF state-variable model. At 2.4 Hz, the predictedwas able to predict the resonant frequencies of thebounce (sprung mass) resonant frequency matchedvehicle accurately. However, the linear 5 DoF state-almost exactly the experimental results (2.44 Hz). Avariable model makes predictions for the unsprung masssimilarly good correlation was obtained for the unsprungresonant frequencies that are too low, irrespective ofmass resonant frequencies (20 Hz predicted comparedwhich suspension stiness regime was used. Clearly someto 19 Hz). Furthermore, the amplitude ratios for theother spring element is acting on the unsprung massessprung and unsprung mass resonant frequencies wereto raise their resonant frequencies. The non-linear modelalmost identical. The model was certainly able to capturehas been used to show that this inuence was mostly duethe transition from the low stiness regime at the bounceto the compliant rubber bushes that were incorporatedfrequency to the high stiness regime above about 4 Hz.at each end of the suspension shock absorbers (viscousFigures 17a and b also show a curve marked Breakdampers, see section 4.3).Free Time %. This curve was a measure of how long

If these shock absorbers were included in the non-during any particular sine wave cycle at any particularlinear model as only their non-linear viscous dampingfrequency the suspension deected suciently for it toelement, the resulting unsprung mass resonant frequen-act as normal leaf springs (not locked together). Thiscies would be much closer to those predicted by the high

curve was plotted against the vertical dB scale, such that stiness regime linear 5 DoF state-variable model, as100 per cent break-free time equalled 10 dB and 50 pershown by rows 5 and 3 respectively in Table 4 (see alsocent equalled 5 dB. The curve shows that even at theFigs 18a and b for the frequency responses). Althoughsprung mass resonant frequency, the maximum amountas a rst approximation these rubber bushes might beof time that the suspension was acting as normal leafconsidered insignicant, their actual eect was to raisesprings was only 35 per cent; for the remaining time thethe unsprung mass resonant frequencies by approxi-suspension leaves were in their locked state. Above 5 Hz,mately 2.4 Hz. The remaining discrepancies between thethe leaves were permanently locked together, whichpredicted resonant frequencies and what could bewould account for the apparent success of the linearaccounted for solely by a combination of suspension5 DoF state-variable model in predicting the shape ofstinesses were probably due to a slight underestimatethe frequency response curves when using the higher sus-of the unsprung masses, which would serve to raise thesepension stiness regime. The partial breakaway state of

frequencies in the non-linear model above what theythe springs would also account for the actual resonant would otherwise be in practice.frequency of the sprung mass being between the two

5 DoF state-variable predictions for this frequency, as 5.4.3 Leaf-spring suspension ride characteristicsthe two stiness regimes would be to some extent aver-

aged to produce an intermediate stiness that resulted The results in Table 4 also show that the sprung mass

resonant frequency was determined by a combination ofin the actual resonant frequency measured.

Also worth noting was that the curve marked F both the low and high stiness regimes of the leaf

springs. Furthermore, the unsprung mass resonant fre-(vehicle front) in Fig. 2a experienced an unusual feature

between 9 and 11 Hz, and this was again shown to be quencies were dominated by the high stiness regime,

and other eects (rubber bushed shock absorbers,present in the equivalent curve in Fig. 17a and the 5 DoF

state-variable predictions. This feature was caused by rubber bump stops, etc.) served to raise their resonant

frequencies still higher.the bounce resonant frequency of the engine on its

resilient mounts. For good ride quality, the sprung mass resonant

Table 4 Experimental and theoretical results

Sprung mass Unsprung massresonant frequency resonant frequency Results derived

Condition and amplitude ratio and amplitude ratio from

1 Experimental results 2.4 Hz and 10.24 dB 19 Hz and 1 dB Experiment2 Low suspension stiness regime 1.9 Hz and 13.5 dB 10.8 H z and 4.5 dB Linear 5 DoF3 High suspension stiness regime 2.8 Hz and2 0 d B 1 7.0 Hz a nd-2 dB Linear 5 DoF4 Low and high stiness regimes, shock

absorbers with rubber bushes 2.4 Hz and 8.5 dB 20.0 H z* and-1 dB Non-linear model5 Low and high stiness regimes, shock

absorbers without bushes 2.4 Hz and 8.5 dB 17.6 H z* and -5 dB Non-linear model

* Average of front and rear unsprung masses.



18/21


Fig. 18 (a) Theoretical sprung mass frequency responses and (b ) theoretical unsprung mass frequency

response for shock-absorbers without compliance

frequency would ideally be between 1.5 and 2 Hz, with 3. Furthermore, the model was able to predict accu-

rately the dynamic frequency response characteristicsthe unsprung mass resonant frequencies being about10 Hz. The damping would be set to allow adequate of the vehicle suspension system, particularly the

transition from the high deectionlow suspensioncontrol of the amplitude ratios of these resonances.

A comparison of the linear 5 DoF predictions with stiness regime at the sprung mass resonant fre-

quency, to the low deectionhigh stiness regime atthe non-linear equivalent in Table 4 suggests that the leaf

springs had the eect of not only increasing the sprung all frequencies above 4 Hz, the latter including the

unsprung mass resonant frequencies.mass resonant frequency from 1.9 to 2.4 Hz (an undesir-

able eect) but also of decreasing very signicantly the 4. The models assumption that the static and dynamic

elements of the suspension friction characteristicsamplitude ratio of this resonance, from 13.5 to 8.5 dB.

Indeed, calculations show that this was equivalent to were equal was supported by the available evidence.

5. The sprung mass resonant frequency was derivedhaving four times as many shock absorbers on the

vehicle. Whereas this reduction in amplitude ratio at the from a combination of both stiness regimes of the

leaf springs, whereas the unsprung mass resonant fre-sprung mass resonant frequency was a desirable eect,it had been achieved at the expense of riding on a near quencies were dominated by the friction-induced

locked-leaf state (higher stiness regime) of thesolid suspension (high stiness regimes) under most

normal road conditions, and the apparent advantages suspension leaves.

6. Even apparently insignicant suspension elements arewere therefore oset by a harsher ride quality with

increased road noise and structural vibration. This latter necessary to predict correctly the vehicle resonant

frequencies (shock absorber rubber bushes, assistorpoint was certainly borne out by the available evidence,

as the vehicle was generally regarded as being very noisy springs).

7. Such non-linear stiness characteristics can result inand tiring to drive.

the vehicle riding on a near-solid suspension under

most normal road conditions, with consequent

increased levels of road noise and vehicle vibration.6 CONCLUSIONS

The following are conclusions:

REFERENCES1. The frequency response characteristics of a 10 ton

truck with multiple-leaf suspension springs has been1 Potter, T. E. C., Cebon, D. and Cole, D. J. Indirect methodsinvestigated and found to have resonant frequencies

for assessing road friendliness of lorry suspensions. Proc.that diered markedly from what might be expectedInstn Mech. Engrs, Part D: J. Automobile Engineering, 1997,

from linear 5 DoF state-variable model predictions of211(D4), 243256.

the system.2 Winkler, C. B. and Hagan, F. A test facility for the measure-

2. A non-linear model of the vehicle and its suspension ment of heavy vehicle suspension parameters. SAE paperhas been created that was able to predict accurately 800906, 1980.the static stiness and hysteresis characteristics of the 3 Haessig, D. A. and Friedland, B. On the modelling and simu-leaf spring suspension system, in particular the two lation of friction. Trans. ASME, J. Dynamic Systems,

Measmt and Control, 1991, 113.distinct stiness regimes of the leaf springs.



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276 J B HOYLE

4 Wong, J. Y. Theory of Ground Vehicles, 1993 (Wiley

Interscience, New York).

5 Fancher, P. S., et al. Measurement and representation of the

mechanical properties of truck leaf springs. SAE paper

800905, 1980.

6 Lindley, P. B. Engineering design with natural rubber. NR

Technical Bulletin, Malayan Rubber Fund Board, Natural

Rubber Producers Research Association, 1966.

APPENDIXFig. 19 Linear damped 5 DoF representation of the vehicle

suspensionLinear damped 5 degree-of-freedom state-variable model

of suspensionstate-variable techniques within MATLAB. The system

is shown in Fig. 19, and the equations of motion areThe vehicle suspension system and principal dynamic

components can be modelled as a linear spring mass presented below. The tie-down strop and axle stand

are ignored.damper system that lends itself readily to analysis using

Body (sprung mass) vertical motion:

yb

Mb=(y

f-y

b-h

bL

f)K

f+(y

r-y

b-h

bL

r)K

r+(y

e-y

b-h

bL

r)K

e

+(ybf-yb

b-hb

blf

)Cf+(yb

r-yb

b-hb

bL

r)C

r+(yb

e-yb

b-hb

bL

r)C

e(21)

Body (sprung mass) angular pitch motion:

hb

Ib=(y

f-y

b-h

bL

f)L

fK

f+(y

r-y

b-h

bL

r)L

rK

r+(y

e-y

b-h

bL

r)L

eK

e

+(ybf-yb

b-hb

blf

)Lf

Cf+(yb

r-yb

b-hb

bL

r)L

rC

r+(yb

e-yb

b-hb

bL

r)L

eC

e(22)

Front axle (unsprung mass):

yf

Mf=(y

b+h

bL

f-y

f)K

f+(yb

b+hb

bL

f-yb

f)C

f+(y

g-y

f)K

t(23)

Rear axle (unsprung mass):

yrM

r=(y

b+h

bL

r-y

r)K

r+(yb

b+hb

bL

r-yb

r)C

r+(y

g-y

r)K

t(24)

where yg

in equations (23) and (24) is the ground displacement system excitation input to the wheels.

Engine:

ye

Me=(y

b+h

bL

e-y

e)K

e+(yb

e+hb

eL

e-yb

e)C

e(25)

These equations can be written as a set of rst-order dierential equations and then represented in state-variable form

as the vector equation:

xb=f(x, u) (26)

where the input is u and the output is

y=h(x, u) (27)

For the linear case, equations (26) and (27) can be written as

xb=Fx+Gu (28)

and

y=Hx+Ju (29)

where for an nth-order system, F is an nn system matrix (in this case n=10), G is an n1 column vector of inputs,

H is an mn output matrix and J is a scaler called the direct transmission term. For the 5 DoF system represented

by Fig. 19 and equations (21) to (25), and ignoring the eects of gravity, matrices F, G, H and J for this system are

as follows:



20/21


System matrix F:

Frow1= [0, 1, 0, 0, 0, 0, 0, 0, 0, 0]

Frow2=C-

Kf+K

r+K

eM

b

, -C

f+C

r+C

eM

b

, -L

fK

f+L

rK

r+L

eK

eM

b

,

-L

fC

f+L

rC

r+L

eC

e

Mb

,K

f

Mb

,C

f

Mb

,K

r

Mb

,C

r

Mb

,K

e

Mb

,C

e

Mb

DFrow3= [0, 0, 0, 1, 0, 0, 0, 0, 0, 0]F

row4=C-

Lf

Kf+L

rK

r+L

eK

eIb

, -L

fC

f+L

rC

r+L

eC

eIb

, -L2

fK

f+L2

rK

r+L2

eK

eIb

,

-L2

fC

f+L2

rC

r+L2

eC

eIb

,L

fK

fIb

,L

fC

fIb

,L

rK

rIb

,L

rC

rIb

,L

eK

eIb

,L

eC

eIbD

Frow5= [0, 0, 0, 0, 0, 1, 0, 0, 0, 0]

Frow6=C

Kf

Mf

,C

fM

f

,L

fK

fM

f

,L

fC

fM

f

, -K

f+K

tM

f

, -C

fM

f

, 0, 0, 0, 0DF

row7=[0, 0, 0, 0, 0, 0, 0, 1, 0, 0]

Frow8=C

Kr

Mr

,C

rM

r

,L

rK

rM

r

,L

rC

rM

r

, 0, 0, -K

r+K

tM

r

, -C

rM

r

, 0, 0DF

row9= [0, 0, 0, 0, 0, 0, 0, 0, 0, 1]

Frow10

=CK

eM

e

,C

eM

e

,L

eK

eM

e

,L

eC

eM

e

, 0, 0, 0, 0, -K

eM

e

, -C

eM

eD

The input matrix G is given by

G=

C0, 0, 0, 0, 0, Kt

Mf

, 0, KtM

r

, 0, 0

DT

The output matrix H derives the displacement responses of the ve individual degrees of freedom and the displacement

of the vehicle body at the positions of the front and rear mounted experimental test accelerometers at distances Laccelf

and Laccelr

from the body centre of gravity respectively, giving seven outputs in total:

H=

1, 0, 0, 0, 0, 0, 0, 0, 0, 0

0, 0, 1, 0, 0, 0, 0, 0, 0, 0

0, 0, 0, 0, 1, 0, 0, 0, 0, 0

0, 0, 0, 0, 0, 0, 1, 0, 0, 0

0, 0, 0, 0, 0, 0, 0, 0, 1, 01, 0, L

accelf, 0, 0, 0, 0, 0, 0, 0, 0

1, 0, Laccelr

, 0, 0, 0, 0, 0, 0, 0, 0

Matrix J is given by a (71) column vector of zeros.

The frequency responses of the system-to-ground inputs were derived using MATLAB. Two sets of results were

generated; the rst (shown in Figs 20a and b for the sprung and unsprung masses respectively) assumed that the

suspension stiness was given by the nominal stiness of the suspension leaf springs (sum of the stinesses of the

individual leaves) and the second (shown in Figs 21a and b) assumed that the suspension stiness was in a locked-up

state, thereby giving a much higher suspension stiness. The resonant frequencies of the sprung and unsprung masses

for the two dierent stiness values are given in Table 5.

It is clear from the two sets of graphs and Table 5 that the stiness of the suspension makes a dramatic dierence

to the response of the vehicle body and the axles. The response of the engine is largely insensitive to that of the rest

of the system. All of the data used in these predictions is presented in Table 1 and Fig. 6, and the two stiness regimes



21/21

278 J B HOYLE

Fig. 20 (a) Sprung mass frequency responses and (b) unsprung mass frequency responses for nominal suspen-

sion stiness

Fig. 21 (a) Sprung mass frequency responses and (b) unsprung mass frequency responses for locked leaf

suspension stiness

Table 5 Lumped mass resonant frequencies

Low suspension High suspensionstiness resonant stiness resonantfrequency and frequency and

Lump ed m ass amplitud e ratio amplitud e ratio

1 Vehicle body centreof grav ity 1.9 Hz a nd 1 3.5 dB 2.8 Hz* an d20 dB

2 Engine 9.3 Hz 9.3 Hz3 Front and rear axles

(average) 9.5 Hz and 4.5 dB 17 Hz and-2 dB

* Average of the two peaks.

were as follows:

Kf=K

Lowf(low stiffness regime) and 5.5K

Lowf(high stiffness regime)

Kr=K

Lowr(low stiffness regime) and 5.5K

Lowr(high stiffness regime)

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