34231

DETC/MECH-34231

Proceedings of the 2002 ASME Design Engineering Technical Conference:27th Biennial Mechanisms and Robotics Conference

September 29–October 2, 2002, Montreal, Canada

REDUCTION OF IMPACT AND VIBRATION IN AN INDUSTRIAL CAM-FOLLOWER SYSTEMUSING THE SPLINEDYNE METHOD: A CASE STUDY

Robert L. NortonDept. of Mechanical EngineeringWorcester Polytechnic Institute

Worcester, MA 01609

Robert P. GordonEngineering and

Implementation GroupThe Gillette Company

Boston, MA 02127

Proceedings of DETC�02 ASME 2002 Design Engineering Technical Conferences

and Computer and Information in Engineering Conference Montreal, Canada, September 29-October 2, 2002

DETC2002/MECH-34231

ABSTRACT

The splinedyne method of cam design is a variation on the poly-dyne method that uses B-spline functions to obtain superior con-trol of follower motion and its derivatives in combination with adynamic model of the follower train to minimize vibration in thefollower at any one design speed. This paper presents the resultsof the application of this technique to an automated assemblymachine cam-follower train whose vibratory behavior before re-design was limiting machine speed and causing increased prod-uct scrap rate due to impacts and vibrations. The improvedsplinedyne system contributed to a measured 14% increase inproduction rate in combination with a 1% reduction in scrap ratefrom that station.

INTRODUCTION

The splinedyne method of cam-follower design, a contraction of“spline” and “dynamic,” is a modern variation on the long estab-lished polydyne method of cam design. The term polydyne is acontraction of "polynomial" and "dynamic." It was coined byThoren, Engemann, and Stoddart (Thoren 1952) to describe acam design method first proposed and implemented by Dudley(Dudley 1948) who was the first to use a dynamic model of acam-follower system to determine a cam profile that would, ineffect, compensate for dynamic vibration of the follower train, atleast at one particular cam speed. In essence it applies the desiredmotion function to the end effector of the cam-follower systemand back-calculates the cam profile function needed to deliverthat motion to the end effector considering the dynamic responseof the follower train based on a lumped parameter model and usesB-splines to control motion.

In the 1950-1970 period, polydyne cams were used in auto-motive pushrod, overhead valve-train systems to control vibra-tions but are seldom used in that application today. Modern valvetrain designs have stiffer and lighter follower trains, which reduc-es the need for polydyne methods. Also, a polydyne cam must

1

be designed for a particular camshaft speed, and in an engine ap-plication where speed is not constant, this proves a limitation. In-dustrial cam-follower systems can have relatively heavy and flex-ible follower trains, and are being required to run at ever higherspeeds that make follower vibrations a significant problem inmany instances. This makes them a candidate for the polydyneapproach, especially since they tend to run at constant speed.

Polydyne cams have another potential problem related to themathematical limitations of polynomial functions. To controlvibration with the polydyne method will usually require creatinga polynomial function that uses a large number of boundary con-ditions. This drives the order of the polynomial up with concom-itant problems of high peak values of acceleration and the poten-tial for oscillation of the displacement function beyond desiredlimits. The automotive valve train polydyne functions of the1950’s typically used polynomials with terms as high as 50thpower (Thoren 1952).

B-spline functions (Schumaker 1969, Sanchez 1980, Schu-maker 1981, MacCarthy 1985, MacCarthy 1988, Tsay 1988)eliminate the problems with high order polynomials and allowvirtually unlimited application of boundary conditions needed fordynamic control while keeping the order of the function reason-able and thus well behaved. B-splines can be thought of as “su-per polynomials” that provide a set of “knots” that can be used tocontrol the function shape while preserving its conformance toany set of desired boundary conditions. In fact, any polynomialfunction can be exactly reproduced with a B-spline, making theformer a subset of the latter.

This paper presents an approach to cam-follower system de-sign that combines the principles of polydyne cam design withthe advantages of B-spline functions, dubbed the “Splinedyne”approach, a combination of “spline” and “dynamic.” (Norton2002a), This paper also describes the application of the spline-dyne technique to a cam-follower system for one particular sta-tion of an automated assembly machine. The same technique hasrecently been applied to several other systems on the same and

Copyright © 2002 by ASME

similar assembly machines with equally successful results. Ingeneral, residual vibration of the end effector tooling duringdwell has been successfully minimized to levels approachingzero, and the motion functions have been customized with B-splines to control velocities in order to limit impact levels whilestill keeping acceleration in good control. Significant increasesin machine speed have been accomplished with simultaneous in-creases in machine efficiency and reduction of scrap rate.

The machine in question was designed over 20 years ago andhas undergone several speedups and modifications since thattime. It now can be thought of as a 3rd-generation design whosespeed is over twice its original design speed. The machine is typ-ical of many automated production machines that have an index-ing chassis to carry the product through the process in nests on aconveyor mechanism. As each nest stops at a workstation, cam-driven tooling performs an operation on the product. A CAD sol-id model of the subject cam-follower mechanism is shown in Fig-ure 1. The cams are on one or more camshafts (not shown) thatrun the length of the machine and are driven in time with the con-veyor motion. Air cylinder springs are typically used to close thecam joint.

FIGURE 1

Cam, follower train, tooling, conveyor nests, and frame of onestation of an automated assembly machine

22

This product assembly process has very tight tolerance re-quirements for positioning of its component parts. Any vibrationof the tooling end effectors during the assembly process can po-tentially compromise accurate placement and thus increase prod-uct scrap rate. It is desired to have the smoothest possible mo-tions of the end effectors and to have minimal residual vibrationduring the dwell portion of the cycle when operations are per-formed on the product.

The function of the mechanism in Figure 1 is simply toclamp the assembly of product parts while a fastening operationis done. If it is vibrating during the dwell’s clamp cycle, the as-sembly may be out of tolerance after fastening. Because of theneed to clamp this assembly, the end effector must be providedwith a small amount of compliance and the cam-follower motionwith a small amount of overstroke, in order to guaranty contactwith sufficient clamping force despite differences in product po-sition from nest to nest due to conveyor and nest tolerances. Thisthen requires that the tooling impact the product on the nest. Anyimpact will, of course, ring the system and cause structural dy-namic vibrations. Thus it is desirable to minimize the impactvelocity between tooling and workpiece. B-spline functions pro-vide an easy way to do this. Including the dynamics of the fol-lower train in the cam design serves to minimize residual vibra-tions in the follower at one design speed. (Dudley 1948) Thissplinedyne approach has been found to be extremely effective incam design for automated machinery as we will show in this casestudy.

DYNAMIC MODELS

Figure 2 shows a schematic version of the mechanism of Figure1. We wish to reduce this to a simple, single-mass single degree-of-freedom (SDOF) dynamic model. (Norton 2002b) Figure 3shows the simplest SDOF lumped parameter model and the free-body-diagram (FBD) for the system of Figure 2. The spring k1and damper c1 represent those characteristics, respectively, of thejoint closure spring (which may be an air spring). The modelingof an air spring is described in (Norton 2002c). The spring k2 anddamper c2 represent, respectively, the stiffness and dampingwithin the links and joints of the follower train. The effectivemass m represents all the moving mass in the follower system.

Its system equation is:

F mx

k s x c s x k x c x F mx

xc c

mx

k k

mx

k

ms

c

ms F

i

i

=

−( ) + −( )− − − =

++

++

= + +

∑ ˙

˙ ˙ ˙ ˙

˙ ˙ ˙

1 1 2 2

1 2 1 2 1 1 (1a)

If we assume that contact is always maintained between camand follower by sufficient joint closure force, the system simpli-fies to that shown in Figure 4 with the equation


-t

-fo-

-

FIGURE 2

Schematic of the industrial cam-follower mechanism of Figure 1

cam

air cylinder

ground (1)

bellcrank (4)I4, k4 and k5

tooling (5) m5

rollerfollower

ground (1)

ground (1)

connecting rod (3) m3, k3

follower arm (2) I2, k2

r4 r5

l3

r1

r2

ra

FIGURE 3

Simplest SDOF model and free-body diagrams (FBD) of a typical industrial cam-follower system

k 1 c 1

x

s

s – x

m

m

k 2 (s – x )

s+ x +

k 1 (s )

k 2 (s – x )

c 1 (s ).

c 2 ( s – x ). .

c 2 (s – x ). .

Fc

k 2 c 2

F i

F i

3

˙ ˙ ˙xc

mx

k

mx

k

ms

c

ms+ + = + (1b)

The follower acceleration ˙x has units of length/sec2. Convert it to an angle base (in degrees) and introduce the camshafangular velocity N in rpm.

length

sec

length

deg

360 deg

rev

rev

60 sec

(2)

2 2

2 2

2

2

2 2=

= =

N

xd x

dtN

d x

d

2

2

22

2

236˙θ

Koster (1974) has shown, and we have confirmed by measurement in industrial cam-follower systems, that the range ovalues of damping ratios in these systems is about ζ = 0.05 t0.10. These are very underdamped systems, and so are potentially subject to significant vibration.

Assuming then that damping is negligible, equation 1b is rewritten without the damping terms.

˙xk

mx

k

ms+ = (3a)

Rearranging to solve for s:

sm

kx x= +˙ (3b)

Substitute equation 2 in 3b

s Nm

k

d x

dx= +36 2

2

2θ(4a)

FIGURE 4

SDOF model and free-body diagram (FBD) of a form-closed industrial cam-follower system

m

k(s – x ) c(s – x ). .

s+ x +

x

s

s – x

m

k c


Differentiate with respect to θ to get the splinedyne cam ve-locity and acceleration functions.

′ = +s Nm

k

d x

d

dx

d36 2

3

3θ θ(4b)

′′ = +s Nm

k

d x

d

d x

d36 2

4

4

2

2θ θ(4c)

Note that the cam acceleration function involves the fourthderivative of the selected follower displacement function, whichwe call "ping." This means that any function selected for the fol-lower displacement should be continuous through at least fourderivatives, velocity, acceleration, jerk, and ping. Since splinefunctions allow control of continuity at the ends of the segmentto any derivative, they provide a useful means to this end and also

lumped mass model(b)

Ffol

cam

r1

meff

link 4

link 3

link 2

m5

A

B

C DO4

O2

lumped masses

Fcyl

(a)

m4 eff

m3

r4 r5

r2

cam

r1

m1

m2 eff

FIGURE 5

Combining lumped masses for the system of Figure 2

44

allow control of the function’s shape and peak value independentof the number of boundary conditions required.

Equation 4a shows that the cam profile will be different forany value of N selected. Thus the dynamic behavior of the sys-tem can be optimized (i.e., vibration minimized) only for one op-erating speed. This is not a severe limitation in this case as as-sembly machines are typically operated at one speed for long cal-endar periods.

MODELING THE SYSTEM

To apply the above models to a system such as that in Figures 1and 2, the distributed mass and stiffness properties must be de-fined and reduced to properly configured lumped parameters asshown in Norton 2002a. Figure 5a shows lumped masses of thevarious elements of Figure 2. The rotating links are modeled aspoint masses at one of their connecting pins based on an equiva-

m4 eff

k4m5

k 2

r4 r5

r2

cam

r1

m1

r1

keff

m3

k3

m2 eff

link 4

link 3

link 2

lumped spring model

cam

A

B

CD

O4

O2

Ffol

(a)

(b)

Fcyl

k5

FIGURE 6

Combining lumped springs for the system of Figure 2


lent mass moment of inertia. This simplification is acceptable forsmall angular rotation of the link, which is typical in these mech-anisms. The lumped masses are combined taking all link ratiosinto account until an effective mass is defined to be acting at thecam follower as shown in Figure 5b. The existence of a CADsolids model of the assembly (Figure 1) makes it easy to deter-mine the mass properties of the individual links needed to calcu-late the effective mass for the dynamic model of Figure 3 or Fig-ure 4 as was done here.

The effective spring rate of the follower linkage is found ina similar fashion as shown in Figure 6. The spring rate of eachelastic element such as the follower arm, connecting rod, andbellcrank must be calculated, measured, or estimated and thencombined properly to obtain an effective spring rate. When CADsolid models of the parts exist and when finite element analysis

FIGURE 7

Timing diagrams of three successive cam designs for Fig 1

(a) Original modified trapezoidal cam function

(b) Improved asymmetric 3-4-5-6 polynomial cam function

(c) Latest B-spline function timing diagram

5

(FEA) software is available, perhaps the best way to obtain anestimate of the individual part’s spring rates is to mesh the solidsmodel in an FEA package and calculate the deflection for a loadapplied at the point where the link joins its neighbor, being care-ful to replicate the boundary conditions correctly.

Absent the availability of these CAD tools, the spring ratescan be calculated using classical beam and column theory as wasdone for this design. If parts exist, then a simple experiment canbe done by supporting the part as it is held in the machine, apply-ing a known load to the pin joint, and measuring the deflectionwith a dial indicator.

The data for the system of Figure 1 was calculated to be:

m

k E

eff

eff

=

=

15 04

5 410 6

.

.

kg(5)

N/m

The effective spring rate and preload of the air cylinder were:

k E

F

spring

i

=

=

3 793 4

603

. N/m(6)

N

CAM TIMING DIAGRAMS

Over the history of this particular machine, there have been threedifferent cam designs applied to this follower train. The timingdiagrams for them are shown in Figure 7. The original designwas a conventional (for the time) modified trapezoidal accelera-tion (MT) double dwell motion (Figure 7a). During an earlier

FIGURE 8

Velocity and acceleration of a modified trapezoidal cam


speedup program, an asymmetric polynomial design was devel-oped that incorporated an increase of stroke coupled with chang-es to the rise and fall durations to control acceleration. This poly-nomial design (Figure 7b) is essentially a modified 3-4-5 polyno-mial double-dwell motion with an added interior boundary con-dition that sets acceleration to zero at other than the midpoint ofthe motion, making it an asymmetric double-dwell 3-4-5-6 poly-nomial. This redesign, along with similar changes to many of theother stations of the machine, allowed a 25% increase in speed.The most recent cam design (Figure 7c) uses a quintic B-splinefunction to control the velocity and acceleration functions in or-der to reduce impact and dynamic forces and has allowed an ad-ditional 14% increase in speed. The follower train design has re-mained essentially unchanged throughout all the cam designchanges. These three cam designs will now be described in moredetail with respect to their dynamic performance, all defined atthe same machine speed.

MODIFIED TRAPEZOIDAL CAM PERFORMANCE

Figure 8 shows (from top to bottom) the kinematic velocity (v),simulated follower velocity ( x ), kinematic acceleration (a), andsimulated follower acceleration ( ˙x ) for the modified trapezoidal(MT) design, simulated at current machine speed. Note the os-cillation in the simulation during the dwell in both velocity andacceleration curves as compared to the theoretical kinematicfunctions. There is vibration in acceleration through more thanhalf of the nominal dwell period.

FIGURE 9

Velocity and acceleration of a 3-4-5-6 polynomial cam

66

FIGURE 10

B-spline functions for rise and fall

POLYNOMIAL CAM PERFORMANCE

Figure 9 shows (from top to bottom) the kinematic velocity (v),simulated follower velocity ( x ), kinematic acceleration (a), andsimulated follower acceleration ( ˙x ) for the asymmetric 3-4-5-6polynomial design. Compared to the MT design in Figure 7, the


FIGURE 11

Velocity and acceleration of a quintic B-spline, splinedyne camfunction over entire interval including dwells

follower velocity is better controlled and is close to the kinemat-ic velocity. The simulated follower acceleration shows oscilla-tion to a lesser degree than the MT design, but still has an unde-sirable level of distortion and residual vibration during the dwell.

SPLINEDYNE CAM DESIGN AND PERFORMANCE

Figure 10 shows the design of the B-spline functions for rise andfall of the splinedyne cam using program DYNACAM (Norton2002a). Note the knots (vertical lines) that control the shape ofthe curve. They are arranged to bias the curves asymmetricallyin order to reduce the velocity as the impact point is approached.The fall motion is the one that impacts the nest. Toward the endof the fall the function is shaped so as to approach the impactpoint slowly with low velocity. Despite a designed increase in thetotal stroke, the peak acceleration is nevertheless controlled toreasonable values.

Figure 11 shows (from top to bottom) the kinematic velocity(v), simulated follower velocity ( x ), kinematic acceleration (a),and simulated follower acceleration ( ˙x ) for the splinedyne camdesign. Note that this is the result of both the superior splinefunction shape and the inclusion of the dynamic response of thefollower train in the cam design. Compared to both of the earlierdesigns, note that the simulated dynamic responses of the follow-er velocity and acceleration are essentially identical to the theo-retical kinematic velocity and accelerations.

Figure 12a shows the splinedyne solution follower accelera-tion superimposed on that of the polynomial design. The differ-

7

ence is striking. The polynomial shows ringing during the dwelland the splinedyne does not. Figure 12b shows the velocity func-tions for the polynomial and splinedyne functions superposed.The splinedyne solution shows no oscillation during the dwell,but the polynomial function does.

Figure 12c shows the displacement functions for the twodesigns. Note that, by design, the splinedyne function (which hasmuch less dynamic error) also has substantially greater displace-ment then the prior polynomial design. There are good engineer-ing reasons for this having to do with other desired changes to theprocess specifications at this station, but they are not germane tothe issues addressed here. The point, however, is that, despite theadditional stroke that increases peak acceleration significantly, itsdynamics are superior, due to the splinedyne approach.

Figure 12d shows the error function between the cam dis-placement command signal (defined at the follower end effector)and the actual displacement of the end effector (according to thesimulation). The significant difference is seen in the dwell. Thepolynomial design shows ringing that lasts for nearly all of thedwell period. The splinedyne solution shows essentially no fol-lower oscillation during the dwell. Since the primary goal of thecam-follower mechanism is to move the end effector to its finalposition and then hold it stationary and quiet during the dwell sothat the desired operation can be accomplished, any oscillationduring the dwell defeats its purpose. Thus the splinedyne solu-tion shows a clear advantage over the others described herein.

EXPERIMENTAL DATA

The splinedyne cam was installed on several machines and willeventually be fitted to all machines of this type. The accelerationat the end effector was measured on one machine with the poly-nomial cam design and on one to which the splinedyne cam hadbeen fitted. Figure 13 shows the results. The acceleration wasmeasured with a Dytran model 3110a piezoelectric accelerome-ter and recorded in a Hewlett Packard 35670a Dynamic SignalAnalyzer. There are 1176 data points in each trace over one rev-olution of the cam. The raw data is averaged over 50 cam revo-lutions, and is also shown as a 10-point moving average trend linefitted to the raw data The traces of both cams are in time phase.

The polynomial data shows more oscillation during thedwells than does the splinedyne data, which is essentially quietafter the ringout of the impact in the first third of the dwell. Therise and fall motion events show more oscillation than was pre-dicted in the simulation probably due to several factors. The sim-ulation is only an approximation of the true behavior and does notinclude the effects of impact nor of other systems that are runningsimultaneously in the machine. There are more than 20 cam-fol-lower trains in this machine and vibrations from any one train canaffect the others. The most significant difference between the twotraces shown in Figure 13 is the quiescence of the second dwellfollowing the fall event and impact ringout of the splinedyne sys-


FIGURE 12 Part 1

Comparison of simulated polynomial and splinedyne system responses - acceleration and velocity

(a)

(b)

tem in comparison to that of the polynomial design. This is theperiod of clamping of the product that is required to be as vibra-tion free as possible. High speed video of both designs confirmedthat the splinedyne system had significantly less residual vibra-tion during the dwell.

88

CONCLUSIONS

The splinedyne approach to cam design has been shown in thiscase study to provide significant advantages in the control of vi-brations and impact forces. B-splines offer significant designflexibility in terms of controlling follower motion by allowinglarge numbers of boundary conditions to be applied in combina-tion with a practical limitation on function order that prevents un-wanted oscillations. Inclusion of the dynamics of the followersystem in order to alter the cam profile design so as to control the


FIGURE 12 Part 1

Comparison of simulated polynomial and splinedyne system responses - acceleration and velocity

(a)

(b)

motion of the end effector provides an additional and powerfultool that can virtually eliminate unwanted vibrations in the out-put motion at any one machine speed. Application of these camdesign principles has been shown to increase productivity by al-lowing operation at higher speeds than were possible before thesetechniques were introduced, and to simultaneously provide sig-nificant reductions in product scrap rates. In addition, acousticnoise levels from stations to which these splinedyne techniqueswere applied have been reduced by as much as 5 dB.

9

REFERENCES

Dudley, W. M. (1948). “New Methods in Valve Cam Design.” SAE QuarterlyTransactions, 2(1), pp. 19-33.

Fawcett, G. F., and J. N. Fawcett, eds. (1978). “Comparison of Polydyne and Non-Polydyne Cams.” Cams and Cam Mechanisms, Jones, J. R., ed.,Institution of Mechanical Engineers: London.

Koster, M. P. (1974). Vibrations of Cam Mechanisms. Phillips Technical LibrarySeries, Macmillan Press Ltd.: London.


FIGURE 13

Experimentally measured accelerations in two machines, one with the polynomial cam and one with the splinedyne cam

MacCarthy, B. L., and N. D. Burns. (1985). “An Evaluation of Spline Functionsfor Use in Cam Design.” IMechE, 199(C3), pp. 239-248.

MacCarthy, B. L. (1988). “Quintic Splines for Kinematic Design.” Computer-Aided Design, 20(7), pp. 406-415.

Norton, R. L. (2002a). Cam Design and Manufacturing Handbook. IndustrialPress: New York.

Norton (2002b): Norton, R. L., C. A. Gillis, C N. Maynard. “Dynamic Modelingof the Typical Industrial Cam-Follower System: Part 1 Single-Degree-of-Freedom Models.” Paper #DETC2002/MECH-34232, Proc. ofASME International Design Engineering Conference, Montreal,Canada.

Norton (2002c): Norton, R. L., C. O’Brien, P. Duperry, J. R. Hall, “Determiningthe Effective Spring Rate of Air Cylinders Used to Close Cam-Follower Joints.” Paper #DETC2002/MECH-34280, Proc. of ASMEIntl Des. Eng Conf., Montreal, Canada.

Sanchez, M. N., and J. G. deJalon. (1980). “Application of B-Spline Functions tothe Motion Specifications of Cams.” Proc. of ASME DesignEngineering Technical Conference, Beverly Hills, CA.

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Schumaker, L. L., ed. (1969). “Some Algorithms for the Computation andApproximating Spline Functions.” Theory and Application of SplineFunctions, Greville, T. N. E., ed., Academic Press: New York, pp. 87-102.

Schumaker, L. L. (1981). Spline Functions: Basic Theory. John Wiley & Sons:New York.

Stoddart, D. A. (1953). “Polydyne Cam Design-I”. Machine Design, January,1953, pp. 121-135.

Stoddart, D. A. (1953). “Polydyne Cam Design-II”. Machine Design, February,1953, pp. 146-154.

Stoddart, D. A. (1953). “Polydyne Cam Design-III”. Machine Design, March,1953, pp. 149-164.

Thoren, T. R., et al. (1952). “Cam Design as Related to Valve Train Dynamics.”SAE Quarterly Transactions, 6(1), pp. 1-14.

Tsay, D. M., and C. O. Huey. (1988). “Cam Motion Synthesis Using SplineFunctions.” Journal of Mechanisms, Transmissions, and Automation inDesign, 110, pp. 161-165.


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