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Pergamon Mech Mach Theory Vol 31 No 4 pp 397 412 1996 Copyright 1996 Elsevier Science Ltd 0094 114X 95 00087 9 Printed in Great Britain All fights reserved 0094 114X 96 15 00 0 00 AN EXPERIMENTAL STUDY OF THE EFFECTS OF CAM SPEEDS ON CAM FOLLOWER SYSTEMS H S YAN and M C TSAI Department of Mechanical Engineering National Cheng Kung University Tainan 70101 Taiwan Republic of China M H HSU Department of Mechanical Engineering Kung Shah Institute of Technology and Commerce Yungkang Tainan 71016 Taiwan Republic of China Received 9 September 1994 received for publication 26 October 1995 Al traet Traditionally in a cam follower system the cam is often operated at a constant speed and the motion characteristics of the follower are determined once the cam displacement curve is designed From the kinematic point of view the approach by varying cam input driving speed is an alternative way for improving the follower motion characteristics Here we show how to find a polynomial speed trajectory for reducing the peak values of the motion characteristics Furthermore constraints and systematic design procedures for generating an appropriate trajectory of the cam angular velocities are developed Design examples are given to illustrate the procedure for getting an appropriate speed trajectory as variable speed cam follower systems Furthermore an experimental setup with a servo controller is developed to study the feasibility of this approach Experimental data show that the results are very close to those of theory NOMENCLATURE a acceleration of the follower A At normalized acceleration of the follower c d e n Ta Tb x y constant parameters h maximum displacement of the follower j jerk of the follower J Jc normalized jerk of the follower s displacement of the follower S normalized displacement of the follower t time for the cam to rotate through angle 0 T Tp Tpj Tpv normalized time v velocity of the follower V Vc normalized velocity of the follower t cam angle rotation for total rise h tim t2 f13 t4 cam rotation angle normalized cam angle of rotation O cam angle of rotation time of cam rotation for total rise h T r4 time of cam rotation ca cam angluar velocity COave average cam angular velocity of a complete cycle tOsm cos2 co co 4 average cam angular velocity in a follower motion period oh the 1st derivative of co 6b the 2nd derivative of to t normalized cam angular velocity t F the 1st derivative of f the 2nd derivative of f INTRODUCTION In a cam follower system the load produced by inertia forces is prone to deflection and creates vibrations and the load introduced by jerks may cause vibrations as well These will affect the operating life of the cam Therefore the design of motion curves to minimize dynamic loading is of importance for high speed cam mechanisms It is well known that the velocity and acceleration curves are required to be continuous and to have smaller peak values In addition the jerk curve should be finite 397 398 H S Yan et al A cam is often assumed to be operated at a constant speed in designing a cam follower system However the motion characteristics of the follower are changed as the cam speed varies Traditionally to achieve the desired motion is an application of synthesis for obtaining new displacement curves which have better dynamic characteristics In this paper we propose an alternative method by varying the cam speeds The concept of using variable speeds in a cam follower system design was seldom studied in the literature Rothbart 1 designed a variable speed cam mechanism in which the input to the cam is the output of a Withworth quick return mechanism Tesar and Matthew 2 derived the motion equations of the follower by considering the case of variable speed cams The criteria for selecting proper angular velocities which will eliminate the discontinuity in motion characteristics of the follower are investigated by Yan et al 3 From the kinematic point of view the objective of this work is to find cam speed trajectory for reducing the peak values of the follower output motion Furthermore constraints and system design procedures for generating a proper trajectory of the cam angular velocities are deve oped Design examples are given to illustrate the procedure for getting a proper angular speed for a given follower system An experimental cam follower system is set up in which a servo motor is controlled to generate the desired speed trajectory for performance evaluations MOTION EQUATIONS For a cam follower system the follower displacement s t is a function of cam rotation angle O t Mathematically it can be expressed as s t f O t 1 where O t is the cam rotation angle at time t The follower velocity v t of the follower is then given by v t f O o t 2 where f O df O dO and co t dO t dt is the cam angular velocity Furthermore the corresponding follower acceleration a t and jerk j t are a t f O co2 t f O dg t 3 j t f O co3 t 3f O co t cb t f O 6J t 4 where f O df2 O dO 2 f O df3 O dO 3 t dco t dt and c3 t dcoZ t dt 2 Equations 1 4 present the relationship between cam input angular velocity co t and follower output motions s t v t a t and j t Obviously if co t is a constant they can be greatly simplified Let h be the total displacement of the follower as the cam rotates an angle in time period 3 Furthermore denote T t r 0 8 and S s h Now we have T 0 1 Vt 0 r 0 1 V0 0 and Se 0 1 Vs 0 h Then equations 1 4 can be rewritten in terms of their normalized forms as follows s t S T g h 5 V T g y t2 T 6 A T g y fl2 T I g y i l T 7 J T g y 3 T 3g 7 t2 T t I T g y T 8 where t2 T dy T d T is the normalized cam angular velocity and V T A T and J T are the normalized velocity acceleration and jerk of the follower respectively The relationship between equations 1 to 8 can be found as s hs 9 h v v 10 T Effects of cam speeds on cam follower systems 399 h a A ll hj j 12 When the cam operates at a constant speed i e f T 1 the normalized velocity Vc T acceleration At T and jerk J T of the follower can be expressed as V T g 7 13 Ac T g 7 14 J T g 15 where 7 T T CRITERIA FOR DESIGN T For a given cam follower system the peak values of the normalized velocity acceleration and jerk resulting from a constant driving speed may possibly be reduced if we properly control its input speed trajectory f T For example to reduce the peak values of normalized velocity f T can be chosen so that I V Tpv l II o Tpv l where Vc has peak values at normalized time Tpv Then from equations 6 and 13 we select that 12 T must satisfy the condition 1 f Tpv 1 16 For the case that we want to reduce the peak values of the normalized acceleration i e IA r l IAc r l were Ao has peak values at normalized time Tp Based on equations 7 and 14 f T should be chosen such that T 1 F 2 Tpa g y Tpa t p tl f 2 Tp 17 g Y Tpa t Note that g 7 Tp must be nonzero Similarly if it requires J Tpj l IJc Tpj l where J has peak values at normalized time Tpj then from equations 8 and 15 we need some II T which satisfies 3g 7 Tpj tl Tpj t Tpj g Tpj Tpj 1 S Tpj 18 1 f 3 Tpj 0 i e the direction of cam speed is not changed As a result design criteria for selecting fZ T to reduce the peak values of follower outputs are a I for the case of reducing the peak value of the normalized velocity l Tpv 1 II for the case of reducing the peak value of the normalized acceleration g TPa T 1 fZ2 T 1 f 2 rpa g T III for the case of reducing the peak value of the normalized jerk 1 fZ3 Tp 3g Tpj fl Tpj Tp g T rpj Tpj g y r j 0 Let equations 5 8 represent the normalized motion characteristics of the follower in the rising period Then the motion characteristics in the falling period are S T 1 g 22 V T g fZ T 23 A T g fZ2 T g T 24 J T g n3 T 3g r n T t T g r t T 25 It is easy to find that the absolute values of the normalized velocity acceleration and jerk in the falling period are equal to those in the rising period respectively Hence we have the following fact If the same displacement curve is used in the rising and falling periods of a follower the functions of fl T in these two periods are identical ANGULAR VELOCITY f T Consider a cam follower system which has a cam providing a cycloidal motion curve where cam input fZ t is a polynomial In the rising or falling period and applying criteria a and g to reduce the peak values of motion curves we choose the following polynomial fZ T Fig 1 t 3 d t i t i T 0 T a T b 1 Fig 1 Polynomial angular velocity in rising or falling period Effects of cam speeds on cam follower systems W 0 0 5 Fig 2 Polynomial angular velocity in dwell period 401 f fr sfr VO3 Aft J T 1 15 1 101 051 001 0 95 t 0 90 I I t t 0 0 25 0 5 0 75 1 00 0 75 0 50 0 25 0 00 I I I 0 25 0 5 0 75 2 50 2 00 1 50 1 00 0 50 0 00 0 25 01 5 0 75 10 5 0 5 10 0 25 0 5 0 75 75 25 0 25 50 t 0 0 25 0 5 0 75 Variable speed Constant speed Fig 3 Cycloidal motion T T T 402 H S Yan et al Table 1 Cycloidal motion Constant Variable angular velocity angular velocity Difference Peak value of V 2 00 1 83 8 5 Peak value of A 6 28 5 97 4 9 Peak value of J 39 48 52 55 33 1 for 0 T Ta for T T Tb for Tb T I f T 1 d 26a fl T 1 d 1 e r Ta X T Tb y 26b f T 1 d 26c where constant parameters d e x y TR and Tb are to be determined Parameter d presents the fluctuation of f T where 1 d 2 27 x y 0 2 4 28 Parameters x and y are determined based on the type of cam displacement curves and the design criteria e Furthermore parameter e subject to design criteria f is given by 30 e Ta Tb 5 29 Apparently we can properly select d Ta and Tb tO obtain the lowest peak values of motion characteristic under the polynomial f T of Fig 1 Since the cycloidal motion curve is of symmetry we let Tb 1 T for simplicity and symmetry In addition when the follower is in the dwell period from design criteria c and g and equations 26 29 we obtain f T Fig 2 as follows fl T 2n 2T 3 3T 2 1 n 30 Under design criterion d and equation 30 we have y T n T 4 2T 3 1 n T 31 and from design criteria c and g we imply T 12n T 2 T 32 8 C0sl 0 ave I I t Time t 0 1 1 t2 1 2 3 1 2 3 g4 Fig 4 Angular velocity of a variable sl ed cam follower system Effects of cam speeds on cam follower systems 403 g 110 105 100 95 90 85 I I I I 0 O 12 0 24 0 36 0 48 0 6 Time s D 35 30 25 20 15 10 5 0 5 J I I 0 12 0 24 I 0 36 I 0 48 0 6 Time s 300 200 100 0 100 o 200 300 I I I 0 12 0 24 0 36 0 48 0 6 Time s 5 000 2 500 0 O 2 500 5 000 U I I O 12 0 24 I 0 36 0 48 0 6 Time s 300 000 150 000 0 0 150 000 300 000 I I O 12 0 24 I 0 36 0 48 0 6 Time s Constant speed Variable speed Fig 5 Cam follower system with cycloidal motion n 0 d 0 1 co vo 100 rpm MMT 31 4 D 404 H S Yan et al l T 12n 2T 1 33 where l n r t 0 5 011 0 15 012 0 25 Time sec offHne theory on line theory measure a Response of motor speed Fig lOa Caption on p 409 408 H S Yan et al to drive the cam follower system The achievement of this angular velocity by the motor can be accomplished most easily by employing a velocity control system 4 An IBM PC AT plug in evaluation board TMS320C30 system board 5 6 is used in the real time experiment setup The hardware configuration of the experimental system is depicted in Fig 9 In addition to digital communications through the AT bus the input output analog signals are accessed through on board A D converters ADC and D A converters DAC These input and output channels are for the feedback signals to the DSP and for the control signals to the controlled plant respectively In the real time control the sampling rate 60 sec is adopted in our experiments so that controller design can be done from the continuous time The controlled 30 25 20 8 t 0 05 0 1 0 15 0 2 0 25 Time scc theory measure b Response of follower displacement Fig lOb Caption on p 409 600 400 200 200 4O0 6000 012 0 1 0 15 Time scc theory measure c Response of follower velocity Fig lOc Caption on p 409 0 Effects of cam speeds on cam follower systems 409 output responses are measured through an on board ADC and DAC and stored on the on board memories The speed of driving motor is picked up from the driver of the motor i e the voltage signal of built in tachometer and fed to a PC486 personal computer The acceleration and displacement signals of the follower can be measured by using the accelerometer PCB 353A34 linear encoder HEIDENHAIN LS404 as shown in Fig 8 The signal from the accelerometer is conditioned by o 0 5 1 1 5 xl04 2 1 5 1 0 5 1 l k 20 0 05 011 0 15 012 0 25 Time see theory measure d Response of follower acceleration Fig 10d i v d xl06 1 5 11 5 41 5 1 vii J 1 50 0J 5 I t li I Ii o11 0 15 0 2 Time see theory measure e Response of follower jerk Fig 10e 0 5 Fig 10 Experimental results of a cycloidal cam follower system n 0 d 0 1 and to e 200 rpm 410 H S Yan et al a power unit PCB mode 480E09 ICP Using least square fit method 7 we can obtain the velocity of follower from the displacement signal and the jerk from the acceleration signal The measured data are then passed back to the PC host for performance evaluation Both of the theoretical and experimental results of n 0 d 0 1 at the average cam speed of 200 and 150 rpm are presented in Figs 10 and 11 respectively Although the fluctuation of the driving speed occurs Figs 10 and 11 demonstrate good agreement between the theoretical and experimental results of each cycle The experimental results show the proposed approach is feasible 170 160 2 150 8 140 130 120 V J it d db 4 I h i i 1 110 0 05 011 0 15 0 2 0 25 013 0 35 Time see off line theory online theory measure a Response of motor speed Fig lla Caption on p 412 j v 30 25 20 15 10 01 0 1 5 i 0 1 0 15 0 2 0 25 0 3 0 35 Time see theory m asure b Response of follower displacement Fig lib Caption on p 412 Effects of cam speeds on cam follower systems 411 40 30 200 100 0 I00 200 300 4OO 5000 o b5 011 0 15 012 0 Time sec theory measure c Response of follower velocity Fig 11c Caption on p 412 k 013 0 5 xlO 1 5 G B e o o 0 5 0 0 5 1 1 5 0 0 5 011 0 15 012 0 25 013 0 15 Time sec theory measure d Response of follower acceleration Fig 1 ld Caption on p 412 412 H S Yan et al xl06 1 fl v 0 8 0 6 0 4 0 2 0 0 2 0 4 0 6 0 8 10 jV i ii I i o 5 o11 o 15 012 0 013 0 35 Time sec theory measure e Response of follower jerk Fig lie Fig I1 Experimental results of a cycloidal cam follower system n 0 d 0 1 and toav e 150 rpm CONCLUSION In this work from the kinematic point of view and b