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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) Altraet-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, co4-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 tF-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 T0, 1, Vt 0, r, 0, 1, V00,/ and Se0, 1, Vs0, 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)il(T) (7) J(T) = g (y)3(T) + 3g (7)t2(T)tI(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 _ f2(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 + f3(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 + f2(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 ffr) 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 + d1 -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-sled 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 -ln . / / 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 150 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 - masure (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 iii 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