Mech. Mach. Theory Vol. 23, No. 1. pp. 1-6, 1988 0094-114X/88 $3.00+0.00 Printed in Great Britain. All rights reserved Copyright © 1988 Pergamon Journals Ltd

O P T I M A L S Y N T H E S I S OF C A M M E C H A N I S M S

W I T H O S C I L L A T I N G F L A T - F A C E F O L L O W E R S

JORGE ANGELES and CARLOS LOPEZ-CAJI]N Robotic Mechanical Systems Laboratory--McRCIM and Department of Mechanical Engineering,

McGill University, Montr6al, Qu6bec, Canada

(Received in revised form 29 April 1987)

Abstract--The synthesis of the cam profile producing a given displacement program of its oscillating flat-face follower, while enclosing a minimum area is presented. The displacement program of the follower, as well as the cam profile, are generated through the use of cubic periodic splines following a method reported previously. The optimum parameters of the cam machanism are obtained via the minimization of the cam-disk area subject to geometric constraints. It is shown that the problem reduces to finding the roots of a quartic equation, which is done explicitly using Ferrari's formula. The procedure is illustrated with a fully-solved example showing its applicability to the automatic design of cam follower mechanisms.

NOMENCLATURE

a = distance between the cam axis of rotation and that of the follower

b = follower arm offset = normalized follower arm offset

f = a limiting factor set by the designer t(x) = tangent of x/2

A = area of the cam disk T -- period of follower motion 0 = see p, 0 below 00 = reference value for 0 0 = angular displacement of the follower

0,~ = angle of the follower at its lowest dwell 0 ' = d0/d0, derivative of 0 with respect to O"= d20/d~, 2, second derivative of 0 with respect

to ¢ p, 0 = polar coordinates describing the cam profile

= normalized p (~b)=a positive definite function such that

0 ~< ~(~) ~< A0 ~, = angular displacement of the cam disk

A0 = maximum amplitude of the follower oscillation

INTRODUCTION

With the upcoming and widespread use of numer- ically controlled (NC) machine tools, a significant effort has been given to the development and imple- mentat ion of integrated CAD/ CAM systems. As a consequence, during the last two decades the manufacture of cams has become more accurate [1]. Research work along this lines, involving the use of computational schemes which deal with problems related to novel approaches for the displacement- program synthesis have been reported [2-6]. This is particularly relevant to the analysis and synthesis of cams, as these are used more often under high operational speeds [7-11]. However, literature related to the synthesis of cam mechanisms with oscillating M.M.T. 23/I--A

flat-face followers is rather scarce, to the best of our knowledge. One can only mention the contribution of Wunderlich [12], who determined the cam profile for the said type of follower in closed form, but without attempting any optimization. His method is more suitable for a graphical construction. The work re- ported herein, which is a continuation of previous work [13-15], deals with a computer algorithm that allows the synthesis of the displacement program, and the computat ion of the opt imum parameters. These, in turn, are used to calculate the polar coordi- nates of the cam profile and thus to generate the instructions program that will guide the tool of a NC machine tool. The approach for determining the follower-displacement program is the same as the one described before [5, 6]. Given the displacement pro- gram, the opt imum parameters of the cam mech- anism are obtained via cam disk area minimization, subject to geometric constraints. This leads to a quartic equation that is solved explicitly. Moreover, criteria are set for discarding unfeasible roots. Once the opt imum parameters have been determined, a set of points is generated, which, in turn, are used as supporting points of a periodic parametric spline that renders the coordinates used to generate the machine-tool program. The software developed is part of an integrated C AD/C AM package for the interactive opt imum design and manufacture of cam mechanisms.

S Y N T H E S I S O F T H E FOLLOWER-DISPLACEMENT PROGRAM

The method that allows to synthesize the follower- displacement program has been described in detail elsewhere [14, 15] and is briefly outlined here for quick reference. Let us consider the layout of a cam

2 JORGE ANGELES and CARLOS L~PEZ-CAJON

c

Fig. 1. Layout of a disk cam with oscillating flatrface follower.

~(~)

1.0

o I Fig. 3. Normal spline.

T

0.S

with an oscillating flat-face follower as shown in Fig. 1. Lines OF and OC are fixed to the reference frame and to the cam plate, respectively, whereas b is a follower offset (segment P-Q), a parameter to be determined. Let p (segment O--R) and 0 be the polar coordinates describing the cam profile, and $ and the angular displacements of the cam plate and the follower, respectively. Moreover, q~ is the sum of a constant Ore, as yet to be determined, and a positive definite function, ~(¢), whose minimum value is 0, its maximum value being A4b, the amplitude of the follower oscillation, i.e.

~ =~m+~(¢) , 0<~(¢)~<a~. O)

On the other hand, let the displacement program be the one shown in Fig. 2. The rise phase, R i, is obtained by a proper scaling plus a rigid-body trans- lation of segment S T of the normal spline x(x), shown in Fig. 3 which, in turn, was synthesized using periodic spline functions [5, 6]. Reflection of S T with respect to x =0.5, plus the two transformations mentioned above, produce the return phase, R 2.

Constant ~m of function 4b(¢), as well as b are to be determined in order to produce a minimum-area cam profile, while satisfying geometric constraints. This procedure is outlined in the next section.

zx~

0

D2

,hi ,b2 ~3 i .

Fig. 2. Follower-displacement program.

SYNTHESIS OF THE CAM PROFILE

In the following all the lengths will be normalized with respect to a, (segment O-"P, Fig. I), which is the distance between the axis of rotation of the cam and that of the follower, and hence never vanishes. The nondimensional quantities fi and G are introduced now, which are defined as

= p/a, 5 = b/a. (2a)

From Fig. 1 one has,

sinO + G fi = sin(~, + ~b + 0) ' (2b)

whereas from polar-coordinate calculus,

p'(O) = p(O___~) = _ p(0) (3) tan y tan(C, + ~ + 0)"

Now, application of the chain rule yields

p'(o) = p'(¢)/o'(¢). (4)

Differentiating equation (2b) with respect to $, one obtains

4 , ' ( ¢ )cos 4, - :[l + ~'(¢) + o'(¢)1 cos(~ + ,# + O)

p '(~k) = sin(~b + ~b + O) (5)

Substitution of equations (4) and (5) into equation (3) yields

tan(~, + t# 4- O) = sin,- 4- 5 ~b cos ~b (1 + O'), (6)

where 4 ' denotes dO/d~,, i.e. ~'(~,), the ~ argument being omitted for brevity. Therefore,

sin(~/4- ~ + 0) ---- (1 + ~ ' ) ( s in ~ + ~')

J~'~ cos ~ ¢ + (sin ¢ + ~2(1 + ¢'Y (7)

Optimal synthesis of cam mechanisms 3

Substitution of equation (7) into equation (2b) leads to

# x/0 '2cos20 + ( s i n 0 -{-5)2(1 "F'0') 2 = 1 + 0 ' . (8)

From equation (8), for a given value of 5, and a prescribed follower-displacement program, 0(0) , one obtains ~ (0). From equation (6), one can readily derive the functional relation 0 = 0(0). Combination of the parametric representations P(0) and 0(0) produces ~ =~(0) , the cam profile being readily obtained as p(O)= a~(O).

To this point, the only remaining problem is to determine the parameters 0m and 5. In order to obtain such values a minimization of the area of the cam disk is undertaken. In fact, the area is given by:

1 ~0o+2~ 1 f0 2" A = ~ j O o /~2d0=~ f i2(0)0 ' (0)d 0. (9)

Differentiating equation (6) with respect to ~ one obtains

0 '3 + 0'2[1 +/~ sin 0(I + 0')]

- 0" c°s 0 (sin 0 +~ I o'(0) = o,~ cos~ o 7-(,i--Fg-~ G)~ ~ - F F - (l + 0'),

(10) where the trigonometric identity

1 cos2(x) =

1 + tan2(x)

has been used. Substituting equations (8) and (lO) into equation (9) leads to

A = Zm + Z2 + Z 3, ( l la)

where ~2n 0,2

Zl = ~0 ~ sin(Ore + °)[sin(Ore + o) + b'] d O

(1 lb)

f [ 0" Z2= - (l + 0 ' ) 2c°s(0~ + ° )

x [sin(0. + o) + 5] dO (1 lc)

f? Z3= - [sin(0~ + o) + 512(1 + 0 ' ) d 0 . ( l ld)

The foregoing equations show an explicit relation- ship between the area, A, and the parameters 0~, and 5. In order to find the values of these parameters that minimize A, the following necessary conditions are introduced:

~A ~A ~O-'-~ = O, ~-~ = O. :' (12a)

After algebraic manipulations, equations (12a) lead to

6 sin 20~ + E cos 20m + b '[-~ sin Or. + fl COS 0m] = 0

'(12b)

r/5 + fl sin 0m + a cos 0m = 0 (12C)

where

f ~ 2x = A, sin O(0) de -- f0 A2 COS O(0) d0

f? - 2 A3 sin a (0) dO (12d)

f? f: fl -- A1 cos o(0) d0 + A2 sin o(0) d0

I? - 2 A3 cos o (0) dO (12e)

=f~x A 1 COS 2O (0) dO --f]" -'t2 sin 2o (0) dO

-- [ ~ A3 cos 2o(0 ) dO (12f)

=f? -I; Al sin 2o(0) d0 A2 cos 2o'(0) d0

- I~ ~ A3 sin 2o(0) dO (12g)

~/= A3 dO (12h)

coefficients Ai (i = 1, 2, 3) being defined as

0 '2 0" A, = 1--~-- 7, A2 = (1 + 0 ' ) - ' - - ' ~ ' A3 = 1 + 0'- (12i)

From equation (12c),

5-- fl sin0m+~' cos0m (13)

Substituting equation (13) into equation (12b), one obtains

Bi cos 2 0m + Be sin 2 0m + B3 sin 0tacos 0m = 0, (14a)

coefficients Bt (i = 1, 2, 3) being defined as

Bl ~- tle -- Otfl, B2 = -- BI, B3=2r/6--f12q-~ 2.

(14b)

Resorting to the trigonometric relationships

2t 1 -- t 2 sin(x) = 1 - - -~ ' cos(x) = 1 + t - ' - - 5 '

x t = t ( x ) - tan-~,

equation (14a) can be rewritten as

t4 + Clt3 + C2t2 + C3t + C4=O, (15a)

where

2B3 4B2 - 2B~ c ,=- - K, c~= B-----7--' C 2B3 3 = - ~ - , ( 7 4 = - - 1 . (15b)

Thus, the problem has been reduced to finding the roots o fa quartic equation, equation (15a), which can be done explicitly, making use of Ferrari's formula [16].

4 JORGE ANGELES and CARLOS LOPEZ-CAJISN

There are, however, geometric constraints that have to be satisfied. By assuming that the cam disk and the follower arm PQ are contained in the same plane, /5 is limited to lie between 0 and 1, i.e. 0 </5 < 1. This guarantees that the vertical com- ponent of the normal force, i.e. the thrust force, will never vanish; that is, the mechanical advantage will never be 0. This will allow us to establish criteria for discarding unfeasible roots.

From equation (8), it can be seen that, as long as 05" is away from - 1 , /5 will have a finite value; therefore the designer should obtain the minimum value of 05', which occurs exactly at @ = (T + ff/3)/2 (half of the return phase), and check that his/her follower-displacement program satisfies such require- ment; otherwise, it will be necessary to modify it.

Furthermore, from equation (8), /52 can be ex- pressed as

where

COS2 05 q(05)- (] + 05')~

/52_= 1 + q(05), (16a)

c o s ~ 4' "] + ~ + 2 (t~ sin 05 i-~-~j. (16b)

~-0.19

-0 .85-

-0.91

- -0.95

55.0 65.0 75.0 85.0

f = 0.9

J

~ f = 0 . 3

Fig. 4. Function q (~b) vs q5.

From the constraints imposed on /5, one can write

q(05) < - 1 +./-2, (17a)

where

f < 1. (17b)

Fig. 5. Graphics workstation.

Optimal synthesis of cam mechanisms 5

Thus, if the designer sets f =fO(< 1), he/she can verify, through a graphic display (for example) that his/her follower-displacement program strictly satisfies the inequality (17a). In Fig. 4, a plot of equation (16b), withf = 0.9, and f = 0.3 is shown for the example presented herein. The attempted design for the latter one being discarded, since q(4) over- comes the imposed limit.

All the software developed for this research was implemented in a SUN workstation with multiple- window capabilities. Three windows are shown in Fig. 5, which display the following in counter- clockwise order: (i) a segment of the main FOR- TRAN program; (ii) the displayed q(4) function and (iii) the normalized cam profile. A brief description of the algorithm for the software developed is outlined next.

ALGORITHM DESCRIPTION

1. Reads the values of 1(11, &, tjs, of Fig. 2, the maximum angular displacement A6 and the value off.

2. Generates the displacement, velocity and accel- eration curves through proper scaling (and reflection, for the return phase) of the normal spline.

3. Calculates the optimum values of 6, 4min, deter- mines pmin and verifies inequality (17a). If this holds, go to step 4: otherwise, abort (this indicates that the displacement program should be modified).

4. Evaluates 6($) and p($). 5. Generates p(6) and its graphical representation

using periodic parametric splines. 6. Evaluates the geometrical properties of the cam

profile. 7. Done.

EXAMPLE

Synthesize a cam follower mechanism for an oscil- lating flat-face follower, which will produce the fol- lower-displacement program appearing in Table 1. The amplitude of the follower oscillations are pre-

Table 1. Prescribed angular-displacement program

Phase Angle of

rotation (JI) Displacement

(4)

D, (dwell) R,(rise) D, (dwell) R,(retum)

36 0” 72 + 30” 72” 0”

180” -30

scribed as 30”. The procedure produced the nor- malized cam profile shown in Fig. 6, the optimum parameters being: q$,, = 56.48” and 6 = -0.6222. The centroid, C (-0.0828, -0.0422) and the centroidal axes of inertia, E, and E2, are indicated in the same figure, E, making a 6.3” angle with the X axis. The normalized geometric properties are: area, 2 = 0.2997, centroidal moments of inertia, 4 = 0.0089 and & = 0.00595. Since all the geometric prop- erties were obtained with the normalized value p, one should multiply by the appropriate scaling factor, for example, assuming a value of tl = 2OOmm, then b = - 124.45 mm the coordinates of the centroid are C (- 16.28, -8.44 mm), the area is A = 11988 mm*, and the centroidal moments of inertia are Z, = 14.24 x 106mm4 and I2 = 9.52 x lo6 mm4.

Y4 E2 \ ‘A 1

4

0.4 =

-0. w -0.

Fig. 6. Normalized cam profile.

2.10s-2 0.83s-’

6-i

Fig. 7. Displacement, velocity and acceleration of the follower.

6 JORGE ANGELES and CARLOS L~PEZ-CAJIJN

Plots of displacement, velocity and acceleration for the example presented herein are shown in Fig. 7.

CONCLUSIONS

An algorithm allowing for a computer imple- mentation, and therefore the interactive opt imum design (and manufacture) of cam mechanisms, has been presented. The software developed, which is user oriented, produces a minimum-size cam disk for an oscillating flat-face follower, moving according to a prescribed angnlar-displacement program. Ad- ditionaUy, it produces relevant geometric properties of the profile, such as its area, its centroid and values of its principal moments of inertia, that are necessary for a static and dynamic analysis of the mechanism.

Acknowledgements--The research work reported here was partly done at the CAD Laboratory of the Graduate Division of the Faculty of Engineering--National Autono- mous University of Mexico (UNAM). It was completed at the Robotic Mechanical Systems Laboratory of the McGill Research Centre for Intelligent Machines (McRCIM). This was possible under NSERC (National Sciences and Engineering Research Council, of Canada) Research Grant No. A4532, FCAR (Foods pour la Formation de Chercheurs et rAide fi la Recherche, of Quebec) Research Grant No. EQ3072 and an Actions-Structurantes (Qu6bec) Research Associateship. The valuable suggestions of Mr Liangcai Zhao, visiting research fellow from People's Republic of China, are highly acknowledged.

REFERENCES

I. B. Grant and A. H. Soni, A survey of cam manufacture methods. J. mech. Des. 101, 455-464 (1979).

2. N. T. Thompoulos and T. W. Knowles, Use of linear programming for cam design. Int. J. Mach. Tool Des. Res. 15, 257-265 (1975).

3. T. Hitoshi, Kinematic design of cam follower systems. Ph.D. Dissertation, Columbia Univ. (1976).

4. M. N6 S~nchez and J. Garcia de Jal6n, Application of B-spline functions to the motion specification of cams. ASME Paper No. 80-DET-28 (1980).

5. J. Angeles, Synthesis of plane curves with prescribed local geometric properties using periodic splines. Corn- put. Aided Des. 15, 147-155 0983).

6. J. Angeles, Sintesis de curvas planas cerradas usando funciones "spline" param6tricas y peri&iicas. Revta ANIAC (M6xico) 2, 53-81 (1983).

7. F. Y. Chen, A survey of the state of the art of cam systems dynamics. Mech. Mach. Theory 12, 201-224 (1977).

8. F. Y. Chen, Mechanics and Design of Cam Mechanisms. Pergamon Press, New York (1982).

9. D. Tesar and G. K. Matthew, The Dynamics, Synthesis, Analysis and Design of Modeled Cam Systems. Lexing- ton, Mass. (1976).

10. J. Chakraborty and S. G. Dhande, Kinematics and Geometry of Planar and Spatial Cam Mechanisms. Wiley, New York (1977).

1 I. Y. Terauchi and S. A. E1-Shakery, A computer-aided method for optimum design of plate cam size avoiding undercutting and separation phenomena--I. Mech. Mach. Theory 18, 157-163 (1983).

12. W. Wundedich, Contributions to the geometry of cam mechanisms with oscillating followers. J. Mechanisms 6, 1-20 (197l).

13. J. Angeles and C. L6pez-Cajfin, Disefio automatizado de mecanismos de leva de disco con segnidor trans- lacional de cara plana. Memoria del IX Congreso de la ANIAC, Le6n, Guanajuato, Mbxico. pp. 91-93 (1983).

14. J. Angeles and C. L6pez-Cajfin, Optimal synthesis of translating roller-follower cam mechanisms with prescribed functional constraints. Proc. 1st Int. Symp. on Design and Synthesis, Tokyo, Japan, pp. 782-787 (1984).

15. J. Angeles and C. L6pez-Caj/m, Optimal synthesis of oscillating roller-follower cam mechanisms with prescribed functional constraints. ASME Int. Conf. Comput. in Engin, 2, 273-277 (1984).

16. S. M. Selby (Ed.), CRC Standard of Mathematics Tables. The Chemical Rubber Co., Cleveland Ohio (1973).

O P T I M I S A T I O N D U M E C A N I S M E S D E L A C A M E D I S Q U E A V E C P O U S S O I R R O T A T I F A, S U R F A C E P L A N E

R/~um6---Les auteurs pr6sentent une m~thoe de synth6se optimale du profil d'une came disque entrainant un poussoir rotatif ~ surface de contact plane. Le programme de d6placement ainsi que le profil sont repr~m~s par des splines cubiques p~riodiques, d'apres une m6thode pr6sent~ allleurs. Les param~tres opfimaux du m~anisme sont calcul6s ~ travers tin probl~me de minimisation de la surface du profil, compte tenu des contraintes g6om6triques. On d~montre que ce probl~me se traduit par une ~luation quartique, dont les racines sont calcul~es avec la formule de Ferrari. Un exemple r~solu en d~tail montre comment appliquer ce processus ~t la conception automatique des m~canismes du type consid6r~.