Mr, cA. Mack. ?'kt, eg, Vol. 27. No. ]. pp. 225-233. 1992 00~I-IIIX/92 t~.00 + 0.00 Printed in G~at hitm. An riots nmrvud Cot~YriSht © 1992 Perllamou Prom pig

TORSIONAL VIBRATIONS OF CAMSHAFTS

ZtYA q~AKA Engineering Faculty, Selfudg University, Makina Mt3hendislil]i B61fimfi, Koeya, Turkey

YOKSEL YILMAZ Mechanical Engineering Faculty, Istanbul Technical University, Tekstil B61., lstanbul, Turkey

(Received 3 June 1991; received for publication II July 1991)

Almract---This study deals with the torsional vibrations on the camshafts in force closed cam mechanisms. On the cam mechanisms with reciprocating followers, the variable torque which is prod,__,¢~__ by force exerted to the cam from the follower, will create torsional vibrations on the camshaft.

A dynamic model of the camshaft was established and some noudime~oual ~ which can be used to characterize dynamic behaviour of a camshaft were obtained. Torsional vibrations of camshath are pummetric and self-induced vibrations. The igoveming nonlinear differential equations were solved numerically. Vibrations of camshafts with • flat-faced follower differ from ores with • roller foUower because of the friction between cam and follower. Vibrations on the camshaft affect the follower motion and contact force.

INTRODUCTION

A cam undergoes rotational motion and is driven by a camshaft in almost all of the cam mechanisms. The cam is subjected to a dynamic force from the follower, therefore, the camshaft to which it is connected is subjected to a dynamic torque. In many cases, angular velocity of the camshaft is assumed to be constant and the follower motion is examined under this assumption. As the velocity of the mechanism and the follower mass increase, the magnitude of the torque acting on the shaft increases and torsional vibrations occur on the camshaft. This causes fluctuations in angular velocity of the camshaft and affects the operation o f the system.

Many articles considering the effects of follower elasticity on the dynamics of a cam mechanism are shown in Refs [!-3]. Elasticity of the camshaft was also considered by some authors, goster [4] and Chen and Polvanich [!] investigated influences of camshaft elasticity on follower motion. Kim and Newcombe [5] examined a cam mechanism with a detailed model. Szakallas and Savage [6] presented some performance charts considering windup on closed and open track cam mechanisms. By using Koster's model, Midha and Tureic [7] analysed the periodic response of a cam mechanism with flexible follower and camshaft. The linearized equations o f motions of a two degrees-of-free- dom system was derived and presented with stability charts in Refs [8, 9].

In this work, firstly, the differential equation of torsional vibration was obtained by means of dynamic analysis of cam mechanisms and some nondimensional parameters are defined. Secondly, with the differential equation being solved numerically, the nature of vibrations and their influences on follower acceleration and contact force were discussed.

DYNAMICAL ANALYSIS

Free body diagrams of the cams with reciprocating roller and flat-faced follower and their followers are shown in Fig. I. The vertical component of contact force is F, and the horizontal one is Fx. In the cams with a roller follower, at being the pressure angle,

Fx -- F, tan ~,, (1)

the F~ component is the friction force in.the cams with a flat-faced follower

Fx ~ F, = ~ r . (2)

For the follower motion, when the D'Alemhert principle is applied, we find

F,-~ F + F. + ma, (3)

225

226 Z~,A ~ and YOlr.~. Yn.ttAz

F s F --~ _ ,Fy__ _ Y

r 0 + S r 0 + S

/

Fig. I. Free body diavams of the cams wilE reciprocating roller and e,-t-f~ed fdlower.

where F is the spring force (F=k,s), F, is sum of external forces like weisht and l~ load , m and a are follower mass and acceleration, respectively.

To prevent jumping, F, should be greater than zero. At the extreme case this should he:

F. + F,> mo., (4)

where

F. = k,h and a . = ~2s~,,

k is the maximum rise of the follower and s~ is the maximum value of s". Then, for the spring stiffness, we find

k, > ~ (,,uozs~, - F.). (5)

In practice, in viewpoint of safety, this value is multiplied by a coefficient ~ = I.! - 2 and the spring is designed in according to this:

1 k, = ~ (, ,~2s~, - F.), (6)

where ). is a safety factor for preventing jumping of the follower due to exceuive loads and vibrations. By multiplying k, by 2, equation (4) is satisfied. In this paper, it is considered that

= 1.5161. The follower acceleration is found by differentiating the follower displacement two times:

a = s"# 2 + s'#',

where (') denotes d/d0. Then, for the F~ component,

F,--ms"02+ms'~+~mco ss.+F, 1

As shown in Fig. !, the torque on a cam with roller follower is

r = Fx(,o + s). (8)

where ro is the base circle radius. Substituting equation ( ! ) and tan ~ -, s'/(ro + s) in equation (8), we find

r = 6". (9)

For the camshaft of a cam with a flat-faced follower (Fig. 1),

r = F,b + 6 ( to + s). (10)

Torsional v i b m ~ of ~ 227

e

Fig. 2. Dynamic model of camslu~ and follower.

Substituting equation (2) and b = s' in equation (10), we find

T = F,[s' + ~(ro + s)]. (I I)

As shown above, it is understood that the torque on the camshaft depends on the vertical component of the contact force, cam slope (s') and in addition the cams with a fiat-faced follower, coefficient of friction and cam size. The torque on the camshaft of a cam with a fiat-faced follower is greater than one with a roller follower.

The dynamic model of the camshaft is shown in Fig. 2. The cam is considered to be an inertia element connected to the input shaft (driving member) by a torsional spring. The mass moment of inertia of this element equals the sum of those of the cam and camshaft. The torsional stiffness of the spring is the same as that of the camshaft. Damping may be neglected. It is assumed that the follower is a mass equipped with a return spring. The follower can be assumed to be rigid since the subject is the camshaft's torsional vibrations [I0].

If the D'Alembert principle is applied for the rotational motion of the cam,

1O" = k(00 - 0) - T, (12)

Substituting equations (9) and (10) in equation (12), for the cams with a roller follower,

!# - k ( Oo - O) - ms',"O 2 - ms'2ff - -~ m~2ss's;- F, 1

and for cams with a fiat.faced follower,

2 . / ' 1 ' ) 1 ( , lO'=k(Oo-O)- ms'O2+ma'~+gmm ss.+F, LI -~,/ . j s +Pro+m), (14)

where O0 is the rotation of input and varies linearly since input is rotating at a constant speed. However, the cam rotation angle 0 is a variable. From the point of view of understanding torsional vibrations of the camshaft, it is more suitable to determine a relative variation of the cam angle 0 with respect to the input angle Oo, instead of the time variation. Hence, let us define relative rotation angle ¢ :

, / ,=Oo-O.

O = Oo-,/,.

0 dO dOdOo=m(i d,/,,'~, dOo = d'-t ---dOo dr" -- "~o/ m = d-T'

, d ' ¢ #=-co dOo2, (is)

where m is angular velocity of input as mentioned before. Dividing each term in equation (13) by I and -m' and writing ~ instead of 0:

I m ,2Xd2¢ k m , ,/

Let us normalize s, s' and s" terms dividing by the maximum rise k: $0 $0,

s=~, s'=-~, s"= T.

06)

228 ZIYA ~ and yCrr~ Yn.tt~

Substituting these normalized terms and rearranging equation (16), we find

. I'S"X 2-] d'$ S'S" / _ d$~ ' = ,l . SS" S" I+M~"~=:J-~o+N'd/-M'-~.2 ~I dOo] MS"-~.2+P-~ (1-2S)" (17)

where

u=--7-' M, N and p nondimensional parameters are inertia, frequency and energy ratios, respectively, s~, is the maximum cam slope.

Using the same mathematical operations used for cams with a flat-faced follower, a similar differential equation is obtained:

( __S'U,d 2, S'U( _ d ~ 2 ,SU U I + M--~,2 J-~o+ N2q/ -- M--~21 dOo/ - AMS,,,.-~2 + p-~(l - AS), (19)

where

U=S'+#S+¢ and size ratio

Equations (17) and (19) are nonlinear differential equations with variable cneffgients. There are no closed form solutions to these. As shown in the equations, torsional vibrations of camshafts are parametric and self-induced vibrations.

These nondimensional parameters can be used to characterize the dynamic behaviour of the camshaft. The first parameter inertia ratio is the proportion of follower's inertia to that of the cam and is a measure of the influence of follower mass on the camshaft motion. The frequency ratio is the proportion of the natural frequency of a camshaft to input angular velocity. It gives us an idea of the shaft rigidity and relative velocity. The energy ratio is the proportion of potential energy of F, force and kinetic energy of the camshaft. The size ratio which is special to cams with only a flat-faced follower, is concerned with the friction coefficient and relative size of the cam. The greatness of this ratio means that the friction or base circle radius is great, this case augments the torque that affects the camshaft.

D Y N A M I C R E S P O N S E OF A C A M S H A F T

Nonlinear differential equations (17) and (19) obtained for the camshaft's vibratory motion were solved numerically by the Runge-Kutta-Nystr6m method[l i]. Figure 3 shows the dynamic responses of camshafts for harmonic, cycloidal and 4-5-6-7 polinomiai cam curves for the roller and flat-faced follower. These are steady-state responses which are obtained after the first three revolutions. The steady-state occurs almost after the first three revolutions for the values of nondimensional parameters in Fig. 3. The cam angles corresponding to the rise and return in the follower motion are equal and they are considered to be 90 ° in the figures. The rise and return motions of the follower are symmetrical, thus responses of the camshaft corresponding to them can be better examined.

As shown in Fig. 3, there exists a vibration with a small amplitude accompanying one with a large amplitude, i.e. windup on the camshaft. The vibration with a large amplitude originated from the torsion of the camshaft by the effect of the torque, the one with a small amplitude takes place with the self-induced mechanism. In the follower motion cycle in the form of dwell-rise-return, cam delays with respect to input by twisting of the camshaft with resisting torque during the rise period and the energy is stored in the camshaft. During the return period, since the cam exerts torque on the shaft, it advances the input by twisting the shaft in the reverse direction. These vibratory motions produce undesirable effects especially on mechanisms with heavy loads and a relatively flexible shaft. The windup phenomenon is more severe on cams with a flat-faced follower. Also vibratory motion increases in the return period. Increasing the inertia ratio increases the

T o n i o ~ l ~ 'b~t iom of camaha~ 229

~p ('!

0.~

• -O.Z

{.)

I ,

(o)

0.2

-0.Z

A A

, (')

0.2

-0 .2

%v LoA

.hie ~A ~ ~ °If- -o.7.

Co)

nA,, ,A {4)

, (,,)

0.1~

0.2

- 0 . 2

{o)

, ( . )

0o4

0.2

+.

-0.;

It- 0.

t,

tz_

e.

F'i& 3. Dymunic reqponm of camshaft for tome cam curvu. (a) Hzuqm~ai¢,, (b) c,~Joidal; (c) 4-5-6-7 polynomial cam curve for roller follower (M - 5, N - 43, p -, 3, p ,, 90°); (d) harmonic; (e) cycloidal;

(f) 4-5-6-7 polynomial cam curve for flat-faced follower (M - 3, N ,, 45,p - 3, ~ ,, 0.2, p =, 90°).

amplitudes. With the increasing of the frequency ratio, as shown by the responses in Fig. ~a ) found with several values of this ratio for the harmonic cam curve, while amplitudes reduce, frequency of self-induced vibrations increases. Vibrations arc not very sensitive to the variation of energy ratio, as shown in Fig. 4(b). In the cams with a flat-faced follower, augmentation of size ratio causes large amplitudes, because this means that the coefficient of friction or cam size rises, thus magnitude of vibration increases.

230 Z~A ~ and YOr~ Yu.tt~z

. )

0.I,

0.2

-0.2

-O.t

N

3S

' " / ~ e5

t A

: ::^~.,, . . . . t

I t

~ t~ t , / , , ,

e.

, (,) I)-I

o, t "~*~ tt,-t

r~ / _ ' , ...... p.t0 t I I %..~-,~

-0.2

\ ,

\ ! - 0 . ~ I t

Fig. 4. Dynamic responses for some values of N and p for harmonic cam curve. (a) M - 5. f -, 3. p ,, 90 °, (b) M = 5. ,v = 4 5 . . e = 9o °.

EFFECTS OF CAMSHAFT VIBRATIONS ON THE F O L L O W E R ACCELERATION AND CONTACT FORCE

The fluctuations on cam rotation affect follower motion and considerable deviations occur in the follower acceleration. These deviations lead to facilitate jumping. Herein, without considering other factors, the effects of camshaft vibrations on the follower acceleration and vertical contact force component were investigated.

In case of angular velocity of the camshaft being constant, theoretical follower acceleration is

a t A ~ ( 0 2 5 - .

One term is added to the acceleration expression since 0 is not constant:

a = 0 2 s ' + ~ s ".

Let us get the deviation in acceleration by substracting the theoretical acceleration from the correct one and normalizing by k and (02,

& a = ( l d0'~2e, d2~'e. ~- o - 0o - (20)

Figure 5 shows deviations in acceleration for cam curves in Fig. 3. The m o u n t of deviation for the cams with a fiat-faced follower is approximately three times more than the others.

Toraio~d v ~ x ' a ~ of ~ 231

.2

-2

-4

, a a/ lO

vvvv vV Vvvvvu',, (a)

2

-2

-4

a a / l O

hA, M,..^,,,, AA _

(b)

e.

Aalto

"v I Vv"' V '-

(o)

All

o.s A ~ A _

!

O.S

-0.

-1

Ai

'vv~ 'V" ';,

(e)

=e.

t

0.$

4 . $

All

Fill. 5. Deviations in the foUower a~ekration for cams in Fig. 3.

L

Deviations in the follower acceleration cause fluctuations in the contact force. If ~ is substituted instead of 0 in vertical contact force component expression (7) and some arrangements are made, normalized f , force is obtained in nondimensional parameters M and p:

(2l)

In Fig. 6, variation of normalized force f , is shown for cam curves in Fig. 5. For no jumping to occur this force should be greater than zero. From the fisures, it can be seen that the risky area is on the middle of motion the cycle. The magnitude o f f , force d e c r e a ~ at the return period on cams with a flat-faced follower. The harmonic curve has more fluctuations in this force in comparison with the other curves.

CONCLUSIONS

In recent years, in the studies considering flexibility of the camshaft, authors often examined influences of this flexibility on the follower motion. In this work, differently, only the torsional

232 ZiYa . ? ~ and YOgs~. Yu.tu~

f

l o •

ee,

2(3

10

f Y

,~ 2e't,

2 0 .

10.

( , )

. . D

e 2s e,

(e)

(b)

I 2S e.

(d)

20

1 0 ¸

( e ) ( r )

Fill. 6. Variation of normalized f, force for cams in Fill. 3.

vibratory motion of the camshaft is treated and therefore a comparatively simple model of the cam mechanism is used. Since elasticity of the follower is more effective on the follower motion and its influence on the camshaft motion can be neglected, the follower is assumed risid. For the vibrations of camshafts of cam mechanisms with roller and a flat-faced follower, nonlinear differential equations are obtained, also four nondimensional parameters are defined. It is seen that these are parametric and self-induced torsional vibrations. Dynamic responses obtained numerically show that vibrations on the cams with a flat-faced follower are more severe than ones with a roller follower. Joining with other factors, fluctuations due to this vibrations increase jumping risk. The stability analysis of this parametric vibration and the variation of dynamic response depending on the nondimensional parameters can be investipted in the further works.

R E F E R E N C E S

I. F. Y. Chen and N. Polvanich, Trans. ASME Jl ~ hld. 97B, 769 (19"/5). 2. F. Y. Chat and N. Polvaaich, TrmJ. ASM£ J! F-nlr~r Ind. fiB, 777 (1975). 3. D. Ardayfio, ASME ~ 76-DET-63 (1977). 4. M. P. Kmter, Trans. ASM£ Jl Engnf Ind. 9"I, 595 (1975). 5. H. R. Kim and W. R. Newcombe, Mech. Mac& Theory 17, 57 (1982). 6. L. E. Szakallas and M. Savap, Trans. ASME Jl Mech. Des. 102, 287 0980). 7. A. Midha and D. A. Turcic, Trans. ASMF. JI Dytmmic Systems Measw. Control 102, 255 (1980). 8. R. I. Zadoks and A. Midlm, Trans. ASME Jl Mech. Tr~sm. Awomn. Des. 109, 210 (I~7). 9. R. I. Zadoks and A. Midha, Trans. ASME Jl Mech, Tronsm. Autumn. Des. IM, 216 (1987).

10. B. Diziolllu, C, etriebbhro.Oynamik, Band 3. F. Viewql und Sohn, Bruanschweill (1906). I1. A. gtwsZilk Ad0onced F.ng/neer/ng Mathematics. Wiley, New York (1985).

Torsional vibratiom of cuos l a~ 233

TORSIONSSCHWINGUNGEN DER WELLE VON KURVENGETRIEBE

~ B e / kraftschl0Jsen K u r v e n ~ b e n mit zeutrisch IM'Ulman St6mel wirkt die dymuniscbe Kraft aM" die Kurvtsmcheibeuwdle ein, die dmch die Bewesunl des EinlpiffqJiedes hetVOrlK'ml~ wird. Inl'oIlle dieser dynambchen Kraf~ treten TondemJchwinjunll~ aM'. Um dam Vefludteu auftmtendeu SchwinlPml~ you Kurvemcheibenwdle zu behsnddn, wird eh~ dym,mmchm Model aufwlxmt, yon dent die Di~. t /a lSJeichu8 der Stsx'Sunj berl~eitet wird. Bei tier L6mnj dieser D i ~ - , ~ l l l b i c h ~ wmkm dn/dimem/oelo~ Pmme~r ~ die ,Is Tr/sheits-, F m q ~ - ~ d E I ~ beaeh:hnet s/rid. Be/der Kurveugetriebe mit Tellerst6uei handelt es sich um e/he andere ~ , d.h. ~ t n / s .

Die an der Wc.~ der Kurvengetriebe entstehende Tors/onsschwingunllen s/nd die Schwinsunsm. Die efludtene nichtlinatre Differeutialgleichun 8 wird druch das numerische Method 8el6st. Die Tors/ouschwiniPmFn fl0uen auf die Kmf~.schlimskeit der Kurvengetr/ebe nesativ tin. Dieser Einfluss w/chst bei Kurvengetrieben mit Tellerstt3-.sei an.