Z E S Z Y T Y N A U K O W E P O L T I E C H N I K I ş Ó D Z K I E J

Z E S Z Y T Y N A U K O W E P O L T I E C H N I K I ş Ó D Z K I E J

SCIENTIFIC BULLETIN OF LODZ TECHNICAL UNIVERSITY
Nr 891, Textiles 60, 2001
ZESZYTY NAUKOWE POLITECHNIKI ŁÓDZKIEJ Nr 891,
WŁÓKIENNICTWO z. 60, 2001
Pages: 63-69
JERZY ZAJACZKOWSKI
Lodz Technical University
Lodz, Zwirki 36, Lodz, Poland
DEPENDENCE OF THE CAM FOLLOWER LOADING
ON THE SYSTEM ELASTICITY
Reviewer: Janusz Lipinski, PhD, DSc
The paper is concerned with the problem of dependence of the
loading of the cam mechanism on the elasticity of its elements. The
mechanisms with rigid elements, with spring mass system, with an
elastic cam shaft and with the elastic rocker shaft are studied. The
motion of the mechanism is described by nonlinear differential
equations. The moment acting on the rocker is calculated and plotted.
1. INTRODUCTION
The cam mechanisms are widely used in the textile machinery.
Vibrations of shafts coupled by mechanisms have been studied in work
[1]. The phenomenon of torsional buckling of shafts has been described
in work [2]. In paper [3] it has been shown that the vibrations are mainly
due the collision of the rotary and oscillatory motion of elements of the
mechanism. It was pointed out that the elasticity of the driving belt
improves the system dynamics. The purpose of this paper is to study the
effect of the cam mechanism elasticity on the rocker loading.
2. EQUATIONS OF MOTION
6
J. Zajaczkowski
The cam mechanisms considered are shown in Figure 1. They are
composed of the rotating grooved cams and the oscillating followers. The
mechanisms with rigid elements (1a), with a spring and a mass (1b), with
an elastic cam shaft (1c) and an elastic follower shaft (1d) are considered.
Figure 1. The cam mechanisms with:(a) rigid elements, (b) spring and
mass, ( c ) elastic cam shaft, (d ) elastic follower shaft.
The equation of the motion of the cam mechanism with rigid
elements, shown in Figure 1a, has the form

 A  B d
 d






2
2
 d 2
d 2 d  d 
d

B
 M m.

  Mr
2
2
 dt
d d  dt 
d

(1a)
The moment of the forces acting on the follower is found to be
Cam follower loading
7
 d 2 d   d   2 d 2  
d2

MB  B
 M r   B


2
2   Mr .
dt 2
 dt d   dt  d  
(2a)
The motion of the cam mechanism with spring and oscillating mass,
shown in Figure 1b, is governed be the equations
d 2x
dx
 d 
m 2  c  kx  m
 e  x   0,
dt
dt
 dt 
2
2
2
2

dx  d  d
 d   d 
 A  me  x 2  B 

  2  2me  x  

 d   dt
dt  d  dt





2
 d  d  d
 me  x   B 
 M m  0.

2
 dt  d d
2
2
(1b)
The moment of the forces acting on the follower is
 d 2
d dx 
d 2


M B   2 e  x   2
me  x   B 2 .
dt dt 
dt
 dt
(2b)
The equations of the motion of the cam mechanism with an elastic
cam shaft, shown in Figure 1c, can be written in the following form
A0
d 2 0
 d 0 d 1 

  s0  1   M m  0 ,
2  D
 dt
dt
dt 
2
2


  2 1
d 1 d 2 1  d1 
 A1  B1  d 1   d 
 B1

 
2

d 1 d 12  dt 
 d 1  

 dt
8
J. Zajaczkowski
d 0 
 d1
 D

  s 1   0   0.
dt 
 dt
(1c)
The moment of the forces acting on the follower is given by
M B1   B1
d 21
.
dt 2
(2c)
The approximate natural frequency is =(s/A0+s/A1) .
The equations of the motion of the cam mechanism with the elastic
follower shaft, shown in Figure 1d, have the form

2
2
2

 d 1   d 2 1
d

d

d



1
1
1
 A1  B1 
 
 B1

 

d 1 d 12  dt 
 d 1   dt 2

 d 1 d 1 d 2  d 1
d 1

 H

 k 1  2 
 Mm  0 ,
dt  d 1
d 1
 dt d 1
B2
 d 2 d1 d1 
d 2 2
  k  2 1   0.
 H 

2
dt
dt d1 
 dt
(1d)
The moment of the forces acting on the follower is
M B1   B1
d 21
 d1 d 2 
 H

  k 1  2 .
2
dt
dt 
 dt
The approximate value of the natural frequency is =(k/B2)
The motor torque is given be the following equation
(2d)

Cam follower loading
T
9
dM m
d 

 C  
  M m.
dt
dt 

(3)
The differential equations (1) describing the motion of the cam
mechanisms can be integrated numerically together with the equation of
the motor torque (3) and the moment (2) of the forces acting on the
follower can be calculated.
2. RESULTS AND DISCUSSION
The function relating the motion of the follower to the motion of the
cam was taken in the form
 
1  cos
 0    2
2 
 
 0
2     2 .
 
S 
The calculations were carried out for =60/180, S=30/180,
=1000/30, T=1/, C=0.1. It was assumed that at the initial moment
the angular speed of the motor was equal to , the motor torque
Mm(0)=0, the deformations of the elastic elements were assumed to be
equal to zero. The total inertia of the rocker was kept unchanged for all
mechanisms. The torques for mechanisms (Figure 1a,b,c,d) are shown in
Figures 2a,b,c,d for (a) A=0.02, B=0.02, (b) A=0.02, B=0.01, e=0.1, m=1,
k= m, c=0.1(mk), A0=0.01, A1=0.01, B1=0.02, s=(5 /(1/A0 +1/A1),
D=0.1(s/(1/A0 +1/A1)), (d) A1=0.02, B1=B2=0.01, k=(15 B2,
H=0.1(B2k), respectively.
Comparing the results shown in Figure 2a,b,c and d, one may see that
the elasticity of the follower shaft has significant influence on the system
behaviour.
10
J. Zajaczkowski
Figure 2. The moment of the forces acting on the cam follower for cam
mechanisms with: (a) rigid elements, (b) spring and mass, ( c ) elastic
cam shaft, (d ) elastic follower shaft.
REFERENCES
1. Zajaczkowski J.: Torsional buckling of shafts coupled by mechanisms.
Journal of Sound and Vibration. 173(4), 449-455 (1994).
2. Zajaczkowski J.: Vibrations of shafts coupled by mechanisms.
Journal of Sound and Vibration. 177(5), 709-713 (1994).
3. Zajaczkowski J.: Torsional vibration of a camshaft. Zeszyty Naukowe
Politechniki ódzkiej Nr 736, Wókiennictwo, z. 53, 99-103, (1996).
Cam follower loading
11
ZALEŻNOŚĆ OBCIĄŻENIA POPYCHACZA
MECHANIZMU KRZYWKOWEGO OD
SPRĘŻYSTOŚCI UKŁADU
Streszczenie
Praca dotyczy zagadnienia zależności obciążenia mechanizmu
krzywkowego od sprężystości jego elementów. Badane smechanizmy
z elementami sztywnymi, ze sprężystym wałem krzywkowym, ze
sprężystym wałem popychacza oraz z układem drgającym złożonym
z masy i sprężyny. Ruch mechanizmu opisany jest za pomoc
nieliniowych równa różniczkowych. Moment działający na
popychacz, wyznaczony po rozwiązaniu równa różniczkowych, jest
przedstawiony wykreślnie.