Page 389 - Cam Design Handbook

THB12 9/19/03 7:34 PM Page 377

CAM SYSTEM DYNAMICS—ANALYSIS 377

-
Fdx F x d ¢ q c = 0 x¢ = dx() dq c
q
c
n
n
Fdq - F ¢ q dt = 0 q ¢ = dq dt
sh 1 sh 1 i
d
Ê d L - d L ˆ dx F dx = Æ mx kx F = 0 (12.12)
˙˙
0
+
+
+
Ë dt x x d˙ ¯ n n
d
Ê d d L d L ˆ ˙˙
k q - ) -
¢
Á - ˜ dq - F x dq =Æ Iq + ( q F x¢ = 0 (12.13)
0
Ë dt dq ˙ dq ¯ c n c c s c i n
c c
Ê d d L d L ˆ
k q - )+
Á - ˜ dq + F = Æ ( q F = 0 .
0
Ë dt dq ˙ dq ¯ i sh s i c sh
i i
where F n = contact force on cam
F nx¢= torque on cam due to contact force
F sh = torque on shaft
Eliminating F n between Eqs. (12.12) and (12.13) and taking two time derivatives of
x(q c) yields
q
mxx ˙˙ ¢ + k xx¢ + q ˙˙ + k (q - ) = 0
I
f c c s c i
x ˙˙ = x¢¢q ˙ 2 +q ˙˙ x¢
c c
whence
2 ˙˙
( I + mx¢ )q + mx x ¢¢¢q 2 ˙ + k xx¢ + q = k q t () (12.14)
k
c c c f s c s i
yields the basic second degree vibration equation characterized as the spring-windup equation.
The equation is consistent with those reported in cam windup studies by Bloom and
Radcliffe (1964) and Koster (1975a and b).
The problem being studied is one of self-induced torsional vibration in the cam system.
The transient response is of interest with the angular velocity of the power source being
a constant. Also the torque response for the open-track system fluctuates about the same
rotational speed as the input system.
Four dimensionless parameters are used to characterize the dynamic response of the
system. These parameters, according to Szakallas and Savage (1980), are the frequency
ratio h, the reflected inertia ratio Q m (measure of nonlinearity of the system), the maximum
drive windup ratio b m , and the maximum radial force ratio g m .
w q f k 2 p
h = l s where w = q ˙ i
b
w 2 pq ˙ i I c q f
f
(
mx¢ ) 2
Q = max
m
I c
Ê q - q ˆ
f
i
b = Á ˜
m
Ë q ¯
f m
F kx mx ˙˙
+
g = y = r .
m
˙
F n Ê q ˆ 2
mx f Á i ˜ (12.15)
Ë q ¯
f