Page 388 - Cam Design Handbook

THB12 9/19/03 7:34 PM Page 376

376 CAM DESIGN HANDBOOK

F sh = torque on shaft, lb/in (Nm)
F y = compressive contact force in the radial direction between the cam surface and the
follower, lb (N)
2
I c = mass moment of inertia of the cam, lb/in /RAD (N/m/RAD)
k r = follower return spring rate, lb/in/RAD (N/m)
k s = drive system flexibility lb/in/RAD (N/m/RAD)
m = follower mass, lb (kg)
Q = variable reflected follower inertia ratio
Q m = maximum reflected inertia ratio
R = radius from cam center to follower roller center, in (m)
T = drive system torque at cam speed, lb/in (N-m)
t = time, sec
x = follower displacement, in (m)
.
x = follower velocity (dx/dt) in/sec (m/s)
2
2
2
2
x ¨ = follower acceleration (d x/dt ) in/sec (m/s )
x¢= cam slope (dx/dq c), in (m)
2
2
x≤= rate of change of cam slope (d x/dq c ), in (m)
x f = final or total displacement of rise, in (m)
y = normalized cam position, in (m)
b = variable cam drive system windup ratio
b m = maximum cam drive system windup ratio
g = variable follower radial force ratio
g m = maximum follower radial force ratio
h = limiting linear frequency ratio
q c = cam angular position, rad
.
q c = cam angular velocity, rad/sec
¨
q c = cam angular acceleration, rad/sec 2
q f = final or total duration of rise, rad
q i = input power source position, rad
.
q i = input power source speed, rad/sec
f = cam surface pressure angle, rad
w b = base natural frequency rad/sec
w l = limiting linear natural frequency, rad/sec
w n = variable natural frequency, rad/sec
The system takes account of camshaft flexibility and leads to a set of nonlinear
equations owing to the fact that the driving-cam function is no longer a known function
of time.
In Fig. 12.10, where q i(t), the input shaft rotating at a known angular rate, is not equal
to the cam angle, q c , or q i π q c . This is due to camshaft flexibility; k s is the camshaft spring
rate and the cam-follower rise is x. A convenient method of formulation is derivable from
Lagrange’s equations as follows:
2 ˙
Iq
T = 1 Mx + 1 ˙ 2
2 2 c
2
V = 1 2 k (q c - ) + 1 2 k x 2
q
s
i
s
-
L = T V.
The follower is constrained by the cam function
q
x = x()
c
where q i = q i(t), by the Lagrange multiplier method