Page 120 - Cam Design Handbook

THB5 8/15/03 1:52 PM Page 108

108 CAM DESIGN HANDBOOK

M = Mass of follower (used in addressing nonrigid follower)
M i,k2(s 2) = B-spline basis functions of order k 2 in the parametric directions, s 2
n = Number of kinematic constraints
N i,k1(f 2) = B-spline basis functions of order k 1 in the parametric directions, f 2
(m)
th
N j,k (x) = m derivative of B-splines of order k
N j,k(x) = B-splines ( j = 1,..., n) of order k
P i,j = One of (n ¥ m) coefficients
r = Follower radius
R j = Rational B-spline of order k
(m)
th
R j,k = m derivative of rational B-spline of order k
s 2 = Axial input displacement of three-dimensional cam
S = Displacement of the follower
S 1 = Follower displacement of three-dimensional cam
th
S (m) = m derivative of the displacement of the follower
t = Time
T = Knot sequence T 1,... , T k+1
U = Approximate solution of differential equation
W j = Weight sequence with positive values
x = Cam rotational angle; x min £ x £ x man.
Y = Displacement of follower (used in addressing nonrigid follower)
Y (1) = Velocity of follower (used in addressing nonrigid follower)
Y (2) = Acceleration of follower (used in addressing nonrigid follower)
Y c = Displacement of cam (used in addressing nonrigid follower)
(1)
Y c = Velocity of cam (used in addressing nonrigid follower)
b = Total range of cam rotation
f 2 = Angular position of three-dimensional cam
t = Normalized time, t = t/T h, where T h is the total time for the rise of h
w = Angular velocity of cam
w n = Natural frequency of the cam-follower system
x = Damping ratio

5.1 INTRODUCTION

In synthesizing motion programs for cams the designer is faced with a variety of prob-
lems. Specific displacement, velocity, and acceleration constraints must be satisfied and
continuity of the derivatives of displacement, at least through the second derivative, should
be preserved. Expressions for displacement frequently must be refined to reduce acceler-
ation peaks or to shift peak values of a particular kinematic parameter away from critical
regions. Sometimes local adjustments must be made to reduce the pressure angle at a
troublesome spot. On occasion, it is desirable to account for nonrigid behavior of the
follower system when synthesizing the cam profile.
In the absence of especially demanding requirements, standard conventional motions
defined by harmonic, modified harmonic, trapezoidal, cycloidal, modified trapezoidal, and
the polynomial functions are applied directly or combined into piecewise functions. Some-
times polynomial functions that satisfy specified boundary conditions can, through manip-
ulation of exponent values, be modified to improve motion characteristics by shifting peak
acceleration values (Rothbart, 1956; Chen, 1982). Although the innovative combination
and modification of standard analytical functions into effective motion programs is often
a possibility, it is not sufficiently general nor is it a practical approach in the face of numer-
ous constraints.