Page 420 - Cam Design Handbook

THB13 9/19/03 7:56 PM Page 408

408 CAM DESIGN HANDBOOK

The methods presented do not rely on rules of thumb. In fact, some solutions provide
motions with low vibration levels even though they violate the commonly accepted design
criterion that jerk must be minimized.

13.4.2 Symbols
a i , b i = Cosine and sine term coefficients, respectively, to be determined in the cam profile
function y, Eq. (13.26)
¯
A 1 , A 2 , A 3 = User-defined coefficients in Eq. (13.22) for performance index P ¯
B i = Coefficients of the sine series terms in the desired follower lift function X, Eq. (13.11)
C i, D i = Cosine and sine term coefficients, respectively, to be determined in the cam profile
¯
function y, Eq. (13.16)
F 2 = w 2 q 0 /(pw) normalized inverse operating speed
F 2h, F 2l = The high and low speed operating limits, respectively, of F 2 for the system being
analyzed
H = Required follower lift at the end of the cam segment
P = Performance index to be minimized to find the optimum cam lift function y over a
¯
range of frequencies, Eq. (13.25)
¯
P = Performance index which when minimized indicates an optimum cam lift function y ¯
at one frequency, Eq. (13.22)
r = The follower displacement relative to its static equilibrium position
R = The residual vibration—the magnitude of r induced by one cycle of the cam profile
as defined by Eq. (13.5)
R s = The steady state follower vibration induced by the cam profile
t = Time
T = pwt/q 0 , normalized time
T i = Normalized time corresponding to X(T i ) and X i
V 1 = Required follower velocity at start of cam segment
V 2 = Required follower velocity at end of cam segment
X(T i ) = Follower displacement at time T i
X i = Desired follower displacement X at time T i
2
2
¯ y = (w 1/w 2)y, normalized cam displacement
¯ y k = Desired normalized cam displacement y¯ at time T k , Eq. (13.23)
()¢= d()/dT
x 1 = Damping ratio for the cam input
x 2 = Damping ratio for the follower motion
q 0 = Angle of cam rotation over which the cam profile segment being analyzed occurs
w = Angular rate of cam rotation
w 1 = Square root of the coefficient of y¯ determining the follower input, Eq. (13.2)
w 2 = No-damping follower natural frequency

13.4.3 Tuned Cam Design
In many applications a cam operates at a constant or nearly constant speed. It is then
theoretically possible to completely control the follower motion by using dynamically
compensated (i.e., tuned) cam designs. To do this, the desired motion is specified and the
required cam profile determined through the equations of motion.
We here consider the problem of designing a cam profile segment to move the follower
exactly from one state of position and velocity to another. To begin, we start with the usual
single-degree-of-freedom linear system of the form