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THB11 9/19/03 7:33 PM Page 318

318 CAM DESIGN HANDBOOK

m
x

k

FIGURE 11.1. A simple mass-spring system.

as the equation of motion. If moved from its equilibrium position by a distance x 0 and
released from rest, the system illustrated in Fig. 11.1 will execute simple harmonic motion
and move in a sinusoidal trajectory given by
x = Acos (w n t + ) f (11.4)
for some amplitude A, frequency w n, and phase shift f. Using this expression in the equa-
tion of motion gives
mx ˙˙ =- mAw n 2 cos(w n t + ) =- kAcos(w n t + ) f . (11.5)
f
The frequency of oscillation, or natural frequency, is thus given by
k
w = (11.6)
n
m
in units of radians/second. Note that this formula works directly when the stiffness and
mass are in SI units (N·m and kg, respectively), but when English units of lbf/in and lbm
are used a conversion is necessary

k
¥
w = 32 2 12 ¥ . (11.7)
.
n
m
The amplitude and phase can be determined from the initial conditions. Since the initial
conditions assumed here are
f
˙
f
x 0 () = Acos ( 0 + ) = x x 0 () =- Asin ( 0 + ) = 0
0
the amplitude and phase are given by
A = x 0 f = 0.
Unlike the amplitude and phase, the frequency of vibration is a property of the system and
depends only upon the mass and stiffness, not the initial conditions. For this reason, it is called