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0593_C13_fm Page 445 Monday, May 6, 2002 3:21 PM

Introduction to Vibrations 445

θ

FIGURE 13.3.2
The simple pendulum.

If B is at rest when t = 0, (that is, ˙ x = 0), then the displacement x becomes:
0
x = x cosω t (13.3.8)
0

In this context, the initial displacement x is the amplitude and ω is the circular frequency.
0
That is,

ω = 2 π = k m (13.3.9)
f

or

f = (12π ) k m and T = 2π m k (13.3.10)

where f is the frequency (or natural frequency) and T is the period.
Consider next the simple pendulum as in Figure 13.3.2. From Eq. (12.2.9), we see that
the equation of motion is:

˙˙
θ +( ) l sinθ (13.3.11)
g
where as before, θ measures the displacement away from equilibrium, g is the gravity
constant, and is the pendulum length. Observe that if θ is small, as is the case with most
pendulums, we can approximate sinθ by the first few terms of a Taylor series expansion
of sinθ about the equilibrium position θ = 0. That is,

+
=−
5
3
sinθθ θ 3 ! θ 5 !−… (13.3.12)
Hence, for small θ, we have:
≈
sinθθ (13.3.13)
The governing equation. Eq. (13.3.11), then becomes:

˙˙ g l θ (13.3.14)
θ +( ) = 0