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Structural Vibration and Dynamics 263

2
−ω .m sin ω+ Usin ω= 0
U
t
k
t

that is
2
[−ω m + ] k U = 0

For nontrivial solutions for U, we must have:

2
[−ω m + k ] = 0

which yields

k
ω= (8.3)
m

This is the circular natural frequency of the single DOF system (rad/s). The cyclic fre-
quency (1/s = Hz) is ω/2π.
Equation 8.3 is a very important result in free vibration analysis, which says that
the natural frequency of a structure is proportional to the square-root of the stiffness
of the structure and inversely proportional to the square-root of the total mass of the
structure.
The typical response of the system in undamped free vibration is sketched in Figure 8.3.
For nonzero damping c, where

m
0 < c = 2 ω = 2 k m ( = critical damping ) (8.4)
< c c c c
we have the damped natural frequency:

ω= ω 1 − ξ 2 (8.5)
d

where

/ (8.6)
ξ= cc c

is called the damping ratio.

u
u = U sin t
U
U t
T = 1/f

FIGURE 8.3
Typical response in an undamped free vibration.