2.4 Time Domain Analysis
15
0
1
2
3
4
5
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
t
x
h (t)
ζ = 0.0
ζ = 0.05
ζ = 0.1
Fig. 2.3 Free vibration solution for ζ = 0.0, 0.05, and 0.1
that of Eq. 2.21 is not since the motion amplitude decreases steadily in time. The
parameters T and ω are referred to as the natural period and the natural frequency
of the oscillator.
Example 2.3 The solid, dotted, and dashed lines of Fig. 2.3 show the free vibration
solutions of SDOF systems with natural frequencies ω = 2 π and damping ratios
ζ = 0.0, 0.05, and 0.1 for the initial conditions x 0 = 2 and ˙
x 0 = 1. All solutions
start at x 0 = 2 and initially have positive slope although it is not highly visible at
the figure scale. The amplitudes decrease at rates which increase with the damping
ratio. The motion is periodic with period T = 1 for ζ = 0 but not for ζ > 0.
2.4.2.2 Amplitude-Phase Representation
An alternative form of x h (t) for ζ < 1 is
x h (t) = a e
−ζ ω t sin
ω d t + ϕ
,
(2.23)
where the amplitude a and the phase ϕ have the expressions
a
2
= x
2
0 +
˙
x 0 + ζ ω x 0
ω d
2
and tan(ϕ) =
˙
x 0 + ζ ω d x 0
ω x 0
.
(2.24)
This follows from the observation that x h (t) of Eq. 2.21 can be written as