2.4 Time Domain Analysis
17
Fig. 2.4 Solutions x h (t),
˙
x h (t) and ¨
x h (t) for ω = π
and ϕ = π/4 (solid, dashed,
and dotted lines)
0
1
2
3
4
5
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
t
x
h (t), ˙
x
h (t)/ω and ¨
x
h (t)/ω
2
2.4.2.3 Damping Estimation
The free vibration displacement of a damped SDOF system is the product of a
periodic function with period T d = 2 π/ω d , the square bracket in Eq. 2.21, and
the exponential function exp(−ζ ω t). Since the square bracket has the same values
at the times t and t + k T d , where k is an integer, we have
x(t)
x(t + k T d )
=
e −ζ ω t
e −ζ ω (t+k T d ) = e
ζ ω (k T d )
so that
δ = ln
x(t)
x(t + k T d )
= ζ ω (k T d ) =
2 π k ζ
1 − ζ 2
2 π k ζ,
(2.25)
where the latter approximations hold for ζ 1. These relationships can be used
to estimate the damping ratio ζ from measurements of the displacement at time
intervals equal to T d or multiple of it. The measurements are used to estimate δ. The
corresponding value of ζ results from Eq. 2.25.
2.4.3 Particular Inhomogeneous Solution, Simple Forcing
Functions
We first construct particular solutions for several simple forcing functions and then
use the resulting solutions to develop a method for finding particular solution for