28
2 Single Degree of Freedom (SDOF) Systems
0
2
4
6
8
10
−20
−15
−10
−5
0
5
10
15
20
t
x(t)
0
10
20
30
40
50
60
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80
−40
−30
−20
−10
0
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t
x(t)
Fig. 2.9 Displacement function x(t) for an undamped SDOF system with natural frequency ω = 6
subjected to a harmonic force with frequency ν = ω = 6 (left panel) and ν = 5.9 (right panel)
2.4.6 Harmonic Force, Damped System
We have seen that the particular solution for undamped oscillators subjected to
harmonic forcing functions can be obtained simply. This section constructs the
particular solution x p (t) for damped SDOF systems subjected to harmonic forces.
The Duhamel integral can be used to find x p (t). The reader is encouraged to
implement this approach.
A direct approach is presented here, i.e., we postulate a functional form for x p (t)
and require that it satisfies the differential equation ¨
x p (t)+2 ζ ω ˙
x p (t)+ω 2 x p (t) =
q/m
sin(ν t). It is shown that the particular solution is
x p (t) = x st r d (ν)
2
−
2 ζ ν/ω
cos(ν t) +
1 − (ν/ω)
2
sin(ν t)
, where
r d (ν) =
1
1 − (ν/ω) 2
2 +
2 ζ ν/ω
2
and x st = q/k
(2.44)
or
x p (t) = x st r d (ν) sin
ν t − ϕ
, where tan(ϕ) =
2 ζ ν/ω
1 − (ν/ω) 2 ,
(2.45)
where r d (ν) is called dynamic amplification factor (DAF).
The particular solution of the previous example does not satisfy the equation of
motion since 2 ζ ω ˙
x p (t) = 2 ζ ω α ν cos
ν t
is the only term which is proportional
to cos
ν t
so that the equation of motion is satisfied if and only if 2 ζ ω α ν = 0
which implies α = 0, i.e., x p (t) = 0. Accordingly, the equation of motion is not
satisfied since its left and right sides are zero and
q/m
sin(ν t).
This observation suggests to augment the trial solution of the previous example
to