Page 59 - Dynamic Loading and Design of Structures
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Figure 2.10 Free vibration with damping due to an initial displacement.
(2.36)
For instance, when per cent, d is equal to 0.2πand the ratio of two consecutive
)
peaks is exp(0.2π=1.87. Thus, a damping ratio of 10 per cent reduces the dynamic
displacements by a factor of 0.534 during each vibration cycle.
2.2.2.2 Forced vibrations
By analogy to the case of forced vibrations in the absence of damping, we now have that a
damped impulse element is
(2.37)
The complete expression in terms of a Duhamel integral can be found through time
integration of the above impulse, to which the effect of initial conditions is subsequently
superimposed. Thus,
(2.38)
As a special case consider F(t)=F1; substitution of the time function f(t)=1 in eqn (2.38) gives
the solution for a suddenly applied and maintained load as
(2.39)
