2.7 Applications
45
2.7 Applications
Our analysis can be applied directly to find the response of SDOF systems to seismic
events via response spectra, find the response of simple systems to moving loads
which may cause resonance, and calculate the displacement of SDOF systems with
nonlinear damping. The following subsections illustrate these applications.
2.7.1 Response Spectra
Consider SDOF systems with parameters (ω, ζ ) which is subjected to a seismic
ground acceleration a(t), 0 ≤ t ≤ τ , so that its displacement x(t) is the solution
of Eq. 2.32, i.e., ¨
x(t) + 2 ζ ω ˙
x(t) + ω 2 x(t) = −a(t), 0 ≤ t ≤ τ , where τ is the
duration of the seismic event.
Denote the displacement, velocity, and acceleration functions of an oscillator
with parameters (ω, ζ ) by x(t; ω, ζ ), ˙
x(t; ω, ζ ), and ¨
x(t; ω, ζ ). The following
response maxima
S d (ω, ζ ) = max
0≤t≤τ
x(t; ω, ζ )
,
S v (ω, ζ ) = max
0≤t≤τ
˙
x(t; ω, ζ )
,
S a (ω, ζ ) = max
0≤t≤τ
¨
x(t; ω, ζ ) + a(t)
(2.78)
are called the displacement, velocity, and absolute acceleration spectra. In
Earthquake Engineering, the displacement spectrum S d (ω, ζ ) is calculated while
S v (ω, ζ ) and S a (ω, ζ ) are approximated by
S v (ω, ζ ) P S v (ω, ζ ) = ω S d (ω, ζ )
S a (ω, ζ ) P S a (ω, ζ ) = ω
2 S d (ω, ζ ) = ω P S v (ω, ζ ).
(2.79)
They are referred to as pseudo-velocity and pseudo-acceleration spectra. These
equations provide the following two expressions of P S v ,
log(P S v ) = − log(T ) + log(2 π) + log(S d )
log(P S v ) = log(T ) − log(2 π) + log(P S a ),
(2.80)
where T = 2 π/ω is the oscillator natural period. The two families of functions
in Eq. 2.80 are lines at ±45 o in the coordinates
log(T ), log(P S v )
which are
indexed by the damping ratio ζ . They are constructed for selected seismic ground
accelerations a(t) and are used to read the values of the displacement and pseudoacceleration spectra for given period T and damping ratio ζ .