GG 454
February 20, 2002
1
Stephen Martel
1 6  1
University of Hawaii
RESPONSE OF STRUCTURES (16)
I
Main Topics
A
Acceleration, velocity and displacement spectra
B
Resonance and natural frequencies
C Response of structures
I I Displacement, velocity, and acceleration spectra
A
Spectra represent parameters as a function of wave frequency (or
period), not time, and reveal the most energetic/forceful waves
B
Examples of shaking vs. time and shaking vs. frequency plots
C Displacement, velocity, and acceleration spectra
y
x
A
λ
A = amplitude
Displacement as a function of position along a wave
(1)
y(x,t) = A sin {(2
π
/
λ
)(x + vt)}where v =
λ
/T = f
λ
, so
λ
= v/f
(2)
y(x,t) = A sin {(2
π
f/v)(x + vt)}
Consider the displacement at a fixed point on the ground (e.g. x = 0)
(3)
y(t) = A sin {(2
π
f/v)(vt)}
or
y = A sin (2
π
f t )
Displacement given as a function of the frequency and wave amplitude
Let
ω
= 2
π
f = angular frequency
(4)
y = [A] sin (
ω
t )
The term in square brackets gives y
max
The shaking velocity of the ground (not
the velocity of the wave) = dy/dt
(5)
y'
= d(A sin (
ω
t))/dt =
[
ω
A] cos (
ω
t)
The acceleration of the ground = dy'/dt
(6)
y''
= d(
ω
A cos (
ω
t))/dt
= [
ω
2
A] sin (
ω
t) = 
ω
2
y
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 Spring '11
 various
 Frequency, Velocity, Wavelength, Stephen Martel

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