Transcript Strategic Plan NSL

Earthquake Engineering
GE / CE - 479/679
Topic 7. Response and Fourier Spectra
John G. Anderson
Professor of Geophysics
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F
x = y-y0
y
m
y0
(x is negative here)
Hooke’s Law
F  kx
k
c
Friction Law
F  cx
z(t)
Earth
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In this case, the force acting on the mass due
to the spring and the dashpot is the same:
F  kx  cx
However, now the acceleration must be
measured in an inertial reference frame,
where the motion of the mass is (x(t)+z(t)).
In Newton’s Second Law, this gives:
mx(t )  zt   kx  cx
or:
mx(t )  cx  kx  mzt 
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So, the differential equation for the forced
oscillator is:
mx(t )  cx  kx  mzt 
After dividing by m, as previously, this equation becomes:
x(t )  2h0 x  02 x  zt 
This is the differential equation that we use to characterize
both seismic instruments and as a simple approximation for
some structures, leading to the response spectrum.
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DuHammel’s Integral
This integral gives a general solution for the response of
the SDF oscillator. Let:
at   zt 
The response of the oscillator to a(t) is:

H t   
1


2 2
x(t)   a 
 0 1 h  t   exp 0 h t   d
1 sin

0
 0 1 h 2  2 
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Let’s take the DuHammel’s integral apart to
understand it. First, consider the response of the
oscillator to a(t) when a(t) is an impulse at time t=0.
Model this by:
at    t 
The result is:
H t 
1


2 2
x(t) 
sin

1
h
t
exp 0 ht 



0
1



 0 1 h 2  2 

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H(t) is the Heaviside step function. It is defined as:
H(t)=0, t<0
H(t)=1, t>=0
This removes any acausal part of the solution - the
oscillator starts only when the input arrives.
1
0
t=0
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This is the result for an oscillator with f0= 1.0 Hz and h=0.05.
It is the same as the result for the free, damped oscillator with
initial conditions of zero displacement but positive velocity.
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The complete integral can be regarded as the result of summing
the contributions from many impulses.

x(t )   a 
0
H t   
0 1  h
2

1

2
sin 0 1  h 2

 t    exp  ht   d
1
2
0
The ground motion a(τ) can be regarded as
an envelope of numerous impulses, each
with its own time delay and amplitude.
The delay of each impulse is τ. The
argument (t- τ) in the response gives
response to the impulse delayed to the
proper start time. The integral sums up all
the contributions.
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Convolutions.
• In general, an integral of the form

x(t )   a bt   d
0
is known as a convolution. The properties of convolutions
have been studied extensively by mathematicians.
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Examples
• How do oscillators with different damping
respond to the same record?
• Seismologists prefer high damping, i.e.
h~0.8-1.0.
• Structures generally have low damping, i.e.
h~0.01-0.2.
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Response Spectra
• The response of an oscillator to an input
accelerogram can be considered a simple
example of the response of a structure. It is
useful to be able to characterize an
accelerogram by the response of many
different structures with different natural
frequencies. That is the purpose of the
response spectra.
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What is a Spectrum?
• A spectrum is, first of all, a function of
frequency.
• Second, for our purposes, it is determined
from a single time series, such as a record
of the ground motion.
• The spectrum in general shows some
frequency-dependent characteristic of the
ground motion.
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Displacement Response Spectrum
•
•
•
•
Consider a suite of several SDF oscillators.
They all have the same damping h (e.g. h=0.05)
They each have a different natural frequency fn.
They each respond somewhat differently to the
same earthquake record.
• Generate the displacement response, x(t) for each.
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Use these calculations to form the
displacement response spectrum.
• Measure the maximum excursion of each
oscillator from zero.
• Plot that maximum excursion as a function
of the natural frequency of the oscillator, fn.
• One may also plot that maximum excursion
as a function of the natural period of the
oscillator, T0=1/f0.
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Definition
• Displacement Response Spectrum.
• Designate by SD.
• SD can be a function of either frequency or
period.
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Assymptotic properties
• Follow from the equation of motion
x(t )  2h0 x  02 x  zt 
• Suppose ωn is very small --> 0. Then
approximately,
x(t )  zt 
• So at low frequencies, x(t)=z(t), so SD is
asymptotic to the peak displacement of the
ground.
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Assymptotic properties
• Follow from the equation of motion
x(t )  2h0 x  02 x  zt 
• Suppose ωn is very large. Then approximately,
 x  zt 
2
0
• So at high frequencies, SD is asymptotic to the
peak acceleration of the ground divided by the
angular frequency.
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Velocity Response Spectrum
• Consider a suite of several SDF oscillators.
• They all have the same damping h (e.g.
h=0.05)
• They each have a different natural
frequency f0.
• They each respond somewhat differently to
the same earthquake record.
• Generate the velocity response, xt  for each.
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Use these calculations to form the
velocity response spectrum.
• Measure the maximum velocity of each
oscillator.
• Plot that maximum velocity as a function of
the natural frequency of the oscillator, f0.
• One may also plot that maximum velocity
as a function of the natural period of the
oscillator, T0=1/f0.
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Definition
• Velocity Response Spectrum.
• Designate by SV.
• SV can be a function of either frequency or
period.
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How is SD related to SV?
• Consider first a sinusoidal function: xt   sin0t 
• The velocity will be: xt   0 cos0t 
• Seismograms and the response of structures
are not perfectly sinusiodal. Nevertheless, this
is a useful approximation.
• We define: PSV  0 SD
• And we recognize that: PSV  SV
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Definition
• PSV is the Pseudo-relative velocity
spectrum
• The definition is: PSV  0 SD
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PSV plot discussion
• This PSV spectrum is plotted on tripartite
axes.
• The axes that slope down to the right can be
used to read SD directly.
• The axes that slope up to the right can be
used to read PSA directly.
2
PSA


• The definition of PSA is
0 SD
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PSV plot discussion
• This PSV spectrum shows results for
several different dampings all at once.
• In general, for a higher damping, the
spectral values decrease.
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PSV plot discussion
• Considering the asymptotic properties of
SD, you can read the peak displacement and
the peak acceleration of the record directly
from this plot.
• Peak acceleration ~ 0.1g
• Peak displacement ~ 0.03 cm
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Absolute Acceleration Response, SA
• One more kind of response spectrum.
• This one is derived from the equations of motion:
x(t )  2h0 x  02 x  zt 
• This can be rearranged as follows:
x(t )  zt   2h0 x  02 x
• SA is the maximum acceleration of the mass in an
inertial frame of reference:
SA  maxx(t )  zt 
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Summary: 5 types of response spectra
• SD = Maximum relative displacement
response.
• SV = Maximum relative velocity response.
• PSV  0 SD
2
• PSA 0 SD
• SA = Maximum absolute acceleration
response
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Here are some more examples of
response spectra
• Magnitude dependence at fixed distance
from a ground motion prediction model, aka
“regression”.
• Distance dependence at fixed magnitude
from a ground motion prediction model, aka
“regression”.
• Data from Guerrero, Mexico.
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Data from Guerrero, Mexico, Anderson and Quaas (1988)
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Main Point from these spectra
• Magnitude dependence.
– High frequencies increase slowly with
magnitude.
– Low frequencies increase much faster with
magnitude.
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Note about ground motion
prediction equations
• AKA “regressions
• Smoother than any individual data.
• Magnitude dependence may be
underestimated.
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Note about ground motion
prediction equations
• Spectral amplitudes decrease with distance.
• High frequencies decrease more rapidly
with distance.
• Low frequencies decrease less rapidly.
• This feature of the distance dependence
makes good physical sense.
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