Transcript 06-SDF-Oscillator_2008

Earthquake Engineering
GE / CEE - 479/679
Topic 6. Single Degree of Freedom Oscillator
Feb 7, 2008
John G. Anderson
Professor of Geophysics
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Note to the students
• This lecture may be presented without use
of Powerpoint. The following slides are a
partial presentation of the material.
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SDF Oscillator
• Motivations for studying SDF oscillator
• Derivation of equations of motion
• Write down solution for cases:
–
–
–
–
Free undamped (define frequency, period)
Free damped
Sinusoidal forcing, damped
General forcing, damped
• Discuss character of results
• Use of MATLAB
• MATLAB hw: find sdf response and plot results
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Motivations for studying SDF
systems
• Seismic Instrumentation
– Physical principles
– Main tool for understanding almost everything
we know about earthquakes and their ground
motions:
• Magnitudes
• Earthquake statistics
• Locations
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Motivations for studying SDF
systems
• Structures
– First approximation for the response of a
structure to an earthquake.
– Basis for the response spectrum, which is a key
concept in earthquake-resistant design.
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y0
m
k
Earth
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F
y
m
k
Earth
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y0
F
(F is negative here)
x = y-y0
y
m
k
Earth
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y0
(x is negative here)
F
(F is negative here)
x = y-y0
y
y0
(x is negative here)
m
Hooke’s Law
F = kx
k
Earth
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Controlling equation for singledegree-of-freedom systems:
Newton’s Second Law
F=ma
F is the restoring force,
m is the mass of the system
a is the acceleration that the system experiences.
2
d x(t )
a(t ) 
 x(t )
2
dt
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Force acting on the mass due to the spring:
F=-k x(t).
Combining with Newton’s Second Law:
mx(t )  kx
or:
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mx(t )  kx  0
John Anderson, GE/CEE 479/679
This is a second order differential equation:
mx(t )  kx  0
The solution can be written in two different ways:
1.
x(t )  A cos( t )  B sin(  t )
2. As the real part of:
x(t )  C exp( i t )
Note that the angular frequency is:
k
 
m
A and B, or the real and imaginary part of C in equation 2, are
selected by matching boundary conditions.
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Frequency comes with two
different units
• Angular frequency, ω
– Units are radians/second.
• Natural frequency, f
– Units are Hertz (Hz), which are the same as
cycles/second.
• Relationship: ω=2πf
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a f = 1 Hz
b
f = 1 Hz
c f = 2 Hz
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Friction
• In the previous example, the SDF never
stops vibrating once started. In real
systems, the vibration does eventually stop.
The reason is frictional loss of vibrational
energy, for instance into the air as the
oscillator moves back and forth.
• We need to add friction to make the
oscillator more realistic.
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Friction
• Typically, friction is modeled as a force
proportional to velocity.
• Consider, for instance, the experiment of
holding your hand out the window of a car.
When the car is still, there is no air force on
your hand, but when it moves there is a
force. The force is approximately
proportional to the speed of the car.
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Friction
• We add friction to the SDF oscillator by
inserting a dashpot into the system.
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F
x = y-y0
y
m
y0
(x is negative here)
Hooke’s Law
F  kx
k
Friction Law
c
F  cx
Earth
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Force acting on the mass due to the spring
and the dashpot:
F  kx  cx
Combining with Newton’s Second Law:
mx(t )  kx  cx
or:
mx(t )  cx  kx  0
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This is another second order differential
equation:
mx(t )  cx  kx  0
We make the substitution:
c
 2h n
m
So the differential equation becomes:
x(t )  2h n x   n2 x  0
The parameter h is the fraction of critical damping, and has
dimensionless units.
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We seek to solve the differential equation:
x(t )  2h n x   x  0
2
n
The solution can be written as the real part of:
x(t )  A exp(  t )
Where:

  n  h 
h
2

1
The real and imaginary part of A are selected by matching
boundary conditions.
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All: h=0.1
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h=0.1
h=0.2
h=0.4
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Forced SDF Oscillator
• The previous solutions are useful for
understanding the behavior of the system.
• However, in the realistic case of
earthquakes the base of the oscillator is
what moves and causes the relative motion
of the mass and the base.
• That is what we seek to model next.
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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 n x   x   zt 
2
n
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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Sinusoidal Input
• It is informative to consider first the response to a
sinusoidal driving function:
z t   Z 0 exp i t 
• It can be shown by substitution that a solution is:
xt   X 0 exp i t 
• Where:
X0

 2
Z0
 n   2  2ih n
2

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
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Sinusoidal Input (cont.)
• The complex ratio of response to input can
be simplified by determining the amplitude
and the phase. They are:
X0

Z0

2
2
n
 2
  2h 
2
n
 2h n 

  tan  2
2 
 n   
1
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2
h=0.01, 0.1, 0.8
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h=0.01, 0.1, 0.8
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Discussion
• In considering this it is important to
recognize the distinction between the
frequency at which the oscillator will
naturally oscillate, ωn, and the frequency at
which it is driven, ω.
• The oscillator in this case only oscillates at
the driving frequency.
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Discussion (cont.)
• An interesting case is when ω << ωn. In this
case, the amplitude X0 approaches zero,
which means essentially that the oscillator
will approximately track the input motion.
• The phase in this case is   0 This means
that the oscillator is moving the same
direction as the ground motion.
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Discussion (cont.)
• A second interesting case is when ω >> ωn. In this
case, the amplitude of X0 approaches Z0.
• The phase in this case is    This means that
the oscillator is moving the opposite direction as
the input base motion.
• In this case, the mass is nearly stationary in
inertial space, while the base moves rapidly
beneath it.
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Discussion (cont.)
• A third interesting case is when ω = ωn. In this
case, the amplitude of X0 may be much larger than
Z0. This case is called resonance.



• The phase in this case is
This means that
2
the oscillator is a quarter of a cycle behind the
input base motion.
• In this case, the mass is moving at it’s maximum
amplitude, and the damping controls the amplitude
to keep it from becoming infinite.
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