John G. Anderson Professor of Geophysics

February 7, 2008 1 John Anderson, GE/CEE 479/679

Earthquake EngineeringGE / CEE - 479/679

Topic 6. Single Degree of Freedom OscillatorFeb 7, 2008

John G. Anderson

Professor of Geophysics


Note to the students

• This lecture may be presented without use of Powerpoint. The following slides are a partial presentation of the material.


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


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


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.


m

Earth

k

y0


m

Earth

k

y0

F

y


m

Earth

k

y0y

x = y-y0

(x is negative here)

F(F is negative here)


m

Earth

k

y0

F(F is negative here)

y

Hooke’s LawF = kx

x = y-y0



Controlling equation for single-degree-of-freedom systems:

Newton’s Second Law

F=maF is the restoring force,

m is the mass of the system

a is the acceleration that the system experiences.

)()(

)(2

2

txdt

txdta &&==


Force acting on the mass due to the spring:

F=-k x(t).

Combining with Newton’s Second Law:

or:

kxtxm −=)(&&

0)( =+kxtxm&&


This is a second order differential equation:0)( =+kxtxm&&

The solution can be written in two different ways:

)sin()cos()( tBtAtx ωω +=

)exp()( tiCtx ω=

1.

2. As the real part of:

A and B, or the real and imaginary part of C in equation 2, are selected by matching boundary conditions.

Note that the angular frequency is:

m

k=ω


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


a

b

c

f = 1 Hz

f = 1 Hz

f = 2 Hz


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.


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.


Friction

• We add friction to the SDF oscillator by inserting a dashpot into the system.


m

Earth

k

y0

F

yx = y-y0


Hooke’s Law

cFriction Law

xcF &=

kxF =


Force acting on the mass due to the spring and the dashpot:

Combining with Newton’s Second Law:

or:

xckxtxm &&& −−=)(

0)( =++ kxxctxm &&&

xckxF &−−=


This is another second order differential equation:

0)( =++ kxxctxm &&&

We make the substitution:

nhm

c ω2=

So the differential equation becomes:

The parameter h is the fraction of critical damping, and has dimensionless units.

02)( 2 =++ xxhtx nn ωω &&&


We seek to solve the differential equation:

The solution can be written as the real part of:

)exp()( tAtx λ=

Where:

The real and imaginary part of A are selected by matching boundary conditions.

( )( )12 −±−= hhnωλ

02)( 2 =++ xxhtx nn ωω &&&


All: h=0.1


h=0.1

h=0.2

h=0.4


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.


m

Earth

k

y0

F

yx = y-y0


Hooke’s Law

cFriction Law

xcF &=

kxF =

z(t)


In this case, the force acting on the mass due to the spring and the dashpot is the same:

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:

or:

( )( ) xckxtztxm &&&&& −−=+)(

( )tzmkxxctxm &&&&& −=++)(

xckxF &−−=


So, the differential equation for the forced oscillator is:

After dividing by m, as previously, this equation becomes:

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.

( )tzxxhtx nn &&&&& −=++ 22)( ωω

( )tzmkxxctxm &&&&& −=++)(


Sinusoidal Input• It is informative to consider first the response to a

sinusoidal driving function: ( ) ( )tiZtz ωexp0=

• It can be shown by substitution that a solution is:

( ) ( )tiXtx ωexp0=

• Where:

( ) nn ihZ

X

ωωωω

ω

222

2

0

0

+−=


Sinusoidal Input (cont.)

• The complex ratio of response to input can be simplified by determining the amplitude and the phase. They are:

⎟⎟⎠

⎞⎜⎜⎝

⎛

−= −

221 2

tanωω

ωωφ

n

nh

( ) ( )2222

2

0

0

2 nn hZ

X

ωωωω

ω

+−=


h=0.01, 0.1, 0.8


h=0.01, 0.1, 0.8


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.


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 This means that the oscillator is moving the same direction as the ground motion.

0→φ


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.

πφ →


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 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.

2

πφ =

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John G. Anderson Professor of Geophysics