earthquake engineering ge cee 479 679 topic 6 single degree of freedom oscillator feb 7 2008
Download
Skip this Video
Download Presentation
John G. Anderson Professor of Geophysics

Loading in 2 Seconds...

play fullscreen
1 / 35

John G. Anderson Professor of Geophysics - PowerPoint PPT Presentation


  • 107 Views
  • Uploaded on

Earthquake Engineering GE / CEE - 479/679 Topic 6. Single Degree of Freedom Oscillator Feb 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.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' John G. Anderson Professor of Geophysics' - bunme


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
earthquake engineering ge cee 479 679 topic 6 single degree of freedom oscillator feb 7 2008

Earthquake EngineeringGE / CEE - 479/679Topic 6. Single Degree of Freedom OscillatorFeb 7, 2008

John G. Anderson

Professor of Geophysics

John Anderson, GE/CEE 479/679

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

John Anderson, GE/CEE 479/679

sdf oscillator
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

John Anderson, GE/CEE 479/679

motivations for studying sdf systems
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

John Anderson, GE/CEE 479/679

motivations for studying sdf systems1
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.

John Anderson, GE/CEE 479/679

slide6

m

y0

k

Earth

John Anderson, GE/CEE 479/679

slide7

F

m

y

y0

k

Earth

John Anderson, GE/CEE 479/679

slide8

F

(F is negative here)

x = y-y0

(x is negative here)

m

y

y0

k

Earth

John Anderson, GE/CEE 479/679

slide9

F

(F is negative here)

x = y-y0

(x is negative here)

m

y

y0

Hooke’s Law

F = kx

k

Earth

John Anderson, GE/CEE 479/679

controlling equation for single degree of freedom systems newton s second law
Controlling equation for single-degree-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.

John Anderson, GE/CEE 479/679

slide11
Force acting on the mass due to the spring:

F=-k x(t).

Combining with Newton’s Second Law:

or:

John Anderson, GE/CEE 479/679

slide12

The solution can be written in two different ways:

This is a second order differential equation:

1.

2. As the real part of:

Note that the angular frequency is:

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

John Anderson, GE/CEE 479/679

frequency comes with two different units
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

John Anderson, GE/CEE 479/679

slide14

a

f = 1 Hz

f = 1 Hz

b

f = 2 Hz

c

John Anderson, GE/CEE 479/679

friction
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.

John Anderson, GE/CEE 479/679

friction1
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.

John Anderson, GE/CEE 479/679

friction2
Friction
  • We add friction to the SDF oscillator by inserting a dashpot into the system.

John Anderson, GE/CEE 479/679

slide18

F

x = y-y0

(x is negative here)

m

y

y0

Hooke’s Law

Friction Law

c

k

Earth

John Anderson, GE/CEE 479/679

slide19
Force acting on the mass due to the spring and the dashpot:

Combining with Newton’s Second Law:

or:

John Anderson, GE/CEE 479/679

slide20
This is another second order differential equation:

We make the substitution:

So the differential equation becomes:

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

John Anderson, GE/CEE 479/679

slide21
We seek to solve the differential equation:

The solution can be written as the real part of:

Where:

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

John Anderson, GE/CEE 479/679

slide22

All: h=0.1

John Anderson, GE/CEE 479/679

slide23

h=0.1

h=0.2

h=0.4

John Anderson, GE/CEE 479/679

forced sdf oscillator
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.

John Anderson, GE/CEE 479/679

slide25

F

x = y-y0

(x is negative here)

m

y

y0

Hooke’s Law

Friction Law

c

k

z(t)

Earth

John Anderson, GE/CEE 479/679

slide26
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:

John Anderson, GE/CEE 479/679

slide27
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.

John Anderson, GE/CEE 479/679

sinusoidal input
Sinusoidal Input
  • It is informative to consider first the response to a sinusoidal driving function:
  • It can be shown by substitution that a solution is:
  • Where:

John Anderson, GE/CEE 479/679

sinusoidal input cont
Sinusoidal Input (cont.)
  • The complex ratio of response to input can be simplified by determining the amplitude and the phase. They are:

John Anderson, GE/CEE 479/679

slide30

h=0.01, 0.1, 0.8

John Anderson, GE/CEE 479/679

slide31

h=0.01, 0.1, 0.8

John Anderson, GE/CEE 479/679

discussion
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.

John Anderson, GE/CEE 479/679

discussion cont
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.

John Anderson, GE/CEE 479/679

discussion cont1
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.

John Anderson, GE/CEE 479/679

discussion cont2
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.

John Anderson, GE/CEE 479/679

ad