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### 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

- 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

- 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

- 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 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

(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

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

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

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

- 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

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

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

Friction

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

John Anderson, GE/CEE 479/679

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

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

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

John Anderson, GE/CEE 479/679

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

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

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

- 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.)

- 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

John Anderson, GE/CEE 479/679

John Anderson, GE/CEE 479/679

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.)

- 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 (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 (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

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