1. February 7, 2008 1John Anderson, GE/CEE 479/679 Earthquake Engineering GE / CEE - 479/679 Topic 6. Single Degree of Freedom Oscillator Feb 7, 2008 John G. Anderson Professor of Geophysics

2. February 7, 2008 2John 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.

3. February 7, 2008 3John 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

4. February 7, 2008 4John 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

5. February 7, 2008 5John 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.

6. February 7, 2008 6John Anderson, GE/CEE 479/679 m Earth k y0y0

7. February 7, 2008 7John Anderson, GE/CEE 479/679 m Earth k y0y0 F y

8. February 7, 2008 8John Anderson, GE/CEE 479/679 m Earth k y0y0 y x = y-y 0 (x is negative here) F (F is negative here)

9. February 7, 2008 9John Anderson, GE/CEE 479/679 m Earth k y0y0 F (F is negative here) y Hooke’s Law F = kx x = y-y 0 (x is negative here)

10. February 7, 2008 10John 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.

11. February 7, 2008 11John 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:

12. February 7, 2008 12John Anderson, GE/CEE 479/679 This is a second order differential equation: The solution can be written in two different ways: 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:

13. February 7, 2008 13John 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

14. February 7, 2008 14John Anderson, GE/CEE 479/679 a b c f = 1 Hz f = 2 Hz

15. February 7, 2008 15John 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.

16. February 7, 2008 16John 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.

17. February 7, 2008 17John Anderson, GE/CEE 479/679 Friction We add friction to the SDF oscillator by inserting a dashpot into the system.

18. February 7, 2008 18John Anderson, GE/CEE 479/679 m Earth k y0y0 F y x = y-y 0 (x is negative here) Hooke’s Law c Friction Law

19. February 7, 2008 19John Anderson, GE/CEE 479/679 Force acting on the mass due to the spring and the dashpot: Combining with Newton’s Second Law: or:

20. February 7, 2008 20John 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.

21. February 7, 2008 21John 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.

22. February 7, 2008 22John Anderson, GE/CEE 479/679 All: h=0.1

23. February 7, 2008 23John Anderson, GE/CEE 479/679 h=0.1 h=0.2 h=0.4

24. February 7, 2008 24John 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.

25. February 7, 2008 25John Anderson, GE/CEE 479/679 m Earth k y0y0 F y x = y-y 0 (x is negative here) Hooke’s Law c Friction Law z(t)

26. February 7, 2008 26John 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:

27. February 7, 2008 27John 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.

28. February 7, 2008 28John 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:

29. February 7, 2008 29John 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:

30. February 7, 2008 30John Anderson, GE/CEE 479/679 h=0.01, 0.1, 0.8

31. February 7, 2008 31John Anderson, GE/CEE 479/679 h=0.01, 0.1, 0.8

32. February 7, 2008 32John 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, ω n, and the frequency at which it is driven, ω. The oscillator in this case only oscillates at the driving frequency.

33. February 7, 2008 33John Anderson, GE/CEE 479/679 Discussion (cont.) An interesting case is when ω << ω n. In this case, the amplitude X 0 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.

34. February 7, 2008 34John Anderson, GE/CEE 479/679 Discussion (cont.) A second interesting case is when ω >> ω n. In this case, the amplitude of X 0 approaches Z 0. 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.

35. February 7, 2008 35John Anderson, GE/CEE 479/679 Discussion (cont.) A third interesting case is when ω = ω n. In this case, the amplitude of X 0 may be much larger than Z 0. 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.