1. Chapter 1 - Vibrations Harmonic Motion/ Circular Motion Simple Harmonic Oscillators –Linear, Mass-Spring Systems –Initial Conditions Energy of Simple Harmonic Motion Damped Oscillations Driven/Forced Oscillations

2. Math Prereqs

3. Identities (see appendix A for more)

4. Why Study Harmonic Motion http://www.kettering.edu/~drussell/Demos/waves/wavemotion.html http://www.falstad.com/mathphysics.html

5. Relation to circular motion Or

6. Math Prereqs Example:

7. Horizontal mass-spring Good model! –Force is linear –Mass is constant –Spring has negligible mass –No losses Hooke’s Law: Frictionless (1D constraint)

8. Solutions to differential equations Guess a solution Plug the guess into the differential equation –You will have to take a derivative or two Check to see if your solution works. Determine if there are any restrictions (required conditions). If the guess works, your guess is a solution, but it might not be the only one. Look at your constants and evaluate them using initial conditions or boundary conditions.

9. Our guess

10. Check

11. The restriction on the solution

12. Any Other Solutions? A A1A1 A2A2 Or

13. Definitions Amplitude - (A) Maximum value of the displacement (radius of circular motion). Determined by initial displacement and velocity. Angular Frequency (Velocity) -    Time rate of change of the phase. Natural Angular Frequency Period - (T) Time for a particle/system to complete one cycle. Frequency - (f o ) The number of cycles or oscillations completed in a period of time. Natural Frequency Phase -  t  Time varying argument of the trigonometric function. Phase Constant -  Initial value of the phase. Determined by initial displacement and velocity.

14. The constants – Phase Angle Case I: Case II: Note phase relationship between x, u, and a

15. General Case A A1A1 A2A2

16. Energy in the SHO

17. Average Energy in the SHO

18. Example A mass of 200 grams is connected to a light spring that has a spring constant (s) of 5.0 N/m and is free to oscillate on a horizontal, frictionless surface. If the mass is displaced 5.0 cm from the rest position and released from rest find: a) the period of its motion, b) the maximum speed and c) the maximum acceleration of the mass. d) the total energy e) the average kinetic energy f) the average potential energy

19. Complex Exponential Solution Check it – it works and is simpler. Phase relationships are more obvious. Implied solution is the real part Are there enough arbitrary constants? What are they? Re Im a jb

20. Dashpot Equation of Motion: Solution Guess: Damped Oscillations Dissipative forces

21. Check

22. Damped frequency oscillation B - Critical damping (=) C - Over damped (>)

23. Relaxation Time Decay modulus, decay time, time constant, characteristic time Time required for the oscillation to decrease to 1/e of its initial value

24. Forced Vibrations Transient Solution – decays away with time constant,  Steady State Solution

25. Resonance Natural frequency make small!!

26. Mechanical Input Impedance Think Ohm’s Law

27. Significance of Mechanical Impedance It is the ratio of the complex driving force to the resulting complex speed at the point where the force is applied. Knowledge of the Mechanical Impedance is equivalent to solving the differential equation. In this case, a particular solution.

28. V Electrical Analogs m s RmRm ElecZ elec Mech Vf Iu L jLjL m jmjm RRRmRm RmRm 1/C 1/j  C s s/j  f

29. How would you electrically model this? m s RmRm f u umum f 1/s RmRm m u umum

30. Transient Response See front cover and figure 1.8.1 (pg 14) Which is transient, which is steady state?

31. Instantaneous Power Think EE

32. Average Power

33. Quality (Q) value Q describes the sharpness of the resonance peak Low damping give a large Q High damping gives a small Q Q is inversely related to the fraction width of the resonance peak at the half max amplitude point.

34. Tacoma Narrows Bridge

35. Tacoma Narrows Bridge (short clip)