1. Chapter 14 - Oscillations
Harmonic Motion Circular Motion
Simple Harmonic Oscillators
Linear - Horizontal/Vertical Mass-Spring Systems
Angular - Simple Pendulum
Energy of Simple Harmonic Motion
Damped Oscillators
Driven Oscillators - Resonance

2. Harmonic

3. Horizontal mass-spring
Hooke’s Law:

4. 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.

5. Our guess

6. The restriction on the solution

7. Vertical Springs

8. The constant – phase angle

9. Definitions
Amplitude - (A, qm) Maximum value of the displacement (radius of circular motion). Determined by initial displacement and velocity.
Period - (T) Time for a particle/system to complete one cycle.
Frequency - (f) The number of cycles or oscillations completed in a period of time
Phase - (wt + f) Time varying argument of the trigonometric function.
Phase Constant - (f) Initial value of the phase. Determined by initial displacement and velocity.
Angular Frequency (Velocity) - (w) Time rate of change of the phase.

10. Relation to circular motion

11. Energy in the SHO

12. Simple pendulum

13. The restriction on the solution

14. Damped Oscillations
“Dashpot”
Equation of Motion
Solution

16. Damped Oscillations
“Dashpot”
Equation of Motion
Solution

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

19. Forced vibrations

20. Resonance
Natural frequency

21. 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.

22. Tacoma Narrows Bridge

23. Tacoma Narrows Bridge (short clip)