1. Short Version : 13. Oscillatory Motion
Wilberforce Pendulum

2. Disturbing a system from equilibrium results in oscillatory motion.
Absent friction, oscillation continues forever.
Oscillation

3. 13.1. Describing Oscillatory Motion
Characteristics of oscillatory motion:
Amplitude A = max displacement from equilibrium.
Period T = time for the motion to repeat itself.
Frequency f = # of oscillations per unit time.
same period T
same amplitude A
[ f ] = hertz (Hz) = 1 cycle / s
A, T, f do not specify an oscillation completely.
Oscillation

4. 13.2. Simple Harmonic Motion
Simple Harmonic Motion (SHM):
2nd order diff. eq  2 integration const.
Ansatz:
angular frequency  

5. A, B determined by initial conditions 
( t )  2
x  2A

6. Amplitude & Phase  C = amplitude  = phase
Note:  is independent of amplitude only for SHM.
Curve moves to the right for  < 0.
Oscillation

7. Velocity & Acceleration in SHM
|x| = max at v = 0
|v| = max at a = 0

8. Application: Swaying skyscraper
Tuned mass damper :
Damper highly damped.
Overall oscillation overdamped.
Taipei 101 TMD:
41 steel plates,
730 ton, d = 550 cm,
87th-92nd floor.
Also used in:
Tall smokestacks
Airport control towers.
Power-plant cooling towers.
Bridges.
Ski lifts.
Movie
Tuned Mass Damper

9. Example 13.2. Tuned Mass Damper
The tuned mass damper in NY’s Citicorp Tower consists of a 373-Mg (vs 101’s 3500 Mg) concrete block that completes one cycle of oscillation in 6.80 s.
The oscillation amplitude in a high wind is 110 cm.
Determine the spring constant & the maximum speed & acceleration of the block. 

10. The Vertical Mass-Spring System
Spring stretched by x1 when loaded.
mass m oscillates about the new equil. pos.
with freq

11. The Torsional Oscillator
 = torsional constant 
Used in timepieces

12. The Pendulum
Small angles oscillation:
Simple pendulum (point mass m):

13. Conceptual Example 13.1. Nonlinear Pendulum
A pendulum becomes nonlinear if its amplitude becomes too large.
As the amplitude increases, how will its period changes?
If you start the pendulum by striking it when it’s hanging vertically,
will it undergo oscillatory motion no matter how hard it’s hit?
If it’s hit hard enough,
motion becomes rotational.
(a) sin increases slower than 
 smaller    longer period

14. The Physical Pendulum
Physical Pendulum = any object that’s free to swing
Small angular displacement  SHM

15. 13.4. Circular & Harmonic Motion
Circular motion:
2  SHO with same A &  but  = 90
x =  R
x = R
x = 0
Lissajous Curves

16. GOT IT? 13.3.
The figure shows paths traced out by two pendulums swinging with different frequencies in the x- & y- directions.
What are the ratios x : y ?
1 : 2
3: 2
Lissajous Curves

17. 13.5. Energy in Simple Harmonic Motion
SHM:
= constant
Energy in SHM

18. Potential Energy Curves & SHM
Linear force:
 parabolic potential energy:
Taylor expansion near local minimum:
 Small disturbances near equilibrium points  SHM

19. 13.6. Damped Harmonic Motion
sinusoidal oscillation
Damping (frictional) force:
Damped mass-spring:
Amplitude exponential decay
Ansatz: 
Real part 實數部份 :
where

20. At t = 2m / b, amplitude drops to 1/e of max value.
is real, motion is oscillatory ( underdamped )
(a) For
(c) For
is imaginary, motion is exponential ( overdamped )
(b) For
= 0, motion is exponential ( critically damped )
Damped & Driven Harmonic Motion

21. 13.7. Driven Oscillations & Resonance
External force  Driven oscillator
Let
d = driving frequency
Prob 75:
( long time )
= natural frequency
Damped & Driven Harmonic Motion
Resonance:

22. Buildings, bridges, etc have natural freq.
If Earth quake, wind, etc sets up resonance, disasters result.
Collapse of Tacoma bridge is due to self-excitation described by the van der Pol equation.
Tacoma Bridge
Resonance in microscopic system:
electrons in magnetron  microwave oven
Tokamak (toroidal magnetic field)  fusion
CO2 vibration: resonance at IR freq  Green house effect
Nuclear magnetic resonance (NMR)  NMI for medical use.