1. I. Harmonic Motion & Resonance
CSUEB Physics 1200
Category: Harmonic Motion, Waves, Sound
I. Harmonic Motion & Resonance
Updated 2012Apr06
Dr. Bill Pezzaglia

2. Outline Harmonic Oscillators Harmonic Motion Energy in Oscillations2
Outline
Harmonic Oscillators
Harmonic Motion
Energy in Oscillations
Resonance

3. A. Harmonic Oscillators3
A. Harmonic Oscillators
Equilibrium
Periodic Motion
Frequency

4. 4
1. Equilibrium
Equilibrium: a system which is not changing with time (no net force).
There are 3 types:
Neutral equilibrium: boring case, if you move the object to another position it will just sit there.

5. b. Unstable Equilibrium5
b. Unstable Equilibrium
if you displace system slightly it changes drastically,
e.g. a ball perched on top of a steep hill

6. 6
c. Stable Equilibrium
if you displace the system, there is a “restoring force” which opposes the change
Strength of restoring force increases with displacement

7. 2. Periodic Motion a) Oscillations:7
2. Periodic Motion
a) Oscillations:
A system displaced from stable equilibrium will oscillate about the equilibrium point
The motion is “periodic” (repeats in time)
The time for one cycle is called the “period”

8. 8
b. Pendulums
Galileo: (1581) showed the period of oscillation depends only upon gravity “g” and length “L” of the string:
Period is INDEPENDENT of:
Mass on end of string
Size (“amplitude”) of oscillation
Acceleration of gravity on earth: g=9.8 meters/second2.
Gravity on moon is 1/6 as strong, so pendulum will go slower!

9. b.2. Physical Pendulum Complex body will behave like a simple pendulum9
Complex body will behave like a simple pendulum
L is distance to CM from pivot
 is moment of inertia about “O” L

10. 10
c. Springs
Hooke’s Law: if you squash a spring by distance x, it will give a restoring force F proportional to x:
Spring Constant “k” tells the stiffness of the spring.
Period of Oscillation for mass m on spring:

11. 3. Frequency a). Definition: Frequency is the rate of vibration11
3. Frequency
a). Definition:
Frequency is the rate of vibration
Units: Hertz=“cycles per second”
Relation to Period:

12. b. Frequency is “Pitch” 12
1600 Scraper across grooved board produces notes (relates frequency of vibration to pitch of sound)
Mathematical Discourses Concerning Two New Sciences (1638) most lucid of the frequency equivalence
Made sound waves visible by striking a wine glass floating in water and seeing the vibrations it made on the water’s surface.
First person to accurately determine frequency of musical pitch was probably Joseph Sauveur ( )
Galileo Galilei ( )

13. c. Toothed Wheels & Sirens13
1819 Cagnaird de la Tour’s siren used to precisely measure frequency of sound (disk with holes spun, air blown across holes)
1830 Savart uses card against moving toothed wheel to equate frequency and vibration
Measures the lowest pitch people can hear is about 16 to 20 Hertz

14. B. Harmonic Motion Displacement is sine wave Velocity is cosine wave14
B. Harmonic Motion
Displacement is sine wave
Velocity is cosine wave
Acceleration is opposite displacement

15. 1. Displacement is a sine wave15
A is the amplitude of oscillation
 is the angular frequency
f is the frequency
P is the period

16. 2. Velocity 16
The maximum speed of the mass on a spring is related to the maximum displacement (amplitude “A”) by the frequency “ f ” :

17. 3. Acceleration 17
The acceleration for a “harmonic” system is proportional to the displacement. The minus sign means it’s a restoring force, such that the system will oscillate.

18. C. Energy in Oscillations18
C. Energy in Oscillations
Kinetic and Potential Energy
Impedance
Damped Oscillations

19. 1. Kinetic and Potential Energy19
Kinetic Energy:
Potential Energy (of displaced spring) where k is the spring constant.
Total Energy is constant. This gives a relation between maximum velocity and amplitude:

20. 20
2. Impedance of a Spring
Impedance “Z” relates the (maximum) force of a spring “F” to the (maximum) oscillation speed of the mass “v”.
Hooke’s Law relates force, spring constant “k” to maximum stretch, or amplitude “A”
Substitute for amplitude (previous page)
Substitute for frequency :
Yields impedance:

21. 21
3. Damped Oscillations
Including friction the oscillations die out with time.
Frictional force is velocity dependent.
Friction parameter “b”

22. 3b. Decay Envelope 22
The solution is an exponentially decaying oscillation:
The decay constant  or time constant  describe how fast the system loses energy. After one time constant the amplitude is reduced by 37%.
Note the frequency  of the system is slightly less than the undamped frequency 0

23. 23
3.c. Critical Damping
Critical Damping: If friction is big enough, frequency =0 and system does not oscillate. [Note the relation of critical friction to the spring impedance!]
If b>bc then it is “overdamped”.

24. D. Resonance Driven Oscillator Resonance Curve Bandwidth and Quality24
D. Resonance
Driven Oscillator
Resonance Curve
Bandwidth and Quality

25. Unwanted Resonance (1850) 25
Angers Bridge: a suspension bridge over the Maine River in Angers, France. Its famous for having collapsed on April 15, 1850, when 478 French soldiers marched across it in lockstep. Since the soldiers were marching together, they caused the bridge to vibrate and twist from side to side, dislodging an anchoring cable from its concrete mooring soldiers died in the river below the bridge.

26. Tacoma Narrows Bridge Collapse (1940)26
“Just as I drove past the towers, the bridge began to sway violently from side to side. Before I realized it, the tilt became so violent that I lost control of the car... I jammed on the brakes and got out, only to be thrown onto my face against the curb... Around me I could hear concrete cracking... The car itself began to slide from side to side of the roadway. On hands and knees most of the time, I crawled 500 yards [450 m] or more to the towers... My breath was coming in gasps; my knees were raw and bleeding, my hands bruised and swollen from gripping the concrete curb... Toward the last, I risked rising to my feet and running a few yards at a time... Safely back at the toll plaza, I saw the bridge in its final collapse and saw my car plunge into the Narrows.” -eyewitness account
Video on Collapse of Bridge:

27. Millennium Bridge (London)27
You’d think that engineers don’t make mistakes like this anymore? Yet again in 2000…. The Millennium pedestrian Bridge opened June 10, Almost immediately it was discovered to resonate when people walked over it, causing it to be close 2 days later.
Video on Millennium Bridge Problems:

28. 1. Driven Oscillator Notes on this part was done on the board in class28
1. Driven Oscillator
Notes on this part was done on the board in class

29. 29
2. Resonant Curve
Notes on this part was done on the board in class

30. 3. Bandwidth and “Q” factor30
3. Bandwidth and “Q” factor
Notes on this part was done on the board in class

31. Notes 31 Demo: Spring, Pendulum Comb or other similar device
Perhaps the “Helmholtz Resonator” should be included here?
Need to finish part “D”.