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

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

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

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.

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

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

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 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!

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

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:

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

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 (1653-1716) Galileo Galilei (1564-1642)

c. Toothed Wheels & Sirens 13 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

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

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

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

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.

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

1. Kinetic and Potential Energy 19 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 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 3. Damped Oscillations Including friction the oscillations die out with time. Frictional force is velocity dependent. Friction parameter “b”

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 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”.

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

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.. 226 soldiers died in the river below the bridge.

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: http://www.youtube.com/watch?v=ASd0t3n8Bnc

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, 2000. 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: http://www.youtube.com/watch?v=gQK21572oSU&feature=related

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

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

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

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