Short Version : 13. Oscillatory Motion Wilberforce Pendulum

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

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

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

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

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

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

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

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. 

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

The Torsional Oscillator  = torsional constant  Used in timepieces

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

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

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

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

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

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

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

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

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

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

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.