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Short Version : 13. Oscillatory Motion

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:

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Short Version : 13. Oscillatory Motion

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  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. • PeriodT = time for the motion to repeat itself. • Frequencyf = # 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 ( long time ) Prob 75: = 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.

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