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Physics 321

Physics 321. Hour 11 Simple and Damped Harmonic Oscillators. Find U and T Equilibrium Equation of motion ω , T Energy plot . Mass on a Massless Spring. y. y. Assume Find U and T Equation of motion ω , T . x. A Shallow Frictionless Bowl. z. y. Assume Find U and T

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Physics 321

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  1. Physics 321 Hour 11 Simple and Damped Harmonic Oscillators

  2. Find U and T Equilibrium Equation of motion ω, T Energy plot Mass on a Massless Spring y

  3. y Assume Find U and T Equation of motion ω, T x A Shallow Frictionless Bowl z

  4. y Assume Find U and T Newton’s Equations x Another Bowl z

  5. Lissajous.nb Example

  6. The equation: A little rearranging: Damped Oscillator A trial solution:

  7. Underdamped: • Critically damped: • Overdamped: Three Regimes

  8. Solution: or Underdamped

  9. Solution: Overdamped

  10. Solution: Critically Damped

  11. DampOsc5_4.nb Example

  12. Physics 321 Hour 12 Driven Harmonic Oscillators

  13. The equation: Let The oscillator wants to oscillate at but the driver forces it to oscillate at . This leads to transient vs steady state behavior! Driven Oscillator

  14. Driven_Osc.nb Example

  15. We assume a solution something like But So we employ a trick… The driving force is the real part of Driven Oscillator

  16. We assume a solution of the form This gives: Driven Oscillator

  17. Conclusion 1: Driven Oscillator

  18. Real parts: Imaginary parts: Driven Oscillator

  19. Conclusion 2: And finally the steady state solution is: Driven Oscillator

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