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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 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 Newton’s Equations x Another Bowl z
Lissajous.nb Example
The equation: A little rearranging: Damped Oscillator A trial solution:
Underdamped: • Critically damped: • Overdamped: Three Regimes
Solution: or Underdamped
Solution: Overdamped
Solution: Critically Damped
DampOsc5_4.nb Example
Physics 321 Hour 12 Driven Harmonic Oscillators
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
Driven_Osc.nb Example
We assume a solution something like But So we employ a trick… The driving force is the real part of Driven Oscillator
We assume a solution of the form This gives: Driven Oscillator
Conclusion 1: Driven Oscillator
Real parts: Imaginary parts: Driven Oscillator
Conclusion 2: And finally the steady state solution is: Driven Oscillator