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Damped SHM

Damped SHM. k. “Natural Frequency”. “Damping Parameter”. (rad/s). (s -1 ). m. “Damping Constant”. b. (kg/s). EOM: damped oscillator. Guess a complex solution:. “trivial solution”. Actually 2 equations:. Imaginary = 0. Real = 0. … also trivial !. Try a complex frequency:. Real.

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Damped SHM

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  1. Damped SHM k “Natural Frequency” “Damping Parameter” (rad/s) (s-1) m “Damping Constant” b (kg/s) EOM: damped oscillator

  2. Guess a complex solution: “trivial solution” Actually 2 equations: Imaginary = 0 Real = 0 … also trivial !

  3. Try a complex frequency: Real Imaginary A, f are free constants.

  4. * amplitude decays due to damping * frequency reduced due to damping

  5. How damped? Quality factor: unitless ratio of natural frequency to damping parameter Sometimes write solution in terms of wo and Q Sometimes write EOM in terms of wo and Q:

  6. 1. “Under Damped” or “Lightly Damped”: Oscillates at ~wo (slightly less) Looks like SHM (constant A) over a few cycles: wo= 1, g = .01, Q = 100, xo = 1 Amplitude drops by 1/e in Q/p cycles.

  7. 2. “Over Damped”: imaginary! part of A Still need two constants for the 2nd order EOM: No oscillations!

  8. Over Damped wo= 1, g = 10, Q = .1, xo = 1

  9. 0 3 “Critically Damped”: …really just one constant, and we need two. Real solution:

  10. Critically Damped wo= 1, g = 2, Q = .5, xo = 1 Fastest approach to zero with no overshoot.

  11. Real oscillators lose energy due to damping. This can be represented by a damping force in the equation of motion, which leads to a decaying oscillation solution. The relative size of the resonant frequency and damping parameter define different behaviors: lightly damped, critically damped, or over damped. +

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