Damped SHM k Natural Frequency m rads Damping
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Damped SHM k “Natural Frequency” m (rad/s) “Damping Parameter” (s-1) “Damping Constant” (kg/s) b EOM: damped oscillator
Guess a complex solution: “trivial solution” Actually 2 equations: Real = 0 Imaginary = 0 … also trivial !
Try a complex frequency: Real Imaginary A, f are free constants.
* amplitude decays due to damping * frequency reduced due to damping
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:
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.
2. “Over Damped”: imaginary! part of A Still need two constants for the 2 nd order EOM: No oscillations!
Over Damped wo = 1, g = 10, Q =. 1, xo = 1
3 “Critically Damped”: 0 …really just one constant, and we need two. Real solution:
Critically Damped wo = 1, g = 2, Q =. 5, xo = 1 Fastest approach to zero with no overshoot.
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. +
