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

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

RealImaginary Try a complex frequency: A,  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 EOM in terms of wo and Q: Sometimes write solution in terms of wo and Q

1. “Under Damped” or “Lightly Damped”: Oscillates at ~  o (slightly less) Amplitude drops by 1/e in Q /  cycles. Looks like SHM (constant A) over a few cycles:  o = 1,  =.01, Q = 100, x o = 1

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

 o = 1,  = 10, Q =.1, x o = 1 Over Damped

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

Fastest approach to zero with no overshoot.  o = 1,  = 2, Q =.5, x o = 1 Critically Damped

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. +