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Published byAshley Shelton Modified over 8 years ago
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k m b Damped SHM “Damping Constant” (kg/s) “Damping Parameter” (s -1 ) “Natural Frequency” (rad/s) EOM: damped oscillator
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“trivial solution” Actually 2 equations: Imaginary = 0 Real = 0 … also trivial ! Guess a complex solution:
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RealImaginary Try a complex frequency: A, are free constants.
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* amplitude decays due to damping * frequency reduced due to damping
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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
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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
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2. “Over Damped”: imaginary! part of A No oscillations! Still need two constants for the 2 nd order EOM:
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o = 1, = 10, Q =.1, x o = 1 Over Damped
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3 “Critically Damped”: …really just one constant, and we need two. Real solution: 0
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Fastest approach to zero with no overshoot. o = 1, = 2, Q =.5, x o = 1 Critically Damped
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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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