Damped Harmonic Oscillator

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Damping Coefficient When a damped oscillator is subject to a damping force which is linearly dependent upon the velocity, such as viscous damping, the oscillation will have exponential decay terms which depend upon a damping coefficient. If the damping force is of the form then the damping coefficient is given by This will seem logical when you note that the damping force is proportional to c, but its influence inversely proportional to the mass of the oscillator. 
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Underdamped OscillatorFor any value of the damping coefficient γ less than the critical damping factor the mass will overshoot the zero point and oscillate about x=0. The behavior is shown for onehalf and onetenth of the critical damping factor. Also shown is an example of the overdamped case with twice the critical damping factor. Note that these examples are for the same specific initial conditions, i.e., a release from rest at a position x_{0}. For other initial conditions, the curves would look different, but the the behavior with time would still decay according to the damping factor.

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Underdamped OscillatorWhen a damped oscillator is underdamped, it approaches zero faster than in the case of critical damping, but oscillates about that zero.

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