The damping coefficient is less than the undamped resonant frequency. The equation is that of an exponentially decaying sinusoid. Underdamped OscillatorWhen a damped oscillator is underdamped, it approaches zero faster than in the case of critical damping, but oscillates about that zero. For other initial conditions, the curves would look different, but the the behavior with time would still decay according to the damping factor. Note that these examples are for the same specific initial conditions, i.e., a release from rest at a position x 0. Also shown is an example of the overdamped case with twice the critical damping factor. The behavior is shown for one-half and one-tenth of the critical damping factor. 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.įor 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. An oscillator is underdamped if is real, which will be true if: b 2 4 m 2 parenleftbigg b 2 m parenrightbigg 2 < k m 2 0 (892) Week 9: Oscillations 417 An underdamped oscillator will exhibit true oscillations, eventually (exponentially) approaching zero amplitude due to damping. 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. The three resulting cases for the damped oscillator are The roots of the quadratic auxiliary equation are Substituting this form gives an auxiliary equation for λ
This is in the form of a homogeneous second order differential equation and has a solution of the form Damped Harmonic Oscillator Damped Harmonic Oscillator
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