Dampled harmonic motion
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Amplitude of successive oscillations
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Consider the case where the damping factor so there are damped oscillations. [Diagram goes here - download the original to see it.] The diagram marks the successive amplitudes of the oscillations. The general solution is [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.] and the two complex conjugate solutions to the equation [Equation goes here - download the original to see it.] which is the auxiliary equation to the differential equation [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.n Hence [Equation goes here - download the original to see it.] The angular frequency is given by [Equation goes here - download the original to see it.] so the period is [Equation goes here - download the original to see it.] Successive peaks occur when [Equation goes here - download the original to see it.]. This occurs after every complete oscillation - that is, after intervals of the period T. Let [Equation goes here - download the original to see it.] represent the first peak, then at [Equation goes here - download the original to see it.] where [Equation goes here - download the original to see it.] is the first value such that [Equation goes here - download the original to see it.] Then the second peak occurs at [Equation goes here - download the original to see it.] and the nth at [Equation goes here - download the original to see it.]; hence [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] So successive peaks maintain a constant ratio. That is, the amplitude of successive oscillations decreases with time in geometric progression with common ratio [Equation goes here - download the original to see it.].
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Contents of
Dampled harmonic motion
1 Damped and forced vibrations
2 Dashpots
3 Critical and non-critical damping
4 Amplitude of successive oscillations
5 Forced oscillations
6 Resonance and frequency response
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