harmonicoscillators[1] - Applied Mathematics 2101...

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Unformatted text preview: Applied Mathematics 2101 Supplement on identities relevant to harmonic oscillators The behavior of forced harmonic oscillator systems (whether L-R-C electronic circuits, spring dashpot mechanisms like shock-absorbers in automobiles, or instruments to detect vibrations like seismometers) is usually represented by the superposition of sinusoids. Superposing two sinusoids of the same frequency It is hard to visualize the superposition of sinusoids unless they have the same zero- crossings. However, the sum of two sinusoids of the same frequency but different amplitudes and phases can be written as a single sinusoid, with phase and amplitude derived from those of the summands. For understanding a response, we may wish to write, for example, A cos ωt + B sin ωt as R cos( ωt- δ ). We use the identity cos( α- β ) = cos α cos β + sin α sin β . Begin by expanding R cos( ωt- δ ) using the identity: R cos( ωt- δ ) = R cos ωt cos δ + R sin ωt sin δ ....
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