Classical Mechanics (Goldstein) · Problem 15

Oscillations — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · forced-vibrations-fourier-transform

Problem

If the generalized driving forces QiQ_i are not sinusoidal, show that the forced vibrations of the normal coordinates in the absence of damping are given by ζi=12π∫−∞+∞Gi(ω)ωi2−ω2e−iωt dω,\zeta_i = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \frac{G_i(\omega)}{\omega_i^2 - \omega^2} e^{-i\omega t}\, d\omega, where Gi(ω)G_i(\omega) is the Fourier transform of QiQ_i defined by Qi(t)=12π∫−∞+∞Gi(ω)e−iωt dω.Q_i(t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} G_i(\omega) e^{-i\omega t}\, d\omega. If the dissipation function is simultaneously diagonalized along with TT and VV, show that the forced vibrations are given by ζi=12π∫−∞+∞Gi(ω)(ωi2−ω2+iωFi)(ωi2−ω2)2+ω2Fi2e−iωt dt,\zeta_i = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{+\infty} \frac{G_i(\omega)(\omega_i^2 - \omega^2 + i\omega \mathcal{F}_i)}{(\omega_i^2 - \omega^2)^2 + \omega^2 \mathcal{F}_i^2} e^{-i\omega t}\, dt, which has the typical resonance denominator form. These results are simple illustrations of the powerful technique of the operational calculus for handling transient vibrations.

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