Author Topic: Resonance phenomena  (Read 2042 times)

Offline Alberto Dominguez

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Resonance phenomena
« on: August 31, 2012, 08:29:21 PM »
Driven Harmonic Oscillators Equation

Periodic force with frecuency of vibration \omega:

F=F_{o}cos(\omega t+\alpha_{o})

Movement equation:

\frac{d^{2}x}{dt^{2}}+2\gamma\frac{dx}{dt}+\omega_{o}^{2}x=\frac{F_{o}cos(\omega t+\alpha_{o})}{m}

\omega_{o}^{2}=\frac{k}{m};\gamma=\frac{b}{2m}

x(t)=x_{h}(t)+x_{p}(t)

Transient solution (depends on initial conditions):

x_{h}(t)=Ce^{-\gamma t}sen(\sqrt{\omega_{o}^{2}-\gamma_{o}^{2}}t+\delta)

Steady State (independent of initial condition):

x_{p}(t)=\frac{F_{o}}{m\sqrt{(\omega_{o}^{2}-\omega^{2})^{2}+4\gamma^{2}\omega^{2}}}sen(\omega t+\alpha_{o}+\beta)

Solution:

x(t)=Ce^{-\gamma t}sen(\sqrt{\omega_{o}^{2}-\gamma_{o}^{2}}t+\delta)+\frac{F_{o}}{m\sqrt{(\omega_{o}^{2}-\omega^{2})^{2}+4\gamma^{2}\omega^{2}}}sen(\omega t+\alpha_{o}+\beta)

When \omega=\omega_{o} the Amplitude is maximum and if b=0 then it is infinite, so a not too great force can collapse an strong structure if this force is applied at this frecuency of vibration.  This is the resonance phenomena.  All materials or structures have several natural frecuencies (\omega_{o}) and they can enter into resonance.  Perhaps the most famous sample is the Tacoma bridge:

« Last Edit: September 02, 2012, 07:39:39 AM by Alberto Dominguez »
Kind regards,

Alberto Domínguez
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