The equation of motion for an undamped harmonic oscillator, with driver $ f=f(t)$ is given by: $ $ \ddot{x}+x=f.$ $ Let the initial conditions be given by: $ $ x(0)=\dot{x}(0)=0.$ $ If $ f=\cos(t)$ then the solution is: $ $ x(t)=\frac{1}{2}t\sin(t).$ $ Hence, a resonance is setup and the energy of the oscillator will grow forever. If $ f=\cos(\omega t)$ where $ \omega\ne1$ , the solution is: $ $ x(t)=\frac{2}{\omega^2-1}\sin\left(\frac{\omega-1}{2}t\right)\sin\left(\frac{\omega+1}{2}t\right),$ $ hence, the energy oscillates about some finite value. My question is, if $ f$ were replaced with some continuous random driver where the frequency profile resmbled that of say gaussian white noise, would the energy of the oscillator grow forever or would is oscillate about some finite value?
Does anyone know of a simple function I could replace $ f$ with to generate a continuous white noise driver?
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