Suppose I have a damped harmonic oscillator which is at rest, sitting comfortably with no initial amplitude, obeying the equation

$$\ddot{x} + \frac{1}{Q}\dot{x} + x = 0$$

where x is the vertical amplitude and Q is the quality factor. At $t = 0$, $x = 0$.

Now, suppose I model my system to include some sort of small pertubation, such as heat. We can model this as random, Gaussian-like vibrations: for example white noise. The equation becomes:

$$\ddot{x} + \frac{1}{Q}\dot{x} + x = N(t)$$

where $N(t)$ is some random number function shaped for a Gaussian distribution.

**Will this noise perturb the oscillator and give it a small amount of amplitude, and can we expect to see a plot like the one below for such a case?**

This is a simulation I ran and I am wondering whether these small random perturbations will set off the oscillator and cause the typical, yet haphazard, sinusoidal behaviour.

## Best Answer

The position of the mass, as a function of time, will simply be a

filteredversion of the random noise 'input' signal. To see this in the frequency domain, take the (magnitude of the) Fourier transform of both sides and rearrange:$$|X(\omega)| = \frac{1}{\sqrt{\left(1 - \omega^2\right)^2 + \frac{1}{Q^2}\omega^2}}|N(\omega)|$$

For $\omega = 1$, we have

$$|X(1)| = Q|N(1)|$$

So, for large $Q$ (highly under-damped), the position function will have a strong sinusoidal component at $\omega = 1$.

For smaller $Q$, the position function should be a 'smoothed' version of the input noise function since frequencies well above $\omega = 1$ fall off at the rate of approximately 40 dB /decade.

A plot of the (magnitude) transfer function $\frac{X(\omega)}{N(\omega)}$ for various values of $Q = \frac{1}{2\zeta}$ looks like: