Will a damped harmonic oscillator, with no initial amplitude, oscillate if there was background “noise”?

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.

The position of the mass, as a function of time, will simply be a filtered version 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: • Maybe add that $X(\omega)$ means the Fourier Transform of $x(t)$, such that a imaginary value for $X(\omega)$ means a phase shift of 90° with respect to the input ($N(t)$). And since PPG talks about noise as input, I think a bode plot or transfer function might be useful as well. – fibonatic Jan 8 '15 at 13:21
• A small side note, I think it is bad practice to not mention the units of any physical quantity, so $\omega=1$ should be $\omega=1\ rad/s$. – fibonatic Jan 8 '15 at 16:48
• @fibonatic, in this case, $\omega$ is normalized which I assumed is evident given the context. Otherwise, the given differential equation requires factors of $\omega_0$. – Alfred Centauri Jan 8 '15 at 17:23

I don't think you really need an answer. The answer is yes and moreover what you have done is a pretty sound model of the effect of noise on the damped oscillator. I'm assuming that you have normalised frequencies so that the oscillator's natural frequency $\omega_n$ is one unit.

The only factor you haven't mentioned and which you seem to have overlooked is the spectral density of the noise $N(t)$. You can either assume that the noise will be very broadband compared to your oscillator's bandwidth, in which case $N(t)$ will be an uncorrelated sequence of numbers with variance given by $\sigma^2 = \eta\,\int_{-infty}^\infty |H(\omega)|^2\,\mathrm{d}\omega$ where $\eta$ is the power spectral density of the noise and $H(\omega)$ the Fourier transform of your oscillator's unit impulse response. Otherwise, you may want to think about the kind of mechanical / other effects that produce the noise, and thus make $N(t)$ a sequence of variables that are correlated by passing Gaussian white (uncorrelated) noise through models of the noise production mechanism, which themselves will likely be systems described by linear ODEs exactly like yours. So either way you will end up feeding Gaussian, uncorrelated noise either directly into your system or through a modified system.

The answer must be yes, because a force is applied then there will be some motion (if there was no motion then Newton's laws would be violated of course).

The applitude will most likely be tiny and the damping term will ensure that it remains tiny. If there was no damping term you could get strong oscillation.

The shape of the graph will most likely be similar to what you suggest based on oscillations at the natural frequency of the system - but I'm afraid I don't understand exactly what you mean for $N(t)$ and that will, of course, have a strong influence on the motion.

This reminds me of the problem with a bridge in London that didn't have damping where the natural frequency was close to the frequency of people walking. They fixed the bridge by putting damping in.

• That was my intuition also. Well, in my code I have implemented $N(t)$ simply as a (random) number, which varies with time. I bring it over to the LHS and I solve the equation: I end up with two equations and I use RK4 to simulate. Also, why the hell are the people in that video purposely trying to make the bridge swing!? – turnip Jan 8 '15 at 11:51
• @PPG - people found it difficult to walk on the bridge as it was moving and found the easiest way to move was to keep time with it.... this of course made the problem worse. There is a whole lot on the web about it - interesting story of what went wrong in the design.... – tom Jan 8 '15 at 11:54