How to mathematically prove that a linear system's vibration frequency is equal to frequency of the harmonic external force?

Question

I was reading about "Vibration" on Wikipedia:

Forced vibration: 'for linear systems, the frequency of the steady-state vibration response resulting from the application of a periodic, harmonic input is equal to the frequency of the applied force or motion, with the response magnitude being dependent on the actual mechanical system.'

I would like to enquire the detailed mathematical proof of this conclusion.

Answer 1

Hints:

1. Fourier transform $t\leftrightarrow \omega$ the damped forced linear oscillator, which is a linear second-order ODE. For more details, see e.g. my Phys.SE answer here.

2. Show that there are at most 3 (possibly complex) frequencies with non-zero amplitudes: 2 characteristic frequencies and the driven frequency.

3. All realistic systems contain dissipation, so the characteristic frequencies will die out. Hence only the driven frequency remains.

Answer 2

Best approached using linear system's theory and transfer function analysis.

A linear system (which can be modeled as a rational polynomial transfer function in either $s$ or $j\omega$) responds to a linear excitation resulting in response having the same frequency however possibly different phase and amplitude.

This can be shown in a very generalized fashion either in the frequency domain or in the time domain by convolution of the time based excitation function and the impulse response of the system.