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

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.

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.

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.