It is a common practise for engineers to try to determine the resonant frequency of a system through a chirp signal.
Given a damped oscillating system with displacement $x$, driven by a chirp signal force, the equation of motion is given by:
$$\ddot{x} + 2\beta \dot{x} + \omega_0^2 x = A\cos\omega(t) t \, ,$$
where for a linear chirp frequency, $\omega(t)$ is given by the following:
$$\omega(t) = const \cdot t + t_0$$
The practise is to examine the amplitude of the system in the frequency space, with the frequency at the largest amplitude being equivalent to the resonant frequency of the system.
What is the mathematical reasoning behind this technique?
EDIT:
I have tried to solve this problem analytically. Please look at this link for details: https://math.stackexchange.com/questions/2835621/how-do-i-solve-the-second-ode-ddotx-2-beta-x-alpha2-f-cos-omegat
