# Damped forced Oscillation with variable external frequency

Consider that we have the following forced vibration with an input frequency $$ω(t)$$ variable in time. $$m\ddot{x}+c\dot{x}+kx = F_0 \sin{(\omega(t) t)}$$

Assuming that the solution must be a harmonic form but with lag some phase angle $$ψ$$ we consider a solution to be:

$$x(t)=X_0 \sin{(\omega(t) t-ψ)}$$

Firstly, Is it true that the lag phase $$ψ$$ is also variable with time and so the particular solution must be as follows? $$x(t)=X_0 \sin{(\omega(t) t-ψ(t))}$$

Secondly, If the first question holds, how is the general solution formulated because the calculations gets really complicated?

$$\psi$$ is frequency-dependent so if frequency is time-dependent, $$\psi$$ will be time dependent. That's assuming that $$\omega$$ is varying slowly enough [$$|\frac{d\omega}{dt}|\ll\omega^2$$] for the solution you give to be approximately correct. In that case you can treat $$\psi$$ as a constant when solving the equation, and put its dependency on $$t$$ in afterwards! But if $$\omega$$ can vary as rapidly as you like, then the solution will not be of the form you have quoted anyway!