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!


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