# Find coefficients in general second order differential equation

Suppose you have a system that can be described via the following equations of motion: $$\ddot{y}+\delta(t)\dot{y}+\alpha(t) y = \gamma\sin(\omega t)$$ The functions $$\delta(t)$$ and $$\alpha(t)$$ are periodic in $$T=\frac{2\pi}{\omega}$$ and can be described via a Fourier series, i.e. $$\alpha(t) = \alpha_0 + \sum_{n=1}^N \alpha_n\exp(in\omega t)$$ $$\delta(t) = \delta_0 + \sum_{n=1}^N \delta_n\exp(in\omega t)$$ Now, given that you know the response (i.e. $$y$$) of the system for all $$\omega$$ for fixed $$\gamma$$, can you infer the coefficients of the Fourier series in $$\alpha(t)$$ and $$\delta(t)$$? How would one proceed in doing so?

• I guess the first thing would be to try to write an analytic solution to the ODE, and Floquet's theorem seems to be relevant here. Aug 9 at 18:30
• Is $y(t)$ also periodic? Over what time domain to you know $y(t)$? Aug 10 at 0:48
• Also I don't quite understand the expansions of $\alpha$ and $\delta$. Are these functions dependent on the frequency $\omega$? If so, it would probably be useful to indicate that by saying $\alpha(t,\omega)$ and $\delta(t,\omega)$. But then $\omega$ simply becomes a parameter, and the question is the same as asking whether they can be identified for a given $\omega$. Aug 10 at 0:50
• Lets say I know the steady state response of y(t), i.e. no transient motion. And yes, its periodic. The assumption is that the expansions for α and δ are the same for the relevant frequency range. For context, you can think of a cantilever type resonator on which a magnetostrictive layer is deposited. Forcing oscillation can be achieved via a magnetic field which results in magnetostriction. At the same time, the mechanical properties magnetostrictive layer changes (ΔE effect), which detunes the resonator periodically. Thus, the magnetic field is the reason for the changes in the parameters.
– Ron
Aug 10 at 6:20
• I guess you can think of the system as a parametric resonator for which the parameters change due to the forced oscillation. In reality you can also observe parametric resonances for $\frac{2\omega_r}{n}$ with $n$ being and integer $\geq1$ and $\omega_r$ being the unperturbed resonance frequency. Conceptually I think you can also say that the oscillation modulates itself, yielding sidepeaks in the spectrum around $\omega$, which can couple into the mechanical resonance of the cantilever if $\omega$ is an integer fraction of the unperturbed resonance frequency.
– Ron
Aug 10 at 7:06