The Problem
I have the following damped mass-spring system in the form of a Linear Time-Varying (LTV) system:
$$\mathbf{M}(t)\mathbf{\ddot{x}}(t) + \mathbf{C\dot{x}}(t) + \mathbf{Kx}(t) = \mathbf{f}(t) \tag{1}$$
where $\mathbf{M}(t)$ is a diagonal time-dependent mass matrix with sinusoidal functions on its diagonal (can be easily decomposed using Fourier series), $\mathbf{C}$ and $\mathbf{K}$ are tridiagonal matrices and $\mathbf{f}(t)$ is the forcing vector.
I would like to find the Laplace transform of Eq.(1), however due to the time dependent term on the left hand side, I am unsure to do this.
My Attempt
What I would normaly do if $\mathbf{M}$ was not time dependent, is that I would easily take the Laplace function to find the transfer function:
\begin{gather}
[\mathbf{M}s^2 + \mathbf{C}s + \mathbf{K}]\mathbf{x}(s) = \mathbf{f}(s) \\
\Rightarrow [-\mathbf{M}\omega^2 + \mathbf{C}j\omega + \mathbf{K}]\mathbf{x}(j\omega) = \mathbf{f}(j\omega)
\end{gather}
What I have done in this case is I have simply taken the average of $\mathbf{M}$ and performed the same as above, however as advised by the literature on the specific physical problem that I am tackling this approximation results to innacurate results.
Therefore, how would I be able to find the Laplace transform of Eq.(1), or at least a method to approximate it better?
Best Answer
while there is no general formula for Laplace transform of a product of two functions, if the functions in ${\bf M}$ are only comprised of sine and cosine, you can use the fact that $$ L[\sin(\omega t) f(t)](s) = -i\frac{L[f](s-i\omega) - L[f](s+i\omega)}{2}$$ and for cosine $$ L[\cos(\omega t) f(t)](s) = \frac{L[f](s-i\omega) + L[f](s+i\omega)}{2}$$