# Impulse response out of a modal model, how is it gotten?

For example, I'm reading this paper:

Modal Synthesis for Vibrating Objects
Kees van den Doel and Dinesh K. Pai

I am confused about the theory source for $$y(t)$$ at page 2:

The impulse response $$y(t)$$ of $$M$$ at location $$k$$ is given by

$$y(t)=\sum_{n=1}^N a_{nk} \exp(-d_n t)\sin(2\pi f_n t)$$

Anyone know where the derivation for this could be found?

More particularly, how does one end up to that from a 3D linear elastic dynamics model used in:

That's the impulse response of a generic damped resonant filter. Derivations abound. If you understand continuous-time frequency-domain analysis, consider a filter with transfer function $$H(s) = \frac{b_1 s + b_0}{s^2 + 2 \zeta \omega_o s + \omega_o^2},$$ and solve for the impulse response. Basically, that one's common enough that you'll find it in a table of Laplace transform pairs, in any text on frequency-domain analysis and splattered all over the web.
• And I think I was asking a bit broader thing, particularly the way this arises from a 3D linear elastic dynamics model $Mx''+Cx'+Kx=f$. obj.umiacs.umd.edu/gamma-umd-website-imgs/pdfs/… Jun 8, 2022 at 16:19