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I'm reading about modal models and related modelling.

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:

Jin, Xutong, et al. "Deep-modal: real-time impact sound synthesis for arbitrary shapes." Proceedings of the 28th ACM International Conference on Multimedia. 2020., p. 3.

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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.

Note that they may be a bit lazy in their notation -- they're assuming the sine wave is not phase shifted; I suspect that for a real system it is. However since they're talking about the sound that the bar makes, and because with sounds the phase of the components often doesn't matter, they're probably discounting this phase shift and they just expect you to do so as well.

The reason this shows up in their analysis is because it's a reasonably good fit to reality: bodies made from elastic material tend to exhibit a number of vibratory modes, the modes tend to be resonant, and they tend to be damped. The only real divergence between the sum that they're presenting is that -- in theory at least -- the number of modes in a thing made from elastic material is infinite.

In practice, though, the effect of the higher-order modes (i.e. modes that result in more complicated deformations of the body) tend deform less, they tend to have higher damping coefficients, and they tend to be at higher frequencies. All of these factors make them easier to ignore -- so for practical work, the number can be truncated.

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  • $\begingroup$ 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/… $\endgroup$
    – mavavilj
    Jun 8, 2022 at 16:19
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    $\begingroup$ I expanded my answer, but the knowledge base for that expansion was me hanging around in the cubicles of mechanical engineers while they did nodal analysis on structures to see if they'd break or -- more importantly -- muck up my control systems. This is not DSP. If your real question is "why do vibrating 3D structures have multiple modes, and how do I find them?", then this question needs to migrate to another group -- engineering.stackexchange.com would be a good candidate. $\endgroup$
    – TimWescott
    Jun 8, 2022 at 17:27

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