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what is the physical interpretation of poles and zeros of a mechanical system? I'm asking things like the meaning if I have only 0 or only complex numbers. And about the number of the poles and zeros, it should be equal? What can I infer about the stability?

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  • $\begingroup$ Do you mean you want to look at the poles and zeros of a mechanical system and make judgements about how it will act? Or do you mean you want to map back from the poles (and possibly zeros) in a transfer function to the motors, gears, wheels, and whatnot in your mechanical system? $\endgroup$ – TimWescott Jan 2 '20 at 19:08
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The poles and zeros of a LTI plant or system in relation to their eigenvalues determine how quickly a system will stabilize or destabilize and if it oscillates.

When looking at these in the complex plane, the Re(a+ib) determine how quickly a system will exponentially stabilize or destabilize (the left vs right half plane) while the Im(a+ib) determine how the system oscillates. (High/low freq and amplitude)

The poles and zeros do not have to be equal, however imaginary systems always appear in pairs namely a+ib and a-ib.

A system is absolutely stable if all poles are at the left half plane =<0.

Other stability descriptions can be named depending on specific criteria such as conditional, marginal, or unstable or chaotic depending on where poles lie.

You can find a lot of infos on the web about this, such as this pdf from MIT. (warning pdf)

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