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?
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)