Dear Riccardo

Please see the attachments sites

Regards

https://ocw.mit.edu/courses/sloan-school-of-management/15-988-system-dynamics-self-study-fall-1998-spring-1999/readings/oscillations.pdf

http://www1.spms.ntu.edu.sg/~ydchong/teaching/04_complex_oscillations.pdf

https://www.johndcook.com/blog/2013/03/04/negative-damping/

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3087393/

http://imsi.rs/ippa_web/21simpozijum/issues/34(1)/IPPA1998341_247_258.pdf

Swing equation for the given system would give you the effect of varying different parameters of system, and by plotting swing curves(stable, unstable(degree of negative damping) or critically stable).....

Let’s assume that m is positive. Otherwise multiply the equation above by -1. What happens if γ or k are negative?

The term γ, when positive, takes energy out of the system. A negative value of γ would be a term that adds energy, a sort of negative damping. The behavior of the solutions is determined by the eigenvalues of the system, that is the roots of the equation: m x2 + γ x + k = 0. If γ2 < 4mk then the eigenvalues are complex and so the solutions have an oscillating component. If γ2 = 4mk then there is one repeated, positive eigenvalue. But if γ2 > 4mk the system has one positive and one negative eigenvalue. The general solution is a linear combination of these two solutions. As time increases, only the exponentially increasing component of the solution matters because the effect of the other component goes to zero.

In my opinion, negative damping means that the poles of the system lies in the right hand side of S-plane. In this case the system can show growing oscillations.

Practically, any amplifying system with positive feed back and timing circuits can show oscillations. It is then has negative damping .

Best wishes