Stability of Solutions to Damped Equations with Negative Stiffness

TL;DR

This paper investigates the stability of mass-spring systems with damping and negative stiffness, showing that variable coefficients can ensure stability despite instability with fixed coefficients, and extends results to nonlinear systems.

Contribution

It provides new conditions on variable coefficients that guarantee stability and challenges the belief that slow eigenvalue changes imply instability.

Findings

Variable coefficients can stabilize systems with negative stiffness.

Slow eigenvalue variation does not necessarily lead to instability.

Results extend to certain nonlinear systems.

Abstract

This article concerns the stability of a model for mass-spring systems with positive damping and negative stiness. It is well known that when the coefficients are frozen in time the system is unstable. Here we find conditions on the variable cofficients to prove stability. In particular, we disprove the believe that if the eigenvalues of the system change slowly in time the system remains unstable. We extend some of our results for nonlinear systems.