Unstable transient response of gyroscopic systems with stable eigenvalues

Gyroscopic conservative dynamical systems may exhibit flutter instability that leads to a pair of complex conjugate eigenvalues, one of which has a positive real part and thus leads to a divergent free response of the system. When dealing with non-conservative systems, the pitch fork bifurcation shifts toward the negative real part of the root locus, presenting a pair of eigenvalues with equal imaginary parts, while the real parts may or may not be negative. Several works study the stability of these systems for relevant engineering applications such as the flutter in airplane wings or suspended bridges, brake squeal, etc. and a common approach to detect the stability is the complex eigenvalue analysis that considers systems with all negative real part eigenvalues as stable systems. This paper studies analytically and numerically the cases where the free response of these systems exhibits a transient divergent time history even if all the eigenvalues have negative real part thus usually considered as stable, and relates such a behaviour to the non orthogonality of the eigenvectors. Finally, a numerical method to evaluate the presence of such instability is proposed.