The stability and stabilization of the motion of non-conservative mechanical systems

Paradoxes of dissipation‐induced destabilization or who opened Whitney's umbrella?

“…We mention interesting connections of this effect to structured pseudospectra [44] and to eigenvalue optimization problems [20]. We did not even consider the effect of nonlinearites, see however [1,15,29]. We mention the closely related effect of discontinuous change of the critical flutter frequency due to small dissipation [14,51] and its connection to the Whitney umbrella singularity at the exceptional points on the eigenvalue surfaces [96].…”

“…www.zamm-journal.org and the vector u = (u(0), ∂ x u(0), ∂ 2 x u(0), ∂ 3 x u(0), u (1), ∂ x u(1), ∂ 2 x u(1), ∂ 3 x u(1)) T . Some authors considered different boundary conditions that depend both on the physical parameters and on the spectral parameter [83,116] The undamped Beck's column is stable for q < q 0 20.05 [21].…”

Asymptotical stability of the motion of mechanical systems with partial energy dissipation

“…Rather complete observations of references on this topic are given in [15][16][17][18]. Most of these studies considered the classical case of full energy dissipation when Rayleigh function R is positive definite on all generalized velocities.…”

A note on the complete instability of linear non-conservative undamped systems

“…For many years, it has been well known that circulatory forces − can destabilize a stable equilibrium of purely potential (conservative) system, and that they can stabilize an unstable potential system [2,4]. Various results concerning the stability problem for circulatory systems can be found in [2][3][4][5][6][7][8][9][10][11][12].…”