Sensitivity Analysis of Hamiltonian and Reversible Systems Prone to Dissipation-Induced Instabilities

• Published in: Matrix Methods: Theory, Algorithms And Applications pp. 31 - 68
• Main Author: Kirillov, Oleg N.
• Format: Book Chapter
• Language: English
• Published: WORLD SCIENTIFIC 01.04.2010
• Subjects: Algebra and Matrices; indefinite damping; reversible system; matrix polynomial; perturbation; Lyapunov stability; Hamiltonian system; multiple eigenvalue; singularity; dissipation-induced instabilities; destabilization paradox
• ISBN: 9812836020; 9789812836014; 9812836012; 9814469556; 9789812836021; 9789814469555
• DOI: 10.1142/9789812836021_0003

Stability of a linear autonomous non-conservative system in the presence of potential, gyroscopic, dissipative, and non-conservative positional forces is studied. The cases when the non-conservative system is close either to a gyroscopic system or to a circulatory one, are examined. It is known that marginal stability of gyroscopic and circulatory systems can be destroyed or improved up to asymptotic stability due to action of small non-conservative positional and velocity-dependent forces. We show that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as “Dihedral angle”, “Break of an edge” and “Whitney's umbrella” that govern stabilization and destabilization as well as are responsible for the imperfect merging of modes. Sensitivity analysis of the critical parameters is performed with the use of the perturbation theory for eigenvalues and eigenvectors of non-self-adjoint operators. In case of two degrees of freedom, stability boundary is found in terms of the invariants of matrices of the system. Bifurcation of the stability domain due to change of the structure of the damping matrix is described. As a mechanical example, the Hauger gyropendulum is analyzed in detail; an instability mechanism in a general mechanical system with two degrees of freedom, which originates after discretization of models of a rotating disc in frictional contact and possesses the spectral mesh in the plane ‘frequency’ versus ‘angular velocity’, is analytically described and its role in the excitation of vibrations in the squealing disc brake and in the singing wine glass is discussed.