Stabilization and destabilization in non‐conservative gyroscopic systems

Abstract

Stability of a linear autonomous non-conservative system in presence of potential, gyroscopic, dissipative, and nonconservative positional forces is studied. The cases when the non-conservative system is close to a gyroscopic system or to a circulatory one, are examined. It is known that the 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. The present contribution shows that in both cases the boundary of the asymptotic stability domain of the perturbed system possesses singularities such as "Dihedral angle" and "Whitney umbrella" that govern stabilization and destabilization. Approximations of the stability boundary near the singularities and estimates of the critical gyroscopic and circulatory parameters are found in an analytic form. In case of two degrees of freedom these estimates are obtained in terms of the invariants of matrices of the system. As an example, the asymptotic stability domain of the modified Maxwell-Bloch equations is investigated with an application to the stability problems of gyroscopic systems with stationary and rotating damping, such as the Crandall gyropendulum, tippe top and Jellet's egg. An instability mechanism in a system with two degrees of freedom, originating after discretization of models of a rotating disc in frictional contact and possessing the spectral mesh in the plane 'frequency' versus 'angular velocity', is described in detail and its role in the disc brake squeal problem is discussed. Bifurcation of the domain of asymptotic stability, singularities on its boundary, and the critical movement of eigenvaluesConsider an autonomous non-conservative system described by a linear differential equation of second orderwhere dot denotes time differentiation, x ∈ R m , and real matrix K = K T corresponds to potential forces. Real matrices D = D T , G = −G T , and N = −N T are related to dissipative (damping), gyroscopic, and non-conservative positional (circulatory) forces with magnitudes controlled by scaling factors δ, Ω, and ν respectively. A circulatory system is obtained from (1) by neglecting velocity-dependent forceswhile a gyroscopic one has no damping and non-conservative positional forcesThe circulatory system (2) possesses a reversible symmetry while the gyroscopic system (3) possesses the Hamiltonian one, which is seen after transformation of equation (1) to the Cauchy formẏ = Ay. As a consequence, det(A − λI) = det(A + λI), so that the eigenvalues of systems (2) and (3) appear in pairs (−λ, λ). Hence, the equilibrium of the circulatory system as well as of the gyroscopic one is either unstable or all its eigenvalues lie on the imaginary axis of the complex plane implying marginal stability, if they are semi-simple. In the presence of all the four forces the Hamiltonian and reversible symmetries are broken and the marginal stability is generally destroyed. Instead, system (1) can be asymptotically stable if its characteristi...

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