Dissipation‐induced subcritical flutter in the acoustics of friction

Abstract

We consider a gyroscopic system under the action of small dissipative and non-conservative positional forces, which has its origin in the models of rotating elastic bodies of revolution in frictional contact such as the singing wine glass or the squealing disc/drum brakes. The spectrum of the unperturbed gyroscopic system forms a spectral mesh in the plane 'frequency versus gyroscopic parameter' with double semi-simple purely imaginary eigenvalues at the nodes. In the subcritical range of the gyroscopic parameter the eigenvalues involved into the crossings have the same Krein signature and thus their splitting due to changes in the stiffness matrix, which break the rotational symmetry of the body, cannot produce complex eigenvalues and, therefore, flutter. We establish that perturbation of the gyroscopic system by the dissipative forces with the indefinite matrix can lead to the subcritical flutter instability even if the rotational symmetry is destroyed. With the use of the perturbation theory of multiple eigenvalues we explicitly find the linear approximation to the domain of the subcritical flutter, which turns out to have a conical shape. The orientation of the cone in the three dimensional space of the parameters, corresponding to gyroscopic, damping, and potential forces, is determined by the sign of an explicit expression involving the entries of both the damping and potential matrices. With the use of a time-dependent coordinate transformation we demonstrate that the conical zones of flutter for the original autonomous system coincide with the zones of the subcritical parametric resonance of the rotationally symmetric flexible body with the load moving in the circumferential direction. Conical zones of the subcritical flutter instability induced by the indefinite dampingWe consider a linear autonomous non-conservative gyroscopic system with dissipationwhere Ω is the gyroscopic parameter, G = diag(J, 2J, . . . , nJ) = −G T and P = diag(ω 2 1 I, ω 2 2 I, . . . , ω 2 n I) = P T are the matrices of gyroscopic and potential forces withand the dissipative, conservative, and non-conservative perturbations with the real matrices D = D T , K = K T , and N = −N T are controlled by the parameters δ, κ, and ν, respectively. The transformation x = Az := exp(−ΩGt)z yields an equivalent to (1) potential system with the periodic perturbation, seeFor n = 2 the matrix N(t) := A −1 NA = N and the periodic stiffness and damping matrices areEquation (1) frequently originates after linearization and discretization of continuous models of rotating flexible bodies of revolution in frictional contact and describes their small oscillations in the stationary frame, whereas (3) describes them in the rotating frame [2][3][4]. Due to the rotational symmetry the eigenvalues ω 2 s , s = 1, 2, . . . , n, of the matrix P are double semi-simple. The distribution of the doublets as a function of s is usually different for various bodies of revolution. For example, ω s = s corresponds to the spectrum of free vibrations of a circular string...