Pervasive damping in mechanical systems and the role of gyroscopic forces

Abstract

AbstractIn this paper, we analyze the stability problem for autonomous non‐conservative mechanical systems in the presence of potential, gyroscopic, and dissipative forces. It is assumed that the dissipation is pervasive, i.e. the matrix of dissipative forces is semi‐positive definite. With this requirement, the celebrated Kelvin–Chetaev theorems cannot be applied. Considering this circumstance, we discuss the role of gyroscopic forces and their contribution to the overall phenomenon. This influence may be both positive and negative (there are some sets in space of parameters, where the asymptotic stability of the motion is broken). Moreover, the gyroscopic stabilization, which is neglected by complete dissipative force, may be saved with pervasive damping. We use a special procedure to prove the asymptotical (or marginal) stability. This procedure applied to considered case study is simpler than common algorithms based on the eigenvalues analysis (Routh–Hurwitz or Lienard–Chipart criteria, etc.). We apply this approach for a stability problem of two mechanical systems. The first of them is the problem of a passive stabilization of permanent rotations of Lagrange gyroscope. It is proved that adding a dashpot to a gyroscope with flattened inertia ellipsoid stabilizes permanent rotations of the gyro with the exception of some “critical” values. The last may be found analytically from special conditions. The second one is a rotary oscillating system with the external harmonic excitation. The absorber attached to the rotating frame stabilizes the periodic oscillations of the system. Also we have found the critical (resonant) value, which may be avoided by appropriate choice of the absorber's mass.