On the separation of fast and slow motions in mechanical systems with high-frequency modulation of the dissipation coefficient

Dynamics of a model mechanical system with ‘fast and strong’ oscillations of the damping coefficient has been analyzed by Fidlin (2005) [6]. He has performed the asymptotic analysis of the equation of motion of this system to conclude that these oscillations produce variation in its effective stiffness. The present paper continues analysis of dynamics of the system in the regimes of motion treated by Fidlin as well as in those left out in his asymptotic solution. The results are compared with the results of the solution of the classical Mathieu equation, which features fast oscillations in the stiffness of a system. The influence of stiffness and damping modulations on the stability of motion of corresponding oscillators is studied. Several engineering applications modeled by the system with oscillations of the damping coefficient are introduced. Analysis of motion of this system exemplifies how the method of direct separation of motions (Blekhman (2000) [7]) can be applied for solving equations with fast oscillating terms depending on the velocities. Some features of the application of the method of direct separation of motions in this case and in the similar ones are highlighted.