Self-excited and Forced Oscillations with Solid Friction: 3rd Report, Piecewise Linear Approximate Solutions in Case of Combined Friction-Velocity Characteristics

Frictional oscillations occurring in a forced self-excited system were treated when the kinetic friction force was shown as a combined characteristic curve of linear and hyperbolic functions of the relative sliding velocity. The authors studied the characteristics and stabilities of the oscillations in the first approximate solution for harmonic oscillation form (among the steady-state oscillation solutions which can be obtained by the averaging method). The study results of various properties of the solution for non-harmonic oscillation solution accompanying the occurrence of limit cycles have been reported previously. Highly accurate, steady-state oscillation solutions (with consideration given to the two motions of slipping and sticking) are obtained with a computer by approximating the characteristic curve of the frictional force and slipping velocity with n broken lines and connecting the successive solution curves on the phase plane in the velocity boundaries of each broken line. The piecewise linear approximation method for obtaining a highly accurate solution which is very similar to the exact solution is explained first. The occurrence forms and the distribution conditions of various steady-state oscillation solutions unobtainable by the averaging method, including the maximum static friction force becoming an isolated point, are described. The resonance characteristics of the system and the effect of the velocity of the moving surface on the characteristics of the steady-state oscillation solutions are reported.