Direct Numerical Integral Method to Determine Periodic Solutions of Nonlinear Systems. Equations of Motion with Discontinuous Functions.

The direct numerical integral method (D.N.I.) which is a kind of shooting method, is presented for determining the periodic solutions of nonlinear systems with discontinuous characteristics. The present method is applied to a system with Coulomb's frictions, a system with bilinear stiffness and damping, and a system with preloaded compliance. By comparing the results of this method and the analytical method, it is found that this method yields very accurate results. In the system with Coulomb's frictions, many kinds of periodic solutions, i.e., slip, stick-slip and lockup solutions are obtained. In the other two systems, superharmonic vibrations, subharmonic vibrations and ultra-subharmonic vibrations are obtained by the D.N.I. Moreover, Liapunov numbers are calculated by applying the algorithm to integrate the corresponding variational equation in the D.N.I.