The contact interaction of two Timoshenko beams

A mathematical model of contact interaction of two geometrically nonlinear S.P. Timoshenko beams is obtained. The transverse alternating load acts on one of the beams. The infinitedimensional problem is reduced to a finite-dimensional one by using the second-order finite difference method. The Cauchy problem obtained is solved by the Runge – Kutta method of 4th order. The contact pressure is determined by B.Ya. Kantor’s method. The analysis of the results is carried out by methods of nonlinear dynamics and qualitative theory of differential equations. The scenario of transition of vibrations of the structure from harmonic to chaotic ones is studied. It is found that the shape of the beams’ vibrations becomes asymmetric in the event of the first bifurcation, but at the same time, there comes a phase synchronization of chaotic vibrations. Maps of the dynamic behavior for both beams are constructed.