Chaotic dynamics of the contact interaction of two beams described of Euler-Bernoulli and Pelekh-Sheremetyev-Reddy hypotheses

Beam structures are widely used as the main elements of structures in aircraft and rocket engineering, mechanical engineering and instrument making. During operation, beams can experience external influences of various types, which leads to contact of the beams. In connection with this, the construction of a mathematical model of the contact interaction of beams is an actual problem. The aim of the work are to construct mathematical models based on the kinematic hypotheses of the first (Euler-Bernoulli) and the third (Pelekh-Sheremetyev-Reddy) approximations, the creation of methods for calculating the highly nonlinear (geometric and constructive nonlinearities) mechanical structures under the action of transversal harmonic loads. The resulting system of nonlinear partial differential equations in a dimensionless form by the method of finite differences of the second order of accuracy reduces to the Cauchy problem, which is solved by the Runge-Kutta method of the fourth order. The convergence of the obtained solutions is investigated depending on the intervals of the partitioning over the spatial and temporal coordinates. A rigid limitation was imposed on the coincidence of the basic functions in chaotic vibrations for n and 2n partitions of the interval of integration over the spatial coordinate. As an example, we consider the nonlinear dynamics of two beams, the gap between which is equal to unity. It is shown that the transition of the beam structure vibrations from harmonic to chaotic occurs through a subharmonic cascade of bifurcations. Chaotic phase synchronization based on the Morlet wavelet is investigated. The values of the highest Lyapunov exponent were calculated by the methods of Wolff, Kantz, and Rosenstein. (paper)