E.Y. Krylova - Ph.D. (Phys.-Math.), Associate Professor, National Research Saratov State University n.a. N.G. Chernyshevsky E-mail: kat.krylova@bk.ru I.E. Kutepov - Ph.D. (Phys.-Math.), Yuri Gagarin Saratov State Technical University E-mail: ilyakutepov@yandex.ru I.V. Papkova - Ph.D. (Phys.-Math.), Associate Professor, Yuri Gagarin Saratov State Technical University E-mail: ikravzova@mail.ru V.A. Krysko - Dr.Sc. (Eng.), Professor, Head of the Department, Yuri Gagarin Saratov State Technical University E-mail: tak@san.ru

In this article we constructed a mathematical model of sloping curvilinear beams with taking into account the connection of fields of temperature and deformation, geometric and structural nonlinearity, involving the influence of external additive white noise. The eq-uations of the motion, boundary and initial conditions for each beam package were got from the energy principle of Ostrogradsky-Hamilton. A mathematical model is a system of differential equations of the hyperbolo-parabolic type. There are no restrictions in the distribution of temperature field along the thickness of the beams of the package, that is, the inhomogeneous two- dimensional heat conduction equation is considered, taking into account the presence of internal heat sources. Boundary conditions of I, II or III kinds can be added to the heat conduction equation. The equations of a motion of a beam element are written on the basis of the hypothesis of Euler-Bernoulli. The conduct interaction is described by Winkler relationship between the compression and the contact pressure. We developed a number of programs allowing to investigate the nature of vibrations of the considered systems by the methods of nonlinear dynamics. With its help such characteristics as deflection, displacement, two-dimensional and three- dimensional phase portraits, showing and Poincare section, a signal, Fourier spectrum, the wavelet spectrum, the highest Lyapunov exponents are built. The desired solution of the differential equations is performed numerically. The solution algorithm is built on the combination of two methods - the final differences method (in spatial coordinates) of the 2nd order accuracy and Runge-Kutta method (in time) of the 4th order. The accuracy of the results is provided by the study of convergence of the solution by the method of Runge depending on the number of breaking points for the spatial coordinates and a time step. At each time step for each grid node on the spatial coordinate the temperature moments and voltages are calculated by the Simpson method. The influence of the stationary temperature field on the non-linear dynamics of the beam package of a zero curvature with the hinge fixing of one end and a rigid closing of another one in the case of the absence of external fluctuations is considered. The initial conditions are taken as zero, the boundary conditions of the first kind are used for the heat equation. We have studied the nonlinear dynamics of the two-layer package and analyze the contact interaction of the beam layers. It is found that in the contact interaction of the beam layers the chaotic mode of vibrations occurs.

 

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