MATHEMATICAL MODELING OF NONLINEAR VIBRATIONS OF A PLATE WITH EXPOSURE TO COLOR NOISE TAKING INTO ACCOUNT OF CONTACT INTERACTION WITH THE BEAM

Abstract

A theory of contact interaction of a plate locally supported by a beam, under the influence of external lateral load and external additive color noise (pink, red, white) was constructed. Also described design is in a stationary temperature field. For the plate, the Kirchhoff kinematic model was adopted; for the beam, Euler - Bernoulli, the physical nonlinearity is taken into account according to the theory of small elastic-plastic deformations. The temperature field is taken into account according to the Duhamel - Neumann theory, and there are no restrictions on the temperature distribution over the plate thickness and the height of the beam. The temperature field is determined from the solution of the three-dimensional (plate) and two-dimensional (beam) heat conduction equations. The theory of B.Ya. Cantor. The heat conduction equations are solved by the finite difference method of the second and fourth order of accuracy. The system of differential equations is reduced to the Cauchy problem by the Bubnov - Galerkin methods in higher approximations and finite differences in spatial variables. Next, the Cauchy problem is solved by the fourth-order Runge - Kutta method and the Newmark method. At each time step, the iterative procedure of I. Birger was applied. The results of a numerical experiment are given. To analyze the results, the methods of nonlinear dynamics were used (construction of signals, phase portraits, Poincare sections, Fourier power spectra and Morlet wavelet spectra, analysis of the sign of Lyapunov indices by three methods: Wolf, Kantz, Rosenstein). The effect of color noise on the contact interaction between the plate and the beam has been studied. It has been established that red additive noise has the most significant effect on the oscillation pattern of the lamellar-beam structure compared to pink and white noise.