O.A. Afonin - Ph.D. (Phys.-Math.), Professor, Department of Mathematics and Modeling, Yuri Gagarin Saratov State Technical University
O.A. Saltykova - Ph.D. (Phys.-Math.), Associate Professor, Department of Mathematics and Modeling, Yuri Gagarin Saratov State Technical University; Senior Research Scientist, 3D-Modeling Laboratory ISHITR, Tomsk Polytechnic University
I.V. Papkova - Dr.Sc. (Phys.-Math.), Associate Professor, Department of Mathematics and Modeling, Yuri Gagarin Saratov State Technical University
A.V. Krysko - Dr.Sc. (Phys.-Math.), Professor, Department Applied Mathematics and Systems Analysis, Yuri Gagarin Saratov State Technical University; Engineer, 3D-modeling Laboratory ISHITR, Tomsk Polytechnic University
Micro and nano electromechanical systems (MEMS and NEMS) are widely used in economically viable industries in Russia, such as in-strument making, medicine, robotics, mechatronics, oil production and refining, aviation and space technology, nanoindustry, military equipment and armament, transport and civil engineering. The «heart» of nano and microelectromechanical sensors is a sensitive ele-ment (SE), on the quality of which the accuracy of the sensor itself depends. Features of the principle of operation and operating condi-tions MEMS and NEMS devices determine the requirements for them such systems must operate in a variety of adverse external influ-ences: dynamic, thermal, noise, vibration and their combinations. The object of the study is a structure consisting of two beams.
In the paper presented, the truth of the chaotic oscillations of a beam structure consisting of two beams is investigated. There is a gap between the beams. An external alternating load acts on one of the proteins. This issue has become relevant recently, and was first put by Professor Rene Lozi from the University of Nice (France). In connection with the fact that it is not possible to obtain analytical solutions for problems of nonlinear dynamics, the question arises of the truth of the solutions obtained. To answer this question, you should use alternative methods at each step of the simulation. That is, the problem is solved by different methods, including the convergence of these methods.
In constructing the mathematical model of the beam structure, various kinematic hypotheses were adopted: the first (the Euler-Bernoulli model), the second (the model of Timoshenko S.P.), the third approximation (the Pelekh-Sheremetyev model). An important issue is the study of the nonlinear dynamics of the contact interaction of two beams described by kinematic models of different approximations. This issue is devoted to this work. Three problems are considered: problem 1 - both beams are described by the kinematic hypothesis of the first approximation; Problem 2 the beam under load is described by the second approximation model, and the second beam by the first approximation model; Problem 3 the loaded beam obeys the third approximation hypothesis, the second beam is the first approximation model.
The equations of motion of the beams, taking into account the hypotheses of the first, second, third order, and also the boundary and initial conditions are obtained from the Hamilton-Ostrogradsky energy principle. The resulting systems of partial differential equations by the finite difference method reduce to a system of ordinary differential equations, the Cauchy problem obtained is solved by Runge-Kutta methods of various orders. The geometric nonlinearity is taken into account by the model of T. Von Karman, the contact interaction by the Winkler model. When solving contact problems, it is necessary to take into account the geometric non-linearity of objects, which brings a significant contribution to the solution, even in the case of small deflections.
In all the problems studied, in a geometrically nonlinear formulation, it was found that the oscillations of the beams occur at the same frequencies. The use of higher-order kinematic hypotheses (Timoshenko and Pelekh-Sheremetyev-Reddy), in the construction of a mathematical model of the contact interaction of two beams, leads to the appearance of new linearly dependent frequencies. It is shown that the use of only the Fourier spectrum for the analysis of frequency characteristics is not sufficient, because of the complexity of the oscillatory process. It is necessary to use additional tools, for example wavelet analysis. The values of the highest Lyapunov exponent are calculated using three different algorithms. The sign of the highest Lyapunov exponent is positive for all problems, which indicates chaotic oscillations of the beams according to Gulik's theory. The truth of the chaotic oscillations of a two-layer packet of beams is proved on the basis of the convergence of the results depending on the number of integration intervals in the method of finite differences, on the method of solving the Cauchy problem, on methods for analyzing problems of nonlinear dynamics, and on the method for calculating the highest Lyapunov exponents.
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