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Dissipative dynamics of geometrically nonlinear Bernoulli−Euler beam under the action of transverse alternating load with white noise added


N.P. Erofeev − Chief of Educational Computing Laboratory, Physics-Technical Faculty, Saratov State Technical University n.a. N.G. Chernyshevskii. Е-mail: V.M. Zakharov − Post-graduate Student, Department of Mathematics and Modeling Saratov State Technical University n.a. N.G. Chernyshevskii. Е-mail: E.Yu. Krylova − Assistant, Department of Mathematical and Computer Modeling, Saratov State Technical University n.a. N.G. Chernyshevskii. Е-mail: V.A. Krysko − Dr.Sc. (Eng.), Head of Department of Mathematics and Modeling, Saratov State Technical University n.a. N.G. Chernyshevskii. Е-mail: I.V. Papkova − Ph.D. (Phys.-Math.), Associate Professor, Department of Mathematics and Modeling, Saratov State Technical University n.a. N.G. Chernyshevskii. Е-mail:

The work is devoted to the study of noise-induced transitions of flexible Euler-Bernoulli beams under the transverse alternating load. Flexible beams are composite structural elements in air, space, ship, instrument, which have to work in environments with time-varying properties (random impact effects, sound pressure). Therefore there is a need for a comprehensive study of such systems behavior affected by external noise and the ascertainment a set of parameters of influence that would characterize a safe and unsafe modes. It\'s constructed a mathematical model of flexible beam vibrations under alternating transverse load with an additive external noise added to the system in form of random variable with constant intensity. Beam material assumed to be isotropic and homoge-neous. Constructed mathematical model of the beam based on the hypothesis of Euler-Bernoulli, and considered a first approxima-tion, but is accurate enough to be able to analyze it. The geometric nonlinearity is taken as the form of T. Karman. System of nonlin-ear differential equations with initial and boundary conditions leads to a system of ordinary differential equations for space coordinate by finite difference method (FDM) with an error O(h2). For time coordinate the differential equations are solved by the Runge - Kutta fourth-order accuracy. Using the FDM a beam can be considered as system with a potentially infinite number of degrees of freedom. To investigate the influence of external noise on the complex oscillations of considered mechanical system it was designed a software package allowing mapping the nature of oscillations as a function of the control parameters. The algorithm allows us to detect on maps the areas for harmonic oscillations of doubling bifurcations period, areas of beam vibrations at independent frequencies and ar-eas of chaos. Establishing the nature of the oscillations is carried out on the basis of signal analysis, modal and phase portraits, Poincaré sections, autocorrelation functions, the analysis signs of Lyapunov exponents, Fourier analysis, wavelet analysis. The overall maps analysis shows that the presence of noise significantly reduces the area of harmonic oscillations and variations of white noise intensity have a little effect on the change in areas of harmonic oscillations. The research was aimed at exploring the possibility to control the state of geometrically nonlinear beams by adding to the load chaotic component in the form of white noise.


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