The Bubnov-Galerkin method is applied to reduce partial differential equations governing the dynamics of flexible plates and shells to a discrete system with finite degrees of freedom. Chaotic behaviour of systems with various degrees of freedom is analysed. It is shown that the attractor dimension of a system has no relationship with the attractor dimension of any of its subsystems. Multibody dynamical systemsChaotic vibrations exhibited by lumped systems with many degrees of freedom (DOF) are rarely investigated. Recently, however remarkable progress has been observed: hydrodynamic processes governed by ordinary differential equations (ODE) have been investigated within this field in [1-3]; finite-dimensional models of Ginzburg-Landau equations discretized with respect to spatial coordinates, multidimensional models of radiophysical systems governing the dynamics of coupled oscillators and generators as well as chains of oscillators and generators have been analysed in [4] and [5][6][7]. Most of the cited works addressed the problem of modeling a continuous system by a lumped (discrete) system governed by ODEs.There is a wide gap however between finite-and infinite-dimensional models exhibiting chaotic dynamics: it begins at one and a half DOF and ends with systems governed by partial differential equations (PDE), e.g. of the Navier-Stokes type. In the DOF interval ½1:5; þ1, of various systems, our attention is focused on structures governed by the von Kármán equation, which also belongs to the class of equations with these properties. This equation plays an important role in both mathematics and mechanics, [8][9][10][11][12].In order to follow qualitative behaviour of various PDEs governing dynamics of continuous systems, a concept of phase space is used and infinite-dimensional systems are replaced by finite-dimensional ones governed by ODEs, [12][13][14].However, two important questions appear which require clarification in order to address properly the stated problem.First, infinite-dimensional systems of equations are truncated in the procedure, i. e. finitedimensional systems are considered. It is arbitrarily assumed that an increase of the number of equations leads (beginning from a certain value) to stabilization of the system properties, and,

No abstract

No abstract

Forced oscillations of flexible plates with a longitudinal, time dependent load acting on one plate side are investigated. Regular (harmonic, subharmonic and quasi-periodic) and irregular (chaotic) oscillations appear depending on the system parameters as well as initial and boundary conditions. In order to achieve highly reliable results, an effective algorithm has been applied to convert a problem of finding solutions to the hybrid type partial differential equations (the so-called von Kármán form) to that of the ordinary differential equations (ODEs) and algebraic equations (AEs). The obtained equations are solved using finite difference method with the approximations 0(h4) and 0(h2) (in respect to the spatial coordinates). The ODEs are solved using the Runge–Kutta fourth order method, whereas the AEs are solved using either the Gauss or relaxation methods. The analysis and identification of spatio-temporal oscillations are carried out by investigation of the series wij(t), wt,ij(t), phase portraits wt,ij (wij) and wtt,ij(wt,ij, wij) and the mode portraits in the planes wx,ij(wij), wy,ij (wij) and in the space wxx(wx,ij,wij), FFT as well as the Poincaré sections and pseudo-sections.

The dependence of the quality factor of nonlinear microbeam resonators under thermoelastic damping for Timoshenko beams with regard to geometric nonlinearity has been studied. The constructed mathematical model is based on the modified cou-Russian Federation 634050 ple stress theory which implies prediction of sizedependent effects in microbeam resonators. The Hamilton principle has yielded coupled nonlinear thermoelastic PDEs governing dynamics of the Timoshenko microbeams for both plane stresses and plane deformations. Nonlinear thermoelastic vibrations are studied analytically and numerically and quality factors of the resonators versus geometric and material microbeam properties are estimated. Results are presented for gold microbeams for different ambient temperatures and different beam thicknesses, and they are compared with results yielded by the classical theory of elasticity in linear/nonlinear cases.

Regular and chaotic dynamics of the flexible Timoshenko-type beams is studied using both the standard Fourier (FFT) and the continuous wavelet transform methods. The governing equations of motion for geometrically nonlinear Timoshenko-type beams are reduced to a system of ODEs using both finite element method (FEM) and finite difference method (FDM) to ensure the reliability of numerical results. Scenarios of transition from regular to chaotic vibrations and beam dynamical stability loss are analyzed. Advantages and disadvantages of various wavelet functions are discussed. Application of continuous wavelet transform to the investigation of transitional and chaotic phenomena in nonlinear dynamics is illustrated and discussed.

No abstract

No abstract