Chaotic dynamics of flexible Euler-Bernoulli beams

Abstract

Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c(2)) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions is carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q(0) and frequency ω(p) of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.

Chaotic vibrations of flexible shallow axially symmetric shells

“…The rigorous discussion of the advantages of the FDM can be found in numerous papers, including [40,41]. The shells considered in the present study Table 7 Fourier frequency power spectra, 2D wavelet spectra, It should be emphasized that in the paper [42], the authors studied the dynamics of geometrically nonlinear beams by means of employing both the FEM in the Galerkin form and the FDM.…”

Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads

“…Onozato et al [20] analyzed chaotic vibrations of a post-buckled L-shaped beam with an axial constraint. Chaotic dynamics of flexible Euler-Bernoulli beams has been studied by Awrejcewicz et al [21]. They analyzed time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor.…”

Clean numerical simulation: a new strategy to obtain reliable solutions of chaotic dynamic systems

“…For more examples of spurious numerical simulations, please refer to Yee et al [19][20] . What's more, analysis of chaotic dynamics were numerically studied in non-linear vibrations of spatial structures such as shells [21][22] , plates [23] and beams [24][25][26] . Qualitative theory of differential equations is widely used in study of chaotic dynamic systems.…”

“…Qualitative theory of differential equations is widely used in study of chaotic dynamic systems. Researchers [23][24][25][26][27][28][29][30][31] investigated chaotic dynamic systems by means of the Poincaré maps, Lyanponuv expoent, phase portraits, Fourier and wavelet power spectra, autocorrelation functions, etc. Therefore, it is necessary to verify reliability of these results for chaotic dynamic systems.…”