Chaotic dynamics of flexible Euler-Bernoulli beams

Journal Article · Sun Dec 15 00:00:00 EST 2013 · Chaos (Woodbury, N. Y.)

DOI:https://doi.org/10.1063/1.4838955· OSTI ID:22251760

Awrejcewicz, J., E-mail: [e-mail] ^[1]; Krysko, A. V., E-mail: [e-mail] ^[2]; Kutepov, I. E., E-mail: [e-mail]; Zagniboroda, N. A., E-mail: [e-mail]; Dobriyan, V., E-mail: [e-mail]; Krysko, V. A., E-mail: [e-mail] ^[3]

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1. Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski St., 90-924 Lodz, Poland and Department of Vehicles, Warsaw University of Technology, 84 Narbutta St., 02-524 Warsaw (Poland)
2. Department of Applied Mathematics and Systems Analysis, Saratov State Technical University, Politehnicheskaya 77, 410054 Saratov (Russian Federation)
3. Department of Mathematics and Modeling, Saratov State Technical University, Politehnicheskaya 77, 410054 Saratov (Russian Federation)

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{sup 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{sub 0} and frequency ω{sub 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.

• OSTI ID: 22251760
• Journal Information: Chaos (Woodbury, N. Y.), Vol. 23, Issue 4; Other Information: (c) 2013 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); ISSN 1054-1500
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGORITHMS
AMPLITUDES
APPROXIMATIONS
ASYMMETRY
ATTRACTORS
BEAMS
BOUNDARY CONDITIONS
BOUNDARY-VALUE PROBLEMS
CAUCHY PROBLEM
CHAOS THEORY
FINITE DIFFERENCE METHOD
FINITE ELEMENT METHOD
LYAPUNOV METHOD
MATHEMATICAL MODELS
PARTIAL DIFFERENTIAL EQUATIONS
PERIODICITY
RUNGE-KUTTA METHOD