Bifurcations and chaos in a periodic time-varying temperature-excited bimetallic shallow shell of revolution

Abstract

The chaotic vibrations of a bimetallic shallow shell of revolution under time-varying temperature excitation are investigated in the present study. The governing equations are established in forms similar to those of classical single-layered shell theory by re-determination of reference surface. The nonlinear differential equation in time-mode is derived by variational method following an assumed spatial-mode. The Melnikov function is established theoretically to estimate regions of the chaos, and the Poincaré map, phase portrait, Lyapunov exponent, and Lyapunov dimension are used to determine if a chaotic motion really appears. Further investigations are developed by means of detailed numerical simulation, and both the bifurcation diagrams and corresponding maximum Lyapunov exponent are illustrated. The influence of static and time-dependent temperature parameters, height parameter of the shell, and damping parameter on the dynamic characteristics is examined. Interesting phenomena such as the onset of chaos, transient chaotic motion, chaos with interior crisis and period window, period-doubling scenario and reversed period-doubling bifurcation leading to chaos, jump phenomena, and chaos suddenly converting to period orbit have been observed from these figures.

Multi-pulse jumping orbits and homoclinic trees in motion of a simply supported rectangular metallic plate

“…For I = I r ≡ 2 3 μ, (16) has periodic orbits in (I, φ) plane. I = I r corresponds to the circle of fixed points.…”

“…Several global bifurcation methods may be used to detect chaos in systems that possess homoclinic or heteroclinic orbits. One method, due to Melnikov, provides conditions under which a homoclinic orbit in the unperturbed system may break under perturbation, allowing the stable and unstable manifolds of the saddle node to intersect transversally which may lead to chaos in the systems [14][15][16]. A second method, developed by Kovačič and Wiggins [17], determines conditions under which a Silnikov-type homoclinic orbit may be present in a perturbed resonant system.…”

Threshold for Chaos of a Duffing Oscillator with Fractional‐Order Derivative

“…e Melnikov method has certain advantages when it is used to analyze the chaotic motion threshold value of multiparameter nonlinear systems. For getting threshold values of the chaotic motion, many scholars have used the analytic Melnikov method to study the necessary condition for chaos of Duffing oscillators without a fractional-order derivative [31][32][33][34][35]. However, due to the complexity of the fractional-order derivative, it is rare to discuss its effect on the chaotic threshold of nonlinear fractional-order systems.…”

Application of the Lyapunov Exponents and Wavelets to Study and Control of Plates and Shells

“…Chaotic dynamics of structural members has been investigated by many researchers [1][2][3][4][5][6][7][8][9][10]. In this work we propose a novel approach to study non-linear vibrations of a plate based on the neural network approach and we analyze dynamics of flexible shells with constant stiffness and density subjected to harmonic load action.…”