Dynamics of a nonlinear multilayer beam structure on elastic foundation: Chaos detection and application to transport engineering technologies

Abstract

The dynamics of a composite consisting of the nonlinear multilayer beam structure, interacting through elastic intermediate layers, under mobile point loading is investigated. This study finds a direct application in transport engineering technologies, more precisely in railways, where the moving point load is the train, and the multilayer beam, the rails interacting with the ballast, the foundation and base layers. From the Lagrange formulations, the system of damping partial differential equations of the model is found, and by considering the non-dissipative case with weak nonlinearity and constant charge they are used to find the eigen modes and the natural vibration frequencies of the system. Then the dissipative case with nonlinearity is studied, with a particular attention carried on the temporal part, which is reduced to a system of coupled nonlinear differential equations, where the first line is forced. This system of equation is used to determine the equilibrium points, after which they are subsequently solved analytically through the multiple time scale method for harmonic resonance case, showing the formation of hysteresis more and more complex as the number of cells increases. The coupled nonlinear equations of the system is next solved numerically, with the transition of the system towards chaos analyzed through the bifurcation diagram and the maximum Lyapunov exponent, which show strong sensitivity to the coupling parameter λ2 as well as the system frequency. The results show for N = 2 and for some parameters the periodic behavior and the crisis for ω = 0.5. When the frequency is low; that is ω = 0.05 the chaotic band is considerably reduced, chaos appearing around the nonlinearity parameter γ2 = 0.5 and also for γ2 > 0.85. The time trace shows chaotic pulses and bursting type behavior, for some choices of the coupling parameter. The synchronization curves are also plotted and it is shown that q2 doesn’t synchronizes with q1 for some frequencies, while for others parameters, they synchronize, but fairly. For N = 3, the dynamics is more complex and the time traces plots show regular impulse for ω = 0.5 and bursting for weak frequency, ω = 0.05.