Routes to chaos in continuous mechanical systems. Part 1: Mathematical models and solution methods

Abstract

In this work chaotic dynamics of continuous mechanical systems such as flexible plates and shallow shells is studied. Namely, a wide class of the mentioned objects is analyzed including flexible plates and cylinder-like panels of infinite length, rectangular spherical and cylindrical shells, closed cylindrical shells, axially symmetric plates, as well as spherical and conical shells. The considered problems are solved by the Bubnov–Galerkin and higher approximation Ritz methods. Convergence and validation of those methods are studied. The Cauchy problems are solved mainly by the fourth Runge-Kutta method, although all variants of the Runge-Kutta methods are considered. New scenarios of transition from regular to chaotic orbits are detected, analyzed and discussed.First part of the paper is devoted to the validation of results obtained. This is why the same infinite length problem is reduced to that of a finite dimension through the FDM (Finite Difference Method) with the approximation order of O(c2), BGM (Bubnov–Galerkin Method) or RM (Ritz Method) with higher approximations. We pay attention not only to convergence of the mentioned methods regarding the number of partitions of the interval [0, 1] in the FDM or regarding the number of terms in the series applied either in the BGM or RM methods, but we also compare the results obtained via the mentioned different approaches. Furthermore, a so called practical convergence of different Runge-Kutta type methods are tested starting from the second and ending with the eighth order.Second part of the work is devoted to a study of routes to chaos in the so far mentioned mechanical objects. For this purpose the so-called “dynamical charts” are constructed versus control parameters {q0, ωp}, where q0 denotes the loading amplitude, and ωp is the loading frequency. The charts are constructed through analyses of frequency power spectra and the largest Lyapunov exponent (LE). Analysis of the mentioned charts indicates clearly that different routes to chaos exist and allow us to control the objects being investigated. In some cases we detect the classical Feigenbaum scenario and we compute also the Feigenbaum constant. This scenario accompanied all problems which we studied. In addition, we detect and illustrate novel scenarios of transition from regularity into chaos including the Ruelle–Takens–Newhouse–Feigenbaum scenario, and the so called modified Pomeau–Manneville scenario.Third part of the paper is devoted to analysis of the Lyapunov exponents. Namely, while investigating evolutions of vibration regimes of a shell associated with an increase of excitation amplitude q0 phase transitions chaos–hyper chaos as well as chaos-hyper chaos–hyper–hyper chaos dynamics are illustrated and studied. Furthermore, for all investigated plates and shells the Sharkovskiy windows of periodicity are detected. In particular, a space-temporal chaos/turbulence is studied.