Analysis of nonlinear chaotic vibrations of shallow shells of revolution by using the wavelet transform

In the present paper, we study complex vibrations of flexible axisymmetric shallow shells under the action of transverse sing-alternating pressure. In addition to the traditional methods of nonlinear dynamics, we for the first time use the wavelet transform to analyze the transition from harmonic to chaotic vibrations. We analyze the use of Gauss-type wavelets (the order of derivatives varies from m = 1 to m = 8) and also the use of the Morlet wavelet (both real and complex). We conclude that the use of the complex Morlet wavelet is preferable to that of the Gaussian wavelets: the more zero moments a wavelet has, the better it describes the complex vibrations of flexible shallow shells.     

The sound field in a hydroacoustic waveguide with an uneven hard bottom

An analytic representation is constructed for a nonaxially symmetric sound field to simulate a hydroacoustic waveguide the bottom of which is hard and has an axially symmetric relief. A numerical analytic method for finding the velocity potential is proposed, for which undetermined coefficients for normal modes are determined from a corresponding infinite system of algebraic equations. The sound fields are studied with for variations of the problem parameters.     

Complex fluctuations of flexible plates under longitudinal loads with account for white noise

This paper describes the influence of intensity of external additive white noise on the nonlinear dynamics of rectangular plates as mechanical systems with an infinite number of degrees of freedom. A new scenario is discovered, which is a combination of the classic Feigenbaum and Pomeau–Manneville scenarios. The classical methods of nonlinear dynamics and wavelet transforms were used to reveal the peculiarities of a modified scenario. The noise-induced transitions are investigated, and it is shown that the noise exposure is accompanied with the transition to chaotic fluctuations with a lower amplitude of the driving load. It is determined that the presence of external fluctuations does not affect the scenario of transition from harmonic to chaotic fluctuations.     

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

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, ω_p}, where q₀ denotes the loading amplitude, and ω_p 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.