Abstract
In this work, chaotic dynamics of flexible spherical axially symmetric shallow shells subjected to sinusoidal transverse load is studied with emphasis put on the vibration modes. Chaos reliability is verified and validated by solving the implemented mathematical model by partial nonlinear equations governing the dynamics of flexible spherical shells and by estimating the signs of the largest Lyapunov exponents with the help of qualitatively different approaches. It is shown how the scenario of transition of the investigated shells from regular to chaotic vibrations depends on the boundary condition. The following cases are considered: (1) movable and fixed simple supports along the shell contours, taking into account shell stiffness (Feigenbaum scenario) and shell damping (Ruelle–Takens–Newhouse scenario), and (2) movable clamping (regular shell vibrations). The presence of dents, the location and character of which essentially depend on the shell geometric parameters, boundary conditions, and the external load parameters, is detected in some regions of the shell surface and discussed.
Highlights
Symmetric spherical shells, which are examples of thin-walled constructions, are widely applied in aviation and rocket industries, shipbuilding, automotive industries, energy-harvesting and chemical industries, fabrication of sensor devices, and bioengineering
The dynamics of flexible shells has a long history in mechanics, and only a selected part of it is considered in the present paper, putting emphasis on the novelty of the present work with respect to the findings in the available literature
The vibration forms as well as the occurrence of located dents exhibited by flexible axially symmetric shallow shells depending on the employed boundary condition are studied
Summary
Symmetric spherical shells, which are examples of thin-walled constructions, are widely applied in aviation and rocket industries, shipbuilding, automotive industries, energy-harvesting and chemical industries, fabrication of sensor devices, and bioengineering. The mentioned structural members are subjected to different external loads and boundary conditions, and it is important to study their nonlinear vibrations. No studies aimed at analyzing the modes of vibration of flexible circle axially symmetric shells subjected to uniformly disturbed harmonic loads have been found in the available, published literature. The vibration forms as well as the occurrence of located dents exhibited by flexible axially symmetric shallow shells depending on the employed boundary condition are studied. The finite difference method (FDM) is employed only to spatial coordinates It reduces the problem governed by PDEs (2.1) of an infinite dimension to the system of ODEs (2.8) with respect to time (the Cauchy problem). The counterpart difference forms of the boundary conditions are as follows: 1. Simple contour movable in the meridional direction
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