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Abstract
This paper presents a study on nonlinear asymmetric vibrations in shallow spherical caps under pressure loading. The Novozhilov’s nonlinear shell theory is used for modeling the structural strains. A reduced-order model is developed through the Rayleigh–Ritz method and Lagrange equations. The equations of motion are numerically integrated using an implicit solver. The bifurcation scenario is addressed by varying the external excitation frequency. The occurrence of asymmetric vibrations related to quasiperiodic and chaotic motion is shown through the analysis of time histories, spectra, Poincaré maps, and phase planes.
Highlights
IntroductionThin-walled structures like plates, panels, shells, and caps are important structural elements in Engineering; their applications can be found in Civil Engineering (roofs, vaults, tensile structures), Aerospace (airplanes, missiles and rockets); Mechanics (membrane based microsensors and energy harvesters)
Thin-walled structures like plates, panels, shells, and caps are important structural elements in Engineering; their applications can be found in Civil Engineering, Aerospace; Mechanics.These structures are strong and lightweight at the same time, but they are extremely sensitive to perturbations, present a complicated instability behaviors and are very difficult to model
In order to develop a reduced-order model (ROM) for studying the cap nonlinear dynamics, the eigenfunctions of the linearized operator are obtained through the Rayleigh-Ritz approach [36]
Summary
Thin-walled structures like plates, panels, shells, and caps are important structural elements in Engineering; their applications can be found in Civil Engineering (roofs, vaults, tensile structures), Aerospace (airplanes, missiles and rockets); Mechanics (membrane based microsensors and energy harvesters). Huang [2] and Weinitsche [3] used Margurre’s theory with possibility of having nonsymmetric buckling They showed how, for deeper caps, the wavelength of the buckling modes was higher compared to shallow caps, and numerical results agreed with the experimental ones available at that time. The asymmetric dynamic buckling of shallow spherical caps was investigated even by Akkas [16], who showed that the asymmetric buckling under step pressure load results in cusps in phase-plane diagrams. Using the multiple-scale perturbation method, results showed that, having integer or quasi-integer ratio between natural frequencies is not a sufficient condition for having internal resonances activation This is due to the body symmetry, which leads to the canceling of some nonlinear coefficient in the ODEs. Experiments were carried out by forcing the specimen using an electromagnetic coil. The superimposition of a static and a dynamic pressure yields to non-periodic and chaotic oscillation related to the activation of asymmetric modes
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