Nonlinear vibrations of doubly-curved cross-ply shallow shells

Abstract

The objective of this work is to study the local and global nonlinear vibrations of isotropic single-layered and multi-layered cross-ply doubly curved shallow shells with simply supported boundary conditions. The study is based-on the full nonlinear partial-differential equations of motion for shells. These equations of motion are based-on the von Kármán-type geometric nonlinear theory and the first-order shear-deformation theory, they are developed by using a variational approach. Many approximate shell theories are presented.We used two approaches to study the responses of shells to a primary resonance: a direct approach and a discretization approach. In the discretization approach, the nonlinear partial-differential equations are discretized using the Galerkin procedure to reduce them to an infinite system of nonlinearly coupled second-order ordinary-differential equations. An approximate solution of this set is then obtained by using the method of multiple scales for the case of primary resonance. The resulting equations describing the modulations of the amplitude and phase of the excited mode are used to generate frequency-and force-response curves. The effect of the number of modes retained in the approximation on the predicted responses is discussed and the shortcomings of using low-order discretization models are demonstrated. In the direct approach, the method of multiple scales is applied directly to the nonlinear partial-differential equations of motion and associated boundary conditions for the same cases treated using the discretization approach. The results obtained from these two approaches are compared.For the global analysis, a finite number of equations are integrated numerically to calculate the limit cycles and their stability, and hence their bifurcations, using Floquet theory. The use of this theory requires integrating 2n + (2n) 2 nonlinear first-order ordinary-differential equations simultaneously, where n is the number of modes retained in the discretization. A convergence study is conducted to determine the number of modes needed to obtain robust results.The discretized system of equation are used to study the nonlinear vibrations of shells to subharmonic resonances of order one-half. The effect of the number of modes retained in the approximation is presented. Also, the effect of the number of layers on the shell parameters is shown.Modal interaction between the first and second modes in the case of a two-to-one internal resonance is investigated. We use the method of multiple scales to determine the modulation equations that govern the slow dynamics of the response. A pseudo-arclength scheme is used to determine the fixed points of the modulation equations and the stability of these fixed points is investigated. In some cases, the fixed points undergo Hopf bifurcations, which result in dynamic solutions. A combination of a long-time integration and Floquet theory is used to determine the detailed solution branches and chaotic solutions and their stability. The limit cycles may undergo...

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

“…The equations of motions for shallow shells subjected to large deflections and moderate rotations, with small strain so that HookeÕs law is verified, were derived by various authors in the case of particular geometries: Donnell (1934) and Evensen and Fulton (1965) for cylinders, Marguerre (1938), Leissa and Kadi (1971) and Alhazza (2002) for curved panels, Mushtari and Galimov (1961) and Koiter (1965) in the general case. A recent work presents a justification of these equations by an asymptotic method (Hamdouni and Millet, 2003).…”

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

“…A complete review of the available mathematical methods is provided by Steindl and Troger [34]. Strategies based upon the application of the multiple scales method directly into the PDE have been proposed by Nayfeh and coworkers [24,35], and has been successfully applied to the cases of non-linear vibrations of buckled beams [36,37], shallow suspended cables [38] and doubly-curved cross-ply shallow shells [23]. Non-linear normal modes (NNMs), defined as invariant manifolds in phase space [39], state a proper framework to embed the influence of all linear modes into a single NNM.…”

“…Leissa and Kadi [21] studied the transition for a shallow shell having a rectangular boundary. Doubly-curved shallow shells have also been investigated, by Shin [22] with the assumption of a single-mode, recently by Alhazza [23] with the direct method proposed by Nayfeh [24], and by Amabili [25] for a number of different geometries.…”

A hierarchical finite element for geometrically non‐linear vibration of doubly curved, moderately thick isotropic shallow shells

“…This can lead to quantitative and even qualitative erroneous results in the analysis of shallow shell non-linear vibrations [15]. Alternatively, an accurate spatial model can be obtained by the ÿnite element method.…”

“…The displacements of the shells were approximated by a power series, which are eigenvectors of the ÿrst and second linear vibration modes. In Reference [15] the geometrically non-linear vibrations of cross-ply shallow shells are investigated by the method of multiple scales and by Galerkin's method, using an approximation series constituted by several linear modes.…”