The present literature review focuses on geometrically non-linear free and forced vibrations of shells made of traditional and advanced materials. Flat and imperfect plates and membranes are excluded. Closed shells and curved panels made of isotropic, laminated composite, piezoelectric, functionally graded and hyperelastic materials are reviewed and great attention is given to non-linear vibrations of shells subjected to normal and in-plane excitations. Theoretical, numerical and experimental studies dealing with particular dynamical problems involving parametric vibrations, stability, dynamic buckling, non-stationary vibrations and chaotic vibrations are also addressed. Moreover, several original aspects of non-linear vibrations of shells and panels, including (i) fluid–structure interactions, (ii) geometric imperfections, (iii) effect of geometry and boundary conditions, (iv) thermal loads, (v) electrical loads and (vi) reduced-order models and their accuracy including perturbation techniques, proper orthogonal decomposition, non-linear normal modes and meshless methods are reviewed in depth.

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Non-linear vibrations of shells: A literature review from 2003 to 2013

Flexural and in-plane vibration analysis of elastically restrained thin rectangular plate with cutout using Chebyshev–Lagrangian method

An approach to the solution of nonlinear forced vibration problem of structural systems reinforced with advanced materials in the presence of viscous damping

Linear and geometrically nonlinear vibrations of the three-layered functionally graded shallow shells with a complex form of the base are studied. It is assumed that outer and inner layers are made of functionally graded materials (FGM) or an isotropic material (metal or ceramic). The first-order shear deformation theory of shallow shells (FSDT) is employed. Effective material properties of layers are varied along the thickness according to a power law. To calculate different mechanical characteristics for different types of lamination schemes, analytical expressions are obtained. The linear problem is solved by combining the Ritz and the R-functions method. Linearization of the nonlinear problem is carried out by a novel original approach. Namely, the initial problem is reduced to solving a sequence of linear problems including those vibrations related to linear, special type of the elasticity problem and the nonlinear system of ordinary differential equations. Validation of the proposed method and the developed software has been examined on test problems for shallow shells with rectangular plan form and different boundary conditions. In order to demonstrate possibilities of the proposed approach, new results for laminated FGM shallow shells of the complex form of the base are presented. Effects of different material distributions, lamination schemes, curvatures, boundary conditions, and geometrical parameters on natural frequencies and backbone curves are reported and analyzed.

Linear and nonlinear free vibration analysis of laminated functionally graded shallow shells with complex plan form and different boundary conditions

In the present paper, a composite finite element is proposed using the assumed natural strain (ANS) method, which is based on the unified strain interpolations and the minimized potential energy. In the proposed formulation, the linearized weak form of compatibility and equilibrium equations is obtained for the geometrically nonlinear analysis. Additionally, the parabolic strain interpolation functions without any shear correction coefficient are imposed into the first-order shear deformation theory (FSDT) to verify the zero shear stresses condition at the top and bottom surfaces. The iterative arc-length method is applied to handle the post-buckling and free vibration responses. For the micromechanical part, the effective elasticity modulus based on the modified Halpin–Tsai micromechanical model is attained to analyze functionally graded graphene platelet-reinforced composite (FG-GPLRC) plates with and without hole. Furthermore, the effective mass density and the Poisson’s ratio are assessed via the rule of mixture. Therefore, the major novelty of this paper is in detecting the nonlinear thermomechanical behavior of GPLRC plates using robust strain-based interpolations. Hence, nonlinear equilibrium paths are plotted to illustrate the effects of different boundary conditions, weight fractions and distribution schemes on GPLRC plates. • A finite element is proposed ANS method. • Modified FSDT is used to establish the governing equations. • Halpin–Tsai micromechanical model is employed to measure the effective elasticity modulus. • Analysis of thermo-mechanically loaded GPLRC plates is performed. 

Linear and nonlinear mechanical responses of FG-GPLRC plates using a novel strain-based formulation of modified FSDT theory

Geometrically non-linear forced vibrations of a shallow circular cylindrical panel with a complex shape, clamped at the edges and subjected to a radial harmonic excitation in the spectral neighborhood of the fundamental mode, are investigated. Both Donnell and the Sanders–Koiter non-linear shell theories retaining in-plane inertia are used to calculate the elastic strain energy. The discrete model of the non-linear vibrations is build using the meshfree technique based on classic approximate functions and the R-function theory, which allows for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries; Chebyshev orthogonal polynomials are used to expand shell displacements. A two-step approach is implemented in order to solve the problem: first a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for expanding the non-linear displacements. Lagrange approach is applied to obtain a system of ordinary differential equations on both steps. Different multimodal expansions, having from 15 up to 35 generalized coordinates associated with natural modes, are used to study the convergence of the solution. The pseudo-arclength continuation method and bifurcation analysis are applied to study non-linear equations of motion. Numerical responses are obtained in the spectral neighborhood of the lowest natural frequency; results are compared to those available in the literature. Internal resonances are also detected and discussed.

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Non-linear vibrations of shallow circular cylindrical panels with complex geometry. Meshless discretization with the R-functions method

Geometrically nonlinear forced vibrations of shells based on the domains with cut-outs are investigated. Classical nonlinear shallow-shell theories retaining in-plane inertia is used to calculate the strain energy; the shear deformation is neglected. A mesh-free technique based on classic approximate functions and the R-function theory is used to build the discrete model of the nonlinear vibrations. This allowed for constructing the sequences of admissible functions that satisfy given boundary conditions in domains with complex geometries. Shell displacements are expanded by using Chebyshev orthogonal polynomials. A two-step approach is implemented to solve the problem: first a linear analysis is conducted to identify natural frequencies and corresponding natural modes to be used in the second step as a basis for nonlinear displacements. The system of ordinary differential equations is obtained by using Lagrange approach on both steps. The convergence of the solution is studied by using different multimodal expansions. The pseudo-arclength continuation method and bifurcation analysis are used to study the nonlinear equations of motion. Numerical responses are obtained in the spectral neighbourhood of the lowest natural frequency. When possible, obtained results are compared to those available in the literature.Copyright © 2011 by ASME

Galyna Pilgun

Papers

Non-linear vibrations of shallow circular cylindrical panels with complex geometry. Meshless discretization with the R-functions method

Analysis of Nonlinear Forced Vibrations of Shallow Shells With Cut-Outs by Using the R-Function Method