Analysis of Geometrically Nonlinear Vibrations of Functionally Graded Shallow Shells of a Complex Shape

Abstract

Geometrically nonlinear vibrations of functionally graded shallow shells of complex planform are studied. The paper deals with a power-law distribution of the volume fraction of ceramics and metal through the thickness. The analysis is performed with the use of the R-functions theory and variational Ritz method. Moreover, the Bubnov-Galerkin and the Runge-Kutta methods are employed. A novel approach of discretization of the equation of motion with respect to time is proposed. According to the developed approach, the eigenfunctions of the linear vibration problem and some auxiliary functions are appropriately matched to fit unknown functions of the input nonlinear problem. Application of the R-functions theory on every step has allowed the extension of the proposed approach to study shallow shells with an arbitrary shape and different kinds of boundary conditions. Numerical realization of the proposed method is performed only for one-mode approximation with respect to time. Simultaneously, the developed method is validated by investigating test problems for shallow shells with rectangular and elliptical planforms, and then applied to new kinds of dynamic problems for shallow shells having complex planforms.

Buckling and free vibration analysis of functionally graded sandwich plates and shallow shells by the Ritz method and the R-functions theory

“…where � i , i ¼ 1, 5 are undetermined components of the solution structure. 23,[30][31][32][33][34][35][36][37] They are expanded in a power series. In formula (27), !…”

“…[26][27][28] The problems of free vibration of laminated shallow shells with complex planform by RFM were considered in Kurpa et al 28 The algorithms of solving geometrically nonlinear vibrations of plates and shallow shells were proposed in various studies. [29][30][31][32][33][34] In the papers, [35][36][37] the method of the R-functions was extended to the problems of vibration of functionally graded shallow shells of arbitrary planforms.…”

Application of the R‐functions in free vibration analysis of FGM plates and shallow shells with temperature dependent properties

“…They have to satisfy at least the geometrical boundary conditions as well. Construction of these functions is carried out by the R‐functions theory [24–30, 32] in case of complicated planform of the shell or complex boundary condition.…”

“…Therefore, combined application of the variational Ritz's method and the R‐functions theory allows to find an approximate solution in an analytical form. This approach was employed earlier for investigation of vibration of plates and shallow shells including laminated and FGM objects [14, 27–30]. Yet, in previous works there were considered temperature independent characteristics of the materials.…”

Free vibrations and static analysis of functionally graded sandwich plates with three-dimensional finite elements

“…Besides, the complexity of analytic solutions for FGM plates which is associated with their geometry different from rectangular, can be successfully overcome by using semi-analytic approaches. For instance, a numerical-analytical approach based on R-functions theory and the Ritz method has been worked out for studying geometrically nonlinear vibrations of functionally graded shallow shells of complex planform in [40]. An approach implementing the differential transform method for free vibration analysis of nonuniform cross-section of functionally graded beams has been presented, e.g.…”