On New Analytic Free Vibration Solutions of Doubly Curved Shallow Shells by the Symplectic Superposition Method Within the Hamiltonian-System Framework

Abstract

<jats:title>Abstract</jats:title> <jats:p>This study presents a first attempt to explore new analytic free vibration solutions of doubly curved shallow shells by the symplectic superposition method, with focus on non-Lévy-type shells that are hard to tackle by classical analytic methods due to the intractable boundary value problems of high-order partial differential equations. Compared with the conventional Lagrangian-system-based expression to be solved in the Euclidean space, the present description of the problems is within the Hamiltonian system, with the solution procedure implemented in the symplectic space, incorporating formulation of a symplectic eigenvalue problem and symplectic eigen expansion. Specifically, an original problem is first converted into two subproblems, which are solved by the above strategy to yield the symplectic solutions. The analytic frequency and mode shape solutions are then obtained by the requirement of the equivalence between the original problem and the superposition of subproblems. Comprehensive results for representative non-Lévy-type shells are tabulated or plotted, all of which are well validated by satisfactory agreement with the numerical finite element method. Due to the strictness of mathematical derivation and accuracy of solution, the developed method provides a solid approach for seeking more analytic solutions.</jats:p>

An analytical closed-form solution for free vibration and stability analysis of curved sandwich panels made of porous metal-foam core and nanocomposite reinforced face-sheets

“…This effect is the main concept of dynamic stability of structures which has been focused in Keshtegar et al, 5 Al-Furjan et al, 28 Mirfatah et al, 38 and Shahmohammadi et al 62 At the following of this section, various boundary conditions and the corresponding approximating functions are presented in equation ( 17). [43][44][45] (1) Simply-supported edges at j 1 = 0, a and j 2 = 0, b (SSSS):…”

“…The other solution in order to overcome to this drawback is employing the weight functions for the components of deformation field which can satisfy various boundary conditions. This approach is employed in this paper by using the weight functions introduced in Sobhy and Zenkour, 43 Li et al, 44 and Sobhy. 45 By considering the present literature, some advantages of this method comparing to the other analytical methods can be observed.…”

Symplectic analytical solution for forced vibration of a multilayer plate system

“…However, the finite element model of the structure can be large; therefore, model reduction is essential to reduce the computational cost and avoid potential numerical issues. 22 As one of the most popular analytical wave propagation methods, the method proposed by Ma et al, 23 based on the symplectic method [23][24][25][26][27][28][29][30] can treat the boundary condition more conveniently and accurately. Compared with the wave and finite element method, the symplectic analytical wave propagation method is based on analytical waves that provide efficient and accurate solutions.…”

A closed-form solution for asymmetric free vibration analysis of composite cylindrical shells with metamaterial honeycomb core layer based on shear deformation theory

“…The governing equations were solved using the Galerkin method. Li, Zhou, and Zheng (2021) presented an analytical procedure for free vibration analysis of doubly curved shells using the symplectic superposition method. Bagheri et al (2021) analyzed the free vibration of an FG cylindrical shell closed with two hemispherical caps based on the first-order theory of shells and Donnell equations and the equations of motion were solved using the generalized differential quadrature method.…”