Abstract
The free vibration characteristic of spherical cap with general edge constraints is studied by means of a unified method. The energy method and Kirchhoff hypothesis are adopted to derive the formulas. The displacement functions are improved based on the domain decomposition method, in which the unified Jacobi polynomials are introduced to represent the displacement function component along circumferential direction. The displacement function component along axial direction is still the Fourier series. In addition, the spring stiffness method forms a unified format to deal with various complex boundary conditions and the continuity conditions at two adjacent segments. Then, the final solutions can be obtained based on the Ritz method. To prove the validity of this method, the results of the same condition are compared with FEM, published literatures, and experiment. The results show that the present method has the advantages of fast convergence, high solution accuracy, simple boundary simulation, etc. In addition, some numerical results of uniform and stepped spherical caps with various geometric parameters and edge conditions are reported.
Highlights
Spherical caps have been widely used in many practical engineering branches, such as pressure vessels, dome-shaped structures, submarines, and nuclear power plants. ese structures usually bear different extreme loads caused by wave, wind, and even earthquakes. e dynamic excitations caused excessive vibration and even resonance in complex environmental conditions. erefore, the analysis of free vibration of spherical caps becomes really meaningful. e related literatures are reviewed below
Gautham and Ganesan [1] conducted a research to deal with the free vibration characteristics of isotropic and laminated orthotropic spherical caps
Based on the first-order shear deformation theory, a semianalytical shell finite element was utilized to investigate the effect of geometric configurations on vibration behavior of spherical caps
Summary
Spherical caps have been widely used in many practical engineering branches, such as pressure vessels, dome-shaped structures, submarines, and nuclear power plants. ese structures usually bear different extreme loads caused by wave, wind, and even earthquakes. e dynamic excitations caused excessive vibration and even resonance in complex environmental conditions. erefore, the analysis of free vibration of spherical caps becomes really meaningful. e related literatures are reviewed below. Wu and Heyliger [4] analyzed the free vibration of spherical caps on the basis of twodimensional first-order shear deformable shell theory. Shi et al [18] analyzed free vibration of double-curved shallow shell structures by means of the improved Fourier series method (IFSM); the excellent convergence and accuracy of the presented method have been proved. Lee et al [23] applied Flugge’s thin shell theory and Rayleigh’s energy method to analyze the free vibration characteristics of the joined spherical-cylindrical shell with general edge constraints. Xie et al [32] combined Flugge’s thin shell theory with the power series method to investigate vibration characteristic of stepped curved shells with different edge conditions. Erefore, a unified method is necessary and meaningful to establish to solve the vibration behavior of uniform and stepped spherical caps with elastic edge conditions. Final solutions are derived on the basis of the Rayleigh–Ritz method
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