Displacement (psychology)

Spherical shell

Physics

Mathematical optimization

Engineering

Mathematical analysis

Finite element method

Civil engineering

Psychology

Economics

Boundary value problem

Vibration

Fourier series

Acoustics

Geometry

Convergence (economics)

Shell (structure)

Mathematics

Economic growth

Psychotherapist

Penalty method

Boundary (topology)

Structural engineering

Rayleigh–Ritz method

International Journal of Mechanical Sciences

A semi analytical approach is employed to analyze the free vibration characteristics of uniform and stepped combined paraboloidal, cylindrical and spherical shells subject to arbitrary boundary conditions. The analytical model is established on the base of multi-segment partitioning strategy and Flügge thin shell theory. The admissible displacement functions are handled by unified Jacobi polynomials and Fourier series. In order to obtain continuous conditions and satisfy arbitrary boundary conditions, the penalty method about the spring technique is adopted. The solutions about free vibration behavior of uniform and stepped combined paraboloidal, cylindrical and spherical shells were obtained by approach of Rayleigh–Ritz. To confirm the reliability and accuracy of proposed method, convergence study and numerical verifications for combined paraboloidal, cylindrical and spherical shell with different boundary conditions, Jacobi parameters, spring parameters and maximum degree of permissible displacement function are carried out. Through comparative analyses, it is obvious that the present method has a good stable and rapid convergence property and the results of this paper agree closely with FEM. In addition, some interesting results about the geometric dimensions are investigated.

Free vibration analysis of uniform and stepped combined paraboloidal, cylindrical and spherical shells with arbitrary boundary conditions