A Numerical Approach for Vibration Analysis of an Axisymmetric Shell Structure With a Nonuniform Edge Constraint

A numerical approach for computing the eigenvalues and eigenfunctions of an axisymmetric shell with a nonaxisymmetric edge constraint is presented. The shell structures are modeled without constraint by an assemblage of axisymmetric shell elements. The constraint at any point along the edge circumference may be imposed by two linear springs acting against the axial and the radial degrees of freedom, and by a torque spring acting against the rotational degree of freedom. The nonuniform constraint is thus represented by the arbitrary distribution of these spring constants per unit length along the circumference. This arbitrary distribution of spring constants is then resolved by a Fourier series expansion. Utilizing the natural modes of the unconstrained shell as the generalized coordinates, the equations of motion which include the effects of a nonuniform constraint are derived. The mass and the stiffness matrices of these equations of motion are used as inputs for solving the linear numerical eigenvalue problem. A circular plate, which can be considered as an extreme case of an axisymmetric shell, is used as a numerical example. For a simply supported circular plate with a sinusoidal variation of rotational edge constraint, the computed results agree well with the data available in the literature.