A semi-analytic method is presented to analyze free and forced vibrations of combined conical–cylindrical–spherical shells with ring stiffeners and bulkheads. First, according to locations of discontinuity, the combined shell is divided into one opened spherical shell and a number of conical segments, cylindrical segments, stiffeners, and bulkheads. Meanwhile, a semi-analytic approach is proposed to analyze the opened spherical shell. The opened spherical shell is divided into narrow strips, which are approximately treated as conical shells. Then, Flügge theory is adopted to describe motions of conical and cylindrical segments, and stiffeners with rectangular cross section are modeled as annular plates. Displacement functions of conical segments, cylindrical segments, and annular plates are expanded as power series, wave functions, and Bessel functions, respectively. To analyze arbitrary boundary conditions, artificial springs are employed to restrain displacements at boundaries. Last, continuity and boundary conditions are synthesized to the final governing equation. In vibration characteristics analysis, high accuracy of the present method is first demonstrated by comparing results of the present method with ones in literature and calculated by ansys. Further, axial displacement of boundaries and open angle of spherical shell have significant influence on the first two modes, and forced vibrations are easily affected by bulkheads and external force.

Wave based method (WBM) is presented to analyze the free and forced vibration of cylindrical shells with discontinuity in thickness. The hull is first divided into multiple segments according to the locations of thickness discontinuity and/or driving points, and then the Flügge theory is adopted to describe the motion of cylindrical segments. The dynamic field variables in each segment are expressed as wave function expansions, which accurately satisfy the equations of motion and can be used to analyze arbitrary boundary conditions, e.g., classical or elastic boundary conditions. Finally, the boundary conditions and interface continuity conditions between adjacent segments are used to assemble the final governing equation to obtain the free and forced vibration results. By comparing with the results existing in open literate and calculated by finite element method (FEM), the present method WBM is verified. Furthermore, the influences of the boundary conditions and the locations of thickness discontinuity on the beam mode frequency and fundamental frequency are discussed. The effects of the direction of external force, location of external point force, and the structural damping on the forced vibration are also analyzed.