A Unified Solution for Free Vibration Analysis of Beam-Plate-Shell Combined Structures with General Boundary Conditions

A semi-analytical method is presented to analyze free vibration response of beam-plate-shell combined structures with general boundary conditions. Based on the beam-plate-shell energy theory, the coupled annular plate-conical-cylindrical-spherical shell with stiffened rings and bulkheads regarded as the theoretical model is constructed. The unified displacement admissible functions of each substructure are expanded as modified Fourier series and auxiliary convergence functions along generatrix direction and Fourier series along circumferential direction. Virtual spring technology is adopted to express the energy stored at the junction of adjacent substructures and both boundaries. The energy variational procedure and Ritz method are used to obtain the vibrational governing equation of the combined structure. The present method provides an analytical way for the vibrational response of complicated combined structures. The convergence, accuracy and reliability are validated by comparing the free vibrational response with those of the references and finite element method. Some numerical examples show effects of different boundary conditions on the free vibration. And the influence of stiffened rings and bulkheads treated as Euler-beams and annular plates is also discussed from quantity, size and spatial distribution, offering a feasible way to design the reinforced structures and optimize the bulkheads in engineering problems.