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An exact and closed-form solution is obtained for free vibration problems of homogeneous isotropic cylindrical shells, which is under arbitrary boundary conditions and with varied initial stresses in different longitudinal sections. First, the cylindrical shell is divided into multiple sub-shells according to their thicknesses and initial stresses. And the displacement functions of the sub-shell’s middle plane are expanded as trigonometric series in circumferential direction. Then, based on the simplified Donnell shell theory, a set of fundamental dynamic equations, which take initial stresses into accounts, is derived through Hamilton’s principle for each sub-shell. Correspondingly, boundary conditions and connection conditions are derived too. These equations and conditions are simplified through setting the displacements varied in harmonic form. Finally, through defining the state vector, the dynamic equations and solution-determine conditions are described as state-space forms and solved conveniently. Numerical examples validate this method. Driving process of analytical solutions show that it is convenient for introducing and dealing with solution-determine conditions and solving the dynamic problems of cylindrical shells with varied initial stresses in different longitudinal sections.

The surface cracks are easy to occur in the solid missile motor grain. The solid missile motor grain with cracks is very dangerous when it is lunched, because the surface cracks might be propagation unstable when the internal pressure enters the surface cracks cavity. In order to analyze the surface crack of solid motor grain, with accelerated aging test, the variation law of mechanics properties of propellant in the storage period is obtained. Under internal pressure and axial acceleration loading, the three-dimension finite element models of the motor grain are established, three-dimension singular crack elements at the tips of surface cracks are established to simulate the cracks propagation. Along with the surface crack propagation, the stress intensity factors of the crack tips in different storage period are calculated to prejudge its stability respectively. The method and conclusions are available for using solid missile motor grain with surface cracks.