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Large Amplitude Axisymmetric Free Vibration of a Closed Two-Dimensional Spherical Pressure Vessel with a Constrained Volume

The nonlinear axisymmetric vibrational modes of a closed two-dimensional spherical pressure vessel with a constrained volume are presented using the fundamental forms of surfaces. The strain energy of the spherical pressure vessel is derived as a quadratic function of the initial Eulerian, added, and total Lagrangian strains. Two-dimensional bilinear elements described in spherical polar coordinates with the symmetric boundary condition along both sides of the meridian line are used for investigating the nonlinear axisymmetric mode shapes of a spherical pressure vessel. Numerical results were obtained by a nonlinear finite element approach and verified in the case of an empty spherical shell for both lower and upper branches. This study showed that the spherical pressure vessel gave a higher nonlinear axisymmetric natural frequency than the empty spherical shell. The elastic modulus has a large effect on the nonlinear axisymmetric frequencies of both the lower and upper branches for the spherical pressure vessel. Changing the thickness and initial internal pressure significantly affects only the lower branch of the spherical pressure vessel at high mode numbers. The results also indicate that the ratios of the nonlinear natural frequencies of the lower to upper branches decrease when the vibrational modes increase.