Frequency analysis of the response of a one-sheet hyperboloid to periodic longitudinal loading

Oksana Paliy, Olga Lukіanchenko


The frequency analysis of the steady state forced vibrations of a thin shell of negative Gaussian curvature is performed. The one-sheet hyperboloid shell is subjected to the action of the periodic longitudinal load. The finite element model for the shell is built using flat quadrilateral shell elements with six degrees of freedom per node. The modal analysis of the shell is performed by the Lanczos method using linear formulation of the problems. Eigenfrequencies and eigenmodes of the shell are also defined using nonlinear formulation of the problems that takes into account the effect of a longitudinal under critical static force preliminary applied to the top edge of the shell. For this purpose the nonlinear statics problems and the natural vibration problems are sequentially solved by the modified Newton-Raphson method and Lanczos method respectively. A modal analysis showes that dependence of the shell eigenfrequencies on the static longitudinal loading is nonlinear. The steady state response of the shell to the periodic longitudinal loading is obtained both by means of direct and modal frequency analysis. It turned out that sensitivity of the hyperboloid of one sheet to the periodic longitudinal loading considerably increases when the static under critical longitudinal loading is taken into account. The responses of the shell are presented in the form of amplitude frequency dependences of the finite element model nodes.


forced vibrations; frequency analysis; finite element method; thin shell; one-sheet hyperboloid; longitudinal loading


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