Speakers
Description
Composite thin-walled shell and plate elements are indispensable components of modern critical structures and technical devices for various purposes. This is due to their efficient material usage and ability to provide the required stiffness in specific directions under operational conditions. Intense cyclic loads cause geometrically nonlinear vibrations in these elements. Preventing resonance during operation requires determining amplitude-frequency characteristics at the design stage. Research in the nonlinear mechanics of thin-walled structural elements began with Kármán's quadratic plate theory, an extension of Kirchhoff-Love’s classical linear theory. Later, some researchers used nonlinear technical theory based on S. P. Timoshenko's shear model. However, these theories do not adequately account for the specific deformation characteristics of composite plates and shells. The refined theory developed by the authors addresses this issue by approximating stress-strain characteristics using finite sums of Legendre polynomials with respect to the coordinate normal to the mid-surface. This method, based on I. Vekua’s ideas and further developed by B. Pelekh and M. Sukhorolskyi, satisfies boundary conditions on the surface. The resulting two-dimensional relations involve minimal-order differential operators while accounting for the material's compliance to transverse shear and compression — key properties of polymer-matrix reinforced composites — for both linear and nonlinear deformation. In Kármán’s theory, known relations describe the dependence between fundamental frequency and vibration amplitude for rectangular plates and cylindrical shells. These nonlinear relations, derived using A. Volmir’s method of integrating over one vibration period, are polynomial or transcendental. However, applying a similar approach in the case of the generalized and aforementioned refined theory leads to the appearance of secular terms in the amplitude-frequency characteristics, which contradict the physical nature of free vibrations. The authors addressed this issue by employing the generalized perturbation method, previously applied by R. Lewandowski to problems of geometrically nonlinear beam vibrations. Both approaches yield close results within certain ranges of physical-mechanical characteristics and geometric parameters of the thin-walled elements considered. However, significant discrepancies in specific cases prompt a more detailed analysis of the mathematical deformation models and their differential operators. Some aspects and results obtained by the authors will be presented for discussion.
