Pilgun G. Methods of analysis of the geometrically nonlinear vibrations of shallow shells and plates with complex form

Object of research is nonlinear elastic mechanical systems, which elements can be shallow shells and plates of arbitrary shape. The aim is the creation of methods and software to research free geometrically nonlinear vibrations of thin-walled constructions elements that can be simulated by shallow shells and plates with arbitrary base and various boundary conditions. Methods of research are multiple use of the R-function theory, variational Ritz's method, Galerkin procedure and numerical methods such as including Runge-Kutta method. The theoretical meaning of the work is the new numerical-analytic method for free nonlinear vibrations of shallow shells and plates with arbitrary form research creating. The practical meaning is algorithms and software developing that allows to automatize nonlinear vibrations of thin-walled shell constructions elements calculation process. The novelty is in following: the new numerical-analytic method for free nonlinear vibrations of isotropic shallow shells with arbitrary shape research based on the R-functions theory and variational methods has been proposed; coefficients for nonlinear differential equation of Duffing type equation to what the original nonlinear boundary-value problem reduces have been obtained; the method based on strain-displacements relationships linearization for free vibrations of flexible plates has been enlarged to those with complex base and various boundary conditions; the new problems about nonlinear free vibrations of thin-walled constructions elements such as blade strip cap, tank shells with pilofacturing holes composed from panels with cuts of arbitrary shape have been also solved; influence of geometrical form and boundary conditions on amplitude to frequency ratios have been studied. The results of this work are used in research engineering and educational process of applied mathematics department in NTU "KhPI" as well. The field is aeronautics, building industry.