Onufrienko O. Researches of the nonlinear vibrations of orthotropic plates with complex form by R - functions method

Object of research is nonlinear elastic mechanics systems, which are under action of outside dynamic loads. The purpose of work is creation of an effective method of research of nonlinear free and forced vibrations of orthotropic elements of thin-walled constructions that can be simulated by plates with complex form and different boundary conditions. General means of researches are joint usage of R-function theory, variational Ritz's method, methods of Galyorkin and Runge-Kutt's. The theoretical meaning of work is creation of an effective method based on R-functions theory, which is universal for geometric forms of considered objects. The amplitude-frequency characteristics for orthotropic plates with complex forms and different boundary conditions were defined. A practical meaning of work is development of algorithms and software, which allows to automating process of calculation of nonlinear vibrations of orthotropic plates constructions. The scientific novelty of work is: the proposed effective method of research of nonlinear vibrations of orthotropic elements of thin-walled constructions that simulated by plates with complex form; this method is based on theory of R-functions and variational methods; the method of reducing equations of motions to Cauchy's task; it allows to research nonlinear dynamics behavior of orthotropic plates with complex forms and different boundary conditions; the task was solved in displacements and in mixed forms with help one-mode and two-modes approximation of deflection functions; coefficients for ordinary differential equations Cauchy's problem were obtained; the new problems about nonlinear free and forced vibrations of elements of thin-walled constructions and aero-engine compressor rotor blade are also solved; influence of geometrical form, outward loads, boundary conditions and type of materials on amplitude - frequency characteristics were researched; the method for investigation of modes stability of nonlinear vibration was proposed; this method is based on thetheory of R-functions, Bubnov-Galyorkin method, limited stability criterion by Lyapunov and Runge-Kutta method. The results of researches are used in scientific researches A.N.Podgorny's Institute for Problems in Machinery NAS Ukraine and are used into educational process of applied mathematics department NTU "KhPI" as well. The obtained results of researches may be used in space branch, industrial construction, motor industry an industrial complex for projection of thin-walled elements.