Timchenko G. Linear and geometrically nonlinear vibrations of laminated plates and shallow shells with complex form

Object of research is nonlinear elastic mechanical systems, which elements can be laminated shallow shells and plates. The aim is the creation of methods and software to research free linear and geometrically nonlinear vibrations of thin-walled constructions elements that can be simulated by composite shallow shells and plates with an arbitrary planform and different boundary conditions. Research Methods are combined using variational methods and the R-function theory (RFM). The theoretical meaning this work is the new numerical-analytic method for free linear and nonlinear vibrations of shallow shells. Developed algorithms and received numerical results which may be used for calculations of the shell's dynamical characteristics are the practical meanings. The results of this work are used in engineering research and educational process of applied mathematics department in NTU "KhPI". The novelty of the work is developeding an effective method for research of free geometrically nonlinear vibrations oflaminated shallow shells and plates with arbitrary shape. The method has been worked for two mathematical statements of the problem: formulated in the context of classical theory and in the context of the improved Timoshenko's theory; the constructive tools of the R-functions theory were developed as structures solutions; to reduce the initial system with partial derivatives to nonlinear system of ordinary differential equations that is to problem by Cauchy the special approach is proposed. New problems of linear and nonlinear vibrations of laminated shallow shells and plates with the complex planform are solved. Influence on frequency and amplitude to frequency ratios is investigated from geometrical and mechanical parameters, different ways of clamping, ways of layer packing and properties of material. The method is also used for solving practical problems. The field is mechanical engineering, aeronautics, building industry.