Methods of Dynamic Design of Structural Elements

This chapter presents methods of dynamic design of structural elements. This chapter shows the main methods used in the dynamic analysis of rods, plates, and shells. The application of this method is described in the examples of solving problems of free vibration of rectangular and circular membranes. The range of problems that can be solved accurately, in a closed form, is limited and, therefore, there are often variational methods used in practice. The variational methods are discussed. The use of integral equations, in particular the reduction of problems with oscillation of one-dimensional structures to the integral equations of Volterra of the second kind and Fredholm of the second kind are demonstrated. The methods used for numerical solutions of problems of dynamics are described. These are the finite element method (FEM) and finite difference method (FDM). In either the FEM or FDM, the whole domain of the partial differential equations (PDEs) requires discretization. The FEM is presented as an example of a plane problem of the theory of elasticity. A diagram of solving a problem of nonstationary oscillations of a cantilever plate is shown. The oscillations of a compressed rod/bar are solved using the finite difference method (FDM). This chapter describes both a general approach to the construction of various types of boundary equations and a spectral method of boundary elements (SMBE) method, which allows to obtain effective solutions of various boundary value problems of structural mechanics. The application of SMBE in this book is further illustrated by solving several problems on the structural analysis of the design of construction on the static and dynamic impacts. The generalized functions in structural mechanics, generalized method of integral transformation (GMIT), delta-transform method (DTM), and compensating loads method (CLM) are also presented in this chapter.