The Special Features of Solution of Dynamic Problems

In this chapter describes the features of solving dynamic methods of boundary equations. It is known that most building materials have damping properties, characterized by the frequency’s independent loss. However, there is a certain dependency of damping parameters (coefficients loss, decrement) on the frequency of the deformation for soils in the experiments. Frequency-dependent losses are typical for a number of modern materials, such as certain plastics. It is therefore a very important task to describe the dynamic behavior of soil and structural materials possessing both frequency-independent and frequency-dependent internal friction. This chapter presents a new way to do the calculation of Hilbert transform through the Fourier transformation of corresponding functions, which simplifies the problem due to extensive Fourier transform tables. The presented method of calculating the Hilbert transform allows to comparatively and easily analyze the aprioristic models of the dynamic systems based on the creation of complex rigidity or complex pliability of system. This method makes it possible to analytically determine one of the components of the complex rigidity in accordance with the experimentally defined other component. A few examples of the calculation of the Hilbert transform of delta-functions, Heaviside function, and a power function are examined. The above-described method of calculating the Hilbert transform is applied to the analysis of dynamic models bases of internal friction, described by the hypothesis of Voigt, Sorokin, and Schlippe-Boc. This chapter is devoted to the construction of Green’s function for an unlimited plate, lying on Winklerian type elasto-viscous base with an arbitrary dependence of parameters of the complex coefficient of bed on the frequency. In this chapter, the problem of construction of the Green’s function for single-dimensional dynamic systems with damping, described by different model frequency-independent and frequency-dependent internal friction, is examined and solved.