Dynamic influences are one of the important factors affecting project decisions during the construction of buildings and structures. In the case of dynamic analysis, the range of possible variations in structural response is very difficult to determine. The complexity is determined not only by the parameters of the load itself (intensity, the law of change over time), but also by the characteristics of the research object (their values and change over time), which is an interconnected system of the structure and the soil base. Dynamic factors of influence can include periodic vibration or shock loads, the effects of explosions that cause a sharp change in pressure on the structure of the building, seismic fluctuations, etc. In many cases, the load is the result of stress waves propagating in the soil base. These factors can act on the natural environment long-term, relatively short-term, short-term and instantaneous. Therefore, the development of refined approaches for assessing the stress-strain state of structures during the interaction of the structure and the soil environment under the conditions of dynamic evolutionary processes remains an actual task today. In order to effectively find a solution to the dynamics problems, it is necessary to possess the entire set of analysis tools, to clearly present the rules and limits of their application. The ability to correctly assess the engineering and geological conditions of the construction site, the properties of the soils, the joint work of these soils with the foundations and structures of the building has always been and remains the basis of reliable and effective engineering design. The given statement of the problem is a theoretical basis to build the numerical approaches to the study of the vibrational processes of heterogeneous spatial bodies during their contact interaction with an elastic-plastic medium. The paper considers the modeling aspects of the structure and the soil elastic-plastic environment interaction under the conditions of dynamic evolutionary processes. The basic geometric and physical equations of the elasticity theory for heterogeneous circular and prismatic spatial bodies are given. In the range of solving the dynamics problem, the peculiarities of the mathematical model implementation, regarding the representation of the variation of kinetic energy as part of the equations of dynamic equilibrium of linear and nonlinear mechanical systems, are considered.