The main goal of deformable solid mechanics is to determine the stress–strain state of parts, structural elements, and their connections. The most accurate results of calculations of this state allow us to optimize design objects. However, not all models can be solved using exact methods. One such model is the problem of a layer with cylindrical embedded supports that are parallel to each other and the layer boundaries. In this work, the supports are represented by cylindrical cavities with zero displacements set on them. The layer is considered in Cartesian coordinates, and the cavities are in cylindrical coordinates. To solve the problem, the Lamé equation is used, where the basic solutions between different coordinate systems are linked using the generalized Fourier method. By satisfying the boundary conditions and linking different coordinate systems, a system of infinite linear algebraic equations is created. For numerical realization, the method of reduction is used to find the unknowns. The numerical analysis has shown that the boundary conditions are fulfilled with high accuracy, and the physical pattern of the stress distribution and the comparison with results of similar studies indicate the accuracy of the obtained results. The proposed method for calculating the stress–strain state can be applied to the calculation of structures whose model is a layer with cylindrical embedded supports. The numerical results of the work make it possible to predetermine the geometric parameters of the model to be designed.

The design of parts of machines, mechanisms, structures and foundations, particularly in the aerospace industry, is closely related to the definition of the stress state of the body. The accuracy of determining the stress state is the key to optimizing the use of materials. Therefore, it is important to develop methods to achieve such goals. In this work, the second main spatial problem of the elasticity theory is solved for a layer with a longitudinal cylindrical cavity with periodic displacements given on the surface of the layer. The solution of the problem is based on the generalized Fourier method for a layer with a cylindrical cavity. To take into account periodic displacements, an additional problem is applied with the expansion of the solution for a layer (without a cavity) in the Fourier series. The general solution is the sum of these two solutions. The problem is reduced to an infinite system of linear algebraic equations, which is solved by the reduction method. As a result, the stress-strain state of the layer on the surface of the cavity and isthmuses from the cavity to the boundaries of the layer was obtained. The conducted numerical analysis has a high accuracy for fulfilling the boundary conditions and makes it possible to assert the physical regularity of the stress distribution, which indicates the reliability of the obtained results. The method can be applied to determine the stress-strain state of structures, whose calculation scheme is a layer with a cylindrical cavity and a given periodic displacement. Numerical results make it possible to predict the geometric parameters of the future structure.

ПРО ЗНИЖЕННЯ ДИНАМІЧНИХ НАВАНТАЖЕНЬ У МЕХАНІЗМІ ПОВОРОТУ ГУСЕНИЧНОЇ МАШИНИ З ГІДРООБ’ЄМНОЮ ПЕРЕДАЧЕЮ

A solution method of dynamic creep-damage problems for bodies which can be modeled as thin-walled beams is presented in the paper. The original initial-boundary-value problem of dynamic creep for bodies under the action of harmonic loading is reduced to two correlated initial-boundary-value problems in two time scales – slow and fast. The first problem (slow time scale) is solved by using the variational principle for the mixed functional, numerical methods of Runge–Kutta–Merson and RFM (Rvachov’s Functions Method). The second problem in fast time scale is solved by using Gaussian numerical method for determining of amplitude values of stresses. Numerical data of solving dynamic creep-damage problem for beams bending with different types of edges are given. The amplitude of the oscillating load has significant effect on the time to rupture of the beam. Increasing the values of the amplitude of this load leads to a decrease in the value of the time to rupture of the beam for all types of edges fixations.

The research results of the mathematical model development of dynamic processes in a peristaltic concrete pump with a hydraulic drive are presented. Cast concrete occupies a leading place in modern construction. Peristaltic concrete pumps are good proved in this area. Developing of devices for performance of such activities is an actual task. One of contemporary efficient directions in its solving is an adequate modeling of dynamic processes in mentioned devices. The model is development in the form of a differential equation for rotation angle of the pump rotor; contains general geometric characteristics, rotor’s mass characteristics, dynamic characteristics of the hydraulic motor, parameters of the hose and a structural mix. A method has been developed for the resistance torque forming to rotation of the pump rotor from rollers deforming the hose. A model of the friction forces during the movement of the structural mix is proposed. An expression is obtained for the resistance forces as result of gravity forces action on the mix particles in the outlet part of the hose. An approach is proposed for using catalog data to represent the torque of a hydraulic motor as a function of the angular velocity of its rotor. Researches of dynamic processes by using the proposed model are carried out. Important technological regularities of the unit functioning have been established, in particular: the mortar speed may have a significant variable component; the mortar speed and the pump performance increase with a decrease in the length of the outlet hose and a decrease in its height, a decrease in the rolling friction coefficients for the rollers, and not using of the side rollers. The conducted research show that an increase of an altitude leads to a decrease in the average speed of the mixture and the frequency of pulsations. In this case the amplitude of velocity pulsations increases. With an increase in the length of a deferent hose, the amplitude of velocity pulsations changes insignificantly. The rejection of side rollers leads to an increase in the average speed of the mixture motion with minor pulsations. The walls of the hose, however, in this case experience more intense bending stresses, which reduce the product life of the hose.