2018
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Item Sensitivity of Stresses to the Forces Acting on the Cast Parts of Freight-Car Bogie(Дніпропетровський національний університет залізничного транспорту ім. акад. В. Лазаряна, Дніпро, 2018) Reidemeister, Alexey G.; Shykunov, Oleksandr A.EN: Purpose. To determine the effect of the force components acting in the axle box and the central spring suspension on the stresses occurring in the side frame of the three-piece bogie. Methodology. To assess the effect of the forces acting on the side frame on the stresses in it, we developed a finite element model of the side frame. After that, we carried out an assessment of its stress-strain state under loading conditions corresponding to the І and ІІІ design modes. According to the results obtained, we determined the stress concentration points in the construction, which are selected as check ones for further studies. Also, as checkpoints we took the points corresponding to the sensor locations when estimating the stresses in the side frame during the tests. At the next stage, we applied unit loads in sequence at the interaction points of the side frame with the boxes and the central spring group. To obtain a more accurate result, the unit forces were balanced by the corresponding forces and moments of inertia. With each loading variant, tensors of stresses arising from the action of unit loads were obtained at checkpoints. On the basis of the stress tensors obtained, we determined the corresponding equivalent stresses – the sensitivity coefficients. Findings. The paper determines the stress sensitivity coefficients in the three-piece bogie side frame to external loads acting on the side frame from the side of axle box and central spring group. Based on the results of the assessment of the coefficients obtained, we determined the forces having the greatest influence on individual sections of the side frame. We estimated the possibility of using the obtained results in optimizing the parameters of the bogie spring suspension to increase the strength and durability of the side frame. Originality. For the first time, the effect of individual components of the forces acting on the side frame on the stresses in it has been estimated. Practical value. The obtained result can be used in the design and optimization of three-piece bogies, to improve the side frame durability. The stress tensors obtained can be used to estimate the effect of complex loading on the side frame strength and durability.Item Stability of Motion of Railway Vehicles Described with Lagrange Equations of the First Kind(Дніпропетровський національний університет залізничного транспорту ім. акад. В. Лазаряна, Дніпро, 2018) Reidemeister, Alexey G.; Levytska, Svitlana І.EN: Purpose. The article aims to estimate the stability of the railway vehicle motion, whose oscillations are described by Lagrange equations of the first kind under the assumption that there are no nonlinearities with discontinuities of the righthand sides. Methodology. The study is based on the Lyapunov’s stability method of linear approximation. The equations of motion are compiled in a matrix form. The creep forces are calculated in accordance with the Kalker linear theory. Sequential differentiations of the constraint equations reduced the equation system index from 2 to 0. The coefficient matrix eigenvalues of the system obtained in such a way are found by means of the QR-algorithm. In accordance with Lyapunov's criterion of stability in the linear approximation, the motion is stable if the real part of all eigenvalues is negative. The presence of «superfluous» degrees of freedom, which the mechanical system does not have (in whose motion equations there are left only independent coordinates) is not trivial. Herewith the eigenvalues and eigenvectors correspond to these degrees of freedom and have no relation to the stability. In order to find a rule that allows excluding them, we considered several models of a bogie, with rigid and elastic constraints of high rigidity at the nodes. In the limiting case of high rigidities, the results for a system without rigid constraints must coincide with the results for a system with rigid constraints. Findings. We carried out the analysis and compared the frequencies (with decrements) and the vibration modes of a threepiece bogie with and without constraints. When analysing the stability of the system with constraints, only those eigenvalues are of interest whose eigenvectors do not break the constraints. The values of these numbers are limits for the eigenvalues of the system, in which rigid constraints are replaced by elastic elements of high rigidity, which allows us to leave the Lyapunov’s criterion unchanged. Originality consists in the adaptation of Lyapunov's stability method of linear approximation to the case when the equations of railway vehicle motion are written in the form of differential-algebraic Lagrange equations of the first kind. Practical value. This written form of the equation of motion makes it possible to simplify the stability study by avoiding the selection of a set of independent generalized coordinates with the subsequent elimination of dependent ones and allows for the coefficient matrix calculation in an easily algorithmized way. Information on the vehicle stability is vitally important, since the truck design must necessarily exclude the loss of stability in the operational speed range.