### STABILITY OF MOTION OF RAILWAY VEHICLES DESCRIBED WITH LAGRANGE EQUATIONS OF THE FIRST KIND

#### Abstract

**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 right-hand 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 three-piece 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.

#### Keywords

#### References

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#### GOST Style Citations

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- Mao, X. A design method for rail profiles based on the geometric characteristics of wheel-rail contact / X. Mao, G. Chen // Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit. – 2018. – Vol. 232. – Iss. 5. – P. 1255–1265. doi: 10.1177/0954409717720346
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- Polach, O. Comparison of Methods Analyzing Bifurcation and Hunting of Complex Rail Vehicle Models / O. Polach, I. Kaiser // Journal of Computational and Nonlinear Dynamics. – 2012. – Vol. 7. – Iss. 4. doi: 10.1115/1.4006825
- Polach, O. Compatibility of the non-linear and linearized stability assessment during railway vehicle design / O. Polach // Vehicle System Dynamics. – 2006. – Vol. 44. – Iss. sup1. – P. 129–138. doi: 10.1080/00423110600869537
- True, H. The Dynamics of a Railway Freight Wagon Wheelset with Dry Friction Damping / H. True, R. Asmund // Vehicle System Dynamics. – 2002. – Vol. 38. – Iss. 2. – P. 149–163. doi: 10.1076/vesd.38.2.149.5617
- Xia, F. On the dynamics of the three-piece-freight truck / F. Xia, H. True // Proceedings of the 2003 IEEE/ASME Joint Rail Conference. – 2003. – P. 149–159. doi: 10.1109/rrcon.2003.1204661

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