DOI: https://doi.org/10.15802/stp2018/148023

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

A. G. Reidemeister, S. I. Levytska

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


railway vehicle; motion stability; differential-algebraic equations

References


Demmel, D. (2001). Vychislitelnaya lineynaya algebra. Teoriya i prilozheniya. Moscow: Mir. (in Russian)

Lazaryan, V. A., Dlugach, L. A., & Korotenko, M. L. (1972). Ustoychivost dvizheniya relsovykh ekipazhey. Kyiv: Naukova dumka. (in Russian)

Orlova, A. M. (2008). Vliyanie konstruktivnykh skhem i parametrov telezhek na ustoychivost, khodovye kachestva i nagruzhennost gruzovykh vagonov. (Avtoreferat dissertatsii doktora tekhnicheskikh nauk). Emperor Alexander I St. Petersburg State Transport University, Saint Petersburg. (in Russian)

Bigoni, D., True, H., & Engsig-Karup, A. P. (2014). Sensitivity analysis of the critical speed in railway vehicle dynamics. Vehicle System Dynamics, 52(sup1), 272-286. doi: 10.1080/00423114.2014.898776 (in English)

Jeon, C.-S., Cho, H.-S., Park, C.-S., Kim, S.-W., & Park, T.-W. (2018). Critical speed of a Korean high-speed train through optimization with measured wheel profiles. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 232(1), 171-181. doi: 10.1177/0954409716662091 (in English)

Gasch, R., Kik, W., & Moelle, D. (1981). Non-Linear Bogie Hunting. Vehicle System Dynamics, 10(2-3), 145-148. doi: 10.1080/00423118108968657 (in English)

Cui, D., Li, L., Jin, X., Xiao, X., & Ding, J. (2012). Influence of vehicle parameters on critical hunting speed based on Ruzicka model. Chinese Journal of Mechanical Engineering, 25(3), 536-542. doi: 10.3901/cjme.2012.03.536 (in English)

Kalker, J. J. (1990). Three-Dimensional Elastic Bodies in Rolling Contact. Dordrecht: Springer. doi: 10.1007/978-94-015-7889-9 (in English)

Mao, X., & Chen, G. (2018). A design method for rail profiles based on the geometric characteristics of wheel-rail contact. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 232(5), 1255-1265. doi: 10.1177/0954409717720346 (in English)

Mazzola, L., Alfi, S., & Bruni, S. (2014). Evaluation of the hunting behavior of a railway vehicle in a curve. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 229(5), 530-541. doi: 10.1177/0954409713517379 (in English)

Moelle, D., Steinborn, H., & Gasch, R. (1979). Computation of Limit Cycles of a Wheelset Using a Galerkin Method. Vehicle System Dynamics, 8(2-3), 168-171. doi: 10.1080/00423117908968592 (in English)

Molatefi, H., Hecht, M., & Kadivar, M. H. (2006). Critical speed and limit cycles in the empty Y25-freight wagon. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 220(4), 347-359. doi: 10.1243/09544097jrrt67 (in English)

Polach, O. (2003). Bogie design for better dynamic performance: example of a locomotive bogie. European Railway Review, 1, 69-77. (in English)

Polach, O., & Kaiser, I. (2012). Comparison of Methods Analyzing Bifurcation and Hunting of Complex Rail Vehicle Models. Journal of Computational and Nonlinear Dynamics, 7(4), 041005. doi: 10.1115/1.4006825 (in English)

Polach, O. (2006). Comparability of the non-linear and linearized stability assessment during railway vehicle design. Vehicle System Dynamics, 44(sup1), 129-138. doi: 10.1080/00423110600869537 (in English)

True, H., & Asmund, R. (2003). The Dynamics of a Railway Freight Wagon Wheelset With Dry Friction Damping. Vehicle System Dynamics, 38(2), 149-163. doi: 10.1076/vesd.38.2.149.5617 (in English)

Fujie Xia, & True, H. (n.d.). (2003). On the dynamics of the three-piece-freight truck. Proceedings of the 2003 IEEE/ASME Joint Railroad Conference. doi: 10.1109/rrcon.2003.1204661 (in English)


GOST Style Citations


  1. Деммель, Дж. Вычислительная линейная алгебра. Теория и приложения / Дж. Деммель ; пер. с англ. Х. Д. Икрамова. – Москва : Мир, 2001. – 435 с.
  2. Лазарян, В. А. Устойчивость движения рельсовых экипажей / В. А. Лазарян, Л. А. Длугач, М. Л. Коротенко. – Киев : Наук. думка, 1972. – 198 с.
  3. Орлова, А. М. Влияние конструктивных схем и параметров тележек на устойчивость, ходовые качества и нагруженность грузовых вагонов : автореф. дис. ... д-ра техн. наук : 05.22.07 / Орлова Анна Михайловна ; Петербург. гос. ун-т путей сообщения. – Санкт-Петербург, 2008. – 32 с.
  4. Bigoni, D. Sensitivity analysis of the critical speed in railway vehicle dynamics / D. Bigoni, H. True, A. P. Engsig-Karup // Vehicle System Dynamics. – 2014. – Vol. 54. – Iss. supp1. – P. 272–286. doi: 10.1080/00423114.2014.898776
  5. Critical speed of a Korean high-speed train through optimization with measured wheel profiles / C.-S. Jeon, H.-S. Cho, C.-S. Park, S.-W. Kim, T.-W. Park // Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit. – 2018. – Vol. 232. – Iss. 1. – P. 171–181. doi: 10.1177/0954409716662091
  6. Gasch, R. Non-linear bogie hunting / R. Gasch, W. Kik, D. Moelle // Vehicle System Dynamics. – 1981. – Vol. 10. – Iss. 2-3. – P. 145–148. doi: 10.1080/00423118108968657
  7. Influence of vehicle parameters on critical hunting speed based on Ruzicka model / D. Cui, Li Li, X. Jin, X. Xiao, J. Ding // Chinese Journal of Mechanical Engineering. – 2012. – Vol. 25. – Iss. 3. – P. 536–542. doi: 10.3901/cjme.2012.03.536
  8. Kalker, J. J. Three-dimensional elastic bodies in rolling contact / J. J. Kalker. – Dordrecht : Springer, 1990. – 314 p. doi: 10.1007/978-94-015-7889-9
  9. 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
  10. Mazzola, L. Evaluation of the hunting behavior of a railway vehicle in a curve / L. Mazzola, S. Alfi, S. Bruni // Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit. – 2014. – Vol. 229. – Iss. 5. – P. 530–541. doi: 10.1177/0954409713517379
  11. Moelle, D. Computation of Limit Cycles of a Wheelset Using a Galerkin Method / D. Moelle, H. Steinborn, R. Gasch // Vehicle System Dynamics. – 1979. – Vol. 8. – Iss. 2-3. – P. 168–171. doi: 10.1080/00423117908968592
  12. Molatefi, H. Critical speed and limit cycles in the empty Y25-freight wagon / H. Molatefi, M. Hecht, M. H. Kadivar // Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit. – 2006. – Vol. 220. – Iss. 4. – P. 347–359. doi: 10.1243/09544097jrrt67
  13. Polach, O. Bogie design for better dynamic performance: example of a locomotive bogie / O. Polach // European Railway Review. – 2003. – No. 1. – P. 69–77.
  14. 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
  15. 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
  16. 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
  17. 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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