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

Dep. «Cars and Car Facilities», Dnipropetrovsk National University of Railway Transport named after Academician V. Lazaryan, Lazaryan St., 2, Dnipro, Ukraine, 49010, tel. +38 (056) 373 15 19, e-mail reidemeister.a@gmail.com, ORCID 0000-0001-7490-7180 Dep. «Foreign Languages», Prydniprovsk State Academy of Civil Engineering and Architecture, Chernyshevsky St., 24 A, Dnipro, Ukraine, 49000, tel. +38 (056) 756 33 56, e-mail svetik23com@ukr.net, ORCID 0000-0001-6725-0280


Introduction
Studies on the railway vehicle motion stability have been under the spotlight since the 1950s.Loss of stability is accompanied by the emergence of large transverse forces that threaten the safety of movement, which prevents from operating cars at high speeds.Among the extensive literature devoted to this issue, we point out [1][2][3][4][5][6][7][8][9][10][11][12][13][14].In accordance with modern concepts, loss of stability is a very complex phenomenon, which near the critical speeds is described by the subcritical Hopf bifurcation.Up to a certain velocity v1 there is only one attractor corresponding to a straight-line motion, then a periodic attractor appears, while the original one remains and disappears at the velocity  2 >  1 .At high velocities, chaotic attractors may appear.
There may be cases when they occur already at the velocity  1 [5].The following methods of motion stability analysis are used [15]: 1) Linearization of the motion equations (Lyapunov's stability criterion of linear approximation [1]); 2) Quasi-linearization; 3) Galerkin-Urabe method [12,13] (quasilinearization by several frequencies, a large amount of computational work is required); 4) «Brute force» method, when one reduces the movement speed and waits for the auto-oscillations to disappear; to determine the unstable limit cycle, one gradually increases the disturbance range [14]; 5) Trajectory tracing method (the motion is assumed to be periodic, and the equation (0) = () is solved; it is not suitable for the study of quasi-periodic and chaotic oscillations).
Despite the obvious unsuitability to analyze the complex picture of the emergence and disappearance of attractors, Lyapunov's stability criterion of linear approximation retains its attractiveness due to its simplicity and ability to do the main thingto evaluate the critical velocity.It is formulated for the systems that describe ordinary differential equations.In the present paper we will extend it to the systems whose motion is defined by Lagrange differential-algebraic equations (DAE) of the first kind.Nowadays, due to the spread of standard integration programs (for example, DASSL), DAE are increasingly used in modeling railway vehicle oscillations, since they make it possible to do both without dependent generalized coordinates and without replacing rigid constraints between the car parts with high rigidity elastic elements.

Purpose
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 structure of the railway vehicle motion equations is as follows: (without nonlinear and non-uniform terms describing the movement along a curve).Here q is the generalized coordinate vector; M is the inertial coefficient matrix; C, B are the rigidity and viscosity matrices; K, F are the matrices describing the wheelrail interaction.Equation ( 1) is obtained if we remove the dependent generalized coordinates from the vector q using the equations of constraints.
When applying the Lagrange equation of the I kind, another approach is used: instead of eliminating the elements of the vector q, they are all remained, the constraint equations are included in the full set of equations describing the system motion, and additional unknowns λ are introduced (in the amount equal to the number of constraint equations) so that all these equations can be solved.The result is the following system of equations: The last expression is the equation of the constraints which the mechanical system is subject to.We will assume that the matrix L is constant (depends neither on time nor on system phase coordinates).The system of equations ( 2) and ( 3) is linear, so its solution is: where the constants j C are found from the initial conditions.The indices j p together with nonzero eigenvectors γ j , j l are solutions of the equation It is possible to understand whether motion is stable or not, by the sign of the real part of the values j p -if there are positive numbers among them, the motion is stable.It is inconvenient to search for numbers j p , equating the determinant of the left matrix to zero.Instead, we reformulate the problem so that the indices j p turn out to be eigenvalues of a certain matrix.From (2) it follows that Multiplying the resulting expression by L and using the fact that 0 Lq  , we get The matrix 1 T

LM L
 is non-degenerate (if the constraint coefficient matrix L has less rows than columns, and the rank is equal to the number of rows, which we assume), therefore Substituting this expression into the original equation, we get 11 Thus, the vector of phase coordinates   T qq satisfies the differential equation us consider how they are related to the eigenvalues and eigenvectors of the original system with constraints, that is, if they satisfy the equation ( 4) with a suitable choice of the vector of Lagrange multipliers j l .We will need an obvious correlation 0 LQ  .Multiplying the left expression by L 21 ( ) ( ) we will get 2 0 jj pL γ  .Therefore, for nonzero j p the vector γ j satisfies the constraint equation 0 j Lγ  .Equation ( 4) is easy to rewrite as 2 ( ) ( ) Thus, with nonzero j p the vectors γ j satisfy the equation ( 4) with   It is not clear whether the vectors   satisfy the equation ( 4) for 0 j p  , but, since these solutions correspond to constant processes that are of no interest, we will not deal with them.
Thus, the stability condition of the system with constraints is as follows: where j p are eigenvalues of matrix A. Let us apply the above theory to the study of stability, natural frequencies and vibration modes of a simplified mechanical system consisting of half a car body and a 3-piece bogie, on which it rests (Fig. 1).We consider the motion only in the horizontal plane.The system consists of (half) the body with a bolster, two side frames and two wheel sets.The body and the bolster are connected by a hinge in the center plate arrangement, the bolster with side frames and the side frames with wheel sets -by elastic elements that prevent relative translational movements in the longitudinal and transverse directions, as well as relative angular movements of hunting of the interacting bodies.
There are no dissipative elements in the system.The degrees of freedom are listed in Table 1.x, y,  indicate small movements of recoiling, swaying and hunting, for wheel sets the coordinate  is chosen so that ()   is a small deviation of the angular velocity of wheel set rotation relative to its axis from the value / Vr (V is the car velocity, r is the wheel radius), corresponding to the undisturbed motion.
T ab le 1

Body
Degrees of freedom Generalized coordinates Body with bolster (bd) x We will be interested in how the frequencies and forms of oscillations of the system without constraints (SF) and systems, whose displacement is subject to the following restrictions, correlate: SCX -it is prohibited to move the bolster relative to the side frames (in the spring suspension openings) in the longitudinal direction; SAJ -it is prohibited to move the pedestal openings of the side frames relative to the wheel set axle journals (side frames are pivotally connected to the wheel sets).
As for system parameters, the meaning of the notation for rigidity coefficients and basic dimensions is clear from Figure 1: the letters m, I with corresponding indices denote the masses and central moments of body inertia, the coefficients in the expressions for the interaction forces are explained below, the capital letters X, Y,  denote the force components and the system body interaction force moments.Without giving a complete derivation of the expressions for the matrices M, L, etc, let us dwell only on certain points that may be of methodical interest.The elements of the matrix C are coefficients for the products of generalized coordinates and their variations in the expression for the virtual work of forces in elastic elements Let us consider the contribution (b)  C to the ma- trix C from the elastic elements that are in axle boxes.The components of the displacement of the side frame pedestal opening relative to the wheel set axle box are combined into a vector They are linear combinations of the generalized coordinates Comparing the expressions ( 6) and ( 7), we get: (contributions from other elastic elements).
In order to prohibit linear movements of the pedestal openings of the side frames relative to wheel set axle boxes, it is necessary to require the fulfillment of the conditions: There are 8 rows in the L matrix, which we get by writing the first two rows of each matrix under each other.Thus, the compilation of a system of equations describing the motion of a mechanical system with constraints does not practically require additional calculationsin our case, the matrices (b) mj D were written out at the stage of working with the system without constraints.

Findings
Let us consider the results of the calculation of the eigenvalues and eigenvectors describing the 3piеce bogie oscillations.Our goal is to understand how the eigenvalues and eigenvectors of SF system with constraints and SCX and SAJ systems without constraints are related.We expect that the results for SF with (b)   x C , (b)   y C  will tend to the re- sults for SAJ, and the results for SF with (b)   x C  to the results for SCX.The subject of the study will be the confirmation of this expectation and a detailed description of the limiting transition nature.
The eigenvalues of the matrix  for the SF and SAJ systems are listed in Table 2.The system parameters correspond to the 4-axle car loaded up to deadweight capacity on 18-100 bogies (with an axle load of 23.5 tf).The motion speed 100 V  km/h.The eigenvalues were ordered by the QR algorithm, so they can be compared only by values.Even without analyzing the eigenvectors, it is clear that the numbers with 9, 11, 14 j  of the SAJ system are the limits for the eigenvalues 25, 27, 29 j  of the SAF system.It seems plausible to assume that large negative numbers of one system go into large negative numbers of the other system, both systems have five such numbers, but the correspondence between them is not obvious.It is not quite clear which of the numbers of the SF system goes into the number 6.29 335i  of the SAJ system.The numbers 9, , 24 j  of SF, except for one pair, apparently correspond to the side frame oscillations on the high rigidity elastic elements in the axle boxes, since these numbers have a large imaginary component.
The study of eigenvectors confirms the conclusions made and allows for some refinements.Let us consider the SAJ system with hinges in axle boxes.Equations of constraints do not violate the first 15 eigenvectors: 1, 2) non-physical solutions, which appeared due to the fact that there are no variables (ws)   m  in the equations of motion, there are only their derivatives; 3, 4, 13) extremely rapidly decaying solutions describing the motion of wheel sets against pseudo-slip (for example, bogie swaying without hunting); For all these vectors, one can find the corresponding eigenvectors of the SF system with close values of the components.Some vectors j γ are shown in Table 3.The vectors 13 γ , 27 γ for a bogie without constraints, with large rigidity of elastic elements in the axle boxes are almost coincide with the vectors 5 γ , 11 γ for a bogie with hinges in boxes.The vector 9 γ (SF) describes the longitudinal oscillations of the side frames relative to the wheel sets, which is incompatible with the constraints to which the SAJ system is subordinate, and it is impossible to find a corresponding vector among the eigenvectors of the latter.The bogie movement is unstable, the eigenvalues 27 p (SF) and 11 p (SAJ) have a positive real part.Wheel sets perform selfoscillations of hunting and swaying (the ratio between the amplitudes y and  is as in the Klingel solution), and the body swaying is twice as large as wheel set swaying. Figure 2    If a rigid longitudinal constraint in the spring suspension is added to the hinges in the axle box (Table 2, column SAJ + SCX), then the oscillation patterns 5, 6, 14, 15 in the SAJ system, which are accompanied by deformations of the spring groups in the longitudinal direction will disappear and four more eigenvectors, corresponding to zero eigenvalues and violating equations of constraints, will be.Other eigenvalues will change slightly.

Originality and practical value
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.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.

Conclusions
1.An effective method for studying the stability of railway vehicle motion, described by the Lagrange equations of the first kind, has been proposed.Stability criterion -the real numbers of exponential functions that satisfy the equations of motion -should not be greater than zero.The indicators themselves can be found as eigenvalues of a certain matrix A, depending on the matrices of physical parameters M, B, F, C, K and the matrix of constraint coefficients L, using the QR algorithm [2, chapter 4].
2. The eigenvectors of this matrix, corresponding to nonzero eigenvalues, satisfy the equations of constraints.The advantage of the proposed method is the easy algorithmization of the motion equation derivation (no need to choose independent generalized coordinates).

.
The eigenvectors of the matrix A, corresponding to the eigenvalues j p , has the form  

11 c , 22 c
for the longitudinal and transverse directions are considered equal to 3.90.The expression for longitudinal sliding additionally contains terms proportional to the velocities(ws) shows how the components of the corresponding eigenvector change as rigidity changes( )