Introduction
Smooth and safe guiding of wheel sets in curves and keeping them stable require a small tolerance for contact geometry at rail head level. Therefore, grinding track surface to maintain proper conditions for rolling is an essential issue for high speed lines. Railway vehicles lateral oscillation represents a major problem for high speed railways, which can be reduced by modifying vehicles and by improving track lateral resistance. Lateral movements cause passenger discomfort, and in some cases because of them the speed average has to be reduced. In time, this could be the source for some trouble. Nowadays, many high speed lines are in service all around the world. On some special high speed lines in Europe, a speed of 330 km/h is possible. In Germany, on some railway lines, traffic is combined: passenger high speed trains during the day and freight trains during night. No doubt, train high speed produce high dynamic forces. Contact forces produced by the powerful acceleration of motor wheels amplify surface alteration in wheel-rail contact area.
Lateral variable dynamic forces existing between the vehicle and the track can be produced by the rail and wheel profile, which are especially designed to control wheel lateral movements, and to improve wheel – rail contact. Lateral suspension and vehicle rolling features can amplify lateral forces, but this type of lateral oscillation can be managed by a special design and an efficient maintenance.
Lateral forces and wheel-rail dynamic, typical for locomotives with two- wheel bogies
Rolling features depend on suspension, permitted clearance between wheel, bogie and vehicle structure, and on motor wheel effect. Track geometry represents a critical element. An efficient maintenance is easier to obtain if the vehicle has a high construction standard, and a big mass per length unity. Other rail or vehicle damages could lead to excessive vehicle swinging, gallop or vertical instability, all those amplifying lateral forces. Under the pressure of these forces, rails can bend, causing temporary or permanent deformation, depending on the intensity of the lateral force.
Permanent deformations are called origin distortions. If these distortions accumulate, because of frequent rolling of poorly maintained rolling stock, the rails shall progressively deteriorate and produce lateral oscillations to other vehicles. This is a vicious circle which leads to wear or even damage of tracks and vehicle suspensions. Lateral forces repeatedly worked on rail by the wheel is never higher than a certain H value, origin distortions do not sum up endlessly, and final distortion is set out in admissible limits.
The limiting value is here represented by the H force value (1), where H = lateral force, P [kN] = static wheel load. This proportion was first designed for ballasted tracks with sleepers, without thermal variations. The H force is considered the rail breaking point where admissible standards of origin distortion are fulfilled, under the pressure of repeated lateral load, which vies with the vertical load P. Prud’homme has later extended his research to consider thermal variation in the welded rail, and used a 0.85 multiplication factor. The final form of Prud’homme limit is shown at (2).
Test results conducted on BR 189 locomotives, on concrete sleepers, welded rail covered with ballast, with dynamic track stabilizers, on high speed lines did not determined important changes for Prud’homme formula, as proved in practice. Figure one shows the second approach based on the proportional external load of the wheel. The horizontal force H which acts in the load centre, at a c height from the railhead, incurs resistance from H1 and (H – H1) horizontal forces which act on the rails. Also, a wheel load variation R=Hc/e appears, where e is the distance between the line centers.
The increased wheel load is represented by Q+R (the dynamic wheel overload), and the external wheel load is represented by Q-R. The proportional external load is calculated as R/Q = Hc/Q.e. A certain value of external load shall result when H reaches the Prud’homme limit on normal gauge. This will not lead to either rolling discomfort, or derailments.
Mathematically speaking, wheel oscillations in curves have an unstable character. Lateral sliding friction of wheels in curves, where the gauge slightly extends compared to standard gauge, adds to oscillations that appear at wheel-rail contact level, where friction forces due to stick-slip phenomenon also interact. Practice showed that lateral movement speed of the wheel in curves remains constant. Also, sliding friction forces are proportional in phase (3): If excitation is harmonic, then and, if there is a column matrix with real elements which defines an excitation mode with “in phase” forces so that at any beat, displacements should be all “in phase” between them, but not compulsory with forces ( represents the alteration of phase between forces and displacements), than the matrix has real elements.
The resonance appears when main frequency of the input signal (when frequency leads to extreme values of power spectral density of excitation) is the same as the frequency of oscillating system. If the input signal features a damped harmonic correlation, an enhancement of vibration system shall appear, acting as a maximum value of quadratic average acceleration when harmonic correlation frequency reaches the frequency of oscillating system. In this case, power spectral density of input signal as shown at (4).
For a dissipative system with dry friction (BR 182 electric locomotive suspension) the movement equation can be presented as in (5), – elastic force, where R is a positive constant. If this is the case, than the equation solutions presented as in (7). If the mechanic system considered has neutral balance, energy equation can be presented as in (8). In this case, speed is positive in the first semi-oscillation and negative in the second and periods can be calculated after determining extreme elongations.
If spectral densities of any stable “” or flexible functions are known, then quadratic average deviation of this function and of its derivative until m order are determined as at (11), where the function is a symmetric function representing the power spectral density of the function, which is stable, ergodic and determinable by the Fourier transform. If the last condition is met, the white noise appears and Fourier integral is divergent.
Identifying the function with the white noise means defining the white noise as any function with the identically null anticipated value, represented by the Dirac symbol distribution. The Green specific function represents the intensity of the white noise.
Bibliography
1. Burstow, M. C., Watson, A. S. and Beagles, M. (2003) Simulation of rail wear and rolling contact fatigue using the Whole Life Rail Model. Proceedings of ‘Railway Engineering 2003’, London 30th April-1st May 2003;
2. Clark, S. L., Dembosky, M. A. and Doherty, A. M. (2003) Dynamic wheel/rail interactions affecting rolling contact fatigue on the British railway system, Proceedings of the World Congress on Railway Research (WCRR 2003), pp 392-399, Edinburgh, September 2003;
3. Dumitru, G. Consideraţii asupra unor aspecte legate de dinamica vehiculelor motoare de cale ferată, Revista MID-CF, no. 1/2008.
4. Dwyer-Joyce,R.S., Lewis, R., Gao, N, Grieve, D.G., (2003), “Wear and Fatigue of Railway Track Caused by Contamination, Sanding and Surface Damage”, Proceedings of 6th International Conference on Contact Mechanics and Wear of Rail/Wheel Systems (CM2003), Gothenburg, Sweden, pp. 211-220;
5. Fletcher, D. I. and Beynon, J. H. (2000) Equilibrium of crack growth and wear rates during unlubricated rolling-sliding contact of pearlitic rail steel, Proc. Instn Mech. Engrs 214 Part F, 93-105;
6. Garnham, J. E. (1991) Crack Initiation in Rolling Contact Fatigue, Final Report, University of Leicester report, February 1991;
7. Ringsberg, J. W. and Josefson, B. L. (2000) Finite element analysis of rolling contact fatigue crack initiation in railheads, J. Rail and Rapid Transport;
8. Sebeşan, I. “Tehnica Marilor Viteze la Vehiculele de Cale Ferata” – Note de curs, 2001;
To be continued in the April issue 2011
[ by Marian Călin, PhD Candidate Eng. UPB., Manager of The National Railway Authority ASFR – AFER; George Dumitru, PhD. Eng., Head of Department of Locomotives & Rolling Stock – The National Railway Training Centre – CENAFER ]
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