Effect of the wheel/rail contact angle and the direction of the
saturated creep force on rail corrugation
X.L. Cui
1, G.X. Chen
1*, W.J. Qian
1, Q. Zhang
1, H. Ouyang
2, M.H. Zhu
1 1State Key Laboratory of Traction Power, Tribology Research Institute, Southwest Jiaotong University, Chengdu, Sichuan 610031, China2School of Engineering, University of Liverpool, Brownlow Street, Liverpool L69 3GH, UK
Abstract
The purpose of present work is to obtain a further understanding of rail corrugation on tight curved tracks. The stability of a wheelset-track system is studied using the finite element complex eigenvalue method. Based on the viewpoint of friction-coupling self-excited vibration causing rail corrugation, an elastic vibration model of the leading wheelset-track system is established. It is assumed that the creep forces at the inner and outer wheels of the leading wheelset are saturated when a vehicle negotiates a tight curved track, so the tangential forces are equal to the normal forces times the dynamic coefficient of friction. The simulation results demonstrate that the saturated creep force can induce self-excited vibration of the wheelset-track system, which is able to cause rail corrugation. The effects of the contact angle and the direction of the saturated creep force on the self-excited vibration of the leading wheelset-track system are studied. Parameter sensitivity analysis shows that the contact angle and the direction of the saturated creep force have significant effects on self-excited vibration of the leading wheelset-track system. Rail corrugation easily occurs when the contact angle is small. The saturated creep force in the lateral direction more easily produces rail corrugation than the saturated creep force in the longitudinal direction of the track does.
Keywords: Rail corrugation, Wear, Self-excited vibration, Contact angle, Traction angle
1. Introduction
Rail corrugation is one of the most serious problems in railway engineering, which causes fierce vibration of the structures of the railway vehicle and noise. It not only influences the comfort of passengers, but also reduces the service life of structural components. Furthermore, serious rail corrugation can lead to derailment accidents. Mild corrugation with small amplitude can be removed by grinding but rails with severe corrugation have to be replaced with new ones. The cost of grinding and replacement work is very high. Nowadays, completely eliminating or suppressing corrugation is still the best solution. In order to prevent the generation of rail corrugation, many studies have been performed all over the world since the end of 19th century. Based on recently
1*Corresponding author. Tel.: +86 28 87634122; fax: +86 28 87634122.
published papers on rail corrugation [1-5], the generation mechanisms of rail corrugation are roughly divided into
two major categories: the transient dynamic interaction and the stick-slip vibration.
In the first category, it is believed that a transient dynamic interaction between the wheel and the rail causes fluctuations of the friction work, and then rail corrugation is generated. In this case, it is assumed that the original rail head is discontinuous. When wheels roll over a rail repeatedly, the unevenness causes fluctuations of contact forces between wheels and a rail, which result in different wear rates of rail surfaces to generate corrugated wear. Grassie and Johnson [6] proposed the first model of the independence of corrugation wavelength on speed based
on differential wear. However, as well as later Frederick [7] and Bhaskar et al [8,9], they failed to explain the
growth of corrugation in the wavelengths of interest (20 - 80 mm) on continuous track model, due to a "limitative" assumption: constant creepage between the wheel and the rail. In this field, Barber [10], Afferrante and Ciavarella
[11,12] pointed out that the short-pith corrugation can exist at a "resonance-free" regime for rails with continuum
supports and at the pinned-pinned resonance regime, as suggested also by Grassie [4], for rails with discrete supports. In other aspects of the corrugation research, Muller [13] proposed that the high lateral rail receptance and
the low vertical rail receptance could also dominate the profile of corrugation development. Igeland and Ilias [14]
firstly considered nonlinear factors in the study of rail corrugation. They found that the shift of contact points and the distribution of wear within the contact area played an important role on rail head wear. Nielsen [15] presented a
nonlinear wear model for a cylinder rolling over a periodically varying surface to investigate the evolution of rail corrugation. He found that the amplitude of corrugation would grow exponentially with the number of cylinder passages. Wu and Thompson [16] numerically investigated the effect of multiple wheel-track passes by using a
frequency domain model. Jin et al. [17,18] developed a more complicated three-dimensional train-track model that
combined Kalker’s rolling contact theory with non-Herzian form, a linear frictional work model and a dynamic model of a half railway vehicle coupled with the curved track. Using this model, they researched the effect of discrete sleepers on rail corrugation.
In the second category, rail corrugation is attributed to stick-slip vibration of the wheelset-track system. Although researchers who accept this viewpoint are fewer than those who accept the previous viewpoint, the impact of this viewpoint continues to date. Clark et al. [19,20] investigated the self-excited vibration of a flexible
wheelset and a discretely supported track system under high creepage conditions. Brockley [21] studied rail
corrugation from the viewpoint of friction-induced vibration and derived a formula expressing the relationship between the corrugation wear and the friction-induced vibration. Matsumoto et al. [22] also proposed that rail
corrugation in curved sections of track was created by stick-slip vibration of the wheel and rail system due to a large creepage, which caused a fluctuation of the normal force between the wheel and rail. Sun and Simson [23]
carried out a comprehensive study on rail corrugation due to wheel stick-slip process on a curved track. Eadie et al. [24] experimentally studied the effect of the friction coefficient on rail corrugation and concluded that a low
friction coefficient between the wheel and rail might suppress rail corrugation. Chen et al. [25,26] studied the
good prediction of rail corrugation frequency, which is very close to the actual frequency of rail corrugation at the tight curved tracks. Kurzeck [27] also conducted an independent study on rail corrugation from the viewpoint of
the mode-coupling in self-excited vibration of a wheel-rail system.
It should be noted that the mode coupling in self-excited vibration of a wheel-rail system used in Ref. [Error: Reference source not found,Error: Reference source not found] is to some extent different from the traditional stick-slip vibration of a wheel-rail system used in Ref. [Error: Reference source not found]. In the mode coupling equations of motion of the wheel-rail system, the friction coupling between the wheel and the rail must be used but the negative friction-speed slope of the saturated creep force may not be used. In the stick-slip equations of motion of the wheel-rail system, no friction coupling between the wheel and the rail is used but the negative friction-speed slope of the saturated creep force must be used. One of the drawbacks of the rail corrugation model established based on the traditional stick-slip vibration of a wheel-rail system is that the model only predicts rail corrugation of frequency 20-80 Hz [Error: Reference source not found], which is far from the actual corrugation frequency of 200-800 Hz. It is a very common physical phenomenon that rail corrugation always firstly occurs on the low rail in a tight curved track. However, this common physical phenomenon appears not to have been explained satisfactorily in the literatures. Most of the rail corrugation models published in the literature consists of a single wheel, a single rail and a series of sleepers. Obviously, this type of models cannot be used to satisfactorily explain the common physical phenomenon of rail corrugation.
On the basis of previous studies, it can be concluded that the formation and the development of rail corrugation are influenced by many factors. The purpose of the present work is to study this issue. In the present paper, according to the viewpoint of friction-coupling self-excited vibration causing rail corrugation, the elastic vibration models of a leading wheelset-track system at different contact angles are built, which consist of a leading wheelset, two rails, a series of sleepers, support springs and dampers. The creep forces between wheels and rails on the leading wheelset are assumed to be saturated, so the tangential forces are equal to the normal forces times the dynamic coefficients of friction. There are different contact conditions between wheels and rails when a vehicle travels on a tight curved track. The finite element complex eigenvalue analysis method is applied to study the effect of the contact angle between the wheel and the rail and the direction of the saturated creep force on self-excited vibration of the wheelset-track system and rail corrugation. Simulation results show that the contact angle has an important influence on self-excited vibration of the wheelset-track system, and then influences the generation of the rail corrugation. The direction of the saturated creep force also has an influence on rail corrugation.
The contents of this paper are as follows. First, a brief review of progress in rail corrugation research is introduced. Next, the establishment of the finite element model of the wheelset-track system and the theoretical basis of analysis are presented. In Section 3, numerical results are given and discussed. Finally, conclusions close the paper.
2.1. The contact model of the wheelset-track system
Dynamic simulations of the vehicle curve negotiation and field measurements all show that when a train travels on a tight curved track, the leading wheelsets of both front and rear trucks have positive angles of attack and the trailing wheelsets have positive or negative angles of attack mainly depending on running speed, as shown in Fig. 1a [Error: Reference source not found]. The simulation results also show that in the tight curved track, the creep forces on the inner and outer wheels of the leading wheelsets are generally saturated, so that the creep forces are equal to the normal forces times the dynamic friction coefficients. However, the creep forces on the inner wheels of the trailing wheelsets are highly unlikely to be saturated. In the present paper, only the leading wheelset together with two rails and a series of sleepers are included in modelling the wheelset-track system. Since the creep force is the resultant force of the longitudinal and lateral creep forces, the direction of the saturated creep force changes in the range of 90° between the lateral and longitudinal directions of the curved track depending on the values of the lateral and longitudinal creep forces. Fig. 1b shows the contact condition of the leading wheelset and the track when a train negotiates a tight curved track [Error: Reference source not found]. In the present paper, contact geometry parameters of the contact model are determined by curve negotiation calculation using the Simpack package. Specific parameters will be presented in Section 2.5.
Fig. 1. Wheel-rail contact model in a tight curved track. (a) wheelset positions. (b) wheel-rail contact condition.
As shown in Fig. 1, FSVL, FSLL and FSVR, FSLR are the left and the right suspension forces, respectively. δL and δR
are the left (outer) and the right (inner) contact angles, respectively. NL and NR are the normal forces at the left
(high) and the right (low) contact points, respectively. FL and FR are the lateral creep forces at the left (high) and
the right (low) contact points, respectively. KRV, KRL are the vertical and lateral stiffness values of each rail
fastener, respectively. CRV, CRL are the vertical and lateral damping values of each rail fastener, respectively. KSV,
KSL are the vertical and lateral combined support stiffness values of each sleeper and the subgrade, respectively.
CSV, CSL are the vertical and lateral combined support damping values of each sleeper and the subgrade,
respectively.
2.2. The finite element model of the wheelset-track system
In the present paper, authors focus on the elastic vibration of the wheelset-track system, which includes rigid body motion of the wheelset-track system. Considering the complexity of the system, a leading wheelset, two rails and a series of sleepers are built in this model. According to the wheelset-track contact model of Section 2.1, a finite element model of the leading wheelset-track system is established, in which the contact angles can be changed easily. The 8-noded hexahedral element (C3D8I) is selected to mesh the model. Each rail has 105280 nodes and 71010 elements. Each sleeper part has 882 nodes and 600 elements. The wheelset part has 68922 nodes and 58176 elements. The overall finite element model of the wheelset-track system is shown in Fig. 2. In the finite element model, the rail mass is 60 kg/m, the total length is 60 m and it is assumed that the rail ends are respectively hinged. The wheel of nominal diameter of 840 mm with worn tread profile is chosen. The wheelset is placed in the central position of rails. This arrangement can decrease the influence of the constraint boundary conditions of rail ends on simulation results. Sleepers are made from concrete and the distance between two sleepers is 600 mm. Since there are a large number of fasteners between the rails and sleepers, the rails are supported by a group of lateral and vertical springs and a group of lateral and vertical dampers at each position of sleepers in contact with the rails. Stiffness values and damping values of the rail fastener are evenly distributed on every node. Similarly, every sleeper is supported by a group of grounded springs and dampers. The distribution of springs and dampers is shown in Fig. 3.
Fig. 2. Overall finite element model of the wheelset-track system. (a) The isometric view of the model. (b) The front view of the model.
Fig. 3 Distribution details of springs and dampers.
2.3. Finite element equations of the friction-induced self-excited vibration of the wheelset-track system
(a) (b)
When a train negotiates a tight curved track, the creep forces at the inner and outer wheels of the leading wheelset probably become saturated [28]. In the present work, the creep forces between wheels and rails are always
assumed to be saturated. In this case, the creep forces are similar to the friction forces and approximately equal to the normal forces times the dynamic coefficient of friction. Abaqus package is used to analyze the dynamic stability of the wheelset-track system. The model is solved using the complex eigenvalue method. The equations of motion of the wheelset-track system are established primarily based on Yuan’s method [29]. Here, the
methodology of the complex eigenvalue analysis is introduced briefly. The equations of motion of the wheelset-track system are generally written as follows:
, (1)
1[?] Y. Sato, A. Matsumoto, K. Knothe, Review on rail corrugation studies, Wear, 252 (2002) 130-139.
5[?] S.L. Grassie, Rail corrugation: characteristics, causes, and treatments, Proceedings of the Institution of Mechanical Engineers Part
F-Journal of Rail and Rapid Transit, 223 (2009) 581-596.
6[?] S.L. Grassie, K.L. Johnson, Periodic microslip between a rolling wheel and a corrugated rail, Wear, 101 (1985) 291-309.
7[?] C. Frederick, A rail corrugation theory, Proceedings of the International Symposium on Contact Mechanics and Wear of Rail-Wheel
Systems, II, 1986, pp. 181-211.
8[?] A. Bhaskar, K.L. Johnson, G.D. Wood, J. Woodhouse, Wheel-rail dynamics with closely conformal contact .1. dynamic modelling and
stability analysis, Proceedings of the Institution of Mechanical Engineers Part F-Journal of Rail and Rapid Transit, 211 (1997) 11-26.
9[?] A. Bhaskar, K.L. Johnson, J. Woodhouse, Wheel-rail dynamics with closely conformal contact .2. forced response, results and
conclusions, Proceedings of the Institution of Mechanical Engineers Part F-Journal of Rail and Rapid Transit, 211 (1997) 27-40.
10[?] J.R. Barber, M. Ciavarella, L. Afferrante, A. Sackfield, Effect of small harmonic oscillations during the steady rolling of a cylinder on a
plane, International Journal of Mechanical Sciences, 50 (2008) 1344-1353.
11[?] L. Afferrante, M. Ciavarella, A. Sackfield, Rolling cylinder on an elastic half-plane with harmonic oscillations in normal force and
rotational speed. Part I: Solution of the partial slip contact problem, International Journal of Mechanical Sciences, 53 (2011) 989-999.
12[?] L. Afferrante, M. Ciavarella, M. Dell'Orco, G. Demelio, Rolling cylinder on an elastic half-plane with harmonic oscillations in normal
force and rotational speed. Part II: Energy dissipation receptances and example calculations of corrugation in the short-pitch range, International Journal of Mechanical Sciences, 53 (2011) 1000-1007.
13[?] S. Muller, A linear wheel-track model to predict instability and short pitch corrugation, Journal of Sound and Vibration, 227 (1999)
899-913.
14[?] A. Igeland, H. Ilias, Rail head corruption growth predictions based on non-linear high frequency vehicle/track interaction, Wear, 213
(1997) 90-97.
15[?] J.B. Nielsen, Evolution of rail corrugation predicted with a non-linear wear model, Journal of Sound and Vibration, 227 (1999) 915-933.
16[?] T.X. Wu, D.J. Thompson, An investigation into rail corrugation due to micro-slip under multiple wheel/rail interactions, Wear, 258
(2005) 1115-1125.
17[?] X.S. Jin, Z.F. Wen, K.Y. Wang, Z.R. Zhou, Q.Y. Liu, C.H. Li, Three-dimensional train–track model for study of rail corrugation, Journal
of Sound and Vibration, 293 (2006) 830-855.
18[?] X.S. Jin, Z.F. Wen, Effect of discrete track support by sleepers on rail corrugation at a curved track, Journal of Sound and Vibration, 315
(2008) 279-300.
where [M] is the mass matrix, which is symmetric and positive definite. [C] is the damping matrix, which can include friction-induced damping effects as well as material damping contribution. [K] is the stiffness matrix, which is asymmetric due to friction. x is the nodal displacement vector. The corresponding eigenvalue equation can be written as follows:
, (2)
where λ is the eigenvalue, and is the corresponding eigenvector.
When both the damping matrix and the asymmetric contributions to the stiffness matrix [K] are neglected, the eigenvalue problem becomes:
, (3)
where [Ks] is the symmetric part of the stiffness matrix, and is an eigenfrequency of the wheelset-track system.
This symmetric eigenvalue problem is solved using the subspace iteration method. The next step is that original matrices are projected in the subspace of real eigenvectors and given as follows:
, (4a)
, (4b)
. (4c)
Then, the complex eigenvalue problem is simplified as follows:
. (5)
Finally, the complex eigenvectors of the original system can be obtained by:
20[?] R. Clark, G. Scott, W. Poole, Short wave corrugations-an explanation based on stick-slip vibrations, Applied Mechanics Rail
Transportation Symposium, 1988, pp. 141-148.
21[?] C.A. Brockley, P.L. Ko, An investigation of rail corrugation using friction-induced vibration theory, Wear, 128 (1988) 99-105.
22[?] A. Matsumoto, Y. Sato, H. Ono, M. Tanimoto, Y. Oka, E. Miyauchi, Formation mechanism and countermeasures of rail corrugation on
curved track, Wear, 253 (2002) 178-184.
23[?] Y.Q. Sun, S. Simson, Wagon–track modeling and parametric study on rail corrugation initiation due to wheel stick–slip process on
curved track, Wear, 265 (2008) 1193–1201.
24[?] D.T. Eadie, M. Santoro, K. Oldknow, Y. Oka, Field studies of the effect of friction modifiers on short pitch corrugation generation in
curves, Wear, 265 (2008) 1212-1221.
25[?] G.X. Chen, Z.R. Zhou, H. Ouyang, X.S. Jin, M.H. Zhu, Q.Y. Liu, A finite element study on rail corrugation based on saturated creep
force-induced self-excited vibration of a wheelset–track system, Journal of Sound and Vibration, 329 (2010) 4643-4655.
26[?] W.J. Qian, G.X. Chen, H. Ouyang, M.H. Zhu, W.H. Zhang & Z.R. Zhou (2014): A transient dynamic study of the self-excited vibration
of a railway wheel set–track system induced by saturated creep forces, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, DOI: 10.1080/00423114.2014.924629.
27[?] B. Kurzeck., Combined friction induced oscillations of wheelset and track during the curving of metros and their influence on
, (6)
where i is the approximation of the ith eigenvector of the original system. The general solution of Eq. (5) is:
, (7)
where αi + ji is the ith eigenvalue. The real part of the eigenvalue is the basis of the judgment of system stability.
It can be seen that when the real part of an eigenvalue is larger than zero, the nodal displacement u(t) will increase with time, which means the vibration of the system is growing and the system will become unstable.
The effective damping ratio ξ is a parameter to measure the propensity of self-excited vibration occurrence. It is defined as:
. (8)
If the effective damping ratio is negative, the system becomes unstable and has a tendency to induce self-excited vibration. Generally speaking, the smaller the effective damping ratio, the more easily the corresponding self-excited vibration occurs.
2.4. Rail corrugation mechanism due to friction-induced vibration
In the field of corrugation investigation, it is generally accepted that fluctuating friction work results in undulant wear of rails [Error: Reference source not found]. Therefore, the friction work rate approach is applied to study the corrugation formation in relation to the self-excited vibration. According to the wear equation of rails proposed by Brockley [Error: Reference source not found], a simplified version of this equation is shown as follows.
, (9)
where w is the wear volume per unit time, K is the wear constant, H is the friction work rate (H = FV), F is the creep force, V is the relative velocity and C is the durability friction work rate. When a vehicle negotiates a tight curved track, it is assumed that the creep forces between the wheels and rails are saturated (F = μN), where μ is the friction coefficient between the wheel and rail, N is the normal contact force. The lateral velocity of the wheelset is V = ψ × v, where ψ is the angle of attack of the wheelset and v is the forward speed of the wheelset. The parameters of μ, ψ and v considered to be constant in the case. Therefore, the lateral velocity of wheels remains constant. The change of the friction work rate depends on the variation of the creep force.
2.5. Nominal parameters of the wheelset-track system
The parameters of the wheelset–track system are presented briefly as follows. The track gauge is 1440 mm. The superelevation of the curved track is 100 mm and the cant of the track is 1/40. The friction coefficient is set as μ = 0.4. It is assumed that the vehicle negotiates a curved track of radius 300 m at a speed of 70 km/h. According to the Simpack simulation results, the suspension forces are FSVL= 88000 N, FSVR= 72000 N, FSLL= 4000 N, FSLR=
3500 N, respectively. In this case, the suspension forces are assumed to be constant when a vehicle negotiates a tight curved track at a speed of 70 km/h. Meanwhile, according to the Simpack simulation results, it can be verified that the creep forces of leading wheelset on both high and low rail are saturated. According to the field measurements [Error: Reference source not found], stiffness values of the rail fastener are set as KRV = 7.8×107 N/
m and KRL= 2.947×107 N/m. Damping values of the rail fastener are set as CRV= 5.0×104 Ns/m and CRL = 5.2×104
Ns/m. The combined support stiffness values of the sleeper and the subgrade are set as KSV = 8.9×107 N/m and KSL
= 5.0×107 N/m. The combined support damping values are set as C
SV = 8.98×104 Ns/m and CSL = 4.0×104 Ns/m.
The material parameters of the wheel and the rail are introduced as follows. The density is set as ρ = 7800 kg/m3,
the Young’s modulusis set as E = 1.9×1011 Pa and the Poisson’s ratio is set as ν = 0.3.
3. Results and discussion
3.1. Mode shape of self-excited vibration of the wheelset-track system on a tight curved track
The finite element complex eigenvalue analysis is thought to be an effective method available to predict propensity of unstable vibration of friction sliding systems [30]. In the present work, the finite element complex
eigenvalue analysis is applied to study the propensity of the self-excited vibration of the wheelset-track system with different contact angles and different directions of the saturated creep force. In the present study, the unstable vibrations whose fundamental frequency is in the range from 50 Hz to 1200 Hz are considered to be responsible for rail corrugation. Fig. 4 shows the mode shape of self-excited vibration of the wheelset-track system. It occurs at a frequency of 501.31 Hz. From Fig. 4, it can be seen that self-excited vibration probably takes place only on the low rail and the corresponding wheel. That suggests that rail corrugation only occurs on the low rail. The conclusion is consistent with field results of railway lines [31].
30[?] H. Ouyang, W. Nack, Y.B. Yuan, F. Chen, Numerical analysis of automotive disc brake squeal:a review, Intermational Journal of
Vehicle Noise and Vibration, 1 (2005) 207-231.
31[?] X.S. Jin, Z.F. Wen, K.Y. Wang, Effect of track irregularities on initiation and evolution of rail corrugation, Journal of Sound and
Fig. 4. Mode shape of self-excited vibration of the wheelset-track system on a tight curved track of radius 300 m: the outer contact angle δL = 37.02o, the inner contact angle δR = 1.53o, lateral creep force is saturated, unstable vibration frequency fR=
499.41 Hz, damping ratio ξ = -0.02614.
3.2. Effect of the contact angle on self-excited vibration of the wheeleset-track system
Why rail corrugation always firstly occurs on the low rail in the tight curved track? It is an issue needed to be studied further. Obviously, the contact angle between the outer wheel and high rail is different from that between the inner wheel and low rail. Fig. 5 shows variations of the contact angles with lateral displacement of the wheelset. In order to study the effect of the contact angle on rail corrugation, we assume that the creep forces at the inner and outer wheels on the leading wheelset are saturated and the direction of the saturated creep force is unchanged. Only the contact angles are changed according to Fig. 5.
Fig. 5. Change of the contact angle with the lateral displacement of the wheelset (rail gauge = 1440 mm).
Fig. 6 shows the evolution of effective damping ξ and the distribution of self-excited vibration frequency fR
for different contact angles. When the outer contact angle δLis 2.65o, there is only one negative damping ratio.
With increasing outer contact angle, the number of self-excited vibration increases obviously. Fig. 7 shows mode shapes of all unstable vibrations when the outer contact angle δLis 6.63o. From Fig. 7, it is found that there are six
probably takes place only on the low rail and the wheel. It is also found that the mode shown in Fig. 7c has a negative damping ratio ξ = -0.02119. This damping ratio is the least among all these six damping ratios of unstable modes and suggests that this unstable mode most easily occurs.
Fig. 6. Evolution of unstable complex eigenvalues for different contact angles on a tight curved track. (a) δL= 2.65o, δR= 5.62o.
(b) δL= 3.01o, δR= 4.63o . (c) δL= 3.44o, δR= 4.25o. (d) δL= 4.56o, δR= 3.21o. (e) δL= 6.63o, δR= 2.56o. (f) δL= 8.41o, δR= 2.42o.
(g) δL= 15.21o, δR= 1.63o. (h) δL= 22.40o, δR= 1.51o. (i) δL= 35.83o, δR= 1.54o.
(b) (c)
(a)
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
R= 5.62o
E ffe ct iv e d a m pi ng r at io
Frequency fR L= 2.65o
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00 E ffe ct iv e d am p in g ra tio
Frequency fR L= 3.01o
R= 4.63o
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
R= 4.25o
E ffe ct iv e d am p in g ra tio
Frequency fR L= 3.44o
(d) (e) (f)
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
R= 3.21o
E ffe ct iv e d a m pi ng r at io
Frequency fR L= 4.56o
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
R= 2.56o
E ffe ct iv e d a m pi ng r at io
Frequency fR L= 6.63o
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
R= 2.42o
E ffe ct iv e d am p in g ra tio
Frequency fR L= 8.41o
(h)
(g) (i)
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
R= 1.54o
E ffe ct iv e d am p in g ra tio
Frequency fR L= 35.83o
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
R= 1.51o
E ffe ct iv e d a m pi ng r at io
Frequency fR L= 22.40o
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
R= 1.63o
E ffe ct iv e d am p in g ra tio
Frequency fR L= 15.21o
Fig. 7. Mode shapes of unstable vibrations when the contact angle δL= 6.63o. (a) unstable vibration frequency fR = 234.24 Hz,
damping ratio ξ = - 0.00390. (b) fR = 268.27 Hz, ξ = - 0.01823. (c) fR = 504.33 Hz, ξ = - 0.02119. (d) fR = 514.86 Hz, ξ = -
0.01055. (e) fR = 517.72 Hz, ξ = - 0.00903. (f) fR = 941.93 Hz, ξ = - 0.00041.
When the outer contact angle becomes 8.41o, there are only two negative damping ratios, whose values are
-0.01778 and -0.01781. Fig. 8 shows the corresponding mode shapes of unstable vibrations. It can be found that self-excited vibration obviously takes place on the low rail and wheel. Almost no unstable vibration on the high rail is found. It should be noted that if the deformation of the high rail is not obvious in the mode shape, the unstable vibration can be ignored. That is because there is always enough structural damping to overcome this small amount of unstable vibration, which is not taken into account in the present complex eigenvalue analysis in order to better understand the mode coupling mechanism of the wheelset-track system. With increasing outer contact angle sequentially, the number of self-excited vibration remains constant and the self-excited vibration most probably takes place on the low rail and the wheel. The mode shapes of unstable vibrations are similar to Fig. 8.
From the results presented in this section, it is known that when the creep forces at the inner and outer wheels of the leading wheelset are saturated, if the contact angle between the wheel and rail is larger than 8.41°, no self-excited vibration occurs on the wheel and the rail, therefore no rail corrugation occurs on the corresponding rail. This provides us a new insight into controlling rail corrugation.
(d)
High rail High rail
Low rail Low rail
(c)
(e) (f)
High rail High rail
Low rail Low rail
(b)
High rail
Low rail
High rail
Fig. 8. Mode shapes of unstable vibrations when the contact angle δL= 8.41o. (a) fR = 505.66 Hz, ξ = - 0.01778. (b) fR = 517.20
Hz, ξ = - 0.01781.
In order to further illustrate the effect of the contact angle on rail corrugation, the contact stresses between wheels and rails are analyzed under different contact angles. Fig. 9 shows the distributions of contact stress between wheels and rails with the contact angles. From Fig. 9, it can be seen that the contact area on the outer rail gradually shifts to the gauge corner of the outer rail with increasing outer contact angle. Furthermore, the contact point between the outer wheel and high rail shifts to the flange root for wheel and to the gauge corner for rail.
Fig. 9. Distributions of the contact stresses under different contact angles. (a) contact angle δL= 3.01o. This figure shows that
the contact details and the contact stresses between the wheel and the rail at the left (high) and the right (low) contact points, respectively. (b) δL= 6.63o. (c) δL= 8.41o. (d) δL= 29.95o.
3.3. Effect of the direction of the saturated creep force on self-excited vibration of the wheelset-track system
According to the Simpack simulation results, the creep forces on the inner and outer wheels of the leading wheelset are approximately saturated. The saturated creep force is equal to the resultant force of the lateral creep force and the longitudinal creep force. However, the lateral creep force and the longitudinal creep force are different when different vehicles negotiate the tight curved track. Depending on the values of the lateral and longitudinal creep forces, the direction of the saturated creep force will change in the range of 90° from the lateral direction of the track to the longitudinal direction (forward direction) of the track in the tight curved track. Here,
δL= 6.63o (b)
(a)
δL= 3.01o
δL= 8.41o (c)
the effect of the direction of the saturated creep force on self-excited vibration of the wheeleset-track system is studied. It is assumed that the contact angles between the wheels and rails are kept constant. The traction angle α is defined as the angle between the lateral direction of the track and the saturated creep force. Fig. 10 shows that the distributions of self-excited vibrations of the wheelset-track system when the traction angle changes in the range of 0o-90o. When the outer contact angle δ
L is 37.02o, the inner contact angle δRis 1.53o and the traction angle
α is 0o, there are two negative damping ratios. With increasing traction angle, the least effective damping ratio
decreases and the number of self-excited vibration decreases as well. When the saturated creep force is in the lateral direction of the track, the corresponding least effective damping ratio is -0.002614. When the saturated creep force is in the longitudinal direction, there is no unstable vibration. It can be observed that with increasing traction angle, the propensity of rail corrugation is reduced. This result also suggests that the saturated creep force in the lateral direction more easily brings about rail corrugation than the saturated creep force in the longitudinal direction does. This explains why rail corrugation more easily occurs in the tight curved track than in the tangential track.
(b) (c)
(a)
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
= 0o
E ffe ct iv e d a m pi ng r at io
Frequency fR L= 37.02o
0 200 400 600 800 1000 -0.03
-0.02 -0.01 0.00
Frequency fR
E ffe ct iv e d am pi n g ra tio
L= 37.02o
= 15o
0 200 400 600 800 1000 -0.03
-0.02 -0.01 0.00
Frequency fR
E ff ec tiv e d a m p in g r a tio
L= 37.02o = 30o
(e) (f)
(d)
0 200 400 600 800 1000 -0.03
-0.02 -0.01 0.00
Frequency fR
E ff ec tiv e d a m p in g r a tio
L= 37.02o = 60o
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
Frequency fR
E ffe ct iv e d am pi n g ra tio
L= 37.02o
= 75o 0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
Frequency fR
E ff ec tiv e d a m p in g r a tio
L= 37.02o = 45o
0 200 400 600 800 1000
-0.03 -0.02 -0.01 0.00
= 90o
E ffe ct iv e d a m pi ng r at io
Frequency fR
L= 37.02o
Fig. 10. Evolution of effective damping ratio for different directions of the saturated creep force, contact angle δL = 37.02o, δR =
1.53o. (a) traction angle α = 0o. (b) α = 15o. (c) α = 30o. (d) α = 45o. (e) α = 60o. (f) α = 75o. (g) α = 90o.
4. Conclusions
In this paper, the effect of the contact angle and the direction of the saturated creep force on rail corrugation are studied from the viewpoint of friction-coupling self-excited vibration of the wheelset-track system. A finite element model consisting of a leading wheelset, two rails, a series of sleepers, support springs and dampers is established and analyzed using the complex eigenvalue method. The following conclusions can be drawn.
(1) When the creep forces at the inner and outer wheels of the leading wheelset are saturated, if the contact angle between the wheel and rail is larger than 8.41°, no self-excited vibration occurs on the wheel and the rail, therefore no rail corrugation occurs on the corresponding rail.
(2) The direction of the saturated creep force plays an important role in self-excited vibration of the wheelset-track system and rail corrugation. It is found that with increasing traction angle, the propensity of rail corrugation is reduced. The saturated creep force in the lateral direction more easily brings about rail corrugation than the saturated creep force in the longitudinal direction does. It suggests that rail corrugation more easily occurs in the tight curved track than in the tangential track.
Acknowledgements
The authors thank the financial support from National Natural Science Foundation of China (No.51275429), the innovation team development plan of the ministry of education (IRT1178) and the Fundamental Research funds for the Central University (SWJTU12ZT01). The fifth author acknowledges the support of Changjiang Scholarship.
References
28[?] V. Garg, Dynamics of railway vehicle systems, Elsevier, 1984.
29[?] Y. Yuan, An eigenvalue analysis approach to brake squeal problem, Proceedings of the 29th ISATA Conference, Automotive Braking
