| Citation: | WANG Jian, MA Xiao-chuan, MA Dao-lin, WANG Ping. Simplified calculation method of critical value of 3D derailment coefficient under quasi-static condition[J]. Journal of Traffic and Transportation Engineering, 2017, 17(5): 71-80.
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Safety is of paramount importance in railway transportation, and ensuring that trains do not derail is the most basic requirement for ensuring transportation safety[1]The openness constraint between the train and the track determines the objective existence of vehicle derailment. Since the development of railway transportation, derailment problems have occurred from time to time[2]However, due to the complexity of the wheel rail system, the mechanism of vehicle derailment has not been fully explained to date. With the development of high-speed and heavy-duty railway transportation, once a train derails, it will cause catastrophic consequences. Therefore, research on derailment is even more important.
As early as 1896, French scientist Nadal derived the force balance conditions for the wheel to climb the track under quasi-static conditions and proposed the famous Nadal derailment criterion. Although the Nadal derailment criterion is still widely adopted by countries around the world, it only considers the two-dimensional force balance of the wheel rail contact patch under quasi-static conditions, ignoring the effects of wheelset heading angle and creep, resulting in the amplification of tangential force, making it somewhat conservative under small wheelset heading angles[1, 3-4]Therefore, many researchers have improved the Nadal derailment criterion through theoretical analysis and experimental research[1-23]Braghin et al. conducted experimental and theoretical research on single wheelset derailment using the BU300 rolling test bench. Based on the experimental and numerical simulation results, they proposed a method for evaluating wheel derailment that can consider the yaw angle of the wheelset. However, in practical applications, it is necessary to measure the longitudinal creep force between the wheel and rail and the yaw angle of the wheelset[4]Zhang Weihua et al. also conducted experimental research on the derailment of a single wheelset using a rolling test bench, and believed that lateral force is not the main cause of wheel derailment, while the wheel load reduction rate is the main factor affecting derailment. The combined effect of load reduction and angle of attack is more likely to cause derailment[5]O'Shea et al. proposed using the theory of multibody dynamics to replace the traditional two-point contact method with a three-point contact method for simulating wheel derailment, and achieved relatively accurate simulation results[6-7]Barbosa analyzed in detail the three-dimensional stress situation of the contact patch under the critical state of wheel derailment, and used FastSim simplified creep theory to consider in detail the influence of wheel rail creep force on wheel derailment. Based on this, a three-dimensional derailment evaluation method considering longitudinal and transverse creep forces between wheel and rail was proposed, and the conservatism of Nadal derailment criterion was improved[8-9]Karmal et al. conducted extensive experimental research and theoretical simulations on the problem of wheel rail climbing using track loading vehicles and dynamic calculations. They analyzed in detail the effects of friction coefficient and wheelset yaw angle on wheel rail climbing and concluded that wheel rail climbing depends on the distance traveled by the vehicle related to the derailment coefficient, rather than the duration related to the derailment coefficient[10-11]At the same time, many scholars have also begun to study the derailment conditions of the entire wheelset. Zhang Hong et al. established a two-dimensional force balance relationship for the entire wheelset based on the Nadal derailment criterion, and provided a method for determining the limit of wheel load reduction rate based on the Nadal derailment coefficient limit[12]However, it failed to consider the influence of wheelset yaw angle; Kuo et al. analyzed the rationality of the Weinstock derailment criterion under two-dimensional conditions, and believed that the Weinstock derailment criterion has certain shortcomings in evaluating wheel derailment. They proposed a new method for evaluating wheel derailment[13]Zeng et al. extended the two-dimensional force balance condition of the wheelset to three dimensions based on a detailed consideration of wheel rail creep and wheelset yaw angle. They proposed using the wheel axle derailment coefficient as the evaluation method for wheel derailment, and proposed an indirect measurement method for wheel rail force. By using test quantities such as axle box acceleration, relative displacement of the primary suspension, and arm strain, the wheel rail force was inverted and identified, and the safety of vehicle derailment was evaluated online[14-16]In addition, a large number of scholars have conducted research on simplifying the calculation of derailment evaluation criteria. Yokose proposed using equivalent friction coefficient to describe the sliding state between wheel and rail based on the experimental results of wheel rail contact mechanics, thus correcting the conservatism of Nadal derailment criterion[17]However, the calculation formula for the equivalent friction coefficient is relatively complex; Zeng Yuqing et al. conducted a detailed derivation of the calculation method for the equivalent friction coefficient and proposed a method for calculating the equivalent friction coefficient using measurable longitudinal, lateral, and vertical forces between the wheel and rail[18]However, it did not directly consider the creep phenomenon between the wheel and rail; Guan Qinghua et al. carefully considered the effects of wheel rail creep and wheelset heading angle on wheel derailment. Based on the variation law of equivalent friction coefficient with wheelset heading angle, they proposed an improved derailment coefficient limit calculation method using Boltzmann inverse curve function to fit the equivalent friction coefficient. The calculation process of derailment coefficient limit was simplified[19-20].
Improving derailment evaluation criteria is often limited in practical applications due to the complexity of calculation methods and limitations in their applicability. This article is based on the derivation idea of Nadal derailment criterion, and establishes a three-dimensional analysis model of the critical state of wheel derailment under quasi-static conditions. It analyzes the influence of wheelset heading angle, friction coefficient, and wheel rail contact angle on the normal force, tangential force, and derailment coefficient under the critical state of derailment, and proposes a more simplified method for calculating the critical value of three-dimensional derailment coefficient based on the consideration of wheelset heading angle.
The wheel rail contact relationship is the link that couples the wheel and the rail, and establishing the wheel rail system coordinate system is the basis for determining the wheel rail relationship. in compliance withFigure 1As shown, with the center of the orbit as the originOTThe longitudinal direction of the line isxTAxis, the horizontal direction of the line isyTAxis, the vertical direction of the line iszTAxis, establish track coordinate systemOTxTyTzTStarting from the center of the wheelsetOWThe longitudinal direction of the wheel axle isxWAxis, the lateral direction of the wheel axle isyWThe vertical direction of the axle iszWAxis, establish wheelset coordinate systemOWxWyWzWTake the center of the left and right wheel rail contact spots as the origin respectivelyOL、ORUnder the critical condition of derailment, the longitudinal motion direction of the wheel contact patch is taken asxL、xRAxis, with the tangential direction of the tread position where the wheel contact patch is located asyL、yRAxis, with the normal direction of the tread position where the wheel contact patch is located aszL、zRAxis, establish coordinate system for left and right wheel rail contact spotsOLxLyLzL、ORxRyRzRThe orbital coordinate system moves along the centerline of the orbit at a certain speed. The wheelset coordinate system moves together with the wheelset, and has translational and rotational degrees of freedom relative to the track coordinate system. The coordinate system of the wheel rail contact spot moves together with the wheelset. The transformation relationship between the wheelset coordinate system and the track coordinate system is
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In the formula:xT、yT、zTThe coordinates in the three directions of the orbital coordinate system are respectively;ψFor the shaking angle of the wheelset;φTo determine the roll angle of the wheelset;xW、yW、zWThey are the coordinates in three directions of the wheelset coordinate system.
Taking the right wheel as an example, the transformation relationship between the contact spot coordinate system and the wheelset coordinate system is
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In the formula:δThe wheel rail contact angle at the right wheel rail contact spot;xR、yR、zRThe coordinates of the right wheel rail contact spot coordinate system are in three directions.
Substituting equation (2) into equation (1), the transformation relationship between the coordinate system of the right wheel rail contact spot and the track coordinate system is
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In the formula:AThe transformation matrix from the coordinate system of the right wheel rail contact spot to the track coordinate system.
Taking the right wheel as an example, under the critical state of steady-state derailment of the wheel, the force at the wheel rail contact spot is shownFigure 2, F、Q、PThe longitudinal, transverse, and vertical forces exerted on the contact patch in the orbital coordinate system, respectively,TxR、TyR、NThey are the longitudinal creep force, transverse creep force, and normal force exerted on the contact patch in the wheel rail contact patch coordinate system.
Under quasi-static conditions, there is a balance of forces on the contact spot
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The derailment coefficient of the wheel under the critical condition of derailment is
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Due to the small yaw angle and roll angle of the wheelset, equation (6) can be simplified as
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According to equation (7), the derailment coefficient at the critical state of wheel derailment is related to creep force and normal force. Under quasi-static conditions, the wheel rail creep rate is
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In the formula:ξxr、ξyr、ξsThey are the longitudinal creep rate, transverse creep rate, and spin creep rate within the contact patch;rWThe actual rolling circle radius of the wheel at the contact spot;rnThe nominal rolling circle radius.
According to equation (8), the creep rate under quasi-static conditions is only related to the geometric parameters of wheel rail contact. After obtaining the creep rate at the critical state of wheel derailment, according to Kalker's linear creep theory, there is a relationship between creep force and creep rate as follows
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In the formula:T′xR、T′yRThey are Kalker linear longitudinal and transverse creep forces, respectively;GSynthesize shear modulus for wheel rail materials;a、bThe major and minor axes of the Hertz contact patch ellipse are respectively;C1、C2、C3All are Kalker creep coefficients, anda、bThe ratio is related.
According to Hertz contact theory, the major and minor axes of the elliptical wheel rail contact patcha、brespectively
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In the formula:σPoisson's ratio;RT、RWThe radii of the rail and wheel profile at the wheel rail contact spot are respectively;m、nCorrection coefficients for the major and minor axes of the ellipse;θFor Hertz contact parameters.
Due to the assumption that there is no relative sliding within the contact patch of the Kalker linear creep force model, it is only applicable in small creep situations. However, in the case of wheel flange contact, due to the increase in creep rate, the contact patch can even reach a fully sliding state. Therefore, it is necessary to correct the linear creep force. Adopting Shen Hedrick Elkins nonlinear creep model[24]Correct the wheel rail creep force using the following calculation method:
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In the formula:TThe synthetic creep force calculated by Kalker's linear creep theory;TFor the modified synthetic creep force;μThe coefficient of friction between the wheel and rail.
According to the calculation model of wheel rail creep force, it can be seen that under the given wheel rail contact parameters and wheelset yaw angle, creep force is only related to the normal force. The vertical force balance of the contact spot can be obtained from equation (5)
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After ignoring the roll angle, we can obtain
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Given the vertical force between the wheel and railPUnder the given conditions, the normal force between the wheel and rail can be obtained through iterative methods, and then the creep force can be obtained.
Taking the Chinese LMA wheel tread and CHN60 rail profile as examples,Table 1The wheel rail contact parameters under the critical state of quasi-static derailment of the wheels are provided. On this basis, the variation laws of the normal force, longitudinal creep force, transverse creep force, and derailment coefficient critical values of the wheel track under different wheel shaking angles and friction coefficients were analyzed. The results are shown inFigure 3~6.
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causeFigure 3It can be inferred that the normal force of the wheel track increases with the increase of the friction coefficient at a negative shake angle; At the positive heading angle, the normal force between the wheel and rail decreases as the friction coefficient increases; When the friction coefficient is the same, the normal force of the wheel track increases rapidly with the increase of the absolute value of the yaw angle under negative yaw angle, while the normal force of the wheel track remains basically unchanged under positive yaw angle.
causeFigure 4It can be seen that the longitudinal creep force increases with the increase of friction coefficient; When the friction coefficient is the same, the longitudinal creep force under negative heading angle increases first and then decreases with the absolute value of heading angle, while under positive heading angle, the longitudinal creep force gradually decreases with the increase of heading angle.
causeFigure 5It can be seen that the lateral creep force increases with the increase of friction coefficient, but at a negative shake angle, the lateral creep force remains negative, playing a role in preventing the wheel from climbing up. Wheel derailment is manifested as slide rail derailment; At the positive heading angle, the lateral creep force is always positive, which helps the wheels climb up. Wheel derailment is manifested as track climbing derailment; When the friction coefficient is the same, the absolute value of lateral creep force under negative heading angle increases rapidly with the increase of heading angle absolute value, while the lateral creep force under positive heading angle remains basically unchanged with the increase of heading angle.
causeFigure 6It can be seen that the critical value of derailment coefficient under negative heading angle increases significantly with the absolute value of heading angle, while under positive heading angle, the critical value of derailment coefficient decreases to some extent with the increase of heading angle. When the heading angle exceeds 1.5 °, the critical value of derailment coefficient remains basically unchanged.
The comparison between the critical values of the three-dimensional derailment coefficient of the wheel under quasi-static conditions and the critical value of Nadal derailment coefficient shows that（Figure 6）The critical value of the three-dimensional derailment coefficient under negative heading angle is significantly higher than that of the Nadal derailment coefficient. This is because the Nadal derailment model only considers the situation of track climbing derailment and does not take into account the hindering effect of lateral creep force on wheel derailment in slide rail derailment; When the shaking angle is small (within 1.5 °), The Nadal derailment coefficient has a certain degree of conservatism compared to the critical value of the three-dimensional derailment coefficient, because when the heading angle is small, the proportion of longitudinal creep force in tangential force is significantly greater than that of transverse creep force（Figure 4 (b)、5(b)）The Nadal derailment model suggests that all tangential forces are provided by lateral creep forces, but when the heading angle is large, the proportion of lateral creep forces in tangential forces reaches over 90%（Figure 5 (b)）The critical values of derailment coefficients for both are basically the same. It can be inferred that using the Nadal derailment coefficient limit is appropriate when the positive heading angle is large, but in the case of small heading angles and negative heading angles, using the Nadal derailment coefficient limit has a certain degree of conservatism.
According to equation (14), the ratio relationship between the Kalker linear composite creep force and 3 times the Coulomb friction force under different wheel shaking angles, friction coefficients, and vertical force conditions is shown inFigure 7It can be inferred that when the wheel is in a critical state of derailment, The Kalker linear composite creep force increases with the increase of the wheelset heading angle, and the Kalker linear composite creep force exceeds three times the Coulomb friction force, indicating that the wheel rail rolling contact exhibits pure sliding.
When the creep force reaches saturation, according to equation (14), the modified composite creep force is
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set upTxRgiveTyRThe angle between them isαThen there is
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Substituting equation (18) into equation (7), the critical derailment coefficient can be expressed as
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The ratio of lateral creep force to longitudinal creep force at the critical state of derailment is
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By substituting equations (7) and (10) into equation (20), we can obtain
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According to equation (21), when the contact parameters at the critical state of wheel derailment are known, the ratio of lateral creep force to longitudinal creep force is only related to the yaw angle and normal force of the wheelset.Figure 8The ratio of lateral creep force to longitudinal creep force under different friction coefficients is given as a function of the wheel shaking angle when the maximum wheel rail contact angle is 65 ° and 70 °, respectively. When the maximum wheel rail contact angles are 65 ° and 70 ° respectively, the calculated values are based on equations (22) and (23)K1They are 84.423 6 and 104.269 6, respectively,K2They are 0.005 5 and 0.005 7 respectively. causeFigure 8It can be seen that under a positive heading angle, the ratio of the two is basically not affected by the friction coefficient, while under a negative heading angle, the friction coefficient has a certain influence on the ratio of the two, but the effect is not significant. This is because under positive heading angles, the normal force of the wheel rail is relatively small and does not change much with the friction coefficient. However, under negative heading angles, the normal force of the wheel rail increases significantly compared to under positive heading angles, and is greatly affected by the friction coefficient. Overall, due to the critical state of derailmentK1Far greater thanK2The ratio of lateral creep force to longitudinal creep force varies strongly linearly with the change of wheelset heading angle, and is less affected by the friction coefficient.
However, in practical applications, due to the difficulty in providing the wheel rail normal force in equation (21), considering the strong linear relationship between lateral creep force and longitudinal creep force, this paper is based on the principle of least squares method[25]Using equation (24)Figure 8Fit the ratio of lateral creep force to longitudinal creep force in the middle
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In the formula:B、CAll are fitting coefficients of linear equations.
The fitting results of the ratio of lateral creep force to longitudinal creep force under different friction coefficients are shown inFigure 9WhenδWhen they are 65 ° and 70 ° respectively, the coefficients in equation (24)BThey are 1.454 3 and 1.952 9, respectively, with coefficientsCThey are 0.263 3 and 0.444 6, respectively.
Thus, the simplified evaluation formula for the three-dimensional derailment coefficient under quasi-static conditions at the critical state of wheel derailment can be obtained as follows:
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The maximum wheel rail contact angle for commonly used wheel treads (LM, LMA, S1002CN, etc.) in China is 70 °, and the commonly used rail profile parameters are as follows:Table 1As shown, according toFigure 9B can be taken as 1.952 9, C is 0.444 6.
To verify the difference between the critical values of derailment coefficients calculated by the simplified formula (equation (25)) and the exact formula (equation (6)) in this article,Figure 10The comparison between the two under different wheel shaking angles and friction coefficients is given,Figure 11The error rate between the simplified formula and the exact formula is given.
causeFigure 10It can be seen that the critical values of derailment coefficients calculated by equations (25) and (6) have a high degree of consistency, especially when the wheelset heading angle is large, there is a good degree of approximation between the two. causeFigure 11It can be seen that as the friction coefficient increases, the error rate between the critical values of derailment coefficients calculated by equations (25) and (6) increases to a certain extent, but both can be controlled within -5% to 5%, which can meet the requirements of engineering applications. This indicates that the simplified formula proposed in this article as a criterion for evaluating wheel derailment under quasi-static conditions is reasonable and feasible, and has high credibility.
(1) It is appropriate to use Nadal derailment coefficient as the evaluation criterion for wheel derailment when the wheel heading angle is large (greater than 1.5 °); In the case of a small positive yaw angle, using Nadal derailment coefficient as the evaluation criterion for wheel derailment is somewhat conservative due to the significant proportion of longitudinal creep force in tangential force being greater than that of transverse creep force; In the case of negative yaw angle, due to the Nadal derailment model not considering the hindering effect of lateral creep force on wheel derailment, its conservatism is relatively high, indicating that it is not suitable for the evaluation of slide rail derailment.
(2) Under the critical condition of quasi-static derailment of wheels, it can be inferred that the ratio of lateral creep force to longitudinal creep force varies strongly linearly with the yaw angle of the wheelset and is insensitive to changes in friction coefficient.
(3) Based on the strong linear variation law of the ratio of lateral creep force to longitudinal creep force with the yaw angle of the wheelset, the least squares method was used for linear fitting, and a simplified calculation formula for the three-dimensional derailment coefficient under quasi-static conditions was proposed. By comparing with the accurate formula, the calculation error of the simplified formula was within ± 5%, indicating that it can meet the requirements of engineering applications.
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WANG Jian, MA Xiao-chuan, MA Dao-lin, WANG Ping. Simplified calculation method of critical value of 3D derailment coefficient under quasi-static condition[J]. Journal of Traffic and Transportation Engineering, 2017, 17(5): 71-80.
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