World Congress on Railway Research 001, Köln, 5-9 November 001 Derailment Safety Evaluation by Analytic Equations Masao UCHIDA*, Hideyuki TAKAI*, Hironari MURAMATSU*, Hiroaki ISHIDA** * Track Technology Division, ** Railway Dynamics Division Railway Technical Research Institute, Japan Summary In estimating the risk of flange-climb derailment, it is generally considered that a wheel will not derail if the derailment coefficient is smaller than the critical value calculated with "Nadal's equation". To estimate derailment coefficients, time-series simulation is often done, but this needs long calculation time. As a quicker method, we propose to estimate the derailment coefficient from analytically calculated lateral and vertical wheel loads, base on analytic equations and measured data. This paper describes the equations used, shows how the derailment coefficient is calculated, and compare with the results achieved by measurements. Keywords: Flange-climb derailment, Vehicle dynamics simulation, Derailment coefficient, Q/ estimation equations, Estimated Q/ ratio 1 Introduction Recently, flange-climb derailments of light-weight commuter trains on tight curves at low speed have tended to increase. This leads to a need for establishing safety evaluation methods against flange-climb derailments. Many factors influences the cause of derailments. In the field of vehicle eng ineering, these are axle load, imbalance between the right and left-side static wheel loads, spring constants of the primary and secondary suspensions, the height of the vehicle center-of -gravity and so on. In the field of track engineering, curve radius, superelevation, twist, track irregularities and so on. Beside these factors, the friction coefficients between the wheels and each rail in both the tread and flange areas, and the train speed also play important roles. To estimate the level of safety against flange-climb derailment, we traditionally use the "d e- railment coefficient" known as "Nadal's equation". When the derailment coefficient is smaller than the calculated value by Nadal's equation, we judge that there is no risk of flangeclimb derailment. On the other hand, in some cases, time-series computer simulations based on vehicle dynamics model may be used to calculate the derailment coefficients for various scenarios. However, because this method needs long calculation time, it is not suitable to evaluate many cases that consist of combinations of numerous conditions. We proposed equations that more easily estimate the wheel loads, lateral forces and d erailment coefficients (lateral force/wheel load, usually described "Q/" in Japan) from various parameters. These equations consist of theoretical analysis of the mechanism of the lateral 1
World Congress on Railway Research 001, Köln, 5-9 November 001 force/wheel load on curves and of field test data. After estimating the lateral force and wheel load of the outside rail, the derailment coefficient is obtained. These equations are improved products of earlier equations that had been proposed u sing numerous data from curving speed tests held in the 1990's. This method was effectively applied to investigate the cause of the unfortunate derailment accident that occurred on the Hibiya line of Teito Rapid Transit Authority (major subway operator in Tokyo) in March, 000 1). This paper expresses the Q/ estimation equations, calculation method of critical Q/, e stimation of safety margin against derailment using the "estimated Q/ ratio" (critical Q/ / estimated Q/), and examples of trial calculations. This study aims at vehicles with bolster-less bogies and air spring secondary suspension. The carbody is conservatively considered as a rigid body. Wheel load estimation equations.1 Wheel load variation by centrifugal force When trains pass through curves, centrifugal forces act on the trains according to the curve radius, superelevation and train speed. This adds to or subtract from the quasi-static outside and inside wheel loads. If the train speed is lower than the balanced speed (= superelevation excess), the wheel load of the outside wheel is less than its static value. On the contrary, if the train speed is higher than the balanced speed (= superelevation deficiency), the wheel load of the outside wheel will be higher than its static value. Considering the mechanism of increased and decreased wheel loads by the effect of centrifugal forces, equations (1) and () are formed to estimate the quasi-static outside and inside wheel loads. o W 0 v gr? C H G v C G...?E G gr G (1) i W 0 v gr? C H G v C...?E () G G gr G o : constant component of wheel load of the outside wheel (kn) i : _: constant component of inside wheel load (kn) static wheel load ratio of outside wheel W 0 : static axle-load (kn) v: train speed (m/s) G: gauge (m) C: track cant (m)
World Congress on Railway Research 001, Köln, 5-9 November 001 R: curve radius (m) g: gravity acceleration (9.8 m/s ) H * G : effective height of center-of-gravity of vehicle (m) (For non-tilt vehicles regardless of anti-roll bars, H * G is set to 1.5 times of the real height of the center-of-gravity). Wheel load variation due to track twist On transition curves, as the track surface is twisted, wheel loads increase and decrease a c- cording to the deformation of the primary and secondary suspension springs. Especially, on the exit transition curve, the wheel load of the outside wheel of the leading axle decreases according to the extension of its primary suspension. At the same time, the wheel loads of the outside wheel of the whole leading bogie decreases according to the extension of the secondary suspension spring. Furthermore, the wheel loads vary by local track irregularities. The static wheel load decrease is expressed by equation (3), considering the mechanism of the wheel load variation due to track twist. 1 8b 4k 1 1b1 t c?e K ta?e k 1 1... (3) 1 k 1 k1b1 K t c c a TC t c k b t a a a TC _ st : Static wheel load decrease due to track twist (kn) K _ : Effective rotational stiffness / wheelset (kn-mm) K' _1 : Effective rotational stiffness / bogie (kn-mm) b: Width between right and left wheel/rail contact points (mm) b 1 : Width between right and left primary suspension springs (mm) b : Width between right and left secondary suspension springs (mm) k 1 : rimary suspension vertical stiffness / axlebox (MN/m) k : Secondary suspension vertical stiffness / bogie side (MN/m) _t c : Track twist between bogie centers (mm) _t a : Track twist between a wheelsets (within bogie) (mm) a: Distance between wheelsets (within bogie) (mm) c: Distance between bogie centers (mm) a TC : Cant gradient t c : Track twist between bogie centers excess cant gradient (mm) t a 3
World Congress on Railway Research 001, Köln, 5-9 November 001 t a : Track twist between wheelbase excess cant gradient (mm).3 Wheel load variation due to torsion of secondary suspension spring When passing through curves, torsion of secondary suspension springs will occur on a bo l- ster-less bogie due to the relative rotational deformation between carbody and bogies. The reaction force F 1 acts laterally on the rail. At the same time, F 1 ' (= F 1 /tan60 ), the vertical component of the reaction force, will act vertically (Figure 1). Track shifting force F 1 that occurs by the torsion of air springs is described in Section 3., Clauses () and (3)..4 Estimation equations for of wheel loads The outside and inside wheel loads can be calculated by the equation (4) and (5), consi dering three factors: centrifugal force, track twist and torsion secondary suspention springs.... 1 (4) tan 60 0 0 F i i... F1 (5) tan60 o : Wheel load of the outside wheel (kn) 4
World Congress on Railway Research 001, Köln, 5-9 November 001 i : Inside wheel load (kn) : Constant component of wheel load of the outside wheel (kn) o i : F 1 : Constant component of inside wheel load (kn) : Static wheel load decrease by track twist (kn) Track shifting force by distortion of air springs (kn) _: Collection coefficient of vertical component of F 1 (See chapter 3.) 3. Lateral force estimation equations 3.1 Turning lateral force due to reaction of the inside friction force When a vehicle is running on a curve, the flange of the leading, outside wheel is in contact with the outside rail and is pushed against its gauge face. Then the inside wheel resists the force with a friction force (= product of wheel load and friction coefficient applied on the tread). This acts as a quasi-static lateral force toward the outside, that is "turning lateral force" (Figure ). Accordingly, the larger the friction coefficient between the inside wheel tread and the rail (nearly inside Q/ ratio _) is, the larger the turning lateral force becomes. The estimation equation for the inside wheel quasi-static component of the lateral force (turning lateral force) is expressed by equation (6). Q i = _i... (6) Q i : Constant component of inside lateral force (kn) _: i : Inside Q/ ratio Inside wheel load (kn) Two values for _ are established for tapered and arc or modified arc wheel profiles, r espectively (Figure 3). These characteristics are obtained by time-series simulations and field test data. 5
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World Congress on Railway Research 001, Köln, 5-9 November 001 3. Track shifting force due to centrifugal force and torsion of secondary suspension springs (1) Track shifting force due to centrifugal force Centrifugal forces act on the train that is running through curves according to the curve r a- dius, superelevation and train speed. It constitutes one part of the quasi-static track shifting force. This force is negative when the train speed is lower than the balanced speed (superelvation excess). On the contrary, this force is positive when the train speed is higher than the balanced speed (superelvation deficiency). () Track shifting force due to torsion of air spring On curves, the torsion of the secondary suspension springs due to the yaw angle b etween carbody and bogies cause a track shifting force. At the position of the leading wheelset of a bogie, this force acts towards the outside rail as shown in Figure 4. (3) Estimation equations for quasi-static component of track shifting force Based on the above mentioned (1) and (), quasi-static component of the track shifting force due to the centrifugal force and torsion of the secondary suspension springs is expressed by the following equation (7). Q AS W v gr C G 0 F 1 7
World Congress on Railway Research 001, Köln, 5-9 November 001 v C kb c 6 W... (7) 0 10 gr G ar _Q AS : Quasi-static component of track shifting force (kn) F 1 : Track-shifting force due to deformation of secondary suspension springs (kn) W 0 : Static axle load (kn) v: Train speed (m/s) G: Gauge (m) C: Superelevation (m) R: Curve radius (m) g: Gravitational acceleration (9.80 m/s ) b : Width between right and left air springs (mm) a: Distance between wheelsets (within bogie) (mm) c: Distance between bogie centers (m) k: Yaw stiffness of secondary suspension spring / bogie (kn/m) (zero for bogies with bolster) _: Modifying coefficient of track shifting force F 1 (4) Modifying coefficient _ of the track shifting force due to torsion of secondary suspension springs According to the result of study on the calculation value of track shifting force F 1 due to the deformation of air springs by estimation equation and, time series simulation (Figure 5), we set modifying coefficient _ per equations (8) and (9) (a) In case of inside Q/ ratio _ 0.50 0.7?@?@?@?@?@?@ R 160 310 R 0.7...?@ 160 R 1000 (8) 150 3.?@?@?@ 1000 R?@?@?@?@?@ (b) In case of inside Q/ ratio _ > 0.50 0.7?@?@?@?@?@?@ R 160 310 R 0.7...?@ 160 R 1000 (9) 150 3.?@?@?@ 1000 R?@?@?@?@? 8
World Congress on Railway Research 001, Köln, 5-9 November 001 In Figure 5, when the curve radius is larger than a certain value, the track shifting force F 1 due to the torsion of the secondary suspension springs becomes negative. Consequently, for the outside wheel, the reaction force of the vertical component of F 1 acts downward. This force reduces the wheel load of the outside wheel. However, such a condition was not found during field tests nor by time series simulation. Therefore, the modifying coefficient for vertical component of F 1 is set per equation (10). _ = 1 ( _ > 0 ), _ = 0 ( _ 0 )... (10) 3.3 Lateral force variation due to track irregularities and impacts at rail joints When there are track irregularities, especially alignment irregularities, variation of track shifting forces occur mainly by inertia forces according to the vehicle vibrations. Shocking variation of lateral forces at rail joints occur. These forces increase with higher train speed. The way of calculations are shown in equations (11) and (1). Q AD 0 3W k... Q ZV (11) Q unsp 500 V... 4 (1) 100 R 100 9
World Congress on Railway Research 001, Köln, 5-9 November 001 _Q AD : Variation of track shifting force (kn) W 0 : Static axle load (kn) _ z : Standard deviation of alignment irregularities (mm) V: Train speed (km/h) k Q : Variation coefficient of track shifting force (1/mm/(km/h)) _Q unsp : Variation of lateral force at rail joints (kn) R: Curve radius (m) _: Effective ratio of variation of lateral force (%) 3.4 Estimation equations of outside lateral forces We estimate the outside lateral forces using equation (13) considering the above mentioned three factors: turning lateral forces by the inside rail friction force, track shifting force due to centrifugal force and torsion of secondary suspension springs, and variation of lateral forces due to track irregularities and impact forces at rail joints. Q o Q i Q AS... Q Q (13) AD unsp Q o : Outside lateral force (kn) Q : Constant component of inside lateral force (kn) i _Q AS : Constant component of track shifting force (kn) _Q AD : Variable component of track shifting force by track irregularity (kn) _Q unsp : Variable component of lateral force at rail joints (kn) 4 Calculation of critical derailment coefficients 4.1 Critical derailment coefficients We usually use the "Nadal's equation" to calculate the critical derailment coefficient. instead of the friction coefficient _, we adopt the equivalent friction coefficient _ e which is a function of the wheel angle of attack,. This is to reflect the difference of the track geometry (curvature) accurately at the wheelset under consideration. This process gives higher critical derailment coefficients than with the common friction coefficient _. 4. Nadal's equation Nadal's equation, which is used for calculating the margin against the flange-climb d erailment quantitatively, is shown as equation (14). Q cri 1 tan e tan e 10
World Congress on Railway Research 001, Köln, 5-9 November 001... (14) (Q/) cri : Critical derailment coefficient _: Wheel flange angle (rad) _ e : Equivalent friction coefficient = f y /N (f y : lateral creep force, N: normal force) 4.3 Approximation of equivalent friction coefficient _ e The equivalent friction coefficient _ e is expressed as equation (15), considering the satur a- tion characteristics of the creep force. e y N... 1 (15) N y _: Index expressing saturation characteristic ( = 1.5) v y : Lateral creep ratio _ : Lateral creep coefficient N: Normal force _ /N 7.0 Considering that v = tan_ sin_ (1 cot _) _ (16) is obtained. _ ( _ is the wheel angle of attack), equ ation e 1.5 7.0... (16) 1.5 7.0 3 _ is the effective value of friction coefficient at the outside wheel flange with co nsidering the forward/rear direction of the tangent force. 4.4 Setting method of angles of attack For vehicles with tapered, arc or modified-arc wheel tread, considering the result of analysis of time-series simulation (Figure 6), the wheel angle of attack _ is calculated by equation (17). a??... T w (17) R (a) For tapered wheel tread T W 90 1?@?@?@ R 90 R a 11
World Congress on Railway Research 001, Köln, 5-9 November 001 T W (b) For modified arc wheel tread 1... (18)?@?@?@?@ R 90 a T W 60 1?@?@?@ R 00 R a T T W W 1 0.7 1 1 0.3...?@?@? 80 R?ƒ00 (19) a 0.0075 R 00 1?@?@?@?@ R 80 a?ƒ _: Attack angle (rad) a: Wheelbase (m) R: Curve radius at the center of leading bogie (m) _ T : Bogie yaw angle (including carbody yaw angle) (rad) _ W : Yaw angle by the bogie staring (rad) _ 1 : Side gap + gauge widening/ at leading axle (m) _ : Side gap + gauge widening/ at trailing axle (m) 1
World Congress on Railway Research 001, Köln, 5-9 November 001 5 Evaluation of the margins to the flange-climb derailments using estimated derailment coefficient ratio 5.1 Estimation of outside derailment coefficients The derailment coefficient of the outside wheel is calculated by equation (0) as the ratio of outside lateral force to wheel load of the outside wheel at the same position. Qo ( Q /... ) o?? (0) o (Q/) o : Derailment coefficient Q o : Outside lateral force (kn) o : Wheel load of the outside wheel (kn) 5. Estimated derailment coefficients ratio We definite an "Estimated derailment coefficient ratio" as the ratio of the critical Q/ shown in Chapter 4 to the Q/ of the outside wheel shown in Section 5.1. This definition is shown in equation (1). This ratio expresses the margin against flange-climb derailment with the base value of 1.0. Q cri ( Q / ) ratio...?? (1) Q o Here (Q/) cri : Critical derailment coefficient (Q/) o : Outside derailment coefficient 5.3 Trial calculation of estimated derailment coefficient ratio (Sensitivity analysis) (1) Comparison of Q/ between estimation and measurement Figure 7 shows result of comparison of Q/ between estimation and measurement. E stimated values are the good approximations of measured data. () Example of estimated Q/ ratio Figure 8 shows an example of calculation result at the section of a circular curve and the exit transition curve, with a calculation interval of 1 meter. The estimated Q/ ratio has a minimum just after entering the exit-side transition curve. (3) Sensitivity analysis of estimated Q/ ratio We calculate the estimated Q/ ratio in various cases with changing p arameters shown in Table 1 as basic values. Figure 9 shows the calculation result. those values are minimum value in the calculation area. 13
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World Congress on Railway Research 001, Köln, 5-9 November 001 Calculation parameters 15
World Congress on Railway Research 001, Köln, 5-9 November 001 6 Conclusions (1) We composed the wheel load estimation equation considering centrifugal forces, track geometry deviation and the deformation of secondary suspension springs. () We established the lateral force estimation equation considering the curve turning lateral force, track shifting force due to centrifugal force and secondary suspension springs, variation of lateral force due to track irregularities. (3) We proposed a calculation method for the critical derailment coefficient considering wheel flange angle, equivalent friction coefficient and wheel angle of attack. (4) We define the "Estimated derailment coefficient ratio" as the ratio of critical Q/ to the outside Q/, and evaluate the margin against the flange-climb derailment. In this report, some parts, such as establishing method for the inside wheel Q/ ratio or modification of track shifting force, were set up provisionally. Hereafter it is necessary to improve the accuracy according to theoretical study and field data. At this stage, this method is only for bolster-less bogies with air spring secondary suspension. We intend to improve the 16
World Congress on Railway Research 001, Köln, 5-9 November 001 equations to apply also to other types of bogies. References 1) Railway accident investigation committee, Ministry of Transportation : Investigation Report of Train Derailment at Naka-Meguro Station on the Hibiya Line of the Teito Rapid Transit Authority (in Japanese), 000.10 17