PAPER Derailment Safety Evaluatin by Analytic Equatins Hideyuki TAKAI General Manager, Track Technlgy Div. Hirnari MURAMATSU Assistant Senir Researcher, Track Gemetry & Maintenance, Track Technlgy Div. Masa UCHIDA General Manager, Research & Develpment Prmtin Div. Hiraki ISHIDA Senir Researcher, Labratry Head, Vehicle Mechanics, Railway Dynamics Div. In estimating the risk f flange-climb derailment, it is generally thught that a wheel will nt derail if the derailment cefficient is smaller than the critical value calculated with "Nadal's equatin." T estimate derailment cefficient, time-series simulatin is ften perfrmed, but this needs a lng calculatin time. As a quicker methd, we prpse t estimate the derailment cefficient frm analytically calculated lateral and vertical wheel lads, based n analytic equatins and measured data. This paper describes the equatins used, shws hw the derailment cefficient is calculated, and cmpares it with the results achieved by measurements. Keywrds : flange-climb derailment, vehicle dynamics simulatin, derailment cefficient, Q/P estimatin equatins, estimated Q/P rati. Intrductin Recently, flange-climb derailments f light-weight cmmuter trains tend t increase n tight curves at lw speed. This creates a need fr establishing safety evaluatin methds against flange-climb derailments. Many factrs influence the cause f derailments. In the field f vehicle engineering, they are the axle lad, imbalance between the right- and left-side static wheel lads, spring cnstant f suspensin, height f vehicle center-f -gravity and s n. In the field f track engineering, they are the curve radius, superelevatin, twist, track irregularities and s n. Besides these factrs, the frictin cefficient between the wheel and each rail, bth in the tread and flange areas, and the train speed als play imprtant rles. T estimate the level f safety against flange-climb derailment, we traditinally use the "derailment cefficient" (lateral frce/wheel lad, usually described "Q/P" in Japan) knwn as "Nadal's equatin." On the ther hand, time-series cmputer simulatin may be used t calculate the derailment cefficient. Hwever, because it needs a lng calculatin time, this methd is nt suitable t the applicatin in wide ranges. This paper describes the Q/P estimatin equatins, calculatin methd f critical Q/P, estimatin f safety margin against derailment using the "estimated Q/P rati" (critical Q/P / estimated Q/P) and examples f trial calculatin. This study targets at vehicles with blsterless bgies and air spring secndary suspensin. The carbdy is cnservatively cnsidered as a rigid bdy. 2. Wheel lad estimatin equatins 2. Wheel lad variatin by centrifugal frce When trains pass thrugh curves, the centrifugal frce acts n the trains accrding t the curve radius, superelevatin and train speed. This increases r decreases the quasi-static lads f utside and inside wheels. If the train speed is lwer than the balanced speed (= superelevatin excess), the wheel lad f the utside wheel is less than its static value. In cntrast, if the train speed is higher than the balanced speed (= superelevatin deficiency), the wheel lad f the utside wheel will be higher than its static value. When the mechanism f the wheel lad increases r decreases wheel lads by the effect f centrifugal frces, the equatins () and (2) are derived t estimate the quasi-static lads f utside and inside wheels. 2 2 W ν ν C HG C P = γ + + 2 gr G G/2 gr G () * W ν 2 ν 2 C HG C Pi = (2 γ ) + (2) 2 gr G G/2 gr G P : Cnstant cmpnent f utside wheel lad P : Cnstant cmpnent f inside wheel lad i γ : Static wheel lad rati f utside wheel W : Static axle-lad, ν: Train speed (m/s), G: Gauge (m) C : Track cant (m), R: Curve radius (m), g: Gravity acceleratin (9.8 m/s 2 ) H * G : Effective height f center-f-gravity f vehicle (m) (Fr nn-tilt vehicles withut anti-rll bars, * H G is set t.25 times the real height f center-f-gravity) QR f RTRI, Vl. 43, N. 3, Sep. 22 9
2.2 Wheel lad variatin due t track twist On transitin curves, as the track surface is twisted, wheel lads increase and decrease accrding t the defrmatin f the primary and secndary suspensin springs. In particular, n the exit transitin curve, the wheel lad f the utside wheel f the leading axle decreases accrding t the extensin f its primary suspensin. At the same time, the lad f the utside wheel f the leading bgie decreases accrding t the extensin f the secndary suspensin spring. Furthermre, the wheel lads vary by lcal track irregularities. The decrease in the static wheel lad is expressed by the equatin (3), t respect the mechanism f the wheel lad variatin due t track twist. Perpendicular frce t the cnstant plane N Reactin Outside Reactin f wheel lad cmpnent tan6 6deg Lateral axle frce by twist f air springs t Fig. Mechanism f wheel lad variatin due t trsin c P = φ + i ' 2 K ta kφ r secndary suspensin springs 8b 2 ' 2 Kφ = kφ = 2kb (3) 2.4 Wheel lad estimatin equatins + 2 2 4kb 2kb 2 2 The utside and inside wheel lads can be calculated 2c by the equatins (4) and (5), in cnsideratin f three factrs, which are the centrifugal frce, track twist and tr- tc = + tc ta = + ta atc a TC sin air suspensins. P : Static wheel lad decrease due t track twist η KΦ: Effective rtatinal stiffness / wheelset (kn-mm) P = P P + F (4) tan6 k' Φ : Effective rtatinal stiffness / bgie (kn-mm) 2b: Width between right and left wheel/rail cntact pints (mm) Pi = Pi P F (5) η tan6 2b : Width between right and left primary suspensin springs (mm) 2b 2 : Width between right and left secndary suspensin P : Outside wheel lad springs (mm) P i : Inside wheel lad k : Primary suspensin vertical stiffness / axlebx (MN/m) : Track shifting frce by the distrtin f air suspensins k 2 : Secndary suspensin vertical stiffness / bgie side (MN/m) η: Cllectin cefficient f vertical cmpnent f (See sectin 3.2.) t c : Track twist between bgie centers (mm) t a : Track twist between wheelsets (within bgie) (mm) 3. Lateral frce estimatin equatins : Distance between wheelsets (within bgie) (mm) 3. Turning lateral frce due t the reactin f inside 2c: Distance between bgie centers (mm) frictin frce a TC : Cant gradient t c : Track twist between bgie centers except cant gradient (mm) t a : Track twist between wheelbase except cant gradient (mm) 2.3 Wheel lad variatin due t trsin f secndary suspensin spring When the train passes a curve, trsin f secndary suspensin springs ccurs n a blster-less bgie due t the relative rtatinal defrmatin between carbdy and bgies. The reactin frce acts laterally n the rail. At the same time, ' (= /tan6 ), the vertical cmpnent f the reactin frce, will act vertically (Fig. ). Track shifting frce that ccurs by the trsin f air springs is described in Sectin 3.2, Clauses (2) and (3). When a vehicle runs n a curve, the flange f the leading utside wheel is in cntact with the utside rail and is pushed against its gauge face. Then the inside wheel resists the frce with a frictin frce (= wheel lad multiplied by the frictin cefficient applied n the tread). This acts as an utward quasi-static lateral frce, that is "turning lateral frce." Accrdingly, the larger the frictin cefficient between the inside wheel tread and the rail (nearly inside Q/P rati κ) is, the larger the turning lateral frce becmes. The estimatin equatin fr the inside wheel quasi-static cmpnent f the lateral frce (turning lateral frce) is expressed by the equatin (6). Qi = κ Pi (6) 2 QR f RTRI, Vl. 43, N. 3, Sep. 22
Inner Q/P rati.7 Basic tread.7.6 =5 =.4 =.4.3.2. Line : Estimatin..2.4.6.8..2.4 Arc r mdified arc tread.6 =5 () Track shifting frce due t centrifugal frce A Centrifugal frce acts n the train that is running thrugh curves accrding t the curve radius, superelevatin and train speed. It cnstitutes a part f the quasi-static track shifting frce. This frce is negative when the train speed is lwer than the balanced speed (superelvatin excess). In cntrast, this frce is psitive when the train speed is higher than the balanced speed (superelvatin deficiency). (2) Track shifting frce due t the trsin f air spring On curves, the trsin f the secndary suspensin springs due t the yaw angle between carbdy and bgies cause a track shifting frce. At the psitin f the leading wheelset f a bgie, this frce acts tward the utside rail. (3) Estimatin equatins fr quasi-static cmpnent f track shifting frce Based n the abve mentined paragraphs () and (2), the quasi-static cmpnent f the track shifting frce due t the centrifugal frce and trsin f the secndary suspensin springs is expressed by the fllwing equatin (7). 2 2 2 ν C ν C 2kb = + = + 2 c 6 QAS W F W (7) gr G gr G ar Q AS :Quasi-static cmpnent f track shifting frce k: Yaw stiffness f air suspensins / bgie (kn/m) (zer fr bgies with blster) : Mdifying cefficient f track shifting frce (4) Mdifying cefficient f the track shifting frce due t the trsin f secndary suspensin springs Accrding t the study result n the calculated value f track shifting frce due t the defrmatin f air springs by the estimatin equatins and, time series simulatin (Fig. 3), we set mdifying cefficient as equatins (8) and (9) Inner Q/P rati.4.3 =.4 =.2. Line : Estimatin..2.4.6.8..2.4 Fig. 2 Mdel f inside Q/P rati Q : Cnstant cmpnent f inside lateral frce i κ : Inside Q/P rati, Pi: Inside wheel lad The values f κ are established fr tapered and arc r mdified arc wheel prfiles, respectively (Fig. 2). These characteristics are btained by time-series simulatin and field test data. 3.2 Track shifting frce due t centrifugal frce and trsin f air suspensins (a) In the case f inside Q/P rati κ < =.7 ( R < 2) (3 R) =.7 (2 R < ) 5 = 3.2 ( R) (b) In the case f inside Q/P rati κ > (8) =.7 ( R < 6) (3 R) =.7 (6 R < ) 5 (9) = 3.2 ( R) In Fig. 3, when the curve radius is larger than a certain value, the track shifting frce due t the trsin f the secndary suspensin springs becmes negative. Cnsequently, fr the utside wheel, the reactin frce f the vertical cmpnent f acts dwnward. This frce reduces the wheel lad f the utside wheel. Hwever, such a cnditin was nt fund during field tests nr by time series simulatin. Therefre, the mdifying cefficient fr vertical cmpnent f is set as equatin (). Lateral axle frce F by twist f air springs 5 5-5 Line : Estimatin.2.4.6.8..2.4 =5 Arc tread = Arc tread =.4 Arc tread =5 Basic tread = Basic tread =.4 Basic tread Fig. 3 Cmparisn f the values f, simulatin and estimatin after mdificatin QR f RTRI, Vl. 43, N. 3, Sep. 22 2
We usually use the "Nadal's equatin" t calculate the critical derailment cefficient. instead f the frictin cefficient µ, we adpt the equivalent frictin cefficient µ e which is a functin f the wheel angle f attack,. η = ( > ), η = ( ) () This is t reflect the difference f the track gemetry (curvature) accurately at the wheelset under cnsideratin. 3.3 Lateral frce variatin due t track irregularities This prcess gives a higher critical derailment cefficient and impacts at rail jints than with the cmmn frictin cefficient µ. When there are track irregularities, especially alignment irregularities, variatin f track shifting frce ccur mainly by the inertia frce accrding t the vehicle vibratins. Shcking variatin f lateral frces ccur at rail jints. These frces increase with higher train speed. The way f calculatins are as shwn by equatins () and (2). QAD = 3WkQσ ZV () (Q/P) cri :Critical derailment cefficient ε 25 V α: Wheel flange angle (rad) Qunsp = + 4 (2) R µ e : Equivalent frictin cefficient = f y /N (f y : lateral creep frce, N: nrmal frce) Q AD : Variatin f track shifting frce 4.3 Apprximatin f equivalent frictin cefficient µ e W : Static axle lad σ z : Standard deviatin f alignment irregularities (mm) The equivalent frictin cefficient µ e is expressed by the equatin (5), t respect the saturatin characteristics V: Train speed (km/h) f the creep frce. k Q : Variatin cefficient f track shifting frce (/mm/(km/h)) µ = µ 22 y / N µ = µ 27. µ Q unsp :Variatin f lateral frce at rail jints ε: Effective rati f variatin f lateral frce (%) 3.4 Estimatin equatins f utside lateral frces We estimate the utside lateral frces by using the equatin (3) in cnsideratin f the abve-mentined three factrs, r the turning lateral frce by the inside rail frictin frce and track shifting frce due t the centrifugal frce and trsin f secndary suspensin springs, and variatin f lateral frces due t track irregularities and impact frces at rail jints. Q = Qi + QAS + QAD + Qunsp (3) Fr vehicles with tapered, arc r mdified-arc wheel treads, when the result f analysis f time-series simula- Q : Outside lateral frce i Q i : Cnstant cmpnent f inside lateral frce Q AS : Cnstant cmpnent f track shifting frce Q AD : Variable cmpnent f track shifting frce by track irregularies Q unsp :Variable cmpnent f lateral frce at rail jints 4. Calculatin f critical derailment cefficient 4. Critical derailment cefficient 4.2 Nadal's equatin The Nadal's equatin, which is used fr calculating the margin against the flange-climb derailment quantitatively, is shwn by the equatin (4). ( Q/ P) cri tanα µ e = (4) + µ tanα e { µ + ( κ ) } 22ν y / N µ + ( 27. ϕ) { } e e 2 e 3 κ ν ϕ (5) µ: Effective value f frictin cefficient at the utside wheel flange : Index expressing saturatin characteristic ( = ) ν y : Lateral creep rati (=tanα sinα (-ct 2 α) ϕ ϕ ) κ 22 : Lateral creep cefficient, N: Nrmal frce, κ 22 /N 27. ϕ: Wheel angle f attack 4.4 Setting methd f angles f attack Attack angle ψ rad.4.3.2. Line : Estimatin Basic tread 2 9 8 6..4.8.2.6 Arc r mdified arc tread Fig. 4 Cmparisn f the values f wheel angle f attack, simulatin and estimatin after mdificatin 22 QR f RTRI, Vl. 43, N. 3, Sep. 22
The derailment cefficient f the utside wheel is cal- tin (Fig. 4) is cnsiderd, the wheel angle f attack ϕ is culated by the equatin (9) as the rati f utside lateral frce t wheel lad f the utside wheel at the same calculated by the equatin (6). a psitin. ϕ = ϕw (6) R Q ( Q/ P) = P (9) (a) Fr tapered wheel treads 9 δ + δ 2 = ( R 9) (Q/P) R : Derailment cefficient Q δ + δ : Outside lateral frce 2 = ( R 9) (7) P : Outside wheel lad (b) Fr mdified arc wheel treads 5.2 Estimated derailment cefficient rati 6 δ + δ2 = ( R 2) R δ + δ2.7 =.3 + (8 R < 2) (8).75 R 2 δ + δ2 = ( R < 8) ϕ: Attack angle (rad) : Wheelbase (m) R: Curve radius at the center f leading bgie (m) ϕ T : Bgie yaw angle (including carbdy yaw angle) (rad) ϕ W : Yaw angle by the bgie steering (rad) δ : Side gap + gauge widening/2 at leading axle (m) δ 2 : Side gap + gauge widening/2 at trailing axle (m) 5. Evaluatin f the margin t flange-climb derailment using estimated derailment cefficient rati 5. Estimatin f utside derailment cefficient We define the "Estimated derailment cefficient rati" as the rati f the critical Q/P shwn in Chapter 4 t the Q/P f the utside wheel shwn in Sectin 5.. This definitin is shwn by the equatin (2). This rati expresses the margin against flange-climb derailment with the base value f.. ( Q/ P) rati = ( Q / P ) ( Q/ P) cri (2) (Q/P) cri : Critical derailment cefficient (Q/P) : Outside derailment cefficient 5.3 Trial calculatin f estimated derailment cefficient rati (Sensitivity analysis) Figure 5 cmpares the estimated and measured values f Q/P, t indicate that estimated values are gd apprximatins f measured data. Figure 6 shws an example f the values f Q/P at the sectin f a circular curve and exit transitin curve, calculated at intervals f m. The estimated Q/P rati has a minimum value just after the vehicle has entered the Wheel lad Lateral frce f uter wheel 5 4 3 2 2 3 4 5 Running speed (km/h) 35 3 25 2 5 5 Inner wheel Outer wheel.4 Measurement Static wheel lad rati f uter wheel.9 Curve radius 6 m Track cant 6 mm Track cant gradient 488 Derailment cefficient f uter wheel.2 2 3 4 5 Running speed (km/h) 2 3 4 5 Running speed (km/h) Fig. 5 Cmparisn f derailment cefficient, ficient, estimatin and measurement.6.4.2.8.6.4.4 QR f RTRI, Vl. 43, N. 3, Sep. 22 23
End f circular curve Cant, Curvature Wheel lad f uter wheel Minimum value End f transitin curve track distrtin between bgie centers track distrtin between wheel base Lateral frce f uter wheel Limit f derailment cefficient Derailment cefficient f uter wheel Fig. 6 Example f the change in the estimated Q/P rati exit-side transitin curve. We calculate the estimated Q/P rati in varius cases. Figure 7 shws the calculatin result. thse values are minimum in transitin curve. 6. Cnclusins () We cmpsed wheel lad estimatin equatins by cnsidering the centrifugal frce, track gemetry deviatin and defrmatin f secndary suspensin springs. (2) We established lateral frce estimatin equatins by cnsidering the curve turning lateral frce, track shifting frce due t centrifugal frce and secndary suspensin springs, and variatin f lateral frce due t track irregularities. (3) We prpsed a calculatin methd fr the critical derailment cefficient by cnsidering wheel flange angle, equivalent frictin cefficient and wheel angle f attack. (4) We define the "Estimated derailment cefficient rati" as the rati f critical Q/P t the utside Q/P, and evaluate the margin against the flange-climb derailment. In this reprt, the methds t estimate the inside wheel Q/P rati and mdify track shifting frce, and sme ther prvisin are set up tentatively. Hereafter, it is necessary t imprve the accuracy based n a theretical study and field data. At this stage, this methd is nly fr blster-less bgies with air spring secndary suspensin. We als intend t imprve the equatins fr applicatin t ther types f bgies. 2... R2m R4m R6m.6.7.8.9.. Static wheel lad rati f uter wheel 2... R2m R4m R6m.4.6.8..2.4.6 Rati f axle spring cnstant 2.. R2m R4m R6m..35.4.45 5.6 Inner Q/P rati 2... R2m R4m R6m 3 5 7 9 Track cant (mm) 2... R2m R4m R6m 2 4 6 8 2 Track cant gradient Fig. 7 Parameter study n estimated Q/P rati 2.. R2m R4m R6m. 2 3 4 5 6 Running speed km/h 24 QR f RTRI, Vl. 43, N. 3, Sep. 22