Spiral length design

Transcription

1 Spiral length design Nazmul Hasan *1 1 SNC-AVAIN INC. Transportation Diision, West Georgia Street, Vanocuer, British Columbia, Canada V6E C9, Tel: , Fax: , Nazmul.Hasan@snclaalin.com eceied: Accepted: Abstract: On the circular cure, lateral acceleration affects comfort. On the spiral, both lateral acceleration and jerk affect comfort. The determination of maximum curing speed on the basis of lateral acceleration only is not critical because the lateral acceleration is proportional to the square of speed whereas jerk is proportional to cubic power of speed. The maximum speed on a cure is thus significantly restricted by the spiral length. In this paper, spiral length is formulated on the basis of both lateral acceleration and jerk. The proposed formula is applied on cures of arious curature and alidated against an earlier formula by the author. On the basis of current choice of alues of lateral acceleration and jerk, a relation between jerk and lateral acceleration is gien that can be used to determine the acceptable alue of jerk for a gien lateral acceleration. A single criterion is proposed to decide the requirement of a spiral cure. Keywords: spiral length, unbalanced supereleation, actual super-eleation, jerk, lateral acceleration. * Corresponding Author 107 Adances in ailway Engineering, An International Journal Vol./ No./ Summer and Autumn 014

2 1. Introduction In an earlier paper (Hasan 010) by the author the spiral length is gien by: Ea Eu V (1) J On circular cure, theoretically lateral jerk is zero since the lateral acceleration is constant. On the other hand, the lateral acceleration is ariable and jerk is constant, equialent to design alue, on the spiral. Eq. (1) considers speed, super-eleation, and jerk to compute the spiral length. It does not consider lateral acceleration explicitly. The comfort criterion is usually gien by lateral acceleration. In this paper, a formula is deried that includes lateral acceleration explicitly in addition to speed, supereleation, and jerk. To alidate, the formula is applied and results are compared with the results gien by Eq. (1). Usually lateral acceleration is gien. A relation is gien to determine the alue of jerk for any gien lateral acceleration. The determination of maximum curing speed on the basis of lateral acceleration only is not critical because the lateral acceleration is proportional to the square of speed whereas jerk is proportional to cubic power of speed. The maximum speed on a cure is thus significantly restricted by the spiral length. The suggested spiral length equation is useful to determine the maximum curing speed on a cure as it contains both terms - lateral acceleration and jerk.. Deriation of spiral length formula In this section, jerk and lateral acceleration refer to axle based alues. Jerk is defined as the rate of change of lateral acceleration. Mathematically a J t J J () () The well known equilibrium super-eleation equation is gien by: Eq Ea Eu 15 Ea 15 Eu Diiding both sides of Eq. (4) by, 15 Eu Ea 15 Eu eplacing on right side of Eq. (5) by Eq. (): Ea J J 15* Eu * Ea J J 15* Eu * * Ea J Eu J 15* * a Ea 15* J * a Eu * J a Ea * J 15* J Eu * a Table 1. Spiral length by Eq. (1) and Spiral ength, (m) (4) (5) V Curature Eq Ea Eu Eu/Ea,Jerk.at 5 10 Super-sharp Very sharp Sharp Mild Mild Adances in ailway Engineering, An International Journal Vol./ No./ Summer and Autumn

3 Changing the unit of speed from m/s to km/h, VEa J 551* J.6* Eu * a (5). Validation The Eq. (6) is alidated by application only. The Eq. (1) and Eq. (6) are applied for three pairs of axle based lateral acceleration and jerk on different curatures in the following three Table (1) ~ (): Usually the spiral length is rounded to nearest 5m. In the aboe tables spiral lengths are not rounded. Thus it appears from the aboe three tables that the Eq. (6) gies the same spiral length as the Eq. (1) which is already an accepted formula. Theoretically Eq. (6) is a better expression as it contains both lateral acceleration and jerk. In fact Eq. (1) and Eq. (6) are same. From Eq. (6), Eq. (1) can be deduced. This is shown below: Ea * Ea * J Eu 15* J Eu * 15* J a t Ea * * t Ea * 15* J Eu 15* J * t Eu Taking out from both sides, Ea 1 15* J * t Eu 15 * J * t Ea Eu Ea Eu t 15* J Vt V Ea Eu V Ea Eu.6.6*15* J Under equilibrium (balanced) speed, both Eq. (1) and Eq. (6) gie a supereleation gradient of 1 in 6V [V(kmph)] under a jerk of 0.0 g/s. Since trains always run with some unbalanced supereleation, the minimum supereleation gradient would be flatter than 1 in 6V. On NS steeper transition gradients of up to 1: 8V are allowed with a minimum of 1 in 600 at speeds lower than 80 km/h (Eseld 001). Table. Spiral length by Eq. (1 ) and Spiral ength, (m( V Curature Eq Ea Eu Eu/Ea,Jerk.at 5 10 Super-sharp Very sharp Sharp Mild Mild Table. Spiral length by Eq. (1 ) and Spiral ength, (m( V Curature Eq Ea Eu Eu/Ea,Jerk.at Super-sharp Very sharp Sharp Mild Adances in ailway Engineering, An International Journal Vol./ No./ Summer and Autumn 014

4 4. elation between jerk and lateral acceleration Usually a comfort alue of lateral acceleration is suggested without any mention of corresponding jerk as comfort criteria. As per UIC Code, the comfort limit is 1.0 to 1.5 m/s range (=0.1g to 0.15g) (UIC Code 1989). According to Eseld (001), the lateral acceleration must in all cases remain below 1.5 m/s, and preferably below 1 m/s. In North America the comfort limit is usually set at 0.1g. Jerk is also an important parameter which affects ride comfort. To use Eq. (6), one needs both lateral acceleration and jerk alue. Using the alues of lateral acceleration and jerk used in current practice, a relation between lateral acceleration and jerk is established hereafter to facilitate the use of Eq. (6). Generally, the limit of comfort refers to alues measured in the carbody s reference system. These are usually measured on the car s floor, on top of the leading and trailing bogie piot, at the connection between bogie and carbody. In literature (TCP 01) and here, the alues are applied on axle basis. Usually the perceied alues on floor are reduced from the axle based alues. Thus, the application of codified comfort alues on axle basis is conseratie. In Tables 1~, the alues of pair of lateral acceleration and jerk e.g. 0.1g and 0.0g/s, 0.1g and 0.04 g/s, and 0.15g and 0.1g/s are taken from TCP (01). TCP does not clear the basis of the choice of the alues of the pair. The spiral lengths calculated in Tables 1~ indicate that the suggested choice by TCP works. Jerk and lateral acceleration are linearly related (cf. Eq. ()). Thus, a relation between lateral acceleration and jerk is established by interpolation between alues of two pairs e.g. 0.1g and 0.0g/s, and 0.15g and 0.1 g/s. The relation stands as: J 0.0g / s 1.4 a 0.1g (7) in which a = lateral acceleration in terms of g e.g. 0.1g, 0.15g etc. As per Eq. (7), for a jerk of 0.04 g/s, theoretically lateral acceleration should be 0.107g instead of 0.1g in Table. For a lateral acceleration of 0.g, jerk should be 0.17g/s. Eq. (7) would help to use Eq. (6) if jerk alues are not gien. Eq. (7) would help to judge the gien alue of jerk for a gien lateral acceleration. 5. Utility of the suggested formula The formula has two other applications other than determining the spiral length. They are: First application: Curing speed model The formula may be manipulated to model curing speed. Second application: equirement of spiral cure In the literature, authors often put some criteria where spiral cures are not required. As for example, Eseld (001) says spiral cures are not used if: - The cure radius is > 000 m; - A calculation shows that no supereleation is necessary; - Between two adjacent cures in the same direction the discontinuity in acceleration remains limited to 0. ~ 0. m/s. In the literature, authors also put some guide line on installation of supereleation. Eseld (001) says if the V (kmph )m( Table 4. Testing of single criterion for the requirement of spiral cure a Eq (mm( Ea (mm( Eu (mm( equirement of SC )m/s ( )m( < 1.4.Not reqd < 4.7.Not reqd > 6.4.Absolute min length reqd > 7.5.Absolute min length reqd Adances in ailway Engineering, An International Journal Vol./ No./ Summer and Autumn

5 calculated supereleation is less than 0 mm it can be disregarded. If the calculated supereleation is 5 mm or less, actual supereleation is not usually installed (TCP 01). In the preceding statements, the calculated supereleation means equilibrium supereleation. Thus, actual supereleation should be less than 0 mm to aoid installation. According to proportion between unbalanced and actual supereleation suggested by me (Hasan 011), the actual supereleation comes out to be 1 mm (=0/1.7) or less to aoid installation. In USA, there are many instances of installation of half inch supereleation. With a minimum supereleation gradient of 1 in 600 (Eseld 001), 1 mm of actual supereleation leads to a spiral length of 7. m. Thus noninstallation of actual supereleation may not call for noninstallation of spiral; it needs to be confirmed from the required length of spiral cure. Usually, the spiral length is rounded to the nearest 5 m. Thus, I would like to propose a single criterion that if the calculated spiral length is less than 5 m, then it is not necessary to install spiral cures. In all other cases at least absolute minimum length of spiral cure shall be installed. The proposed criterion for requirement of spiral cure is analyzed in Table Future research In this paper, ride comfort is estimated by quasi-static formula, which has limitations. The formula considers only ertical stiffness of suspension ia suspension factor; in fact, ride comfort is affected by many other ehicle parameters. It would be ery expensie to measure ride comfort by a simple accelerometer or track geometry car on a cure with the proposed spiral length. Currently, there are sophisticated, alidated computer software programs that estimate ride comfort. These programs also can predict derailment safety by computing lateral load/ ertical load (/V) ratios. Use of any suitable software is recommended to estimate ride comfort and safety. This would certainly help to alidate the work and increase confidence in it. 7. Conclusion The spiral length deried is different in its form from all current equations. The equation may be manipulated to model curing speed. The spiral length is gien by VEa J 551* J.6* Eu * a For a gien lateral acceleration, jerk is to be determined by the following relation: J 0.0g / s 1.4 a 0. 1g ength is suggested as a criterion for the requirement of installation of spiral cure. If the calculated spiral length is less than 5 m, then the spiral is not required. 8. Notations a = axle based lateral acceleration in m/s ; Ea = actual super-eleation in mm; Eq = equilibrium super-eleation in mm; Eu = unbalanced super-eleation in mm; J = axle based jerk in m/s; = spiral length in m; = radius of cure in m; t = time to traerse spiral in s; V = speed in km/h; = speed in m/s. 9. eferences - Eseld, C. (001). Modern railway technology, MT- Productions, the Netherlands: 8, 9, Hasan, N. (010). Spiral length design, Proc., 010 ASME/ASCE/IEEE Joint ail Conference, Urbana- Champaign, I - Hasan, N (011). Passenger track cure design criteria: Comfort criteria, equialent comfort criteria, and applications, Proc., 011 ASME/ASCE/IEEE Joint ail Conference, Pueblo, CO - International Union of ailways (UIC) Code 70. (1989). ayout characteristics for lines used by fast passenger trains. International Union of ailways, nd edition, Paris, France: 8, 9. - Transit Cooperatie esearch Program (TCP). (01). Track design handbook for light rail transit. eport 155, National Academy Press, Washington, DC. 111 Adances in ailway Engineering, An International Journal Vol./ No./ Summer and Autumn 014