241
ON METHODS USED FOR SETTING THE CURVE FOR RAILWAY TRACKS
Srihari Palli1, Rakesh Chandmal Sharma2*, Sunil Kumar Sharma3, Vinod Babu Chintada4
1Dept. of Mech.Engg., AITAM, Tekkali, AP, India.
2Mech. Engg. Dept., Maharishi Markandeshwar (Deemed to be University), Mullana, Ambala, India.
3Dept. of Mech. Engg., Amity School of Engg. and Tech., Amity University, Uttar Pradesh, Noida, India.
4Dept. of Mech. Engg., GMRIT, Rajam, India.
E-mail: [email protected]1, [email protected]2 [email protected]3, [email protected]4
*corresponding author
ABSTRACT: The regular efforts to enhance vehicle velocities and obtain even higher speeds on the railways need high quality track. The measure of track quality is the quality of the curves linking intersecting but multiple directional straight lines. Presently, transition curves are provided on track suitable for higher speeds to negotiate the contact stresses and dynamic effects generated due to lateral acceleration. Despite the less literature available on the area, methods exist for planning transition curves taking account of the dynamics of movement. Most of them are based on the effects or non-compensated lateral acceleration changes in such acceleration and the shock of sudden displacement, the resulting track curves being adapted to the proscribed basic geometry in most cases on a trial and error basis or by insertion of circular arcs. In an attempt to establish new ways of planning, this paper proposes a method for developing the curvature functions that are so important from a dynamic-point of view from a combination not just of the two fundamental factors- lateral acceleration and shock-but also taking account of geometric conditions and desired deflection angle.
KEY WORDS: Contact stresses, Lateral acceleration, Non-compensated lateral acceleration, Transition curves
I. INTRODUCTION
Looking back at the history of the human race it is clear to see that mankind's urge to move from place to place is as old as the species. Even in the days when there were as yet no vehicles for the purpose, the constant toing- and- froing between much frequented locations carved paths ("roads") that in themselves made movement easier. After a while vehicles were invented making travel an even simpler proposition and the roads and railways on which these vehicles moved were developed, vehicles and infrastructure subsequently adapting to each other as time went by.
The advent of motive power was a major milestone in the history of transport as in almost all other sectors of industry. As vehicles have become capable of carrying heavier loads and moving at increasingly higher speeds, track quality as regards carrying capacity and geometry has equally had to be improved in order to keep pace [1-3].
Track-bound transport systems may be short of historical perspective but their development period after inception is identical with that of road transport. Higher speeds require appropriately designed track geometry [4-5]. This applies to the railway (as a track-bound transport system), since road vehicles have a certain degree of manoeuvrability on the surface provided for their movement whereas the characteristics of rail movement are entirely determined by track geometry. Speed variation is the only possibility but even then only to a certain extent because of the inertia of large masses [6-11].
II. METHODS OF SETTING A CIRCULAR CURVE Following methods are used for setting curves
a. Long Chord Offset Method
In Fig. 1 T1 T2 is the long chord of a R radius. C is long chord length and it is divided in eight uniform sections . T1A, BC, CD etc., all section having a length xC8 . PW is parallel line to the long chord and let offsets considered form points R, Q & P are O1, O2 & O3. Versine V from the chord C is given as:
8R V C
2 or
8R
T
DS T1 2 (1)
Offset O1 from the line PW is given by the formula:
128R C 2R RS 2R O x
2 2 2
1
(2) Or,
128R RC C
2 (3)
242 From formula (i) and (ii) values of V1, V2, V3 etc. Can be calculated as follows:
16V 15 8R C 16 15 128R
C 8R V C
2 2
2
1
(4)
V16 12 128R
C 16 12 128R
2C 8R V C
2 2 2
2
(5)
V16 7 8R C 16
7 128R
3C 8R V C
2 2 2
3
(6)
Figure 1: Long Chord Offset Technique a. Quartering Versine Technique
This is used for constructing curves of short length of nearly 100 m. The position of two tangent points (T1 and T2) is initially located and the distance among them measured. The versine (V) is finally evaluated, using the expressions
R V 125C
2 (mm) (7)
Meanwhile, V = versine in mm, C = chord in metres and R is radius in metres or,
R V 1.5C
2 (inches) (8)
Figure 2: Quartering Versine Technique
b. Chord Deflection Technique
This technique is mostly used in constructing the curves. In the Figure. 3, T1 is the tangent point and A, B, C and D etc.
be the successive points on the curve. The length of chords T1 A, AB, BC and CD be
x
1, x
2, x
3 and the chords are picked of uniform lengths and let their value be c. The last chord may be of different length and let their value be c1.Figure 3: Chord Deflection Method 1st offset
2R c 2R A x
A1 21 2 2nd offset
2R c 2R
x 2R
x B x
B1 1 2 22 2 3rd offset
2R c 2R
x 2R
x C x
C1 1 3 23 2 Last offset
2R c 2R cc 2R x 2R
x N x N
2 1 1 2 n n 1 n
1
2R c c c1 1
(9)
Non-compensated lateral acceleration, Cant, Cant deficiency, Roll acceleration
A railway vehicle moving over circular curve is under inertial centrifugal acceleration which is proportional to its velocity and curvature. This lateral acceleration leads to discomfort to the passengers and its value beyond a limit
243 results in the lateral instability of the vehicle. In order to take care of a part of this effect the curve track is assigned cant or superelevation S, defined as “the vertical difference in positions of the two rails of a track, measured at centerline of the railheads (G)", in other words the track on curve is provided inclination with the cross-level angle α (Fig. 4).
Fig. 4: Geometry of railway vehicle on curve
With this inclination, the gravitational acceleration g develops a component parallel to the plane of rail, taking care of a part of the centrifugal acceleration.
Net difference among the two components is defined as non-compensated lateral acceleration of the vehicle expressed as:
sin
0 acos g
a (10)
For small angle α, cosα =1.
a0 is expressed as:
G gS R
a0v2 (11)
The cant deficiency is defined as:
R E g
v DS
.
.2 (12)
Circular curve track sections without cant may be laid provided that the non-compensated lateral acceleration and its time derivative are below the permissible values for related track geometry.
Circular curves track with cant and transition curve with or without cant are essential to be laid when the abovementioned condition is not fulfilled. In this case, the following parameters are required to be considered for planning of curved track.
• cant (S);
• time derivative of cant (dS/dt);
• length derivative of cant (dS/dl);
• cant deficiency ( S−);
• time derivative of cant deficiency ( dS−/dt )
• roll acceleration ()
• time derivative of roll acceleration()
The Federal Railroad Administration’s (FRA) Curved-Track Class definitions are summarized in Table 1 (Appendix A) [12]. The geometry of track curvature and superelevation can generally be obtained from track charts. If accuracy is desired, a direct measurement of curvature and superelevation is recommended.
CEN [13] has specified limiting values of these parameters for the safe travel of railway vehicle on circular and transition tracks as mentioned in Table 2 (Appendix A). Few other transportation agencies i.e.KÖZDOK [14] for Hungarian railways and ÖBB GB [15] for German railways have also specified their limiting values of these parameters.
Conclusions:
In this article the techniques of laying circular curves have been discussed. Limiting values of these parameters with each type of curve have also been evaluated. With this article, authors have provided a method of design the transition
244 curves, a problem which is of interest to track designers.
The advantage of the suggested method is that it permits to decide a compound curvature function consists of different parts for given geometrical conditions. The technical terms like non-compensated lateral acceleration cant, cant deficiency and roll acceleration have been discussed and their expressions have derived and international standards used for them have been discussed.
III. REFERENCES
1. Sharma, R.C. et al. 2018. Sensitivity analysis of three-wheel vehicle's suspension parameters influencing ride behavior. Noise and Vibration Worldwide, 47(7-8): 272-280.
2. Sharma, R.C. et al. 2020. Analysis of generalized force and its influence on ride and stability of railway vehicle, Noise and Vibration Worldwide, 51(6): 95-109.
3. Sharma, R.C. et al. 2017. Modernization of Railway Track with Composite Sleepers, Int. J. Vehicle Structures
& Systems, 9(5): 321-329. http://dx.doi.org/10.4273/ijvss.9.5.10.
4. Palli, S. et al. 2018. A Review on Dynamic Analysis of Rail Vehicle Coach, International Journal of Vehicle Structures and Systems, 10(3): 204-211. https://doi.org/10.4273/ijvss.10.3.10
5. Bhardawaj, S. et al. 2019. A Survey of Railway Track Modelling, Int. J. Vehicle Structures & Systems, 11(5):508-518.
6. Sharma, R.C. et al. 2016. Evaluation of passenger ride comfort of Indian rail and road vehicles with ISO 2631- 1 standards: Part 1-Mathematical Modeling. Int J Veh Struct Syst, 8(1), 7–10.
7. Sharma, R.C. 2016. Evaluation of passenger ride comfort of Indian rail and road vehicles with ISO 2631-1 standards: Part 2-Simulation. Int J Veh Struct Syst, 8(1), 1–6.
8. Sharma, S.K. and Sharma, R.C. 2018. Simulation of quarter-car model with magnetorheological dampers for ride quality improvement. Int J Veh Struct Syst, 10(3): 169–173. DOI: 10.4273/ijvss.10.3.03.
9. Sharma, R.C. et al. 2018. Rail Vehicle Modelling and Simulation using Lagrangian Method. Int J Veh Struct Syst; 10(3): 188–194. DOI: 10.4273/ijvss.10.3.07.
10. Sharma, R.C. 2017. Ride, eigenvalue and stability analysis of three-wheel vehicle using Lagrangian dynamics, Int. J. Veh. Noise Vibr. 13 (1):13–25.
11. Palli, S., Sharma, R.C., Rao P.P.D. 2017. Dynamic behaviour of a 7 DoF passenger car model, Int. J. Veh.
Struct. Syst. 9 (1): 57–63.
12. Singh, G. et al. 2020. Comparative Measurements of Physical and Mechanical Properties of AA6082 Based Composites Reinforced with B4C and SiC Particulates Produced via Stir Casting. Article in press.
13. CEN ENV 13803-1:2002, Railway Applications-Track alignment design parameters -Track gauges 1435 mm and wider - Part 1: Plain line, 2002.
14. KÖZDOK. 1983. Track alignment design regulation of national public railways (TADR), Budapest.
15. ÖBB GB. 2004. Track Technology Superstructure - Technical Principles B 50 – Part 2: line management of tracks.
245 Appendix A:
Table 1. Condensed FRA curved-track safety standards [7]
Safety Standard Track class
1 2 3 4 5 6
Speed Limits Fraight, km/h
16 40 64 96 128 176
Passengers km/h 24 48 96 128 144 176
Gauge,
mm G
mm G
...
...
1422.4 1422.4 1422.4 1422.4 1422.4 1422.4
1466.8 1466.8 1466.8 1460.5 1460.5 1447.8
Alignmenta...mm 127.0 76.2 44.5 38.1 15.9 09.5
Rail Profile r ...mm
88.9 76.2 50.8 38.1 25.4 12.7
mm
Vr ... 44.5 38.1 31.8 25.4 19.4 12.7
Cross levelVc ...mm 50.8 44.5 31.8 25.4 19.1 12.7
c ...mm
76.2 50.8 44.5 31.8 25.4 12.7
c ...mm
76.2 50.8 44.5 31.8 25.4 15.9
a: Mid offset of alignment from a18.9 m chord touching the alignment at both chord-ends.
r: Mid offset of a rail from a 18.9 m chord touching the rail at both chord-ends.
c: Deviation from zero cross level at any point along the track.
Vr: Deviation from designated elevation on spiral.
Vc: Variation in cross level on spirals in any 9.45 m.
c: Difference in cross level between any two points less than 18.9 m apart along the track.
Table 2: CEN guidelines of limiting values for different parameters of railway transition curve track [9]
S.N. Parameter Governing expression Limiting value
1. Non-compensated lateral
acceleration(a0) 2 0,lim
2
0 1500
. 6 .
3 gs a
R
a v a0,lim(m/s2)0.980665-1.0787
(circular) 2. Time derivative of non-
compensated lateral acceleration )
(a0
lim , 0 3
0 800 a
R
a v a0,lim(m/s3)0.3597-0.5886
(circular)
lim , 0 3
0 23 . a
L R
a v a0,lim(m/s3)0.3597-0.5886
(clothoid)
lim , 0 3
0 30 . a
L R
a v a0,lim(m/s3)0.3597-0.5886
(cosine)
3. Cant (S) - Slim(mm)160180
0 1 1 )
lim(mm
s for tracks adjacent to
passenger platforms 4. Time derivative of cant
6 lim
. 3
.
dt dS L v S dt
dS (mm/s) 50 60
lim
dt
dS (clothoid)
70 55 ) mm/s (
lim
dt
dS (cosine)
5. Transition length derivative of cant
6 lim
. 3
.
dl
dS dt dm v dl
dS (mm/m) 2.25 2.5
lim
dl dS
6. Cant deficiency (S)
lim 2
) ( .
802271 .
11
m S
R
S v ( )lim(mm) =150-165
S
7. Time derivative of cant deficiency )
/ (dS dt
6 lim
.
3
dt dS dl dS v dt
dS (mm/s) 55 90
lim
dt dS
8. Angular roll acceleration(
)2 lim 2
2 ()
19440
dl l S d
v lim(1/s2)0.1
9. Angular roll jerk(
)3 lim 3
3 ()
69984
dl l S d
v lim(1/s3)0.2
246 Authors Profile:
Srihari Palli received M. Tech. from Indian Institute of Technology Roorkee, India in 2005. He is an Associate Professor in the Department of Mechanical Engineering, Aditya Institute of Technology and Management Tekkali, India. His research interests include Vehicle Dynamics, Design Optimization, Computer Integrated Manufacturing, Product Life Cycle Management, Digital Manufacturing, Manufacturing Process Management, Finite Element Analysis, Machine &
Structural Dynamics. He has a cumulative 15 years of academic and industrial experience.
Rakesh Chandmal Sharma received his M. Tech from the Institute of Technology, Banaras Hindu University, Varanasi, India in 1998 and PhD from the Indian Institute of Technology Roorkee, Roorkee, India in 2010. He is Professor in Mechanical Engineering Department, Maharishi Markandeshwar (Deemed to be University), Mullana, Ambala, India. His research interests include vehicle dynamics, mechanical vibrations and machine design. He has 22 years of academic experience.
Sunil Kumar Sharma did Ph.D. from Indian Institute of Technology Roorkee, Roorkee in 2017.
His research interests include vehicle dynamics. His research interests include rail vehicle dynamics. He is Asst. Professor in Dept. of Mech. Engg, Amity School of Engg. and Tech., Amity University, Uttar Pradesh, Noida, India.
Mr. Vinod babu Chintada is working as Assistant Professor in the department of mechanical engineering, GMR Institute of Technology, Rajam. He obtained B.Tech (Mechanical Engineering) from JNTUK, Kakinada and ME (Machine Design) from Andhra University, Visakhapatnam. His area of specialization includes surface coatings, composite materials. He is expertise in synthesis of nanomaterials.
