J. Cent. South Univ. Technol. (2011) 18: 587-592

DOI: 10.1007/s11771-011-0735-1

Optimization of superelevation runoff model for cycling tracks

CHENG Jian-chuan(程建川)^{1, 2}, DU Xiao-yu(杜小玉)¹, SHI Jin-jun(施金君)¹,

ZHANG Yun-long(张云龙)³, LI Fang(李方)¹

1. School of Transportation, Southeast University, Nanjing 210096, China;

2. Jiangsu Provincial University Enterprise Base for Graduate Education on Roadway Safety Design and Evaluation,

Nanjing 210005, China;

3. Department of Civil Engineering, Texas A&M University, TX77843, USA

? Central South University Press and Springer-Verlag Berlin Heidelberg 2011

Abstract: To improve the possible superelevation runoff models for the cycling track design, at first, two existing representative superelevation runoff models used in China were investigated and fitted. Then, an optimization methodology was proposed, which was focused on the track geometry itself, without the consideration of the physical characteristic of the cyclist, assuming that less vertical curvature values correspond to less riding time. The riding performance formulae were obtained with the variables of riding time, riding velocity and vertical curvature of cycling track. Finally, with the refined adjustment on the vertical curvatures with the help of cycling track design software and considering the effect of horizontal alignments, the optimized models were finalized. It is clearly seen that these optimized models take the form of quartic parabola and are verified to achieve 0.005-0.021 s improvement in the event of 200 m time trial.

Key words: cycling track; superelevation runoff; optimization design; track geometry

1 Introduction

The current velodrome track geometry requirement issued by Union Cycliste Internationale (UCI) in 2010 is relatively flexible [1]. It only has guidelines such as “the length of the track must lie between 133 m and 500 m”, and “the inner edge of the track shall consist of two curves connected by two parallel straight lines. The entrance and exit of the bends shall be designed so that the transition is gradual”. However, it does not regulate the parameters of the curve like the radius and the type of transition curve from straight line to curve, as well as the banking method (hereinafter as superelevation runoff model) of cycling track, compared with the ordinary cycling track [2], highway and street [3-5]. Therefore, these variables lead to varied designs of cycling tracks and finally affect the racing performance [6-7]. However, there are few literatures evaluating the quality and effectiveness of different superelevation runoff models. In this work, to simplify the relationship of track geometry and racing performance, the horizontal alignment parameters such as the radius of circular curve and the transition curve type were not considered, and only the superelevation runoff model was investigated. Two existing cycling tracks with different horizontal alignments were selected for the superelevation runoff model optimization. The relationships among the superelevation runoff model, vertical curvature of cycling track and riding performance were studied.

2 Existing superelevation runoff models

The superelevation runoff model refers to the process of transition from the normal superelevation with 13° cross slope to the full superelevation with 45° cross slope, as shown in Fig.1. Based on the design documents and field surveying data of some existing velodromes, it was found that some of these cycling tracks were different in superelevation runoff, which seemed to suggest that there was not a superelevation runoff model widely recognized [8-9]. In this work, the trajectory of 200 m time trial race was chosen as the typical riding track because it is closely related to the superelevation runoff model. Fig.1 shows the horizontal alignment and cross section of the cycling track and the dashed line is the simplified trajectory of 200 m time trial. Fig.2 shows a 3D-view of the cycling track.

Fig.1 Horizontal alignment and cross sections of cycling track

Fig.2 3D-view of cycling track

2.1 Laoshan cycling track

Laoshan cycling track (Laoshan Velodrome, used for Beijing Olympic Games in 2008), well recognized with its high quality design and construction and known for the many world records created there, is elliptical-basin shaped, with a length of 250 m and a width of 7.5 m. In order to find its superelevation runoff model, MATLAB was used to generate curves for matching the surveying data [10]. In practice, it was hard to find such a whole curve that could fit into the superelevation runoff well. There always left a maximum error point, 35 m from the starting line. After many attempts, two-segment cubic parabolas, 0-35 m and 35-70 m intervals from the starting line, respectively, were finally found to be most suitable, as shown in Fig.3.

2.2 Nanjing cycling track

Nanjing cycling track (located at Nanjing Sport Institute, China) has a saddle-shaped design and its track length and width are the same as those of Laoshan Velodrome [11]. Similarly, MATLAB was used to find a curve fitting the surveying data well, and the result indicated that the seventh power parabola model was the most appropriate, as shown in Fig.4.

Both Laoshan cycling track and Nanjing cycling track are praised by athletes, but there still exist some problems such as short weightlessness, short overweight and “touching track surface” caused by inappropriate superelevation runoff models for these tracks. Besides these problems, inappropriate design of superelevation runoff would also result in longer riding time and lower riding velocity. It is necessary to analyze and improve these superelevation runoff models to achieve more comfortable riding and shorter riding time. The related horizontal alignment parameters of these tracks are listed in Table 1.

3 Methodology for superelevation runoff optimization

3.1 Theory of kinematic calculation

When calculating the riding velocity and riding time, it is assumed that the riding bicycle is perpendicular to the track, and the riding force is constant, which is confirmed according to the actual racing situations, and in a fixed small interval the riding process could be considered as uniformly accelerated motion [12-14]. The riding resistance includes air resistance, rolling resistance, slope resistance and accelerating resistance. Riding torque equation could be established based on the following dynamic model [15]:

                     (1)

                         (2)

                               (3)

                                (4)

                 (5)

where F_f is the rolling resistance, F_w is the air resistance, F_θ is the slope resistance, F_a is the accelerating resistance, r is the radius of bicycle wheel, T_f is the rolling couple, F_n is the normal reaction force, b is the acting force displacement of F_n, f is the coefficient of rolling resistance, c is the coefficient of air resistance, ρ is the density of air, A is the projected area  of riding direction, v is the velocity, m is the total mass of bicycle and athlete, θ is the slope of track, dw/dt is the accelerating angular velocity, dv/dt is the accelerating velocity, and I is the moment of inertia. With this kinematic equation, the riding velocity and riding time can be acquired.

Fig.3 Superelevation runoff fitting graphs of Laoshan velodrome track: (a) 0-35 m; (b) 35-63 m

Fig.4 Superelevation runoff fitting graph of Nanjing velodrome track

Table 1 Horizontal alignment of Laoshan and Nanjing cycling tracks

The vertical curvature of track is calculated using the following equations:

                                  (6)

                        (7)

where R_v is the vertical radius, K is the vertical curvature, y=f(x), y is the height and x is the distance along track.

3.2 Existing superelevation runoff models

Besides Laoshan (superelevation runoff) model and Nanjing model, a quartic parabola model was tested to analyze the racing performance after many attempts. It is assumed that, 1) less vertical curvature values correspond to less riding time, and 2) the physical characteristic of the cyclist is not taken into consideration. Therefore, equations with variables, such as riding time, velocity and vertical curvature of cycling track, could be established to evaluate the models for the typical trajectory of 200 m time trial race. The results of existing superelevation runoff models are shown in Table 2 and Fig.5. The assumption of starting velocity of    15 m/s and starting acceleration of 0.15 m/s² are made for the convenience of calculation when considering the actual race situation. In Table 2, t means the riding time of the last 200 m, v₁ means the starting velocity of the last 200 m in time trial qualification race, and v₂ means the finishing velocity of the last 200 m. For simplification, only the situation of one track corner with a tangent-transition-curve is illustrated.

Table 2 Riding time and velocity of existing superelevation runoff models

Fig.5 Vertical curvature of existing superelevation runoff models

From Table 2, the quartic parabola model was found to be able to reach faster velocity and take shorter riding time than the others. But from Fig.5, the vertical curvature of the quartic parabola modal was worse than the others at the first 15 m section. This means when athlete rides on this kind of track, he or she could feel uncomfortable, which could cause unnecessary psychological burden. So, the conclusion is that the quartic parabola model is a better fit than Laoshan and Nanjing models, but it needs to be further optimized to reduce the vertical curvature.

3.3 Methodology for model optimization

Based on the conclusion that the quartic parabola model would make riding faster, the methodology for model optimization was that keeping this advantage while eliminating the high value of vertical curvature. This research did not focus on what extent the vertical curvature reduced would not affect the athlete. Therefore, these existing models are used as references, that is, as long as the vertical curvature of the optimization model is smaller than that of the existing models, it can be considered reasonable. Taking Laoshan model as example, the graphs of two kinds of existing models and their vertical curvature could be adjusted by the cycling track design software, as shown in Fig.6.

Fig.6 Graphs (a) and vertical curvatures (b) of two kinds of existing superelevation runoff models

By locating the worst section of the quartic parabola model on the graph of vertical curvature and marking the same part on the superelevation runoff model graph, improved quartic parabola models could be generated by the cycling track deign software, as shown in Fig.7. Then, it should be calculated again to confirm if it is improved. If not improved, further optimization needs to be done and repeated until meeting the requirements.

4 Optimized superelevation runoff models

4.1 Optimized model based on horizontal alignment of Nanjing cycling track

From the fitting process of Nanjing superelevation runoff model, it was found that this track used a partial superelevation runoff which had a constant value on circular curve. The length of superelevation section is  45 m which is different from 60 m of Laoshan model. Fig.8 shows the vertical curvatures of quartic parabola, Nanjing and Optimized-1 models based on the horizontal alignment of Nanjing cycling track.

From Fig.8, it is found that the Optimized-1 (model) could attain lower vertical curvature value than both the quartic parabola and Nanjing models, hence the Optimized-1 model is more reasonable. The corresponding riding velocity and riding time of these models are listed in Table 3.

Fig.7 Improved quartic parabola model

Fig.8 Vertical curvatures of superelevation runoff models based on horizontal alignment of Nanjing cycling track

Table 3 Riding time and velocity of superelevation runoff models based on horizontal alignment of Nanjing cycling track

The above data state that when riding on the track with the Optimized-1 model in 200 m time trial race, the riding velocity can be higher than that of the quartic parabola model or Nanjing model, and the riding time can be shortened by 0.005 s and 0.021 s, respectively, compared with those of the quartic parabola and Nanjing models.

4.2 Optimized model based on horizontal alignment of Laoshan cycling track

Laoshan model is a whole superelevation, that is, the curve section of track is not constant. Fig.9 shows the comparison of vertical curvatures among the quartic parabola, Laoshan and the Optimized-2 models all based on the horizontal alignment of Laoshan cycling track.

Fig.9 Vertical curvatures of superelevation runoff models based on horizontal alignment of Laoshan cycling track

From Fig.9, it is obvious that the vertical curvature of the Optimized-2 model is more reasonable than that of the quartic parabola. The data in Table 4 shows that the riding velocity and riding time of the Optimized-2 model are close to those of the quartic parabola and better than Laoshan model, which can be verified by the riding time saved in the last 200 m. So, the Optimized-2 model could be accepted.

Table 4 Riding time and velocity of superelevation runoff models based on horizontal alignment of Laoshan cycling track

4.3 Crosstabulation analysis

Because of the different horizontal alignments of cycling track used, it was difficult to compare these improved superelevation runoff models. Therefore, it was necessary to make a crosstabulation analysis to find out the applicability of these models. The horizontal alignment of Laoshan track was taken as an example to analyze the optimized superelevation runoff models.

In Fig.10, it is obvious that the vertical curvature of the Optimized-2 is more reasonable than that of the quartic parabola or the Optimized-1 model. It should be noted that the Optimized-1 is a partial superelevation runoff model, which makes the vertical curvature of the first 4 m even higher. The data in Table 5 also shows that the riding velocity and riding time of both the Optimized-1 and the Optimized-2 model are close to the quartic parabola and much better than those of Laoshan model. These optimized models could save nearly 0.02 s in the last 200 m and the velocity could be 0.03 m/s faster. So, it could be concluded that the Optimized-2 is more suitable for the whole superelevation runoff model.

Fig.10 Vertical curvatures of superelevation runoff models of Laoshan track by crosstabulation analysis

Table 5 Riding time and velocity of superelevation runoff models of Laoshan track by crosstabulation analysis

The same analysis based on the horizontal alignments of Nanjing and the velodrome for 2010 Asian Games at Guangzhou, China was made and concluded. The result states that the Optimized-1 model is better than other models under the condition of partial superelevation runoff and the Optimized-2 model is suitable for the whole superelevation runoff.

5 Conclusions

1) For a fixed track horizontal alignment, superelevation runoff model of track is the key factor affecting the racing performance. With the assumption that less vertical curvature values correspond to less riding time, the quartic parabola model is verified to be more reasonable than the existing cubic and seventh power parabola models.

2) Two improved superelevation runoff models (the Optimized-1 and the Optimized-2) are obtained for two corresponding different horizontal alignments.

3) The Opitmized-1 model is more suitable for partial superelevation runoff and the Opitmized-2 is most suitable for whole superelevation runoff supported by the crosstabulation analysis.

Acknowledgements

The authors would like to thank Mr. KONG Fan-xing, engineer of China Second Railway Survey and Design Co. Ltd, China for his help in computer aided cycling track design and superelevation runoff models testing.

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(Edited by YANG Bing)

Foundation item: Project(BZ2008056) supported by Jiangsu International Cooperative Research Program in 2008, China

Received date: 2010-05-06; Accepted date: 2010-07-20

Corresponding author: CHENG Jian-chuan, Associate Professor, PhD; Tel: +86-25-83790385; E-mail: jccheng@seu.edu.cn