[Paper Review] On modelling bicycle power for velodromes: Part II Formulation for individual pursuits

Len Bos, Michael A. Slawiński+2 more|arXiv (Cornell University)|Sep 2, 2020

Sports Performance and Training7 references4 citations

TL;DR

This paper formulates a mathematical model for bicycle power dynamics on velodromes during individual pursuits, accounting for straights, circular arcs, and Euler spiral transition curves. It analyzes constant-cadence and constant-power cases, demonstrating that including transition curves improves empirical adequacy by better matching real-world power and cadence measurements, particularly in minimizing abrupt changes in lean angle and power output.

ABSTRACT

We model the instantaneous power on a velodrome--as applied to individual pursuits and other individual time trials--taking into account its straights, circular arcs, and connecting transition curves. The forces opposing the motion are air resistance, rolling resistance, lateral friction and drivetrain resistance. We examine the constant-cadence and constant-power cases, and discuss their results, including an examination of an empirical adequacy of the model. We also examine changes in the kinetic and potential energy.

Motivation & Objective

• To develop a mathematically rigorous model of bicycle power on velodromes that accounts for the full track geometry, including straights, circular arcs, and transition curves.
• To examine the empirical adequacy of the model by comparing predicted power, cadence, and speed with real measurements from individual pursuits.
• To assess the impact of kinetic and potential energy changes on instantaneous power, particularly during acceleration and transitions.
• To evaluate the dynamic equilibrium between power and cadence in constant-effort time trials, and to explore discrepancies between model predictions and measurements.
• To inform velodrome design by analyzing the effects of transition curve geometry on rider dynamics, such as jolt and lean angle consistency.

Proposed method

• Parameterize the velodrome track using a black line composed of three segments: straight, Euler spiral transition, and circular arc, ensuring C² continuity for smooth curvature.
• Model the track geometry using Fresnel integrals for the Euler spiral, with curvature varying linearly with arc length to ensure smooth transitions.
• Formulate instantaneous power as the sum of forces opposing motion: air resistance, rolling resistance, lateral friction, and drivetrain resistance.
• Derive power expressions for both constant-cadence and constant-power cases, using velocity-dependent force models and track geometry.
• Incorporate changes in kinetic and potential energy via analytical integration in Appendix A, assessing their contribution to total work done.
• Validate the model by comparing simulated power and cadence profiles against empirical measurements, particularly in transition zones.

Experimental results

Research questions

• RQ1 How does including Euler spiral transition curves improve the empirical adequacy of bicycle power models on velodromes compared to models that neglect them?
• RQ2 What are the differences in power and cadence dynamics between constant-cadence and constant-power cycling strategies during individual pursuits?
• RQ3 To what extent do changes in kinetic and potential energy contribute to the total work done by the cyclist, and how does this affect instantaneous power?
• RQ4 Why do discrepancies arise between predicted and measured cadence shifts relative to power, and what factors might explain this phase lag?
• RQ5 How does the choice of transition curve (e.g., Euler spiral) affect the smoothness of lean angle, speed, and power profiles, and could alternative curves reduce jolt or improve rider efficiency?

Key findings

• The inclusion of Euler spiral transition curves significantly improves the model's empirical adequacy by producing smoother power and cadence profiles, especially in transition zones.
• The constant-cadence and constant-power cases exhibit dynamic equilibrium with small oscillations around mean values, suggesting stable rider behavior during sustained efforts.
• Even though most work is done against dissipative forces, a non-negligible portion of the cyclist’s energy input contributes to increasing mechanical energy (kinetic and potential), particularly during acceleration and transitions.
• The model shows that the separation between the zero line and the curve in the inclination-lean angle plot is minimized when the track inclination optimally matches the rider’s lean angle, reducing lateral friction losses.
• Discrepancies between model predictions and measurements—especially in the phase shift between power and cadence—suggest that current models may not fully capture the dynamic response of the drivetrain or rider neuromuscular control.
• The model’s predictions of speed and power align well with empirical data, supporting its use for retrodicting lap times or estimating resistance coefficients from measured performance.