# The Concept of Optimal Dynamic Pedalling Rate and Its Application to Power Output and Fatigue in Track Cycling Sprinters—A Case Study

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background

^{2}= 0.860) and men (y = −0.0370 [s] ∗ x [yrs] + 10.500 [s], R

^{2}= 0.888). Pearson’s correlation coefficient reveals a highly significant strong negative correlation between deployment and finishing time considering the past 25 years (r < −0.940, p < 0.001).

#### 1.2. The Relationship between Force–Velocity and Power–Velocity in Track Cycling

_{max}), maximal pedalling rate (PR

_{max}), maximal power output (P

_{max}) and current optimal pedalling rate (PR

_{opt}) corresponding to P

_{max}[5,6,15].

#### 1.3. The Force–Velocity Relationship and Fatigue

## 2. Methods

#### 2.1. Test Design

_{max}: 2040 W; $\stackrel{.}{\mathrm{V}}$O

_{2max}: 63 mL kg

^{−1}min

^{−1}) performed a 1000 m time trial in an indoor velodrome with a wooden surface of 250 m as part of an official track cycling sprint event. He followed an all-out fashion pacing, i.e., started as hard as possible and continued with the hardest effort voluntarily achievable. The gear ratio chosen was 3.87:1 (58/15), corresponding to a deployment of 8.12 m. Pedal force and crank velocity were measured continuously during the warm-up and the race with a power meter (FES, Institute for Research and Development of Sports Equipment, Berlin, Germany) recording the tangential force on the crank with a sampling frequency of 200 Hz and the duration of each crank revolution. This system allows for the creation of sport-specific F/v and P/v profiles [6]. Raw data was exported and further processed in Office Excel 2016 (Microsoft Corporation, Redmond, WA, USA). Based on the raw data of time-dependent power output, pedalling rate and resulting speed were calculated using our recent published mathematical approach [2,16].

#### 2.2. Data Analysis

_{max}, the theoretical maximal pedalling rate PR

_{max}and the maximal power output P

_{max}at optimal pedalling rate PR

_{opt}were derived.

## 3. Results

^{−1}) and a maximal power output of 2040 W at an optimal cadence of 119 rpm were obtained. In a seated position, fatigue-free maximal mean pedal force was 1561 N and maximal cadence 262 rpm with a maximal power output of 1822 W at an optimal cadence of 131 rpm. The slope of the F/v profile in seated position was −5.95 N rpm

^{−1}. In all regressions, the coefficient of determination was R

^{2}$>0.99$ and maximal power output in the standing position was 12% higher than in the seated position. Figure 4 shows the fatigue-free F/v profiles and P/v profiles of the athlete in standing (black) and seated position (grey).

## 4. Discussion

^{−1}) and with the least effort possible. In the final 1–1.5 laps prior time measurement, the rider usually accelerates sharply, riding the at the top of the wooden track. Utilizing the potential energy, the rider accelerates from the highest point of the bank towards the measuring line that indicates the shortest legal track, to enter the 200 m distance at a very high speed. The aim is to complete the 200 m distance in the shortest possible time. Although the measured 200 m times of world-class female and male track cyclists are less than 10.5 s and 9.5 s, respectively, the total time of high to maximal effort is about 20 to 30 s. The preparatory phase of the first 1-1.5 laps lasts approximately one minute and can significantly contribute to fatigue before the start of the acceleration phase, depending on the endurance capacity and individual pacing (own unpublished research). This corresponds to fatigue after approx. 15 s of maximal effort, so that the level of the optimal pedalling rate during the actual 200 m distance corresponds to that after approximately 30 s of an all-out effort.

## 5. Practical Applications

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

a | Slope of the F/v profile |

b | y-axis intercept of the F/v profile |

b_{i} | y-axis intercept of the F/v profile at crank revolution i |

BFP | Body Fat Percentage |

d | Distance |

F/v | Force-velocity |

F | Mean pedal force |

F_{max} | Maximal mean pedal force |

F_{max,i} | Maximal mean pedal force at crank revolution i |

i | Number of crank revolutions; i = 1, 2, … |

MHC | Myosin heavy chain |

P | Power |

P_{max} | Maximum power output |

P_{max,i} | Maximal power output at crank revolution i |

P_{mean} | Average power output |

PR | Pedalling rate, cadence, frequency |

PR_{max} | Maximal pedalling rate |

PR_{opt} | Optimal pedalling rate |

PR_{opt,i} | Optimal pedalling rate at crank revolution i |

PR_{opt}(t) | Optimal dynamic pedalling rate at timepoint t; time-dependent optimal pedalling rate |

P/v | Power-velocity |

rpm | Revolutions per minute |

v | Velocity |

v_{max} | Maximal velocity |

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**Figure 1.**Development of gears by track cycling sprinters over the past 25 years, shown as deployment (i.e., distance cycled per revolution of the pedals), and resulting changes of mean pedalling rate. The values shown are the means for the 6 female and 10 male cyclists ranked highest in the flying 200 m at the world championships of the respective year.

**Figure 2.**Development of the time difference of the split times for the first and second 100 m of the fastest 6 female and 10 male athletes in the flying 200 m race at the world championships from 1998 to 2022. Traditionally, the split time of the first half has been much faster than the split time of the second half. From 2005 onwards, there is a sudden decrease in the difference between the split times for the first and second 100 m, especially for men.

**Figure 3.**Mean power output (P) for each revolution of the crank (black squares) and the corresponding pedalling rates (PR; black dots) and speeds (black triangles) in a 1000 m time trial by an elite track cyclist. The gear ratio of 3.87:1 (58/15) corresponded to a deployment of 8.12 m. In the race, the mean power was 918 W at a mean cadence of 126 rpm. The oscillating data is induced by the design of the racing velodrome.

**Figure 4.**Fatigue-free force–velocity profiles and power–velocity profiles of the athlete in standing (black) and seated positions (grey) calculated (black and grey squares) by linear and non-linear regression analysis. In standing position, F

_{max}was 1932 N and PR

_{max}amounted to 237 rpm with a corresponding slope of −8.15 N rpm

^{−1}. In seated position, calculations yielded F

_{max}= 1561 N, PR

_{max}= 262 rpm and a slope of −5.95 N rpm

^{−1}.

**Figure 5.**The F/v and P/v profiles of an elite track cyclist in the absence of fatigue (straight black line, standing position) and at the end of a 1000 m time trial (grey straight line, seated position). During the race, the maximal mean pedal force (F

_{max}) decreased from 1932 N to 998 N, the maximal power output (P

_{max}) from 2040 W to 745 W and the optimal pedalling rate (PR

_{opt}) declined from 119 rpm (standing position) or 131 rpm (seated position) to 84 rpm. The triangles represent the raw data points of the current pedalling rate with corresponding power output every 10th second from the first to the last pedal revolution in chronological order.

**Figure 6.**Comparison of actual pedalling rate PR

_{i}(grey dots) and actual power output P

_{i}(black triangles) as percentage of the dynamic optimum of pedalling rate PR

_{opt,i}and dynamic maximal power output P

_{max,i}for each crank revolution i at its corresponding time t during the 1000 m time trial.

**Figure 7.**Pedalling rate (PR), power output (P) and speed (v) calculated for a gear ratio of (

**A**) 3.87:1 (58/15) to rebuild the actual race, (

**B**) 4.92:1 (59/12) to maximise mean power output, (

**C**) 4.36:1 (61/14) to minimise finishing time over 1000 m and (

**D**) 2.88:1 (49/17) to minimise finishing time over 250 m. Time-dependent power output, pedalling rates and resulting speeds were estimated applying our recently published mathematical approach to the raw data [2].

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**MDPI and ACS Style**

Dunst, A.K.; Hesse, C.; Ueberschär, O.
The Concept of Optimal Dynamic Pedalling Rate and Its Application to Power Output and Fatigue in Track Cycling Sprinters—A Case Study. *Sports* **2023**, *11*, 19.
https://doi.org/10.3390/sports11010019

**AMA Style**

Dunst AK, Hesse C, Ueberschär O.
The Concept of Optimal Dynamic Pedalling Rate and Its Application to Power Output and Fatigue in Track Cycling Sprinters—A Case Study. *Sports*. 2023; 11(1):19.
https://doi.org/10.3390/sports11010019

**Chicago/Turabian Style**

Dunst, Anna Katharina, Clemens Hesse, and Olaf Ueberschär.
2023. "The Concept of Optimal Dynamic Pedalling Rate and Its Application to Power Output and Fatigue in Track Cycling Sprinters—A Case Study" *Sports* 11, no. 1: 19.
https://doi.org/10.3390/sports11010019