#
Optimal Football Pressure as a Function of a Footballer’s Physical Abilities^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{1}is the mass of the players foot, m

_{2}is the mass of the deformed portion for the ball during impact, m

_{3}is the mass of the remainder portion of the ball, k

_{1}and c

_{1}are the stiffness and damping of the players foot, k

_{3}and c

_{3}are the stiffness and damping of the ball between the two ball masses, and f is the force the foot applies to the front end of the football. The mass of the football, m, is related to masses m

_{2}and m

_{3}through mass ratio $\mu $ as

_{1}, and ball, ω

_{3}, can be written as functions of the mass and stiffness as

_{1}and z

_{2}are the damping ratios for the foot and ball, respectively. The force the foot applies to the front end of the football, f, can be defined by the nonlinear function as

_{2}and ${\overline{x}}_{1}$ is the displacement of the foot at the time of initial impact with the ball, such that when x

_{2}is larger than ${\overline{x}}_{1}$ (the otherwise case in Equation (7)) the foot is no longer in contact with the front end of the ball.

## 3. Results

^{4}/lb

^{2}${a}_{1}=-0.0223$ in

^{2}/lb and ${a}_{2}=0.0009$. While the resulting function does not monotonically decrease over the high end of the pressure range considered, there is a significant overall trend that the damping decreases as the pressure increases. The results should be considered only valid over the pressure range tested. The r-squared value for this model is 96.5%.

## 4. Discussion

_{1}(p) = −SP(p), f

_{2}(p) = −TC(p) + FR(p)

## 5. Conclusions

## Conflicts of Interest

## References

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**Figure 1.**Proposed lumped mass representation of a foot striking a soccer ball with linear stiffness and viscous damping.

**Figure 2.**Photographs of (

**a**) pumping balls to pressure; (

**b**) video recording ball dropped at 3.084 m (10 ft) height; and (

**c**) experimentally determined average damping ratio as a function of pressure (experimentally measured mean damping ratio—red circles; Equation (9)—black curve).

**Figure 3.**Kicking of football MATLAB Simulink model and resulting plot indicating: deformation of the ball; velocity of foot (red), front (blue) and back (black) of ball; and force of the foot on the ball.

**Figure 4.**The effect of ball pressure on the: speed amplification (final velocity of ball divided by initial foot velocity (black); time the foot is in contact with the ball (green); and maximum impact force of the foot and ball (red), for (

**a**) engineering units and (

**b**) normalized to response at 0.69 bar pressure.

Parameter | Variable | Value | Units |
---|---|---|---|

mass of ball | m | 0.43 | kg |

mass ratio | μ | 0.1 | -- |

ball frequency | ω | 150 | Hz |

ball damping ratio | z_{3} | 12 | % |

foot mass | m_{1} | 2.27 | kg |

foot frequency ^{1} | ω | 0.5214 | Hz |

foot damping ratio | z_{1} | 100 | % |

foot displacement I.C. | x_{10} | −30.48 | cm |

Foot velocity I.C. | ${\dot{x}}_{10}$ | 20.32 | m/s |

shoe stiffness | k_{2} | 10 × k_{1} | -- |

^{1}corresponds to a foot at the end of a 3 ft long leg/pendulum. I.C.—initial condition.

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**MDPI and ACS Style**

Christenson, A.; Cao, P.; Tang, J.
Optimal Football Pressure as a Function of a Footballer’s Physical Abilities. *Proceedings* **2018**, *2*, 236.
https://doi.org/10.3390/proceedings2060236

**AMA Style**

Christenson A, Cao P, Tang J.
Optimal Football Pressure as a Function of a Footballer’s Physical Abilities. *Proceedings*. 2018; 2(6):236.
https://doi.org/10.3390/proceedings2060236

**Chicago/Turabian Style**

Christenson, Andrew, Pei Cao, and Jiong Tang.
2018. "Optimal Football Pressure as a Function of a Footballer’s Physical Abilities" *Proceedings* 2, no. 6: 236.
https://doi.org/10.3390/proceedings2060236