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Adjusting a Momentum-Based Golf Clubhead-Ball Impact Model to Improve Accuracy^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Data

#### 2.2. Momentum-Based Clubhead-Ball Impact Model

#### 2.2.1. Equations of Impulse and Momentum

#### 2.2.2. Clubhead Face Geometry

#### 2.2.3. Prediction Errors for Nominal Parameters

#### 2.3. Impact Model Augmentation

## 3. Results and Discussion

## 4. Conclusions

## Conflicts of Interest

## References

- Petersen, W.; McPhee, J. Comparison of impulse-momentum and finite element models for impact between golf ball and clubhead. In Science and Golf V: Proceedings of the World Scientific Congress of Golf, Phoenix; Energy in Motion, Inc.: Mesa, AZ, USA, 2008. [Google Scholar]
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**2016**, 147, 407–412. [Google Scholar] [CrossRef] - Winfield, D.; Tan, T. Optimization of clubhead loft and swing elevation angles for maximum distance of a golf drive. Comput. Struct.
**1994**, 53, 19–25. [Google Scholar] [CrossRef] - Tanaka, K.; Teranishi, Y.; Ujihashi, S. Finite element modelling and simulations for golf impact. Proc. Inst. Mech. Eng. Part P: J. Sports Eng. Technol.
**2012**, 227, 20–30. [Google Scholar] [CrossRef] - McNally, W.; McPhee, J.; Henrikson, E. The golf shaft’s influence on clubhead-ball impact dynamics. Proceedings
**2018**, 2, 245. [Google Scholar] - McNally, W.; Henrikson, E.; McPhee, J. A continuous analytical shaft model for fast dynamic simulation of the golf swing. Sports Eng.
**2019**, 22, 20. [Google Scholar] [CrossRef]

**Figure 4.**Change of Center of Mass of the golf ball and the resulting moments from the normal force (the figure is not drawn to scale).

**Table 1.**Nominal (measured) properties of the clubhead and golf ball. The reference frames used are provided in Figure 1.

Parameter | Value | Description |
---|---|---|

${m}_{c}$ | 204 | Mass of the clubhead (g). |

${}^{CoF}r{}_{C/CoF}$ | $[\begin{array}{ccc}-42.3& 2.9& 1\end{array}]$ | Position of the center of mass (C) of the clubhead with reference to Center of Face (CoF) expressed in CoF reference frame (mm). |

${}^{C}I{}_{C}$ | $\left[\begin{array}{ccc}2843& -733& 530.2\\ -733& 5679& 0.5\\ 530.2& 0.5& 4085\end{array}\right]$ | $\mathrm{Inertia}\text{}\mathrm{tensor}\text{}\mathrm{of}\text{}\mathrm{the}\text{}\mathrm{clubhead}\text{}\mathrm{expressed}\text{}\mathrm{in}\text{}\mathrm{center}\text{}\mathrm{of}\text{}\mathrm{mass}\text{}\mathrm{coordinate}\text{}\mathrm{frame},\text{}\left\{C\right\}$ (g·cm^{2}). |

${m}_{b}$ | 45.9 | Mass of the golf ball (g). |

${I}_{b}$ | $83.63{1}_{3\times 3}$ | Inertia matrix of the golf ball (g·cm^{2}). |

${}^{I}r{}_{b}$ | $[\begin{array}{ccc}-21.3& 0& 0\end{array}]$ | $\mathrm{The}\text{}\mathrm{impact}\text{}\mathrm{position}\text{}\mathrm{relative}\text{}\mathrm{to}\text{}\mathrm{the}\text{}\mathrm{center}\text{}\mathrm{of}\text{}\mathrm{mass}\text{}\mathrm{of}\text{}\mathrm{golf}\text{}\mathrm{ball}\text{}\mathrm{in}\text{}\left\{I\right\}$ coordinate frame (mm). |

$e$ | 0.83 | Coefficient of restitution. |

Loft | ${9.8}^{\circ}$ | Loft angle. |

Bulge | 305 | Bulge (mm). |

Roll | 305 | Roll (mm). |

Velocity Type | Component | Shaft A | Shaft B | Shaft C | |||
---|---|---|---|---|---|---|---|

Mean | STD | Mean | STD | Mean | STD | ||

Linear velocity | Speed (mph) | −0.73 | 1.41 | −0.49 | 1.35 | −0.58 | 1.31 |

Launch angle (deg) | −1.18 | 0.88 | −0.94 | 1.03 | −0.85 | 0.85 | |

Azimuth angle (deg) | −0.86 | 0.48 | −0.94 | 0.54 | −0.89 | 0.53 | |

Angular velocity | Back-spin (rpm) | 741 | 450 | 741 | 479 | 790 | 401 |

Side-Spin (rpm) | 187 | 178 | 187 | 210 | 187 | 166 |

Velocity Type | Component | Shaft A | Shaft B | Shaft C | |||
---|---|---|---|---|---|---|---|

Mean | STD | Mean | STD | Mean | STD | ||

Linear velocity | Speed (mph) | 0.027 | 1.35 | 0.018 | 1.29 | 0.02 | 1.25 |

Launch angle (deg) | −1.98 | 0.89 | −1.73 | 1.03 | −1.72 | 0.85 | |

Azimuth angle (deg) | −1.07 | 0.49 | −1.15 | 0.55 | −1.1 | 0.53 | |

Angular velocity | Bactablek-spin (rpm) | 111 | 452 | 110 | 476 | 95 | 399 |

Side-Spin (rpm) | 34 | 173 | 30 | 209 | 24 | 163 |

Parameter | Shaft A | Shaft B | Shaft C |
---|---|---|---|

$\Delta {m}_{c}\text{}\left(\mathrm{g}\right)$ | 2.54 | 0.88 | 1.18 |

$\Delta {}^{I}r{}_{b}\text{}\left(\mathrm{mm}\right)$ | $\left[\begin{array}{ccc}0& 0.48& -0.07\end{array}\right]$ | $\left[\begin{array}{ccc}0& 0.48& -0.07\end{array}\right]$ | $\left[\begin{array}{ccc}0& 0.52& -0.076\end{array}\right]$ |

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**MDPI and ACS Style**

Danaei, B.; McNally, W.; Henrikson, E.; McPhee, J.
Adjusting a Momentum-Based Golf Clubhead-Ball Impact Model to Improve Accuracy. *Proceedings* **2020**, *49*, 47.
https://doi.org/10.3390/proceedings2020049047

**AMA Style**

Danaei B, McNally W, Henrikson E, McPhee J.
Adjusting a Momentum-Based Golf Clubhead-Ball Impact Model to Improve Accuracy. *Proceedings*. 2020; 49(1):47.
https://doi.org/10.3390/proceedings2020049047

**Chicago/Turabian Style**

Danaei, Behzad, William McNally, Erik Henrikson, and John McPhee.
2020. "Adjusting a Momentum-Based Golf Clubhead-Ball Impact Model to Improve Accuracy" *Proceedings* 49, no. 1: 47.
https://doi.org/10.3390/proceedings2020049047