Simulation of the trajectory of a punted rugby ball taking into account the asymmetrical pressure distribution caused by the seams

Abstract

This paper describes the effect of the seams of a rugby ball on the side force and the flight trajectory of the punted kick. Measurement of the aerodynamic force on a non-spinning rugby ball reveals that the side force coefficient depends on the position of the seam as well as the angle of attack. It was found from pressure-sensitive paint measurements that the seam of the ball is the trigger for initiating low pressure when the seam is situated around 60° from the stagnation point. The flight trajectory of the fluctuating ball can be obtained by numerically integrating the six degree-of-freedom non-linear equations of motion. It was shown that a slower spinning ball fluctuates from side to side during flight because of the asymmetrical pressure distribution on the sides of the ball.

Graphical Abstract

C _Y :

Side force coefficient

D :

Drag (N)

g :

Gravitational acceleration (m s⁻²)

I _L :

Moment of inertia of the ball on its longitudinal axis (kg m²)

I _T :

Moment of inertia of the ball on its transverse axis (kg m²)

L :

Lift (N), or rolling moment (Nm)

(L _a , M _a , N _a ):

(x _b , y _b , z _b ) Components of the aerodynamic moment (Nm)

M :

Pitching moment (Nm)

m _b :

Mass of the ball, 0.42 (kg)

m _ij :

Euler-angle transformation matrix

N :

Yawing moment (Nm)

(P, Q, R):

(x _b , y _b , z _b ) Components of the angular velocity vector (s⁻¹)

(U, V, W):

(x _b , y _b , z _b ) Components of the velocity vector (m s⁻¹)

\( \vec{V} \) :

Velocity vector (m s⁻¹)

V _b :

Volume of the ball, 4.8 × 10⁻³ (m³)

Y :

Side force (N)

(X _a , Y _a , Z _a ):

(x _b , y _b , z _b ) Components of the aerodynamic force (N)

(X _E , Y _E , Z _E ):

Inertial coordinate system (m)

(x _b , y _b , z _b ):

Body-fixed coordinate system (m)

\( \left( {\hat{x}_{b} ,\hat{y}_{b} ,\hat{z}_{b} } \right) \) :

(x _b , y _b , z _b ) Components of the unit vector in the body-fixed coordinate system

α :

Angle of Attack (°)

Θ:

Pitch angle (°)

θ _wt :

Angle between \( \vec{V} \) and \( \hat{x}_{b} \) (°)

ρ:

Air density (kg m⁻³)

Φ:

Roll angle (°)

Ψ:

Yaw angle (°)

\( \vec{\omega } \) :

Angular velocity vector (revolutions per second)

(χ, ξ, η, ζ):

Quaternion parameters

( )₀ :

initial conditions

This work is supported by Grant-in-Aid for Young Scientists (A).

Authors and Affiliations

1. Faculty of Education, Art and Science, Yamagata University, Yamagata, Japan

   Kazuya Seo

2. Department of Aeronautics and Astronautics, Tokai University, Hiratsuka, Japan

   Osamu Kobayashi

3. Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba, Japan

   Masahide Murakami

4. Department of Aerospace Engineering, Tohoku University, Sendai, Japan

   Daisuke Yorita, Hiroki Nagai & Keisuke Asai

Corresponding author

Correspondence to Kazuya Seo.

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