The Effect of Instrumentation on Sports Balls During Flight, Exemplified by A Smart AFL Ball: Model and Experiment ^†

Abstract

Sensors incorporated in a sports ball for data collection can affect the properties of a ball, specifically the spin rate of a ball if the mass distribution (moments of inertia, MOI) is altered. This paper provides a method for assessing the MOIs of a smart ball by means of spin rate data, collected from a gyroscopic sensor. The critical elevation angle of the angular velocity vector defines the separatrix condition, which decides whether the angular velocity vector precesses about the axis with the greatest MOI or with the smallest MOI. The critical elevation angle can be directly determined from the experimental of the angular velocity data, and, together with the ratio of precession speed to angular velocity, applied to calculating the three MOIs. In the smart AFL ball used for the experiments, the critical angle was 13.5°, and the ratio of the two small MOIs was 1.014.

1. Introduction

Smart balls are becoming more popular with the technical improvements of lightweight batteries, miniaturised electronics and long-range wireless data transfer. In addition to electronics considerations, putting instrumentation into a ball can affect the dynamics of the ball:

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The mass of sports balls is regulated and must not be affected by electronic items;

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The mass concentration, i.e., the centre of mass (COM), must remain in its original location, as the ball rotates about the COM during flight, and any COM offset can influence the ball’s aerodynamics;

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The mass distribution, i.e., the moments of inertia (MOI) should not be changed too much, as the MOI affects the rotational behaviour of the ball (spin rate).

It is therefore obvious that any instrumentation of a ball has to be designed carefully and tested extensively. The influence of a changed MOI can be assessed easily if the ball is instrumented with a gyroscopic sensor (gyro). If the spin axis moves (i.e., precesses) with respect to the body coordinate system (BCS, i.e., the ball’s, or more precisely, the gyro sensor’s coordinate system), then either the ball wobbles or the instrumentation has an effect on the MOI. Furthermore, the three MOIs of spherical sports balls are not necessarily identical even without instrumentation. Seams, valves, laces, and material distribution affect the MOIs.

The aim of this paper is to “reverse-engineer” the conditions of a ball’s MOIs from gyro data.

2. Model

The three MOIs about the axes of the BCS are denoted I₁, I₂, and I₃. The instantaneous angular velocity vector (spin axis) is denoted ω₁, ω₂, and ω₃. The intersection lines of the moving (precessing) ω-vector and the ball’s surface are defined by the intersection of two ellipsoids, based on the conservation of two physical properties [1]:

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Conservation of angular momentum, H, defines the angular momentum ellipsoid:

$$H^2={(I_1{\omega }_1)}^2+{(I_2{\omega }_2)}^2+{(I_3{\omega }_3)}^2=const.$$

(1)

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Conservation of rotational kinetic energy, T, defines the kinetic energy ellipsoid:

$$2T=I_1{\omega }_1{}^2+I_2{\omega }_2{}^2+I_3{\omega }_3{}^2=const.$$

(2)

The position of the unit ω-vector has to satisfy two conditions, derived from re-arranging Equations (1) and (2):

$$\frac{{\omega }_1{}^2}{{(H/I_1)}^2}+\frac{{\omega }_2{}^2}{{(H/I_2)}^2}+\frac{{\omega }_3{}^2}{{(H/I_3)}^2}=1$$

(3)

and

$$\frac{{\omega }_1{}^2}{2T/I_1}+\frac{{\omega }_2{}^2}{2T/I_2}+\frac{{\omega }_3{}^2}{2T/I_3}=1.$$

(4)

If follows from Equations (3) and (4) that

$$\frac{2T}{I_{1,2,3}}={\left(\frac{H}{I_{1,2,3}}\right)}^2.$$

(5)

The two ellipsoids (H- and T-) intersect, and the curve of intersection, the so-called polhode, is the path of the angular velocity vector with respect to the BCS. In prolate spheroids (oval balls, such as for rugby, Australian rules football and American football), the ω-vector can rotate about the long (major) axis or about the short (minor) one. Figure 1 exemplifies this behaviour.

Subtracting Equation (3) from Equation (4) and multiplying the result by H² yields the polhode equation

$${\omega }_1{}^2{I}_1\left(I_1-\frac{H^2}{2T}\right)+{\omega }_2{}^2{I}_2\left(I_2-\frac{H^2}{2T}\right)+{\omega }_3{}^2{I}_3\left(I_3-\frac{H^2}{2T}\right)=0.$$

(6)

For real solutions of Equation (6), the three coefficients cannot have the same sign, and

$$I*=\frac{H^2}{2T}$$

(7)

where I* is defined as follows: if I₁ > I₂ > I₃, then I₁ ≥ I* ≥ I₃.

Projecting the polhode equation, i.e., Equation (6), on the ω₁–ω₃ plane yields

$${\omega }_1{}^2{I}_1(I_1-I_2)+{\omega }_3{}^2{I}_3(I_3-I_2)=2T(I*-I_2).$$

(8)

If I* = I₂, then we obtain the separatrix condition

$${\omega }_1{}^2{I}_1(I_1-I*)+{\omega }_3{}^2{I}_3(I_3-I*)=0,$$

(9)

namely, the decision condition that determines whether the ω-vector rotates about the axis with the greatest MOI (I₁) or about the axis with the smallest MOI (I₃). This principle excludes a stable rotation about the axis with the intermediate MOI (I₂)

Equation (9) delivers the critical angle of the ω-vector, namely θ_crit, from

$$\frac{{\omega }_3{}^2}{{\omega }_1{}^2}={\tan }^2{\theta }_{crit}=-\frac{I_1(I_1-I*)}{I_3(I_3-I*)}=-\frac{I_1(I_1-I_2)}{I_3(I_3-I_2)}$$

(10)

and

$${\theta }_{crit}={\tan }^{-1}\sqrt{-\frac{I_1(I_1-I_2)}{I_3(I_3-I_2)}}.$$

(11)

If the pitch angle of the ω-vector with respect to the ω₁–ω₂ plane is smaller or greater than θ_crit, then the spin axis precesses about the minor (with the greatest MOI I₁) or major ball axis (with the smallest MOI I₃), respectively.

According to Fuss and Smith [2],

$$\frac{\Omega }{{\omega }_R}=\left(\frac{I_3-I_{1,2}}{I_{1,2}}\right)\cos \left(\frac{\pi }{2}-{\theta }_{crit}\right),$$

(12)

where ω_R and Ω are the resultant angular velocity, i.e., √(ω₁² + ω₂² + ω₃²), and the precession rate, respectively. The precession rate is the speed at which the ω-vector moves about the minor or major axes. From Equation (12), when plotting Ω/ω_R against cos(π/2 – θ_crit), the gradient of the regression fit corresponds to (I₃ − I_1,2)/I_1,2 [2]. Ω/ω_R and cos(π/2 − θ_crit) are determined experimentally from a ball instrumented with a gyro.

Solving Equation (11) for I₂ yields

$$I_2=\frac{I_1{}^2+I_3{}^2\left({\tan }^2{\theta }_{crit}\right)}{I_1+I_3\left({\tan }^2{\theta }_{crit}\right)}.$$

(13)

If I₁ and I₃ are known, or at least the ratio I₁/I₃ or I₃/I₁, then I₂ can be determined directly, or relative to I₁. If θ_crit = 0° (ω-vector in ω₁–ω₂ plane), then Equation (13) reduces to I₂ = I₁. If θ_crit = 90° (ω-vector perpendicular to the ω₁–ω₂ plane) then tan²θ_crit = ∞, and Equation (13) reduces to I₂ = I₃.

3. Experiment

The experiments were carried out with the smart AFL ball developed by, and described in, Fuss and Smith [2] and Fuss et al. [3]. The angular velocity ω and its three components (ω₁, ω₂, and ω₃.) with respect to the BCS were measured with a gyro. The data from these two references were analysed further. The precessing ω-vectors were reconstructed in 3D in AutoCAD 2000 (Autodesk, San Rafael, CA, USA) with respect to the ball, as well as plotted on an elevation (pitch) vs. azimuth (yaw) diagram (where the yaw angle is the position of the ω-vector projected onto the ω₁–ω₂ plane, and the pitch angle θ is the included angle between ω-vector and ω₁-ω₂ plane). The task was to find ω-vectors, precessing very close to, and on either side of, the separatrix condition, in order to determine θ_crit. From Equation (13), I₂ was estimated and compared to I₁.

4. Results

Figure 2 shows ω-vectors at different θ-angles, precessing about the major axis of the ball (Figure 2a–c) and the minor axis (Figure 2d).

Figure 3 explains how θ_crit was determined, which was approximately 13.5°. From Fuss and Smith [2], by applying Equation (12), (I₃ − I_1,2)/I_1,2 was 0.42, resulting in I_1,2/I₃ of 1.724. When normalizing the MOIs to I₃, then I₃ = 1, and I_1,2 = 1.724. Yet, if θ_crit > 0°, then I₁ ≠ I₂. Finding I₂ from Equation (13), considering that I_1,2 = (I₁ + I₂)/2, delivers the following results: I₁ = 1.736, I₂ = 1.712, and I₁/I₂ = 1.014. The latter ratio teaches that I₁ is only 1.4% greater than I₂.

5. Discussion

There are other ways of determining the MOI of an object about a specific axis, such as the pendulum method. The method used in this study, however, has the advantage that the behaviour of the ω-vector changes immediately when crossing the separatrix polhode.

It has to be borne in mind that the axes, about which an object spins and its ω-vector precesses, are not identical. In the ball shown in Figure 2c, the ω-vector precesses about the major axis while the ball spins about the minor axes with a slightly elevated and tilted ω-vector. The latter fact defines the kick as a drop punt. Thus, the critical angle θ_crit cannot be used for classification of, and distinction between, drop punts and torpedo punts.

That I₁ and I₂ turned out to be different, even only slightly, is an effect of the instrumentation as well as of the valve. The instrumentation was protected by an electronics box with a rectangular cross section such that I₁ > I₂. The valve alone (two semi-cylindrical pieces of rubber) had the same effect (I₁ > I₂.). Nevertheless, even if that small difference in MOI can be ignored mechanically, it does make a difference in terms of the ω-vector’s behaviour. The latter is characterized by changing the axis about which it precesses. These principles should be known to researchers and practitioners dealing with smart balls, in order to interpret the sensor data correctly.