Using Physics Simulations to Find Targeting Strategies in Competitive Bowling Simon Ji Shouzhuo Yang Wilber Dominguez Cacey Bester

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Using Physics Simulations to Find Targeting Strategies in

Competitive Bowling

Simon Ji, Shouzhuo Yang, Wilber Dominguez, Cacey Bester

October 14, 2022

Abstract

This article demonstrates a new approach to ﬁnding ideal bowling targeting strategies through

computer simulation. To model bowling ball behaviour, a system of 5 coupled diﬀerential equa-

tions is derived using Euler’s equations for rigid body rotations. We used a computer program

to demonstrate the phases of ball motion and output a plot that displays the optimum initial

conditions that can lead to a strike. When the bowler is modeled to be imperfect, it is shown that

some targeting strategies can lead to higher strike rates due to the “miss room” created by the

inhomogeneity of the oil pattern.

1 Introduction

Bowling remains as one of the most popular sports in the U.S. , with over 45 million people participating

regularly in bowling as of 2017. (4) With millions of dollars at stake every year in national competitions,

signiﬁcant research has been done to understand how players can achieve higher scores. Due to the

complexity of the calculations and the vast number of variables that can aﬀect the ball’s trajectory,

most of the research has focused on statistical analysis from empirical data instead of theoretical

modeling. For example, the 2018 U.S. Bowling Congress (USBC) Equipment Speciﬁcations Report

used “37 bowlers with a full range of revolutions per minute (RPM) rates” rather than a computer

model.(3)

Literature on quantitative analysis of bowling physics is rare due to the many parameters involved,

but has been attempted by Frohlich, Hopkins and Huston over the past few decades.(5)(2)(6) Frohlich

and Huston created mathematical models that took into account the eﬀects of a weighted core within

the bowling ball, and provided simulation results for a small sample of parameter values, including

eﬀects of varying radius of gyration (RG), center of gravity (CG) oﬀset from the geometric center

of ball and initial angular velocity. The simulations demonstrated the qualitative eﬀects of changing

certain variables, but only assumed simple friction proﬁles.

The purpose of this paper is to demonstrate bowling target strategies through a simulation that

samples a large number of possible initial conditions, and explores the eﬀects of realistic friction

proﬁles based on modern competition oil patterns. The program can then provide the user with the

best possible starting positions. These results would be much easier to obtain compared to empirical

methods, and will be useful for both competitors and tournament oil pattern designers, who would

quickly ﬁnd the ideal way to tackle the lane condition. We verify the success of the simulation by

testing the likelihood of a strike with changing entry positions and lane oil patterns.

arXiv:2210.06753v1 [physics.pop-ph] 13 Oct 2022

2 Methods

2.1 Equations of Motion

The equations of motion derived in this section describe rigid body rotation using a rotating frame of

reference ﬁxed to the ball. The derivations use a novel approach through Euler’s equations, as they

are only dependent on the principal moments of inertia, and not the non-diagonal terms in the inertia

tensor; this is important as the full inertia tensor of a reference frame ﬁxed to space cannot be easily

determined simply based on the published Radius of Gyration (RG) and diﬀerential values of each ball.

Previous theories derived by Hopkins assumed a perfectly uniform, spherical ball (2), while theories

from Frohlich required knowing the non-diagonal terms in the inertia tensor of the bowling ball. (5)

The CG oﬀset (distance from the center of gravity to the geometric center of the ball, must be below

1mm to be competition-legal (5)) is assumed to be zero for this study in order to simplify calculations.

Parameters Explanation

v0, θ0

Initial velocity and angle of the ball. θ0is measured relative to the

y-axis, typically ranging from 0 to 5 degrees.

ωx,0, ωy,0, ωz,0

Initial angular velocities imparted on the ball. A skilled bowler typi-

cally generates around 300-400 revolutions per minute (5), while some

bowlers using the two-handed style are able to generate up to 600

revolutions per minute. The direction of rotation varies depending

on bowling style, and is typically determined using measurements of

“positive axis point”.

Ix0, Iy0, Iz0

The principal moments of inertia, measured in a rotated frame so that

the highest inertia occurs at the y’ axis. These values are diﬀerent

depending on the bowling ball, and can be derived using three values

given by the manufacturer: Radius of Gyration (RG), Diﬀerential

(Diﬀ), Intermediate Diﬀerential (Int. Diﬀ). Diﬀerent combinations

lead to diﬀerent ball trajectories.

Iy0=m(RG)2

Ix0=m(RG+Diﬀ)2

Iz0=m(RG+Diﬀ+Int. Diﬀ)2

Kinetic friction coeﬃcient between the ball’s surface and the lane.

When the lane is completely dry, this value is roughly 0.2. When oil

is applied to the lanes, µdecreases to around 0.04. (7) Exact val-

ues will depend on the lane conditions, oil thickness and ball surface

characteristics.

Angle between the major moments of inertia axis y’ and the lane axis

y. (See ﬁgure 1) This value would change based on the bowler’s style

and the way ﬁnger holes are drilled into the ball.

x0, y0

Starting Position of the ball. x0is easily adjusted by the bowler; y0is

typically close to zero; however, for rare cases when the lane friction

is too high, professional bowlers sometimes choose to “loft” the ball

and increase y0by 3 to 5 meters. This allows a lower total contact

time with the lane, hence decreasing hook.

mMass of bowling ball. Most league bowlers use masses between 6.3 to

7.3 kg (14 to 16 lbs).

r Radius of bowling ball. A typical ball has a radius of 10.85 cm.

Table 1: List of parameters used to derive the ball trajectory.

Figure 1: Axis Deﬁnitions with respect to the lane and the bowling ball. The x-axis of a bowling

lane is typically measured in boards. A USBC-approved bowling lane has 39 boards, each measuring

2.74cm. The y’ axis is aligned with the minimum moment of inertia axis of the core.

Ignoring air resistance, the only force of interaction between the ball and the lane is at the contact

surface. The direction of the friction force on the ball is therefore purely dependent on the contact

surface velocity ~

vb, as illustrated in ﬁgure 2.

Figure 2: Surface interactions between the ball and the lane along the ˆxand ˆydirections.

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UsingPhysicsSimulationstoFindTargetingStrategiesinCompetitiveBowlingSimonJi,ShouzhuoYang,WilberDominguez,CaceyBesterOctober14,2022AbstractThisarticledemonstratesanewapproachtondingidealbowlingtargetingstrategiesthroughcomputersimulation.Tomodelbowlingballbehaviour,asystemof5coupleddierentialequa-t...

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Using Physics Simulations to Find Targeting Strategies in Competitive Bowling Simon Ji Shouzhuo Yang Wilber Dominguez Cacey Bester

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