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THE EFFECT OF ROUGHNESS ELEMENTS

ON THE MAGNUS CHARACTERISTICS

OF

ROTATING SPHERICAL PROJECTILES

THESIS

Presented to the Graduate Council of the

North Texas State University in Partial

Fulfillment of Requirements

For the Degree of

MASTER OF SCIENCE

by

Michael A. Smith

Denton, Texas

December, 1982

Smith, Michael A., The Effect of Roughness Elements on

the Magnus Characteristics of Rotating Spherical Pro-

jectiles. Master of Science (Physical Education),

December, 1982, 87 pp., 8 tables, 3 illustrations, bibli-

ography, 23 titles.

Thirty trials of each of three roughness conditions

were examined. The first condition consisted of a baseball

pitched so that two of the roughness elements opposed the

flow. The second condition consisted of a pitched base-

ball with four of the roughness elements opposing the flow.

The third consisted of a pitched uniformly rough sphere.

The conclusions were that roughness elements increase

horizontal flight deviations when a baseball rotates about

a vertical axis; roughness elements on the surface of a

baseball may cause a decrease in the encountered drag

forces; linear velocity has a dominating effect on the

trajectory of a spinning baseball; previously developed

mathematical models do not adequately predict flight devia-

tions.

TABLE OF CONTENTS

LIST OF TABLES .. . . .,. .. , ...

LIST OF ILLUSTRATIONS . .-.......... ..

Chapter

I. INTRODUCTION........ ..... . ..

Purpose of the StudyDelimitations of the StudyLimitations of the StudyDefinition of Terms and Symbols

Pagev

vi

1

II. REVIEW OF LITERATURE........

III. PROCEDURES.... . . . . . . .

InstrumentationTesting ProceduresData Acquisition ProceduresStatistical Analysis

IV. RESULTS . . . . . . . . . .

Environment and ProjectilesResults

V. SUMMARY AND CONCLUSIONS . .

13

24

. . . . . . . 33

. . . . . . . . 44

IntroductionProceduresResultsConclusionsRecommendations

APPENDICES . . . . . . . . . 52

A. Formulas for the Computation of AngularVelocity . . . . . . . . . . . . . . . . . . . 53

B. A Basic Program to Compute Angular and LinearKinematics of a Ball in Flight ... .. ..... 58

iii

APPENDICES Page

C. Formulas of Rubinow and Keller (1963) to aNumerical Approximation of HypotheticalLanding Points....... . . . . . . . . . . 75

D. A Basic Program Utilizing Numerical Approxi-mation to Predict Values for the LandingPoints of Projectiles . . .. . . . . . . .. 77

E. Summary Table for the Analysis of VarianceConducted on the X-Component of FlightDeviations . . . . . . . . . . . . . . . . . . 81

F. Summary Table for the Analysis of VarianceConducted on the Z-Component of FlightDeviations.. ...... . . . . . . . . . . . 82

G. Summary Table for the Analysis of VarianceConducted on the Z-Component of AngularVelocity. ....-.. . . . . . . . . . . . . . 83

H. Summary Table for the Analysis of VarianceConducted on the X-Component of the LandingPoints. ........ . ........ 84

I. Summary Table for the Analysis of VarianceConducted on the Z-Component of the LandingPoints. .-....-.. . . . . . . . . . . . . . 85

BIBLIOGRAPHY.-.....-........................ 86

iv

LIST OF TABLES

Table Page

I. Mean X- and Z- Flight Deviations for Eachof the Roughness Conditions . . . . . . . . 34

II. Mean Component Initial Angular Velocitiesfor Each of the Roughness Conditions . . . 35

III. Mean Component Displacements for Each ofthe Roughness Conditions . . . . . . . . . 37

IV. Mean Hypothetical Component Displacementsfor Each of the Roughness Conditions . . . 38

V. Correlation Coefficients Between ActualLanding Points and Hypothetical LandingPoints. . . ..-.. . . . . . . . . . . . . 39

VI. R and R-Squared Values Resulting from theMultiple Linear Regression Used to PredictComponent Landing Points for Each of theRoughness Conditions . . . . . . . . . . . 40

VII. Entry Order of the Independent Variables forEach of the Step-Wise RegressionAnalyses . . . . . . . . . . . . . . . . . 41

VIII. Pearson Product-Moment Correlation CoefficientsCalculated Between the Components ofLinear Velocity ........................ 42

V

LIST OF ILLUSTRATIONS

Figure Page

1. Boundary Layer Conditions . . . . . . . . . . 4

2. Location of Testing Instruments . . . . . . . 27

3. Roughness Conditions .... ...... . . . . . 29

vi

CHAPTER I

INTRODUCTION

A projected object which is both translating and ro-

tating will deviate from a hypothetical parabolic pathway

due to the presence of air resistance (Magnus, 1853). The

deviation in flight is caused by the spin of the object and

is called the Magnus or Robins effect (Magnus, 1853;

Robins, 1842; Jacobsen, 1973). Robins (1842) noted that

the spin imparted to cannon balls resulted in accuracy

losses. Magnus (1853) successfully demonstrated, with ro-

tating bodies, the presence of pressure inequalities and a

resulting eccentric force.

One of the factors influencing the motion of a solid

object through a fluid is the nature of the boundary layer

of the fluid surrounding the object. There exists five

forms of boundary layer flow; i.e., creeping flow, laminar

flow, turbulent flow, partially turbulent flow and super-

sonic flow.

Creeping flow occurs with a non-viscous fluid and is

characterized by no boundary layer separation. Martin

(1955) found that the Magnus effect for creeping flow was

caused by an assymetrical fluid boundary layer. That is,

the portion of a rotating body that is moving in relative

1

2

opposition to the incoming fluid carries with it a thicker

fluid boundary layer than does the portion which is moving

in the same direction as the incoming flow. This, in

effect, creates a new body on which the point of applica-

tion of the resultant fluid drag force is eccentric to the

center of gravity. The resultant eccentricity of the force

causes the object to deviate from a normal path.

Laminar flow exists when the boundary layer separation

point(s) occurs in a region of the object not presented to

the flow. Briggs (1959) studied lateral deflections of

baseballs and smooth spheres caused by the Magnus effect in

primarily laminar flow. He stated, that in the case of the

baseballs, the spin imparted to the ball causes an in-

equality of pressures and affects the general flow field

around the body. This, in accordance with the Bernoulli

principle, causes a flight deviation from a hypothetical

parabolic path. Briggs found a negative Magnus effect oc-

curred for smooth spheres. This effect, to date, has not

been completely explained.

In turbulent flow, boundary layer separation occurs in

the region of the object presented to the flow. It is

achieved when the Reynolds number exceeds an experimentally

determined critical value. This critical value is achieved

under conditions of high fluid velocity and/or motion of

the object through a highly viscous fluid (Daily &

Harleman, 1966).

3

Partially turbulent flow occurs when an immersed body

is assymetrical or a symetrical body is rotating in a

fluid. Until a critical Reynolds number is reached, the

boundary layer around a nonspinning cylinder separates at

approximately 82 degrees from the forward stagnation point

(Jacobsen, 1973; Krahan, 1956). The separation point moves

downstream to about 130 degrees as the Reynolds number in-

creases and the fluid flow becomes turbulent. The location

of the separation points are affected when the Reynolds

number approaches a critical value and an immersed cylinder

begins to rotate. On the portion of the cylinder opposing

the flow, the separation point moves toward the forward

stagnation point. The separation point of the side moving

in the same direction remains located at 82 degrees, and

the fluid flow in that portion of the body remains laminar.

This causes a resultant lift force in the opposite direc-

tion to the classical Magnus effect (Jacobsen, 1973).

Supersonic flow differs from turbulent flow in that

shock waves are created. As the critical Mach number is

exceeded, the flow in the region of the shock waves becomes

non-isentropic. As this critical Mach number is exceeded,

drag forces increase (Hughes & Brighton, 1967). Efforts

to reduce drag forces at supersonic speeds include the

"swept-wing" design of aircraft (Daily & Harleman, 1966).

The separation points and fluid flow conditions are as

illustrated in Figure 1.

4

0

00 '

C.C 0

0ah.0

C..)- 1

- 0a .

0

4)

00

N 0 0

LL r-U..

(U C.C 0

E(Uo.

5

Of the five fluid flow conditions only laminar, tur-

bulent and partially turbulent flow are likely to be en-

countered in sports situations. Turbulent flow primarily

occurs in ballistic activities. For example, Jacobsen

(1973) reviewed the literature pertaining to ballistics and

concluded that the dynamic stability of a bullet is depen-

dent upon the location of the center of gravity, shape and

angular velocity about the longitudinal axis. Borg (1980)

discussed the potential use of an immersed revolving

cylinder to aid in the steering of large ocean vessels.

The literature pertaining to partially turbulent flow is

scarce. Partially turbulent flow is rarely encountered al-

though its presence has been recorded (Briggs, 1959).

In the majority of sport settings, the fluid flow is

laminar. In addition, in many activities spin is purposely

imparted to a ball to cause flight deviations.

Barnaby (1978) discussed the importance of spin in

tennis and stated that

On any surface, the flight of the ball through theair can and must be controlled by spin whenever theball is played vigorously enough so that gravity willnot do the job. That is why topspin should be thefirst new skill acquired by an intermediate.

Gray (1974), in discussing the experimental results of

Plagenhoef (1970), claimed that if no spin was imparted to

a typical served ball, the ball would land well beyond the

opponent's baseline. In the game of soccer, a predominance

of sidespin is often used to project the ball around

6

obstacles. This "bending ball" has been described by

Miller (1979). In the game of golf, the inclination of the

clubface (i.e., its loft) causes the ball to be projected

with a backspin (Hay, 1978; Davies, 1949). The magnitude

of the spin can approach 8000 revolutions per minute and

cause an increase in the range of the ball up to 50 percent

(Chase, 1981; Davies, 1949). Both desirable and undesi-

rable slices or hooks also result from the Magnus effect.

Chase (1981) explains that when the spin imparted to the

golfball has an axis of rotation deviating from horizontal,

deviation in flight will occur to an amount which is de-

pendent upon the deviation of the axis.

In baseball, the ability of a pitcher is perhaps the

most important factor in determining the success of a team.

Reiff (1971) claims that 65 to 85 percent of winning base-

ball depends on a pitcher's ability. The quality of a

pitcher is, in turn, dependent upon an ability to both vary

the linear velocity of the pitches and to deviate the ball

from normal projectile flight. In most situations, this

deviation is accomplished by imparting spin to the ball to

create a Magnus effect. An indication of the amount that a

ball can deviate from normal projectile flight has been

reported by Briggs (1959). Lateral deflections ranging

from 6.1 inches to 26.0 inches for a spinning baseball in a

six foot drop across windtunnel streams were reported.

Selin (1959) reported maximum vertical deviations of 2.4

7

feet and horizontal deviations of 2.1 feet for a variety of

spinning pitches. Attempts have been made to mathe-

matically model a pitched baseball. Krieghbaum and Hunt

(1978) developed a mathematical model to simulate pitches

of various linear and angular velocities. The results of

their approximated deviations from normal projectile flight

yielded maximum values more than 50 percent lower than

those reported by either Selin (1959) or Briggs (1959). A

factor not considered in the Krieghbaum and Hunt model was

the roughened surface of the ball. Hay (1978) claims that

the variables effecting the drag force on a projectile are

the velocity of the flow, the surface area of the body, the

characteristics of the fluid involved, the cross-sectional

area presented to the flow, and the smoothness of the sur-

face of the body. The effects of the smoothness of the

body on drag forces have been examined by Watts and Sawyer

(1975). In discussing the aerodynamics of a knuckleball,

the authors stated:

There are two possible mechanisms for the erraticlateral force that causes the fluttering flight of theknuckleball. A fluctuating lateral force can resultfrom a portion of the strings being located just atthe point where the boundary layer separation occurs.A far more likely situation is that the ball spinsvery slowly, changing the location of the roughnesselements (strings), and thereby causing a nonsymmetricvelocity distribution and a shifting of the wake.

The differences between the reported mathematical and

experimental data indicate that a need exists to further

8

examine the Magnus effect for rotating spheres and to in-

clude a consideration of the roughness of the spheres.

Purpose

The purpose of this study was to examine the effect of

roughness elements on the Magnus characteristics of ro-

tating spheres.

Delimitations of the Study

The delimitations in the analysis of experimental

pitching performances included the following.

1. A baseball pitching machine was used to approxi-

mate the motion of a pitched baseball.

2. Data extracted off the film records of thirty

trials for each of three conditions were analyzed:

a. A baseball rotating with four roughness ele-

ments opposing the flow

b. A baseball rotating with two roughness ele-

ments opposing the flow

c. A rotating sphere of uniform roughness

Limitations of the Study

1. Normal cinematographical and data extraction

limitations were recognized.

2. The limitations of numerical approximation sam-

pling rates of a 64 Kilobyte Tektronics 4052

computer were recognized.

9

3. Angular velocity was assumed to remain constant

throughout the flight of the ball.

4. Laminar fluid flow was assumed.

Definitions of Terms and Symbols

Angle of attack - The angle between the velocity vector and

the angular velocity vector

Numerical Approximation - A mathematical technique used to

estimate a solution of unsolvable differential equa-

tions which involves a solution for the integral of an

equation using geometric means. A sufficiently small

time increment, together with known constants and

initial values for variables are used in the equa-

tions. The solutions represent new initial values for

the variables. This process is repeated, usually by

computer, using these new initial values for the

variables. The accuracy of such approximations is de-

termined by the sampling frequency, or how many times

the process is repeated.

r = radius of the ball

p = kinematic viscosity of the fluid

p = density of the fluid

m = mass of the ball

t = time

A = change in any variable (i.e., At = change in time)

|VI = magnitude of velocity

10

R = Reynolds number = p V r

V = velocity vector (subscripts indicate direction)

= angular velocity vector (subscripts indicate direction)

= acceleration vector (subscripts indicate direction)

d = any distance

8 = angle of attack

F = lift force

F = drag force

F = force due to gravity

F = FL + FD + FG + total force

CHAPTER BIBLIOGRAPHY

Barnaby, J. Ground strokes in match play. Garden City,New York: Doubleday and Company, Inc., 1978.

Borg, J. L. New twist to steering. Ocean Industry, June1980, 15 (6), 56-61.

Briggs, L. J. Effect of spin and speed on the lateral de-flection (curve) of a baseball and the Magnus effectfor smooth spheres. American Journal of Physics, 1959,27, 589-96.

Chase, A. A slice of golf. Science 81, July-August 1981,90-91.

Daily, J. W., & Harleman, D. R. F. Fluid dynamics.Reading, Massachusetts: Addison-Wesley PublishingCompany, Inc., 1966.

Davies, J. M. The aerodynamics of golf balls. Journal ofApplied Physics, September 1949, 20 (9), 821-828.

Gray, M. R. What research tells the coach about tennis.American Alliance of Health, Physical Education andRecreation, 1974.

Hay, J. G. The biomechanics of sports techniques. Engle-wood Cliffs, New Jersey: Prentice Hall, Inc., 1978.

Hughes, W. F., & Brighton, J. A. Fluid dynamics. NewYork, New York: McGraw Hill Book Company, 1967.

Jacobsen, I. D. Magnus characteristics of arbitrary ro-tating bodies. (AGARDograph, No. 171), 1973.

Krahn, E. Negative Magnus force. Journal of AeronauticalScience, April 1956, 377-378.

Krieghbaum, E. F., & Hunt, W. A. Relative factors in-fluencing pitched baseballs. In Landry, F. & Orban,W. A. R. (Eds.). Biomechanics of Sports andKinanthrophometry, 6, 1978, 227-236.

Magnus, G. On the deflection of a projectile. PoggendorfsAnnalen der Physik und Chemie, 1853, 88, (1).

11

12

Martin, J. C. An experimental correlation between the flowand Magnus characteristics of a spinning ogive-nosecylinder. Unpublished doctoral dissertation, Univer-sity of Notre Dame, August, 1971.

Miller, L. Mastering soccer. Chicago, Illinois: Contem-porary Books, Inc., 1979.

Reiff, C. G. What research tells the coach about baseball.American Alliance for Health, Physical Education andRecreation, 1971.

Robins, B. New principles of gunnery. London, England:1842.

Selin, C. An analysis of the aerodynamics of pitchedbaseballs. Research Quarterly, 1959, 30, 232-240.

Watts, R. G., & Sawyer, E. Aerodynamics of a knuckleball.American Journal of Physics, 1975, 43, 960-963.

CHAPTER II

REVIEW OF LITERATURE

In many sports activities the performer and/or sport

implement is projected into the air. According to Hay

(1978), the factors affecting projectile flight are the

velocity of the flow, the surface area of the body, the

characteristics of the fluid involved, the cross-sectional

area presented to the flow, and the smoothness of the sur-

face of the projectile. The fluid resistance forces may be

of sufficient magnitude to cause appreciable deviations

from a parabolic flight path. The observed airborne motion

of a badminton shuttlecock provides an obvious example of a

non-parabolic trajectory (Hay, 1978). In some activities

the sport implements are projected in such a way to pur-

posely cause flight deviations. These flight deviations

commonly result from imparted rotations. For example, in

the game of soccer, the sidespin of the ball permits it to

travel around obstacles (Miller, 1979; Hay, 1978).

One of the major factors influencing the fluid force

acting on a projectile is the nature of the boundary layer

of fluid surrounding the object. There exists five forms

13

14

of boundary layer flow; i.e., creeping flow, laminar flow,

turbulent flow, partially turbulent flow and supersonic

flow.

Creeping flow is characterized by no boundary layer

separation. It occurs only in a non-viscous fluid or at

extremely low velocities (Jacobsen, 1973). Martin (1955)

found that the Magnus effect for creeping flow was caused

by an assymetrical fluid boundary layer. That is, the por-

tion of a rotating body that is moving in relative opposi-

tion to the flow carries with it a thicker fluid boundary

layer than does the portion moving in the same direction as

the flow. This creates a new body upon which the point of

application of the resultant fluid drag force is eccentric

to the center of gravity of the body. This eccentricity

of the resultant force causes the object to deviate from a

hypothetical parabolic path.

Laminar flow exists when the boundary layer separation

occurs in a region of the object not presented to the flow.

Briggs (1959) studied lateral deflections of baseballs and

smooth spheres caused by the Magnus effect in primarily

laminar flow. He stated, that in the case of the base-

balls, the spin imparted to the ball causes an inequality

in pressures and affects the general flow field around the

body. This, in accordance with the Bernoulli principle,

causes the deflection from a hypothetical parabolic path.

15

Briggs, however, found that an unexplained negative Magnus

effect occurred for smooth spheres.

In turbulent flow, boundary layer separation occurs in

the region of the object presented to the flow. For a

sphere this separation occurs at an angle of 130 degrees

from the forward stagnation point (Jacobsen, 1973). Tur-

bulent flow is achieved when the so-called Reynolds number

exceeds a critical value. This critical Reynolds number is

achieved under conditions of high fluid velocity and/or

motion of the object through viscous fluid (Daily & Harle-

man, 1966).

Partially turbulent flow occurs only when the immersed

body is assymetrical or a symetrical body is rotating in a

fluid. Under laminar flow conditions, boundary layer

separation occurs at 82 degrees from the forward stagnation

point. When the Reynolds number approaches the critical

value and a symetrical body begins to spin, the boundary

layer separation in the portion of the body moving in rela-

tive opposition to the flow moves downstream to 130 de-

grees. The portion of the body moving in the same direc-

tion as the flow remains laminar with boundary layer

separation occurring at 82 degrees from the forward stag-

nation point. This causes a resultant lift force in the

opposite direction to the classical Magnus effect (Jacob-

sen, 1973). This force is commonly referred to as the

negative Magnus effect (Briggs, 1959; Jacobsen, 1973).

16

When the velocity of the flow becomes extremely high,

the flow can become supersonic. Flow becomes supersonic

when a critical Mach number is exceeded. As this Mach

number is exceeded, shock waves are created within the flow

which causes the flow to become non-isentropic. In super-

sonic flow, fluid drag increases due to the large fric-

tional forces (Hughes & Brighton, 1967). Daily and Harle-

man (1966) discussed efforts in aviation to reduce drag

forces at supersonic speeds by the use of "swept-wing"

designs.

Of the five fluid flow conditions only laminar, tur-

bulent and partially turbulent flow are likely to be en-

countered in sports situations. Turbulent flow primarily

occurs in ballistic activities. For example, Jacobsen

(1973) reviewed the literature pertaining to ballistics and

concluded that the dynamic stability of a bullet is depen-

dent upon the location of the center of gravity, shape and

angular velocity about the longitudinal axis. Borg (1980)

discussed the potential use of an immersed revolving cylin-

der to aid in the steering of large ocean vessels. The

literature pertaining to partially turbulent flow is

scarce. Partially turbulent flow is rarely encountered

although its presence has been recorded (Briggs, 1959).

In the majority of sport settings, the fluid form is lami-

nar. Moreover in many activities spin is purposely im-

parted to a ball to cause flight deviations.

17

Barnaby (1978) discussed the importance of spin in

tennis and stated that

On any surface, the flight of the ball through theair can and must be controlled by spin whenever theball is played vigorously enough so that gravity willnot do the job. That is why topspin should be thefirst new skill acquired by an intermediate.

Gray (1974), in discussing the experimental results of

Plagenhoef (1970), claimed that if no spin was imparted to

a typical served ball, the ball would land well beyond the

opponent's baseline. In the game of soccer, a predominance

of imparted sidespin is often used to project the ball

around obstacles. This so-called "bending ball" has been

described by Miller (1979). In the game of golf, the in-

clination of the clubface (i.e., its loft) causes the ball

to be projected with a backspin (Hay, 1978; Davies, 1949).

The magnitude of the spin can approach 8000 revolutions

per minute and cause an increase in the range of the ball

up to 50 percent (Chase, 1981; Davies, 1949). Both de-

sirable and undesirable slices or hooks also result from

the Magnus effect. Chase (1981) explains that when the

spin imparted to the golf ball has an axis of rotation de-

viating from horizontal, deviation in flight will occur to

an amount which is dependent upon the deviation of the

axis.

In baseball, the ability of a pitcher is perhaps the

most important factor determining the success of a team.

18

Reiff (1971) claims that 65 to 85 percent of winning base-

ball depends on a pitcher's ability. The quality of a

pitcher is, in turn, dependent upon an ability to both vary

the linear velocity of the pitches and to deviate the ball

from normal projectile flight. In most situations, this

deviation is accomplished by imparting spin to the ball to

create a Magnus effect. An indication of the amount that a

ball can deviate from normal projectile flight has been

reported by Briggs (1959). Lateral deflections ranging

from 6.1 inches to 26.0 inches for a spinning baseball in

a six foot drop across windtunnel streams were reported.

Selin (1959) reported maximum vertical deviations of 2.4

feet and horizontal deviations of 2.1 feet for a variety of

spinning pitches. Watts and Sawyer (1975) examined the

factors causing the unpredictable motion of a knuckleball.

The authors stated that

There are two possible mechanisms for the erraticlateral force that causes the fluttering flight ofthe knuckleball. A fluctuating lateral force canresult from a portion of the strings being locatedjust at the point where the boundary layer separationoccurs. A far more likely situation is that the ballspins very slowly, changing the location of the rough-ness elements (strings), and thereby causing a non-symmetric velocity distribution and a shifting of thewake.

Attempts have been made to mathematically model both

the path of a pitched baseball and the forces acting on a

sphere in a fluid. Krieghbaum and Hunt (1978) developed a

19

mathematical model for a pitched baseball. Using the

parameters described by Selin (1959) the following model

was constructed:

dV = K Cd Vx + C1 (V cosB + Vz sinAsinB)

dV = K C V + C1 (V sinBcosA - V cosB)y d y z x

and dV = K C V + C1 (V sinBsinA - V sinBcosA)z d z x y

where K = pVAdt/(2m).

The displacement in each direction during the interval

dt was given as dS. = (V. + dV.) dt where j = x, y or z.J J J

The sum of the dS. terms over the time of flight of theJ

ball was given as the total displacement in the direction

of j. The model took the following factors into account:

A = horizontal angle - defined as the "angle between the

direction of the pitch and the axis of rotation

measured in a counterclockwise direction from the

direction of the pitch to the axis of rotation"

(Krieghbaum & Hunt, 1978).

B = vertical angle - defined as the "angle between a verti-

cal line to the center of the ball and the axis of

rotation measured from the upward-directed vertical to

the axis of rotation" (Krieghbaum & Hunt, 1978).

Cd = coefficient of drag

C1 = coefficient of lift

Fd = drag force

F 1 = lift force

20

F = gravitational force

dt = change in time

m = mass of the ball

V = velocity (subscripts indicate direction)

A = area presented to the flow

p = density of the air

g = acceleration due to gravity

This model yields deviations over 50 percent less than

those reported by either Selin (1959) or Briggs (1959).

Two possible causes for the difference between actual and

hypothetical values for flight deviation are the exclusion

of a roughness factor in the model and no consideration for

the magnitude of angular velocity. The necessity for in-

cluding a roughness factor has been implied by Hay (1978).

Oseen, as referenced by Rubinow and Keller (1963),

developed a formula for the drag force acting on a totally

immersed translating sphere. The developed equation was

Fd = -6lrpV(l+3R/8) where Fd is the force of drag, r is the

radius of the sphere, p is viscosity, Vis the velocity

vector and R is the Reynolds number p IVr/ where p is the

density of the air.

In the above equation the first term -6TrV was calcu-

lated by Stokes, as referenced by Rubinow and Keller (1963).

From this information, Rubinow and Keller (1963) developed

a mathematical model for the forces acting on a rotating

21

and translating sphere. Their results yielded the fol-

lowing formula:

Ft = Fd + Fl = -6'3rpV(l+3R/8) + Trr3p(0 X V)

where 0 is the angular velocity vector. These relation-

ships were confirmed by Hess (1968). This model differs

from that of Krieghbaum and Hunt (1978) in that it provides

a more comprehensive consideration of the factors affecting

the nature of the fluid and, in particular, it takes into

account the magnitude of angular velocity.

In summary, a review of literature revealed that very

few scientific studies have been concerned with the fluid

forces acting on a pitched baseball. Attempts have been

made to measure flight deviations caused by the Magnus

effect and to model the forces acting on spheres and base-

balls travelling through a fluid. Discrepancies between

experimentally and mathematically obtained deviations in-

dicate a need to include a roughness factor in the models.



Borg, J. L. New twist to steering. Ocean Industry, June1980, 15 (6), 56-61.

Briggs, L. J. Effect of spin and speed on the lateraldeflection (curve) of a baseball and the Magnus effectfor smooth spheres. American Journal of Physics,1959, 27, 589-96.



Davies, J. M. The aerodynamics of golf balls. Journal ofApplied Physics, September, 1949, 20 (9), 821-828.



Hess, S. Coupled translational and rotational motions ofa sphere in a fluid. Zeitschrift Fur Naturfonschung,1968, 23A, 1095-1101.

Hughes, W. F., & Brighton, J. A. Fluid dynamics. NewYork, New York: McGraw Hill Book Company, 1967.


Krieghbaum, E. F., & Hunt, W. A. Relative factors in-fluencing pitched baseballs. In Landry, R. & Orban,W. A. R. (Eds.). Biomechanics of Sports and Kinan-throphometry, 6, 1978, 227-236.

22

23


Miller, L. Mastering soccer. Chicago, Illinois: Con-temporary Books, Inc., 1979.


Rubinow, S. I., & Keller, J. B. The transverse force on aspinning sphere moving in a viscous fluid. Journal ofFluid Mechanics, 1963, 11 (3), 447-459.

Selin, C. An analysis of the aerodynamics of pitched base-balls. Research Quarterly, 1959, 30, 232-240.


CHAPTER III

PROCEDURES

The purpose of this study was to examine the effects

of roughness elements on the Magnus characteristics of

rotating spheres.

Instrumentation

Cinematographical Instrumentation

A high-speed 16mm motion picture camera (Teledyne

Camera Systems, Model DBM-54) was used to obtain film

records of each trial. The camera was positioned 2.2

meters above the level at which the ball was to be pitched.

Appropriate levelling techniques were used to ensure that

the optical axis of the camera corresponded with a verti-

cal axis. The operating speed of the camera was 500 frames

per second. Temporal scales were obtained by means of a

timing light generator used in conjunction with the motion

picture camera. In order to make it possible to determine

the release angle of the ball, a mirror was placed thirty

centimeters to the right, and at the same vertical height,

of the anticipated initial flight of the ball. The mirror

was oriented at approximately 0.8 radians. One number

coded card was included within the field of view of the

24

25

camera and recorded on film for each trial. The number was

subsequently used to identify the trial.

Pitching Machine

An automatic pitching machine (Jugs Curveball Pitcher,

JoPaul Industries, Inc., Tualatin, Oregon) was used to

deliver balls towards a measurement board. The height of

release of the ball was fixed at 2.59 meters which corre-

sponded to the average height of release of a pitched ball

(Williams, 1971).

Projectiles

The projectiles used during the testing session were

ten baseballs (Rawlings R. 0., St. Louis, Missouri) and a

rubber sphere of uniform roughness.

Measurement Board

A board was anchored 17.1 meters from the release

point of the ball. This distance corresponded to the dis-

tance from a pitchers mound to home plate. A cartesian

coordinate system was marked on the board. The origin of

the coordinate system was located at the expected horizon-

tal landing point of the ball neglecting all fluid and

gravitational forces and at ground level. The board was

coated with chalk which permitted the determination of the

contact point of the ball.

26

Testing Procedures

All of the trials for the study were conducted during

a one-day filming session. The testing location was the

Men's Gymnasium of the North Texas State University

(Denton, Texas). The location of the testing instruments

was as shown in Figure 2. The filming session occurred at

a time when all air conditioning and heating systems were

inoperative and the building was closed. These conditions

were necessary to minimize extraneous air turbulence.

Prior to and at thirty-minute intervals during the

filming session, the temperature and barometric pressure at

the testing site were determined and recorded. These re-

corded values were converted at a later time to approximate

values for both the density and kinematic viscosity of the

air.

To assist in determining angular velocity, the balls

were marked with black dots approximately 0.5 centimeters

in diameter. To ensure that at least one point on the ball

was visible at all times, the landmarks were placed such

that the largest distance between any two marks was ap-

proximately four centimeters.

Ten trials at each of three pitching machine settings

for each of three roughness conditions were recorded on

film. The first condition consisted of a baseball pitched

in a manner that two of the roughness elements opposed the

incoming flow. The second condition consisted of a pitched

K

(0.

N

0 w

-- EEN UU

N .CL

27

U,.x

x~

Q

UE EN 0)

N

N

to

Vl

V

E

r%

r.

U)

U)

4--i

04-)

cj

U

0

0

2r

28

baseball with four of the roughness elements opposing the

flow. The third consisted of a pitched rotating uniformly

rough sphere. An illustration of the roughness conditions

is shown in Figure 3. Any trials in which the pitched ball

did not strike the measurement board were eliminated from

this study. After each trial, the location of the ball

with respect to the origin of the measurement board was

recorded.

Data Acquisition Procedures

For each successful trial, the initial portion of the

flight of the ball was analyzed with the aid of a Lafayette

16mm Analyzer (Lafayette Instrument Co., Lafayette,

Indiana) in conjunction with a Numonics Electronic Graphics

Digitizer (Model 1200, Numonics Corp., North Wales,

Pennsylvania), which was interfaced to a Tektronix 4052

Graphics Calculator (Tektronix Inc., Beaverton, Oregon).

The motion of the ball for three frames in which the pro-

jected image in the mirror was visible was analyzed. The

cartesian coordinates of each of the following landmarks

on the ball were digitized and recorded for all film

frames:

1. Any three landmarks on the circumference of the

ball

2. Any three landmarks on the circumference of the

ball's projected image in the mirror

29

sN

p O

0 4-4..

pU..

N U)

04 0

O

/ -

M N

.. N

-N

O O

O -IX 4-

U

ci

i

U)U)

fa

PO

aO

30

3. One landmark on the surface of the ball.

The three landmarks on the circumference of the ball

were used to determine the trajectory of the center of

gravity of the ball in the X-Y plane. The three landmarks

on the circumference of the ball's projected image on the

mirror were used to determine the trajectory of the center

of gravity of the ball in the X-Z plane. The derivative of

each of the component displacements of the ball was com-

puted to determine the initial linear velocity of the ball

in the component directions. The angular velocity of the

ball was computed from the displacement history of the

landmark. The values of barometric pressure and tempera-

ture were used to calculate the density of the air and the

kinematic viscosity of the air as prescribed by the Hand-

book of Chemistry and Physics (CRC, 1978). These variables

were then used in a computer program which used numerical

approximation techniques to give predicted values for

deviation from normal projectile flight.

The formula used to compute angular velocity appears

in Appendix A. A listing of the computer program used to

compute the angular and linear velocity of the balls appears

in Appendix B. The mathematical development of the mathe-

matical model developed by Rubinow and Keller (1963) for

numerical approximation of hypothetical landing points

appears in Appendix C. A listing of the computer program

31

utilizing this numerical approximation to predict values

for the landing points of the projectiles appears in

Appendix D.

Statistical Analysis

A one way analysis of variance was used to determine

if differences existed in component flight deviations be-

tween roughness conditions. A statistical analysis

utilizing mixed design repeated measures analysis of vari-

ance procedures with the roughness conditions serving as

the non-repeating factor was conducted to determine whether

or not interactions existed between roughness conditions

and the method of determining component landing points.

The actual and hypothetical component landing points served

as dependent variables. Pearson product-moment correlation

coefficients were computed to ascertain the relationship

between the component landing locations for each condition

and the hypothetical landing locations. Multiple linear

regression was used to determine if the component landing

points could be accurately predicted using statistical

techniques. The component initial and angular velocities

for each condition were entered as the independent mea-

sures. Pearson product-moment correlation coefficients

were computed between the component initial linear veloci-

ties for each of the roughness conditions.


CRC. Handbook of chemistry and physics. West Palm Beach,Florida: CRC Press, 1978.

Rubinow, S. I., & Keller, J. B. The transverse force on aspinning sphere moving in a viscous fluid. Journal ofFluid Mechanics, 1963, 11 (3), 447-459.

Williams, T. S., & Underwood, J. The science of hitting.New York, New York: Simon and Schuster, 1971.

32

CHAPTER IV

RESULTS



rotating spheres.

Environment and Projectiles

The temperature and barametric pressure were deter-

mined during the data collection session. Both remained

constant and yielded values of 1.84 X 10 mPl and

1.193 kg/m3 for air viscosity and air density respectively.

The ten baseballs used during the testing session ranged

in mass from 146.5 grams to 150.75 grams (M = 148.56 grams)

with radii of 3.689 centimeters. The rubber sphere of

uniform roughness had a mass of 94.78 grams and a radius of

3.598 centimeters.

Results

The mean component flight deviations of the projec-

tiles for each of the roughness conditions are shown in

Table I. All of the component deviations are expressed in

differences between the component landing points on the

33

34

measurement board and hypothetical component landing points

calculated by assuming flight in the absence of air re-

sistance.

TABLE I

MEAN X- AND Z- FLIGHT DEVIATIONS FOR EACHOF THE ROUGHNESS CONDITIONS

Condition Flight DeviationaX Z

Two roughness elements 45.83 41.70

Four roughness elements 48.54 55.51

Uniformly rough sphere 4.82 146.83

aDeviation measured in centimeters

Statistical analyses revealed significant differences

(p<0.05) between both of the baseball conditions and the

uniformly rough sphere for both the X- and Z- flight devia-

tions. Summary tables for the statistical analyses appear

in Appendices E and F.

The flight deviations in the X- direction for the

baseball conditions were due to a relatively large Magnus

effect whereas for the uniformly rough sphere this effect

was minimal. The mean component angular velocities for

each condition are shown in Table II. The results show

that the resultant axis of rotation for all conditions was

primarily oriented in the Z- direction. A statistical

35

TABLE II

MEAN COMPONENT INITIAL ANGULAR VELOCITIES FOREACH OF THE ROUGHNESS CONDITIONS

Angular VelocityaCondition X Y Z

Two roughness elements -715.8 -346.1 1651.8

Four roughness elements -452.4 -346.7 2534.1

Uniformly rough sphere -236.2 277.2 1425.6

aAngular velocity measured in radians per second

analysis revealed a significant difference (p<0.05) between

the baseball with four roughness elements opposing the flow

and the uniformly rough sphere. The finding of significant

differences between both baseball conditions and the uni-

formly rough sphere for the X- component of flight devia-

tion, and the lack of a significant difference between the

baseball with two roughness elements and the uniformly

rough sphere for the Z- component of angular velocity in-

dicates that differences in the Magnus effect were not

primarily due to differences in angular velocity. A sum-

mary table for the statistical analysis of the angular

velocities appear in Appendix G.

The mean magnitude of the initial linear velocity for

the baseball with two roughness elements, the baseball with

four roughness elements and the uniformly rough sphere were

36

34.93 meters per second, 36.04 meters per second and 44.28

meters per second respectively. The greater velocity mag-

nitude for the uniformly rough sphere could have con-

tributed to a reduced Magnus effect for this condition.

However, the Reynolds number for this magnitude of velocity

was approximately one-third of the critical value. This

suggests that a classical Magnus effect should have been

present. The reported small flight deviation in the X-

direction was therefore probably not primarily due to a

greater linear velocity but instead to the absence of

roughness elements.

The reported flight deviations in the Z- direction can

in part be attributed to differences in the masses of the

baseballs and the uniformly rough sphere. That is, the

drag forces acting on the relatively smaller mass of the

uniformly rough sphere prolonged the time of flight and

therefore extended the gravitational effects. However, if

the differences in mass are normalized, the Z- component of

flight deviation for the baseballs should have been 1.93

times that recorded. It is therefore possible that the

roughness elements increased the turbulence of the boundary

layer which decreased the fluid drag force and thus de-

creased the time of flight. A decrease in the time of

flight would reduce the gravitational effects and therefore

the flight deviation.

37

The mean component landing points with respect to the

origin established on the measurement board for each of the

roughness conditions are shown in Table III.

TABLE III

MEAN COMPONENT DISPLACEMENTS FOR EACHOF THE ROUGHNESS CONDITIONS

Condition Linear Displacementa

x z

Two roughness elements -26.05 79.75

Four roughness elements -25.12 80.24

Uniformly rough sphere - 2.66 56.00

Displacements measured in centimeters

The mean component landing points as calculated using the

formulae developed by Rubinow and Keller (1963) for each of

the roughness conditions are shown in Table IV.

A statistical analysis revealed significant differ-

ences (p<0.05) between the actual and hypothetical X-

components of the landing points for the baseball with four

roughness elements and for the uniformly rough sphere. A

significant interaction was found between the roughness

conditions and the methods for determining the Z- com-

ponent of the landing point. An analysis of the simple

main effects revealed a significant difference between the

methods of determining the Z- component of the landing

38

TABLE IV

MEAN HYPOTHETICAL COMPONENT DISPLACEMENTS FOREACH OF THE ROUGHNESS CONDITIONS

Linear DisplacementaConditions

X Z

Two roughness elements -44.61 73.75

Four roughness elements -63.25 98.14

Uniformly rough sphere -42.34 171.34

aDisplacements measured in centimeters

points for the uniformly rough sphere. This indicates that

for this condition, the equations developed by Rubinow and

Keller (1963) provide a poor prediction. Summary tables of

the statistical analyses appear in Appendices H and I.

Pearson product-moment correlation coefficients cal-

culated between the actual and hypothetical component

landing points for all of the conditions indicate the exis-

tence of an overall positive relationship. However, the

finding of some significant differences between the actual

and hypothetical component displacements suggests that the

predictive equations are of questionable practical value

for the conditions examined in this study. There exists

the possibility that a roughness factor should be included

in the mathematical model. The correlation coefficients

39

between the actual and hypothetical component landing

points are shown in Table V.

TABLE V

CORRELATION COEFFICIENTS BETWEEN ACTUAL LANDINGPOINTS AND HYPOTHETICAL LANDING POINTS

Conditions Component of Landing PointsN X Z

Two roughness elements 28 .4568* .4467*

Four roughness elements 29 .6404* .8770*

Uniformly rough sphere 28 .7545* -.3111

*

p<0.05

Multiple linear regressions were used to determine if

the component landing points could be accurately predicted

using statistical techniques. The component initial linear

and angular velocities for each condition were entered as

the independent measures. The results of the analyses ap-

pear in Table VI. The results indicate that the regression

analyses provided a better prediction of the actual landing

points than did the model developed by Rubinow and Keller

(1963). The order of entry of the independent variables

into the regression analyses is shown in Table VII. The

results show, that with the exception of the two roughness

elements condition, the X- component of linear velocity

provides the greatest contribution towards predicting both

40

TABLE VI

R AND R-SQUARED VALUES RESULTING FROM THE MULTIPLELINEAR REGRESSION USED TO PREDICT COMPONENT

LANDING POINTS FOR EACH OF THEROUGHNESS CONDITIONS

Condition R2 R

X-direction




Z-direction




components of the landing points. The consistent relation-

ship between the X-component of linear velocity and the X-

component of the landing points, and the relationship be-

tween the Z-component of linear velocity and the Z-

component of the landing point for the baseball with two

roughness elements reflect the anticipated landing loca-

tions when projectiles are released in a given direction.

Pearson product-moment correlations calculated between the

X-components of linear velocity and the Z- and Y- component

41

TABLE VII

ENTRY ORDER OF THE INDEPENDENT VARIABLES FOR EACHOF THE STEP-WISE REGRESSION ANALYSES

Condition Linear Velocity Angular VelocityX Y Z X Y Z

X-Displacement Prediction

Two roughness elements 1 4 2 5 3 6

Four roughness elements 1 5 4 3 6 2

Uniformly rough sphere 1 2 3 4 5 6

Z-Displacement Prediction

Two roughness elements 6 2 1 4 3 5

Four roughness elements 1 3 4 2 5

Uniformly rough sphere 1 2 3 4 5 6

linear velocities revealed, for the baseball with four

roughness elements that there existed concomitant increases

in all of the component velocities. That is, for this con-

dition the pitching machine consistently projected the ball

either rapidly upward to the right or slowly downward to

the left. For the uniformly rough sphere, a significant

relationship was found between the X-component of linear

velocity and the Z-component of linear velocity. That is,

for this condition, the pitching machine consistently pro-

jected the ball either upward and to the right or downward

42

and to the left. The correlation coefficients describing

the relationships between the component initial linear

velocities are shown in Table VIII.

TABLE VIII

PEARSON PRODUCT-MOMENT CORRELATION COEFFICIENTSCALCULATED BETWEEN THE COMPONENTS

OF LINEAR VELOCITY

Component VelocityComponent Velocity X Y

Two Roughness Elements

Y 0.4191

Z 0.5873* 0.3098

Four Roughness Elements

Y 0.7336*

Z 0.4351* 0.3373

Uniformly Rough Sphere

Y 0.2781

Z 0.8248* 0.2203

*p<0.05



Rubinow, S. I., & Keller, L. B. The transverse force on aspinning sphere moving in a viscous fluid. Journal ofFluid Mechanics, 1963, 11 (3), 447-459.

43

CHAPTER V

SUMMARY AND CONCLUSIONS

Introduction



rotation spheres.

The motion of a totally immersed and translating ob-

ject is affected by both gravitational and fluid forces

(Hay, 1978). For an object which is also rotating, one

component of the fluid forces is the lift caused by the

Magnus or Robins effect (Magnus, 1953; Robins, 1842). One

of the major factors influencing the fluid forces acting

on a totally immersed object is the nature of the boundary

layer surrounding the object. Of the various fluid flow

conditions, only laminar, turbulent or partially turbulent

flow are encountered in the majority of sport situations.

Laminar flow exists when the boundary layer separation oc-

curs in a region not presented to the flow (Jacobsen,

1973). In turbulent flow, boundary layer separation occurs

in the region of the object presented to the flow. Par-

tially turbulent flow, present only when the immersed

object is assymetrical or rotating, is characterized by

laminar flow in one region of the object and turbulent flow

44

45

in another region. This type of flow can cause a resultant

lift force in the opposite direction to the classical

Magnus effect (Jacobsen, 1973).

In many sports, the performer imparts a spin to the

sport implement to cause flight deviations. In baseball,

the pitcher uses the Magnus effect to deviate the ball

from a hypothetical parabolic pathway. Briggs (1959) re-

ported horizontal flight deviations of up to twenty-six

inches for a spinning baseball. The effects of roughness

elements on the flight of a baseball with little or no

angular velocity have been described by Watts and Sawyer

(1975).

Rubinow and Keller (1963) and Kreighbaum and Hunt

(1978) developed mathematical models to describe both the

trajectory of a pitched sphere and the nature of the acting

fluid forces. However, it would appear as though no at-

tempts have been made to experimentally validate the

models.

Procedures

Thirty trials of each of three roughness conditions

were recorded on film. An appropriately oriented plane

mirror ensured that both superior and lateral perspectives

of each of the conditions were simultaneously recorded.

The first condition consisted of a baseball pitched in

such a manner that two of the roughness elements opposed

46

the incoming flow. The second condition consisted of a

pitched baseball with four of the roughness elements op-

posing the flow. The third consisted of a pitched uni-

formly rough sphere. The initial linear velocities of the

projectiles varied from 27.3 meters per second to 51.9

meters per second and angular velocity varied from 127.8

radians per minute to 11558 radians per minute. Any trials

in which the pitched ball did not strike a measurement

board were repeated. Any trials in which both images of

the ball were not visible for three successive film frames

were eliminated from the study. After each trial, the

location of the landing point of the ball with respect to

the origin of the measurement board was recorded. During

the testing session, recordings were made of the ambient

barometric pressure and temperature.

The motion of the ball for each successful trial was

analyzed. The cartesian coordinates of each of the fol-

lowing landmarks or locations were digitized and recorded

for three film frames:

1. Any three points on the circumference of the ball;

2. Any three points on the circumference of the

projected image of the ball in the mirror;

3. One marked point on the surface of the ball.

The three points on the circumference of the ball were used

to determine the trajectory of the center of gravity of

the ball in the X - Y plane. The three points on the

47

circumference of the balls in the mirror were used to de-

termine the trajectory of the ball in the Y - Z plane.

The derivative of each of the component displacements of

the ball were computed to determine the initial linear

velocity of the ball in the component directions. The an-

gular velocity of the ball was computed from the displace-

ment history of the marked point on the surface of the ball.

The values of the barometric pressure and temperature were

used to compute the density and kinematic viscosity of the

air. The components of initial linear and angular velo-

city, the mass and radius of the balls and the density and

kinematic viscosity of the air were input into a computer

program which utilized numerical approximation techniques

to derive predicted values for deviation from normal pro-

jectile flight.

A one way analysis of variance was used to determine

if differences existed in component flight deviations be-

tween roughness conditions. A statistical analysis

utilizing mixed design repeated measures analysis of vari-

ance procedures with the roughness conditions serving as

the non-repeating factor was conducted to determine whether

or not interactions existed between roughness conditions

and the method of determining component landing points.

The actual and hypothetical component landing points served

as dependent variables. Pearson product-moment correlation

coefficients were computed to ascertain the relationship

48

between the component landing locations for each condition

and the hypothetical component landing locations. Mul-

tiple linear regression was used to determine if the com-

ponent landing points could be accurately predicted using

statistical techniques. The component initial linear and

angular velocities for each condition were entered as the

independent measures. Pearson product-moment correlations

coefficients were computed between the component initial

linear velocities for each of the roughness conditions.

Results

Statistically significant differences were found be-

tween both of the baseball conditions and the uniformly

rough sphere for both the X- and Z- flight deviations. The

flight deviations in the X-direction for the baseball con-

ditions were found to be due to the presence of a rela-

tively large Magnus effect. The differences in flight

deviations between the baseball conditions and the uni-

formly rough sphere could not be accounted for by differ-

ences in the Z-component of angular velocity and were

therefore attributed to differences in the roughness of the

surface of the conditions. The differences in the Z-

component of flight deviation can be attributed, in part,

to the relatively small mass of the uniformly rough sphere

and an increase in boundary layer turbulence for the base-

ball conditions.

49

Statistically significant differences were found be-

tween some of the actual and hypothetical component landing

points despite a predominance of significant relationships

between these parameters. It would therefore appear as

though the prediction formulae developed by Rubinow and

Keller (1963) take into account appropriate input para-

meters but provide inadequate estimations of flight devia-

tions.

A prediction of the actual component landing points

using multiple linear regression techniques provided better

estimations than the Rubinow and Keller model (1963) al-

though the practical usefulness of the derived results is

questionable.

Conclusions

Based on the results of the study, the following con-

clusions appear to be warranted.

1. The presence of roughness elements on the surface

of a baseball increases the horizontal flight

deviation when the baseball rotates about a verti-

cal axis.

2. The presence of roughness elements on the surface

of a baseball may increase the turbulence of the

boundary layer and therefore cause a decrease in

the encountered drag forces.

50

3. Linear velocity is the parameter which has the

dominating effect on the trajectory of a spinning

baseball.

4. The mathematical model developed by Rubinow and

Keller (1963) does not adequately predict flight

deviations.

Recommendations

Based on the results of this study, an additional

examination of the effect of roughness elements on the

Magnus characteristics of rotating spheres would seem ap-

propriate. In particular, future attention should be

directed towards minimizing all sources of measurement

error. The recommendations for future studies are there-

fore made:

1. An examination of the effect of roughness elements

on the Magnus characteristics of rotating spheres

using a wind tunnel to control fluid velocity, a

motor drive to control the angular velocity of the

spheres and transducers to directly measure the

lift and drag forces;

2. An examination of the effects of varying roughness

configurations on flight deviations of those pro-

jectiles used in sports in which the rules permit

either modifications or variations of the surface

of the projectiles;

51

3. A determination of the optimal combinations of

the linear and angular velocity components for

achieving the objective of specific skill which

involves the use of a projected sport implement.

APPENDICES

APPENDIX A

FORMULAS FOR THE COMPUTATION OF ANGULAR VELOCITY

Given that the equation for a plane through P1= (x1 ,yi,zi)

with some normal vector <a,b,c> is

a(x-x) + b(y-y1 ) + c(z-z 1 ) = 0

A normal vector <a,b,c> given three points

P1 = (xiy 1 ,zi) , P 2 = (x2,y2,z 2 ) and P 3 = (x 3 ,y3,z 3 )

can be found with the cross product of P1P2 and P1 P 3 :

i j k

N = P1P 2 x P1P 3 = x2-x1 y2-y1 Z2-Z1X 3 -X1 Y3-Y1 Z3-Z1

= [(y2-y1) (z3-z 1 ) - (y3-y7) (z2 -z1 )] i

+ [ (x3 -x1 ) (Z2-Z1) - (x2-x 1 ) (z3-zi)] j

+ [ (x 2 -x 1 ) (y3-yL) - (x 3 -x 1 ) (y2-y1)] k.

Letting N = <a,b,c> where:

(y2-yl) (z 3 -zi) - (y3-y1) (Z2-Zl) = a

(x 3 -xi) (Z2-Zi) - (x2-x 1 ) (z3-z1 ) = b

(x2-x 1 ) (y3-yi) - (x 3 -x 1 ) (y2-yl) = c

the equation for the plane containing P1, P2 , P3 becomes

a(x-x1 ) + b(y-y1) + c(z-z 1 ) = 0

53

54

The three points P1, P2 and P3 also define a circle with

the center at (xc 'c'zc). The intersection of the per-

pendicular bisectors, that lie in the same plane with

P1 , P2 and P3 , of any two chords connecting two of the

points define that center (xc 'c'zc). The midpoints of

the chords PiP2 and PiP3 are given by

M2 = X+ X2 y1+ 2 Z1 + Z2)22 ' 2

and M3 = (Xi+ X 3 y+ y3 Z1 +Z3)2 ' 2 ' 2

The dot product of two vectors that are perpendicular to

each other is 0. Thus, a perpendicular bisector of

P1P2 , M2C, dotted with P1 P2 is 0:

(1) P1 P2 ' M 2 C = (x 2 -x1 ) (X+X2 - x c)+ (y2-yi) (Y12Y2 y c

+ (Z2-Zi) (ZiZ2 - zc) = 0

Also, P1 P3 dotted with M3C, a perpendicular bisector of

PiP3 , is 0:

(2) P1 P3 - M3 C = (x 3 -x 1 ) (Xi+X3 - x)2 c

+ (y3-y1 ) (y1 y23 -c + (z 3-z1 ) (Zl+Z3 - zc) = 0

where C = (xc ,yc,zc)

If the point C lies on the same plane with P1,P2 and P3

(3) a(x -x 1 ) + b(yc-y1 ) + c(zc-z1 ) = 0.

55

Simplifying equation (1)

X 2 X 1 - (x 2 -x)x 2 + Y2 y-2 2 X) c+ 2 (y2 Y1) YC

+ Z2 2 -z12 (z2-zl)z = 02 -c

2 2X 2 2Y2 2- Z 2_ Z 2

-(X2-X1) X - (y2-y1)y - (z 2 -z 1 ) = X2 2 - 1 _ Y22_yl2- 22-c c c 2 2 2

Multiplying by -l

(X2-Xi)xc + (y2-yl)y + (Z2-z1)zc= x 2 _ 1 2 y22y 2+ z22-z1 2

X22 -X12 + Y22_Y2 + Z22_Z1 2

= 2 2 2X2-x1

(y2-yl)yc + (Z2-Zi)ZCX2-x1

Letting Qi =X2 2-X 1 2 + y22_Y12 + Z2-Z12

2 2 2X2-X1

x = QiC(y2-Yi) yc -X2-X1

(Z2-z)X2-X X

zC

Letting P1,= y2-yl and R1, = z2-zi

x2-x1 x2 -x 1

(4) xc= Q - Piy - R1zc

we get

Substituting (4) into (2) and simplifying we get

P1P 3 M3C = (x3 x1 )[X1+3 _ (Q 1_Py -Riz)

+ (y 3 -yi) 2(1y3

+ (z 3 -z 1 ) (Zl+Z3 -z )2 C

xC

we get

_ c

56

X32-x12

2 - (x 3 -x 1 ) Ql+ (x 3 -x 1 ) Piyc+ (x 3 -x 1 ) Rizc

+ 2 2 - (y3 y)yC+(Z32i)- (z 3 -z)zc=O

x 2_ 2 3 _2 2 2- 2

2 - (x3-x 1)Q1+ 2 + 2 = (y3-y1)yc-(x3x1)P1Yc

+ (Z3-Z1)ZC - (x3-x1)Rizc

X32_X 12 Y32_Y2 Z32 -Z1 2

2 - 3-xi)Qi+ 2 + 2

(y3-yl) - (x 3-x)P1

[(z3-zl)-(x3-x1)RlIzc

(Y3-yi) - (x3-x 1 )P 1

Yc

Letting

x32_ 12 _y32_y,2 Z32-Z12

(2X (x3-x1)Q1+2Z

(y3-yl) - (x 3 -x 1 )P 1

and P2 = (z3-z 1 ) - (x3-x1)R1(y3-yi) - (x3-x 1 )P1

(5) yc = Q2-P2zc

Substituting (4) and (5) into (3) we get

a[Qi-Pi(Q2-P2zc) - Rizc - xi] + b(Q2-Pzzc-yi)

+C(ZC-Zi) = 0

Q2 =

we get

57

aQ1 - aP1 Q 2 + aPiP2zc - aRizc - axi + bQ2 - bP2zc - by1

+ CZc - CZ1 = 0

aQ1 - aP1 Q 2 - ax1 + bQ 2 - by, - cz1

= (aR 1 - aP1 P 2 + bP2 - C)Zc

aQ1 - aP 1 Q 2 - ax1 + bQ 2 - by, - cz1 _aRt- aP1 P2+ bP 2 - c-c

Replacing zc in (5) we get

aQ1-aPiQ 2-ax1 +bQ 2-by1 -cz1Yc = Q2-P2 [ aR1 -aP1P 2 +bP 2 -c

and, replacing zc and yc in (4) we get

=[ aQl-aPQ2-axl+bQ2-by,-cz1aRi-aPiP2+bP2-c

R (aQlaPlQ2 axlI+bQ 2 bylczl)aR1 -aP 1 P2 +bP 2 -c

Thus, the center of the circle containing P1 , P2 and P3

on it's circumference is

C = (xc ,Yc,zc) .

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APPENDIX C

FORMULAS OF RUBINOW AND KELLER (1963) TO A NUMERICALAPPROXIMATION OF HYPOTHETICAL LANDING POINTS

(1) F =-6iryV(l+3pIvr) = -6irryV - 4-r pIVlV

(2) (3) FL = ir 3 p(Qxv) and F = mg

where

r = radius of the ball Q = angular velocityp = density of the air V = linear velocityp = viscosity of the airm = mass of the ballg = acceleration due to gravity = -9.8066 m/sec 2

(4) (5) (6) (7) Letting B = -6rrp, C = (-4) ('rpr 2 ), D = rrr3 p

and E = mg

(8) (9) (10) FD = BV+CIVIV

(11) Ftotal

FL = D (I2xV) I = E

= BV + CjVIV + D(QxV)+E

By Newton's Second Law F = ma

(12) a total= V + IVIV + - (QxV)+ttl m m m m

AVBy definition a - where t = time A = > change in

thus AV = a(At)

AV _ Bat + CAtiviv + DAt ( + EAtm m m m

75

76

(13) AV= [BV+Cj VIV+D(QxV)+E] -t

(14) Letting velocity V at time = i be Vi and taking

At sufficiently small

V. = V.+(BV.+C jV IV+D (ExV)+E)-i+At i im

in component form

(15 ) V . =V . + [BV .+CV.IV .+D(Q V -. V )]--tx xi+At xi xi i xi y z z ly m

(15 ) V . = V .+[BV .+CIV. IV . +D(Q V -Q v )]Aty yi+At yi yi i yi z x x z m

At(15 ) V . +B + V . +D(Q V -Q V )+E]--z zi+Atvzi+[Bzi+CIi z xy y x m

Letting S be displacement and As be change in displacement

(16) As=VavAt => As=(Vi+Vi+At) Atav 2

(17) Si+AtS.+AS

Therefore at time = i + At

(18 ) Sxi+At = S .+ASx xat xl

(18y) Syi+At = S .+ASyl y

(18) Si+t = S .+ASz

77

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APPENDIX E

SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE CONDUCTED ON

THE X-COMPONENT OF FLIGHT DEVIATIONS

Source SS df MS F

A 3.4257 2 1.7128 18.1937*

S/A 7.7199 82 0.0941

Comp 1/2 0.0143 1 0.0143 0.1520

Comp 1/3 2.3546 1 2.3546 25.011*

Comp 2/3 2.7808 1 2.7808 29.5380*

*p<0 .0 5

aA = Roughness conditions

S = Trials

Comp 1/2 = comparison between the baseballs with tworoughness elements opposing the flow andthe baseballs with four roughness elementsopposing the flow

Comp 1/3 = comparison between the baseballs with tworoughness elements opposing the flow andthe uniformly rough sphere

Comp 2/3 = comparison between the baseballs with fourroughness elements opposing the flow andthe uniformly rough sphere

81

APPENDIX F

SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE CONDUCTED ON

THE Z-COMPONENT OF FLIGHT DEVIATIONS

Source SS df MS F

A 18.1879 2 9.0940 30.980*

S/A 24.0706 82 0.2953

Comp 1/2 0.3153 1 0.3153 1.074

Comp 1/3 15.4739 1 15.4739 52.714*

Comp 2/3 11.6047 1 11.6047 39.533*

*p< 0 .0 5

aA = Roughness condition

S = Trials

Comp 1/2 = comparison between the baseballs with tworoughness elements opposing the flow andthe baseballs with four roughness elementsopposing the flow



82

APPENDIX G

SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE CONDUCTEDON THE Z-COMPONENT OF ANGULAR VELOCITY

Sourcea SS df MS F

A 21738821.68 2 10869410.84 3.5343*

S/A 252185686.52 82 3075435.20

Comp 1/2 11095091.91 1 11095091.91 3.6080

Comp 1/3 1292997.25 1 1292997.25 0.4200

Comp 2/3 20052136.09 1 20052136.09 6.5200*

*p<0.05


S = Trials

Comp 1/2 = comparison betweenroughness elementsthe baseballs withopposing the flow

the baseballs with twoopposing the flow andfour roughness elements



83

APPENDIX H

SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE CONDUCTED

ON THE X-COMPONENT OF THE LANDING POINTS

Source SS df MS F

A 14161.31 2 7080.65 1.19

S/A 480330.57 81 5930.01

B 42037.54 1 42037.54 23.43*

AxB 3657.65 2 1828.82 1.02

BxS/A 145308.44 81 1793.93

A at Bi 10446.02 4 2611.50 3.36*

A at B2 7372.93 4 1843.23 0.60

B at Al 4819.29 1 4819.29 4.16

B at A2 18834.45 1 18834.45 23.78*

B at A3 22041.45 1 22041.45 6.43*

*p< 0 . 0 5


B = Method of determining landing points

S = Trials

APPENDIX I

SUMMARY TABLE FOR THE ANALYSIS OF VARIANCE CONDUCTEDON THE Z-COMPONENT OF THE LANDING POINTS

Source SS df MS F

A 39722.91 2 19861.46 3.15

S/A 510832.87 81 6306.58

B 73626.72 1 73626.72 16.30*

AxB 117133.52 2 58566.76 12.97*

BxS/A 365778.26 81 4515.78

A at Bi 11307.88 4 2826.97 8.08*

A at B2 145548.55 4 36387.14 7.19*

B at Al 588.25 1 588.25 0.13

B at A2 3927.88 1 3927.88 1.36

B at A3 186244.11 1 186244.11 29.92*

*p< 0 .0 5


B = Method of determining landing points

S = Trials

85

BIBLIOGRAPHY

Books


CRC. Handbook of chemistry and physics. West Palm Beach,Florida: CRC Press, 1978.




Hughes, W. F., & Brighton, J. A. Fluid dynamics. New York,New York: McGraw Hill Book Company, 1967.


Miller, L. Mastering soccer. Chicago, Illinois: Con-temporary Books, Inc., 1979.


Robins, B. New principles of gunnery. London, England:1842.

Williams, T. S., & Underwood, J. The science of hitting.New York, New York: Simon and Schuster, 1971.

Unpublished Materials


86

87

Articles

Borg, J. L. New twist to steering. Ocean Industry, June

1980, 15 (6), 56-61.

Briggs, L. J. Effect of spin and speed on the lateral de-flection (curve) of a baseball and the Magnus effectfor smooth spheres. American Journal of Physics,1959, 27, 589-96.


Davies, J. M. The aerodynamics of golf balls. Journal ofApplied Physics, September 1949, 20 (9), 821-828.

Hess, S. Coupled translational and rotational motions of asphere in a fluid. Zeitschrift Fur Naturfonschung,1968, 23A, 1095-1101.

Krahn, E. Negative Magnus force. Journal of AeronauticalScience, April 1956, 377-378.

Krieghbaum, E. F., & Hunt, W. A. Relative factors in-fluencing pitched baseballs. In Landry, F. & Orban,W. A. R. (Eds.). Biomechanics of Sports andKinanthrophometry, 6, 1978, 227-236.

Magnus, G. On the deflection of a projectile. PoggendorfsAnnalen der Physik und Chemie, 1853, 88, (1).

Rubinow, S. I., & Keller, J. B. The transverse force on aspinning sphere moving in a viscous fluid. Journal ofFluid Mechanics, 1963, 11, (3), 447-459.

Selin, C. An analysis of the aerodynamics of pitched base-balls. Research Quarterly, 1959, 30, 232-240.


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