Ale/ Igo, fY63 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
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Ale/
Igo, fY63
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-
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
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)
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.
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 lateraldeflection (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.
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.
Jacobsen, I. D. Magnus characteristics of arbitrary ro-tating bodies. (AGARDograph, No. 171), 1973.
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
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: Con-temporary Books, Inc., 1979.
Reiff, C. G. What research tells the coach about baseball.American Alliance for Health, Physical Education andRecreation, 1971.
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.
Watts, R. G., & Sawyer, E. Aerodynamics of a knuckleball.American Journal of Physics, 1975, 43, 960-963.
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
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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
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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
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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
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78
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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
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
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
aA = Roughness conditions
S = Trials
Comp 1/2 = comparison betweenroughness elementsthe baseballs withopposing the flow
the baseballs with twoopposing the flow andfour roughness elements
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
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
aA = Roughness conditions
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
aA = Roughness conditions
B = Method of determining landing points
S = Trials
85
BIBLIOGRAPHY
Books
Barnaby, J. Ground strokes in match play. Garden City,New York: Doubleday and Company, Inc., 1978.
CRC. Handbook of chemistry and physics. West Palm Beach,Florida: CRC Press, 1978.
Daily, J. W., & Harleman, D. R. F. Fluid dynamics.Reading, Massachusetts: Addison-Wesley PublishingCompany, Inc., 1966.
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. New York,New York: McGraw Hill Book Company, 1967.
Jacobsen, I. D. Magnus characteristics of arbitrary ro-tating bodies. (AGARDograph, No. 171), 1973.
Miller, L. Mastering soccer. Chicago, Illinois: Con-temporary 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.
Williams, T. S., & Underwood, J. The science of hitting.New York, New York: Simon and Schuster, 1971.
Unpublished Materials
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.
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.
Chase, A. A slice of golf. Science 81, July-August 1981,90-91.
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.
Watts, R. G., & Sawyer, E. Aerodynamics of a knuckleball.American Journal of Physics, 1975, 43, 960-963.