U.S. patent application number 11/211537 was filed with the patent office on 2007-03-01 for method for predicting ball launch conditions.
This patent application is currently assigned to Acushnet Company. Invention is credited to William Gobush.
Application Number | 20070049393 11/211537 |
Document ID | / |
Family ID | 37056048 |
Filed Date | 2007-03-01 |
United States Patent
Application |
20070049393 |
Kind Code |
A1 |
Gobush; William |
March 1, 2007 |
Method for predicting ball launch conditions
Abstract
A method and computer program for predicting a golfer's ball
striking performance is disclosed. In order to generate a predicted
trajectory and predicted ball launch conditions, it is desirable to
determine several properties of the golf equipment and a player's
swing. For instance, pre-impact swing conditions of a golfer are
observed using a monitoring system. Additionally, it is desirable
to determine the properties of the golf ball and golf club.
Moreover, the effect of the shaft properties and the slippage
between the golf club and the golf ball may also be considered. In
this manner, a more accurate determination of a golfer's predicted
trajectory and ball launch conditions may be determined without
requiring the golfer to actually strike a golf ball.
Inventors: |
Gobush; William; (North
Dartmouth, MA) |
Correspondence
Address: |
BINGHAM MCCUTCHEN LLP
3000 K STREET, NW
BOX IP
WASHINGTON
DC
20007
US
|
Assignee: |
Acushnet Company
|
Family ID: |
37056048 |
Appl. No.: |
11/211537 |
Filed: |
August 26, 2005 |
Current U.S.
Class: |
473/131 ;
473/219; 473/282; 473/351; 473/407; 473/409 |
Current CPC
Class: |
A63B 69/3632 20130101;
A63B 69/3658 20130101 |
Class at
Publication: |
473/131 ;
473/282; 473/409; 473/351; 473/219; 473/407 |
International
Class: |
A63B 69/36 20060101
A63B069/36; A63B 53/00 20060101 A63B053/00; A63B 37/00 20060101
A63B037/00; A63B 57/00 20060101 A63B057/00 |
Claims
1. A method for predicting a golfer's ball striking performance,
comprising: determining a plurality of pre-impact swing properties
for the golfer based on the golfer's swing with a golf club, the
plurality of pre-impact swing properties including an impact
location, an orientation of a golf club head, and the golf club
head speed; determining a plurality of equipment properties
including a plurality of golf ball properties and plurality of golf
club properties, the plurality of golf ball properties including a
coefficient of restitution at a plurality of velocities and a time
of contact at a plurality of velocities, and the plurality of club
properties including a center of mass of the club head, a center of
a club face, and a moment of inertia; determining the slippage
between the golf club and the golf ball based on the plurality of
ball properties, the plurality of club properties, and the
plurality of pre-impact swing properties; and generating a
predicted trajectory and a plurality of predicted ball launch
conditions of the golf ball if struck with the golf club based on
the slippage, the plurality of equipment properties, and the
plurality of pre-impact swing properties.
2. The method according to claim 1, further comprising determining
the properties of a shaft of the golf club on the impact of the
golf ball with the club head, the properties of the shaft including
a longitudinal force component and a torque component.
3. The method according to claim 1, wherein the determining the
slippage comprises determining a first slip period between the golf
club and the golf ball, a stick period between the golf club and
the golf ball, and a second slip period between the golf club and
the golf ball.
4. The method according to claim 3, wherein the determining the
slippage comprises computing each time step in the first slip
period, the stick period, and the second slip period in microsecond
time intervals.
5. The method according to claim 4, wherein the computing each time
step is based on a transverse force of the golf ball, a coefficient
of friction of the golf ball, and a normal force of the golf
ball.
6. The method according to claim 1, wherein the predicted
trajectory includes at least one of distance, flight path, landing
position, and final resting position.
7. The method according to claim 1, wherein the plurality of
predicted ball launch conditions includes at least one of side
spin, back spin, rifle spin, azimuth angle, launch angle, and
velocity.
8. A method for predicting a golfer's ball striking performance,
comprising: determining a plurality of pre-impact swing properties
for the golfer based on the golfer's swing with a golf club, the
plurality of pre-impact swing properties including an impact
location, an orientation of a golf club head, and the golf club
head speed; determining a plurality of equipment properties
including a plurality of golf ball properties and plurality of golf
club properties, the plurality of golf ball properties including a
coefficient of restitution at a plurality of velocities and a time
of contact at a plurality of velocities, and the plurality of club
properties including a center of mass of the club head, a center of
a club face, and a moment of inertia; determining the effect of
properties of a shaft of the golf club on the impact of the golf
ball with the club head, the properties of the shaft including a
longitudinal force component and a torque component; and generating
a predicted trajectory and a plurality of predicted ball launch
conditions of the golf ball if struck with the golf club based on
the properties of the shaft, the plurality of equipment properties,
and the plurality of pre-impact swing properties.
9. The method according to claim 8, further comprising determining
the slippage between the golf club and the golf ball based on the
plurality of ball properties, the plurality of club properties, and
the plurality of pre-impact swing properties.
10. The method according to claim 8, wherein the properties of the
shaft further includes at least one of a shear force, a bending
moment, density, shear modulus, and Young's modulus.
11. The method according to claim 8, wherein the determining the
slippage comprises determining a first slip period between the golf
club and the golf ball, a stick period between the golf club and
the golf ball, and a second slip period between the golf club and
the golf ball.
12. The method according to claim 8, wherein the predicted
trajectory includes at least one of distance, flight path, landing
position, and final resting position.
13. The method according to claim 8, wherein the plurality of
predicted ball launch conditions includes at least one of side
spin, back spin, rifle spin, azimuth angle, launch angle, and
velocity.
14. The method according to claim 11, wherein the determining the
slippage comprises computing each time step in the first slip
period, the stick period, and the second slip period in microsecond
time intervals.
15. The method according to claim 8, further comprising: modifying
at least one of the plurality of equipment properties; generating
another predicted trajectory and another plurality of predicted
ball launch conditions of the golf ball if struck with the golf
club based on the at least one modified equipment property.
16. The method according to claim 15, wherein the plurality of
equipment properties comprises the golf club center of mass, the
golf club weight distribution, the center of the golf club face,
the moment of inertia of the golf club, and the friction
coefficient of the golf club face.
17. The method according to claim 15, wherein the modifying
comprises using one or more different golf balls.
18. A method for predicting a golfer's ball striking performance,
comprising: determining a plurality of pre-impact swing properties
for the golfer based on the golfer's swing with a golf club, the
plurality of pre-impact swing properties including an impact
location, an orientation of a golf club head, and the golf club
head speed; determining a plurality of equipment properties
including a plurality of golf ball properties and plurality of golf
club properties, the plurality of golf ball properties including a
coefficient of restitution at a plurality of speeds, and a time of
contact at a plurality of speeds, and the plurality of club
properties including a center of mass of the club head, a center of
a club face, and a moment of inertia; determining the slippage
between the golf club and the golf ball based on the plurality of
ball properties, the plurality of club properties, and the
plurality of pre-impact swing properties; determining the effect of
properties of a shaft of the golf club on the impact of the golf
ball with the club head, the properties of the shaft including a
longitudinal force component and a torque component; and generating
a predicted trajectory and a plurality of predicted ball launch
conditions of the golf ball if struck with the golf club based on
the slippage, the plurality of equipment properties, and the
plurality of pre-impact swing properties.
19. The method according to claim 18, wherein the determining the
slippage comprises determining a first slip period between the golf
club and the golf ball, a stick period between the golf club and
the golf ball, and a second slip period between the golf club and
the golf ball.
20. The method according to claim 18, wherein the properties of the
shaft further includes at least one of shear force, a bending
moment, density, shear modulus, and Young's modulus.
Description
FIELD OF THE INVENTION
[0001] The present invention relates to a method and computer
program for determining golf ball launch conditions. More
specifically, the present invention relates to a method and
computer program that is capable of predicting golf ball trajectory
and launch conditions without requiring a golfer to strike a golf
ball.
BACKGROUND OF THE INVENTION
[0002] Over the past thirty years, camera acquisition of a golfer's
club movement and ball launch conditions have been patented and
improved upon. An example of one of the earliest high speed imaging
systems, entitled "Golf Club Impact and Golf Ball Monitoring
System," to Sullivan et al., was filed in 1977. This automatic
imaging system employed six cameras to capture pre-impact
conditions of the club and post impact launch conditions of a golf
ball using retroreflective markers. In an attempt to make such a
system portable for outside testing, patents such as U.S. Pat. Nos.
5,471,383 and 5,501,463 to Gobush disclosed a system of two cameras
that could triangulate the location of retroreflective markers
appended to a club or golf ball in motion.
[0003] Systems such as these allowed the kinematics of the club and
ball to be measured. Additionally, systems such as these allowed a
user to compare their performance using a plurality of golf clubs
and balls. Typically, these systems include one or more cameras
that monitor the club, the ball, or both. By monitoring the
kinematics of both the club and the ball, an accurate determination
of the ball trajectory and kinematics can be determined.
[0004] On Jul. 6, 2004, U.S. Pat. No. 6,758,759, entitled "Launch
Monitor System and a Method for Use Thereof," issued. This
application described a method of monitoring both golf clubs and
balls in a single system. This resulted in an improved portable
system that combined the features of the separate systems that had
been disclosed previously. On Dec. 5, 2001, the use of fluorescent
markers in the measurement of golf equipment was disclosed in U.S.
patent application Ser. No. 10/002,174.
[0005] Monitoring both the club and the ball requires complicated
imaging techniques. Additionally, complicated algorithms executed
by powerful processors are now required to accurately and precisely
determine club and ball kinematics. Systems such as these are often
complicated and require significant research and development,
increasing their cost. Despite their ability to monitor a golf club
and ball, these systems are typically unable to quickly determine
which combination of club and balls produces the best outcome for a
particular player. In the past, the only way to accomplish this was
to test a golfer with a variety of different clubs and/or balls,
and then monitor which combination resulted in the most desirable
ball trajectory.
[0006] The need for a mathematical tool for evaluating golf club
performance is dictated by the large number of club design
parameters and initial conditions of the impact between club head
and ball. Without such a tool, it is not feasible to make
quantitative predictions of the effects of a given design change on
the ball motions and shaft stresses.
[0007] For example, in stereo mechanical impact, as described in
U.S. Pat. No. 6,821,209 to Manwaring et al., the final velocities
and spin rates can be related to the initial values of these
quantities without consideration of details of the phenomena that
occur during the short time of contact of the ball and the club,
i.e., about 500 microseconds. However, by eliminating the
consideration of details of the contact between the club and the
ball, the stereo mechanical impact approach includes simplifying
assumptions, which include: (1) that the three components of the
relative velocity of recession of the ball from the club head can
be related to those of the approach of the club to the ball, as
measured at the impact point, by "coefficient of restitution" and;
(2) that the shaft can be considered completely flexible, like a
stretched rubber band, as far as the dynamics of impact are
concerned, so that no dynamic changes occur in the force or torque
that it exerts on the club head during the impact.
[0008] The stereo mechanical approximation problem involves a set
of 12 simultaneous linear algebraic equations in the 12 unknown
components of motion of the ball and club after impact. The known
quantities in these equations are the initial conditions, i.e.,
club head motions and impact point coordinates, and the many
mechanical parameters of the club head and golf ball, e.g., masses,
mass moments of inertia, centers of mass, face loft angle, and face
radii of curvature. The explicit algebraic expressions are
described in U.S. Pat. No. 6,821,209 to Manwaring et al.
[0009] The stereo mechanical approximation has drawbacks, however,
because: (1) the effects of the shaft on the impact, although
small, are not negligible, and it is desirable to obtain
quantitative measures of these effects for shaft design purposes;
(2) shaft stresses cannot be computed in any realistic manner; (3)
the explicit algebraic expressions obtained are still too complex
to permit assessments to be made of the effects of design parameter
changes except by working out many specific cases with the aid of a
computer; and (4) the coefficient of restitution approximation may
not be accurate because the sliding and sticking time of the ball
at the impact point is not taken into account. In addition, the
coefficient of restitution approximation is poor because different
amounts of stress wave energy may be "trapped" in the shaft under
different impact conditions.
[0010] In an effort to improve the accurate modeling of the contact
between the club and the ball, a model published by Ralph Simon,
titled "The Development of a Mathematical Tool for Evaluating Golf
Club Performance," ASME Design Engineering Conference, New York,
May 1967 (pages 17-35) was improved and updated mathematically. In
addition, the modeling may also be implemented by a golf ball model
described in the paper titled "Spin and the Inner Workings of a
Golf Ball," by W. Gobush, 1995, in a book titled Golf the
Scientific Way, Editor A. Cochran, Aston Publishing Group,
Hertfordshire. Both models were shown to give roughly equivalent
results on studies of a golf ball hitting a steel block.
[0011] Therefore, a continuing need exists for a monitoring system
that is capable of determining the trajectory and launch conditions
of a golf ball without requiring a golfer to strike the golf ball.
Moreover, a continuing need exists for a monitoring system that
includes software that reduces the complexity associated with
fitting a golfer with golf equipment. Furthermore, a continuing
need exists for a monitoring system that more accurately predicts a
golfer's ball striking performance.
SUMMARY OF THE INVENTION
[0012] According to one aspect, the present invention comprises a
method for predicting a golfer's ball striking performance. The
method includes determining a plurality of pre-impact swing
properties for a golfer based on the golfer's swing with a golf
club. The plurality of pre-impact swing properties may include, for
example, an impact location, an orientation of a golf club head,
and the golf club head speed.
[0013] The method also includes determining a plurality of
equipment properties that may include a plurality of golf ball
properties and plurality of golf club properties. It is desirable
for the plurality of golf ball properties to include a coefficient
of restitution at a plurality of velocities and a time of contact
at a plurality of velocities. Furthermore, it is desirable for the
plurality of club properties to include a center of mass of the
club head, a center of the club face, and a moment of inertia.
[0014] Additionally, the slippage between the golf club and the
golf ball is preferably determined. The slippage may be based on
the plurality of ball properties, the plurality of club properties,
and the plurality of pre-impact swing properties. The slippage may
be determined by computing each time step, in microsecond time
intervals, for a first slip period, a stick period, and a second
slip period between the golf club and the golf ball. It is desired
that each time step is based on at least a transverse force of the
golf ball, a coefficient of friction of the golf ball, and a normal
force of a golf ball.
[0015] A predicted trajectory and a plurality of predicted ball
launch conditions of the golf ball if struck with the golf club may
then be generated. The predicted trajectory and ball launch
conditions are based on, for example, the slippage, the plurality
of equipment properties, and the plurality of pre-impact swing
properties.
[0016] In one embodiment, the method further comprises determining
the properties of the shaft of the golf club on the impact of the
golf ball with the club head. The properties of the shaft may
include a longitudinal force component and a torque component.
[0017] The predicted trajectory may include at least one of
distance, flight path, landing position, and final resting position
of the golf ball. In addition, the plurality of predicted ball
launch conditions may include at least one of side spin, back spin,
rifle spin, azimuth angle, launch angle, and velocity.
[0018] According to another aspect, the present invention comprises
a method for predicting a golfer's ball striking performance. The
method includes determining a plurality of pre-impact swing
properties for a golfer based on the golfer's swing with a golf
club. The plurality of pre-impact swing properties may include, for
example, an impact location, an orientation of a golf club head,
and the golf club head speed.
[0019] The method also includes determining a plurality of
equipment properties that may include a plurality of golf ball
properties and plurality of golf club properties. It is desirable
for the plurality of golf ball properties to include a coefficient
of restitution at a plurality of velocities and a time of contact
at a plurality of velocities. Furthermore, it is desirable for the
plurality of club properties to include a center of mass of the
club head, a center of the club face, and a moment of inertia.
[0020] In addition, the effect of properties of a shaft of the golf
club on the impact of the golf ball with the club head may be
determined. The properties of the shaft preferably include a
longitudinal force component, a torque component, a shear force, a
bending moment, density, shear modulus, and Young's modulus.
[0021] A predicted trajectory and a plurality of predicted ball
launch conditions of the golf ball if struck with the golf club
based on the properties of the shaft, the plurality of equipment
properties, and the plurality of pre-impact swing properties may
then be generated. In one embodiment, the predicted trajectory
includes at least one of distance, flight path, landing position,
and final resting position. Moreover, the predicted ball launch
conditions include at least one of side spin, back spin, rifle
spin, azimuth angle, launch angle, and velocity.
[0022] In another embodiment, the method further comprises
determining the slippage between the golf club and the golf ball
based on the plurality of ball properties, the plurality of club
properties, and the plurality of pre-impact swing properties. The
slippage may be determined based on a first slip period between the
golf club and the golf ball, a stick period between the golf club
and the golf ball, and a second slip period between the golf club
and the golf ball. Furthermore, the slippage may be determined by
computing each time step in the first slip period, the stick
period, and the second slip period in microsecond time
intervals.
[0023] According to this aspect of the present invention, the
method further comprises modifying at least one of the plurality of
equipment properties and then generating another predicted
trajectory and another plurality of predicted ball launch
conditions of the golf ball if struck with the golf club based on
the at least one modified equipment property. In one embodiment,
the plurality of equipment properties that may be modified
comprises the golf club center of mass, the golf club weight
distribution, the center of the golf club face, the moment of
inertia of the golf club, and the friction coefficient of the golf
club face. Alternately, one or more different golf balls may be
used.
BRIEF DESCRIPTION OF THE DRAWINGS
[0024] Further features and advantages of the invention can be
ascertained from the following detailed description that is
provided in connection with the drawings described below:
[0025] FIG. 1 is a flow chart showing exemplary steps according to
one embodiment of the present invention;
[0026] FIG. 2 is a flow chart showing exemplary steps according to
another embodiment of the present invention;
[0027] FIG. 3 is a diagram showing five different coordinate
systems B, C, D, P, and Q according to one embodiment of the
present invention; and
[0028] FIG. 4 is a diagram showing an exemplary image according to
one embodiment of the present invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0029] The present invention relates to a method and computer
program for determining the velocity components and spin rate
components of a golf ball. By acquiring pre-impact measurements of
the club speed, rotational rate, and ball hit location as input
into a computer program, along with pertinent club features, e.g.,
moment of inertia, and ball features, e.g., normal and transverse
forces, a computer program can predict the resulting trajectory and
launch conditions of the golf ball. One advantage of the present
invention is that the need for a ball monitor to evaluate the ball
launch conditions is eliminated, allowing the analysis to be used
to optimize a club design specific to a particular golfer.
[0030] It is desirable for the method and computer program of the
present invention to be used in combination with a monitoring
system. In one embodiment, the monitoring system may only include
the ability to monitor the kinematics of a club, i.e., pre-impact
swing properties. For example, the present invention may be used
with the monitors disclosed in U.S. patent application Ser. No.
10/759,080 ("the '080 application"), and U.S. Ser. No. 10/770,457
("the '457 application"), the entireties of which are incorporated
herein. As described in these applications, it is possible to
measure club impact data with a single camera. The utilization of
the predictive method of the present invention along with a club
monitoring device such as the ones disclosed by the '080
application and the '457 application provides a relatively
inexpensive monitoring system for fitting a golfer to a specific
club or ball based on their particular swing characteristics.
[0031] According to one aspect, the present invention is capable of
predicting a golfer's ball launch conditions and ball trajectory
without actually monitoring the golf ball. In other words, by
inputting a plurality of characteristics, such as pre-impact swing
properties, club properties, and ball properties, the present
invention predicts a ball's launch conditions and trajectory
without requiring a golfer to actually strike a ball.
[0032] FIG. 1 is a flow chart showing exemplary steps according to
one embodiment of the present invention. As shown in FIG. 1, in one
embodiment a golfer's ball trajectory and ball launch conditions
may be generated by determining a plurality of pre-impact swing
properties for a golfer based on the golfer's swing of a golf club.
In this embodiment, a golfer swings a golf club and the golfer's
swing properties are determined by using any monitoring device
known to those skilled in the art. The pre-impact swing properties
may include, but are not limited to, impact location, angular
velocity, linear velocity, orientation of the golf club head, club
head speed, club head rotational rate, and the like.
[0033] In one embodiment, the pre-impact swing properties may be
determined by having a golfer swing a golf club in front of a
monitoring system. The golfer may swing the club any desired number
of times in order to generate accurate pre-impact swing properties.
In other words, it may be desirable to take an average of a
predetermined number of swings in order to obtain pre-impact swing
properties that are substantially free of deviations caused by
human error during a swing. Typically, the standard deviation of
the kinematics of a golfer's swing does not change after about 10
swings. Thus, the pre-impact swing properties are preferably based
on about 1 or more swings of a golf club. More preferably, the
pre-impact swing properties are based on about 10 or more swings of
a golf club. Most preferably, the pre-impact swing properties are
based on about 30 or more swings of a golf club. In another
embodiment, the pre-impact swing properties are preferably based on
between about 1 and about 30 swings of a golf club.
[0034] It is also desirable to determine a plurality of equipment
properties, e.g., golf club properties and golf ball properties.
The golf club and golf ball properties may be determined according
to any equipment or method known to those skilled in the art.
Preferably, the golf ball properties that are determined include,
but are not limited to, the coefficient of restitution of the ball
at a plurality of velocities, the time of contact at a plurality of
velocities, and the spin at a plurality of velocities and loft
angles.
[0035] Additionally, the golf club properties that may be
determined include, but are not limited to, the geometric center of
the club face, the center of mass of the club head, the distance
from the hosel to the center of mass of the club face and/or the
center of mass of the club head, effective density of shaft
material, the effective shear modulus for torsion about the shaft
axis, the effective Young's modulus for the shaft material, and the
outer and inner diameters of the shaft in two directions at the
hosel end.
[0036] Thus, a golfer is only required to swing a golf club once to
determine a predicted trajectory of the golf ball and the ball
launch conditions. The predicted trajectory may include
characteristics such as distance, flight path, landing position,
final resting position, and the like. Moreover, the ball launch
conditions may include the side spin, back spin, rifle spin,
azimuth angle, launch angle, velocity, and the like.
[0037] In another embodiment, the method described above may be
performed using a computer program comprising computer
instructions. Any computer language and/or compiler may be used to
create the computer program, as will be appreciated by those
skilled in the art. Furthermore, the computer instructions may be
executed using any computing device. The computing device
preferably includes at least one of a processor, memory, display,
input device, output device, and the like. Moreover, the computer
instructions may be stored on any computer readable medium, e.g., a
magnetic memory, read only memory (ROM), random access memory
(RAM), disk, optical device, tape, or other analog or digital
device known to those skilled in the art.
[0038] In determining or predicting the trajectory and ball launch
conditions, the present invention provides the advantage of
specifically accounting for slippage between the golf club and the
golf ball. When a golf club strikes a golf ball, there is a first
slip period that occurs, i.e., the friction forces between the club
and the ball do not prevent motion between the two. Shortly after
the initial club-ball impact, as the club and the ball deform
slightly, friction forces cause the golf club and the golf ball to
stick together, resulting in a stick period. During the stick
period, the golf club and golf ball are locked together, and there
is a substantially small amount of relative motion between the two
objects. As the club and the ball begin to return to their original
shapes, the golf ball and club undergo a second slip period. In a
manner similar to the first slip period, the golf club and ball
once again experience motion relative to one another.
[0039] As a skilled artisan will recognize, the trajectory of the
ball is significantly affected by the slippage between the golf
club and the golf ball, i.e., the first and second slip periods and
the stick period. Accordingly, the present invention accounts for
the slip and stick periods by integrating Newtonian equations that
account for the duration of the slippage at each time step in the
collision process, as described in more detail below. Thus, the
present invention computes each time step in the collision process
between the club and the ball in microsecond time intervals. As a
result of the Hertzian theory of deformation, the time of contact
varies inversely with the 1/5 power of velocity, i.e., the time of
contact between the club and the ball varies as the -0.2 power of
the relative velocity. Thus, in one embodiment the time of contact
may vary between about 300 and about 700 microseconds.
[0040] Another advantage of the present invention relates to
accounting for the effects of the club shaft on the trajectory and
ball launch conditions. As described above, prior art monitoring
systems have used methods or computer programs that assume that the
shaft can be considered completely flexible, like a stretched
rubber band, as far as the dynamics of impact are concerned, so
that no dynamic changes occur in the force or torque that it exerts
on the club head during the impact. However, although the effects
of the shaft on the impact are small, they are not negligible.
Therefore, it is desirable to measure these effects to determine
how they affect the kinematics of a golf ball. The method for
determining these effects, and the calculations involved, are
described in more detail below.
[0041] According to another aspect, the present invention may be
used to assist in golf club design, as illustrated by the exemplary
steps in the flow chart shown in FIG. 2. For instance, in one
embodiment the ball properties and club properties may be input
into the computer program of the present invention. Pre-impact
swing properties may also be input, although the source of the
pre-impact swing properties may vary. In one embodiment, the
pre-impact swing properties may be from a golfer. However in
another embodiment, a machine or mechanical device may be used to
swing a golf club.
[0042] After the pre-impact swing properties have been input, the
present invention may be used to determine a predicted trajectory
and predicted ball launch conditions. In embodiments where a
mechanical device is used to swing a golf club, the club properties
may be held constant and the ball properties may be varied to
determine which of a plurality of golf balls results in an optimal
trajectory and ball launch conditions for the given club.
Conversely, the ball properties may be held constant, and the club
properties e.g., center of mass, moment of inertia, and friction
coefficient, may be manipulated to determine the optimal design for
a club to achieve a desired trajectory and ball launch conditions.
Alternately, the geometric center and/or center of mass of the club
may be varied in order to design a club that is more forgiving,
i.e., is able to achieve a desired trajectory even when a ball
strikes near, for example, the toe or heel of the club head. As
will be appreciated by those skilled in the art, this embodiment of
the present invention may be useful with, for example, golf club
and/or golf ball design and manufacturing.
[0043] In an embodiment where a player swings a club, the present
invention may be used to vary club properties, e.g., weight
distribution, center of mass, and moment of inertia, to design a
club that results in the optimum trajectory and ball launch
conditions for a particular player's swing. Alternately, the club
properties may be held constant, and the ball properties may be
varied for a plurality of different balls. This allows a player to
determine which of a plurality of balls results in an optimal
trajectory and ball launch conditions, all without actually
striking a golf ball.
[0044] As described herein, the formulation of a useful
mathematical tool for evaluating golf club performance necessitates
making step-by-step calculations of the detailed changes in time of
the forces and motions at various points in the golf ball/golf club
system during the course of impact. The analytical formulations for
these calculations and a brief description of the calculations are
discussed below after the input information is described.
Club Analysis Input Data
[0045] According to one aspect of the present invention, the
predictive model of impact between the club and the ball requires
the different body coordinate systems involved in the computer
analysis to be defined. In FIG. 3, the five coordinate systems
describing the position of the club and ball at the instant of
impact are shown.
Initial Variables Before Impact
[0046] The first set of variables are CYOB, CZOB, which are the y
and z coordinates of the initial ball contact point on the club
face as measured from a coordinate system designated as C with its
origin at the center of the clubface. The x-axis is perpendicular
to the tangent plane of the club face and the z-axis is parallel to
the inscribed lines on the club face directed positively toward the
toe of the club. Additionally, the y-axis forms the right handed
system.
[0047] The second set of variables are DXVOH, DYVOH, DZVOH, DXSOH,
DYSOH, DZSOH. These are the velocity and spin components of the
club at impact as measured at the D-system origin shown in FIG. 3.
The y-axis of this system is directed positively upward along the
shaft axis and the y, z plane of the D-system is taken to be
parallel to the z-axis of the C-system, thus defining the D-system
z-axis. Moreover, the x-axis is chosen to form a right handed
system.
[0048] The third set of variables are DXFOH, DYFOH, DZFOH, DXLOH,
DYLOH, DZLOH. These variables represent, for example, the three
initial components of force and torque at the hosel end of the
club, just prior to impact. The forces and torques are about an
order of magnitude less than the forces applied by the club head to
the ball, as explained in more detail below with respect to the
shaft channel. The three initial components of force and the three
initial components of torque exerted by the shaft on the hosel end
of the club head are experimentally determinable by measuring
extension, shear, torsion, and bending strains in the shaft at its
hosel end as a golfer swings the club.
[0049] The fourth set of variables are CXOQ, CYOQ, CZOQ, which are
the three components of the Q-system origin and represent the
center of the club head with respect to the C-system in C-system
coordinates. In one embodiment, at impact, the Q-system is
initially the coordinate system in which the camera makes its
measurements.
[0050] The fifth set of variables is the matrix TMQC, TMQD, which
represent the rotational transform matrices between the coordinate
system Q and the C and D body systems, respectively.
Club Head Characteristics
[0051] In one embodiment, the input should also include the
constants that describe the club and its frictional properties.
Moreover, the ball properties are also needed. The variables are
denoted as follows: W.sub.h, CHIXX, CHIYY, CHIXY, CHIXZ, CHIYZ.
These variables represent, for example, the weight of the club head
and the inertia matrix in the C-system coordinates. The clubface
curvature in the y and z directions and friction constants for
sliding perpendicular and parallel to the inscribed lines on the
clubface as measured in the C-system are represented by the
variables Cury, Curz, Cfry, Cfrz.
Shaft Characteristics
[0052] The input accuracy of variables of the shaft may have a
smaller impact than other club variables since the greatest
computed value of shaft force obtained for the maximum impact
condition is about 300 lbs. In one embodiment, the peak value of
the force between the club head and the ball is about 3500 pounds
for any position on the clubface for a typical driver impact.
Furthermore, the relative influence of these former forces on the
ball velocities is much less than the ratio of 300 lbs to 3500
lbs.
[0053] The following input shaft characteristics are RHOS, GS, ESX,
ESY, ESZ, which represent the effective density of shaft material,
the effective shear modulus for torsion about the shaft axis, and
the three values of the effective Young's modulus for the shaft
material, i.e., for tension and bending in two directions. The
other shaft characteristics are DXODS, DXIDS, DZODS, DZIDS, which
represent the outer and inner diameters of the shaft in two
directions at the hosel end.
GolfBall Characteristics
[0054] Due to the complex nature of the material make-up of a golf
ball, the force deformation equation is based on parameters of the
Hertzian theory. In one embodiment, these parameters may be
determined by, for example, measuring the contact time and
coefficient of restitution. In another embodiment, the parameters
of the force law may be determined by using the finite element
solution using the basic material constants. An exemplary model
that may be used is based on the following equation:
F(X)=KN(X/a).sup.1.5(1+.alpha.(VN/a)) where:
[0055] X is the ball deformation in a direction normal to the
applied force at an instant of time during contact;
[0056] KN=the normal force constant;
[0057] a is the radius of the ball; and
[0058] .alpha. is a damping constant that varies with the inverse
of the normal velocity of the deformation, VN, and is given by the
equation .alpha. = .alpha. 1 + ( .alpha. 2 VN ) . ##EQU1##
[0059] Typical values for a two piece constructed golf ball are,
for example, KN=20616 Lbs, a=0.84inches, .alpha..sub.1=0.000123,
and .alpha..sub.2=0.221.
[0060] The transverse force deformation in the Y and Z plane of
contact are given by the force equations, for example:
F(Y)=KT(X/a).sup.0.5(Y/a)(1+AT(Y/a).sup.2)
F(Z)=KT(X/a).sup.0.5(Z/a)(1+AT(Z/a).sup.2) where:
[0061] KT=the transverse force constant.
[0062] For an isotropic surface, the parameters measured for a two
piece ball are, for example, KT=70000 pounds, a=0.84, and AT=1100.
A coefficient of friction to account for the slipping period at the
beginning and end of impact is also desirable. On a typical steel
surface, this coefficient varies between about 0.2 and about
0.3.
[0063] In one embodiment, these forces may be measured with
transducers, as described in the article "Impact Measurements on
Golf Balls," pages 219-224 in Science and Golf, edited by A. J.
Cochran, published by E. and F. N. Spoon, London, 1990.
Alternately, a second method of measuring the normal force is
measuring the time of contact and coefficient of restitution as
described in U.S. Pat. No. 6,571,600 to Bissonnette et al., the
entirety of which is incorporated herein.
[0064] According to the method of the present invention, the
parameters in the equation describing normal force may be fitted by
a nonlinear least squares method by measuring the coefficient of
restitution and contact time at a measured series of impact
velocities. The parameters of the transverse force may be
determined, for example, by measuring the spin rate of different
balls striking a lofted steel block at a series of angles and
speeds in addition to the use of piezoelectric force transducers
described above.
[0065] Over a period of ten years, several books entitled Science
and Golf were written. The books describe typical methods for
measuring forces with transducers, as described in "Impact
Measurements on Golf Balls", pages 219-224 in Science and Golf, by
A. J. Cochran, published by E. and F. N. Spoon, London, 1990.
Experimental methods for measuring the coefficient of sliding
friction are described in "Experimental Determination of Golf Ball
Coefficients of Sliding Friction", pages 510-518 in Science and
Golf, edited by M. R. Farally and A. J. Cochran, published by Human
Kinetics, 1999. Also, measurements are discussed in a paper titled
"Friction coefficient of golf balls", by Gobush, 1996 in the
Engineering of Sport, Editor Haake, Blackwell Science, Oxford. A
finite element analysis can also be employed to model the normal
and transverse forces. A description of this method is described in
"Use of Finite Element Analysis in Design of Multilayer Golf
Balls", pages 473-480 in Science and Golf, edited by M. R. Farally
and A. J. Cochran, published by Human Kinetics, 1999. Any of these
known methods may be used in combination with the present
invention.
Equations of Motion
[0066] Exemplary explicit equation of motions for the ball and club
head are represented below by four vector equations. For example,
the ball's motion during impact is represented by the following
equations: (W.sub.b/g)(dV.sub.b/dt)=F.sub.b
(d/dt)(Ib.times..omega.b)/g=r.sub.b.times.F.sub.b
[0067] In one embodiment, the club head motion may be represented
by, for example, the following equations:
(W.sub.h/g)(dV.sub.h/dt)=-F.sub.b-F.sub.s
(d/dt)(Ih.times..omega.h)/g=-r.sub.hb.times.F.sub.b-r.sub.hs.times.F.sub.-
s-L.sub.s
[0068] In the above equations W.sub.b and W.sub.h are the weights
of the ball and club head, respectively. Ib/g and Ih/g are the
respective mass moments of inertia of the ball and the club head
respectively. Additionally, V.sub.b and V.sub.h are the center of
mass velocity vector of the ball and club head, respectively, and
.omega.b and .omega.h are the angular velocity vector of the ball
and club head, respectively. Moreover, the quantity F.sub.b is the
instantaneous vector force exerted by the club head on the ball;
F.sub.s and L.sub.s are the instantaneous vector force and torque,
respectively, exerted by the club head on the hosel end of the
shaft; r.sub.b and r.sub.hb are the vector positions of the impact
point from the center of mass of the ball and from the center of
mass of the club head, respectively; and r.sub.hs is the vector
position of the hosel end of the shaft center line, as measured
from the center of mass of the club head.
Description of Shaft Model for Forces and Torque Exerted on a Club
Head
[0069] According to one aspect of the present invention, a finite
element model of shaft dynamics requires solving eight differential
equations for each 1-inch segment of the shaft since two
integrations with respect to time are required for each of the
longitudinal displacement wave, the torsional displacement wave,
and the bending waves, in the two perpendicular directions. This
makes 304 equations for a 38 inch shaft, as compared with 12
equations for the club head and 12 equations for the ball. Using
304 equations may be necessary to obtain a detailed determination
of the stresses developed by the shaft for the purposes of shaft
design, however this requires complicated calculations. According
to one embodiment of the present invention, it is possible to
simplify the representation of the shaft as the input impedances of
infinitely long mechanical transmission lines for tension, torsion,
and bending waves. In one embodiment, this representation is an
accurate approximation because no significant wave amplitudes are
reflected back to the hosel end of the shaft before termination of
the contact between the golf ball and the club face.
[0070] In this embodiment, the shaft is assumed to be infinitely
long and to have a constant cross section along its length. Both of
these assumptions are sufficient for the purposes of analyzing the
effects of the shaft on the golf ball impact with the head. This is
because the ball does not remain in contact with the club head long
enough for any appreciable amount of stress wave energy in the
shaft to return to its hosel end after reflection at the grip end.
Furthermore, the shaft cross-section is usually uniform for a few
inches above the hosel and then tapers gradually enough to avoid
any appreciable back reflections of stress wave energy along the
length of the shaft (tapering in small steps would have about the
same effect as continuous tapering for the range of wavelengths of
the stress waves of interest).
[0071] Under the assumption of an infinite and uniform shaft,
rectilinear motion of the hosel end of the club head in the
direction of the shaft axis generates a compressive stress wave in
the shaft whose stress amplitude is proportional to the velocity of
this motion. Similarly, torsional motion of the hosel end of the
clubhead about the shaft axis generates a torsional stress wave
whose stress amplitude is proportional to the angular velocity of
this torsional motion. The electrical analogue of these mechanical
motions is that of the input impedance of an effectively long
uniform electrical transmission line, for which the voltage is in
phase with and proportional to the current, i.e., a pure
resistance.
[0072] For motions of the hosel end of the club head perpendicular
to the shaft axis, the analysis is more complicated. For these
motions, it is necessary to solve the standard equation for the
bending of a shaft with superimposed tension, which is a fourth
order partial differential equation in space and second order in
time. It is desirable for the bending wave equation to be solved
twice for each time step, since there are two perpendicular
directions of transverse motion. The boundary conditions for each
direction are the known displacements and known changes in angular
orientation of the hosel end of the club head; these quantities are
known from the solutions of the club head equations of motion. The
solutions of the bending wave equations yield values for the
bending moments and for the shear forces on the shaft at its hosel
end. The negative of these forces and moments are preferably
entered into the club head equations, in combination with the
negatives of the forces and moments exerted on the golf ball.
[0073] In this embodiment, the longitudinal force component in the
shaft axis direction on the hosel end, F.sub.sy, is set equal to
the appropriate input impedance multiplied by the longitudinal
velocity, DVSY. This is represented by, for example: F sy = A
.function. ( ( ESY .times. RHOS ) ( g ) ) .times. ( DVSY ) ##EQU2##
where:
[0074] A is the cross-sectional area of the shaft and is equal to
.pi.((DXODS)(DZODS)-(DXIDS)(DZIDS))/4;
[0075] ESY is Young's modulus for tension and bending in three
directions; and
[0076] RHOS is the shaft material density. The torque component
along the shaft axis is represented by, for example, the equation:
L sy = DYIS .function. ( ( GS .times. RHOS ) ( g ) ) .times. ( DWSY
) ##EQU3## where:
[0077] DWSY is the angular velocity at the hosel end of the
shaft;
[0078] GS is the shear modulus; and
[0079] DYIS is the area moment of inertia of the shaft cross
section about the shaft axis.
[0080] Additionally, the partial differential equation for the
propagation of the bending waves for a uniform shaft beginning at
the hosel end (y=0) and extending indefinitely in the +y direction
for a deflection in the z direction may be represented, for
example, by the equations: E .times. I .times. .differential. 4
.times. Z .differential. y 4 - T .times. .differential. 2 .times. Z
.differential. y 2 + .rho. .times. .times. A g .times.
.differential. 2 .times. Z .differential. t 2 = 0 ##EQU4##
.differential. Z .function. ( y , 0 ) .differential. t = 0
##EQU4.2## and ##EQU4.3## Z .function. ( y , 0 ) = 0 ##EQU4.4##
where:
[0081] E is Young's modulus;
[0082] I is the section area of moment of inertia about the
x-axis;
[0083] T is the tensile force along the y-axis;
[0084] .rho. is shaft density;
[0085] A is the cross section area; and
[0086] g is the acceleration of gravity.
[0087] The force T is the sum of the centrifugal tensile force
resulting from the swing just prior to impact and the dynamic force
at y associated with the longitudinal waves generated by
impact.
[0088] After the bending wave equation is solved by using Laplace
transforms, the shear force and bending moment components at y=0
may be determined using, for example, the equations: F sz
.function. ( t ) = E sx .times. I sx .function. ( 0 ) .times.
.differential. 3 .times. Z .function. ( 0 , t ) .differential. y 3
##EQU5## L sx .function. ( t ) = E sx .times. I sx .function. ( 0 )
.times. .differential. 2 .times. Z .function. ( 0 , t )
.differential. y 2 ##EQU5.2## A similar solution for bending in the
x direction results in the quantities F.sub.sx(t) and L.sub.sz(t)
for use in the equations of motion. Program Specific
Occurrences
[0089] As shown in FIG. 3, five different coordinate systems
labeled B, C, D, P, and Q are used according to one embodiment of
the present invention. The B, C, and D systems are described above
in the input section. The origin of the Q-system, OQ, is at the
center of mass of the club head and the initial origin of the
P-system is also at this point. Moreover, the QZ axis is parallel
to the CZ axis and the QX axis makes an angle equal to the negative
of the loft angle with the CX axis, and the QY axis makes an angle
equal to the complement of the lie angle with the DY axis.
[0090] In one embodiment, the orientation in space of the P
coordinate system is determined by the angular position of the club
head at the instant of initial contact with the golf ball.
Thereafter, the P-system remains fixed in this orientation. All
changes in positions and orientations are accumulated in the
P-system because the P-system is a fixed system in space.
[0091] In addition, the Q-system coincides with the P-system at the
instant of initial contact with the ball, but thereafter follows
the club head in a stepwise fashion as the position and orientation
of the club head changes during the time of contact with the ball.
At the beginning of each integration time interval, the origin of
the Q-system is at the center of mass of the club head and the axes
are oriented with respect to the club head, as described above. The
Q-system then remains temporarily fixed in space while the
integration needed to determine the changes in positions and
orientation of the club head and the ball over the time interval is
computed. When the integrations are successfully completed, the
Q-system is transposed to the new position and orientation of the
club head and all dependent variables converted to the new Q-system
in preparation for the next integration step. Thus, the moment of
inertia tensor in the Q-system does not have to be recomputed each
time but the effects of its rate of change must be taken into
account.
[0092] In one embodiment, the small angle transformation theory is
sufficiently accurate for transformations between successive
orientations of the Q-system, transformations between the P and the
Q-systems, and between the B and the C-systems. The maximum
orientation differences involved are only about a few degrees.
[0093] According to one aspect of the present invention, the
effects of friction on the club face may be computed as follows.
For each time interval, the normal and transverse components of the
ball force are computed as described above. As long as the absolute
value of each transverse force component does not exceed the
coefficient of friction for that direction multiplied by the normal
force component, the ball rolls without sliding. If a given
transverse force becomes excessive, the ball contact point slides
in such a direction as to decrease the transverse distortion just
enough to remove the excess transverse force. The transverse force
component is thus either less than or about equal to, in absolute
magnitude, the applicable coefficient of friction times the normal
force. There may be two friction coefficients such as Cfry for
motion perpendicular to the inscribed lines, and Cfrz for motion
parallel to these lines.
[0094] In one embodiment, the position of the B-system origin is
changed at the end of each integration time step in accordance with
the amounts of rolling and sliding of the ball on the club
face.
General Computational Procedure for Each Time Interval
[0095] In one embodiment, it is desirable to solve for the motions
of the club head as subjected to the steady forces and torques on
the hosel end of the head by the shaft (actually by the golfer
through the medium of the shaft). The steady motions at the hosel
end are subtracted from the total motions there to obtain the
motion inputs to a computer program subroutine, referred to
hereinafter as the SHAFT subroutine. The SHAFT subroutine outputs
the dynamic forces and torques exerted on the shaft. The difference
between the total motion of the club head at the ball contact point
and the motion of the ball are used to compute the three components
of the force on the ball distortion, from which the three
components of the force on the ball are obtained. These forces and
motions may be corrected for sliding as previously explained. The
linear and angular accelerations of the club head are then
determined from the resultant forces and torques on the head. These
are the steady forces and torques on the club head minus the
dynamic forces and torques on the ball, minus the dynamic forces
and torques on the shaft. This leads to a computation of the total
motions at the hosel end of the club head, at which point the
computation is repeated. Preferably, at least about two iterations
of the process are performed, and additional ones as necessary,
until successive results agree within a predetermined error limit.
More preferably, at least about three iterations of the process are
performed, and most preferably at least about five iterations are
performed.
[0096] According to one aspect, if this criterion cannot be met
after ten iterations, the process is repeated and the time step
halved. Completion of a successful integration over a given time
interval is followed by the transformations previously described in
preparation for integration over the following time interval.
EXAMPLES
[0097] In order to determine the accuracy of the present invention,
the calculated data was compared with actual data. Accordingly,
four golfers hit ten shots with a driver club that was measured on
a ball club monitor using the method of photogrammetry, an example
of which is described in U.S. Pat. No. 6,758,759, the entirety of
which is incorporated herein. Each club included about 5 or 6 marks
for analyzing the correspondence between adjacent cameras so that
accurate triangulation of marked positions on the club in space
could be determined. Furthermore, each ball included 12 circular
marks to measure its position after and before impact to determine
ball cub head hit position and ball speed, launch angle, and spin
rate. An exemplary image acquired according to this setup is shown
in FIG. 4. As shown in the table below, the predicted ball velocity
was accurate to within about 1 foot per second of the actual ball
velocity, on average. Additionally, the predicted launch angle and
side angle were accurate to within about a half a degree. The
predicted and actual spin rate was also accurate, as shown below.
TABLE-US-00001 Ball velocity Launch angle side angle (fps)
(degrees) (degrees) model measured model measured Model measured
Golfer 1 224 225 10.3 10.1 0.5 1.3 Golfer 2 238 237 8.1 7.3 -0.2
-0.3 Golfer 3 224 222 7.7 8.7 -4.7 -4.7 Golfer 4 223 223 6.95 8.8
2.33 3.1 model experiment model experiment model experiment average
227.25 226.75 8.2625 8.725 -0.5175 -0.15 Std. 7.182154 6.946222
1.439546 1.144188 2.985391 3.336165 backspin sidespin roll spin
Model (rpm) measured Model (rpm) measured Model (rpm) measured
Golfer 1 2678 2706 29 328 9 44 Golfer 2 2638 3460 115 755 139 143
Golfer 3 1680 2560 194 -258 103 200 Golfer 4 2470 3310 229 280 130
208 model experiment model experiment model experiment average
2366.5 3009 141.75 276.25 95.25 148.75 Std. 466.4544 442.4975
89.01451 415.2577 59.5 75.59266
[0098] Although the present invention has been described with
reference to particular embodiments, it will be understood to those
skilled in the art that the invention is capable of a variety of
alternative embodiments within the spirit of the appended
claims.
* * * * *