U.S. patent application number 16/522703 was filed with the patent office on 2019-11-14 for golf ball dimple plan shape.
This patent application is currently assigned to Acushnet Company. The applicant listed for this patent is Acushnet Company. Invention is credited to Michael R. Madson, Nicholas M. Nardacci.
Application Number | 20190344124 16/522703 |
Document ID | / |
Family ID | 68463790 |
Filed Date | 2019-11-14 |
![](/patent/app/20190344124/US20190344124A1-20191114-D00000.png)
![](/patent/app/20190344124/US20190344124A1-20191114-D00001.png)
![](/patent/app/20190344124/US20190344124A1-20191114-D00002.png)
![](/patent/app/20190344124/US20190344124A1-20191114-D00003.png)
![](/patent/app/20190344124/US20190344124A1-20191114-D00004.png)
![](/patent/app/20190344124/US20190344124A1-20191114-D00005.png)
![](/patent/app/20190344124/US20190344124A1-20191114-D00006.png)
![](/patent/app/20190344124/US20190344124A1-20191114-D00007.png)
![](/patent/app/20190344124/US20190344124A1-20191114-D00008.png)
![](/patent/app/20190344124/US20190344124A1-20191114-D00009.png)
![](/patent/app/20190344124/US20190344124A1-20191114-D00010.png)
View All Diagrams
United States Patent
Application |
20190344124 |
Kind Code |
A1 |
Madson; Michael R. ; et
al. |
November 14, 2019 |
GOLF BALL DIMPLE PLAN SHAPE
Abstract
The present invention relates to golf balls having improved
packing efficiency and aerodynamic characteristics and a high
degree of dimple interdigitation. In particular, the present
invention relates to a golf ball including at least a portion of
dimples having a plan shape defined by low frequency periodic
functions having high amplitudes. The present invention is also
directed to methods of developing the dimple plan shape geometries,
as well as methods of making the finished golf balls with the
inventive dimple patterns applied thereto.
Inventors: |
Madson; Michael R.; (Easton,
MA) ; Nardacci; Nicholas M.; (Barrington,
RI) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
Acushnet Company |
Fairhaven |
MA |
US |
|
|
Assignee: |
Acushnet Company
Fairhaven
MA
|
Family ID: |
68463790 |
Appl. No.: |
16/522703 |
Filed: |
July 26, 2019 |
Related U.S. Patent Documents
|
|
|
|
|
|
Application
Number |
Filing Date |
Patent Number |
|
|
15953785 |
Apr 16, 2018 |
10369418 |
|
|
16522703 |
|
|
|
|
15228502 |
Aug 4, 2016 |
9943728 |
|
|
15953785 |
|
|
|
|
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
A63B 37/0074 20130101;
A63B 37/08 20130101; A63B 37/0017 20130101; A63B 37/0007 20130101;
A63B 2037/087 20130101; A63B 37/0019 20130101; A63B 37/0006
20130101; A63B 37/0012 20130101; A63B 37/002 20130101; A63B 37/0021
20130101 |
International
Class: |
A63B 37/00 20060101
A63B037/00 |
Claims
1. A golf ball having a generally spherical surface and comprising
a plurality of dimples on the spherical surface, wherein at least a
portion of the plurality of dimples have a non-circular plan shape
defined by a periodic function mapped along a circular closed path,
wherein the periodic function is selected from a periodic square
wave function defined by the equation: f ( x ) = s + a .pi. ( sin (
.pi. px ) + sin ( 3 .pi. px ) 3 + sin ( 5 .pi. px ) 5 + sin ( 7
.pi. px ) 7 ) ##EQU00019## and a periodic sawtooth function defined
by the equation: f ( x ) = s - a .pi. ( sin ( .pi. px ) + sin ( 2
.pi. px ) 2 + sin ( 3 .pi. px ) 3 + sin ( 4 .pi. px ) 4 )
##EQU00020## where s is the sharpness factor of the periodic
function and is defined by a constant value of from 5 to 60; a is
the amplitude of the periodic function and is from 0.1 to 1; and p
is the period of the periodic function and is from 3 to 6; wherein
the maximum distance of any point on the plan shape from the
circular closed path is about 0.015 inches to about 0.050 inches;
and wherein the surface of said dimples having a non-circular plan
shape includes a protruding center portion.
2. The golf ball of claim 1, wherein the protruding center portion
lies below the nominal chord plane of the dimple.
3. The golf ball of claim 1, wherein each one of the dimples having
said non-circular plan shape comprises a first dimple profile and a
second dimple profile, the first dimple profile having a first edge
angle (.PHI..sub.EDGE1) and a second edge angle (.PHI..sub.EDGE2)
and the second dimple profile having a third edge angle
(.PHI..sub.EDGE3) and a fourth edge angle (.PHI..sub.EDGE4),
wherein at least two of .PHI..sub.EDGE1, .PHI..sub.EDGE2,
.PHI..sub.EDGE3 and .PHI..sub.EDGE4 have different values.
4. The golf ball of claim 3, wherein at least two of
.PHI..sub.EDGE1, .PHI..sub.EDGE2, .PHI..sub.EDGE3 and
.PHI..sub.EDGE4 have values that differ by 1.0.degree. to
6.0.degree..
5. The golf ball of claim 3, wherein:
.PHI..sub.EDGE1=.PHI..sub.EDGE2, .PHI..sub.EDGE3=.PHI..sub.EDGE4,
and .PHI..sub.EDGE1.noteq..PHI..sub.EDGE3.
6. The golf ball of claim 3, wherein, for each one of the dimples
having said non-circular plan shape, the average of the edge angles
of all of the dimple profiles of that dimple is 11.degree. to
16.degree..
7. The golf ball of claim 3, wherein, for each one of the dimples
having said non-circular plan shape, the difference between the
maximum edge angle and the minimum edge angle of all of the dimple
profiles of that dimple is 0.5.degree. to 6.0.degree..
8. The golf ball of claim 3, wherein, for each one of the dimples
having said non-circular plan shape, the difference between the
maximum edge angle and the average of the edge angles of all of the
dimple profiles of that dimple is 3.00.degree. or less.
9. The golf ball of claim 1, wherein the plurality of dimples
having said non-circular plan shape have a Degree of
Interdigitation DOI of about 0.05 to about 0.50, DOI being defined
by the following equation: DOI = 1 n k = 1 n ( R 1 + R 2 D - 1 ) k
##EQU00021## where R.sub.1 is the maximum radial distance of a
first dimple having a first center, R.sub.2 is the maximum radial
distance of a second dimple having a second center, D is the
distance between the first center and the second center, and the
first dimple is adjacent to the second dimple on the surface.
10. The golf ball of claim 1, wherein 50 percent or more of the
dimples on the golf ball are dimples having said non-circular plan
shape.
11. The golf ball of claim 1, wherein 80 percent or more of the
dimples on the golf ball are dimples having said non-circular plan
shape.
12. The golf ball of claim 1, wherein the circular path has a
radius of from 0.025 inches to 0.150 inches.
Description
CROSS-REFERENCE TO RELATED APPLICATIONS
[0001] This application is a continuation-in-part of U.S. patent
application Ser. No. 15/953,785, filed Apr. 16, 2018, which is a
continuation-in-part of U.S. patent application Ser. No.
15/228,502, filed Aug. 4, 2016, now U.S. Pat. No. 9,943,728, the
entire disclosures of which are hereby incorporated herein by
reference.
FIELD OF THE INVENTION
[0002] The present invention relates to golf balls having improved
packing efficiency and aerodynamic characteristics and a high
degree of dimple interdigitation. The improved characteristics are
obtained through the use of specific dimple arrangements and dimple
plan shapes. In particular, the present invention relates to a golf
ball including at least a portion of dimples having a plan shape
defined by low frequency periodic functions having high
amplitudes.
BACKGROUND OF THE INVENTION
[0003] Aerodynamic forces acting on a golf ball are typically
resolved into orthogonal components of lift (F.sub.L) and drag
(F.sub.D). Lift is defined as the aerodynamic force component
acting perpendicular to the flight path. It results from a
difference in pressure that is created by a distortion in the air
flow that results from the back spin of the ball. Due to the back
spin, the top of the ball moves with the air flow, which delays the
separation to a point further aft. Conversely, the bottom of the
ball moves against the air flow, moving the separation point
forward. This asymmetrical separation creates an arch in the flow
pattern, requiring the air over the top of the ball to move faster,
and thus have lower pressure than the air underneath the ball.
[0004] Drag is defined as the aerodynamic force component acting
opposite to the ball flight direction. As the ball travels through
the air, the air surrounding the ball has different velocities and,
thus, different pressures. The air exerts maximum pressure at the
stagnation point on the front of the ball. The air then flows over
the sides of the ball and has increased velocity and reduced
pressure. The air separates from the surface of the ball, leaving a
large turbulent flow area with low pressure, i.e., the wake. The
difference between the high pressure in front of the ball and the
low pressure behind the ball reduces the ball speed and acts as the
primary source of drag.
[0005] Lift and drag, among other aerodynamic characteristics of a
golf ball are influenced by the external surface geometry of the
ball, which includes the dimples thereon. As such, the dimples on a
golf ball play an important role in controlling those parameters.
For example, the dimples on a golf ball create a turbulent boundary
layer around the ball, i.e., the air in a thin layer adjacent to
the ball flows in a turbulent manner. The turbulence energizes the
boundary layer and helps it stay attached further around the ball
to reduce the area of the wake. This greatly increases the pressure
behind the ball and substantially reduces the drag.
[0006] The design variables associated with the external surface
geometry of a golf ball, e.g., surface coverage, dimple pattern,
and individual dimple geometries, provide golf ball manufacturers
the ability to control and optimize ball flight. However, there has
been little to no focus on the plan shape of a dimple, i.e., the
perimeter or boundaries of the dimple on the golf ball outer
surface, as a key variable in achieving such control and
optimization. In particular, since the bifurcation created by the
plan shape of a dimple creates a large transition from the external
surface geometry, it is considered to play a role in aerodynamic
behavior. As such, there remains a need for a dimple plan shape
that maximizes surface coverage uniformity and packing efficiency,
while maintaining desirable aerodynamic characteristics.
SUMMARY OF THE INVENTION
[0007] The present invention is directed to a golf ball dimple
having a perimeter defined by a low frequency periodic function
along a simple closed path according to the following function:
Q(x)=F.sub.path(l,scl,x)*F.sub.periodic(S,a,p,x)
where F.sub.path is a path function of length l, with scale factor
scl, defined along the vertices x, and F.sub.periodic is a periodic
function with sharpness factor s, amplitude a, and period p defined
at the vertices x; wherein the period p is about 15 or less, for
example, about 12 or less, and the perimeter has an amplitude A
such that the maximum absolute distance of any point on the
perimeter from the simple closed path is about 0.015 inches to
about 0.050 inches. In one embodiment, the periodic function is
selected from a sine, cosine, sawtooth wave, triangle wave, square
wave, or arbitrary function. In another embodiment, the path
function is any simple closed path that is symmetrical about two
orthogonal axes, for example, a circle, ellipse, or square. In
still another embodiment, the golf ball dimple has a Degree of
Interdigitation DOI of about 0.05 to about 0.50. In yet another
embodiment, the perimeter has an amplitude A such that the maximum
absolute distance of any point on the perimeter from the simple
closed path is about 0.025 inches to about 0.050 inches. In yet
another embodiment, each one of the dimples having a perimeter
defined by a low frequency periodic function along a simple closed
path comprises a first dimple profile and a second dimple profile,
the first dimple profile having a first edge angle
(.PHI..sub.EDGE1) and a second edge angle (.PHI..sub.EDGE2) and the
second dimple profile having a third edge angle (.PHI..sub.EDGE3)
and a fourth edge angle (.PHI..sub.EDGE4), wherein at least two of
.PHI..sub.EDGE1, .PHI..sub.EDGE2, .PHI..sub.EDGE3 and
.PHI..sub.EDGE4 have different values.
[0008] The present invention is further directed to a golf ball
having a substantially spherical surface, including a plurality of
dimples on the surface, wherein at least a portion of the plurality
of dimples have a plan shape defined by a low frequency periodic
function along a simple closed path according to the following
function:
Q(x)=F.sub.path(l,scl,x)*F.sub.periodic(S,a,p,x)
where F.sub.path is a path function of length l, with scale factor
scl, defined along the vertices x, and F.sub.periodic is a periodic
function with sharpness factor s, amplitude a, and period p defined
at the vertices x, wherein the period p is about 15 or less, for
example, about 12 or less, the plan shape has an amplitude A such
that the maximum absolute distance of any point on the plan shape
from the simple closed path is about 0.015 inches to about 0.050
inches, and the portion of the plurality of dimples have a Degree
of Interdigitation DOI of about 0.05 to about 0.50, for example, of
about 0.10 to about 0.30. In one embodiment, the plan shape has an
amplitude A such that the maximum absolute distance of any point on
the plan shape from the simple closed path is about 0.025 inches to
about 0.050 inches. In another embodiment, the periodic function is
selected from a sine, cosine, sawtooth wave, triangle wave, square
wave, or arbitrary function. In still another embodiment, at least
a portion includes about 50 percent or more of the dimples on the
golf ball. In yet another embodiment, each one of the dimples
having a non-circular plan shape defined by a low frequency
periodic function along a simple closed path comprises a first
dimple profile and a second dimple profile, the first dimple
profile having a first edge angle (.PHI..sub.EDGE1) and a second
edge angle (.PHI..sub.EDGE2) and the second dimple profile having a
third edge angle (.PHI..sub.EDGE3) and a fourth edge angle
(.PHI..sub.EDGE4), wherein at least two of .PHI..sub.EDGE1,
.PHI..sub.EDGE2, .PHI..sub.EDGE3 and .PHI..sub.EDGE4 have different
values.
[0009] The present invention is further directed to a golf ball
including an outer surface having a plurality of dimples arranged
in a dimple pattern thereon, wherein at least a portion of the
plurality of dimples arranged in the dimple pattern have a
non-circular plan shape defined by a low frequency periodic
function along a simple closed path and the portion of the
plurality of dimples arranged in the dimple pattern have a Degree
of Interdigitation DOI of about 0.05 to about 0.40, for example, of
about 0.10 to about 0.30. In one embodiment, the periodic function
is selected from a sine, cosine, sawtooth wave, triangle wave,
square wave, or arbitrary function. In another embodiment, the low
frequency periodic function has a period p of about 15 or less. In
still another embodiment, the low frequency periodic function of
the non-circular plan shape has a period p equal to the number of
neighboring dimples. In yet another embodiment, the low frequency
periodic function of the non-circular plan shape has a period p
that is a scalar multiple of the number of neighboring dimples. In
still another embodiment, the non-circular plan shape is defined by
a low frequency periodic function according to the following
function:
Q(x)=F.sub.path(l,scl,x)*F.sub.periodic(S,a,p,x)
where F.sub.path is a path function of length l, with scale factor
scl, defined along the vertices x, and F.sub.periodic is a periodic
function with sharpness factor s, amplitude a, and period p defined
at the vertices x. In yet another embodiment, each one of the
dimples having a non-circular plan shape defined by a low frequency
periodic function along a simple closed path comprises a first
dimple profile and a second dimple profile, the first dimple
profile having a first edge angle (.PHI..sub.EDGE1) and a second
edge angle (.PHI..sub.EDGE2) and the second dimple profile having a
third edge angle (.PHI..sub.EDGE3) and a fourth edge angle
(.PHI..sub.EDGE4), wherein at least two of .PHI..sub.EDGE1,
.PHI..sub.EDGE2, .PHI..sub.EDGE3 and .PHI..sub.EDGE4 have different
values.
[0010] In a particular aspect of the above embodiments, the surface
of one or more of the dimples having a non-circular plan shape
includes a protruding center portion. In a further particular
aspect, the protruding center portion lies below the nominal chord
plane of the dimple. In another further particular aspect, the
protruding center portion intersects with the nominal chord plane
of the dimple.
BRIEF DESCRIPTION OF THE DRAWINGS
[0011] 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:
[0012] FIG. 1 illustrates the waveform of a sawtooth wave periodic
function approximated by a Fourier series for use in a dimple plan
shape according to the present invention;
[0013] FIG. 2 illustrates the waveform of a triangle wave periodic
function approximated by a Fourier series for use in a dimple plan
shape according to the present invention;
[0014] FIG. 3 illustrates the waveform of a square wave periodic
function approximated by a Fourier series for use in a dimple plan
shape according to the present invention;
[0015] FIG. 4 illustrates the waveform of an arbitrary periodic
function for use in a dimple plan shape according to the present
invention;
[0016] FIG. 5 illustrates a dimple plan shape produced in
accordance with the present invention;
[0017] FIG. 6 is a flow chart illustrating the steps according to a
method of the present invention;
[0018] FIG. 7 illustrates a dimple plan shape produced in
accordance with the present invention having a centroid and a
maximal radial distance;
[0019] FIG. 8 illustrates a neighboring dimple pair produced in
accordance with the present invention;
[0020] FIG. 9 is a graphical representation illustrating dimple
surface volumes for golf balls produced in accordance with the
present invention;
[0021] FIG. 10 is a graphical representation illustrating preferred
dimple surface volumes for golf balls produced in accordance with
the present invention;
[0022] FIGS. 11A-11F illustrate various embodiments of a golf ball
dimple plan shape defined by a sawtooth wave periodic function
along a circular path;
[0023] FIGS. 12A-12F illustrate various embodiments of a golf ball
dimple plan shape defined by a square wave periodic function along
a circular path;
[0024] FIGS. 13A-13F illustrate various embodiments of a golf ball
dimple plan shape defined by an arbitrary periodic function along a
circular path;
[0025] FIGS. 14A-14F illustrate various embodiments of a golf ball
dimple plan shape defined by an arbitrary periodic function along
an arbitrary path;
[0026] FIG. 15 illustrates a golf ball dimple pattern constructed
from a plurality of dimple plan shapes according to the present
invention;
[0027] FIG. 16 is a schematic diagram illustrating a method for
measuring the edge angle of a dimple;
[0028] FIG. 17A illustrates a golf ball dimple plan shape defined
by a periodic square wave function mapped along a circular path in
accordance with an embodiment of the present invention;
[0029] FIG. 17B illustrates two dimple profiles of a dimple having
the plan shape shown in FIG. 17A in accordance with an embodiment
of the present invention; and
[0030] FIG. 18A illustrates a golf ball dimple plan shape defined
by a periodic square wave function mapped along a circular path in
accordance with an embodiment of the present invention; and
[0031] FIG. 18B illustrates two dimple profiles of a dimple having
the plan shape shown in FIG. 18A in accordance with an embodiment
of the present invention.
DETAILED DESCRIPTION
[0032] The present invention is directed to golf balls having
improved aerodynamic performance due, at least in part, to the use
of non-circular dimple plan shapes. In particular, the present
invention is directed to a golf ball that includes at least a
portion of its dimples having a plan shape defined by low frequency
periodic functions having high amplitudes. The present invention is
also directed to methods of generating the dimple plan shape
geometries, as well as methods of making the finished golf balls
with the inventive dimple patterns applied thereto. Further, the
present invention is directed to a parameter for quantifying the
measure of interdigitation or interlockability of neighboring
dimples produced in accordance with the present invention.
[0033] Advantageously, the dimple plan shapes produced in
accordance with the present invention combine a low period of
oscillation with a high degree of deviation to form plan shapes
defined by low frequency, high amplitude periodic functions. When
the inventive plan shapes are applied to dimples on a golf ball,
the resulting dimple patterns exhibit a high degree of
interlockability or interdigitation of neighboring dimples. This,
in turn, provides for improved dimple packing efficiency and
increased surface coverage. As a result, the present invention
provides a golf ball manufacturer the ability to fine tune golf
ball aerodynamic characteristics by controlling the external
surface geometry of the golf ball.
[0034] In addition, while golf balls with circular plan shape
dimples are indistinguishable from one another in the market place,
the plan shapes of dimples according to the present invention are
unique in appearance. For example, in one embodiment, the low
frequency periodic functions defining the plan shapes of the
present invention provide perimeters having a distinct appearance.
In turn, the plan shapes of the present invention provide for golf
ball surface textures having distinct visual appearances as well as
golf balls having improved aerodynamic characteristics.
Dimple Plan Shapes
[0035] The present invention contemplates dimples having a
non-circular plan shape defined by low frequency, high amplitude
periodic functions along a simple closed path. In particular, golf
balls formed according to the present invention include at least
about 10 percent or more of dimples having a plan shape defined by
low frequency, high amplitude periodic functions. In another
embodiment, the golf balls formed according to the present
invention include at least about 25 percent or more of dimples
having a plan shape defined by low frequency, high amplitude
periodic functions. In still another embodiment, the golf balls
formed according to the present invention include at least about 50
percent or more of dimples having a plan shape defined by low
frequency, high amplitude periodic functions. By the term, "plan
shape," it is meant the shape of the, perimeter of the dimple, or
the demarcation between the dimple and the outer surface of the
golf ball or fret surface.
[0036] According to the present invention, at least one dimple is
formed using a simple closed path, i.e., a path that starts and
ends at the same point without traversing any defining point or
edge along the path more than once. For example, the present
invention contemplates dimples formed using any simple cycle known
in graph theory including circles and polygons. In one embodiment,
the simple closed path is any path that is symmetrical about two
orthogonal axes. In another embodiment, the simple closed path is a
circle, ellipse, square, or polygon. In still another embodiment,
the simple closed path is an arbitrary path. In this aspect, a
suitable dimple shape according to the present invention may be
based on any path that starts and ends at the same point without
intersecting any defining point or edge.
[0037] The present invention contemplates the use of periodic
functions to form the dimple shape including any function that
repeats its values at regular intervals or periods. For the
purposes of the present invention, a function f is periodic if
f(x)=f(x+p) (1)
for all values of x where p is the period. In particular, the
present invention contemplates any periodic function that is
non-constant, non-zero.
[0038] In one embodiment, the periodic function used to form the
dimple shape includes a trigonometric function. Examples of
trigonometric functions suitable for use in accordance with the
present invention include, but are not limited to, sine and cosine.
Indeed, the waveform of a cosine periodic function may be used to
form a dimple shape in accordance with the invention. The cosine
wave suitable for use in accordance with the present invention has
a shape identical to that of a sine wave, except that each point on
the cosine wave occurs exactly 1/4 cycle earlier than the
corresponding point on the sine wave.
[0039] In another embodiment, the periodic function suitable for
use in forming a dimple shape in accordance with the present
invention includes a non-smooth periodic function. Non-limiting
examples of non-smooth periodic functions suitable for use with the
present invention include, but are not limited to, sawtooth wave,
triangle wave, square wave, and cycloid. In one embodiment, a
sawtooth wave is suitable for use in forming a dimple shape in
accordance with the present invention. In particular, a dimple in
accordance with the present invention may have a shape based on a
non-sinusoidal waveform that ramps upward and then sharply
drops.
[0040] In another embodiment, a triangle wave is suitable for use
in forming a dimple shape in accordance with the present invention.
The triangle wave suitable for use in forming a dimple shape in
accordance with the present invention is a non-sinusoidal waveform
that is a periodic, piecewise linear, continuous real function. In
yet another embodiment, a square wave is suitable for use in
forming a dimple shape in accordance with the present invention.
For example, the square wave suitable for use in forming a dimple
shape in accordance with the present invention is a non-sinusoidal
periodic waveform in which the amplitude alternates at a steady
frequency between fixed minimum and maximum values, with the same
duration at minimum and maximum.
[0041] In this aspect of the invention, any of the above-mentioned
periodic functions may be constructed as an infinite series of
sines and cosines using Fourier series expansion for use in forming
a dimple shape in accordance with the present invention. In
particular, the Fourier series of a function, which is given by
equations (2)-(5), is contemplated for use in forming the dimple
shape according to the present invention:
f(x)=1/2a.sub.0+.SIGMA..sub.n=1.sup..infin.a.sub.n
cos(nx)+.SIGMA..sub.n=1.sup..infin.b.sub.n sin(nx), (2)
where:
a 0 = 1 .pi. .intg. - .pi. .pi. f ( x ) dx ( 3 ) a n = 1 .pi.
.intg. - .pi. .pi. f ( x ) cos ( nx ) dx ( 4 ) b n = 1 .pi. .intg.
- .pi. .pi. f ( x ) sin ( nx ) dx ( 5 ) ##EQU00001##
and n=1, 2, 3 . . . .
[0042] In addition, the following Fourier series are contemplated
for use in forming the dimple shape in accordance with the present
invention.
TABLE-US-00001 TABLE 1 FOURIER SERIES OF NON-SMOOTH PERIODIC
FUNCTIONS Periodic Function Fourier Series Sawtooth wave 1 2 - 1
.pi. n = 1 .infin. 1 .pi. sin ( n .pi. x L ) ##EQU00002## Triangle
wave 8 .pi. 2 n = 1 , 3 , 5 .infin. ( - 1 ) ( n - 1 ) 2 n 2 sin ( n
.pi. x L ) ##EQU00003## Square wave 4 .pi. n = 1 , 3 , 5 .infin. 1
n sin ( n .pi. x L ) ##EQU00004##
[0043] For example, FIG. 1 illustrates the waveform of a sawtooth
wave approximated by a Fourier series. In particular, FIG. 1
illustrates a sawtooth wave 4 approximated by a four-term Fourier
series expansion for use in forming a dimple shape in accordance
with the present invention. The four-term sawtooth expansion can be
described by:
f ( x ) = s - a .pi. ( sin ( .pi. px ) + 1 2 sin ( 2 .pi. px ) + 1
3 sin ( 3 .pi. px ) + 1 4 sin ( 4 .pi. px ) ) . ##EQU00005##
In addition, FIG. 2 illustrates the waveform of a triangle wave
approximated by a Fourier series. FIG. 2 illustrates a triangle
wave 6 approximated by a four-term Fourier series expansion for use
in forming a dimple shape in accordance with the present invention.
Further, FIG. 3 illustrates the waveform of a square wave
approximated by a Fourier series. For example, FIG. 3 illustrates a
square wave 8 approximated by a four-term Fourier series expansion
for use in forming a dimple shape in accordance with the present
invention. The four-term square wave expansion can be described
by:
f ( x ) = s + a .pi. ( sin ( .pi. px ) + 1 3 sin ( 3 .pi. px ) + 1
5 sin ( 5 .pi. px ) + 1 7 sin ( 7 .pi. px ) ) . ##EQU00006##
[0044] While the above examples demonstrate four-term Fourier
series expansions, it will be understood by those of ordinary skill
in the art that more than or less than four terms may be used to
approximate the non-sinusoidal waveforms. In addition, any method
of approximation known to one of ordinary skill in the art may be
used in this aspect of the invention.
[0045] In yet another embodiment, the present invention
contemplates arbitrary periodic functions, or linear combinations
of periodic functions for use in forming a dimple shape in
accordance with the present invention. Accordingly, in one
embodiment of the present invention, an arbitrary periodic function
may be created using a linear combination of sines and cosines to
form a dimple shape in accordance with the present invention. In
this aspect, FIG. 4 illustrates the waveform of an arbitrary
periodic function contemplated by the present invention. As shown
in FIG. 4, the arbitrary wave 10 represents a linear combination of
sines and cosines.
[0046] According to the present invention, the plan shape of the
dimple may be produced by projecting or mapping any of the
above-referenced periodic functions onto the simple closed path. In
general, the mathematical formula representing the projection or
mapping of the periodic function onto the simple closed path is
expressed as equation (6):
Q(x)=F.sub.path(l,scl,x)*F.sub.periodic(S,a,p,x) (6)
where F.sub.path represents the simple closed path on which the
periodic function is mapped or projected with length (l), scale
factor (scl), defined along the vertices (x), and F.sub.periodic is
any suitable periodic function with sharpness factor (s), amplitude
(a), and period (p) defined at the vertices (x).
[0047] In one embodiment, the projection may be described in terms
of how the path function is altered by the periodic function. For
example, the resulting vector Q(x) represents the altered
coordinates of the path. Indeed, the "path function" contemplated
by the present invention includes any of the simple paths discussed
above.
[0048] In this aspect of the invention, the resulting vector Q(x)
may also be a suitable path for a dimple plan shape according to
the present invention. That is, the resulting vector Q(x) could
itself become a path to which another periodic function is mapped.
Indeed, any of the periodic functions disclosed above may be mapped
to the resulting vector Q(x) to form a dimple plan shape in
accordance with the present invention.
[0049] The length (l) and scale factor (scl), may vary depending on
the desired size of the dimple. However, in one embodiment, the
length is about 0.150 inches to about 1.400 inches. In another
embodiment, the length is about 0.250 inches to about 1.200 inches.
In still another embodiment, the length is about 0.500 inches to
about 0.800 inches.
[0050] The variable, F.sub.periodic, of equation (6) will vary
based on the desired periodic function. The sharpness factor (s) is
a scalar value and defines the mean of the periodic function.
Generally, small values of s produce periodic functions that
greatly alter the plan shape, while larger values of s produce
periodic functions having a diminished influence on the plan shape.
Indeed, as will be apparent to one of ordinary skill in the art,
once an amplitude value is chosen, the sharpness factor (s) may be
varied depending on the desired amount of alteration to the plan
shape. In one embodiment, the sharpness factor (s) ranges from
about 5 to about 60. In another embodiment, the sharpness factor
(s) ranges from about 8 to about 55. In still another embodiment,
the sharpness factor (s) ranges from about 15 to about 50.
[0051] The amplitude of the plan shape, A, is defined as the
absolute value of the maximum distance from the path during one
period of the periodic function. The amplitude of the function, a,
affects the dimple plan shape in the opposite sense as sharpness
factor (s). In this aspect, the sharpness factor (s) and amplitude
(a) parameters are both used to control the mapped periodic
function used to define Q(x). For example, the sharpness factor (s)
and amplitude (a) parameters control the severity of the perimeter
of the final plan shape. Indeed, sharpness and amplitude parameters
are both used to "tune" the final plan shape geometry. That is, the
plan shapes of the present invention can be tailored to maximize
neighboring dimple interdigitation, thus improving packing
efficiency and surface coverage.
[0052] In one embodiment, the function amplitude (a) ranges from
about 0.25 to 10. In another embodiment, the amplitude (a) ranges
from about 0.5 to 5. In still another embodiment, the amplitude (a)
ranges from about 1 to 3. For example, the amplitude (a) may be
about 1.
[0053] The amplitude of the plan shape, A, can be expressed as the
maximum distance of any point on the plan shape from the path. FIG.
5 illustrates a plan shape constructed in accordance with the
present invention, wherein a periodic function is mapped along a
simple closed path. The absolute distance, d, of any point on the
plan shape 15 from the path 20 (represented by a dashed line) is
defined by the following equation (7):
d= {square root over
((x.sub.circle-x.sub.plan).sup.2+(y.sub.circle-y.sub.plan).sup.2)}
(7)
where d is a directed distance calculated along a line from the
plan shape centroid through corresponding points on the plan shape
and path. The amplitude of the plan shape, A, is expressed as the
maximum value, d.sub.max, for all calculated distances, d.
[0054] High amplitude periodic functions are contemplated for use
in forming a dimple shape in accordance with the present invention.
That is, the present invention contemplates plan shapes having a
high degree of deviation from the path. In one embodiment, the
amplitude of the dimple plan shape is such that the maximum
absolute distance (d.sub.max) of any point on the plan shape from
the simple path is greater than 0.015 inches. In another
embodiment, the amplitude of the dimple plan shape is such that the
maximum absolute distance (d.sub.max) of any point on the plan
shape from the simple path is greater than 0.025 inches. In still
another embodiment, the amplitude of the dimple plan shape is such
that the maximum absolute distance, (d.sub.max) of any point on the
plan shape from the simple path is greater than 0.035 inches. In
yet another embodiment, the amplitude of the dimple plan shape is
such that the maximum absolute distance, (d.sub.max), of any point
on the plan shape from the simple path is greater than 0.050
inches.
[0055] The period (p) refers to the horizontal distance required
for the periodic function to complete one cycle. As will be
apparent to one of ordinary skill in the art, the period may vary
based on the periodic function. However, in one embodiment, the
present invention contemplates periodic functions having a period
of less than about 15. In another embodiment, the present invention
contemplates periodic functions having a period of less than about
12. In still another embodiment, the present invention contemplates
periodic functions having a period of less than about 10. In yet
another embodiment, the present invention contemplates periodic
functions having a period of less than about 8. For example, the
present invention contemplates periodic functions having a period
of less than about 5.
[0056] The period of the wave function is inversely proportional to
the function frequency. Indeed, the frequency refers to the number
of periods completed over the path function. For example, the
frequency of a periodic function having a period (p) is represented
by 1/p. In one embodiment, the present invention contemplates low
frequency periodic functions. That is, the present invention
contemplates periodic functions having a frequency of about 1/15 or
more. In one embodiment, the periodic function has a frequency of
about 1/12 or more. In another embodiment, the periodic function
has a frequency of about 1/10 or more. In still another embodiment,
the periodic function has a frequency of about 1/8 or more. In yet
another embodiment, the periodic function has a frequency of about
1/5 or more.
[0057] Accordingly, by manipulating the variables of equation (6),
the present invention provides for golf ball dimples having various
plan shapes defined by low frequency and high amplitude periodic
functions. Indeed, the plan shapes of the present invention have a
low period of oscillation (for example, p less than about 15) and a
high degree of deviation from the path that leads to large values
of d.sub.max (for example, greater than about 0.015 inches). By
using the low frequency and high amplitude periodic functions
disclosed herein, the present invention provides for dimple plan
shapes and dimple patterns with a high degree of interlockability
or interdigitation of neighboring dimples and, thus, high packing
efficiency and surface coverage.
[0058] FIG. 6 illustrates one embodiment of a method of forming a
dimple plan shape in accordance with the present invention. For
example, step 101 includes selecting the simple closed path on
which the periodic function is to be projected. In this aspect, the
present invention contemplates the use of any of the simple closed
paths discussed above. Step 102 includes selecting the desired
periodic function. Indeed, any of the periodic functions disclosed
above are contemplated in this aspect of the invention.
[0059] At step 103, the amplitude, sharpness, period, or frequency
of the periodic function is selected based on the desired periodic
function and path. In one embodiment, the present invention
contemplates dimple plan shapes defined by a low frequency, high
amplitude periodic function. That is, the plan shapes of the
present invention have a low period of oscillation (for example, a
period, p, less than about 15) and an amplitude that leads to large
values of d.sub.max (for example, greater than about 0.015 inches).
Accordingly, the amplitude, sharpness, period, or frequency should
be selected such that the values are in accordance with the
parameters defined above.
[0060] At step 104, the variables selected above, including the
path, periodic function, amplitude, sharpness, and period, are
inserted into equation (6), reproduced below:
Q(x)=F.sub.path(l,scl,x)*F.sub.periodic(S,a,p,x) (6)
[0061] The resultant function is then used to project the periodic
function onto the simple closed path in order to generate the
dimple plan shape. The resultant function will vary based on the
desired path and periodic function. For example, if the desired
periodic function is a cosine function, F.sub.periodic may be
represented by equation (8), depicted below:
f(x)=s+a*cos(p*.pi.*x). (8)
[0062] As discussed above, the resultant dimple plan shape (e.g.,
the resulting vector Q(x)) may also be used as the path to which
another periodic function is mapped. For example, a periodic
function having a different period or a different periodic function
may be projected onto the resultant dimple plan shape to form a new
dimple plan shape in accordance with the present invention.
[0063] After the dimple plan shape has been generated, at step 105,
the plan shape can be used in designing geometries for dimple
patterns of a golf ball. For example, the plan shape paths
generated by the methods of the present invention can be imported
into a CAD program and used to define dimple geometries and tool
paths for fabricating tooling for golf ball manufacture. The
various dimple geometries produced in accordance with the present
invention can then be used in constructing a dimple pattern that
maximizes interdigitation with neighboring dimples, which in turn
leads to high surface coverage uniformity and improved dimple
packing efficiency.
Degree of Interdigitation
[0064] The low frequency, high amplitude plan shapes described by
the present invention produce dimple patterns having a high degree
of interlockability or interdigitation of neighboring dimples. In
this regard, the present invention is further directed to golf ball
dimple patterns having a high degree of interlockability or
interdigitation of neighboring dimples, thus producing dimple
patterns with high packing efficiency and surface coverage. In
particular, the interdigitation or interlockability of neighboring
dimples in the dimple patterns according to the present invention
(and golf balls made with such dimple patterns thereon) is
quantifiable. Indeed, the dimple patterns according to this aspect
of the present invention are associated with a parameter referred
to as the Degree of Interdigitation (DOI). As described herein, the
DOI is a value that quantifies the capability of the dimples of the
present invention to interlock with each other when configured in a
dimple pattern. According to the present invention, the DOI of each
plan shape is based on the maximal radial distance of each dimple
plan shape and the Dimple Penetration Coefficient of each
neighboring dimple pair.
[0065] In determining the overall DOI, the maximal radial distance
of each dimple plan shape is calculated. As shown in FIG. 7, the
maximal radial distance, R, is the distance between the centroid,
C, and any point on the plan shape. In one embodiment, the maximal
radial distance is determined by rotating the dimple plan shape to
the pole of the golf ball surface and then projecting the plan
shape onto a plane located at the golf ball center with a normal
parallel to a line running from the center of the golf ball through
the pole. The centroid of the plan shape may then be calculated
using the following equations:
x centroid = 1 A .intg. xdA ( 9 ) y centroid = 1 A .intg. ydA ( 10
) ##EQU00007##
where A is the area of the plan shape and .intg. xdA and .intg. ydA
are the first moments of the area with respect to the y and x axes,
respectively. Upon calculating the centroid of the plan shape, the
maximum radial distance of any point on the plan shape from the
centroid is determined according to the following equation:
R= {square root over
((x.sub.plan-x.sub.centroid).sup.2+(y.sub.plan-y.sub.centroid).sup.2)}
(11)
[0066] The present invention contemplates a maximal radial distance
of about 0.050 inches to about 0.250 inches for each dimple plan
shape. In another embodiment, the maximal radial distance is about
0.100 inches to about 0.220 inches for each dimple plan shape. In
still another embodiment, the maximal radial distance is about
0.110 inches to about 0.200 inches for each dimple plan shape. In
yet another embodiment, the maximal radial distance is about 0.120
inches to about 0.190 inches for each dimple plan shape.
[0067] After the maximal radial distance is determined for each
dimple plan shape, the Dimple Penetration Coefficient (DPC) is
calculated for each neighboring dimple pair. As described herein,
the DPC may be determined by dividing the sum of the maximum radial
distances of both neighboring dimples by the distance between the
neighboring dimple three-dimensional centers on the golf ball
surface and subtracting one.
[0068] Accordingly, to calculate the DPC, one must first determine
all of the neighboring dimple pairs. For example, neighboring
dimples may be determined by drawing two tangency lines from the
center of the first dimple to a potential neighboring dimple. Then,
a line segment is drawn connecting the center of the first dimple
to the center of the potential neighboring dimple. If no line
segment is intersected by another dimple or a portion of a dimple,
then those dimples are considered neighboring dimples.
[0069] In this aspect, after the neighboring dimple pairs are
determined, the distance between the dimple centers of each
neighboring dimple pair may be calculated. For example, FIG. 8
shows a pair of neighboring dimples produced in accordance with the
present invention having centers C.sub.1 and C.sub.2, respectively.
The distance between center C.sub.1 and center C.sub.2 is
represented by D. In one embodiment, the distance, D, may be
calculated for each neighboring pair via the following
equation:
D= {square root over
((x.sub.c1-x.sub.c2).sup.2+(y.sub.c1-y.sub.c2).sup.2+(z.sub.c1-z.sub.c2).-
sup.2)} (12).
[0070] As shown in Equation (12), the distance, D, is determined by
taking the square root of the sum of the differences in center
distances of a first neighboring dimple and a second neighboring
dimple along the x, y, and z axes.
[0071] According to the present invention, after the distance
between the dimple centers of a neighboring dimple pair is
calculated, the DPC is calculated for each neighboring dimple pair.
In one embodiment, the DPC is calculated by using the following
equation:
DPC = R 1 + R 2 D - 1 ( 13 ) ##EQU00008##
where R.sub.1 is the maximum radial distance of the first
neighboring dimple, R.sub.2 is the maximum radial distance of the
second neighboring dimple, and D is the distance between the
neighboring dimple centers. The maximum radial distance, as
described above, is the distance between the centroid of the dimple
and any point on the plan shape. As shown in FIG. 8, the maximal
radial distance of the first neighboring dimple, R.sub.1,
represents the distance between the centroid, C.sub.1, and any
point on the plan shape of the first neighboring dimple projected
to the golf ball surface. The maximal radial distance of the second
neighboring dimple, R.sub.2, represents the distance between the
centroid, C.sub.2, and any point on the plan shape of the second
neighboring dimple projected to the golf ball surface.
[0072] The present invention contemplates DPC values ranging from
about 0.5 to about -0.1. In another embodiment, the DPC value for
each neighboring dimple pair ranges from about 0.3 to about -0.05.
In still another embodiment, the DPC value for each neighboring
dimple pair ranges from about 0.1 to about -0.02. In yet another
embodiment, the DPC value for each neighboring dimple pair ranges
from about 0.05 to about 0. As will be appreciated by those of
ordinary skill in the art, positive values for the DPC indicate a
greater degree of interdigitation between neighboring dimple pairs,
while negative values for the DPC indicate no interdigitation.
[0073] After the DPC is calculated for the neighboring dimple pair,
the Degree of Interdigitation (DOI) may be calculated. As described
above, the DOI is a parameter for quantifying the measure of
interlockability of neighboring dimples produced in accordance with
the present invention. In one embodiment, the DOI is calculated
according to the following equation:
DOI = 1 n k = 1 n DPC k ( 14 ) ##EQU00009##
where n is the number of possible neighboring dimple pairs and
DPC.sub.k is the individual dimple penetration coefficient for each
neighboring dimple pair.
[0074] In this aspect, the present invention contemplates dimple
plan shapes and dimple patterns having DOI values less than 0.50
and greater than zero. In one embodiment, the dimple plan shapes
and dimple patterns of the present invention have DOI values
ranging from about 0.01 to about 0.40. In another embodiment, the
dimple plan shapes and dimple patterns of the present invention
have DOI values ranging from about 0.05 to about 0.30. In still
another embodiment, the dimple plan shapes and dimple patterns of
the present invention have DOI values ranging from about 0.10 to
about 0.20. Without being bound by any particular theory, the high
degree of dimple interdigitation created by the present invention
minimizes land area spacing and gives a more uniform distribution
of surface coverage for improved aerodynamic symmetry.
[0075] The foregoing procedure may be repeated for any dimple pair
on the ball. However, as one of ordinary skill in the art would
readily understand, golf balls formed according to the present
invention include at least about 10 percent or more of dimples
having a plan shape defined by low frequency, high amplitude
periodic functions and a DOI value greater than zero. In another
embodiment, the golf balls formed according to the present
invention include at least about 25 percent or more of dimples
having a plan shape defined by low frequency, high amplitude
periodic functions and a DOI value greater than zero. In still
another embodiment, the golf balls formed according to the present
invention include at least about 50 percent or more of dimples
having a plan shape defined by low frequency, high amplitude
periodic functions and a DOI value greater than zero.
Dimple Patterns & Packing
[0076] The present invention allows for improved dimple packing
over previous patterns so that a greater percentage of the surface
of the golf ball is covered by dimples. In particular, due to the
high degree of interlockability or interdigitation of neighboring
dimples, each dimple having a plan shape in accordance with the
present invention is part of a dimple pattern that maximizes
surface coverage uniformity and packing efficiency.
[0077] In one embodiment, the dimple pattern provides greater than
about 80 percent surface coverage. In another embodiment, the
dimple pattern provides greater than about 85 percent surface
coverage. In yet another embodiment, the dimple pattern provides
greater than about 90 percent surface coverage. In still another
embodiment, the dimple pattern provides greater than about 92
percent surface coverage.
[0078] In this aspect, the golf ball dimple plan shapes of the
present invention can be tailored to maximize surface coverage
uniformity and packing efficiency by selecting a period for the
periodic function that is a scalar multiple of the number of
neighboring dimples. For example, if the number of neighboring
dimples is 4, the present invention contemplates a dimple plan
shape having a period of 8 or 12. In another embodiment, the period
is equal to the number of neighboring dimples. For example, if the
dimple plan shape is constructed using a period of 5, the present
invention contemplates that the dimple will be surrounded by 5
neighboring dimples.
[0079] FIG. 15 illustrates an example of a dimple pattern created
in accordance with the present invention. In particular, FIG. 15
illustrates a golf ball dimple pattern 110 made up of dimple plan
shapes (represented by 115) defined by low frequency, high
amplitude periodic functions and produced in accordance with the
present invention. As shown in FIG. 15, the plan shapes 115 are
formed using a square wave function mapped along a circular path
with period (p) of 6, sharpness factor (s) of 10, and amplitude (a)
of 1. In this embodiment, the dimple pattern 110 is further defined
to have a DOI of about 0.018. As can be seen by FIG. 15, the high
degree of interlockability or interdigitation of the dimple plan
shapes 115 allows for a dimple pattern 110 that maximizes surface
coverage uniformity and packing efficiency.
[0080] While the plan shapes of the present invention may be used
for at least a portion of the dimples on a golf ball, it is not
necessary that the plan shapes be used on every dimple of a golf
ball. In general, it is preferred that a sufficient number of
dimples on the ball have plan shapes according to the present
invention so that the aerodynamic characteristics of the ball may
be altered and the packing efficiency benefits realized. For
example, at least about 30 percent of the dimples on a golf ball
include plan shapes according to the present invention. In another
embodiment, at least about 50 percent of the dimples on a golf ball
include plan shapes according to the present invention. In still
another embodiment, at least about 70 percent of the dimples on a
golf ball include plan shapes according to the present invention.
In yet another embodiment, at least about 90 percent of the dimples
on a golf ball include the plan shapes of the present invention. In
still another embodiment, all of the dimples (100 percent) on a
golf ball may include the plan shapes of the present invention.
[0081] While the present invention is not limited by any particular
dimple pattern, dimples having plan shapes according to the present
invention are arranged preferably along parting lines or equatorial
lines, in proximity to the poles, or along the outlines of a
geodesic or polyhedron pattern. Conventional dimples, or those
dimples that do not include the plan shapes of the present
invention, may occupy the remaining spaces. The reverse arrangement
is also suitable. Suitable dimple patterns include, but are not
limited to, polyhedron-based patterns (e.g., icosahedron,
octahedron, dodecahedron, icosidodecahedron, cuboctahedron, and
triangular dipyramid), phyllotaxis-based patterns, spherical tiling
patterns, and random arrangements.
Dimple Dimensions
[0082] The dimples on the golf balls of the present invention may
include any width, depth, depth profile, edge angle, or edge radius
and the patterns may include multitudes of dimples having different
widths, depths, depth profiles, edge angles, or edge radii.
[0083] Since the plan shape perimeters of the present invention are
noncircular, the plan shapes are defined by an effective dimple
diameter which is twice the average radial dimension of the set of
points defining the plan shape from the plan shape centroid. For
example, in one embodiment, dimples according to the present
invention have an effective dimple diameter within a range of about
0.05 inches to about 0.300 inches. In another embodiment, the
dimples have an effective dimple diameter of about 0.080 inches to
about 0.250 inches. In still another embodiment, the dimples have
an effective dimple diameter of about 0.100 inches to about 0.225
inches. In yet another embodiment, the dimples have an effective
dimple diameter of about 0.125 inches to about 0.200 inches.
[0084] The surface depth for dimples of the present invention is
within a range of about 0.003 inches to about 0.025 inches. In one
embodiment, the surface depth is about 0.005 inches to about 0.020
inches. In another embodiment, the surface depth is about 0.006
inches to about 0.017 inches.
[0085] The dimples of the present invention also have a plan shape
area. By the term, "plan shape area," it is meant the area based on
a planar view of the dimple plan shape, such that the viewing plane
is normal to an axis connecting the center of the golf ball to the
point of the calculated surface depth. In one embodiment, dimples
of the present invention have a plan shape area ranging from about
0.0025 in.sup.2 to about 0.045 in.sup.2. In another embodiment,
dimples of the present invention have a plan shape area ranging
from about 0.005 in.sup.2 to about 0.035 in.sup.2. In still another
embodiment, dimples of the present invention have a plan shape area
ranging from about 0.010 in.sup.2 to about 0.030 in.sup.2.
[0086] Further, dimples of the present invention have a dimple
surface volume. By the term, "dimple surface volume," it is meant
the total volume encompassed by the dimple shape and the surface of
the golf ball. FIGS. 9 and 10 illustrate graphical representations
of dimple surface volumes contemplated for dimples produced in
accordance with the present invention. For example, FIGS. 9 and 10
demonstrate contemplated dimple surface volumes over a range of
plan shape areas. In one embodiment, dimples produced in accordance
with the present invention have a plan shape area and dimple
surface volume falling within the ranges shown in FIG. 9. For
example, a dimple having a plan shape area of about 0.01 in.sup.2
may have a surface volume of about 0.20.times.10.sup.-4 in.sup.3 to
about 0.50.times.10.sup.-4 in.sup.3. In another embodiment, a
dimple having a plan shape area of about 0.025 in.sup.2 may have a
surface volume of about 0.80.times.10.sup.-4 in.sup.3 to about
1.75.times.10.sup.-4 in.sup.3. In still another embodiment, a
dimple having a plan shape area of about 0.030 in.sup.2 may have a
surface volume of about 1.20.times.10.sup.-4 in.sup.3 to about
2.40.times.10.sup.-4 in.sup.3. In yet another embodiment, a dimple
having a plan shape area of about 0.045 in.sup.2 may have a surface
volume of about 2.10.times.10.sup.-4 in.sup.3 to about
4.25.times.10.sup.-4 in.sup.3.
[0087] In another embodiment, dimples produced in accordance with
the present invention have a plan shape area and dimple surface
volume falling within the ranges shown in FIG. 10. For example, a
dimple having a plan shape area of about 0.01 in.sup.2 may have a
surface volume of about 0.25.times.10.sup.-4 in.sup.3 to about
0.35.times.10.sup.-4 in.sup.3. In another embodiment, a dimple
having a plan shape area of about 0.025 in.sup.2 may have a surface
volume of about 1.10.times.10.sup.-4 in.sup.3 to about
1.45.times.10.sup.-4 in.sup.3. In yet another embodiment, a dimple
having a plan shape area of about 0.030 in.sup.2 may have a surface
volume of about 1.40.times.10.sup.-4 in.sup.3 to about
1.90.times.10.sup.-4 in.sup.3.
[0088] Since, as discussed above, the dimple patterns useful in
accordance with the present invention do not necessarily include
only dimples having plan shapes as described above, other
conventional dimples included in the dimple patterns may have
similar dimensions.
Dimple Profile
[0089] In addition to varying the size of the dimples, the
cross-sectional profile of the dimples may be varied. The
cross-sectional profile of the dimples according to the present
invention may be based on any known dimple profile shape. In one
embodiment, the profile of the dimples corresponds to a curve. For
example, the dimples of the present invention may be defined by the
revolution of a catenary curve about an axis, such as that
disclosed in U.S. Pat. Nos. 6,796,912 and 6,729,976, the entire
disclosures of which are incorporated by reference herein. In
another embodiment, the dimple profiles correspond to polynomial
curves, ellipses, spherical curves, saucer-shapes, truncated cones,
trigonometric, exponential, or logarithmic curves, and flattened
trapezoids. For example, the dimple profile may be defined by a
conical shape, such as that disclosed in U.S. Pat. No. 8,632,426,
the entire disclosure of which is incorporated by reference
herein.
[0090] The profile of the dimple may also aid in the design of the
aerodynamics of the golf ball. For example, shallow dimple depths,
such as those in U.S. Pat. No. 5,566,943, the entire disclosure of
which is incorporated by reference herein, may be used to obtain a
golf ball with high lift and low drag coefficients. Conversely, a
relatively deep dimple depth may aid in obtaining a golf ball with
low lift and low drag coefficients.
[0091] The dimple profile may also be defined by combining a
spherical curve and a different curve, such as a cosine curve, a
frequency curve or a catenary curve, as disclosed in U.S. Patent
Publication No. 2012/0165130, which is incorporated in its entirety
by reference herein. Similarly, the dimple profile may be defined
by a combination of two or more curves. For example, in one
embodiment, the dimple profile is defined by combining a spherical
curve and a different curve. In another embodiment, the dimple
profile is defined by combining a cosine curve and a different
curve. In still another embodiment, the dimple profile is defined
by combining a frequency curve and a different curve. In yet
another embodiment, the dimple profile is defined by combining a
catenary curve and different curve. In still another embodiment,
the dimple profile may be defined by combining three or more
different curves. In yet another embodiment, one or more of the
curves may be a functionally weighted curve, as disclosed in U.S.
Patent Publication No. 2013/0172123, which is incorporated in its
entirety by reference herein.
[0092] Dimple cross-sectional profiles have two edge angles
(.PHI..sub.EDGE), one at each of the two ends of the profile where
the profile meets the dimple perimeter. As a result of having a
plan shape defined by mapping a periodic function along a simple
closed path, a dimple of the present invention has at least two
cross-sectional profiles wherein at least one edge angle of one
profile is different from at least one edge angle of another
profile. Further, depending on the period p, a single
cross-sectional profile of a dimple of the present invention may
have an edge angle on one side of the cross-sectional profile that
is different from the edge angle on the other side of the same
cross-sectional profile. Thus, each dimple of the present invention
comprises at least a first dimple profile having a first edge angle
(.PHI..sub.EDGE1) and a second edge angle (.PHI..sub.EDGE2) and a
second dimple profile having a third edge angle (.PHI..sub.EDGE3)
and a fourth edge angle (.PHI..sub.EDGE4), wherein at least two of
.PHI..sub.EDGE1, .PHI..sub.EDGE2, .PHI..sub.EDGE3 and
.PHI..sub.EDGE4 have different values. Preferably, at least two of
.PHI..sub.EDGE1, .PHI..sub.EDGE2, .PHI..sub.EDGE3 and
.PHI..sub.EDGE4 have values that differ by 0.5.degree. to
6.0.degree., or values that differ by 1.0.degree. to 6.0.degree..
In a particular embodiment, one of the following is true:
[0093] a) .PHI..sub.EDGE1=.PHI..sub.EDGE2, [0094]
.PHI..sub.EDGE3=.PHI..sub.EDGE4, and [0095]
.PHI..sub.EDGE1.noteq..PHI..sub.EDGE3,
[0096] b) .PHI..sub.EDGE1.noteq..PHI..sub.EDGE2, [0097]
.PHI..sub.EDGE3=.PHI..sub.EDGE4, and [0098]
.PHI..sub.EDGE1=.PHI..sub.EDGE3,
[0099] c) .PHI..sub.EDGE1.noteq..PHI..sub.EDGE2, [0100]
.PHI..sub.EDGE3=.PHI..sub.EDGE4, and [0101]
.PHI..sub.EDGE1.noteq..PHI..sub.EDGE3,
[0102] d) .PHI..sub.EDGE1.noteq..PHI..sub.EDGE2, [0103]
.PHI..sub.EDGE3.noteq..PHI..sub.EDGE4, [0104]
.PHI..sub.EDGE1=.PHI..sub.EDGE3, and [0105]
.PHI..sub.EDGE2.noteq..PHI..sub.EDGE4, or
[0106] e) .PHI..sub.EDGE1.noteq..PHI..sub.EDGE2, [0107]
.PHI..sub.EDGE3.noteq..PHI..sub.EDGE4, [0108]
.PHI..sub.EDGE1.noteq..PHI..sub.EDGE3, and [0109]
.PHI..sub.EDGE2.noteq..PHI..sub.EDGE4.
[0110] Depending on the frequency and amplitude of the periodic
function defining the dimple plan shape, a dimple of the present
invention may include an infinite number of different edge angle
values. For purposes of the present invention, edge angles are
generally considered to be the same if they differ by less than
0.25.degree.. It should be understood that manufacturing variances
are to be taken into account when determining whether two
differently located edge angles have the same value. The location
of the edge angle along the dimple perimeter shape should also be
taken into account. Preferably, the edge angles of a dimple of the
present invention that do not have the same value differ by
0.5.degree. to 6.0.degree., or differ by 1.0.degree. to
6.0.degree..
[0111] In a particular embodiment, the average of the edge angles
of all of the dimple profiles of a single dimple of the present
invention is from 11.degree. to 16.degree..
[0112] In another particular embodiment, the difference between the
maximum edge angle and the average of the edge angles of all of the
dimple profiles of a single dimple of the present invention is
3.00.degree. or less.
[0113] In another particular embodiment, the difference between the
minimum edge angle and the average of the edge angles of all of the
dimple profiles of a single dimple of the present invention is
3.00.degree. or less.
[0114] For purposes of the present disclosure, edge angle
measurements are determined on finished golf balls. Generally, it
may be difficult to measure an edge angle due to the indistinct
nature of the boundary dividing the dimple from the ball's
undisturbed land surface. Due to the effect of coatings on the golf
ball surface and/or the dimple design itself, the junction between
the land surface and the dimple is typically not a sharp corner and
is therefore indistinct. This can make the measurement of
properties such as edge angle (.PHI..sub.EDGE) and dimple diameter,
somewhat ambiguous. To resolve this problem, edge angle
(.PHI..sub.EDGE) on a finished golf ball is measured as follows, in
reference to FIG. 16. FIG. 16 shows a dimple half-profile extending
from the dimple centerline 310 to the ball's undisturbed land
surface 330. A ball phantom surface 320 is constructed above the
dimple as a continuation of the land surface 330. A first tangent
line T1 is then constructed at a point on the dimple sidewall that
is spaced 0.003 inches radially inward from the phantom surface
320. T1 intersects phantom surface 320 at a point P1, which defines
a nominal dimple edge position. A second tangent line T2 is then
constructed, tangent to the phantom surface 320, at P1. The edge
angle (.PHI..sub.EDGE) is the angle between T1 and T2. The dimple
diameter is the distance between P1 and its equivalent point at the
opposite end of the dimple profile. The dimple depth is the
distance measured along a ball radius from the phantom surface of
the ball to the deepest point on the dimple. The dimple surface
volume is the space enclosed between the phantom surface 320 and
the dimple surface 340 (extended along T1 until it intersects the
phantom surface).
Golf Ball Construction
[0115] The dimples of the present invention may be used with
practically any type of ball construction. For instance, the golf
ball may have a two-piece design, a double cover, or veneer cover
construction depending on the type of performance desired of the
ball. Other suitable golf ball constructions include solid, wound,
liquid-filled, and/or dual cores, and multiple intermediate
layers.
[0116] Different materials may be used in the construction of the
golf balls made with the present invention. For example, the cover
of the ball may be made of a thermoset or thermoplastic, a castable
or non-castable polyurethane and polyurea, an ionomer resin,
balata, or any other suitable cover material known to those skilled
in the art. Conventional and non-conventional materials may be used
for forming core and intermediate layers of the ball including
polybutadiene and other rubber-based core formulations, ionomer
resins, highly neutralized polymers, and the like.
EXAMPLES
[0117] The following non-limiting examples demonstrate plan shapes
of golf ball dimples made in accordance with the present invention.
The examples are merely illustrative of the preferred embodiments
of the present invention, and are not to be construed as limiting
the invention, the scope of which is defined by the appended
claims.
Example 1
[0118] The following example illustrates golf ball dimple plan
shapes defined by a low frequency, high amplitude sawtooth wave
periodic function mapped to a circular path. Table 2, depicted
below, describes the mathematical parameters used to project the
periodic function onto the simple closed path.
TABLE-US-00002 TABLE 2 PLAN SHAPE PARAMETERS OF EXAMPLE 1 Path
Circular Periodic Function Sawtooth Wave (2-term Fourier expansion)
Function (f(x)) General Fourier Series: f ( x ) = s + a .pi. k = 1
.infin. sin ( k .pi. px ) k ##EQU00010## 2-term Fourier Expansion:
f ( x ) = s + a .pi. ( sin ( .pi. px ) + sin ( 2 .pi. px ) k )
##EQU00011## Sharpness Factor, s about 5 Amplitude, a about 1
[0119] FIGS. 11A-11F demonstrate the golf ball dimple plan shapes
produced in accordance with the parameters of Table 2.
Specifically, FIG. 11A shows a dimple plan shape 30 defined by a
sawtooth wave function approximated by a two-term Fourier series
having period, p=3, mapped to a circular path. FIG. 11B shows a
dimple plan shape 31 defined by a sawtooth wave function
approximated by a two-term Fourier series having period, p=4,
mapped to a circular path. FIG. 11C shows a dimple plan shape 32
defined by a sawtooth wave function approximated by a two-term
Fourier series having period, p=5, mapped to a circular path. FIG.
11D shows a dimple plan shape 33 defined by a sawtooth wave
function approximated by a two-term Fourier series having period,
p=6, mapped to a circular path. FIG. 11E shows a dimple plan shape
34 defined by a sawtooth wave function approximated by a two-tern
Fourier series having period, p=7, mapped to a circular path. FIG.
11F shows a dimple plan shape 35 defined by a sawtooth wave
function approximated by a two-term Fourier series having period,
p=8, mapped to a circular path.
Example 2
[0120] The following example illustrates golf ball dimple plan
shapes defined by a low frequency, high amplitude square wave
periodic function mapped to a circular path. Table 3, depicted
below, describes the mathematical parameters used to project the
periodic function onto the simple closed path.
TABLE-US-00003 TABLE 3 PLAN SHAPE PARAMETERS OF EXAMPLE 2 Path
Circular Periodic Square Wave (4-term Fourier expansion) Function
Function (f(x)) General Fourier Series: f ( x ) = s + a .pi. k = 1
, 3 , 5 .infin. sin ( k .pi. px ) k ##EQU00012## 4-term Fourier
Expansion: f ( x ) = s + a .pi. ( sin ( .pi. px ) + sin ( 3 .pi. px
) 3 + sin ( 5 .pi. px ) 5 + sin ( 7 .pi. px ) 7 ) ##EQU00013##
Sharpness about 8 Factor, s Amplitude, a about 1
[0121] FIGS. 12A-12F demonstrate the golf ball dimple plan shapes
produced in accordance with the parameters of Table 3. In
particular, FIG. 12A shows a dimple plan shape 40 defined by a
square wave function approximated by a four-term Fourier series
having period, p=3, mapped to a circular path. FIG. 12B shows a
dimple plan shape 41 defined by a square wave function approximated
by a four-term Fourier series having period, p=4, mapped to a
circular path. FIG. 12C shows a dimple plan shape 42 defined by a
square wave function approximated by a four-term Fourier series
having period, p=5, mapped to a circular path. FIG. 12D shows a
dimple plan shape 43 defined by a square wave function approximated
by a four-term Fourier series having period, p=6, mapped to a
circular path. FIG. 12E shows a dimple plan shape 44 defined by a
square wave function approximated by a four-term Fourier series
having period, p=7, mapped to a circular path. FIG. 12F shows a
dimple plan shape 45 defined by a square wave function approximated
by a four-term Fourier series having period, p=8, mapped to a
circular path.
Example 3
[0122] The following example illustrates golf ball dimple plan
shapes defined by a low frequency, high amplitude arbitrary
periodic function mapped to a circular path. Table 4, depicted
below, describes the mathematical parameters used to project the
periodic function onto the simple closed path.
TABLE-US-00004 TABLE 4 PLAN SHAPE PARAMETERS OF EXAMPLE 3 Path
Arbitrary Periodic Function Arbitrary Function (f(x)) f ( x ) = s +
a ( cos ( .pi. px ) 3 * sin ( .pi. px ) + sin ( 7 .pi. px ) 7 )
##EQU00014## Sharpness Factor, s about 8 Amplitude, a about 2
[0123] FIGS. 13A-13F demonstrate the golf ball dimple plan shapes
produced in accordance with the parameters of Table 4. In
particular, FIG. 13A shows a dimple plan shape 50 defined by an
arbitrary periodic function having period, p=3, mapped to a
circular path. FIG. 13B shows a dimple plan shape 51 defined by an
arbitrary periodic function having period, p=4, mapped to a
circular path. FIG. 13C shows a dimple plan shape 52 defined by an
arbitrary periodic function having period, p=5, mapped to a
circular path. FIG. 13D shows a dimple plan shape 53 defined by an
arbitrary periodic function having period, p=6, mapped to a
circular path. FIG. 13E shows a dimple plan shape 54 defined by an
arbitrary periodic function having period, p=7, mapped to a
circular path. FIG. 13F shows a dimple plan shape 55 defined by an
arbitrary periodic function having period, p=8, mapped to a
circular path.
Example 4
[0124] The following example illustrates golf ball dimple plan
shapes defined by a low frequency, high amplitude arbitrary
periodic function mapped to an arbitrary path. Table 5, depicted
below, describes the mathematical parameters used to project the
periodic function onto the simple closed path.
TABLE-US-00005 TABLE 5 PLAN SHAPE PARAMETERS OF EXAMPLE 4 Path
Arbitrary Periodic Function Arbitrary Function (f(x)) f (x) = s + a
(cos(.pi.px) * sin(.pi.px).sup.2 - abs(sin(.pi.px))) Sharpness
Factor, s about 8 Amplitude, a about 0.727
[0125] FIGS. 14A-14F demonstrate the golf ball dimple plan shapes
produced in accordance with the parameters of Table 9. In
particular, FIG. 14A shows a dimple plan shape 60 defined by an
arbitrary periodic function having period, p=3, mapped to an
arbitrary path. FIG. 14B shows a dimple plan shape 61 defined by an
arbitrary periodic function having period, p=4, mapped to an
arbitrary path. FIG. 14C shows a dimple plan shape 62 defined by an
arbitrary periodic function having period, p=5, mapped to an
arbitrary path. FIG. 14D shows a dimple plan shape 63 defined by an
arbitrary periodic function having period, p=6, mapped to an
arbitrary path. FIG. 14E shows a dimple plan shape 64 defined by an
arbitrary periodic function having period, p=7, mapped to an
arbitrary path. FIG. 14F shows a dimple plan shape 65 defined by an
arbitrary periodic function having period, p=8, mapped to an
arbitrary path.
Example 5
[0126] The following example illustrates golf ball dimple plan
shapes defined by a low frequency, high amplitude square wave
function mapped to a circular path. The square wave function is
approximated by a four-term Fourier series. Table 6 below describes
the mathematical parameters used to project the periodic square
wave function onto the simple closed path.
TABLE-US-00006 TABLE 6 PLAN SHAPE PARAMETERS OF EXAMPLE 5 Path
Circular Periodic Square Wave (4-term Fourier expansion) Function
Function (f(x)) General Fourier Series: f ( x ) = s + a .pi. k = 1
, 3 , 5 .infin. sin ( k .pi. px ) k ##EQU00015## 4-term Fourier
Expansion: f ( x ) = s + a .pi. ( sin ( .pi. px ) + sin ( 3 .pi. px
) 3 + sin ( 5 .pi. px ) 5 + sin ( 7 .pi. px ) 7 ) ##EQU00016##
Sharpness 8 Factor, s Amplitude, a 1 Period, p 5
[0127] FIG. 17A illustrates a golf ball dimple plan shape 70
produced in accordance with the parameters of Table 6 and having a
0.175 inch nominal dimple diameter. FIG. 17B illustrates profiles
A-A (dashed line) and B-B (solid line), which are cross-sectional
profiles of the dimple at the locations indicated in FIG. 17A.
Profiles A-A and B-B are defined by the polynomial function
y=x.sup.2. Profile A-A has two ends 70-1 and 70-2 where profile A-A
meets the dimple perimeter. Profile A-A has an edge angle,
.PHI..sub.EDGE1, at end 70-1, and an edge angle, .PHI..sub.EDGE2,
at end 70-2. Profile B-B has two ends 70-3 and 70-4 wherein profile
B-B meets the dimple perimeter. Profile B-B has an edge angle,
.PHI..sub.EDGE3, at end 70-3, and an edge angle, .PHI..sub.EDGE4,
at end 70-4. In this example, .PHI..sub.EDGE1 is about
11.00.degree., .PHI..sub.EDGE2 is about 8.900, .PHI..sub.EDGE3 is
about 9.85, and .PHI..sub.EDGE4 is about 9.85. Thus, in this
example: .PHI..sub.EDGE1.noteq..PHI..sub.EDGE2,
.PHI..sub.EDGE3=.PHI..sub.EDGE4,
.PHI..sub.EDGE1.noteq..PHI..sub.EDGE3, and
.PHI..sub.EDGE2.noteq..PHI..sub.EDGE3.
Example 6
[0128] The following example illustrates golf ball dimple plan
shapes defined by a low frequency, high amplitude square wave
function mapped to a circular path. The square wave function is
approximated by a four-term Fourier series. Table 7 below describes
the mathematical parameters used to project the periodic square
wave function onto the simple closed path.
TABLE-US-00007 TABLE 7 PLAN SHAPE PARAMETERS OF EXAMPLE 6 Path
Circular Periodic Square Wave (4-term Fourier expansion) Function
Function (f(x)) General Fourier Series: f ( x ) = s + a .pi. k = 1
, 3 , 5 .infin. sin ( k .pi. px ) k ##EQU00017## 4-term Fourier
Expansion: f ( x ) = s + a .pi. ( sin ( .pi. px ) + sin ( 3 .pi. px
) 3 + sin ( 5 .pi. px ) 5 + sin ( 7 .pi. px ) 7 ) ##EQU00018##
Sharpness 8 Factor, s Amplitude, a 1 Period, p 6
[0129] FIG. 18A illustrates a golf ball dimple plan shape 80
produced in accordance with the parameters of Table 7 and having a
0.175 inch nominal dimple diameter, such that the dimple profile
has a maximum radial distance of 0.175/2. The average radial
dimension for dimple plan shape 80 is about 0.0765 inches,
resulting in an effective dimple diameter of about 0.153
inches.
[0130] FIG. 18B illustrates profiles A-A (dashed line) and B-B
(solid line), which are cross-sectional profiles of the dimple at
the locations indicated in FIG. 18A. Profiles A-A and B-B are
defined by the polynomial function y=x.sup.3-3/5x.sup.2. Profile
A-A has two ends 80-1 and 80-2 where profile A-A meets the dimple
perimeter. Profile A-A has a radial distance of about 0.068 inches
from the center line to end 80-1 and from the center line to end
80-2. Profile A-A has an edge angle, .PHI..sub.EDGE1, at end 80-1
of about 14.2.degree., and an edge angle, .PHI..sub.EDGE2, at end
80-2 of about 14.2.degree.. Profile B-B has two ends 80-3 and 80-4
wherein profile B-B meets the dimple perimeter. Profile B-B has a
radial distance of about 0.085 inches from the center line to end
80-3 and from the center line to end 80-4. Profile B-B has an edge
angle, .PHI..sub.EDGE3, at end 80-3 of about 11.1.degree., and an
edge angle, .PHI..sub.EDGE4, at end 80-4 of about 11.1.degree..
[0131] As shown in FIG. 18B, the surface of the dimple includes a
protruding center portion, the highest point of which lies below
the ball phantom surface and also below the nominal chord plane of
the dimple.
[0132] In a particular aspect of this example, a dimple produced
from dimple plan shape 80 and having a cross-sectional profile
defined by the polynomial function y=x.sup.3-3/5x.sup.2, has an
average edge angle of about 12.5.degree., a maximum edge angle of
about 14.2.degree., and a minimum edge angle of about
11.0.degree..
[0133] Notwithstanding that the numerical ranges and parameters
setting forth the broad scope of the invention are approximations,
the numerical values set forth in the specific examples are
reported as precisely as possible. Any numerical value, however,
inherently contain certain errors necessarily resulting from the
standard deviation found in their respective testing measurements.
Furthermore, when numerical ranges of varying scope are set forth
herein, it is contemplated that any combination of these values
inclusive of the recited values may be used.
[0134] The invention described and claimed herein is not to be
limited in scope by the specific embodiments herein disclosed,
since these embodiments are intended as illustrations of several
aspects of the invention. Any equivalent embodiments are intended
to be within the scope of this invention. Indeed, various
modifications of the invention in addition to those shown and
described herein will become apparent to those skilled in the art
from the foregoing description. Such modifications are also
intended to fall within the scope of the appended claims. All
patents and patent applications cited in the foregoing text are
expressly incorporate herein by reference in their entirety.
* * * * *