Patent Application
20210197029
The present invention relates to golf balls having a polyhedra design that can yield lower drag coefficient than a dimpled sphere. The drag reduction is applicable to other sports equipment in general having a bluff body, such as the head of a golf club or a bike helmet.
For the past 100 years the vast majority of commercial golf balls have used designs with dimples. For the purpose of this invention, a dimple generally refers to any curved or spherical depression in the face or outer surface of the ball. The traditional golf ball, as readily accepted by the consuming public, is spherical with a plurality of dimples, where the dimples are generally depressions on the outer surface of a sphere. The vast majority of commercial golf balls use dimples that have a substantially spherical shape. Some examples of such golf balls can be found in U.S. Pat. Nos. 6,290,615, 6,923,736, and U.S. Publ. No. 20110268833.
It is well established that such depressions can lower the drag coefficient of a ball compared to that of a smooth sphere at the same speed. The drag coefficient is a dimensionless parameter that is used to quantify the drag force of resistance of an object in a fluid. The drag force is always opposite to the direction in which the object travels. The drag coefficient for a golf ball is defined as CD=2*Fd/(ρ*U2*A), where Fd is the drag force, p is the density of the fluid in which the object is moving, U is the speed of the object and A is the cross sectional area. For a sphere the cross-sectional area is πD2/4, where D is the diameter of the ball.
In dimpled spheres (shown by dashed and dotted black lines) the drag crisis happens at a much lower critical Reynolds number (Re<100,000). In general, a dimpled sphere will have a 50% or more reduction in drag compared to a smooth sphere in the range of Reynolds number from 100,000 to 250,000. A golf ball in flight during a driver shot can experience a Reynolds number ranging from 220,000 at the beginning of the flight to 60,000 at the end of the flight. It is therefore very important that a golf ball design can achieve a very low drag coefficient in that range. The critical value of the Reynolds number, as well as the attained minimum drag coefficient in the post-critical regime depend on the dimple geometry and arrangement. In general, as the aggregate dimple volume, measured as the sum of a dimple volume of each dimple of the plurality of dimples, decreases the critical Reynolds is increased and the drag coefficient in the post-critical regime increases. As the aggregate dimple volume approaches zero the drag curve approaches that of a smooth sphere. Drag is one of the two forces (the other being lift) that influence the aerodynamic performance of a golf ball. To increase the carry distance on a driver shot, the distance a golf ball travels during the flight, the dimple design is a balance between achieving the lowest critical Reynolds number and the lowest drag coefficient in the post-critical regime.
However, dimples are the only configuration known to provide a golf ball having a reduced drag coefficient.
Accordingly, it is an object of the invention to provide a golf ball having minimal drag coefficient. It is a further object of the invention to provide a golf ball having reduced drag coefficient compared to a golf ball having only dimples. It is another object to provide a golf ball with minimal drag coefficient that is non-dimpled. In one embodiment, the golf ball has a plurality of flat faces with sharp edges and points that collectively form a polyhedron. These and other objects of the invention, as well as many of the intended advantages thereof, will become more readily apparent when reference is made to the following description, taken in conjunction with the accompanying drawings.
In describing certain illustrative, non-limiting embodiments of the invention illustrated in the drawings, specific terminology will be resorted to for the sake of clarity. However, the invention is not intended to be limited to the specific terms so selected, and it is to be understood that each specific term includes all technical equivalents that operate in similar manner to accomplish a similar purpose. Several embodiments of the invention are described for illustrative purposes, it being understood that the invention may be embodied in other forms not specifically shown in the drawings.
The present invention is directed to a golf ball design based on polyhedra that can have reduced drag coefficient compared to a dimpled sphere. In one embodiment, a family of golf ball designs are made up of convex polyhedra whose vertices lie on a sphere. A polyhedron is a solid in three dimensions with flat polygonal faces, straight sharp edges and sharp corners or vertices.
The body 110 defines a circumscribed sphere 102, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 104 of the body 110. The sphere and the diameter provide a reference for the size of the golf ball. The Rules of Golf, jointly governed by the R&A and the USGA, state that the diameter of a “conforming” golf ball cannot be any smaller than 1.680 inches. For the purpose of a golf ball the diameter of the circumscribed sphere is at least 1.68 in. The vertices 122a, 122b of the polyhedron are the only points 104 on the polyhedron that lie on the sphere. Any point along the edges 124a, 124b or on the faces 120a, 120b of the polygons lies below the surface of the circumscribed sphere.
The golf ball body 110 is a polyhedron that is made from first faces 120a and second faces 120b. As shown, the first faces 120a have a first shape, namely pentagons, and the second faces 120b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 120a and 150 hexagons 120b (a hexagon-to-pentagon ratio of 12.5:1), each having corners or points 122a, 122b connected by boundaries such as straight lines or edges 124a, 124b. In various other embodiments, other quantities and/or ratios of such pentagons 120a and hexagons 120b can be used. However, the number of polygons and the angle between them determine when the drag coefficient will start to drop and how low it will become. In general, as the number of faces is increased the drag crisis occurs at higher Reynolds number and the drag coefficient decreases. The first and second faces 120a, 120b form the pattern 116.
The edges 124 are sharp, in that the faces are at an angle with respect to one another.
Each face 120 is immediately adjacent and touching a neighboring face 120, such that each edge 124 forms a border between two neighboring faces 120 and each point 122 is at the intersection of three neighboring faces 120. And, each point 122 is at an opposite end of each linear edge 124 and is at the intersection of three linear edges 124. Accordingly, there is no gap or space between adjacent neighboring faces 120, and the faces 120 are contiguous and form a single integral, continuous outer surface 112 of the ball 100.
A golf ball usually has a rubber core and at least one more layer surrounding the core. The pattern 116 is formed on the outermost layer. The pattern is based on an icosahedron shown in
In the icosahedron 170 five equilateral triangles 180 meet at each of its twelve vertices 182. The 12 pentagons 120a of the golf ball 100 shown in
Table 1 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the golf ball. Table 1 lists the coordinates x, y, and z of all of the vertices 122 of golf ball 100. Faces are constructed by connecting the vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 104 and are with respect to the center of the sphere. The golf ball 100 contains 320 vertices 122, 480 straight edges 120 and 162 polygonal faces 120.
The particular polyhedron can be any suitable configuration, and in one embodiment is a class of solids called Goldberg polyhedra that comprises convex polyhedra that are made entirely from a combination of rectangles, pentagons and hexagons. Goldberg polyhedra are derived from either an icosahedron (a convex polyhedron made from 12 pentagons), or an octahedron (a convex polyhedron made from 8 triangles) or a tetrahedron (a convex polyhedron made from 4 triangles). An infinite number of Goldberg polyhedra exist, as shown in https://en.wikipedia.org/wiki/List_of_geodesic_polyhedra_and_Goldberg_polyhedra.
However, it will be recognized that the invention can utilize any convex polyhedron (that is, a polyhedron made up of polygons whose angle is less than 180 degrees), though in one embodiment on such polyhedron is a convex polyhedron with sharp edges, flat faces forming a single plane, and adjacent faces having an angle of less than 180 degrees between them.
The particular configuration (Goldberg polyhedral 162 faces and icosahedral symmetry) can only be achieved with a combination of hexagons and pentagons. However other geometries with around 162 faces may be possible to do using only pentagons or only hexagons. Other embodiments of the invention can include a pattern with various geometric configurations. For example, the pattern can be comprised of more or fewer of hexagons and pentagons than shown. Or it can comprise all hexagons, all pentagons, or no hexagons or pentagons but instead one or more other shapes or polyhedrals having flat faces and sharp edges. One other shape can be formed, for example, by splitting each hexagon into 6 triangles or each pentagon into 5 triangles, which provides a similar drag coefficient. One embodiment can include any of the Goldberg polyhedra with a combination of pentagons and hexagons or even a convex polyhedral made of triangles or squares.
It is further noted that flat faces give lower drag and have the uniqueness of not being dimples (curved indentations). The flat faces only provide points of the faces that lie on the circumscribed sphere. The sharp edges are defined by the angle between two adjacent faces. In addition, the edges forming the boundaries between the two adjacent faces are flat and not excessively rounded. An example of a non-sharp edge is shown in
As the number of polygons increases (i.e. from 162 faces to 312 faces) the angle between the faces increases too and approaches 180 degrees. For the embodiments that are based purely on polyhedral shape such as those shown in
In
The body 210 defines a circumscribed sphere 202, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 204 of the body 210. The vertices 222a, 222b of the polyhedron are the only points 204 on the polyhedron that lie on the sphere. Any point along the edges 224a, 224b or on the face of the polygons lies below the surface of the circumscribed sphere.
The golf ball body 210 is a polyhedron that is made from first faces 220a and second faces 220b. As shown, the first faces 220a have a first shape, namely pentagons, and the second faces 220b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 220a and 180 hexagons 220b (a hexagon-to-pentagon ratio of 15:1), each having corners or points 222a, 222b connected by boundaries such as straight lines or edges 224a, 224b. In various other embodiments, other quantities and/or ratios of such pentagons 220a and hexagons 220b can be used. The first and second faces 220a, 220b form the pattern 216. The edges 224 are sharp, in that the radius of curvature of an edge is less than 0.001 D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use.
The geometric shape of the embodiment illustrated in
Each face 220 is immediately adjacent and touching a neighboring face 220, such that each edge 224 forms a border between two neighboring faces 220 and each point 222 is at the intersection of three neighboring faces 220. And, each point 222 is at an opposite end of each linear edge 224 and is at the intersection of three linear edges 224. Accordingly, there is no gap or space between adjacent neighboring faces 220, and the faces 220 are contiguous and form a single integral, continuous outer surface 212 of the ball 200.
The pattern is based on an icosahedron shown in
Table 2 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the golf ball. Table 1 lists the coordinates x, y, and z of all of the vertices 220 of polyhedron of golf ball 200. Faces are constructed by connecting the vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 204 and are with respect to the center of the sphere. The golf ball 200 contains 380 vertices 222, 570 straight edges 220 and 192 polygonal faces 220.
From a visual perspective the above designs of
The graph reveals that as the number of faces increases the drag crisis shifts towards a higher Reynolds number and the CD in the post-critical regime decreases. This feature can be taken into advantage when designing a golf ball to tailor the needs of a golfer. For an amateur golfer a golf ball in flight during a driver shot can experience a Reynolds number ranging from 180,000 at the beginning of the flight to 60,000 at the end of the flight. For a professional golfer a golf ball in flight during a driver shot can experience a Reynolds number ranging from 220,000 at the beginning of the flight to 80,000 at the end of the flight.
In other words, the range of Reynolds number experienced by a golf ball for an amateur golfer is lower than that for a professional golfer. A golf ball with more polygon faces might suit the needs of a professional golfer while a golf ball with less polygon faces might suit the needs of an amateur golfer.
The advantage of a design based on a convex polyhedron such as golf ball 200 with respect to a dimpled golf ball is now discussed. A comparison of the drag curve of the polyhedron with 192 faces, namely golf ball 200, against a dimpled sphere is shown in
A comparison of the drag curve of the golf ball 100 against the same dimpled sphere is shown in
The embodiment shown in
In another embodiment of the present invention a convex polyhedron is shown in
It is important to note that the polyhedra described above and shown in
However, golf ball geometries that are based on any convex polyhedra and either include or omit dimples are contemplated. In particular, for any convex polyhedra at least one of the faces may include one or more dimples. For example,
However, in this embodiment each face 520 of the polyhedron contains one dimple 560. The dimples 560 have a substantially spherical shape and are created by subtracting spheres 570 from each of the faces 510 of the polyhedron 500. Table 3 lists the coordinates x, y and z of the center of the spheres 570 along with the diameter d of the spheres 570. The coordinates x, y, z and the diameter of the spheres d are normalized by the diameter D of the circumscribed sphere 504. The coordinates x, y, and z are with respect to the center of the circumscribed sphere. The total dimple volume ratio, defined as the aggregate volume removed from the surface of the polyhedron divided by the volume of the polyhedron, is 0.317%.
As shown, the dimples 560 are each located at the center of each 510. However, the dimples 560 can be positioned at another location such as offset within each face 510, or overlapping two or more faces 510. In addition, while spherical dimples 560 are shown, the dimples 560 can have any suitable size and shape. For example, non-spherical depressions can be utilized, such as triangles, hexagons, pentagons. And the dimples need not all have the same size and shape, for example there can be dimples with more than one size and more than one shape.
The effect that the addition of dimples has on the drag coefficient is now discussed. A comparison of the drag curve of this embodiment against that of golf ball 100 is shown in
The body 610 defines a circumscribed sphere 602, which is the smallest sphere containing the polyhedron, and is shown with a dashed line about the outermost points 604 of the body 610. The vertices 622a, 622b of the polyhedron are the only points 604 on the polyhedron that lie on the sphere. Any point along the edges 624 or on the face of the polygons lies below the surface of the circumscribed sphere.
The golf ball body 610 is a polyhedron that is made from first faces 620a and second faces 620b. As shown, the first faces 620a have a first shape, namely pentagons, and the second faces 620b have a second shape different than the first shape, namely hexagons. All of the faces are flat and each such face forms a single plane. In one embodiment, there are 12 pentagons 620a and 300 hexagons 620b (a hexagon-to-pentagon ratio of 25:1), each having corners or points 622a, 622b connected by boundaries such as straight lines or edges 624a, 624b. The first and second faces 620a, 620b form the pattern 616. The edges 624 are sharp, in that the radius of curvature of an edge is less than 0.001 D, where D is the diameter of the circumscribed sphere. Ideally all edges should be the sharpest possible. A radius of curvature of 0 is the case with the sharpest possible edges, though golf ball edges might become a little rounded during manufacture or follow use.
The geometric shape of the embodiment illustrated in
Each face 620 is immediately adjacent and touching a neighboring face 620, such that each edge 624 forms a border between two neighboring faces 620 and each point 622 is at the intersection of three neighboring faces 620. And, each point 622 is at an opposite end of each linear edge 624 and is at the intersection of three linear edges 624. Accordingly, there is no gap or space between adjacent neighboring faces 620, and the faces 620 are contiguous and form a single integral, continuous outer surface 612 of the ball 600.
The pattern is based on an icosahedron shown in
Table 4 lists the coordinates of all the vertices and how each face is formed, which defines the geometry of the polyhedron. Table 4 lists the coordinates x, y, and z of all of the vertices 622 of polyhedron of golf ball 600. Faces are constructed by connecting the group of vertices with straight lines. The coordinates x, y, z are normalized by the diameter D of the circumscribed sphere 604 and are with respect to the center of the sphere. The golf ball 600 contains 620 vertices 622, 930 straight edges 624 and 312 polygonal faces 620.
Each of the face 620 of the polyhedron contains one dimple 690. The dimples 690 have a substantially spherical shape and are created by subtracting spheres 670 from the face 620 of the polyhedron. Table 5 lists the coordinates x, y and z of the center of the spheres 670 along with the diameter d of the spheres 670. The coordinates x, y, z and the diameter of the spheres d are normalized by the diameter D of the circumscribed sphere 604. The coordinates x, y, and z are with respect to the center of the circumscribed sphere.
A comparison of the drag curve of this embodiment against that of a commercial golf ball, namely the Bridgestone Tour is shown in
Other sizes and shapes of dimples are contemplated by the present invention, and any suitable size and shape dimple can be utilized in the golf balls of
The following documents are incorporated herein by reference. Achenbach, E. (1972). Experiments on the flow past spheres at high Reynolds numbers. Journal of Fluid Mechanics. Harvey, P. W. (1976). Golf ball aerodynamics. Aeronautical Quarterly. J. Choi, W. J. (2006). Mechanism of drag reduction by dimples on a sphere. Physics of Fluids, 149-167. Ogg, S. S. (2001).
It is further noted that the description and claims use several positional, geometric or relational terms, such as neighboring, circular, spherical, round, and flat. Those terms are merely for convenience to facilitate the description based on the embodiments shown in the figures. Those terms are not intended to limit the invention. Thus, it should be recognized that the invention can be described in other ways without those geometric, relational, directional or positioning terms. In addition, the geometric or relational terms may not be exact. For instance, edges may not be exactly linear, hexagonal, pentagonal or spherical, but still be considered to be substantially linear, hexagonal, pentagonal or spherical, and faces may not be exactly flat or planar but still be considered to be substantially flat or planar because of, for example, roughness of surfaces, tolerances allowed in manufacturing, etc. And, other suitable geometries and relationships can be provided without departing from the spirit and scope of the invention.
In addition, while the invention has been shown and described with boundaries formed by straight edges 124, 224, 324, 424, 524, 624 and points 122, 222, 322, 422, 522, other suitable boundaries can be provided such as curves with or without rounded points. And the boundaries need not be sharped, but can be curved. And other suitable shapes can be utilized for the faces. And, while the invention has been described with a certain number of faces and/or dimples, other suitable numbers of faces and/or dimples, more or fewer, can also be provided within the spirit and scope of the invention. Within this specification, the various sizes, shapes and dimensions are approximate and exemplary to illustrate the scope of the invention and are not limiting. The faces need not all be the same shape and/or size, and there can be multiple sizes and shapes of faces.
The sizes and the terms “substantially” and “about” mean plus or minus 15-20%, and in one embodiment plus or minus 10%, and in other embodiments plus or minus 5%, and plus or minus 1-2%. In addition, while specific dimensions, sizes and shapes may be provided in certain embodiments of the invention, those are simply to illustrate the scope of the invention and are not limiting. Thus, other dimensions, sizes and/or shapes can be utilized without departing from the spirit and scope of the invention.
Use of the term optional or alternative with respect to any element of a claim means that the element is required, or alternatively, the element is not required, both alternatives being within the scope of the claim. Use of broader terms such as comprises, includes, and having may be understood to provide support for narrower terms such as consisting of, consisting essentially of, and comprised substantially of. Accordingly, the scope of protection is not limited by the description set out above but is defined by the claims that follow, that scope including all equivalents of the subject matter of the claims. Each and every claim is incorporated as further disclosure into the specification and the claims are embodiment(s) of the present disclosure.
The foregoing description and drawings should be considered as illustrative only of the principles of the invention. The invention may be configured in a variety of shapes and sizes and is not intended to be limited by the embodiment shown. In addition, the statements made with respect to one embodiment apply to the other embodiments, unless otherwise specifically noted. For example, the statements regarding
Numerous applications of the invention will readily occur to those skilled in the art. Therefore, it is not desired to limit the invention to the specific examples disclosed or the exact construction and operation shown and described. Rather, all suitable modifications and equivalents may be resorted to, falling within the scope of the invention.
This application claims the benefit of U.S. Provisional Application No. 62/616,861, filed Jan. 12, 2018, the entire contents of which are incorporated herein by reference.
| Filing Document | Filing Date | Country | Kind |
|---|---|---|---|
| PCT/US2019/013052 | 1/10/2019 | WO | 00 |
| Number | Date | Country | |
|---|---|---|---|
| 62616861 | Jan 2018 | US |
