Patent Publication Number: US-2017361179-A1

Title: Adjustable golf putter head having teeth

Description:
BACKGROUND OF THE INVENTION 
     This invention relates to a golf putter. More particularly, this invention relates to a golf putter head assembly that includes a mechanism for adjusting a loft angle of the front face of the putter head. 
     When a conventional putter strikes a golf ball, the ball slides forward on the putting green with little or no spin. Because of the large (backwards directed) sliding friction between the ball and the green, this sliding motion is erratic and irregular, causing the ball&#39;s trajectory to deviate from its intended direction and speed. This frictional force does, however, cause the ball to both gain forward spin and lose forward speed at a continuously increasing rate. This process proceeds until pure rolling sets in when the point of contact between the ball and the green is instantaneously at rest. (The forward linear motion of the contact point is then exactly canceled by the backward rotational motion.). From this time on, the friction force becomes very small (because the rolling friction force is small) and so the ball continues to roll at a slowly decreasing speed. During this rolling phase of the motion, the ball&#39;s trajectory is smooth and regular because the green exerts almost no frictional force on the ball. 
     It is obviously highly desirable to eliminate the initial sliding phase of the ball&#39;s motion, which can last for several feet. There have therefore been a number of attempts to design and construct putters that claim to accomplish that. For reasons that will be explained below, none of these putters can do more than place a small amount of forward spin onto a struck ball, and all have unintended negative consequences. 
     OBJECTS OF THE INVENTION 
     It is an object of the present invention to provide an improved gold putter head assembly. 
     It is a further, more particular object of the present invention to provide a golf putter head assembly which imparts a rolling motion to a golf ball immediately upon striking of the ball with the putter head. 
     Another particular object of the present invention is to provide such a golf putter head assembly which incorporates means for adjusting the loft angle of the front face of the putter head. 
     These and other objects of the present invention will be apparent from the drawings and descriptions herein. It is to be noted that any single embodiment of the invention may not achieve all of the objects of the invention, but that every object is attained by at least one embodiment. 
     SUMMARY OF THE INVENTION 
     The present invention aims to present a class of putters that cause a struck golf ball to immediately start moving forward with a perfect rolling motion, with no unintended negative consequences. This is accomplished by incorporating three properties into the design of the putter: (1) The face of the putter head is inclined forward at a relatively large initial angle; (2) the face of the putter head incorporates a plurality of short forward facing teeth; (3) the face of the putter head can be manually rotated forward or backward such that the final loft angle can be set at the optimal value for rolling, as determined by the characteristics of the golfer, the putting club, the putting green, and the golf ball. The purpose of the initial loft angle is to impart spin to the struck ball by impacting it above its center of mass, thus creating a torque on the ball. The purpose of the teeth is to direct the applied force forward so that it does not push the ball downward. The purpose of the final loft angle adjustment is to create the optimal loft angle, by taking the above characteristic variables into account, so that the struck ball acquires the correct spin for pure rolling. 
     Each of these three properties has been tried before for other purposes, but what is novel here, in addition to the combination, is the theoretical determination of the range of useable teeth angles, the design of the loft angle adjustment mechanism, and the use of this mechanism to experimentally determination and set the final loft angle. Previous attempts that used a putter face inclined at a large forward angle forced a struck ball downward, into the green. Previous attempts that used a putter face with teeth assumed that such teeth would provide increased friction between the face and the ball, or a lifting force on the ball, so that the ball would acquire forward spin. The forward spin possibly created by these effects is insignificant. Previous attempts to design a rotatable putter face used impractical or ineffective mechanisms. 
     If a flat putter face moving horizontally is inclined at a forward angle A 0  when it strikes a spherical golf ball of uniform density, and if the consequent force exerted on the ball is in the horizontal direction, then there is a unique value of A 0  (A 0 =arcsin(0.4)=23.58°) that causes the struck ball to execute pure rolling when it rebounds off the face. The problem is that none of the assumptions listed in the previous sentence are correct.
         (1) Essentially no golfer produces a club swing that causes the putter head to move in a perfectly horizontal direction as it strikes a ball. Players almost always hit a ball during a small downswing or upswing.   (2) Golf balls are never perfectly spherical. They are covered with a plurality of dimples that are necessary for a flying ball to minimize air drag and maximize lift. It is, furthermore, impossible for these dimples to be uniformly distributed over the spherical surface of the ball.   (3) Essentially no golf balls have uniform density. Two-piece balls typically have an interior central sphere covered by a thin outer shell. Three-piece balls typically have a solid rubber or liquid core covered by elastic windings and an outer shell. Four-piece balls are also available.   (4) The force on a ball created by a flat forwardly inclined putter face will not be in the horizontal direction. It will instead be directed towards the center of the ball, in the direction perpendicular to the compression plane on the ball created by the impact.       

     The present invention overcomes these problems by incorporating two elements into the putter head in addition to the forwardly inclined face. The first new element is the incorporation of forward facing teeth, whose angles and directions are chosen so that the impact force is mainly directed forward. The second new element is the incorporation of a means to rotate the putter face (with attached teeth) in the forward or backward direction. It is crucial to have the ability to make this loft angle adjustment in order to create the desired pure rolling motion arising from every impact. This is because the value A of the correct final loft angle will depend on the detailed characteristics of the following relevant variables. 
     (1) The swing trajectory of the golfer, which determines the direction of the putter head velocity in the vertical plane when it strikes a ball. Relative to the horizontal direction, this direction can be either positive (upswing hit) or negative (downswing hit). 
     (2) The moment of inertia (MOI) of the golf ball about an axis through it&#39;s center of mass (COM). If the ball were a perfect sphere of radius r, mass m, and uniform density, this MOI would be I0=2 mr 2 /5. As noted above, a realistic golf ball is neither perfectly spherical nor uniformly dense, so the actual MOI is different than I0, often substantially different, and this means that the optimal face angle can be substantially different from A 0 . 
     (3) The size and orientations of the teeth attached to the putter&#39;s face. These teeth are necessary to direct the force applied by the putter onto the ball towards the forward direction. There is a range of teeth angles that can accomplish this, but the applied force is never perfectly horizontal because it depends on the intricate details of the interaction between the teeth and the ball. 
     (4) Other properties of the putter, such as shaft length and shaft orientation and location, and other properties of the putting green, such as grass height and density. 
     It is, of course, not possible to take these variables into account analytically in order to determine mathematically the optimal value A of the loft angle. This is where the ability to adjust this angle comes into play. By hitting a chosen ball on a putting green using a range of loft angles, a given golfer can determine the loft angle that most perfectly suits his stroke. Once this angle is so determined and locked in place, the golfer&#39;s struck balls will move forward with close to a pure rolling motion immediately after leaving the putter face. 
     A golf club head assembly in accordance with the present invention comprises a body member having a top-forward angled front face, the face comprising a plurality of generally forward-facing teeth. The golf club head assembly further comprises a loft angle adjustment mechanism attached to the body member for adjusting a loft angle of the face. 
     In the golf club head assembly one or more of the plurality of teeth are positioned on the face to contact a golf ball above a center of mass of the ball. The teeth each comprise a top forward face surface having an angle B 1  from a horizontal plane and a bottom forward face surface having an angle B 2  from the horizontal plane, wherein B 1  and B 2  are chosen such that a net vertical force exerted on the ball from the teeth is approximately zero when the teeth strike a ball. Preferably, the following solution equation is satisfied: 
       Sin(2 B 1)Cos( B 1)=Sin(2 B 2)Cos( B 2)5. 
     The ordered pair (B 1 , B 2 ) is preferably one of the following: (90°, 0°), (60°, 13.2°), (45°, 45°), (30°, 40.6°), and (15°, 57.7°). 
     Pursuant to a feature of the present invention, the loft angle adjustment mechanism includes a hosel element releasably attached to a bar rotatably mounted to an upper side of the body member parallel to the face. The bar is rotatably journaled or mounted in a pair of bar supports attached to the body member. 
     Pursuant to another feature of the invention, the loft angle adjustment mechanism includes set screws residing in threaded holes in the bar supports, the set screws operable to releasably lock the bar in a selected orientation relative to the body member, where the selected orientation of the bar corresponds to and determines the loft angle of the front face. 
     The bar preferably has circular cross-sections at toe and heel ends and a polygon cross-section (e.g., hexagonal) in at least part of the section of the bar between the toe and heel ends. A bottom end of the hosel element is then formed with a hole having a polygonal shape geometrically similar to the polygon cross-section of the bar so that the hosel element is slidable on the bar in a toe-heel direction and is rotatably fixed to the bar. 
     The hosel element may be provided with a set screw disposed in a threaded hole in the hosel element, for releasably locking the hosel element to the bar. 
     The hosel element may include a lower section attachable to the bar and an upper section into which a shaft is insertable, the lower section and the upper section being rotatably connected to one another in order to adjust a lie angle of a respective golf club. 
     The golf club head assembly according to claim  13 , wherein the lower section and the upper section of the hosel element are releasably locked together at a desired relative angle by a threaded bolt inserted in a threaded hole residing in an overlapping portion of the lower section and the upper section. 
     The loft angle is typically set to a value to optimally eliminate sliding friction at a start of a putting motion beginning with striking of the ball. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a schematic perspective view of a golf putter head assembly in accordance with the present invention. 
         FIG. 2  is a schematic front elevational view, partly in cross-section, of a shaft attachment and integral loft-angle adjustment mechanism included in the golf putter head assembly of  FIG. 1 . 
         FIG. 3  is a schematic front elevational view of a mounting bar included in the shaft attachment and integral loft-angle adjustment mechanism of  FIGS. 1 and 2 . 
         FIG. 4  is an end elevational view of the bar of  FIG. 3 . 
         FIG. 5  is a side elevational view, partly in cross-section, of a bar-mounting post included in the shaft attachment and integral loft-angle adjustment mechanism of  FIGS. 1 and 2 . 
         FIG. 6  is side elevational view, partly in cross-section, of the shaft attachment of  FIGS. 1 and 2 . 
         FIG. 7  is a diagram of a putter head striking a ball, showing impact of a conventional putter head with a golf ball, with an exerted force F being horizontal and directed towards the center of the ball. 
         FIG. 8  is a diagram of vectors pertaining to the motion of a golf ball struck by a club such as a putter, where v is the linear speed of the COM, w is the rotational speed about the horizontal axis through the COM, and f is the sliding friction force between the ball and the green. 
         FIG. 9  is a force diagram of an off-center impact on a golf ball showing a horizontal force F on a ball at a distance h above the center. 
         FIG. 10  is a diagram of a putter head striking a ball, the putter head having a rib or tooth on a vertically oriented front face, showing imparted force above center of the ball. 
         FIG. 11  is a diagram of a putter head striking a ball, the putter head having a forwardly inclined front face striking a ball of radius r a distance h above center, where the loft angle a is arcsin(h/r). 
         FIG. 12  is a diagram similar to  FIG. 11 , showing a force vector impinging on the ball owing to the forwardly inclined putter face, the force being directed towards the center of the ball. 
         FIG. 13  is a diagram of a putter head striking a ball, the putter head having a forwardly angled front face and an array of horizontal ribs or teeth. 
         FIG. 14  is a diagram of one of the teeth on the front face of the putter head shown diagrammatically in  FIG. 13 , with a representation of geometrical parameters, namely angles A, B 1 , B 2 , C, and D and side lengths a, b, and c, and where d is the vertical height of the tooth. 
         FIG. 15  is a diagram depicting force vectors f 1  and f 2  exerted on a golf ball by a tooth or rib of the putter head of  FIG. 13 . 
         FIG. 16  is a diagram of six alternative front face designs for the putter head of  FIG. 13 . 
         FIG. 17  is a diagram of another six alternative front face designs for the putter head of  FIG. 13 . 
         FIG. 18  is a diagram similar to  FIG. 13 , showing the first design of  FIG. 16  (B 1 =90°, B 2 =0°) juxtaposed to a circle representing a golf ball at a moment of impact. 
         FIG. 19  is a schematic perspective view of another putter head assembly in accordance with the present invention. 
         FIG. 20  is a schematic rear and side perspective view of a shaft attachment and integral loft-angle adjustment mechanism included in the golf putter head assembly of  FIG. 19 . 
         FIG. 21  is a schematic front and side perspective view of the shaft attachment and integral loft-angle adjustment mechanism of  FIG. 20 . 
     
    
    
     DETAILED DESCRIPTION 
     A preferred embodiment of a golf putter head assembly having a loft-angle adjustment mechanism is shown in  FIG. 1 . The assembly includes two parts, a putter head body  1  with an attached handle-like structure  20  on top, and a hosel component  22  connected thereto. 
     As shown in  FIG. 6 , hosel component  22  includes an upper section  9  and a lower section  8 , adjustably fastened to one another via a bolt  12   a , and further includes a slideable annulus-like member  2  that is attached to the bottom of the lower hosel section  8 . 
     Handle-like attachment structure  20  includes a pair of handle supports or posts  5  which are vertically oriented cylindrical sections attached near the toe and the heel of the putter head body  1 . Handle-like attachment structure  20  further includes a horizontally oriented rotatable solid cylindrical section or handle bar  6  lying above the top of the putter body  1 , parallel to a front face  24  thereof, the bar  6  being inserted into supports or posts  5 . Bar  6  has a circular cross-section except for a hexagonal length or portion  7  that extends from near a center of the bar  6  to near the heel-end of the bar. A central hole  10  in the annulus member  2  is hexagonal and concentric with hexagonal portion  7  of horizontal bar  6 , and has an inner diameter that is slightly larger than the diameter of this hexagonal portion  7 . The annulus  2  can therefore slide parallel to and along bar  6 . 
     Hosel  22  is carries annulus member  2  at a lower end. The inclination of upper hosel section  9  is adjustable relative to lower hosel section  8  so that the lie angle can be adjusted as desired. 
     As the cylindrical bar section  6  is rotated within the supports  5 , the loft angle (orientation of front face  24 ) increases or decreases. After the desired loft angle is set, the section can be locked into place. This device is described in greater detail below. 
     PRIOR ART 
     Macera (U.S. Pat. No. 4,664,385) discloses a putter head whose flat face is forwardly inclined, preferably from 20° to 25°, and most preferably at the angle a 0 =23.58° from the vertical. Macera claims that this putter will cause a struck ball to roll smoothly forward. This claim is correct if the following four conditions are satisfied. (See the DETAILS—THEORY section below.)
         (1) The struck golf ball has uniform density.   (2) The struck golf ball is perfectly spherical.   (3) The putter head is moving parallel to the putting surface when the ball is struck.   (4) The putter head exerts a perfectly horizontal force on the struck ball throughout the impact.       

     The problem is that these conditions are essentially never satisfied in practice. If (1) and (2) were satisfied, the moment of inertia (MOI) of the ball would be 10=2 mr 2 /5, where m is the mass and r is the radius of the ball. However, golf balls never have uniform density. As noted in the above SUMMARY section, contemporary balls have two, three, or four interior spherical sections of different densities. This can change the ball MOI and the optimal forward face angle dramatically. (For example, if most of the ball weight were concentrated in a central sphere of radius r/2, the MOI would decrease to approximately I0/4, and the optimal forward face angle would decrease to approximately 5.74°. Increasing the MOI by a factor of 2 increases the optimal angle to 53.13°. (Good golfers prefer a small MOI to get more spin and longer drives, whereas poor golfers prefer a large MOI to reduce hooking and slicing.) The ball&#39;s deviation from a perfect sphere also changes the optimal angle, but this is a much smaller effect. 
     As also noted above, the putter head is rarely moving perfectly horizontally when the ball is struck. This can have a small or large effect on the value of the optimal face angle, depending on the mechanics of a golfer&#39;s putting stroke. Finally, a putter with a smooth forwardly angled face will not produce a horizontal force on a struck ball. The force will instead be directed towards the center of the ball, in the direction perpendicular to the flat tangential compression plane created on the ball by the impact. This effect therefore not only changes the optimal angle in a large and uncontrollable way, it also causes the struck ball to be pushed into the putting surface during the impact. (Macera acknowledges that he made the assumptions (1) and (3), but not (2) and (4).) The current inventive putter solves these problems by having teeth that direct the applied force forward, and by having a loft angle adjustment mechanism. These devices together make it possible to impart the desired rolling motion on a struck ball immediately after the impact. 
     Ma (U.S. Pat. No. 5,348,301) also discloses a putter head whose face is forwardly inclined, preferably from 3° to 15° from the vertical. The disclosed face has a plurality of lateral grooves. Ma claims that this putter will shorten the impact time, thus reducing the negative effect of the trembling of a golfer&#39;s hands during the impact. This claim is not correct because the struck ball leaves the club face well before an impulse can travel from the head to the player&#39;s hands and back. Ma also claims that the grooves on the face serve to increase the force of rotation on the ball. This claim is also not correct because of the reasons stated above. (Unlike the teeth in the inventive putter described herein, grooves cannot change the direction of the applied force.) 
     Rife (U.S. Pat. Nos. 5,618,239 and 5,709,616) discloses a putter face having a plurality of identical symmetrical adjacent V-shaped grooves perpendicular to the face. Some embodiments of these grooves depict tooth-like structures on the face similar to the teeth on the putter described herein. Rife claims that these grooves grip a struck ball creating a lifting action that creates overspin and eliminates the need for different face loft angles. This claim is not proved, but even if correct the effect is insignificant. Any possible created lifting force causing overspin is negligible compared to the large forward force causing forward sliding of the ball. 
     For example, a 12-foot putt requires an initial forward ball speed of about 12 ft/sec (more or less depending on the characteristics of the grass in the putting surface). This requires that the putter exert an average forward force of about 40 lbs on the struck golf ball, a force that compresses the ball between 0.025″ and 0.035″, depending on the hardness of the ball. In order for the Rife putter to impart on the ball enough spin w for rolling (w=v/r), the upward force created by the teeth penetration would have to be about 15 lbs (more or less depending on the MOI of the ball). It is not possible for the upward force caused by the small penetration of the ball by the teeth to create a force that is nearly that large. Such a force would cause the ball to shoot up 5″ in the air. The amount of spin actually created by the teeth penetration is miniscule. (In general, the average forward force needed to create a forward speed v is F=my/t, where t is the impact time; and the average upward force needed to create a rotation speed w is f=qmrw/t, where q specifies the MOI of the ball as I=qmr 2 . Pure rolling requires v=wr, and thus requires an average vertical force of f=qF. This is f=2F/5 for the uniform density value q=2/5. In general, the relation f=qF requires that the upward average force f is the substantial fraction q of the average horizontal impact force F and is therefore much too large to be obtainable by a lifting action created by grooves or teeth.) 
     The Rife putter thus cannot come close to creating an initial pure rolling motion. If an appropriate embodiment of the grooves disclosed by Rife is combined with the inclined putter face disclosed by Macera, the result would be a putter head that superficially resembles the head disclosed herein, but, without the specific suitable teeth angles and the ability to adjust the loft angle, such a putter also could not create an initial pure rolling motion. Note in this connection that the purpose of the teeth in the putter face disclosed herein is to direct the applied force on the ball into the forward direction in order to create spin from the applied torque, whereas the purpose of the grooves in the Rife putters is to create spin from a lifting action. The torque created by the herein disclosed mechanism is easily large enough to create the desired spin because it is in the same direction as the applied forward force, whereas the lift mechanism is not nearly large enough because it is in the direction perpendicular to the applied forward force. 
     Swash (U.S. Pat. No. 5,637,044) discloses a putter head whose face contains a plurality of grooves, which are triangular in some embodiments and superficially resemble the teeth of the head disclosed herein. Swash claims that one advantage of this construction is that it enables spin to be imparted to a struck ball if the putting stroke has an upward element. This claim is correct but it is a very small effect and any small spin created by an upswing is in the opposite direction of the spin necessary for forward rolling. The effective loft created by an upswing is backwards, causing an impact (and applied torque) below the center of the ball that results in the creation of a backward spin. Spin in the correct direct for rolling could be created if a ball were struck during a downswing, but the amount of this spin could not be nearly large enough for pure rolling because the effective forward loft angle would be much too small. 
     Miesch et al (U.S. Pat. No. 4,964,641) disclose a putter face that includes a mechanical pattern created by an electrical discharge machining process. In a preferred embodiment, this pattern defines a plurality of pyramidal pits, which can superficially resemble the teeth of the putters described herein. Miesh et al claim that this construction increases the friction between the club and a ball, a correct claim, and that it imparts overspin to the struck ball. No explanation or proof of this second claim is given and it is not conceivable to us how it could be correct. 
     Other putter embodiments have been disclosed which are claimed to impart forward spin, sometimes enough spin for pure rolling, but, for the reasons stated above, we are not aware of any that actually perform as claimed. This is presumably why, as far as we know, none of these putters are on the market. 
     A number of mechanisms have been disclosed that provide for an adjustable putter face loft angle. Some examples are the following. 
     Britton (U.S. Pat. No. 6,203,433) discloses a putter head in which the face loft angles can be changed by changing attachable faceplates. 
     Fisher (U.S. Pat. No. 6,287,225) discloses a putter head in which the face loft angles are changed by suitably bending a flexible metal tube that connects the head to the shaft. 
     Dombrowski (U.S. Pat. No. 6,964,616) discloses a putter head in which the face loft angles are changed by changing an angled sleeve within the head. In an embodiment, the loft angle can be varied between 0° and 8°, in 2° increments. 
     Lucas (U.S. Pat. No. 7,316,622) discloses a putter head in which the face loft angles are changed by changing a collet within the head into which a faceplate support is locked in placed. 
     Lee et al (US 2007/0004533) discloses a putter head in which the face loft angles are changed by rotating a hosel attached to a small rotatable disc. 
     These patents contain references to other attempts at designing a putter head with an adjustable face angle. No attempt that we are aware of discloses a mechanism having the simplicity, strength, or accuracy as the one that is described herein, and none is proposed for use in imparting an immediate forward rolling motion on a struck ball. 
     Putters with forwardly inclined faces, putters with faces supporting attached teeth, and putters with adjustable face loft angles have thus been previously disclosed. What is original in the putters disclosed herein is the recognition that a flat forwardly inclined face does not impart significant spin and has unintended negative effects, the recognition that appropriately designed teeth can direct the applied force on a struck ball into the forward direction, the recognition that an often large loft angle adjustment is necessary to impart an immediate forward rolling motion on a struck ball, and a mechanism that accomplishes this adjustment in a simple and effective way. 
     DETAILED DESCRIPTION 
     1. Theory 
     A vertically oriented horizontally moving conventional putter striking a golf ball exerts a force directed towards the center of the ball (See  FIG. 7 ). The impact results in an applied force F on the center of mass (COM) of the ball, but it exerts no torque about the COM. The ball therefore acquires an initial horizontal linear speed V (slide) but zero initial angular speed W (spin). 
     The ball therefore starts its motion with a pure forward slide. This sliding motion is undesirable because it causes the ball to skip and become deflected by irregularities in the green. This sliding is immediately opposed by a friction force f pointing backwards at the (bottom) point of contact between the ball and the grass. After a time t, this force causes the linear speed of the ball to decrease from its initial value V to a smaller value v(t)=v. Also, the friction force exerts a torque T=rf about the COM, which causes the angular speed to increase from its initial value W=0 to a larger value w(t)=w. (r is the radius of the ball. See  FIG. 8 .) 
     As the ball moves forward after the impact, v continues to decrease and w continues to increase until v=r·w, at which point pure rolling sets in and the point of contact between the ball and the green is instantaneously at rest. (When v=r·w, the forward linear motion of the contact point is exactly canceled by the backward rotational motion.) From this time on, the friction force becomes very small (the rolling friction force is small) and so the ball continues to roll. During the rolling phase of the motion, the ball&#39;s trajectory is smooth and regular because the green exerts very little frictional force on the ball. 
     It is obviously highly desirable to eliminate the initial sliding phase of the ball&#39;s motion, which can last for several feet. To see how to accomplish this, consider a horizontal impact force F exerted on a ball at a distance h above the center of the ball. (See  FIG. 9 .) This force imparts an initial horizontal linear speed V to the ball, and the torque T=hF arising from the force imparts an initial angular speed W to the ball. 
     Because T(t)=hF(t) at all times t during the impact, the linear speed V and angular speed W acquired by the ball after the impact satisfy the equation: 
     
       
      
       hmV=IW,  
      
     
     where I=2 mr 2 /5 is the moment of inertia of the ball. (Constant ball density is assumed.), Therefore, independently of the values of the impact force F(t), the speeds V and W are related by: 
         V= 2 Wr   2 /5 h.    
     This relation makes it clear how to impact the ball so that it begins rolling immediately. If the impact is made at a height h=2r/5 above the center of the ball, then V=rW, which is the condition for rolling. This height corresponds to a distance H=r+h=7D/10 above the bottom of the ball, where D=2r is the diameter of the ball. A ball struck at this point will execute pure rolling motion throughout its entire trajectory. There will be no initial sliding phase, with it&#39;s awkward skipping and veering away from the desired direction towards the hole. 
     One way to accomplish such an impact is to use a putter with a suitable forward-facing extended element instead of the conventional flat forward surface. (See  FIG. 10 .) If the putt is executed such that the forward element strikes the ball at a distance h=2r/5 above its center, then the desirable rolling motion will result. 
     There are, unfortunately, two serious problems with this putter. The first problem is that it is extremely difficult to hit the ball at the correct height. The second problem is that, once the forward element strikes the ball, it tends to slip upwards, resulting in an uncontrolled motion of the struck ball. 
     An alternative putter design would appear to solve these problems. The forward face of this putter is inclined forward at an angle a. Such a putter will strike the ball at a height h=r sin(a), independently of the height of the putter head above the green. If a is optimally chosen such that sin(a)=2/5=0.4 (a=23.58 degrees), then h will have the desired value of 2r/5. See  FIG. 11 . 
     Unfortunately, this putter does not work. Not only does it fail to impart the desired rolling motion, it forces the ball downward, into the green. This is because the inclined face of the putter, which strikes the ball tangentially, exerts a vector force on the ball, which is directed towards the center of the ball. See  FIG. 12 . 
     The exerted force on the ball therefore does not exert a significant torque about the COM. The initial motion of the ball is thus mainly pure sliding, just as with a conventional putter. Furthermore, the downward component of the exerted force causes the ball to move downward, into the grass, during the impact. This results in an extremely uncontrollable putt. 
     The current invention is a putter that combines the advantages of the leading edge head and the forward inclined head, without the disadvantages of either. The putter strikes the ball at the desired location and with no slippage, independently of the height of the head during impact, and the impact force is directed forward, causing the ball to start rolling immediately, with no downward motion. The idea is to have the forward face inclined forward at the optimal angle A, so that impact occurs at the optimal height h=r*sin(A), with attached forward facing teeth, chosen so that the impact force is directed forward. 
     Upon impact, one, or at most two, of the teeth strike the ball at close to the optimal height, and the impact force is close to horizontal. The disadvantages of the above-described putters are thus avoided. The impact height is automatically correct, and there is no slippage because, if the impacting tooth starts to slide upward, the tooth below it will come into contact with the ball and stop the sliding. The impact force is directed forward, creating the desired torque that imparts the correct initial spin, with no vertical component to push the ball downward. See  FIG. 13 . 
     2. Geometry of Teeth 
     In general, the putter teeth are constructed according to the diagram shown in  FIG. 14 . 
     The inclination angle is A. B 1  is the tooth angle above the horizontal axis, and B 2  is the tooth angle below the horizontal axis, so that the forward tooth angle is B=B 1 +B 2 . The other angles inside of the triangular tooth are C=90°+A−B 1  and D=90°−A−B 2 . The overall dimension of the tooth will be specified by the value of the vertical height d. The sides of the tooth are then a=d/cos(A), b=a*cos(B 2 +A)/sin(B), and c=a*cos(B 1 −A)/sin(B). 
     Although there is a range of tooth geometries which give rise to an improved bell rotation upon impact, there are optimal choices that can be arrived at as follows. Consider the (exaggerated) impact, depicted in  FIG. 15 , of a tooth on a ball at height h above the COM. The upper face of the tooth will exert a compressional force f 1  on the ball, and the lower face of the tooth will exert a compressional force f 2  on the ball. See  FIG. 15 . In general, force f 1  will have an upward pointing vertical component and a forward pointing horizontal component, and force f 2  will have a downward pointing vertical component and a forward pointing horizontal component. The magnitude and direction of the vertical component of the net vector force f 1 +f 2  depends on the values of the angles B 1  and B 2 . However, a net vertical component is undesirable because, if it points downward it will tend to push the ball into the grass, and if it points upward it will tend to push the ball into the air. The optimal choice of B 1  and B 2  is therefore such that the net vertical component is zero. It turns out that there is a limited range of values for this optimal choice. 
     The relative magnitudes of the forces f 1  and f 2 , I can be determined using the stress-strain relation σ=Yε, where σ is the stress (force/area), ε is the strain (fractional length change δl/l), and Y is the Young&#39;s modulus for the golf ball material. The condition that the vertical components of f 1  and f 2  cancel can be derived from the geometry of the above figure. As long as the loft angle A is not too large (less than 30°), this condition is approximately equivalent to the equality: 
       Sin(2 B 1)Cos( B 1)=Sin(2 B 2)Cos( B 2). 
     For each value of B 1  between 0° and 90°, there are two value of B 2  between 0° and 90° that satisfy the equation. One of these values of course is B 2 =B 1 , and the other value is given in Table I. Notice that, as B 1  and B 2  vary between 0° and 90°, apart from the smaller values of B 1  or B 2 , the sum B varies in the limited range between 71° and 72°. 
     
       
         
           
               
               
               
             
               
                 TABLE I 
               
               
                   
               
               
                 B1 
                 B2 
                 B1 + B2 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
            
               
                 90 
                 0 
                 90 
               
               
                 80 
                 1.7 
                 81.7 
               
               
                 75 
                 3.7 
                 78.7 
               
               
                 70 
                 6.4 
                 76.4 
               
               
                 60 
                 13.2 
                 73.2 
               
               
                 50 
                 21.4 
                 71.4 
               
               
                 45 
                 25.9 
                 70.9 
               
               
                 40 
                 30.6 
                 70.6 
               
               
                 30 
                 40.6 
                 70.6 
               
               
                 20 
                 51.6 
                 71.6 
               
               
                 15 
                 57.7 
                 72.7 
               
               
                 10 
                 64.4 
                 74.4 
               
               
                 0 
                 90 
                 90 
               
               
                   
               
            
           
         
       
     
     The putter-head cross-sections for some of these angles are shown in  FIG. 16 . for various values of the loft angle. 
     Because the final loft angle is adjustable, the above teeth angles are not the only possibilities for providing the desired immediate rolling motion. Although the above angles are preferred, other reasonable choices can be accommodated by an appropriate choice of loft angle. Some examples are shown in  FIG. 17 . 
     There is one preferred value of the angles that produces a purely forward force on a any struck golf ball for any value of the loft angle, namely B 1 =90°, B 2 =0°. The putter-head and ball cross-sections for these angles are shown in  FIG. 18 . 
     3. Adjustments 
     In the above THEORY and GEOMETRY OF TEETH sections, it was assumed that the putter face was moving horizontally at the moment of impact with the ball, and that the ball was perfectly spherical and had uniform density. With these assumptions, the optimal face loft angle is A 0 =23.58°. As explained above, these assumptions are almost always false. It is therefore very important to have the ability to make a loft angle adjustment in order to create the desired pure rolling motion arising from every impact. The magnitude of the necessary loft angle adjustment can be large. In general, the actual MOI of a golf ball can be expressed as I=qmr 2 , where q is a dimensionless number. (q=2/5 if the ball is perfectly round and has uniform density.) Then, if the putter head trajectory and the applied force were perfectly horizontal during the impact, the previous analysis (given in the above THEORY section) implies that the linear speed v and rotational speed w of the struck ball are related by v=qwr 2 /h, where h is the vertical distance between the center of the ball and the impact point. The condition v=wr for pure rolling then reads h=r/q, so that the loft angle that imparts immediate pure rolling is A=arcsin(q). For the uniform density case (q=2/5) this gives A=A 0 =23.58°, whereas A=53.13° if q=4/5 and A=11.54° if q=1/5. If all of the ball&#39;s weight were concentrated in a central sphere of radius r/2, q would be 1/10 and A would be only 5.74°. The large spread in golf ball MOIs thus produces a large spread in optimal face loft angles. (Good golfers prefer a small MOI to get longer drives, whereas poor golfers prefer a large MOI to reduce hooking and slicing.) Having the ability to adjust the loft angle therefore provides for a significant improvement in putter performance. 
     A preferred embodiment of the inventive putter head with an adjustable loft angle is described in the SUMMARY above. More details will be given here, with reference to  FIGS. 1-6 . 
     The preferred putter head embodiment consists of three parts, putter head body  1  (with a forwardly angled face  24  containing forward pointing elongate teeth or ribs  26 , and with a relatively wide flat top surface, not separately designated), handle-like structure  20  attached to the top surface, and a hosel component  22 . As described above, handle structure  20  includes central rotatable bar  6  inserted between two end supports or posts  5 , while hosel component  22  includes lower section or base  8  attached to, and slidable on, part of the bar  6 . The body  1  can have a variety of different overall shapes as long as the front face  24  can accommodate the above-described teeth  26  and the top surface of body  1  can accommodate handle structure  20 . In preferred embodiments, the body  1  can also accommodate inserted weights  4  in order to provide the head with a desired overall weight and moment of inertia. 
     Handle supports or posts  5  are basically vertically oriented cylindrical sections with inward facing circular holes  5   a  near the tops of the posts to receive or mount the bar  6 . Handle supports or posts  5  reside in vertically oriented circular holes (not illustrated) in the top surface of the putter head body  1 , with both holes located near the front face  24 , one near the toe and one near the heel of the putter head. Lower portions  28  of these supports  5  contain vertically oriented central threaded holes  13  to accommodate threaded set screws  13   a  that can be turned up to lock the bar  6  in place, or turned down to release the bar and thereby enable adjustment of the loft angle of putter front face  24 . The tops of these screws  13   a  can have an upward-facing concave shape to increase the contact area with the bar. Alternatively, tips  30  of screws  13   a  may be provided with resilient friction pads (not shown) to enhancing fixing of bar  6 . 
     The horizontally disposed bar  6  has three sections. A toe-facing section  6   a  is cylindrical, with a diameter slightly smaller that of the inward facing hole  5   a  in the handle support  5  near the toe, so that the bar  6  can be rotated within said hole. A heel-facing section  6   b  is also cylindrical, with a diameter slightly smaller that of the inward facing hole  5   a  in the handle support  5  near the heel, so that the bar can be rotated within that hole or recess. Central section  7  of the bar  6 , located between these cylindrical sections  6   a  and  6   b , is hexagonal in cross-section. The heel-facing end of this section lies just outside of the handle support  5  near the heel, and the toe-facing end of this section lies near the center of the bar  6 . The bar  6  can thus be rotated within the support holes  5   a  when the locking screws  13   a  are not engaged. 
     As indicated above, lower section  8  of hosel component  22  has a horizontal-facing hexagonal hole  10  that accommodates the hexagonal section  7  of the handle bar  6 . Upper section  9  of hosel component  22  is provided with a vertical round hole or bore  11  that accommodates a putter shaft  3 . The size of the hexagonal hole  10  in the lower hosel section  8  is slightly larger than the size of the hexagonal section  7  of the bar so that the hosel part  8  can slide on the bar. The hosel annulus or sleeve  2  can be locked in place on the bar section  7  by means of a vertically oriented setscrew  14   a  that resides in a vertically oriented threaded hole  14  in the upper lip of the lower hosel section  8 . The vertical hole  11  in the upper section  9  of the hosel component  22  is preferably threaded internally so that a externally threaded shaft  3   a  can be screwed into it. The upper and lower sections  9  and  8  of the hosel  22  can be rotatable connected by a split section with a horizontal threaded hole  12 , so that a threaded screw  12   a  in the hole can be engaged to lock the two hosel sections  9  and  8  together at a chosen lie angle. 
     The preferred putter head  1  described above is adjustable in three ways. The loft angle is adjusted by rotating the bar  6  in either the clockwise or counter-clockwise direction. The lie angle is adjusted by rotating the upper section  9  of the hosel part  22  relative to the lower section  8 . The heel-toe location of the hosel  22  and attached shaft  3  is adjusted by sliding the lower hexagonal open section  8  over the hexagonal surface section  7  of the bar  6 . After these adjustments are made, the loft angle is locked in place by engaging the set screws  13   a , located in the threaded holes  13  in the bottom of the body, onto the handle bar  6 ; the lie angle is locked in place by engaging the screw  12   a  located in the threaded hole  12  through the overlap between the upper and lower sections of the hosel  2 ; and the heel-toe hosel location is locked in place by engaging the set screw  14   a  in the hole  14  in the lip of the lower hosel section  8 . 
     The specific embodiment of the loft adjustability mechanism described above can be changed in various ways. An example of one such alternative embodiment is shown in  FIGS. 19-21 . A rotatable mounting bar  106  to which a hosel  108  is attached has a circular cross section along its entire length, with a shallow longitudinal slit  110  and a threaded section  112  at a heel-facing end. The threaded section  112  screws into a threaded hole  114  in a raised section  116  at a top or upper side of the heel end of a putter head body  118 , and an opposite, unthreaded, end  120  of the mounting bar  106  is rotatably inserted into a circular hole  122  in a raised section  124  at the toe end of the putter head body  118 . A transverse outer face  126  of the threaded bar section  112  has a central hexagonal hole  128  that is exposed at the opening (not separately enumerated) of the threaded hole  114  in the raised heel section  116 . The mounting bar  106  can thus be rotated using a hexagonal Allen wrench, and can be locked in place with a set screw (not shown) in the same way as in the embodiment described above with reference to  FIGS. 1-6 . A lower section  130  of the hosel  108  has a cross-sectionally circular hole  132  through which the mounting bar  106  passes, and a set screw  134  that extends into the slit  110  in the bar. This set screw  134  is used to lock the hosel  108  onto the bar  106  at a chosen location. With the hosel  108  locked onto the bar  106 , the lie angle of the putter face  136  can be adjusted by rotating the bar  106  with the Allen wrench. 
     The preferred protocol for choosing the optimal loft angle, for a given golfer and golf ball, is the following. For a given club head, with a given tooth structure and adjustable loft angle as described above, the golfer should make a series of putts with the loft angle set at a given value. This should be done for each angle value in a suitable range. To start, the angles should differ by a relatively large value. (For example, the angles can vary between 10° and 50°, in steps of 10°.) After the preferred value is found among these angles, the values around this preferred value, in smaller steps, should be tried. (For example, if 30° was the angle chosen in the first range, then 25°, 30°, and 35°, should be compared.) Finally, after the preferred value is found in this second range, the values around this preferred value, in even smaller steps, should be tried. (For example, if 25° was the angle chosen in the first range, then 23°, 24°, 25°, 26° and 27° should be compared.) In this way, the loft angle that produces the optimal ball spin after an impact can be determined. The golfer can then retain this angle for all of his play. 
     The above procedure will provide the golfer with a club that significantly improves his putting performance. If putter heads with different teeth geometries are also available, the procedure can be implemented for several of these choices. In some cases, this can lead to even better performance, but in most cases it will be sufficient to work with a single given tooth structure since the loft adjustability will take this structure into account. 
     Although the invention has been described in terms of particular embodiments and applications, one of ordinary skill in the art, in light of this teaching, can generate additional embodiments and modifications without departing from the spirit of or exceeding the scope of the claimed invention. Accordingly, it is to be understood that the drawings and descriptions herein are proffered by way of example to facilitate comprehension of the invention and should not be construed to limit the scope thereof. 
     REFERENCE NUMERALS IN THE FIGURES 
     1. Preferred embodiment of putter head with teeth and adjustable loft angle.
         ( 1 ) Putter head body with teeth.   ( 2 ) Hosel mechanism.   ( 3 ) Shaft.   ( 3   a ) Threaded end of shaft.   ( 4 ) Inserted weight.   ( 5 ) Bar supports or posts.   ( 6 ) Rotatable bar.   ( 6   a ) Circular section of bar facing putter toe.   ( 6   b ) Circular section of bar facing putter heel.   ( 7 ) Hexagonal section of bar.   ( 8 ) Lower section of hosel attachment.   ( 9 ) Upper section of hosel attachment.   ( 10 ) Hexagonal opening in hosel attachment.   ( 11 ) Threaded hole in upper hosel section for insertion of shaft.   ( 12 ) Threaded hole through hosel sections.   ( 13 ) Threaded holes in bar supports.   ( 14 ) Threaded hole in lower hosel section lip.   ( 12   a ) Bolt for connecting and adjusting the angle between the upper and lower hosel sections.   ( 13   a ) Set screws for locking bar into supports.   ( 14   a ) Set screws for locking hosel to bar.