US 20070049393 A1 Abstract A method and computer program for predicting a golfer's ball striking performance is disclosed. In order to generate a predicted trajectory and predicted ball launch conditions, it is desirable to determine several properties of the golf equipment and a player's swing. For instance, pre-impact swing conditions of a golfer are observed using a monitoring system. Additionally, it is desirable to determine the properties of the golf ball and golf club. Moreover, the effect of the shaft properties and the slippage between the golf club and the golf ball may also be considered. In this manner, a more accurate determination of a golfer's predicted trajectory and ball launch conditions may be determined without requiring the golfer to actually strike a golf ball.
Claims(20) 1. A method for predicting a golfer's ball striking performance, comprising:
determining a plurality of pre-impact swing properties for the golfer based on the golfer's swing with a golf club, the plurality of pre-impact swing properties including an impact location, an orientation of a golf club head, and the golf club head speed; determining a plurality of equipment properties including a plurality of golf ball properties and plurality of golf club properties, the plurality of golf ball properties including a coefficient of restitution at a plurality of velocities and a time of contact at a plurality of velocities, and the plurality of club properties including a center of mass of the club head, a center of a club face, and a moment of inertia; determining the slippage between the golf club and the golf ball based on the plurality of ball properties, the plurality of club properties, and the plurality of pre-impact swing properties; and generating a predicted trajectory and a plurality of predicted ball launch conditions of the golf ball if struck with the golf club based on the slippage, the plurality of equipment properties, and the plurality of pre-impact swing properties. 2. The method according to 3. The method according to 4. The method according to 5. The method according to 6. The method according to 7. The method according to 8. A method for predicting a golfer's ball striking performance, comprising:
determining a plurality of pre-impact swing properties for the golfer based on the golfer's swing with a golf club, the plurality of pre-impact swing properties including an impact location, an orientation of a golf club head, and the golf club head speed; determining a plurality of equipment properties including a plurality of golf ball properties and plurality of golf club properties, the plurality of golf ball properties including a coefficient of restitution at a plurality of velocities and a time of contact at a plurality of velocities, and the plurality of club properties including a center of mass of the club head, a center of a club face, and a moment of inertia; determining the effect of properties of a shaft of the golf club on the impact of the golf ball with the club head, the properties of the shaft including a longitudinal force component and a torque component; and generating a predicted trajectory and a plurality of predicted ball launch conditions of the golf ball if struck with the golf club based on the properties of the shaft, the plurality of equipment properties, and the plurality of pre-impact swing properties. 9. The method according to 10. The method according to 11. The method according to 12. The method according to 13. The method according to 14. The method according to 15. The method according to modifying at least one of the plurality of equipment properties; generating another predicted trajectory and another plurality of predicted ball launch conditions of the golf ball if struck with the golf club based on the at least one modified equipment property. 16. The method according to 17. The method according to 18. A method for predicting a golfer's ball striking performance, comprising:
determining a plurality of pre-impact swing properties for the golfer based on the golfer's swing with a golf club, the plurality of pre-impact swing properties including an impact location, an orientation of a golf club head, and the golf club head speed; determining a plurality of equipment properties including a plurality of golf ball properties and plurality of golf club properties, the plurality of golf ball properties including a coefficient of restitution at a plurality of speeds, and a time of contact at a plurality of speeds, and the plurality of club properties including a center of mass of the club head, a center of a club face, and a moment of inertia; determining the slippage between the golf club and the golf ball based on the plurality of ball properties, the plurality of club properties, and the plurality of pre-impact swing properties; determining the effect of properties of a shaft of the golf club on the impact of the golf ball with the club head, the properties of the shaft including a longitudinal force component and a torque component; and generating a predicted trajectory and a plurality of predicted ball launch conditions of the golf ball if struck with the golf club based on the slippage, the plurality of equipment properties, and the plurality of pre-impact swing properties. 19. The method according to 20. The method according to Description The present invention relates to a method and computer program for determining golf ball launch conditions. More specifically, the present invention relates to a method and computer program that is capable of predicting golf ball trajectory and launch conditions without requiring a golfer to strike a golf ball. Over the past thirty years, camera acquisition of a golfer's club movement and ball launch conditions have been patented and improved upon. An example of one of the earliest high speed imaging systems, entitled “Golf Club Impact and Golf Ball Monitoring System,” to Sullivan et al., was filed in 1977. This automatic imaging system employed six cameras to capture pre-impact conditions of the club and post impact launch conditions of a golf ball using retroreflective markers. In an attempt to make such a system portable for outside testing, patents such as U.S. Pat. Nos. 5,471,383 and 5,501,463 to Gobush disclosed a system of two cameras that could triangulate the location of retroreflective markers appended to a club or golf ball in motion. Systems such as these allowed the kinematics of the club and ball to be measured. Additionally, systems such as these allowed a user to compare their performance using a plurality of golf clubs and balls. Typically, these systems include one or more cameras that monitor the club, the ball, or both. By monitoring the kinematics of both the club and the ball, an accurate determination of the ball trajectory and kinematics can be determined. On Jul. 6, 2004, U.S. Pat. No. 6,758,759, entitled “Launch Monitor System and a Method for Use Thereof,” issued. This application described a method of monitoring both golf clubs and balls in a single system. This resulted in an improved portable system that combined the features of the separate systems that had been disclosed previously. On Dec. 5, 2001, the use of fluorescent markers in the measurement of golf equipment was disclosed in U.S. patent application Ser. No. 10/002,174. Monitoring both the club and the ball requires complicated imaging techniques. Additionally, complicated algorithms executed by powerful processors are now required to accurately and precisely determine club and ball kinematics. Systems such as these are often complicated and require significant research and development, increasing their cost. Despite their ability to monitor a golf club and ball, these systems are typically unable to quickly determine which combination of club and balls produces the best outcome for a particular player. In the past, the only way to accomplish this was to test a golfer with a variety of different clubs and/or balls, and then monitor which combination resulted in the most desirable ball trajectory. The need for a mathematical tool for evaluating golf club performance is dictated by the large number of club design parameters and initial conditions of the impact between club head and ball. Without such a tool, it is not feasible to make quantitative predictions of the effects of a given design change on the ball motions and shaft stresses. For example, in stereo mechanical impact, as described in U.S. Pat. No. 6,821,209 to Manwaring et al., the final velocities and spin rates can be related to the initial values of these quantities without consideration of details of the phenomena that occur during the short time of contact of the ball and the club, i.e., about 500 microseconds. However, by eliminating the consideration of details of the contact between the club and the ball, the stereo mechanical impact approach includes simplifying assumptions, which include: (1) that the three components of the relative velocity of recession of the ball from the club head can be related to those of the approach of the club to the ball, as measured at the impact point, by “coefficient of restitution” and; (2) that the shaft can be considered completely flexible, like a stretched rubber band, as far as the dynamics of impact are concerned, so that no dynamic changes occur in the force or torque that it exerts on the club head during the impact. The stereo mechanical approximation problem involves a set of 12 simultaneous linear algebraic equations in the 12 unknown components of motion of the ball and club after impact. The known quantities in these equations are the initial conditions, i.e., club head motions and impact point coordinates, and the many mechanical parameters of the club head and golf ball, e.g., masses, mass moments of inertia, centers of mass, face loft angle, and face radii of curvature. The explicit algebraic expressions are described in U.S. Pat. No. 6,821,209 to Manwaring et al. The stereo mechanical approximation has drawbacks, however, because: (1) the effects of the shaft on the impact, although small, are not negligible, and it is desirable to obtain quantitative measures of these effects for shaft design purposes; (2) shaft stresses cannot be computed in any realistic manner; (3) the explicit algebraic expressions obtained are still too complex to permit assessments to be made of the effects of design parameter changes except by working out many specific cases with the aid of a computer; and (4) the coefficient of restitution approximation may not be accurate because the sliding and sticking time of the ball at the impact point is not taken into account. In addition, the coefficient of restitution approximation is poor because different amounts of stress wave energy may be “trapped” in the shaft under different impact conditions. In an effort to improve the accurate modeling of the contact between the club and the ball, a model published by Ralph Simon, titled “The Development of a Mathematical Tool for Evaluating Golf Club Performance,” ASME Design Engineering Conference, New York, May 1967 (pages 17-35) was improved and updated mathematically. In addition, the modeling may also be implemented by a golf ball model described in the paper titled “Spin and the Inner Workings of a Golf Ball,” by W. Gobush, 1995, in a book titled Therefore, a continuing need exists for a monitoring system that is capable of determining the trajectory and launch conditions of a golf ball without requiring a golfer to strike the golf ball. Moreover, a continuing need exists for a monitoring system that includes software that reduces the complexity associated with fitting a golfer with golf equipment. Furthermore, a continuing need exists for a monitoring system that more accurately predicts a golfer's ball striking performance. According to one aspect, the present invention comprises a method for predicting a golfer's ball striking performance. The method includes determining a plurality of pre-impact swing properties for a golfer based on the golfer's swing with a golf club. The plurality of pre-impact swing properties may include, for example, an impact location, an orientation of a golf club head, and the golf club head speed. The method also includes determining a plurality of equipment properties that may include a plurality of golf ball properties and plurality of golf club properties. It is desirable for the plurality of golf ball properties to include a coefficient of restitution at a plurality of velocities and a time of contact at a plurality of velocities. Furthermore, it is desirable for the plurality of club properties to include a center of mass of the club head, a center of the club face, and a moment of inertia. Additionally, the slippage between the golf club and the golf ball is preferably determined. The slippage may be based on the plurality of ball properties, the plurality of club properties, and the plurality of pre-impact swing properties. The slippage may be determined by computing each time step, in microsecond time intervals, for a first slip period, a stick period, and a second slip period between the golf club and the golf ball. It is desired that each time step is based on at least a transverse force of the golf ball, a coefficient of friction of the golf ball, and a normal force of a golf ball. A predicted trajectory and a plurality of predicted ball launch conditions of the golf ball if struck with the golf club may then be generated. The predicted trajectory and ball launch conditions are based on, for example, the slippage, the plurality of equipment properties, and the plurality of pre-impact swing properties. In one embodiment, the method further comprises determining the properties of the shaft of the golf club on the impact of the golf ball with the club head. The properties of the shaft may include a longitudinal force component and a torque component. The predicted trajectory may include at least one of distance, flight path, landing position, and final resting position of the golf ball. In addition, the plurality of predicted ball launch conditions may include at least one of side spin, back spin, rifle spin, azimuth angle, launch angle, and velocity. According to another aspect, the present invention comprises a method for predicting a golfer's ball striking performance. The method includes determining a plurality of pre-impact swing properties for a golfer based on the golfer's swing with a golf club. The plurality of pre-impact swing properties may include, for example, an impact location, an orientation of a golf club head, and the golf club head speed. The method also includes determining a plurality of equipment properties that may include a plurality of golf ball properties and plurality of golf club properties. It is desirable for the plurality of golf ball properties to include a coefficient of restitution at a plurality of velocities and a time of contact at a plurality of velocities. Furthermore, it is desirable for the plurality of club properties to include a center of mass of the club head, a center of the club face, and a moment of inertia. In addition, the effect of properties of a shaft of the golf club on the impact of the golf ball with the club head may be determined. The properties of the shaft preferably include a longitudinal force component, a torque component, a shear force, a bending moment, density, shear modulus, and Young's modulus. A predicted trajectory and a plurality of predicted ball launch conditions of the golf ball if struck with the golf club based on the properties of the shaft, the plurality of equipment properties, and the plurality of pre-impact swing properties may then be generated. In one embodiment, the predicted trajectory includes at least one of distance, flight path, landing position, and final resting position. Moreover, the predicted ball launch conditions include at least one of side spin, back spin, rifle spin, azimuth angle, launch angle, and velocity. In another embodiment, the method further comprises determining the slippage between the golf club and the golf ball based on the plurality of ball properties, the plurality of club properties, and the plurality of pre-impact swing properties. The slippage may be determined based on a first slip period between the golf club and the golf ball, a stick period between the golf club and the golf ball, and a second slip period between the golf club and the golf ball. Furthermore, the slippage may be determined by computing each time step in the first slip period, the stick period, and the second slip period in microsecond time intervals. According to this aspect of the present invention, the method further comprises modifying at least one of the plurality of equipment properties and then generating another predicted trajectory and another plurality of predicted ball launch conditions of the golf ball if struck with the golf club based on the at least one modified equipment property. In one embodiment, the plurality of equipment properties that may be modified comprises the golf club center of mass, the golf club weight distribution, the center of the golf club face, the moment of inertia of the golf club, and the friction coefficient of the golf club face. Alternately, one or more different golf balls may be used. 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: The present invention relates to a method and computer program for determining the velocity components and spin rate components of a golf ball. By acquiring pre-impact measurements of the club speed, rotational rate, and ball hit location as input into a computer program, along with pertinent club features, e.g., moment of inertia, and ball features, e.g., normal and transverse forces, a computer program can predict the resulting trajectory and launch conditions of the golf ball. One advantage of the present invention is that the need for a ball monitor to evaluate the ball launch conditions is eliminated, allowing the analysis to be used to optimize a club design specific to a particular golfer. It is desirable for the method and computer program of the present invention to be used in combination with a monitoring system. In one embodiment, the monitoring system may only include the ability to monitor the kinematics of a club, i.e., pre-impact swing properties. For example, the present invention may be used with the monitors disclosed in U.S. patent application Ser. No. 10/759,080 (“the '080 application”), and U.S. Ser. No. 10/770,457 (“the '457 application”), the entireties of which are incorporated herein. As described in these applications, it is possible to measure club impact data with a single camera. The utilization of the predictive method of the present invention along with a club monitoring device such as the ones disclosed by the '080 application and the '457 application provides a relatively inexpensive monitoring system for fitting a golfer to a specific club or ball based on their particular swing characteristics. According to one aspect, the present invention is capable of predicting a golfer's ball launch conditions and ball trajectory without actually monitoring the golf ball. In other words, by inputting a plurality of characteristics, such as pre-impact swing properties, club properties, and ball properties, the present invention predicts a ball's launch conditions and trajectory without requiring a golfer to actually strike a ball. In one embodiment, the pre-impact swing properties may be determined by having a golfer swing a golf club in front of a monitoring system. The golfer may swing the club any desired number of times in order to generate accurate pre-impact swing properties. In other words, it may be desirable to take an average of a predetermined number of swings in order to obtain pre-impact swing properties that are substantially free of deviations caused by human error during a swing. Typically, the standard deviation of the kinematics of a golfer's swing does not change after about 10 swings. Thus, the pre-impact swing properties are preferably based on about 1 or more swings of a golf club. More preferably, the pre-impact swing properties are based on about 10 or more swings of a golf club. Most preferably, the pre-impact swing properties are based on about 30 or more swings of a golf club. In another embodiment, the pre-impact swing properties are preferably based on between about 1 and about 30 swings of a golf club. It is also desirable to determine a plurality of equipment properties, e.g., golf club properties and golf ball properties. The golf club and golf ball properties may be determined according to any equipment or method known to those skilled in the art. Preferably, the golf ball properties that are determined include, but are not limited to, the coefficient of restitution of the ball at a plurality of velocities, the time of contact at a plurality of velocities, and the spin at a plurality of velocities and loft angles. Additionally, the golf club properties that may be determined include, but are not limited to, the geometric center of the club face, the center of mass of the club head, the distance from the hosel to the center of mass of the club face and/or the center of mass of the club head, effective density of shaft material, the effective shear modulus for torsion about the shaft axis, the effective Young's modulus for the shaft material, and the outer and inner diameters of the shaft in two directions at the hosel end. Thus, a golfer is only required to swing a golf club once to determine a predicted trajectory of the golf ball and the ball launch conditions. The predicted trajectory may include characteristics such as distance, flight path, landing position, final resting position, and the like. Moreover, the ball launch conditions may include the side spin, back spin, rifle spin, azimuth angle, launch angle, velocity, and the like. In another embodiment, the method described above may be performed using a computer program comprising computer instructions. Any computer language and/or compiler may be used to create the computer program, as will be appreciated by those skilled in the art. Furthermore, the computer instructions may be executed using any computing device. The computing device preferably includes at least one of a processor, memory, display, input device, output device, and the like. Moreover, the computer instructions may be stored on any computer readable medium, e.g., a magnetic memory, read only memory (ROM), random access memory (RAM), disk, optical device, tape, or other analog or digital device known to those skilled in the art. In determining or predicting the trajectory and ball launch conditions, the present invention provides the advantage of specifically accounting for slippage between the golf club and the golf ball. When a golf club strikes a golf ball, there is a first slip period that occurs, i.e., the friction forces between the club and the ball do not prevent motion between the two. Shortly after the initial club-ball impact, as the club and the ball deform slightly, friction forces cause the golf club and the golf ball to stick together, resulting in a stick period. During the stick period, the golf club and golf ball are locked together, and there is a substantially small amount of relative motion between the two objects. As the club and the ball begin to return to their original shapes, the golf ball and club undergo a second slip period. In a manner similar to the first slip period, the golf club and ball once again experience motion relative to one another. As a skilled artisan will recognize, the trajectory of the ball is significantly affected by the slippage between the golf club and the golf ball, i.e., the first and second slip periods and the stick period. Accordingly, the present invention accounts for the slip and stick periods by integrating Newtonian equations that account for the duration of the slippage at each time step in the collision process, as described in more detail below. Thus, the present invention computes each time step in the collision process between the club and the ball in microsecond time intervals. As a result of the Hertzian theory of deformation, the time of contact varies inversely with the ⅕ power of velocity, i.e., the time of contact between the club and the ball varies as the −0.2 power of the relative velocity. Thus, in one embodiment the time of contact may vary between about 300 and about 700 microseconds. Another advantage of the present invention relates to accounting for the effects of the club shaft on the trajectory and ball launch conditions. As described above, prior art monitoring systems have used methods or computer programs that assume that the shaft can be considered completely flexible, like a stretched rubber band, as far as the dynamics of impact are concerned, so that no dynamic changes occur in the force or torque that it exerts on the club head during the impact. However, although the effects of the shaft on the impact are small, they are not negligible. Therefore, it is desirable to measure these effects to determine how they affect the kinematics of a golf ball. The method for determining these effects, and the calculations involved, are described in more detail below. According to another aspect, the present invention may be used to assist in golf club design, as illustrated by the exemplary steps in the flow chart shown in After the pre-impact swing properties have been input, the present invention may be used to determine a predicted trajectory and predicted ball launch conditions. In embodiments where a mechanical device is used to swing a golf club, the club properties may be held constant and the ball properties may be varied to determine which of a plurality of golf balls results in an optimal trajectory and ball launch conditions for the given club. Conversely, the ball properties may be held constant, and the club properties e.g., center of mass, moment of inertia, and friction coefficient, may be manipulated to determine the optimal design for a club to achieve a desired trajectory and ball launch conditions. Alternately, the geometric center and/or center of mass of the club may be varied in order to design a club that is more forgiving, i.e., is able to achieve a desired trajectory even when a ball strikes near, for example, the toe or heel of the club head. As will be appreciated by those skilled in the art, this embodiment of the present invention may be useful with, for example, golf club and/or golf ball design and manufacturing. In an embodiment where a player swings a club, the present invention may be used to vary club properties, e.g., weight distribution, center of mass, and moment of inertia, to design a club that results in the optimum trajectory and ball launch conditions for a particular player's swing. Alternately, the club properties may be held constant, and the ball properties may be varied for a plurality of different balls. This allows a player to determine which of a plurality of balls results in an optimal trajectory and ball launch conditions, all without actually striking a golf ball. As described herein, the formulation of a useful mathematical tool for evaluating golf club performance necessitates making step-by-step calculations of the detailed changes in time of the forces and motions at various points in the golf ball/golf club system during the course of impact. The analytical formulations for these calculations and a brief description of the calculations are discussed below after the input information is described. Club Analysis Input Data According to one aspect of the present invention, the predictive model of impact between the club and the ball requires the different body coordinate systems involved in the computer analysis to be defined. In The first set of variables are CYOB, CZOB, which are the y and z coordinates of the initial ball contact point on the club face as measured from a coordinate system designated as C with its origin at the center of the clubface. The x-axis is perpendicular to the tangent plane of the club face and the z-axis is parallel to the inscribed lines on the club face directed positively toward the toe of the club. Additionally, the y-axis forms the right handed system. The second set of variables are DXVOH, DYVOH, DZVOH, DXSOH, DYSOH, DZSOH. These are the velocity and spin components of the club at impact as measured at the D-system origin shown in The third set of variables are DXFOH, DYFOH, DZFOH, DXLOH, DYLOH, DZLOH. These variables represent, for example, the three initial components of force and torque at the hosel end of the club, just prior to impact. The forces and torques are about an order of magnitude less than the forces applied by the club head to the ball, as explained in more detail below with respect to the shaft channel. The three initial components of force and the three initial components of torque exerted by the shaft on the hosel end of the club head are experimentally determinable by measuring extension, shear, torsion, and bending strains in the shaft at its hosel end as a golfer swings the club. The fourth set of variables are CXOQ, CYOQ, CZOQ, which are the three components of the Q-system origin and represent the center of the club head with respect to the C-system in C-system coordinates. In one embodiment, at impact, the Q-system is initially the coordinate system in which the camera makes its measurements. The fifth set of variables is the matrix TMQC, TMQD, which represent the rotational transform matrices between the coordinate system Q and the C and D body systems, respectively. In one embodiment, the input should also include the constants that describe the club and its frictional properties. Moreover, the ball properties are also needed. The variables are denoted as follows: W The input accuracy of variables of the shaft may have a smaller impact than other club variables since the greatest computed value of shaft force obtained for the maximum impact condition is about 300 lbs. In one embodiment, the peak value of the force between the club head and the ball is about 3500 pounds for any position on the clubface for a typical driver impact. Furthermore, the relative influence of these former forces on the ball velocities is much less than the ratio of 300 lbs to 3500 lbs. The following input shaft characteristics are RHOS, GS, ESX, ESY, ESZ, which represent the effective density of shaft material, the effective shear modulus for torsion about the shaft axis, and the three values of the effective Young's modulus for the shaft material, i.e., for tension and bending in two directions. The other shaft characteristics are DXODS, DXIDS, DZODS, DZIDS, which represent the outer and inner diameters of the shaft in two directions at the hosel end. Due to the complex nature of the material make-up of a golf ball, the force deformation equation is based on parameters of the Hertzian theory. In one embodiment, these parameters may be determined by, for example, measuring the contact time and coefficient of restitution. In another embodiment, the parameters of the force law may be determined by using the finite element solution using the basic material constants. An exemplary model that may be used is based on the following equation:
X is the ball deformation in a direction normal to the applied force at an instant of time during contact; KN=the normal force constant; a is the radius of the ball; and α is a damping constant that varies with the inverse of the normal velocity of the deformation, VN, and is given by the equation
Typical values for a two piece constructed golf ball are, for example, KN=20616 Lbs, a=0.84inches, α The transverse force deformation in the Y and Z plane of contact are given by the force equations, for example:
KT=the transverse force constant. For an isotropic surface, the parameters measured for a two piece ball are, for example, KT=70000 pounds, a=0.84, and AT=1100. A coefficient of friction to account for the slipping period at the beginning and end of impact is also desirable. On a typical steel surface, this coefficient varies between about 0.2 and about 0.3. In one embodiment, these forces may be measured with transducers, as described in the article “Impact Measurements on Golf Balls,” pages 219-224 in According to the method of the present invention, the parameters in the equation describing normal force may be fitted by a nonlinear least squares method by measuring the coefficient of restitution and contact time at a measured series of impact velocities. The parameters of the transverse force may be determined, for example, by measuring the spin rate of different balls striking a lofted steel block at a series of angles and speeds in addition to the use of piezoelectric force transducers described above. Over a period of ten years, several books entitled Equations of Motion Exemplary explicit equation of motions for the ball and club head are represented below by four vector equations. For example, the ball's motion during impact is represented by the following equations:
In one embodiment, the club head motion may be represented by, for example, the following equations:
In the above equations W Description of Shaft Model for Forces and Torque Exerted on a Club Head According to one aspect of the present invention, a finite element model of shaft dynamics requires solving eight differential equations for each 1-inch segment of the shaft since two integrations with respect to time are required for each of the longitudinal displacement wave, the torsional displacement wave, and the bending waves, in the two perpendicular directions. This makes 304 equations for a 38 inch shaft, as compared with 12 equations for the club head and 12 equations for the ball. Using 304 equations may be necessary to obtain a detailed determination of the stresses developed by the shaft for the purposes of shaft design, however this requires complicated calculations. According to one embodiment of the present invention, it is possible to simplify the representation of the shaft as the input impedances of infinitely long mechanical transmission lines for tension, torsion, and bending waves. In one embodiment, this representation is an accurate approximation because no significant wave amplitudes are reflected back to the hosel end of the shaft before termination of the contact between the golf ball and the club face. In this embodiment, the shaft is assumed to be infinitely long and to have a constant cross section along its length. Both of these assumptions are sufficient for the purposes of analyzing the effects of the shaft on the golf ball impact with the head. This is because the ball does not remain in contact with the club head long enough for any appreciable amount of stress wave energy in the shaft to return to its hosel end after reflection at the grip end. Furthermore, the shaft cross-section is usually uniform for a few inches above the hosel and then tapers gradually enough to avoid any appreciable back reflections of stress wave energy along the length of the shaft (tapering in small steps would have about the same effect as continuous tapering for the range of wavelengths of the stress waves of interest). Under the assumption of an infinite and uniform shaft, rectilinear motion of the hosel end of the club head in the direction of the shaft axis generates a compressive stress wave in the shaft whose stress amplitude is proportional to the velocity of this motion. Similarly, torsional motion of the hosel end of the clubhead about the shaft axis generates a torsional stress wave whose stress amplitude is proportional to the angular velocity of this torsional motion. The electrical analogue of these mechanical motions is that of the input impedance of an effectively long uniform electrical transmission line, for which the voltage is in phase with and proportional to the current, i.e., a pure resistance. For motions of the hosel end of the club head perpendicular to the shaft axis, the analysis is more complicated. For these motions, it is necessary to solve the standard equation for the bending of a shaft with superimposed tension, which is a fourth order partial differential equation in space and second order in time. It is desirable for the bending wave equation to be solved twice for each time step, since there are two perpendicular directions of transverse motion. The boundary conditions for each direction are the known displacements and known changes in angular orientation of the hosel end of the club head; these quantities are known from the solutions of the club head equations of motion. The solutions of the bending wave equations yield values for the bending moments and for the shear forces on the shaft at its hosel end. The negative of these forces and moments are preferably entered into the club head equations, in combination with the negatives of the forces and moments exerted on the golf ball. In this embodiment, the longitudinal force component in the shaft axis direction on the hosel end, F A is the cross-sectional area of the shaft and is equal to
ESY is Young's modulus for tension and bending in three directions; and RHOS is the shaft material density.
DWSY is the angular velocity at the hosel end of the shaft; GS is the shear modulus; and DYIS is the area moment of inertia of the shaft cross section about the shaft axis. Additionally, the partial differential equation for the propagation of the bending waves for a uniform shaft beginning at the hosel end (y=0) and extending indefinitely in the +y direction for a deflection in the z direction may be represented, for example, by the equations:
E is Young's modulus; I is the section area of moment of inertia about the x-axis; T is the tensile force along the y-axis; ρ is shaft density; A is the cross section area; and g is the acceleration of gravity. The force T is the sum of the centrifugal tensile force resulting from the swing just prior to impact and the dynamic force at y associated with the longitudinal waves generated by impact. After the bending wave equation is solved by using Laplace transforms, the shear force and bending moment components at y=0 may be determined using, for example, the equations:
As shown in In one embodiment, the orientation in space of the P coordinate system is determined by the angular position of the club head at the instant of initial contact with the golf ball. Thereafter, the P-system remains fixed in this orientation. All changes in positions and orientations are accumulated in the P-system because the P-system is a fixed system in space. In addition, the Q-system coincides with the P-system at the instant of initial contact with the ball, but thereafter follows the club head in a stepwise fashion as the position and orientation of the club head changes during the time of contact with the ball. At the beginning of each integration time interval, the origin of the Q-system is at the center of mass of the club head and the axes are oriented with respect to the club head, as described above. The Q-system then remains temporarily fixed in space while the integration needed to determine the changes in positions and orientation of the club head and the ball over the time interval is computed. When the integrations are successfully completed, the Q-system is transposed to the new position and orientation of the club head and all dependent variables converted to the new Q-system in preparation for the next integration step. Thus, the moment of inertia tensor in the Q-system does not have to be recomputed each time but the effects of its rate of change must be taken into account. In one embodiment, the small angle transformation theory is sufficiently accurate for transformations between successive orientations of the Q-system, transformations between the P and the Q-systems, and between the B and the C-systems. The maximum orientation differences involved are only about a few degrees. According to one aspect of the present invention, the effects of friction on the club face may be computed as follows. For each time interval, the normal and transverse components of the ball force are computed as described above. As long as the absolute value of each transverse force component does not exceed the coefficient of friction for that direction multiplied by the normal force component, the ball rolls without sliding. If a given transverse force becomes excessive, the ball contact point slides in such a direction as to decrease the transverse distortion just enough to remove the excess transverse force. The transverse force component is thus either less than or about equal to, in absolute magnitude, the applicable coefficient of friction times the normal force. There may be two friction coefficients such as Cfry for motion perpendicular to the inscribed lines, and Cfrz for motion parallel to these lines. In one embodiment, the position of the B-system origin is changed at the end of each integration time step in accordance with the amounts of rolling and sliding of the ball on the club face. General Computational Procedure for Each Time Interval In one embodiment, it is desirable to solve for the motions of the club head as subjected to the steady forces and torques on the hosel end of the head by the shaft (actually by the golfer through the medium of the shaft). The steady motions at the hosel end are subtracted from the total motions there to obtain the motion inputs to a computer program subroutine, referred to hereinafter as the SHAFT subroutine. The SHAFT subroutine outputs the dynamic forces and torques exerted on the shaft. The difference between the total motion of the club head at the ball contact point and the motion of the ball are used to compute the three components of the force on the ball distortion, from which the three components of the force on the ball are obtained. These forces and motions may be corrected for sliding as previously explained. The linear and angular accelerations of the club head are then determined from the resultant forces and torques on the head. These are the steady forces and torques on the club head minus the dynamic forces and torques on the ball, minus the dynamic forces and torques on the shaft. This leads to a computation of the total motions at the hosel end of the club head, at which point the computation is repeated. Preferably, at least about two iterations of the process are performed, and additional ones as necessary, until successive results agree within a predetermined error limit. More preferably, at least about three iterations of the process are performed, and most preferably at least about five iterations are performed. According to one aspect, if this criterion cannot be met after ten iterations, the process is repeated and the time step halved. Completion of a successful integration over a given time interval is followed by the transformations previously described in preparation for integration over the following time interval. In order to determine the accuracy of the present invention, the calculated data was compared with actual data. Accordingly, four golfers hit ten shots with a driver club that was measured on a ball club monitor using the method of photogrammetry, an example of which is described in U.S. Pat. No. 6,758,759, the entirety of which is incorporated herein. Each club included about 5 or 6 marks for analyzing the correspondence between adjacent cameras so that accurate triangulation of marked positions on the club in space could be determined. Furthermore, each ball included 12 circular marks to measure its position after and before impact to determine ball cub head hit position and ball speed, launch angle, and spin rate. An exemplary image acquired according to this setup is shown in
Although the present invention has been described with reference to particular embodiments, it will be understood to those skilled in the art that the invention is capable of a variety of alternative embodiments within the spirit of the appended claims. Referenced by
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