## No full-text available

To read the full-text of this research,

you can request a copy directly from the author.

An overview of the application of physics to the game of golf is given. The golf swing is modelled as a double pendulum. This model and its variations have been used extensively by researchers in determining the effect that various swing parameters have on clubhead speed. These results as well as examples of three-link models are discussed. Kinematic and kinetic measurements taken on the recorded downswings of golfers as well as force measurements are reviewed. These measurements highlight differences between the swings of skilled and unskilled golfers.
Several aspects of the behaviour of a golf ball are examined. Measurements and models of the impact of golf balls with barriers are reviewed. Such measurements have allowed researchers to determine the effect that different golf ball constructions have on their launch parameters. The launch parameters determine not only the length of the golf shot but also the behaviour of the golf ball on impact with the turf. The effect of dimples on the aerodynamics of a golf ball and the length of the golf shot is discussed. Models of the bounce and roll of a golf ball after impact with the turf as well as models of the motion of a putted ball are presented.
Researchers have measured and modelled the behaviour of both the shaft and the clubhead during the downswing and at impact. The effect that clubhead mass and loft as well as the shaft length and mass have on the length of a golf shot are considered. Models and measurements of the flexing of the shaft as well as research into the flexing of the clubface and the effects of its surface roughness are presented. An important consideration in clubhead design is its behaviour during off-centre impacts. In line with this, the effects that the curvature of a clubface and the moments of inertia of the clubhead have on the launch parameters and trajectory of an off-centred impacted golf ball are examined.

To read the full-text of this research,

you can request a copy directly from the author.

... The analysis is extended in Sec. III to the standard constant-torque driven double pendulum model 1,12,13 to explain why the optimum mass is, in practice, nearer 200 g. The sensitivity of the swing efficiency and driving distance to wrist-cock angle, shaft length, shaft mass, release delay, and wrist torque is also investigated. ...

... The simplest description of the model golf swing is in terms of two phases. 1,12,13 During the first phase, the arms and the club are accelerated with shoulder and wrist torques such that the wrist-cock angle remains constant. In the second phase the wrist-cock angle is no longer constrained; the club is released and allowed to swing away from the hub. ...

... Deceleration before impact is a common characteristic of amateur swings. 12 The reduction in the downswing angle accompanying the use of wrist torque may well be useful for players with long shafted clubs ͑also for beginners attempting to maintain a "one-piece swing" with a large wrist-cock angle͒. As the shaft length increases, the total downswing angle also increases, and for golfers with short arms, natural-release swings with long-shafted drivers may be anatomically impossible. ...

A non-driven double pendulum model is used to explain the principle underlying the surprising efficiency of the golf swing. The principle can be described as a parametric energy transfer between the arms and the club head due to the changing moment of inertia of the club. The transfer is a consequence of conservation of energy and angular momentum. Because the pendulum is not driven by an external force, it shows that the golfer need do little more than accelerate the arms with the wrists cocked and let the double pendulum transfer kinetic energy to the club head. A driven double pendulum model is used to study factors affecting the efficiency of a real golf swing. It is concluded that the wrist-cock angle is the most significant efficiency-determining parameter under the golfer's control and that improvements in golf technology have had a significant impact on driving distance.

... This results in an enhancement of the coefficient of restitution (COR) due to the close match between the fundamental frequency of vibration of the clubface and that of the ball. Many literatures explain about the 'trampoline effect' that increases the COR of a golf ball after being hit by a golf club [1][2][3]. (1) ...

... The results proved that the fundamental flexing mode (around 4.4 kHz in this case) of the clubface is much higher than the frequency of impact between a golf ball and a golf club (between 800Hz to 1300Hz; see Penner [3]). In addition, the results show good agreement with the theory for calculating the fundamental flexing frequency by assuming the clubface as a clamped plate. ...

... At this frequency, the clubface vibrates as part of the rigid body motion of the clubhead. This mode is similar to the one reported by Penner[3]. He reported in the review of literature that the 1 st mode of a 250-300cc titanium clubhead is about 1200 Hz, while a 150 cc stainless steel clubhead had a fundamental natural frequency at 1800 Hz. ...

A complete analysis on the modal analysis of a golf clubface has been presented to investigate 'trampoline effect'. The availability of titanium golf drivers with thin, flexible faces opens up the opportunity to design the club so that it works together with the elasticity of the ball to provide the 'trampoline effect' or 'spring' from the clubface. Coefficient of restitution enhancement results from a close match of the fundamental flexing frequency of vibration of the clubface to a natural frequency for the compressive mode of oscillation. Experimental modal analyses were performed on a driver clubface to determine the frequency and modes of vibration. For the fundamental flexing mode, good agreement is found between the results of experimental modal analyses and the theory.

... Investigations suggest that clubhead speed at impact is the primary factor in determining the length of a shot and, each percentage gained in clubhead speed will result in a corresponding percentage increase in hitting distance (Penner, 2003;Fradkin et al., 2004;Wiseman & Chaterjee, 2006;Arnold, 2010). This is providing all other launch parameters remain equal. ...

... This has been examined in players of all abilities. With clubhead speed seen as a key factor in determining ball velocity and influencing driving distance (Penner, 2003;Fradkin et al., 2004;Wiseman & Chaterjee, 2006;Arnold, 2010), the need to understand what kinematic variables relating to the golfer contribute to generating clubhead speed is important. ...

... Further, high level amateur golfers who generate faster clubhead speeds may be taller, and have longer upper limbs when compared to those with higher golfing handicaps (Yoon et al., 1998;Kawashima et al., 2003). However, whilst having longer upper limbs may be advantageous for creating clubhead speed, the related moment of inertia may be such that is effects clubhead speed (Penner, 2003). Interestingly, upper limb length explained only 5 % of the variance in clubhead speed in previous work (Keogh et al., 2009) with a greater amount of variance explained by a high level amateur golfer's physical characteristics such as strength and rotational power. ...

... In these marketing efforts, the club being promoted is often described to possess the necessary characteristics for best golf performance (Hume, Keogh, & Reid, 2005). Researching efforts for designing golf clubs has been centered on increasing overall drive length and reducing the effects of off center impacts on ball flight (Penner, 2003). One characteristic often indicated and believed to effect driver performance is the degree of loft on the clubface, which is considered to be the angular difference between the club face and the ground (Penner, 2001a). ...

... Different researchers have reported different launch angles to maximize the total distance of the drive, with launch angles typically being around 11° (Lamb, 2012) or 12° (Kai, 2008). Regardless of the optimal launch angle, it has been reported that as impact speed increases, the amount of dynamic loft decreases (Penner, 2001a;Penner, 2003). Penner (2001b) found for a designated club head speed, increasing the loft on the face of the club head would result in a lower launch speed, higher launch angle, and increased backspin for the golf ball (Penner, 2001b). ...

... The drivers used for this study possibly did not contain a large enough difference in loft from the low to high degree to produce an alteration in launch angle. Launch angle, unlike descent angle, is generally not affected by the level of spin and club head speed because the measure of launch angle is taken at the point of impact as it leaves the club before maximal spin is induced (Penner, 2003). ...

... This results in an enhancement of the coefficient of restitution (COR) due to the close match between the fundamental frequency of vibration of the clubface and that of the ball. Many literatures explain about the 'trampoline effect' that increases the COR of a golf ball after being hit by a golf club [1][2][3]. (1) ...

... The results proved that the fundamental flexing mode (around 4.4 kHz in this case) of the clubface is much higher than the frequency of impact between a golf ball and a golf club (between 800Hz to 1300Hz; see Penner [3]). In addition, the results show good agreement with the theory for calculating the fundamental flexing frequency by assuming the clubface as a clamped plate. ...

... At this frequency, the clubface vibrates as part of the rigid body motion of the clubhead. This mode is similar to the one reported by Penner[3]. He reported in the review of literature that the 1 st mode of a 250-300cc titanium clubhead is about 1200 Hz, while a 150 cc stainless steel clubhead had a fundamental natural frequency at 1800 Hz. ...

A complete modal analysis of a golf clubface has been presented in order to investigate the 'trampoline effect'. The availability of titanium golf drivers with thin, flexible faces opens up the opportunity to design the club so that it works together with the elasticity of the ball to provide the 'trampoline effect' or 'spring' from the clubface. Enhancement of coefficient of restitution results from a close match of the fundamental flexing frequency of vibration of the clubface to a natural frequency for the compressive mode of oscillation. Experimental modal analyses were performed on a driver clubface to determine the frequencies and vibration modes. For the fundamental flexing mode, good agreement is found between the results of experimental modal analyses and the theory.

... 8 Besides having a great influence on the energytransfer efficiency, the mechanical properties of the ball also influence the generation of ball spin that is acquired during impact between the club face and the ball. 7,9,10 Initially, the ball will start to slide along the club face, and as the friction between the ball and club face increases, the ball will start to roll. 11 The spin rate is directly related to the coefficient of friction, where the cover hardness is negatively correlated with the spin rate. ...

... 11 The spin rate is directly related to the coefficient of friction, where the cover hardness is negatively correlated with the spin rate. 7,9,10 However, the stiffness of the core correlated positively with the spin and it was suggested that a core with a high stiffness serves to compress the cover against the club face, resulting in a greater friction compared to a ball with a softer core. 7,10 The spin rate after impact is of great importance for the carry distance because it affects the ball's aerodynamics; the dimples on the surface cause local flow separation and shear-layer instability resulting in an intense turbulence, which reduces the drag force by approximately 50% compared to a smooth-surfaced ball. ...

... Consequently, a considerably great back spin could reduce the carry distance, owing to a higher trajectory than is optimal, as well as a reduction in carry distance caused by the spin rate's negative impact on the drag force. 9 Hence, based on this reasoning, the significantly reduced carry distance in the 4°C group than in the temperature groups of 18°C and 32°C is not only a result of a lower ball speed but could, to some extent, also be related to the relatively higher spin rate. However, further research is warranted to investigate the influence of ball temperature on spin rate and launch angle. ...

The purpose of this study was to investigate the effect of ball temperature on impact ball speed and carry distance during golf drives in a blind randomized test design. The balls were exposed to a temperature-controlled environment (4 °C, 18 °C, 32 °C, and 46 °C) for 24 h prior to the test and each temperature group consisted of 30 balls. The 120 drives were performed by an elite male golfer (handicap: 0.0) in an indoor driving range. All drives were measured by a Doppler-radar system to determine the club-head speed, launch angle, spin rate, ball speed, and carry distance. Differences between the groups were investigated using a one-way analysis of variance. The results indicated that ball-speed and carry-distance differences occurred within the four groups (p < 0.001 and p < 0.01, respectively). The post hoc analyses showed that the ball temperatures of 18 °C and 32 °C had greater ball speeds and carry distances than balls at 4 °C and 46 °C (all p < 0.05). The intervals for the between-group differences were 0.6–0.7 m s⁻¹ and 2.9–3.9 m for ball speed and carry distance, respectively. Hence, the results showed that ball temperature influences both the ball speed and the carry distance. Based on the findings in this study, standardization of ball temperature should be factored into governing body regulation tests for golf equipment.

... Mather and Jowett [3] describe the same initial deflection followed by a "rebounding" of the club head. Penner [5] also supports the deflection progression during the downswing proposed by Milne and Davis. He proposes that if shaft flexibility is matched to a player's swing, the springing back of the shaft can help add to club head velocity at impact. ...

... One of the goals of this thesis is to develop a shaft fitting method. Based on the observations of Milne and Davis [2] and Penner [5], a valuable shaft fitting method needs to match shaft flexibilities to specific golfers such that club head velocity and the dynamic loft of the club at impact are optimized to yield maximum ball carry. ...

... The oscillating deflection progression of the club head is consistent with published results [2,3,5]. The shape of the club shaft at impact is also a very important part of the information that can be extracted from the solution of the equation of motion as it determines the dynamic loft added to the club face. ...

This study investigated the relationships between golfer hub path trajectories and interaction kinetics, and club behavior. An equation of motion describing a flexible golf club system was derived and solved to yield time and club position deflection behavior during the downswing. This equation was applied to three diverse subjects whose kinematic and kinetic information was used to drive the simulation. It was determined that there is a relationship between the timing of the maximum interaction torque and the increase in normal force applied to the club and club head deflections. Also, it appears that there is a correlation between degree of radius reduction directly before impact and shaft deflection behavior. The timing of both torque and normal force are directly related to changes in hub path radius thus the effect of hub path geometry on club deflection behavior is secondary. Based upon these findings, a method for fitting shafts to specific swing characteristics was developed that optimized predicted carry distance. These results are based upon limited subjects.

... This moves the boundary layer separation point further behind the ball reducing drag force resulting in longer ball trajectories for identical impact conditions. This model for D-plane impact and resulting ball flight is empirically validated using Doppler technology and high-frame rate video capture [12,13,14]. ...

... Ball elevated on a tee allows the longer, less lofted drivers (c) to create impact after the low-point. The greater velocity component normal to the clubface (vCM, ꓕ) imparts greater ball compression resulting in greater carry distances for shots further from the green (d)[12,13,14]. ...

... Manipulation of D-plane orientation relative to neutral creates a variety of ball flights (b). For a right-handed golfer vCM oriented to the right: pull-draw (A), pull (B), pull-fade (C); vCM oriented along target line: straight-draw (D), straight (E), straight-fade (F); and vCM oriented left: push-draw (G), push (H), push-fade (I)[12,13,14]. ...

The golf swing is a complex motion that requires accuracies and precisions orders of magnitude greater than other ball sports. Thus, an improved understanding of club and ball material mechanics during the golf swing would assist the coach and athlete to select optimized equipment that would complement an individual's unique swing character. Additionally, this would allow R&A and USGA governing bodies to impose conformance criteria that encourage equipment innovations yet maintain the game primarily as a test of the athlete's skill and not technological superiority of the equipment.

... The golf swing is a precise movement, comprised of a complex sequence of events that are ideally brought together at the point of impact to meet the main requirements of an effective golf swing; distance and direction [1][2][3][4][5][6]. Assuming that a golfer strikes the ball accurately, the distance that the ball travels will be a function of the velocity of the clubhead at the point of impact; however variations in launch angle and spin resulting from different club use, can also contribute to this factor [1,[7][8][9][10][11][12]. Although the golf swing may appear to be a benign activity, significant physiological effort and precise neuromuscular control is required to accelerate the clubhead to more than 160 km/h in 1/5 th of a second [4,[13][14][15][16][17]. ...

... Early research modeled the downswing as a planar double pendulum, which comprised an upper segment representing the arms and a lower segment representing the club [71]. The upper segment pivoted at a central hub that was approximately located between the shoulders, whilst the lower segment was prevented from moving more than 90° either way by a hinge system that interconnected the two segments at the hands and wrists [8,72]. Although Cochran and Stobbs [71] believed this model ( Figure 2) was a good mathematical analogue of the professional golfer's downswing, several researchers have since improved this model by allowing the hub to accelerate both horizontally and vertically [73], including a third segment representing the trunk [74] and incorporating more realistic muscle dynamics [75]. ...

The modern golf swing is a complex and asymmetrical movement that places an emphasis on restricting pelvic turn while increasing thorax rotation during the backswing to generate higher clubhead speeds at impact. Increasing thorax rotation relative to pelvic rotation preloads the trunk muscles by accentuating their length and allowing them to use the energy stored in their elastic elements to produce more power. As the thorax and pelvis turn back towards the ball during the downswing, more skilled golfers are known to laterally slide their pelvis toward the target, which further contributes to final clubhead speed. However, despite the apparent performance benefits associated with these sequences, it has been argued that the lumbar spine is incapable of safely accommodating the forces they produce. This notion supports a link between the repeated performance of the golf swing and the development of golf-related low back injuries. Of the complaints reported by golfers, low back injuries continue to be the most prevalent, but the mechanism of these injuries is still poorly understood. This review highlights that there is a paucity of research directly evaluating the apparent link between the modern golf swing and golf-related low back pain. Furthermore, there has been a general lack of consensus within the literature with respect to the methods used to objectively assess the golf swing and the methods used to derived common outcome measures. Future research would benefit from a clear set of guidelines to help reduce the variability between studies.

... The swing mechanics literature expanded with the organization of conferences on Science and golf in 1990, 1994, 2002and 2008. The proceedings to 1999 and other literature to 2001 have been nicely reviewed by Penner (2003), whose account will be largely relied upon. However, the shoulder-arm-club swing model set up by Turner & Hills (1999), see below, is of special note, since it demonstrated the capability of such a model to replicate reasonable swings, provided that the driving torques are suitably chosen. ...

... As soon as the right arm of a right-handed golfer bends to allow a realistic arm-swing length, the swing geometry is much more that of the three-link model than of the two-link one, as shown in figure 2. As evident in the figure, the shoulder-arm-club swing presumes the shoulders to rotate around a fixed hub, the left arm (of the right-handed player) to rotate around the left shoulder joint restrained by a limit stop and the club to rotate around the wrist joint, again restrained by a limit stop. Important observations by Penner (2003) include the following. ...

A perspective of the golf swing is defined, and current knowledge of swing mechanics is reviewed. The implications of previou research are collected. A conventional arm–club simulation model is set up and used to show that an apparently important resul from the literature is incorrect. The swing model is fitted, through a parametric optimization process, to data relating t expert golfers. The kinematic deficiencies of the arm–club model are pointed out and a shoulder–arm–club simulation mode with shaft flexibility is developed. Parameters of the shoulder–arm–club model for best matching the expert-golfer data ar found by optimization. The torques generated by the golfers are deduced. The shoulder–arm–club model is then applied for findin whether or not improvements to the observed torque histories are possible, with a positive result. A pattern for the optima use of available driving torques is established. Scaling of the problem to help the understanding of the relationship betwee large and small players is studied through dimensional analysis. Several contributory conclusions are drawn.

... Club-head velocity has been shown to be related to physical and anthropometric characteristics such as trunk flexibility and height (Kawashima et al., 2003;Fradkin et al., 2004Doan et al., 2006Wells et al., 2009). Theoretical and mathematical modelling approaches have been formulated to describe characteristics of the golf swing from a mechanical standpoint (Penner, 2003). The influence of variables such as wrist hinge angle, club release angle, and moments applied by the shoulders and wrists throughout the swing were assessed by altering a double pendulum model which simulated a real golf swing (Miura, 2001;Penner, 2003). ...

... Theoretical and mathematical modelling approaches have been formulated to describe characteristics of the golf swing from a mechanical standpoint (Penner, 2003). The influence of variables such as wrist hinge angle, club release angle, and moments applied by the shoulders and wrists throughout the swing were assessed by altering a double pendulum model which simulated a real golf swing (Miura, 2001;Penner, 2003). Furthermore, previous kinematic investigations have shown that variables including and increased angle of the backswing (Reyes & Rittendorf, 1998), delayed club release (Pickering & Vickers, 1999), greater rotational torque applied at the arms centre of rotation (Jorgensen, 1994), have all been associated with increases in club-head velocity. ...

Golfers are able to attain a competitive advantage when they are able to achieve long hitting distances from the tee. Club-head velocity is perhaps the most commonly reported kinematic variable in the golfing scientific literature. This study aimed to identify 3-D kinematic aspects of the golf swing linked to the generation of club-head velocity using regression analyses. Maximal golf swings were obtained from fifty golfers using an eight camera motion capture system operating at 500 Hz. Full body three-dimensional kinematics were obtained. Multiple regression modelling was used to identify the discrete 3-D kinematic parameters associated with the development of club-head velocity. Two biomechanical parameters; sagittal plane wrist velocity and peak transverse plane torso rotation (Adj R2=0.58, p≤0.01) were obtained as significant predictors of club-head velocity. The findings from this study therefore suggest these parameters are the strongest contributors to ball velocity and potentially overall driving performance. It is conceivable based on these observations, that golfers may benefit from exposure to coaching and conditioning techniques geared towards the improvement of these parameters in order to improve their driving distance.

... Studies by Daish (1972), Rojas and Simon (n.d.), and Cochran and Stobbs (1986) have mathematically estimated that rolling will occur at about 70% of the initial speed of the ball, and the skid distance accounts for about 20% of the putt length. Models by Alessandrini (1995) and Penner (2003) have been created assuming pure rolling motion for the entirety of the putt, however, more valid models by Perry (2002) and Lorensen and Yamrom (1992) have accounted for skidding effects. These models made the assumption that contact between the ball and the surface was maintained for the duration of the putt. ...

In putting, a common assumption is made that a performance improvement will be made when the ball begins to roll more quickly following the impact with the club. Previous research has reported that distance control accounts for 80% of putting performance. Other research has claimed that maximum percentage of roll in a putt will increase predictability and accuracy; however, this has never been directly studied. An experimental study was conducted to investigate the relationship between putter loft angles, roll ratio, and distance control on both an artificial putting turf surface and natural putting green. Four identical blade putters with different loft angles (-1, 1, 3, 5 degrees) were tested using a high speed camera and a mechanical putting machine to determine the skid length of a large number of putts. The final ball position was recorded to ascertain the distance variability of each club at each putt length and quantify each club's roll ratio. A ball ramp was used to quantify the variability attributed to the putting surface. The roll ratios of each club were found to be significantly different from each other (p<0.05) with the exception of the three and five degree putters. The negatively lofted putter in this study showed the greatest roll ratio and most consistent final distance. Recommendation of negative delivered loft could lead to changes in putter fitting philosophy.

... In fact, the human component plays a better and dominant role in the games of golf or cricket (the case of a batsman). The physical theories developed 25 to this effect still appear to be inadequate from this point of view. ...

The role played by the pendulum and its variants in the development of mathematical sciences is reviewed. In particular, the case of a pendulum viz. harmonic oscillator in physics is highlighted in the context of mathematical, conceptual, conventional and engineering disciplines besides the one in mathematical physics. It is pointed out that in all these applications the concept of pendulum is used as a vehicle of knowledge mainly on the basis of structural analogy. Finally, its role is an example of the consciousness-manifesting phenomenon, will be discussed.

... Note that when the mass offset is zero, the side spin of the ball is zero and the expression for the launch velocity of the ball is the same as that of a one-dimensional analysis [27]. ...

This paper seeks to address the implications on putting a golf ball with an off-center mass by analyzing the effect of unbalanced mass of ball on its impact and subsequent rolling. We present the general formulation of a rigid golf ball rolling with slip that is able to transition to rolling friction on an arbitrary surface. Particular attention is given to the effects of the offset center of mass on the golf ball’s path. An experimental setup based on a USGA Stimpmeter is used to calibrate the position of contact point as the ball rolls on the green. The trajectories of the ball due to the mass imbalance were studied by numerically solving the equations of motion during putting. Theoretical predictions show that a mass imbalance has little effect on the launch conditions of the ball. However, on a level green a mass offset center of 0.2 % of the ball’s radius can impact the path of the ball with the consequences of missing the hole in a 5.8 m putt. Changing golf ball trajectories with mass offset center has implications on the development of balls and putting.

... On cherchera à comprendre les interactions de la balle avec le milieu dans lequel elle évolue. Dans l'air de nombreuses études sont consacrées au sillage de la balle et aux effets qui en résultent : les trajectoires courbées, obtenues lorsque la balle tourne sur elle-même, observées au tennis [32,33,34,35], au football [36,37,38,39], au golf [40,41,42] ou les trajectoires flottantes obtenues lorsqu'elle est frappée sans rotation au baseball [43,44], au football [45,46] et au volleyball [47,48]. ...

Physics tends to understand the world and to find the laws which govern it. Physics of Sports consists in observing with a physicist’s eye the phenomena which happen on the fields. These fields are wide and this study is built on several points which capture our attention : - records sports : to understand the records of speed or strength sports, we focus on muscle contraction mechanisms. We extract the dynamics of a simple gesture at Bench Press and understand it through simple mechanics laws and microscopic model of muscle contraction. - badminton trajectories : we solve the equations of motion of a particle which undergoes gravity and aerodynamic drag (proportional to the velocity sqaure, for high Reynolds numbers) and extract an analytical expression of the range. At each time the launching velocity is higher than the terminal velocity (which is the constant velocity of vertical falling of the projectile under gravity and drag), one observes a Tartaglia triangular shape of the trajectory. It is the case for most of sports balls, for fireworks, for firemen water jets, for cannonballs... For all these trajectories, the range saturates with the initial velocity : even if you hit stronger, the ball will not go further. We show several applications of this property. For exemple in sport, as there is no use to play on a field which is larger than the useful length, the size of sports fields is bound to be linked to the maximal range of the associated ball, from table tennis to golf. - balls impacts : we study soccer kicks and focus on the way of kicking the ball. The toe poke is said to be more efficient than the push pass. We perform experiments which show that the ball velocity does not depend on the shape of the impactor, but only on the velocity of kicking. We then discuss some ways to enhance the impactor velocity by using joints or elasticity of the launcher.

... There was also a similar negative and moderate, but nonsignificant correlation for the driver containing the low kick point. Increased spin imparted on the ball was associated with lower launch angles, and this finding supports previous research 28,39 where elite golfers who aim to maximise clubhead speed off the tee lowered their launch angles and imparted greater spin on the ball when attempting to maximise driving distance. ...

In golf, many parameters of the driver can be modified to maximise hitting distance. The main objective of this study was to determine whether drivers fitted with shafts having high and low kick points would alter selected swing parameters and related launch conditions. In total, 12 elite male golfers (handicap score = 1.2 ± 1.8) had three shots analysed for two drivers fitted with 'stiff' shafts with differing kick point location. Stiffness profiles of these shafts were also measured. Five swing and related launch parameters were measured using a real-time launch monitor. The locations of the low and high kick points on each shaft during the golf swing (the dynamic kick points) were confirmed via motion analysis. The driver fitted with the shaft containing the high kick point displayed a more negative (steeper) angle of attack (p < 0.01), a lower launch angle (p < 0.01) and an increased spin rate (p < 0.01) when compared to a driver fitted with a low kick point shaft. It is possible that the attack angle differed between drivers due to the greater amount of shaft bending found late in the downswing (80% of the downswing and just before impact). Future work is needed in this under-researched area to determine why these differences occurred.

... Since the golf shot is one of the most difficult biomechanical motions in sport to execute, a detailed understanding of the mechanics of the swing would be beneficial to the golfer and teacher (Vaughn, 1979). Traditional and standard methods of biomechanical studies of golf swings have employed models of varying degrees of sophistication (Budney and Bellow, 1979; 1982; Jorgensen, 1970; Lampsa, 1975; Neal and Wilson, 1985; Penner, 2003; Vaughn, 1979; Williams, 1967) to perform kinetic analyses of the golfer. Generally, these models were limited to one or two rigid link (double pendulum) systems and constrained the motion to two dimensions. ...

A work and power (energy) analysis of the golf swing is presented as a method for evaluating the mechanics of the golf swing. Two computer models were used to estimate the energy production, transfers, and conversions within the body and the golf club by employing standard methods of mechanics to calculate work of forces and torques, kinetic energies, strain energies, and power during the golf swing. A detailed model of the golf club determined the energy transfers and conversions within the club during the downswing. A full-body computer model of the golfer determined the internal work produced at the body joints during the downswing. Four diverse amateur subjects were analyzed and compared using these two models. The energy approach yielded new information on swing mechanics, determined the force and torque components that accelerated the club, illustrated which segments of the body produced work, determined the timing of internal work generation, measured swing efficiencies, calculated shaft energy storage and release, and proved that forces and range of motion were equally important in developing club head velocity. A more comprehensive description of the downswing emerged from information derived from an energy based analysis. Key PointsFull-Body Model of the golf swing.Energy analysis of the golf swing.Work of the body joints dDuring the golf swing.Comparisons of subject work and power characteristics.

... By increasing the launch angle, the golfer may increase carry which results in greater carry distance. This is supported by simulation work from Penner (2003) [28] who found that when playing with a modern golf ball, relatively higher launch angles and spins (10° and 60 rpm) results in greater driving distance. ...

Golf is a popular leisure and competitive activity for individuals with disabilities. The current golf handicap system does not take into account the possible challenges of playing golf with any form of physical disability. The aim of this study was to examine golf driving performance measures, comparing golfers with various types of physical disabilities to able-bodied golfers. Through drive shot ball launch analysis this study compared amputees (single leg, below and above knee), deaf, visually impaired, polio, Les Autres and arthrogryposis golfers to able-bodied golfers with similar golf handicaps. Twenty-seven able-bodied (handicap category 3, 12.4 ± 7.0) and 15 disabled (handicap category 3, 18.2 ± 9.2) hit 10 drives each. Able-bodied golfers presented longer but less accurate drives (208.1 m carry, 4.6 m lateral deviation), and concomitant higher club head and ball velocity than disabled golfers (157.6 m carry, 6.0 m lateral deviation) [p<0.001]. The apparent difference in outcome performance cannot be fully accounted for by the small difference in golf handicap score, thus disabled golfers appear to be penalised/ disadvantaged by the current golf handicap classification rules. [ahead of print]

... Previous golf reviews have covered areas such as the general physics of the golf swing [3], muscle activity during the golf swing [4], the biomechanics of the swing [5], and golf injuries [6]. None of these reviews have specifically focused on the use of modeling and simulation to understand the dynamic aspects of the golf swing, and to our best knowledge, no review of golf models has been published previously. ...

Golf is one of the most popular sports worldwide. Scientific research into golf has grown substantially over the past four decades. This article reviews the biomechanical models of the golf swing, focusing on how these models can aid understanding of golf biomechanics and the fitting of golf clubs to individual players. It is shown that models range in complexity from the conventional double pendulum model to full-body simulations that include sub-models. The usefulness of any model or simulation is ultimately determined by the assumptions included and the model's complexity. The article summarizes the established areas of golf swing modeling and simulation, discusses the assumptions made by those models, and identifies areas where more research and further development are needed. © 2008 John Wiley and Sons Asia Pte Ltd

... It should be emphasised that the overall purpose of the golf swing is to develop a maximal amount of kinetic energy that is transferred directly to the golf ball (Nesbit & Serrano, 2005). Indeed, the displacement of the golf ball (from tee to eventual point of rest), has been shown to be a direct function of linear club head velocity (Penner, 2003;Hume et al., 2005). Despite this, it should be noted that increasing the club head velocity alone might not result in an overall increase in the distance achieved during ball flight, because other factors, such as spin, may affect the accuracy and distance of the shot (Hume et al., 2005). ...

A proficient golf swing is composed of a sequence of highly complex biomechanical movements and requires precisely timed and coordinated body movements to achieve great distance and accuracy. The aim of the current study was to identify the key physiological and biomechanical variables that relate to golf drive performance. Eighteen golfers (handicap 11±6 strokes, playing experience 18±15 years), volunteered to take part in the study. Drive distance and accuracy were measured directly. Balance was assessed using a modified stork test and hand-eye coordination was assessed using a 3D maze. Average balance duration of both legs (r= 0.563; p=0.015), left leg (r= 0.620; p=0.006) and right leg (r= 0.488; p=0.044) were all significantly correlated to drive distance. Hand-eye coordination was significantly negatively correlated to total drive distance (r=-0.600 p=0.008), but was not associated significantly with the centre of hit between the clubface and ball. Several parameters were found to have significant relationships to golf drive distance in a group of amateur golfers. Therefore, training regimes could include tasks that aim to improve hand-eye coordination and balance.

... Shaft bending stiffness is an aspect of the golf receiving the most attention among the scientific community (Penner, 2003). However, the role of shaft stiffness, more commonly referred to shaft flex, on golfing performance is not yet completely understood. ...

The role of shaft stiffness on the golf swing is not well understood. Studies in which golfers hit balls with clubs of varying shaft flex have reported changes in ball distance. The results of mathematical models suggest that shaft stiffness affects only the orientation of the clubhead at impact, not the speed of the clubhead, but there are no experimental results validating these findings. The purpose of this study was therefore to experimentally examine the influence of shaft stiffness on clubhead kinematics at ball impact. Forty golfers hit 10 balls with each of five drivers varying in shaft stiffness from 'Ladies' to 'Extra-Stiff', in a double-blind study design. The motions of three reflective markers attached to the clubhead were captured with a high-speed motion analysis system. At ball impact, shaft stiffness had a statistically significant influence on clubhead speed for 27 subjects, on loft angle for 11 subjects, and on lie angle for all 40 subjects. No effect was observed on face angle, in to out path angle, or attack angle. These results show that shaft stiffness can affect ball launch conditions by altering clubhead speed and/or loft angle.

... The shaft then deforms significantly in the toe-down direction at the transition from backswing to downswing as a consequence of the inertia of the clubhead and the golfer applying forces and moments at the grip end to accelerate the club. During the downswing, the shaft straightens and then reaches a bent forward shape just before impact (Figure 1; Penner, 2003). Previous research suggested that this forward bending is a consequence of the offset position of the centre of gravity of driver clubheads relative to the centreline of the shaft (Horwood, 1994;MacKenzie & Sprigings, 2010). ...

The aim of this study was to quantify and explain the effect of shaft stiffness on the dynamics of golf drives. Twenty golfers performed swings with two clubs designed to differ only in shaft bending stiffness. Wrist kinematics and clubhead presentation to the ball were determined using optical motion capture systems in conjunction with a radar device for capturing ball speed, launch angle, and spin. Shaft stiffness had a marginally small effect on clubhead and ball speeds, which increased by 0.45% (p < 0.001) and 0.7% (p = 0.008), respectively, for the less stiff club. Two factors directly contributed to these increases: (i) a faster recovery of the lower flex shaft from lag to lead bending just before impact (p < 0.001); and (ii) an increase of 0.4% in angular velocity of the grip of the lower flex club at impact (p = 0.003). Unsurprisingly, decreases in shaft stiffness led to more shaft bending at the transition from backswing to downswing (p < 0.001). Contrary to previous research, lead bending at impact marginally increased for the stiffer shaft (p = 0.003). Overall, and taking effect sizes into account, the changes in shaft stiffness in isolation did not have a meaningful effect on the measured parameters, for the type of shaft investigated.

... Broadie [1] statistically analysed the importance of variations in direction and distance on the golf course and how they contribute to the score. Numerous books and papers discuss the physics of the golf swing, the club-ball impact and the flight of the ball [2][3][4][5][6]. The flight of the ball is determined by the speed, position and orientation of the golf club at impact. ...

A two-part experimental study was conducted in order to better understand how the delivered face angle and club path of a golf club influences the initial launch direction of a golf ball for various club types. A robust understanding of how these parameters influence the ball direction has implications for both coaches and club designers. The first study used a large sample of golfers hitting shots with different clubs. Initial ball direction was measured with a Foresight Sports camera system, while club delivery parameters were recorded with a Vicon motion capture system. The second study used a golf robot and Vision Research camera to measure club and ball parameters. Results from these experiments show that the launch direction fell closer to face angle than club path. The percent toward the face angle ranged from 61% to 83%, where 100% designates a launch angle entirely toward the face angle.

... The golf swing has been often matter of research in the last years (for an excellent review paper see [10]). Different models have been considered from a classical double-pendulum up to a full body representation of a player ( [4], [8], [11]). ...

The paper deals with the analysis the golf swing using a triple pendulum model and a simple class of feasible trajectories for the joints. The basic idea stems from the analysis of motions in sports. Many tasks in sport are not specified in term of equivalent joint trajectories but mainly with boundary conditions of joint position and velocity (e.g. weight lifting, punching, swinging in golf). In such tasks even if the 'optimal' trajectory is not a-priori known, some properties can be however inferred and therefore translated in the shape of the representing functions. For the golf swing, the influence of the joints' trajectory as well as that of the initial posture is analyzed with respect to the applied torques and impact velocity.

... This would be indicative of a less efficient transfer of energy from club head to the ball, possibly caused by the increased vertical club shaft angle at ball contact. The direction of this change was unexpected as it has previously been identified that the CoR of a golf ball increases as club head speed reduces (Penner, 2002) and so the club head speed would have been expected to be less than the theoretical 80% calculated. Conversely, it appears that greater than 80% of full club head speed is required to propel the ball to 80% distance. ...

During practice and competition, golfers are required to use submaximal effort to hit the ball a given distance, i.e., perform a partial shot. While the full golf swing has undergone extensive research, little has addressed partial shots and the biomechanical modifications golfers employ. This study investigates the biomechanical changes between full and partial swings, and determines if the partial swing is a scaled version of the full swing. Using a repeated measures design, 13 male golfers completed a minimum of 10 swings in the full and partial swing conditions, whilst club, ball, kinematic, and kinetic parameters were recorded. Large and statistically significant reductions in body motion (centre of pressure ellipse: 33.0%, p = 0.004, d = 2.26), combined with moderate reductions in lateral shift (25.5%, p = 0.004, d = 0.33) and smaller reductions in trunk rotation (arm to vertical at top of backswing: 14.1%, p = 0.002, d = 2.58) indicate golfers favour larger reductions in proximal measures, combined with diminished reductions as variables moved distally. Furthermore, the partial swing was not found to be a scaled version of the full swing implying a new approach to coaching practices might be considered.

... During the ball's flight, mechanical kinetic energy transfer continues, and makes the ball rise. Then the energy is transferred to the object that it eventually impacts (3,4). ...

... The role of shaft bending during the golf swing has been subject to much debate, with claims that it may facilitate loading and unloading of the shaft like a spring [1], and counterclaims that 'centrifugal stiffening' [2] and the damping provided by the hands [3] would prevent this from occurring. Shaft bending stiffness has been the focus of scientific investigation [4]; however, despite universal acceptance that the club shaft bends during the backswing and the downswing with a recoil as the clubhead approaches impact with the ball, the role of varying shaft stiffness on golfing performance is not yet fully understood [5]. ...

There is much debate around the role of shaft stiffness in the dynamic response of the club shaft during the golf swing. This study used a novel complex analysis to investigate within- and between-golfer differences in shaft strain patterns for three shaft stiffnesses. Twelve right-handed male golfers, with a handicap less than or equal to five, hit six shots with three driver clubs which differed only in shaft stiffness. Clubs were instrumented to record the shaft strain in the lead/lag and toe/heel directions. The analysis combined these perpendicular components into a single complex function, which enabled the differences between two swings to be characterised by a scale and a rotation component. Within-golfer strain patterns were found to be significantly more consistent than between-golfer, p < 0.01. Whilst some golfers displayed more similar patterns than others, there were no clear groups of golfers with similar patterns of shaft strain. Between the clubs, shaft strain patterns differed in the scale component, p < 0.01, rather than the rotation, p = 0.07.

... Since the publication of the landmark book by Cochran and Stobbs (1968), a large literature on the scientific and statistical analysis of golf has developed. Recent surveys include Penner (2003), Farrally et al. (2003) and Hurley (2010). Statistical analysis of amateur golfers was done in Riccio (1990). ...

The game of golf involves many different types of shots: long tee shots (typically hit with a driver), approach shots to greens, shots from the sand, putts on the green, and others. While it is easy to determine the winner in a golf tournament by counting strokes, it is not easy to assess which factors most contributed to the victory. In this paper we apply an analysis based on strokes gained (previously termed shot value) to assess the performance of golfers in different parts of the game of golf. Strokes gained is a simple and intuitive measure of the contribution of each shot to a golfer's score. Strokes gained analysis is applied to extensive ShotLink data in order to rank PGA TOUR golfers in various skill categories and to quantify the factors that differentiate golfers on the PGA TOUR. Long game shots (those starting over 100 yards from the hole) explain about two-thirds of the variability in scores among golfers on the PGA TOUR. Tiger Woods is ranked number one in total strokes gained and he is ranked at or near the top of PGA TOUR golfers in each of the three main categories: long game, short game and putting. His dominance is a result of excelling at all phases of the game, but his long game accounts for about two-thirds of his scoring advantage relative to the field. A similar approach is used to rank PGA TOUR courses in terms of overall difficulty and difficulty in each part of the game. A preliminary analysis shows that the recent change in the groove rule for irons by the United States Golf Association (USGA) has had almost no impact on scores from the rough.

The aim of this study was to identify and characterise individual differences in launch conditions measured from the same hole during four rounds of a professional golf tournament. Launch data from the 18th tee at the 2009 Dubai World Championship were used for the analysis. Self-organising maps (SOMs) were chosen to visualise the potentially non-linear relationship among the launch variables. Several distinctly different types of drives were identified on the output map. Drives which carried the furthest were not necessarily associated with the highest rates of ball speed. As indicated by carry distance, the longest drives had backspin rates of roughly 2700 rpm, a launch angle of 11 degrees, a straight or slightly left-to-right curving ball flight (for right-handers), and reached an apex of about 36 m. These values are specific to the 18th hole at the Dubai World Championship and differ from the general launch recommendations found in the literature.

This paper describes the application of the first fully wireless network of inertial sensors for full-body 3-D motion capture -- the Orient-2 system, to real-time on-body analysis of the golf swing. The golfer-club system is modelled as a double pendulum. The factors which affect the efficiency of the golf swing are outlined, such as the length of the back swing, the wrist-cock angle, swing plane and club-head speed and methods are described for monitoring them using a wireless network of Orient-2 specks. The implementation of the Speckled Golfer System is described and two motion rules are illustrated for executing efficient golf swings.

Golf provides numerous examples of common physical phenomena which can be elucidated through mathematics. This notes provides a simple introduction to mathematical modeling in golf, by briefly describing a few of the many ways mathematics can be used to understand or improve the golf drive. First we describe the doublependulummodel of a golf swing, which is a simple but useful model of the mechanical systemconsisting of the golfer and the golf club, used to accelerate the club head. Secondwe consider the basicmechanics of the energy and momentumtransfer which takes place when the club head impacts the golf ball. Finally we describe the three basic forces—gravity, drag, and lift—which determine the ball’s trajectory after it is struck by the club. © 2010 by The Mathematical Association of America (Incorporated).

Current methods of shaft fitting are only partially successful at matching players with optimal equipment. This could be due to player adaptation. Twenty-four players hit drives into a net with clubs of different shaft flexes. This was repeated with vibration applied. Club kinematics were stable across flex conditions with no vibration, and it is probable that players varied the application of torques during the downswing to compensate for changes in club mechanics. With vibration, for eleven players, club head speed and grip speed at impact increased with flex. This suggests these players could not apply desired torques, perhaps due to noise in proprioceptive feedback caused by vibration.

The emergence of accurate MEMS inertial sensors motivates the design of miniature inertial measurement units (IMU) for applications well outside the field of inertial navigation. One promising application concerns novel sports training systems with inertial sensors embedded directly in sports equipment. This paper describes the theory, design, and evaluation of a miniature, wireless IMU that precisely measures the dynamics of a golf club used in putting. The design consists of a complete six degree-of-freedom IMU composed of MEMS accelerometers and angular rate gyros with an integrated microprocessor and RF transceiver. The resulting sensor system has negligible mass (25g relative to 490g for the putter) and is a mere 13mm in diameter allowing it to fit wholly within the shaft of the club at the grip end. The measurement theory enables the computation of the position, velocity, and orientation of the club head at the opposite end of the shaft during the entire putting stroke. Experiments reveal that the three-dimensional position and orientation of the club head can be resolved to within 3mm and 0.5°, respectively. These achievements yield a highly accurate, portable, and inexpensive sensor system to support golf swing training, custom club fitting, and club design.

Resource Letters are guides for college and university physicists, astronomers, and other scientists to literature, websites, and other teaching aids. Each Resource Letter focuses on a particular topic and is intended to help teachers improve course content in a specific field of physics or to introduce nonspecialists to this field. The Resource Letters Editorial Board meets at the AAPT Winter Meeting to choose topics for which Resource Letters will be commissioned during the ensuing year. Items in the Resource Letter below are labeled with the letter E to indicate elementary level or material of general interest to persons seeking to become informed in the field, the letter I to indicate intermediate level or somewhat specialized material, or the letter A to indicate advanced or specialized material. No Resource Letter is meant to be exhaustive and complete; in time there may be more than one Resource Letter on a given subject. A complete list by field of all Resource Letters published to date is at the website www.kzoo.edu/ajp/letters.html. Suggestions for future Resource Letters, including those of high pedagogical value, are welcome and should be sent to Professor Roger H. Stuewer, Editor, AAPT Resource Letters, School This Resource Letter provides a guide to the literature on the physics of sports, updating Resource Letter PS-1, published 25 years ago Ref. 17. The intent is to suggest literature for anyone curious about the basic physics of particular sports, for physics teachers searching for sports examples to augment their teaching, and for physicists contemplating research on unsolved sports-related questions. © 2011 American Association of Physics Teachers.

A narrative review of dynamic models of golf phenomena is presented, as well as current technologies for measuring the motions of a golfer, club, and ball. Kinematic and dynamic models of the golf swing are reviewed, including models with prescribed motions or torques as inputs, and predictive dynamic models that maximize an objective (e.g., driving distance) to determine optimal inputs or equipment designs. Impulse–momentum and continuous contact dynamic models for clubhead–ball and ball–ground impacts are described. The key observations from 172 cited references are extracted and presented, along with suggestions for future research.

A common belief in the golf community is that a lighter shaft allows the golfer to swing the club faster. From a mechanical point of view, reducing the mass of the shaft would result in a faster swing. However, a golfer is not a purely mechanical system, and so it is simplistic to assume that identical loads will be applied when swinging different clubs. Therefore, the purpose of this study was to test the hypothesis that golfers behave similar to a mechanical model when swinging clubs of varying mass. A torque driven model estimated the effects caused by the addition of 22 g to the shaft. Twelve golfers hit balls with a standard driver as well as a driver fitted with the same 22 g increase in mass. Club kinematics were collected with a high-speed motion capture system. The model predicted a 1.7 % lower club head speed for the club with additional mass. One subject showed a similar reduction (1.4 %), but one subject showed an increase in club head speed by 3.0 %. Ten subjects did not show any significant differences. These results suggest that golfers do not respond to changes in club mass in a mechanically predictable way.

Background: To understand an effective golf swing, both swing speed and impact precision must be thoroughly and simultaneously examined. The aim of this study was to perform both swing speed test and impact precision test to ascertain what swing type determines an effective impact.
Methods: Seven golfers from a college team (handicap: 0–12) were recruited to complete a swing speed test and impact precision test using a 5-iron club. A force plate and electromyography (EMG) system were used to collect data in the swing speed test to compare the difference between two motion sequences. High speed video cameras were used to determine the displacement of rotation center for impact precision test.
Results: The results showed a significant difference (p

Magnus gliders are spinning toys displaying spectacular looped trajectories when launched at large velocity. These trajectories originate from the large amplitude of the Magnus force due to translational velocities of a few meters per second combined with a backspin of a few hundred radians per seconds. In this article, we analyse the trajectories of Magnus gliders built from paper cups, easily reproducible in the laboratory. We highlight an analogy between the trajectory of the glider and the trajectory of charged particles in crossed electric and magnetic fields. The influence of the initial velocity and the initial backspin on the trajectories is analyzed using high speed imaging. The features of these trajectories are captured by a simple model of the evolution of the Magnus and drag forces as a function of the spin of the gliders. The experimental data and the modeling show that the type of trajectory—for instance, the occurrence of loops—depends mostly on the value and orientation of the initial translational velocity regardless of the value of the backspin, while the maximum height of the apex depends on both the initial translational velocity and initial backspin.

Objective : The purpose of this study was to analyze the correlation between physical factors (X-factor, X-factor stretch) and club factors (club speed, ball speed, club path, smash factor, vertical launch angle, spin rate, flight time, total length) during impact and it affect on the total distance of the ball during a golf driver swing. Background : There were not enough studies that analyzed the correlation between physical factors(X-factor, X-factor stretch) and club factors(club speed, ball speed, club path, smash factor, launch angle, spin rate, flight time, total length) during a purpose swing to increase total distance. Method : For this study, 9 right handed professional male golfers (KPGA) were chosen. The test subject group used their own drivers and each took a total of 10 swings. These swings consisted of 5 purpose swings to increase total distance and 5 normal swings. Results : The purpose swing to increase total distance showed larger physical factors(X-factor, X-factor stretch) compared to a normal swing however the results were not statistically significant. Total distance increased during a purpose swing as a result of ball and club speed. Conclusion : The results showed that club factors, ball speed and club speed contributed the most in affecting the total distance of the ball during a purpose swing.

Research has evaluated golf ball lie as means to compare various management techniques, determine the influence of nitrogen fertility, and separate turfgrass species and cultivars. However, little work has been performed to determine the impact of golf ball lie on playability, and therefore, the objective of this study was to evaluate the effects of golf ball lie on golf shot characteristics such as carry distance, backspin, ball speed, smash factor, club speed, and accuracy. Two golfers differing in handicaps struck balls using a seveniron from various ball lies created by ranging mowing heights from 0.5 to 4.0 inches in Agrostis stolonifera L., Poa pratensis L., and Festuca arundinacea Schreb. Ball lie was measured as the percentage of golf ball exposed within and above a turf canopy using digital image analysis, and golf shot characteristics were measured using a launch monitor. Mowing heights of the turf and ball lies were significantly correlated (P < 0.001, r² = 0.863). Ball lie significantly affected carry distance (P < 0.001, r² = 0.357) and ball speed (P < 0.001, r² = 0.517), both exhibiting logarithmic relationships. Smash factor was linearly related to ball lie (P < 0.001, r² = 0.518). Comparison of the regression coefficients for models specific to each golfer indicated that carry distance, backspin, ball speed, and smash factor were affected equally for both golfers. However, accuracy of golf shots was only significantly related to ball lie for the better golfer (P = 0.004, r² = 0.158). Results from this study confirm conventional wisdom concerning the effects of golf ball lie on golf shot characteristics. © 2015 American Society of Agronomy and Crop Science Society of America.

The drag coefficient and Magnus coefficient are different along with the geometric parameters of target object when calculating and simulating the trajectory. Generally, the coefficients are measured in wind-tunnel via repeated experiments, and only for object which have the same shape and geometric parameters. This paper proposes a numerical iterative algorithm based on hierarchical optimization method. First, all feasible solutions are calculated with the cost function of carry distance. Then a gradient cost function was added to get the optimal coefficient. In the experiment section, golf projectile motion was employed to verify the algorithm. The result shows that the algorithm is low cost and low error, and just relies on the carry distance of different initial velocity, is more versatility in application.

Golf is one of the most popular sports worldwide. Scientific research into golf has grown substantially over the past four decades. This article reviews the biomechanical models of the golf swing, focusing on how these models can aid understanding of golf biomechanics and the fitting of golf clubs to individual players. It is shown that models range in complexity from the conventional double pendulum model to full‐body simulations that include sub‐models. The usefulness of any model or simulation is ultimately determined by the assumptions included and the model's complexity. The article summarizes the established areas of golf swing modeling and simulation, discusses the assumptions made by those models, and identifies areas where more research and further development are needed.

The purpose of this study was to kinematically analyze the differences between short(2.17 m) and long(10.94 m) putting stroke motions. Thirteen male professional golfers were participated in this study. Experiment was conducted on the artificial grass mat in the gymnasium. Kinematic data were collected by the 60 Hz Kwon3D motion analysis system. Differences were compared by SPSS paired t-test and one-way ANOVA. Duncan was used for post-hoc test and a

The aim of this project was to design a golf wedge capable of increasing backspin for the amateur golfer. This was accomplished by embedding a metal lattice structure behind the clubface to allow the face to elastically deform slightly upon impact. This would increase contact time between the club and ball. The mechanism of spin generation was discussed and the relationship between contact time and spin rate was established. The design was enabled by using additive manufacturing, which allowed for the generation of a metal lattice structure. An appropriate control and prototype were designed to minimize run time and material usage due to limited machine capacity. Various lattice topologies were generated and analyzed with finite element analysis. Design validation build in plastic revealed that these were not feasible due to support material generation, so X topology was used instead. After printing, player testing was conducted. The prototype design underwent plastic deformation during testing, and resulted in a significantly lower spin rate than the control. The design outlined in the report is not recommended unless changes to prevent plastic deformation are made and more testing is performed. Economic justification for the production of additive manufacturing golf club designs is made in case future designs prove viable. Future work involves earlier consideration of design for manufacturability given the constraints of the selective laser melting (SLM) machine and better testing using an automated process such as a golf swing robot.

Thin-wall composite spheres (TWCSs) are very common in both natural and artificial structures. Their response to mechanical loading was investigated in the past almost solely in the limit of infinitesimal deformations. We examine, within the framework of finite deformation elasticity, the mechanics of incompressible TWCSs with neo-Hookean core and shell phases subjected to general homogeneous displacement and traction boundary conditions. We derive explicit general forms for the displacement and the pressure fields in both phases in terms of a power series about the shear and the tension magnitudes and the shell volume fraction. The predictions of the analytical solutions are analyzed and compared with corresponding results of finite element simulations for TWCSs with different ratios between the phases shear moduli. In addition to an extension of the work of Weil and deBotton [22] from simple shear to general homogeneous boundary conditions, we modify the power series solution and provide a reliable solution for any combination of phases shear modulus. We demonstrate that a relatively small number of terms in the series is required for a good agreement with the numerical simulations up to a stretch ratio of 1.5 when considering the local fields, and up to a stretch ratio of 2 when considering the average fields. The analysis emphasizes the interaction between the shell and the core and reveals the different roles of the coating under different boundary conditions. We highlight interesting similarities and dissimilarities between the spatial distributions of the local stresses and the variations of the average stresses developing in TWCSs with stiff and soft shells, under displacement and traction boundary conditions.

The intricate physical interaction between the golfer and golf club has driven the development of numerous dynamic golf swing models over the past few decades. Within these golf swing models, the shaft is represented analytically, and simplifications are often made to strike a balance between the demanding computation required for simulating full swings and running optimizations. Despite recent progress in modeling golfer biomechanics, there has been a marked lack of experimental validation surrounding the dynamic behaviour of the analytical shaft models used in golf swing simulations. The purpose of this study was to evaluate the simulated behaviour of an advanced, continuously-parameterized analytical golf shaft model using experimental data collected in constrained mechanical property experiments as well as golf swing motion capture experiments. In the constrained experiments, the model predicted the experimental shaft’s static deflection and bending frequency within a relative error of 1%. In simulations of full swings, the model authentically replicated the unique bending patterns for each of the 20 golfers in the motion capture experiment. The root mean square errors for the shaft droop, lag, and twist angles for all swings (\(N=200\)) were \(1.14^{\circ }\), \(0.83^{\circ }\), and \(0.85^{\circ }\), respectively, and the clubhead speed at the time of impact was predicted to within \(-\,2.9\) to 0.6%. Empowered by symbolic computation, the model is computationally efficient enough for fast simulations of full swings (\(<1\) s CPU time) on a desktop computer, thereby enabling golf shaft design optimization.

Changes to the Rules of Golf for 2019 allow a golfer to make a stroke from the putting green with the flagstick unattended in the hole. Such an action would previously incur a penalty. This work provides a model framework for the interaction among the ball, hole, and flagstick. The model is based in part on earlier work by Holmes, who examined the ball without the flagstick as allowed under the former rules. The interaction of the ball with the hole/flagstick area is evaluated for whether the ball is holed and for the final distance from the hole in the case of a miss. The flagstick is found to be of mixed benefit, performing more as a backstop as the impact with the flagstick becomes more central (head-on). Glancing blows at higher input angles are found to cause missed shots at lower velocities. Putts that are too fast to be made whether the flagstick is in or out of the hole are found to rest closer to the hole with the flagstick present. The implications are that shots for which the line of putt is better known (increasing the odds of central impact) and where speed may not be well controlled (e.g., greenside pitch shots) benefit from having the flagstick in the hole.

This project aims to investigate the effect impacting a high speed golf ball on TiN coated metal plate, a simulated golf club head. It was found that the surface coating caused greater deformation of the golf ball. It can be said that a club with this coating will absorb less energy from the impact. The result of this is that the energy will that was not absorbed by the club threw deformation will remain in the golf ball. This extra energy will be transformed into two forms of kinetic energy. The other will be used up by the forces acting on the ball during flight. This research is important as large golf club and golf ball companies have great interest and motivation to further their understanding of new ways to improve their golf clubs.

The impact between a clubhead and a golf ball along with the resulting
flight and run of the ball after landing is considered. The clubhead
loft which results in the maximum drive distance and its dependence on
the initial clubhead speed is then determined.

The impact of the clubhead of a driver with a golf ball is modeled. The effect of the convex clubface of a driver on the flight of the golf ball is considered and the dependence of the optimum curvature of the clubface on the volume, mass, and impact speed of the clubhead is determined. (C) 2001 American Association of Physics Teachers.

The run, which includes both the bounce and the roll, of a golf ball landing on turf is modeled. The effect of launch speed, impact angle, backspin, and green firmness on the run for a variety of golf shots is considered. It is found that the dominant factor that determines the length of the run, in the case of drives, is the impact angle. It is also found that for high-lofted iron shots, where the golf ball is given sufficient backspin, the ball may, for firm enough greens, initially bounce forward before running backwards.
PACS No.: 01.80+b

The motion of a rolling golf ball on a sloped golf green is modeled. The resulting calculated path of a golf ball is then used, along with a model of the capture of the golf ball by the hole, to determine the resulting launch conditions required for a successful putt. Estimates of the probability of making certain putts are also presented.
PACS No.: 01.80+b

Nonrealistic rendering techniques that enhance viewer perception of golf green topography and how that topography affects putting are discussed. Conventional computer graphics and numerical analysis techniques are combined with application-specific modeling and analysis algorithms to enhance a viewer's understanding of the golf green and model golf ball trajectories and putting difficulty. The coding of the system in LYMB and implementation of the system in broadcasts of two golf tournaments are described

Three-dimensional kinematics and kinetics for a double pendulum model golf swing were determined for 6 subjects, who were filmed by two phase-locked Photosonics cameras. The film was digitally analyzed. Abdel-Aziz and Karara's (1971) algorithm was used to determine three-dimensional spatial coordinates for the segment endpoints. Linear kinematic and kinetic data showed similarities with previous studies. The orientation of the resultant joint force at the wrists was in the direction of motion of the club center of gravity for most of the downswing. Such an orientation of the force vector would tend to prevent wrist uncocking. Indeterminate peak angular velocities for rotations about the X axis were reported. However, these peaks were due to computational instabilities that occurred when the club was perpendicular to the YZ plane. Furthermore, the motion of the club during the downswing was found to be nonplanar. Wrist uncocking appeared to be associated with the resultant joint torque and not the resultan...

A wind tunnel technique has been developed to measure the aerodynamic forces acting on golf balls over a wide range of Reynolds number and spin rate. Balls with round dimples and hexagonal dimples have been investigated. The dimples are found to induce a critical Reynolds number behavior at a lower value of Reynolds number than experienced by a smooth sphere and beyond this point, unlike the behavior of a sand-roughened sphere there is little dependence of the forces on further increases in Reynolds number. A hexagonally-dimpled ball has a higher lift coefficient and a slightly lower drag coefficient than a conventional round-dimpled ball. Trajectories are calculated using the aerodynamic data and the ranges are compared with data obtained from a driving machine on a golf course.

This paper presents an example of a two-point boundary-value problem which can be used to motivate the study of numerical techniques for solving such problems by undergraduates. The problem is referred to as the putting problem and illustrates the shooting method for two-point boundary-value problems. The differential equations are developed and numerical results are given.

This paper reviews the available literature on family management of childhood diabetes and highlights gaps in the current knowledge base. Four aspects of family management of childhood diabetes are discussed: coping (how the family adjusts to living with a child who has diabetes), compliance (how the family manages the child's diabetes on a daily basis), communication (how the family learns from interactions with the health provider), and context (how the family environment "sets the stage" for managing diabetes).

With the personal computer the undergraduate physics student can investigate interesting physical systems. This is illustrated with a study of golf drives predicted from experimentally measured lift and drag coefficients which give forces nearly proportional to the square of the velocity of the ball. Their interplay gives a surprising linear dependence of range on initial velocity, in good agreement with empirical formulas but in contrast with the quadratic dependence discussed in introductory physics courses. Spin damping is easily introduced and found to decrease the range for a given launch angle. The exploration of quantitative and qualitative differences between the trajectories produced by measured drag and lift forces and those predicted by a model with a linear velocity dependence illustrates the scientific method at work.

The equations of motion for a golf ball interacting with a hole on a putting surface are found. A computer is used to predict the outcome of the ball's encounter with the hole as a function of the impact parameter and the initial velocity of the ball at the rim. If the ball does not skid or bounce when it collides with the opposite rim, it is predicted that a uniform ball must travel at 1.626 m/s or less to be captured; that the British ball is easier to sink than the slightly larger American ball; that a ball with a larger moment of inertia is more difficult to sink; and that bouncing and skidding (factors that vary from green to green) result in capture at greater speeds. Experimental studies support these predictions.

An experimental method of measuring the aerodynamic drag of a golf ball in free flight under typical playing conditions is
described and the results obtained are discussed. The method depends on deriving a curve in which the ‘carry’ of the ball
is plotted against initial ball velocity off the tee, so that only these two parameters need to be measured. A main conclusion
is that the variation of the drag coefficient with Reynolds number is such as to make the drag vary linearly rather than as
the square of the velocity (over the relevant spees), with a consequent advantage to long hitters.

This article describes the computer simulation of the three-dimensional projectile motion of a spinning golf ball subjected to aerodynamic effects. Linear lift and drag forces are employed in the mathematical model and shown to be physically valid for launch conditions typical of a drive with a 1-wood. The effects of sidespin and wind are also included in the model, and an analytical solution of the dynamic equations of motion is obtained. This solution has been encoded in a fortran-77 program to provide rapid computation of trajectories, and graphical plots are presented that clearly illustrate the resulting ``slicing'' or ``hooking'' motion. Finally, the effects of various combinations of sidespin and crosswind on the final ball position are presented. Specifically, it is shown that a ball hit with sidespin into a crosswind can fly farther than it could in conditions of no wind.

This paper explores the interesting problem of projectile motion without
the vacuum idealization. Particular attention is paid to golf ball
trajectories with and without lift. No lift trajectories with linear and
quadratic drag are considered first. Then, trajectories with lift and
linear drag are investigated. Projection angles for maximum range are
determined for all these cases. Computer solutions are used throughout,
with a Runge-Kutta routine used for all cases except for the well-known
closed solution for the no lift, linear drag projectile.

The objective was to optimize the post-impact conditions of a golf driver and golf ball in order to minimize dispersion of a drive for off-center hits. When a golf ball is struct by a clubhead at a point away from the “sweetspot” on the clubface, spin is induced on the ball which makes it curve during flight. Over the years, clubmakers have made the clubface slightly convex in shape in order make off-center hits land as close to the fairway as possible. A general, three-dimensional impact model using principles of momentum conservation on rigid bodies was used to extract the spin and velocity vectors of the ball after impact. An aerodynamic model including drag, lift and skin friction was then used to obtain the landing position of the ball. An optimization routine was used to calculate the optimum clubface shape to minimize golf ball dispersion from the middle of the fairway, for off-center hits. In order to have an accurate simulation of impact on a computer, the mass and inertia tensor of the clubhead was experimentally measured. The clubface shape relative to the center of gravity of the clubhead was also measured. The impact computer program and golf ball trajectory computer programs were verified by using the golfing robot “Iron Byron” to hit various shots on the clubface of a particular driver. The computer simulation yielded results of ball landing positions from the middle of the fairway that fell within the standard deviations of those experimentally measured. The optimization routine was used to find the optimum clubface shape to minimize golf ball dispersion for off-center hits, for a particular driver swung at a certain initial velocity in a certain direction. The clubface shape was modeled in the impact program as an ellipsoid. After running the optimization program, the average dispersion from the middle of the fairway for the worst possible off-center hit locations was reduced on the average from 8.075 to 2.198 yards. The optimization program demonstrated its effectiveness in determining the optimum clubface to minimize dispersion of off-center hits.

The Lagrangian method is used to obtain two coupled differential equations describing the motion of a simple model of the swing of a golf club. These equations are simplified by a special treatment of the gravitational torques and are put in a form such that different constant torques applied by the golfer give solutions differing only in a scale factor. The equations are solved numerically for various suitable boundary conditions. It is shown that the clubhead speed achieved for a swing with a constant torque applied by the golfer increases with the hindrance to the uncocking of the wrists during the swing, and that the backswing of the club may be decreased substantially with a correspondingly small decrease in clubhead speed.

This volume gives and excellent survey of our present knowledge of
molecular processes in stellar and proto-stellar objects. It reviews
molecular physics in stellar environments and is intended to bridge the
gap between astrophysicists and chemists. The topics range from the
theoretical to the computational and include observational data. Among
the topics treated are questions of stellar evolution, the determination
of physical properties and structures , and the chemical composition of
stellar protospheres. Opacity is studied in the context of various types
of stellar and proto-stellar objects.

The present work is concerned with the flow past spheres in the Reynolds number range 5 × 104 [less-than-or-eq, slant] Re [less-than-or-eq, slant] 6 × 106. Results are reported for the case of a smooth surface. The total drag, the local static pressure and the local skin friction distribution were measured at a turbulence level of about 0·45%. The present results are compared with other available data as far as possible. Information is obtained from the local flow parameters on the positions of boundary-layer transition from laminar to turbulent flow and of boundary-layer separation. Finally the dependence of friction forces on Reynolds number is pointed out.

The inward pull motion at the impact stage of the golf swing commonly observed with expert players was investigated in this paper. First, a model of nonconcentrated rotation was studied. It was found that, for a mass rotating around a pivot, if the pivot is moved in the direction opposite to the direction of centrifugal force of the mass, the kinetic energy of the mass could be increased. The increase is a result of the mutual action of the two governing factors of the system, which are the centripetal force and the pull velocity. A special type of equation of motion governs this phenomenon
, and the parameter in the second term of the left-hand side of the equation ξ˙ characterizes its behaviour. The phenomenon is called the parametric acceleration, following the parametric excitation of vibration problems also governed by a similar equation. Second, the golf swing was investigated using the above finding. In the golf swing, the club is accelerated by the hands in a tangential direction. Theoretically, the additional acceleration of the clubhead could be achieved by pulling the club in the radial direction at impact stage, assuming that the centrifugal force of the clubhead is fully developed. To test this idea, an emulation using a modified double-link model was carried out. It was shown that the clubhead velocity could be increased substantially by the inward pull motion of the club at the impact stage, at which point no other means of acceleration is available. Discussions include the actual movement of the body for the inward pull, the efficiency of the pull motion and application to other sports.

The perception of ‘feel’ during a ball-implement impact is considered a significant determinant in equipment selection. Previous studies in golf have found that the perceived time for which the ball and clubface are in contact is a factor in the ‘feel’ of the shot. This factor appears to have become more significant with the development of the latest metal ‘woods’. The purpose of this study was to investigate whether golfers’ perceptions of impact duration correspond to measured values or whether the perceptions are created by other factors.
A technique has been developed to measure the duration of impact by creating an electrical circuit in which the ball and clubface form a ‘switch’, completing the circuit whilst contact is maintained between the two bodies. Measurements were taken of the duration of impact between five different types of clubhead and two different constructions of golf ball. Further tests, also reported in this paper, investigated the effect of both clubhead speed at impact and ball compression on the impact duration.
The results suggest that the ball has a greater effect on impact duration than the type of clubhead with lower compression balls producing longer impact durations than higher compression balls and two piece balls producing shorter impact durations than three piece, wound balls. It was also found that the duration of impact decreased as the clubhead speed at impact was increased. Finally, results suggest that there is no correlation between the perception of the golfer and the actual duration of impact and therefore other factors are responsible for creating this perception.

The primary aim of this computational study was to investigate the effect of positioning the ball so that contact between clubhead and ball takes place at the point in the downswing where the clubhead achieves its maximum horizontal component of velocity. The double pendulum model of the downswing was employed and computational results were obtained for a range of ‘release angles’ (the release angle determines the stage in the downswing at which the wrist joint is allowed to turn freely). The position of the wrist joint and the direction of motion of the clubhead at the instant at which the clubhead makes contact with the ball were also determined. Furthermore, clubhead velocity was found to increase if the release is delayed (the so-called ‘late hit’). The energy supplied by the golfer was also investigated and, in particular, its variation with release angle is studied. Using a detailed perturbation analysis of the equations of motion, the results show that for a swing using a ‘natural release,’ the energy supplied by the golfer was a minimum.

This paper investigates how the physical characteristics of a golf club head affect its performance, with the aim of developing a superior club head. The physical characteristics investigated were the magnitude of the moment of inertia and the location of the centre of gravity. The performances were classified as the release velocity, the spin rate of the ball and the size of the uniform restitution area. In the numerical analysis, several kinds of club head were modelled with different moments of inertia and different locations of the centre of gravity. For every club head model, the ball was hit at various impact points and the release velocity, spin rate and size of the uniform restitution area were calculated. The results are as follows. The bigger the moment of inertia Izz about the vertical axis or the shallower the depth of the centre of gravity, the less side spin rate caused. Izz has more influence on the side spin rate than the depth of the centre of gravity. The shallower the depth of the centre of gravity, the more back spin rate caused. The size of the uniform restitution area increases in proportion to the increase in Izz, or in proportion to the decrease of the depth of the centre of gravity.

This paper presents a study on the optimization of loft and swing elevation angles on the impact between the clubhead of a driver and a golfball in order to maximize the distance of a drive. Computer programs were written to simulate the collision between the golfball and clubhead as well as the golfball in flight. A general, three-dimensional impact model using principles of momentum conservation on rigid bodies was used to simulate the impact between the golfball and clubhead to extract the spin and velocity vectors of the ball after impact. An aerodynamic model was then used to simulate ball flight in order to obtain the landing position of the ball. The results of the golfball landing positions generated by the computer simulations were compared to experimental data of golfball landing positions of shots hit by the golfing robot ‘Iron Byron’. The computer models were then used to calculate the optimal loft and swing elevation angles for a particular swing speed, clubhead mass, and golfball aerodynamic properties by making use of a nonlinear optimization routine. Also, the relationship between the maximizing distance for various driver loft angles and swing elevation angles is discussed.

A dynamic model of the golf swing is used to analyze the effect changes of “swing weight” and club type have on the forces, power, and work exerted by a golfer. The dynamic model demonstrates the significance of various equipment types as well as some modifications to the equipment. The model is also used to show how club head speed and the forces exerted by the golfer are changed by a slight modification to the kinematics of a real golf swing. On the basis of mathematical analysis, it is shown that the effect of drag on a golf swing is negligible, and that adding weight to the handle of a club to maintain “swing weight” has little effect on the forces exerted by a golfer. It is also shown that for the same club head speed, a graphite driver requires less effort than an ordinary driver.

Current marketing of golf clubs places great emphasis on the importance of the correct choice of shaft in relation to the golfer. The design of shafts is based on a body of received wisdom for which there appears to be little in the way of hard evidence, either of a theoretical or experimental nature. In this paper the behaviour of the shaft in the golf swing is investigated using a suitable dynamic computer simulation and by making direct strain gauge measurements on the shaft during actual golf swings. The conclusion is, contrary to popular belief, that shaft bending flexibility plays a minor dynamic role in the golf swing and that the conventional tests associated with shaft specification are peculiarly inappropriate to the swing dynamics; other tests are proposed. A concomitant conclusion is that it should be difficult for the golfer to actually identify shaft flexibility. It is found that if golfers are asked to hit golf balls with sets of clubs having different shafts but identical swingweights the success rate in identifying the shaft is surprisingly low.

How Golf Clubs Really Work and How to Optimize Their Design

- F D Werner
- R Greig

Werner F D and Greig R C 2000 How Golf Clubs Really Work and How to Optimize Their Design (Jackson: Origin Inc.)

- M G Reyes
- A Mittendorf

Reyes M G and Mittendorf A 1999 Science and Golf III ed Farrally and Cochran (Leeds: Human Kinetics) pp 13–19

- D C Winfield
- T Tan

Winfield D C and Tan T E 1996 Comput. Struct. 58 1217–24

- E J Sprigings
- R Neal

Sprigings E J and Neal R J 2001 Sports Eng. 4 15–21

- A B Turner
- N Hills

Turner A B and Hills N J 1999 Science and Golf III ed Farrally and Cochran (Leeds: Human Kinetics) pp 3–12

- M B Stanczak
- L D Lemons
- D Beasley
- J A Liburdy

Stanczak M B, Lemons L D, Beasley D E and Liburdy J A 1999 Science and Golf III ed Farrally and Cochran (Leeds:
Human Kinetics) pp 440–5

- K Watanabe
- S Kuroki
- M Hokari
- S Nishizawa

Watanabe K, Kuroki S, Hokari M and Nishizawa S 1999 Science and Golf III ed Farrally and Cochran (Leeds: Human
Kinetics) pp 29–39

- S Ujihashi

Ujihashi S 1994 Science and Golf II ed Cochran and Farrally (London: E & FN Spon) pp 302–8

- K Aoki
- Y Nakayama
- T Hayasida
- N Yamaguti
- M Sugiura

Aoki K, Nakayama Y, Hayasida T, Yamaguti N and Sugiura M 1999 Science and Golf III ed Farrally and Cochran
(Leeds: Human Kinetics) pp 446–56

- W Gobush

Gobush W 1990 Science and Golf I ed Cochran and Farrally (London: E & FN Spon) pp 219–24

- J Mather

Mather J S B 2000 The Engineering of Sport: Research, Development, and Innovation ed Subic and Haake (London:
Blackwell) pp 61–8

- M Hubbard
- L W Alaways

Hubbard M and Alaways L W 1999 Science and Golf III ed Farrally and Cochran (Leeds: Human Kinetics) pp 429–39

- P C Chou
- D Liang
- Yang J Gobush

Chou P C, Liang D, Yang J and Gobush W 1994 Science and Golf II ed Cochran and Farrally (London: E & FN Spon)
pp 296–301

- D R Budney
- D Bellow

Budney D R and Bellow D G 1990 Science and Golf I ed Cochran (London: E & FN Spon) pp 30–5

- R Maltby

Maltby R 1990 Golf Club Design, Fitting, Alterations and Repair (Newerk: Maltby)

- S H Johnson
- B Lieberman

Johnson S H and Lieberman B B 1996 The Engineering of Sport ed Haake (Rotterdam: Balkema) pp 251–6

- S H Johnson
- E A Ekstrom

Johnson S H and Ekstrom E A 1999 Science and Golf III ed Farrally and Cochran (Leeds: Human Kinetics) pp 519–25

- C J Dillman
- G Lange

Dillman C J and Lange G W 1994 Science and Golf II ed Cochran and Farrally (London: E & FN Spon) pp 3–13

- L D Lemons
- M B Stanczak
- D Beasley

Lemons L D, Stanczak M B and Beasley D 1999 Science and Golf III ed Farrally and Cochran (Leeds: Human
Kinetics) pp 423–8

- A Mittendorf
- M Reyes

Mittendorf A and Reyes M G 1997 http://www.Golfphysics.com
Miura K 2001 Sports Eng. 4 75–86