A Three-Dimensional Forward Dynamic Model of the Golf Swing
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A Three-dimensional Forward Dynamic Model of the Golf Swing by Daniel Johnson A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Master of Applied Science in Systems Design Engineering Waterloo, Ontario, Canada, 2015 © Daniel Johnson 2015 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Abstract A three-dimensional (3D) predictive golfer model can be a valuable tool for investigating the golf swing and designing new clubs. A forward dynamic model for simulating golfer drives is presented, which includes: (1) a four degree of freedom golfer model, (2) a flexible shaft model based on Rayleigh beam theory, (3) an impulse-momentum impact model, (4) and a spin rate controlled ball trajectory model. The input torques for the golfer model are provided by parameterized joint torque generators that have been designed to mimic muscular inputs. These joint torques are optimized to produce the longest ball carry distance for a given set of golf club design parameters. The flexible shaft model allows for continuous bending in the transverse directions, axial twisting of the club and variable shaft stiffness along its length. The completed four-part model is used for examining the following parameters of interest in club design by performing simulation experiments: clubhead mass, clubhead centre of mass location, clubhead moment of inertia, shaft flexibility, and clubhead and shaft aerodynamics. Analysis of the experiments led to the following recommendations for golf club design: 1. The clubhead mass should continue to be around 200g. 2. The centre of mass of the clubhead should be as close to the face as possible. 3. Shaft flexibility should be tuned for an individual golfer, depending on their particular swing. 4. Clubhead and shaft aerodynamic drag have a significant effect on the ball carry and clubhead orientation, and should be minimized during the club design process. Finally, suggestions are made for future research which can be performed in this area. iii Acknowledgments I would like to thank all the people who helped make this thesis possible. In particular I'd like to thank my supervisor, Professor John McPhee for his support, ideas, and sug- gestions throughout the process of creating this work. I'd also like to thank Dr. Joydeep Banerjee for his help with modeling clubhead and shaft aerodynamics, Dr. Steve Quin- tavalla for support in modeling ball aerodynamics, and Dan Wilson for help debugging the impact model and providing me with insight into how golfers think about their swings. I'd also like to acknowledge Professor David Clausi and Dr. Chad Schmitke for reading this thesis and providing invaluable suggestions for improving this work. I would like to acknowledge the Natural Sciences and Engineering Research Council of Canada and the Government of Ontario for providing funding support during the comple- tion of this thesis. iv Table of Contents List of Figures ix List of Tables xiii 1 Introduction1 1.1 Motivation and Goals..............................2 1.2 Project Outline.................................2 1.3 Contributions..................................3 1.4 Document Structure..............................4 2 Background and Literature Review5 2.1 Golf Swing....................................6 2.1.1 Biomechanics of the Golfer.......................6 2.1.2 Muscle Dynamics and Passive Joint Torques.............8 2.1.3 Club Flexibility............................. 14 2.1.4 Impact.................................. 18 v 2.1.5 Ball Aerodynamics........................... 23 2.2 Golf Driver Design............................... 26 2.2.1 Club Mass................................ 26 2.2.2 Clubhead Moment of Inertia...................... 27 2.2.3 Clubhead Centre of Mass Position................... 27 2.2.4 Shaft Flexibility............................. 28 2.3 Golfer Models.................................. 29 2.3.1 Early Golfer Models.......................... 29 2.3.2 Modern Golfer Models......................... 30 2.3.3 Other Golfer Models of Interest.................... 33 2.4 Opportunities.................................. 34 3 Golfer Model 36 3.1 Biomechanical Golfer Model.......................... 36 3.1.1 Active Muscle Torques......................... 39 3.1.2 Passive Muscle Torques......................... 43 3.1.3 Control of the Swing.......................... 46 3.1.4 Validation................................ 46 3.2 Flexible Club Model.............................. 47 3.2.1 Flexible Shaft.............................. 47 3.2.2 Clubhead................................ 50 vi 3.2.3 Clubhead and Shaft Aerodynamics.................. 52 3.2.4 Validation................................ 57 3.3 Combined Club and Golfer Model....................... 58 3.4 Impact Model.................................. 58 3.4.1 Validation................................ 67 3.5 Ball Aerodynamics............................... 68 3.5.1 Validation................................ 71 3.6 Optimal Control................................ 71 3.6.1 Objective Function One: Maximum Clubhead Velocity....... 72 3.6.2 Objective Function Two: Ball Carry................. 74 3.7 Complete Model................................. 76 3.8 Implementation................................. 76 4 Results and Discussion 80 4.1 Results for the Default Parameters...................... 81 4.2 Effect of Golfer Strength............................ 90 4.3 Effect of Clubhead Mass............................ 94 4.4 Effect of Clubhead Centre of Mass Position.................. 97 4.5 Effect of Clubhead Moment of Inertia..................... 103 4.6 Effect of Shaft Flexibility............................ 104 4.7 Effect of Club Aerodynamics.......................... 108 vii 4.8 Model Limitations............................... 110 4.8.1 Limitations of the Golfer Model.................... 111 4.8.2 Limitations of the Club Model..................... 114 4.8.3 Limitations of the Impact Model................... 115 4.8.4 Limitations of the Ball Aerodynamic Model............. 115 5 Conclusions and Future Work 117 5.1 Project Summary................................ 117 5.2 Recommendations................................ 119 5.3 Future Research................................. 119 References 123 APPENDICES 130 A Shaft Characteristics - Polynomial Fits 131 viii List of Figures 2.1 An illustration of muscle activation and deactivation dynamics [1]......8 2.2 Force-length curve for active portion of a human muscle [1]......... 10 2.3 Full force-length curve for a muscle including passive and active components. 11 2.4 Normalized force-velocity curve for a typical muscle [1]............ 11 2.5 General shape of the passive joint moment [2]................. 13 2.6 Illustraction of clubhead droop......................... 15 2.7 Typical flexing pattern of the club during the swing.............. 16 2.8 Top view of an open clubface orientation (left) and a closed clubface orien- tation (right)................................... 19 2.9 Illustrating the \gear effect” for a toed shot.................. 21 2.10 Top view of the effect of side spin on the flight of the golf ball........ 24 2.11 Flight paths of the ball illustrating the effect of backspin on the flight of the golf shot..................................... 25 3.1 Golfer model with four degrees of freedom indicated............. 38 ix 3.2 A comparison of muscle torques generated by Equation 3.1 to a typical muscle activation curve............................. 41 3.3 Sample joint torque curve incorporating velocity scaling........... 42 3.4 Contour plot illustrating the passive torques for the shoulder joint [3].... 44 3.5 Fitting curves for the passive forces at each joint in the golfer model.... 45 3.6 Flexible beam model as proposed by Shi et al. [4]............... 48 3.7 Flexible club properties provided by a club manufacturer........... 50 3.8 Diagram illustrating the frames of reference and measurements for the rigid body clubhead.................................. 51 3.9 Cd values for the clubhead as modeled for different combinations of clubhead speed and yaw angle............................... 53 3.10 An illustration of the relevant frames and velocities for calculating the aero- dynamic loads on the clubhead......................... 54 3.11 The combined golfer and club model in the early part of a swing. The wrist has yet to break................................. 59 3.12 Side view of the impact model illustrating impulses, frames of reference, and the clubhead ellipsoid.............................. 63 3.13 Top view of impact mode illustrating impulses, frames of reference, and the clubhead ellipsoid................................ 64 3.14 Free body diagram of the ball in flight..................... 69 3.15 Illustrating the required angles for calculating the penalties in objective function 3.37................................... 73 x 3.16 A plot of objective function 3.38 for constant X = 200 with Zmax = 10... 75 3.17 The complete golfer-club, impact, and carry model.............. 77 4.1 Ball trajectory for swing with default parameters............... 82 4.2 Clubhead centre of mass speed for swing with default parameters...... 82 4.3 Clubhead kinematics for swing with default parameters............ 84 4.4 Golfer joint angles for swing with default parameters............. 85 4.5 Golfer joint torques for