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THESIS
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This is to certify that the
thesis entitled

ANALYSIS OF A GOLF SWING WITH A KINEMATIC,
THREE-LINK PENDULUM MODEL

presented by
BENJAMIN CURTIS LIN

has been accepted towards fulfillment
of the requirements for the

m. 5 degree in Mach” [all Enjl‘nmj

WKW

Major Professor’s Signature

‘L/lz/o‘l‘

Date

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ANALYSIS OF A GOLF SWING WITH A KINEMATIC,
THREE-LINK PENDULUM MODEL

By

Benjamin Curtis Lin

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Mechanical Engineering

2004

Abstract

ANALYSIS OF A GOLF SWING WITH A KINEMATIC,
THREE-LINK PENDULUM MODEL

By
Benjamin Curtis Lin
The work contained within this thesis discusses the formulation of the three-link
pendulum model analyzing the golf downswing. The relationship was bridged by
formulating a unique, mathematical model using kinematically prescribed equations for
the shoulder, arm, shaft-release and wrist-roll motions. Lagrange’s Equations were
invoked to solve for the equations-of-motion governing the swing profile.

The analysis of sensitivity measurements of the applied moments in the wrist joint

with respect to the I12 and I23 inertia terms showed that the gradient was negative and
decreasing at impact. The oscillatory nature Was dictated by the kinematic assumption
and the rapid change of the sensitivities indicated that the kinematics are required to be
refined to investigate this behavior more closely. Further testing with the model
indicated that there was small variance in the maximum applied moments when only
principal inertia terms were used. However, a large change in the applied moments was
observed when each inertia term was scaled according to a sizable increase in the club
head.

The shoulder rotation study proved that more club head velocity is generated at
impact if the final angular displacement of the shoulder link is increased. Additionally, a
reduction of the maximum applied wrist-release moment and the total angular impulse in

the shaft link is achieved by increasing the final angular displacement.

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Preface

The work presented within this thesis is the culmination of my combined
education and research efforts at Michigan State University while pursuing a Masters
Degree in Mechanical Engineering. My prior knowledge of golf was due, in partial, to a
personal hobby and the exposure to the mechanics during my studies. The academic
work involved with studying topics in mechanical engineering has allowed the analysis of
the game from a theoretical perspective. I am amazed at how science and mathematics
can be used to explain such an advanced topic. Although this thesis represents my effort
that I have strived for during my stay in East Lansing, MI, there are individuals that have

guided me through and I would like express my sincere gratitude for their direction and

expertise.

1. Dr. Tom Mase, Composite Materials and Structures Center, Michigan State
University, for the initiation of the project, expertise on subject matter, direction
and leadership.

2. Dr. Ron Averill, Mechanical Engineering, Michigan State University, for serving

directly as my thesis advisor, direction, and leadership.
3. Dr. Alan Haddow, Mechanical Engineering, Michigan State University, for being
a member of the thesis committee, direction, and the Lab Coordinator for the

Mechanical Vibrations Lab where I worked as a Teaching Assistant.

4. My family for their constant words of encouragement and faith.

iii

Table of Contents

List of Tables ........................................................................................................ vi
List of Figures ...................................................................................................... vii
Key to Symbols and Abbreviations ..................................................................... x
Chapter 1 Introduction .......................................................................................... 1
1.1 Background Information ................................................................................ 1
1.1.1 Modeling the Golf Swing .................................................................... 2
1.2 Literature Review ............................................................................................ 4
1.2.1 Measuring the Motion of the Golf Swing ........................................... 4
1.2.2 Two-Link Pendulum Utilizing a Kinematically Driven
Downswing ......................................................................................... 5
1.2.3 Two-Link Pendulum Models Investigating Club Head Mass
Distribution ......................................................................................... 6
1.2.4 Three-Link Pendulum Model to Investigate Applied Hub Torque
Effect ................................................................................................... 8
1.2.5 Three-Link Pendulum Model to Investigate the Wrist-Release
Timing ............................................................................................... 12
1.2.6 Four-Link Pendulum Model Addressing Golfer Skill Level ............ 15
1.3 Thesis Problem Statement ............................................................................. 16
1.4 Methodology ................................................................................................... 18
Chapter 2 Formulation of the Mathematical Model ........................................ 21
2.1 Generalized Coordinates: w, a ,6, y ............................................................. 21
2.2 Description of the Kinematically Driven Downswing ................................. 23
2.2.1 Shoulder Link Prescribed Motion ..................................................... 24
2.2.2 Arm Link Prescribed Motion ............................................................ 25
2.2.3 Shaft Link Prescribed Motion ........................................................... 26
2.2.4 Club Head Rotation Prescribed Motion ............................................ 29
2.3 Derivation of Position and Velocity Vectors ................................................ 31
2.4 Squares-of-Velocities ..................................................................................... 35
2.5 Kinetic Energy for the Shoulder, Arm, and Shaft Links ........................... 37
2.5.1 Club Head Angular Velocity Vector ................................................. 38
2.6 Derivation of Lagrange’s Equations for the Kinematic Swing .................. 40
2.7 Special Cases of Simple Pendulum Motion ................................................. 43
2.7.1 Case I: Mt)‘=‘a(t)EO, ,6(t) Prescribed ............................................... 43
2.7.2 Case 11: M050, aft)E,B(t) Prescribed ............................................... 45
2.7.3 Case 111: 1140501 t)E,6(t) Prescribed .................................................. 47
2.8 Comparing the Kinematically Driven Downswing Model ......................... 49

iv

Chapter 3 Implementation of the Model and Presentation of the Golf

Downswing Modeling Results ........................................................... 52
3.1 Model Parameters .......................................................................................... 52
3.2 Computer Coding of the Kinematic Downswing and the
Generalized Forces ......................................................................................... 54
3.3 Club Head Inertia Components and the Effect on the Applied
Moments .......................................................................................................... 56
3.3.1 The Applied Wrist-Roll Moment as a
Function of Club Head Inertia ......................................................... 56
3.3.2 The Applied Wrist-Release Moment as a
Function of Club Head Inertia ......................................................... 61
3.3.3 Additional Inertia Component Studies .............................................. 68
3.4 The Effect of the Range-of-Motion of the Shoulder in the Golf Swing ..... 75
Chapter 4 Conclusions and Recommendations ................................................. 84
4.1 Summary and Validity of the Model ............................................................ 84
4.2 Future Work and Recommendations ........................................................... 88
Appendix A ........................................................................................................... 91
A.1 Derivation of Squared Velocity Equation (2.32) ............................... 91
A2 Derivation of Squared Velocity Equation (2.33) ............................... 91
A3 Derivation of Squared Velocity Equation (2.34) ............................... 92
AA Total Kinetic Energy ........................................................................... 93
A5 Differentiation of Total Kinetic Energy for the Shoulder Link
Generalized Force ............................................................................... 93
A6 Differentiation of Total Kinetic Energy for the Arm Link
Generalized Force ............................................................................... 95
A7 Differentiation of Total Kinetic Energy for the Wrist-Release
Generalized Force ............................................................................... 96
A8 Differentiation of Total Kinetic Energy for the Wrist-Roll
Generalized Force ............................................................................... 97
A9 Wrist-Roll Sensitivity Equations ........................................................ 98
A. 10 Wrist-Release Sensitivity Equations ................................................. 99
Appendix B ......................................................................................................... 100
B.1 Plots of the Generalized Coordinates vs. Time ................................. 100
8.2 Plots of the Generalized Forces and
Mass Center Velocities vs. Time ...................................................... 101
B3 Swing Plots of the Pendulum Positions for
Three Impact Configurations ................................................................... 102
References ........................................................................................................... 104

3.4

3.6.

 

 

 

 

 

 

 

List of Tables

Table Page
2.1 Comparison of Applied Moments with Other Model Types ....................................... 51
3.1 Three-Link Parameters in SI Units .............................................................................. 53
3.2 Kinematic Parameters Used for the Inertia Component Study .................................... 56
3.3 Calculated Mass and Inertia Data for the 470 cc Driver .............................................. 69
3.4 Principal Inertia Terms for the 350 cc and the 470 cc Drivers (kg-m2) ....................... 70
3.5 Results of the Inertia Component Case Studies ........................................................... 73

3.6 Percent Increase in the Maximum Moments in the Wrist
for the Case Studies ..................................................................................................... 74
3.7 Kinematic Parameters Used in the Calculation of the Angular Impulse

in the Shoulder and Shaft Links ................................................................................... 82

vi

List of Figures

Figure Page
1.1 Sketch of Two-Link and Three-Link Pendulums for the Golf Swing ........................... 3
1.2 Orientation of the Wrist-Cock Angle, .Q ..................................................................... 18
2.1 Three-Link Generalized Coordinates and System Parameters .................................... 22
2.2 Free-Body Diagram of the Shoulder Link ................................................................... 32
2.3 Free-Body Diagram of the Shoulder and Arm Links ................................................... 33
2.4 Free-Body Diagram of the Shoulder, Arm and Shaft Links ........................................ 34
2.5 Free-Body Diagram of the Entire Three-Link Pendulum ........................................... 35
2.6 The Orientation of the Club Head Angular Velocity Vector, a) ................................. 39
2.7 Free-Body Diagram for Case I .................................................................................... 44
2.8 Free-Body Diagram for Case 11 .................................................................................. 46
2.9 Free-Body Diagram for Case 111 ................................................................................. 48
2.10 Assumed Motion for Ref. 3 ...................................................................................... 49
3.1 Wireframe of a Golf Club Head .................................................................................. 54
3.2 Wrist-Roll Moment Sensitivity as a Function of the I n Inertia Component ............. 57
3.3 Wrist-Roll Moment Sensitivity as a Function of the I 22 Inertia Component ............ 58
3.4 Wrist-Roll Moment Sensitivity as a Function of the I 12 Inertia Component ............ 59
3.5 Wrist-R011 Moment Sensitivity as a Function of the I 13 Inertia ............................... 60
3.6 Wrist-Roll Moment Sensitivity as a Function of the I 2,: Inertia Component ............ 61
3.7 Wrist-Release Moment Sensitivity as a Function of the I n Inertia Component ....... 62

vii

 

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3.8 Wrist-Release Moment Sensitivity as a Function of the 122 Inertia Component ....... 63

3.9 Wrist-Release Moment Sensitivity as a Function of the I33 Inertia Component ....... 64

3.10 Wrist-Release Moment Sensitivity as a Function of the I 12 Inertia Component....65

3.11 Wrist-Release Moment Sensitivity as a Function of the I 13 Inertia Component ..... 66
3.12 Wrist-Release Moment Sensitivity as a Function of the I 23 Inertia Component ..... 67
3.13 Generalized Forces in the Wrist for the 350 Cubic Centimeter Driver .................... 70
3.14 Generalized Forces in the Wrist for the 470 Cubic Centimeter Driver .................... 71
3.15 Generalized Forces in the Wrist using the Principal Inertia Terms (350 cc) ............ 72
3.16 Generalized Forces in the Wrist using the Principal Inertia Terms (470 cc) ............ 73
3.17 Standard Impact Configuration of the Three-Link Pendulum

and Geometrical Proof .............................................................................................. 76
3.18 Incomplete Shoulder Rotation of the Three—Link Pendulum

and Configuration Geometrical Proof ....................................................................... 77
3.19 Advanced Golfer Impact Configuration of the Three-Link Pendulum

and Geometrical Proof .............................................................................................. 78
3.20 Club Head Velocity vs. Shoulder Finish Angle ........................................................ 79
3.21 Maximum Applied Moment vs. Shoulder Finish Angle ........................................... 80
3.22 Total Angular Impulse in Linkages vs. Shoulder Finish Angle ................................ 83
B.1 Angular Displacements of the Generalized Coordinates vs. Time .......................... 100
B2 Angular Velocities of the Generalized Coordinates vs. Time .................................. 100

viii

Key to Symbols and Abbreviations

Time Constants:

ta Total time for downswing

t, Time at wrist—release

tw Time at wrist-roll

tb Time interval between impact and wrist-release
tc Time interval between impact and wrist-roll

Kinematic Constants:

P Shoulder link kinematic constant
R Arm link kinematic constant

S Shaft link kinematic constant

W Club head kinematic constant
Greek:

a0 Initial angle of arm link

a Arm link prescribed motion

05: Finish angle of arm link

,60 Initial angle of shaft link

,6 Shaft link prescribed motion

,6; Finish angle of shaft link

7:, Initial angle of club head

7 Club head prescribed motion

7f Finish angle of club head

1%, Initial angle of shoulder link

I// Shoulder link prescribed motion
t/If Finish angle of shoulder link
Qa Generalized force in arm link
Q5 Generalized force in shaft link
Q, Generalized force in club head
QV Generalized force in shoulder link

 

 

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Figure Page

B3 Angular Accelerations of the Generalized Coordinates vs. Time ............................ 101
B4 Generalized Work vs. Time ..................................................................................... 101
B5 Mass Center Velocity vs. Time ................................................................................ 102
B6 Swing Plot for the Standard Impact Configuration .................................................. 102
8.7 Swing Plot for the Intermediate Impact Configuration ............................................ 103
B8 Swing Plot for the Advanced Impact Configuration ................................................ 103

ix

 

 

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CHAPTER 1

Introduction

1.1 Background Information

The basic components of a golf club consist of a club head, typically made from
steel or other lightweight materials such as titanium, a shaft made of steel or carbon fiber,
and a rubber or leather wound grip. Golf club designers differentiate the characteristics
of the golf club based on the physical attributes of the golfer and the manner in which the
golf club is swung. The latter part is related to the golfer’s skill level and their athletic
ability.

The way the golf swing varies among individuals adds to the diversity of
designing golf equipment because a wide range of skill levels exist. To help with the
understanding of component function, scientists and researchers have analyzed different
types of golf swings from an engineering mechanics standpoint in order to gain insight
between the golf swing and the interaction of the human body. The information has
helped golf club manufacturing companies design equipment that is more forgiving —
suitable for beginners and intermediate golfers who have yet to achieve their own optimal
level of golf swing consistency. In the past, touring professionals and people that golfed
on a consistent basis took notice in the types of clubs they used; however, nowadays
custom fitting of golf clubs has enabled the hobby golfer to increase their level of play.

Recently, a custom golf club maker and personal out-fitter developed “Game-
Improvement Clubs” and devised a systematic approach for determining the proper

equipment for the individual. Maltby [1] has outlined the major steps in the fitting

 

 

 

 

 

 

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process. The information in Maltby’s book is somewhat technical in nature, but written
for distribution to club-fitting outfits, sole proprietors that want to start their own
business, or for individuals whose interest is piqued. The literature has pertinence based
on a factor called the playability factor which involves the location of the center-of-
gravity of the club head. Since the diameter of the golf ball is regulated to 1.68 inches in
diameter, the playability is simply a measurement of the golf ball’s center-of-gravity with
respect to the center-of—gravity of the club head. “Playable” clubs are those that have a
low center-of-gravity with respect to the ball’s center-of-gravity and typically, clubs that
do not, tend to be sensitive to off-center impacts.

The goals of the work contained herein were to analyze the mechanics of the golf
swing from a mathematical standpoint and work to generate a model such that the
characteristics of the equipment and the golf downswing could be parameterized. If the
golf downswing were standardized, that is, if there were such a thing as a perfect swing
and configuration, how would the difficulty to swing a golf club be expressed in a
technical nature? Insight was sought regarding the characteristics of the golf club head
and the applied torques in the golf swing. With the help of engineering mechanics, the
work in this thesis analyzed a modern type of golf downswing with a representative

model.

1.1.1 Modeling the Golf Swing

Typically there are two general compound pendulum models that have been used
to describe the golf downswing: a two-link and a three-link pendulum. Figure 1.1 depicts
a basic sketch of how the pendulums may be orientated during the downswing. In the

general case, the figure uses arbitrary angles, with the Greek letter 19 assigned with

 

 

 

 

 

 

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numbered subscripts for each link. With a two-link pendulum, the arm and the shaft are

represented. The human shoulder entity is added in the improved three-link model. As

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opposed to the two-link model, the shoulder rotates at the central hub or the spine and
_ represents human torso forces, which are valuable sources capable of transferring energy
from the golfer to the golf club. In both types of models, the compound pendulum rotates
about a fixed hub. Adding the shoulder link allows the evaluation of the degree of
shoulder turn in a golf swing — a crucial aspect to make a solid golf shot. The left arm
(for a right-handed golfer), is represented, spanning from the end of the shoulder blade to
the wrist joint connection. For simplicity, in golf downswing modeling, the left arm is
kept straight and does not bend at the elbow. Lastly, the golf club is used as the third
linkage as in the double pendulum model.

The two-link pendulum benefits from simplicity, the governing equations and

solutions are found in many engineering textbooks, but the relationship between the

 

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human torso and the golf swing is omitted. Another aspect of the golf swing that has not
been investigated in depth is the interaction of the hips and legs. The Author felt that the
most complete mathematical model would involve the hip interaction, the shoulder, arm
and golf club, however, the level of detail of the results would probably be too complex

to make meaningful deductions.

1.2 Literature Review

This section will give attention to the works by sports researchers and scientists
who have analyzed the golf swing and subject matter. Since golf integrates the field of
biomechanics and engineering mechanics, many have used the previously described two

and three-link pendulum models to describe the golf swing.

1.2.1 Measuring the Motion of the Golf Swing

Cochran and Stobbs [2] have used high-speed photography of golfers to generate
drawings of the golf swing. They recorded the backswing and downswing of
professional golfers to find the position of the golf club head at certain time steps based
on the movement of the left hand. Knowing the positions of the left hand and the club
head, they showed how the golf club rotates in a circular are by connecting these two
points with a straight line. This information allowed the motion of the golf club to be
described as a function of time, which introduced a new way to model the golf swing
with the two-link pendulum. Expanding on this method of using high-speed
photography, researchers have measured the motion of the golfer’s shoulders, arms,
wrists and the golf club head during the downswing. Utilizing the experimental data, the

kinematic motion is then estimated as a continuous function of time for the shoulder, arm

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and golf club. These kinematic functions were then used as prescribed motions for the
linkages and inverse dynamics was used to solve Newton—Euler equations for the required
applied moments responsible for the motion.

Assigning a kinematic motion to the linkages as a function of time is the best way
to model the golf downswing because the timing aspects of the linkages are prescribed,
meaning that different types of golf swings can be modeled. Typically, the motion of the
hands is modeled in two stages — before wrist-release and after wrist-release time. The
release time is specified based on the angular position of the wrist and describes the
snapping-like action of the hands to generate club head speed leading up to the point of

impact with the golf ball.

1.2.2 Two-Link Pendulum Utilizing a Kinematically Driven Downswing

Winfield and Soriano [3] have discussed the modeling of a flexible beam element
with a tip mass driven by two rotational degrees of freedom. This information is useful
because the kinematic motions of the linkages were prescribed functions of time. The
golf downswing of a test subject was approximated from data recorded using a motion

analysis system resulting with the following prescribed kinematic functions:

a=—0.747 Sin(10.472t)+7.830(+2.356 OSISO.3 seconds (1.1)
,B=-—l.570 03130.1 seconds (1-2)
,6 = —0.446 sin(15.708 t) + 7.003 t- 1.570 0.1 S t S 0.3 seconds (1.3)

The rotational motion of arepresented the motion of the golfer’s shoulder and arm while
the motion of ,6 represented the hands. Notice, the motion of ,6 does not start to change

until after t = 0.1 seconds, which describes how the release time of the golf club is

 

 

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designated to occur at a specific time step. Newton—Euler equations-of—motion were
solved by method of integration with a Newmark integration scheme and a time step of
one millisecond was required for convergence. The work focused on finding the
deflections of the golf club head during the downswing and showed that the position of
the tip mass was ahead of the center-of—mass of the shaft at the time of impact.
Interestingly, the work concluded that the forces required to provide the kinematic
motions caused the fundamental frequency of the golf shaft to change during the

downswing, meaning that the stiffness of the golf club varies as a function of time.

1.2.3 Two-Link Pendulum Models Investigating Club Head Mass Distribution

In an effort to investigate the mass distribution of a club head on performance,
Whittaker [4] has investigated two main categories of club head designs — traditional
blades and newer cavity back designs. The blade iron has a small hitting surface area and
heel-toe weighting, making it a less forgiving design. The concentration of mass is
directly in the center of the club head. When the golf ball is struck in the “sweet-spot”
correctly, the impact response is small and the golfer experiences a small, negligible
amount of vibration. However, the golfer is penalized with an awkward feeling and
large, vibrational forces due to the eccentric impact are transmitted to the golfer when the
ball is miss-hit. On the contrary, a cavity-back design has an increased amount of mass
along the edges of the golf club and thus more perimeter weighing. This design has an
enlarged sweet-spot, and typically the club face is oversized relative to a blade design.
An increase in perimeter weighing affects the moment-of-inertia of the club head, and it

is less likely to twist with an off-center impact with the golf ball.

 

 

 

 

 

 

 

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The work detailed three models and used a finite element program called DADS
(LMS International, Belgium) to dynamically simulate a two-pendulum model. Model
One simulated a rigid shaft whereas Model Two simulated a flexible shaft. In Model
Three, no golf shaft was used. In all three models, off-center shots were tested and the
paper reports information that is common knowledge - blades are less forgiving with off-
center golf ball impacts. The main differentiation in the results was small, and the work
concludes that there was no major difference between cavity backed clubs versus a blade
designs when comparing on—center shots. An important note that the paper makes is that
in the First Model (rigid golf shaft), no major difference exists between the cavity and
blade design in terms of distance and drift in the golf shot.

In a similar type of work, Nesbit et al. [5] discussed the effects of inertia tensors
on iron golf club heads and the influence on the golfer and golf club. The inertia tensors
of various five-iron club heads were measured using solid modeling techniques. Using
the software package ADAMS (MSC Software, Santa Ana, CA), a computer model of a
golf club was built with input characteristics for the inertia properties. The golf club was
modeled with a flexible shaft of three dimensional beam elements and an appropriate
material model for golf shaft damping characteristics. The shaft was joined to a rigid
club head having mass and inertia assigned for the selected club head designs. Kinematic
data obtained from a recorded golf swing was used to drive the model and the impact of
the club head and ball was modeled by a spring-damper function.

The analysis determined that varying inertia tensors did not have a large effect on
the ability to swing the club regarding on-center impacts. According to the paper, for an

eccentric impact (outside of the center-of—gravity of the club head), the first principal

 

 

 

 

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inertia component had the greatest effect on the transmitted torques and the club head
deflections. Regarding eccentric impacts below the center-of—gravity of the club head,
the third principal inertia component had the greatest effect on the forces and club head
deflections. Linear forces were unaffected with varying the inertia tensor of the club
head. This model explored new ground in the simulation world regarding the club head,
but did not model the human counter parts as a function of the inertia tensors (shoulder or
arm); rather the reaction forces were calculated in the shaft and the club head. Also, a
relationship between the products-of-inertia of the club head and the wrist-release torque
was not detailed in the paper.

The main conclusion of the paper does report a relationship between the club head
inertia and the effect on the golfer and equipment. Inference is given to suggest that in
designing club heads, it is desirable to select inertia values to give more impact
performance in all directions, meaning that consistency between impacts on the toe
versus the heel of the club head would be met. The Author felt that this means
consistency of inertia terms could be included as a design specification. Also, the paper
touched briefly on how golf club head club head could be designed so that each club

within the set has uniform mass and inertia properties.

1.2.4 Three-Link Pendulum Model to Investi gate Applied Hub Torque Effect

Recently, Turner [6] created a numerical procedure with computer simulation that
used a rigid, three-link pendulum consisting of the shoulder, left arm, and golf club
swinging in a single plane of motion. The work simulated a backswing plus a
downswing and focused on finding the relationship between the applied torques and how

to produce a well-timed golf swing. Contrary to the work in Ref. 3 the forward dynamics

 

 

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problem of a three-link pendulum model was solved by writing Newton-Euler equations-
of-motion. This method requires constraint equations to be written for each individual
link, describing the positions of the center of mass. This helps to simplify the number of
unknown entities in the equations-of-motion. Following are the key equations from this
paper.

Newtonian Equations (Conservation of Linear Momentum):

x: mixiz—Fim+Fl.x n—lZiZl (1.4)
m" x; 2" F,u (15)
y: ml. y, =—E+,y+Fi).—mig n—lZiZl (1.6)
m,,.v,, = -F.,. - mg (1.7)

Eulerian Equations (Conservation of Angular Momentum):

[(6, =C, —C,.+l +17“); sinQ +Fims, sin6, —F,.+,),s,.c036,. n—lZiZl (1.8)

In 6,, = C" + Fmrn sin 6" — Fnyrn cos 6,, (1.9)
Holonomic Constraint Equations:

xi—ncosl9iz0 (1.10)

.v.-r.-Sl°n9.-=0 (1‘11)

For the model in this paper, in Eq. 1.1 l, the coordinates x,- and y,- locate the mass center

for the link with respect to the fixed, inertial frame of reference and r, is the distance

 

 

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between the proximal end and the mass-center distance. For n links, the system of

algebraic constraint equations are written and solved at each time step:
xi—ricosQ=x,_,+s,._,cosQ_, nZiZZ (1.12)
yj—nsinQ=y,_,+s,._,sinQ-, nZiZZ (1.13)

The method of this type of solution requires Equations (1.12) and (1.13) to be each
differentiated twice with respect to time to help alleviate the number of unknowns in the

equations-of-motion.

O. C. .2 O. C. .2
xi+ri Q san +r; Q cosQ = x.-;—s,_,Q szn Q_, —s,._,Q .-_;cosQ_, (1.14)
O. O. O 2 O. O. .2
”vi—r, QcosQ +rl. Q sinQ = y,_,+s,._,QcosQ_, —s,._,l9 ,-_1sinQ_, (1.15)

Using Equations (1.14) and (1.15) to reduce the number of unknowns in the equations-of-

motion, the accelerations x, , y. , and Q were solved for and the new positions and

velocities were then found by using a first-order time stepping scheme at the next time
step. A time step of one-half millisecond was required for convergence of the model.
Seeking a relationship between the backswing and downswing applied torques,
the work concluded that a constant set of torques in each set resulted in a “good” golf
swing. From the information gained within the literature, the Author feels that the output
and orientation of the linkages were used to determine the best configuration of the golf
swing based on a specification of the applied torques required to satisfy the equations-of-
motion. The paper reports the two sets of applied moments were closely related, meaning
that a well-timed golf swing was sensitive to the values with the torque applied to the

shoulder link being the most influential. In the golf downswing the shoulder link has the

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task of aligning the arm and golf club links into the correct positions, and this was
accessed with the specification of applied hub torque.

The information within the paper concluded that if the applied wrist torque was
too low or high, a correction could be made by adjusting the arm and shoulder torques in
proper proportion. In golf terms, this means that the model simulated what happens when
the wrist torque is too high causing the club head to reach impact too early which is a
common problem. The solution for this is to turn the shoulders faster, meaning that more
torque should be applied to the shoulder to correct for the position of the wrist joint
during the downswing, reinforcing that a correct shoulder turn during the downswing is
important to ensure proper form and tempo of the modeled downswing.

Additionally, the work investigated what happens when the arms are swung away
from the body with out-of-plane motion. This is important because the key to making a
strong powerful golf shot is to keep the swing compact, reducing the number of degrees-
of-freedom. When the arms are swung incorrectly by an improper arm torque, then there
is an increase of the total moment-of—inertia of the system and the hub motion is slowed
resulting in energy consumption from erratic arm movement, proving that it is important
to keep the arms close to the body providing a more compact and consistent swing. The
syndrome is termed, “flying elbow” and is common among beginners and it affects the
accuracy of the golf shot.

The optimum result required two sets of constant torques for the arm and the
shoulder. Turner acknowledged that the applied torques in the real world are far from

constant, but in this model, the best profile of the simulated golf swing required a

ll

constant torque input for the shoulder and the arm. Also, the correct specification of the
applied shoulder moment was essentially the main conclusion of this paper.

The Author felt that it is not practical in the real golf swing to maintain a constant
set of torques because the idea to keep in mind is to accelerate through impact, therefore
the set of applied torques by the shoulder and arm would not be constant in a more
representative model of the golf downswing. In conclusion, this work stated that the
model could be useful for comparing the positions from the simulated output to

stroboscopic pictures for correcting the hand and arm positions in the downswing.

1.2.5 Three—Link Pendulum Model to Investigate the Wrist-Release Timing

The objective of this work by Sprigings and Mackenzie [7] was to determine how
the delayed wrist-release in a golf swing affects the club head velocity. It is known that a
delayed wrist-release will provide more club head velocity at impact, but the work here
set forth to determine how much and to what degree. The work focused finding how
much the wrist torque affected the club head velocity at impact, and to quantify the main
sources of mechanical power in the downswing. In theory, Jorgenson [8] determined the
delayed wrist-release during the downswing will result in faster club head speeds by
making the energy transfer occur later, by restraining the wrist-release.

The modeling utilized a rigid, three-link pendulum representing the shoulder, arm,
and the golf club. Applied torque functions that reflected the properties of human muscle
were substituted into the equations-of-motion. These functions are documented in the
work by Sprigings and Neal [9]. The instantaneous torque associated with human muscle

can be described by the following equation:

12

_I_ ._
T27711(11[l—e T]|:_E)AW_I__1§)_:| (1'16)
(1) + (l)

[HUI

This function is a prediction from the Hill model structure, which is a force-velocity
approach to describing human arm movement by Niku and Henderson [10] and Sprigings
[11]. In Eq. 1.16, T represents the instantaneous value produced by the torque generator,
Tnm is the maximum isometric torque (allowable limit of the joint), (am is the maximum
angular velocity of the associated joint, (ois the instantaneous joint angular velocity, Fis
a shape factor controlling the relationship between the torque and the velocity curve, I is
the elapsed time from the activation of the torque, and Tis the activation time constant.
For the study in Ref. 7, the values for Tm,” were 180 N-m, l20-N-m, and 60 N-m for the
spine, shoulder and wrist respectively.

Three separate cases were setup and deductions were made according to benefit.
The focus of the modeling was constraining of the wrist torque by resisting the rotational
motion of the club by the wrist joint. Case One did use any amount of resistive wrist
torque and the simulation aimed at a natural wrist-release. Case Two delayed the wrist-
release timing by applying a resistive torque, then after the time of wrist-release, an
actively applied wrist torque was used to accelerate the club head to impact. Case Three
was similar to the second in that a resistive wrist torque was used to delay the release, but
after the release point, an active wrist torque was not present. Following are the key
equations from this paper.

Newtonian Equations (Conservation of Linear Momentum):

Fm = Zmrari (117)

13

F\.n=:mi(afl+g) (1.18)

i=1
Eulerian Equations (Conservation of Angular Momentum):

[nan : Ftnr2n-I 31" 6n — Fl'nr2n—I COS 6n + Frn+lr2n 31716

n+1

(1.19)
—F\'n+lr2n COS 6n + Cn — Cn+l
Constraint Equations:
n—I
am = |:Z—(2',.L,~ sin Q — (0,-le cos Q — anrzm sin Q, — wnzrzn_, cos Q] (1.20)
i=l
11-] 7
a)?! 2|: (ail? COSHI —a)i~Li Sinai +anr2n—l 6086!: —a)"2r2"_, sing" ):I (121)
i=1

In Eq. 1.20 and 1.21, F is the external component of the force on the link, a is the
linear acceleration of the mass center, or is the absolute angular acceleration of the link, r
is the length between the link’s proximal end and the mass center distance, 6 is the
angular displacement of the link measured in a counter-clockwise sense, and L is the
entire length of the link. In Eq. 1.19, the applied couples, C", were the muscle torque
terms previously described.

The computer software MATHEMATICA (Wolfram Research, Inc. Champaign,
IL, USA) was used to find the expressions for the angular accelerations, or of links 1-3.
A fifth-order Runge-Kutta-Fehlberg algorithm was used to integrate the equations-of-
motion. The results showed that a combination of using a delayed wrist technique plus a
resistance of the natural release by the wrist, followed by an actively applied wrist torque
provided the maximum club head velocity at impact. Also, the work concluded that the

major source of mechanical power came from the muscles crossing over the arm joint,

14

followed by the muscles in the wrist, and lastly by the muscles enabling the rotation of
the torso.

The model explores new ground in that it takes into account the behavior of the
human muscles and emulates the characteristics into the model. This work confirmed
earlier studies by Jorgenson [8] and Lampsa [12] on the correct application of active wrist
torque timing in that it would produce approximately a one percent gain in club head
velocity if the torque was delayed to about seventy milliseconds before impact. The
studies by Sprigings and Mackenzie attempted to find a relationship between an
excessively applied wrist torque, with the characteristics of human muscle properties, and
how it influences club head velocity in the downswing. The Author felt that a driving
wrist torque is good to prevent deceleration of the golf club, which is a common problem;
however, an excessive application may penalize golf shot accuracy due to misalignment
of the golf club face at impact. The modeling of the three cases analyzed the wrist-

release torque based on time, but omitted the analysis of the effect of the shoulder link.

1.2.6 Four-Link Pendulum Model Addressing Golfer Skill Level

In a recent paper, Iwatsubo et al. [13] addressed the dynamic simulation capability
of the three different types of double pendulum models. They used two types of
pendulum models: a four-linkage device and a two-linkage device. The four-linkage
device is an alteration of the arm link in that it considered the elbow joint as a degree-of—
freedom. This means that the shoulder, left upper-arm, forearm, and the golf club shaft
were taken as the individual links. The other pendulum was a conventional two-link

model.

15

As stated, the work presented three types of models. Model One utilized the two-
link pendulum where the arm had a constant length and the elbow was not considered a
degree-of-freedom. Model Two also used the two-link pendulum where the arm had a
constant length, but the length was calculated based on the weighted arithmetic average
distance based on the kinetic energy at each time step. Model Three used the complex,
four-bar linkage pendulum with a variable virtual link changing the length of the arm in
terms of the distance between the neck joint (hub) and the wrist joint. In each model an
elastic shaft was used and modeled as a cantilever beam.

This model accessed the different skill levels of golfers and added the
functionality of the degree-of-freedom of the elbow joint. Sometimes, average golfers
bend their elbow too much during the downswing, and the Author felt that the variable
virtual upper-arm and forearm length modeling was what the work was attempting to
simulate. Also, by varying the distance of the virtual link as a function of time, golfer
skill can be evaluated. Comparing an intermediate golfer with an elite golfer was done
with the variation of the length between the neck and wrist joints. Prior to wrist—release,
the virtual length of the intermediate golfer changed, whereas for the elite golfer, it was
constant in time. After the time of wrist-release, for the elite golfer, the total length of the
virtual link was extended fifty percent, representing how a good golfer would fully
extend the left arm during the downswing in a large sweeping motion to produce a good

golf shot.

1.3 Thesis Problem Statement

In an effort to reach the ultimate goal of fully understanding the relationship

between the club head inertia terms and the applied moments exerted by the golfer’s

16

 

E—l

 

 

 

wrist, the work in this thesis progressed in that direction by contributing a unique type of
kinematic downswing model. In this work, a rigid, three-link pendulum model was
developed with general club head inertia terms. Also, the kinematic motion of the wrist-
roll motion was added and attempted to describe how a golfer would rotate the wrists to
square the golf club face to the ball just prior to impact. The dynamic collision of the
club face and the golf ball impact was not considered. The purpose of creating this
model was to see if analyzing the sensitivity of the applied moments in the golfer’s wrist
would provide information regarding the golf downswing if the motions of the wrist were
prescribed.

Aside from the sensitivity measurements, the inertia parameters were varied
directly. The inertia values for a 350 cubic centimeter club head were obtained from a
simulation output, and the maximum moments in the wrist were calculated for a
particular downswing motion. In comparison, the inertia values were scaled accordingly
by considering a sizable increase of the club head to 470 cubic centimeters and as before,
the effect on the applied moments was observed. Last, the principal inertia values for
both second order tensors were calculated and then the maximum moments in the wrist
were calculated. A presentation of the results is provided in Chapter 3.

Relating the golfer’s kinematic impact position, the range-of—motion of the
golfer’s upper body was modeled by changing the final angular position of the shoulder
link. The resultant effect on the applied moments, the total angular momentum, and the
club head velocity were calculated and observed and the differences were observed based

on three different impact configurations of the pendulum.

l7

1.4 Methodology

Specifying the motion of the linkages was accomplished with kinematics, and the
angles-of—rotation of the pendulum were defined as the generalized coordinates. Using
this method, a textbook golf swing was described by finding the appropriate positions in
time of the compound pendulum.

In this model, the kinematic motion of the pendulum was described in three
stages. At the top of the swing, the shoulder, arm and shaft are all at initial angles

relative to the vertical. These are the fixed initial angles and can be prescribed for any

   

Wrist-Cock Angle
Q

    
  

Shaft Link /
\ Arm Link

K Wrist Joint
Figure 1.2: Orientation ofWrist-Cock Angle, .0.

type of golf swing type based on the golfer’s skill level. During the downswing, the
shoulder and arm are given prescribed sinusoidal functions that have time dependency.
The golf shaft however is dependent on the motion of the arm link, which is a dependent
on the shoulder link. The details of this model will be given in Chapter 2. In the time
period before the wrist-release (stage 1), the golfer holds the club in a manner where the
left wrist (for a right-handed golfer) is fixed such that the angle between the respective

arm and shaft, defined as the wrist-cock angle, is constant. A sketch is provided in Fig.

18

1.2. The release of the angle .(2can be specified in time and exists as a model parameter.
Elite golfers hold the club such that the angle is fixed until an optimum time — found by
hours of practice and instruction.

As the downswing continues, based on the inertia of the golf club, there would be
a natural release point where the wrist joint releases the golf club and .Q will increase as
the arm and club shaft become collinear as the shaft link rotates through more angular
displacement. This represents the second stage. The kinematic profile of the golf shaft
will have a different motion based on the reduced time period and the current angle of the
golf shaft at the time of wrist-release. This second kinematic motion for the golf shaft
will carry through until impact. Equations and theory of this kinematic assumption are
discussed in Chapter 2.

The third and final stage of the downswing describes how a player rotates the golf
club about the longitudinal axis of the shaft to square the club face to the ball at impact.
Similar to the prescribed motion of the golf shaft, this motion is also described in two
time intervals designated by a point in time called the wrist-roll time. It has been
researched by Milne and Davis [14] in their investigation of the role of the golf shaft in
the downswing that an optimum time is about fifty milliseconds prior to impact. This
was determined by measuring the motion of the downswing of a collegiate athlete on the
golf team. Prior to the impact time, the angle of the club face with respect to the ball is
defined as a constant. After the wrist-roll time, this angle is described similarly to the
shaft release kinematic function, operating in the reduced time period.

The benefits of prescribing kinematic functions for the motion of each link allow

the specification of critical input parameter such as the finish angle of the shaft link, the

19

initial and final angles of the shoulder link, and the specific time steps of the wrist-release
and wrist roll times. By definition, the finish angle of the shaft link is aligned with the
vertical axis at the time of impact. By specifying the configuration of the shoulder link,
the arm link finish angle is calculated and the benefit enables different impact
configurations to be analyzed. This is proven geometrically in Chapter 3 in the section
that describes the modeling to find the effect of the range-of-motion of the shoulder link.
Chapter 2 begins with the development of the kinematic downswing model and
leads into the derivation of the governing equations-of-motion responsible for the
kinematic motion of the links. Chapter 3 discusses the method of computer coding, the
model parameters that were gathered from the literature search, and the subsequent
modeling results. A summary of the main conclusions and recommendations for future

work is the subject of Chapter 4.

20

CHAPTER 2

Formulation of the Mathematical Model

The mathematical model for the compound, three-link pendulum is described
within this chapter. Modeling the golf downswing was done with prescribed, time-
dependent generalized coordinates. Generalized forces, in this case applied moments,
needed to achieve the prescribed motion were computed from equations derived from

Lagrange’s Equations.

2.1 Generalized Coordinates: 1!, g, Q, y

The three-link pendulum model used to describe the golf downswing is a
conservative system with holonomic constraints, implying the number of generalized
coordinates used to describe the motion is equal to the number of degrees-of-freedom
Greenwood [15]. Also, the system is modeled as frictionless and the constraints are
rheonomic, which means the generalized coordinates are explicit functions of time [15].
It is possible to characterize the configuration of the compound pendulum that has four
degrees-of—freedom with position and length; however, given the description above, and
with the assumption of the time-dependency of the angles, it is advantageous to define
the angles of rotation as the generalized coordinates.

Figure 2.1 shows the three-link system parameters and the orientation of the

generalized coordinates. Define the shoulder link to have mass m], length L], and mass

center from the hub axis point, (11. The time-dependent generalized coordinate, Wt)

21

describes the motion of the shoulder link during the downswing. Similarly, define the

arm link to have mass m2, length L2, and distance to mass center from the end of the

d1 Ll

. .
Arm Link. V“) .
mass = ml \ a(t)

1.3Y/ d

Shoulder Link,
mass = m l

I

Btt)

a Y (t) ( Shaft Link.
mass -
Club Head
mass = m 4

Figure 2.1: Three-Link Generalized Coordinates and System Parameters.

shoulder link, d2. The time-dependent generalized coordinate, oft) will describe the
motion of the arm link. Next, define the golf shaft to have mass m3 inclusive of golf grip,
length L3, and distance from the butt-end of the shaft as (13. The time-dependent
generalized coordinate, ,B( t ) describes the motion of the golf shaft. The wrist-roll motion,
is the rotation of the golf shaft about the longitudinal centerline of the golf shaft so that
the club head is square to the target at impact is defined as )(t). The next section will

describe the kinematic assumption of each generalized coordinate.

22

2.2 Description of the Kinematically Driven Downswing

This section will describe the formulation of the time-dependent, generalized
coordinates Mt), art), ,8“), and )(t) and the orientation will be described with figures of
the free-body diagrams of each individual link starting in Section 2.3. Mase [16] has
specified a way to model the downswing of a two-link pendulum model by specifying the
coordinates as a function of time. Accordingly, the two-link pendulum models in Refs. 3
and 5 utilized kinematic prescribed downswing functions approximated from
experimental data to drive the linkages.

The downswing starts with the assumption that the golfer has just completed the
backswing and is about to initiate the motion to impact the golf ball. The time constant,
ta will be the total time for the downswing. During the downswing, the wrists will
naturally release, but depending on the skill of the golfer, the release time will vary.
Allow the constant, I = I, to describe the time at which the wrists initiate release of the
golf club. Similarly, the rotation of the golf club about the shaft centerline to square the
club head at impact is defined as the wrist-roll. Define this time as I = I...

Having these two finite times located during the downswing, it is convenient to
define two separate time periods that characterize each phase of the downswing. Recall
that in the first phase of the downswing the wrist-cock angle, {2, is constant. Define tb as
the time interval between impact and the time at which the wrists release: n, = ta - t,.
Also, in the last portion of the swing, let tc define the time interval between impact and
the wrist-roll time: t, = ta - t... Having these key time constants (ta, 1,, t,,., tb, and t.)

described, the kinematics of the downswing will be described in the next section.

23

2.2.1 Shoulder Link Prescribed Motion

Recall, 1W) is the generalized coordinate that describes the motion of the shoulder
link. There are two fixed states in this downswing: the position at the beginning of the
downswing and the impact state. Let the constant 11/0 be the angle that the shoulder link
makes with the vertical axis passing through the hub axis at the start of the downswing
and 11/be the angle that the shoulder link makes with the vertical at impact. Assume that

a time-dependent sine function adequately describes the shoulder motion over this time

period, which takes the following form:

W(t)=P[t—L'-Sin(-fl1]:|+i/lfl (2.1)
71' id

To find the kinematic constant P, allow the finish angle of the shoulder link at impact to

be a function of WI), so:

t 71'!
= t =P t -i 7' —“ + :>
Wf W( a) [a If SHI( t ]] Wu

(1

P = [L I V“ j (22)

Differentiating Eq. 2.1 with respect to time to find the angular velocity and acceleration:

y}(z) = P[I—cos[-:[:t)] (2.3)

1;;(t) = Plsin(fl] (2.4)

24

Physically, the fundamental form of Eq. 2.3 represents a golf downswing well
because the velocity of the shoulder link starts from rest at the top of the swing and then

reaches a maximum at impact.

2.2.2 Arm Link Prescribed Motion

The angle a(t) will describe the motion of the arm link. Similar to the shoulder
link, the arm link is defined to have two defining position parameters at the onset of the
downswing and impact. To remain consistent in the nomenclature, let 04, be the angle the
of the arm link at the beginning of the downswing and off he the angle at impact. Both

angels are measured from the vertical, given the configuration of the shoulder and arm

link. Consider the following form of the kinematic downswing for the arm:

(I

a(t) =R[t—tisin[fl]]+an (2.5)
71' I

To find the kinematic constant R, consider of,- as a function at the time of impact:

I . 7”
(II = a(tu)= R|:tu —ism[—H+a§, 2
71' I

(I

 

l“)
R = ' (2.6)

(I

Differentiating Eq. 2.5 with respect to time to find the angular velocity and acceleration:

an): R|:I—cos[f—:H (2.7)

;(t)=Rt£sin[fl) (2.8)

25

2.2.3 Shaft Link Prescribed Motion

The kinematics of the golf shaft motion during the downswing requires a slightly
more sophisticated description to account for having the wrist cocked initially. During
the downswing, the profile of the shaft is assumed to be a rigid body motion. Prior to the
wrist-release, the arm and the club essentially rotate as a single rigid body, thus
simplifying the kinematics in the pre-wrist release phase.

As described before, there are two phases describing the motion of the shaft link
during the downswing. Sprigings [7], in his work studying the delayed wrist-release, has
stated that the wrist-cock angle of elite golfers, like Tiger Woods and Ernie Els measures
approximately 900 during the initial stage of their downswing. By definition of

conventional angle sense, flfl) is a counter-clockwise measurement and is a negative

angle. Geometrically, the angle that ,B(t) makes with the vertical is: (1(1) - 7r+ [2, and in

the first part of the downswing, where t < t,:
,6(t)=a(t)—7r+.(2 0<t$t, (2.9)

Substituting Eq. 2.5 for a(t):

7! I

(I

fl(t)=[R[t—tisin[fl]]+au:|—II+Q O<rStr (2.10)

Differentiation with respect to time, the angular velocity and acceleration are

respectively:
M0550) 0<tSt, (2.11)
23(1):}541) 0<t$t, (2,12)

26

Now, consider the second phase of the downswing for the shaft link. As the wrist
unhinges, the wrist-cock angle will increase in a counter-clockwise direction. At this
point, the arm and the shaft link become two distinct rigid bodies with identical initial

conditions. To find the value of flfl) at the time of wrist-release, recall that the shaft is

still in pre-release phase, therefore: ,B(t,.) = 01b) - 7! + .(2. Substituting for t, and aft) :

 

II t

(I

.B(tr)=a(tr)—II+Q=|:R[I, —tisin[m' ]]+a(,]—II+Q (2.13)

Therefore, the shaft release angle is a direct function of the arm link at the time of wrist-
release and the wrist-cock angle. The kinematic constant R is given in Eq. 2.6.
Enforcing position, velocity, and acceleration continuity of the arm link after the wrists
unhinge (t > t,), the prescribed motion of the shaft link must retain the kinematic
constraints of the arm in terms of angular velocity and angular acceleration as the wrist is
released. To ensure Eqs. 2.5, 2.7, and 2.8 follow continuity at t = t, assume the following

equation for ,B(t) after release time:

fl(t—t,)= 5|:t—t, —tisirz[fl—(t:’)]]+fl(r,)+a(t,)(t—t,)

(2.14)

Eq. 2.14 includes two extra terms and was obtained by maintaining angular velocity and
acceleration continuity of the arm link prescribed motion at the time of shaft release.
Applying the same methodology for finding prior constants P and R, S is found by

enforcing 16m): 0. Substituting for ta and t1, and rearranging Eq. 2.14 for S gives:

27

——+a’(t,)+%a’(t,)th (2.15)

Noting that subsequent time steps after the wrist—release, the angular velocity and
acceleration of the shaft link is inclusive of the continuity relationships of the arm link at
wrist-release stage, moreover:

,fi(t—t, ) = SjI—cos[fl—(t:—t’—)]J+o(t,)+EU, )(t —t,) t > t, (2.16)

[b

kin—1,): S£[sin[££t—_tr—)—]:|+;(t,) t> 1, (2.17)

1,, 1,,

To reiterate the derivation of the shaft link motion during the golf downswing,
divide the motion into two phases, pre—wrist-release and post-wrist-release. In the first
phase, the shaft angle is determined from the arm link measurement with the vertical and
the wrist-cock angle. The wrist-cock angle is essentially a constraint, reducing the
generalized coordinates. As the wrist unhinges, the shaft is based on the second
kinematic assumption and the angle at which the shaft makes with the vertical, or ,B(t,).
The kinematic profile in the second phase is prescribed for the shaft link until the time of
impact between the club head and the golf ball.

Using this type of kinematic impact configuration and plotting the compound
pendulum positions for three different impact configurations, which are displayed in Figs.
B6, B7, and B8, it is noted that there are some differences between a real type of golf
swing and what this model is attempting to predict based on orientation of the pendulum
at impact. The first major difference, as shown in Ref. 3, is that the club head position is

ahead of the mass center of the shaft link, which indicates the limitation of a rigid body

28

model in predicting the real motion of a golf swing. Secondly, in a typical impact
configuration of a real golf swing, there is a small degree of misalignment, but in this
model, since the final orientation of the arm link is calculated and based on the
configuration of the shoulder link at impact, there is a larger degree of misalignment as
seen in the plots in Appendix B. This shows that the kinematic impact position will

require additional work in future models that are created.

2.2.4 Club Head Rotation Prescribed Motion

Similarity of the club head rotation to the shaft link motion is noted and can be
described in two phases. At the top of the backswing, the golfer has set the hands,
configured the angle of the wrist-cock and is about to apply the moments to initiate the
downswing. Given that the orientation of the club in the hands is fixed, consider the
angle of the club face relative to the ball to be at some initial angle, )3. Simply, this angle
remains a constant and is not a function time constants until the time at which the golfer
will rotate the golf club about the centerline of the shaft.

It is convenient to define another time period similar to the time period after
wrist-release, and in this case, the squaring of the club head occurs in a smaller time
period relative to the wrist-release period, 1),. It has been researched by Milne and Davis
[14] that the club head rotates approximately 900 in fifty milliseconds prior to impact
determined from high speed photography. Define the time period that the wrists start to
rotate the golf club as to = 1,, - t,,.. In the period where t < I", the angle is a constant that is

prescribed due to any orientation that the golfer holds the club:

7(t) = yo tStw (2.18)

29

Consider the time just after the wrist start to roll or (I,.+5t). Applying the same

kinematic assumption:

II t(

7(I)=W[t—tw—Lsin[£(—t:t‘—‘)H+yo t>tw (2.19)

By definition, at the time of impact, the club face will be square to the ball and the angle

equal to y:

t

(.

. ”(a -t...)
70“): y] =W tu —tw —sm —— +7” (2.20)

Rearranging and solving for the constant, W:

 

 

W = 7, — 7" (2.21)
t 71' I —I .
[ta - t“, — L.Sil1[(—‘i “ )H
7r tr
Recall that the time period 1,. = 1,, - 1,, so the sine term vanishes and the constant is:
W = [ml] (2.22)
1.

Time differentiation of Eq. 2.19 results in the angular velocity and acceleration:

i4!) = W [I — cos[flt—t_—t“—)—]] t> I“. (2.23)

t t

( (‘

33(1) = w £[sin [MN 1 >1, (2.24)

30

This concludes the derivation of the time dependency of the generalized
coordinates that describes the motion of the shoulder, arm, golf shaft and club head
rotation during the downswing. Given these kinematic assumptions, it is possible to
depict the golf downswing characteristics based on inputs such as the orientation of the
angles of each rigid link and timing attributes of the wrist-release or wrist-roll. The
benefits of having this kinematic assumption will allow the formulation of generalized
force expressions using Lagrange’s Equations. However, since the kinematics have been
arrived at, it is fitting that the next step to finding the equations—of-motion shall use the
principal of work-energy requiring the position and velocity vectors of the rigid links to
be found, which is the topic of the next section. From this, the driving forces in the

swing are ascertained.

2.3 Derivation of Position and Velocity Vectors

The following section outlines the derivation of the position and velocity vectors
of the three—link pendulum. Consider Fig. 2.2, which is a free-body diagram of the hub

and attached shoulder link used to derive the position and velocity vectors from the

A A A

inertial frame of reference, {cI , e3, e3 }. The following nomenclature for the mass

center of the position and velocity vectors will be as standard convention r and v .
respectively. The time differentiation of the position vector described with an over-bar
and dot notation designates mass center values. Recall d) and L) are the lengths to the
center-of—mass for the shoulder link and the total length, respectively, and t// is the angle-

of—rotation of the link. The position vector from the hub axis in terms of the inertial

coordinate frame is then:

31

1?: d, [cost/l e,- singl/ 63) (2.25)

 

 

(D)

Figure 2.2: Free—Body Diagram of the Shoulder Link.

Time differentiation of Eq. 2.25 results in the velocity of the mass center of the shoulder

link:

1:: I; = —I//d, (sing! e,+ cost// 82) (2.26)

Figure 2.3 shows a free-body diagram used for deriving the position and velocity

vectors of the arm link with respect to the inertial frame.

r2 = L, (cosy! e,-— sinI/I 621+ (12(cosa e,+ sin a (’2) (2.27)

32

Ezra =—t//L, (sinI/I e,+cost// e_,]+afd2 {-sina'eficosaez) (2.28)

3
(D >
to

 

 

Figure 2.3: Free-Body Diagram of the Shoulder and Arm Links.

Figure 2.4 is used to derive the position vector from the inertial reference frame to the

mass center of the shaft link.

33

1?: L, (cost/1 e,- sinw e2)+ L2 (cosa e,+ sina 62)
(2.29)

+d3 (cosfi 61+ sinfl 62]

13:6, =—tz/L, (sing/I e,+cosifle2)+aL2 (—sin0te,+cosae2)
(2.30)

+fld3 {—sinfl e,+cos,Be2]

 

Figure 2.4: Free-Body Diagram of the Shoulder, Arm and Shaft Links.

The club head is treated as a point mass attached to the end of the shaft link and

the free-body diagram is shown in Fig. 2.5. The following vector equation describes the

velocity of the club head:

34

v4 = -I//L, [sinl/I e,+cost// czj+aIL2 {—sina e,+cr).9062) (2 31)

+ ,BL, {—sin ,6 e,+ cosfl a.)

 

m4 3 .-
B

Figure 2.5: Free-Body Diagram of the Entire Three-Link Pendulum.

2.4 Sguares-of—Velocities

The derivation of the generalized forces using Lagrange’s Equations requires the
total kinetic energy of the system. With the position and velocity vectors defined for
each link, it is convenient to find the squares of each velocity equation. The
mathematical details leading to the development of the subsequent equations representing
the squares-of—velocity equations are detailed in Appendix A. Starting with the shoulder

link, squaring and expanding Eq. 2.24 and using the trigonometric identity

cos2 11/ + sin2 11/ = I , will yield the scalar equation:

35

V,2 =d22 W2 (2.32)
Squaring Eq. 2.26 and using the trigonometric addition formula:

cos I]! + a = cos 1;! cos a — sin a sin a

will yield the scalar equation:

v22 = L,2 t/I"+ (122 az_ 2d3L, I/lacos(tfl + or) (2.33)

Squaring Eq. 2.21 and using the three trigonometric identities:
cos 1;! + or = cos t// cos a — sin I// sin 0
cos I// + ,6 = cos 11/ cos )6 — sin I/l sin ,8
cos a — ,5 = cos a cos ,6 + sin a sin ,3

will yield the scalar equation:

v," = L,‘2 1/12+ L22 a2+d32 ,6"—2L,L, wacos(y/+a)

. . . . (2.34)
—2d_.L, wflwdwflhzflz “(WSW—r3)

Similarly, squaring Eq. 2.31 and using the trigonometric additional formulas will result in
the last velocity equation required to formulate the kinetic energy for the three-link

pendulum:

v42 = L,2 1/12+L22 012+ L32 flz—ZL,L, wacos(w+a)

. . . . (2.35)
-21,le y/flcos(l//+,B)+2L2L3 aflcos(a—,B)

36

2.5 Kinetic Energy for the Shoulder, Arm, and Shaft Links

The total kinetic energy is required for finding the Lagrangian. Define the
individual kinetic energy of the shoulder, arm, and golf shaft as T1, T2, T3, respectively.
The sum of the individual kinetic energies is the total kinetic energy (excluding the club
head):

3
Tina)! = 27; :77 +72 +7;

i=1

From elementary dynamics, for any rigid body motion, the kinetic energy of a pendulum

rotating about a fixed point is:

T=lmv2 +—]—I(02
2 2

where v and 07 represent the linear and angular velocity of the mass center. Utilizing
this general form of the kinetic energy for the rotating pendulum and substituting the
squares of the velocity found from Eqs. 2.32, 2.33, and 2.34 into this general form, the

kinetic energy for the three-link pendulum is defined in terms of system parameters:

_1 .2 2 I .2'
T: E [fl/(I'l- +§Vll:]
+ 5%[1112 W2+d22 az—ZdzL, élo'cos(t//+a)]+éazfz] (2.36)

h

+ 1 Lfif”;a2+d,2,62-21,1,1i/£zcos(y/+a) +1 -,
—’1l3 . O C C —
[2 —2d_,L, t/Iflcos(y/+fl) +2d3L2 aficos(a—,B) 2

i)

 

37

Each bracketed term represents the kinetic energy equations for the shoulder, arm and

shaft links respectively where T, ,1—2 and I; represent the moment-of—inertia about the

A

mass center of each link about the e3 axis, which is perpendicular to the page. The

kinetic energy of the club head due to the wrist rotation about the centerline of the shaft is
not included in Eq. 2.36 because derivation of the rotation vector of the club head is

required.

2.5.1 Club Head Angular Velocity Vector

In the initial part of the thesis, the golf club head was modeled as a point mass
attached to the end of the rotating shaft link. Recall the rotation of the golf club head is
dictated by the generalized coordinate, Kt). Assume that the shaft link bisects the center-
of—mass of the club head, as shown in Fig. 2.6 showing the orientation of the club head

angular velocity vector. Assigning two rotating unit vectors is required to relate the

A

vector mathematically in terms of the inertial frame of reference. Let e5 point in the

A

direction of ,6 and 6, point in the longitudinal direction of the shaft link.

Describing the angular velocity vector of the club head in terms of the rotating
unit vectors that are attached to the point mass is accomplished by taking the time
derivative of the position vector relating the rotating coordinates to the inertial frame of

reference fixed at central hub axis:

a): -).Ie,-,d’ej = —y[cosfle,)—y[sinflezj—fiej (2.37)

Define the scalar components of this velocity vector as:

38

 

2182

 

 

Figure 2.6: The Orientation of the Club Head Angular
Velocity Vector.

Greenwood [15] has shown that the three-dimensional rotational portion of the

kinetic energy for a rigid body rotation in tensor notation can be represented as:

r... = giwr -lIllwl

With the following notation of DR describing the rotational kinetic energy of the club

head, the equation is written in terms of the velocity scalar components, namely:

T4” = (0,2 In+ (of I22 + (1)32T33— 204a), I12 — 2w,ai,Ii,1— 2022(1), I23 (2.38)

39

This general expression for the rotational portion of the kinetic energy of the club head is
written in terms of the club head’s inertial components, [I] and the numbered indices
denote the particular component of the inertia tensor. Keeping the same nomenclature,

let the kinetic energy of the club head be defined as T, which is expressed as:

.—

T _ 1m L,2w2+L22a2+Lf,63-2L,L21//acos(w+a)
4" _ 4

[2 -2L,L, tibcosw + ,6) + 2111., 5.13.0..(02- ,6)

(2.39)
}’2 COS2 fliu+ 72 sz fligz-t-flz T33

:2)” cosflsinfliiz-2,.lecosflii.e—2,.Bj’sirzfliz.e

 

Eq. 2.39 results from substituting the translational velocity of the club head, 1!,2 from Eq.

2.35 and the vector components from the club head angular velocity vector into the
general form of the rotational kinetic energy shown in Eq. 2.38. Since the last required
kinetic energy equation has been derived, it is possible to generate the generalized force

expressions that drive the three-link pendulum using Lagrange’s Equation

2.6 Derivation of Lagrange’s Eguation for the Kinematic Swing

Early in the development of this model, the approach to finding the equations-of-
motion through Newton’s second law was attempted; however the equations became
complex in regards to algebraic coupling. Therefore, the decision to solve for the
equations-of-motion through work-energy methods was explored. The Lagrangian1

approach reduces equation complexity by removing constraints and provides a systematic

 

' As explained by Thomson (1993) in [17], Joseph LaGrange (1736-1813), was a French
mathematician that lived during the French Revolution. He is famous for formulating a treatment for
complex mechanical systems from the scalar quantities of kinetic energy. potential energy and virtual work.

40

approach to formulating the equations-of—motion through partial differentiation of
potential and kinetic energy equations with respect to generalized coordinates.

In general, Lagrange’s Equations may be written as:

(it aé dq, Eq.

1

Q,-

For simplicity, in the treatment of the golf downswing, air resistance and gravity are
neglected. Jorgenson [8] has shown that the elimination of gravity results in a 6%
increase in the applied torques, which is considered negligible. Pickering [20] has
performed a computational study by formulating expressions for the applied torques in a
two-link model using Lagrange’s Equations and has determined that omitting the
gravitational force is satisfactory for analysis. Therefore, the expression for the

generalized force can be written as:

 

d 8T 8T
:1“ - 7:9
I aql q,

Here, q, are the generalized coordinates, T is kinetic energy, and Q) are the generalized
forces. For the three-link pendulum, neglecting potential energy, the Lagrangian, L = T -

V, is the sum of the kinetic energies of the shoulder, arm and shaft as shown in Eq. 2.36.

Let Q, , Qa ,Qfl , and Q7 represent the forces in the shoulder, arm, wrist-release, and the

wrist-roll respectively.

Following the systematic process of differentiation of the kinetic energy equation
shown in Eq. 2.36 with respect to shoulder, arm and the shaft link will result in the

equation-of—motion for each individual linkage of the pendulum. Details of the partial

41

differentiation of a: 1 8T , and E)_T_ are provided in Appendix A. Following the

aqi d! aq' do,

 

systematic process of partial differentiation of the total kinetic energy, the following

equations of motion were then formulated for the three-link pendulum:

Qv :J/[mldlz +mzl.,2 +In,L,2 +mJIf +I/:l-Zt/cos(t/I+Ct)[m_,d_,l1 +m,l,L, +mleL_,]
_Ecm(V/+IB)[m3d,L, +171411L3]+o(i;/+ot)sin(l//+a)[mgd_,l., +m,L,L2 +mJL,l?]

(2.40)
+' ix+b}in(1u+fl)lm.d.l. +m.I.L.l _i&..~t,.(.,,+a)[m,d,z, was +m.1.1.l

-iubsin(w+ Alma. +m.I.L.l

Q0 =—;/cos(tll+a)[m2d2L, +1713ng +1714L,l?]+gr[n12d22 +mjl,_,2 +m4172 +T2]
+,.Qcos(a—fl)[m_,d_,L2 +mJIQL,]+t/./[t/./+ casing/1+ a)[m?d_,L, +m‘,L,L, +m4L,L_,]

(2.41)
56h." PJS’"(a—fl)lm_.d.l. ”2.1.1.1-iw3rsin(w+a)lwizh WM? +444]

+£I,.Bsin(a-,B)[rn,d_,l,z +m4L2L3]

Qfl =—l/”/cos(i,1/+,B)[mjdjl7 +m4L,L_,]+thcos(a—,B)[m3d3Lz +m4L2L3]

+,.Q[rn,d32 +m4L32 + T3 )fli/[tifl b)sin(yl+fl)[m3djlfl +m4L,L,]
—c.r£c.r—b)sin(a—,6)[m,djlfi +m4LZL_, ] —l/./,.Bsin(y/+,B)[In,d,l., +m4L,L_,] (2.42)

—c.r,.3sin(a-fl)[m3d3l? +m4L2L3 ] +2 2'2 (cosflsinfl) Til—2 5/2 (cosflsinfl) T22+2,.QT13

.2 __ 00 _ 00 _
+2y (c052 fl—sinz ,6)112—27cosfl113—27sinfllze

42

Q7 —_- 2(;cosz ,B- 2bj/cosflsinflJT/w 2(;sinz ,6 + 2,3 j/cosflsin fljizz
—4(.};cosflsinfl+ b j/cosz fl+lbysinz fljil2_ 2(2cosfl— ,82 sinfljin (2-43)

—2(:6.sinfl+fl.2 cosfljlzs

This concludes the derivation of the generalized forces in the three-link pendulum
model. To compare the validity of the expressions for the equations-of—motion, special

cases of the compound pendulum were considered as a check.

2.7 Special Cases of Simple Pendulum Motion

The special cases of simple pendulum motion were devised to verify the
generalized force equations that were presented in the previous section. By locking all
but one degree-of-freedom, the generalized force equations reduce to simple pendulum

motion, defined by the Newtonian mechanics. One must simply calculate the moment-

of-inertia for the link that is rotating and compare 1 Q to the linear, generalized force
equations previously described with “locked” kinematic variables prescribed (zero
velocity and acceleration). In Case 1, the pendulum consists of the shaft link swinging at
the wrist joint. In Case 11 and Case 111, the pendulums are composite, meaning that they

are combined at the wrist and shoulder joints, respectively.

2.7.1 Case I: MUEattEO, fin) Prescribed
This pendulum case is constrained such that the shoulder and arm are fixed and
the golf shaft is swinging as a single pendulum with a point mass equivalent to m4

attached at the end of the shaft link. The angles t//(t) and an) are set to be identically

43

zero, implying that first and second time derivatives are zero. For Case 1, since the wrist-

release is not considered, the motion that is prescribed to the shaft assumes the form of

 

 

 

/ Shaft Link. mass = m3
d 3

 

/ Club llead. mass = m4

1
_,_.

Figure 2.7: Free-Body Diagram for Case I.

 

Eq. 2.1 with the generalized coordinate, flfl) and the appropriate constants, ,6" and ,Bf
substituted. Consider the free-body diagram of the shaft link and the point mass

orientated in Fig. 2.7. To find the equation—of—motion of the single pendulum with mass

m3, and moment-of-inertia about the center—of-mass T3, which rotates about the fixed

pivot point with motion ,B(t), Newton’s second law states:

EM =IendI6

Using the parallel axis theorem to find the moment-of—inertia of a bar rotating about its

end:
1““, = [émflf + m,d_,2 + m, L_,2:| = [TH m_,d,2 + m, L32]

By summing moments about the fixed pivot point and substituting for the moment-of-

inertia lend:

M, 2|:713m3df + m,d_,2 + m, L,2 ]fl 2 [AW m,d_,2 + m, L,2 ],B (2.44)

Consider the expression for Q,, in Eq. 2.42 and apply the conditions of Case I:

11/5 02500, which implies all first and second time derivatives will equate to zero in Eq.

2.42. Therefore, the modified equation-of-motion reduces to:

Q, = [T3+ m,d_,2 + m,L_,2 ],B (2.45)
Equation 2.45 is denoted by a primed superscript and indicates the conditions of Case I
are implemented resulting in equivalence between the closed form solution and the

specially derived equation-of—motion.

2.7.2 Case 11: MNEO, 040350) Prescribed

In this special case, the shoulder link is fixed and the combined pendulum consists
of the arm, golf shaft and point mass rotating about the shoulder joint. The pendulum
swings with the kinematic motion of oft) described in Eq. 2.4. The free-body diagram of
the combined arm and shaft pendulum is pinned at the shoulder joint and fixed at the

wrist joint and is shown in Fig. 2.8. The arm has length L2, distance to mass center, d2,

45

mass m2, and the moment-of—inertia about the center of the link I2. Using the parallel

axis theorem, the moment-of—inertia about the pivot point is:

 

 

 

9/

Ml

/Arm Link. mass = m2

Wrist Joint. Fixed

2"“:

/ Shaft Link. mass = m 3

Club Ilead Mass = m

 

/ 4

Figure 2.8: Free-Body Diagram for Case II.

In“, =[I2+m,d,2 +ém,L,2 +m, (L2 +d_,)2 +m4L,2]

Using Newton’s second law to find the equation-of-motion:

ZM =I,m,a

M1+M2 :[i2+m2d22 “Li-”"13“? +d~*2)+m“(L2 +113]

a (2.46)

46

The conditions of Case 11 require a E ,6 because the rotational degree-of—freedom

in the wrist joint is constrained. After manipulating Eqs. 2.42 and 2.43 by performing
algebra and factoring common terms that are multiplied by the mass elements, the

following equivalent equation-of-motion is found:

Q0, +Q, {12mg 2.2.4.1., (L, +d,2)+m, (L, + 2.2 )]a (2.47)
The primed generalized forces indicate that the special conditions of Case U have been
applied to Eqs. 2.41 and 2.42, and since Eqs. 2.46 and 247 are equivalent, it is thus
proven that the combined pendulum of the arm and shaft link can be reduced down to a
simple pendulum motion swinging at the shoulder joint and when the correct conditions

of constraint are applied

2.7.3 Case 111: MUEanEflt) Prescribed

The combined pendulum is composed of the entire three-link pendulum: the
shoulder, arm, golf shaft, and point mass rotating about the central hub axis. The motion
of the pendulum is governed by l/Ifl), and is equivalent to Eq. 2.1. The free-body-
diagram (Fig. 2.9) of the combined pendulum is pinned at the hub and the rotational
degrees-of—freedom in the shoulder joint and wrist joint are constrained.
Using the parallel axis theorem, the moment-of—inertia about the fixed pivot point for the
composite link is found:

_ Ti-lvm,al,‘2 +T2+m,(L, +d22)+I3

end

+m_,(L, +1, +d,2)+m,(L, +1, +L_,2)

47

Using Newton’s second law to find the equation-of-motion:

 

  
  

 

Shoulder Link. mass = m 2

Shoulder Joint. Fixed

 

 

Wrist Joint. Fixed
I J/
—i j M |

/ Shaft Link. mass = m3

 

 

/(‘|ub Head Mass = m4

Figure 2.9: Free-Body Diagram for Case 111.

2M = lend W
M,+M,+M,

_ i,+m,al,2 +iz+m, (L, +d,")+13 (2.48)

+m_,(L, +1., +d,2)+m, (L, +1., +1.3)

Due to the presence of t/Iand the derivatives in the equations of motion, there are

nonlinear acceleration terms still present. Therefore the model was checked by using the

48

prescribed motion and locking all degrees of freedom. The maximum moments were
evaluated and the sum was compared to the Eq. 2.48. In this special case, trigonometric
substitutions are required to be made with the full form of the equation of motion in order
to have equivalence with the special closed form case.

In conclusion, the three special case studies represent modifications to the three-
link pendulum model derived. It was proven that using an elementary approach, the
closed form solutions are reduced forms of Eqs. 2.41, 2.42 and 2.43. The closed form
case equations represent modified forms of the complete equations-of-motions that

govern the driving forces derived using Lagrange’s Equations.

2.8 Comparing the Kinematically Driven Downswing Model

The assumed motion for the motion of this model was plotted and the results and

are displayed in Appendix B. This type of motion was unique in that it refinement of the

 

6
g 5
4 —
2 alt)
QA 3
.Z’. '3
E3 3: 2
2 I _
E“ [3(t)
< O

-1

-2

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

time (s)

Figure 2.10: Assumed Motion For Ref. 3.

49

kinematic shaft-release assumption by considering the continuity relationships of the arm
link. In comparison to the assumed motion for the arm link in Ref. 3, which is plotted in
Fig. 2.10, overall, the motions of the arm link look the same by observing the shape of
the curves. However, there is some difference noted in the prescribed motion of the shaft
link as the curve spikes in Fig 2.10 after the time for wrist-release. It is noted that this
motion was measured experimentally and the function in Eq. 1.2 was estimated from a
real golf swing. In conclusion, more work is required to further access whether a smooth
function is desired in kinematic golf downswing modeling as opposed to what is actually
measured from the golfer’s swing.

By calculating the required, applied moments that are responsible for the
kinematic motion, an additional comparison can be made. In this model, the applied
moments were functions of the system parameters and the time dependent generalized
coordinates. The maximum applied moments were compared for a golf swing typical of
an average type of golfer. The results were then compared to previous three-link models
found from the literature search. Table 2.1 shows that the calculated applied moments in
this model are relatively in the same neighborhood as the results found in Refs. 6 and 7,
which were specified torque values that were required to satisfy the equations-of—motion
in these models. Some discrepancies may exist due to selection of the three-link system

parameters used in this model and due to the added mass from the club head entity.

50

Chapter 3 starts with a discussion of the model parameters that were gathered

from the literature survey and then discusses the modeling techniques to help ascertain

Table 2.1: Comparison of Applied Moments with
Other Model Types.

 

Author: A.B. Turner [6]
Modclflpe: Three Link Pendulum
Hub Torque: 84 N-m
Arm Torque: 73 N—m
Wrist Torque; 15.0+14.7303-5.8602+0.276293
Author (s): SprinLirlgs and Mackienzie [7]
Model Type: Three Link Pendulum
Hub Torque: l 10 N-m
Arm Torque: 90 N-m
Wrist Torque: 30 N—m
Calculated Moments
Hub Torque:l90.5() N-m
Arm Torqueml 1.21 N-m
Wrist Torquezl52.53 N-m

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

meaningful information between the generalized forces required providing for the
kinematic motion and the components of the club head inertia tensor. Furthermore, the
results of the study devised to find the effect of the range-of-motion of the shoulder link

are then presented.

51

CHAPTER 3

Implementation of Model and Presentation

of the Golf Downswing Modeling Results

This chapter will describe the simulation of the swing using the three-link
pendulum model. Microsoft Excel (Microsoft Corporation, Redmond, WA) was chosen
for the ease of the treatment of modeling parameters, coding of the kinematic swing

functions, and the expressions for the generalized forces.

3.1 Model Parameters

The characteristics of the modeled human body and were obtained from the
literature search and Table 3.1 details the non—kinematic, three-link parameters. The
properties of the human shoulder and arm were found from an electronic resource.
Madsen [18] is a Professor of Biomechanics at the University of Auburn (Auburn, AL)
and designed a special exercise for his undergraduate students to calculate the mechanical
properties of their bodies based on their height, weight, and the geometrical
characteristics of their appendages. The data presented on the website was for a 180-
pound male who stands six feet tall and showed the length, mass, and moment-of-inertia
of the upper arm and the fore-arm. The values for the club head inertia elements were

taken for a simulated 350 cubic centimeter driver head from Mase [22].

52

Table 3.1: Three-Link Parameters in SI units.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Description Parameter Value Units Source

Distance to mass center of shoulder link d , 0.09525 m Calculated
Length of shoulder link L, 0.1905 m Madsen [18]
Mass of shoulder link m, 2 kg Madsen [18]
Moment-of-inertia of shoulder link II 0.3 kg-m2 Turner [3]
Distance to mass center of arm link (i; 0.150 m Madsen [18]
Length of arm link L3 0.560 m Madsen [18]
Mass of arm link mg 3.6 kg Madsen [18]
Moment-of-inertia of arm link T2 0.0858 kg~m Madsen [18]
Length to mass center of shaft link d3 0.5715 111 Measured 9-17-03
Length of shaft link L3 1.143 m Measured 9-17-03
Mass of shaft link m3 0.1 kg Measured 9-17-03
Mass moment-of-inertia of shaft link T3 0.01 l kg-m Calculated 9-17-03

Club Head Properties - - - -
Club head mass m, 0.200 kg Mase [22]
Moment-of—inertia (l,l component) T), 2.4845E-4 kg-m2 Mase [22]
Moment-of—inertia (2,2 component) 132 2.9408E-4 kg-m2 Mase [22]
Moment-of-inertia (3.3 component) T33 4.7906E-4 kg-m2 Mase [22]
Product-of-inertia (1.2 component) T12 2.66E-5 kg-m2 Mase [22]
Product-of-inertia (1.3 component) T13 1.596E-5 fimz Mase [22]
Product-of-inertia (2,3 component) i 2.. -2.802E-5 kg-m2 Mase [22]

 

 

 

 

 

 

To show the orientations of the directions of the club head inertia components, a
Wireframe sketch is provided in Fig. 3.1. Some assumptions that were made include the
following. The distance to the mass center for the shoulder link is taken to as one-half
the distance of the total length of the link. It is noted that a golf shaft is not a prismatic
cylinder with a constant cross-section; however for simplicity it is assumed that the
equation for the moment-of—inertia for a cylinder is a good approximation. Therefore, the

moment-of—inertia for a standard-length golf shaft measuring 1.143 meters was used. The

53

distance to the mass center was measured by a counterweight and measuring the balance

point with a ruler.

 

 

Figure 3. l: Wireframe of a GolfClub Head.

3.2 Com uter Codin of the Kinematic Downswin and the Generalized Forces

 

The kinematic downswing formulas governing the motion .of the three-lever
pendulum were coded into a file in Microsoft Excel. Since the generalized coordinates
are a function of time, this allowed straightforward input of the kinematic equations for
each generalized coordinate.

Recall the kinematic assumptions for the golf shaft link and the wrist rotation
occur at two time steps, I, and t,,., respectively. Built-in, user-manipulated algorithms
were utilized in Microsoft Excel. The “’IF’ function compares two numbers based on
equality or inequality and this is called the truth test. Depending on the results, the
function will execute one of two calculations: if the initial test is true, the first action is

performed, if the first test is false, the second action is performed. The calculation of ,6(t)

54

and fit) was performed in the following manner where commas show the three arguments
for the IF statement:

0 ,6: IF I < t, , execute Eq. 2.9 , execute Eq. 2.14

0 y: IF t < t“. , execute Eq. 2.18 , execute Eq. 2.19

Therefore, the appropriate equations for fin) and fit) were typed as arguments of the IF
formula and Microsoft Excel performed the computation of the algorithm based on the
value of time, I. Once the formulas for the kinematic swing were encoded properly, the
expressions for the generalized forces were then typed into the spreadsheet. The initial
stage of compound pendulum modeling was focused on finding the position of linkages

in time to produce a textbook golf swing at the address and impact states.

3.3 Club Head Inertia Components and the Effect on the Applied Moments

Recall, the wrist-release and wrist-roll moments are functions of the club head
inertia elements as shown in Eqs. 2.2 and 2.43. The goal of building this kinematic
model was to see how the inertia components influence the applied moments and this is
performed by computing a sensitivity measurement. The sensitivity measurements entail
treating a club head inertia term as a variable and taking the partial derivative of the
expressions with respect to the inertia component in question. The results are functions
dependent on the generalized coordinates and time. Furthermore, the sensitivity
functions were plotted against the total time for the downswing. Table 3.2 outlines the

kinematic input parameters used in this study.

55

Table 3.2: Kinematic Parameters Used for the Inertia Component Study.

 

 

 

 

 

 

 

 

 

 

 

 

 

Downswing Time (I a) 0.300 s
Wrist-Release Time (I ,) 0.100 s
Wrist-Roll Time (I w ) 0.180 s
Shoulder Finish Angle (y 0) 900
Initial Arm Angle (a o ) - 1 35°
Shoulder Link Kinematic Constant (P) 5,6] (rad/s)
Arm Link Kinematic Constant (R) 5.74 (rad/s)
Shaft Link Kinematic Constant (S ) 12,72 (rad/s)
Wrist-Roll Kinematic Constant (W) 15.71 (rad/s)
Club Head Impact Velocity (v 4 ) 47.07 (m/s)

 

3.3.1 The Applied Wrist—Roll Moment as a Function of Club Head Inertia

In this model, the shaft link bisects the center of the club head and internal club

A

head forces along the e} direction are zero. Therefore the applied wrist-roll moment is

not a function of the 733 inertia component. Treating the club head’s moment-of—inertia
terms as variables, Eqs. 3.1 and 3.2 are the sensitivity functions that were obtained by

taking partial derivatives of Eq. 2.43.

 

 

a .. . .

?7 =2[7coszfl—2flycosflsinfl:' (3.1)
all!

a .. . .

Q7 =2[7sin2fl+2flycosflsinfl] (3.2)
8722

The sensitivity functions were analyzed in terms of the global and local extrema
to draw comparisons throughout the analysis and these points are labeled on the

following figures presented throughout. Equation 3.1 was plotted against time showing

56

the sensitivity of the wrist-roll moment in terms of the I 11 inertia component. For

comparison with other sensitivity functions subsequently presented, extrema were labeled

 

 

1500 \

I787
\
1000

\
/"‘

0 ‘ _-362 J
-5(X)

0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time (seconds)

Figure 3.2: Wrist-Roll Moment Sensitivity as a Function of the I n Inertia Component.

on the graphs and for this basic study, either the maximum or minimum gradient were
used to analyze the influence of the inertia terms.

Next, Eq. 3.2 was plotted in Fig. 3.3 which shows the sensitivity of the applied

wrist-roll moment as a function of the 722 inertia component. The shape of this
sensitivity function is noticeably different than the function plotted in Fig. 3.2; however
the graph shows that the function still oscillates. Both of the functions in Figs. 3.2 and
3.3 are sometimes positive or negative and this is based on the kinematic assumption. By

comparing the extrema of the sensitivity functions in these two graphs, this analysis

indicates the I 1, inertia component may possibly have more influence on the wrist-roll
torque. However, in light of the oscillatory nature, more work is required to investigate

how the sensitivities are affected when the kinematic assumption is changed.

57

 

1500

1000
500 1044 \

0 4* \

22 -500 I
-1000 -l448
-1500

-2000 .
0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

 

 

 

 

e

 

Q)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time (seconds)

Figure 3.3: Wrist-Roll Moment Sensitivity as a Function of the 122 Inertia Component.

The applied wrist-roll sensitivity functions for the products-of—inertia were then
considered. Eqs. 3.3, 3.4, and 3.5 show the partial differentiation of Eq. 2.43 with respect
to each product-of—inenia term. To ensure completeness in the analysis, the time interval

prior to wrist-release was considered. Notice, Eq. 3.3 is the only function dependent on

both the wrist-release and wrist-roll generalized coordinates, and since y(t) was defined

as a constant in the time interval 0 < t < 0.180 seconds, this implies that y E y 20.

Therefore, Eq. 3.3 equates to zero in this time interval. Since the applied wrist—roll
moment is a function of the prescribed motion for the shaft link and the wrist-roll motion,

the time period prior to wrist-release was considered when plotting Eqs. 3.4 and 3.5.

 

 

d—Qy = —4 [;cosflsin fl+ ,5 ;’C05‘2 ,8— :8 g’sin2 ,6] (3.3)
3112

.. . 2
3?)! = —2 [flcosfl— ,3 Sinfl] (3.4)
8113

58

3Q,
a I 23

 

=—2[,.dsinfl+,b cosfl] (3-5)

Figure 3.4 shows the plot of the sensitivity of the wrist-roll moment as a function

of the 712 inertia component. The sensitivity is negative and decreasing leading up to the
point of impact. The maximum positive value of this sensitivity function was calculated

to be 2,359 and the value at the time of impact was -3,640. Compared to sensitivity

-3640

 

0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

time (seconds)

Figure 3.4: Wrist-Roll Moment Sensitivity as a Function of the 112 Inertia Component.

functions previously shown, this sensitivity function resulted in the calculation of the

largest extrema values. Also, near the time of impact, the gradient changes very rapidly.

These two facts may indicate that the 712 inertia term is a critical component in terms of
the wrist-roll moment, but different kinematic assumptions should be tested to see if this

inertia term is the most critical.

Considering the effect of I 13 inertia component on the wrist-roll torque, Eq. 3.4

was plotted against time and is presented in Fig. 3.5. The global maximum and

59

minimums were calculated to be 438 and -1,897 respectively. As opposed to the

sensitivity function plotted in Fig. 3.4, this sensitivity curve is more dispersed close to the

1000
500

o

9.9; -500
0113 -1000
-1500
-2000

-2500
0.00 0.03 0.05 0.08 0.10 0.13 0.15 0.18 0.20 0.23 0.25 0.28 0.30
time (seconds)

 

Figure 3.5: Wrist-Roll Moment Sensitivity as a Function of the I13 Inertia Component.

time of impact as opposed to the function plotted in Fig. 3.4. The global minimum is less

and the function approaches the neutral state. This shows that the oscillatory nature of

the sensitivity functions are not the same regarding the 712 and the I :3 inertia terms near

impact.

Plotting Eq. 3.5, the sensitivity of the wrist-roll in terms of the 723 component is
shown in Fig. 3.6. The maximum gradient was observed at impact and was calculated to
be -2,416 and the global maximum was calculated to be the 1,077. Before the un-cocking
of the wrists occur, the function hardly changes and for the most part, the sensitivity does

not actively change until approximately 0.140 seconds.

Looking at Figs. 3.4 and 3.6, which analyze the I 12 and the I2: inertia terms

respectively, there is some resemblance of the sensitivity functions near the time of

60

impact. By comparing the value of the function at the final state, it seems the 712 inertia

term may have more influence on the wrist-roll moment. However, based on the sharply

1500

6Q
4 _500

5’23 -1000
-1500
-2000
-2500
-3000

0.00 0.03 0.05 0.08 0.10 0.13 0.15 0.18 0.20 0.23 0.25 0.28 0.30

 

-24l6

time (seconds)

Figure 3.6: Wrist-Roll Moment Sensitivity as a Function of the [23 Inertia Component.

changing gradients and the oscillatory nature of these curves, a future, more in-depth

study is required to look at how these functions are affected based on a different assumed

motion for the wrist-roll.

3.3.2 The Applied Wrist-Release Moment as a Function of Club Head Inertia
The effect of the club head inertia tensor on the applied shaft-release moment was
studied. By partially differentiating Eq. 2.42 with respect to each moment-of—inertia term

of the club head inertia tensor, the following sensitivity functions were found.

a '2

git—Zr (cosflsinfl) (3.6)
all]

a '2

—_Qi=—27 (cosflsinfl) (3.7)
3122

61

a ..
fl = 2 ,6 (3.8)

a I 23

Notice that Eqs. 3.6 and 3.7 are equal in form, but opposite in sign, therefore there
is definite symmetry among these two equations. Equation 3.6 was plotted against time

in Fig. 3.7, and it is worthwhile to compare the wrist-release sensitivity function with the

 

0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

time (seconds)

Figure 3.7: Wrist-Release Moment Sensitivity as a Function of the In Inertia Component.

wrist-roll sensitivity function because both are implicitly related to the I n inertia term.

3
Analyzing Fig. 3.2, the function, Q’ is positive whereas the function —Qf— is negative

din Bin

 

in Fig. 3.7 in the same respective time intervals where the maximum gradients are

observed. This means that the In inertia component has opposite effects on the wrist-

release and the wrist-roll moments in this particular operating range for the assumed

motions.

62

By plotting Eq. 3.7, the symmetry is definitely apparent between Figs. 3.7 and 3.8

and hence the minimum gradient in Fig 3.7 is equal and opposite when compared to the

700
600
500
400
39} 300

8122 200
100

0

-100

-200

0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

 

time (seconds)

Figure 3.8: Wrist-Release Moment Sensitivity as a Function of the I 22 Inertia Component.

maximum gradient in Fig. 3.8. When comparing Figs. 3.3 and 3.8, interesting behavior is

7

 

noted. When the function

 

. . . . 3Q”
_ in Fig. 3.3 becomes negative, the function _
a I 22 a I 22

in Fig.

3.8 starts to become positive. These figures are relatively comparable in this respective
time range and it was observed from the data in the spreadsheet that there was a time

difference of eight milliseconds. This analysis may show that there is another tradeoff

between the wrist-release and the wrist-roll sensitivity functions in terms of the I22
inertia component.

As opposed to the wrist-roll moment, the wrist-release moment is a function of

the In inertia term and Fig. 3.9 shows the sensitivity plot. Notice, this sensitivity

measurement is a function of the angular acceleration of the shaft link only. Therefore

63

the function changes abruptly at the time of wrist-release. The sensitivity function is

always positive and the shape of the curve appears more gradual over the entire

450

400

350

300

£191 250
(3" I 3 3 200
150

100

50

0

0.00 0.03 0.05 0.08 0.10 0.13 0.15 0.18 0.20 0.23 0.25 0.28 0.30 0.33

 

time (seconds)

Figure 3.9: Wrist-Release Moment Sensitivity as a Function of the I33 Inertia Component.

downswing as opposed to the previous sensitivity functions thus far presented.
The last sensitivity study analyzed the effect of the product-of—inertia terms on the
wrist-release moment. Equations 3.9, 3.10, and 3.11 represent the sensitivity functions

for this last group of measurements.

a&

_ = 2y (ms2 ,B—sin2 ,8) (3.9)

3112

a ..

fl = —2 ycosfl (3.10)
8113

a ..

gz—Zysinfl (3.11)
8123

64

3Q”
8I12

versus time. The first

 

Figure 3.10 shows the wrist-release sensitivity plot

major observation can be related to the wrist-roll sensitivity function plotted in Fig. 3.4.

 

1500

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1370'
1000
@9500
6712 o \ II
-500 \A!
-1000

 

0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30
time (seconds)

Figure 3.10: Wrist-Release Moment Sensitivity as a Function of the I 12 Inertia Component.

7

aim

 

After analyzing the data, the after t=0.278 seconds, the sensitivity function becomes

3Q,
3 I12

 

negative whereas the sensitivity function becomes positive after #0280 seconds.

This difference in time of two milliseconds is considered negligible. Also, both functions
start to change rapidly leading up to the point of impact. Also, by comparing the
gradients at the time of impact, there seems to be more influence on the wrist-roll

sensitivity versus the wrist-release sensitivity.

Figure 3.1] shows the sensitivity plot of the wrist-release in terms of the In

inertia term. Unlike before, in regards to the wrist-roll sensitivity function, the wrist-

65

release sensitivity function does not follow the same general pattern as the function
plotted in Fig. 3.5. This function appears more periodic in the time period that was

plotted, but the maximum gradient is much larger than the minimum gradient.

500

400

300

200
53—99 100
a I 13 0
400
-200
-300
-400
0.l8 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

time (seconds)

 

Figure 3.1 l: Wrist-Release Moment Sensitivity as a Function of the I 13 Inertia Component.

Figure 3.12 shows the plot of the wrist-release sensitivity function in terms of the

I23 inertia component. This sensitivity curve is very gradual compared to the previous
function shown in Fig 3.6 and the major difference exists with the fact that the function in
Fig. 3.12 is zero at the time of impact whereas a negative gradient was calculated at
impact for the curve in Fig. 3.6. Also, the global maximum is much less. However, both
sensitivity functions start to increase initially, reach a maximum, then decrease.

When comparing the sensitivity curves, it was shown in this study that the global
extrema for the wrist-roll sensitivity functions were higher in magnitude versus the global

extrema for the sensitivity of the wrist-release. Also, the analysis showed that there was

interesting phenomena regarding the sensitivity of the wrist-roll in terms of the I 22 and

66

the I23 inertia components because the gradients were negative and non—zero at the time

of impact. In this study, based on the minimum value of these sensitivity functions, the

 

-100
0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30

time (seconds)

Figure 3.12: Wrist-Release Moment Sensitivity as a Function of the I 23 Inertia Component.

I12 and the I23 inertia components appear to be the most influential terms for the wrist-

roll moment. In terms of the wrist-release moment, the I12 inertia term has more of an
affect than the other inertia terms when comparing the global extrema of the sensitivity
functions.

However, because of the behavior of all the sensitivity measurements and the
basic, limited study that was presented, the work represents only a small portion of the
insight on how the moments in the wrist are affected by the club head inertia. This is
because the analysis was unable to explain why the functions are positive or negative at
the final time step. This study was limited because only one assumed motion was
considered. More importantly, this type of analysis is directly related to the kinematics

governing a simplified wrist motion in the downswing; therefore, there are many

67

parameters that can be varied that affect the sensitivity functions. Additionally, the
highly nonlinear behavior of the sensitivity functions suggests that more in-depth studies
which are well devised must be performed in order to bridge the desired relationships
between the inertia of the club head and the moments in the wrist. However, there was
progress made in this topic of study, and there is room for improvement.

The relationship between the sensitivities and the value of the inertia term is could
be accessed because it is possible for a club head mass distribution to result in negative
production-of-inertia values. To consider the effect of the value and magnitude of the
inertia component, the sensitivity equations could be normalized by the numerical value
of the inertia component. Lastly, to analyze how the sensitivities change in terms of the
assumed motion, the kinematic parameters could be varied. A good, future test would be
look at the total time of the downswing and the wrist-release and wrist-roll times as
variables of the total downswing. For example, an argument could be made that if the
downswing time is longer for a particular golfer’s downswing motion, then the overall
wrist-release period and wrist-roll period would be extended and therefore the motion
would be more gradual. This may result in smoother sensitivity functions as opposed to
the sharply changing gradients seen in some of the curves presented. Further refinement
of the kinematic model, along with creatively devised parameter studies are required to

analyze the oscillatory nature of the sensitivity curves.

3.3.3 Additional Inertia Component Studies
This section will present the results of the inertia component studies in which the
inertia parameters were changed directly. The mass and the inertia data for a 350 cubic

centimeter driver was provided and shown in Table 3.1. Currently, at this juncture, golf

68

club driver heads have reached the size of 470 cubic centimeters and are approximately

thirty-four percent larger than the size that was provided in this work. Based on this

Table 3.3: Inertia Data for the 470 cc Driver.

 

In 3.33E-04 kg-m2

 

112 3575-05 kg-mz

 

I 1.1 2145—05 kg-m2

 

} 32 3.941304 kg-m2

 

j, , 45513-05 kg-m2

 

 

 

 

j 11 5095-04 kg-mz

 

scaling factor, simple tests with the kinematic model were devised in which each inertia
parameters was scaled up by this percentage. The applied wrist-release and wrist-roll
moments were then analyzed to investigate the effect. Additionally, it was fitting to use
the model with the principal inertia terms, and in this case, the test is assuming a perfect
club with zero products-of-inertia. Therefore, the principal terms were calculated by
solving for the eigenvalues of the second order inertia tensors for the 350 cubic
centimeter driver and the 470 cubic centimeter driver. Table 3.3 shows the data for the
470 cubic centimeter driver and it is noted that this is an approximation to the outcome of
the inertia terms, strictly by a scaled percentage. Therefore, this model may not
accurately describe the correct mass and subsequent inertia terms of a correctly drawn
club head model with solid modeling. However, for the purposes in this model, the
approximation will suffice.

Next, the principal inertia terms were calculated for the 350 cubic centimeter

driver head and the 470 cubic centimeter driver head. This data is tabulated (Table 3.4)

69

and was used in place of the original inertia parameters and subsequently the moments in

the wrist were found. The curves for all four case studies will be presented subsequently.

Table 3.4: Principal Inertia Terms for the 350cc and 470 cc Drivers (kg-m2)

 

By using the inertia parameters with the 350 cubic centimeter driver head, and the

same kinematic parameters shown in Table 3.2, the affected portions of the wrist-release

Generalized Force (N-m)

0.00
-0.05
-0.10
-0.15

-0.20

 

-0.25
0.000 0.050 0.100 0.150 0.200 0.250 0300 0350

time (seconds)

Figure 3.13: Generalized Forces in the Wrist for the 350 Cubic Centimeter Driver

moment and the wrist-roll moments were analyzed during the simulated downswing time.

The results of the outputs are shown in the Fig. 3.13, and the final results of all of the

70

additional tests can be found in Table 3.5. It is noted that the wrist-roll torque becomes

negative late in the downswing. This is based on the assumed motion and indicates that

Generali7cd Force (N-m)
e
o
e

 

41.30
0.000 0.050 0.| 00 0.150 0.200 0.250 0300 0350

time (seconds)

Figure 3.14: Generalized Forces in the Wrist for the 470 Cubic Centimeter Driver

refinement of the wrist-roll motion may be required in future work. Since the applied
wrist-roll torque behaves sinusodially and erratically, further analysis is required. This
nature of the assumed motion directly affects the outcome of the sensitivity curves shown
earlier.

Using the mass and the driver head data for the 470 cubic centimeter model, the
generalized forces in the wrist were again analyzed. The output is shown in Fig. 3.14.
As a result of the scaling of the inertia parameters, the maximum applied moments in the
wrist joint increase as expected. Also, the overall oscillatory shape of the curves

remained the same with no substantial change.

71

Next, the principal inertia terms for the 350 cubic centimeter driver were used and

the applied moment curves were analyzed. In this case, the product-of-inertia terms were

0.25

0.20
E 0.15
2 _
i: Qyit)
LE 0.10
'8 _
N
L: Qfilt)
S 0.05
5
O

 

-0.05
0.000 0.050 0.100 0.l 50 0.200 0.250 0300 0350

time (seconds)
Figure 3.15: Generalized Forces in the Wrist using the Principal Inertia Terms (350 cc).

eliminated. As a result, two things occur. For the wrist-release moment, the curve looks
somewhat smoother near the time of impact (red trace). The reason this observation was
made was because the oscillatory behavior in the red trace shown in Fig. 3.15 does not
change as much as opposed to the red trace in Figs. 3.13 or 3.14. More importantly, the
applied torque was non-negative. Therefore, using this type of model, it was found that
non-zero product—of—inertia terms results in a negative applied wrist-roll torque. This
matter is required to be addressed in future models, but it may suggest instability in the
model. Noting that the behavior of the torque output relies solely on the assumed motion,
this suggests that the motion may need to be refined in a more correct model. As a result

of using only the principal terms in this case study, the maximum applied wrist-release

72

moment increased a very small amount whereas the maximum applied wrist-roll moment

decreased. The final results can be observed in Table 3.5.

G e G
I— ‘— 5.3
G '1! C

Generalized Force (N-m)

P
c
u-

0.00

 

-0.05
0.000 0.050 0.100 0.150 0.200 0.250 0300 0.350

time (seconds)
Figure 3.16: Generalized Forces in the Wrist using the Principal Inertia Terms (470 cc).

Last, the principal inertia terms were used for the 470 cubic centimeter driver
head and the generalized forces in the wrist can be observed in Fig. 3.16. The shape of
the curve is very similar to that of Fig. 3.15, however, as expected the maximum values
changed. The maximum applied wrist—release moment increased and the maximum
applied wrist-roll moment decreased slightly. The percentage increase was then

calculated and shown in Table 3.6.

Table 3.5: Results of the Inertia Component Case Studies.

Maximum 0, Minimum 0 ,7 Maximum 0 7 Minimum 0 7

Case Study (N-m) (N-m) (N-m) (N-m)

 

73

Table 3.6: Percent Increase in the Maximum Moments in the Wrist for the Case Studies.

 

Percent Increase in Percent Increase in

Case Study Comparison Max Moment (0 ,3) Max Moment (0 r)

 

 

 

 

 

 

 

350 cc vs. 470 cc 33.74 34.39
350 cc vs. Principal Terms 1.84 3.17
470 cc vs. Principal Terms 229 3.37

 

The maximum wrist-release torque increased when the principal terms and the
inertia terms of the 470 cubic centimeter driver were used. This was a direct result of an
increase in the first and third principal terms. However, the percentage increase was very
small in the big picture as shown in Table 3.6. Using this type of model, it was shown
that the product-of—inertia terms do not have a significant influence on the wrist-roll
torque, but in terms of the shape of the torque curve, there is a large effect. The curves
become more sinusoidal when the product terms are used, and the fact that the applied
wrist-roll moment becomes negative and oscillatory near impact suggests that the wrist-
roll motion is required to be analyzed further. Perhaps a sine wave is not the best way to
assume the motion. As a result of using the inertia data for the 470 cubic centimeter
driver, the percent increase was substantial and roughly speaking, is equivalent to the
overall size increase in the driver head over the 350 cubic centimeter club head. This
makes sense because the inertia terms were scaled directly by thirty-four percent. A
repeat of this test using actual mass and inertia data from a simulated output from the
same source would provide the best comparison because the inertia terms may not all be
affected.

The assumed motion is very critical because it dictates how applied moments in

the wrist will behave, which in turn will affect the sensitivity of these moments. The

74

oscillatory motion of all curves that were presented resulted in an inconclusive study in
which the motion of the downswing, while refined in this unique type of kinematic
model, may require additional work to ascertain why negative torques were calculated
and furthermore which kinematic parameters affect the sensitivities and the applied

moments.

3.4 The Effect of the Range-of-Motion of the Shoulder in the Golf Swing

Different impact configurations of the golf swing were analyzed in three scenarios
based on the total range-of—motion of the shoulder link. Enforcing the alignment of the

shaft link with the vertical axis at the time of impact was completed by setting the

kinematic constraint of ,6 (ta) 5 0°. Therefore, the modeling ascertained the effects of an

incomplete shoulder rotation at the time of impact versus a standard and advanced impact
configurations of the shoulder link with the shaft link aligned correctly in each case.

The impact configurations for each type of golf swing is visually displayed in
Figs. 3.17, 3.18 and 3.19 which show the various types of impact configurations of the

compound pendulum to model different types of golf swings. In the standard case (Fig.

3.17), l/lf makes a 900 angle with vertical line bisecting the hub axis at the time of

impact. Geometrically speaking, a right triangle is formed by the shoulder and arm links
to align the shaft in the proper location. Assigning 9' and 77 as inclusive angles of the
right triangle, the simple geometrical proof is given to show how the finish angle of the
arm is a function of the finish angle of the shoulder link and the lengths, L, and L2.

With an incomplete shoulder rotation, the golfer would dip the left side of the

body (for a right-handed golfer) and prevent a proper hip turn; subsequently the shoulder

75

 

 

 

 

 

 

—a, + n = Zr—

2

LI _ L2
sin 4' — sin I!!!

L
(If = —Arc sin[Z-’— sin WI]

2

L
(If = —Arc sin[—’—]
L2

 

 

 

 

 

.11—

Figure 3.17: Standard Impact Configuration of the Three-Link Pendulum and
Geometrical Proof.
rotation would be less than 90" at the time of impact and the transfer of energy from the
shoulder muscles would be less due to a lower angle of rotation. This type of impact
configuration is typical of intermediate golfers whose joint flexibility has not allowed
them to utilize the benefit of the range-of-motion of the shoulder in the golf swing. This
type of impact configuration is expressed in Fig. 3.18.

Figure 3.19 shows an advanced impact configuration where the shoulder link
rotates through an angle greater than 90°. Here the configuration of the pendulum is
showing how an elite golfer takes advantage of the proper torso rotation allowing for the
full range-of-motion allowed by the joints in the neck and the shoulder. The geometrical

proof is provided to show how the calculation of the arm link is affected by the increase

76

I I.l wf+n+§=7r

 

 

—a + —a-5
a n f '7 2
.. -__ - --..- L1 _ L2
‘1’; I sing“ singl/f
1"
a _ A . L, .
f af —— rcsm Z—smwl
2
1r— C
L2
L3
_Jl__

 

 

Figure 3.18: Incomplete Shoulder Rotation of the Three-Link Pendulum and
Geometrical Proof.

in VI]- . Thus, if the finish angle of the shoulder link is specified as an input parameter,

the arm finish angle can be found by considering the geometry of the shoulder and the
arm links.

The devised study looking into the effect of the range-of—motion of the shoulder
link analyzed the previously described impact configurations by setting the shoulder
finish angle, 04 at 70°, 90°, and 110°. The defining factors were the club head velocity
at impact, the required linear generalized work terms in the shoulder and arm links and
the total generalized work term in the shaft link inclusive of the rotational portion.
Additionally, the total angular impulse in the shoulder link and the shaft links was found

by integration of the generalized force expressions.

77

w, +17+§=7r

 

 

It

—a +' +b=——

f ,7 2
L, _ L,
sing Sinl/lf

L
a, = —Arc sin£ Ll sinwf]

2

 

 

 

 

.JL_..

Figure 3.19: Advanced Golfer Impact Configuration of the Three-Link
Pendulum and Geometrical Proof.

Generating more club head velocity at impact, advanced golfers will rotate the
hips more resulting in a higher range-of-motion of the torso and essentially the shoulder
link. At impact, this is called an open stance where the hips pivot a large amount rotating
the entire upper body from the waist up including the shoulders. By simulating this type
of impact as shown in Fig. 3.19, and using the model, it was found that when the shoulder
link undergoes higher angular displacements, the club head velocity, V4 increases (Fig.
3.20). Also, the maximum applied wrist-release moment decreases in the shoulder link
when t//f> 90° (Fig. 3.21). The reduction in the maximum release torque of the shaft link
is a result of the resultant increase in the angular acceleration of the shoulder link.
Analyzing the equation of motion, the acceleration term with the highest magnitude

containing the angular acceleration of the shoulder link is:

78

 

 

 

 

 

48
3 47.70
E
I: 47.5
0
El
1; 47 46.80 ‘
3
"g 45 5 46.34
2
E
2
L2 46

45.5

70 90 110
‘1’ f (degrees)

Figure 3.20: Club Head Velocity vs. Shoulder Finish Angle.

—t,1/c0s((//+,B)[m_,d_,L, +n14L,L,]. Notice the minus sign in front of the acceleration
term hence the overall reduction of the wrist-release moment Qfl. Looking at the data in

the spreadsheet, this term containing the angular acceleration of the shoulder link
contributes the most to the reduction in the wrist—release torque of the shaft link.

With an increase in the range—of—motion of the shoulder link, the maximum
applied moment in the shoulder link increases subsequently. However, this makes sense
fundamentally because the amount of work done by a rotating rigid bar is directly related
to the angular displacement. According to Redford [23] who is a specialist in increasing
golfer conditioning, it is important to focus specifically on the dynamic flexibility of the
joints involved in the golf swing. The statement means that the human muscles are more
capable of optimum output if the joints they are connected to are conditioned to operate

in a greater range-of—motion because of higher preload when the torso is rotated and the

79

Max Moment (N—m)

I Maximum Applied Moment in Shoulder (N-m)
I Maximum Applied Moment in Arm (N-m)
I Maximum Applied Moment in Wrist (N-m)

 

S

8

N
O

 

 

103.53 107.73 11121

53.27 53.66

70 90 110
Wr (degrees)

52.86

 

Figure 3.21: Maximum Applied Moments vs. Shoulder Finish Angle.

80

muscles crossing over the shoulder and neck are flexed at the onset of the downswing.
More importantly, this information leads to the important fact that stretching exercises
which increase the flexibility of the torso and the shoulder so the muscles operate at the
their optimum potential. Also, according to an web article written by written Woods and
Daniel [24] and published on the Golf Digest website, the ability for maintaining
flexibility and increasing the range-of—motion of the torso is important through stretching
exercises. This is because warming up the body before practicing allows increased blood
flow and suppleness of the muscles crossing over the torso, neck, and shoulder and will
prevent the restriction of the rotational motion of the upper body. These muscles are all
valuable sources capable of transferring mechanical energy (torque) in the golf swing.
The article then continues to discuss that professional golfers create a significant amount

of torque in the golf swing that maintaining flexibility of the hips is important as well.

This makes sense physically because the hip—pivot motion in the downswing is what
controls the spinal cord and the shoulders. In reality, an increase in the generalized force
in the shoulder is not a drawback but a direct result of the increase in the range-of-motion
of the shoulder link which is dependent on the flexibility of the joints in the neck,
shoulders and the hips. Therefore, big muscles, like the Pectoralis Major and the
Deltoids which work the shoulder joints, are more capable creating the moment in the
shoulder link than small muscles crossing over the wrist joint in generating faster club
head speeds.

Since the calculation of the work done was performed in the model, further post-
processing entailed the calculation of the total angular impulse. Greenwood [15] has
stated that the total angular impulse acting on a system due to external applied forces can

be expressed as:

A

This important mathematical formula states that the total angular impulse M is
equivalent to the integration with respect to time of the total external applied moments M.
Also, Greenwood further explains that the total angular impulse is equivalent to the

change in angular momentum from state 1 to state 2:

M=H,—H,

Applying this equation individually, the equations of motion were integrated with
respect to time with Mathcad (Mathsoft, Cambridge, MA) to find the total angular

impulse in each link of the pendulum. Table 3.7 documents the kinematic constants used

81

in the calculation and the results are presented in Fig. 3.22, which correspond well to the

results for the calculation of the maximum applied moment in the shaft link (Fig. 3.21).

Table 3.7: Kinematic Parameters Used in the Calculation of the Angular
Impulse in the Shoulder and Shaft Links.

 

 

 

 

 

 

 

 

 

Kinematic Parameters y, = 70° y, = 90° y, = 110°
Shoulder Link Kinematic Constant (P) (rad/s) 4.07 5.24 6.4
Arm Link Kinematic Constant (R) (rad/r) 6.77 6.7 6.77
Shaft Link Kinematic Constant (S) (rad/s) 95255 9.63 9.5255
Wrist-Roll Kinematic Constant (W) (rad/s) 13.09 13.09 13.09
13(1, ) "“1 -3.810 -3.811 -3.81
”(’1)"“’/‘ 3.38 3.35 3.38
0(1, ) rad / s°
61.39 60.73 61.39

 

 

 

 

 

So the study shows that an increase in the range-of-motion of the shoulder link reduces
the required applied maximum shaft release moment and the total angular momentum
required to swing the golf club decreases accordingly. Fig. 3.19 shows that the angular
impulse in the arm link is slightly lower when the shoulder link rotates to 90° in the
standard position as opposed to the other two impact states. The total angular impulse is
equivalent to the area underneath the curve of Eq. 2.48 when plotted against time. It was
observed that the value of the applied wrist-release moment at the time of impact for the
arm link was negative and the lowest value when Vlj=90°. This explains why the total
angular impulse (total net area) in the arm link is lower in the standard impact case.

In conclusion, the parameter study investigating the range-of—motion of the
shoulder link showed why it is beneficial to rotate the shoulder though higher angular

displacements in the golf downswing as seen in the increase in club head velocity at

82

I Shoulder Link I Arm Link I Wrist
I9. .2 I Q. d: in. d:

 

Angular Impulse (N-m-s)

 

 

 

70 90 110
Wf (degrees)

Figure 3.22: Total Angular Impulse in Linkages vs. Shoulder Finish Angle.

at impact and a reduction of the required maximum applied moment and total angular

impulse (momentum) in the golf club.

83

CHAPTER 4

Conclusions and Recommendations

4.1 Summary and Validity of the Model

The complexity of the golf swing was reduced by creating a rigid, three-link
pendulum model to investigate the effect of the club head entity and the range-of-motion
of the shoulder in the golf downswing. The solution utilized the techniques of inverse
dynamics and kinematic assumptions governing the motion of the linkages were
specified. The detailed development of these key equations is contained in Chapter 2.

Utilizing Lagrange’s Equations, the required applied moments to provide for the
kinematic motions of each link were developed and are equivalent to the equations-of-
motion for the compound, three-link pendulum. The computer coding of the kinematic
equations and the expressions for the generalized moments were implemented through a
basic spreadsheet program, Microsoft Excel (Microsoft Corporation, Redmond, WA),
which was capable of the analysis once the problem statement was clearly understood.
Using Microsoft Excel, simple, built-in, user-defined functions were utilized and
algorithms were created based on the different stages of the golf shaft in the downswing
regarding the shaft release motion and the rolling motion by the golfer’s wrists. This
reflected the modeling of how the golf club is swung by the individual and was
developed through the creation of the kinematic assumptions of the respective time

dependent generalized coordinates.

84

The goal of this work was to create a kinematic downswing model that was
unique to what has been published in this field. The additional wrist-rotational motion
was added and attempted to describe how the golfer would square the club head to the
target prior to impact between the club face and the golf ball. Contribution to the
ultimate goal of establishing a relationship between the wrist-release and wrist-roll
moments was attempted with the sensitivity measurements of the applied moments in the
wrist joint. These functions are time-dependent functions, implicitly related to the club
head inertia elements. However, due to the complexity of the sensitivity curves and
difficulty interpreting the oscillating nature of the sensitivity output, more studies need to
be devised to uncover hidden relationships that this model fell short in accessing.
Furthermore, the relationship is solely represented to how the motion of the downswing is
assumed. Based on the analysis in Chapter 3, there exist points to be addressed in terms

of future models that will be created studying this relationship.

The sensitivity functions of the wrist-roll in terms of the I12 and I2; inertia terms
showed that the function was negative and the oscillatory behavior did not depict that the
functions would revert back to the neutral state. Also, the sensitivity curves are basically
showing how difficult the moments in the wrist are to apply based on the motion, but this
must be taken lightly due to the limited study because only one type of motion was
assumed and tested. This fact suggests that more studies are required to look into this
matter regarding the kinematics and the influence on the sensitivity of the wrist moments.
Furthermore, the fact that the applied wrist-roll moment was negative in the time just
before impact suggests that the kinematic motion of the wrist-roll motion needs to be

analyzed to help understand why the moment curve becomes negative and highly

85

sinusoidal in short time intervals before the time of impact. The torque output does not
really apply in a real world situation because it does not make physical sense as to why a
negative wrist-roll torque is required. This matter needs to be addressed in future models
that are created. The way the applied wrist-roll moment and the applied moment in the
shoulder behave suggests that there are some issues in the assumed motion and that the
stability of the model needs to be addressed as well. While the assumed motion was
compared to the motion as published in Ref. 3, the validation of the kinematic model was
not fully addressed. In order to do this, a model with more sophisticated algorithms may
be required, such as computer coding of the kinematic functions in Microsoft Visual
Basic (Microsoft Corporation, Redmond, WA). This would allow comparison to the
output in this analysis. The computer coding in another software program is the likely
path in continuing work of this thesis. The ultimate relationship between the interrelated
circle of dependent entities such as the assumed kinematic motion driving the swing, the
system parameters, the equations-of motion of the pendulum, and the sensitivity
measurements all contribute to the effect on the simulated output of the applied torque in
the linkages, but more importantly, the kinematic assumption must be evaluated with
another computer code as the first step.

By performing additional inertia parameter tests in which the inertia values were
scaled directly, this model showed that there was not much change when the products-of-
inertia are added into the model. However, there was a significant change when all
inertia terms were scaled. This was done by comparing the inertia values for a sizable
increase in the club head. When using just the principal inertia terms, the wrist-roll

torque output was still oscillatory in nature, however, near the time of impact, the torque

86

was non-negative. Future tests can be performed in which one inertia component is
varied and the maximum moments in the wrist observed once additional insight is
obtained regarding how the two influential cross terms affect the wrist-roll torque. Once

further insight is obtained regarding the oscillatory nature in terms of the assumed

motion, then further information regarding the sensitivity in terms of the I12 and I 23
inertia terms on a sole basis would provide insight if the terms are negative or positive. It
is noted that the sign of the inertia term should be taking into consideration when
performing partial differentiation for the sensitivity measurements, and in this model, the
sensitivity equations were only normalized by the multiplication constant and did not
consider the sign of the inertia term.

In an effort to quantify the importance of including the shoulder link and the
modeled effect of the muscles, a study was devised to investigate the effect of the range-
of—motion of the shoulder during the golf downswing through different kinematic impact
positions. However, the assumed position at the time of impact will require future work,
to illustrate, the arm link was not inline with the shaft link in this model, but would be in
a more representative impact position of the golf swing. Also, due to the rigid shaft link,
and not accounting for the elasticity of the golf shaft, the proper club head position was
not modeled. As a result, the rigid body model was unable to accurately predict the
position of the club head as shown in Ref. 3.

However, the general idea is to emulate professional golfers and observe how
they increase the range that their upper body moves through in the golf swing to generate
torque with their muscles and ultimately the club head velocity at impact. In the modern

golf swing, the ability to provide this great power lies in two concepts — increased

87

athleticism through strength training, however in this model, the flexibility of the joints
was loosely described in this model by specifying the total angular displacement of the
shoulder link. The suggestions made within this thesis have more pertinence for the
hobby golfer, since increased flexibility and proper stretching are viable avenues to
pursue for golf game improvement. Using this model it was shown with engineering
mechanics why the increased range-of—motion of the shoulder results in a faster club head
speeds at impact. The first important conclusion is that the amount of work done is
dependent on the angular displacement of the shoulder link and was seen to increase
when the shoulder displaces through higher angles. This suggestion relates to increased
flexibility of the human joints and is common knowledge stressed by PGA golfing
professionals and golf teachers. An increased range-of—motion of the upper body allows
the muscles to perform at their optimum effectiveness and therefore more work output is
observed and was shown in the study. As a result of the increased range-of—motion of the
shoulder link, it was proven that the maximum applied shaft release moment decreased as
well as the total angular impulse which is equivalent to the total change in angular
momentum in the golf club. More importantly, it was also shown that the generation of
more club head speed at impact was the primary benefit of the shoulder link rotating
through higher angles at the time of impact. The results obtained from this basic study
can be used in future models to investigate the validity or research how the wrist-release

moment is related to the shoulder link prescribed motion.

4.2 Future Work and Recommendations

Regarding the overall model, in terms of the applied moments that the muscles in

the shoulder and upper back must provide, more in-depth literature searches should be

88

performed to research how the kinetics of the muscles behave with certain joint motion.
According to Trew and Everett [25], there does not exists a direct way to measure the
force generated with a joint motions. However, there are many mathematical approaches,
such as Isokinetic Dynamometry. In general, Ref. 25 provides an excellent starting
ground for investigating this subject matter more closely for someone who is interested in
learning the basics of evaluating human muscle movement. This paper was cited from an
electronic source, and in general shows why the evaluation of the human body is difficult
to measure and provides a discussion of the techniques that are practiced. Still, the
importance lies in the fact that future models can be related to the behavior of the human
muscles. For instance, studying the measured, kinetic torque outputs of the muscles
involved in the golf swing and comparing with the torque outputs obtained from a
simulated, prescribed kinematic motion is one area that can be investigated to somehow
validate the correctness of the assumed motion of the golf downswing. This would allow
the assessment and refinement of the kinematic motion, which is an area that requires
attention in this model due to the negative, generalized force in the shoulder link.

The kinematic impact position in this model was insufficient in describing the
actual position a golfer would make with the upper body, arm, and the angle of the wrist-
cock. This was because the shaft and the arm were misaligned at the time of impact;
however in a real golf swing, the arm and the golf shaft are collinear with a small degree
of misalignment. This shows that to represent the impact position in future models, the
geometry and constraints of the finish angle of the arm and the shaft linkages should be

revaluated in terms of the kinematic impact configuration.

89

The sensitivity of the applied moments was affected by the assumed kinematic
motion. More studies should be performed with a more correct model in terms of the
prescribed wrist-roll motion. This was because the negative wrist-roll torque that was
calculated with this model. As mentioned before, the overall downswing time could be
made as the primary variable which dictates the time at which the wrist-release and wrist-
roll occur. In general, if someone has a slower downswing motion, then the wrist-release
and wrist-roll time intervals would be longer and thus the motion associated would be
more gradual. This recommended work may show interesting results in terms of the
sensitivity measurements which are affected. As an example, an argument could be made
that the wrist—release time for average golfers accounts for sixty percent of the
downswing time whereas for elite golfers, the time interval would only be forty percent.
First, a good initial test would be to vary the downswing time to see what happens to the
global extrema and if there is a possibility for minimization by just the downswing time
variable. Next, based on this time, the wrist-release or wrist-roll time can be varied to see
what happens to the oscillations of the sensitivity function and whether there is
smoothing of the curve. This may show some additional insight into how the oscillatory
nature of the sensitivity curves change by varying the kinematic parameters instead of the
analysis in this model, which just presented how the sensitivities are related to the inertia
properties of the club head. Overall, the model showed that more work is needed but will
help to establish the relationship between the sensitivity of the applied moments in the
golfer’s wrist by varying both the kinematic parameters and if the sensitivities take into
consideration of the value of the inertia term. This can be done by sealing the sensitivity

function and by taking the sign of the inertia term into consideration.

90

Appendix A

A.l Derivation of Squared Velocity Equation (2.32)
Starting with the given Equation (2.26):
F, = 1:, = —t1/d, (sinw e,+ cost}! 62] (2.26)

2

v, =|:-lI/d, (sinll’ €1+C03W 92]] = dlz W2

v" : (1,2 [I] (2.32)

. 1/2
iv] i=[d12 W2]

A.2 Derivation of Squared Velocity Equation (2.33)

(’11.! (A.1.1)

Starting with the given Equation (2.28):

12 2r; = —t,uL, [sing/l e,+cos(z/ 62 ] +a'd_. {-sina e,+cosa 62] (2.28)

2 . A A . A A 2
v2 = [—t//L, [Sim/I e,+ cost/I 62 ] + adz {—sina e,+cosa' e2 )]

o
7

1122 = V2 L,2 +2d2L, wa(sint//sina—costycosa)+d22 a“

1122 2 W2 L,2 — ZdZL, wa(cost//cosa— sint/lsina) + (122 a2

12,2 = L,2 122+ d22 a2— 2d,L, t/;£ZCOS(w+a) (2.33)

. , . . 1/2
Iv, |=[1//2 L,2 +d22 az-ZdZL,1;/acos(1/I+a):| (A.2.1)

91

A.3 Derivation of Squared VelocflEquation (2.34)

Starting with the given Equation (2.30):

v, = if, = -t//L, (Shit/l e,+cost// e, )+aL2 {—sina e,+ cosa' e2)
(2.30)

+ ,Bd‘, {—sinfl e,+ (03,6 62)

2 (11L, (Sim/I e,+c0st1/ 621+ a'L2 {—sina' e,+c0sa e2]

VJ.

 

 

L+1341,» [‘Sinfl e1+ C0516 62]
113 Lfi +2L,L2 111a(s1nwsina-coswosa) i

11,2 = +2a’j L, Jibhin t// sin ,6 - cos View ,5)

 

 

+2d_,L_, (1,5(sin asin )6 + cos acos ,6) + L22 a2 + (I32 ,32

l _

L," 01" + L22 a2+ (1,2 fl2 — 2L,L, t/la’(cost//C0sa — sing/sin a)

v; = —2d3L, 1;],‘5Q‘05 t/lcos ,6 — sin t/lsin ,6)

 

 

+2d‘, L2 or ,8 (cos acos ,6 + sin a sin ,3)

L2 .2+L2a:2+d2 .Z—ZL .drcos +a
2;: 111/ 2 .fl 11414 (W ) (2.34)

—2d_,L,11.1,0C(13(u/+,B)+2d3L2 Erbcosm- fl)

NZ

(1% L,2 + L,2 a2+df 02—2“, u.lc.rcos(t//+a)

—2d_,L, tilbcos(1,1/+,B) +2d3L2 dbcosflt—fl)

iv; l= (A.3.l)

92

 

 

Since the club head is treated as a point mass, the velocity of the mass center for
the club head is equivalent to the velocity at the end of the shaft link. This is

found by substituting d3=L3 in Equation (A3. 1 ):

l/2

3L2+Lfai3+Lf .Z—ZLL .dcos +a
111 1 . 73 12V (11I ) (A32)

iv4 i:

-214le wflcos(w+ ,6) +2L2L3 aficosm—fl)

A.4 Total Kinetic Energy

The Total Kinetic Energy results from adding Equation (2.36) and (2.39):

P] .2 2 [’2‘
TzLEmIW (I, +§W I]

,—

2

o2 o o o ‘2-
+ ém—iL’ZW +d22a —2d2L,1//acos(1//+a)]+%a 12]
l.

.2 .2 .2 o o
1 My +L22a +dffl —2L,L,1//acos(1//+a)

+ —m,
—2d.,L, tuflcos(t//+,6)+2d3L‘2 aflcos(a-fl)

 

’— .2 .2 .2 o o
+ 1 lel// +L22a +L.,2,6 —2L,L2t1/acos(1//+a)]
—m4

—2L,L, pbcosfi/Nrfl) +2L2L‘, dbcosM—fl)

l.

.2 _ .2 _ .2_
7 6052,3111-1-7 Sin2fl122+fl I33
2

—2;' cosflsinflIn-2,.85’cosflI13—2,.Bi/sinflI23

A.5 Differentiation of Total Kinetic Energy for Shoulder Link Generalized Force

17.; = m,d,2 13+ I1 171+ 1112 [L2 (II—(12L, drcos(1//+a)]
811/
+m_,. L,21/./—- L,L_, cotcos(t//+ a)—d~,L,,.6cos(1//+,B)] (A.5.l)

l-

 

+m4 L,2 1;!— L,L2 dcos(1//+ a)—m4L,L_, Bcos(t//+fl):l

h

93

 

8_T = mzdzL,t1./c.t'sin((1/+a)

.—

+m

L,L,y}51sin(yx + a) + d_,L,1/./,fisin(tu + mi (A52)

1- —d

1‘‘

r— —

 

 

+m4 L1L2 (Iltitsin(t//+a)+ L,L_, t/Olbsin(t/l+,5)

 

51— 8]: =t//|:m,d,2 +m2L,2 +m,L,2 +m4L12 +I1]
d1 ‘
611/

—acos(t//+a')[m2d2L, +m3L,L2 +m4L1L2]

—,§c0s(t/I+fl)[m3d3L, +m4L,L_,] (A.5.3)

 

+5411.» £1]si11(1//+ a)[m_,d,L, +m_,L,L, +111,L,L,]
+,.B(t/.I+ b] sin(t1/+ fl)[m3d3L, + m4LlL3]

The generalized force in the shoulder link is formulated:

 

Q __.1__ er ”a:
W (II 31;! By!

Q11! =t;[m,d,2 +m2L,2 +m_,L,2 +m4L,2 +I1]

—.0;cos (y/ +a)[mzd2L, +m3L,L2 +m4L,L2]

—,.B.c0s(t//+,6)[m3d3L, +m4L,L_,] (A.5.4)
+c.tr(1/./+ drjsin (w+a)[m2d2L, +m3L,L2 +m4L,L2]

+80.” bjsinw + ,B)[m,d_,L, + m4L,L,]

-t/./c.Zsin(t//+a)[m2d2L, +m3L,L2 +m,,L,L2]

—t//,Bsin (t/I+fl)[m_,d3L, +m4L,L3]

94

 

A.6 Differentiation of Total Kinetic Energy for Arm Link Generalized Force

 

£91.sz [c122 6.1—0.1d2L,cos(V+a')]+C.rI2
8a
+m3 L22 dr— LIL, (Ilcos(t,y+a')+djL2 :8cos(a'—,B)i (A.6.l)

_ .1

— —

+m4 L22 cir— L,L2 0.1cos(t//+a)+L2L3 bcos(a—,B)

1- u-

 

 

:1 : "12(12LI1/1asirz(1// + a) + m_,L,L2 V70.lSin(V/ + a)
a

-m.,d_,L2 dbsin (a — ,3) + m4 LIL, t/Iasin(t// + a) (A.6.2)

-m4L_,L3 dbsifia—fl)

 

(1 3T “
El: ' i: 'Wc0s(y/+a)[m2(12L, +1113L,L2 +1n4L,L2]
8a

+0!|:m2dz2 +m3L22 +m4L22 +I2]
+ ,Bcos(a— ,6)[m,d,L, + m,L,L_,] (A.6.3)

+t.u(1/./+ d!)sin(t//+ a)[m_,d2L, +m3LIL2 + m4LIL2]

— 8(3— fl) sin (a - fl) [111.112, + 1114L2L_, ]

95

The generalized force in the arm link is formulated:

d 3T 3T
QG=Z[ .]_5_
I aa a

Q, =—t;c0s(1//+a)[m2d2 L, +m L,L2 +m,L,L2]

 

W|:1d +111,,2L +mL2+12]+flc0s(a' ,B)[mdL +111LL]

+l//(l,l.l+ ajsin( ()1/1+a' )[1112 d, L, +m ,,2LL +m LL] (A.6.4)

—,°6[o°1— b)s1n(a- fl..)[m d L, +m,L2 L]

—t.uc.rsin (1/1 + a)[m2d,L, + 1n_,L,L, +111,L,L2 ]

+aflsin(a ,B)[m,,2dL +11142LL,]

A.7 Differentiation of Total Kinetic Energy for the Wrist Generalized Force

2:— : m, [(132 b- (13141 V.’C05(W+fl) +d3L2 £¥C05(a_’6)]+ L b
313

+111, [L32 ,0— L,L, tileos(t// + ,5) + L,L, drc‘os(a'— fl)] (A.7.l)

+2bI33— 2 7(‘OSfiI13- 2 7S1)? ,6I23

3—2 = m, [11,; 1335141114. ,6) + d_,L, ammo—11)]

+111, [L,L,1/./,.BSI”(V’+IB)+L2L3&'b3ilz(a_’8)] (A7 2)

02 .2 .2

—2y cosflsinflIn+27 cosflsinflI22+27 (sinzfiwcos2 ,6)I12

+2;Bjmin,8I1_1— 2b}cosflI23

96

 

 

5’— °T. =,.B.|:m,d,2+m,L,2+I3+2I33]
d1 ‘ '

3,3
—t;cos(ty+,6)[m,d3L, + m,L,L,]+ gco.s(a—fl)[m,d,L, +m,L2L,]

+1110.” fl) sin ((11 + ,6)[m_,d,L, + m, L, L, ] (A.7.3)

 

—£z(£z- fl) s1n(a—fl)[m_.d,L, + m,L,L,]
—2(.}:Cosfl — ,8 }sin 0) 1,1- 2[;sin,6 + 8 ;’cos fljiy

The generalized force in the shaft link is formulated:

 

Q :1 8T __B_T_:
fl (II ab 8,8
-t,.1;cos(t,//+ ,6)[m,d,L, + m,L,L,]+ ooic0s(a—,B)[m3d,L, +1114L2L3]

+,.6.(m,d,2 + m,L,2 + I3J+ tI/[t/l+ fl) sin(tfl+ fl)[m,d,L, + m,L,L_,]
—£1[51—,6)sin(a— fl)[m,d,L, + m,L,L,] (A.7.4)

—1//,8sin(w+,6)[m,d,L, +n14L,L,]—a,6sin(a—,B)[m3d,L2 +m,,L2L,]
.2 _ .2 _ oo_
+27 (c0.s,6sin,6)l11—2}’ (cosflsinfl)122+2,6133

2

+2} ((032 ,B—sin2 ,5)I12—2;c0s,BI1.1—2.}:sinflI23

A.8 Differentiation of Total Kinetic Energy for the Wrist-Roll Generalized Force

a_T—2;/coszfliu+2ySInzflI22—4}C0S,BSinflI12

a 1’ (A81)

—2,Bc0s,3I13—2flsinflI23

97

 

i[°7:]=[2;cosz fl—4b}cosflsinfl:|In
(If a},

+[2;sin2 fi+ 4,.82'cosflsin ,5] 722
—4 ;c‘os,6sinfl- ,0 ;/sin2 ,B+ ,0 i/cosz ,3] I12 (A-8-2)

K. .2 ‘_
—2 flcosfl—fl sinfl I12

.0 O 2 _
—2 ,Bsinfl+,6 cosfl—i 12.1
i. _.

 

 

— E 0 (A.8.3)

The generalized force in the club head is formulated:

Q :1 8T —a—T:
7 (It a} By

2|:.}./cosz ,5 — 2:6}cosflsin ,6] In

 

 

 

 

 

+2 ;sin2 ,8 + 2,0 ;/c05,6sin ,3] I22

: (A.8.4)
—4 L}/c0s,Bsin,B+ flycosz ,3 — ,6 ysinz ,3]I12

_oo o 2 1 _
—2 ,Bcosfl—fl sinfl I13

:00 o 2 : _
—2 fisinfl+fl 005,6 12.1

L _
A.9 Wrist-Roll Sensitivity Equations
aQr .. 2 . . .

=2 ycos fl—Zflycosflsmfl (A.9.l)

8 I11

98

 

 

 

 

8% = 2[.}./sin2 ,3 + 2 ,6 2cosflsin ,3] (A92)
3 122
a_Q7 : —4 |:;cos‘,8sinfl+ :6 2cos2 ,3 — ,0 ;’sin2 ,6] (A93)
8 I12
3Q .. . 2

_ 7 = '2 [1560315 - fl sin fl] (A94)
3 113
39 .. . 2

_ 7 = —2 [,5 sin ,8+ ,8 cos ,8] (A95)
3 12.1

A. 10 Wrist-Release Sensitivity Equations

8 '2

31:27 (cosflsinfl) (A-IO-l)
3111

a '2

—?fl—=—27 (cosflsinfl) (A-IO-Z)
3122

a ..

& ._. 25 (A.10.3)
0133

a '2

g = 2 y (c052 ,6— sin2 ,6) (A-10-4)
8112

a ..

fl : —2 ycosfl (A. 10.5)
311.1

a ..

fl : -2 7sinfl (A.10.6)
0123

99

APPENDIX B

8.1 Plots of the Generalized Coordinates vs. Time2

(rad)

T
T
T
T

 

Angular Displacement

0.00 OJD 0.04 0.“ 0.“ 0.10 0.12 0.14 0.16 0.10 0.20 0.2 0.24 0% 0.28 0.30 032

time (seconds)

Figure B. l: Angular Position of the Generalized Coordinates vs. Time.

40.11)

35.0) _.
>‘ 0
.g 3am V
—‘ A 25.1!) —
3’ S a.
a a 20.00
a, 15.00 .
c B
<2 10.00 _

5.00 i1

 

0.“)
000 am 0.04 0.15 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.3 0.24 0.% 0.28 0.30 032

time (seconds)

Figure B.2: Angular Velocity of the Generalized Coordinates vs. Time.

 

2 Figures B. l , B2 and B3 were plotted using the kinematic parameters shown in Table 3.2.

100

Generalized Force (N-m)

,§
8

 

 

350.00 ..
2‘ V
‘5 A 300.1!)
‘0‘ N 25000 ‘—
0 J) , on
> E a
g § 200.00
30 150.00 —
c O.
< 100.00 9
50.0) * __..
0.“) 3"
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32
time (seconds)
Figure 3.3: Angular Acceleration of the Generalized Coordinates vs. Time.
B.2 Plots of the Generalized Forces and Ma_ss Center Velocities vs. Time
120.00 0.300
100.00
0.200 _
80.00
a Q"
60.00 0.100 2' —
3' Qt:
40.00 E _
20.00 g (Total)
0 .__..
0.00 -o.100 5‘5 Qt,
-2000 (Rofitiona')
-0.2000 Q1

-40.00

 

-60.00 -0.300
0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

time (seconds)

Figure 8.4: Generalized Forces vs. Time.3

101

50.00

40.00

30.1!)

20.00

Mass Cemter Velocity (m/s)

10.00

 

0.”
0.00 0.13 0.04 0.“ 0.“ 0.10 0.12 0.14 0.16 0.18 0.20 02 0.74 0.3 0.28 030 0.32

time (seconds)

Figure 8.5: Mass Center Velocity vs. Time

B.3 Swing Plots of the Pendulum Positions for Three Impact Configurations

 

Figure 8.6: Swing Plot for Standard Impact Configuration.

102

 

Figure 8.7: Swing Plot for Intermediate Impact Configuration.

 

Figure 8.8: Swing Plot for Advanced Impact Configuration.

103

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[2]

[3]

[4]

[5]

[7]

l8]

[9]

[10]

[ll]

[12]

l13l

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105

 

 

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