FINITE ELEMENT ANALYSIS OF

FORCE TRANSFER FROM WRIST TO

ELBOW FOR A CRICKET PLAYER

Dissertation submitted to

Visvesvaraya National Institute of Technology, Nagpur In partial fulfilment of requirement

For the award of degree of

Master of Technology

in Computer Aided Design and Computer Aided

Manufacturing

By

Aayush Kant

Under Guidance of

Dr. P M Padole

Department of Mechanical Engineering Visvesvaraya National Institute of Technology

Nagpur 440010 (India)

2013

FINITE ELEMENT ANALYSIS OF

FORCE TRANSFER FROM WRIST TO

ELBOW FOR A CRICKET PLAYER

Dissertation submitted to

Visvesvaraya National Institute of Technology, Nagpur In partial fulfilment of requirement

For the award of degree of

Master of Technology

in Computer Aided Design and Computer Aided

Manufacturing

By

Aayush Kant

Under Guidance of

Dr. P M Padole

Department of Mechanical Engineering Visvesvaraya National Institute of Technology

Nagpur 440010 (India)

2013 Visvesvaraya National Institute of Technology(VNIT)

Department of Mechanical Engineering Visvesvaraya National Institute of Technology

Nagpur 440010 (India)

2013

CERTIFICATE This is to certify that project work entitled FINITE ELEMENT ANALYSIS OF

FORCE TRANSFER FROM WRIST TO ELBOW FOR A CRICKET

PLAYER is a bonafide work done by Mr Aayush Kant (Enrolment No

MT11CDM003), at Mechanical Engineering Department, Visvesvaraya National

Institute of Technology, Nagpur in partial fulfilment of the requirements for the award

of degree Master of Technology in CAD-CAM. The work is comprehensive,

complete and fit for evaluation.

P. M. Padole Project guide

Dept. of Mechanical Engineering

Forwarded by- Dr. I. K. Chopde Head Department of Mechanical Engineering VNIT, Nagpur Date:

Department of Mechanical Engineering Visvesvaraya National Institute of Technology

Nagpur 440010 (India)

2013

D E C L A R A T I O N

It is hereby declared that this dissertation, entitled FINITE ELEMENT

ANALYSIS OF FORCE TRANSFER FROM WRIST TO ELBOW FOR A

CRICKET PLAYER which is being submitted to Visvesvaraya National Institute

of Technology, Nagpur for the award of the Degree of Master of Technology in

Computer Aided Design and Computer Aided Manufacturing is a bonafide report

of the work carried out by me under the guidance of Dr P. M. Padole, Professor,

Mechanical Engineering Dept, VNIT, Nagpur. The material contained in this report

has not been submitted to any university or institution for the award of any degree.

Mr Aayush Kant (MT11CDM003)

Dept of Mechanical Engineering VNIT, Nagpur

Department of Mechanical Engineering Visvesvaraya National Institute of Technology

Nagpur 440010 (India) 2013

ACKNOWLEDGEMENTS

An equation means nothing to me unless it represents a thought of God

-Srinivasa Ramanujam

My first sense of deepest regard and heartfelt thankfulness is to the Supreme

Lord, for His innumerable gifts, for having defined my existence itself. He is to be

remembered and glorified at every moment.

A heartfelt acknowledgement and gratitude to my supervisor Prof P M

Padole who guided the whole work throught thick and thin, providing invaluable

assistance and constant encouragement, which saw the project to its completion. His

regular enquiries as to if I was enjoying myself kept me enlivened through the course

of the work. Most importantly I am grateful to him and his son for having lent their

own cricket bat for experiments, letting me see no signs of reservations from their

side for the same.

I would also like to thank Dr I K Chopde, Head of Mechanical Dept, VNIT,

Nagpur, for the facilities provided which were very much helpful to me. I would like

to thank Prof (Mrs.) Rashmi V Uddanwadikar, whose guidance in various subject

matters was very valuable. Also I would like to acknowledge the support shown by

Prof A B Andhare, who provided me with the setup for the modal analysis

experiments. Prof Prasad Kane also guided me in the same. A lot of help was

provided by Mr Chaitanya Sargaonkar, CAD lab, VNIT regarding software

support, allowing me to run many simulations on a system in the server room in much

shorter time than my personal laptop. My sincerest thanks to him.

I express my indebtness to my classmates who enlivened the two years I spent

with them completing my masters.

Most importantly I feel a deep sense of gratitude towards those special friends

with whome I stayed through these two years. They provided me a place to stay which

I would not hesitate to regard as my second home.

i

FINITE ELEMENT ANALYSIS OF FORCE

TRANSFER FROM WRIST TO ELBOW FOR A

CRICKET PLAYER

ABSTRACT

A number of games like cricket, tennis, baseball, etc have developed a lot due

to the extensive research done in the sporting equipments. The main aim of this

project was decided as the evaulation of stresses in the hands of a cricket batsman.

Very less literature has been found to attempt such an analysis, although it can be of

great use, like predicting the location of injury, predicting the performance of the

safety wear being used by the batsman, etc. Also one of the aims of this work is to

study in detail the variation in the ball exit velocity with respect to the impact location

on the blade. Finite Element Modeling is used as an approach to predict the exit

velocity of the ball. The three simulations with various velocities of bat and ball are

considered. The results confirm the existence of sweet spot, and indicate the same

location where minimum amplitude of vibration is expected. A study on the reaction

forces on the hand due to both the bat swing as well as ball impact is done. It is seen

that reaction forces are minimum for sweet spot impact. The load on the hand is

observed to be a dynamic load, occurring for a period almost 5 times the impact

period of ball. A study is also performed on the stress distribution in the hand of the

batsman, due to these reaction forces.

ii

LIST OF TABLES

Table 3.1: Time Period of Oscillations of Bat..............................................................16

Table 3.2: The Dimensions of Various Parts of Bat.....................................................18

Table 3.3: Values of the Parameters at Various Cross Sections..................................20

Table 4.1: Orthotropic Properties of Willow Wood.....................................................25

Table 4.2: Material Properties of Ball..........................................................................26

Table 4.3: Calibration of Bat Model............................................................................27

Table 4.4: Comparison of FEA and Experimental Results..........................................28

Table 4.5: Comparison of the Modal Results with Previous Authors..........................30

Table 4.6: Variation of BEV and Maximum Stress on Bat with Impact Location......35

Table 4.6: Variation of Ball Exit Velocity and

Maximum Stress on Bat with Impact Location (17-35 Impact)...........36

Table 4.6: Variation of Ball Exit Velocity and

Maximum Stress on Bat with Impact Location (40-40 Impact)...........37

Table 5.1: A Comparison of Important Parameters

of the Cricket and Baseball Bats..........................................................51

Table 5.2: Reaction Forces, 17 35 Impact at 0.2m From Shoulder...........................56

Table 5.3: Reaction Forces, 17 35 Impact at 0.25m From Shoulder.........................57

Table 5.4: Reaction Forces, 17 35 Impact at 0.3m From Shoulder...........................58

Table 5.5: Reaction Forces, 17 35 Impact at 0.35m From Shoulder.........................59

Table 5.6: Reaction Forces, 17 35 Impact at 0.4m From Shoulder...........................60

Table 5.7: Reaction Forces, 17 35 Impact at 0.45m From Shoulder.........................61

Table 6.1: Material properties for bone and ligament (ACL)......................................71

Table 6.2: Pressures for 0.3 m Impact location, 17 35 case......................................74

Table 6.3: Pressures for 0.35 m Impact location, 17 35 case....................................74

Table 6.4: Pressures for 0.4 m Impact location, 17 35 case......................................74

Table 6.5: The maximum stresses developed in the hand

varying with time after impact for various impact locations................79

iii

Table 6.6: Comparison of the stresses in various

parts of hand for three different impacts..............................................79

iv

LIST OF FIGURES

Figure 1.1: Nomenclature of a Cricket Bat....................................................................2

Figure 2.1: Bending Mode Shapes for Baseball Bat..............................................7

Figure 2.2: Location of Minimum Amplitude of Vibration ..........................................7

Figure 2.3: The Skeletal Structure of the Hand............................................................11

Figure 2.4: Internal Composition of Femur Bone........................................................12

Figure 3.1: Measurement of the Centre of

Gravity of the Bat (Inset is a zoomed view).........................................15

Figure 3.2 Measuring Moment of Inertia of Bat .........................................................16

Figure 3.3: Measuring the Coordinates........................................................................19

Figure 3.4: The Parameters Defining a Cross Section of the Bat.........................19

Figure 3.5: Bat Suspended With Strings......................................................................21

Figure 3.6: Close-up of the Sensor Setup.....................................................................21

Figure 3.7: The Accelerometer.....................................................................................22

Figure 4.1: Solid Model of Bat and Ball, Sketches Highlighted..................................25

Figure 4.2: Meshed Model of Bat and Ball..................................................................26

Figure 4.3: The First Two Bending Modes of Vibration of the Bat.............................28

Figure 4.4: The First Mode with One Node at the Constraint Point, at 9.102 Hz....29

Figure 4.5: The Second Mode with Two Nodes, at 117.92 Hz....29

Figure 4.6: The Third Mode with Three Nodes, at 538.78 Hz.....................................29

Figure 4.7: Location of Sweet Spot..............................................................................31

Figure 4.8: Impact Locations Considered for Simulation............................................33

Figure 4.9: BEV Graph for 0-35 Impact..................................................................... 35

Figure 4.10: Stress Graph for 0-35 Impact.................................................................. 35

Figure 4.11: BEV Graph for 17-35 Impact..................................................................36

Figure 4.12: Stress Graph for 17-35 Impact................................................................ 36

Figure 4.13: BEV Graph for 40-40 Impact................................................................. 37

Figure 4.14: Stress Graph for 40-40 Impact.................................................................37

v

Figure 4.15: Impact of Ball with a Stationary Bat and

Post-Impact Wave Propagation in the Bat...........................................38

Figure 4.16: Variation of COR with Speed for Two Different Balls...........................42

Figure 4.17: Comparison of RBM and FEA results for 0-35 impact...........................42

Figure 4.18: Comparison of RBM and FEA results for 17-35 impact.........................43

Figure 4.19: Comparison of RBM and FEA results for 40-40 impact.........................43

Figure 4.20: Rebound velocities obtained using experiments by David,et al..............44

Figure 5.1: Motion of A Bat.........................................................................................47

Figure 5.2: Plot of V and v/s Time...........................................................................49

Figure 5.3: Plot of resolved forces v/s Time................................................................49

Figure 5.4: Plot of net force and direction v/s Time....................................................49

Figure 5.5: Plot of torques and couple v/s Time..........................................................49

Figure 5.6: Modelling Variation to Find the Reaction Forces.....................................54

Figure 5.7: Graphical Results, 17 35 Impact at 0.2 m From Shoulder......................56

Figure 5.8: Graphical Results, 17 35 Impact at 0.25 m From Shoulder....................57

Figure 5.9: Graphical Results, 17 35 Impact at 0.3 m From Shoulder......................58

Figure 5.10: Graphical Results, 17 35 Impact at 0.35 m From Shoulder..................59

Figure 5.11: Graphical Results, 17 35 Impact at 0.4 m From Shoulder....................60

Figure 5.12: Graphical Results, 17 35 Impact at 0.45 m From Shoulder..................61

Figure 5.12: Graphical Results, 17 35 Impact at 0.45 m From Shoulder..................62

Figure 5.14: Graph of Impact Forces on the

Left Hand for All Impact Locations, 17 35 Analysis........................63

Figure 5.15: Plot of Maximum Forces Experienced

on Left and Right Hands v/s Impact Location.....................................64

Figure 5.16: Graph of Impact Forces on Right

Hand for All Impact Locations, 40 40 Analysis................................65

Figure 5.17: Graph of Impact Forces on Left

vi

Hand for All Impact Locations, 40 40 Analysis................................66

Figure 5.18: Plot of Maximum Forces Experienced on Left

and Right Hands v/s Impact Location (for 40 40 analysis)...............67

Figure 6.1: Beam Representation of the Forearm to Estimate the Stresses.................69

Figure 6.2: The Model of Hand....................................................................................71

Figure 6.3: Patch Independent Meshing at the Wrist...................................................72

Figure 6.5: Patch Independent Meshing at the Elbow..................................................73

Figure 6.6: Stress Distribution at Time t = 2 ms..........................................................75

Figure 6.7: Stress Distribution at Time t = 4 ms..........................................................75

Figure 6.8: Stress Distribution at Time t = 6 ms..........................................................75

Figure 6.9: Stress Distribution at Time t = 9.5 ms.......................................................76

Figure 6.10: Maximum Stress in Wrist Ligaments, at Time t = 6.5 ms.......................76

Figure 6.11: Maximum Stress in Elbow Ligaments, at Time t = 9.5 ms.....................77

Figure 6.12: Normal Bending Stresses for the Forearm Bones....................................77

Figure 6.13: Region of Stress concentration................................................................78

CONTENTS

ABSTRACT.i

LIST OF TABLES ........................................................................................................................... ii

LIST OF FIGURES ........................................................................................................................ iv

CONTENTS ................................................................................................................................ vii

CHAPTER 1 PROJECT OVERVIEW ........................................................................................ -1 -

1.1 INTRODUCTION ............................................................................................................ - 2 -

1.2 AIMS AND OBJECTIVES ................................................................................................. - 4 -

CHAPTER 2 LITERATURE REVIEW ....................................................................................... - 2 -

2.1 DYNAMIC BEHAVIOUR OF A BAT ................................................................................. - 6 -

2.2 FINITE ELEMENT MODELING OF BAT ........................................................................... - 8 -

2.3 REACTION FORCES ON THE WRIST............................................................................... - 9 -

2.4 MODELLING OF HAND ............................................................................................... - 10 -

2.5 WORK PROPOSED ...................................................................................................... - 13 -

CHAPTER 3 EXPERIMENTAL METHODOLOGY .................................................................... - 6 -

3.1 MASS AND CENTRE OF GRAVITY ................................................................................ - 15 -

3.2 MOMENT OF INERTIA ................................................................................................ - 15 -

3.3 COORDINATE MEASUREMENT ................................................................................... - 18 -

3.4 EXPERIMENTAL MODAL ANALYSIS ............................................................................. - 20 -

CHAPTER 4 FINITE ELEMENT ANALYSISBAT BALL IMPACT ........................................... - 15 -

4.1 BAT MODELLING ........................................................................................................ - 24 -

4.1.1 Procedure for Modelling ..................................................................................... - 24 -

4.1.2 Calibration of model ........................................................................................... - 27 -

4.2 MODAL ANALYSIS ....................................................................................................... - 28 -

4.2.1 Comparison with Experimental Results .............................................................. - 28 -

4.2.2 Comparison with Previous Authors .................................................................... - 29 -

4.2.3 Location of Sweet Spot ....................................................................................... - 30 -

4.3 IMPACT SIMULATION ................................................................................................. - 31 -

4.3.1 Procedure for Simulation .................................................................................... - 31 -

4.3.2 Verification of Impact Simulation ....................................................................... - 39 -

4.4 SUMMARY .................................................................................................................. - 45 -

CHAPTER 5 REACTION FORCES ON HAND ....................................................................... - 24 -

5.1 REACTION FORCES DUE TO BAT SWING .................................................................... - 47 -

5.1.1 Work Done by Rod Cross .................................................................................... - 47 -

5.1.2 Swing Force Calculations for a Cricket Bat .......................................................... - 50 -

5.2 FORCES DUE TO IMPACT ............................................................................................ - 54 -

5.2.1 Model Variations ................................................................................................. - 54 -

5.2.2 Discussion of Results for Forces on Hand ........................................................... - 55 -

CHAPTER 6 STRESSES INDUCED IN HAND ....................................................................... - 47 -

6.1 EXPECTED VALUE OF STRESSES .................................................................................. - 69 -

6.2 MODELLING OF THE HAND ........................................................................................ - 70 -

6.3 SIMULATION AND RESULTS ....................................................................................... - 71 -

CHAPTER 7 CONCLUSION ................................................................................................. - 69 -

APPENDIX I ........................................................................................................................... - 86 -

APPENDIX II .......................................................................................................................... - 88 -

REFERENCES ......................................................................................................................... - 82 -

CHAPTER 1

PROJECT OVERVIEW

- 2 -

1.1 INTRODUCTION

Cricket is a very popular bat-and-ball sport played in many countries all over

the world. It is believed to have originated inthe18th century in England. The main

equipment used in cricket is the cricket bat and ball. The laws of cricket limit the size

of the bat to not more than 96.5 cm in length and the blade to be not more than 10.8

cm wide at its widest part [1]. Cricket bats typically weigh from 1.1 to 1.4 kg though

there is no restriction on the weight of the bat [2].

The cricket bat consists of a handle, shoulder, blade and toe as shown in

Figure 1. Until recently, the quality of the cricket bat was based only upon the grain

structure of the face of the blade. The modern tendency is to use heavy bats since

most bats men believe that a heavier bat allows them to hit the ball farther. The blade

of the bat is commonly made of Kashmir or English willow [2].

Figure 1.1: Nomenclature of a Cricket Bat

Cricket is one of the sports, where the constantly evolving rules, do not allow

much scope for use of technological advancement to improve the game. For instance,

in baseball the bats can be either made up of solid wood or hollow aluminium barrel,

where latter gives an increased ball velocity upon impact. In 1979, when Dennis

Lillee walked onto the pitch using an aluminium bat, named the ComBat, a few

overs into the game the opposing English captain complained to the umpire that the

bat was damaging the ball. The bat was replaced with its orthodox wooden

- 3 -

counterpart. And soon a new rule was into place, the bat must be made of wood [3], [4].

These and many such constraints on the bat are aimed at protecting the integrity of the

game. However, within the parameters enforced by the rule, there is still a scope for

technological research to improve the design to give the batsmen an additional

advantage. It is a known fact that many of the players have a set of bats to be used for

test matches, and another set of bats to be used for one day matches and 20-20s. Other

than personal experience and preference, there is no method for the batsman to select

an appropriate bat for an appropriate version of the game, or situation within the same

game [5]. A design variation in the cricket bat will improve the efficiency of the shot

as well as reduce the effort the player has to put into playing the shot.

One of the parameters of paramount importance is the sweet spot which is

discussed in detail later on in the course of the work. Sweet spot is a location

primarily identified by the batsman as the best location on the bat with which the ball

can come in contact. Physicists and researchers alike have given various

interpretations of what is the sweet spot, three of them being considered as

simultaneously valid.

The purpose of this work is to study the parameters involved in characterising

a cricket bats performance, especially the location of the sweet spot, and try and find

out a relation between its three interpretations. The work will also focus its attention

on the forces involved in the process of swinging a cricket bat and hitting the ball.

Thus the total reaction forces on the batsmans hand will be studied and this will be

used for calculating the stresses developed in the hand during the course of playing

certain shots. . As far as the author could research, not much work has been done to

study the stresses developed in the hands of a batsman. This is surprising considering

the contribution such a study would lend in designing the bat, the handle of the bat,

the grip, and also study the probable causes of injury to a batsman. The work being

presented, in regard to the stress analysis of the hand, is quite elementary in its

assumptions, but at the same time presents a basic picture, which can be worked on

further, as a future scope.

- 4 -

1.2 AIMS AND OBJECTIVES

The main aims of this project are as follows:

1. Study the relationships between the three interpretations of sweet spot.

2. Study the variation of reaction forces of a cricket bat-ball impact on the

batsmans hand, with respect to the impact location of the ball on the bat.

3. Study the stress distribution on an approximate model of the batsmans hand,

and identify the location of maximum stresses.

These aims are accomplished using Finite Element Software, ANSYS Workbench.

Also the Finite Element Analysis (FEA) results are verified for certain simulations as

far as possible.

CHAPTER 2

LITERATURE REVIEW

- 6 -

2.1 DYNAMIC BEHAVIOUR OF A BAT

Majority of researches that were studied to gain a proper background on the

subject, were found to concern with various other sports like baseball, golf and tennis.

However these were also found to be of use, since all these sports involve and impact

of ball with a certain device, which varies from sports to sports. The dynamic details

of these devices may vary, but their basic principles and mechanics will remain the

same. Moreover, it is also observed that a baseball shot is very similar to a pull shot

in cricket. Similarly it can be seen that a golf drive is very similar to a straight drive of

cricket. Pull shot is a shot which involves the maximum energy of the batsman and

hence is a shot of interest.

Van Zandt [6] did a significant research on baseball. The standard theory of

beams was modified to consider the non-uniform section of a baseball bat. The

normal modes for bending vibrations of the bat were calculated and the collision

problem was solved by including the effect of vibrations of the bat. One interesting

result from this work is the calculation of ball exit speed as a function of impact

position along the length of the bat. The work also showed that impacts at any point

other than the node will yield a relatively low performance, thus establishing a region

which is called the sweet spot.

The sweet spot has three interpretations. The sweet spot is commonly known

as the location on the bat that produces maximum batted-ball velocity. Often, it is also

understood as the location on the bat that produces no sting to the batters hands. A

third interpretation of the sweet spot is the location on the bat where minimum

amplitude of fundamental vibrations is obtained. Physicists also think of the sweet

spot as the optimal location on the bat that produces best overall results.

A vibrational analysis of a bat helps to understand the region of minimum

amplitude of vibration. The Figure 2.1 shows first three bending modes of vibration

for baseball bat, as studied by Sutton and Sherwood [7]. However when the impact

occurs, it lasts for roughly 0.001s [8]. Thus only the first two modes of vibration are

activated. The higher modes of vibration therefore do not play much role in an impact.

It can be seen that the region between the second node of the first mode and the third

node of the second mode is the region where the amplitude of the vibration will be

- 7 -

minimum if both modes are excited simultaneously. This is because of the close

location of both these nodes as shown in Figure 2.2 [9].

Figure 2.1: Bending Mode Shapes for Baseball Bat

Figure 2.2: Location of Minimum Amplitude of Vibration

Nathan [10] established a very important principle, which has been used by

most of the researchers in future in their studies of impact of hand held instruments. It

was shown in this work, that any effect of clamping action of hands at the end of the

bat is felt at the impact point only after the ball leaves the bat. Therefore, for all

- 8 -

testing and modelling purposes, a free-free boundary condition of the bat is a

legitimate approach. A video demonstration of a baseball player hitting a home run,

even though his bat slips from his hand is found on the website of Nathan [11], which

establishes this claim further. The same conclusion is further claimed by Noble and

Eck [12].

Many researchers have studied the various interpretations of the sweet spot

and have tried to find a correlation between them. Hariharan and Srinivasan [13] have

carried out a similar investigation for a cricket bat and have found that the region of

minimum amplitude of vibration indeed the region where the batted ball-speed is the

maximum. This study was performed using FEM software ANSYS/LS DYNA, and

compared to modal analysis done using ANSYS.

David Theil, et al [14] in their research established the relationship between the

jerk a batsman experiences at his wrist to the location where the ball impacts on the

wrist. Using 3 axis accelerometers mounted on the bat and the wrists, ball strikes were

recorded for defensive drives along the ground. It is found that the sweet spot impacts

give low levels of vibrations in the wrist sensors.

2.2 FINITE ELEMENT MODELING OF BAT

Mustone et al [15] developed a finite element model to predict the performance of

baseball bats. Mass, moment of inertia and centre of gravity were calculated

experimentally. The sweet spot on the bats was obtained using a hitting machine.

Frequencies of the bats were obtained by conducting modal tests on the bats.

HyperMesh was used for building the geometries of the wood and aluminium bats,

LS-DYNA was used for analysis, and LS-Post and the ETA Postprocessor were used

for post-processing. The bat models were calibrated by comparing the mass, MOI,

centre of gravity and the vibrational properties (natural frequencies) of the bats with

experimental values. Differences between wood and aluminium bats in terms of

performance and contact time were studied. Two important conclusions from their

study were:

- 9 -

There was no difference in the batted-ball velocity when the bat was given a

purely rotation or purely translational motion towards the ball.

Modal Analysis was an effective tool for the calibration of FEA models to

predict their batted-ball performance.

2.3 REACTION FORCES ON THE WRIST

When the reaction forces on the hand of the batsman is to be considered, it is

important to understand that there are two independent causes of these reaction forces.

First cause is the impact of the ball on the bat, which results in a jerk on the wrist.

And second is the rotation of bat itself. Studies concerning themselves to finding out

these reaction forces are not much, but one researcher Rod Cross has done much

study in regards to both these subject matters.

In one of his works [9], Cross has experimentally studied the force waveforms,

using piezoelectric ceramic discs taped onto the outer surface of the baseball bat

handle. The experimental apparatus gave the force output not in terms of the force,

but a force waveform in Volts. Moreover, the experiment was conducted by swinging

the bat to a stationary, suspended baseball, with the ball not being struck with the

usual vigour normally associated with the game of baseball. In the words of Cross,

the results obtained by the experiment may not give quantitatively accurate results but

a qualitatively same result, since the physics is independent of the reference frame.

Yet, the qualitative conclusions obtained through the course of the study are worth

mentioning. The hand is found to exert a significant force on the bat even when the

ball has not left contact with the bat. The effect of this phenomenon is not much when

concerned with the motion of the ball, but it has significant effect on the behaviour of

the bat during and after the impact.

The dominant vibrations induced in a hand-held bat are heavily damped

versions of the fundamental and second modes of a free bat. The vibration amplitude

of both modes is sufficiently large to cause pain for any impact out-side the sweet

spot zone. The amplitudes of these modes are minimised for impacts at the respective

barrel nodes, but an impact at either one of these modes will excite the other mode.

- 10 -

An impact at the fundamental node is about optimum since the amplitudes of both the

fundamental and second modes remain small. For impacts outside the sweet spot

zone, the second mode amplitude is largest under the right hand (for a right-handed

batter) and smallest near the node under the left hand. The node of the fundamental

mode is also located under the left hand for a hand-held bat.

The sweet spot of a wood baseball bat was found to represent an impact zone

of width about 3 cm where the force and the impulse (i.e., the time integral of the

force) transmitted to the hands are both minimised.

In another work [16], interestingly done almost a decade later, using very

simple mechanics, Cross studied the force due to the bat swing motion. A right

handed player swinging a baseball bat of known configurations was filmed from the

top using a high speed camera. The almost spiral motion that the bat undergoes is then

studied by splitting the video into constituent frames, and analysing the velocity of the

bat. Using centripetal and tangential accelerations, the net force being exerted by both

the hands is obtained. The couple being applied by the two hands is obtained by

studying the angular acceleration and the torques acting on the bat. The net force

being exerted by both the hands and the couple are used to find out the net force being

exerted by each hand.

Due to the similarity of a baseball shot to a pull shot of cricket, as discussed

previously, it is possible to extend this technique to a pull shot and use the values

obtained by Cross, with small modifications, for a study of the swing of a cricket bat.

This has been discussed in detail in Chapter 5.

2.4 MODELLING OF HAND

The internal mechanical structure of the hand is not a simple one. Various

components like bone, muscles, ligaments and tendons are involved in providing a

mechanical support. The manners in which these components interact with each other

are also very complex. Through the skeletal system the bones form the backbone of

the hand, quite literally. The bones are further supported by muscles which act not

only like actuators, helping in the relative motion between the bones, but also

- 11 -

transmitting forces, as a good mechanical support system. The tendons are like

connecting linkages between the bones and muscles, having very complex geometries

and equally complex material composition. The ligaments perform the same functions

between bones. In order to reduce the complexity of the situation at hand, the focus

was directed towards only bones and ligaments.

The skeletal structure of the hand is shown in Figure 2.3. It shows the various

bones and their interactions. The complexity at the wrist is worth noting. The wrist

region along consists of 8 bones together called as the carpals. From these carpals

emerge the metacarpals which are located in the region of the palm. The metacarpals

lead to the digits, or phalanges, which are the portions of the five fingers. The carpals,

at the other end are connected to two bones which form the forearm of the human

arm. These two bones are called the radius and the ulna. The radius and ulna through

ligaments at the elbow joint are connected to the humerus, the long bone from the

elbow to the shoulders.

Figure 2.3: The Skeletal Structure of the Hand

Each bone in itself is a complex structure as shown in Figure 2.4. The bone consists of

various components. Primarily, the bone is an inhomogeneous composite material

having anisotropic material properties, depending not only on its composition but also

- 12 -

distribution within a structure. All bones have a dense cortical shell and less dense

cancellous inner component. The bone surface is surrounded by perisoteum, a

membrane providing a network of blood vessels and nerves. The bone, like any good

composite material, has strength higher than either of the two main components, the

softer component prevents the stiff one from cracking and the stiff one prevents the

softer one from yielding [17].

Figure 2.4: Internal Composition of Femur

The ligament is a fibrous band of soft tissue joining two bones of a joint. It has

three biomechanical functions, (i) to resist external load, (ii) to guide relative motions

of the two bones, and (iii) to control maximum range of joint motions. The ligaments

are known to buckle under compression and support no load. Therefore only tensile

properties are relevant [17]. The ligament is generally taken as a viscoelastic material,

absorbing more energy and requiring more force to rupture as the loading rate is

increased. Most of the material properties available in research papers are regarding

Anterior Cruciate Ligament, (ACL), present in the human knee.

- 13 -

2.5 WORK PROPOSED

On the basis of the literature survey done, a strategy is devised to accomplish

the goals of the project, as discussed in the previous chapter. This would comprise the

following steps:

(i) Experimentation to measure the characteristics of a cricket bat.

(ii) Generating a physical model of the bat, and calibrating it against the actual

bat using the characteristics experimentally obtained.

(iii) Performing a Modal analysis using FEA, and comparing the results to

experimental modal analysis. Also locate the sweet spot.

(iv) Using the calibrated model to simulate ball impacts at different locations

for different bat and ball velocities, thus verifying the sweet spot to be a

location which gives maximum batted ball velocity.

(v) Using the simulation to calculate the reaction forces on the wrist, verifying

that sweet spot impacts give minimum jerk to the batsman.

(vi) Using the technique developed by Cross to obtain the force required to

swing a bat.

(vii) Modelling the hand and simulating a stress analysis of the same.

CHAPTER 3

EXPERIMENTAL METHODOLOGY

- 15 -

3.1 MASS AND CENTRE OF GRAVITY

There are no laws which govern the mass of the bat, but usually bats used by

cricketers weigh around 1.1 to 1.4 kg. The mass of the bat was measured to nearest

0.001 kg using a digital weighing machine. The bat weighed 1.360 kg.

The centre of gravity is the point where the whole weight of a body is

supposed to act. It is possible to hold a body just by giving support at the centre of

gravity. Thus a simple way to locate the centre of gravity of any body is to try and

balance it on a thin edge. The point, over which the body can be balanced, is the

location of the centre of gravity. Figure 3.1 shows the bat being studied, balanced at

the centre of gravity. The centre of gravity was found to be located at a point 23 cm

from the shoulder of the bat.

Figure 3.1: Measurement of the Centre of Gravity of the Bat (Inset is a zoomed view)

3.2 MOMENT OF INERTIA

The moment of inertia by definition is the resistance to the angular

acceleration of an object. The moment of inertia of a body is dependent on the mass

of the body as well as the distribution of the mass along the length of the body. The

ASTM standards for measuring the moment of Inertia are defined for a baseball, in

the standard F1881. For measuring the moment of inertia the bat needs to be

suspended such that it is allowed to oscillate freely about its tip. For this purpose the

bat may be clamped at the handle.

- 16 -

In this study, the bat was tied using a thread of a very short length, so as not to

affect the time period. The bat is given a small angle rotation (less than 15 ), to allow

the small angle approximation to hold valid. Care has to be taken that bat undergoes

oscillation in its own plane, and does not oscillate like a conical pendulum. The

Figure 3.2 shows the readings being taken for a suspended cricket bat.

Figure 3.2 Measuring Moment of Inertia of Bat

Time taken for 20 oscillations is observed using stopwatch, and averaged over

three readings to reduce errors. The Table 3.1 shows the time periods for the three

readings, and their averaged value.

Table 3.1: Time Period of Oscillations of Bat Sr No. Time for 20 oscillations (s) Time period (s)

1 33 1.65

2 32 1.6

3 30 1.5

Average Time Period 1.58

- 17 -

The moment of Inertia of the bat can be calculated using the formula given in Eq 3.1.

=

(Eq 3.1)

Where,

Ipivot = Moment of Inertia about the pivot point,

T = Time Period,

Mbat = Mass of the bat,

g = Acceleration due to gravity, and

s = Distance of the pivot from the Centre of Gravity.

Substituting the values in the Eq 3.1, as,

T = 1.58 s

Mbat = 1.36 kg

g = 9.81 m/s2

s = length of handle + distance of CG from shoulder

= 0.32 + 0.23 = 0.55 m

Thus, the moment of inertia is obtained as,

=1.58 1.36 9.81 0.55

4

= 0.4640 kgm2

In order to calculate the moment of inertia about the centre of mass, the parallel axis

theorem can be used as stated in Eq 3.2.

= + (Eq 3.2)

Where, all symbols hold their normal meanings.

- 18 -

Thus, the moment of inertia about centre of mass is obtained as,

ICM = 0.4640 (1.36 0.552)

= 0.0526 kgm2

3.3 COORDINATE MEASUREMENT

In order to model the cricket bat, it is required to know the dimensions, and

coordinates of the critical points on the bat. For this purpose, the task of finding out

the coordinates of the bats was undertaken.

First of all, the lengths of various parts of the bat were measured. This has

been tabulated and shown in Table 3.2.

Table 3.2: The Dimensions of Various Parts of Bat Length of handle 0.255 m

Length of Shoulder curve 0.042 m

Length of blade of bat 0.533 m

Length of toe 0.02 m

Total length of bat 0.85 m

Diameter of Handle 0.036 m

Width of the blade (assumed constant) 0.106 m

This was done by keeping the bat on a flat surface, above a sheet of drawing

paper, and measuring the heights of the points on the surface using a set square, by a

simple geometric procedure. The sheet of paper was used for directly recording the

measurements, such that a projection image to the scale of 1:1 was obtained on it. The

sheet is shown in Figure 3.3.

- 19 -

Figure 3.3: Measuring the Coordinates

Measurements were made considering the top edge of blade as the origin. X axis was

considered as being along the blade, Y axis as perpendicularly out of the sheet, as

shown in Figure 3.3. At different values of X coordinates, the parameters defining the

cross section of the bat were measured. These parameters are a, b and c, as

shown in Figure 3.4. Parameter c was used to calculate the height of the top central

spline, h, using the formula given in Eq 3.3

= + 5.3 (Eq 3.3)

Figure 3.4: The Parameters Defining a Cross Section of the Bat

The Table 3.3 shows the data obtained through this procedure:

X axis

Z axis

width/2

- 20 -

Table 3.3: Values of the Parameters at Various Cross Sections X A b c H

0 1.4 2.7 5.5 4.2

10 1.8 3.6 5.6 5.4

20 1.9 4.2 5.8 6.6

22.5 1.8 4.4 5.7 6.5

25 1.8 4.5 5.7 6.6

27.5 1.7 4.6 5.7 6.7

30 1.7 4.8 5.8 7.2

32.5 1.7 4.7 5.8 7.1

35 1.7 4.7 5.8 7.1

37.5 1.6 4.7 5.8 7.1

40 1.5 4.6 5.7 6.7

42.5 1.4 4.4 5.6 6.2

45 1.4 4.1 5.5 5.6

47.5 1.1 3.6 5.5 5.1

50 1 3.1 5.4 4.1

52.5 0.7 2.6 5.4 3.6

At each section, 4 data points have been considered. They are the four vertices

of the quadrilateral shown in Figure 3.4, which defines a cross section of the bat. The

coordinates of these four data points can be easily obtained using the parameters. The

x coordinate is directly available. Z coordinate is zero for the mid-section data points

and equal to half the width (i.e. 0.053 m) for the edge data points. And Y coordinates

are either a, b or h depending on the location of the data point. Thus for each section

coordinates of four data points are obtained.

The coordinates of the data points for the cylindrical handle are easily

obtained using the width and length mentioned in Table 3.2.

3.4 EXPERIMENTAL MODAL ANALYSIS

Modal analysis is a procedure, which describes a structure in terms of its

dynamic characteristics. The procedure involves inducing vibrations in a structure by

applying a dynamic load to it. This can be done by hitting it with an impact hammer

or by using a shaker. The response of the structure is recorded as a time trace using an

accelerometer. In Modal Analysis, this time-domain response is then transformed into

the frequency domain by using Fast Fourier Transformation (FFT), and the Frequency

- 21 -

Response Functions (FRFs) are computed. The FRFs show the response of the

structure in terms of the frequency domain. The FRFs also have peaks in the

frequency plot which correspond to the peaks in the time trace. These peaks

correspond to the natural frequencies of the structure. Therefore, the FRFs can be

used to obtain the natural frequencies of the structure directly. Also by obtaining

many FRFs along the length of the structure, it is possible to obtain the mode shapes,

and hence the location of the nodes of the structure directly.

However, in this study, the purpose of experimental modal analysis is to use it

as a tool in order to verify the model. Numerical modal analysis using FEA can be

used to obtain the mode shapes and node locations.

The bat under consideration is suspended using strings at the end. This

maintains a free-free boundary condition. The accelerometer is placed near the blade

end of the bat. Impacting is done using a hammer at several locations, keeping the

accelerometer at the same location. Figure 3.5 and 3.6 show the setup for the modal

analysis.

Figure 3.5: Bat Suspended With Strings.

Figure 3.6: Close-up of the Sensor Setup.

Magnetic pick up

Accelerometer Sensor

- 22 -

The accelerometer sensor must be bound tightly to the surface of the object being

analysed, otherwise, the vibrations will not be transferred properly to the sensor, and

it will give erroneous results. For this purpose the manufacturer provides a magnetic

pickup on which the sensor can be mounted through a screw arrangement. This

ensures a proper transmission of vibrations. The magnetic pickup can be easily

mounted on metallic machine parts, but not on a wooden bat. For this reason, a small

steel washer was bonded on the bat using a strong adhesive, in this case commercially

available Quickfix. The magnetic pickup was mounted on it, and sensor was fixed to

the magnetic pickup.

The frequencies are measured up to a span of 800 Hz. The spectrum readings

on the screen are observed. The cursor is moved to the peak of the FRF. The cursor

directly gives the frequency at the point. This is noted to be the first bending mode of

vibration for the cricket bat. The Figures 3.7 and 3.8 show the accelerometer and the

screen recordings.

Figure 3.7: The Accelerometer

Figure 3.8: Screen Readings

The first bending mode of vibration is obtained at 225 Hz.

CHAPTER 4

FINITE ELEMENT ANALYSIS BAT-BALL IMPACT

- 24 -

4.1 BAT MODELLING

4.1.1 Procedure for Modelling

Using the data points calculated in the last chapter, it is possible to obtain a

solid model of the bat. This sub-section delineates the procedure which is used for

obtaining such a model.

The modelling and analysis was carried out on the commercially available

software ANSYS Workbench 14.5. ANSYS Workbench 14.5 provides a platform on

which various analyses can be done, and even combined with each other. ANSYS

workbench combines the strength of the core product solvers of ANSYS with the

project management tools necessary to manage the project workflow. In ANSYS

Workbench, analyses are built as systems, which can be combined into a project. The

project is driven by a schematic workflow that manages the connections between the

systems.

The solver module used for this work was ANSYS Explicit Dynamics, which

uses an AUTODYN solver. In dynamic problems, always an explicit solver is

preferred over an implicit solver. This has certain reasons. The differences between

implicit and explicit solver is discussed separately in Appendix 1.

An Explicit Dynamics analysis system provides its own modelling software

called ANSYS Design Modeller. The modelling of the cricket bat was done using

ANSYS Design Modeller.

Using the data points calculated a sketch profile is created for the middle

longitudinal section of the bat. A similar profile is created for the edge of the bat. The

two profiles are combined to create the volume of half of the bat, using Skin/Loft

feature. A solid with its top and bottom surfaces fitting through both the profiles is

obtained. This solid is mirrored about the central section to create the complete blade

of the bat. Proper dimensions are then used to create the handle of the bat.

Modelling of ball is done as a simple sphere, with radius 3.58 cm. The value

for the radius of the ball was selected such that its mass and circumference lie within

- 25 -

the range defined by the laws [18]. Figure 4.1 shows the solid models of bat and ball.

The two sketches which were made using the data points are highlighted.

Figure 4.1: Solid Model of Bat and Ball, Sketches Highlighted

The solid model once created is used for meshing. The meshing operation is

performed using ANSYS Mechanical. Prior to meshing operation, it is required to

define the material models which will be used for defining the bat and ball.

A cricket bat is typically made of willow wood. The cricket bat being used

was specifically a Kashmir willow bat. Wood is a linear orthotropic elastic material.

So its material properties are different in the three principal directions, longitudinal,

radial and transverse. Moreover wood being a natural material, its density varies from

species to species. The orthotropic properties used are listed in Table 4.1.

Table 4.1: Orthotropic Properties of Willow Wood Density (kg/m3)

Young's Modulus (GPa)

Poisson's Ratios Shear Modulus

(GPa)

Ex Ey Ez xy yz xz Gxy Gyz Gzx

650 13.3 0.883 7.06 0.015 0.6 0.16 1.33 0.133 1.33

To define the material properties of ball a simple elastic model is not sufficient. There

is a time-dependent energy loss associated with deforming of ball. Thus to define the

- 26 -

dynamics of mechanical behaviour during and after contact, a visco-elastic material is

defined. The visco-elastic material is associated with a time dependent shear modulus

and a constant bulk modulus, K. Hence it represents a strain rate dependent elastic

behaviour. Eq 4.1 shows the time dependency of the shear modulus.

() = + ( )() (Eq 4.1)

Where,

G = Long term shear modulus,

G0 = Instantaneous (short term) shear modulus,

= viscoelastic decay constant.

The material properties used to define the ball are summarized in Table 4.2.

Table 4.2: Material Properties of Ball

Density (kg/m3) 814

Instantaneous shear modulus (MPa) 41

Long term shear modulus (MPa) 11

Bulk Modulus (MPa) 69

Decay Constant (s-1) 10500

The meshing is done using patch conforming method for 4 node tetrahedral elements.

The bat is meshed using 2203 elements and the ball using 831 elements. Figure 4.2

shows a meshed model.

Figure 4.2: Meshed Model of Bat and Ball.

- 27 -

4.1.2 Calibration of model

In order to check whether the model represents the actual bat closely, certain

parameters of the model need to be checked against the actual bat. This process is

called as modelling. In chapter 3, these parameters were defined and estimated using

experimental analyses for the actual bat. These parameters are the mass of the bat,

location of centre of gravity of the bat, the mass moment of inertia of the bat and

natural frequencies.

The values of these parameters for the model are recorded and compared with

experimental values. Although a mention of the first mode of vibration is done, a

detailed procedure of modal analysis and the resulting conclusions on the validity of

the model are explained in the next section.

A comparison between the calibration parameters for the model and the actual bat are

presented in Table 4.3.

Table 4.3: Calibration of Bat Model

Property Unit Experimental

value FEA value

Mass (kg) 1.360 1.347

Location of Centre of mass

(mm) from the shoulder

230 241

Moment of inertia (kgm2)

about centre of mass

0.0526 0.044

First mode of vibration (Hz)

Free-Free boundary 223 218.45

It can be seen that the model gave good results when compared with experimental

values. Thus, the model is validated.

- 28 -

4.2 MODAL ANALYSIS

4.2.1 Comparison with Experimental Results

A very important means of calibration of the model is modal analysis. The

experimentally measured fundamental bending frequencies are compared to those

obtained using FEA.

In the workbench schematic, a new system for modal analysis is generated

with the same geometry and material properties as earlier generated for explicit

dynamic analysis.

The same model is used. A free-free boundary condition is simulated by not

giving any constraints to the model. Ten modes are extracted within the frequency

range of 0 2000 Hz. Out of the ten obtained mode shapes, first six mode shapes will

give 0 frequency. That is because these represent the six possible rigid body motions,

three displacements and three rotations about the principal axes. The next frequencies

indicate the bending modes about XY plane and about XZ plane. Out of these the

bending modes about XY plane are of interest for this case. Hence, the first two

frequencies of the modes of vibration are obtained as 218.45 Hz and 733.54 Hz.

Figure 4.3 shows the first two bending modes about XY plane, and Table 4.4

compares the FEA and experimental results for the fundamental mode.

Figure 4.3: The First Two Bending Modes of Vibration of the Bat

Table 4.4: Comparison of FEA and Experimental Results.

FEA analysis Experimental analysis

218.45 Hz 225 Hz

218.45 Hz

733.54 Hz

Node 1 Node 2

Node 1

Node 2

Node 3

- 29 -

4.2.2 Comparison with Previous Authors

A few of the previous authors who have done a modal analysis for cricket bat

have used a fixed-free condition, like a cantilever beam. Although, it is understood by

many authors that a free-free condition is to be considered for modelling purposes, a

few are still of the belief that proper modal results will be given for a cantilever

boundary condition. For the sake of checking the modal results obtained, a fixed-free

modal analysis is performed, to compare the results with these authors.

The top surface of the cylindrical handle is given a fixed support. Again ten

modes are extracted within the same range of frequencies. The three mode shapes

obtained are shown in figures 4.4 to 4.6.

Figure 4.4: The First Mode with One Node at the Constraint Point, at 9.102 Hz

Figure 4.5: The Second Mode with Two Nodes, at 117.92 Hz

Figure 4.6: The Third Mode with Three Nodes, at 538.78 Hz

The results are compared with two authors, John and Li [2], and Hariharan and

Srinivasan [13]. The comparison is shown in Table 4.5.

- 30 -

Table 4.5: Comparison of the Modal Results with Previous Authors

MODE FEM model

John & Li

Hariharan

1 9.1018 13.44 -

2 117.92 125.67 164.228

3 538.78 474.26 665.963

4 993.3 842.45 -

Although differences can be seen, it is evidently due to the different

geometrical models used by different researchers. The modal results can change

easily, even with slight changes in the mass distribution of the model.

4.2.3 Location of Sweet Spot

As has been discussed in the Chapter 2, section 1, the location of the sweet

spot can be obtained as the location of minimum amplitude of vibration. If there is a

single mode of vibration this would be the nodes of the mode. But in general more

than one mode is excited at a time. In a cricket bat, it has been seen that the first two

modes of vibration can be expected.

An impact of ball on a cricket bat lasts for 0.001 s. Therefore, the

corresponding frequency is calculated as,

=1

=1

0.001= 1000

Thus an impact of duration of 0.001 s can excite a natural frequency of at most

1000 Hz. All vibrations at a higher frequency would not be allowed to happen, since

by the time they return to the impact location, the ball has still not lost contact with

the bat. So the ball absorbs the vibration. However, the lower frequency vibrations

would have a larger time period, and hence by the time they would return to the

impact location, the ball would have lost contact, and the bat would be free to vibrate.

This is the reason why higher modes of vibration are not considered.

So it can be understood that the modes of vibration which are important in this

analysis are the modes which are lower than 1000 Hz, as calculated above. This

- 31 -

allows only the first two bending modes of vibration. Hence it can be calculated that

when the bat vibrates after an impact with the ball, it vibrates only on the first two

bending modes.

When these two modes are simultaneously activated, the region of the

minimum vibration can be seen to be the region between the closest located nodes of

the two vibrating modes. Figure 4.7 shows this region for the cricket bat model which

is being analysed. This is the region between the second node of the first mode and

the third node of the second mode.

Figure 4.7: Location of Sweet Spot

Thus the location of the sweet spot can be identified as between 32.4 cm to

38.8 cm from the shoulder of the bat.

4.3 IMPACT SIMULATION

4.3.1 Procedure for Simulation

The modelling of bat and ball are already done. The model was made in

Design Modeller. The meshing was also already done in ANSYS Mechanical. The

next step is to give the boundary conditions, initial conditions and solver settings.

The simulation is decided to be done for three conditions. The first condition

will be when the bat is considered stationary, and ball impacts on it with a velocity of

35 m/s. Throughout this work, this situation will be referred to as a 0-35 impact.

Mode 2

Mode 1

32.4 cm 38.8 cm

Shoulder of Bat

- 32 -

However, this is not a realistic situation. In a realistic situation, both bat and ball are

moving towards each other with certain velocity.

In this work, an analysis of Pull shot is attempted. A pull shot is a shot in

cricket where the batsman rotates his bat in an almost semi-circle, actually a spiral, to

impact the incoming ball. For a pull shot to be played the ball must be a short delivery

coming towards the batsman on the leg side, at an almost waist height or slightly

above. Such a delivery if left untouched by the batsman would mostly be declared a

wide. However, during situations which require aggressiveness, a properly executed

shot would pay off more than leaving it. The ball will be impacted a little after the bat

is in front of the batsman. This will direct the ball over the square leg, usually for a

boundary, or six. The ball may also be directed in between the square leg and fine leg,

with a slightly delay in executing the shot. The shot is very similar to a baseball shot,

which has been taken as the prototype for analysis.

It has already been discussed in the Chapter 2, Section 2, that a bat given a

linear velocity, as well as a bat given a rotational velocity does not produce difference

in the batted ball velocity. So a second simulation is done with bat given a linear

velocity of 17 m/s, and ball a linear velocity of 35 m/s. This situation is referred in

this work to as a 17-35 impact. In reality, a cricket bat will have velocities around this

value. Ball velocity will depend on the bowler, but usually medium pacers will bowl

around 35 to 37 m/s. A fast bowler may deliver bowls up to 39 to 42 m/s. So the

second situation, the 17-35 impact is a realistic situation. In order to verify the results

with a previous author, a third simulation, which is not very realistic is done. In this

simulation the bat is given velocity of 40 m/s and ball also 40 m/s. This is referred to

as a 40-40 impact.

It is to be noted, that as such no clear data is available on the bat velocity at

the time of impact in cricket. However, an analysis on the bat swing in a baseball

shot, done by Cross [16], the bat velocity is graphically shown to reach a maximum of

almost 17 m/s towards the end of the swing. Therefore this value has been taken for

analysis as a realistic value.

In each of the three situations which are to be simulated, there is a further sub-

division done based on the impact location. In each case the ball is impacted on the

- 33 -

blade of the bat at varying distances from the shoulder of the bat. Throughout the

remaining work, the impact location is measured as distance from the shoulder of the

bat. The impact location is varied from 20 cm to 45 cm in steps of 5 cm. Thus impacts

are simulated at 20 cm, 25 cm, 30 cm, and so on till 45 cm from the shoulder of the

bat. Figure 4.8 shows the various impact locations considered for simulation purposes.

Figure 4.8: Impact Locations Considered for Simulation

As has already been discussed, a free-free simulation of bat has been

considered as a legitimate approach by many researchers, to simulate an impact

between ball and bat. This is mainly because a cricket bat is not held as firmly by the

batsman to consider a clamped boundary condition. The hands allow the bat to vibrate

though they do damp the vibrations much faster, and a free bat.

Initially the ball and the bat are kept at a small distance apart. This ensures that

no automatic contact is generated by the system. As the simulation progresses, the

ball will move towards the bat, and at a point of time a contact will be generated

between the two bodies. This contact needs to be modelled. Hence an auto-detection

of contact is switched on for a face/face contact. The contact detection algorithms

allowed are of two types, trajectory or proximity based. Proximity based algorithms

encapsulate the external region by contact detection zone. If a node of another body

enters into this zone, a penalty based force is applied to repel it. However an

additional constraint is applied to the analysis time step when this contact detection

algorithm is selected. The time step is constrained such that a node cannot travel

through a fraction of the contact detection zone size in one cycle.

Origin

20 cm 45 cm

- 34 -

Trajectory based contact detection, on the other hand, does not impose any

constraint on the analysis time step and therefore often provides the most efficient

solution. The trajectory of nodes and faces included in frictional or frictionless contact

are tracked during the computation cycle. If the trajectory of a node and a face

intersects during the cycle a contact event is detected.

Trajectory based contact detection gives two choices for the formulation

algorithm of the contact, decomposition response or penalty. The decomposition

response algorithm can give rise to large hourglass energies and result in energy

errors. A penalty based approach detects a contact and responds by applying a local

penalty force to push the node, which would have penetrated, back to the surface. In

process the linear and angular moment is always conserved. But the kinetic energy

may not be conserved. A penalty based algorithm is more suited for simulating the

contact between a bat and ball. A frictionless contact type is selected, to allow some

sliding between the ball and bat.

The ball is given an initial velocity of 35 m/s, and bat is considered stationary

for the 0-35 impact simulation. For other simulations also the initial conditions are

appropriately defined by defining the proper velocities for bat and ball.

An average nodal Tet-Pressure integration is selected as it does not exhibit

volume locking and allows large deformation. An exact hex integration is selected

over as it allows for warped elements. This is done to allow for the deformation in the

ball upon impact. A 1pt Gauss hex integration gives hourglassing errors on

encountering warped faces. Damping controls are set at default values, as is

recommended.

The impact of ball on bat lasts for 0.001 s. So the solution time is allowed to

run to 0.004 s. This is done in order to see the vibrational effects in the bat, and the

exit velocity of the ball.

The simulation is done for all impact locations in all three situations of velocities.

Tables 4.6 to 4.8 show the exit velocity of the ball and resulting maximum stresses in

the bat for each of the three situations for each impact location. Figure 4.9 to 4.14

show the line graph of the corresponding tables preceding them.

- 35 -

Table 4.6: Variation of BEV and Maximum Stress on Bat with Impact Location

Impact Location (0-35 impact)

Ball Exit Velocity Maximum Stress

(m) (m/s) (MPa)

0.2 10.36 66.9

0.25 13.71 65.2

0.3 15.62 48.2

0.35 16.16 25.8

0.4 13.53 47.8

0.45 4.81 68.15

4

6

8

10

12

14

16

18

0.2 0.25 0.3 0.35 0.4 0.45

Vel

ocit

y (m

/s)

Impact Location (m)

Ball Exit Velocity

BEV

20

30

40

50

60

70

80

0.2 0.25 0.3 0.35 0.4 0.45

Str

ess

(MP

a)

Impact Location (m)

Max Stress

Max Stress

Figure 4.9: BEV Graph for 0-35 Impact

Figure 4.10: Stress Graph for 0-35 Impact

- 36 -

Impact Location Ball Exit Velocity Maximum Stress

(m) (m/s) (MPa)

0.2 31.463 99.89

0.25 31.8855 81.95

0.3 33.251 58.006

0.35 33.1925 42.69

0.4 30.3225 58.24

0.45 23.5715 87.77

20

22

24

26

28

30

32

34

0.2 0.25 0.3 0.35 0.4 0.45

Vel

ocit

y (m

/s)

Impact Location (m)

Ball Exit Velocity

BEV

Figure 4.11: BEV Graph for 17-35 Impact

35

45

55

65

75

85

95

105

0.2 0.25 0.3 0.35 0.4 0.45

Vel

ocit

y (m

/s)

Impact Location (m)

Max Stress

Max Stress

Figure 4.12: Stress Graph for 17-35 Impact

Table 4.6: Variation of Ball Exit Velocity and Maximum Stress on Bat with Impact

Location (17-35 Impact

- 37 -

Impact Location Ball Exit Velocity Maximum Stress

0.2 58.725 157.3

0.25 60.18 145.7

0.3 61.11 102.5

0.35 62.11 76

0.4 59.17 92.7

0.45 47.79 139.1

45

47

49

51

53

55

57

59

61

63

65

0.2 0.25 0.3 0.35 0.4 0.45

Vel

ocit

y (

m/s

)

Impact Location (m)

Ball Exit Velocity

BEV

60

70

80

90

100

110

120

130

140

150

160

0.2 0.25 0.3 0.35 0.4 0.45

Str

ess

(MP

a)

Impact Location (m)

Max Stress

Max Stress

Figure 4.13: BEV Graph for 40-40 Impact

Figure 4.14: Stress Graph for 40-40 Impact

Table 4.6: Variation of Ball Exit Velocity and Maximum Stress on Bat with Impact

Location (40-40 Impact)

- 38 -

Fig

ure

4.1

5: I

mpa

ct o

f B

all

wit

h a

Sta

tion

ary

Bat

and

Pos

t-Im

pac

t W

ave

Pro

paga

tion

in

the

Bat

- 39 -

Figure 4.15 shows some of the frames of a simulation of 0-35 impact. It can be

seen from all the simulations that for sweet spots impacts, the exit velocity of the ball

is maximum and the stresses in the bat are minimum. From these graphs it can be

estimated that the sweet spot location is around 0.35 m from the shoulder of the bat.

This is very much in agreement with the range of 32.4 to 38.8 cm obtained using

modal analysis.

4.3.2 Verification of Impact Simulation

In order to put faith in the values obtained using FEA, an independent analysis

needs to be done in order to corroborate the results. Two methods of verification are

used. One is an independent analysis, and another is verification by referring to

previous authors.

Till now the mechanics of the impact of bat and ball are being considered

keeping the dynamic nature of the bodies into consideration. This is the actual

situation. The ball experiences a lot of compression at the time of impact, and also the

bat vibrates. There is another technique in mechanics which is called rigid body

analysis. Rigid body analysis considers the bodies to be rigid in nature. Thus no

consideration of vibrations or compressions in structure is made.

The validity of rigid body analysis has been checked by many researchers

previously. Cross [19] studied the model for a baseball bat. A rigid body model will

work best if the duration of the collision is greater than the vibration period of the

fundamental mode of the freely supported beam. This is because in such a situation by

the time the vibration returns to the location of impact, the impacter would still not

have left the contact with the beam. Hence, the energy of vibration is given back to

the impacter. Thus the beam beahves almost like a rigid body. However this is not

true for a cricket bat and ball, since the duration of impact is shorter than the time

period of two of the modes of vibrations. Hence the bat vibrates. Thus Cross explains

that the rigid body model should be followed by independent estimates of energy

stored in the vibrational modes. An experimentalist could measure the amplitudes of

the various modes after the collision is over, calculate the energy in the modes, and

then use the conservation equations to calculate the rebound speed of the ball.

However, it would be simpler to measure the rebound speed directly. In this sense, a

- 40 -

rigid body model of the collision is of limited use in determining the rebound speed of

the ball.

Cross suggested measuring the vibrational energy and then using it to work

out the conservation of energy. But another way of doing the same thing is by

estimating a proper coefficient of restitution and using that coefficient of restitution to

conserve energy.

Precisely the same procedure has been explored by David, et al [5]. A

procedure of conservation of linear and angular momenta, as well as the coefficient of

restitution is used to define the exit velocity of the ball after impacting the bat. A

detailed discussion on the derivation of the mathematical model can be seen in

Appendix II. The ball exit velocity as derived using the model is described by Eq 4.2.

v =

()()

(Eq 4.2)

Where,

v1a = Velocity of ball after impact,

v1b = Velocity of ball before impact,

v2b = Velocity of bat before impact,

m1 = Mass of ball,

m2 = Mass of bat,

B = Impact location of ball from the Centre of mass of bat,

I0 = Moment of Inertia about center of mass,

e = Coeffiecient of Restitution, and

2b = Angular Velocity of the bat before impact.

- 41 -

Out of these all constants for this analysis are known, except for Coefficient of

Restitution. This needs to be found, and requires a little research into previous

literature.

David, et al [5], determined the value of coefficient of restitution using

experimentation. A bowling machine was used to launch a cricket ball on a stationary

cricket bat, and high speed video cameras were used to determine the inbound and

rebound velocities of ball and the recoil velocity of bat. The coeffient of restitution

was determined using its definition, as described by Eq 4.3.

=

(Eq 4.3)

The values of coefficient of restitution (COR) obtained were 0.511, 0.519 and

0.509 for three different bats. However the authors declared their procedure of

measurement of COR as erraneous. It was cited that the value of COR will vary along

the length of the bat. Hence the measured value of COR being at the center of mass of

the bat, would be lower to the value at the impact points.

However another indepenedent research in the course of an MS degree

by Singh [20], estimated the COR of a cricket ball using experimentation. Singh

explained that the COR of a ball decreases with increased speeds of impact. This is

also studied by Chauvin and Carlson [21] for baseball and softballs. The thesis by

Singh yields a graph which shows the variation in COR for two different balls at

different speeds. The graph is reproduced just for representational purpose in this

work properly referenced to original author, as Figure 4.16. The present work

concerns itself with ball impacting the bat at velocities upto 35 m/s 40 m/s (i.e. 78.3

mph 89.5 mps). The value for COR assumed in this work is directly read from the

graph. Moreover, the variation of the COR with impact location is not considered for

simplicity, as the entire procedure is just a verification tool, not an actual analysis.

The value of COR taken in this work is 0.45.

- 42 -

Figure 4.16: Variation of COR with Speed for Two Different Balls [20]

Substituting the constants thus derived into Eq 4.2, the values of the exit

velocity of the cricket ball after impacting the bat, at various locations along the

length of the bat, are obtained. The values obtained using Rigid Body Model (RBM)

are graphically compared with the results obtained using FEA in Figures 4.17 to 4.19,

for 0-35, 17-35 and 40-40 impacts respectively. It can be seen from the graphs that the

comparison between the two analytical models is good. Thus the FEA results are

verified.

Figure 4.17: Comparison of RBM and FEA results for 0-35 impact

4

6

8

10

12

14

16

18

0.2 0.25 0.3 0.35 0.4 0.45

Vel

ocit

y (m

/s)

Impact Location (m)

FEM

RBM

- 43 -

Figure 4.18: Comparison of RBM and FEA results for 17-35 impact

Figure 4.19: Comparison of RBM and FEA results for 40-40 impact

45

50

55

60

65

70

0.2 0.25 0.3 0.35 0.4 0.45

Vel

ocit

y (m

/s)

Impact Location (m)

FEM

RBM

22

24

26

28

30

32

34

0.2 0.25 0.3 0.35 0.4 0.45

Vel

ocit

y (m

/s)

Impact Location (m)

FEM

RBM

- 44 -

Once the FEA results are verified against Rigid Body Model, the next step is

verifying the results with the previous authors. Some of the authors have previously

by various methods estimated the ball exit velocity for an impact of ball on a cricket

bat.

David James, et al [5], performed experiments to determine the rebound

velocity of ball upon impacting a stationary cricket bat. The ball was fired using a

bowling machine at a fairly constant speed, such that the impacting velocity would be

27.7 m/s. This is fairly lower than the velocity of 35 m/s considered for this work. Yet

the results are quite comparable. Hence a comparison is made to verify the 0-35

impacting situation. The Figure 4.20 shows the graph obtained by David, et al. The

spots indicate an experimental results, and the continuous line indicates the RBM

prediction. The range of rebound velocities are 7 m/s 12 m/s. This is quite close to

the range of 5 m/s 16 m/s, obtained using FEA in this work. The results obtained in

this work are higher, as expected, since the initial ball velocities are higher.

Figure 4.20: Rebound velocities obtained using experiments by David,et al [5]

Hariharan and Srinivasan [13], used FEA to simulate a 40-40 impact. The

rebounding velocities obtained by them are in the range of 70m/s 80m/s. The

present work for a 40-40 impact simulation has given the results of rebounding

velocities in the range of 50 m/s 65 m/s. The results are different due to the

differences in the bat being considered.

- 45 -

4.4 SUMMARY

The chapter was aimed at modelling a cricket bat and simulating the impact of

cricket ball on the bat. The modelling was done using ANSYS Workbench. The

model was prepared, and relevant material properties for bat and ball were applied.

The model was calibrated with the experimental results. A free free modal analysis

was performed to obtain the natural frequencies of the bat. This was verified against

experimental modal analysis to calibrate the model. Also a verification was done by

comparing the results for a fixed free modal analysis against the results obtained by

previous authors. The modal analysis was also used to estimate the location of sweet

spot.

The impact of ball on the bat was simulated. The exit velocities of the ball

were plotted against the impact locations, for three different simulations, consisting of

three different initial conditions. The results were verified using Rigid Body Method.

Also a verification was done by comparing the results with the results obtained by

previous authors. The verifications could confirm the validity of the model.

CHAPTER 5

REACTION FORCES ON HAND

- 47 -

5.1 REACTION FORCES DUE TO BAT SWING

When a bat is swung by the batsman, the speed with which the bat is swung

determines the force that the bat exerts on the hand. Determination of this force can be

done by a simple mechanical analysis of a swing process. This work is already done

by Cross [16]. The work done by Cross is first explained in the section 5.1.1. This is

followed by a discussion of application of this research in the present analysis.

5.1.1 Work Done by Rod Cross [16]

The discussion in this section is completely based on a work previously done,

as is referenced in the title itself. A baseball bat was swung by a batter at an imaginary

ball. The batter was instructed to swing the bat as fast as possible in a horizontal

plane at waist hight, after releasing the bat from shoulder. A single video camera

operating at 25frames per second (fps), exposure time of 2 ms, was mounted at a 4 m

height above the head of the batter. The

bat was an 840 mm long Louisville

Slugger R161 wood bat, weighing 871 g.

The center of mass of the bat was located

560 mm from the knob end, the barrel

diameter was 66.7 mm, and its moment

of inertia about an axis through the center

of mass,Icm, was measured to be 0.039

0.001 kg m2.

The video was converted into

frames, and each frame was de-interlaced

to obtain the two images. The images

were used to locate the position of the bat

in each frame. Figure 5.1 shows the resulting motion of bat generated by Cross. The

figure shows the positions of the bat generated at 20 ms intervals. The bat is shown as

a solid straight line. The locus of barrel end and knob end are shown in blue and red

respectively. The location of center of mass is shown as a solid dot marked CM. The

inner circle denotes the locus of the instantaneous center of curvature of the locus of

the center of mass. The radius is calculated as the distance from the instantaneous

Figure 5.1: Motion of A Bat

- 48 -

center of curvature to the center of mass. At every instant the long axis of the bat

makes an angle, say , with the laboratory X axis. The angural velocity, , of the bat

with respect to the laboratory frame of reference would then be . At every step

in the motion of the bat, it is possible to calculate /. Also by calculating the

distance covered by the center of mass, through every step of the bat swing, it is

possible to get the linear velocity of the center of mass, V. Thus using these

calculations, and V are plotted against time, t. This graph is shown in Figure 5.2.

The net force that the batter is applying on the bat can be resolved into two mutually

perpendicular forces, the centripetal force, , and the tangential force,

. Through the data available, these both have been calculated and as shown

in Figure 5.3, plotted against time. The net force being applied by the batsman is the

resultant of these two perpendicular forces. The resultant force, F, and the angle, ,

between the resultant force and the axis of the bat is plotted in Figure 5.4.

To summarize, till now the total force being applied by the batsman on the bat

with both of his hands is obtained, along with its direction of action. It is desired to

know the force due to each of these hands. To find out this, the couple applied by the

two hands on the bat needs to be known.

The resultant force being generated by both the hands generates a torque

which causes the bat to rotate, or swing through its path. If the angle between the

radius vectore R, and longitudinal axis of the bat is , the resulting torque due to the

resultant force, F, can be expressed as the sum of torque due to centripetal force, A,

and torque due to tangential force, B. Since an axis through center of mass of the bat

is being considered, = ( ), and = ( ).

Here, d is the distance from the point of application of the resultant force to

the center of mass of the bat. Actually the force is being applied by two hands,

distributed over a length of almost 200 mm along the handle, but it is regarded as a

point force at 100 mm from the knob end. Thus d is calculated to be 0.46 m.

In addition to this torque, the hands apply a couple C. Hence by Newtons

Second law, as applied to rotating bodies, Eq 5.1 is obtained.

+ + =

(Eq 5.1)

- 49 -

In Eq 5.1, A and B can be easily calculated, as described above. Also ICM is

know. From the graph of v/s t, can be easily calculated. Hence the equation

is used to get the value of C, the coup