Simulation of a Badminton Racket A parametric study of racket design parameters using Finite Element Analysis. Master’s thesis in Applied Mechanics ELIAS BLOMSTRAND MIKE DEMANT Department of Applied Mechanics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2017 MASTER’S THESIS IN APPLIED MECHANICS Simulation of a Badminton Racket A parametric study of racket design parameters using Finite Element Analysis. ELIAS BLOMSTRAND MIKE DEMANT Department of Applied Mechanics Division of Solid Mechanics CHALMERS UNIVERSITY OF TECHNOLOGY Göteborg, Sweden 2017 Simulation of a Badminton Racket A parametric study of racket design parameters using Finite Element Analysis. ELIAS BLOMSTRAND MIKE DEMANT © ELIAS BLOMSTRAND, MIKE DEMANT, 2017 Master’s thesis 2017:52 ISSN 1652-8557 Department of Applied Mechanics Division of Solid Mechanics Chalmers University of Technology SE-412 96 Göteborg Sweden Telephone: +46 (0)31-772 1000 Cover: Illustration of a smash sequence for a badminton racket. Chalmers Reproservice Göteborg, Sweden 2017 Simulation of a Badminton Racket A parametric study of racket design parameters using Finite Element Analysis. Master’s thesis in Applied Mechanics ELIAS BLOMSTRAND MIKE DEMANT Department of Applied Mechanics Division of Solid Mechanics Chalmers University of Technology Abstract Badminton, said to be the worlds fastest ball sport, is a fairly unknown sport from a scientific point of view. There has been great progress made to get from the old wooden rackets of the 19th century to the light-weight high performance composite ones used today, but the development process is based on a trial and error method rather than on scientific knowledge. The limited amount of existing studies indicate that racket parameters like shaft stiffness, center of gravity and head geometry affect the performance of the racket greatly. These studies have either been physical test with limit number of data points or simplified computer simulations. In this work a parametric study of racket models is performed with the purpose to find key racket design parameters using Finite Element Analysis, FEA. Eleven racket models are simulated using one smash and one clear swing. One is the reference model, based on a modern isometric racket, and the other ten have one parameter modified to see how this alters the response. In addition to the swing tests, the sweet spots of the racket head is analysed and the eigenmodes of the racket are investigated. The results show that the previously mentioned parameters have a noticeable effect on racket performance. A low shaft stiffness may give higher shuttle velocities if used correctly, and an inconsistent shuttle speed and trajectory if used incorrectly. High center of gravity has a less inconsistent effect but instead would require more effort to swing. Noticeable is that the Y-shaped head introduced by Prince, which since has been removed from the market, perform very well in some aspects. These results show that FEA should be seen as a useful tool in badminton racket development and with further work on the field advances can be made that would make badminton the definitive fastest ball sport in the world. Keywords: Badminton, Racket, Composite, FEA, FEM, LS-Dyna, Smash, Clear, Sweet spot i ii Preface This thesis treat the possibility to adapt an CAE method for design and evaluate badminton racket. The study is performed at the engineering consultant firm FS Dynamics with the sports equipment company Salming as the end costumer. The project was carried out during the spring semester 2017 corresponding 20 week for each of the two participants. The final examination was done at the department of Applied Mechanics at Chalmers University of Technology with Martin Fagerström as the accountable examiner. Acknowledgements First and foremost our gratitude goes towards FS Dynamics who gave us the opportunity to work with this project together with the end costumer Salming. Many thanks to our supervisors at FS-Dynamics, Björn Andersson and Mattias Wångblad, for providing us with valuable feedback and guidance throughout the project. Further we like to thank Linus Lindgren for helping us with several issues in the software LS-Dyna. An additionally gratitude towards Maxine Kwan and John Rasmussen for providing us with the motion capture data. Finally a special thank to our examiner at Chalmers University of Technology, Martin Fagerström. iii iv Contents Abstract i Preface iii Acknowledgements iii Contents v 1 Introduction 1 1.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4 The Game of Badminton . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.4.1 Badminton Racket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4.2 Shuttlecock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4.3 Badminton Strokes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4.4 The Player . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Theory 5 2.1 Racket Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.1 Racket Mass and Swing-Weight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1.2 Racket Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1.3 String Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Key Parameters for Racket Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 Challenges Due to Individual Player Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4 Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.1 Mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4.2 Hourglass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4.3 Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Composite Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5.1 FE-model of Composite Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.6 Motion Capture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Method 13 3.1 Used Software Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.2 Racket Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3 CAD Model for the Racket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.3.1 CAD Model for the Head and the Shaft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3.2 CAD Model for the Net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3.3 CAD Model for the Handle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3.4 CAD Model for the Shuttle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4 Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4.1 Static Bending Test for Material Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.4.2 Composite Material for Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4.3 Wood Material for Handle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4.4 Material for Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.4.5 Cork Material for Shuttle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.5 The Finite Element Model of the Racket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.5.1 Defining Part Identities (PID) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.5.2 Meshing Procedure and Used Element Formulations . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.5.3 FE-model of the Badminton Racket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.5.4 Alternations for Model Used in the Implicit Eigenmode Analysis . . . . . . . . . . . . . . . . . . 22 3.5.5 Alternations for Rigid FE-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.6 Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 3.7 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.7.1 Simulation Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.7.2 Execute Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.7.3 Eigenfrequency Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.7.4 Deflection Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 v 3.7.5 Sweet Spot Analysis for Swing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.7.6 Sweet Spot for Clamped Racket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.7.7 Time Window Analysis for Shuttle Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.8 Method for Post-process and Organise Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.8.1 Post-processing Result in Meta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.9 Post-processing in MATLAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.9.1 Comparison of Input and Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.9.2 Deflection of the Racket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.9.3 Shuttle Motion and Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.9.4 Sweet Spot Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 Result 29 4.1 Racket Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Static Deflection Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Mesh Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.4 Frequency Response and Eigenfrequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.5 Clear - Racket Behaviour and Parameter Influence . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.5.1 Deflection of Racket During Swing and Hit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.5.2 Racket Velocity at Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.5.3 Momentum During Stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5.4 Contact Time and Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.5.5 Shuttle Velocity and Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.5.6 Repositioned Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4.6 Smash - Racket Behaviour and Parameter Influence . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.6.1 Deflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.6.2 Racket Velocity at Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.6.3 Momentum During Stroke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.6.4 Contact Time and Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.6.5 Shuttle Velocity and Direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.6.6 Repositioned Ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.7 Sweet Spot Analysis for Clamped Racket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5 Discussion 46 5.1 Racket Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.2 Calibration of FE-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 5.3 The Natural Frequency Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.4 Racket Responses for Key Parameter Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.4.1 Centre of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 5.4.2 Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.4.3 Shaft Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.4.4 Head Stiffness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4.5 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 5.5 FE-model and Material Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 5.6 Assumptions and Other Sources of Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6 Conclusion 52 7 Future Work 54 References 55 A Tables I B Sweet Spots V vi 1 Introduction Badminton is said to be the worlds fastest ball game [1] where the increase in the pace of the game has mainly been dependent on new available materials [2]. However, the effect from different parameters such as stiffness, weight distribution, tension in the strings etc. on the speed of the game is yet poorly understood [3]. Manufacturers work by improving on the latest model of rackets based on player feedback and experience, rather than using scientific methods [4]. This trial and error type of development does not guarantee the best design and is probable to occasionally fail. With greater scientific understanding of which parameters that affect the performance of the badminton racket, and in which way, the development process could be made more accurate. By identification of the impact of each parameter, rackets could also be further personalised. This is of interest for manufacturers of high-performance rackets to provide players with an optimised racket. Salming Sports is a manufacturer of premium sports equipment and acknowledged actor in the squash business with clothes, shoes and rackets. With similar demands on equipment in the sport of badminton the company has begun to expand into this market as well. But before they start manufacturing badminton rackets they want deeper insight into what makes a racket high performing. Salming base their products on quantitative facts, whereby it is important to have an understanding of the racket dynamics and the performance due to different parameters. FS Dynamics is a Computer Aided Engineering (CAE) supplier based in Scandinavia which has been tasked by Salming to provide the CAE analysis of the racket mechanics. This project is a collaboration between FS Dynamics and Salming and is conducted as a master thesis at the Department of Applied Mechanics at Chalmers University of Technology. 1.1 Purpose The purpose of this thesis is to identify key parameters pertaining to racket dynamics using Finite Element Analysis (FEA) and to develop a greater understanding of their influence. The FEA will be performed using LS-Dyna as an explicit FEA solver. The result can be used as a scientific basis in the racket design process, and will be used to show if FEA is a valid tool in badminton racket design and optimisation. The end goal of this thesis is not to develop a racket, but to identify and define a comprehensive summary of the governing racket dynamic parameters. 1.2 Objective The objective of this project is to identify key properties of the racket design and how they change the characteristics of the racket. The resulting insight will give more information to the racket design process and further demonstrate the usability of CAE within badminton racket development. To complete this, the following milestones have been set: · Identify key properties that govern the badminton racket characteristics. · Have data for a smash and a clear swing to add the motion into the FE-analysis. · Define suitable geometry and material data to include in the model. 1 1.3. LIMITATIONS CHAPTER 1. INTRODUCTION · Define and validate the FE-model of a badminton racket. · Simulate the defined motions using the explicit solver in LS-Dyna. · Perform a parametric study of the variations on the racket model and evaluate the impact of the racket design parameters. 1.3 Limitations The work in this project is carried out by two persons and is limited to 20 weeks. These limitations have been set for the objectives to be achievable: · String properties are not evaluated, i.e. the same properties and pre-loading for the strings are used for all models. · Geometrical variations is limited to variations of the so called isometric racket shape, see Figure 1.2. · All variations of racket parameters are kept within ranges allowed by the rules set by Badminton World Federation [5]. · The simulation is limited to the two strokes smash and clear. · Each stroke is limited to one movement pattern each, i.e. different players and individual variation is not considered. · No material data is available, thus the material properties is chosen using calibration methods. · The FEA does not contain any air resistance. 1.4 The Game of Badminton The game of badminton has been played since the middle of the 19th century and in 1992 it became part of the Olympic Summer Games [6]. The objective of the game, as stated in the Badminton Laws [5], is to score points by landing the ball, named shuttlecock and shown in Figure 1.3, on the opposition’s part of the court. The shuttle may only be hit using a badminton racket, see Figure 1.2 for an illustration of a badminton racket. Dividing the court is a net which the shuttle has to pass over before hitting the ground of the adversary, see Figure 1.1. Any attempt to hit the shuttle where it does not pass over the net or does land outside of the opponent’s court is a foul which grants the opponent the point. As the shuttle passes over the net the opponent must prevent the ball from touching the ground on their side by shooting it back over the net and, hopefully, hitting the ground of the first player’s court to score a point. Each side is only allowed one touch of the shuttle before it must pass over the net. The game can be played as single or double with one or two players per side respectively. Figure 1.1: A badminton court. 2 CHAPTER 1. INTRODUCTION 1.4. THE GAME OF BADMINTON Figure 1.2: Modern badminton racket, isometric shape. It consists of the handle, shaft, head and string bed. Figure 1.3: Shuttlecocks. To the left a plastic shuttle and to the right a feathered shuttle. 1.4.1 Badminton Racket The badminton racket originally was constructed by fitting a net made out of parchment or catgut to a wooden frame [7]. Since then the sport has developed to a fast paced game, much due to the improvements in racket design. The modern racket is made primarily out of a composite material [2] which allows for a stiff, lightweight construction. The strings are usually made of a polymer which can hold a strong tension, normally between 90-140 N [8]. There are two head shapes in use: an oval shape and an isometric shape, seen in Figure 1.2 where the traditional oval shape has been stretched to become more square. Isometric is the used abbreviation of the full name ”Isometric square head shape” which the manufacturer Yonex named it, and claimed it gave a bigger sweet spot than the oval shape. While many parts of the racket have been modernised with the development of the sport the handle is still made out of wood. Racket Regulations The design of the racket is limited by regulations defined by the Badminton World Federation [5]. These specify that: 1. the overall length may not be longer than 680 mm. 2. the overall width may not be wider than 230 mm. 3. the overall length of the stringed area may not be longer than 280 mm. 4. the overall width of the stringed area may not be wider than 220 mm. 5. the stringing pattern shall be generally uniform, particularly not less dense in the center than elsewhere. 6. the stringed area may extend into an area connecting the head with the shaft, provided that: (a) the width of the extended area does not exceed 35 mm. (b) the overall length of the stringed area does not then exceed 330 mm. The stringed area is defined as the area within the head i.e. the string bed. There are more rules regarding the racket, but these do not apply to this study. 1.4.2 Shuttlecock The ball used in badminton is called a shuttle, short for shuttlecock, which has two significant parts: the feather cone and the sturdy cork base. The cone feather structure is either made from real feathers or a more durable plastic skirt, as shown in Figure 1.3. While the real feather type is more expensive than the synthetic it also has less variations in flight characteristics [9], which is why it is normally used in competitive games. The structure of the shuttle makes it aerodynamically stable with high drag properties, a necessity for an opponent to be able to hit the flying projectile which has been recorded to have a maximum speed of 113 m/s [10]. 3 1.4. THE GAME OF BADMINTON CHAPTER 1. INTRODUCTION Clear Smash Figure 1.4: The clear and smash badminton stroke. 1.4.3 Badminton Strokes In a game of badminton, several distinct badminton strokes are used which each has its specific purpose. Two of the most common strokes are the clear and the smash stroke, see Figure 1.4. The first is a generally used as a defensive move where the shuttle is given a high upwards velocity, arcing over the court, and is only reachable by the opposing player as it comes down again at the end of the court, furthest away from the net. From here it is usually easy for the opponent to hit the shuttle but hard for him to make an aggressive counter attack. This together with the fact that the player hitting the clear stroke will have time to re-position himself during the extra time it takes for the ball to return down to the court makes the clear a good defensive shot. The second stroke, the smash, is an attacking move where the ball is hit with high velocity in a steep angle, giving the opponent less time to react and try to return the shuttle. There are more types of badminton strokes, but the aforementioned have been chosen to be used in this study due to the frequency of use and that they require a lot of power from the racket to be executed well. 1.4.4 The Player A game of badminton contains a lot of variation where the player needs to make split second decisions on how to reach the shuttle before it touches their court and return it disadvantageously to the opponent, in the way the player personally sees as the optimal way. This action is greatly influenced by the player style, if he or she is a offensive or defensive player. An offensive player may try to position themselves more aggressively where he can return the shuttle before the opponent has re-positioned himself, using a more forceful swing than the defensive player would use. To quantify measurable properties and parameters of a badminton racket which influence the performance, it is vital to understand the game and be mindful of these existing different types of players. Another property of the player which has a high impact on the game is the experience level of the player. The amount of experience defines whether a player is novice, recreational or an expert. The intention of this thesis is not to evaluate the difference between an inexperienced player and an expert, thus the focus will be to compare response and results with skilled players and their patterns. Although one observation by Kwan et. al. [4] is that recreational players tend to hit the shuttlecock in a later state having a positive deflection on the racket while an expert hits the shuttlecock with a negative deflection, see Figure 2.1. The distinction between an attacking or defending player also becomes more prominent with more experienced players. In the studies by Lee et al. [11] and Tong et al. [12] the pattern regarding preferable strokes and the efficiency of these is measured for players from different matches in professional championships. The conclusion is that the most efficient and preferred stroke when trying to obtain a point, also referred as a kill shot, is the smash followed by the net shot. For the return strokes the net, drop and clear shot is the most efficient and thereby used shots. It is further discovered that male and female players prefers different techniques. The pattern among male players is dominated by smash and net shot whereas female players prefers drop and clear shots [11]. The characteristics of the strokes can be divided into power and control and by linking racket parameters to these actions an understanding how to design the racket to influence the most critical parts of badminton is achieved. 4 2 Theory The technical development of badminton has increased the speed of the game since its inception. But rarely are these improvements based on scientific data, instead these are the result of trial and error. This study therefor uses data from studies on similar fields, e.g. tennis, in addition to the limited number of badminton studies. In this section, the current theories regarding racket properties are introduced. These, together with the necessary theory for the FE-model and motion capture, serve as the theoretical base for the method used in this project. 2.1 Racket Properties Previous studies have mainly focused on the effect on racket performance based on weight, stiffnesses and string properties as variables. The following section provides a summary of the theories and results that define the foundation on which this study was based on. 2.1.1 Racket Mass and Swing-Weight A modern racket weighs about 90 g with its centre of gravity (COG) located along the shaft axis, typically roughly half-way along the shaft. The position of the COG is shifted along the axis of the shaft by the manufacturers, which they claim changes the performance of the racket. It is said to be ”head-heavy” if the COG is closer to the head and ”head-light” if it is instead closer to the handle. The ”head-heavy” model is marketed towards attacking players with the reason that it should give a more forceful swing, while its counterpart is angled towards the defending players as it should be easier to make fast reactive moves when the COG is closer to the hand of the player. The reason why manoeuvring the racket is easier with a COG closer to the handle is that it lowers the racket’s moment of inertia (MOI), given that the racket is held at the handle. The MOI determines how much torque is needed to give a rotational acceleration to the racket. It is calculated from the mass of each infinitely small point of the racket and the distance of the point from the rotational axis, which during the swing is at the handle. Thus, when the COG, also known as the centre of mass, is shifted towards the handle the MOI decreases and less torque is needed to swing the racket. As mentioned in Section 1.4.1, the mass of the racket have decreased drastically by changing from the old wooden rackets to the modern composite rackets. As a consequence, the rackets can be swung faster with less effort. However as Hsieh et. al. observe in a study with two different rackets with the masses 100g and 85g, the heavier racket is faster to swing which indicates that weight reduction may only be beneficial until a certain point [13]. Based on this observation further evaluation is needed regarding how the head speed and the shuttle post impact speed correlates to the mass. Is the increased speed due to a heavier racket adequate to compensate for the extra energy needed to move the racket? The link between MOI, also referred to as swing-weight, and the velocity of the racket at impact needs to be further examined. Mitchell et. al. [14] imply that a correlation between MOI and the pre-impact racket speed exist but the result contradict each other between the participating players. Due to the low amount of players in their study and the diverging result, further evaluation to capture the correlation is needed. Nevertheless, the data is enough to indicate a possible theory. During a study by Cross et. al. [15] the result indicated that the swing speed of a racket with constant mass decreased as the moment of inertia increases, regardless if the player performs the swing using the forearm only or by also include the upper arm. However the magnitude of the influence was greater for a swing preformed 5 2.1. RACKET PROPERTIES CHAPTER 2. THEORY Figure 2.1: Deflection of a racket during swing. Negative deflection at first when handle accelerates, at the end positive deflection. solely by the forearm. This is due to the fact that the total MOI for a forearm swing is dominated by the racket swing-weight. To calculate the MOI for a badminton stroke the entire player-racket system needs to be considered to obtain the true value. This is a complex procedure since several individual aspects as play style, technique and pure bio-mechanical properties affect the result. For example, the rotation of elbow, shoulder and wrist can differ. Additionally, more or less wrist action is used by the player. Regarding the measurement of the swing-weight the usual approach is to measure the moment of inertia in a laboratory environment with the reference axis close to the end of the handle [15]. The true axis is dependent on the individual anatomic and play style, thus the result using the measured swing-weight might need to be calibrated to fit a specific player. A conclusion from these studies is that changes in racket mass or baseball bat mass only have a small effect on the resulting shuttle speed. Although even small effects can have significant outcomes, improvement on a professional level is rarely measured in big numbers when it comes to accelerations or time etc. but can be the central difference from loosing to winning a point or even a match. As an example, Brody [16] showed that a tennis player is relatively sensitive to alternations of properties affecting the moment of inertia. Players in the study capture differences as low as 2.5% and further differences as small as 5% regarding the polar moment (the rackets ability to resist torsion). 2.1.2 Racket Stiffness The elasticity of the racket is primarily inherited from the shaft’s properties as it is where most of the bending of the racket occurs as the racket is swung [17]. To swing the racket a translating and/or rotating force is applied to the handle by the hand. Because the shaft is elastic, the distance between handle and head leads to a delay of the force affecting the head of the racket. Thus, the shaft will bend backwards initially during a swing, see Figure 2.1. The timing window of the impact during a smash was identified in a study with both ranked and unranked players [17]. It was found that there was a 20 ms window in which the positive acceleration was at its maximum and the study participants hit the shuttle within 87 % of this time. The window was shown to shift position in relation to the start of the forward motion of the swing with different racket stiffness. Thus, a relatively small shift of stiffness can have an impact on optimal swing timing. In Kwans research to design the worlds best badminton racket [4] the the hypothesis is that the optimal racket stiffness could be identified as where stroke variation is minimal (maximising control) but with stiffness as flexible as possible (maximising power). It is also noted that the influence of elasticity can provide an additional 2-3 m/s to the racket head speed at impact. This contribution is more noticeable for novice players as their maximum head speed is generally lower than for advanced players, who in the studies achieve head speeds of 35 m/s. For advanced players, the greater variation in strokes due to higher compliance affects the control too much, leading to them preferring stiffer rackets due to the extra head speed gained from elasticity not outweighing the loss off control [17]. The rest of the racket, the head and handle, also have stiffness’s which affect the performance of the racket. These have not been studied as extensively as the shaft stiffness but are likely to impact both the energy transfer to the shuttle and to the hand. 6 CHAPTER 2. THEORY 2.2. KEY PARAMETERS FOR RACKET DESIGN Figure 2.2: Example of locations of specific tennis racket spots. CM = center mass, C = maximum COR, P = center of percussion, V = vibration node and D = dead spot. [20] 2.1.3 String Properties To show how the string bed affects the shuttle a measurement of the coefficient of restitution (COR) is often used. The relative speed of the shuttle and racket before and after impact are used to give the quotient COR as: COR = v2 − V2 V1 − v1 where v are shuttle speeds and V racket speeds. Pre-impact speeds are indicated by the number 1, and post-impact by number 2. It was shown for a stationary racket that the COR value decreased with greater string tension in the range of 60-150 N in a FE-simulation study [3]. In this study, a ball was dropped on a racket with the rim of the head fixed. A physical study with a low number of participants indicated that string tension is not the only factor relevant for fast shuttle speeds [8]. The study found that more experienced players can somehow consistently achieve similar speeds with string tension in the range of 70-130 N. Power is not the only property of the racket effected by the string tension. Control of the out-bounding ball from a tennis racket depends on the string tension [18]. It was shown that low string tension lead to problems controlling the speed of the shuttle post impact. The same researchers also noted in another study that 28% of the participants were sensitive to changes in string tension of the tennis racket [19]. The effects of impact between shuttle and racket, for example the shuttle velocity, is dependant on where on the string bed contact is made. The existence and locations of sweet spots on the bed is the main factor of this response [20]. Sweet spots are points or zones on the string bed where a characteristic of the impact is maximised when the shuttle collides into it, see Figure 2.2. The sweet spots can be defined as: the location of the maximum COR, the centre of percussion and the vibration node. The last two pertain to the vibrations and forces affecting the hand post-impact and are not studied in this thesis. As the shuttle connects with the racket it is accelerated forward. This transfer of energy into the shuttle also induces a force on the racket which leads to vibrations in the racket handle. The noise produced during impact is an audial feedback of the racket [18]. This sound could indicate the efficiency of the racket and if the shuttle hit a sweet spot. The frequency of the feedback however is very dependent on string tension. It also results in a vibrations in the hand which can be used to sense the effectiveness of the hit. 2.2 Key Parameters for Racket Design Previous studies indicate that specific racket design parameters are especially interesting to study for their impact on racket performance. These are: · center of gravity, which is the main factor in determining the MOI of the racket. Studies show that this affects the swing properties more than other factors. · weight, which should not affect the swing as much as COG. A greater weight should mean greater force applied to the shuttle resulting in higher outbound velocities. · shaft stiffness, which determines how the shaft flexes during the swing. A less stiff shaft can lead to higher shuttle velocities but also a weak racket. · head stiffness, which like the shaft stiffness impacts the tip deflection of the racket. Also may impact the shuttle velocity after impact. · head shape, which affects the location of sweet spots on the string bead and the stiffness of the head. 7 2.3. CHALLENGES DUE TO INDIVIDUAL PLAYER CHARACTERISTICS CHAPTER 2. THEORY The racket head shape is not the same for all racket sports. This could be because the sports differ in aspects like ball and court size, but could also be due to a history of development dominated by non-data based development. In Figure 2.3 three head shapes are shown where the isometric is the predominant one. The Y-shaped head was an attempt from the sports accessory company Prince which seems not to be available on the market anymore. The oval shape is less common than the isometric but is currently available on the market and is advertised to have be more powerful but at the cost of a smaller sweet spot. Isometric Y-shaped Oval Figure 2.3: Common racket sport head shapes. 2.3 Challenges Due to Individual Player Characteristics Badminton and other ball sports are high in variation due to the individual player aspects affecting the circumstances of the game. The shuttle is hit from different angles in different areas of the court at various speeds. On top of this, the player hitting the ball has his individual playing style; how he holds his racket, how fast he swings and how well he manages to hit the ball with the center of the rackets stringed area etc. As such, finding the optimal design of a racket could be described as pursuit to find the racket which performs best in all possible circumstances, which may be weighted by the likely-hood of that circumstance. Another step of optimisation would be to design the racket to fit a certain player type, as suggested in Kwan’s study to design the best racket [4]. This is done to an extent in the modern racket industry, where head-heavy rackets are marketed to aggressive smashing players which are supposed to give a higher shuttle velocity and head-light rackets are more manoeuvrable giving higher control and a better chance to defend against hard smashes. 2.4 Finite Element Method The Finite Element Method, FEM, is a numerical method commonly used to solve engineering problems. A model representing a physical geometry is divided into a finite amount of elements, a mesh. By putting this model in a simulation environment the model behaviour is evaluated. This is done by calculating how the applied load and boundary conditions affects each element. The resolution and reliability of the FE-model partly depends on the number elements and how they are organised. In most cases, but not all, a finer mesh gives a more precise result and facilitate iterative calculations, however the computational cost increase as the time to finish a simulation increases. A case where finer mesh gives a less reliable result is where the elements are smaller but organised so that the quality of the mesh is reduced. 2.4.1 Mesh The quality of a mesh can be defined in several ways, one part is to evaluate the geometric shapes of the elements and compare them to the ideal shape. Together with mesh convergence studies and other bench marking aspects as element formulations and adaptiveness etc., the quality can be defined. It is needed to have reliable results. By knowing the most weak sections in the model regarding element quality the user is able to pick the areas used for data extraction more carefully. For the element shape, some of the more common parameters are aspect, skewness and warping for each included element. Figure 2.4 to 2.6 illustrate schematic pictures of how the quality regarding aspect, skewness and warping are calculated for shell elements in Ansa. The aspect is a measurement of how equal the length and height are for triangular and quadratilateral elements, whereas skewness defines how far the element deviates from a equilateral triangle and quadrilateral. Regarding the warping angle, it is calculated by using the node distance h in Figure 2.6 which is calculated from the middle plane corresponding to the ideal orientation of the element, i.e. with the nodes orientated in the same 2D-plane. 8 CHAPTER 2. THEORY 2.4. FINITE ELEMENT METHOD b Aspect ratio 3a 2b = Aspect ratio = max(a,b) min(a,b) a a b Figure 2.4: Definition of shell el- ement aspect according to the Pa- tran criteria in Ansa. Skew angle = 90° - < 30° Figure 2.5: Definition of shell el- ement skewness according to the Patran criteria in Ansa. Warping angle =max[arcsin(h/l)]<5° h l Figure 2.6: Definition of shell el- ement warping according to the Patran criteria in Ansa. The quality parameters are calculated slightly different for solid elements, see Figures 2.7 and 2.8. For the hexa elements, a brick element of first order which is has an equivalent shape of a cube and is defined using 6 faces and 8 nodes, Ansa check each surface for warping using the same definition as for shell elements. The surface which performs worst regarding the warping criterion define the number assigned to the volume element. The aspect is measured similar to the shell element formulation although as another dimension is included the min and max length is used to calculate the worst aspect. Finally the shape factor is used to determine the skewness where the volume of the perfect cube is compared to the volume of the element and for the perfect shape the factor is equal to 1. In the figure 2.8 this is illustrated for tetrahedrons as the procedure is similar [21]. Aspect = max(h)/min(h)i i hi Figure 2.7: Definition of volume element aspect according to the Patran criteria in Ansa. Shape factor = V/V, Perfect tetra V/V = 100 V V0 Figure 2.8: Definition of volume element skewness according to the Ansys criteria in Ansa. 2.4.2 Hourglass The finite element discretisation is based on assumptions and simplifications which introduces minor errors that can be negligible or serious depending on whether the assumptions are valid or not. One important behaviour is the hourglass effect for elements using reduced integration. For shell elements this is when an element is used with only one integration point in the shell plane. A reduced shell element has five hourglass modes which all are a form of un-physical deformation occurring in the FE-model due to the lack of integration points. These modes are so called zero energy modes which imply that they can be triggered without any use of strain energy. Figure 2.9 illustrate the effect from hourglassing for a shell mesh. As can be reviewed the mesh has a distorted deformation which is non-physical. Mesh Hourglassing in mesh Hourglassed element Figure 2.9: The effect of hourglassing in a 2D shell mesh. By using fully integrated elements, this unwanted behaviour would be prevented. But using fully integrated elements is time consuming. Hence, in many cases hourglassing is allowed but controlled using various methods to artificially add the energy corresponding to an hourglass mode. These energies acts as nodal counter forces preventing the unwanted deformations [22]. 9 2.5. COMPOSITE MECHANICS CHAPTER 2. THEORY 2.4.3 Elements When using the software LS-Dyna for a model consisting of shell elements the standard type of element is Belytschko-Tsay element (Element type 2). The element is based on the kinematic relation of Mindlin-Reissner theory where plane stress is assumed. Comparing to the more simple theory of a Kirchhoff plate, the shear strains are considered. This is analogous to the Timoschenko beam theory illustrated in Figure 2.10 where the deformation is explained by mid plane displacement w and the rotation θ giving the shear strains. The cross section still have the initial linear shape but can be rotated around the point located on the mid plane. For Kirchhoff plate theory the through thickness deformation is perpendicular to the mid surface, thus no shear strains are captured [23]. w(x) (x)x x w Figure 2.10: The kinematic assuption for a Timoschenko beam. The element is reduced using one in-plane integration point, see an illustration in Figure 2.11, meaning that it is computational fast but introduce the risk of hourglassing. Thus, this behaviour needs to be controlled when using reduced elements. An error check using fully integrated elements for a similar model could be needed to control the difference in the results between a mesh of reduced and fully integrated elements to satisfy a good accuracy. Further, by using the Belytschko-Tsay formulation the quality of the mesh may influence the result and cause unreliable results. Thus, it is important that correct limits for the quality is defined in a proper way, especially for warping [23]. For shell elements, Stelzmann recommend only one other type of element and that is the fully integrated shell element formulation using 2x2 integration points in the shell plane. The formulation prevent the existence of hourglassing modes but its downside is a more expensive calculation with an increase around a factor of 2-2.5. The formulation does not completely resolve the problem with warping but this can be countered by using hourglass control with the parameter IHQ set to 8 in LS-Dyna, although it gives an even more expensive calculation in terms of time [24]. Fully integrated element Reduced element Figure 2.11: Illustration of a fully integrated element and the corresponding reduced element. 2.5 Composite Mechanics A composite material consists of fibres and a matrix holding these fibres in place. Thus, the properties of a specific composite material is determined by the amount and characteristics of these two parts. A unidirectional- ply (UD-ply) is a sheet with fibres aligned in the same direction. Thus, a layer of composites are modelled using a combination of UD-plies where the fibres are orientated different for each layer. In Figure 2.12, an example is given where the orientation offset is based on the x-axis meaning that 0◦ is along the x-axis while 90◦ is perpendicular. A notation commonly used in fibre composites is index L for the direction along the fibres and T for the transversely direction, i.e. L-direction in the figure for the first ply is along the x-axis while for the 90◦ ply it is along the y-axis. 10 CHAPTER 2. THEORY 2.5. COMPOSITE MECHANICS y x z90°45°0° Figure 2.12: Composite Layup and Orientation. Because of the schematics of the composites with fibres orientated in different directions it is an orthotropic type of material, meaning that the material properties are directional. For linear elastic models of the material Hooke’s law can be used to calculate the Cauchy stress under the assumption of small strains. Using index notation the equation is defined as: σij = Eijklεkl (2.1) As the stress and strain tensors are symmetric they can be reduced to 6x1 vectors when using Voigt format. Moreover orthotropic materials have the same properties in the normal and reversed normal directions for certain planes. This means it has the same stiffness regardless if it is in compression or tension. Furthermore, for this project the composite plies were considered to be transversely isotropic meaning that the plane perpendicular to the fibre direction is isotropic, i.e the properties perpendicular to the longitudinal directions are the same regardless of orientation within the plane. Given that the Young’s modulus for the transversely and longitudinal directions are known, ET and EL, together with the major Poisson’s ratio νLT then the minor Poisson’s ration νTL is given by [25]: νTL = νLT ET EL (2.2) Given that there exists a plane of isotropy the shear modulus for this plane can be calculated with Equation (2.3). It is rare to have all material parameters available and several of these parameters are fairly complicated to model as they require additional parameters to define. Nevertheless as stated in the Composite Mechanics course at Chalmers [26] the value of νTT ′ is normally in between 0.4 − 0.45 for some carbon-epoxy materials. GTT ′ = ET 2(1 + νTT ′) (2.3) 2.5.1 FE-model of Composite Materials A composite layup is a complex structure with plies stacked in different orientations and with different materials. Hence the FE-model require a proper definition of the composite to capture the correct behaviour. Regardless if shells or solids are used to model the composite layup, the option remains if each ply is to be model with a corresponding element or a through thickness integration point using only one element, see Figure 2.13. If efficiency is prioritised or only the elastic response is of interest, the integration points method is recommended to lower the computational time needed. The downside is that delamination will not be captured properly as a contact condition needs to be defined between elements to obtain the separation behaviour [27]. Composite layup Integration points Figure 2.13: FE-model of composite materials. 11 2.6. MOTION CAPTURE CHAPTER 2. THEORY 2.6 Motion Capture Motion capture is a technique to capturing movements of people or objects using a sensor, typically a camera. It is used widely to capture physical movements and transfer them to computer animations. It has also become a standard tool in sports, where for example the Hawk-Eye system is used to help the referees make the correct calls in badminton games [28]. Motion capture has been used to capture the motion and behaviour of a badminton racket in previous badminton studies at Aalborg University [4] [29] [30]. These recordings were made using a system by Qualisys containing eight cameras recording at 500 Hz and saved to C3D, coordinate 3D, files. To follow the movement of the racket five markers were placed on the racket, see Figure 2.14. The position of these were recorded and compared to data from strain gauges positioned on the racket showing that the method gave reliable results. Figure 2.14: A motion capture sequence with five markers on a badminton racket. 12 3 Method The objective of the study was to find how racket design parameters impact the performance of a badminton racket. A condensed view of the method can be seen in Figure 3.1. Create motion from motion capture data Create FE-model from given data Apply motion to the FE-model Simulate racket stroke using motion Get result with current parameters Change racket parameter Interpret parameter performance impact Figure 3.1: Overview of the method used in this study. 3.1 Used Software Products To create the model and process the results in this study several computer software were used. The usage of these are detailed below and are shown in the flow chart of Figure 3.2. Mokka Export motion capture data MATLAB DWG TrueView CATIA Ansa Create net geometry Create handle geometry 3D Mold file Import frame geometry LS-Dyna Create FE-model Create node movements Meta Completed simulation MATLAB Export simu- lation data Processed simu- lation data Figure 3.2: Overview of the usage of software. Mokka An open-source software which can be used to analyse motion capture data C3D files. Was used to export the motion data to a comma-separated value format which was processed in Matlab. MATLAB A commercial software used for numerical computations. Was used to processes the motion data and to process the data simulation results outputted from Meta. DWG TrueView A free software used for looking at CAD, computer aided design, drawings. Was used to accurately reproduce the net geometry from the drawing. CATIA A commercial software used for CAD. Was used to create the handle geometry. Ansa A commercial software used for pre-processing the FE-analysis. Was used to create the FE-model which was based upon the imported 3D geometries. LS-Dyna A commercial software used for FE-analysis. Was used as the solver for the FE-problem. Meta A commercial software used for post-processing the FE-analysis. Was used to export data from the simulations that ran in LS-Dyna. It is made by the same company that makes Ansa. 13 3.2. RACKET MOTION CHAPTER 3. METHOD Figure 3.3: Visualisation of the clear swing motion, going from red to blue as time increases. Figure 3.4: Visualisation of the smash swing mo- tion, going from red to blue as time increases. 3.2 Racket Motion The FE-simulation of a moving badminton racket required knowledge about the swing motion. This data had already been recorded in studies at Aalborg University. From these studies, data for two types of badminton strokes was provided: the clear (Figure 3.3) and the smash (Figure 3.4). It was recorded with the Qualisys system described in Section 2.6. The look of the two swings are very similar, where the major difference is found in the vertical movement of the clear swing. This is due to a restriction in the original smash study where the shoulder movement was kept to a minimum. The movement data was exported from Mokka to a format readable in MATLAB. It contained xyz-position data for each marker on the racket, see Figure 3.5, during about one and a half second, where the swing motion was roughly 0.6 seconds long. This data was smoothed to remove small jitters in the data which most likely is due to error in the capture process, see Figures 3.7 and 3.8. By removal of the small fast movements, the FE-analysis became more stable and visually looked more like the movement of a swing. The movement was applied to the racket by controlling translation and rotation of the bottom node, see Figure 3.6. The motion capture for the clear stroke also contained data for the post impact shuttle speed, which was almost 50 m/s right after impact. Figure 3.5: Markers used in the motion capture. At- tached on the outside of the frame and handle. Figure 3.6: Node used to control model movement. Positioned in the center of the handle bottom. 3.3 CAD Model for the Racket The finite element model was based on a provided 3D CAD model defining the geometry for the shaft and the head of the racket. Furthermore, a 2D blueprint was supplied with a definition of the stringed area. Regarding 14 CHAPTER 3. METHOD 3.3. CAD MODEL FOR THE RACKET Figure 3.7: The xyz-position of bottom marker, original data and smoothed data (overlapping). Figure 3.8: Original and smoothed x-position data, zoomed in. the handle no schematic definition was given. Also supplied were three different physical prototypes, all with the same geometry but different material layup. This geometry and chosen material properties defined the original model, thus changes in responses due to parameter alternations were evaluated against this model. 3.3.1 CAD Model for the Head and the Shaft In Figure 3.9 the CAD model for the original model, an isometric shaped racket, is illustrated. This CAD-model, which was given by Salming, was possible to import into Ansa. It is the part where alterations were made to evaluate the different geometrical influences in the parametric study. Figure 3.9: CAD model of the frame. 3.3.2 CAD Model for the Net The geometry of the net was added to the frame in Ansa using a 2D-drawing defining the stringed area, see Figure 3.10. Here a small error was revealed as the drawing include one more horizontal string than the psychical model. The strings were chosen to be based on the drawing as this was easiest and the racket was already deviating from the prototypes due to lack of material data. The string bed consists of 22 vertical and 23 horizontal string sections. The strings had a diameter of 1 mm giving it an cross sectional area of 0.72 mm2. Figure 3.10: The resulting stringed geometry, taken from the 2D-drawing. 3.3.3 CAD Model for the Handle For the handle, see Figure 3.11, as mentioned above no schematics were accessible. Thereby, the geometry was estimated by measurements. The handle is a stiff volume with the purpose to provide the player with an firm and controlling grip. The shape is optimised to suit the hand with a low weight. The bottom half of the handle is hollow reducing the weight contribution to the racket. Furthermore the handle has a plastic plug at the bottom and a plastic cone that is glued across the cone-shaped surface of the handle. These were neglected in 15 3.4. MATERIAL PROPERTIES CHAPTER 3. METHOD the project due to the lack of material data and presumably their influence on the elastic behaviour of the racket is negligible. Furthermore, to obtain a good grip the handle surface is covered with a grip tape. This part was excluded for this project as it was assumed to have little impact on the structural responses of the racket. As the hand was not accounted for in the FEM model, there was no reason for the grip tape to be included which otherwise would damp the vibrations transferred into the hand. Figure 3.11: CAD model of the handle. 3.3.4 CAD Model for the Shuttle As the simulations including the shuttle only considered the small window before and after hit, the aerodynamics of the ball was neglected. Thus, the otherwise complex geometry of the shuttle is simplified to a ball, see Figure 3.12. This model had the same radius as the cork part of the shuttle with an equal weight. With other words no CAD model was provided and the ball was created directly in Ansa. Figure 3.12: CAD model of the simplified shuttle. 3.4 Material Properties The modern badminton racket consists of several different materials. The frame, handle, strings and other small components are all made of different materials that is carefully chosen to be the optimal material for each parts purpose. To fully capture the true model response it was important to model the materials with the right properties. Even though several different prototypes with varying material composition was provided by Salming, no material data was known. Hence, the project focused on a general method for evaluation badminton racket, but due to the lack of data a full validation for the provided racket was not possible to achieve. To get realistic properties the material data was gathered from the material database CES Toolbox. Further, as it was of great importance to obtain a correct center of gravity for the racket the density presented below are not directly taken from CES but tweaked to fit the given prototype. The presented material properties following in this section defines the original model. Thus, if an evaluation of a 10% stiffer material was made it was with these values as reference. 3.4.1 Static Bending Test for Material Verification Even though the total weight and its distribution was modelled correctly, properties as damping and stiffness of the racket are equally important to capture the racket behaviour properly. A theory of why badminton is such a high velocity ball sport is that it depends on the flex of the racket. Thus, the racket material behaviour regarding stiffness was calibrated using a simple bending test. The provided prototypes were evaluated for bending stiffness by clamping the handle and applying a load at the top of the racket upwards using a small baggage scale. The result was then used as a reference when simulating the bending test for the FE-model with the chosen material. In Figure 3.13, the boundary condition for the simulations are illustrated with the clamped handle defined in blue and the applied load as the red arrow. The FE-model is described in detail in Section 3.5. 16 CHAPTER 3. METHOD 3.4. MATERIAL PROPERTIES Figure 3.13: Boundary condition for the static deflection test. 3.4.2 Composite Material for Frame When looking at advertisement and selling arguments for badminton rackets the main focus is usually on the shaft and frame. Different tweaks regarding stiffness and geometry is common and the properties of the badminton frame is frequently debated. It is visualised by the abundance of different layups and material compositions for the badminton racket hinting that it is hard to find the best design. Consequently, the main focus of the material evaluation revolve around the properties of the composite structure. The racket consists of two different composites structures, one for the head and one for the shaft. The head was modelled with a stiff material with the following properties: EL = 160 GPa ET = 12 GPa GLT = 7.1 GPa ρ = 900 kg/m3 νLT = 0.3 While the shaft was modelled with a less stiff material with the following properties: EL = 70 GPa ET = 8.9 GPa GLT = 7.1 GPa ρ = 1400 g/cm3 νLT = 0.3 Using Equations (2.2) and (2.3) with νTT ′ = 0.4, the shear modulus and the minor Poisson’s ration in the transverse isotropic plane was obtained. For the head material νTL = 0.0225 and GTT ′ = 4.3 GPa while the shaft material had νTL = 0.0384 and GTT ′ = 3.2 GPa. No method to measure the composite layup was available, hence a layup of 0 and 45 degree plies where chosen with the static test as a reference. Both the head and the shaft was defined with a layup of (0/± 45/0)S , as a result from the calibration test, with a total of 8 plies. Further, the ply thickness for the shaft was set as 0.15 mm and for the head 0.225 mm. 3.4.3 Wood Material for Handle According to Salming, attempt has been made with handles consisting of carbon fibre. The reason why to this point these rackets is rejected by many players is due to the harsh vibrations that come with the carbon fibre handle. The material used for the handle was therefore chosen to be wood as it is the current standard. The following properties was used: E = 20 GPa ρ = 720 kg/m3 ν = 0.3 3.4.4 Material for Strings Just as for the racket frame, there are an abundance of different choices regarding strings. As the strings influence was beyond the scope of the project the material properties were fixed for all simulated models with the exception of the density which was altered for some models when the mass was modified. The strings were modelled with the properties of Nylon: E = 23 GPa ρ = 800 kg/m3 ν = 0.3 3.4.5 Cork Material for Shuttle Due to the simplification of the shuttle, the feather or synthetic material was neglected and only the cork material was modelled. Regarding the Poisson’s ration there was a large interval for the value in CES varying between 0.05-0.45 i.e. from almost fully compressible to incompressible. According to Silva et al., due to the 17 3.5. THE FINITE ELEMENT MODEL OF THE RACKET CHAPTER 3. METHOD structure of the cork material, the cell walls building the cork structure will fold when compressed. Consequently cells are not expanding in other directions and the resulting behaviour is that the material are close to fully compressible with a small Poisson’s ration [31]. Another way to illustrate this is by simply evaluate a glass bottle with a stopper made of cork, a wine bottle for instance. The cork do not bend above the edge of the bottle even if it have been compressed to fit the bottleneck, which further verifies the compressibility. The following values for the cork material were used: E = 30 MPa ρ = 700 kg/m3 ν = 0.05 3.5 The Finite Element Model of the Racket The following section describes the procedure used to model the racket prior to the simulation. Using the graphical interface in Ansa the FE-models were defined, suitable for the LS-Dyna environment, both when it comes to the computational domain but also the racket response when exposed to the swing simulation. 3.5.1 Defining Part Identities (PID) The CAD files for the frame and handle was loaded as stl-files and assembled in Ansa. As mentioned in Sections 3.3.2 and 3.3.4, the net and the shuttle had no 3D geometry given and where directly defined in Ansa. Due to the fact that the different parts and sections of the model required different mesh procedure, both in terms of element type and form, the domain was divided into different properties with belonging part identification (PID) and color. In Figure 3.14, an illustration of the distribution of PIDs can be reviewed and the definition of each PID is listed in Figure 3.15. Not only did the PID definition facilitate the meshing procedure, it also provided more options when boundary conditions were defined together with material definitions. For instance, one parameter to evaluate was to vary the center of mass which easily was done assigning the assigned PID materials different densities. Figure 3.14: Part definition by color in Ansa. Figure 3.15: List of PIDs in Ansa. 3.5.2 Meshing Procedure and Used Element Formulations A combination of solid, shell and truss elements was used to create a mesh of the racket. The wooden handle was modelled using first order shell elements with node based thickness. The shuttle was modelled with solid elements using the hexablock meshing procedure in Ansa. The advantage with this method is that the mesh structure can be controlled simply by alter the controlling boxes. Further the geometry was copied well without the use of tetrahedron or pyramid elements which both are less accurate types of elements. The head and shaft consists of composite material with a hollow structure with a constant wall thickness for each part. Further, the composite was modelled using first order shell elements with one element in the thickness direction. Thus, the layup was modelled using integration points for each composite ply. Furthermore, the shell elements were of reduced type to prioritise the efficiency. Finally the net was meshed with truss elements, a two node element that only support axial loading i.e. only translation DOFs are defined. The reason for using truss elements was based on the characteristics of the strings which, without tension, only provide support to axial tension. Further different attempts were made to introduce a pretension into the strings and the most simple way in LS-Dyna was to add the pretension using dynamic stress relaxation defined in the truss element card, which was another argument to use that element 18 CHAPTER 3. METHOD 3.5. THE FINITE ELEMENT MODEL OF THE RACKET formulation. In total, the number of elements was 4167. In Table 3.1 the type and number of elements are listed in detail for each PID. A mesh convergence study was done to evaluate the accuracy and the results are presented in Section 4.3. Part Element shape Type of Element Number of Elements Handle Quadratic Belytschko-Tsay(Reduced) 368 Handle Triangular Belytschko-Tsay(Reduced) 19 Shaft Quadratic Belytschko-Tsay(Reduced) 360 Shaft Triangular Belytschko-Tsay(Reduced) 1 Frame Quadratic Belytschko-Tsay(Reduced) 1340 Frame Triangular Belytschko-Tsay(Reduced) 19 Net 2D-bar Truss 1844 Ball Hexahedon Fully integrated, 8 node with rotations. 216 Total number of elements 4167 Table 3.1: Table with detailed information of the elements included in the meshed domain. The quality of the mesh regarding shape of the elements was measured using the parameters aspect, skewness and warping. Besides pure quality parameters, a minimum length was defined to prevent to large amounts of time steps and elements in the explicit simulation and also to prevent using very small elements close to larger ones. Table 3.2 consists of the limiting values for the quality of the mesh. Regarding aspect, the ideal case would be to have the value 1 which is also true for the warping when calculation for solid elements. For the skewness and shell warping angle the ideal value would be 0. Type of Element Quality param. Value Shell Aspect 3 - Skewness 30◦ - Warping 7 - Min. Length 1mm Volume Aspect 3 - Skewness 0.5 - Warping 10 - Min. Length 1mm Table 3.2: Table of the minimum values defining the limit for the mesh quality. 3.5.3 FE-model of the Badminton Racket Having a defined computational domain the properties of the model could be defined. Both in term boundary condition and contact behaviours and also part specific properties such as material and pre-stresses. Consistent Units One of the more important settings for the model was to ensure that the units were correctly defined and used in a proper way. That would otherwise lead to big flaws in the results from the simulations. For this project, the model was defined using the metric system with the units ms, MPa, mm, N and g. As a reference steel is defined by LS-Dyna support for each type of unit combinations with ρ = 7.83 · 10−3 and E = 2.07 · 105, i.e. every material used in the model need to have the density and Young’s modulus defined in the same manner [32]. Material Definition The material in the FE-model was defined using two different material cards in LS-Dyna. For the solid parts and the net, an elastic material was modelled using the material MAT01 in LS-Dyna in which Youngs modulus, density and Poisson’s ratio were defined. For the more complex composite material MAT116 for composite layup was used in which the different orthotropic properties were defined. 19 3.5. THE FINITE ELEMENT MODEL OF THE RACKET CHAPTER 3. METHOD This means that the material has different properties depending on orientation and consequently the material definition regarding orientation was crucial to model the composites correctly. For the shaft this meant that the direction of the longitudinal fibres was aligned with the shaft axis, see Figure 3.16, whereas for the head the orientation was set to be tangential to the inner oval circle defining the head shape of the racket, see Figure 3.17. To ensure that this orientation updated properly, i.e. following the deformation of the racket, invariant node numbering was used [27]. Figure 3.16: Definition of material orientation for the shaft. Figure 3.17: Definition of material orientation for the head. Properties Card and Element Type in LS-dyna As mentioned in Section 2.5.1 there are several ways of model a composite layup. To avoid the need of user defined integration rules and part specific definitions for each layer the *PART COMPOSITE option in LS-DYNA was used. The part environment gives a simple overview of properties for the layup regarding orientation, thickness, material and identification for each ply. The total amount of integration points and thickness of the elements included in the part are the sum of all entries [33]. Due to the extent of elements available in LS-Dyna the selection of elements used for the model was based on results from a previous thesis [34]. In a bench marking study of steel-composite structures, Andersson and Larsson [34] tested the efficiency and accuracy for the most recommended element formulations by performing a simulation test of a bending cantilever. For shell elements, the fully integrated element had almost twice the calculation time compared to the reduced element but with similar accuracy comparing to an analytic result. Hence, the reduced element was recommended and was also the one used for this project with an hourglass formulation to prevent hourglass modes. According to their test, using the reduced constant stress elements (ELFORM 1) an significant overestimation of the stiffness was obtained while the fully integrated elements (ELFORM 3) followed the analytic behaviour better. Thus the fully integrated element was recommended and even though bending will be a minor behaviour regarding the shuttle model this was the formulation used in this parametric study as well [34]. Damping The damping was modelled based on the eigenfrequencies, f , of the model. Using the knowledge from the eigenfrequency simulations, a frequency range which included the lowest eigenfrequencies was set for which the vibrations was damped. LS-Dyna recommend that using this procedure the ratio fmax/fmin should not exceed 300, thus the limit for this ratio was set with a margin to 250 for this project. Further this approach will slightly decrease the dynamic stiffness of the model. But since this error was similar for all models no increase of stiffness was made as it was the relative value between the models that were interesting. The damping ratio, which is defined as the fraction of critical damping, was set to 0.02. LS-Dyna recommend that no ratios equal or higher than 0.05 should be used for this type of damping [33]. Boundary Conditions - Motion The main objective for the FE-model was to capture the behaviour of the racket when swung in different badminton strokes. To enable the model to follow the motions given from the motion capture, the lowest nodes of the handle were defined as rigid with a rigid connection to an extra node placed in the middle, see Figure 3.18. This made it possible for the model to capture all motions including translations and rotations. This approach introduce local stress concentrations around the rigid nodes but since the results gathered from the simulations only consists of the elastic response with the stress distribution falling outside the scope, this does not affect the sought results. Further, the applied motions were placed far from areas that were of interest to evaluate. 20 CHAPTER 3. METHOD 3.5. THE FINITE ELEMENT MODEL OF THE RACKET Figure 3.18: Illustration of the nodes controlling the prescribed motion. The controlling node was prescribed in the translational and rotational degrees of freedom using the interpolated data for the motion capture data. By defining the curves controlling the prescribed motions as a separate file that was included in the model it was possible to, with ease, switch to different motions without any further alternations. Regarding the shuttle no motion or boundary condition was applied at start but the possibility was left open to move the ball using the restart environment in LS-Dyna. By defining the card *DEFORMABLE TO RIGID the shuttle could be altered between being rigid and deformable to prevent any internal stresses and deformation to be introduce when the ball was to be moved into position before impact. As the ball was supposed to be hit with different orientations when simulating the strokes these features was desired preventing the shuttle to move away from the expected location when releasing the boundary condition. Contact Due to the different included parts there were several connections needed to be defined. The different parts of the racket had to be coupled with contact conditions but also the ball to net contact needed a definition. For the contact between the net and the frame it was assumed to be a tied contact between the inner frame surface and the most outer nodes for the net, i.e. there was no sliding between these parts. A fairly accurate assumption as the net in reality is strung in small holes around the frame. The only slide possible in the physical model was in and out of the hole, this will not alter the distance between each part of the net. Further, the contact between the shaft and the handle was modelled with three connections, which is illustrated in Figure 3.19, preventing the shaft to slide out from the handle. This way, the force will be transferred in the top connection creating a stress concentration assumed to leave the interested results unaffected. Figure 3.19: Illustration of the nodes defining the contact between shaft and handle. Pretension of the Strings As mentioned in Section 3.5.2 when the strings are attached to a badminton racket they are initialised with a pretension of 90-120 N depending on the player skill and requests. The racket in this project was modelled with a tension corresponding roughly 105 N, i.e. in the middle of the two extreme values. The truss element card in LS-Dyna model this is defined using a dynamic relaxation of the stress in the strings, i.e. a pre-stress of 140 MPa was introduced in the model. 21 3.6. PARAMETER STUDY CHAPTER 3. METHOD 3.5.4 Alternations for Model Used in the Implicit Eigenmode Analysis Apart from the above defined FE-model, two others was used to capture some other behaviours that were needed for the interpretation of the simulated results. The first one was the model used to calculate the eigenmodes for the a clamped racket. All boundary conditions were removed, except the clamped bottom of the handle, including the shuttle and belonging contact definitions. Further, the damping was removed as it was unnecessary since the eigenvalues was calculated using an implicit static simulation. Apart from these simplifications the same pre-stress and material properties were used. 3.5.5 Alternations for Rigid FE-model The third model was a rigid FE-model for the racket with the purpose to capture the motion behaviour for troubleshooting but also as a reference when analysis the deflection data. The model was made rigid using the LS-Dyna material MAT20 for all parts. To be able to control the time step in the simulation, one element separated from the racket model was defined with an elastic material as LS-Dyna is unable to define the time step using MAT20. This was defined with the element length of 7 mm, Young’s modulus of 210 GPa, a viscosity of 0.3 and the density of 115 kg/m3 giving a time step of 4.46 · 10−3 ms. The contacts between the different racket sections were defined using rigid connections leading to a racket that moves as one rigid body. When prescribed a motion for a rigid body in LS-Dyna, the rotations are defined around the coordinate system for the center of gravity of the affected part. To get around this a small controlling sphere, see Figure 3.20 was created in the bottom of the handle with the middle point coinciding with the controlling node for the deformable model. By setting a heavy weight for the sphere and a low weight for the rigid racket, the combined rigid body had a center of gravity in the center of the sphere. The boundary conditions were then defined on this sphere resulting in the same reference system as for the deformable model. Hence, the same motions are applied for the rigid model as for the deformable model. Figure 3.20: Illustration of controlling ball for rigid model. 3.6 Parameter Study To find the impact of the design parameters of the racket, several models were analysed in the parameter study, see Table 3.3. The chosen parameters in the study were taken from conclusions in previous studies. These are: · Head shape · Center of gravity · Weight · Shaft stiffness · Head stiffness The models in the study are the original racket, which is based on a typical modern isometric racket with data from the bending test, and modifications upon it. Each modification focuses on one parameter being altered while keeping the other as in the original model. This isolation of parameter change was done to isolate the impact it has on the racket performance. For all parameters, excepted shape, the change was both to a greater and a lesser value. A greater value of centre of gravity was defined to as the point moving closer to the 22 CHAPTER 3. METHOD 3.7. SIMULATIONS Model Head shape Center of gravity* Weight Shaft stiffness Head stiffness Original Isometric 301.1mm 84.4g 70 GPa 160 GPa Stiff shaft Isometric ±0 ±0 98 GPa ±0 Weak shaft Isometric ±0 ±0 42 GPa ±0 Stiff head Isometric ±0 ±0 ±0 224 GPa Weak head Isometric ±0 ±0 ±0 96 GPa High CoG Isometric 331.8mm ±0 ±0 ±0 Low CoG Isometric 270.2mm ±0 ±0 ±0 High mass Isometric ±0 93.3g ±0 ±0 Low mass Isometric ±0 76.3g ±0 ±0 Prince Y-shaped ±0 ±0 ±0 ±0 Oval Oval ±0 ±0 ±0 ±0 Table 3.3: The racket models which were simulated and their parameter values. When shown ±0 this means no deviation from the Original model. *from bottom of racket. head and further away from the shaft. The head shapes chosen are the ones described in Section 2.2: isometric, y-shaped and oval. The parameters were altered within different intervals. For the centre of gravity initial simulations showed that small alternations gave large differences. Thus, the interval was set as 10% lower or higher than the original position. The weight for existing rackets varies within small ranges. Consequently, the evaluated interval was defined as 10% lower or higher than the original. For the shaft stiffness initial simulations indicated that a higher variation was needed. Thus, the head and the shaft stiffness was evaluated for values 40% higher and lower than the original Young’s modulus. 3.7 Simulations The defined models were exposed to different simulations to obtained the necessary data for the parametric study. The simulations are defined over a specified time domain, thus the used time step required a definition. Each simulation explained in this section had a specific purpose as the combination of these defines the foundation on which the characteristics of a badminton racket could be defined. 3.7.1 Simulation Settings The simulations were calculated using an explicit method due to the dynamic environment of a badminton stroke. The time step for the simulations was governed by the wave propagation in an element according to Equation (3.1), where l denotes the minimum element length for each element. ∆tmin = l c (3.1) The parameter c corresponds to the wave propagation in a 3D-continuum, denoted cs in Equation (3.2) or the speed of sound in the material which is calculated as ct in Equation (3.2). Which c that LS-Dyna use depends on the effected element. For solids and shell elements, the value cs is used whereas for the truss elements, the value ct is used. cs = √ E(1 − ν) (1 + ν)(1 − 2ν)ρ ct = √ E ρ (3.2) Consequently, the time step for each simulation was controlled by a single element, i.e. the smallest element that had the lowest stiffness and the highest density or the worst combination of these gives the lowest time step. With other words, to reduce the simulation time the smallest element was needed to be scaled up or the material properties had to be changed. Due to this limitation, a slight change in the time step was expected for the simulations where the model used different shapes that could lead to smaller elements or the models with alternated material properties. 23 3.7. SIMULATIONS CHAPTER 3. METHOD As the models were using large amounts of time steps, well above 300 000, it was recommended to use double precision in the simulations to prevent round-off deviations. Even if the errors for each time step were extremely low, the amount of steps could lead to misleading results. In terms of simulation time this adds about 30 % which was considered to be an accepted amount for a greater accuracy. No mass scaling was used as even small mass differences could influence the result noticeably. The data, including deformations, accelerations, forces etc., was sampled with a frequency of 100 times per millisecond and the states which later were evaluated in Meta were sampled ones every other millisecond. Racket model Rigid analysis Eigen analysis Flexible analysis Clear swing Smash swing Clear swing Smash swing No shuttle Shuttle 1 Shuttle 2 Shuttle ... No shuttle No shuttle Early Original Late No shuttle Shuttle 1 Shuttle 2 Shuttle ... Original Late Later Sweet spot analysis Figure 3.21: The simulations which are run for each racket model. 3.7.2 Execute Simulation For each racket model several simulations were run. These consisted of an eigenfrequency analysis, a rigid body analysis and a flexible body analysis. The rigid body analysis was simulated twice using two different swing patterns and the same was done for the flexible body, which in addition to the normal swing also simulated 7 shuttle impacts. Additional sweet spot analysis and changed impact timing of shuttle 1 was also performed, see Figure 3.21. The normal swing simulation and the shuttle impact simulation are identical up to the time when the racket strikes the shuttle. The time needed to run the several impacts could hence be greatly reduced by using the information from the normal swing. The normal simulation was run until 10 ms before impact where it was stopped and a restart file was created, which enables the simulation to restart from that state. This information was stored to be used for running each ball simulation. The normal simulation was then started from the restart state and ran until finished. To simulate the impact between the shuttle and racket the shuttle needed to be positioned. The time between the restart state and impact was used to move the shuttle into position, without introducing any stresses to it. This means that all flexible body simulations did contain the shuttle and that the normal simulation intentionally missed it. 3.7.3 Eigenfrequency Analysis The modal response for the racket was obtained by an eigenfrequency analysis. The different modes oscillates at different frequencies. During for instance a hit the experienced response is a combination of these modes. Furthermore, each mode has an individual shape which was evaluated to fully understand the influence of the vibration regarding the subjective feeling when using the racket. Li et al. [35] suggest that for tennis rackets it is preferential to design with the target to obtain the lowest bending mode which include the highest amplitude and consequently the most amount of energy. Further it is showed that hitting a ball at the dead spot, i.e. where the lowest rebound effects are, give the peak in feedback for the player [35]. Hence, part of the players judgement on whether the stroke is performed correctly is governed by the vibrations of the racket. These studies indicate that eigenmodes is important to understand the characteristics of a racket, thus evaluating the influence from different parameters and how they affect the vibrations of the racket was desirable. 24 CHAPTER 3. METHOD 3.7. SIMULATIONS The simulation was done using the implicit solver. Based on the earlier mentioned FEM-model a simplified version was defined suitable for the implicit eigenmode analysis which is described in Section 3.5.4. The modal analysis was then performed on the clamped racket giving the vibrations of the racket. 3.7.4 Deflection Analysis Badminton is said to be the fastest ball sport played and a part of the explanation of this is the deflecting behaviour of the shaft. Due to the slim design, the shaft deflects when accelerated. This phenomena creates a spring effect during the stroke which could lead to higher velocities of the ball. Thus, it was interesting to evaluate the deflection behaviour of the racket to see how the shuttle was effected for different flex. For each type of stroke simulated the deflection analysis requires the full swing to analyse the deformation of the racket. Thus for each parameter, a total simulation of the stroke was needed. Consequently, the method for evaluating the parameters needed to be defined prior to the simulations to estimate the required calculation time needed to enable all simulations to be executed. The method to obtain the deflections was to compare the deformable racket with a rigid model, i.e. for each swing and geometry a rigid model had to be simulated. However, the rigid model, consisting of the rigid material MAT22 in LS-Dyna, was much more time efficient to compute. 3.7.5 Sweet Spot Analysis for Swing The velocity and path of the shuttle after a stroke largely depends on the string bed. The tension of the strings, geometry of the frame and the density of the strings are all parameters controlling the response from the stringed area, not to mention where on the stringed area the contact to the shuttle occurs. Thus, depending on the desired outcome when hitting the shuttle there could be different optimum characteristics of the strings. With this in mind the sweet spot of the simulated racket was analysed during the swing to predict the response for the different parameters. The sweet spot is defined as the area on the strings with the highest COR-value, calculated according to Equation 2.1.3. In Figure 3.24, a typical result from a sweet spot analysis is illustrated. The most yellow part shows the area where the COR-value is at its peak, i.e. this area is the sweet spot for this specific racket. As illustrated in Figure 3.22, to capture the spread of the sweet spot the last part of the swing for each racket was simulated using different contact positions for the shuttle. With this setup the ball behaviour after hit was analysed by observing the velocity before and after hit together with the path of the shuttle. One theory was that, as in a lot of sports, the contact time governs the amount of force that is transferred between the two object in contact. Thus, the contact time for the shuttle-to-strings contact was analysed for the different strokes and shuttle hits. By subtracting the rigid deformations and the deflections from the simulated stroke without any shuttle, the deflections due to the shuttle contact was obtained. Figure 3.22: Illustration of the dis- tribution of the shuttle impacts for the swing. Figure 3.23: Illustration of the shuttle distribution for sweet spot simulation. 7 8 9 9 10 10 11 11 11 12 12 12 12 13 13 13 13 14 14 14 vmax = 14.5 vmin = 6.2 Original 5 6 7 8 9 10 11 12 13 14 15 Figure 3.24: A typical result of dif- ferent rebounding velocities from the sweet spot analysis. 3.7.6 Sweet Spot for Clamped Racket To get a better resolution and understanding of the COR and re-bounce behaviour depending on impact location, another sweet spot analysis was made using a clamped racket without any applied motion. This simulation neglects the deformations otherwise occurring due to the swing. Thus, is gives a better picture of 25 3.8. METHOD FOR POST-PROCESS AND ORGANISE DATA CHAPTER 3. METHOD the string bed behaviour if the racket is undeformed which is more realistic when including strokes with a much lower accelerations such as drop strokes. In Figure 3.23, the distribution of the tested impact locations is illustrated using a total of 26 different locations. The simulations were done using all the different rackets and each simulation with a new ball is initialised with a impact speed of 20 m/s. The amount of simulations required to obtained the data was 286 st. 3.7.7 Time Window Analysis for Shuttle Impact Due to the deflection behaviour of a badminton racket, if the results indicate that the deflection is affected by the different parametric alternations an additional simulation was of interest. By moving the ball backwards and forward in the path of the racket swing an earlier or later shuttle impact would be simulated. This wound be done in a similar way as for the sweet spot analysis for the swing with restart files. Although the simulation was limited to the centred shuttle to minimise the amount of simulations. The time of impact and how the shuttle position should be changed was depending on the racket head velocity and the racket deflection as the peak values should influence the shuttle the most. 3.8 Method for Post-process and Organise Data The parameter study generated a great amount of data which needed to be post-processed for the parametric impact. To prevent faults occurring from human errors during the post-processing a script was written and used. It ensured that the data was extracted from the study in the same way for each simulation. The data it extracted was: · the position of specific points on the racket throughout the swing. · the position of the ball center node, to capture the post-impact velocity. · the data for the contact between the ball and the net, to get the amount of time of contact. · the force applied to the controlling nodes, to find the force needed to swing the racket. · images of the racket from the simulation. · videos of the racket from the simulation. · data and videos of the racket’s eigenfrequencies and eigenmodes. The chosen specific points on the racket can be seen in Figure 3.25. These are divided into three subdivisions: head, shaft and handle and given a name containing three letters for the shaft and five letters for the head and handle. For the head points, the first letter is h for head, the middle three indicating if positioned on top, middle or bottom of the head and the last letter if it is on the left, right or middle of the head width. The shaft uses a similar naming convention; s for shaft, then a number which incrementally grows with the distance from the handle and the last letter if on the left or right side. The handle uses the same type as the head, but the first letter was swapped to a g for grip to prevent confusion. On the shaft and handle, the points are placed in pairs. This is to work around that the shaft and parts of the handle are hollow which means that there are no points and therefore no position data there. By having the node pairs a value for the virtual point between them can be positioned. 3.8.1 Post-processing Result in Meta The software used to post-process the simulation results was Meta. It allows to record the actions used to post-process a result and can then repeat the same actions on other results. This was combined with Meta’s Python API which allows the recorded actions to be combined with Python code into a batch script. The batch script was triggered to run when a new result was generated and extracted the wanted data. The data extracted by the Meta batch script was still too much to be handled manually. It was therefore imported into MATLAB to compare data and present it in figures and tables. 26 CHAPTER 3. METHOD 3.9. POST-PROCESSING IN MATLAB Figure 3.25: The points on the racket from which data was extracted in Meta. Figure 3.26: The coordinate system used to describe the deflection of the racket. 3.9 Post-processing in MATLAB In MATLAB the final post-processing of the simulation results was made. The data put into MATLAB was: · nodal positions of racket in rigid and flexible body simulation. · shuttle position. · force data for contact between shuttle and racket. 3.9.1 Comparison of Input and Output The most essential result from the model was that it behaved similarly to the real racket. To evaluate this, the motion capture input data from the C3D files was compared to the node motion in the FE-simulation. The most interesting node to follow was the node at the top of the racket, as it is furthest away from the motion controlled bottom node. 3.9.2 Deflection of the Racket The deflection of the racket in the simulation was defined as the difference in nodal position between the rigid and flexible model. As both models were swung using the same motion pattern, if the flexible body did not deflect it would perfectly co-inside with the rigid body. The deflection was calculated using three degrees of freedom: forward, side and rotational deflection, see Figure 3.26. Any deflection along the direction of the racket axis was neglected and seen as the result of the racket being bent forward or sideways. The forward and sideways deflection were calculated by taking the vector result subtracting nodei,rig − nodei,flex and then projecting it onto the forward and sideways directional vectors. The rotational deflection was only calculated for the head of the racket, which is where the greatest rotation would be found. This was done first by aligning the racket’s rigid and flexible centre head nodes, hmidm, with each other. Then, by examining the angle between the vectors pointing to the rigid and flexible left side head nodes, hmidl, and the angle between the vectors pointing to the rigid and flexible right side head nodes, hmilr, two angles were found. These were then averaged, as they always were very similar, to get one value of the rotational deflection of the racket about its axis. 27 3.9. POST-PROCESSING IN MATLAB CHAPTER 3. METHOD 3.9.3 Shuttle Motion and Contact Movement data for the shuttle was gathered from the simulation. This was used to find the speed and the trajectory of the shuttle. The angle of the trajectory was only measured in the horizontal plane, i.e. only looking how much to the left or right the shuttle went and disregarding upwards or downwards. This was done to make it easier to understand the difference in trajectory and as this was deemed to be the most crucial angle of error. The trajectory was deemed to be straight, having 0◦ offset, when it followed the trajectory of shuttle 1 for each separate FE-model. This was used as it was hard to decide a true 0◦ offset, where knowledge would be needed of where the player in the recorded motion was actually aiming. The trajectory of the other shuttles, shuttle 2 - shuttle 7, are then evaluated against the trajectory of shuttle 1 to find the angular offset. The information about the contact between shuttle and the racket was given by the contact force data, which was nonzero when there was contact. 3.9.4 Sweet Spot Analysis For each racket 26 shuttle impacts were performed in the sweet spot analysis. The position data for the shuttle post impact was used to get the velocity of it. This was used to create a table containing the average, standard deviation, minimum and maximum value of the shuttle velocity for each racket model. The data was also used to create plots to create contour plots for the area within the outer ring of impact locations. To calculate the values between the data points the cubic interpolation technique inside MATLAB’s griddata function was used. 28 4 Result In this section the results from the simulations described in Section 3.7 are presented. The first part includes the validation for the global behaviour regarding motion and stiffness. Sequentially, the vibration behaviour for the different models are evaluated in terms of eigenfrequency. The two simulated strokes are evaluated for the different parameters and how these impact the performance separately. Throughout the report, the racket denoted original racket is defined as the reference racket. Hence, the alternations are measured against this racket. This means that the data in each table is the difference compared to the original racket and the absolute value given is for the original racket only. Absolute values for all racket simulations can be found in Appendix A, but as the FE-model was created without knowledge of the true material data only the relative impact of each parameter is considered. 4.1 Racket Motion The applied motions were from two separate motion captures described in Section 3.2; one of a clear swing and one of a smash sw