4. Equipment Rules for Tennis Rackets ______________________________ 3 4.1 Qualities of modern tennis rackets ______________________________ 4
5. Kinematical analysis of the Tennis game___________________________ 9 6. Bio-mechanical analysis of the Tennis game ______________________ 12
7. Theoretical grounds of alternative phase models. __________________ 15
7.1 General conditions __________________________________________ 15
7.2 Collision of mechanical objects ________________________________ 17
7.3 Loading the tennis racket_____________________________________ 21
9. Grounds for setting up the principle concept and
Reference list ___________________________________________________ 46
In some ways tennis has become a truly high tech sport. Advances in racket design and composition have changed the speed of the game from the days of wooden frames, and made the game easier and more enjoyable at the recreational level.
Television coverage of matches makes use of sophisticated statistical analysis to explain the outcome of matches in terms of winners, errors, serving percentages and ball placement Players study the statistical tendencies of opponents in an effort to get an edge.
But at the most basic level of understanding there is no clear picture of what happens to the tennis ball during the course of a tennis match. The radar guns give us an initial velocity of the serves, but beyond that very little is really known about what happens, when players strike the ball in the game. New types of strikes result in complicated rotation of the ball. For example:
Increasing service speed of ball shootingTop spin resulting in maximum upper rotation
Slice resulting in simultaneous bottom and side rotation
Tennis rackets nowadays need to be constantly re-developed. Design and manufacturing procedures linked with the most essential equipment a tennis player can have, are subjected to aerodynamical change. This book seeks to study tennis from a new perspective – the perspective of design aerodynamics, setting out to determine what we can learn about the flight of the tennis ball. When the ball hits the racket during play, some dynamic stresses and deformations occur with the load periodically changing [1]. This results in oscillations (vibrations) in the racket and under certain conditions the phenomenon “resonance” might occur during which the stress and deformations might suddenly rise to dangerous values. When no resonance conditions exist the periodically changing stress might lead to damages in the racket caused by gradually forming cracks in the material [1].
2. Project Identification and DefinitionThe wider popularity of tennis as a sport game on a world-wide scale, the increasing quality of the play of both professional and amateur players has set grounds for expanding the production of aids and accessories, such as tennis rackets, for this game.
There is a wide variety of tennis rackets presently available on the market and these could be basically assessed on the basis of the following characteristics:1. Type of material used for manufacture
2. Size and dimensions of racket head
3. Handle length
4. Racket weight
5. String tensioning
6. Degree of damping and vibrations
The first 5 characteristics are quantitative and could easily be determined and controlled using common measuring equipment. The sixth characteristic, - vibration resistance, is the appealing element. The value of these vibrations is important and is hard to determine with formulas due to the complexity of the friction process, the friction and the air resistance losses [2].
Tennis Ball data such as, the ball’s size, the ball speed and the rate at which the ball spins have to be evaluated as well as other distinguishing shape features.
The tennis ball is roughly a 63.5-mm (2.5-inch) sphere. It has a uniform outer surface, a continuous hourglass shaped seam and a felt-like fuzz. The ball has to be white or yellow in colour. It has to be more than 56.7-grams (two ounces) and less than 58.5-grams (two and one-sixteenth ounces) in weight. The average professional tennis player serves the ball at 120kmph and the spin of the ball during a serve can reach around 1000 rpm [2].
The seam of the ball is a very important issue to deal with. International equipment regulations dictate for the minimum seams to appear in the surface of a ball. Furthermore, these should be stitchless. The seam makes the surface of the tennis-ball very porous and non-uniform.
My interest on the seam feature and its consequences led me to start simulating the tennis-ball. Computers are used extensively throughout aeronautics to aid in the design and analysis of aircraft, watercraft, and even sports equipment. Scientists have developed mathematical models, which emulate the physics of fluids [3]. These mathematical fluid models are programmed into software applications, which can be used on the computer.
An engineer can take an object, like an aeroplane or ball, and use the fluid model to analyse how this object would perform in "flight". It is like having a virtual wind tunnel. There is a special name for this type field of work. It's called "Computational Fluid Dynamics" or "CFD" for short. We decided not to include the whole simulation procedure step by step in this section. We would like though, to feature some of the set up made.
Figures 1 and 2 above show two different cases. Comparing them, we could find differences in the ball’s speed and spin.
Figure 1 Figure 2 4. Equipment Rules for Tennis Rackets
Rackets failing to comply with the following specifications are not approved for play under the Rules of Tennis:
a. The hitting surface of the racket shall be flat. It should consist of a pattern of crossed strings connected to a frame and alternately interlaced or bonded where they cross. The stringing pattern shall be generally uniform, and in particular not less dense in the centre than in any other area. The strings shall be free of attached objects and protrusions, other than those utilized solely and specifically to limit or prevent wear and tear or vibration, and which are reasonable in size and placement for such purposes [4].
b. For professional play, the frame of the racket shall not exceed 73.66-cm (29 inches) in overall length, including the handle, as from 1st January 1997. For nonprofessional play, the frame of the racket shall not exceed 73.66-cm (29 inches) in overall length including the handle, as from 1st January 2000. Until 1st January 2000, the maximum length of a racket for non-professional play shall be 81.28-cm (32 inches). The frame of the racket shall not exceed 31.75-cm (12 1/2 inches) in overall width. The strung surface shall not exceed 39.37-cm 15 1/2 inches in overall length, and 29.21-cm (11 ½ inches) in overall width [4].
c. The frame, including the handle, shall be free of attached objects and devices, other than those utilized solely and specifically to limit or prevent wear and tear or vibration, or to distribute weight. Any objects and devises must be reasonable in size and placement for such purposes.
d. The frame, including the handle, and the strings, shall be free of any device, which makes it possible to change materially the shape of the racket. Or to change the weight distribution in the direction of the longitudinal axis of the racket, which would alter the swing moment of inertia, during the playing of a point [5].
Being familiar with the basic performance characteristics of modern tennis rackets contributes to the most suitable choice of a racket for both the beginner and the professional player considering his individual nature (physical characteristics, skills, manner of playing, etc.) and the requirements in the competition (court hardness, ball brand, etc.). This knowledge also contributes to derive the best maximum of a given racket. Or, in the language of “profies” – to “squeeze the maximum capabilities out of a racket” [6]. To achieve this leading players even order specially made rackets with some modifications to the basic model.
On the other hand, the progress in the development of rackets results in changes of the manner and skills of the player. The striking technique for a wooden racket is very much different to that of a racket made of modern composite materials. Every revolutionary change to the rackets results in modifications in striking techniques [6].
Modern rackets are manufactured and tested mainly to provide the following basic mechanical and dynamic customer performance characteristics:Racket Length
This is the distance from the tip of the racket to the end of the handle. A common tendency is to increase the length of the racket to provide more powerful strike and dynamic inertia. To prevent uncontrolled increase of racket lengths ITF approved a document at its congress in Lozane in 1997 according to which professional players can use rackets of maximum 29 inches. Long Body rackets are made of 28, 28.5 and 29 inches length [6].
This parameter is:
− directly proportional to the length, weight, dynamic inertia and power; − inversely proportional to body rigidity, manageability and control.
String surface area – head size
This is the surface area of the string flat measured on the inside of the rim. This is measured in square centimetres or square inches. Various companies offer rackets of 500 to 750 sq. inches for this parameter [6].
This characteristic is:
− directly proportional to the length, dynamic inertia, power and control;
− inversely proportional to the rigidity of the string area flat, body rigidity and manageability.
Balance
This is the distribution of the static weight of the racket along its entire length. It is an indication of the distance measured from the low end of the handle to the cross line where the racket would stand horizontally if we supported it at this cross point. Higher values indicate increased weight on the head on the account for the weight on the handle and lower values indicate the opposite balance of weight. The lightest racket heads for adults have a balance figure of 300 and the heaviest heads have a balance of 380-mm [6].
This parameter is:
− directly proportional to the length, weight, dynamic inertia and power;
− inversely proportional to manageability and control.
Weight
This is the static weight of the racket measured in grams. Kid’s rackets are the lightest. Their weight starts from 200 gr. Professional rackets are the heaviest. Their factory weight values come up to 367 – 370 gr [7]. Professional players often attach additional weights to the head of the racket to change its balance and add to the weight of the racket.
This parameter is:
− directly proportional to the length, dynamic inertia and power; − inversely proportional to manageability and control.
Knit
This is the number and positioning of the string knit openings in the body of the racket, which determine the density and structure of the string area. According to the number of knit openings modern rackets usually have between 16 and 20 longitudinal strings and 18 to 22 transverse strings though there are some exceptions. According to the positioning of the string openings some rackets feature uniform and other, nonuniform and denser in the striking zone (centre) string area [7].
This parameter is:
− directly proportional to the rigidity of the string area, the dynamic inertia and control
− inversely proportional to tensioning and power.
Tensioning
This is the tension applied to the string when knit. Tensioning is usually performed within 18 to 35 kg. Tension is “equal” when longitudinal and transverse strings are identically tensioned and is “differential” when longitudinal strings are tensioned heavier than transverse ones. This difference is usually between 1 and 2 kg and is defined by the shape of the head of the racket [7]. Rackets featuring larger differences in the length of longitudinal and transverse strings are usually knit using different tension.
This parameter is:
− directly proportional to the rigidity of the string area and control
− inversely proportional to knit, dynamic inertia and power.
String area rigidity
This is the pressure pliability of the knit string area not counting the racket body (rim) elasticity. This property depends on the knit, tensioning and the shape and size of racket head. This is resultant characteristic and is an exact indication of the racket knit. It is more important to know the final result of the knit and not the exact value of tensioning applied to the string when knit. The most pliable surface has a figure of 51 for this parameter and the hardest one comes up to 75 kg [7].
This property is:
− directly proportional to the knit, tensioning, body rigidity and control; − inversely proportional to the surface area of the string flat, the dynamic inertia and power.
Racket body rigidity
This is the pressure pliability of the racket body. This property depends on the manufacturing materials and technology and the shape of the racket body. For professional rackets this figure is between 50 and 75 kg [7].
This property is:
− directly proportional to the dynamic inertia, manageability and control;
− inversely proportional to the length, head size and power.
Dynamic inertia
This is a set of energetic characteristics of the racket indicating how the racket transfers the energy from the hand during the movement. Until very recently this property was disregarded from measurements and was only determined and sensed by the player himself. Now that computer measurement has been introduced into the tennis sport this is a characteristic that is being followed very closely and is the latest feature established so far. It is dependent and functionally incorporates all other racket properties indicating the result of the combination between them. Satisfactory incorporation of opposite parameters is achieved by distributing the mass and elasticity along the racket mainframe. Leading racket brands are made lighter to provide better control and manageability but at the same time they provide much better ball acceleration characteristics due to the improved quality of the dynamic inertia. This makes the racket powerful and precise (control and manageability) simultaneously. This is achieved by improving the balance characteristics and implementing new materials, such as titanium or carbon fibre compounds. This parameter usually varies within 290 and 360 units [8].
This parameter is:
− directly proportional to the length, string surface area, balance, weight, body rigidity and power;
− inversely proportional to the knit, tensioning, string area rigidity, manageability and control.
Manageability
This is the ability of the racket to obey to the movement of the player’s hand. The aim is for the player to exercise minimum effort to change the position of the racket and the string area surface. This is measured within 50 and 100 gr [8].
This property is:
− directly proportional to the body rigidity and control;
− inversely proportional to the length, head size, balance, weight and dynamic inertia.
Power
This is indicated by the portion of energy being transmitted from the racket to the ball or at minimum effort from the hand to have the racket transmitting maximum energy to the ball. Modern rackets have figures for this parameter between 25 and 80 units [8].
This property is:
− directly proportional to the length, head size, balance, weight, body rigidity and dynamic inertia;
− inversely proportional to the knit, tensioning, string area rigidity, manageability and control.
Control
This is the ability of the racket to provide precisely (and quickly) the right direction, trajectory and rotation to the ball. Considering the relative share of unprovoked mistakes this is a very essential parameter. For modern racket this parameter varies within 40 and 80 units [8].
This property is proportional to:
− directly proportional to the area and size of the head, to the knit, tensioning, string area rigidity, body rigidity and manageability;
− inversely proportional to the length, balance, weight, dynamic inertia and power.
The following comparative table of the basic characteristic parameters of most commonly used professional rackets indicates how manufacturers achieve the combination between opposite characteristics and what are the trends in the development of these qualities.
Table 1: The comparative table of designated basic characteristics of some of the widely spread professional rackets shows how manufacturers combine opposite qualities and what is the tendency in the development of these qualities[9].
1 2 3 4 5 6 7 8 9 10 Model Balance Weight TensioString flat Body ning hardness hardness Dynamic ManagePower Control inertia abilityFisher
Pro 1 316 367 28/28 75 60 340 60 29 70
Pro98 316 362 28/27 63 65 317 75 49 59
Pro 105 322 332 29/28 67 54 317 75 29 73
Classic 98 323 341 25/24 49 69 328 68 57 39
R.Titanium 500 335 305 29/27 65 60 304 84 33 69
R.Titanium 700 375 286 29/28 60 60 313 78 38 64
Head
Classic 600 316 342 28/27 51 58 316 55 55 45
Prestige Tour 660 325 340 29/28 69 55 326 69 27 74
Extrema Compet. 365 284 29/28 63 58 312 79 34 69
Radical Tour 690 326 334 29/28 58 58 321 73 38 63
Radical Titanium 630 335 314 28/28 65 63 310 80 35 66
Titanium S 2 363 265 29/28 61 71 308 85 42 59
Titanium S 6 380 259 30/29 51 69 316 76 51 51
Prince
Thunder Lite 335 294 29/28 61 60 318 75 36 66
Michael Chang 320 333 28/28 54 64 330 67 48 52
M.Chang Titanium 330 331 29/28 55 66 347 55 55 45
Precision Titanium 330 330 29/28 60 61 340 60 42 59
Precision 720(L.B.) 325 339 28/28 57 64 339 61 47 53
Wilson
Pro Staff 6.0 308 364 27/26 49 62 316 76 48 47 Pro Staff 6.1 310 366 28/27 70 68 340 60 40 59 Pro Staff 5.0 Carbon 306 345 28/28 56 74 325 70 55 43 Pro Staff 6.6 Carbon 340 300 29/29 58 60 310 80 42 61 Hammer 4.3 Carbon 360 275 29/28 63 71 310 80 45 57 Hammer 5.3 360 324 28/27 74 57 356 80 42 61 Hammer 6.4 373 277 28/27 52 73 321 73 55 46 Hammer 7.1 314 331 27/26 65 60 314 69 43 59
Yonex
Super RD 7 316 326 28/26 59 58 296 89 38 60 Super RD 500 334 303 28/26 57 58 297 89 37 62 Titanium 600 365 304 29/27 56 67 345 57 55 47
From Kinematical point of view tennis can be defined as “non-cyclical locomotive” movement, accompanied by impact interaction between the additional final link in the kinematic chain of the body (the racket) and the ball. The impact interaction results from the dynamic load on the muscles moving the arm with the racket and also, from the static load on the muscles involved in the grip. In other words, the player holds the racket tightly in his hand and moves along the court and strikes the ball with the racket [10]. The movement of the player is different in amount and intensity and is aimed at choosing the most favourable
position for the player to accomplish the strike.
h0
maximum (or optimum) speed and the right flying direction. h1 The speed of the ball after the strike is dependent on the Figure 3 speed of the striking link (the racket) and the ball, and also
on the ratio between their weights and extent of linking
between the racket and the body of the player.
The interaction between the ball and the racket is a complicated process itself, which undergoes two characteristic stages: from the beginning of the strike the striking impulse deforms the ball, it flattens until its speed becomes equal to 0. The elastic forces of the ball then restore its shape and its speed increases to a value, which can vary dependent on the speed of the striking body.
The strikes involved in tennis, both between the ball and the racket and between the ball and the playground can be referred to the so-called “incompletely elastic” collisions [10]. In these the restoration coefficient is lower than 1 (Figure 3) and dependent on the position of the hand when it hits the ball strikes can be either direct (forehand) or back (backhand).
h0 – the height before the strike (the ball drops from a height of h0); h1 – bouncing height (after the strike)A major significance for both types of striking positions is given to the vector of the speed of the racket at the instance of the strike relative to the common centre of gravity of the ball (C.C.G). Two types of strikes are distinguished dependent on this: central and tangential strike. When the striking impulse goes through the C.C.G of the ball its movement after the strike will be only progressive. The strike in this case is referred to as direct [11]. Such type of strike is accomplished always when the string area flat is perpendicular to the direction of racket movement during the strike.
When the striking impulse does not go through the C.C.G. of the ball this results in a striking impulse momentum. This causes a spinning motion around the C.C.G to be added to the otherwise progressive ball movement. Such kind of strike is referred to as tangential (oblique) [11]. The angular speed of the ball is equal to the ratio between the striking impulse momentum and the moment of inertia of the ball relative to its CCG. Such strikes result always when the string area flat surface is at an angle smaller or larger than 90° from the racket movement direction during the strike.
ababFigure 4 Figure 5
Some role in adding the spinning motion of the ball is played by the friction between the ball and string surfaces. While the two bodies are in contact the racket rubs into the ball following its movement thus increasing its angular speed additionally [11].
The tangential strikes can be classified into two major groups dependent on the direction of rotary motion induced to the ball, when the strike is accomplished under the level of the shoulder.
In such strikes it is possible to induce spinning motion to the ball in relation to the horizontal axis and in the direction of the progressive movement of the ball (top-spin movement, Figure 4)
or into the direction opposite to the progressive movement of the ball (slice movement, Figure 5).2a – direction of rotation relative to the progressive ball movement during topspin strikes;
2b – direction of rotation relative to the player during topspin strikes;
3a – direction of rotation relative to the progressive ball movement during slice strikes;
3b – direction of rotation relative to the player during the slice strike.
Inducing various additional spinning motions to the ball – topspin, slice, topspinside, side slice and sidespin, results in a change of the flying trajectory and different bouncing characteristics when the ball contacts the racket string surface [12]. This makes it harder for the opposite player to respond to and makes the tennis game more spectacular. Therefore, it is necessary for all tennis players to learn how to perform spinning strikes and the calculation and measurement of the speed, flying trajectory and racket vibrations are a subject of constant investigation [12].
6. Bio-mechanical analysis ofA major role for the initial ball Figure 6 shooting and the subsequent bouncing is
played by the simultaneous interaction of the muscles involved in the racket grip and the blow of the arm.
From mechanical viewpoint the motor organs of the tennis player can be regarded as comprising kinematic pairs united in several kinematic chains. The kinematic chain “arm”, which performs the blow on the ball, is made up of several links: trunk, armpit, arm and racket [13]. D