The equipment used in tennis, which includes the racquet, balls, strings, the court ...... slower serve with more spin on a clay court causes the ball to kick up or ...
Chapter 8 Equipment and Advanced Performance Stuart Miller1 and Rod Cross2 1. International Tennis Federation 2. University of Sydney 1 Introduction The equipment used in tennis, which includes the racquet, balls, strings, the court and various items of clothing has evolved quite dramatically over the years. In this chapter, we will concentrate primarily on physical properties of racquets, balls, strings, court surface and shoes as they relate to performance. 2 Tennis Racquets Advances in material technology have caused a design revolution in the last 30 years, which has undoubtedly contributed to a change in playing style. Theory Modern racquets, most of which are composite materials, are now lighter and stiffer than they were even five years ago, but much more so than the wooden racquets used until the 1970’s. The maximum dimensions according to the Rules of Tennis are 73.66 cm (29 inches) long and 31.75 cm (12.5 inches) wide, while the hitting surface can be no more than 39.37 cm (15.5 inches) long and 29.21 cm (11.5 inches) wide. Old wooden racquets were all 9 inches wide and 27 inches long, but materials technology has allowed racquets with bigger heads to be developed while retaining the strength necessary to withstand the forces associated with impact (see figure 8.1). Professional players now use racquets about 25.4 cm (10 inches) wide and recreational players use racquets about 27.9 cm (11 inches) wide.
Figure 8.1. Wood and graphite tennis racquets.
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Racquet mass has also changed substantially, largely due to the availability of lighter, stiffer materials at reasonable prices. Wooden racquets were traditionally made from ash and beech, which is a relatively dense material (700-820 kg·m-3), and sufficient material was needed to give them the strength to withstand the forces generated on the head by the strings. Thus, wooden racquets typically weighed 400g, whereas today’s versions can be as little as 250g. In addition to being lighter and, therefore, easier to swing, they are also stiffer, which means that they vibrate less (i.e. don’t bend as much), but at a higher frequency. This is of no practical consequence if the ball is hit near to the middle of the strings, but there is a noticeable difference in the amount of vibration between wood and graphite racquets for impacts near the tip or throat. Reduced vibration provides a more comfortable feel and also adds to racquet power, since less energy is wasted in bending the racquet. Thus more energy is returned to the ball. Table 8.1 summarises the average characteristics of wood and composite racquets. Characteristic Wood Composite Length (cm) 69 70 Mass (g) 380 295 2 Head size (cm ) 439 680 Stiffness (Hz) 100 150 2 Swingweight (kg·cm ) 450 310 Table 8.1. A comparison of average characteristics between wood and composite racquets. Application The major benefits to players of the above changes are the ability to swing a modern lighter racquet faster and the greater resistance to twisting, so being more forgiving for off-centre impacts. Any player realises that the simplest and least energy consuming way to win a point is by serving an ace, and Haake et al. (2000) showed that the number of good returns decreases as serve speed increases, especially over 100mph (Figure 8.2). In general, serve speeds continue to increase and, although the increasing size and strength of top players undoubtedly contributes to this trend, racquet technology also plays its part.
Number
50 45 40 35 30 25 20 15
Number returned in court Number of aces
10 5 0 83
93
102 111 126 138 151 159 Speed (mph)
Figure 8.2. The relationship between serve speed and number of good returns (from Haake et al., 2000).
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All other things being equal, if a player exerts the same amount of energy swinging a light racquet and a heavy one, the lighter one will have a higher speed at the moment of impact. It is not just the mass of a racquet that influences the speed with which it can be swung. The distribution of that mass is also important. For example, if all the mass of the racquet were concentrated in the handle, a player would find it easier to swing than one in which all the mass was concentrated in the tip of the racquet head. This concept, commonly known as moment of inertia or swingweight, is important in the selection of a racquet. Using this concept, a longer racquet generally has a greater swingweight than a shorter one, so making it less manoeuvrable. To compensate, manufacturers usually reduce weight when increasing length. Mitchell et al. (2000) showed that a moment of inertia is inversely related to swing speed. Brody (2000) found that good players are able to distinguish swingweight differences of 2.5%, and 5% differences in the polar moment of inertia (the resistance to twisting around the long axis, which generally increases as the width of the racquet head increases). Of the two variables, swingweight is the larger by a factor of 10, so players are actually more sensitive to changes in polar moment of inertia in relative terms. Therefore, it is not the absolute change that governs a player’s ability to detect changes in racquet characteristics, but the relative change. Thus, while a player could easily detect a change in polar moment of inertia from 30-35 kg·cm2 (a 16% change), the same absolute change in swingweight (say from 300-305 kg·cm2) could not be detected, as the 1.5% change is under the detectable threshold of 5%. 2.1 The sweet spot of a racquet Theory The sweet spot of a racquet is usually identified by players as the location that feels best when they hit the ball. There is a sensation of effortless power combined with sweetness, in the sense that there is no jarring and no vibration. There are two separate sweet spots on a racquet that would account for this.
One is called the vibration node: This is located close to the middle of the strings, about 16 cm from the tip. If a ball impacts at this point, then the racquet does not vibrate at all. Racquet vibrations result when the ball misses this spot, and are especially noticeable for impacts near the tip or throat of the racquet and with flexible racquets. Stiff racquets do not vibrate as much since they are harder to bend, but they vibrate more rapidly.
The other sweet spot is called the centre of percussion. This is located typically a few centimetres closer to the handle than the vibration node. A ball impacting at the centre of percussion causes the racquet to rotate suddenly about an axis near the wrist. At other impact points, the racquet will rotate suddenly about an axis further from the wrist such that the handle either slams into the hand or tries to pull out of the hand, resulting in shock or jarring.
There is another point on the strings that might qualify as a sweet spot, but it does not always feel good to players and it is not a fixed point. This should be called the maximum power point since it is the point from which the ball comes off the racquet at the highest speed. Typical locations of the three sweet spots are shown in Figure 8.3.
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Figure 8.3. The sweet spots of a modern racquet (from Brody et al., 2002). Racquet manufacturers like to say that their sweet spot is larger than a spot, and some will say that their spot is almost as big as the whole racquet head. What they mean is that the shot feels “pretty good” over most of the string plane and the ball comes off the strings with almost the same speed as it does at the actual maximum power point. They are correct, but it is mainly because graphite racquets are quite stiff so they do not vibrate very much. Many players attach lead tape to the head of the racquet, which changes the distribution of mass and makes it feel differently. If the mass is added towards the tip of the racquet, then it will become harder to swing, but will generate a higher ball speed by moving the area of maximum power nearer the tip (Brody, 2002) and towards the centre of the head for groundstrokes (Cross, 2001a). Attaching mass at the tip is more effective in increasing power than adding mass at any other point. Adding mass at the mid-point of the head generates a racquet that is more resistant to twisting and therefore reduces the errors produced due to off-centre impacts. Thus, this strategy would be appropriate for players who have problems with locating the ball correctly on the racquet. Application If the racquet is travelling a lot faster than the ball at impact (as it does in a serve or smash) then the maximum power point is typically about 10 cm from the tip. Adding extra lead tape to the tip shifts the maximum power point even closer to the tip since the extra weight is located where it is needed most. Serving such that the impact point is near the tip gives the player an added height advantage, but it will cause the racquet to vibrate. In terms of racquet selection, there are principles that can be applied. For example, weaker players should choose lighter racquets as they can be manoeuvred more easily. Lighter racquets are also advantageous for players who have no control problems as they allow an increased swing speed (Brody, 1998) and are less forgiving to off-centre impacts. Stronger players have a wider choice of racquets, as they are able to manoeuvre heavier examples, although the advantages of using a heavier racquet are relatively small. Better players do not necessarily need racquets with maximum power (which are normally used by recreational players). Top players can use racquets with smaller heads, flexible frames and high string tensions as the detrimental effects on power are
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fairly small and they are able to generate more power simply by swinging the racquet faster. Higher string tension improves control because the ball spends less time on the strings and the racquet does not rotate as much during impact with the ball, thereby misdirecting the ball. As professionals tend to hit the ball in the middle of the strings, they do not need a large head to minimise the errors caused by off-axis impacts. By contrast, less able players should select a racquet that has an enlarged head towards the handle and a stiff frame, and it should be strung at low tension (Brody, 1981). While taller players may find that they could sacrifice racquet length for increased manoeuvrability, the opposite is true for shorter players, as longer racquets result in greater reach, but at the expense of manoeuvrability. Increased racquet length has the advantage that it allows the ball to be hit at a higher point, which results in a higher service percentage (Brody, 1998). Modern racquets have contributed to a reduction in the number of mishits and errors. Simply using a racquet with a larger head will result in players being less likely to miss the ball or hit it off the frame. As peak power is now nearer the centre of the racquet, which is the preferred impact point, any mishits will tend to fall short (but still in play), as power is lower at any point on the racquet face away from the centre of the strings (Brody, 2002). Brody has also shown that returning the ball in same direction from which it came can reduce errors. That is, as the ball approaches and leaves the racquet perpendicular to the string plane, it always goes in the correct direction. When changing ball direction, the ball impacts at a non-normal angle and direction is dependent on both racquet head angle and racquet speed. A faster racquet will generate a ball direction closer to racquet direction; a slower racquet increases the angle between the ball and the racquet direction. As modern racquets are swung faster, they generate fewer directional errors than older racquets. 2.2 Frame stiffness Theory Modern graphite frames are much stiffer than old wooden frames. Since they are more difficult to bend, they vibrate with less amplitude, but greater frequency. An interesting result is that they are stiffer before adding the strings. One might expect that they get stiffer when the strings are inserted, but they become more flexible. The reason is that the strings pull on the frame in such a way that any slight bending of the frame is assisted and enhanced by the tension force in the strings (Cross, 2001b). Imagine bending a bow (as in bow and arrow) to insert the bowstring. The tighter the string, the more the bow bends. It’s the same with a racquet. If you bend the racquet by hand, then add the strings, the frame will bend even more. Old wooden racquets vibrate at about 90Hz. A modern racquet without strings vibrates at around 150Hz. When the strings are added, the vibration frequency drops to about 142Hz. In other words, the frame becomes more flexible. Application By virtue of their increased stiffness, modern racquets have made a contribution to the increased ‘speed’ of the game, albeit a relatively small one. When the ball hits the racquet, less energy is lost bending the racquet, and more is available to be returned to the ball. Players who are comfortable with a racquet having a particular vibration frequency should be aware that it is the vibration frequency of the strung racquet that is important. Most players tap the strings to see (or hear) how tight they are. One can also tap the frame, with the handle near one ear, to hear how stiff the frame is. It
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works best if the handle is held lightly at a point about 15 cm from the butt end so that the frame vibrations are not damped too quickly. There is a big difference in vibration frequency – even in modern racquets – between stiff and flexible graphite frames. 2.3 Grip firmness Theory Grip firmness has little, if any, effect on the speed at which a ball can be hit. Using a wooden racquet, Elliott (1982) found no significant difference in ball speed for an impact in the middle of the strings between light, medium and tight grips, but there was a slight increase in ball speed with grip tightness for impacts at other positions on the string plane. He concluded that coaches should advocate a firm grip. Cross (1999) found that, for graphite racquets, ball speed is independent of grip firmness. This is because the bending wave resulting from ball/racquet impact takes 5-6ms to propagate from the impact point to the handle and back and, although the returning wave may affect ball speed, the ball (contact time 5ms) has already left the strings. As the bending wave’s journey takes about 8ms in wooden racquets, the same should apply. Application Modern racquets have undoubtedly facilitated a different playing style – one characterised by more power and spin. While the increase in racquet power would favour those players who rely on a fast serve, the extra spin afforded by greater head width favours baseliners who can now hit more successful passing shots and lobs. Most of today’s top players favour a western grip, an open stance and a doublehanded backhand. The wider racquet head has contributed to this trend, but is not solely responsible. Other factors, such as the lower weight of the racquet, are also important (although baseliners typically use heavier racquets). The racquet characteristics that are appropriate for a particular playing style may not be appropriate for a different style. In addition, the physical characteristics of the player should influence the type of racquet selected. In the past, coaches have often suggested that a firm grip helps to add the weight of the arm to the shot, thereby transferring extra momentum and energy to the ball. It is clear from the experiments described above that this is not the case. In general, players should be encouraged to grip the racquet as firmly as necessary, but not so tight as to restrict free movement of the arm. Similarly, it is likely that a tight grip will not help to reduce racquet rotation for impacts near 3 or 9 o’clock. A ball hitting at high speed near 3 o’clock will exert a larger torque than can be applied by the human hand, so a tight grip is unlikely to reduce racquet rotation during the impact itself. 3. Strings Excellent considerations of strings have been provided by Brody (1987) and especially chapters 26-35 of Brody, Cross and Lindsey (2002). Although parts of this section are based on the latter, readers are directed to these texts for a more complete description and analysis than is given here. 3.1 Tighter strings give less power and more control; looser strings give more power and less control Theory This is a consistent finding among authors such as Ellis et al. (1978). In terms of power, strings recover about 95% of stored energy, the ball about 55%, so it is better to store energy in the strings if a hard hit is wanted. Decreasing string tension allows them to store more energy, so generating higher ball speed (Brody, 2002a). This
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relationship only holds true at tensions down to about 40lb, below which the energy lost in stretching and moving the strings is greater than that lost by not compressing the ball (Cross and Lindsey, 2002). A disadvantage of reducing string tension is that lower string tension increases rotation about the long axis for off-centre impacts since the ball remains on the strings for a longer time. This may not be so important for top players, as they are less susceptible to off-centre impacts (Cross, 2000). Application String plane stiffness (the relationship between the magnitude of applied force and movement of the strings perpendicular to their plane) is the key to power. To increase power, players should reduce string plane stiffness, which can be achieved by decreasing string tension, using more elastic strings, decreasing string density and having longer strings (i.e. a wider racquet). All of these will result in faster ball rebound speed (Brody, 2002a). However, the increase in power is quite small, typically only 1 or 2%. The same result can be achieved by hitting the ball 1 or 2% harder. In terms of tension, things are not as well understood, but Brody (2002b) believes that off-centre impacts hold the key. For a racquet strung at lower tension, the racquet twists more than one strung at higher tension due to the greater dwell time. Greater racquet twist will result in a greater error in rebound angle. Interestingly, a clamped racquet may generate the opposite effect (e.g. Knudson, 1993). As the rotating racquet condition bears a greater resemblance to reality, players should take it that higher string tension is associated with greater control. Brody (2002b) also notes that the outgoing ball speed is reduced in off-axis impacts, which may compound the control problem. 3.2 The same string tension will play differently on racquets with different head sizes Theory/Application This seems incongruous – the strings are the same tension, aren’t they? Yes, but again it is the string plane stiffness that is important. When comparing racquets with different head sizes, string densities and patterns, comparisons of string tension can be misleading and string plane stiffness should be used. As a general rule, a racquet with a larger head will have greater string plane deformation for a given string tension. String plane deformation is positively related to dwell time (i.e. the time that the ball and strings are in contact) and so is racquet speed at contact. So, the shortest dwell time occurs for a racquet with low string plane deformation swung quickly, which is typically the scenario for professional play. Short dwell times are often associated with a perception that the ball has come off the racquet more quickly than for longer dwell times. 3.3 String gauge has little effect on performance Theory/Application For the same impact conditions, a ball will rebound in almost the same way off a racquet strung with any material. Strings will return about 95% of the energy that they store during impact independently of gauge. While thinner gauge strings stretch more during the stringing process than thicker strings of the same material, they do not necessarily stretch further during an impact with the ball (after the racquet is strung). All strings get stiffer the further they are stretched, and thin strings get stiffer faster. Increased elasticity – a desirable trait in strings – means a string will play better. Natural gut has the highest elasticity (actually lowest dynamic stiffness) of all string types and this is why it is preferred by top players. Up to about 50lb tension, there is not much difference between string types, but above this (the tensions at which racquets are strung) natural gut will extend more than other types, so resulting in a lower average string plane stiffness and a ‘softer’ impact (i.e. one in which the force of
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impact is absorbed a longer time; Cross, 2002b). Gut also retains its tension better than other string types, although this isn’t much of an advantage if your racquets are being strung on a daily basis! 3.4 String performance and age Theory/Application Interestingly, tests have shown that strings do not lose performance with age (Cross and Lindsey, 2002). It appears that the decrease in string tension (and, therefore, increase in power) that is associated with time almost exactly offsets any other losses that may occur. As was mentioned earlier in this section, top players restring their racquets frequently. Given that age has no effect on string power, what is the point? One reason may be to avoid the loss of control that is associated with looser older strings (Brody, 2002a). Alternatively, loose strings sound and feel different. Good players are very sensitive to the sound and feel of their equipment, even if it makes very little difference to the actual performance. 3.5 String tension and gauge on ball spin Theory/Application An interesting phenomenon exists with respect to the effect of string tension and gauge on ball spin. Data collected in a laboratory using a head-clamped racquet show that thin and thick strings generate essentially the same spin and that low string tension generates essentially the same spin as high string tension (Goodwill, 2002). Anecdotal evidence of many players, however, would suggest otherwise, and further tests on hand-held racquets would help to resolve the issue. In a related study, Putnam and Baker (1984) found that a diagonally strung racquet had no influence on spin generation as compared to a conventional string pattern. Up (or down in this case) to a point, the roughness of a string will not affect the amount of spin that can be generated, unless the coefficient of friction between the string and the ball is less than 0.3 (Cross, 2000). To try put this value into context, the coefficient of friction between the ball and court surface ranges between about 0.50 and 0.75. 3.6 Breakages Theory/Application Breakages are mainly caused by two mechanisms: stretching and notching. Stretching often occurs when the ball impacts the strings near the frame, as this generates a greater tension increase than for an impact at the centre of the racquet head (Brody, 2002b). Notching occurs when strings move relative to each other. More commonly it is a main string that breaks first since the main strings wear at a single point whereas the crosses wear over a longer distance due to the greater sideways movement of the mains when a ball is hit with topspin or backspin. It is possible, according to Brody (2002b), that higher string tension is associated with extended string life, as this allows less movement between strings. Summary From a player perspective, key adages relating to strings are Cross (2002b): Tighter strings give less power and more control; looser strings give more power and less control; String gauge does not affect power; String power remains constant with age; String tension has little effect on ball spin (for a head-clamped racquet).
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3.8 Effect of string tension on rebound angle Theory Even though we have shown that string tension has very little effect on ball speed and spin, this is not the case for the angle at which the ball comes off the strings (see, e.g., Bower and Sinclair, 1999; Brody and Knudson, 2000). Unexpected changes in rebound angle can upset an elite player, so consistency is important. The effects of errors in the speed and angle at which the ball leaves the strings demonstrate the importance of controlling angle. An increase in ball speed of 1% from that at which the ball would travel from baseline to baseline would send it about 0.45 m beyond the baseline. An increase of 1º in angle will add nearly 2 m to the original distance. While good players will be able to compensate for the error after one or two such mistakes, when this is multiplied by the number of times elite players may change racquets during a long match, the importance of matching racquets is clear. Application The change in tension from 40 to 70lb is somewhat extreme, although the principle will still hold over a narrower range of tensions, but to a lesser extent. A swung, rather than clamped, racquet will result in a smaller difference in rebound angle between tensions. Nevertheless, a ball hit with strings at a lower tension will pass over the net at a greater height if the racquet is swung at the same speed and angle than one strung at a higher tension. This effect can become quite obvious when a player breaks a string, in which case the ball often flies upwards into the back fence. 4 Balls Although important, the racquet is not the only piece of equipment that influences the nature of the game. Balls are also key determinants in how tennis is played. The characteristics of ball/racquet impact, the way that balls fly through the air and the changes in their properties with respect to use all affect play. Theory Ball/racquet impact is an inefficient process. During impact, which lasts about 5ms, most of the ball’s kinetic energy is stored – in the form of elastic energy – in the strings and the ball itself. As aforementioned, the strings are relatively efficient and return about 95% of the stored elastic energy back into kinetic energy. The ball, however, is much less efficient. At an impact speed of 7m·s-1 onto concrete, about 55% of energy is returned (this speed corresponds to that used in the ITF ball approval test). Until recently, much less was known about ball characteristics at the higher impact speeds associated with ball/racquet impact during play. An extensive study of these ‘dynamic’ impact characteristics by Miller and Messner (in press) showed that only 28% of energy is returned at an impact speed of 25 m·s-1 (onto concrete) and 16% at 45 m·s-1 (see Figure 8.4).
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Coefficient o f restitutione ()
0.80
ITF Approval test conditions
0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 5
10
15
20
25
30
35
40
45
50
-1
Vin (m·s )
Figure 8.4. The relationship between impact speed and coefficient of restitution (from Miller and Messner, in press). Application Players thus experience diminishing returns when they swing the racquet faster, as a smaller amount of the energy stored is returned to the ball. The actual values for a ball impacting the strings of a racquet will be somewhat higher than shown in Figure 8.4, as the strings will absorb and return some of the energy stored during impact (in the -1 order of 25% at 45 m·s ) and since the ball compresses about half as far on the strings as it does on a concrete slab. As a general rule, bigger balls will experience greater resistance (drag) when flying through the air. With its larger diameter, the Type 3 (larger) tennis ball (6.985-7.302 cm) generates greater drag than any other ball type, which gives players more time to react, resulting in longer rallies and less errors. At altitude (1219m or 4000ft), the Type 3 ball plays almost identically to a regular sized ball at sea level, as the decrease in drag due to the lower air density is offset almost exactly by the greater drag of the larger ball due to its diameter. As far as the felt is concerned, its properties change with use. During early use, it tends to become more ‘fluffy’ and then becomes smoother after a number of games. As the surface area exposed to the oncoming air influences drag force, the ball will experience a rise in resistance at the start of its ‘life’ and then a reduction when the felt becomes shorter and less prominent. Thus, a ball that has been used for a game or two may fly more slowly through the air compared to one that has been in play for 5 or 6 games. On its own, this suggests that players who rely on a fast serve should select well-used balls. However, there are other factors to consider. Firstly, in addition to the drag force that resists the ball’s progress towards the opponent, there is a force that acts at right angles to it called lift. The term ‘lift’ is somewhat misleading in tennis, because although it can indeed help the ball stay in the air longer, it is far more often used by players in the opposite way – to force the ball back down to the ground more quickly. The direction in which ‘lift’ acts is dependent on the direction of rotation of the ball. Most shots in tennis nowadays are played with large amounts of topspin, which acts to force the ball back down to the court surface. Therefore, players can hit the ball harder and it will still fall into play, giving passing shots a greater likelihood of succeeding (see Figure 8.5).
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Figure 8.5. The effects of spin on ball trajectory. For a ball hit from the baseline at a height of 1 m above the court surface, the approximate trajectories are shown for a ball with topspin (A) and backspin (C) at 1900 revolutions per minute compared to a ball with no spin (C) (after Brody, 1987). So, back to the point. The magnitude of the lift force and, therefore, the speed with which a player can hit the ball and still have it fall into play is dependent on several factors, including the roughness of the surface of the ball. As the ball becomes more worn, the surface roughness decreases and, therefore, the lift force for a given amount of spin also decreases. Thus, for the same speed and spin, a worn ball will travel further than a new one and is more likely to go long. Although the precise nature of the relationship between felt properties and lift force has yet to be established, it is reasonable to speculate that, to some extent, the benefit of worn balls slowing down less is offset by the necessary reduction as a result of the decrease in lift force. Secondly, Miller and Messner (in press) showed that a worn ball cannot be hit as quickly as a new one, will slow down less and can generate less lift force. So, is there clear advice that can be offered to players when selecting a ball for serving? The answer is no, as the benefits gained from new balls as compared to worn ones are offset to some (unquantified) extent by other shortcomings. What is still of interest to scientists is exactly what happens to ball properties during the time between it being new and worn.
Coefficient of restitution (e)
Although there has yet to be an extensive study of the aerodynamics of tennis balls, Mehta and Pallis (2001) found a range of drag coefficients (a measure of the resistance to motion independent of ball size) for a small number of new and used balls of 0.55-0.67 at speeds between 45 and 150 mph, a range of approximately 20%. 0.60 0
0.55
50 0.50
100
0.45
150 300
0.40
0 (50)
0.35
(100) 0.30
(150) 20
25
30
35
40
45
(300)
-1
Vin (m·s ) Figure 8.6. The effects of simulated wear on the relationship between vin and e (from Miller and Messner, in press).
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In terms of the effects of wear on coefficient of restitution, Miller and Messner found that use causes balls to become less efficient at recovering stored elastic energy. Figure 8.6 shows coefficient of restitution (e) plotted against impact velocity for a selection of 12 different tennis ball brands. Each ball was tested when new and after 50, 100, 150 and 300 impacts at speeds of approximately 20 and 40 m·s-1. It is not unusual to hear players talk about balls being ‘dead’ or ‘lively’, or feeling good but what does this mean? Research has resulted in the development of a theory that ‘feel’ is the result of two factors – ball stiffness (the amount of force necessary to compress the ball by a certain amount)and coefficient of restitution (Haake and Goodwill, 2002). Current thinking is that balls that are both stiff and leave the racquet more quickly are perceived to ‘feel’ better by players than those that are less stiff and are slower off the racquet. This may be because a shorter contact time or the sound associated with a shorter contact time generates the perception that the ball has been hit harder. 5 Court-related considerations Theory/Application Brody (1984) was the first to describe the physics of a bouncing tennis ball. Brody showed that the ball slides along a court surface during the whole of the “bounce period” if it is incident at an angle less than about 18º to the horizontal. All fast serves and all fast groundstrokes approach the court at angles typically between 11 and 16º and with sufficiently little spin that they slide throughout the bounce. In this situation, the ball slows down by an amount that increases as the angle of incidence increases and as the coefficient of sliding friction (COF) increases. The formula for the ball speed, when it slides along a court surface, is: vx2/vx1 = 1 - COF (1 + COR) tan(θ1) where
(1)
vx2 = the horizontal component of the ball speed after the bounce, vx1 = horizontal speed before the bounce θ1 = angle of incidence COR = coefficient of restitution = vertical speed after bounce vertical speed before bounce
The coefficient of restitution is typically between 0.75 and 0.85 for an oblique bounce, while the coefficient of friction between a tennis ball and a tennis court is a number typically between 0.5 (for a fast court) and 0.7 (for a slow court). This means that at an angle of incidence of 11º, the ball will slow down by about 20% on a fast court and by about 28% on a slow court. At an angle of incidence of 16º, the ball will slow down by about 30% on a fast court and about 40% on a slow court. The friction force acting on the bottom of an obliquely-bouncing ball causes it to slow down, and this also causes it to spin. If the ball has no spin before it bounces, it will have topspin after the bounce. The amount of spin increases with the angle of incidence and with the coefficient of friction, and it increases with the incident ball speed. Thus a ball approaching the surface at a steeper angle will acquire more topspin than one approaching at a shallower angle. Similarly, a faster-hit ball will acquire more topspin off the surface than a slower one. Brody’s theoretical analysis showed that if a ball approached the surface at an angle of over 18º, the friction between the two surfaces would generate enough spin in the ball for it to start rolling along the surface before it bounces off it. Brody’s calculated that rolling would start when friction caused the ball to lose 40% of its horizontal
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speed. The implication was that any court that has sufficient friction to cause at least a 40% loss of horizontal speed in the ball is as slow as it is possible for a court to be (as a ball essentially does not slow down any more during its short contact time with the surface once it has started rolling). However, Cross (2002c; 2003) found that balls can lose up to 60% of their horizontal speed before they start rolling (figure 8.7). This means that some slow courts are actually much slower than was thought. Friction is also the cause of what is referred to by players and commentators as ‘bite’ in a kick serve. The large vertical forces exerted on the ball during the bounce acts to increase the friction between it and the surface. Therefore, the bottom of the ball can momentarily stop on the surface, and the rest of the ball distorts in the horizontal direction due to its momentum. When the ball starts to rebound off the surface, the vertical forces decrease and the ball is then ‘released’, kicking off the surface (remember that kick can also result from a heavy topspin serve, due to the steep angle of incidence). The angle of incidence at which a ball starts to grip the surface is inversely related to the coefficient of friction between the ball and surface.
(a) 1
(b) 1
0.9
0.9
Smooth concrete
0.8 P800
x1
0.7 /v
0.6
0.7
0.5
Clay
v
x2
v
x2
/v
x1
0.8
RAce
0.4
0.6
P150
0.5
0.3 0
10 20 30 40 Angle of incidence
0.4
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0
10 20 30 40 Angle of incidence
50
Figure 8.7. Measured ratio of VX2/VX1 against angle of incidence, θ1, for (a) Clay, Rebound Ace and a smooth concrete slab, and (b) P800 and P150 grades of emery paper bonded to a smooth concrete slab. Straight-line segments are fits to the data assuming that the speed ratio is 1.0 when the angle of incidence is zero, as predicted by equation 1. Figure 8.8 shows that the bounce of surfaces can be different, even though all are hard when compared to a tennis ball. For example, on a clay court the ball makes a small depression, so the ball is deflected upwards by an accumulation of loose surface material at the front edge of the ball. The same effect cannot be used to explain the enhanced coefficient of restitution (COR) found on hard courts or on concrete. The significance of the enhanced COR on some surfaces is that the ball bounces at a steeper angle and it bounces to a greater height. A ball bouncing with COR = 0.85 bounces 47% higher than on a surface with COR = 0.7, as the bounce height being proportional to the square of the vertical bounce speed.
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Clay court
1
1
Rebound Ace court v = 15 to 30 m/s
0.8 0.7 0.6 10
1
0.9 COR
COR
0.9
0.7
v = 15 to 35 m/s 1
15
20 25 (degrees)
0.8
0.6 10
30
15
20
25
30
35
(degrees)
Figure 8.8. Measured values of coefficient of restitution on clay and Rebound Ace. Each data point represents one bounce. A high-speed serve on a clay court takes about 0.02 seconds longer to arrive at the baseline than it does on a grass court since it bounces at a slightly lower speed. This generates two interesting questions: Can a player tell the difference in court speed in this case? Why do male players serve more slowly on clay than on grass? The -1 average first serve speed for men at the French Open is about 25km·hr slower than at Wimbledon. The answer to the second question may be due to the trade-off between serve speed and spin, in that more spin can be generated in slower serves. A slower serve with more spin on a clay court causes the ball to kick up or sideways at an awkward angle, making it more difficult to return. The result is usually fewer aces and longer rallies on clay courts. The extra effort required to serve at top speed and play long rallies may not be worth the effort. 6 Footwear Theory/Application Tennis is unusual in terms of elite level sport in that it is played on many different surface types (grass, clay, acrylic), each of which has different characteristics not only in terms of interaction with the ball, but also the player. As the way in which a tennis court feels to a player is dependent on the nature of the interaction between the two surfaces in contact, it is clear that footwear plays an important role in tennis. Such is the variety of tennis surfaces, the problem of providing appropriate footwear is not one that can be solved with a single solution, so shoe manufacturers have developed shoes suited to different surfaces. Shoes influence two distinct aspects of tennis: injury and performance. From a design perspective, these factors are not mutually exclusive, that is they are inter-dependent to a certain extent, an inter-dependence that can result in a design ‘conflict’. It’s clear that the frictional characteristics of the shoe/surface interaction need to strike an appropriate balance between being too high and too low, each of which has its own problems. Tennis movement patterns are relatively complex, involving forward, backward, sideways and rotational movement at a variety of speeds. Tennis is, therefore, demanding on footwear and selection of the correct shoe-surface combination is crucial to striking an appropriate balance between performance enhancement and minimisation of injury risk.
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Shoe/surface friction characteristics are primarily determined by the two materials in contact, but the surface roughness and tread pattern of the shoe must also be considered. A selection of tread patterns is shown in Figure 8.9. While the sole of the shoe on the right of this group may be appropriate for a surface such as carpet, it would prove disastrous on clay, as the loose particles on the surface would act like tiny ball bearings and lower the COF to a point where a player’s ability to accelerate and decelerate would be severely affected. A shoe designed for clay courts should have a treaded sole, so that the loose particles can be caught within the tread so that the COF is determined to a greater extent by the nature of the surface underlying the particles.
Figure 8.9. A selection of tennis shoe tread patterns. A potential conflict exists between the optimisation of the frictional characteristics of the shoe/surface interaction when moving in a straight line and when turning. While sufficient friction to start and stop quickly is necessary when moving in a straight line, minimum friction is beneficial when pivoting, thus allowing rotation to occur as quickly as possible. The soles of some shoes contain a circular pattern under the head of the first metatarsal (the ball of the foot) to facilitate turning, but its effectiveness on different tennis surfaces is not known. Too much friction while turning can transfer rotational forces from the shoe/surface interface to the menisci and collateral ligaments of the knee. Thus, shoes have the dual function of enhancing performance and reducing the risk of injury. There are two major injury types in tennis (as with most sports): acute and chronic. Acute injuries are ones that occur due to a single incident, whereas chronic injuries are due to repetitive activity. Examples of acute injury include tennis leg and lateral ankle ligament sprains. Tennis leg is a rupture of the gastrocnemius muscle where it merges with the Achilles tendon (Peterson and Renström, 2001). It is caused by forced dorsiflexion from a position of extreme plantarflexion with a fully extended knee. This mechanism is dependent on the knee being extended, which results in all of the momentum being absorbed by the muscle that controls dorsiflexion (the gastrocnemius). By encouraging players to land with the knee at least slightly flexed, some momentum can be absorbed by other knee flexors, such as the hamstrings, placing less strain on the gastrocnemius.
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Sprains to the ligaments on the lateral aspect of the ankle often occur during a change of direction when moving laterally (side to side). For example, a player moving to the left would exert a lateral force to the left to decelerate. If the shoe ‘bites’ the surface, then it is not uncommon for momentum (mass × velocity) to be sufficient to force the subtalar joint (a part of the ankle complex) into excessive inversion, so rupturing the ligaments on the lateral side of the ankle. To combat the incidence of lateral ligament damage in sports such as basketball, which not only involve the movements that can cause such injuries but are also invasive, thus adding the possibility of injury due to contact between players, high-cut shoes are often worn, the effects of which are not well known. The fashion in tennis, however, has been to wear low-cut shoes, so an alternative method of prevention could be taping or bracing the ankle. How effective these systems are in reducing injury is questionable. Both taping and bracing work by reducing the range of motion of the ankle, but this may be counter-productive if it limits performance. Miller and Barlow (2001) have also shown that, following a short period of activity, the support properties of these systems deteriorate, thus reducing their effectiveness for a long duration activity such as a tennis match. During heel-toe running on tennis surfaces at 3.83 m·s-1, peak impact forces between the ground and the shoe of 2-2.3 times bodyweight were recorded by Dixon and Stiles (2002). During a tennis match, a player’s body will experience many such impacts, so the shoe must be effective at reducing their potentially harmful effects. A major issue in shoe selection, therefore, is the potentially conflicting requirements of attenuation of the repetitive impact forces and maintenance of foot stability (rearfoot control) with no exacerbation of movement at the subtalar joint. It has been argued that, because tennis performance is dependent on many factors other than friction, the focus of shoe construction (and, by implication, shoe selection) should be on the reduction of the rate of lower extremity injury (Gerritsen et al., 2002). To summarise, these authors have suggested that tennis shoes could:
Increase comfort; Provide lateral stability; Help reduce impact forces; Alter the pressure distribution (presumably on the plantar surface of the foot); Provide rotational and translational friction.
There is often a conflict between what might be considered the two most important (of the many) biomechanical functions of a sports shoe – impact attenuation and rearfoot control. That is, the softer-soled shoes that attenuate impact forces effectively also allow the rearfoot to move relatively freely, excessive motion of which causes the eversion injuries discussed above. It is also known that, in general, shoes lose around 30% of their impact attenuation properties after modest use (the equivalent of about 500 miles), so they should be replaced at appropriate intervals. Sanderson (1981) suggested that simply ensuring that footwear is durable and well-fitting can minimise foot injuries. Calluses, friction blisters, heel bruising and tennis toe (haemorrhage beneath the toenails) can all easily be prevented if the shoes are the correct size. There is evidence that playing on surfaces such as clay is associated with a reduced risk of lower extremity injury. As well as being one of the softer tennis surfaces, clay also allows controlled sliding, and it has been speculated that the latter characteristic plays a more important role in reducing injury (Gerritsen et al., 2002).
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How to select a tennis shoe Although selection of the ideal shoe is not necessarily a straightforward issue, as it should ideally be based on the individual’s lower limb biomechanics, there a number of factors can be used as a general guide. 1. The potential conflict between the requirements of impact attenuation and rearfoot control may be overcome by the selection of shoes with stiffer heel counters, a less compliant (firmer, harder) midsole and a wider heel base at the outsole. All of these tend to control rearfoot motion without necessarily enhancing impact forces or exacerbating subtalar joint movement. 2. Change shoes at regular intervals. This will prevent the exacerbation of impact forces due to the loss of their impact-absorbing qualities. 3. Ensure shoes are the correct size. Feet tend to get bigger as the day goes on (due to supporting the body’s weight), so the late afternoon is generally the best time to try shoes on. 4. Select the shoe that is appropriate to the surface being played on. For example, a smooth-soled shoe is unlikely to be effective on clay, whereas on some surfaces a shoe with a large contact area will be an advantage. Summary We have shown how materials technology has facilitated major changes in racquet characteristics, making them lighter, yet more powerful and more forgiving to mishits and how players can apply principles to tailor racquets to their playing characteristics. It was also shown that tighter strings give less power and more control compared to looser ones, and that, in relative terms, string tension has the greatest effect on ball rebound angle off the racquet. Power is not affected by either the age or gauge of strings. Ball characteristics tend to change with use. New balls tend to fly through the air more slowly for the first few games due to the increased fuzziness of the felt, after which they speed up as the felt starts to wear away. The ball also influences the ‘speed’ of the court via the friction exerted between it and the surface. Finally the interaction between the shoe and surface determines the ability of the player to accelerate, decelerate and turn. Too little or too much friction can be hazardous to players. A very low-friction surface will tend to make players slip over, while a very high-friction surface can result in forces being transferred from the shoe to the ankle and knee. Again, principles can be applied when buying shoes, both in terms of maximising performance and minimising injury risk. Where possible, shoes should be matched to the surfaces on which they will be used. Acknowledgements The authors wish to thank Bob Haines for supplying the data on wooden racket characteristics in table 8.1. References Bower, R. and Cross, R. (2003) Player sensitivity to changes in string tension in a tennis racquet, Journal of Science and Medicine in Sport, March (in press). Bower, R., and Sinclair, P. (1999). Tennis racquet stiffness and string tension effects on rebound velocity and angle for and oblique impact. Journal of Human Movement Studies, 37, 271-286. Brody, H. (1981). Physics of the tennis racquet II: the “sweet spot”. American Journal of Physics, 41, 816-819. Brody, H. (1984). That’s how the ball bounces. The Physics Teacher, 494-497. Brody, H. (1987). Tennis Science for Tennis Players. Philadelphia: The University of Pennsylvania Press.
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Brody, H. (1998). Improving your serve. In The Engineering of Sport: Proceedings of the 2nd International Conference on the Engineering of Sport (edited by S. Haake), pp. 311-316. Oxford: Blackwell Science. Brody, H. (1998). Are the new lightweight racquets for you? Private communication. Brody, H. (2000). Player sensitivity to the moments of inertia of a tennis racquet. Sports Engineering, 3, 145-148. Brody, H. (2002). The Tennis Racquet. In Handbook of Sports Medicine and Science: Tennis (edited by P.A.F.H. Renström), pp. 29-38. Oxford: Blackwell Science. Brody, H. (2002a). The Tennis Racquet. In Handbook of Sports Medicine and Science: Tennis (edited by P.A.F.H. Renström), pp. 29-38. Oxford: Blackwell Science. Brody, H. (2002b). Overview of Strings. In The Physics and Technology of Tennis (edited by H. Brody, R. Cross and C. Lindsey), pp. 239-249. Solana Beach: Racquet Tech Publishing. Brody, H. and Cross, R. (2001). Proposals to slow the serve in tennis Brody. In Tennis Science and Technology: Proceedings of the 1st International Congress on Tennis Science and Technology (edited by S. Haake and A. Coe), pp. 261-268. Oxford: Blackwell Science. Brody, H., Cross R. and Lindsey, C. (2002). Sweetspots. In The Physics and Technology of Tennis (edited by H. Brody, R. Cross and C. Lindsey), pp. 51-59. Solana Beach: Racquet Tech Publishing. Brody, H., Cross R. and Lindsey, C. (2002). The Physics and Technology of Tennis. Solana Beach: Racquet Tech Publishing. Brody, H. and Knudson, D. (2000). A model of tennis stroke accuracy relative to string tension, International Sports Journal, 4, 38-45. Cross, R. (1999). Impact of a ball with a bat or racquet. American Journal of Physics, 67, 692-702. Cross, R. (2000). Effects of friction between the ball and strings in tennis. Sports Engineering, 3, 85-98. Cross, R. (2000). Flexible beam analysis of the effects of string tension and frame stiffness on racquet performance. Sports Engineering, 3, 111-122. Cross, R. (2001a). Customising a tennis racquet by adding weights. Sports Engineering, 4, 1-14. Cross, R. (2001b). Why bows get stiffer and racquets get softer when the strings are added. American Journal of Physics, 69, 907-910. Cross, R. (2002a). Racquet power. In The Physics and Technology of Tennis (edited by H. Brody, R. Cross and C. Lindsey), pp. 159-167. Solana Beach: Racquet Tech Publishing. Cross, R. (2002b). Natural Gut Construction and Play. In The Physics and Technology of Tennis (edited by H. Brody, R. Cross and C. Lindsey), pp. 266-267. Solana Beach: Racquet Tech Publishing. Cross, R. (2002c). Grip-slip behaviour of a bouncing ball. American Journal of Physics, 70, 1093-1102. Cross, R. (2003). Measurements of the horizontal and vertical speeds of tennis courts. Sports Engineering (in press). Cross, R. and Lindsey, C. (2002). String Properties. In The Physics and Technology of Tennis (edited by H. Brody, R. Cross and C. Lindsey), pp. 271-282. Solana Beach: Racquet Tech Publishing. Dixon, S.J. and Stiles, V.H. (2002). Impact absorption of shoe-surface combinations in tennis. Report to the ITF Technical Department (200102). Biomechanics Research Group, University of Exeter. Dowell, L.J., Smith, F., Miller, G., Hope, A. and Krebs, G. (1987). The effect of angle of incidence on rebound deviation of a tennis ball. Journal of Human Movement Studies, 13, 69-73. Elliott, B.C. (1982). The influence of grip tightness on reaction impulse and rebound velocity. Medicine and Science in Sports and Exercise, 14, 348-352.
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Ellis, R., Elliott, B. & Blanksby, B. (1978). The effect of string type and tension in jumbo and regular sized tennis racquets. Sports Coach, 2, 32-34. Gerritsen, K.G.M., Nigg, B.M. and Wright, I.C. (2002). Shoes and surfaces in tennis: injury and performance aspects. In Handbook of Sports Medicine and Science: Tennis (edited by P.A.F.H. Renström), pp. 39-45. Oxford: Blackwell Science. Goodwill, S.G. (2002). Spin generation – ball impact on a tennis racquet. Presentation to the ITF Foundation, June. Haake, S. and Goodwill, S. (2002). ‘Feel’ in relation to the dynamic characteristics of tennis balls. In The Engineering of Sport 4 (edited by S. Ujihashi and S. Haake), pp. 168-175. Oxford: Blackwell Science. Haake, S., Rose, P. and Kotze, J. (2000). Reaction time testing and grand slam tiebreak data. In Tennis Science and Technology (edited by S.J. Haake and A. Coe), pp. 269-276. Oxford: Blackwell Science. Hatze, H. (1998). The centre of percussion of tennis racquets: a concept of limited applicability. Sports Engineering, 1, 17-26. Mehta, R. and Pallis, J. (2001). The aerodynamics of a tennis ball. Sports Engineering, 4, 177-189. Miller, S. and Barlow, J. (2001). Effects of ankle brace and tape support on foot and ankle motion on basketball specific performance. In Proceedings of the XIX International Symposium on Biomechanics in Sports (edited by J.R. Blackwell), pp. 223-226. Exercise and Sport Science Department, University of San Francisco: San Francisco. Miller, S. and Messner, S. (in press). On the dynamic coefficient of restitution of tennis balls. Paper submitted to the 2nd ITF Tennis Science and Technology Congress. Mitchell, S.R., Jones, R. and King, M. (2000). Head speed vs. racquet inertia in the tennis serve. Sports Engineering, 3, 99-110. Moore, B.C.J. (1986). Frequency selectivity in hearing. London: Academic Press. Peterson, L. and Renström, P. (2001). Sports Injuries: Their Prevention and Treatment. London: Martin Dunitz. Putnam, C.A. & Baker, J.A.W. (1984). Spin impacted to a tennis ball during impact with conventionally and diagonally strung racquets. Research Quarterly for Exercise and Sport, 55, 261-266.
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