Racquets Match Point Box 1.2 continued Groundstrokes. A different result is obtained fo 「 a groundstroke where heavy both Figures l.8 and l.9. For example, racquets have an advantage over light rac- doubling the racquet weight from 200 quets in terms of ball speed 0 仟 the rac- grams t0 400 grams will increase the quet for a given effort. The result is shown rebound power from O. ワ to 0.44 for an in Figure l.9. Players rarely use maximum impact with a 57-gram tennis ball. effort to hit a groundstroke, so it was Serve. Experiments show that swing assumed in these calculations that 」 oe speed decreases as SWing 、 Neight increases used half his maximum effort to swing his for a maximum effort SWing, as in a first racquet at half his maximum swing speed. serve, according tO the graph shown in 比 was assumed a 0 that the ball was Figure l.8. Everyone has a different maxi- approaching at 40 mph and that 」 oe hit mum SW ・ ing speed fO 「 any given SWing- the ballin the middle ofthe strings.ln that weight, but if the swingweight is doubled, case there is a ー O mph increase in outgo- then the swing speed will decrease by ing ball speed when the racquet weight is about ワ percent for all players, at least for increased from 200 tO 400 grams. racquets between IOO grams and 500 Alternatively, 」 oe could decide to swing the grams in weight. Doubling the racquet 400-gram racquet upwards at a steeper weight from grams to 20 grams will angle than the 200-gram racquet, in which make no measurable difference at all to case the outgoing ball speed 0 仟 bOth rac- swing speed, but ワ percent is the average quets would be about the same, but the measured result for almost all racquets Of 400-gram racquet would generate more practicalinterest and for a range Of differ- topspin. ent players 0f different strengths and abilities. Adding the racquet speed to 70 the rebound speed gives the serve speed results shown in Figure l.8. The serve speed is a maxlmum at a racquet weight 50 of about 300 grams, but there is on ツ a tiny drop in serve speed at 250 grams or at 400 grams. Consequently, there is almost nothing tO be gained by using a heavier or lighter racquet, at least in terms Of maximum serve speed. Groundstroke 80 0.8 Joe'S outgoing ball speed 0.6 (JöM0d pu コ 0q9 区 ) dY 4 0 0 60 Rebound power ()P scale) HaIf 」 oe ・ s max racquet speed 0 30 lncoming ball speed = 40 mph 90 汚 0 200 250 300 350 400 450 500 Racquet weight (gm) Figure 1.9 Effect 0 「 weight on groundstoke speed. 20 -0.2 23

Copyright ◎ 2005 by Rod Cross and C 「 aw 「 0 「 d Lindsey Racquet Tech Publishing ()n imp 「 int ofthe USRSA) 330 Main St. Vista, Ca 鼾 0 「 nia 92084 WWW. racquettech.com All rights reserved. No pa け ofthis publication may be reproduced, stored in 0 retrieval system, 0 「 transmitted, in any form 0 「 by any means, elec- tronic, mechanical, photocopying, recording 0 「 otherwise, withOUt the express written consent Ofthe publisher. Library of Congress Control Number: 2005905393 Cover design and illustrations by Kristine Thom lnside chapter title photos by Ron Waite, Ph0tosportacular Printed in the United States Of America ISBN-1 3 : 9 / 8-0-9 / 22 / -593-4 ISBN-10 : 0-9 / 22 / 59-3-2

Chapter One Match Point Box 1.2 Maximum Effort Power Even though rebound power provides a are shown in Figures に 8 and に 9 fo 「 a play- convenient measure Ofthe intrinsic POW ・ er er whO we call 」 oe. 」 oe is strong, but not of a racquet, and even though players tend super strong. He iS using various racquets tO adjust their swing speed tO suit the shOt that vary in weight and swingweight, but that they are trying to make, the question each is 69 cm long, and each has a balance point 353 mm from the end ofthe handle, arises as tO hO 、Ⅳ much power a given rac- SO the racquets are equally head-héavy. quet can deliver when the player uses 訓 the effort he or she can muster. 旧 that The racquets are strung with the same case, not on レ rebound power, but a 0 string at the same tension and have the swingweight, will determine the final ball same headsize and stiffness. 旧 that case, speed. That is the sort Of power that ten- the swingweight is proportional tO the niS authorities are worried about. げ some- racquet weight. げ the racquet weight is one can make a racquet that iS SO power- doubled, then so is the swingweight. fulthat 150 mph serves become common, Rebound power SO increases as racquet then the game oftennis will be in trouble. weight increases, but it is not exactly pro- に is doubtful that this will ever happen. portional tO racquet weight. に is relatively easy tO calculate rebound power for an That is not to say that a seven-foot giant will not come along one day and belt impact in the middle ofthe strings because down every first serve at 150 mph or so. energy losses due tO frame vibrations can then be ignored. The result is shown in But what if your average six-foot weal<ling could do it? Even if he could, then it is likely that 95 % of first serves would then be a fault because the margin for error 、 wou 旧 90 shrink almost tO zero. have done some inter- esting calculations concern- ing the variation Of racquet power with racquet weight 70 or swingweight for any play- 比 60 er, based on the assumption that the player exerts max- lmum or near maxlmum effort. These calculations Serve 0.8 0 maximum serve speed 0.6 ( 」 0M0d pu コ 09a 比 ) dY っム 0.4 80 」 oe maximum racquet speed 0 Rebound Power ()P scale) -0.2 50 90 巧 0 加 0 250 300 350 400 450 500 Racquet weight (gm) Figure れ 8 Effect 可 weight on serve speed. 22

Strings the ranges 0f higher tensions caused by ball impact, and it is less stiff than Other strings at these tensions as a result. There are only four different strmg materials in common use. ln order fror れ softest t0 stiffestthey are: gut, nylon, polyester, and Kevlar. These groups have very little, if any, overlap in measured stiffness values fror れ one group tO anoth- er. There is a tiny bit of overlap between nylon and polyester (but only for a very few strings) , as new manufacturing processes have allowed polyester tO be softened. Within each category, there is a range Of variance, but nothing as significant as the leap between separate categories. Table 2.1 shows the stiff- ness ranges for all strings on the market ⅲ 2005. TabIe 2 」 Dynamic Stiffness and Tension Loss Material Gut Nylon/Zyex Polyester Kevlar Lengthwise Stiffness Range for a ” string (lb/in. ) 90- Ⅱ 9 日 6-242 円 9-320 470-9 田 Tension LOSS (after 200 secs & 5 50 ⅱ d impacts) 6 」 -9.2 旧 6.5- .2 旧 7.9- 田 .9 lb 5.6-9.5 旧 lfmeasured with string twice as long, the string would stretch twice as 徊ら and the stiffness ん e Ⅳ ou 旧 be わ 0 鳫 but each string Ⅳ ou 旧 maintain its relative stiffness tO the 0 曲 e 「 strings. The stiffness values are experimentally determined in the lab, and each and every string is a bit different. Because stiffness varies with the string tension and length Of the string, each string is measured at the same starting tension and length. For each string, the following parameters can be determined: peak impact force, tension rise, dwell time, string elongation, string deflection (per- pendicular tO string) , impact tension IOSS, and lengthwise and sideways stiff- ness. These values are published by the USRSA ()S Racquet Stringers Association) twice a year, and USRSA members have access [ 0 the data bOth on the internet and in printed materials. A qualified USRSA racquettechnician can help you find the stiffness Of string tO meet your needs or can suggest a hybrid mix of different strings for the mains and crosses. 67

Chapter Two Match Point Box 2 」 Rise in StringTension During a Shot When a string is installed in a racquet, it tend tO cut the ball to pieces. The tension needs tO be stretched tO arrive at the in a nylon string rises by about 30 pounds desired tension. Nylon strings need to be during each impact with a ball (for a hard stretched by about percent, gut to hit) before it drops back to its normal about 7 percent, and polyester strings, value after the ball leaves the strings. 比 being stiffer, need t0 be stretched by on レ rises by about 20 pounds with natural gut, about three or four percent. By contrast, a about 40 pounds with polyester, and about steel string would stretch by on レ about 60 pounds with KevIar ActuaIly, the tension O. 2 percent at the same tension. Steel is drops back to a slightly lower value than 〇 K in a guitar or a piano because it does- normal after every shot. After a few hun- n't need tO stretch any farther when it is dred shots the tension will be one or two used. Tennis strings need to stretch farther pounds lower as a result Of microscopic every time they are used to hit a ball, oth- damage tO the string. erwise they would feel too stiff and would (a) String at 60 旧 (b) Clamped (c)After one hour (d) Tension rise 60 60 60 48 〇 0 77 88 82 さ ゞ mph 60 60 Gut Stretch = 0.9 ” 60 Gut NyIon Po ツ Gut NyIon Po ツ Gut NyIon Po ツ Po ツ NyIon Figure 2.7 Nylon stretches the most during stringing 砒 60 pounds, then gut, then polyester ( の . A 〃 three materials are clamped 砒 the same tension (b). ハな e 「 an hour gut is 砒 the highest tension,then nylon, and then poly (c). 50 厄「 would seem that gut 居 0 ⅵ / stiffstring—it 居 the second st 立 to pu 〃 tO tension, and it is now at the highest tension. But, when you hit the strings with ロて e 0 「 0 / 20 mph serve, the strings ⅲ〃 rise in tension (d). P01y will increase the most, nylon second, and gut the least. So, even though gut starts at the highest tension, ends up at the lowest tension, and vice versa 「 po / /. Gut stretches the most, gains the least tension, and is the SO な e 立 m ロこ e ⅱロ / at ロ〃 tensions 亡 / p ロ / 厄「 hitting ロ tennis ba 〃 . 72

Chapter Two GENERAL RULES OF DWELL TIME tip is lighter than the throat. strings and about 5.5 ms near the throat. That'S because the ms near the racquet tip, about 5 ms ⅲ the middle 0f the ・ Contact time for a bounce on a handheld racquet is about 生 5 racquet. is a big increase in the speed 0f the ball or the speed of the ball or the racquet, although there is a slight decrease if there ・ Contacttime does not depend significantly on the speed of the Contact time increases if the mass Of the racquet increases. ・ Contact ⅱ 1 れ e increases if the mass Of the ball increases. mcreases ・ Contact time decreases if the stiffness Of the string plane ・ Contact time decreases if the stiffness Of the ball increases. 86 playing conditions, 丿 ou 「 n ロ / 0 「 SPorts Sciences, 23 , / 65 ー / ( 2005 ). R. Bower and R. Cross, String tension effects on tennis b ロ〃 rebound speed and accuracy during Science and Medicine ⅲ 5P0 鵬 6 , / 20 ーロ / ( 2003 ). R. Bower and R. Cross, Player sensitivity tO changes ⅲ string tension ⅲ 0 tennis racquet, 丿 n /. 0 「 3 , 刀 9-230 ( 20 〇の . R. Cross, C. Lindsey and D. Andruczyk, 比 borato 「 / testing oftennis strings, 5P0 s Engineering, performance, Sports Engineering, 3 , / / / - ロ 2 , ( 200 の . R. Cross, 日 exible beam analysis 0 「 the effects 0 「 string tension and frame stiffness on racket Jenkins, W00dhead, Cambridge, ( 2003 ). R. Cross, MateriaIs and Tennis Strings, Chapter 8 ⅲ Materials in sports equipment, Ed. M. H. Brody, Tennis Science 厄「 Tennis Players, University 0 「 Pennsylvania Press, / 987. So na Beach, USA ( 2002 ). H. Brody, R. Cross and C. Lindsey, The Physics and Technology ofTennis, Racquet Tech Publishing, んわ e 「 Reading

Chapter One Bu mperguard gm Stri ngsl 5 gm Grommets gm Raw frame 227 gm Paint and decals 田 gm HandIe 30 gm Grip 20 gm End cap gm TotaI racquet weight = 340 gm Figure れ The weights Ofthe individual components 0 「 tOt ロ / racquet weight. About ha / 「 0 「 the raw frame weightis just glue (resin) weight. The weight 0f most racquets varies from about 250 grams t0 about 360 grams, with the majority being around 300 grams. Figure 1.11 shows the components that contribute tO the overall weight Of the racquet. The trend over the years has been toward lighter racquets because most players prefer racquets that are easy [ 0 SWing and easy tO carry ontO and around the court. On the Other hand, professional players tend t0 prefer heavy racquets because heavy racquets are generally more powerful and because professional players are generally fitter and stronger than recreational players. There is no correct or 、、 best" racquet weight. Even some professionals play very well with light racquets. ln that respect you should play with a racquet that feels just right ⅲ terms 0f its weight. Obviously, if you weigh more than 180 pounds, you can swing a heavier racquet than a young child. There is no offi- 32

Chapter One ing launch platform instead of a stationary one. For example, a ball might rebound from the middle of the strings at 40 percent of the speed with which it landed (rebound power of O. (0). So, if a ball hits the middle of the strings of a stationary, free-standing or hand-held racquet at 60 mph, it rebounds at about 24 mph (Figure 1. の . If the racquet hits the ball instead (Figure 1.5 ) , the rebound speed will still be 24 mph, but the exit speed will be 84 mph because the rebound occurs off a platform moving at 60 mph. 60 mph 」 ust before i mpact 0 mph ()t rest) Rebou nd power = 0.4 24 mph recoil lncoming ball speed = 60 mph Racquet speed = 0 mph Relative impact speed = 0 + 60 = 60 mph Rebound speed = 0.4 x 60 = 24 mph Exit speed = 24 + 0 = 24 mph Just after impact Figure れ 4 Ba 〃 hitting stationary racquet. 0 mph ()t rest) 60 m ph 84 mph before impact Rebound power = 0.4 」 u 式 after impact Exit speed = 24 + 60 = 84 mph Rebound speed = 0.4 x 60 = 24 mph Relative impact speed = 0 + 60 = 60 mph Racquet speed = 60 mph lncoming speed = 0 mph Figure れ 5 Racquet hitting stationary ba ″ .

Racquets Match Point Box 1.6 CustomizingWeight, BaIance, and Swingweight Changing the Balance P0int A rule Of thumb is that 5 grams oflead tape added t0 the tip or the end 0f the handle will shift the balance point by about 5 mm, and grams will shift the balance point by about mm. げ you add grams on one side Of the balance point and grams an equal distance on the other side, then the balance point will not shift at all. The latter technique allows you tO increase the weight and the swing- weight without shifting the balance point. Consider a racquet of mass M (gm) with a balance point at distance B (mm) from the extreme end 0f the handle. Suppose you add m (gm) at a distance D (mm) from the extreme end of the handIe.The new balance point will shift toward the extra = x 340 / 330 = 9.3. x = ( 690 ー 350 ) / ( 320 + ) and D = 690 then For example, ifM = 320 , m = , B = 350 mass by a distance x given by extra gm at D = 20 mm, then 9.3 mm toward the tip. げ you put the The new balance point therefore shifts x = ( 20 ー 350 ) / 330 = ー IO.O mm meaning that the new balance point shifts 9.0 mm toward the handle. Changing the Swingweight The on レ way to decrease the swingweight Of a racquet is tO reduce its mass. That is not normally possible unless the racquet was previously customized by adding lead tape.lfyou add m grams at a point D mm from the extreme end ofthe handle, then the increase in swingweight is given by m x (D (O) x (D OO ) 川 OO,OOO in the usual kg ・ cm2 units. For example, if m = gm and D = 690 mm, then the increase iS x 590 x 590 月 00,000 = 34.8 kg ・ cm2 げ you added the gm at D = IOO mm, there would be no lncrease in swingweight at all. That is, no increase about a swing axis IOO mm from the end of the handle where swingweight is normally measured. But players swing their racquet about an axiS near their 、 wrist or near their elbow (depending on the particular stroke) SO there W ・ ou 旧 be a noticeable increase in swingweight as far as the player is con- cerned. ting through the strings and lowering their tension. ln that case all you need tO dO is tO replace the grommets. Professional players usually customize their racquets so each feels the same. The point of this is that even if you buy two nominally identical racquets, they 57

Chapter Fo 町 and 、一、 Traiectory Spin is the same 0 幵 any string, whether it is SO 幵 0 「 stiff, sticky 0 「「 ough , 0 「 thick 0 「 thin. Spin depends on ツ on the speed ofthe racquet head in 0 direction pa 「 0 Ⅱ e い 0 the string plane. SPIN BASICS The modern game Of tennis at top levels is dominated by the amount Of spin that players now impart to the ball. Spin plays an important role in all ball sports, for the same basic reasons. That is, the bounce angle Off a surface iS affected by the amount and direction of ball spin before the bounce, and the flight through the air is also affected by the amount and direction of ball spin. A ball hit with topspin curves down 0m0 the court more quickly than a ball hit without any spin, and it will hit the court at a steeper angle. AS a result, the ball will bounce up higher. lt may also bounce at a steeper angle, but it might even sh00t forward at a lower angle. lt depends on the angle Of incidence and the amount 0f spin. Bounce height depends only on the vertical speed of the ball after the bounce. The bounce angle depends on bo 市 the vertical speed and the horizontal speed after the bounce. The reason that players like hitting with topspin is that it makes it easier tO hit the ball hard withoutthe ball flying over the baseline. ConsequentIy it is easi- er い pass a player at the net or t0 10b over his or her head. The modern game Of singles tennis has evolved intO one where 1 れ OSt top players just run back and forth atthe baseline, trying to keep the ball in play or waiting for an opportu- nity t0 hit a winner. ln the good old days, players came to the net more often 1 19