Dynamic properties of tennis balls

Dynamic properties of tennis balls

R. Cross

Physics Department, University of Sydney, Sydney, NSW 2006, Australia

AbstractMeasurements are presented on the dynamic properties of several different tennis balls,during impacts on a force plate. The force on a tennis ball rises rapidly to about half itsmaximum value in the ®rst 200 ls of the impact due to compression of the cloth coverand the rubber wall near the impact point. The wall then collapses inwards, resulting ina sudden decrease in ball stiffness. Results are presented on the force waveform, theimpact duration, ball compression and coef®cient of restitution as a function of ballspeed.

Keywords: tennis ball, hysteresis, force plate, coef®cient of restitution

Introduction

The properties of tennis balls are rigidly speci®edby the rules of tennis, more so than most other ballsused in major sporting events. Even so, a widevariety of tennis balls with different physicalproperties is manufactured for the consumer. InEurope, tennis balls tend to be more expensive andmore durable, since the consumer expects them tolast for several months. In the United States, ballstend to be less expensive and less durable since USplayers generally prefer to use new balls after a fewsets. Some balls are manufactured to near mini-mum legal size to save on materials costs, whileothers are made to near maximum legal size tosatisfy consumer demand.

There is concern that different balls also playdifferently, not only because the rules allow somevariation in the mass, diameter and coef®cient ofrestitution, but also because the rules are notspeci®c regarding ball properties under actualplaying conditions. The compressibility of a tennisball is speci®ed and tested under static conditions,and the coef®cient of restitution, e, is speci®ed and

tested for a low speed collision with a concrete slab.When dropped from a height of 100 inches(2.54 m) onto a concrete slab, an approved ballmust bounce to a height between 53 and 58 inches(1.35±1.47 m). The advantage of this test is that itis easily implemented by both the manufacturersand the testing authorities. In principle, it shouldalso be relatively easy to measure and specify thecoef®cient of restitution at higher ball speeds, butthe required apparatus has not yet been developedto a point where standards can be simply andreliably speci®ed or enforced.

Furthermore, there is interest in possible alter-ation of the rules concerning ball properties, inorder to make the sport more attractive to specta-tors and television audiences. For example, the ballspeed could be reduced by making the ball larger orby lowering the coef®cient of restitution. The lattermethod is used to reduce the ball speed in top levelsof competitive squash. An alternative solution fortennis balls might be to tailor the coef®cient ofrestitution to decrease, as the ball speed increases,by a speci®ed amount. This already happens, butthe amount is not speci®ed. One might suspect thata heavier ball would also be slower, but this is notnecessarily the case. Even though a heavier ballmay come off the strings at a lower speed, thesubsequent drag force through the air will result in

Ó 1999 Blackwell Science Ltd · Sports Engineering (1999) 2, 23±33 23

Correspondence Address:Rod Cross, Physics Department, University of Sydney, Sydney,NSW 2006, Australia. Fax: +61 2 9351 7727.E-mail: [email protected]

a relatively small deceleration compared with thedeceleration of a light ball. These effects arecurrently under examination.

In this paper, results are presented on theproperties of several types of tennis ball, theobjective being to provide a more scienti®c basison which ball properties can be evaluated. For thispurpose, the dynamic behaviour of each ball wasmeasured by bouncing it vertically off a force platecontaining an array of large area piezo elements.This is obviously not the same as a bounce off thestrings of a racquet, but it provides a valid measureof the dynamic properties of the ball and it providesdata of direct relevance to the bounce of a ball offthe court surface.

Apparatus

A measurement of the force acting on a ball when itimpacts with a rigid surface provides data on theelastic properties of a ball under dynamic condi-tions. These properties can be quite different fromthose measured during a static compression. Theapparatus used in this experiment to measure ballproperties under dynamic conditions is shown inFig. 1. The upper part of Fig. 1 shows the rotatingwheel ball launcher, and the lower part shows theforce plate used to measure the force on the ball.An upper force plate was also used, to measure therebound speed of the ball.

The incident and rebound ball speeds weremeasured under conditions where the ball wasincident vertically downwards on the lower forceplate, passing through a vertical tube of internaldiameter 75 mm. Two horizontal, circular metalplates, with 80 mm diameter holes, were mountedat the base of the tube so that the ball could passthrough the 80 mm holes. Several small piezoswere mounted between the plates to form a forceplate in order to detect the time of arrival of therebounding ball. No spin was imparted to the ball.The ball did not rebound straight back through thetube, but always rebounded at a small angle to thevertical, thereby striking the upper force plate.A horizontal laser beam passing through holes nearthe bottom of the tube was used, in conjunction

with a photodiode, to record the time at which theball passed the beam. The time at which the ball®rst contacted the lower force plate, and the time atwhich it rebounded, were also recorded. The lowerforce plate was located 38.3 cm below the upperforce plate. After making small corrections for thegravitional acceleration downwards and the decel-eration of the rebounding ball, the incident andrebound ball speeds could be determined to withinabout 2%. However, the ball did not alwaysrebound with the same speed when the incidentspeed was kept constant, so all results presentedbelow are averages over two or three separatemeasurements. These measurements were madeafter `conditioning' a new ball by bouncing itseveral times at high speed on the force plate. Thevariability in rebound speed and rebound angle,

Figure 1 Apparatus used to measure the dynamic properties of atennis ball. The grounded circuit board was used as anelectrostatic shield, since the ball charged electrically duringcompression. The Zn paste provided good electrical contactand minimized mechanical cross-talk between the four piezos.

Dynamic properties of tennis balls · R. Cross

24 Sports Engineering (1999) 2, 23±33 · Ó 1999 Blackwell Science Ltd

also commonly observed in the standard 100 inchdrop test, can be attributed to (a) the effect of theseam (b) the fact that balls are not required to beperfectly spherical and (c) the wall thickness is notrequired to be perfectly uniform.

The lower force plate was constructed by theauthor, since force plates with a suitably highfrequency response are not available commercial-ly. Commercial force plates are designed primarilyfor biomechanical applications and suffer fromresonances in the plates for short duration impacts(Cross 1999a). The apparatus shown in Fig. 1 isquite simple, although it was developed aftermany trial and error iterations to obtain a reliableresponse. The main component is an array of foursquare piezoelectric ceramic plates, each of di-mensions 50 ´ 50 ´ 4 mm. Each plate has twosilvered electrodes bonded to and covering each ofthe large surfaces on opposite sides of the plate.The four piezos were arranged in a large squareof dimensions 100 ´ 100 ´ 4 mm, connected elec-trically in parallel, and were mounted on a steelbacking plate of dimensions 140 ´ 140 ´ 25 mm.The dimensions of the array were chosen so thatthe contact area of the ball remained less than thearea of the array. A single, suitably large areapiezo was not available commercially. A similarsystem was described recently for testing baseballs(Hendee et al. 1998). The latter authors did notgive any details of the frequency response of theirsystem. The force plate shown in Fig. 1 providedan accurate response, free of plate resonances, forimpacts of duration between 100 ls and 100 ms,as measured by bouncing a steel ball on the plateand by walking on the plate. The low frequencyresponse was limited by the RC time constant ofthe system, where C represents the capacitanceof the piezos and R � 10 MW is the resistance ofthe voltage probe used to record the outputsignal.

Theoretical model of ball impacton a rigid surface

The dynamics of the bounce of a ball can bepredicted approximately by assuming that it obeys

Hooke's law F � ±kx, where F is the force actingon the ball, x is the ball compression and k is theeffective spring constant of the ball. This leads tothe result that F vs. t is a half-sine waveform ofduration

s � p��m=k

p�1�

where m is the mass of the ball (Cross 1999b).There is no energy loss in the ball in this case.However, it is shown below that the force on atennis ball is a strongly nonlinear function of theball compression, and the dynamics cannot beexpressed simply in any analytical form. Neverthe-less, it is also shown below that Eq. (1) provides auseful estimate of the impact duration, providedthat k is interpreted as a time±average value of theratio F/y where y is the displacement of the centreof mass of the ball.

When a ball of mass m impacts vertically on arigid surface at speed v1, it experiences an impulsiveforce, F, which is typically 100±1000 times largerthan mg. The force is given by F � mdv/dt wherev � dy/dt is the velocity of the centre of mass(CM) of the ball and y is the vertical displacementof the CM of the ball. A measurement of F vs. t cantherefore be used to obtain y vs. t by numericalsolution of the equation

d2y=dt2 � F=m; �2�

assuming that at t � 0, y � 0 and dy/dt � v1.A plot of F vs. y represents a dynamic hysteresiscurve, analogous to the static hysteresis curveobtained when one plots F vs. ball compressionunder static conditions (Brody 1979). The rules oftennis currently specify bounds for such a statichysteresis curve, but do not refer directly todynamic hysteresis measurements. Typical F vs. twaveforms observed with the force plate are shownin Fig. 2, together with the corresponding y vs. tand F vs. y curves computed from Eq. (2).

The area enclosed by a dynamic hysteresis curve,for a complete compression and expansion cycle,represents the loss of energy during the collision.

R. Cross · Dynamic properties of tennis balls


The ball compresses during the collision, convert-ing the initial kinetic energy to potential energy,and then expands to convert the potential energyback to kinetic energy. However, energy is dissi-pated in the ball during this process, with the resultthat the rebound speed, v2, is less than v1. Thereduction in kinetic energy is given byI

F dy � m�v21 ÿ v2

2�=2; �3�

which can be equated to the energy dissipated inthe ball. Eq. (3) is easily derived from the relationF � mdv/dt � m(dv/dy) (dy/dt) where v � dy/dt,but the circumstances are unusual in that (a) theball is not a rigid body and (b) the force is applied ata point on the ball that remains stationary duringthe impact. As a result, no work is done by F in

changing the total energy of the ball, so the totalenergy after the collision (including the energydissipated in the ball) is the same as the total energybefore the collision. A ball can rebound in either acompressed or elongated state depending on therate at which the ball recovers from the compres-sion and depending on the amplitude of anyoscillations excited by the impact. All balls studiedin this paper rebounded in a slightly compressedstate, so that y remained ®nite when F dropped tozero at the end of the impact.

Effects of cloth and rubberon initial impact force

As shown in Fig. 2, the force on a tennis ball risesrapidly during the ®rst 0.2 ms of the impact, to avalue that is typically about half the maximum force

Figure 2 A sample of results obtained with three different balls, including a Dunlop Airloc ball (previously unused, low pressure ordepressurized over time but still very ®rm), Wilson (brand new, pressurized) and a well-used, old, soft, depressurized Slazenger ball.



at high ball speeds. This effect can be attributed tocompression of the cloth and underlying rubber ina small region surrounding the initial impact point.The impact properties of the cloth cover wereinvestigated separately by covering a 6-inch(15.24 cm) diameter spherical aluminium ball, ofwall thickness 3 mm, with standard tennis ballcloth of thickness about 3 mm. The mass of theball plus the cloth was 532 g. Without the cover,the impact duration of the ball on the piezo arraywas 0.8 ms, as shown in Fig. 3. The addition of thecover increased the impact duration to 5 ms for alow speed collision (20 mm drop height) or to 4 msat a slightly higher ball speed (60 mm drop height).High speed impacts were not investigated in orderto avoid possible damage to the ceramic piezos.Despite the low speed of these impacts, theresulting force on the cloth was within the rangeof the impact forces acting on a tennis ball at highspeed.

The impact properties of the rubber were inves-tigated by a similar procedure, using several strips of

3-mm thick rubber cut from a pressurized tennisball and glued side-by-side to the bottom section ofanother 6-inch diameter aluminium sphere of wallthickness 3 mm. As shown in Fig. 3, the impactduration in this case was 3 ms for a 20-mm drop,and 2 ms for a 60-mm drop, indicating that therubber is stiffer than the cloth by a factor of aboutfour. Also shown in Fig. 3 are the correspondingimpacts when several strips of rubber, with clothattached to the rubber, are glued to the aluminiumball to form a 6-mm thick composite covering. Thecombination of the cloth and the rubber is softerthan the rubber or the cloth alone, and the impactdurations are correspondingly longer.

The impact force waveform for the uncoveredaluminium ball is a half-sine wave, indicating thatthe compression of the ball is linearly proportionalto the applied force. The force waveform for thecloth-covered ball is an exponentially rising andfalling bell-shaped curve, indicating that the forceis exponentially proportional to the compression ofthe cloth. An analysis of the hysteresis curves forthe cloth-covered ball shows that, during thecompression phase,

lne�F� � 2:98y� 0:84 �4�

where F is the force in Newton, and y is thedisplacement in mm of the centre of mass of theball. This relation was found to describe the clothcompression accurately in the range 3 < F < 1000 N.For example, a force of 202 N will compress thecloth to half its original thickness (i.e.y � 1.5 mm). As a good approximation, one canassume that the displacement y is equal to thecompression of the cloth since the compression ofthe aluminium was negligible compared with thatof the cloth. Similarly for the rubber, the force isexponentially proportional to the compression ofthe rubber, at least in the range 3 < F < 1000 N,where

lne�F� � 6:06y� 1:94: �5�

The combination of the rubber and the cloth wasfound to satisy the relation

Figure 3 Force vs. time waveforms for a 6 inch (15.24 cm)diameter aluminium ball dropped onto the force plate from aheight of 20 mm or 60 mm, as labelled with a pre®x. The ballwas either completely covered in 3-mm thick cloth (20C and60C waveforms), or partly covered in 3 mm thick rubber (20Rand 60R waveforms), or partly covered with a 6-mm thickcomposite layer of rubber and cloth (20RC and 60RCwaveforms). For the curve labelled Al, the ball was droppedfrom a height of 10 mm to contact directly on the aluminiumsurface of the ball.



lne�F� � 2:0y� 1:20; �6�

which is consistent with the fact that, for anygiven F, the total compression is equal to thecompression of the rubber plus the compression ofthe cloth.

During the ®rst 0.2 ms of the collision of a tennisball with a rigid surface, the ball speed remainsessentially constant since the change in momen-tum, ò F dt � mDv is negligible compared with theinitial momentum. At an initial speed of 15 ms±1,the centre of mass moves a distance of only 3 mmduring the ®rst 0.2 ms. A logarithmic plot of theF vs. t curves in Fig. 2 shows that F increasesexponentially with t during this time. Similarly, alogarithmic plot of the F vs. y curves in Fig. 2shows that

lne�F� � 2:12y� 0:28 �7�

for compressions up to y � 2.5 mm (or F up to265 N), the constants varying only slightly fordifferent balls. For a given compression, this forceis a factor of approximately two lower than thatgiven by Eq. (6). This is consistent with the factthat, for any given y � v1t, the contact area of thetennis ball is a factor of about two smaller than thatof the larger diameter aluminium ball.

Subsequent force on ball

For compressions larger than about 1 or 2 mm,there is a sudden transition from a high to a lowstiffness state commencing about 0.1±0.2 ms afterthe initial contact, depending on the ball speed (seeFig. 2). This suggests that the wall starts to bend atthis time since it is much easier to bend rubber thanto compress it. The transition point does not occurat a ®xed compression or at a ®xed value of theimpact force. At low ball speeds, the transitionoccurs at a force F ~ 50 N. As the ball speedincreases, the transition occurs at a higher forceand at earlier times.

Experimental results indicate that the wall de-forms approximately as shown in Fig. 4. The mostdirect result is obtained by visual inspection of the

interior of a ball that is cut in half and compressedby hand on a ¯at surface. The force waveform of aball half, when dropped or thrown on the piezoarray, is very similar to that of a complete ball,indicating that deformation of the bottom half ofthe ball is responsible for the general featuresobserved in the force waveform. A small verticalforce applied around the perimeter of a ball halfresults in the contact area deforming into arelatively ¯at, circular disk. As the force is in-creased, the wall buckles as shown in Fig. 4. Whenthe force is suf®ciently large, a hemispherical ballturns inside-out. The formation of an interiorbubble in the ball can be attributed to a horizontalcomponent of the applied force, transmittedthrough the wall to the contact area, resulting inan unstable buckling of the contact region undercompression. This effect is probably assisted by thefact that the initial contact area compresses in avertical direction and will therefore tend to bounceoff the surface while the rest of the ball is stillmoving towards the surface. Such an effect wouldhelp to explain why the transition to a low stiffnessstate occurs earlier in time and at a larger force asthe incident ball speed increases.

A dynamic test of the buckling effect wasperformed by mounting a 2-mm thick, 13 mmdiameter piezo disk on top of the lower force plate,as shown in Fig. 5. The small disk was surroundedby a 2-mm thick, 80-mm square piece of circuitboard containing a 15-mm diameter hole for thesmall piezo, so that the ball impacted on a ¯atsurface and so that the small piezo was not subjectto bending. The large piezo array responded to the

Figure 4 Deformation of a ball about 2 ms after initial contact,showing the formation of an internal bubble.



total impact force, and the small piezo sampledthe force acting on a 13-mm diameter section at thebottom of the ball. The sampled location under theball could be varied, depending on where the balllanded. Typical results are shown in Fig. 6, indi-

cating that the vertical force on the initial contactarea is a minimum when the total force on the ballis a maximum.

As shown in Fig. 2, a force peak is sometimesobserved at the end of the contact period, especiallyat high ball speeds and with old or depressurizedballs. This peak is due to the internal bubble`popping' back out of the ball at the end of theimpact, and was observed on both the large piezoarray and the small piezo at the centre of the array.However, the waveform from the small piezo wasfound to be noisy at high ball speeds due to the factthat the small piezo bounced off the force plate andlost electrical contact with the lower surface whenthe force on the small piezo dropped to zero.

Pressurized vs. unpressurized balls

Tennis balls can be classi®ed as either pressurizedor unpressurized (or `pressureless') depending onwhether the air pressure inside the ball is higherthan or equal to atmospheric. Pressurized balls areusually packaged in a pressurized can to preventgradual leaking of the air through the ball. How-ever, some pressurized balls, such as the DunlopAirloc ball, do not need to be stored in a pressur-ized can, since they are basically a thick-walledunpressurized ball ®lled slightly above atmosphericpressure to increase the coef®cient of restitutionslightly. There is no internal metallised coating in atennis ball to prevent the air leaking out. As the airgradually leaks out of an Airloc ball, its coef®cientof restitution decreases, but it remains within thespeci®ed limits even when the internal pressuredrops to atmospheric. The internal pressure of apressurized ball is typically 6±12 psi above atmo-spheric pressure, PA, where PA � 14.7 psi �101 kPa. Upressurized balls are required to havethe same mass, external diameter, coef®cient ofrestitution, and static stiffness as a pressurized ball,within permitted small variations, and are manu-factured using a thicker wall to increase the wallstiffness. The density of the rubber compound istherefore lower than the rubber used in pressur-ized balls. A pressurized ball has a rubber wallthickness of about 3 mm, whereas unpressurized

Figure 5 Arrangement used to measure the force at selectedpositions under the ball.

Figure 6 Evidence for the deformation shown in Fig. 4. Wave-form (a) shows the total force acting on the ball at an impactspeed of about 7 ms±1. Waveform (b) shows the force on a13-mm diameter piezo located on the force plate directly belowthe ball centre, and waveform (c) shows the force on the smallpiezo when the initial impact point is 20 mm away from thecentre of the small piezo.



balls have a rubber thickness of about 4±4.5 mm.The rubber in a pressurized ball is a high densitymaterial with a high ®lling content, low polymercontent and has a relatively low gas permeability inorder to maintain the excess pressure for as long aspossible. Both types of ball are covered with ayellow cloth material, about 3 mm thick, gluedonto the rubber. The difference between pressur-ized and unpressurized balls can be demonstratedby drilling a small hole in the ball or by inserting apin through the wall of a pressurized ball to releasethe compressed air. This has no observable effecton an unpressurized ball, but a large effect on apressurized ball. A depressurized ball is noticeablysofter and has a lower coef®cient of restitution,immediately on release of the air.

The air inside a pressurized ball cannot exert anynet force on the ball. The pressure has the effect ofincreasing the stiffness of the rubber, in the sameway that a rubber band or a string or membrane isstiffer for transverse displacements when it isstretched and under tension. For an internalpressure of 80 kPa, and an internal diameter of3 cm, it is easy to show that the wall tension is1200 Nm±1 and that the wall stiffness is increasedby about 30% as a result of the internal pressure. Inpractice, unpressurized balls tend to be slightlystiffer than pressurized balls, and the impactduration is correspondingly shorter.

Typical results

Measurements of the peak force, contact duration,ball compression and the coef®cient of restitution,e, were made for a variety of pressurized andunpressurized tennis balls, as a function of ballspeed, v1. For this purpose, a ball launcher wasconstructed using two counter-spinning wheels ofvariable speed. The maximum ball speed waslimited to 17 ms±1 with this apparatus. Within thisrange of ball speeds, all new balls tested were verysimilar in performance. Consequently, results foronly two balls are presented in this paper. One setof results is given for a Dunlop Airloc ball, partlybecause this ball (like unpressurized balls) can betested over a period of several months without any

observable change in properties. The ball had amass of 58.1 g and a diameter of 67.0 mm. Theother ball was a Wilson `US Open' pressurized ball,of mass 56.5 g and diameter 65.0 mm, testedimmediately after opening a can of new balls.

Results for the Dunlop ball are shown in Figs 7and 8, and results for the Wilson ball are shown inFigs 9 and 10. Figures 7 and 9 show the peak force,Fmax, and the contact duration, s, as a function ofball speed. Figures 8 and 10 show e and theparameter ymax as a function of ball speed. ymax

represents the maximum displacement of the CM,calculated from the measured force waveform andball speed, as described above.

Figure 7 The maximum force, Fmax, and the impact duration, s,as a function of the incident ball speed, v1, for the DunlopAirloc ball.

Figure 8 The maximum displacement of the CM, ymax, and e asa function of v1 for the Dunlop Airloc ball.



The standard 100 inch drop test corresponds toan incident ball speed of 7.05 ms±1, and the rangeof e at this speed, speci®ed in the rules of tennis, is0.728 < e < 0.762. Both of the balls tested have verysimilar values of e, but the Wilson ball was slightlysofter (both in terms of the qualitative feel of theball and the measured F/y ratio) with the resultthat, for any given ball speed, the peak force wassmaller, ymax was larger and s was longer, each byabout 10%.

Discussion

The actual change in ball diameter, or the dynamiccompression was not measured in this experiment,

but this is likely to be about 70% larger than thedisplacement of the CM. Qualitatively, it wasfound that balls with a low static stiffness also havea low dynamic stiffness. However, even if the actualdynamic compression, x, could be measured, itwould be dif®cult to make a quantitative compar-ison between static and dynamic measurements ofball stiffness or hysteresis. During the initial stagesof the ball compression, the dynamic stiffness, F/y,is almost an order of magnitude larger than thestatic stiffness, F/x. The rules of tennis specify thatfor a static load of 18 lb (8.165 kg), the ball shallhave a forward deformation, x, of more than 0.220of an inch (5.59 mm) and less than 0.290 of an inch(7.37 mm). Consequently, the static stiffness, k,must be in the range 10.9 < k < 14.3 kNm±1 duringthe compression phase. If we take the Wilson ballin Fig. 2 as an example, then the dynamic stiffnessF/y � 87.5 kNm±1 at t � 0.2 ms and F/y �34.1 kNm±1 at maximum compression (i.e. att � 1.7 ms). The latter ®gure is a measure of theaverage slope of the F vs. y hysteresis curve. It istherefore a reasonable measure of the time averagedynamic stiffness and is also consistent with theobserved duration of the impact, as indicated byEq. (1) (s � 4.1 ms). Similarly, the impact dura-tions shown in Figs 7 and 9 are also consistent withEq. (1) if k is interpreted as Fmax/ymax.

Differences in the coef®cient of restitutionbetween different balls may translate to differencesin rebound speed when a ball impacts on aracquet. Differences in ball stiffness or contactduration may translate to differences in therebound angle off a racquet. For example, supposethat a racquet is swung towards an incoming ballwith an angular velocity of 30 rad s±1 and the ballcontacts the strings at normal incidence. If thecontact duration is 4 ms, and the average angularvelocity of the racquet during the collision is25 rad s±1, then the ball will leave the racquetafter the racquet has rotated through an angle of0.1 radian or 5.7°. Neglecting the effects of ballspin, the ball will leave the racket at an angleapproximately normal to the strings. If the contactduration was 5 ms, the racket would rotate by 7.2°during this time. A difference in rebound angle of

Figure 9 The maximum force, Fmax, and the impact duration, s,as a function of v1 for the Wilson ball.

Figure 10 The maximum displacement of the CM, ymax, and e asa function of v1 for the Wilson ball.



1.5°, extended over a travel distance of 20 m, willchange the ball displacement by 0.5 m. Such aneffect should be apparent to good players as adifference in the ball, rather than an error ofjudgement on their part. It is also likely thatplayers could pick the difference between the twoballs by the feel (i.e. the hardness or softness asindicated by squeezing the ball by hand) and bythe sound of the balls.

One of the interesting differences betweenpressurized balls and unpressurized balls is thatthey sound different. This effect does not show upin the force waveforms, but is obvious when thesound of the impact is recorded by means of amicrophone located near the impact point. Fig-ure 11 shows the impact force waveform and asimultaneous measurement of the microphoneoutput for a pressurized ball (Wilson) and anunpressurized ball (Tretorn Plus). The microphonewas located 15 cm from the impact point. Bothballs generate a sound waveform that correspondsroughly to the compression and expansion of theball, and the waveform for both balls also contains a1.2-kHz frequency component (period 0.8 ms).The high frequency component is more stronglydamped in the pressurized ball, which thereforesounds duller. The denser rubber of the Wilsonball appears to damp the vibrations more rapidly.This was not studied in detail, but examination offorce waveforms, such as those in Fig. 3, wouldprovide a useful estimate of the dynamic hysteresislosses in the different rubber compounds.

Conclusions

With the aid of a simple force plate, it is possible toobtain a large amount of information on thedynamic properties of tennis balls. The standard100 inch bounce test, together with static com-pression tests, have provided suf®cient informationto date to regulate the game of tennis, and willcontinue to do so for some time into the future.Measurements of the type described in this paperare perhaps more of purely scienti®c interest, butalso provide data that may help to shape future

developments in the speci®cation and testing of ballproperties. For example, the standard static com-pression test dating from the 1930s is somewhatoperator dependent and could possibly be replacedwith a simpler, more relevant and more reliableimpact duration test or a dynamic compression testas described above.

Acknowledgements

The author would like to thank Andrew Coefrom the ITF for providing the balls used in thesetests.

Figure 11 Output of a microphone located 15 cm from a ballimpacting on the force plate, and the corresponding forcewaveform, for an unpressurized (Tretorn) and a pressurized(Wilson) ball. In both cases, the ball was dropped onto theforce plate from a height of 50 cm.



References

Brody, H. (1979) Physics of the tennis racket. AmericanJournal of Physics, 47, 482±487.

Cross, R.C. (1999a) Standing, walking, running and jump-ing on a force plate. American Journal of Physics, 67, 304±309.

Cross, R.C. (1999b) The bounce of a ball. American Journalof Physics, 67, 222±227.

Hendee, S.P., Greenwald, R.M. & Crisco, J.J. (1998) Staticand dynamic properties of various baseballs. Journal ofApplied Biomechanics, 14, 390±400.



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Dynamic properties of tennis balls