The sweet spots of a tennis racquet

Rod Cross

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

AbstractMeasurements are presented on the behaviour of a hand-held tennis racquet when itimpacts with a tennis ball. It is shown that an impulse is transmitted through the racquetto the hand in about 1.5 ms, with the result that the hand and the forearm both have astrong in¯uence on the behaviour of the racquet even while the ball is still in contactwith the strings. Regardless of the impact point, the racquet head recoils as a result ofthe impact and an impulsive torque is applied to the hand, causing the hand to rotateabout an axis through the wrist. The impulsive forces on the hand, arising from thistorque, do not drop to zero for any impact point, even for an impact at either of the twosweet spots of the racquet. Forces on the hand arise from rotation, translation andvibration of the handle. For an impact at the vibration node, only the vibrationalcomponent is zero. For an impact at the centre of percussion, the net force on the handor forearm is zero since the forces acting on the upper and lower parts of the hand arethen equal and opposite.

Keywords: centre of percussion, hand, rotation, sweet spot, tennis

Introduction

The sweet spot of a tennis racquet is oftenidenti®ed, especially by manufacturers and theiradvertising agents, as the impact point that impartsmaximum speed to the ball. This is not a well-de®ned point on the racquet. It can be locatedanywhere on the longitudinal axis between the tipand throat, depending on the incident speed of theball (Brody 1997; Cross 1997). Alternatively, thesweet spot of a tennis racquet can be de®ned as theimpact point that minimizes the impulsive forcestransmitted to the hand. In 1981, Brody noted thatthere should be two such spots, one correspondingto a vibration node and one corresponding to thecentre of percussion (COP). For a conjugate pointnear the end of the handle, both spots are close to

the centre of the strings, so it is dif®cult todistinguish one from the other in terms of thequalitative feel of the impact. To date, there havebeen no de®nitive experiments to distinguish thetwo points in terms of measured reaction forces onthe hand. The sweet spots described by Brody werede®ned primarily in terms a racquet that is freelysuspended, with no restraining force acting on thehandle. For a freely suspended racquet, the COP isnot a unique point on the strings since there is nounique conjugate point (i.e axis of rotation) in thehandle. The present work examines the effects ofthe hand on the two sweet spots and provides awell-de®ned location for the COP in terms of theimpulsive force acting on the forearm. As shownbelow, the relevant location is the end of thehandle.

There has been debate for many years as towhether the hand plays a signi®cant or a negliblerole in determining the dynamics of the impact of aracquet and ball. The collision of a tennis racquetwith a tennis ball can be modelled (Leigh and Lu

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Correspondence Address:Rod Cross, Physics Department, University of Sydney, Sydney,NSW 2006, AustraliaE-mail: cross@physics.usyd.edu.au

1992; Brody 1995, 1997) by assuming that the handplays no role during the impact, in which case theracquet can be regarded as being freely suspended.The main arguments presented to support thismodel are that (a) the hand has only a small effecton the vibration frequency of the racquet and (b)the ball will leave the strings before the impulse istransmitted along the racquet frame to the hand.Experimental studies of the effect of the hand arenot entirely consistent with this model. Elliott(1982) and Watanabe et al. (1979) studied theeffects of grip ®rmness on the coef®cient ofrestitution (COR). In these studies, a ball wasprojected onto the strings of a racquet and the ballrebound speed was measured under various gripconditions and for impacts at several differentlocations on the strings. It was found that gripconditions have a negligible effect on the COR forimpacts near the centre of the strings, even underextreme conditions where the racquet is eitherfreely suspended or the handle is rigidly clamped.However, it was also found that the COR increasedslightly with grip ®rmness for off-centre impacts.

Four new approaches have been adopted in thispaper to examine the effect of the hand on theracquet (a) by measuring the propagation delay ofthe impulse along the racquet; (b) by comparingmeasured values of the handle velocity of a freelysuspended racquet with those of a hand-heldracquet; (c) by measuring the reaction forces onthe hand and (d) by measuring the velocity of theforearm. It is shown below that an impulse istransmitted through the racquet to the hand inabout 1.5 ms, with the result that the hand has astrong in¯uence on the behaviour of the racquet,even while the ball is still in contact with thestrings. The rotation axis and the vibration node inthe handle are both shifted, from their locations ina freely suspended racquet, to points under or closeto the hand. The reaction forces on the hand do notdrop to zero for an impact at the vibration node,nor for an impact at the centre of percussion. Theforces vary from one point to another under thehand, being negative at some locations and positiveat others. This is because the racquet applies animpulsive torque to the hand, causing the hand to

rotate about an axis through the wrist. The netforce acting on the hand is dif®cult to measure, buta good indication is provided by measuring thevelocity of the forearm, at a point close to the wrist,during and after the impact.

An issue that is not directly addressed in thispaper is whether the hand plays a signi®cant role indetermining the outgoing speed of the ball. If theleading edge of the pulse re¯ected from the handarrives back at the ball just as the ball is leaving thestrings, then the ball will be largely unaffected bythe hand. Theoretical and experimental resultsrecently obtained by the author support thishypothesis and the results will be described else-where.

Experimental techniques

All of the measurements presented in this paperwere made using a 1990 vintage Wilson graphitecomposite racquet of mass 370 gm and length685 mm. All measurements were made underconditions where the racquet was initially at rest,the ball was incident at low speed in a directionperpendicular to the strings and the ball impactedat a point on the central axis passing through thehandle and the centre of the strings. These condi-tions are rarely encountered during normal play,but the physics of the collision between a ball and aracquet does not depend strongly on the speed ofthe ball or the racquet and is independent of thereference frame in which the collision is studied.The experimental conditions were therefore chosento simplify the data collection process as far aspossible and to ensure that the impact conditionswere reproducible.

In order to measure the propagation delay andhandle velocity, ®ve piezoelectric disks were at-tached to the racquet: one in the centre of thestrings, and one each at 24 cm, 17 cm, 12 cm and1 cm from the end of the handle. The piezoelements, in the form of circular disks of diameter19 mm, 0.3 mm thick, were extracted from piezobuzzers commonly available from electronicsshops. The disk on the strings was glued withepoxy resin and three other disks were taped ®rmly

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to the handle to avoid independent vibration of theelements themselves. The piezo at the far end ofthe handle was glued to a ¯at wall of the rectan-gular cross-section cavity inside the handle, inorder to avoid the additional and variable responsedue to the pressure of the hand on a piezo elementmounted on the outside of the handle. The piezo at1 cm was therefore located under the base of thehand, and the piezo at 12 cm was located justbeyond the index ®nger. The piezo disks were verylight in weight (1.8 gm) and had no observableeffect on the properties of the racquet, as evidencedby the fact that the signal observed from any onedisk was not effected by adding or removing any orall of the other disks.

A brass electrode bonded to one side of eachpiezo disk and the silvered electrode on the otherside were connected to 10 MW oscilloscope probesvia very light connecting leads. It was necessary totape the connecting leads to the racquet handle atpoints close to the piezo elements in order to avoidany spurious response as a result of independentmotion of the leads. The piezo outputs wereobserved directly, in order to monitor the racquetacceleration, and were also integrated with a simpleRC circuit, of time constant 100 ms (R � 1 MW,C � 0.1 lF), in order to monitor the racquetvelocity. The output of a piezo is directly propor-tional to the applied force and is therefore propor-tional to the acceleration of the disk. All piezoswere connected to give a positive output whencompressed, and all traces in this paper wererecorded on a DC-coupled digital storage oscillo-scope, pretriggered several ms prior to the impactin order to record the zero level of the correspond-ing acceleration or velocity waveforms. The out-puts were not calibrated to determine the absoluteacceleration or velocity since the only measure-ments of interest were the time delays betweenwaveforms and the relative velocity at differentpoints on the racquet. The technique of using anintegrated piezo signal to measure racquet velocityhas not previously been described as far as theauthor is aware. The validity of the technique wascon®rmed by an independent velocity measure-ment, obtained by differentiating the racquet dis-

placement waveform, as measured by thedisplacement of a small capacitor plate attachedto the racquet frame, relative to a parallel ®xedplate.

The racquet was suspended vertically by a 60-cmstring tied to the handle, or held in the normalfashion by hand but with the strings in a horizontalplane. Tests with other lengths of string con®rmedthat the length chosen was adequate for thepurpose of simulating the response of a completelyfree racquet, and that the restoring force of thestring was negligible during and for at least 30 msafter the impact. A tennis ball was dropped orthrown at low speed, from a distance of about10 cm, onto the strings near the tip or throat of theracquet or directly onto the piezo disk in the centreof the strings. The results of these measurementsare shown in Figs 1±4.

In order to measure the reaction forces on thehand, a separate experiment was performed using a9-mm diameter piezo disk, of thickness 0.3 mm,located at a point on the handle underneath thehand. A second 9-mm diameter disk was located onthe strings to provide a reference signal for timingpurposes and was attached to the strings by meansof re-usable adhesive putty so that it could be easilyrelocated to several different points on the strings.To minimize bending of the piezo on the strings, itwas bonded with epoxy to a 0.5-mm thick, 10-mmsquare sheet of epoxy ®breglass. The piezo underthe hand was taped to the handle with clearadhesive tape so that it could easily be relocatedto different points under the hand on relatively ¯atparts of the handle. The observed signals werefound to be accurately reproducable even afterrelocating the piezo many times. However, the areaunder the tip of the little ®nger was too close to theknob on the end of the handle to generate reliableresults.

Measurements of hand forces using force sensingresistors have previously been made by Knudsonand White (1989). They reported considerablevariability in the magnitude of the observed im-pulsive forces, which they attributed to variations inimpact location and racquet velocity. In the presentexperiment, these variations were minimized since

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the initial racquet velocity was zero and theimpulsive forces were measured at selected impactlocations.

Despite the low speed of the impacts studied,qualitatively similar results can be expected duringhigh speed impacts. The phenomena described inthis paper are almost entirely linear up to the elasticlimits of the racquet, strings and ball. The onlynonlinear process of any signi®cance during thecollision of a ball with a racquet relates to theeffects of hysteresis in the ball. The coef®cient ofrestitution and the duration of the impact variesslightly with ball speed (Brody 1979), but this willhave no effect on the transmission time of a pulsealong the handle or on the resulting effect of thehand. The vibration amplitude of the fundamentalmode remains zero for an impact at the node,regardless of the ball speed.

Transit time of an impulse along the handle

The transit time of an impulse from the impactpoint to the hand has not previously been measuredfor a tennis raquet. It can be obtained from the timedelay between the signal recorded on the stringsand the signal recorded at a point on the handleclose to the hand. Results are shown in Fig. 1 for animpact at the centre of the strings, and in Fig. 2 foran impact on the strings near the tip of the racquet.

In both cases, the racquet was hand-held andstationary prior to the impact. In Fig. 1, thefundamental vibration mode of the frame is excitedwith very low amplitude since the impact occursclose to a node for this mode. Motion of the handleis therefore due almost entirely to rotation andtranslation of the racquet, the vibrational compo-nent being negligible.

Figure 1(a) shows the direct piezo signal detect-ed when the ball is dropped onto the piezo in thecentre of the strings, and Fig. 1(b) shows thewaveform of the handle velocity (i.e the integratedacceleration waveform) measured simultaneously ata point 12 cm from the end of the handle. Thetraces in Fig. 1 were triggered 4 ms before the ballimpacted the strings. The ball exerts a force on thestrings that is approximately a half-sine pulse ofduration 7.4 ms, at least for this low impact speedtest. The negative polarity waveform from thepiezo located on the upper surface of the handle,indicates that the handle de¯ected downwards, inthe same direction as the incident ball. From therelative magnitude of the velocity waveforms atother positions along the handle, it was concludedthat the racquet pivoted about a point near the endof the handle, immediately on arrival of the impulseat the hand. The transit time of a pulse from thecentre of the strings to the point 12 cm from the

Figure 1 Measurement of handle velocity for a hand-heldracquet when a ball is dropped onto the centre of the strings.Traces show (a) the direct output of a piezo disk at the centre ofthe strings and (b) the handle velocity waveform at a point12 cm from the end of the handle.

Figure 2 Measurement of pulse propagation for a hand-heldracquet when a ball is dropped on the strings near the tip.Traces show the direct outputs of piezo disks located at (a) thecentre of the strings and (b) 12 cm from the end of the handle.The time integral of waveform (b) is shown in Fig. 4.

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end of the handle was 1.5 ms, as indicated by thetime delay between the two corresponding wave-forms in Fig. 1. The racquet handle thereforebegins to move well before the ball leaves thestrings and reaches a maximum velocity just afterthe ball leaves the strings.

The pulse propagation time is much faster thanpreviously estimated (Brody 1997). Brody estimatedthe propagation time from an analysis of thefundamental vibration period. In Fig. 1, vibrationalmotion of the racquet frame is not readily apparentsince the impact occurred close to a node of thefundamental mode and since higher frequencymodes are excited with relatively small amplitudeand attenuate more rapidly than the fundamentalmode. Nevertheless, the impact excites a broadspectrum of frequency components and the resul-tant motion of the handle represents a superpositionof all transverse waves excited by the impact. Thefundamental mode of vibration of the hand-heldracquet had a measured frequency of 102 Hz and awavelength of about 0.8 m, with nodes about 15 cmfrom each end of the racquet. The wave speed ofthis mode is therefore about 80 m s)1. The prop-agation time from the centre of the strings to theend of the handle, a distance of 0.53 m, is thereforeabout 6.5 ms for the fundamental mode. The nextvibration mode has a theoretically predicted fre-quency of 276 Hz, a wavelength of about 0.52 mand a velocity of about 143 m s)1. For this frequen-cy component, the transit time from the centre ofthe strings to the end of the handle is 3.6 ms. Theinitial motion of the handle, 1.5 ms after the ball®rst contacts the strings, cannot simply be explainedin terms of the ®rst few vibration modes, nor purelyin terms of rigid body rotation. For an in®nitely stiffracquet, one would expect zero delay between theinitial impact and motion of the handle. The shortdelay must therefore represent the combined effectsof all high frequency transverse waves generated bythe impact. Because of the increased stiffness of aracquet for short wavelength vibrations, high fre-quency transverse waves in a racquet, or any othersolid beam, propagate faster than low frequencytransverse waves (Cross 1997, 1998). The high wavespeed through the strings also contributes to the

short delay time. The velocity is approximately2f L » 300 m s)1 where f » 500 Hz is the vibrationfrequency of the strings and L » 0.3 m is the stringlength. The propagation delay from the centre ofthe strings to the frame is therefore about 0.5 ms,accounting for 1/3 of the observed delay and about1/3 of the transit distance.

The above interpretation is supported by theresults shown in Fig. 2, where the ball impacted thestrings near the tip of the racket. The piezo on thecentre of the strings responds mainly to vibrationsof the strings at 500 Hz, and the piezo on thehandle generates a waveform representing theacceleration of the handle at that point. The stringvibrations are not seen in Fig. 1 since the force onthe piezo due to compression of the ball is muchlarger than the force due to the string vibrations. InFig. 2, the fundamental mode at 102 Hz is seenclearly, but higher frequency components appear atthe beginning of the handle acceleration trace, aftera propagation delay of 1.5 ms. A similar effect hasalso been observed with a baseball bat. An estimateof the propagation time along a baseball bat, basedon the fundamental mode frequency, indicates thatthe ball should leave the bat well before the impulsearrives at the hand. In fact, measurements showthat the impulse arrives at the hand before the ballleaves the bat (Cross 1998).

Measurements of handle velocity

Given that an impulse propagates to the hand wellbefore the ball leaves the strings, one would expectthat the reaction force from the hand should have asigni®cant effect on the motion of the racquet evenwhile the ball is still in contact with the strings.This effect was investigated by comparing thehandle velocity for a freely suspended racquet withthat of the same racquet when it was hand-held.The results are shown in Figs 3 and 4, respectively.

Figure 3 shows the handle velocity at severalpoints along the handle for a freely suspendedracquet and for an impact 8 cm from the tip of theracquet. The absolute values of the handle velocitywere not calibrated, but the relative velocities werepreserved by recording and displaying all signals at

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the same sensitivity. The vibrational components ofthe velocity traces at the 24 cm and 17 cm locationsare in phase, but the traces at the 17 cm and 12 cmlocations are 180° out of phase, indicating that avibration node exists at a point about 15 cm fromthe end of the handle, as expected for the funda-mental mode of oscillation of a beam that is free atboth ends (Cross 1997).

All of the traces in Fig. 3 have an obvious DC aswell as an AC component, except for the trace at17 cm where the DC component is close to zero. Itcan be inferred from these results that the DCcomponent is zero at about 16 cm, at which pointthe velocity due to rotation is equal and opposite tothe velocity due to translation. The racquet there-fore rotates about an axis located about 16 cm fromthe end of the handle. The racquet has a measuredmoment of inertia Icm � 0.017 kg m2 for rotation

about the CM. For an impact 8 cm from the tip ofthe racquet, the conjugate point (i.e. the actual axisof rotation) is expected to be located 16 cm fromthe end of the handle, as observed. At least, that isthe case immediately after the arrival of the impulseat the handle and for a short period after the ballleaves the strings. On a longer time scale, the DCcomponent of the traces in Fig. 3 drifts slowly,partly as a result of the weak restoring force due tothe string suspension and partly due to the 100 mstime constant of the integrator.

The corresponding handle velocity traces for ahand-held racquet are shown in Fig. 4. The gainsettings and drop heights were held constant to

Figure 3 Measurements of the handle velocity for a freelysuspended racquet initially at rest. The ball impacted thestrings at low speed 8 cm from the tip of the racquet, and thehandle velocity was measured at points 24, 17, 12 and 1 cmfrom the end of the handle, as indicated. The velocity isnegative if the handle moves in the same direction as theincident ball. Figure 4 Measurements of the handle velocity for a hand-held,

horizontal racquet that is initially at rest. The ball was droppedon the strings 8 cm from the tip of the racquet and the handlevelocity was measured at points 24, 17, 12 and 1 cm from theend of the handle, as indicated. The velocity is negative if thehandle moves in the same direction as the incident ball(downwards).

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compare the relative magnitudes of the handlevelocity, and the traces were all triggered at thesame ®xed time before the arrival of the pulse at thecentre of the strings. However, the gain for the1 cm position is double the gain at other locationsin order to show the oscillations more clearly. Thevibrational mode of the racquet is similar to that ofa freely suspended racquet, but the frequency isslightly lower, the vibrations are more stronglydamped and the vibration node shifts to a pointsomewhere under the hand, judging from the factthat the oscillations at 1 cm and 12 cm are about180° out of phase. The vibration amplitude de-creases towards the end of the handle with a slightphase shift along the handle. The axis of rotation,for an impact near the tip of the racquet, shifts fromthe 16 cm position for a free racquet to a positionabout 5 cm from the end of the handle. The axis ofrotation is established immediately on arrival of theimpulse at the handle, 1.5 ms after the ball ®rstcontacts the strings, as evidenced by the traces inFigs 1±4. For an impact between the centre of thestrings and the throat of the racquet, the axis ofrotation was observed to be close to the end of thehandle, regardless of whether the racquet was freeor hand held, as was expected since the impact isnear the centre of percussion.

Figure 5 (and also Fig. 8) shows schematicallythe results of the above measurements. The shift inlocation of the node in the handle can be explainedqualitatively by the fact that the vibration ampli-tude of the handle is reduced when it is hand-held,thereby approximating the behaviour of a racquetthat is pivoted or clamped at the handle end. Theshift of the axis of rotation is explained by the factthat the end of the handle does not translate freelybut is constrained by the inertia of the hand andforearm. Both of these effects have also beenobserved with a hand-held baseball bat (Cross1998).

A shift in the location of the vibration node andthe lowering of the frequency can be roughlymodelled if one assumes that the hand acts as anadditional mass loading the end of the handle. Thevibration frequencies in Figs 3 and 4 are, respec-tively, 109 Hz (free) and 102 Hz (hand held).

Figure 6 shows the effects of adding 40 g and80 g masses to the end of a freely suspendedracquet. The vibration frequency drops from109 Hz with no additional mass to 103 Hz withan additional 40 g mass and to 100 Hz with an 80 gmass. The observed frequency shift when theracquet is hand held can therefore be modelled bythe additional 40 g, as noted previously by Brody(1995), but the shift in the node location is notcorrectly simulated by the additional 40 g mass.The vibration node shifts from 15 cm to a point12 cm from the end of the handle when a 40 g massis added, and it shifts even further towards the endof the handle when an 80 g mass is added, since thevibrations at 17 cm and 12 cm are then in phase. Itis therefore possible to simulate the shift in nodelocation and the lowering of the frequency byadditional masses, but both effects cannot besimulated simultaneously with the same additionalmass.

The axis of rotation also shifts towards the end ofthe handle as additional mass is added to a freelysuspended racquet, but the actual shift observedwhen the racquet is hand-held can only be simu-lated by adding a mass in excess of 80 g to the endof the handle. As shown in Fig. 4, a hand-held

Figure 5 Vibration modes of a free and a hand-held racquet.

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racquet de¯ects downwards during the impact, at aposition 12 cm from the end of the handle, but thede¯ection is upwards in Fig. 6 even with 80 gadded to the handle. The large mass required toshift the rotation axis can be attributed to the effectof the hand and arm on the racquet dynamics,provided the dynamics are modelled correctly. Asshown by Casolo and Ruggieri (1991), the effectivemass of the forearm is less than the actual masssince the racquet applies an impulsive force to theend rather than the centre of the arm, the other endof the arm being pivoted at the elbow. Further-more, the arm is not rigidly attached to the handle,due to the ¯exibility of the wrist. Consequently, theeffect of the hand and the arm cannot be simulatedcorrectly simply by adding a ®xed mass to the endof a freely suspended racquet. The dynamics of thesituation can be modelled as shown in the followingSection.

Effect of the arm on racquet dynamics

A simple model of the effect of the arm on racquetdynamics, consistent with the above observations,is shown in Fig. 7. The racquet is approximated asa beam of mass M and length L connected by apivot joint to the forearm, which is represented as abeam of mass MF and length LF. It can be assumedthat the other end of the forearm is pivoted aboutthe elbow, but it is assumed for simplicity that theelbow does not translate during the impact. Theimpact of a ball on the racquet can be representedby an impulsive force, F, applied at a distance bfrom the racquet CM, the CM being located adistance h from the end of the handle. The handlewill exert an impulsive force FR on the forearm,resulting in a reaction force )FR on the handle.The equations of motion are then

F � FR �MdV=dt �1�

Fbÿ FRh � Icmdx=dt �2�

andFRLF � IFdxF=dt �3�

where V is the velocity of the CM of the racquet, Iis the moment of inertia of the racquet about itsCM, IF is the moment of inertia of the forearmabout the elbow, x is the angular velocity of theracquet and xF is the angular velocity of theforearm. The velocity of the pivot joint at the wristis given by

Figure 6 Effects of adding 40 and 80 g masses to the end of thehandle for a freely suspended racquet. The traces show thevelocity waveforms at a point 12 cm from the end of the handle,for an impact near the tip of the racquet. The phase differencebetween the top and bottom traces varies as a function of timesince the vibration frequencies are slightly different.

Figure 7 Model used to evaluate the effect of the forearm whena ball impacts at a distance b from the racquet centre of mass.The racquet is pivoted at the wrist and the forearm is pivoted atthe elbow.

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VP � LFxF � hxÿ V �4�

In these equations, it is assumed that x is measuredin an anticlockwise sense (as in Fig. 7) and that xF

is measured in a clockwise sense as appropriate foran impact near the tip of the racquet (as in Fig. 8).Similarly, V is taken as positive when the racquetmoves downwards in Fig. 7 and VPP is taken aspositive when the pivot joint moves upwards.

Since the racquet exerts a force on the forearm atthe pivot joint, an effective mass of the forearm,ME, can be de®ned by the relation FR � MEdVPP/dt � MELFdxF/dt, So from eqn 3, ME � IF=L2

F.For example, if the forearm is approximated as auniform beam, then IF �MFL2

F=3 so ME �MF=3.Equations (1)±(4) can be combined to show thatdV/dt � x dx/dt where

x � Icm �MEh�h� b�Mb�ME�h� b� �5�

Now consider a point on the racquet located adistance x to the right of the CM where the racquetvelocity is V ) xx. This point will coincide withthe axis of rotation of the racquet if its velocityremains constant during the impact, or if dv/dt �x dx/dt. Consequently, the location of the conju-gate point, i.e. the axis of rotation, is given byeqn 5. This relation reduces to the well-knownexpression x � Icm/(Mb) when ME � 0, corre-sponding to a freely suspended racquet (Brody1979). It can also be seen from eqn 5 that if x � h,then b � Icm/(Mh) which is the same result thatone obtains for a freely suspended racquet. Thisparticular value of b de®nes the centre of percus-sion, since FR � 0 for this value of b, and theracquet behaves as if it were completely free. For

any other impact point, the forearm constrains themotion of the racquet, and the conjugate point isshifted closer to the end of the handle than for afree racquet, regardless of whether the conjugatepoint lies within the handle or beyond the end ofthe handle. This effect is shown schematically inFig. 8

The results of the previous Section can bemodelled with the measured parameters M �0.37 kg, Icm � 0.017 kg m2, h � 0.33 m andb � 0.27 m, corresponding to an impact 0.08 mfrom the tip of the racquet. When ME � 0 thenx � 0.17 m, meaning that the axis of rotation islocated 16 cm from the end of the handle when theracquet is freely suspended. However, whenME � 0.6 kg, x � 0.29 m, so the axis of rotationis shifted to a point 4 cm from the end of thehandle. Both calculations are consistent with theobserved results. This value of ME is consistentwith an approximate estimate of the mass of theforearm, about 1.8 kg, but x does not dependstrongly on ME when ME is larger than the mass ofthe racquet. For the same racquet parameters, thecentre of percussion (COP) is located atb � 0.14 m, assuming that the axis of rotationcoincides with the end of the handle. This locatesthe COP 5 cm from the centre of the strings, asillustrated in Fig. 9.

According to the above theoretical model, thereaction force FR acting on the end of the forearmshould be zero for an impact at the centre ofpercussion. Previously, it has been assumed that foran impact at the COP, the reaction force on thehand would be zero (Brody 1979, 1981). One mightexpect that the force on the hand should beessentially the same as the force on the forearm.However, the measurements presented in the

Figure 8 Schematic diagram comparingthe motion of a free and a hand-heldracquet when a ball is dropped near thetip or throat of the racquet, showingthe racquet position before the impact(thin line) and after the impact (thickline). For a hand-held racquet, the axisof rotation shifts to a point closer to thewrist.

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following section surprisingly show that the forcesacting on different parts of the hard can be quitelarge, even when the force on the forearm is zero.The result is easily interpreted in terms of the netforce on the hand. This will remain zero if theforces on the upper and lower parts of the hand areequal and opposite, even if these forces vary withtime. Recent measurements of the hand forces byHatze (1998) are consistent with the results de-scribed below, but Hatze concluded that the COPwas of limited signi®cance since the forces ondifferent parts of the hand vary with time.

Forces acting on the hand

In order to measure the forces on the hand, a smallpiezo was located on the handle, underneath thehand, as described in the section on ExperimentalTechniques. The impulsive forces acting on thehand were measured at three different points underthe hand, and for four different impact points onthe strings, as indicated in Fig. 9. The racquet washeld ®rmly by the right hand in a stationaryposition with the strings in the horizontal plane,and a tennis ball was dropped onto the strings froma height of 20 cm. The results of this experimentare shown in Fig. 10. Absolute values of the forcewere not calibrated, but the relative magnitudes canbe compared with the 150 mV positive signal

recorded when the handle was gripped ®rmly bythe hand (or ± 150 mV when the grip was released).The grip waveform decayed to zero with a timeconstant of 70 ms, representing the discharge timeconstant of the 7 nF piezo through the 10 MWvoltage probe. This component of the force wave-form therefore decayed to zero prior to each impactmeasurement. The forces shown in Fig. 10 there-fore represent the change in the force at each pointas a result of the impact. The largest impulsiveforce signal was )80 mV, representing the ®rstnegative peak recorded at position d for an impactat the tip.

The results obtained for an impact at the centreof the strings are easiest to interpret since theracquet frame does not vibrate in that case. Theforce on the hand is observed to increase at the baseof the index ®nger (waveform b) and decrease at thebase of the little ®nger (waveform d), during andafter the impact, indicating clearly that the handlemoves towards the base of the index ®nger andaway from the base of the little ®nger. With respectto the hand, the racquet therefore rotates about anaxis that is located between the index and little®ngers. Since the force on the middle ®ngerremains small at all times, the axis of rotationwithin the hand is located almost exactly in themiddle of the hand. The actual axis of rotation ofthe racquet in the laboratory frame may be differ-

Figure 9 The force acting on the hand wasmeasured at three points b, c and d, for fourimpact positions on the strings, as labelled.The distance from the Centre to the COPwas 5 cm.

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ent since the hand itself rotates about an axisthrough the wrist and it can translate as a result ofmotion of the forearm.

An off-centre impact results in vibration of theframe, as well as translation and rotation of theframe. The forces on the hand resulting fromvibration increase as the impact point movesfurther from the centre of the strings. The `DC'component of the force waveforms is qualitativelysimilar to that observed for an impact at the centreof the strings, regardless of the impact point. Theracquet therefore rotates within the hand about anaxis that is near the centre of the hand, regardless ofthe impact point. If the axis of rotation in the

laboratory frame was located further up the handletowards the racquet head, the handle would moveaway from the base of the index ®nger, not towardsit. Consequently, for all of the results shown inFig. 10, the axis of rotation in the laboratory frameis either in the middle of the hand or shifted to apoint close to or beyond the end of the handle. Foran impact at the throat, the DC component ofwaveform (d) is signi®cantly smaller than at otherimpact locations, and is close to zero for the ®rst10 ms, indicating that the axis of rotation of theracquet was close to the end of the handle.

An interesting feature of the results in Fig. 10 isthat there is a phase shift of about 90° between the

Figure 10 Forces acting on the handat points b, c and d, as labelled inFig. 9. Waveform a represents theforce of the ball on the strings, asmeasured by a small piezo on thestrings. The vertical gain of the os-cilloscope used to record waveformsb, c and d was either 20 mV or 50 mVper division, as labelled. A positiveforce indicates an increase in the forceon the hand. Zero impulsive forcecorresponds to the horizontal part ofeach trace prior to impact.

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vibrational components of waveforms b and d. Thiseffect is presumably associated with the fact thatthese waveforms are recorded on the same side ofthe handle but on opposite sides of the vibrationnode under the hand. The node itself is not clearlyapparent but is close to the centre of the hand,regardless of impact point, judging by the reducedvibration amplitude of waveform c. For an un-damped standing wave, there is a phase shift of 180°between any two points on opposite sides of a node.For a damped standing wave, the phase shift isgenerally less than 180° since the phase angle variescontinuously from + 90° to ) 90° from one anti-node to the next. The phase jumps discontinuouslyby + 180° at a node only when the damping is zero.

The results in Fig. 10 are consistent with thoseobtained by Knudson and White (1989) and byHatze (1998) who also found that the forceincreases at the base of the index ®nger anddecreases at the hypothenar eminence, during theimpact, corresponding to an impulsive rotation ofthe racquet in the same (expected) sense as ob-served in this paper.

There is no impact point on the strings wherethe forces on the hand are zero everywhere. Thevibrational component is zero at all points underthe hand for an impact at the vibration node, andthis node clearly quali®es as a sweet spot in terms ofthe qualitative `feel' of the racquet. The smallestDC forces acting on the hand occur for impactsbetween the centre of the strings and the COP, andthe largest forces occur for an impact at the tip ofthe racquet. The COP has no special signi®cancewith respect to the forces acting on different partsof the hand. However, it is of major signi®cance indetermining the force on the forearm, as describedin the following Section.

Impulsive motion of the forearm

Since the forces acting on the hand vary from onepoint to another, a measurement of the force actingat a single point under the hand does not provide avalid indication of the total force of the handle onthe hand (or of the hand on the handle). Inprinciple, one could sum the forces on the hand at

many different points under the hand to determinethe total force, but this is not a practical proposi-tion. Alternatively, a reasonable assumption is thatwaveform b represents the net force on the upperpart of the hand and waveform d represents the netforce on the lower part of the hand. The net forceon the hand is then given approximately by the sumof waveforms b and d. Measurements of theimpulsive motion of the forearm indicate that thisis indeed a good approximation.

In order to measure the impulsive motion of theforearm, a 19-mm diameter piezo was strapped, inwrist-watch fashion, to the forearm around thewrist as shown in Fig. 11. The piezo itself wasattached with adhesive tape to a 2-mm thick, 25-mm diameter ®breglass disk in order to avoidbending of the piezo and in order to provide anchorpoints for the band around the wrist. The output ofthe piezo was integrated with a 100-ms timeconstant integrator to measure the velocity wave-form. The acceleration waveform is more dif®cultto interpret since the largest component is due tovibration of the racquet and arm. The polarity ofthe acceleration signal is therefore dominated bythe polarity of the vibration component. Thevelocity waveform provides a less ambiguous indi-cation of the response of the arm due to rotationand translation. The vibration component is not®ltered out, but integration acts to attenuate theamplitude of the high frequency components of thewaveform. The velocity measurement was testedfor reliability in a number of ways, including simplemotion of the arm up or down without the racquet

Figure 11 Arrangement used to measure the velocity of theforearm using a piezo strapped to the wrist. The impact pointson the strings were the same as those shown in Fig. 9.

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and attaching the piezo to a vibrating, cantileveredmechanical arm.

The velocity of the forearm was measured underconditions where the racquet handle was held bythe right hand, using an eastern forehand grip withthe strings in the horizontal plane, and a tennis ballwas dropped from a height of 20 cm onto a smallpiezo attached to the strings. The same waveforms,proportionally larger in amplitude, were observedfor impacts at ball speeds up to 15 m s)1. Theobserved effects were therefore independent of ballspeed up to this limit. The racquet and arm wereinitially stationary, so the observed velocity of theforearm corresponds purely to the impulsive mo-tion generated by the impact of the ball on theracquet. The velocity of the forearm was measuredfor ®ve different impact points, relocating the piezofor each drop so the ball landed directly on thepiezo to generate a reference signal for timingpurposes. The results of this experiment are shownin Fig. 12. The absolute values of the forearmvelocity were not calibrated, but the results aredisplayed at the same sensitivity for each drop toprovide a comparison of the relative amplitude andpolarity of the velocity in each case. The polaritywas chosen so that a positive velocity correspondsto motion of the forearm vertically upwards,opposite the direction of the incident ball. One ofthe waveforms in Fig. 12 corresponds to a droponto the handle, at a point midway between thehand and the strings.

The most signi®cant effect indicated by thewaveforms in Fig. 12 is that there is almost noinitial motion of the forearm for an impact at theCOP. For an impact at the tip or centre of thestrings, the forearm moves initially in the oppositedirection to the incident ball. For an impact at thethroat of the racquet or on the handle, the initialmotion of the forearm is in the same direction asthe incident ball. Several other effects are alsoobvious from these waveforms:1 Within about 20 ms of the impact, the velocity of

the forearm drops to zero and reverses sign atmost impact locations. After the ball leaves theracquet, one might expect the racquet andforearm velocity to remain constant. Such a

result is predicted from eqns 1,2,3,4, since ifF � 0 then FR � 0. The experimental resultsindicate the presence of other forces acting onthe forearm. In order to hold the racquet in asteady horizontal position prior to the impact,the upper arm exerts a force on the forearm andthe forearm exerts a force on the wrist to keep it

Figure 12 Velocity of the forearm at several different impactpoints, when the racquet is initially stationary and a ball isdropped onto the strings or the handle at the points indicated.A positive velocity corresponds to motion of the forearmvertically upwards. The horizontal line through each waveformis the zero velocity baseline. The drop height (20 cm) andvertical sensitivity were the same for each impact.

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locked in position. The results indicate that theseforces may act as restoring forces to return theforearm to the horizontal position with a re-sponse time of order 10 ms. If this is the case,then the rapid response of the muscles in the armis perhaps faster than one might expect intuitive-ly. However, the tendons connecting muscle tobone act as passive springs, with a spring constantof order 105 N m±1 (Alexander 1992). Acting ona mass of order 1 kg, the half period of oscillationwould be of order 10 ms, as observed.

2 There is a signi®cantly longer propagation delayin the response of the forearm, compared withthe delays shown in Figs 1, 2 and 10. This can beattributed partly to the fact that the velocityincreases slowly from zero even if the appliedforce increases rapidly. For an impact on thestrings lasting 7 ms, the velocity of the forearmshould reach a maximum about 9 ms after theball ®rst contacts the strings, assuming that theimpulsive force on the forearm is delayed byabout 2 ms. An additional delay might be intro-duced by the response time of the hand to rotateabout an axis through the wrist. The long delayobserved for an impact at the tip of the racquetappears to be due to the fact that the rotational,translational and vibrational components of theforearm velocity sum to zero for the ®rst halfcycle of oscillation. The ®rst half cycle is positive

for an impact at the throat or the COP, so the®rst half cycle should be negative for an impact atthe tip.

3 Vibrations in the racquet frame result in asigni®cant vibration of the forearm. This doesnot alter signi®cantly the vibration frequency ofthe racquet, since the frequency is determinedprimarily by wave re¯ection at the end of thehandle rather than the end of the arm. Thissituation can be compared with the more obviousexample of a piano wire where the frequency isdetermined by the mass and length of the wire,not the whole piano. In the present case, thehand acts to shift the vibration node in the handlecloser to the end of the handle, thereby increas-ing the wavelength of the fundamental mode anddecreasing the vibration frequency slightly.However, the situation is probably complicatedby the fact that some wave re¯ection occurs atthe hand, as well as the end of the handle, andthis will act to decrease the effective length of theracquet.

Summary

A summary of the effects observed in this paper,during and immediately following an impact, ispresented in Fig. 13. These drawings are based onthe observations that (i) the axis of rotation of a

Figure 13 A summary of the observedrotation and translation of a hand-heldracquet. The drawing at the left showsthe positions of the racquet, hand andforearm prior to impact. F1 denotes theforce acting on the upper part of thehand. F2 denotes the force acting on thelower part of the hand. F3 denotes theforce acting on the forearm near thewrist and is approximately equal toF1 + F2. The black circle indicates theaxis of rotation of the racquet in thelaboratory frame. The dashed verticalline represents the position of the longaxis through the handle, prior to im-pact.

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hand-held racquet, in the laboratory frame, lieswithin the hand for an impact at the tip of theracquet; (ii) measurements of the forces on thehand show that the axis of rotation, in thelaboratory frame, lies within the hand or beyondthe end of the handle and (c) the forearm is notde¯ected for an impact at the COP. On a longertime scale, starting 10±20 ms after the impact,recoil of the racquet and internal forces in the armmodify the initial response of the forearm. Theseeffects are not included in Fig. 13.

The total force on the hand was not measured,but one would expect that it is at least qualitativelysimilar to the force on the forearm since the forceon the forearm is transmitted from the handle viathe hand and wrist. The relation between the forceson the hand and forearm clearly depend on thebiomechanical linkages, and could be determinedin principle by an independent experiment andmodelled by connecting springs. However, a rea-sonable interpretation of the above observations isthat the racquet exerts a torque on the hand, asrepresented by the forces F1 and F2 in Fig. 13. Theforce F1 is represented by waveform b in Fig. 10and the force F2 is approximately equal to andopposite waveform d, since if the racquet handlemoves away from the base of the little ®nger itmoves towards the tip of the little ®nger on theopposite side of the handle.

The net force on the hand, F1 + F2, is transmit-ted to the forearm as the force F3 shown in Fig. 13.This interpretation is qualitatively consistent withthe results in Fig. 10. For example, for an impact atthe COP, waveforms b and d are approximatelyequal and opposite, indicating that there is essen-tially no net force on the hand. In fact, if theforearm remains at rest and if the hand rotatesabout an axis through the wrist, then the centre ofmass of the hand will translate slightly as a result ofits rotation about the wrist. As a result, thecondition for the forearm to remain at rest is thatF1 must be slightly larger than F2. For an impact atthe tip or centre of the racquet, the DC componentof waveform d is signi®cantly larger in magnitudethan waveform b, indicating that there is a net forceon the hand acting in a direction opposite the

direction of the incident ball. For an impact at thethroat of the racquet, the DC component ofwaveform d is signi®cantly smaller than b, at leastfor the ®rst 10 ms following the initial impact,indicating that there is a net force on the handacting in the same direction as the incident ball.

Conclusions

The primary purpose of this work was to determineexperimentally whether the hand has a signi®cantor a negligible effect on the dynamics of thecollision between a tennis ball and racquet. Theresults show that the hand plays a more signi®cantrole than previously suspected since an impulse istransmitted from the strings to the hand well beforethe ball leaves the strings. The effect of the hand onthe outgoing ball speed was not investigated;however, it was found, by comparing hand-heldand freely suspended racquets, that (a) the vibrationnode in the handle is shifted to a point under thehand; (b) the axis of rotation of the racquet isshifted to a point under the hand or close to the endof the handle; (c) the vibrational forces on the handand forearm are zero for an impact at the vibrationnode in the centre of the strings and (d) the forceon the forearm is minimized for an impact at thecentre of percussion (the force is not zero since asmall vibrational component is present). Theseimpact points are the well-known sweet spots of atennis racquet, but their signi®cance in relation tothe forces acting on the hand and the forearm hasnot previously been studied in any detail.

For a freely suspended racquet, the location ofthe COP is not uniquely de®ned since there is nounique axis of rotation in the handle. The COP cantherefore be located anywhere on the strings,depending on which axis one chooses in the handle.In the case of a hand-held racquet, the location ofthe COP can be de®ned uniquely in terms of theimpulsive motion of the forearm, at least for a shortperiod during and immediately following the initialimpact. The relevant axis of rotation in the handlethen passes through the end of the handle, at leastwhen the racquet is held by one hand. This maynot be the case for a two-handed stroke. On a

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longer time scale, motion of the forearm isdetermined by restoring forces within the arm aswell as by the effects of the impact of the racquetwith the ball.

Acknowledgements

It is a pleasure to acknowledge the assistance givenby Prof. H. Brody.

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