Sports Eng (2010) 12:87–98 DOI 10.1007/s12283-009-0032-5
ORIGINAL ARTICLE
Comparison of a finite element model of a tennis racket to experimental data Tom Allen • Steve Haake • Simon Goodwill
Published online: 5 December 2009 Ó International Sports Engineering Association 2009
Abstract Modern tennis rackets are manufactured from composite materials with high stiffness-to-weight ratios. In this paper, a finite element (FE) model was constructed to simulate an impact of a tennis ball on a freely suspended racket. The FE model was in good agreement with experimental data collected in a laboratory. The model showed racket stiffness to have no influence on the rebound characteristics of the ball, when simulating oblique spinning impacts at the geometric stringbed centre. The rebound velocity and topspin of the ball increased with the resultant impact velocity. It is likely that the maximum speed at which a player can swing a racket will increase as the moment of inertia (swingweight) decreases. Therefore, a player has the capacity to hit the ball faster, and with more topspin, when using a racket with a low swingweight. Keywords Ball Finite element High-speed video Impact Racket Spin Tennis
1 Introduction Tennis racket materials have changed over the years, from wood to aluminium alloy to fibre composites [1–3] and these developments have changed the way in which the game is played. Advances in racket technology, especially developments in materials, have allowed players to hit
T. Allen (&) S. Haake S. Goodwill Faculty of Health and Wellbeing, Sports Engineering Research Group, Centre for Sport and Exercise Science, Sheffield Hallam University, Sheffield, UK e-mail:
[email protected]
shots faster and with greater accuracy [4], effectively increasing the speed of the game [5]. Manufacturers began experimenting with composite materials in 1970s [2, 3], mainly due to their high stiffness-to-weight ratios, in comparison to metals. Currently, the majority of rackets are manufactured from composite lay-ups as this allows materials to be precisely placed for desired stiffness and mass distributions. The reduction in the mass and the increase in the structural stiffness of tennis rackets, dating from 1870s to 2007, have allowed serve speeds to increase by approximately 17.5% and the reaction time available to the receiver to reduce by approximately 15% [6]. Finite element (FE) techniques have been used by previous authors to further the scientific understanding of tennis equipment [7–11]. An earlier FE model by Allen et al. [7] was successfully validated as a good approximation of a head-clamped tennis racket for oblique spinning impacts. The frame of a head-clamped racket can essentially be thought of as infinitely heavy as it cannot be displaced during an impact with a ball. Headclamped rackets are used for analysing the effect of stringbed properties, such as string type and tension [12]. However, rigidly clamping a racket by the head is clearly not a good representation of how it will be supported during play, especially as the frame of the racket cannot be displaced or deformed. Brody [13] demonstrated that the frequency response of a freely suspended tennis racket is similar to that of a handheld tennis racket. Therefore, freely suspending a racket is currently the best representation of how it will be supported during an actual tennis shot. Previous authors have found that, for impacts normal to the face on the long axis of a freely suspended racket, the rebound velocity of the ball is dependent on impact location and is lowest at the tip and highest in an area near the throat [4, 14, 15].
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Goodwill and Haake [15] produced spring damper models for normal impact on rigid and flexible rackets. The flexible racket model showed closer agreement with the experimental data than the rigid racket model. The rigid racket model overpredicted the rebound velocity of the ball, for impacts offset from the geometric stringbed centre (GSC) along the longitudinal axis. In addition, by using only impacts normal to the face on the long axis of the stringbed, the models by Goodwill and Haake [15] were less representative of a typical elite player’s shot [16]; thus they went onto investigate oblique spinning impacts. Goodwill and Haake [17] analysed the oblique impact of a tennis ball with no inbound spin on a freely suspended racket. The inbound angle was set at 36° to the stringbed normal and the inbound velocity was in the range from 15 to 40 m s-1. All of the impacts were reported to be at the GSC, as this was stated to be where players typically hit the ball during play. The rebound velocity and topspin of the ball both increased with the inbound velocity, whilst the rebound angle remained essentially constant. Analysis of tennis shots from elite players has highlighted that the ball can have spin rates of around 300–550 rad s-1 prior to impact with the racket [18, 19]. Therefore, the impacts would have been more representative of a typical tennis shot from an elite player if the balls had been projected onto the racket with initial spin. The aim of this paper is to produce and validate an FE model of a freely suspended tennis racket. The freely suspended racket model will be an extension of the headclamped racket model produced by Allen et al. [7] in ANSYS/LS-Dyna 10.0. ANSYS/LS-Dyna is an Explicit FE solver which can be applied to a variety of different impact scenarios, including sporting applications [11, 20–22]. The frame of the freely suspended racket will have the capacity to displace and deform during an impact with the ball. Initially, the freely suspended racket model will be validated for impacts normal to the face at a variety of locations on the stringbed. The aim of simulating impacts normal to the face at a variety of locations will be to provide a rigorous validation of the model. Following the comparison with experimental data for impacts normal to the face, the model will be validated for oblique spinning impacts close to the GSC. An oblique spinning impact close to the GSC is a good representation of a typical tennis shot from an elite player [16].
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2.1.1 Frame The racket modelled had an overall length of 0.68 m and a head size of 0.35 9 0.27 m (Fig. 1). These dimensions are representative of a modern tennis racket frame [6]. The freely suspended racket model was based on the headclamped racket model published by Allen et al. [7], with some major modifications to the frame of the racket, as detailed below. (1) No constraints were applied to the frame in the freely suspended racket model. Having no constraints allowed the frame of the racket to displace during an impact. (2) The rigid body material model (MAT_RIGID) was changed to a linear elastic material model (MAT_ELASTIC) to enable deformation of the racket frame to be simulated. A linear material model was considered to be adequate due to the relatively small deformations of a racket during an impact with a ball. (3) The frame geometry was also separated into three parts, i.e. the handle, throat and head (Fig. 1). With this model, the mass distribution of the racket can thus be adjusted by changing the shell thicknesses and densities of the handle, throat and head sections. International Tennis Federation (ITF) branded rackets (carbon-fibre construction) were used for the laboratory-based validation experiments. Therefore, the mass and mass distribution of the racket in the FE model was set to correspond to the ITF branded racket, as shown in Table 1. The mass, balance point and mass moment of inertia (swingweight) of the ITF racket were taken from Goodwill [23]. The polar moment of inertia (twistweight) of the ITF racket was obtained experimentally using bifilar suspension theory [24]. 2.1.2 Stringbed The most complex task of the FE model was simulating the interwoven stringbed. A load of 150 N was applied to a rigid cylinder attached to both ends of every string during the dynamic relaxation phase of the simulation. The purpose of the applied load was to produce an initial contact
2 FE model 2.1 Description of the model The model ultimately consisted of three parts: (1) the frame, (2) stringbed of the racket and (3) the ball.
Fig. 1 Finite element model racket geometry with three separate sections
Comparison of a finite element model of a tennis racket to experimental data Table 1 Racket mass distribution in the finite element model
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Part
Mass (kg)
Balance point from butt (m)
Mass moment of inertia about butt (kg m2)
Polar moment of inertia (kg m2)
Handle
0.098
N/A
0.00098
0.000021
Throat
0.090
N/A
0.00622
0.000125
Head
0.162
N/A
0.04331
0.001446
Complete racket FE model
0.348
0.324
0.05111
0.001592
ITF racket
0.348
0.325
0.05337
0.001550
force at the intercepts of the individual strings in the interwoven stringbed. The convergence tolerance for dynamic relaxation was 0.06. The ends of the strings were tied to the frame during the transient phase of the simulation to produce a strung racket. The tied contact between the strings and racket was set to initiate at a simulation time of 0.00135, 0.00015 s before the ball impacted the stringbed. Full details of the method used to define stringbed tension are given in Allen et al. [7]. A linear elastic material model (MAT_ELASTIC) was used for the strings, with a Young’s modulus of 7.2 GN m-2, a density of 1,100 kg m-3, and a Poisson’s ratio of 0.3 [9]. A coefficient of friction (COF) of 0.4 was defined between the ball and stringbed [25]. Previous work, using the FE model of a stringbed published by Allen et al. [9], indicated that increasing the COF between the ball and strings from 0.4 to 0.6 has little influence on the rebound characteristics of the ball [26]. Reducing the COF to 0.2 caused the model to overpredict the rebound topspin of the ball in comparison to experimental data.
the racket. Two versions were created from the base model to encompass the large range of values of racket stiffness typically found (Table 3). Previous authors have also used the natural frequency of a tennis racket to determine the required structural stiffness for the frame in an analytical model [15, 23]. The natural frequencies of tennis rackets dating from 1870s to 2007 are within the range of 70– 190 Hz [6]. The natural frequency of the ITF racket used in the laboratory experiment was 134 Hz [23]. Modal analysis was undertaken on the FE model of the racket frame, for different values of effective modulus. An effective modulus of 20 GN m-2 resulted in a natural frequency of 135 Hz, which is within 1% of the value of 134 Hz for the ITF racket. The reason for using two values of natural frequency was to determine the effect of racket stiffness on the rebound characteristics of the ball. Using two values of effective modulus to produce rackets with natural frequencies that bracket the ITF racket justifies the use of an isotropic material model for simulating an anisotropic composite lay-up.
2.1.3 Ball 3 Experimental methods The ball consisted of a pressurised rubber core and felt cover; full details of the ball model and its validation can be found in Allen et al. [10]. The initial velocity and spin of the ball were defined using INITIAL_ VELOCITY_GENERATION. 2.2 Details of the simulations Table 2 shows the details of the material models and elements which were used for the main parts of the model. The FE model acted as a base for the geometry and mass of
Tennis balls were projected from a modified pitching machine (BOLA) onto the freely suspended ITF racket, using the impact rig detailed in Choppin [27] (Fig. 2a). The bespoke impact rig was specifically made for analysing the impact of a tennis ball on a racket. The balls were projected onto an initially stationary racket, as the impacts were all conducted in the laboratory frame of reference [4]. The racket was supported at the tip, from a small horizontal pin, with its longitudinal axis vertical. The pin was located underneath the tip of the frame between the two central
Table 2 Material model, type of elements and number of elements for the main parts of the finite element model Part
Material model
Type of elements
Number of elements
Ball (felt cover)
Foam (MAT_LOW-DENSITY_FOAM)
Reduced integration 8-node brick (SOLID164)
21,600
Ball (rubber core)
Hyperelastic (MAT_OGDEN_RUBBER)
Reduced integration 8-node brick (SOLID164)
21,600
Strings
Linear elastic (MAT_ELASTIC)
Reduced integration 8-node brick (SOLID164)
29,891
Racket frame
Linear elastic (MAT_ELASTIC)
Belytschko-Tsay formulation shell (SHELL163)
27,410
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Table 3 Natural frequencies of the two racket frame models with different values of effective modulus Racket
Effective Poisson’s Natural frequency modulus ratio (Hz) (GN m-2)
FE model: low structural stiffness 10
0.3
96
FE model: high structural stiffness 70
0.3
253
ITF carbon-fibre racket
N/A
134
N/A
The natural frequency of the ITF Carbon-fibre racket was taken from Goodwill [23]
Fig. 3 Impact positions on the stringbed for the validation of the freely suspended racket model for impacts normal to the face
Table 4 Impact locations for the impacts normal to the face on a freely suspended racket (mean ± SD)
Fig. 2 a Impact rig used for simulating impacts on a freely suspended tennis racket. b Relative camera positions for measuring the trajectory of a tennis ball in 3D (Modified from Choppin [27])
main strings. Using a pin to support a racket is a technique which had been used by previous authors [14, 15, 23]. Three identical rackets were used for the experimental testing, all strung at 289 N (65 lbs). Non-spinning impacts normal to the face were simulated at four different impact positions on the stringbed, labelled: centre, off-centre, tip and throat (Fig. 3; Table 4). The inbound velocity of the balls in the impacts normal to the face was in the range from 10 to 40 m s-1. Oblique spinning impacts were simulated at nominal inbound speeds of 20 and 30 m s-1 and a nominal angle of 25° to the z axis, on a plane parallel to the x and z axes (refer to axes on Fig. 2a). Changing the frame of reference from the court to the laboratory means that the ball should have backspin prior to impact to represent a topspin shot [12] (Fig. 4). The inbound spin of the oblique impacts was in the range from 100 rad s-1 of topspin to 500 rad s-1 of backspin. The nominal impact location of the oblique impacts was the centre of the stringbed, although the impacts were slightly offset from the long axis of the racket to compensate for the horizontal displacement of the ball whilst it remains in contact with the strings [17]. The impacts were captured using two synchronised Phantom V4.3 high-speed video cameras, operating at 1,900 fps and an exposure time of 0.2 ms. The two cameras were positioned on separate sides of the impact rig to provide three dimensional (3D)
Impact location
Horizontal distance from the stringbed centre (mm)
Vertical distance from the stringbed centre (mm) (? = towards tip)
Centre
13 ± 7
Off-centre
31 ± 10
Throat
18 ± 8
-55 ± 16
Tip
13 ± 11
49 ± 7
8±7 4±7
coordinates of the ball and racket, as detailed by Choppin [27] (Fig. 2b). The impacts were recorded as bitmap images and analysed using Richimas v3 image analysis software. A detailed description of Richimas v3 is given in Goodwill and Haake [12]. The 2D positions of the ball were obtained manually from each camera using Richimas. The pairs of 2D coordinates obtained using Richimas were converted into global 3D coordinates (camera frame of reference) using a freely available MATLAB Toolbox which was developed by Bouguet [28]. The 3D calibration was undertaken using a checkerboard, as developed by Zhang [29] and applied to tennis impact testing by Choppin [27]. To measure the impact position on the stringbed, a transformation matrix was used to convert the 3D coordinates of the ball into the racket frame of reference, with the origin at the GSC (Fig. 5). The origin was located at the GSC by obtaining the global 3D coordinates of three white markers at known locations on the frame of the racket (Fig. 5). Impact was assumed to initiate at the first instance when the ball’s perpendicular distance (Z) from the stringbed (XY plane at Z = 0) was less than its radius (33 mm). The horizontal (x) and vertical (y) impact positions (relative to the GSC) were obtained from the position of the ball at the first point of contact with the stringbed. For full details of the method used to reconstruct tennis ball to racket impacts in 3D using two high-speed video cameras, refer to Choppin [27]. The spin of the balls from the oblique impacts was assumed to be about the y axis, which is top/back spin relative to the racket. Ball spin was calculated in 2D using
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Fig. 4 Diagram to show that the ball should impact the racket with backspin in the laboratory frame of reference to represent a topspin shot
Fig. 6 Diagram to show the method used for calculating the top/back spin of a tennis ball, by calculating the change in h over time (T). B is an intercept of the markings on the ball and A is the GSC. h was calculated using trigonometry from the radius of the ball (R) and the horizontal distance between A and B(X)
Fig. 5 Racket position showing throat and side markers and axis coordinate system. B is an intercept of the markings on the ball
markings, which were drawn on the felt (Fig. 5). The process involved using Richimas v3 to obtain the coordinates of the geometric ball centre (GBC) see point A on Fig. 6 and a marker, which is the intercept of lines on the ball, see point B on Fig. 6. The radius of the ball and the distance (X) were then used to obtain the angle h. The process was repeated to obtain four angles, before and after impact. The spin of the ball was calculated from the gradient of the angle time data. To ensure the highest possible accuracy with this method, the spin was calculated independently from both cameras, and the mean value was used to compare with the model. The root mean squared error (RMSE) between the between the spin measured from the two cameras was 21 rad s-1 before impact and 11 rad s-1
after impact. A difference of 21 rad s-1 equates to 21% at 100 rad s-1 and 4% at 500 rad s-1. Table 4 shows the calculated impact locations for the impacts normal to the face. The RMSE between the resultant and z velocities (normal to stringbed) for all the impacts normal to the face in the laboratory-based experiment was 0.008 m s-1 for inbound and 0.04 m s-1 for rebound. As the RMSE is very low the impacts were considered to be normal to the face of the racket and the resultant velocities were analysed against the FE models. Table 5 shows the calculated velocities and angles before impact and the impact locations on the stringbed for the oblique impacts. A repeatability study was undertaken to assess the level of human error in the manual tracking method. An impact with low, medium and high inbound spin was selected and analysed ten times (Table 6). The impacts had a nominal inbound velocity of 20 m s-1. The uncertainties in the measured values are similar to those reported by Goodwill and Haake [17]. FE simulations with the initial conditions shown in Table 7 were undertaken to correspond to the laboratorybased experimental data. The impact positions on the stringbed were identical to the mean values in Table 4 for the impacts normal to the face and Table 5 for the oblique impacts. The initial conditions of each impact and the
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Table 5 Inbound velocities, angles and impact locations for the oblique impacts on a freely suspended racket (mean ± SD) Nominal inbound velocity (m s-1)
20
30
Nominal inbound angle (°)
25
25
Calculated inbound velocity (m s-1)
18.0 ± 0.5 28.0 ± 0.4
Calculated inbound angle (°)
23.7 ± 1.3 22.9 ± 0.9
Horizontal distance from the stringbed centre (mm)
9 ± 16
Vertical distance from the stringbed centre 9 ± 12 (mm) (? = towards tip)
15 ± 10 8 ± 11
material properties of the racket were set using the tennis design tool (TDT). The TDT is a parametric modelling programme which was produced in Visual Basic 2005 [26].
4 Results The aim of this investigation was to compare an FE model against experimental data with the intention of determining the effect of tennis racket structural stiffness. Two racket models were produced to bracket the ITF racket in terms of structural stiffness. The stiffness of the racket was measured from the fundamental frequency. Results from the simulations showed the following key findings. Firstly, for impacts normal to the face, the structural stiffness of the racket frame only affected the rebound velocity of the ball for impacts in the throat region. Secondly, for oblique spinning impacts at the GSC, the structural stiffness of the racket frame did not affect the rebound velocity, angle nor spin of the ball. In addition, the rebound velocity, angle and spin of the ball all decreased as the inbound backspin increased; whilst, the rebound velocity, angle and spin of the ball all increased with the inbound velocity of the ball. These results will be explained in detail below. Figure 7 shows a comparison of the FE model with the experimental data for the impacts normal to the face. The results are expressed as the resultant velocity of the ball after an impact with the racket. There are four nominal
impact positions on the stringbed. There are two sets of data from the FE model corresponding to different racket frame stiffnesses. The rebound velocity of the ball was slightly lower for the off-centre impacts in comparison to those at the centre. The rebound velocity was lowest for the tip impacts and highest for those at the throat, in agreement with Goodwill and Haake [14, 15] and Kanda et al. [30]. Figure 7c shows that four of the tip impacts, which had an inbound velocity below 20 m s-1, had a larger rebound velocity than expected from the trend of the rest of the data (see highlighted data points). Three of these four impacts were closer to the GSC in comparison to the mean impact location, in both the vertical and horizontal directions. The remaining impact had an offset distance from the long axis of the stringbed which was less than the mean value. Raising the effective modulus of the racket frame in the FE model increased the rebound velocity of the ball for the throat impacts whilst having a negligible effect on those at the other locations, in agreement with Goodwill and Haake [15]. The discrepancy between the two models increased with inbound velocity, which can be accounted for by energy losses due to racket frame vibrations [15]. The model with the lower structural stiffness will deform more, particularly at high impact speeds, resulting in a decrease in the rebound velocity of the ball. The FE model of the racket with the effective modulus of 10 GPa, was in relatively good agreement with the experimental data for all four of the impact locations on the stringbed. The model with the higher effective modulus of 70 GPa slightly overpredicted the rebound velocity of the ball for the throat and off-centre impacts when the inbound velocity was greater than 20 and 25 ms-1, respectively. The overprediction of rebound velocity for the off-centre impacts is likely to be due to the model underpredicting the deformation of the racket in torsion. Figure 8 shows a comparison of the FE model with the experimental data, for the oblique impacts at the two inbound velocities. As with the impacts normal to the face, there are two sets of data for the FE model corresponding
Table 6 Results of a repeatability test for impacts with low medium and high inbound spin Low spin (-5 rad s-1)
Medium spin (252 rad s-1)
High spin (530 rad s-1)
0.1 (0.4%)
0.1 (0.6%)
0.1 (0.5%)
Resultant rebound velocity (m s )
0.1 (1.0%)
0.1 (0.8%)
0.1 (1.5%)
Inbound angle (°) Rebound angle (°)
0.3 (1.4%) 0.3 (0.9%)
0.5 (2.0%) 0.5 (3.1%)
0.4 (1.4%) 0.9 (18.1%)
Resultant inbound velocity (m s-1) -1
Inbound spin (rad s-1)
9 (176%)
8 (3.2%)
Rebound spin (rad s-1)
9 (9%)
4 (38.5%)
8 (18.9%)
Impact distance from long axis (mm)
1 (56%)
2 (17.4%)
2 (13.0%)
Impact distance from short axis (mm)
1 (9%)
1 (19.8%)
1 (2.7%)
(value) = SD as a percentage of the mean
21 (3.9%)
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Table 7 Initial conditions used in the FE model to simulate an impact between a tennis ball and freely suspended racket Inbound velocity (m s-1)
Inbound angle (°)
Inbound backspin (rad s-1)
Number of impact positions
Impacts normal to the face
10, 20, 30 and 40
0
0
4
Low velocity oblique impacts
18
23.7
0, 200 and 400
1
High velocity oblique impacts
28
22.9
0, 200 and 400
1
Fig. 7 Ball rebound velocity for impacts normal to the face on a freely suspended racket. a Centre, b off-centre, c tip and d throat
to different racket stiffnesses. All of the impacts were close to the GSC. The rebound speed of the ball decreased with increasing inbound backspin and was lower for the impacts at 18 m s-1 than those at 28 m s-1. Three of the impacts in Fig. 8b had a rebound speed which was lower than expected from the trend of the rest of the data (see highlighted data points). All three of these impacts had: (1) an inbound speed which was lower than the mean speed and (2) an impact location which was further from the GSC in comparison to the mean impact location, in both the horizontal (X) and vertical (Y) directions. Goodwill and Haake [17] state that the rebound characteristics of the ball are highly dependent on the impact position, as a result of the non-uniformity of the stringbed. The resultant rebound
speeds obtained from the two FE models were in good agreement with the experimental data for both inbound speeds. There was only a very small difference in the rebound speeds obtained from the two FE models of different racket stiffnesses. The negligible effect of racket frame stiffness on the rebound speed of the ball was in agreement with the results obtained for the impacts normal to the face at the centre of the stringbed. Figure 9 shows that the rebound angle of the balls (relative to the racket normal) decreased with increasing inbound backspin. The rebound angles were virtually identical for both inbound velocities when the balls had a negligible amount of inbound spin, in agreement with Goodwill and Haake [17]. However, the rebound angle
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Fig. 8 Ball rebound velocity for oblique impacts on a freely suspended racket. a 18 m s-1 and 24°, b 28 m s-1 and 23°
Fig. 9 Ball rebound angle for oblique impacts on a freely suspended racket. a 18 m s-1 and 24°, b 28 m s-1 and 23°
decreased more with increasing inbound backspin when the inbound velocity of the balls was 18 m s-1, in comparison to 28 m s-1. As with rebound speed, there was very little difference in the results obtained from the two FE models. The FE models were both in relatively good agreement with the experimental data, although the models slightly underpredicted the rebound angle of the ball by a few degrees, for both inbound velocities. Goodwill and Haake [17] found rebound angle to increase with string tension. Therefore, it is likely that the FE model underpredicted rebound angle because the structural stiffness of the stringbed was too low. Figure 10 shows that the rebound spin of the balls decreased with increasing inbound backspin. The rebound spin was lower for the inbound velocity of 18 m s-1, and it decreased more with inbound backspin. As with rebound velocity and angle, there was very little difference in the results obtained from the two FE models. The FE models were in good agreement with the experimental data for inbound backspins which were lower than approximately
200 rad s-1. At higher inbound backspins, the models slightly underpredicted the rebound spin of the balls.
5 In-depth analysis of an oblique spinning impact An investigation was then undertaken to ascertain how and why the rebound properties of the ball changed with inbound spin. An impact at 28 m s-1 and 23°, with 200 rad s-1 of backspin was selected for analysis. These values of inbound velocity and backspin were considered to correspond well with those employed in play [16]; these impacts were also in good agreement with the trend of the experimental data. Figure 11 shows how the horizontal and vertical forces acting on the ball and its horizontal velocity and spin change throughout the impact. The horizontal and vertical planes are defined as being parallel and normal to the face of the stringbed, respectively. The vertical force shows a nonlinear rising portion because (1) the internal pressure of a ball increases with deformation [11] and (2) the tangential stiffness of the stringbed increases with
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Fig. 10 Ball rebound spin for oblique impacts on a freely suspended racket. a 18 m s-1 and 24°, b 28 m s-1 and 23°
Fig. 11 a Force, b horizontal velocity and c spin obtained from the FE model, throughout an impact at the centre of a freely suspended racket with an inbound velocity of 28 m s-1, angle of 23° and with 200 rad s-1 of backspin (70 GPa/253 Hz)
contact area and displacement [14]. The initial horizontal force was negative which means that the force was acting in the opposite direction to the horizontal motion of the ball. The negative horizontal force caused a decrease in the horizontal velocity of the ball and an increase in its topspin. At approximately 2.25 ms the ball reached its minimum horizontal velocity and maximum topspin. The contacting region of the ball then ‘gripped’ the stringbed and the ball deformed forwards storing energy [31]. Approximately 0.25 ms later the ball lost its ‘grip’ with the stringbed and the horizontal force reversed sign. The reverse in the sign of the horizontal force caused an increase in the horizontal velocity of the ball and a decrease in its topspin. This reversal of the horizontal force occurs when the spin rate of the ball exceeds that associated with rolling; this is commonly referred to as ‘over-spinning’. The horizontal force acting on the ball then converged towards zero. Once the
horizontal force equalled zero, there was no further change in the horizontal velocity or spin of the ball, which implied that the ball was rolling off the stringbed. There was no horizontal force at the end of the impact despite a non-zero vertical force because the ball was rolling. A further investigation was undertaken with the FE model to substantiate the interesting findings in Fig. 11. This can be referred to in three parts: (1) How does changing the COF between the ball and stringbed influence the results? (Fig. 12) (2) Does the structural stiffness of the stringbed (Fig. 13) or (3) the ball, alter the findings further (Fig. 14)? The impact conditions were identical to those detailed above. The horizontal force reversed sign and the ball was rolling off the stringbed for all the impacts, in agreement with the results shown in Fig. 11. Figure 12 shows the effect of altering the COF between the ball and stringbed. Values of 0.2 and 0.6
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Fig. 12 a Force, b horizontal velocity and c spin obtained from two FE models with different values of ball-to-string COF
Fig. 13 a Force, b horizontal velocity and c spin obtained from two FE models with different stringbed stiffness. The low and high stiffness strings had a Young’s modulus of 5.4 and 9.0 GPa, respectively
Fig. 14 a Force, b horizontal velocity and c spin obtained from two FE models with different stiffness balls. The stiffness of the low stiffness ball was 20% lower than the original model and the stiffness of the high stiffness ball was 20% higher than the original model
were used for the COF in the model. The COF had no noticeable effect on the vertical force. The initial horizontal force was larger when the COF was 0.6, causing the ball to ‘overspin’ earlier in the impact. The horizontal force was also larger when it reversed sign, resulting in the ball leaving the stringbed at a slightly higher horizontal velocity and with very slightly less topspin. Figure 13 illustrates the effect of changing the structural stiffness of the stringbed. The Young’s modulus of the
strings was decreased and increased by 25% to produced values of 5.4 and 9.0 GPa, respectively. The vertical force was larger on the rising portion for the high stiffness stringbed. The initial horizontal force was also very slightly larger for the stiffer stringbed and it reversed sign earlier in the impact. The horizontal force was very similar in magnitude when it reversed sign for both stringbeds. Therefore, the ball rebound from both stringbeds with a very similar horizontal velocity and topspin, in agreement with Goodwill et al. [12].
Comparison of a finite element model of a tennis racket to experimental data
Figure 14 shows the effect of changing the structural stiffness of the ball. The modulus of the rubber core was increased by 20% to produce a ball with high structural stiffness and decreased by 20% to produce a ball with low structural stiffness. The maximum vertical force was very slightly higher for the stiffer ball. The horizontal force was also very slightly larger for the stiffer ball when it reversed sign. Overall, the different between the horizontal and vertical forces acting on the two balls was marginal; therefore both balls rebounded from the stringbed with the same horizontal velocity and topspin.
6 Discussion When simulating impacts normal to the face, the FE model was in relatively good agreement with the experimental data for all four of the impact locations used in this investigation. The FE model was in better agreement with the experimental data when the effective modulus of the racket was 10 GPa as opposed to 70 GPa, due to the natural frequency of the 96 Hz racket being closer to that of the ITF racket. Increasing the structural stiffness of the racket resulted in an increase in the rebound velocity of the ball for impacts at the throat, in agreement with Goodwill and Haake [15] and Kanda et al. [30]. The structural stiffness of the racket had only a very marginal effect on the rebound velocity of the ball at the other impact locations. The small effect of two very different values of effective modulus provides justification for the use of an isotropic material model to simulate a composite lay-up. When simulating oblique spinning impacts, the two racket models of different stiffness were both in good agreement with the experimental data, in terms of the rebound velocity of the ball. The two models were also in relatively good agreement with the experimental data for rebound angle and spin; although, they did slightly underpredict the rebound angle of the ball for the entire range of inbound backspins. The models also underpredicted the rebound spin of the ball for inbound backspins greater than approximately 200 rad s-1. However, it was difficult to precisely determine the accuracy of the FE model due to the uncertainty in experimentally measuring both inbound and rebound spin (*20 rad s-1). The stiffness of the racket frame had very little influence on the rebound characteristics of the ball. The difference between the two models was much lower than the scatter in the experimental data. This agreed with the results obtained for impacts normal to the face close to the GSC. The GSC corresponds to a node point of the racket and hence exhibits very low vibration of the fundamental mode. It is likely that racket stiffness will have a greater influence on the rebound characteristics of the ball for impacts away
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from the GSC, particularly in the throat region, as found with the impacts normal to the face. The results indicate that the ball was ‘over-spinning’ at around the mid-point of the impact. Goodwill and Haake [12] also found the ball to be ‘over-spinning’ when performing experimental impacts on a head-clamped tennis racket. The results also indicate that the ball was rolling off the stringbed at the end of the impact. Therefore, adjusting the COF between the ball and stringbed did not have a large effect on the rebound characteristics of the ball. This investigation has indicated that the structural stiffness of a tennis racket does not have an influence on the rebound characteristics of the ball when simulating a typical groundstroke from an elite player, i.e. an oblique spinning impact at the GSC. However, the rebound velocity and topspin of the ball both increased with the inbound velocity of the ball. Therefore, for a set inbound ball velocity, a player can increase the resultant impact velocity of a shot by simply swinging their racket faster. Mitchell et al. [32] have shown that during a serve, swing speed increases as racket swingweight decreases; therefore, the effect of the mass of the racket on the impact should also be considered. For a constant swing speed, less momentum will be transferred to the ball if a player uses a lighter racket [6, 33]. Haake et al. [6] analysed the effect of racket mass in the range from 0.29 to 0.36 kg on serve speeds. They concluded that the ball is launched faster as the mass of the racket decreases. Miller [33] states that decreasing racket mass below 0.25 kg would be counterproductive, as the transfer of momentum to the ball would be less effective. If swing speed also increases with decreasing swingweight for groundstokes, as would be expected, then it is likely that a player will be able to hit the ball faster and with more topspin when using a lighter racket. This indicates that the use of composite materials in tennis rackets has indeed increased the speed of the game. Further research into the effect of racket mass on oblique spinning impacts and the relationship between swing speed and swingweight should be undertaken to strengthen this hypothesis.
7 Conclusion An FE model of a freely suspended tennis racket was compared against experimental data for both normal to the face and oblique impacts. When simulating impacts normal to the face, the FE model of the racket with the natural frequency of 96 Hz had the best agreement with the experimental data. The results from the FE model showed that the stiffness of the racket had no notable effect on the rebound characteristics of the ball for oblique impacts at the GSC. The structural stiffness of the racket had no effect
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on the rebound characteristics of the ball because the GSC corresponds to a node point of the racket. The results from the FE simulations indicated that the ball was ‘over-spinning’ during the oblique impacts. It would have been very difficult to measure ‘over-spinning’ using a conventional laboratory-based experiment alone. This is the first FE model with the capability to accurately simulate oblique spinning impacts, at different locations on the stringbed of a freely suspended racket. The model can now be used to determine the influence of different racket parameters, such as mass and swingweight, on the game of tennis. Acknowledgments The authors would like to thank Prince for sponsoring the project. They would also like to thank Terry Senior, Simon Choppin and John Kelley.
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