Sport Science

 Tennis racquets

      It is desirable for a chosen tennis racquet to help provide three important quantities: power, control, and consistency.  Each of these is highly correlated with the tension in the racquet’s strings. 

• (1) For a given swing speed and racquet weight, the lower the string tensions, the greater is the power (hit ball speed) provided by the racquet.  That is because when a tennis ball impacts a string bed, a bed with lower string tensions will indent more during the impact, causing the ball to compress less and therefore dissipate less energy.  (Energy dissipation from indented strings is much less that from compressed balls.  A typical impact is shown in Fig. 3.4.) 
• (2) The increased indention of strings with lower tension decreases shot control because, for all impacts except those at the exact center of the racket face, it creates greater rebound deviations.  The created disruptive tangential force differentials increase when the indentation increases.  See Fig. 3.5.  (3) A player’s consistency increases when the sweet spot of his racquet’s face increases in size, and the sweet spot size increases when the string tensions decrease.  The sweet spot for a typical forehand stroke is illustrated in the computer-generated power contour plot given in Fig. 3.6.  Lighter areas correspond to larger rebound speeds for a given inbound ball speed and racquet speed.  The size of these lightest areas increase when the tensions decrease.      

Given the above-described importance of string tensions in tennis performance, it is important for a player to choose the tension values that are appropriate for his game.  This is difficult because, within minutes after a racket is strung to a requested tension T, the actual string tension within the string bed is almost always less than T, and subsequently the tension continues to decrease as a consequence of usage, climate changes, and from the intrinsic weakening that occurs as the internal chemical bonds within the strings break.  The effect can be substantial – for many string brands in common use, a 10% reduction in string tension within a week is not uncommon.

When a racquet is strung, a stringing machine is used to pull each string to the desired tension, and it is assumed that, because of slippage through the grommets and around the outer frame sides, all of the strings end up with the same tension. Among the many problems that arise from this stringing protocol and its variations are the following. 

• (1) Each applied tension changes the shape of the frame (more or less, depending on the racket and on the clamping devices on the stringing machine), and therefore changes the tensions on each of the previously tensioned main and cross strings. 
• (2) The slippage mechanism is not perfect because of the tension forces themselves and the consequent friction forces. 
• (3) As the cross strings are stretched to their desired tension value, they encounter substantial friction and elastic resistance from the main strings that they pass over and under, and this causes the final achieved cross and main string tensions to differ from their intended values.  (As the cross strings are tensioned, the main strings are stretched and their tensions are therefore increased, and, as the main strings relax, the cross strings are shortened and their tensions are therefore reduced.) 
• (4) When the racket clamps are released after the racket is strung, the frame shape is further changed and a large release shock is encountered, causing still further tension changes.  Because of these effects, the actual tensions in a racket given to a player by a stringer are almost always very different from the requested tensions, and very few players are even aware of this.

After a racket is strung, the string tensions continue to decrease over time even if the racket is not used to strike a ball.  When a string is stretched to a given tension and held in place, the bonds within the string material continue to break as long as the string is held under tension.  This is illustrated in Figure 3.7 for a 16-gauge string.  The data, taken over 3 days (4320 min), are obtained by stretching a string attached to a strain gauge to a fixed length of 12”.  The initial tension of 60 lbs decreases to 53 lbs after 10 min, to 51 lbs after 60 min, and to 48 lbs after 1440 min (1 day).  The string tensions in a strung racket will not decrease this fast because of the frame flexibility, but this rapid tension decrease illustrates how rapidly fixed strings degrade, and this monotonic decrease appears to persist forever. Data we have taken on a variety of strings over periods as long as three months show a continual monotonic decrease with no positive horizontal asymptote. 

Because of this tension degradation, it is important for a player to be able to determine the actual tensions in his racquet.  To address this, a variety of tension-measuring devices are available, but none of these devices are practical because they are either too destructive, too complicated to use, or too inaccurate.  We have therefore invented a tension-measuring device that does not have these problems.  A photograph of this device is shown in Fig. 3.8. (This product was developed in collaboration with our partner Division, LLC.)

In addition to the individual string tensions in a racquet, another important property of a racquet face is the elasticity of the string bed. We have measured the force F required to compress the string beds of various tennis rackets, using a 2.5” diameter solid ball, to distances x between 0.05” and 0.50”.  The data confirm that, to an excellent approximation, the forces increase linearly with the compression distances (F = -Kx), but the elastic constants K, as expected, depend on the racket and string, and, because of the tension degradations described above, they also depend on the time between stringing and measurement.  Typical data on a racket newly strung to 60 lb with 16 gauge strings are given in Figure 3.9.  The vertical axis is the applied force in pounds, and the horizontal axis is the measured displacement in inches.  The elastic constant is the slope K  = 134.6 lb/in, and the RSQ correlation coefficient is 0.998, which confirms that the linear fit is a good one. After two days, K decreased to 126.8 lb/in, and after two months, K decreased further to 118.6 lb/in.

The above string tension and string bed elasticity studies involved static measurements.  We also perform dynamic measurements by impacting racquets with spherical metal elements and with tennis balls.  A metal impacting device is shown in Fig. 3.10.  A racquet in a holder set up for ball impacts from a cannon is shown in Fig. 3.11.  These devices are used to measure and compare the performance and sweet spot sizes of racquets.

In addition to the laboratory measurements described above, we have performed extensive (finite element) modeling of the impacts between a ball and a racquet.  A predicted flattening sequence of a tennis ball striking a solid surface is shown in Fig. 3.12, and a photo of such an impact is shown in Fig. 3.13.  A predicted indentation of a string bed during an impacted is shown in Fig. 3.14.