# First, an ACES haiku:

What Purpose Does a Tennis String Serve?

When a tennis racket is impacted with a ball, the racket strings catch the ball and propel it back out of the racket.

Many sports involve striking a ball with various stiff props, such as bats, clubs, and paddles, but the strings of a racket are meant to deform under the ball's impact before sending the ball out again. Other sports, such as lacrosse and jai alai, involve catching a ball with a net or basket and using it to sling the ball, but the net of a tennis racket is not pliant enough to be used in the same way. Because the strings are kept under a tension load of about 50-65 lbs., the net of a tennis racket is stiff when compared to the net used in, for example, lacrosse. A tennis racket impacts a ball rather than slings it. When compared to a baseball bat, however, tennis strings are soft. They are meant to stop a ball, grip it, and propel it outward again.

The Theory Behind a Tennis Racket

Unless it impacts in a perfectly elastic collision, a moving tennis ball will lose energy when it collides with another object. This can be seen in Figure 1, where a tennis ball is bouncing on a surface. Each time the ball bounces, it deforms and loses energy to the collision, and is unable to regain its previous heights.

(www.rit.edu/~andpph/exhibit-8.html; taken by Andrew Davidhazy)

Figure 1. A bouncing tennis ball.

A tennis racket is designed to return as much energy to a tennis ball as it can when the two impact. The racket does this through the use of flexible strings. Ideally, the strings deform elastically under the force of the ball's impact, then fully recover their shape and tension before impact. That is, ball impact stretches the strings, which then exert a force on the ball as they restore themselves to their original position. By using strings that deform under impact, it is hoped that the tennis ball does not deform, and thus does not lose energy. Theoretically, the strings should be fully elastic, to return all the energy that goes into string deformation back to the ball, but in reality the strings are likely to be at least slightly viscoelastic, and will undergo plastic deformation not only after each impact but as soon as they are strung.

The effectiveness of a racket's rebound can be quantitatively measured as the coefficient of restitution, covered in "Power vs. Control" (below).

(www.rit.edu/~andpph/exhibit-8.html; taken by Andrew Davidhazy)

Figure 2. A ball rebounding off a racket. The strings are intended to deform so that the tennis ball does not.

Impact Dynamics

We are designing for a string that performs well even when it is impacted many times; therefore, we are interested in the dynamics between a tennis string and a tennis ball. How should we analyze the dynamics of a ball impact on a tennis racket? For simplicity we have decided to first study the dynamics between a ball and a single string in tension. Of course, a racket consists of many strings weaved together to form a net spanning the head, but examining a single string will give us a starting point for more complicated calculations.

Figure 3. A single string being carrying a load F at its center.

Figure 1 shows the effects of an impact on a string at the string's center. The string has initial length Lo and initial tension To, which is, ideally, the stringing tension (if no tension loss occurs). The center of the string has been deflected a distance y, and the string, which now feels a tension of T, has stretched to length L. Lo and To are related to L and T by

where k is the spring constant of the string. We are more interested, however, in the spring constant of the string perpendicular to its axis, which will tell us how much the string center deflects under an applied force. The applied force is described by

and the perpendicular spring constant by

which are taken straight from Figure 3. Since the deflection of the strings is typically going to be much less than the length of the strings (especially when weaved into the net), we can make approximations for T, L, and consequently the perpendicular spring constant, which is then given by

.

For this approximation, that is, for small deflection y, the perpendicular spring constant is dependent only on Lo and To, and is independent of k, the string's spring constant -- in two dimensions, its elastic modulus. The perpendicular stiffness is then the same for all strings of the same length and stringing tension. This relationship shows us the importance of stringing tension as well.

Power vs. Control

More compliant nets, that is, strings with lower elastic moduli and/or string tension, give a racket more "power," enabling a player to a return a ball at higher speeds, because the strings deform, rather than the ball. Strings will fully recover after impact if they are deformed in their elastic region, allowing the ball to lose less energy and rebound with greater velocity, whereas the ball is unable to regain energy lost from its own deformation.

Often, power is quantitatively defined as the coefficient of restitution, which gives a measure of the elasticity of the collision between a ball and racket by taking the ratio of the relative velocities between the two colliding objects. This value varies between zero (a completely inelastic collision) and one (a completely elastic collision). It is calculated with the equation

where c is the coefficient of restitution, v1 is the linear velocity of the racquet mass center before impact, s1 is the linear velocity of the ball before impact, v2 is the linear velocity of the racquet mass center after impact, and s2 is the linear velocity of the ball after impact.

Since a tennis racket is strung with a specific tension range, the compliance of a net can also be adjusted by using strings of higher or lower elastic moduli. The more flexible a string is, however, the less durable it tends to be. Flexible strings often break sooner and more easily than stiffer strings do. A string consisting solely of nylon, for example, is much more compliant than a string consisting solely of kevlar, and breaks after less playing time.

In tennis, "power" is also a trade-off with "control," the ability to direct a ball where the player intends the ball to go. The amount of control that a player has is determined by several factors determined by the player, such as consistency and timing, but it can be affected by the racket as well.

Figure 4. Timing error, which results in an error in ball direction.

If a ball strikes a string precisely at the string's center, it will rebound in the direction it came from, but if the ball strikes the string away from the string's center, it will rebound with an angular error given by

where L is the length of the string, T is the tension of the string, z is the distance of the impact point from the center of the string, delta v is the change in the ball's velocity, m is the mass of the ball, and C is a constant. [2] From this equation it can be seen that the amount of angular error with which a ball rebounds is inversely proportional to the string tension -- or, a racket with more control (less angular error) has less power (higher string tension).

What's most important when buying a racket?

 Power 23% Control 54% Balance 11% Head Size 6% Handle Length 6%
(taken from www.wilsonsports.com)

Because more players feel that control is more important, we will be focusing on a string that can be strung elastically at higher tension without undergoing significant tension loss, giving a player more control.

Other Considerations: String Vibration

String vibration is also of concern to players. The regions of the tennis racket head where little or no vibration is felt by someone holding the racket when they are impacted with a ball are called the racket's "sweet spots." Stiffer materials will vibrate for a longer time when plucked than softer materials do. Besides causing discomfort and perhaps injury to a player, vibrating strings may be dissipating energy that could otherwise be returned to the ball.

Other Considerations: Spin

A balls that is spinning in its direction of motion when it strikes the ground will lose less speed than a ball that is not; consequently, a ball traveling with topspin will seem to bounce up from the ground with greater speed than it should. This difference in rebound velocity is due to the amount of energy the ball loses when it strikes the ground. When the traveling ball strikes the ground, friction with the ground will slow down the ball and cause it to begin rotating in the direction of motion, but a ball striking the ground with initial topspin is already spinning in its direction of motion and thus loses less energy. If it is spinning quickly enough, friction with the ground may actually be in the direction of the ball's linear motion, causing it to bounce up at an even greater velocity. [3]

The amount of spin that a racket gives a ball is largely determined by the surface texture of the strings. Strings that have higher friction with tennis balls are better able to grip and impart spin to them.

String Materials: Natural vs. Synthetic

Today's tennis strings are made from either cow gut or from combinations of polymers, which are sometimes reinforced with titanium. Tennis strings made from natural gut remain the most favored and also most costly tennis strings on the market. Strings made from polymers are called "synthetic gut" and advertised for their "gut-like playability." Tennis aficionados often claim that synthetic gut still cannot match the performance of natural gut.

Natural gut strings are obtained from the serosa of cow intestine and consist mostly of collagen fibers twisted together. Gut is significantly more flexible than many of the polymers being used in synthetic strings, and as a consequence some players string their gut at tensions reaching up to 70 lbs. in order to maintain control. Although gut is less durable than many synthetic strings, gut proponents claim that it maintains its tension and "bounce" for a longer amount of time -- that is, it will likely break before a synthetic string will, but its performance life is longer.

Figure 5. (Cross et al.) Tension loss in natural gut and polyester strings.

As can be seen in Figure 5, tension loss begins immediately after stringing. The rate of tension loss seems to decrease over time (seen clearly in polyester), but stress relaxation may continue throughout the lifetime of the string. Natural gut retains its tension much better than polyester strings do, but even in gut the tension loss is significant: one day after stringing, the tension in the gut strings has decreased from 28 kg (12.7 lb) to 23 kg (10.5) lb.

Strings also lose tension after they are impacted with an object (such as a tennis ball or hammer). Immediately after impact, the tension peaks -- increasing a certain amount dependent on the transverse stiffness of the string, then dropping to an amount lower than the tension before impact. The first few impacts make the greatest difference in the tension of the string, with subsequent impacts causing a tension loss of successively smaller steps. [1]

Table 2. (Cross et al.) Tension gain during a hammer impact in strings of various materials commonly used in string manufacturing.

 String Material Tension Gain During a Hammer Impact Natural Gut 9 kg (4.1 lb) Nylon 18 kg (8.2 lb) Polyester 22 kg (10 lb) Kevlar 45 kg (20.5 lb)

String Designs: Monofilament vs. Multifilament

String manufacturers use a variety of string designs to imitate gut:

 - very good resilience - not as durable - expensive - sensitive to weather - good resilience - semi-durable - middle price range - okay resilience - durable - low price

These string designs emulate gut with varying degrees of success, and none play quite like gut does. Besides the differences in the properties of the materials used, from a design point of view, natural gut is ordered on several more length scales than synthetic strings are. Only with a great deal more processing (and cost) could the structure of synthetic polymers look like that of gut.

Designing a Unique Tennis String

When designing a tennis string, the various performance traits and the string properties that affect them should be kept in mind. Obviously, not every desirable trait can be maximized, and one often must be sacrificed for another. The goal of ACES is to produce a string with optimum playability and minimal sacrifice of any particular performance trait.

Which playability traits (control, power, feel, vibration, etc.) should be given more weight? Some players prefer a more powerful racket, and some might prefer one with good control. ACES will aim to design a string that matches natural gut in control, but does not sacrifice a great amount of power.

[1] Cross R et al., Laboratory testing of tennis strings, Sports Engineering 3: 219-230, 2000.

[2] Brody H, Physics of the tennis racket, Am J Phys 47(6), June 1979.

[3] Brody H, Tennis Science for Tennis Players, University of Pennsylvania Press, 1987.