This article was originally written by Wilmot H. McCutchen.
Text was taken from Archive.org.
It’s included purely for reference purposes, as many requests come in for this URL each month. There is no commercial intent.
Objective Evaluation Concepts
and Answers to Frequently Asked Questions
Here’s an interesting question: why don’t the touring pros use the same racquets that their sponsors sell to the public? The pros heavily customize to add mass and swingweight, even though the paint job would lead you to believe that they are playing with the same racquet you can buy. Why is such a fix necessary?
It has been estimated that half of all players over 30 suffer from tennis elbow. So it’s important to have good information because the consequences of a bad racquet choice are worse than the waste of $300. But how can a player make an informed choice? Racquet ads are not very informative, and often deliberately misleading. You can’t rely on what the pros pretend to use. Playtests depend on the personal opinion of the testers, and like all subjective tests are suspect for obvious reasons.
What’s needed is a set objective performance criteria, some quantifiable and meaningful terms instead of ad hype and subjective playtests. Here are some scientific concepts and real performance criteria for tennis racquets:
Sweet Spot | Moment | Torque | Torsion | Polar Moment | Impulse Reaction | Shock | Work | Shoulder Pull | Shoulder Crunch | Elbow Crunch | Wrist Crunch | Impact Force
Other commonly used criteria:
Power | Maneuverability | Control
Evaluation Criteria for Racquet Performance
1. The Sweet Spot is known to cognoscenti as the “center of percussion.” It is dependent on the location of the axis of rotation at the hand in the stroke (see formula for sweet spot location). A high sweet spot, i.e. a center of percussion close to the tip of the racquet is good because it means low Impact Force. See Sweet Spot Rankings for the 2002 survey of 167 racquets.
The sweet spot is a point, not an area, although some refer to the “sweet spot” on the racquet being “large.” Another alias for the center of percussion is “center of oscillation.” The “sweet” area on the string bed is where the racquet’s bounce is maximized, and this has nothing to do with the center of percussion.
The sweet spot is determinative of the force from impact: the higher the sweet spot, the lower the Impact Force acting at the racquet’s mass center, and the more positive the Impulse Reaction at the hand. If the balance is close to the hand, this will also mean low Torque and therefore less stress on the arm.
Generally speaking, the higher the sweet spot on the racquet face (i.e. the longer the distance (q) from the hand to the sweet spot) the better. However, because racquets are of different lengths, it is misleading to compare q values directly, because q is the distance of the center of percussion from the axis of rotation, and for a long racquet a high q may still mean a low sweet spot. So what counts is the distance of the sweet spot from the tip. The formula for finding the center of percussion on any racquet is
q = I / Mr
[q is the distance in centimeters from the axis of rotation to the center of percussion; M is racquet mass in kilograms; r is the distance in centimeters from the axis of rotation to the mass center, or balance point, of the racquet; and I is racquet swingweight about the axis of rotation (also known as moment of inertia or rotational inertia or Second Moment) in the units measured on the Babolat Racquet Diagnostic Center (kg.cm2). The axis of rotation is 7 cm from the handle end on the forehand, 5 cm on the serve, and the Babolat RDC measures swingweight about an axis 10 cm from the handle end, so the RDC swingweight must be converted using the Parallel Axis Theorem].
2. Moment is the turning force pivoting the racquet head down when you hold the racquet parallel to the ground. Moment, in Newton.meters, is a measure of how heavy the racquet feels to hold up parallel to the ground (not merely the weight of the racquet, but this weight multiplied by its lever arm).
Moment should be especially important for juniors and ladies. A light racquet having a balance point far from the hand may have a larger Moment than a heavy racquet with a head-light balance, so merely knowing the weight of a racquet is not enough, and may be misleading.
It is deplorable that ignorant consumers are being enticed to buy dangerous racquets by misleading sales ploys like inviting them to pick up the racquet by the wrong end to see how light it is. This “pick up appeal” pitch is causing tennis to lose players and popularity at an alarming rate. What counts is Moment, not weight. Moment is the racquet’s weight times its lever arm, which is the distance to the balance point from the axis of rotation (this axis is at the middle of the hand, 7 cm from the handle end). Weight in the metric system is the mass of the racquet, in kilograms, times the acceleration due to gravity (9.81 meters/s2), and the distance to the balance point is in meters. The unit of measure of torque is the Newton.meter (1 Newton.meter = 0.7375 foot pounds of torque = the same force you would feel holding a 3.6 ounce weight at the end of a meter stick). The lower the Moment, the better.
Moment is key for two reasons: (1) a racquet with a high Moment is bad because it is hard to hold up and to position for volleys and returns, especially for juniors and ladies; and (2) Moment multiplied by Torque gives the Torsion, which is the screwdriver twist about the racquet’s handle centerline resulting from impacts, even impacts on the centerline. High Torsion is bad for tennis elbow. See the discussion below on tennis elbow. Moment Rankings for the 2002 survey of 167 racquets.
Torque and Impulse Reaction are the two “resultant forces” from an “eccentric impact,” such as the impact of a racquet with a ball. This is called an “eccentric impact” because the two mass centers (ball center and racquet balance point) do not move along the same line to a point of collision. When you hit the Sweet Spot, Impulse Reaction is zero, but there is still some Torque.
3. Torque is a bending force resulting from impact, that causes the hand to bend back and then catapult forward. This torque is measured about the axis of rotation of the racquet in the stroke, which is 7 cm from the handle end on the forehand (First, Third, and Fourth Benchmark Conditions), and 5 cm on the serve (Second and Fifth Benchmark Conditions) — approximately the middle finger location. Note that Torque is not the screwdriver twist of the handle (which will be called Torsion, or Longitudinal Torque), but the bending back of the racquet. (See formula for Torque, see derivation of formula for Torque). Torque winds up a catapult in the wrist, which flings the racquet forward after the ball has gone. The stronger this catapulting force, the worse the whipsawing stress cycle resulting from the stroke, and thus the worse for tennis elbow — so high Torque is bad.
Some loss of energy could be expected from the conversion of Torque into the subsequent forward catapulting force, due to absorption of Torque in the bending of the racquet frame and stretching of the muscles, so it will be difficult to quantify the catapult effect. Note, however, that a stiff racquet (high Flex number) will not absorb as much of this bending force, and therefore a stiff racquet is a risk factor for tennis elbow.
Expert players tend to prefer the slim, more flexible racquets, which absorb Torque in frame bending and thereby reduce the catapulting force that flings the racquet forward after impact. The “widebody revolution” (of the late 80’s) never caught on among the pros, for good reason. Stiff racquets may be good for power, but they are bad for tennis elbow. Note that Torque depends on dwell time, and the shorter the dwell time the worse the Torque. Flexible racquets may have the advantage of increasing dwell time, although no proof of this is presently available.
Note that Torque is a different twisting force from Moment, although both are measured in the same metric torque unit (Newton.meters). The difference is in the axis of the twist. Moment is about an axis of rotation parallel to the ground, while Torque is about the axis determined by the path of the ball and the attitude of the racquet face (for ground strokes, an axis perpendicular to the ground). For all racquets, in all benchmark conditions, we assume an impact on the centerline with the racquet face perpendicular to the ground. Torque Rankings for the 2002 survey of 167 racquets.
4. Torsion or Longitudinal Torque is the screwdriver twisting force around an axis running up the handle. Such a force arises even from impacts on the centerline. This criterion is the cross product of Moment and Torque. As we assume a centerline hit in our calculations under both benchmark conditions, the additional screwdriver twist from an off-centerline hit is not evaluated. See the discussion below on whether weighting at 9 and 3 o’clock on the head is good. Torsion from a centerline hit is simply the cross product (vector product) of Moment and Torque. That means you multiply Torque by Moment and then multiply by the sine of the angle between these two vectors. For groundstrokes (assuming the axis of rotation is perpendicular to the ground and the racquet is parallel to the ground), the angle is 90 degrees so the sine is 1. For the serve, there is no Torsion because there is no Moment when the racquet is pointing straight up.
Both high Moment and high Torque contribute to high Torsion. For a right-handed forehand, Torsion would be a twist in the clockwise direction. This twist winds up the racquet to release in a sudden handle twist in the opposite direction (counterclockwise) once the ball leaves. The magnitude of this second twist depends on racquet stiffness (stiff is bad).
Torsion also results from impacts off the centerline, the amount of torsion being the Impact Force (in Newtons) times the distance, in meters, from the centerline to the point of impact. For off-center impacts, a low Impact Force (i.e. a high Sweet Spot) is best.
5. Impulse Reaction is a push (positive Impulse Reaction) or pull (negative) on the axis of rotation (the hand) resulting from impact (see formula for Impulse Reaction, also see derivation of formula for Impulse Reaction). Impacts above the center of percussion (Sweet Spot) result in a pull on the hand; below is a push. A positive Impulse Reaction is better because it means less Impact Force (the rankings of Impulse Reaction [positive good] and Impact Force are exactly the same).
Impulse Reaction is measured in units of force, because it is a translational force (straight ahead push or pull) on the axis of rotation (at the hand). The unit of measurement of force in the metric system is the Newton (1 Newton = 0.2248 pounds, or 3.6 ounces, of force). For impacts above the Sweet Spot (center of percussion), Impulse Reaction is a pull from the player to counter the yank on the hand (negative value). For impacts below the center of percussion, Impulse Reaction is a push against the hand (positive value). Right on the center of percussion (sweet spot), there is no Impulse Reaction at all (zero value).
Our directional convention is that positive is toward the net, so a pull is a negative Impulse Reaction. Positive is better than negative because positive Impulse Reaction adds more speed to the racquet during the impact, while negative Impulse Reaction tends to bring the head under the ball as it yanks the arm forward.
The Impulse Reaction rankings are the same as the rankings for Impact Force. The higher the sweet spot, the more positive (less negative) the Impulse Reaction, and the lower the Impact Force.
6. Shock loading of the racquet results from a sudden change in the racquet’s kinetic energy on impact, which produces an internal energy load on the racquet, which is expressed as frame vibration. Shock also determines Shoulder Crunch , Elbow Crunch, and Wrist Crunch (see formula for Shock, see derivation of formula for Shock). This change in kinetic energy is how much the racquet slows down when it slams into the ball, while kinetic energy is converted into internal or potential energy. Shock is measured in joules (the same metric unit as work, heat and energy). Although the term “shock” has no generally accepted definition in engineering, for our purposes we will call Shock the difference between the initial and final kinetic energy of the racquet. See note.
Before impact, you put energy into the racquet to get it up to speed for the collision, and during the impact you put in a little more energy to aim the shot. After the ball leaves, the racquet mass center (balance point) moves at a slower speed, and this means a loss of its kinetic energy (kinetic energy = 1/2 mass times velocity squared). The ball gets some of this lost energy (the same for all racquets under all benchmark conditions), and the rest becomes internal energy, wasted in bending the frame.
If the frame is stiff and light, the frame bending energy will not be absorbed by the material of the frame but will but will have to be dumped into the arm holding on to the racquet. Don’t place any reliance on string buttons to save your arm. Damping gadgets on the strings are too small in mass to do much besides reduce residual string vibration, which is a minor annoyance, and damping gadgets in the frame must be expected to handle an energy load of the magnitude determined by the design of the racquet.
The most effective vibration damper is a large particulate handle end weight (e.g. a bag of shot or sand), which serves to divert and dissipate the energy. Also effective at vibration damping is the Pro Kennex Kinetic system, which has particulate weights in the racquet head, and the Wilson Triad system, which absorbs the energy in special frame joints.
Better than damping is prevention of Shock by proper weight distribution in the racquet (head-light and heavy overall).
High Shock is bad also because it means high Wrist Crunch, Elbow Crunch, and Shoulder Crunch. That is so because the formulas for these criteria derive from Shock. See the Shock Rankings for the 2002 survey of 167 racquets.
7. Work is the energy required to produce a certain ball speed with the racquet (see formula for Work, see derivation of formula for Work). Work measures the energy efficiency of the racquet, so low Work is good. High Work is bad because the player has to swing harder to get the same result. Work quantifies a racquet’s power: the less work the player has to put in to get the required ball speed in the allotted time for the stroke, the more powerful the racquet. Of course, a player may put in lots of effort and get lots of ball speed, especially with high swingweight racquets, but the power comes from the player, not the racquet.
In the evaluations, head size, flex, string tension, and ball bounce are comprised in a standard bounce or elasticity (coefficient of restitution) of the racquet/ball system (0.85), so Flex and Head Size — which are said to affect bounce, and therefore “power” according to the popular understanding — are not used independently for Work or “power” evaluations. We assume that all racquets are strung such that they have the same bounce.
Work, like Shock, is measured in terms of joules of energy, and the formula for Work (and kinetic energy) is 1/2 Mv2 (M is racquet mass in kilograms and v is the linear velocity of the racquet mass center (balance point) just before impact, in meters per second). The Work done during the impact, i.e. the player’s grip, is not counted, because studies show that it does not add materially to the speed of the ball.
It turns out that head-heavy racquets require a lot more Work to hit the ball fast, which is bad. They are also hard on the wrist, elbow, and shoulder, which is worse. Head-light and heavy racquets with substantial swingweight (like the Prince Original Graphite OS) are the most efficient and therefore most powerful (best payoff for the effort, i.e. lowest Work). See the Work Rankings for the 2002 survey of 167 racquets.
8. Shoulder Pull is the force (in the metric unit of Newtons, a Newton being about a quarter of a pound) exerted by the shoulder muscles in opposing the centrifugal force acting on the racquet as it moves around the shoulder in the swing resulting from the player’s Work (see formula for Shoulder Pull, see derivation of formula for Shoulder Pull). This opposing force is called a “centripetal” force because it acts toward the axis of rotation (here the shoulder socket); Shoulder Pull is equal and opposite to the centrifugal force while the racquet is getting up to speed for the impact, and reaches its maximum the instant before impact, which is where we measure it. After impact, this centripetal force continues, but the offsetting centrifugal force is reduced because the racquet has slowed down. The excess centripetal force becomes a radial compressive force known as Shoulder Crunch.
The formula for centripetal force is Mv2/R (where M is racquet mass in kilograms, v is the linear velocity of the mass center, in meters/second, and R is the distance, in meters, from the racquet mass center to the axis of rotation, here the shoulder). Note that, in rotation, the mass center linear velocity (v) decreases as the balance gets more head-light, so head-light balance can mean low Shoulder Pull, even if the racquet is heavy. The variable v is squared in the formula for centripetal force, so a light racquet having a head-heavy balance may still have a large Shoulder Pull, despite its light weight, due to its distant mass center and consequent high mass center velocity in rotation. That’s bad. See the Shoulder Pull Rankings for the 2002 survey of 167 racquets.
9. Shoulder Crunch is the change in the centrifugal force acting on the racquet, a change that occurs due to the impact slowing the racquet down, thus creating a sudden excess in centripetal force at the shoulder. Before, the centripetal force and centrifugal force were in equilibrium, but suddenly there is an excess centripetal force. This is effectively a muscle spasm in the shoulder muscles. The formula for finding Shoulder Crunch derives from Shock (see derivation of formula for Shoulder Crunch).
This excess centripetal force is a radial compressive force known as Shoulder Crunch, and is measured in units of force (Newtons in the metric system, pounds in the English system, 1 Newton = 0.225 lb.). Once the Shock is known, Shoulder Crunch follows by a simple calculation: Shoulder Crunch = (2/R)(Shock) (where R = distance of the racquet’s mass center from the axis of reference, here the shoulder). See the derivation. R will be equal to the distance from the player’s hand to his shoulder (0.61 m) plus the distance from the hand to the balance point of the racquet (0.01*r). Please note that it is only the Shoulder Crunch due to the racquet that is calculated here — there is also mass in the arm swinging the racquet, and it slows down too, but for purposes of racquet comparisons we assume that it is the same arm for every racquet, as if one player were trying out all of them. Shoulder Crunch Rankings for the 2002 survey of 167 racquets.
10. Elbow Crunch is the excess centripetal force acting at the elbow, an excess that occurs because on impact the racquet slows down, so its centrifugal force drops. The centripetal force of the muscles attaching to the elbow and the centrifugal force of the racquet in its swing had been balanced before the impact, but the sudden slowdown creates what is effectively a muscle spasm. The muscle continues to contract as if it still had a full load, so it suddenly shortens and yanks on the tendons that attach it to the elbow. This yank (Elbow Crunch) is a cyclic stress which, repeated over time, may be a contributing cause to tissue failure. Like Shoulder Crunch, it is calculated by multiplying Shock by 2/R’ (where R’ is the distance of the racquet’s mass center from the elbow, and is equal to the distance from the hand to the elbow (0.36 m) plus the distance from the hand to the balance point (0.01*r)). See the derivation of Elbow Crunch. Elbow Crunch is larger than Shoulder Crunch because the elbow is closer than the shoulder to the mass center of the racquet, so R’ is smaller than R. Elbow Crunch is measured in units of force (Newtons, 1 Newton = 0.225 lb. or 3.6 oz.). Elbow Crunch Rankings for the 2002 survey of 167 racquets.
11. Wrist Crunch is derived the same as Elbow Crunch, only the new distance R” is measured from the mass center to the wrist, not the elbow. R” is equal to the distance from the wrist to the racquet axis of rotation (0.08 m) plus the distance from the axis of rotation to the balance point (0.01*r). Wrist Crunch Rankings for the 2002 survey of 167 racquets.
12. Impact Force is the change in the racquet’s momentum on impact, divided by the time it occurs (0.010 seconds, the dwell time). It is the force (measured in Newtons) appearing at the mass center (balance point) upon impact with the ball, which for comparison purposes we assume to be 16 cm from the tip, approximately at the center of the strings, for all racquets. For a smooth followthrough, and for low resultant stresses on the arm, the Impact Force should be low. See the derivation of a formula for Impact Force. Impact Force correlates exactly with Impulse Reaction in the rankings, so the Impulse Reaction rankings are Impact Force rankings as well. The higher the sweet spot, the lower the Impact Force. Impact Force Rankings for the 2002 survey of 167 racquets.
If we multiply the Impact Force by the lever arm on which it operates (i.e. the distance from the hand to the balance point), we get exactly the same Torque as calculated by the formula for Torque. So the benefit of a high sweet spot will be offset by a head-heavy balance, because a head-heavy balance means a long lever arm for the Impact Force to operate on, and thus a severe bending of the hand on impact, with concomitant damage to the elbow from the whiplash mechanism.
13. Tip Speed is the velocity of the racquet tip just before impact. A low tip speed means that the swing need not be as violent to achieve the same ball speed, and therefore easier to control and more accurate. Derivation of Tip Speed formula.
14. Polar Moment is the racquet’s rotational inertia about its longitudinal axis: its resistance to a screwdriver twist. This should be high. Shoulder weighting, such as by Wilson’s Perimeter Weighting System, or by lead tape at 9 and 3 on the racquet head, increases Polar Moment. So does larger racquet head size. Measurement of Polar Moment so far is not available, so it is not used in the evaluations done on this site. Torsional stability would be increased by high Polar Moment.
Non-Scientific Evaluation Criteria
There is another vocabulary that one frequently encounters in discussions of tennis racquets:
Maneuverability is vague jumble of Moment and swingweight, with a meaning varying from player to player. Some understand maneuverability to be another name for swingweight, so for them a maneuverable racquet is easy to slap at tough gets. Others understand it to be Moment, and a maneuverable racquet is one that is easy to get in position for quick reaction strokes like volleys and returns. There is a difference between Moment and swingweight, despite the common misunderstanding that high swingweight necessarily implies a head-heavy balance and therefore high Moment. It is possible to have a racquet that has a low Moment and a high swingweight (e.g. the tailweighted Hammer). The confusion with regard to the term “maneuverability” has resulted in the unjust charge that high swingweight is bad, when the problem is actually high Moment.
Power. The weaker the player, the stronger his lust for a racquet, at any price, that promises to improve his “power.” No term features so prominently in racquet ads, yet has so little clarity of meaning. It could mean:
(1) racquet bounce (i.e. high coefficient of restitution);
(2) high swingweight (a racquet which allows you to load up a lot of angular momentum so that it will not bounce off the ball); or
(3) low Work (an efficient racquet, which requires the least player effort to achieve a given ball speed, or, which produces the greatest ball speed with a given player effort).
It appears that the common understanding of “power” is (1), high coefficient of restitution.
Proponents of stiff materials make the claim that their racquets are “powerful” because the stiff frame recovers in time to catapult the ball forward. Where is the experimental confirmation of this claim? I have seen none, and my invitation has been extended for over four years. Professor Howard Brody, however, says this:
“When a racket flexes, most of the energy that goes into racket frame deformation is not returned to the ball. … In the literature of tennis, there is evidence that there is little difference in racket response between a free racquet and a racket with its handle firmly clamped, for ball impacts along the long axis of the racket. When a ball hits a racket, it produces a transverse wave that travels along the racket, and then is reflected both from the tip and from the butt end. If the wave that is reflected from the butt end arrives at the impact point after the ball has departed, the ball will have no knowledge of how the handle is secured. The propagation time of such a wave can be estimated by measuring the frequency of the lowest mode of free oscillation of the racquet (about 150 Hz) and separation between the nodes (0.4 m). This gives a velocity of about 120 m/s, and a round trip propagation time to the butt and back of 8 ms, which is considerably longer than the dwell time on the strings. This might explain why the free racket and the clamped handle data taken in the lab show little difference.” H. Brody, “The Physics of Tennis III. The Ball-Racket Interaction,” 65 Am. J. Phys. 981, 982 (Oct. 1997).
And, in any case, the strings are the major component in racquet bounce. Maybe the advantage of the stiff frame is that it does not flex as much initially, thus requiring the strings to stretch more on impact. Anecdotally, stiff frames with large heads are known to be bouncy, with a pronounced trampoline effect. Control, however, suffers as bounce increases, particularly with large heads. Expert players tend to prefer low “power” racquets because they don’t need any help putting pace on the ball, and they have learned the value of accurate placement.
In science, power is measured in watts. A watt is one joule of energy/work/heat/effort per second, and one horsepower is 746 watts. Power is the rate of doing work. Of course, a racquet cannot be given a horsepower rating. The player/racquet system has power, with the player providing the effort and the racquet providing the interface with the ball to deliver that player effort. So if, consistent with this scientific meaning, we consider a powerful racquet to be one can achieve a certain ball speed with the least player effort per unit time, and we limit the time of the stroke, what power then becomes is the inverse of Work: low Work means high power. In the June 1999 Racquet Evaluations, Power was thus defined, but the reaction has been unfavorable because this is not the popular understanding of the term “power.” Instead, the new term will be “Efficiency.” Those who understand will know how to use the information provided by the Efficiency rankings.
Control — everybody wants it, but nobody knows how to measure it. Just what is “control,” exactly? According to the USRSA, power and control are two ends of a continuum, so high power is low control, and vice versa. This makes sense, provided that you assume that the composite of head size/flex that determines racquet bounce is synonymous with “power.”
But if power and control are mutually exclusive, it requires some ingenuity to devise an ad campaign emphasizing control that will lure the crypto-macho consumer. My personal favorite is Head’s “Control Your Rage” ad for its “titanium” racquets, the one that has a settings on the cross bar for “Destroy – Annihilate – Humiliate” (the opponent, presumably). See, you’ve pandered to the lust for power, and you’ve introduced some quaint notion of the need for moderation and sportsmanship in its use. The control setting, however, gets no wimpier than “Destroy.”
Another meaning of “control” might be how easy the racquet is to wield, but now we have some confusion with “maneuverability” and all of its uncertainties. “Stability” is another vague term often heard, which seems to connote “control,” but according to the USRSA, stability is just high swingweight, i.e. the opposite of “maneuverability” — and there we are again back trying to understand just what is meant by “maneuverability.” The idea of controlling the shot by your effort during the impact is refuted by the above quote from Professor Brody.
Macro Criteria for Meaningful Racquet Rankings
Five macro criteria comprise the results under the relevant performance criteria. These macro criteria are Efficiency, Elbow Safety, Shoulder Safety, Wrist Safety, and Dexterity.
The rankings under the performance criteria (Torque, Shock, etc.) for each of the five benchmark conditions were compiled and then weighted according to the comparative magnitude of the forces. See the Weighting Factors.
Efficiency
Which racquets produce the required ball speed with the least effort. The higher ranking (low numbers are better, just like player rankings) racquets under the macro criterion of Efficiency require the least effort. Work ranks under the five benchmark conditions were weighted and summed to get a composite score, which was then sorted for Efficiency rank.Elbow Safety
See what causes tennis elbow. The macro criterion of Elbow Safety comprises the evaluation criteria of 1.5*Elbow Crunch, Torque, Shock, 0.5*Moment, and 0.5*Flex. Moment and Flex were weighted half as much as Torque and Shock, which are more important. Flex serves to ameliorate Torque by absorbing the bending energy and (possibly) increasing the dwell time, and the great weight and preponderance of anecdotal evidence now supports the conclusion that stiff racquets are a risk factor for tennis elbow. Moment is not as severe a force, but it is present for longer during play and stresses the elbow as the arm holds up the racquet. Moment also factors into Torsion. Elbow Crunch was weighted 1.5 times Torque and Shock, because it is even more important. Each of these evaluation criteria (Torque, etc.) produced composite rankings when the results under the five benchmark conditions were compiled, weighted, summed, and sorted. These composite rankings then were compiled, weighted, summed, and sorted to produce rankings under the macro criterion of Elbow Safety.Shoulder Safety
The composite rankings for Shoulder Safety used an equally weighted mix of: Shoulder Pull, Shoulder Crunch, Impact Force, and Torque.Wrist Safety
The macro criterion of Wrist Safety comprises a weighted mix of: 1.5*Wrist Crunch, Moment, Torque, and 0.5*Flex. Moment stresses the wrist nearly constantly, and Wrist Crunch and Torque produce bigger but more infrequent stresses on impact. Flex is ameliorative of Torque, with the more flexible racquets (low flex number) absorbing more of the bending force produced by impact.Dexterity
The macro criterion of Dexterity comprises the evaluation criteria of Moment, Swingweight, and 0.5*Weight. Dexterity rankings are intended to reflect how easy the racquet is to wield before impact. Moment and Swingweight are weighted equally, and Weight is half as important because it is already comprised in Moment. This macro criterion should be of interest principally to weaker players. Expert players should disregard Dexterity because the good racquets tend to score low on it.Overall Rankings
A weighted mix of: Efficiency, 1.5*Elbow Safety, Shoulder Safety, Wrist Safety, and 0.5*Dexterity were summed to produce overall rankings for racquets to be used by beginners and weak players. For strong players, the Dexterity macro criterion was disregarded.
Overall Racquet Rankings for Weak Players 2002
Overall Racquet Rankings for Experts 2002
Rankings under the Macro Criteria
Efficiency | Elbow Safety | Shoulder Safety | Wrist Safety | DexterityManufacturer Report Card
(how each racquet ranks under all of the evaluation criteria)
Babolat | Blackburne | Cayman | Dunlop | Fischer | Gosen | Head | Prince | Pro Kennex | Slazenger | Topspin | Volkl | Weed | Wilson | YonexRankings Under the Evaluation Criteria:
These are the most useful rankings because they come directly from formula calculations.
Moment | Torque | Shock | Work | Shoulder Pull | Shoulder Crunch | Elbow Crunch | Wrist Crunch | Impact Force | Sweet Spot | Flex | Swingweight | WeightRacquet Specs according to U.S. Racquet Stringers Association of 167 racquets currently available, including price, head size, string pattern, and flex.
Weighting Factors used in the composite rankings
Frequently Asked Questions
What Causes Tennis Elbow?Many professional researchers are still looking for an answer. Damage to the tendon attaching the extensor carpi radialis brevis (ECRB) muscle to the elbow is the cause of the pain, but the cause of this cause is a mystery. However, it is fairly certain that this type of damage is the result of repetitive stresses, such as hitting a tennis ball.
Producing causes of tennis elbow may include the following mechanisms, which are offered here for comment and further investigation:
(1) Elbow Crunch is a sudden shortening of the ECRB due to impact (explained at greater length above under Elbow Crunch). This effectively is a muscle spasm that stresses the tendons.
(2) On impact, the resultant Torque twists the racquet head back, while Moment is dragging the head down, and the hand is holding the racquet steady. The resultant twist of the handle (Torsion, or Longitudinal Torque) is clockwise for a right-handed forehand. This twist winds up a catapult. When the ball leaves the racquet, the catapulting force is counterclockwise for the right-handed forehand. The two opposite screwdriver twists in a short time give a severe stress cycle to the extensor carpi radialis brevis muscle that attaches the middle of the hand to the elbow, even for a dead-center hit.
(3) The back-and-forth catapulting stress cycle of Torque from impact twisting the racquet back, followed by catapulting the racquet forward when the ball leaves, aggravates the handle twist cycle mechanism discussed above under (2). The extensor carpi radialis brevis muscle is anchored at the elbow and at the metacarpal (hand) bone of the middle finger, on the index finger side. The resultant Torque from impact is a twist backward that tends to yank this muscle as the middle finder is extended. On impact, this muscle is either straining (on the backhand) or slack (on the forehand). On the backhand, the first twist yanks this straining muscle, further stressing the tissues attaching it to the elbow. Then the muscle suddenly loses resistance but continues to work against the combined stress, so it suddenly shortens after impact, giving an even more severe yank to the elbow (cf. the discussion below on Elbow Crunch). For the forehand, the muscle is slack on impact, so the catapulting stress cycle cracks the muscle like a whip, stressing the points of attachment at the wrist and elbow. Elbow straps help because they damp the whip effect.
(4) Shock becomes internal energy, which expresses itself as frame vibration, and this vibration is transmitted to the arm holding on to the racquet unless it is damped somehow. (The correct term is damped, not “dampened.”) In the old wood racquets, vibration disappeared quickly because it was damped by the flex of the wood, but the new stiffer and lighter frames do a poor job of damping, so they efficiently transfer the subtle shaking to the arm. Undamped high frequency frame vibration can stealthily sabotage the elbow, so the price of power may be pain. Vibration of the frame shakes the extensor carpi radialis brevis muscle that attaches the middle of the hand to the elbow. This causes cyclic stressing of the tendons at the lateral epicondyle, where the fat half of this long teardrop-shaped muscle attaches. Cyclic stressing is how you break a coathanger by bending it back and forth. Eventually, with enough stress cycles, fatigue can cause tissues to snap, even without any tremendous force.
What you don’t want if you are concerned about the risk of tennis elbow is a stiff, high-Torque, high-Moment, high-Shock racquet. That means a light, head-heavy racquet.
Poor stroking technique is frequently accused, conveniently diverting scrutiny from racquet design, but, as the calculations on this site prove, risk factors for tennis elbow include: (1) light racquet weight and (2) head-heavy balance. Stiff frames are also bad. What is good for minimizing elbow damage is low Shock, low Elbow Crunch, low Torque, and low Moment.
See the foregoing discussion of tennis elbow. Wrist Crunch (the muscle spasm at the wrist resulting from impact) is even larger than Elbow Crunch, so it would be relatively more important than other risk factors. Again, light and head-heavy racquets should be avoided.
No, a lightweight racquet is a dumb idea, as pro customizers attest. Weight is not bad. You need weight to return a “heavy” ball (lots of pace and spin). Wimpy racquets can’t put much pace on the ball if you don’t have time to develop a long stroke, such as when you are stretched wide. Pete Sampras uses a racquet that is 14 oz. and evenly balanced, and when he is going for a putaway, he chokes down so the swingweight is even higher. Andre Agassi uses one that is 13.2 ounces and 5/8 inch (5 points) head-light. Mark Philippoussis uses one that is 13.5 ounces and is 3/4 inch head light. Lest you think that these heroic sticks are as unwieldy as the sword of Goliath, remember that the lightest wood racquet was 13 ounces. Ladies and children used them.
Maybe, in the short space that you have to execute your stroke, you might swing the wimpy racquet a little faster — but swing speed is not the key.
Momentum, not energy, and not force, is what counts in a collision (Conservation of Momentum is the principle), and in computing momentum the racquet’s mass is just as important as its velocity (momentum = mass times velocity). Readers with baseball experience know what happens when you try to hit a hardball home run with a softball (i.e. lightweight) bat. A softball bat cannot hit a hardball very far because it doesn’t bring enough mass to the collision, and therefore its momentum on impact is low.High Tip Speed is bad for accuracy because it is harder to time a violent swing precisely. Even if you succeed in increasing the Tip Speed enough to offset the racquet’s lack of mass, the shot will be hard to place.
Aside from the foregoing performance considerations, there is the even more important question of safety. Light racquets are bad for tennis elbow.
Most racquet customers and their stringers know little, and care less, about the difference between weight, Moment, and swingweight. “Pick up appeal” (how light the frame is when you pick it up in the pro shop) is the predominant criterion (after cosmetics) for the ignorant. An epidemic of elbow and other arm injuries has been the result. Tennis is losing players at an alarming rate, and slowly declining in popularity. It’s all because of the fundamental mistake of amateurs regarding racquet weight, a mistake that some racquet salesmen apparently have chosen to exploit for their short-term profit.
The touring pros know better. They add weight when they customize their racquets. A more massive (heavier) racquet will crush majestically through the ball instead of bouncing off, which makes it more comfortable on impact and more accurate. See the Official Rules of the ATP Tour regarding racquets. This little secret vexes the sponsors that pay them lots of money to pretend to play with granny sticks, so you won’t hear much about it. See page 8 of the June 1996 issue of Stringer’s Assistant (published by the US Racquet Stringer’s Association) for some data on pro customized racquets.
Do the Rankings Change if the Game is Slower?
No. As the pace increases, the differences between racquets expand, but the rankings hold. For the Second and Fifth Benchmark Conditions (110 mph and 70 mph serve, respectively) the rankings are exactly the same, and likewise the rankings for the Third and Fourth Benchmark Conditions. The racquet that is better at Centre Court is also better at the country club.
One often hears the specious riposte: “Pro racquets are like race cars.” Etc. The point being that the pro racquet must be therefore inappropriate for recreational play. A Ferrari is better than a Yugo, even in the slow lane, but you really notice the difference in the fast lane. And every player sometimes hits with pace, especially on serves and returns, which is when it’s important to be holding the right racquet.
An interesting fact is that the higher the center of percussion, the lower the force acting at the racquet’s mass center upon impact (Impact Force, see derivation). The “center of percussion” is the real “sweet spot.” Proper weight distribution can raise the center of percussion significantly, and the higher the Sweet Spot, the better.
But a racquet with a relatively high center of percussion (such as the appropriately named Hammer) is not necessarily good. Even though the Impact Force will be less due to the high sweet spot, the resulting Torque will be higher because the mass center where this force acts is far from the hand, giving the Impact Force a longer lever arm. Comparative calculations for the light and head-heavy racquets prove that their Torque and Shock will be high, even with a high sweet spot, and so will the Work, Shoulder Pull, Wrist Crunch, Shoulder Crunch, and Elbow Crunch. What you want is high sweet spot together with head-light balance and adequate mass. Such a combination can be achieved by means of a large tailweight.
The center of percussion is a point along the racquet’s length; it is not a “spot” having an area. Do not be misled by deceptive advertising suggesting that some manufacturer has succeeded in expanding the sweet spot from a point to an area, so that you can get sweet spot performance nearly everywhere on the racquet face. Another misconception is that there is no shock if the impact is at the center of percussion. One resultant force from impact (Impulse Reaction) is reduced to zero, but the other (Torque) still exists, and there is still some Shock, Work, Shoulder Pull, Shoulder Crunch, Wrist Crunch, and Elbow Crunch. Even when you hit the sweet spot dead on, you feel something.
The area concept of the sweet spot concerns mapping where the coefficient of restitution (a measure of the bounce, or elasticity, of the racquet) exceeds a certain arbitrary value. It is a plot of elasticity, with the more elastic region being inside the sweet area. Manufacturers who claim a large sweet area (they call it a sweet spot) are only claiming to have succeeded in making their racquets bouncier, or more elastic. A good string job (lower tensions, thinner gauge, springy string) can also increase bounce. In the benchmark conditions used for evaluation on this site, the elasticity is assumed to be the same (0.85) for all racquets at the point of impact, which is assumed to be at 16 centimeters from the tip for all racquets because this is the center of the head of a standard-length (27″) racquet, which most of us were trained on.
The upside of more elasticity (bounce) is less Shock, Work, Shoulder Crunch, Wrist Crunch, and Elbow Crunch. The downside — especially with a large head racquet — is that your shots are less accurate. Those who can’t aim their shots anyway won’t know the difference in accuracy, but experts prefer low power (less bouncy) racquets for cleaning the lines.
The variables affected in the formulas by string tension are dwell time (t) and coefficient of restitution (c). Dwell time (t) is the length of time the ball stays on the strings. Coefficient of restitution (c) is the measure of the elasticity of the collision between the ball and the racquet (high c means more elastic, livelier bounce).
Longer dwell time (high t) means lower Torque and Impulse Reaction on impact, which means better accuracy. Dwell time decreases with increasing string tension, which is bad. And, after a point, as string tension increases, coefficient of restitution goes down. Lower coefficient of restitution (low c) means higher Shock, Work, Shoulder Pull, Elbow Crunch, and Shoulder Crunch.
The conventional thinking is that loose strings give more “power” (presumably, this means higher c) and tight strings give better “control” (presumably, higher t and therefore less resultant forces from impact). Dr. Jack Groppel, in his interesting book, High Tech Tennis (available from the USPTA bookstore), has shown that these variables are not affected as presumed. See pp. 25-27. Although one might think that the ball would stay on the strings longer when the tension is higher (because it is flattened more), dwell time decreases with increasing string tension, which is not good for accuracy. And the relationship between string tension and coefficient of restitution (c, or racquet bounce) is not linear, especially with midsized racquets, where there is a pronounced peak of bounciness at 60 pounds for gut and at 50 pounds for nylon. For oversized racquets, it is true up to 70 pounds of tension that lower tensions mean higher bounce for both gut and nylon. There seems to be no consensus among the pros. Borg strung as tight as possible (85 lb.) on specially fortified racquets with two extra plies, while McEnroe preferred very low tension (44 lb).
Anecdotal evidence suggests that there is truth in the rule that tight strings give better control, but it isn’t for the reason that dwell time increases. With a loose racquet, especially a big head racquet, an off-center hit will deform the string bed more severely than it would a tight, small-head racquet (like Pete Sampras uses), and therefore there will be less certainty as to the path of the rebounding ball. Thanks to Greg Raven for pointing this out. And Ronald Yepp points out that with a tight racquet, the ball is flattened more, so topspin is easier to produce. This would be particularly true where the head is small. Pete Sampras is a case in point: he can generate amazing topspin on his second serve using his heavy, small-head, tightly strung (75 lb.) racquet. Bjorn Borg is another. Spin gives greater control, and greater spin is possible with tight strings.
Recommendation: for power, use a midsized racquet strung with natural gut at 60 pounds because this tension gives the maximum bounce. If control is your main concern, and your stroke puts a lot of top on the ball, string very tight and use small-head racquets. Restring often because string tension decreases quickly, as Crawford Lindsey has shown.
Flexible racquets (low flex number) absorb more of the Torque from impact, with the energy going into bending the material thus reducing the risk of injuries. Anecdotal evidence from expert players is that flexible racquets also perform better, possibly by increasing dwell time. It therefore appears that the present stampede to stiffer and stiffer frame materials is motivated not by safety or performance considerations, but by a foolish desire for more “power.”
Flex, or frame stiffness, is comprised in the uniform coefficient of restitution (0.85) stipulated in the benchmark conditions, along with string type, string tension, ball bounce, and bounce-boosting innovations such as Power Holes, Rollers, etc. The assumption was made that this mix of factors produced a coefficient of restitution that was the same for all racquets tested. When better data becomes available as to the effect of stiffness on racquet bounce, with the other factors controlled, then a more exact understanding may be possible. For now, we must plod along with a rough but reasonable assumption that all racquets are strung such that they have the same bounce.
Frame stiffness is measured on the Babolat RDC as flex numbers, which represent deflection under a 25 kilogram load on the string bed. High flex numbers mean stiff frames. A stiff frame would have the effect of flattening the ball more (if used with tight strings and a small head), thus making topspin easier to impart, and it would allow the strings to do their job better because the frame would not be deformed so much on impact.
The downside of stiff frames, as many case histories on the Tennis Warehouse Racquets Board attest, is that they feel bad and probably aggravate the risk of injuries. Stiff frames may result in shorter dwell time, thus higher Torque.
Bigger is better for maintaining control. A large handle size gives more area to apply friction and a wider radius to apply the frictional force in order to resist racquet twisting about its longitudinal axis (Torsion, or Longitudinal Torque) on off-center hits. Handle size is the circumference (distance around). It can be increased by adding an overgrip (e.g. Tournagrip � or moleskin), or by building out the handle under the grip with tape or shims. Bigger handles should also be better for preventing blisters.
Big servers, however, prefer smaller handles. The best thing would be a handle that for groundstrokes had a large circumference at the forefinger, and for serves, a small circumference at the forefinger when you choke down: i.e. a coke-bottle-shaped grip, or rounding off the bevels about 4 cm up the handle to give a tapering smaller circumference at that spot. There is no reason but herd mentality why handles have remained uniformly octagonal for so long. This smaller circumference on the serve allows the racquet to cock back farther on the backswing.
To slow down the men’s game, and thus hopefully to increase its entertainment value for the unsophisticated, the rulers of tennis want to change the balls.
Particularly troubling is the lack of prior consultation with the pros who will be using these balls. Which of them are in favor? True, it is rare to find any touring pros who might be called formally educated, so naturally they must expect their opinions to be ignored and their objections overruled. If they don’t want to play, there are plenty of ambitious youngsters eager to take their place. However, even though there might be no reason for courtesy or compassion, wouldn’t it be economically prudent to prolong the careers of the marquee players, and not to increase the already alarming rate of injuries among the recreational players?
The ITF has authorized a ball with a 15% greater diameter to be used “on an experimental basis.” The intention is that the bigger ball will meet more air resistance, therefore play will be slower. Fluffing the nap (felt covering of the ball) will increase diameter and drag, but apparently the intention of the ITF is to require ball manufacturers to mold a larger rubber core. See the diameter test for the “slow ball” in Appendix 1 to the Rules of Tennis — fluffed nap will not keep the ball from falling through the bottom hole of the testing apparatus.
The larger diameter of the rubber core, even if the weight of rubber remains the same, will result in a higher rotational inertia for the ball. That means a “heavy” ball because players will be able to impart a lot of angular momentum (spin). Angular momentum is the product of the rotational inertia and the rotation speed, and the higher rotational inertia permits a much “heavier” ball at the same spin rate. High angular momentum of the ball on impact will aggravate Torsion (screwdriver twist on the handle), causing more stress on the arm of the receiver.
Another problem with bigger balls: they will radically change the game in the same way that the “spaghetti string” racquet did, by giving junkballers an edge. The ITF banned (retroactively) the spaghetti strings which imparted such extreme spin. The same reasoning should ban these balls.
Yet another problem with bigger balls: if the same ball weight (57 grams) is to be maintained, the rubber of the bigger ball must be made thinner to stretch over the larger surface. Thinner rubber means that the air will leak out easier, and higher air pressure will be needed to maintain the same ball bounce. These balls will go flat faster. They will also be less bouncy in actual pro-level play because of higher hysteresis loss from more air being compressed. These will be soft balls.
Presently, for professional tournament play, a ball must bounce more than 53 inches and less than 58 inches when released from a height of 100 in. That means that the coefficient of restitution for the ball itself (apart from the racquet) is between 0.73 and 0.76. It should be noted that the 100 in. drop height does not approximate the speed of a pro serve, so for testing the hysteresis loss from the bigger ball this test would be inadequate. Using softer balls, having a bounce at the low end of this range (low c), means higher Shock, Shoulder Pull, Work, Shoulder Crunch, Wrist Crunch, and Elbow Crunch for the players.
As you can see from the formulas, heavier balls (high b) means both higher resultant forces from impact (Torque and Impulse Reaction), and higher Shock, Shoulder Pull, Work, Shoulder Crunch, and Elbow Crunch. With heavy balls, the game becomes more painful and less accurate. See the formulas.
Club players can take a lesson here, especially those who play on clay, where the balls get heavier as play goes on. Change balls often to protect your arm. Tennis balls are a bargain, so leave them on the court.
Head-light is better, no question. A head-light racquet (balance point closer to the hand than the midpoint of the racquet’s length), has significantly lower Moment, resultant forces from impact (Torque and Impulse Reaction), Shock, Work, Shoulder Pull, Shoulder Crunch, Wrist Crunch, and Elbow Crunch. And it can have high mass (M) and high swingweight (I), but low Moment, with a handle end counterweight. That’s good, remember.
In the formulas, the key variable is r (the mass center radius, or the distance from the axis of rotation to the balance point). Head-light balance means that r is small. When r is small, r2 will be tiny, and the key coefficient in the formulas, Mr2/I (which, as astute students will note, is equal to r / q because q = I / Mr) in the formulas for Torque and Shock, will be small, which is good. The linear velocity of the mass center is critical, and when the mass center is close to the hand (small r), its linear velocity (v) in rotation will be smaller than when it is distant. A distant mass center goes much faster in rotation — remember the carousel at the playground? So head-light is the smart choice. Head-light (low r) with a high sweet spot (high q) is the really smart choice for reducing the risk of tennis elbow. That means a racquet with a large handle end weight (~5 ounces). This handle end weight customization produces significant improvement: check this.An important additional benefit of head-light balance is that Moment is less, so the racquet is easier to position for volleys and returns, and is not so heavy to hold up all afternoon. Moreover, with a low Moment, the Torsion from impact will be small, so the racquet will be easy on the elbow. Head-heavy racquets, on the other hand, increase the risk of tennis elbow because of their high Moment and high Torque (therefore high Torsion), their high Elbow Crunch, and their high Shock.
More mass is definitely better. More swingweight (moment of inertia) is also definitely better. The touring pros, in customizing their racquets, add mass and increase swingweight, because they know from personal experience what really works. Their customized racquets bite on the ball more, so they are able to generate heavy spin on their forehands and serves. Pete Sampras’ heavily customized Wilson Pro Staff 85 (a modification of the legendary St. Vincent ProStaff, which is no longer in production) weighs 14 ounces, about the same as the old woodies, but much heavier than the heaviest racquets marketed to the public these days.
Chain store customers, and even those who buy at pro shops, demand lighter racquets — completely the opposite of the pros! A candid observer must find it somewhat incredible that even though the racquet makers pay the pros lots of money to display what appears to be the same racquet they are selling to consumers, in reality the racquet is not at all the same in weight or swingweight. Racquet manufacturers tout light weight as if it were something good, when in fact they should be putting a warning label on their racquets advising buyers that they are increasing their risk of disabling injury if they insist on banging away with a wimp stick. The heavier, the better. If 14 ounces sounds big to you, consider that even ladies and juniors used to play with wood racquets that weighed that much, and Don Budge won the Grand Slam with a racquet that weighed a whole pound.
Mass (in kilograms, symbol M) is a measure of the racquet’s “inertia,” a word that means essentially its resistance to change. The change resisted by mass is change in linear (straight-line) velocity. More mass means that the racquet will not slow down so much on impact. A short, controlled swing of a heavyweight racquet can hit the ball harder than a frantic flail of a featherweight. Same as bats in baseball: if you want home run power, bring a heavy bat to the plate. Babe Ruth’s long bat weighed 52 ounces. You want a racquet that will hit through the ball, instead of bouncing off. Pete Sampras’ second serve has an incredible amount of spin on it because his racquet can bite on the ball due to its high inertia on contact.
Mass is not the same thing as weight, but weight units such as ounces may readily be converted into mass units of kilograms, using the conversion factor of 1 ounce = 0.02835 kg.
Swingweight (symbol I) is another measure of the racquet’s inertia, but it is a different kind — rotational inertia, or a resistance to change in a racquet’s angular velocity about an axis of rotation. A useful way to understand swingweight is as the energy storage potential of a racquet. Just like flywheels, racquets store up the player’s effort. A racquet with a high swingweight (I in the formulas) requires more effort to swing, but will not lose much angular velocity on impact, and will snap through the ball more, biting for more spin, especially if the strings are tight and the head is small.
Angular velocity means how much of a circle is swept out every second, in radians per second. There are 2p radians in a circle so an object having an angular velocity of 6.28 radians/s will rotate at one revolution per second.
It is most important to realize that swingweight has meaning only with respect to a specified axis of rotation. A swingweight number where you don’t know the axis of rotation is of no use. The published swingweight numbers of the USRSA come from measurements on their Babolat Racquet Diagnostic Center (RDC), which measures swingweight about an axis of rotation 10 cm from the handle end. In play, this is not the axis of rotation of the racquet, so you will have to use the Parallel Axis Theorem to convert the RDC swingweight figure to a value for I that can be used in the formulas.
On the forehand, the axis of rotation is 7 cm from the handle end, which is between the ring and middle fingers. Hold a racquet and waggle it to see that this is true. On the serve, where most top players use a choked-down grip over the butt cap, the swingweight will be higher because the axis of rotation has moved to 5 cm from the handle end. In other words, swingweight will change depending on how low you grip the racquet. Notice when Pete Sampras is going for a forehand putaway, he chokes way down over the butt cap to increase the swingweight.
The unit of measurement of swingweight in tennis is kg.cm2 (kilograms times centimeters squared), which is the unit for the swingweight measurements of the Babolat Racquet Diagnostic Center (RDC), presently the industry standard measuring device. Other scientific names for swingweight are moment of inertia, rotational inertia, and Second Moment. More swingweight is good for accuracy and comfort (low Torque, Shock, Shoulder Crunch, and Elbow Crunch).
A racquet with a high swingweight takes more effort to whip around the axis of rotation, but on impact that investment pays off in better speed and accuracy. “Maneuverable” is a term loosely used to describe a racquet with a low swingweight (although there seems to be some confusion between swingweight and Moment as this term is used). Maneuverability in a racquet, under this definition, is not good, because high swingweight is good.
Two racquets that weigh the same (have the same mass) may have very different swingweights because of the way this mass is distributed. More mass to the head of the racquet, such as by adding lead tape, will increase swingweight. Although more swingweight is good, head-heavy balance is bad for your arm, so lead head tape fixes one problem (higher I) but aggravates another (higher r). Increasing r means a higher Shock, Torque, Moment, Work, Shoulder Pull, Shoulder Crunch, Wrist Crunch, and Elbow Crunch. Note that the effect of r is squared in the Shock and Torque formulas, but the effect of I is not. So lead tape to the head should be counterbalanced somehow with a tailweight. The tailweight will not affect the swingweight materially because it will be close to the axis of rotation.
The Effect of Shoulder Weighting
Adding weight to the 9 and 3 o’clock positions of the head (shoulder weighting) adds swingweight, but not as much as adding the same amount of weight to the tip. Shoulder weighting also increases the Polar Moment of the racquet, which may be necessary to counter the “heavy ball” encountered in top echelon tennis when the opponent hits with a lot of pace and topspin. The heavy ball tends to rotate the racquet, even when the impact is dead center, thus making it more difficult to put your own top on the ball and giving an uncomfortable screwdriver twist (Torsion). A racquet with a lot of rotational inertia about its longitudinal axis will not be pushed around so much by the impact and will be more dominant in play. Shoulder weighting should be offset by a tailweight at the handle end, so the balance is head-light.
No. The increased length of string to be stretched in a large racquet head gives a more pronounced give to the string bed, and therefore presumably a longer dwell time (t) and a more pronounced trampoline effect (higher coefficient of restitution c), both of which are good. But the trade-off is that accuracy on off-center hits may be worse because the string bed is more deformable, and therefore the path of the rebounding ball is less certain. Also, the ball is not flattened against the strings as much, so it tends to just roll down the face when you stroke for topspin. Pete Sampras plays with an extremely small head racquet (85 square inches in area). The wood racquets were even smaller (65 sq in), and the tubular metal racquets that Jimmy Connors used were smaller still, and had a head size like a squash racquet. These world number ones are persuasive authority against big heads.
Another downside of big heads is that, due to their large width, there is a bigger chance for a badly off-center impact. With the ball so far from the centerline, the shot is a loser anyway, so better to let it miss than to have it hit way off to the side and cause a severe jolt.
If there is a weighting system at 9 and 3 o’clock, such as shoulder weighting by lead tape, this jolt can be minimized, but better not to let it occur in the first place. Accept that you will have to learn to hit the ball better, and don’t rely on “forgiveness” to improve your game.
The consensus among physical therapists seems to be that big heads are a risk factor for tennis elbow. The pros who make their living winning tournaments do not favor them. The conclusion must be that big head racquets are not better.
Sometimes. If the impact point is at its standard distance from the hand (i.e. where it would be using a 27-inch long racquet), extra-long performance is worse on groundstrokes but better on the serve. Moment is increased by the extra length, which is bad for the reasons discussed above. There is no need to adjust your striking point on the serve because you do better with the impact point at the usual distance from the hand, even though that moves the impact point lower on the face than the center of the strings. Two-handers should definitely consider an upgrade to an extra-long.
Yes. The axis of rotation on the two-handed shot is between the hands, which is higher up the handle than the axis on the one-handed shot. That’s good, because decreasing r and d in the formulas improves performance under all criteria, and you get those desired decreases by shifting the axis of rotation up the handle. Many players use a two-handed backhand with good results, but few use the cross-handed forehand of Seles and Gambill. Pancho Segura’s two-handed baseball style forehand was, in its day, considered the best shot in the game because it was so deceptive and powerful. A two-handed shot has the additional advantage of allowing a much heavier racquet to be used.
Damping doo-dads on the strings damp only residual string bed vibration, and do not really protect the arm by damping frame vibration. Adding more mass to the head in the form of a damping gadget is a bad idea because it increases r in the formulas and therefore worsens performance, so the damper should be light. Pete Sampras’ string damper is just a cable grommet, and Andre Agassi uses a rubber band.
It is helpful to establish some benchmark conditions so that abstract formulas can yield some meaningful numbers for racquet comparisons. Benchmark conditions stipulate values in the formulas for b, c, t, s1,and s2, and establish common impact point and axis of rotation locations for all racquets. With these stipulations, comparisons are possible using racquet measurements from the USRSA to determine M, r, d, and I in the formulas.
The First Benchmark Condition is a 70 mph return (s2) of a shot received at 20 mph (s1) (its speed parallel to the ground, after bouncing), with duration of impact on the strings (dwell time, t) 0.010 seconds, coefficient of restitution (c) 0.85, axis of rotation 7 cm from handle end, and impact at the center of the string bed (16 cm from the head tip and on the centerline of the racquet). See the latest research figures on ball speed.
The Second Benchmark Condition is a 110 mph serve, with duration of impact (dwell time) 0.010 seconds, coefficient of restitution 0.85, axis of rotation 5 cm from handle end, and impact at the center of the string bed (16 cm from the head tip and on the centerline of the racquet).
The Third Benchmark Condition is a 40 mph volley received at 40 mph, with duration of impact on the strings (dwell time, t) 0.010 seconds, coefficient of restitution (c) 0.85, axis of rotation 7 cm from handle end, and impact at the center of the string bed (16 cm from the head tip and on the centerline of the racquet).
The Fourth Benchmark Condition is a 50 mph return of a serve received at 50 mph, with duration of impact on the strings (dwell time, t) 0.010 seconds, coefficient of restitution (c) 0.85, axis of rotation 7 cm from handle end, and impact at the center of the string bed (16 cm from the head tip and on the centerline of the racquet).
The Fifth Benchmark Condition is a 70 mph serve, with duration of impact on the strings (dwell time, t) 0.010 seconds, coefficient of restitution (c) 0.85, axis of rotation 5 cm from handle end, and impact at the center of the string bed (16 cm from the head tip and on the centerline of the racquet).
Under all benchmark conditions, the mass of the ball (b) is 0.057 kg (57 grams, or about 2 ounces), and string tension, string type, racquet stiffness, etc., are all comprised in the uniform coefficient of restitution (c = 0.85 for all racquets). This figure of 0.85 is typical for impacts at the center of the strings. See H. Brody, “The physics of tennis, III. The ball – racquet interaction” Am. J. Phys. 65, 981 (1997). With the foregoing stipulations fixing constants, the only variables remaining in the formulas are M, r, I, and d, which are calculated using the racquet measurements.
The axis of rotation used for the First, Third, and Fourth Benchmark Conditions is 7 cm from the handle end, where it would be on the forehand. For the Second and Fifth Benchmark Condition (first and second serve), the axis of rotation is 2 cm lower (5 cm from the butt end) because top players typically choke down over the butt cap on the serve to increase swingweight. Michael Change, it appears, does not, so his axis of rotation on the serve would be the same as on the forehand (7 cm) and the swingweight, r and d in the calculations would have to be adjusted accordingly.
Dwell time is assumed to be 0.010 seconds, which is a realistic figure based on available information, but it may vary slightly with string tension, frame flex, and other factors. Roland Sommer, the inventor of the kinetic system used in the ProKennex line, has convincing lab results, using electrical contact timing, that show the dwell time is typically in the 10 millisecond range, not the 4 milliseconds previously accepted on the basis of high speed film. The difference is probably due to the difficulty of cameras seeing through the edge of the frame to the string bed.
Only high school algebra is required to understand the formulas given for evaluating racquet performance, and the derivations of these formulas. The concepts from which the formulas are derived are first year physics and the simple related engineering regarding an “eccentric impact.”
Racquet data are from measurements published in the official magazine of the US Racquet Stringers Association, Racquet Tech, and are repeated at this site by permission of USRSA. The racquets evaluated presently total 167, and all are presently offered for sale. All data are for strung and gripped racquets, measured on the Babolat Racquet Diagnostic Center by USRSA technicians.
How Do I Get Some Idea if a New Racquet is Good?
You can use the published specs for strung and gripped racquets to compute a value for the Quality Index, using the simple formula Mr2/I . M is the mass in kilograms, r is the balance in centimeters, and I is the swingweight in kg.cm2, about an axis 10 cm from the butt. These specs are available from sources such as Tennis Warehouse. The lower the value of the Quality Index obtained by this formula, the better.
The measurements to ask for in order to evaluate objectively a racquet’s performance by means of the formulas are:
M = racquet mass, in kilograms (1 ounce = 0.02835 kg), after the racquet has been strung and gripped.
r = the mass center radius, which is the distance from the axis of rotation to the mass center (balance point). The axis of rotation is 7 cm from the handle end (First Benchmark Condition, anyway). In lieu of that, simply the distance of the strung and gripped racquet balance point from the butt end can be adapted for use in the formulas.
I = swingweight (moment of inertia) about the stipulated axis of rotation for the given benchmark condition, in kg.cm2, for the racquet when strung and gripped. Published measurements, and swingweight measured on the Babolat RDC, are with regard to an axis 10 cm from the handle end. Note that there is no such thing as an absolute swingweight — swingweight only has meaning with respect to a specified axis of rotation, so swingweight numbers are useless unless you know where the measurement was taken. The axis of rotation for the First Benchmark Condition is 7 cm from the handle end, where it would be on the forehand. The swingweight on the serve, where there is a lower axis of rotation (5 cm from the handle end), as well as the swingweight for the forehand, may easily be found from the published swingweight measurement by using the Parallel Axis Theorem.
d = distance, in cm, from axis of rotation to point of impact (16 cm from the tip for all racquets under both benchmark conditions). This is easily derived from the overall racquet length. The axis of rotation for the First Benchmark Condition is 7 cm from the handle end and the impact point is 16 cm from the tip, so subtract 23 cm from the racquet’s total length to find d for the First Benchmark Condition. Subtract 21 for the Second Benchmark Condition.
It critical that a standard axis of rotation be agreed upon so that racquets may be compared objectively. We fix the axis of rotation for the First Benchmark Condition at 7 cm from the handle end, where it would be on the forehand. Cf. H. Brody, “The physics of tennis, III. The ball – racquet interaction” Am. J. Phys. 65, 981 (1997). Of course, the formulas still work whatever the axis of rotation might be (as in the case of the Second Benchmark Condition, where the axis is 2 cm lower).
Mass (M) may be found by weighing the racquet and converting ounces to kilograms (1 ounce = 28.35 grams or 0.02835 kilograms, so multiply your scale reading in ounces by 0.02835). The distance r may be found by experiment, balancing the racquet on a knife blade, then measuring the distance from that point to a point 7 cm from the handle end. To find d, subtract 23 cm from the length of the racquet under the First Benchmark Condition, and 21 cm for the Second Benchmark Condition. Measuring swingweight (I) requires a special device, such as the Babolat Racquet Diagnostic Center or the AccuSwing from Alpha Sports.