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:
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.
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 and/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.
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 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. Alpha also makes a devices for measuring swingweight, called the AccuSwing. 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).
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.
Dampening doo-dads on the strings dampen only residual string bed vibration, and do not really protect the arm by dampening frame vibration. Adding more mass to the head in the form of a dampening gadget is a bad idea because it increases r in the formulas and therefore worsens performance, so the dampener should be light. Pete Sampras' string dampener 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.
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 USRSA. The lower the value of the Quality Index obtained by this formula, the better.
The four 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 (RDC) or the AccuSwing from Alpha Sports.