Introduction
Don Hightower, a.k.a. sysop, a.k.a. President of Tennis Warehouse, first introduced the concept of proportional stringing to this message board. Don in turn has given credit in a previous post for having learned about this idea from UliKuehnel, Boris Becker's former stringer. Don really deserves the kudos for being the one to have spread the word about proportional stringing to the masses.
The basic concept of proportional stringing is to decrease the reference stringing tension of the strings as you move away from the center of the racquet. This results in a larger sweet spot, and also works to decrease the shock of off-center hits on the racquet.
Don's original descriptions of proportional stringing involved decreasing the reference string tensions by up to 10 lbs. maximum for the strings closest to the racquet frame. I got interested in using proportional stringing mainly to decrease the shock of off-center hits on a hybrid string combination of 18g Ashaway Kevlar mains and Technifibre NRG2 crosses. Strung uniformly at 60lbs., this string combination had good playability but did generate nasty shocks on off-center ball hits at the tip and throat ends of the string bed. Proportional stringing seemed like the answer, but I felt that for shock reduction purposes, especially with the stiff kevlar, that more tension reduction would be useful. So I reduced the tensions proportionally by up to 30lbs. for the cross strings at the head and throat ends of the string bed. The result was a much smoother ball strike pattern on this string bed and good reduction of the shocks of these off-center hits.
This made me wonder, just what are the physics that underlie proportional stringing? What sort of maximum possible improvement can we achieve with this method? Would it be possible to use proportional stringing to make the sweetspot to be as big as the entire racquet string bed? Surely the methods for proportional stringing would have to be different for different racquet string bed patterns and different string combinations. Could there be a systematic method to figure out proportional stringing patterns for all types of racquets and string combinations?
Sweet Spots
Because proportional stringing is basically about manipulating the sweet spot to our best advantage, before we can even begin to talk about it, we have to deal with the somewhat inconvenient fact that all tennis racquets have THREE sweet spots.
Howard Brody's research, summarized in his book "Tennis Science for Tennis Players", is the main original source which developed this concept of three sweet spots. Rod Cross of the University of Sidney has expanded somewhat on some of Brody's ideas, and there are multiple copies posted on the web of an article he wrote about these three sweet spots on a tennis racquet. Wilmot McCutchen's racquetresearch.com website gives an extensive treatment of one of the three sweet spots.
The First Sweet Spot
The First Sweet Spot, the Center of Percussion (COP) is usually located somewhere on the string bed. Its location can be changed by manipulation of the total weight and weight distribution of the racquet. The COP is the point on the string bed that, when struck by a ball, will result in the least amount of transmitted racquet handle deflection to the hand gripping the racquet. The farther away from the COP that the ball hits, the more the racquet handle deflection (or shock) that the player will feel.
Commercial racquets have COP’s located anywhere from 14 to 19cm from the tip of the racquet. This puts the COP in either the middle or the lower half of the racquet string bed. Since most players make the majority of hits on the upper half of their racquet, the higher up that one can make the COP the "sweeter" these hits will feel.
Racquetreseach.com focuses intensively on this single issue of how the total weight and weight distribution of tennis racquets affects the shock forces delivered to a tennis player. The main sweet spot involved in such a model of shock forces is the COP however, because the effects on racquet performance of the other two sweet spots are ignored, the real life performance of tennis racquets do not always follow the predictions of racquetresearch.com.
A couple more points about the COP. Although tailweighting a racquet (as advocated by racquetresearch.com) will raise the COP higher up the head of the racquet, this is not the most elegant way to get a tennis racquet with a high COP (because of the concurrent problems with increasing the Dead Spot, see section below). It appears that the main purpose of the Prince "Triple Threat" system (which has been dissed as simply a gimmick to use something fancier than lead tape) is in fact to get a higher COP by manipulating the weight distribution along the racquet in a more clever fashion than simply adding additional weight to the butt end of the racquet. In fact, the Prince Triple Threat racquets seem to have won the prize on racquetresearch.com for the highest collective group of COP’s of any current racquets on the market. This has been done while keeping the overall weights of these racquets fairly light, as well as keeping most of these racquets around the even balanced range. A close examination of the equations which relate the COP to the weight distribution along a tennis racquet shows that it is possible to do all of this by concentrating more of the weight of the racquet at both the tip of the head as well as at the butt cap end. This means, of course, that weight in the middle of the racquet has to be removed in order to keep both the overall weight of the racquet relatively light and the overall balance of the racquet around the even-balance point.
The Third Sweet Spot and the Dead Spot
I'm going to jump ahead to a brief treatment of what Brody called the Third Sweet Spot, because this topic blends naturally into this critique of racquetresearch.com.
According to Howard Brody, the point of maximum power of a tennis racquet will be located at the center of mass of the racquet. This is the Third Sweet Spot and it will at the balance point of the racquet.
The entire principle of head heavy racquets, what Wilson calls the "Hammer" line (even though some Hammers are now virtually even-balanced) is to move this center of power towards the tip.
As a point of fact, the center of balance is located, even on the most head-heavy racquets, in the mid to lower half of the racquet string bed. Head light racquets have the center of balance located somewhere in the throat area.
For oversized racquets, especially racquets with an elongated racquet head, the chance of making ball strikes in the lower half of the string bed increases. If the racquet is also head heavy, the center of mass will be located well within this lower half of the elongated racquet head. I believe this is the reason that people sometimes feel a second (unwanted) "hot spot" in the lower half of these head heavy, oversized racquets.
At the very tip of a racquet, there will also be what Rod Cross called a Dead Spot - this is where the racquet flex causes it to absorb most if not all of the ball strike energy. Brody pointed out that a ball strike at the tip of the racquet will cause the ball to stop dead in its tracks, since the racquet will absorb almost all of its energy (for coaches, this is a great way to "catch" the returning ball from the student with the racquet); the racquet will then snap back and hit the ball again - this explains the sound of two ball strikes that you can sometimes hear whenever somebody makes an off-center hit at the tip of the racquet.
What happens when you make a racquet extremely head light? A couple years ago, hoping to improve its performance, I followed the principles outlined in racquetresearch.com and heavily tailweighted a Prince ThunderliteLongbody racquet. The result was the appearance of a fairly large area at the tip of the racquet where the ball did not bounce off the racquet very well. Instead, at this spot the racquet seemed to absorb a great deal of the ball strike energy, resulting in a diving board - like effect where the entire racquet shook with a low frequency SPROINNNNGGG! vibration. This, I believe was due to the presence of a large Dead Spot at the tip of the racquet. Adding weight to the head of the racquet or otherwise making the racquet less head light decreased this tendency to generate a diving board effect.
Neither Brody nor Cross address this issue of what makes a Dead Spot bigger or smaller. My theory is that the Dead Spot size and the amount of energy dissipated by the racquet at the Dead Spot are probably related to the location of the center of mass and the flexibility of the racquet. The Dead Spot probably gets bigger and deader as the center of mass moves away towards the tail end. More flexible racquets also probably have bigger and deader Dead Spots.
You can do a simple experiment to prove the relationship of the balance point of the racquet and the size/severity of the Dead Spot. Get a big heavy metal washer or bolt (2 oz. or more) and tape it onto the butt end of the racquet. Bounce a ball at the very tip of the racquet head. The ball bounce will definitely be quite "dead" at this spot. Now take this same chunk of metal and tape it firmly to the tip of the racquet head. The ball will all of a sudden bounce quite well at the very tip of the racquet head.
And so, although racquetresearch.com's main themes of making a racquet more arm-friendly by making it more headlight may have some validity, because of the effects of the Third Sweet Spot and the Dead Spot, there are limitations to how headlight one can make a racquet without degrading its performance.
The Second Sweet Spot
The Second Sweet Spot is the one that we are interested in. Brody called it the Node of the first harmonic. Brody described this sweet spot only as the spot on the string bed that generated the fewest vibrations in a tennis racquet when struck by a ball. Brody did not, unfortunately give any descriptions of how to find this nodal point, based on the physical properties of the racquet. The only method that he described was trial and error, by bouncing a ball off the string bed repeatedly.
I think that the Second Sweet Spot is simply the area on the string bed described by the maximum amount of string bed deflection allowed by the combination of main and cross strings. This Second Sweet Spot will also be the point where the maximum ball bounce occurs when the racquet head is fixed in place (to isolate the effects of the other sweet spots) because ball bounce is determined mainly by the amount of energy stored by string bed deflection. In a standard oval shaped racquet head with a symmetrical string pattern, the Second Sweet Spot will be at the center of the racquet head.
Just about all racquets today have string patterns where the spacing between the strings widen for the strings that are closer to the rim of the racquet. The main purpose is to decrease the string density in the periphery of the string bed so that more string bed deflection will be possible in the periphery. This will have the effect of increasing the apparent Second Sweet Spot size.
Racquets with low overall string density patterns typically achieve the lower string density by having wider gaps between the strings in the periphery of the string bed. In these low string density racquets, the central strings will usually still have a spacing similar to racquets with a high overall string density patterns.
Tear drop shaped racquets move the Second Sweet Spot higher up the head of the racquet. This occurs because the cross strings at the tip of the racquet will be longer than they would have been in a standard oval shaped racquet, and so this will allow more string bed deflection at the tip.
Fan shaped racquets take this a step further - not only do they benefit from the longer cross strings at the tip of the racquet, but they also have a much lower string bed density at the tip of the racquet than they do at the throat because the spacing between all of the main strings fan out at the tip of the racquet. These racquets are typically described as having high sweet spots (the Second Sweet Spot), and as being very forgiving on off-center hits at the very tip of the racquet (the Dead Spot). However, control would be compromised in such racquets because both the spin potential and the amount of bounce from the string bed will vary greatly across the string bed.
Semi- rectangular shaped racquets, like the Yonex isometric patterns, will have larger Second Sweet Spots because more strings away from the center of the racquet string bed will be of nearly the same length as the center strings.
A Universal Method of Proportional Stringing
So, with all of these different string bed patterns and racquet head shapes, and different types of strings, is there a way to do proportional stringing under all conditions?
The first thing is to understand that what proportional stringing does is to increase the Second Sweet Spot size by maximizing the amount of string bed deflection in the periphery of the racquet string bed. The second thing to realize is that proportional stringing does this by taking advantage of the fact that the amount of string deflection increases as string tension decreases.
What we need, therefore, is accurate data on the physical properties of strings for the following relationships:
1. How do different strings vary in the amount of deflection (for a given string tension and a given incoming striking force) for different lengths?
2. How do different types of strings vary in the amount of deflection (for a given string length and a given incoming striking force) for different string tensions?
3. How does the amount of string deflection (for a given string length, a given string tension, and a given incoming striking force) vary depending on where along its length the string was struck?
Let's consider first the case of a standard oval shaped racquet string bed. If the ball strikes the center of the racquet head, great, the ball has hit the Second Sweet Spot where the strings are longest, and each string that deflects will deflect at its maximum.
If the ball strikes the periphery of the racquet, let's say, at either 12, 3, 6, or 9 o'clock, the ball will impact two types of strings:
1. Two or three of the LONGER main or cross strings, struck close to their anchoring grommets near the racquet frame. Let's call these strings the Perpendicular Strings, since these strings are perpendicular to the part of the
racquet frame closest to the ball strike.
2. Two or three of the SHORTER main or cross strings, struck in the middle of these strings. Let's call these strings the Parallel Strings since these strings are parallel to the part of the racquet frame closest to the ball strike.
Deflection of the perpendicular strings will be less than it would be if the ball had struck that string in the middle of the string. This is because the shorter half of the perpendicular string will not stretch as much because of its shorter length and the longer half of the perpendicular string cannot stretch more to compensate.
For ball strikes that are VERY CLOSE to the racquet frame, the amount of string stretch from the shorter half of the impacted string will be so much less than the amount of string stretch from the longer half of the impacted string that the string deflection will NOT BE STRAIGHT DOWN; instead, the anchored short segment of the string will redirect the string deflection slightly towards the anchoring points (the grommets/frame). The rebound of the perpendicular strings will be thus be slightly tilted away from these anchoring grommets. And so balls that strike close to the rim of the racquet head will bounce off the string bed not straight up but in a direction slightly tilted towards the center of the racquet. Anybody who has ever jumped on a trampoline understands that the same thing happens when you land close to the rim of the trampoline - the resulting bounce will send you towards the center of the trampoline.
When manufacturers widen the spacing between the strings at the periphery of the string bed, the main effect is to allow for increased string deflection of these Perpendicular Strings by having fewer of the Parallel Strings to support these Perpendicular Strings in the periphery.