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09012012, 07:46 AM  #1 
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Combining weight, balance and swingweight
The relation between weight (mass) and force is something most people have a natural feel for: You have to apply more force to accelerate a heavy ball than a light one when throwing it. The formula is also straight forward:
F=m*acc When you move in a curve, as when swinging a tennis racquet, you have something called swingweight that is supposed to tell you how it feels. The problem, however, is that most people don't know what swingweight really is. And even worse, the values you get from manufacturers and resellers are only valid when you rotate the racquet around a point 10 cm up the handle, a type of swing that rarely occurs in tennis. I will try to shed some light on this and propose a curve where you can compare different racquets for different types of swings. Say that we have a racquet where we apply a force F at the handle and swing it around a point p. The the swing radius r says if the swing is short or long. When swinging we are interested in accelerating the racquet head: If we look at the relation between F and the acceleration of the head acc we can define an equivalent mass me that tells us how much force you have to apply to get a certain acceleration, i.e. how heavy the racquets feels for different kinds of swings: me=F/acc We can then plot me and compare different racquets. But instead of plotting me against r I will plot it against 15/r. I that way we will get a convenient scale where 0 means moving it without rotation (a block) and 1 means whipping it around the wrist (5 cm outside the handle). In the figure below I have plotted two very different racquets: Wilson BLX Pro Staff 90, 357g, 8 pts HL, sw 327 Wilson BLX Cierzo Two, 278g, 8 pts HH, sw 350 As you can see the Pro Staff is heavier for long swings (as expected), but it is also heavier for shorter swings despite that Cierzo has a higher swingweight. The reason is that the swingweight doesn't the the full story, even for a short swing. The diagram therefore gives you a way to compare these two racquets for different swings.  For those who want hear some more details we need to define some lengths: You can then calculate me in terms of the swing radius r: Where m is the weight and sw is the swingweight of the racquet. I have used d=40 cm (i.e. 50 cm from the but) in the diagram above. For those who want to play around with the figures I have an Excelsheet that you can download here Edit: There is an alternative and better version with a new excel sheet presented in post 40 The "proof" of the equation can be downloaded here /Sten __________________________________________________ _________ racquetTune, stringBed and swingTool racquet apps for the iPhone/iPad. Last edited by stoneage; 09202012 at 08:56 AM. Reason: clarification 
09012012, 05:28 PM  #2 
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Interesting!
Is there an ideal personal curve, you think, which will give a nice feel of the forces needed in the stroke. Let me put it differently. Say you like this much force on your shoulder and this much force on your wrist. Can the related curve be translated to the ideal racketspecs? 
09012012, 11:33 PM  #3  
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No, I think that would be to push the significance of the curve a little to far. Different people will still like different racquet behavior. You will also vary the radius during the swing. But maybe you could take a racquet you like and use the curve as the basis when buying or customizing other racquets. I would use the curve in the first post to say that the Pro Staff 90 is "always heavier, especially for long swings" something that is not obvious when you look at the weight, swingweight and balance. 

09022012, 02:50 AM  #4 
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I am having trouble filling in the formula. Could you provide an example of the calculations?
And I don't get the 15/r part. I am trying to figure out which specs influences the shape/steepness of the curve in which way. Thanks! 
09022012, 06:31 AM  #5  
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Or do you mean how I derived the formula? I didn't include that since I didn't want to scare people with to much math to begin with. But it is coming, I just have to type it out so it looks nicer. Quote:
r is infinite when you move the racquet without rotation, but that is not so easy to show in a curve. 1/r = 0 when r is infinite, so thats easier to plot . At the other end I picked 15 cm as a reasonable shortest swing radius. Since r is measured from 10 cm up the handle where the force is applied, r=15 means 5 cm outside the end but. Which is somewhere around the wrist. So when r=15 then 15/r=1. You can of course use another point as reference, r=10 or r=20 if you like, as long as you don't go all the way to r=0 when the calculation becomes irrelevant. 

09022012, 09:56 AM  #6 
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I understand the 15/r part. Thanks.
I don't mean how you got the formula figured out, but I am having trouble filling in the numbers and then get the correct me. I think I do it in the wrong units. Let's say I have a racket with the following specs: 335g, 32cm balance, sw 330. Can you show me the math to calculate the me? Thanks again! 
09112012, 08:37 AM  #7 
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Reposted below for you DEH
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Head TGK 238.1 16x19 a little shortened, 384g, 31.88 cm balance, 366sw Babolat Tonic Gut 16g / ZX Monogut Natural 16g 61/49 lbs. Last edited by TaihtDuhShaat; 09112012 at 08:54 AM. 
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09112012, 10:49 AM  #8 
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Edit:
Now I'm remembering I have tried similar experiments in the past. The result is an extremely spin friendly setup (extreme loopy groundstrokes, great arcing topspin serves), and a flexy feel to the impact. The racquet needed a lot of wrist added before contact on each stroke because the racquet head lagged behind so much (a polarized, disconnected feel between the tip and the handle), like a barbell. It was difficult to hit accurate flat shots, volleys were very grabby and hard to pull off, and I had to prepare earlier for overheads.
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Head TGK 238.1 16x19 a little shortened, 384g, 31.88 cm balance, 366sw Babolat Tonic Gut 16g / ZX Monogut Natural 16g 61/49 lbs. Last edited by TaihtDuhShaat; 09112012 at 11:00 AM. 
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09112012, 11:58 AM  #9 
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Maybe it is easier to do it the other way, add 20 or 30 gram at the center and check if your timing is off. IOW that the rackethead travels too fast.

09122012, 01:49 AM  #10 
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Next step
Just when you though you understood it, I am going to hit you with an alternative curve
Well, actually it is showing the same thing and I think it is an improvement. Instead of showing the equivalent mass, I am showing the equivalent mass for the racquet divided by the equivalent mass for a reference frame. And as the reference I am using a 70 cm long 320 g heavy* racquet with a completely even mass distribution and balance. By doing this it will be easier to see the difference between racquets. And the slope of the curve now also means something: A rising slope means that it is relatively heavier to swing at a short radius compared to the neutral racquet. And inverse for the falling curve. Three examples, first our old friends the Pro Staff and the Cierzo: Then Johns two racquets: Finally an answer to DEH's question of taking travlerajm's racquet and make it 320 g. With balance 32 cm and swingweight 305 they look similar: A new excel sheet is available here for your amusement Digest and tell me what you think. /Sten *Note. I picked the weight of the reference racquet to be 320 g as it is an average of the racquets in the tenniswarehouse database. The value, however, is not so important since it will only shift the scale of the yaxis up or down. Last edited by stoneage; 09122012 at 01:56 AM. 
09122012, 02:17 AM  #11 
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Give me a couple of hours

09122012, 02:33 AM  #12 
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Thanks for posting all the good info. Here is what I got. I have tune my daughter's racquet (she is 15) for a long time using Travlerajm's method and it seem to work very well. I ended up with 324g with a balance of 32.7 and swingweight of 310. She was using a windsheild style forehand and it seemed that the shots where very spinny but not a lot of power. I tried all kinds of strings and tensions and I could not get any more power out of it. So we changed here swinging style to a Djokovic style swing which took about a month to do. Now that racquet will not work at all. The balls fly all over the place. No control at all. Tried more strings and no luck. So I tried another racquet. It was an nTour Wilson and it has some weird specs so it could be fake I don't know. Well she started hit a little better. So I just started adding lead to the top of the racquet and it got better and better and better. I added 18 grams at 12. So now her shots are controlled with a lot of spin and power. The specs are 314g with a balance of 34 and 337 swingweight. Not really close to the other specs. So I was just trying to figure out what is so different between the two styles of hitting and the curve on the racquet map. Here is a link for some WTA specs.
http://www.hdtennis.com/grs/pro_racq...anceopen.html Look at Sloane Stevens racquet. Thanks again. 
09122012, 06:26 AM  #13 
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What puzzles me is that the lighter Travlerajm racket has virtually the same MgR/I and the same shape of the curve. But if you add weight at the center of a racket, the shape of the curve also remains the same, but MgR/I increases.

09122012, 12:46 PM  #14  
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Quote:
My guess is that with her old racket the rackethead was feeling light to her in relation to the static mass and balance. Resulting in high rackethead speed and thus easier spin, but less plow. Djokovic drives thru the ball flatter which would be harder to do when you don't feel enough weight in the rackethead. Last edited by JohnB; 09122012 at 12:53 PM. 

09152012, 09:50 PM  #15 
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whoever's a physics phd should write his dissertation on this stuff.

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09192012, 02:53 PM  #16 
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possible mistakes in spreadsheets
@stoneage Thanks for your many technical posts. This is an interesting attempt to develop a new way of looking at the weight parameters of racquets.
However, I believe there is a mistake in the formula in your effective_mass.xlsx spreadsheet. There is an extra factor of 2 in the denominator there. When you take that out, the plots become monotonic with 15/r and you get much larger effective masses on the right side. In the eqv_mass_norm.xlsx spreadsheet, I don't understand the expression for the reference racquet that you use in the denominator there. Can you explain how Ip for the reference racquet equals 320 (1033 + 50*r + r*r) ? Using I[end] = (M*L^2)/3 for a uniform rod, I get Ip = 320 (70*70/3 + r*r) for the reference racquet. This makes the curves look quite different. Anyways, thanks for posting something that was motivating enough to get me to finally delurk! 
09202012, 08:54 AM  #17  
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Quote:
Quote:
Ip=M*L^2/12+M*(c+r)^2 and c for an evenly balanced rod is L/210 = 25 so I get Ip=M*70*70/12+M*(25+r)^2= M*(1033+50*r+r*r) Also remember that the parallel axis theorem can only be used in relation to the MOI around the center of mass, so you can't calculate Ip the way you did, even if r would have been to the endpoint of the racquet. 

09202012, 10:32 AM  #18 
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@stoneage Thanks for your response explaining the correct Ip for the uniform reference racquet. My mangling of the parallel axis theorem and inability to figure out your correct expression might be more embarrassing than your extra factor of 2! I'll get over it someday.
Now that (hopefully) the math is checked out, we can talk about the interpretation of the model and its results. The "effective mass" was an interesting idea, but I wasn't comfortable with the mixing of linear and angular forces. I didn't know what to make of effective masses of 500 (or 700) grams at small values of r. So the "mass" numbers were no more intuitive than swingweights. The normalization was an improvement, certainly, but since it's now dimensionless quantity, there's no concept of "mass" anymore. And since me/me0 = Ip/Ip0, you only need starting equations 5 and 6 in your proof. Equations 14 aren't needed, and we're just plotting the ratios of moments vs. radius of swing. Still, this is pretty cool. You can plot the moments of two racquets and say something like Racquet A takes 21% less effort to swing on serves than Racquet B, and 11% less effort on forehands. Just need to pick values of r that represent the serve and the forehand. (Let's ignore the fact that the serve is a complex motion involving multiple radii at different phases.) I think that would be about as readily understandable as you can get. And it's a big improvement over just taking the ratios of the conventional swingweights. Your fundamental insight is that 10cm from the butt is the wrong axis of rotation to care about, and looking at longer radii shows that swingweight numbers understate the differences in swing effort. After we revolutionize the reporting of swingweight, we can tackle the endless argument whether racquet speed or weight is more important when the racquet hits a tennis ball. 
09212012, 02:22 AM  #19  
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Quote:
If I keep the reference racquet as before an plot a curve of Ip/Ipref it will look the same as the Me/Meref (not surprise). The advantage is that we longer need the equations as background. While we are at it we might as well plot it against the the swing radius r instead of 15/r since it is easier to understand. The reason for using 15/r was that Me had no relevance for r=0, whereas MOI has. So instead we get a figure like this: It shows the same relation as the first figure, but with short swings to the left instead of to the right. The advantage is that you can see the actual values of the swing radius, which is more intuitive. This curve looses a part for very long radii (probably not so interesting). But instead shows the difference between r=0, where you measure swing weight and the wrist at r=15. 

09212012, 09:57 AM  #20 
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toward a more useful number for "racquet heaviness"
Yes, plotting r instead of 15/r is much better. An alternative option for plotting is to plot the ratio of the racquet moments to each other, avoiding the need for a reference broomstickalthough I like the idea of an official reference broomstick, maybe goldplated and enshrined somewhere.
An example of the swingweightratiooftworacquets plot: In this plot it's easy to see the Pro Staff has roughly equal swingweight at the wrist, roughly 810% more at the elbow, roughly 1314% more at the shoulder. That's why your arm will fatigue faster with that racquet, even though it's the "lower" swingweight racquet at the traditional reference axis. I really like the results of this thread, even if all we have done is a) think about a realistic axis of rotation, and b) use the parallel axis theorem. This is significant because the traditionallydefined swingweight axis clearly doesn't give us the best number to predict arm fatigue, or even indicate how easy or hard it feels to swing a forehand! If it ever worked for such purposes, it was a coincidence owing to racquets under comparison being very close in mass distribution. If I had to pick a single replacement without further study, I'd choose swingweight at the elbow radius, say r=40. That radius applies to armstressing portions of the groundstrokes and the middle phase of the serve. And "tennis elbow" is the epidemic, not "tennis wrist" or "tennis shoulder" although those are bad too. Maybe the traditional axis was intended to be more relevant to the effects of ballimpact, since it represents the axis where torque is applied to resist the ball, but those dynamics depend more specifically on mass distribution than the static swingweight number can capture. We're lucky that TWU is now full of better measurementspower potential, plowthrough, etc.which quantify and summarize impact effects. What I'd like to see developed is the most realistic number for predicting arm fatigue, so if I had that number, and I knew what my limit was, I'd know whether the racquet was too heavy or not. This thread shows the significance of swingweight on a more realistic axis of rotation, which is the torque required for a given angular acceleration. Right now, I'd use the swingweight at r=40 (or maybe an average of the swingweights at r=15, 40, and 60) as my magic number for "racquet heaviness." For me, the next theoretical step would be to consider the maximum centrifugal force caused by the racquet, which would occur at the point of contact of a serve, and try to figure out if that tells me anything different about racquet heaviness. If the racquet speed is high enough, I wonder if centrifugal force outweights torque as a cause of arm fatigue. 

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