Why balance matters for maneuverability separately from swingweight, and an alternative metric to mgr/i

[zoingy]

Retraction: Why I'm stupid & mgr/I is fine as a foundational metric

I was sitting on the train the other day, and I was thinking "Wait a sec, if you were able to apply a constant backward acceleration to the buttcap, of magnitude of, oh say, g -- wouldn't that mean in the non-inertial reference frame of your hand, the entire racquet would be feeling a fictitious force of g in the forward direction? So wouldn't it then be experiencing a forward angular acceleration of...mgr/I?"

Yeah ok I guess I was wrong re: the mgr/I critique - I foolishly operated under the assumption that pure hand pushing/pulling wouldn't result in any significant contribution to the linear component of the racquet speed. That's sort of how it feels subjectively anyway. But as an extreme example, if your balance point was right at your hand, your backward acceleration is mostly just going to pull the entire racquet back, not rotate the tip forward. Which is the intuition behind adding weight a bit further up the handle to increase mgr/I - it allows the racquet to resist your pull linearly, so that the com can actually act as more of a pivot point for the tip to come through.

I still think there's something to the subjective experience of adding weight at the buttcap though, so I'll continue exploring that.

I'll still leave the original post down below.


Original Post

(Hi all, I've tried to do my due diligence in familiarizing myself with prior discussion on this subject, but I'm sure there are things I've missed or misinterpreted. Any feedback is welcome! tldr my claim is: balance matters in the way everybody used to think it would, even though it's not immediately obvious why, and it tidily explains some areas where mgr/I implies strange results)

Maneuverability, and the physics of mgr/I

Many feel that adding weight to the handle helps the racquet head come through faster, despite the fact that adding weight cannot decrease swingweight.

mgr/I is a metric commonly used to describe this effect (despite the fact that weight at the buttcap doesn't affect mgr/I -- more on this later). mgr/I has some nice physical interpretations as the acceleration produced by gravity at a horizontal pendulum position, the square root of which is the frequency of a racquet swinging as a pendulum by the buttcap (often misquoted as directly related to the motion of a double pendulum, but there's no closed-form solution for double pendulum motion wrt time). It captures some aspect of how fast the racquet head comes through, that isn't quite captured by swingweight alone.

But in actual swings, the timescales and distances involved should suggest that gravity is negligible with regards to producing racquet head speed.

For example, the work done by gravity in the forward swing on a forehand up to contact may be estimated by mgh ~= .340 kg * -9.81 m/s^2 * .3m = -1 J. While the racquet tip at contact can reach 80mph on atp forehands, implying a com of ~32cm from the buttcap rotating about a point -10cm from the buttcap is moving at around 42 cm / 78.58 cm * 35.7 m/s = 19.1 m/s. So the purely linear component of the kinetic energy at contact is around .340 kg * (19.1 m/s)^2 = 124 J. The player has to put in 125 J of kinetic energy plus however much for the rotational component, and only has to overcome a measly -1 J of work done by gravity.

Yes, mgr/I just uses a model that describes pendulum motion, and not a direct assertion that gravity should be a truly significant part of the swing. But the relevance of this pendulum motion model is predicated upon there being a force acting on the racquet center of mass, and a non-accelerating pivot point. We can see there's no force acting upon the com that mg would have relevance towards. And the backwards acceleration of the "pivot point" of the hand is actually quite relevant to how the racquet head comes through.

 

[zoingy]

Backward movement for a bigger forward movement

There's an important concept in fishing where beginners are taught to abruptly stop at the end of their cast, in order for the rod tip to come through. Similarly, you can drag a whip around in circles over your head indefinitely, but it won't crack until that traveling loop is created by sharply reversing the direction you're dragging the whip.

There's no direct equivalent in tennis, but the same concept is reached by the idea of swinging directly towards and through contact in that inside out fashion (or, on the serve, swinging up towards the ball rather than in a big arc towards the opponent's side). This causes (among other things) a slightly more linear path, which means your hand will naturally run out of room to go & therefore will decelerate the racquet handle right before contact.

But wait, how do we know this actually happens? You can take a look at this bit of motion tracking I did for a Fed forehand***. See how right before contact, there's a decrease in hand speed, and a massive increase in racquet head speed.

It may not be immediately intuitive why you'd want the handle to decelerate, but consider tapping one end of a stick.† The stick will rotate about its center of mass, and the other side of the stick will move. If the stick's center of mass is closer to your tap, then the other side will actually move more than the side you've tapped.

This amplification effect can happen multiple times within the structure of fishing rods and whips, and it's why their thin taper is so important to their behavior*****. It's also an important part of why the kinematic chain of swings and throws work.


The maneuverability metric to complement swingweight

Anyway, with regard to racquet specs, we just have to consider the case of the floating stick. If swingweight describes how difficult it is to swing the contact point around your hand, then to complement that, we want to construct a metric that describes how difficult it is for the racquet head to swing itself around.

And because a backwards acceleration on one end of the stick results in rotation about the center of mass, forward acceleration is induced at the contact point at a ratio of (distance from com to contact point) / balance

Which describes basically the same thing as balance in reverse, but if I had to give it a name, I'd call it the "whip ratio". It has the nice property of being unitless and orthogonal to other measures of mass & moment.†† It also allows you to account for extended racquets and factors that change the distance of the contact point.

Usefulness is purely speculative at this point, but there's some ways to make it interesting:

• Smaller headsizes and hoops that are larger towards the top then normal (i.e. Yonex, head gravity/boom), may move the stringbed center up, increasing the whip ratio.
• Balance is measured by distance from the buttcap, so if you kept a 32cm balance when extending the racquet, you move the com closer to the handle, increasing the distance from the com to the stringbed center, and increasing the whip ratio.
• If you wanted to assume center of percussion is a good enough metric for the sweet spot, you can use (recoil weight) * / (balance^2 * mass), which sort of adjusts for how polarization can move the sweet spot up by increasing effective mass further from the com.
• A more "liquid" swing with more wrist action (as opposed to a more pushy swing with less wrist action) would result in a higher tip speed relative to throat speed, also making higher contact points more desirable. Having such a swing could both increase the whip ratio and increase the benefit from it.

Things that this explains

Adding weight at the buttcap doesn't increase mgr/I, despite many claiming that this weight affects how the racquet comes through. But it does lower the balance, which many also claim affects maneuverability. The above attempts to explain why -- essentially maneuverability is about both the ease of rotating the racquet head around the handle, and the ease of manipulating the racquet tip by moving the handle around the center of mass.

mgr/I is increased when adding weight at the top of the handle. This makes total intuitive sense in the context of a pendulum, where that mass is around the ideal r that increases the gravitational force's torque (scales linearly with r) more than it increases the moment of inertia about the buttcap (scales with r^2). This doesn't make as much intuitive sense in the context of swinging a racquet, where adding mass doesn't increase any force that would cause the racquet to swing. Balance & whip ratio don't have this issue.

Ohbh players appear to dislike sluggish racquets, because the wrist is not in as advantageous of a position to rotate the racquet head through (when compared to the forehand, or the two-hand system in the 2hbh). Therefore, a racquet head that comes through by itself more quickly is desirable, but ideally not at the expense of swingweight. Therefore, a higher whip ratio may be desired, and may be achieved via a lower balance and smaller head. This may explain why Gasquet can have such a sluggish mgr/i -- his extended racquet gives him a fairly high whip ratio, especially if you adjust for center of percussion with his high recoil weight.

The same com-rotation principle also determines how much of a natural "racquet flip" you get from pulling the handle forward. A higher swingweight and looser wrist allows it to happen, and a lower balance point means the head flips around more given a small movement at the handle. This may contribute to why Fed seems to get so much natural racquet lag, why Dimitrov's forehand lag looks a bit different, and why Djokovic/Nadal/Sinner/Alcaraz appear to almost "place" their racquet head in the takeback by comparison.


Relevant works

http://whip.creatingspeed.com/ - this tends to corroborate what I've said here, and it has some nice simulation examples. There's also a section on "pendulum acceleration" that's unrelated to mgr/I - notably there's no gravitational force involved at all.

This video on wrist action - I don't necessarily agree with the way he says things (it's legitimately a really tricky subject to explain without people taking it the wrong way), but it touches upon the same idea. So does PM, though I'm far from being a fan. It may just be one of those things that may be true, but it's hard to word in the right way, and you can't tell everybody to do it intentionally without there being crippling side-effects.

*** It's not perfect since it just measures movement in the camera image plane, but I think it gets the point across.
† If you've seen some spear scenes in Chinese martial arts movies, you may recall how the spear wielder may manipulate the shaft in one direction to make the tip point in the opposite direction
***** Fishing rods also utilize the fact that increasing acceleration loads the rod elastically, and the removal of that acceleration allows the rod to unload, in a wave that amplifies itself starting from the bottom. But elasticity, while very relevant to tennis swings, isn't strictly required for the amplification effect to work - after all, chain whips can still crack. So we'll leave that elasticity part to be addressed by swingweight.
†† Why doesn't it depend on the mass or rotational inertia of the racquet? It's because your ability to manipulate the racquet around its com isn't really limited by the racquet mass.

 

[Boxbeams4ever]

Okay I've tried reading all of that, and got thru almost all of it, but I don't think most of it stuck and still lost on parts of it.

I'm far from qualified on the physics, all I know is yes the static balance matters, but it's far from being the best measure by itself to gauge maneuverability. We need a better overall measure for that, nothing will be perfect since racquets all come in different shapes and weighting locations. And I'm luckily very good at that myself.

I don't recall many or most people saying that adding weight to the handle makes it more maneuverable, although I actually agree. I had thought there was still contention about that.

All I know is, for a standard length racquet, the most maneuverable balance point seems to be centered around Feds preferred at 9 pts or 31.5cm above the handle.

And I've never read but believe a tennis racquet has 3 points at which it wants to rotate linearly:

-the bottom of the buttcap(due to your wrist)
-the middle of your hand(mostly due to your forearm and that's the middle of where you're holding the thing)
-the balance point of the racquet itself

How it wants to rotate and how quickly is affected by the other measures like SW, Mgr/I, other pendulum type effects

Sometimes I've made racquets swing quicker by adding a tiny amount of weight at 12 o'clock, other times it's close to my hand or top of the handle, or bottom of the buttcap for that backward acceleration of the pivot point you might be confused about, you have to tune it yourself for each racquet

 

[zoingy]

Thanks for reading! It's certainly a bit of a slog - conciseness was never my strong point but I think this topic deserves a lot of words haha.

I'm far from qualified on the physics, all I know is yes the static balance matters, but it's far from being the best measure by itself to gauge maneuverability. We need a better overall measure for that, nothing will be perfect since racquets all come in different shapes and weighting locations. And I'm luckily very good at that myself.

Yeah I think if there's one thing I wanted to say, it's the claim that swingweight and balance together almost fully describe maneuverability. And it turns out that Roman Prokes also says something of the sort, and who am I to disagree with him?

Though I'd probably go so far as to say that static weight doesn't matter much outside of its contribution to swingweight and balance - after all, I don't think people can easily tell the difference between a ball of lead weighing 320 grams and one weighing 330 grams. I'd probably also go so far as to say that high twistweights shouldn't significantly reduce maneuverability given good technique, but that's a different subject.

If there's a second thing I want to say, it's the claim that mgr/I wanted to describe what balance already did, but using a physical model that gives it more flaws than balance (which actually has a nice non-obvious physical interpretation)

I don't recall many or most people saying that adding weight to the handle makes it more maneuverable, although I actually agree. I had thought there was still contention about that.

Good point, maybe I should have said that most people that try adding handle weight seem to find that it makes the racquet more maneuverable.

There's certainly disagreement though, and if I were to represent my impression of it in the most uncharitable light, I'd say it sounds like

But in all seriousness, at the end of the day, what people experience & their results are what matters. Not that we should never question our experiences, but surely a theoretical model that disagrees with the subjective experiences of many racquet customizers (and dare I say thousands of years of weapon customization?) should at least explain why all those people are wrong.

All I know is, for a standard length racquet, the most maneuverable balance point seems to be centered around Feds preferred at 9 pts or 31.5cm above the handle.

And I've never read but believe a tennis racquet has 3 points at which it wants to rotate linearly:

-the bottom of the buttcap(due to your wrist)
-the middle of your hand(mostly due to your forearm and that's the middle of where you're holding the thing)
-the balance point of the racquet itself

How it wants to rotate and how quickly is affected by the other measures like SW, Mgr/I, other pendulum type effects

Sometimes I've made racquets swing quicker by adding a tiny amount of weight at 12 o'clock, other times it's close to my hand or top of the handle, or bottom of the buttcap for that backward acceleration of the pivot point you might be confused about, you have to tune it yourself for each racquet

Oops the "confusion" was supposed to be purely rhetorical - edited!

Interesting that you've made racquets swing quicker from adding 12'o clock weight - similarly I've also felt that too low of a swingweight isn't enough to fully load the elasticity in my wrist & forearm, reducing my ability to generate speed. It's kind of like having high acceleration & handling, but a limited top speed. Wonder if that aligns with your experience!

 

[Khanh PV]

I have tried customization for nearly two years and I can sum up that its not this complex, just like this:
_ Old school swing (push forward, locked wrist, slow swingspeed), skillful S&V fast courters: swingweight not important, it prefers mainly even racket weight distribution (more at lower hoop and middle handle) and light enough. Some old guards here self-mistake that they need heavy rackets bla bla but I found out that they only need a racket simulating the feel wooden rackets, so not really heavy but enough weight in the middle is what they actually want. The result is maneuverability for net play and groundstrokes, little to none topspin added, dependence on fast balls and fast courts for speed and penetration to finish balls. Most suitable for older recreational players.
_ Drive topspin groundstrokes, more follow through, less ball slapping or windshield swing, moderately faster courters (Federer, Djokovic, Murray, Wawrinka...): Really need high swingweight for their ball redirecting style, different weights and headlight balances are varied based on their preferences, but usually lots of weight at hoop (Federer Wawrinka have 1 hbh or Federer likes all court playstyle: more weight at the bottom of handle, a bit at middle of racket for stability if needed; Djokovic or Murray prefer baseline: may require the same or higher swingweight, at the same time only need weight at bottom of handle though, eliminate as much weight at upper part of handle as possible). The result is deeper, heavier and more accurate topspin ball rallies or put-away volleys, most comfortable if the rallies have speed and moderate bounce to redirect balls, but not necessary because higher swingspeed is more accesible now to kill softer and shorter sit-ups or stupid moonballs. Definitely not for majority of recreational players.
_ Hybrid of the above two types (Sampras, Becker, Agassi...): high swingweight for aggressive S&V like Sampras, Becker) or also extremely much weight in the middle handle and lower hoop for stability and flatter swing style. The result is fast penetrating balls with topspin added, fastest and heaviest serves ever if having good swing (Sampras, Becker), but only suitable for fast balls and fast courts, not much for current tennis. Better for good recreational players with preference for solid flatter shots.
_ Crazy swing, almost slapping (Sinner, Kyrgios) or extreme windshield swing (Nadal, Alcaraz), a lot of clay courters, doubles specialists (Bryan brothers): Except older generation's players are affected by "high swingweight" old trend, almost all use no weight at 3&9 and as low swingweight as possible for higher swingspeed. Completely not need weight at handle because what to counterbalance if you dont have weight at tip , maybe just a little at bottom of handle if that is a racket like Nadal's one. If doubles, add much more for racket head maneuverability. The result is rallies with highest topspin potential: heavy shots, easier to choose angles.


The last setup is not for the most consistent tennis if courts are not slow enough, the opponents are higher in skill level and the player is not equipped with appropriate technique (see Nadal, Alcaraz...). However, I think it is now the best for current recreational players because the amateur level is lower, therefore has rally-friendlier tennis for everyone to play this customization style.

 

[Donmikan]

I have tried customization for nearly two years and I can sum up that its not this complex, just like this:
_ Old school swing (push forward, locked wrist, slow swingspeed), skillful S&V fast courters: swingweight not important, it prefers mainly even racket weight distribution (more at lower hoop and middle handle) and light enough. Some old guards here self-mistake that they need heavy rackets bla bla but I found out that they only need a racket simulating the feel wooden rackets, so not really heavy but enough weight in the middle is what they actually want. The result is maneuverability for net play and groundstrokes, little to none topspin added, dependence on fast balls and fast courts for speed and penetration to finish balls. Most suitable for older recreational players.
_ Drive topspin groundstrokes, more follow through, less ball slapping or windshield swing, moderately faster courters (Federer, Djokovic, Murray, Wawrinka...): Really need high swingweight for their ball redirecting style, different weights and headlight balances are varied based on their preferences, but usually lots of weight at hoop (Federer Wawrinka have 1 hbh or Federer likes all court playstyle: more weight at the bottom of handle, a bit at middle of racket for stability if needed; Djokovic or Murray prefer baseline: may require the same or higher swingweight, at the same time only need weight at bottom of handle though, eliminate as much weight at upper part of handle as possible). The result is deeper, heavier and more accurate topspin ball rallies or put-away volleys, most comfortable if the rallies have speed and moderate bounce to redirect balls, but not necessary because higher swingspeed is more accesible now to kill softer and shorter sit-ups or stupid moonballs. Definitely not for majority of recreational players.
_ Hybrid of the above two types (Sampras, Becker, Agassi...): high swingweight for aggressive S&V like Sampras, Becker) or also extremely much weight in the middle handle and lower hoop for stability and flatter swing style. The result is fast penetrating balls with topspin added, fastest and heaviest serves ever if having good swing (Sampras, Becker), but only suitable for fast balls and fast courts, not much for current tennis. Better for good recreational players with preference for solid flatter shots.
_ Crazy swing, almost slapping (Sinner, Kyrgios) or extreme windshield swing (Nadal, Alcaraz), a lot of clay courters, doubles specialists (Bryan brothers): Except older generation's players are affected by "high swingweight" old trend, almost all use no weight at 3&9 and as low swingweight as possible for higher swingspeed. Completely not need weight at handle because what to counterbalance if you dont have weight at tip , maybe just a little at bottom of handle if that is a racket like Nadal's one. If doubles, add much more for racket head maneuverability. The result is rallies with highest topspin potential: heavy shots, easier to choose angles.


The last setup is not for the most consistent tennis if courts are not slow enough, the opponents are higher in skill level and the player is not equipped with appropriate technique (see Nadal, Alcaraz...). However, I think it is now the best for current recreational players because the amateur level is lower, therefore has rally-friendlier tennis for everyone to play this customization style.

How are Nadal and Alcaraz in the same group, when their racket specs are completely different?

 

[Khanh PV]

How are Nadal and Alcaraz in the same group, when their racket specs are completely different?

I categorized based on their swingstyle and only in a relative way. I also explained Nadal case in separate sentences as you can see, but I am sure that my analysis here is logical, not in a stupid way without evidence

 

[timmyy]

What's your academic background?

 

[Khanh PV]

I work at a local center teaching tennis so I have tried many types of customization to see why superstars are playing with those rackets. Of course all of them have high skills but I have wondered why they do not perform all technique like Nadal cannot flick backhand, Djokovic cannot volley smooth like Federer or Federer has weak backhand. Finally, I found out that the difference lies in other factors such as physics, mentality and definitely their rackets. I spent time to replicate relatively their setups and the analysis above was through my watching them playing and my experience on court.

 

[K1Y]

I have tried customization for nearly two years and I can sum up that its not this complex, just like this:
_ Old school swing (push forward, locked wrist, slow swingspeed), skillful S&V fast courters: swingweight not important, it prefers mainly even racket weight distribution (more at lower hoop and middle handle) and light enough. Some old guards here self-mistake that they need heavy rackets bla bla but I found out that they only need a racket simulating the feel wooden rackets, so not really heavy but enough weight in the middle is what they actually want. The result is maneuverability for net play and groundstrokes, little to none topspin added, dependence on fast balls and fast courts for speed and penetration to finish balls. Most suitable for older recreational players.
_ Drive topspin groundstrokes, more follow through, less ball slapping or windshield swing, moderately faster courters (Federer, Djokovic, Murray, Wawrinka...): Really need high swingweight for their ball redirecting style, different weights and headlight balances are varied based on their preferences, but usually lots of weight at hoop (Federer Wawrinka have 1 hbh or Federer likes all court playstyle: more weight at the bottom of handle, a bit at middle of racket for stability if needed; Djokovic or Murray prefer baseline: may require the same or higher swingweight, at the same time only need weight at bottom of handle though, eliminate as much weight at upper part of handle as possible). The result is deeper, heavier and more accurate topspin ball rallies or put-away volleys, most comfortable if the rallies have speed and moderate bounce to redirect balls, but not necessary because higher swingspeed is more accesible now to kill softer and shorter sit-ups or stupid moonballs. Definitely not for majority of recreational players.
_ Hybrid of the above two types (Sampras, Becker, Agassi...): high swingweight for aggressive S&V like Sampras, Becker) or also extremely much weight in the middle handle and lower hoop for stability and flatter swing style. The result is fast penetrating balls with topspin added, fastest and heaviest serves ever if having good swing (Sampras, Becker), but only suitable for fast balls and fast courts, not much for current tennis. Better for good recreational players with preference for solid flatter shots.
_ Crazy swing, almost slapping (Sinner, Kyrgios) or extreme windshield swing (Nadal, Alcaraz), a lot of clay courters, doubles specialists (Bryan brothers): Except older generation's players are affected by "high swingweight" old trend, almost all use no weight at 3&9 and as low swingweight as possible for higher swingspeed. Completely not need weight at handle because what to counterbalance if you dont have weight at tip , maybe just a little at bottom of handle if that is a racket like Nadal's one. If doubles, add much more for racket head maneuverability. The result is rallies with highest topspin potential: heavy shots, easier to choose angles.


The last setup is not for the most consistent tennis if courts are not slow enough, the opponents are higher in skill level and the player is not equipped with appropriate technique (see Nadal, Alcaraz...). However, I think it is now the best for current recreational players because the amateur level is lower, therefore has rally-friendlier tennis for everyone to play this customization style.

what does this last customization style look like? You said low sw as possible, so no weight in the hoop.. So only maybe a bit of weight in the handle? Kyrgios plays highish static, low balance and low sw.

 

[Khanh PV]

what does this last customization style look like? You said low sw as possible, so no weight in the hoop.. So only maybe a bit of weight in the handle? Kyrgios plays highish static, low balance and low sw.

Maybe I mistook a little at Kyrgios case, please let him aside, because as I said I categorized them by their swingstyle then only gave a relative summary for their racket choice. If I categorize players by weight distribution, there will be more various cases.