# Hanging Off Mathematically Quantified!

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I couldn't find anything on the net to quantify the effects of hanging off on lean angle and corner speed so I sat down and derived the equation my self.

Effective lean angle = ArcTangent [Riders Mass/Mass of Bike and Rider X Shift Distance/CG of Bike and Rider + Tan of Original Lean Angle]

For a rider who's mass is 25% of the total bike and rider mass and a horizontal weight shift of 25% of the CG height, I came up with the following numbers:

Lean angle Effective Lean Angle. % Increase Speed

30 deg. 33 deg. 6%

45 deg. 47.5 deg. 4.4%

50 deg. 52.3 deg. 4.3%

55 deg. 57.0 deg 3.8%

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To me, this makes a lot more sense. First, the change in lean required sounds far more plausible than going from 52 to 39 degrees from hanging off. And that the effect of hanging off is reduced as lean angle increases is logical to me.

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I sent this from my iPhone and the tables were compressed for some reason, hope it's readable.

Speed in a corner varies with the square root of the tan of the lean angle.

Some observations I made from studying the equations:

At racing speeds and angles of lean, hanging off yields reductions in lean angle on average of about 2.5 to 3 degrees and conversely, increases in corner speed of just over 4%.

This is significant on a racetrack, an increase of 4% in each turn on a racetrack could make the difference between first place and midpack.

Approximately 96% of cornering speed comes from leaning the bike, so hanging off is not a substitute for leaning the bike over but gives you a small edge when the tires are at their traction limit.

Getting low on the bike when hanging off lowers bike/rider CG and makes the weight shift even more effective.

This is probably the only area of riding where a heavy rider has a technical advantage as his extra weight increases the mass ratio (ratio of rider weight to combined weight of bike and rider).

There is no point in hanging off unless you are already leaning the bike to it's traction limits, simply leaning in more will increase corner speed.

Modern Sportbike tires have coefficient of frictions of 1.2 which is good for 50 degree lean angle before the traction limit is reached. Race tires are beyond 1.3.

Tan[lean angle] = lateral g loading on the tire which is numerically equivalent to the friction coefficient.

Example: A sport touring tire has a maximum coefficient of friction of 1.0, what is maximum lean angle? Tangent 45 deg = 1.0 so 45 degrees maximum.

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I'm a bit confused with the statement that lowering the CoG makes hanging off even more effective. Do you mean that hanging off is more effective if CoG is low, or do you mean that hanging off is more effective if you climb down on the side of the bike to keep CoG as low as possible?

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To me, this makes a lot more sense. First, the change in lean required sounds far more plausible than going from 52 to 39 degrees from hanging off. And that the effect of hanging off is reduced as lean angle increases is logical to me.

Hi Eirik, that picture of the 39 degrees was driving me crazy and motivated me to set up the equilibrium equations and solve them. It took me several days because I wanted to make sure I wasn't making any false assumptions, an easy trap to fall into.

It should have been 49 degrees not 39 according to my calculations.

Greatest effect was at 0 degrees, the rider hanging off gave 3.6 equivalent lean angle. I tried this on the freeway the other night and it was amazingly accurate.

A newbie friend of mine was trying to convince me that hanging off was more effective than

leaning. We took a fast sweeper together, my speed was 126 mph indicated, pure lean while he managed 90 mph hanging off the side. He is afraid to lean his bike over and thinks that hanging off will make him fast. I tried to point out that Ben Spies drags his elbows mainly because he he is leaning to about 55 degrees on racing tires, then hanging off for another 2 to 2.5 degrees equivalent. Without the big lean, he'd be in the beginner class....lol!

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I'm a bit confused with the statement that lowering the CoG makes hanging off even more effective. Do you mean that hanging off is more effective if CoG is low, or do you mean that hanging off is more effective if you climb down on the side of the bike to keep CoG as low as possible?

Either way the net effect is the same, a lower overall CG and less lean angle for a given speed.

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Thanks again for explaining and the speed examples

The reason I asked about lowering the CoG is due to what we've said earlier, that raising the centre of gravity in itself reduce the need for lean due to tyres having width.

However, the examples you have given regarding the amount of bike lean saved by hanging off support my stipulation that body movement in itself (which creates forces) may be the biggest benefit from hanging off. And your 126/90 mph example reminds me of an article way back in Performance Bikes when tester Trev fell off at 55 mph due to running out of cornering clearance despite hanging way off, while John went around at 57 mph, barely skimming a peg while not hanging off dramatically. Not exactly the same scenario, but it shows that skill is more important than hanging off

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Yes, that's true and I've settled down now that I've proven to myself that the advantage of hanging off, while critical in roadracing where hundredths of a second may separate the racers, is not that big a deal out on the streets. I am going to start incorporating weight shift into my riding style but plan to go easy and make the change gradually.

You Are correct, the tire width effect rewards a high CG while weight shifting rewards a lower CG. There is a crossover point somewhere.

All this stuff is such a compromise anyway, raise the CG and the bike flicks into a turn more rapidly but then experiences excessive pitch forces under hard braking so there is an equilibrium point there like everywhere else.

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You are so correct that it's all a compromise, which is why there are as many styles as there are racers, yet they all lap more or less at the same pace. So if you gain in one era, you usually have to give up something in another. There's not a lot of win-win out there, you just have to find wins that are bigger than the losses for an overall gain.

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erm, at the expense of sounding stupid...

I have a few questions:

do bigger/heavier riders have a higher overall COG?? (assuming same height)

Heavier = faster going in when hanging off ?

The last time i did physics was junior high...sorry.

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Does the heavier rider have a fat bum and thick legs, or skinny legs and huge chest and shoulders? Too many variables, I think. Heavier riders will make up a bigger percentage of the bike/rider combination, though. And compress the suspension more (unless one upgrades with stiffer springs). This will lower CoG, but if that's more or less than the change brought by the rider himself is beyond my knowledge.

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Does the heavier rider have a fat bum and thick legs, or skinny legs and huge chest and shoulders? Too many variables, I think. Heavier riders will make up a bigger percentage of the bike/rider combination, though. And compress the suspension more (unless one upgrades with stiffer springs). This will lower CoG, but if that's more or less than the change brought by the rider himself is beyond my knowledge.

Im guessing the same , too many variables.

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I agree with Eirik, is the heavier rider short and rotund or tall and lanky? I guess there are some simple rules that can be applied for anyone though, hang as far off as you can and get as low as possible when you do to maximize the effects. Again as Eirik pointed out you may reach a compromise in bike handling or stability if you hang off to far.

One cool thing I learned from the equations, it doesn't matter whether you have a small, light bike or a heavier sportbike, the dynamics of equilibrium in the turn dictate that the only factor determining how far you can lean is the tires coefficient of friction with the pavement. If you ride a heavy and a light weight sportbike with the same tires, both will lean to the same maximum limit. Lighter bike of course, will flick better and handle transitions better like through a chicane.

This will give me new confidence when I take my ZX-14 out and wring it's neck, I was under the mistaken impression that big bikes can't carve, not true!

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I drew the equation out on a chalk board and took a picture for all of you math geeks. The units of Mr amd Mb+r don't matter as long as you are consistent as they are pure ratios. So use either KG or pounds. Same thing with ^CG and CG, use inches or mm or whatever units you like as they are once again ratios and the units will cancel out.

The effective lean angle is the combined effects of the original lean angle plus the effect of hanging off.

The effect is not huge and decreases slowly with increased angles of lean. For Mr/Mr+b ratios of .25 and ^Cg/CG ratios of .25 which seem reasonable, the effective lean is about 3.6 degrees if the bike is held vertical (0 degrees original lean angle) and by the time you reach 55 degrees lean, you gain only about 2 degrees i.e equivalent of 57 degrees.

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Here's a video that shows the effect at moderate lean. There will now be many telling how the rider doesn't do things right, but that's a nother discussion

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I went through one of my favorite sweepers the other day whilst hanging all of the way off and lost over 10 mph. I credit this to being scared to death of falling off the side of the bike.

Of course that would go away if you do it enough and become comfortable with it but for my road and street riding, I 've decided that the benefits of a full hang off are not warranted.

What I am doing is using a modified, less radical weight shift, moving my butt over into the turn a few inches and moving my head down and ouside towards the mirror. Over time I'll gradually become accustomed to this and can then move further down. I've been riding too many years with one style to make radical changes in my positioning overnight. My orthodontist once told me he could move a tooth from one side of my mouth to the other but it would take a long time.

As the equations don't lie, merely hanging off the side of a bike will not automaticaly move you up from the B group to the A group, there's a lot more to riding than that apparently..

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The effect is not huge and decreases slowly with increased angles of lean.

Centurion,

Without a free body schematic is hard for me to understand your equation.

It seems to me that any angle deviation achieved by hanging off, has no reason to decrease with increased angles of lean, as long as the rider's body position remains the same respect to the bike.

I can't see, either in your equation or in my mind, how the height of the CG location has any effect in the effective lean angle for a similar distance that the rider's CG is shifted.

Would you mind to explain these a little further?

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Think of a pole that's balanced on end. If the pole is vertical and you hang a weight to one side of it, the pole need to shift to the opposite side or it will fall over. Now take a horizontal pole. Whether the weight sit on top or hang from below, it will not matter anything; the load is the same as long as it's the same distance from the end. It's the same when hanging off a bike; the closer you get to horizontal, the less the effect.

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Correct, close to horizontal any additional increment of lean tends to move the riders CG more downward than laterally.

Conversely at vertical, no lean, angular displacement moves rider CG more laterally than downward.

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The effect is not huge and decreases slowly with increased angles of lean.

Centurion,

Without a free body schematic is hard for me to understand your equation.

It seems to me that any angle deviation achieved by hanging off, has no reason to decrease with increased angles of lean, as long as the rider's body position remains the same respect to the bike.

I can't see, either in your equation or in my mind, how the height of the CG location has any effect in the effective lean angle for a similar distance that the rider's CG is shifted.

Would you mind to explain these a little further?

Lnewqban, please send me an email at jweber@superiorairparts.com and I will be happy to draw out the free body diagram and attach it to an email for you. The problem I'm running into is, there doesn't seem to be anyone else on the internet who has done the math on this let alone backed it up with hard data. In fact, I can't find any equations for this except for mine. Generally in this case, it is customary to have a peer review of the new theory or equation to determine it's validity.

To prove or disprove my equation would require a motorcycle equipped with onboard telemetry, lean angle sensor and GPS speed data.

Several simplifying assumptions were made when setting up the equations. First, the effects of tire width on effective lean angle were ignored since this would presumably effect lean angle equally whether or not you were hanging off the machine. I need to further validate this assumption.

The second assumption and one I also have to examine further, is that the weight shift was in the horizontal plane. However any error introduced by this effect would tend to be, if anythng , overly optimistic not conservative.

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One last assumption, the weight shift was purely lateral whereas it is possible to move over and downward, thus lowering the combined CG slightly. My intuition is telling me that if I add in these extra variables, the equations are going to become much more difficult to derive yet have little effect on the actual outcome.

I've posted my equations on several other forums and it has created a shitstorm of controversy. 500 years ago everyone knew the world was flat, if you believed otherwise you were a lunatic or a heretic. If you transported someone from the 15th century to modern day and told them the earth was actually round, they would not believe you because it would require a massive paradigm shift in the way they perceived their world.

This is the problem I'm running into with the whole weight shift thing. Riders are so convinced that weight shift is the key to fast cornering, that when I show them that it accounts only for about 4% of their cornering speed, the rest being attributable to lean angle, they don't or won't believe it.

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Mike Hailwood would have agreed

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Good point, I dragged up some old videos from the Kevin Scwantz, Mick Doohan era of the 90's. They didn't hang off the old 500cc two stroke GP bikes but the speeds don't look that much slower than the modern era. I'm sure Stoner is running faster lap times on the same tracks but I would be surprised if it were much more than a few seconds per lap. Maybe I'm wrong though, only way to find out would be official historical lap times, same tracks but different eras. Traction control and much better tire and suspension technology is going to play a huge role.

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In 1990, Gardner did a 1:33.9 race lap around PI on the NSR500. That is roughly 3 seconds off the fastest race lap ever, set in 2008 by Hayden IIRC. If you consider the billions spent on developing tyres, suspension, chassis, electronics, brakes plus the extra power and driveability of the modern 4-strokes, it's a pretty slim improvement.

Check this out - pretty spectacular even if I do not understand a word

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Another interesting video. Schwantz at times seems to lean enough to wedge his knee between the road and fairing.