What is "Wheel Flop?"

“wheel flop factor,” the distance that the center of the front wheel axle is lowered when the handlebars are rotated from the straight ahead position to a position 90 degrees away from straight ahead

by whoever erroneously put this on wikipedia which has been copied and pasted numerous times

I have yet to see how this formula was derived, but “wheel flop” is supposedly a length usually calculated using the following formula:

ƒ = b sin θ cos θ

where ƒ is “wheel flop,” b is trail (because t is always time I guess) and θ is head tube angle.

Graphically, it can be represented in the following way, where “wheel flop” is the height of a point from the ground, which is the intersection of the steering axis, and a perpendicular line that passes through the tire contact point:

Note that this calculated measurement doesn’t actually correspond to anything at all. It certainly does not correspond with the change in front end when the bars are turned 90 degrees. If you’re thinking that a tire isn’t a 2D disc, you’re right, but factoring in the tire as a torus just reduces the radius by half the tire width as radius will be to the tire center, and the tire center nearest the ground is the radius of the tire away from the ground whether or not it’s 90 degrees because the cross section is a circle.

For the standard road geometry, using a naive with the wheel as a disc, this is ~1.5mm (actually ~1.0 when the tire is modeled as a torus), off by an order of magnitude from the ~15.6mm given by the formula. When turned 90 degrees, two things happen. First the tire contact point is not a vertical line from the axle, it’s a line that touches the ground at the head tube angle (assume the effect on HTA from height differences is negligible, because it is). The second is that the fork effectively loses offset because the offset is orthogonal to the plane which lowers the axle. These two effects more or less cancel each other out.

If you want to go by literally by the definition given on wikipedia, then yes, the axle drops ~14.6mm (not a typo, it’s not ~15.6mm!). How exactly this is relevant to normal handling when the fork crown only drops by ~1.5mm is a mystery. The change in axle height does make a difference if you have front panniers where the center of mass is centered on the axle I suppose.

When you turn the bars 90 degrees, the fork crown actually raises by ~5.8mm. Is this the secret to low-trail front luggage handling? Well no, not really. The change in axle height remains exactly the same, meaning this change really doesn’t do much at all when the mass is fork mounted. The distance from the ground to the axle when straight ahead is a function of wheel radius (it is the wheel radius). The distance when turned 90 degrees is still a function of radius and HTA, as before. The axle drops the same ~14.6mm, and clearly isn’t scaling with trail. Actually it drops move because the HTA is slackened more. So the effect this has on fork mounted luggage is dubious.

Wheel flop is directly proportional to “f” (f = b sin a cos a).

BQ

Now lets so something rather silly and set HTA to 45 degrees. Silly for any kind of normal bike, but not entirely unheard of on choppers.

What’s going on here? I have no idea because we still haven’t been able to pin down exactly what wheel flop is. Trail, the way it’s normally calculated, is 0, so wheel flop is 0 (anything times 0 is… 0), but axle drop is (I’m not actually going to correct for HTA because of how silly this is). Axle drop is a ridiculous ~98mm (it’s not the height shown between axles in the picture, it’s the radius minus the height at 90 degrees), although approximations are failing here. The fork crown rises ~70mm and again, the effects of moving the frame really shouldn’t be ignored, but they are.

Clearly it’s not the same as 0 trail, 0 wheel flop at 73 degrees. So does “wheel flop” succeed at describing the amount the front end is raised or lowered by turning the bars 90 degrees? It doesn’t look like it. Does it succeed at describing the amount the front axle is raised or lowered by turning the bars 90 degrees? Doesn’t seem like it. What is wheel flop? Don’t ask me. Roughly proportional in the range of normalish angles and trail figures. Maybe. Does it even transfer between different combinations of HTA, wheel size and offset? Who knows. Directly proportional? Judge for yourself.

You might be thinking that I must be wrong, the front end dips when the bars are turned. Yes, that’s actually true. 90 degrees isn’t the lowest point, and 90 degrees is pointless since it’s impossible to ride with the bars turned 90 degrees. “Frame sink” is a real thing, and seems to be largely a function of offset, as for a given angle, there’s a greater rising effect the longer the offset. Add in frame tilt if you want to really complicate the problem.

The low-trail geometry’s lowest point is only ~1.3mm lower, compared to the normal road geometry at ~4.4mm. Yet the bars are also not turned the same. These numbers aren’t very big compared to “wheel flop” though, and “frame sink” is about gravity acting on the fork crown, not mass fixed to the forks. As might be expected, for a given rake, a steeper angle seems to reduce it, a shallower angle seems to increase it (for 70mm offset, 71 degrees is ~2.5mm and 75 degrees is ~0.5mm, 45mm offset, ~6.5mm and ~2.7mm respectively).

Turning the handlebar 10 degrees, frame sink is a small fraction of a millimeter, ~0.1mm for the low-trail geometry, ~0.2mm for the standard road geometry. At 20 degrees it’s ~0.5mm and ~0.9mm respectively. Changing HTA also has the expected effect.

[I]magine the bike’s top tube being held in a clamp so it stays level. As you tum the handle-bars, the front wheel will lift off the ground.

BQ

I tried my best to confirm results experimentally as described, except empirically instead of imagining, but it’s difficult without a perfectly flat reference and a very stiff workstand. It was difficult to try to take any measurement with any semblance of precision, but the results at the very least didn’t seem to contradict expected values. It also seemed to confirm that 90 degrees is in fact not the lowest point.

Let’s try a different experiment though. We know that as trail is reduced so is wheel flop and frame sink. What if trail is kept constant at 30mm?

Recall that axle sink is the amount the center of the front axle sinks when the bar is turned 90 degrees. It would seem like a decent proxy for front loading because lowriders would have their center of mass right about there. But it’s almost purely a function of HTA and wheel radius. Offset and trail don’t really factor in. The similar effect from a handlebar bag should be a function of HTA and distance in front of the steering axis when talking purely of the potential energy from the mass of the bag and height.

Flop doesn’t seem to correlate with 90 degree frame sink, which is a useless metric, nor does it correlate well with max frame sink, which is pretty similar for all the example geometries, but varies in terms of how much the bar is turned. It does seem to correlate well with smaller values of frame sink more likely to be experienced when riding, but these values are very small, in the range of a tenth to a half of a millimeter.

In comparison axle sink is relatively very large. One would expect a pronounced effect from front loading if center of mass as at the axle (although closer to frame sink levels if near the steering axis). Perhaps it is partially negated by the mass damping it adds.

Next is to examine if this correlation also fits at for standard road geometry.

Correlation is also good, within the set, but the ratio starts to drift. Next is high-trail.

For the example geometries, 70-76 HTA, 30-70mm trail, and radius of 336mm, “wheel flop” seems to be close to proportional to the low angle frame sink. Maybe it’s a decent approximation, but the actual frame sink values are very very low. Even the high flop designs are only ~1mm of drop for the bars turned 20 degrees. As far as I can tell, any load forward of the and attached to the steering axis adds significant flop component not affected by trail or rake, just head tube angle and relative position to the steering axis (handlebar bag bikes should have 72+ HTA, lowriders should be rearward on bikes with shallow HTA).

Since these values are low and require nuance, it was decided it would be better to create a more complete model to consider how these factors combine, and also consider the effects of handlebar bags, focusing on minor differences between realistic low-trail front geometries.

First thing to note is that the 10 degree values are ~4 times the 5 degree values. This is not a big surprise since (1-cos(10°))/(1-cos(5°)) ~= 4. There’s no major surprising interactions, at least at these two values, and the effects all scale in a trigonometric relation to the steering angle.

“Axle sink” the the change in height of the center of the front axle (10 degree bar rotation), as noted before, is almost purely dependent on HTA and wheel radius. Steeper angles result in less. This is somewhat relevant for lowriders, as they are often mounted with center of mass near the front axle so the wheel and dropout can bear as much weight as possible. If we are to believe that sink has a significant effect in handling, then steeper angles result in less sink. Sink can be reduced to levels closer to frame sink by moving the load closer to the steering axis. While frame sink is lower on low-trail designs, they also have high offsets requiring the load be moved more to be put at the steering axis.

“Frame sink” measured here as the change at the top of the headtube (10 degree bar rotation), is what best corresponds to “wheel flop” at low steering angles actually used in steering a bike. It is extremely related to trail, more than “wheel flop” would imply. There’s only about a 5% difference between the two 56mm geometries compared to a ~10% difference in “wheel flop” showing the limited predictive power of “wheel flop” across a range of head tube angles when all things are considered. It was closer however, when the wheel was approximated as a disc, rather than approximated as a torus. It is worth noting that “frame sink” for the road geometries are very similar when trail is corrected, and those bikes tend not to carry additional bar or front pannier loads. In general, it is very low compared to the other ones, but also not stabilized by inertia.

“Handlebar bag sink” is what it sounds like, it’s the change in height of the handlebar bag when the bars are turned (10 degree bar rotation). I think it is what most people try to describe when they talk about “wheel flop” for low-trail geometry. It is a very poor predictor of handlebar sink across head tube angles. There is a relationship of ~1 degree of HTA being equivalent to ~10mm of trail. For a given HTA, less “wheel flop” results in somewhat less handlebar bag sink, but not nearly as much as implied by comparing wheel flop numbers. Major proponents of low-trail geometry (BQ) tend to almost always test bikes in sizes when 73 degree HTA and nothing else is ever used.

It does seem to be affected by frame sink since it varies with frame sink for any given HTA, but there’s an even larger component of sink due to rotating around the steering axis when the bars are tuned, much like the axle does, which appears dominated by HTA. However, as noted before, the added mass also has inertia. Neither frame sink nor axle sink alone is a good predictor of handlebar bag sink, but combined, they appear to get close.

Having established three different kinds of sink, some very crude experiments were performed. First, it was established that frame sink occurs. Two piece of wax paper were put under the front tire of a particularly floppy bike (~70mm trail, ~23mm “wheel flop”), the bars were turned slightly off center, and weight was applied by pressing down on the top tube near the headset. Bag sink was also observed by filling a handlebar bag with random objects until it weighed precisely not empty but not particularly heavy. Repeated trials showed there was little else of note despite the extreme precision and repeatability of the testing methodology, except that predicted maximum sink steering angle seemed to line up pretty well with the actual results.

Next the same tests were performed without the wax paper. It was noted that the slightly filled handlebar bag easily overcame the friction of the tire causing the wheel to flop. However, frame sink (what “wheel flop” correlates to) could not overcome the friction of the tire, even with significant weight on the top tube near the headset. Increased weight merely increased force on the tire, increasing the resistive frictional force. However, it was observed that when the front of the bike was slightly unweighted, but the tire was still touching the ground, the fork flopped easily even without a handlebar bag because of all of the mass forward of the steering axis (bars, brakes, wheel, rack, etc.). A similar behavior is frequently observed when holding a bike in a workstand.

These experiments were entirely unenlightening except that they show there is a component that causes the wheel to flop that has nothing to with calculated “wheel-flop” or frame sink which can overwhelm frame sink under the right conditions. The relative contributions on handling while actually riding are unknown.

It is unclear exactly what “wheel flop” actually is or measures. It appears to be a rough proxy for “frame sink” but does not translate very well between sizes. HTA angle would appear to have a large effect on sink associated front loading. If sink is in fact what is being minimized, then it would seem that maintaining a steeper HTA is nearly as important minimizing bag sink. However, the same fact that front loads are attached to the fork that causes a large portion of this sink may also be partially damped by the added inertia, unlike pure frame sink which acts through the headset. This would also imply weight applied through the handlebar to the forks may not be the same as weight applied through the headset to the forks. It may also be some other mechanism is at work in front loading low-trail geometry, other than the oft given explanation of sink or flop. The modeling here only looked at the bike when vertical, since this is the explanation frequently given for wheel flop.

Different mechanisms are at work when the bike is actually leaned. This can be observed when walking a bike holding only the saddle to steer, and has been observed by those studying the stability of bicycles. A bike may be steered by the saddle to go more or less in a straight line at walking speeds. According to the “wheel flop” theory, it is because there is less weight on the front end, allowing the trail to still overcome “wheel flop” at low speeds. It may be worth experimenting trying to isolate “bag sink” from “frame sink” when walking the bike such by loading the frame rather than the fork and seeing how it affects straight line stability at walking speeds.

A lot of people will be tempted to say that even if wheel flop doesn’t directly correlate to anything, even if the explanations for how wheel flop work don’t actually make sense, it’s still a useful approximation of something. However this isn’t really right. If it only works for comparing geometries with 73 HTA, it doesn’t tell us anything the trail figure doesn’t. Low flop will just mean the same thing as low trail. If wheel flop doesn’t work between different HTA and HTA is held constant, wheel flop is basically just trail times some constant value. The only other thing flop tells us is for a constant trail figure, a steeper HTA has less lowering (which means low flop bikes should have 74-76 HTA if possible given other constraints). The usefulness of comparing flop values when both HTA and trail are variable is dubious at best. Flop gives the illusion of having some other value to optimize to counter the narrative of the optimal mid trail values. When those pushing the wheel flop narrative all ride medium-large bikes with 73 HTA, all that is being said is that for their purposes, they prefer a low (not generally accepted as optimal) trail geometry and using a narrative about the frame lowering and an indecipherable arcane formula to imply flop is some non-trail value to be optimized (despite just being a rescaled trail given a constant HTA) to obfuscate what they really mean. That is mid trail isn’t optimal for their purposes, but they don’t want to fight the narrative that mid trail is optimal head on. Low trail 73 HTA geometry should just be called what it is. High trail isn’t better than mid trail because trail isn’t something to be maximized. There’s no need for sophistry saying 0.28*trail needs to be minimized. Mid trail 73 HTA was arrived at through trial and error as a goldilocks value. Low trail 73 HTA is just a different goldilocks value for a different set of parameters.

So what exactly is wheel flop? I’m still not sure. So how does low-trail geometry work? Voodoo magic.

Postscript:

I know some people who read this end up thinking frame sink is the real wheel flop (not the point of this article) or want more of an explanation about how low-trail works, not why “wheel flop” is wrong and doesn’t make sense. Bicycles seem to balance by getting the wheels under mass. This is true when rolling along, this is true when trackstanding, and this is true when turning when you consider centrifugal force not just gravity. Bikes achieve self-stability by turning to bring the wheel under this force vector, which means turning the wheel in the direction the bike is leaning.

Front loading can cause flop when the bike is leaned, this has nothing to do with wheel flop or anything, it just seems to be mostly related to the fact that there is weight (including handlebars, wheel, etc) forward of the steering axis. Weight forward of the steering axis encourages turn-in. Weight behind the axis counteracts turn-in, likely leading to the instability described in BQ(?) when front panniers were mounted on the steering axis (the actual center of mass was likely lower/behind the sloping steering axis because heavy things sit on the bottom of the bag, not in the middle). This would also imply that a handlebar bag could be canceled out by pannier mass behind the steering axis (something not supported by wheel flop/frame sink theory), but this remains untested and my experience says lots of weight on the front can lead fairly heavy steering, stable or not.

Trail acts like lever that causes the ground to push the wheel behind the steering axis. With 0 trail, this doesn’t happen. Both of these have the effect that more relative lean to the force vector (vertical gravity when not turning) the more turn-in, and this is probably related to the idea of weight-shift responsiveness. Note that these phenomenon have to do with lean, it’s not just gravity working to destabilize steering while caster effect stabilizes steering on a vertical bicycle as per the wheel flop explanation. These are not identical effects so even when you compensate for one with the other, handing will likely feel somewhat different.

Low-trail/front-loading can largely be explained without wheel flop. Wheel flop just gives an alternative number to trail to make “more better” since people already believe there exists an optimal trail figure. This isn’t to say that trail and front loads are identical and don’t differ in subtle ways, the way they respond, or the ranges they work, but they can be substituted to some extent. (There’s also gyroscopic effects, which cause turn in due to procession). When talking about handling characteristics, human self-stabilization also plays a big role because bipedal walking is inherently unstable. Humans can balance on stilts, which are just plain unstable, but it has to be learned because stilts respond differently than legs. Riding a mid/high-trail road bike after captaining a low-trail tandem (which has the added complication of a long wheelbase affecting the amount of turning to bring the wheels under center of mass), feels weird and makes it seem unstable. That doesn’t mean it actually is unstable. It’s also worth noting that bars/stems also have an impact on handling, and humans can tolerate a decent range of handling characteristics, just like they can tolerate different length stilts (or tall bikes). Hoods, tops and drops all have different weight distribution and reach and drop and such, but they’re all controllable (as an aside, aversion to short stems has to do with drop bar ends being inline with the steering axis, having lots of weight on them, and being narrow). We are mostly concerned if typical conditions are very easy to control after learning, and extreme conditions are controllable enough.

Unfortunately, this doesn’t prescribe any sort of optimal trail figure other than what has been empirically determined to produce good handling, or what is tried and true. Then again, neither does wheel flop. As far as I can tell wheel flop has simply been used to justify existing geometries, not really create novel ones at this point in time. Saying wheel flop doesn’t make sense is not very helpful. Neither is saying stick with traditional low/mid/high-trail geometries. The only possibly interesting point is that weight behind the steering axis (panniers) might be able to counteract weight in front of the steering axis (handlebar bag) which means adjustable pannier rails rather than just hooks might be useful for tuning handling. This is probably the main point where plain old turn-in theory contradicts wheel-flop-justifying-French-low-trail theory and maybe merits more research.

Postscript 2:

There seems to be a new article marketing a book describing how bicycles “really work.” There’s still no real explanation about how wheel flop works, and I’m not about to spend money to see if this book actually explains it, and I’m not in the business of spending money just to debunk things. The bulk of this article was written to try and figure out what wheel flop actually was because of inadequate and incorrect descriptions of it. It was not meant to debunk from the outset with the agenda of selling high trail bikes. I think low trail bikes are a thing, but highly skeptical that wheel flop is the reason for it or that it offers any real predictive power. The article makes little sense to me, but it seems like it may be trying to target this article because it says:

There is some confusion about what wheel flop is: Some people believe it’s when the fork turns while the bike is stopped, because the weight of the handlebars rotates around the inclined steerer axis. That’s a different factor that doesn’t have a catchy name yet.

and I said:

“Handlebar bag sink” is what it sounds like, it’s the change in height of the handlebar bag when the bars are turned (10 degree bar rotation). I think it is what most people try to describe when they talk about “wheel flop” for low-trail geometry. It is a very poor predictor of handlebar sink across head tube angles.

As far as I can tell, few people think “wheel flop” is when the weight of the handlebars rotate around the inclined steerer axis, and the title of this article posits confusion about what wheel flop is. I did not mean that I think that’s what wheel flop is. What I meant when I said that is that when people describe the phenomenon of adding weight to the front causing the handlebars wanting to turn, they ascribed it to “wheel flop” (frame sink explanation). This was not a particularly common opinion at the time this article was originally written as most parroted the frame sink or axle sink (which was said to be equal to frame sink) explanation.

Stability means many things to many people, and the article referenced seeks to debunk intuitive notions without seeming to tackle actual phenomenon, or examine the problem in a logical way. It brings up self centering of steering. A perfectly self centered (locked) steering will go straight, but not be stable (upright). It will fall over. A bicycle needs to self-steer and be self-correcting to remain upright on its own. The caster effect is real, but it causes the bike to steer into the turn, not self center, when there is a lateral force, such as when the bike leans due to weight shift (something low trail is purported to be insensitive to). Centrifugal force makes it so the apparent forces on the wheel are different when actually turning. This is not a wheel flop (formula) or vertical frame sink phenomenon. This will happen on a “bike” with a vertical headtube (0 wheel flop from formula, 0 vertical frame sink). When it comes to not falling to the outside of a turn, initiating effective lean is extremely important for being able to make a tight turn, as lean (or at least offsetting CG from vertical) is what allows the bicycle to not fall over when there are centrifugal forces (or not have to go straight to not fall over). Rider steering inputs can easily overcome “power steering” effects.

There are some other issues in the article, but ultimately, despite acknowledging that bicycle geometry is complex, it does not seem to examine bicycle geometry in a complex way and doubles down on the frame sink explanation (this article has deconstructed the bike lowering argument into several independent measures, which are independently influenced by different sets of bicycle geometry measurements) for the wheel flop formula without further explanation, when the “power steering” can actually be explained by other factors, and seems it may actually be more strongly influenced by other factors. It simply does not follow that just because we have a “wheel flop” formula that doesn’t seem to directly correspond to any physical characteristic of a bike, that it is a dominant factor in bicycle handling (IIRC Foale did the math and says this is a relatively small factor in motorcycle handling, which use geometries with higher flop and are far heavier). It follows the naive approach that wanting to go straight is the same as stability (I have been trying to develop a glossary/vocabulary for handling characteristics that don’t just lump things into stable/unstable buckets, but it’s a daunting task). It does not acknowledge the role of self-steering in stability. Worst of all, they claim they don’t deal with dry theory but want you to buy an entire book that claims to explain how bicycles really work.

Postscript 3:

The above two geometries, 73 degree HTA with 35mm trail, and 13 degree HTA also with 35mm trail, both have wheel flop factors of 10mm. Decide for yourself if wheel flop factor is silly.

I previously said I would not buy volumes of BQ just to disagree with the contents, but I did, to my dismay. For all the coy and vague internet references to BQ articles having the answers about wheel flop, if only you buy or track down a copy, the articles were for the most part, not enlightening, and essentially have little to no value in terms of frame design and frame building. Admittedly, I do like the historical technical documentation aspect of BQ.

You really don’t need to consider wheel flop factor for road/rando frame building. Stick with a reasonable head tube angle then decide on your trail figure for your desired handling profile (this is not debunking low trail geometry, low trail geometry is in fact a real thing). The article with measurements and observed characteristics is vaguely useful for this, but this has nothing to do with the technical wheel flop factor, and no useful conclusions can be drawn about wheel flop factor from them. Unfortunately the observations are vague, not standardized and highly subjective, without even meaningful relative reference points. I do like the fact that they contain geometry measurements though, so thumbs up for that.

I think I’ve tracked down where the silliness of wheel flop factor came from, and I believe it came from a very silly spherical-cow type assumption. Unfortunately I need to buy a book to check it, but I again, don’t want to pay for something just to disagree with it.

I’ve already done more than my due diligence investigating wheel flop factor. This started as an investigation into wheel flop because I didn’t truly understand it (I was curious how inclines would impact wheel flop factor, but this would involved first understanding wheel flop factor on flats) and no one on the internet seemed to understand it beyond parroting other people and stating it as fact. There were assertions that it makes sense just because it does, that there was secret knowledge hidden in arcane texts that could not and would not be freely shared, and someone smarter than me or you devised wheel flop factor so it should not be questioned. I found that it really doesn’t hold up to scrutiny. Maybe the guy who came up with wheel flop forgot more than I’ll ever know (I always hated this phrase), but I’m pretty sure he forgot some pretty important stuff when devising his model. I’ll take someone who knows a thing, or at least knows he doesn’t know a thing, over someone who forgot everything and knows nothing.