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:

If you disagree with how flop is shown here, you can blame the folks at BQ

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.

What about low-trail geometry?

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?

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.

You have to settle for CAD diagrams as I’m too lazy to derive formulas

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?

HTAOffsetFlopMax Frame Sink10 Deg Frame Sink20 Deg Frame Sink90 Deg Frame SinkAxle Sink
7086.739.641.3000.14230.5187-9.40020.26
7369.558.391.3060.12430.4585-3.52414.68
7652.187.041.3130.10470.3915-2.6439.981
trail = 30mm, radius = 336mm, HTA in degrees, all others in mm

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.

HTAAxle sink10 Deg Axle Sink20 Deg Axle Sink
7020.260.59262.311
7314.680.43291.687
769.9810.29631.154
Same as above, independent of trail

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.

HTAOffsetFlopMax Frame Sink10 Deg Frame Sink20 Deg Frame Sink90 Deg Frame Sink
7062.3018.04.4300.26921.024-1.045
7344.6815.74.4720.23460.89791.618
7626.9513.14.5110.19740.76033.461
trail = 56mm, radius = 336mm, HTA in degrees, all others in mm

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

HTAOffsetFlopMax Frame Sink10 Deg Frame Sink20 Deg Frame Sink90 Deg Frame Sink
7049.1422.56.8510.33751.2953.456
7331.3019.66.9280.29401.1345.530
7613.3716.47.0040.24720.95876.746
trail = 70mm, radius = 336mm, HTA in degrees, all others in mm

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.

HTA/Trail10 Bag10 Axle10 Frame5 Bag5 Axle5 Frame
73/560.8700.4070.1710.2190.1030.043
73/700.9310.4070.2230.2340.1030.056
73/300.7600.4070.0780.1910.1030.020
73/350.7800.4070.0960.1960.1030.024
73/400.8010.4070.1130.2010.1030.029
72/300.7990.4550.0790.2010.1150.020
72/350.8210.4550.0970.2060.1150.025
72/400.8420.4550.1150.2120.1150.029
71/300.8380.5050.0800.2110.1270.021
71/350.8600.5050.0990.2160.1270.025
71/400.8830.5050.1170.2220.1270.030
71/560.9560.5050.1790.2400.1270.045
71/701.0220.5050.2350.2570.1270.059
Sink in mm at 10 degrees for various geometries, assumptions have been modified

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.

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