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Wheel flop calculation


Why does it matter?

How good a bike rides is largely driven by its steering characteristics, by the feeling you have while riding a straight line or around bends.

So, what is wheel flop then?

I came across this term on Peter Verdone’s website where you can find profound information about frame building and bike design. I understand wheel flop as a geometric dimension that influences the steering characteristics of a bike. When you turn the handlebar, the front of the bike will lower. I use the lowering distance in mm when the steering is turned by 90°.
In fact, several factors affect steering characteristics, these are the most important in my experience:

1. Front wheel trail
Any force on the contact patch which is parallel to the axle will induce a torque and affect the steering. This can be stabilizing factor, when centrifugal or inertia forces tend to straighten the steering angle. In this case the effect is heavily dependent on velocity (proportional to v²). Sideways forces also occur when the bike is leaned and the force vector on the front wheel is not parallel to the plane of the wheel any more. This can lead both to under- and oversteering.
Trail also straightens the steering angle under braking forces.

2. Gyroscopic forces on front wheel
They create a stabilizing torque counteracting any tilt movement perpendicular to rotation axis, e.g. steering input or leaning of bike. The effect is also proportional to v², and proportional to the moment of inertia of the front wheel. Bigger, heavier front wheels generate higher forces. Almost no stabilizing effect at low speeds.

3. Load on front wheel
High load increases steering effort and can feel sluggish.

4. Tire friction
Friction on the ground impedes turning the handlebar at very slow speeds. More importantly, it creates a torque to follow the tire’s “natural” curved track, depending on the tilt angle of the tire. This is heavily dependent on tire pressure, width, carcass and tread design.

5. Wheel flop
Wheel flop as defined above creates torque that increases steering input. The torque is minimal at small steering angles, but increases with the angle getting bigger. It can feel destabilizing when riding at low speeds where you have to steer to keep your balance or to get around obstacles. Proportional to load on front wheel, independent from speed.

When we look at riding at low speeds, for instance climbing a steep hill or navigating a narrow trail, the stabilizing factor 2 is minimal. The destabilizing effect of wheel flop becomes more noticeable as steering angles can be quite big.

Geometry

Wheel diameter, head tube angle and fork rake determine the amount of wheel flop. A bit less obvious, the cross section of the tire is also important. A wide tire with a big curvature radius lowers the actual wheel flop.
The easiest way to determine the actual amount is a graphic solution, e.g. in a 2D CAD drawing. I also went through a numeric solution, which involves a bit more trigonometry than you might expect at a first glance. The results for four different bikes are shown below. 

Results and discussion

The race bike and the vintage MTB show striking similarities. This is not completely surprising, as fork rake and head tube angle are close. Both have very low amounts of actual wheel flop (corrected for tire curvature radius). Uncorrected, both are in the 5mm range. The MTB goes down corrected to just 3,2mm due to the bigger curvature radius of the tire. The front of this bike lowers just about 3mm when turning the handlebar a full 90°. The effect on the steering is probably barely noticeable.

Looking at the “modern” geometries of the current MTB and the allroad bike, we see much larger amounts of wheel flop. Interestingly, the tire curvature radius reduces the amount by 22 or 24 percent respectively. The 2.4” MTB tire which is used in both cases plays an important role here. Running a narrow tire on one of these geometries would make a difference – I would expect a more “floppy”, unstable feeling when riding very slowly.
The current MTB and the allroad bike are two of my actual bikes, the results of conversions of rather old MTB frames. My riding experience is that the allroad bike feels more floppy than the MTB when climbing slowly, despite the slightly lower wheel flop value. Possible explanations: different weight distribution due to lower stack height and steeper seat tube angle on the allroad bike, wider handlebar on the MTB, higher inertia of the suspension fork compared to the lightweight rigid fork of the allroad bike.

The modern setups also show larger trail values, they have higher gyroscopic forces due to the larger wheels and bigger, heavier tires, which also create more friction on the ground. So all parameters tend to create higher feedback forces in the steering system. Probably it is no coincidence that handlebar widths have grown massively with the geometry evolving in this direction. 

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