Wheel contact dynamics

Even though traction in the wheels is one the major performance improvements felt by the cyclist after changing his frame or wheel setup, it's not commonly highlighted by the marketing departments to try to sell a rigid frame. One of the reasons for this practice is that the abilty to maintain traction of a frame can't be as easily presented to the public as a BB stiffness test because there is a subtle line between a frame that gives good traction and one that is too flexy vertically. Altough this is the typical practice, some manufacturers highlight frame's ability to maintain traction. One recent example is the Cannondale SuperSix Evo presented last year. Cannondale has a long term experience using carbon properties to build softail frames with its ZeroPivot technology so it seems that they decided to apply some of this knowledge in the road side of things. Peter Denk developed a road version of this technology, called it Speed Save and said this about it:

 "We’re not using Speed Save for rider comfort, but more for the performance of the bike,” said Denk. “The suspension in a Formula One car isn't to keep the driver comfortable… [it's] to help keep the wheels and the tires connected to the track... We want the frame and fork to have some compliance to help the wheels stay in contact with the pavement better and to give that sensation of floating."

Let's analyze this paragraph in depth to understand if the loss of contact between the tire and the road can be detrimental to the performance of the bike. If the wheel loses contact with the ground, it will accelerate because there is no friction force and, consequently, the wheel will slip when it makes again contact with the ground because its linear velocity will be bigger than the linear velocity of the bicycle. Slipping means increased friction so we should consider both aspects of the loss of contact. If the energy disipated reducing wheel linear velocity to bicycle's velocity is bigger than the energy saved because the friction is 0 during the period of time that the wheel is not in contact with the ground, the loss of traction will be detrimental. We should also consider that if the cyclist can`t develop all his power because the bicycle is continually bouncing and he isn't comfortable, the reduced traction will be also detrimental. As you can imagine, evaluate this complex system takes some major engineering efforts, it's not an easy task.

I decided to start with something simple and add complexity later. The model that I've developed is a spring-damper system with one of the ends of the spring-damper following the road profile and the other one attached to a mass equivalent to the normal reaction force in the wheel that depends on the weight distribution of the bicycle. I have considered this equivalent mass to be constant but if I finally solve the dynamic equations presented in the Physics II post, I will be able to also consider the variations in the normal force induced by the pedalling motion. Another simplification that I've included for the moment is to consider the horizontal velocity of the bicycle constant. Considering that the road profile has a sinusoidal shape, the governing equation of the system is the following one:

Where N is the normal reaction force in flat terrain and static conditions, g is the acceleration of the gravity, c is the damping term, k the spring stiffness, R the radius of the unloaded wheel, A is the difference between the highest and lowest point of the road profile, w the angular frequency of the profile and v the horizontal velocity of the bicycle. x(t) and l(t) are the distance between the lowest point of the road profile and the wheel axle and the contact between the ground and the wheel respectively measured along the normal to the road profile in each point. As you can see there is no stiffness terms that depends on the frame/fork but I will try to include them after doing some future work about rear ends and forks stiffness.

Next step, do the first test. I have found some problems here to find stiffness and damping values of the wheel-tire sytem. Finally I have found a paper with these values of a road tire nevertheless I don't know neither what tire is nor the inflating pressure but I will use them as orientative values. Another thing to point out is that the model allows the spring to be longer that its unloaded length, something that it's impossible for a wheel. I will include later a variable stifness and damping that has a null value when the stretch is positive to solve this. To prevent the loss of contact between the wheel and the ground I have chosen the parameters cautiously. Finally, I have calculated the maximum angular frequency to ensure that there is only a contact patch between the tire and the ground in static conditions and I have used it for this first test, to do this I have considered that the distance between the center of the contact patch and the equivalent force of the pressure distribution in the contact is approximately equal to the semi-length of the contact patch. The values for this first test are the following ones:

And finally, the graphics. You can see that the maximum force is more than 1.6 times bigger than the static force. We'll see how frame's role affects this maximum value

The wheel and the road profile for the test, note the contact patch. The length of the irregularities is 9.2cm

That's all for today. Greetings

PD I have fixed a small error