Prof, your analysis is all built on the equation: T/r = ma But this model is forgetting that when a wheeled vehicle accelerates, there are two types of inertia to overcome: linear inertia of the riser+board+wheel moving forward and angular inertia of the wheel spinning. This is equation (2) above. I had been making the same mistake up until yesterday, forgetting about the angular term, but I called up a mechanical engineering buddy of mine to confirm and yeah, it exists and it is non trivial so you have to incorporate it in your model to get accurate results. Not all of that torque is spent accelerating you forward, some of it is spent getting the wheel to spin. Imagine a stationary, big, heavy, ball bearing. It takes effort to get it spinning even though there is basically no friction and it’s not going anywhere.
Here’s an article explaining the subject, if you don’t believe me.
Or take a look at this article, where in the section about wheel diameters they explain why wheel diameter makes it harder to accelerate on non-electric longboards, powered by your feet and gravity, and where wheel diameter has therefore no affect on torque.
This inertial effect is a big reason why the Abec11 Flywheels that we love to use on our boards was developed in the first place. Those guys realized that big wheels are heavy and take a while to accelerate, so why not replace the heavy urethane at the center of the wheel that’s not really doing anything with a hollow plastic core?
Another way to think about it is in energy terms. There is kinetic energy stored in the wheels when they are spinning, like literal flywheels. The energy of the system is KE = 1/2 * m*v^2 / 2 + 1/2 Iw^2 When you apply power to the system, the goal is to increase velocity, so you want to as much of that power to go into increasing the first term as possible. Any power that goes into the second term is “wasted”.
Prof, I really respect all the models you have developed. I’m coming from the same place, technical rigor is the only way to really understand our machines and get the most out of them. Please consider my argument with an open mind. If I’m wrong do continue to engage and help me see the gap in my logic. But also consider if you might have missed something as well. Here’s a thought experiment for you. If your wheels were magically gained 10 pounds per wheel, but the rider dropped 40 pounds of bodyweight, would the board accelerate at the same rate? If not, does your model have any means of capturing that effect?