## Monday, March 18, 2013

### Wheel moment of interia: introduction

One issue with wheelsize is angular momentum. Jan Heine claims that 622 mm rims (700C) are best for up to 32 mm tire width, then 650B are good up to 42 mm, then "26-inch" are best for larger size. The idea is the wider tires yield larger mass and also yield a larger rolling radius relative to narrower tires at the same rim radius. This results in more angular momentum. Turning requires changing angular momentum, so more angular momentum creates more stability: more reluctance of the bike to change direction.

It's been commonly asserted that trail is what controls bike stability, not angular momentum of the wheels (the "gyroscopic effect"). Trail's important, for sure, and you can make bikes where trail is the only contributor to stability (for example, ski-bikes), but the detailed analysis by Andy Ruina at Cornell has shown that angular momentum and trail both contribute, as well as center of mass. Bikes are a complex dynamic system.

Angular momentum is an abstraction of Newton's simple laws of motion which applies to rotation about a center-of-mass. When an object has a velocity v and inertial mass m, classically it has a momentum mv. The rate of change of this momentum is the applied force F. Force and velocity (and momentum) are vectors, so in three dimensions, there's three components, and the rule applies to each component separately. The magnitude of a momentum vector is the square root of the sum of the squares of the components -- the same formula which is used to calculate distance in 3-dimensional space. An applied force may change the magnitude of momentum or its direction or both. If the force is constantly applied perpendicular to the momentum vector then the direction of that vector changes but the magnitude remains constant. For a spinning wheel at constant speed this corresponds to changing the heading of the bicycle (assuming flat ground). Force needs to be applied to the rim perpendicular to its spin direction.

This situation is more readily analyzed using torque instead of force, angular momentum instead of translational momentum, and angular velocity instead of translational velocity. Instead of distance/time, the units of translational velocty, angular velocity has units of radians/second, where radians are unitless so this is just /second. Torque is proportional to force multipled by distance (using the component of force in the appropriate direction). In the case of this rotation-based analysis, inertial mass is replaced by moment of inertia. Angular momentum equals the moment of inertia multiplied by the angular velocity. It's easy to see that the units of angular momentum are distance-squared multipled by mass. For a point mass, the angular momentum is the mass multipled by the distance from the rotation axis. For distributed mass, calculus is used, treating the distributed mass as a series of differential point masses.

So I'm interested in the relative angular momentum of wheels. Angular momentum of wheels is dominated by the rim, tire, rim strip or plugs (if any), and inner tube (if any). The hub is at such a small radius to be virtually irrelevent. The spokes extend from inner to outer radius and end up contributing about 1/3 as much on a per-gram basis as the rim, tire, etc. (this can be derived using integral calculus).

Angular momentum depends on moment of inertia but also on angular velocity. Jan Heine's initial analysis on the subject looked only at moment of inertia, but he later corrected himself by recognizing that smaller wheels spin faster, offsetting some of their inertial advantage. So the proper approach is to compare the angular momentum of different wheels when the bike is moving at the same speed. This will correlate to how quicky the bikes handle, assuming the bikes are built with the same trail.