Originally Posted by CanAutM3
The flaw in your example resides in the fact that you consider the masses limited to be directly over the axles.
Take your example of the two mass car but instead of combining both masses over one axle, combine them right in the middle of the two axles. You would also reduce I to zero but would maintain the 50-50 weight distribution.
As another example, let's assume a car of a given weight and polar moment. If the axles are positioned at equal distance from the centre of gravity, the car has a 50-50 weight distribution. Moving the front axle forward (assuming negligeable mass for the axles themselves) will shift the weight distribution towards the rear axle, but the polar moment around the CG remains unaltered.
In both examples one parameter could be changed without impacting the other.
Good counterpoints. Although I would say these are somewhat special cases. Let me restate. You stated polar moment and axle weight distribution are not related. My counterpoint showed that in some cases they certainly are related, although I over generalized to which you provided counterpoint.
Arbitrary changes in the density (or mass) distribution in a vehicle will change its axle weight and will change it polar moment as both are sums (integrals) of the density distribution. Also, a more even axle balance (at a given fixed weight) produces a lower I whereas a more unbalanced axle ratio will produce a lower I (that comes from the simple formula I provided prior).