From: Jason Brett:
So with axle flex you’re usually dealing with some form of three-point beam bending. The deflection of the shaft is — as you’ve pointed out — affected by both length and thickness. The thickness comes into play as the “moment of inertia” of the beam… for a square shaft the formula is proportional to the fourth power of thickness. Doubling the thickness will result in a 2^4 increase in resistance to bending, and therefore deflection should be reduced by a factor of 16.
Length, however is a cubic factor. Reducing the unsupported length (from bearing to bearing) by a factor of two will result in an 8 fold reduction in deflection. It might actually be a bit higher than that because these formulas assume a point load and a uniform beam… your beam is thicker where the wheels are and the load is distributed across that width. But the general concept applies…. shorter is better, and cubic factors make a huge difference! Simply reducing your unsupported span from 7″ down to 5″ will give you (5/7)^3 of your current deflection… about 36%. Reduce it to 4″ and you’ll be down to 18%.
Do BOTH… reducing your unsupported shaft length to 4″ AND doubling the shaft thickness and you’ll be at 1-2% of your current deflection… assuming, of course that load remains constant. If you aren’t able to compress either the tread on the wheels, the balls, or move the support on the other side of the balls, then you’re still going to have the same amount of deflection… because something has to give somewhere! This brings up the third factor in deflection… the load. Perhaps a little less “squeeze” will be sufficient to reduce your shaft flex. Keep in mind, however, that load is only linearly proportional to deflection… you’d have to reduce the load by 50% to cut deflection in half.