You'll almost definitely have heard 29er riders saying just how much better their bikes roll over obstacles on the trail. "I carried so much more speed through that rough section!", or something similar. This is probably the key reason that riders and manfacturers give for having a bigger wheel size... But what does this mean, and just how much better do they perform this action?
The diagram below (Fig. 1) shows the height of a square-edge obstacle, and the angle of attack vis-à-vis the wheel:
The angle of attack itself doesn't really tell you much without applying basic trigonometrical functions to to break it down into horizontal and vertical force vectors. In a simplified form without friction or deformation, if a wheel runs into a vertical obstacle higher than the axle height, it will stop you instantly (horizontal force / vertical force = infinity). Conversely, if an obstacle has zero height it will not slow you down at all (horizontal force / vertical force = 0). On Fig. 3, you can see how the force vector varies as obstacle height increases for each wheel size (the higher the Tan (Angle of Attack), the more it will slow you down):
This graph clearly shows that the relative efficiency is not consistent across all obstacle heights. The larger the obstacle, the larger the effect the wheel size will have. So it is impossible to say that one wheel is x% more efficient over square-edged hits than any other size without saying the size of obstacle, tire size, and tire pressure etc etc.
Once again I want to make it very clear that these numbers are based on wheels that do not deform at all, and that are rolling over perfectly square-edged obstacles, which is obviously not realistic. So let's have a quick look at some real world factors that significantly complicate the situation.
Tire deformation helps to absorb the impact of hitting a square-edge obsticle. This not only reduces the shock that is transferred to the frame and rider, but also makes the wheel roll more efficiently over an obstacle by effectively reducing the angle of attack when it absorbs it. The more the tire absorbs the obstacle the better, so actually lower pressure tires roll over obstacles like this more efficiently (unless you get a snake bite!).
Tire size is an important factor... for example you could realistically have a larger outside diameter running a very high volume tire on 26" wheels than a small volume tire on a 650b wheel. In this situation the 26" wheel would roll over things better than 650b, so tire height should be considered if analysing options.
We have indeed confirmed that big wheels roll over obstacles better than small wheels, and help maintain momentum as a result. But frame geometry and axle path also play a factor if the frame has suspension, as the suspension can help absorption of obstacles and make the bike roll over them better. The slacker the head angle or more rearward the axle path, the better a bike will roll over an obstacle if all other factors are equal.
Plus there is one very very significant factor that none of these numbers take into account...We can bunny hop over things! This is why you should never listen to arguments taken from automotive industry as the car can't be thrown around independently of the driver.
If this second installment of wheel physics hasn't boggled you even more than the first part, the third blog post will tackle contact area and grip. Woop!