X and O in angular bearings and support span

[Juanda]

TL;DR

I can't find a source giving the derivation of the support span.

The title mentions angular bearings although the concept applies to tapered bearings as well.

They can be assembled in different configurations which results in different properties.
(Picture from this video although the pictures I think are from SKF)

The support span (it might also be called with other terms but that's how I found it described on this page) is very important for bearings. For the 3 configurations, the tandem will have the same support span as "straight" bearings, the back-to-back or O configuration will have the greatest, and the face-to-face or X will have the smallest (even negative as in that picture?).

I have seen in several places some rules of thumb to choose one configuration or the other. I'll focus on X and O. Basically, O is stiffer because of the increased support span. Therefore:

1. Looking for stiffness? Choose O.
2. Looking for compliance? Choose X which is less stiff.

I suspect the underlying concept of making that true is related to load paths but I can't bridge the conceptual gap.
(Picture's source)

I tried using a free-body diagram which clearly shows how the support span makes it more rigid in the case of deep groove bearings or other "straight" bearings but I don't see how the angular bearings change the support span.

In the previous picture, it's evident that by increasing the distance between the bearings, the support span increases which gives the assembly more leverage to counter input forces so it'll be stiffer.
I tried doing a similar diagram with angular contact bearings and I don't see how the equivalent support span grows or shortens depending on the angle of the bearings. The implication is that it is equivalent to "straight" bearings being placed at a different location but I don't see why.
Does it just fall out from the algebra once the equations of equilibrium are in place?
Also, what would be the implications of an X configuration where the support span is negative because the X crosses as in the previous picture?

*NOTE: I know that bearings are usually preloaded so, as the input force increases, all the balls would still have a reaction. To be precise, some balls would see their contact pressure decrease and others would see it increase. I focused only on the increment of pressure which would be equivalent to not preloaded bearings as an attempt to keep the thread clearer.

 

[Baluncore]

I suspect the underlying concept of making that true is related to load paths but I can't bridge the conceptual gap.

The arrangement is described by the normal to the contact surfaces, but it is the contact surfaces that explain the behaviour and leads to understanding.

The balls cannot move in the direction of the inner or outer race, but they can roll along slightly different tracks between the races.

When the contact surfaces of two separated bearings fall on a common circle, there is room for some variation of the axial alignment about the centre of the circle. To confuse things, that is called the X configuration.

When the contact surfaces of two separated bearings fall on two cones with coincident apexes, there is less available variation of axial movement. To maintain the confusion, that is called the O-configuration.

 

[Juanda]

The arrangement is described by the normal to the contact surfaces, but it is the contact surfaces that explain the behaviour and leads to understanding.

This part is key. I didn't realize it.
It's even shown in the pictures from one of the pages I linked but I didn't see it that way on the first read.

There it shows an O configuration but there is also a kind of X drawn which can lead to confusion. At least, it confused me.

When the contact surfaces of two separated bearings fall on a common circle, there is room for some variation of the axial alignment about the centre of the circle. To confuse things, that is called the X configuration.

When the contact surfaces of two separated bearings fall on two cones with coincident apexes, there is less available variation of axial movement. To maintain the confusion, that is called the O-configuration.

Do you know of a source to read more about that? I feel I'm getting there but I need more information. I had hoped this topic would be explained in detail in "Shigley's Mechanical Engineering Design" almost like anything else that comes to my mind in relation to mechanical engineering but I didn't find it there so I opened this post.

 

[Baluncore]

You are not the first, and will not be the last, to be confused by the X and O designation.

I do not have a good source. I try to avoid using the X or O designation, by understanding the principles.