# Correct beam theory for calculating radial force in bearing due to a bending moment in beam?

I am designing a cylindrical shaft that has a round disc at its top. I am hoping to seat this shaft in 2 radial bearings and couple to a motor. The image below gives an overview of the loading diagram. There is an axial force applied when the system is static, so I am aiming to calculate the equivalent radial force in the system to specify the radial bearings adequately to handle both radial and axial forces.

Can someone point me toward the correct beam theory to use for calculating what radial force the applied downward force would create on the bearings?

I have checked overhanging beam theory and cantilever beam theory, which point to a moment like this creating zero shear force, which I believe would mean 0 radial force, which doesn't make a lot of sense to me when I consider how the shaft would react to this downward force applied at a distance from the spin axis.

• Don't forget that with rotation, unless the force moves with the rotor, it produces cyclic loading (load and unload once per rev) if you care about life of part. Take a look at how ball bearings are actually assembled- forces on balls will not be purely in radial directions. Axial loads can disassemble some bearings installed in incorrect orientations. If you don't care about these details, then it's just a moment problem with a vertical reaction axially.
– Abel
May 10 at 12:14
• In a beam, the force is applied to a section. However, in your case, the "beam" is circular disk so the beam has fixed height but variable width. Since the width at the point of applied force at the end in 0, the force basically acts on 0 area section, which would lead to infinite displacements and stresses. I think you should be more specific about hoe the force is actually applied on the disc. May 10 at 14:55
• Thank you for the reply guys. May 10 at 16:42
• @TomášLétal: The force is a point loading, and acts downwards as shown in the diagram. Can you give me more info about what still needs to be defined with the way this force is applied to help with the calculation? Thank you May 10 at 16:51

The sketch shows the distribution of static reactions (C & V). Hope this helps.

# Depending on the mass of the disk

and its rotational speed we will have to calculate the procession period and secondary forces caused by that.

$$\Omega = \frac{\tau}{ L} = \frac{\tau}{I\omega}$$

• $$\Omega= procession$$
• $$\tau = torque= 4.5Nm$$
• $$I= moment\ of\ inertia\ of\ disk$$
• $$L= angular\ momentum$$

We set this aside and check the forces on the shaft as if it's not turning for now.

The forces are $$150N\ axial$$ and a cantilever $$moment\ of\ 4.5NM$$. Let's call the distance between the two bearings, D.

The bearings should take the vertical load of 150n and the lateral force of

$$F_{lateral}=4.5Nm/D$$

This lateral force will be causing a reaction to the left on top, and an equal reaction to the right at the bottom bearings.

# effect of the procession.

After plugging in the mass and speed of the disk we find the procession. The vertical load will remain the same at 150n. But the horizontal force on the bearings will alternate changing the direction of the horizontal force $$F_{lateral}$$ in a circular path with the speed of $$\Omega$$.

• Thanks for the answer. Is precession a concern if the axial load is only applied when the shaft and the bearings are stationary? May 11 at 13:57
• No, precession would apply only when the disk rotates. and the faster the disk's w the less the effect. May 11 at 15:58