Axle + Bearing + Preload Calcs
Axle Calcs
W started with an axle with a diameter of 30 mm as a base. Based on the FOS calculated from this diameter, we would then size up or down the axle diameter.
The forces applied to the axle are coming from the contact patch through the tire, rim, hub, bearings, and finally on the axle.
First, we are going to find the forces acting on the axle from the bearings from bump. To do this you are going to represent your axle as a beam, your bearings as two pin supports at their respective locations, and your bump force acting on the beam, wherever your contact patch is located.
This is now a simple statics problem. You can ignore axial forces for now. Solve for B_y using sum of moments at A.
Then solve for A_y using sum of forces on the y axis.
Now, we are going to make the the axle into a cantilevered beam. The upright is going to act as a fixed support in this case, and the reaction forces from the bearings are going to act on the cantilevered beam.
This is just a simple statics problem as well. Solve for the moment at the fixed support using the reaction forces from the first step.
Since the moment at the fixed support, is the biggest moment, we used that to calculate the maximum bending stress on the axle.
I represents the second moment of area for a cylinder, and c represents the radius of the axle. You now have your max bending stress for axle from bump!
The process for finding the maximum stress from brake force is almost the same. The only difference is the force will be coming from where the hub and axle connect, not from the contact patch. (Ask Advait later for explanation). This just changes how the forces are distributed between the two bearings but the idea is the same.
The last bending stress comes from the steering force. From the math and intuition, this force applies the most stress because of the long moment arm being made up of the radius of the tire.
Multiply the applied force from the contact patch by the length of the radius and you will have your moment needed for the bending stress equation.
Now, you can find the axial stress coming from this steering force. This is very simple, it is just the applied force over the cross sectional area of the axle.
The axial force should be split between the two bearings, but at the moment I am not sure about how to find that distribution. For this case we are just saying all of the steering force will contribute to the stress.
Finally, you add up your three bending stress and axial forces to get your theoretical max stress. There could be instances where certain stresses cancel out, so adding them up is a conservative figure.
Preload Calcs
To maximize bearing efficiency, you need to apply a certain amount of preload on them. SKF provides preload specifications on all (i think) of their bearings in the bearing product page.
For our bearings, SKF specs 100 N of preload. Since we have two, I decided to do 200 N. Is this valid? Maybe or maybe not, but it makes sense to me.
To find the preload needed that will apply 200 N while going through the max load case I started with the following equation:
F_sep will be twice the applied force. This gives us a factor of safety of two for separation to account for any errors and simplifications. We will calculate for C, the join constant, and solve for F_PL.
The equation for C is:
Finding the stiffness for the bolts and grips is not complicated but tedious. You need to make sure you are plugging the correct values into the equations I am going to provide. I will try my best to make it as clear as possible. This is also mostly from Shigley’s so referencing the book while reading this will probably help a lot!
This will be the k_bolt from the joint constant equation. In our case l_t was zero because there no threads within the members. All threads are outside of the joint (hopefully that makes sense). E is the elastic modulus for the material of your bolt. TPI is threads per inch. d_nom is the nominal diameter of your threads.
You have to calculate the stiffness of each member inside the joint. In this joint the member are two bearings, the upright, and the hub. You then put in each stiffness into the second equation to get the total stiffness of the grip. D is the diameter of the bolt head, in this case axle head. t is the width of the member. α is the frustum angle, Shigley’s just recommends you use 30 degrees for this.
Now, you plug in al your values into the first equation shown. All these values are in imperial so your preload force will be in pounds-force (lbf). Now, I added 200 N, converted to lbf, to the calculated preload. This gives us a preload that will keep the keep the joint from separating from the max axial load while keeping 200 newtons of preload during max axial load all with a FOS of two. Now that we have this value, we can easily find the torque spec with the following equation:
K_T is the nut factor. It is a value that can be calculated, but I just used the conservative value of 0.30 gathered from the table. There amore values online that could be more specific, but this is good enough for my case. You just calculated the torque spec for the spindle nut!