Akshay Harlalka
Curious Builder
Selecting a bearing for Leg Curl Gym Equipment
Bearings are everywhere. This week while at the gym, I was using the seated leg curl machine. I noticed the bearings which were supporting the shaft of the machine and decided to be do my Seek and Geek report on that. I wanted to learn the process that engineers might have gone through to determine the size of the bearing. Before one can select a bearing, it important to understand the loads that the bearing must support and the type of operating conditions it will be subjected to. For the seated leg curl machine, the criteria to determine the size of the bearing should be based on the static load conditions because the bearing of the seated leg curl machine is not rotating continuously nor is it subjected to high dynamic loads. [1] I decided to do some preliminary machine analysis to find the bearing reaction forces. Below is the picture of the complete machine
The Leg Curl Gym Equipment
Close up view of the pivot shaft and kinds of forces acting on it
Before I could begin the analysis of the FBD, I noticed that there were a couple of unknown forces which I had to determine. To determine these forces, I used a moment balance equation. I measured the force (using digital spring scale) that a user would apply to lift a known amount of weight. I had to assume the weight of the component body which I estimated to be 20 kg acting through the pivot. Applying moment balance about the pivot P for the below figure,
Side View: FBD of the shaft with key forces and critical distances
To lift 20 lbs of the weighing blocks, user had to apply 3.8g of force. Using these values and the parameters like a’, b’ c’ and theta, I was able to find the weight of the big block.
One this was figure out, I decided to apply moment and force balance equations to the front view of the shaft to determine bearing reaction forces. As there were no loads in the axial direction, I got only two equations and two unknowns (Br1 and Br2). The maximum weight that the machine was designed to lift was 260 lbs! So, I calculated the force that the user will have to apply to lift that.
Front View: FBD of the shaft with key forces and critical distances .
Model Validation Procedure
I considered this to be the extreme condition and used the moment balance about the first bearing reaction point (Br1). The equation is below. Next, I applied force balance equation along the Z axis. So, the maximum forces acting at the bearing reaction points were 1080 N and 1755 N. Now under these forces, the maximum allowable contact pressure for rolling element bearings should be around 4000 MPa. [1] This constraint will help us determine the correct size of the bearing for the application.
Material Assumptions for the balls and the raceway Material:
Chrome Steel SAE 52100 Poisson’s Ratio: 0.29
Elastic Modulus: 200 GPa
Ultimate Tensile Strength: 2240 MPa
The machine did not introduce can axial/thrust loads on the bearing. So, the correct type of bearing would be the one which can take good amount of radial loads and minimal axial loads just to protect against unexpected forces. The bore of the bearing should be 20 mm for it to work with the shaft. The Conrad type deep grove ball bearing was considered to be best suited for this purpose. While also well-suited for high speeds, these were the cheapest compared to other types of bearings and still met the functional requirements. Using the Hertz contact excel sheet, I carried out few iterations for what would be the minimum size of the balls to carry this load. Safety factor was assumed to be 1.5. Therefore, required load was now 2632 N. Load was assumed to be distributed over 3 balls so each ball was assumed to carry 1/3rd of the load (approx. 877 N)
It was determined that the minimum ball size in terms of diameter should be 14 mm. Now, I also followed the bearing selection procedure outlined in the catalogs to check if my estimations for the size of the ball are correct. The equivalent static load rating for the bearing is given by;
In this case, Fa/Fr < e, as the axial load is close to zero. Therefore, Po = Fr Po = 1755 N Introducing a safety factor of 1.5, the required basic load rating is 1.5*1755= 2632 N Using the catalog of Timken bearings [2] for this load capacity, I determined that 6004 bearing would be most suited as it had a load rating for 5000 N and a bore diameter of 20 mm The ID and OD of this bearing also seemed to match with the bearing on the gym equipment!
It turns out that the diameter of the balls for 6004 bearing is close to 12.5 mm. While this is close to 14 mm which was previously estimated, the 6004 bearing is designed to carry 5000 N compared to 2632 N with which I did my calculations. Various uncertainties/unknown parameters could have led to the difference in estimates like radius of raceway curvature and the number of balls actually supporting the radial load. However, I was happy that my estimates for the bearing size seemed to match those in the real application.
Closing the loop
The size of the bearing I estimated seemed to perfectly match with what was installed in the equipment! This was pretty satisfying for me!
ID= 20 mm, OD = 42 mm
