I'd like to represent the rolling elements of a bearing with the cylindrical spring connector, but I'm not sure how to define the distributed radial stiffness of the spring. Ultimately, I'd like to recover the load on an individual rolling element to determine whether the supporting structure is sufficiently stiff.
I have a linearized spring stiffness for the rolling element. I have a discrete number of rolling elements in the bearing. Therefore, I can divide the raceways up evenly by the number of rolling elements to allow me to recover the load on an individual element. The problem I am having is calculating the distributed stiffness for the cylindrical faces.
What area do I use? The area of the inner surface? The outer? The average of the two? The projected area of the inner surface?
The documentation is no help, "Springs defined in the common areas are obtained by projection. You define spring stiffness and preload forces per unit areas."
I ran a test study and it is just as ambiguous:
The model is set up as follows: Both parts are plain carbon steel.
Outer sleeve is OD 2mm, ID 1mm, 1mm axial. Siding restraint on each end and fixed on the OD.
Inner pin is OD 0.5mm, 1mm axial. Sliding restraint on each end.
Inner pin has a split line through the center on each end. This line has a restraint applied normal to the line that prevents rotation of the pin.
Radial compression spring defined between the pin and sleeve. Radial distributed stiffness is 1.00E+12 N/m/m^2
Mesh size is .06837765mm.
Force of 1 Newton total split between each end of the pin, applied radially along the end face split line.
This study returned an average displacement of 0.0017011mm for the entire surface of the pin, with a max displacement of 0.0017085mm and a min of 0.0016937mm.
So pretty close to rigid body displacement. With the given load and the resultant displacement the effective spring rate is
How though, does that relate to the linearized rolling element stiffness? I tried using the sinusoidal distribution for half the area but the result stiffness falls between the predicted stiffness from the inner and outer surface areas. However it's not the average.
The outer surface predicts an effective stiffness in the loaded direction of
and the inner surface predicts an effective stiffness in the loaded direction of
So what is going on here?