Cylindrical Roller Parts

Cylindrical rollers are a type of rolling element used in roller bearings. Compared to ball bearings, roller bearings have a higher load capacity.

The roller in a rolling-element bearing is considered stiff in the sense that the geometry does not change globally, but Hertzian contact is taken into account. The deflection and contact between two cylinders with parallel axes are well documented. As long as the contact surfaces are convex, theoretical solutions exist for both deformation and contact pressure. For concave contact, an approximation called "Contact between a rigid cylinder with flat-ended and an elastic half-space" is used for the deformation, while the contact pressure is theoretically exact [1–2].

Forces

It is assumed that the rollers have contact in the whole roller width, i.e. no crowning. Equation for the Hertz contact width b:

 

and displacement of cylinders, delta:

 

where F is the force, nu is the Poisson ratio, Emod is Young's modulus, L is the length of the cylinder and R is the radius of the cylinder. To simplify the equations, the variable Ered is introduced as:

 

With the preceding equations, delta as a function of the contact force can be solved. However, the simulation time can be improved if the equations are rewritten as shown following. The error in deformation delta is normally less than 1% and in contact stress even better.

The relation between contact force and indentation depth can be written as:

 

where with good accuracy:

 

and:

 

Finally, the load indentation depth constant can be written as:

 

With the notation used in the source code:

 

Contact Stress

The Hertz contact stress for a cylinder contact is:

 

Limitations

For certain types of roller elements, a theoretical solution may not exist, and design optimizations such as crowning may be required to alter their behavior. In such cases, the bearing manufacturer can typically provide information on the nonlinear stiffness and Hertz contact stresses. Alternatively, Finite Element software can be used to determine the relationship between load and roller deformation.

The ideal rotational speed of the roller is explicitly calculated, meaning that friction forces do not drive the rotation of the rollers. However, this limitation is usually only relevant for large bearings when smearing effects must be considered as the roller element moves from an unloaded zone to a loaded zone.

Figure 1: Roller part.

References

[1]  Wikipedia. "Contact Mechanics." https://en.wikipedia.org/wiki/Contact_mechanics.

[2]  Johnson, K. L. Contact Mechanics. Cambridge University Press, 1985.

d	Value: Type: Diameter (m) Description: Bearing inner (bore) diameter
D	Value: Type: Diameter (m) Description: Bearing outer diameter
B	Value: Type: Distance (m) Description: Bearing width
d1	Value: Type: Diameter (m) Description: Bearing innerring wall outer diamter
D1	Value: Type: Diameter (m) Description: Outer diameter for roller
F	Value: Type: Diameter (m) Description: Inner diameter for roller
s	Value: Type: Distance (m) Description: Cage clearance
dRoller	Value: Type: Diameter (m) Description: Roller diameter
dLength	Value: Type: Distance (m) Description: Roller length
wallWidth	Value: Type: Distance (m) Description: Wall thickness of outer ring
angle	Value: Type: Angle_deg (°) Description: Angle
numberOfRollers	Value: Type: Integer Description: Number of rollers
ftf	Value: F / (2 * F + 2 * dRoller) Type: Real Description: Cage defect frequency (FTF)
clearance_z	Value: Type: Length (m) Description: Clearance in z direction
clearance_dr	Value: Type: Length (m) Description: Clearence in radial direction
ny	Value: 0.3 Type: Real Description: Poisson's ratio
Emod	Value: 210000000000.0 Type: ModulusOfElasticity (Pa) Description: Young's Modulus
outerRingDefect	Value: false Type: Boolean Description: if = true, then an outer ring defect at 12 o´clock is present
addForceOuterRingDefectAmplitude	Value: 0 Type: Force (N) Description: Magnutude of outer ring defect force. Only valid if outerRingDefect = true
animateRollers	Value: true Type: Boolean Description: Animate rollers
animateRollerForce	Value: true Type: Boolean Description: Animate roller forces
animateOuterRingForce	Value: false Type: Boolean Description: Animate outer ring forces