CONTACT CALCULATIONS OF A BALL IN A BEARING RACE



Herzian contact (sphere-cup contact)

Following calculator has been developed to calculate contact stress of sphere (ball) in a circular race. This type of situation is generally seen at the contact region of ball bearings. The schematic representation of the contact is given in the figure.

This calculator is an additional calculation tool to Hertzian contact calculator. The formulas used for the calculations are given in the "List of Equations" section. See the "Supplements" section for the link to contact calculation example of a ball bearing.



Calculator:

INPUT PARAMETERS
Parameter Symbol Sphere Circular Race Unit
Poisson's ratio v1,v2 ---
Elastic modulus E1,E2
Radius of objects R1,R2
R1,R3
Force F


RESULTS
Parameter Symbol Object-1 Object-2 Unit
Maximum Hertzian contact pressure pmax ---
Maximum shear stress τmax ---
Rigid distance of approach of contacting bodies d ---
Semimajor axis of contact ellipse a ---
Semiminor axis of contact ellipse b ---

Note: Use dot "." as decimal separator.



Definitions:

Ball Bearing: A type of rolling bearing to reduce the friction and support axial and radial loads. There are different kind of designs such as angular contact ball bearing, axial ball bearing and deep-groove radial bearing.

Modulus of elasticity (Young’s modulus): The rate of change of unit tensile or compressive stress with respect to unit tensile or compressive strain for the condition of uniaxial stress within the proportional limit. Typical values: Aluminum: 69 GPa, Steel: 200GPa.

Poisson’s ratio: The ratio of lateral unit strain to longitudinal unit strain under the condition of uniform and uniaxial longitudinal stress within the proportional limit.

Shear stress: A form of a stress acts parallel to the surface (cross section) which has a cutting nature.

Stress: Average force per unit area which results strain of material.

List of Equations:

Parameter/Condition Symbol Equation
Semimajor axis of contact ellipse a Equation for Semimajor axis of contact ellipse
Semiminor axis of contact ellipse b Equation for Semiminor axis of contact ellipse
Maximum Hertzian contact pressure Pmax Equation for Maximum Hertzian contact pressure
Maximum shear stress** τmax Equation for Maximum Shear Stress
Rigid distance of approach of contacting bodies d  Equation for Rigid distance of approach of contacting bodies
Coefficients in equation for locus of contacting points A Equation for Coefficients in equation for locus of contacting points
Coefficients in equation for locus of contacting points i B Equation for Coefficients in equation for locus of contacting points
--- γ Equation for Coefficient Gamma
--- K  Equation for Coefficient K
Elliptic Integral E(e) Equation for Elliptic Integral in Contact Calculations
Elliptic Integral K(e) Equation for Elliptic Integral in Contact Calculations
--- e Equation for e in Contact Calculations
Ratio of b to a k Equation for k in Contact Calculations
--- A/B Equation for A_B in Contact Calculations
--- na Equation for na in Contact Calculations
--- nb Equation for nb in Contact Calculations
--- nc Equation for nc in Contact Calculations

Note: ** Equation is obtained by curve fit to Table 4.1 of Ref-2

Supplements:

Link Usage
Hertzian Contact Calculator Calculates the Hertzian contact parameters such as contact pressure, shear and Von Misses stresses for spherical and cylindrical contact cases.
Contact Stresses in a Steel Ball Bearing An example on calculation of contact stresses in a steel ball bearing.

Reference: