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.

Contact Stress of Ball Calculator:

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


RESULTS
Parameter Obj.-1 Obj.-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:

$$a=1.145\cdot { n }_{ a }\cdot { (F\cdot K\cdot \gamma ) }^{ 1/3 }$$
$$b=1.145\cdot { n }_{ b }\cdot { (F\cdot K\cdot \gamma ) }^{ 1/3 }$$
$${ P }_{ max }=0.365\cdot { n }_{ c }\cdot { [F/({ K }^{ 2 }{ \cdot \gamma }^{ 2 })] }^{ 1/3 }$$
$${ \tau }_{ max }=\quad {\sigma }_{ c }(0.3906{ k }^{ 5 }-1.1198{ k }^{ 4 }+1.2448{ k }^{ 3 }-0.7177{ k }^{ 2 }+0.2121k+0.3)$$
$$d=0.655\cdot { n }_{ d }\cdot { ({ F }^{ 2 }\cdot { \gamma }^{ 2 }/K) }^{ 1/3 }$$
$$A\quad =\quad \frac { 1 }{ 2 } \cdot (\frac { 1 }{ { R }_{ 1 } } -\frac { 1 }{ { R }_{ 2 } } )$$
$$B\quad =\quad \frac { 1 }{ 2 } \cdot (\frac { 1 }{ { R }_{ 1 } } +\frac { 1 }{ { R }_{ 3 } } )$$
$$\gamma =\frac { (1-{ \upsilon }_{ 1 }^{ 2 }) }{ { E }_{ 1 } } +\frac { (1-{ \upsilon }_{ 2 }^{ 2 }) }{ { E }_{ 2 } } $$
$$K=\frac { 1 }{ \frac { 2 }{ { R }_{ 1 } } -\frac { 1 }{ { R }_{ 2 } } +\frac { 1 }{ { R }_{ 3 } } } $$
$$E(e)=\int _{ 0 }^{ \pi /2 }{ \sqrt { 1-{ e }^{ 2 }\cdot \sin ^{ 2 }{ (\varphi ) } } d\varphi } $$
$$K(e)=\int _{ 0 }^{ \pi /2 }{ \frac { d\varphi }{ \sqrt { 1-{ e }^{ 2 }\cdot \sin ^{ 2 }{ (\varphi ) } } } } $$
 $$e=\sqrt { 1-{ (b/a) }^{ 2 }}  $$
 $$k=\frac { b }{ a }  $$
$$\frac { A }{ B } =\frac { K(e)-E(e) }{ (1/{ k }^{ 2 })\cdot E(e)-K(e) } $$
$${ n }_{ a }=\frac { 1 }{ k } \cdot { (\frac { 2\cdot k\cdot E(e) }{ \pi } ) }^{ 1/3 }$$
$${ n }_{ b }={ (\frac { 2\cdot k\cdot E(e) }{ \pi } ) }^{ 1/3 }$$
$${ n }_{ c }=\frac { 1 }{ E(e) } \cdot { (\frac { { \pi }^{ 2 }\cdot k\cdot E(e) }{ 4 } ) }^{ 1/3 }$$

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

List of Parameters :

Symbol Parameter
a Semimajor axis of contact ellipse
b Semiminor axis of contact ellipse
Pmax Maximum Hertzian contact pressure
τmax Maximum shear stress
 d Rigid distance of approach of contacting bodies
 A Coefficients in equation for locus of contacting points
 B Coefficients in equation for locus of contacting points
 E(e) Elliptic Integral
 K(e) Elliptic Integral
 k Ratio of b to ao of b to a


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: