FBG.Hertzian Contact Calculator*

Ellipsoid-To-Ellipsoid Contact

The contact stiffness \( k \) is given by an exponential function: \[ F(\delta) = k \delta^{1.5} \] Therefore, the exponent is always 1.5. The contact stiffness is defined as: \[ k = \frac{\sqrt{8}}{3} \left( \frac{\xi(\theta)}{\psi(\theta)} \right)^{1.5} \frac{\hat{E}}{\sqrt{\frac{1}{R_{11}} + \frac{1}{R_{12}} + \frac{1}{R_{21}} + \frac{1}{R_{22}}}} \]

Definitions:

• \( R_{11}, R_{12} \): Radii of the first ellipsoid.
• \( R_{21}, R_{22} \): Radii of the second ellipsoid.
• \( \theta \): Angular variable measured in radians.

Effective E-Modulus

The effective modulus \( \hat{E} \) is given by:\[ \frac{2}{\hat{E}} = \frac{(1 - \nu_1^2)}{E_1} + \frac{(1 - \nu_2^2)}{E_2} \]

Cylinder-To-Cylinder (Parallel) Contact

For the force \( F \) and displacement \( \delta(F) \):\[ \delta = \frac{4F}{\pi \hat{E} L} \left( 1.8864 + \ln \left( \frac{L}{2a} \right) \right) \]The contact half-width \( a \) is calculated as:\[ a = \sqrt{\frac{8FR(1 - \nu^2)}{\pi \hat{E} L}} \]

Exponential Function Fit:

The user is required to set the expected force \( F_{\omega} \), which helps in creating a suitable exponential function.