Hertz contact: Practical implementations

Construction Fundamentals

Introduction

Hertz contact properties like stiffness and stress can be predicted if the contact conditions are well known. Therefore the contact should not be over constrained: only “contact points” (in fact circular or elliptical shaped areas) are considered to be useful.

Generic – rules of thumb

  • Equivalent Young’s modulus: $E=\left(\frac{1-\nu_1^2}{2E_1}+\frac{1-\nu_2^2}{{2E}_2}\right)^{-1}$
  • Stiffness similar to: $C\sim\sqrt[3]{r}\sim\sqrt[3]{F}\sim\sqrt[3]{E^2}$
  • Trade-off between flat on flat (most stiff, high force capacity) and kinematically constrained designs (no over-constrained design, no hysteresis / micro slip).

Ball – rules of thumb

  • Ball contact area: circular (radius $a$).
  • For large stiffness use large radius, e.g. by using a segment instead of complete ball.

Convex roller – rules of thumb

  • Convex roller contact area: elliptical (half axes $a$ and $b$)
  • Keep convex roller radius ($R$) max 25 times larger than roller radius ($r$). Larger ratios imply a line contact and thus; over-constraining the contact.
  • For large stiffness use large roller radius $r$, e.g. by using a segment instead of complete roller.

Ball equations

$a=\sqrt[3]{\frac{3F_{local}r}{2E}}$

Circle radius

$\delta_{local}=\frac{a^2}{r}$ 

Approach

$\sigma_{av}=\frac{F_{local}}{\pi a^2}$ 

Average local stress

$\sigma_{max}=\frac{3F_{local}}{2\pi a^2}$ 

Maximum local stress

Convex roller equations

$\omega=\frac{R}{r} $ 

Radius-ratio < 25

$a=\omega^\frac{11}{24}\cdot\sqrt[3]{\frac{3F_{local}r}{2E}}$ 

Long half axis

$b=\omega^\frac{-4}{24}\cdot\sqrt[3]{\frac{3F_{local}r}{2E}}$ 

Short half axis

$\delta_{local}=\frac{1}{2}\left(\frac{a^2}{R}+\frac{b^2}{r}\right)$ 

Approach

$\sigma_{av}=\frac{F_{local}}{\pi ab}$ 

Average local stress

$\sigma_{max}=\frac{3F_{local}}{2\pi ab}$ 

Maximum local stress

On flat

Hertz Contact On Flat

$F_{local}=F_{axial}$

$\delta_{axial}{=\delta}_{local}$

$C_{axial\ ball}=\frac{F_{axial}}{\delta_{axial}}=\frac{rF_{axial}}{a^2}=\sqrt[3]{\frac{4}{9}rF_{axial}E^2}$

$C_{axial\ con.\ rol.\ }=\frac{F_{axial}}{\delta_{axial}}=\frac{2F_{axial}}{\left(\frac{a^2}{R}+\frac{b^2}{r}\right)}$
Note: these are the average stiffnesses. The local (or maximum) stiffness is:
$C_{axial\ local}=\frac{dF_{axial}}{d\delta_{axial}}=\frac{3}{2}\cdot C_{axial}$
In terms of maximum allowable stress the local stiffness is:
$C_{axial\ local}=\pi\cdot\sigma_{max}\cdot r$

$C_{tangential\ local}=\frac{2\left(1-\nu\right)}{2-\nu}C_{axial\ local} $

On v-slot (angle $\alpha$)

Hertz Contact On v-slot angle α

$F_{local}=\frac{F_{axial}}{2}\cdot\frac{1}{\sin{\left(\frac{\alpha}{2}\right)}} $

$\delta_{axial}=\delta_{local}\sin{\left(\frac{\alpha}{2}\right)}$

$C_{axial\ ball}=\frac{F_{axial}}{\delta_{axial}}=2\left(2\sin{\left(\frac{\alpha}{2}\right)}\right)^{-\frac{1}{3}}\cdot\sqrt[3]{\frac{4}{9}rF_{axial}E^2}$

$C_{axial\ con.\ rol.\ }=\frac{F_{axial}}{\delta_{axial}}=\frac{2F_{axial}}{\left(\frac{a^2}{R}+\frac{b^2}{r}\right)\sin{\left(\frac{\alpha}{2}\right)}} $
Note: these are the average stiffnesses. The local (or maximum) stiffness is:
$C_{axial\ local}=\frac{dF_{axial}}{d\delta_{axial}}=\frac{3}{2}\cdot C_{axial}$

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