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
$F_{local}=F_{axial}$
$\delta_{axial}{=\delta}_{local}$
$C_{axial\ ball}=\frac{F_{axial}}{\delta_{axial}}=\frac{rF_{axial}}{a^2}=\sqrt[3]{\frac{4}{9}rF_{axial}E^2}$
$C_{axial\ con.\ rol.\ }=\frac{F_{axial}}{\delta_{axial}}=\frac{2F_{axial}}{\left(\frac{a^2}{R}+\frac{b^2}{r}\right)}$
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 $\alpha$)
$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:
$C_{axial\ local}=\frac{dF_{axial}}{d\delta_{axial}}=\frac{3}{2}\cdot C_{axial}$