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.

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Generic – rules of thumb
  • Equivalent Young’s modulus: E=(\frac{1-\nu _1^2}{2E_1}+\frac{1-\nu _2^2}{2E_2})^{-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 _{max}=\frac {F}{\pi a^2}

Local maximum stress

Convex roller equations

\omega =\frac {R}{r}

Radius-ratio < 25

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

Long half axis

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

Short half axis

\delta _{local}=\frac {1}{2}(\frac{a^2}{R}+\frac{b^2}{r})

Approach

\sigma _{max}=\frac {F}{\pi ab}

Local maximum 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}}{(\frac{a^2}{R} +\frac{b^2}{r})}

On v-slot (angle α)
Hertz Contact On v-slot angle α

F_{local}=\frac{F_{axial}}{2}\ast \frac{1}{\sin (\frac{\alpha }{2})}
\delta _{axial}=\delta _{local}\sin (\frac{\alpha }{2})
C_{axial {\ }ball}=\frac{F_{axial}}{\delta _{axial}}=\frac{F_{axial}}{\frac{a^2}{r} \sin (\frac{\alpha }{2})}=\frac {1}{\sin (\frac{\alpha }{2})}\ast \sqrt[3]{\frac {4}{9}rF_{axial}E^2}
C_{axial{\ }con.rol.}=\frac{F_{axial}}{\delta _{axial}}=\frac{2F_{axial}}{(\frac{a^2}{R}+\frac{b^2}{r})\sin (\frac{\alpha }{2})}

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