HERTZ CONTACT – UNIVERSAL POINT

Engineering Fundamentals

INTRODUCTION

Universal calculation of Hertz point contact between two arbitrarily curved bodies under load F and each body has radii in x and y direction of which the largest r is called Ri and the smallest is called ri.

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Equations

Young’s Modulus, Poisson ratio, small radius, large radius per body: E,\nu ,r,R

Equivalent Young’s Modulus
E=\left(\frac{1-\nu _1^2}{2E_1}+\frac{1-\nu _2^2}{2E_2}\right)^{-1}

Equivalent large radius: R_i are the large radii of both bodies
R=\frac{R_1{\cdot}R_2}{R_1+R_2}

Equivalent small radius r_i are the small radii of both bodies
r=\frac{r_1{\cdot}r_2}{r_1+r_2}

Curvature ratio, always > 1
\omega =\frac {R}{r}

Half long axis of contact ellipse; if 1 < \omega < 25
a=2^{-\frac{1}{3}}\ {\cdot}\ \omega ^{\frac{11}{24}}\ {\cdot}\ \left(\frac{3Fr}{E}\right)^{\frac {1}{3}

Half long axis of contact ellipse; if 25 < \omega < 1e5
a=\frac{2^{\frac {5}{3}}}{\pi }\ {\cdot}\ \omega ^{\frac {8}{24}}\ {\cdot}\ \left(\frac{3Fr}{E}\right)^\frac {1}{3}

Half short axis of contact ellipse
b= 2^{-\frac{1}{3}}\ {\cdot}\ \omega ^{-\frac{4}{24}}\ {\cdot}\ \left(\frac{3Fr}{E}\right)^{\frac {1}{3}}

Approach of bodies
\delta =\frac{a^2}{2R}+\frac{b^2}{2r}

Maximum stress, Average stress
\sigma _{max}=\frac{1.5F}{\pi ab} , \sigma _{av}=\frac {F}{\pi ab}

Average stiffness
C_{av}=\frac {F}{\delta }

Impact force: v = velocity, m = mass
F_{impact}=v_{moving\ body}\sqrt{m_{moving\ body}C_{av}}

Hertz contact - Universal point

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