Six-digit auxiliary values by Heinrich Hertz and calculated from them four-digit elliptic coefficients \(\mu\), \(\nu\), \(\mu\nu\), \(2K/\pi\mu\) (Q2707115)

The contact of two elastic rounded bodies is characterized by the so-called pressure ellipse with axes \(2a\) and \(2b\) given by expressions \(2a=4,72\cdot 10^{-3}\cdot \mu\cdot(Q/\Sigma\rho)^{1/3}\) and \(2b=4,72\cdot 10^{-2}\cdot\nu \cdot(Q/ \Sigma\rho)^{1/3}\) which one can find in many technical books. Here \(\mu, \nu\) are elliptic coefficients, and \(Q\) is the contact loading force. In 1885 H. Hertz introduced an auxiliary value for the phenomenon description, namely \(\cos\tau= (1/r_{11}+ 1/r_{21}- 1/r_{12}-1/r_{22})/(1/r_{11} +1/r_{12}+1/r_{21}+ 1/r_{22})\), where \(r_{ij}\) are curvature radii of the bodies in contact \((i,j=1,2)\). He had drawn up, with these formulae, a table for values of \(\mu\) and \(\nu\) corresponding to \(\tau=90^\circ\), \(80^\circ,\dots,20^\circ\), \(10^\circ\), \(0^\circ\). These values are insufficient for technical applications, and many authors completed the Hertz's table with other values. In particular, in the present paper the authors give a new table for values of \(\mu,\nu\), \(\mu \nu\) and \(2K/\pi\mu\), corresponding to \(\cos\tau\), with six exact digits, \(K\) being the elliptic integral of the first kind.