Generalized Hertz Contact
Background
From 1881-1882, Heinrich Hertz published his papers on the topic
of curved bodies in contact with each other over a small patch. It was a
brilliant piece of work. Since then, hundreds of papers and books have
been written on the subject, many of them focusing on particular
areas of interest. And with the advent of FEA, solutions to these
problems have been enhanced with analysis of real-world components.
Still, there is a need to understand the basic equations and to be able
to apply them to simple geometries. This may be for sizing
a component, to see if the stress on a part is going to
be too high, or to verify the result from a full FEA.
The equations for the simple 2D case (e.g., two cylinders parallel
to each other in contact) and the simple 3D case (e.g., two spheres
being pressed together) are straightforward, and the equations are
easily understood and used. These cases are commonly presented in engineering handbooks and machine design textbooks.
The Problem
Unfortunately, the problem of general contact is not so
straightforward. Examples of this include two cylinders with different
radii that are
perpendicular to each other and ball bearings moving in a race. These
situations are called generalized Hertz contact, or elliptical contact,
since their contact patch is often a small ellipse. While Hertz
discussed the generalized problem, he stopped right in the middle of
deriving his equations. He jumped over details and presented the
final results in
terms of tables of coefficients. Omitted parts include: details
of the equations, the steps necessary to calculate the tables, and how
to use elliptic integrals (which are readily available in mathematical
tables). In
addition, Hertz's tables only listed ten values which are insufficient
for
accurate work. To make things worse, it was later verified that there
were errors in Hertz's
original tables, which were subsequently copied by other authors.
While it would be helpful to have a complete derivation of the
equations
and a complete listing of all the equations (including the omitted
equations), these can be nearly impossible to find. It seems that authors who had the
opportunity to fill in the gaps in Hertz's papers introduced
different notations, provided tables of coefficients that may or may not
be in
error, omitted complex equations, and introduced additional
approximations (i.e., errors), etc. In addition, the maximum stress
that occurs below the surface is difficult to calculate and is usually
omitted. This makes it
very difficult for an engineer to use the equations to solve contact problems with confidence.
The Solution
Starting with Hertz's papers, I derived his missing equations,
verifying them against other sources when possible. In the near future,
I will add them to this page in the form of a PDF file along with a summary
of the relevant literature. Meanwhile, I have written a Python script
to calculate the tables of Hertz's coefficients which can be easily
imported into a spreadsheet and used for these calculations. The
advantage of
this script is that the user can choose how extensive or limited
they want the table to be (or compare with Hertz's original tables).
Future additions and verifications are planned. I welcome feedback on this work.
Download Hertz contact python script