**Paper: A mechanistic model of spherical particle entrapment
in elliptical contacts.**

**Author:
****George
K. Nikas**

** Published
in: **Proceedings of the
Institution of Mechanical Engineers, Part J: Journal of Engineering
Tribology, 2006, 220(6), 507-522

** Errata:**
In Eqs. (25), second equation from the top (case

**Abstract**

The dynamic interaction of solid,
spherical particles with the counterfaces of concentrated, dry or
lubricated, conforming or nonconforming, rolling-sliding elliptical
contacts is mathematically analysed. Criteria for the entrapment or
rejection of particles are postulated, based on the resolution of
mechanical and fluid forces on a particle for given contact geometry and
operating conditions. Three-dimensional maps of the various force
components on a particle are presented for a typical application,
together with maps of the zones where a particle would be
“conditionally” entrapped, “weakly” rejected or “unconditionally”
rejected, where the previous terms are defined in a comprehensive
evaluation system in the model to assess the risk of damage of the
contact from the presence of contaminants. This study is concerned with
the process of particle *entrapment*, meaning that a particle must
be in contact with both counterfaces of a concentrated contact, and not
*entrainment* (particle wandering in the lubricant, which has been
studied by the author in [reference]). To the best of the author’s
knowledge, this is the first study explicitly dealing with the general
case of *elliptical* contacts. Elliptical contacts cover both
circular contacts and, taken to the limit, can also simulate line
contacts; thus, the model’s applicability and validity as a design tool
is generalized. In summary, the purpose of the paper is to expand
previous models developed for simpler geometries and to present an
effective computational tool for the simulation of particle entry into
elliptical contacts and the resulting risks of contact damage.

**Some
figures from this work**

Figure 1 shows
a spherical particle of centre K in contact with the counterfaces
(surfaces 1 and 2) of an elliptical contact at points A and B. After
calculating the geometry of the contact during operation, that is,
functions *z*_{1}(*x*, *y*) and *z*_{2}(*x*,
*y*) as in Fig. 1, then, given the* *coordinates (*x*_{0},
*y*_{0}) of the particle’s centre K, contact points A and B
are geometrically located, and the particle diameter and *z*-coordinate
of K (*z*_{0}) are calculated.

**Fig. 1.** Contact
surfaces and particle forces.

This is followed by a mechanical
force analysis on the particle, involving the normal contact forces *N*_{1}
and *N*_{2}
(see Fig. 1), the frictional forces *T*_{1} (or -*T*_{1})
and *T*_{2}_{ }(or -*T*_{2}),
and the resultant fluid force on the particle, *F*^{(f)}.
Force equilibrium dictates the resultant force on the particle and the
direction of that resultant force dictates the probability of entrapment
or rejection of the particle at the given position, based on special
criteria that are comprehensively developed in the article. The process is
repeated for hundreds of positions (*x*_{0},
*y*_{0}).

An example of
the *x*-component of the resultant force on a particle (force *F*_{x})
is shown in Fig. 2. What Fig. 2 shows is a distribution of force *F*_{x}
in the solution domain, which is an area of size 16*D _{x}*

**Fig. 2.** Example of
the *x*-component (*F*_{x}) of the
particle resultant force.

The terms "entrapment", "weak rejection", "unconditional rejection", as well as "rejection", "potential rejection" and "conditional entrapment" are defined in the article and are used to deal with the uncertainties involved, which are owed to the assumptions of the model, that is, that particles are spherical and rigid. Figures similar to Fig. 2 are presented in the article for other mechanical and fluid force components.

Based on the force and entrapment analysis, maps of the particle entrapment risk in the elliptical contact are constructed for visualisation of the risk zones. Such a map is shown in Fig. 3.

**Fig. 3.** Force vector (*F*_{x}, *F _{y}*) (shown
with distributed arrows) and
contour map of particle diameters with labels 20, 40 … 300 for particle diameter in
micrometres.

The elegance of this presentation lies in the fact that the cause as well as the "strength" of particle entrapment or rejection are both realised by looking at the angle and length of the force arrows in Fig. 3.

Among other useful results in the article is the effect of the friction coefficients of the counterfaces on particle rejection, which is useful not only for assessing the risk of damage from debris particles but also for evaluating the effect of surface coatings in protecting the contact from potentially harmful contaminants by preventing their entry in the first place.