Advances in Mathematical Physics

Volume 2018, Article ID 8242614, 10 pages

https://doi.org/10.1155/2018/8242614

## Minimization of Stress State of a Hub of Friction Pair

^{1}Azerbaijan Technical University, H. Javid 25, AZ1073 Baku, Azerbaijan^{2}Institute of Mathematics and Mechanics of the Academy of Sciences of Azerbaijan, B. Vahabzade 9, AZ1141 Baku, Azerbaijan^{3}Azerbaijan State University of Economics, Istiglaliyyat 6, AZ1001 Baku, Azerbaijan

Correspondence should be addressed to Vagif M. Mirsalimov; za.mmi@vomilasrim.figav

Received 14 December 2017; Revised 12 April 2018; Accepted 14 May 2018; Published 2 July 2018

Academic Editor: Eugen Radu

Copyright © 2018 Vagif M. Mirsalimov and Parvana E. Akhundova. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The function of displacements of external contour points of a friction pair hub that could provide minimization of stress state of a hub was determined on the basis of minimax criterion. The problem is to decrease stress state at that place where it is important. The rough friction surface model is used. To solve a problem of optimal design of friction unit the closed system of algebraic equations is constructed. Increase of serviceability of friction pair parts may be controlled by design-engineering methods, in particular by geometry of triboconjugation elements. Minimization of maximum circumferential stress on contact surface of friction unit is of great importance in the design stage for increasing the serviceability of friction pair. The obtained function of displacements of the hub’s external contour points provides the serviceability of friction pair elements. The calculation of friction pair for oil-well sucker-rod pumps is considered as an example.

#### 1. Introduction

Operation efficiency of a friction pair of machines depends in considerable degree on stress state of the friction pairs. For example, typical operational failure of hubs of oil-well sucker-rod pump is appearance of plastic deformations on the internal contour. When operating, the plunger rubs against the hub’s surface. The surface layer of the hub’s metal is heating. In the process of operation of the friction pair “a hub-a plunger” under repeatedly reciprocating motion of the plunger there arises force interaction between contacting surfaces of the hub and plunger; there occur friction forces that cause wear of mating materials. In its parts there arises stress-strain state caused by the action of force and thermal loads.

According to classic theories of strength, in the simplest case maximum normal stress is responsible for failure of friction pair parts. Consequently, the value of maximum normal peripheral stress achieved in the material may be considered responsible for strength failure of friction pair materials. Increase of serviceability of friction pair parts may be controlled by design-engineering methods, in particular by geometry of triboconjugation elements. At present there are no solutions of tribomechanics problems on construction geometry of the surface of friction pair parts such that the stress field created by it prevented failure or occurrence of irreversible deformations of the materials of contact pair elements. Minimization of maximum circumferential stress on contact surface of friction unit is of great importance in the design stage for increasing serviceability of friction pair. Obviously, the lower the stress state of the hub, the higher its operation life. To improve the efficiency of details of friction pair the optimal design is important [1–11].

The goal of the study is to develop a mathematical model for a hub-plunger pair that allows determination of optimal function of displacements of the points of the external contour of the hub under given operation modes of the plunger.

#### 2. Problem Statement

It is known that real treated surfaces are not absolutely smooth but always have irregularities of technological character. Such micro- or macroscopic irregularities form the rough surface. Despite the very small sizes of the irregularities, they affect the different service properties of triboconjugation [12–15]. In the operation process of friction pair, on the internal surface of the hub, on the area of contact with the plunger, there acts surface thermal source caused by external friction of the plunger against the hub’s wall in the course of repeatedly reciprocating motion of the plunger. As a result of such interaction, there happen friction forces reducing to wear of mating materials and increase of hub’s and plunger’s temperature. We adopt the parameters of the function of displacements of the hub’s external surface points as controlling variables.

As a mathematical model of the problem on reduction of hub’s stress level we assume differential equations of thermoelasticity.

To determine the stress distribution and contact pressure in the elements of friction pair, a wear-contact problem on pressing the plunger into the hub’s surface must be considered.

Let in some unknown area a plunger be pressed into the hub internal surface. The shear modulus and Poisson’s ratio of the plunger and hub are various.

It is considered that on the external surface the hub has some displacements. The function of these displacements is unknown beforehand and is to be defined. It is assumed that the plane strain conditions are fulfilled. We simulate the hub and plunger by an isotropic elastic homogeneous body. It is assumed that the operation modes of the friction unit at which residual deformations arise are inadmissible. The loading conditions are considered as quasistatic.

Assign the hub of a friction pair to polar system of coordinates having chosen the origin at the center of concentrical circles , of radii and , respectively (Figure 1). It is assumed that the internal contour of the hub is close to circular one. Let us consider some realization of the hub and plunger’s rough internal surface.