OPTIMAL DESIGN OF THE FRICTION SURFACE PROFILE OF A PLUNGER AND BARREL PAIR WITH LASER-HARDENED SURFACE

ABSTRACT

This study investigates the optimal design of the friction surface profile of a plunger and barrel pair, with surface hardness increased by laser treatment. The aim is to minimize the uneven distribution of contact pressure on the friction surface, thereby ensuring symmetrical wear of the barrel and its maximum durability by selecting the appropriate microgeometry. The newly proposed methodology aims to optimize the contact pressure distribution on the friction surface and ensure the long-term durability of the plunger and barrel components. The mathematical model developed is based on the boundary conditions of the contact problem and the differential equations of thermoelasticity. Using the least squares principle, the optimal contact pressure values and corresponding parameters were calculated, and a microgeometry that ensures symmetrical wear was determined. The proposed method is also supported by calculations to determine the optimal value of the contact pressure. The results demonstrate that this methodology is effective in optimizing the contact pressure of friction pairs and enhancing the durability of the materials.

АННОТАЦИЯ

В данном исследовании изучается оптимальная конструкция профиля поверхности трения пары плунжер-цилиндр с повышением твердости поверхности за счет лазерной обработки. Целью является минимизация неравномерного распределения контактного давления на поверхности трения, что обеспечивает симметричный износ ствола и его максимальную долговечность за счет выбора соответствующей микрогеометрии. Новая предложенная методология направлена ​​на оптимизацию распределения контактного давления на поверхности трения и обеспечение длительной долговечности компонентов плунжера и цилиндра. Разработанная математическая модель основана на граничных условиях контактной задачи и дифференциальных уравнениях термоупругости. Используя принцип наименьших квадратов, были рассчитаны оптимальные значения контактного давления и соответствующие им параметры, а также определена микрогеометрия, обеспечивающая симметричный износ. Предложенный метод также подкреплен расчетами по определению оптимальной величины контактного давления. Результаты показывают, что данная методология эффективна для оптимизации контактного давления пар трения и повышения долговечности материалов.

Ключевые слова: лазерная закалка, плунжер, ствол, поверхность трения, контактное давление, оптимизация, микрогеометрия.

Keywords: laser hardening, plunger, barrel, friction surface, contact pressure, optimization, microgeometry.

1. Introduction

Based on the analysis of the stress-deformation state of the friction pair components (laser-hardened plunger and barrel) in various mechanisms and machines, it becomes clear that deformations and specific contact pressures are unevenly distributed across the contact surface. The uneven distribution of contact pressure on the friction surface of the barrel leads to uneven wear of the barrel. In this regard, the newly designed friction device requires minimal uneven pressure distribution on the friction surface, specifically for the laser-hardened plunger and barrel [1].

The durability of barrel-type components in the contact pair depends on the uniformity and symmetry of the wear. Wear, in turn, is determined by the contact pressure on the friction surface. The issue of selecting the microgeometry of the friction surface that ensures symmetric wear of the barrel and, consequently, its maximum durability has not yet been solved by computational methods [2].

Based on the model of a rough friction surface, a theoretical analysis will be conducted to determine the microgeometry of the plunger and barrel surfaces, which will ensure a nearly uniform distribution of contact pressure. With uniform pressure distribution, the operating life and wear of the contact pair are reduced [3].

Thus, by selecting the microgeometry of the friction surface, we will reduce the concentration of contact pressure. The least squares principle [33] is applied. Let the optimal value of the contact pressure on the friction surface be denoted as . The value of  is initially unknown and will be determined during the optimization process [4].

As a mathematical model, we accept the differential equations of thermoelasticity for the contact problem with the corresponding boundary conditions. The control variables the parameters of the microgeometry of the friction surface, i.e., the outer surface of the plunger and the inner surface of the barrel. The contact pair is modeled as an isotropic homogeneous body. We will relate this to a polar coordinate system rθ and choose the origin of coordinates at the center of concentric arcs with radius 𝑅₀ and 𝑅, and lengths 𝐿₀ and 𝐿. We assume that the previously unknown internal contour of the barrel 𝐿′ is approximately circular, and this can be expressed in the form of the optimization process of the function (𝜃). Similarly, the previously unknown external contour of the plunger is also approximately circular, and during the optimization process, the function H₁() can be determined [5].

2. Methodology

To solve the optimization problem, we first assume that the functions H () and H₁() are arbitrary and known, and that each of them has the form of a segment of a Fourier series. Based on this, we solve the contact problem related to the compression of the plunger onto the surface of the barrel. In this paper, the contact pressure for the arbitrary functions H () and H1(), which describe the roughness of the friction surface, has been determined [6-8].

The obtained contact pressure formula shows that the pressure is linearly dependent on the coefficients  and  ​ of the Fourier series of the function H () and the coefficients of the Fourier series of H₁ ().

Symbolically, the contact pressure for the friction device can be written as ‘barrel and plunger.’

                                            (1)

olanlar. müstəqil dəyişən  və 4m + 2 parametr  və s. funksiyasıdır. Biz zamanı sərbəst parametr kimi qəbul edirik. Bu  parametrləri sabitdir (ümumiyyətlə, zamandan asılıdır), lakin əvvəlcədən məlum deyil və müəyyən edilir.

The independent variables are  and the 4m + 2 parameters  and so on. We consider time as a free parameter. These parameters  are constant (generally time-dependent), but they are initially unknown and are determined during the process.

Based on the explanation provided, a series of calculations are performed to determine the unknown parameters. The interval [1; 2] of the variable  is divided into MMM segments, where M > 4m +2.

                                                (2)

Thus, it is required to find such values for the unknown parameters that will provide the best constant value for the contact pressure function (2). In other words, the most probable values of the unknown parameters must be determined.

The deviations of the contact pressure values from the desired constant will be .

                                           (3)

The least squares principle states that the most probable values of the parameters will be those for which the sum of the squares of the deviations ​ is minimized, that is

                                                         (4)

For any time, we consider  (k = 1,2, m) and ​ as independent variables. By equating the partial derivatives of the left-hand side with respect to these variables to zero, we obtain a system of 4m + 3 unknowns, resulting in a total of 4m + 3 equations. Since the function  is linear with respect to the unknown parameters, the formulation and solution of the system are greatly simplified.

Here, the function  does not depend on the unknown parameters, while the function  is linearly dependent on the parameters ​. For convenience, we introduce new notations for the unknowns (where ).  ​

 the function can be expressed in the following form:

                                                       (5)

Then we will have it.

We find the partial derivatives with respect to the unknown parameters and set them equal to zero

.                                 .                                        .                                          .                                        (7)

.                                 .                                        .                                          .                         

Now if we introduce the abbreviations:

                            (8)

          (-1, -1) = M     

This system of linear equations (7) can be written in the normal system form

          (9)

.                            .                              .                           .                                .

.                            .                              .                           .                                .

Using the abbreviated notations from (8), the coefficients for the unknowns in the normal system defined in (9) depend on time, the mechanical properties of the barrel and plunger, the compression rate, and other factors. Due to the complexity of the formulas for the functions , they are not provided explicitly. They are determined from the algebraic system derived from the primary contact conditions [9, 10].

Conclusion.

Based on the above, the normal system (9) should be solved for fixed time values. The solution at the initial moment of time is of most interest for the designer, i.e., at t = 0. In this case, the definition of the functions  is significantly simplified.

By using the Gauss method for the normal linear system (9) for a fixed time, and selecting the element applied to the wellbore pump in the computer, we determine the required parameters of the microgeometry of the plunger and barrel, as well as the optimal value of the contact pressure . The calculation results for a speed of V = 0,2 m/s at t = 0. In the calculations, it is assumed that  = 4; M = 30. It should be noted that the value of  can be selected in advance based on the condition for ensuring the carrying capacity of the contact pair. However, as shown by the calculations, in this case, the sum of the squares of the deviations is larger. By determining the unknown optimal value of the contact pressure, the sum of the squares of the deviations ​ decreases, meaning the search results are refined.

References:

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