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ISSN : 1598-6721(Print)
ISSN : 2288-0771(Online)

The Korean Society of Manufacturing Process Engineers Vol.16 No.5 pp.119-125
DOI : https://doi.org/10.14775/ksmpe.2017.16.5.119

Lubrication Behavior of Slider Bearing with Square Pocket Surface

Do-Hun Chin^*, Kwang-Heui Kim^**, Moon-Chul Yoon^**#

^*Department of Automotive & Mechanical Engineering, Kookje University
^**Department of Mechanical Design Engineering, Pukyong National University

Corresponding Author : mcyoon@pknu.ac.kr82-51-629-6160, 82-51-629-6150

Received 20170721 Review 20170823 Accepted 20170911

Abstract

In this paper, the characteristics and load carrying capacity of square pocket surfaces on a slider bearing are discussed for the thin film effect by the square pocket slider bearing. To study the lubrication, a Reynolds equation is used in this paper for the analysis of the slider bearing characteristics with square pocket surfaces. For numerical analysis, the central differencing scheme finite difference method is used. In a slider bearing with square pocket surfaces, the simulation dependent parameters such as pressure and load carrying capacity of the bearing can be acquired from the independent parameters, the slope of the slider bearing and number of pockets on the upper slider. These results can be acquired by the programmed softwar,e and they can be analyzed and stored in a sequential data file for later analysis. Furthermore, their pressure and load capacity distribution can be displayed easily by using the developed program with the Matlab GUI.

Key Words : Central Difference Scheme , Load Carrying Capacity , Square Pocket , Slider Bearing

사각 포켓형상 표면을 갖는 슬라이더 베어링의 윤활거동

진 도훈^*, 김 광희^**, 윤 문철^**#

^*국제대학교 기계자동차공학과
^**부경대학교 기계설계공학과

초록

키워드 : 중앙차분법 , 부하용량 , 사각 포켓 , 슬라이더베어링

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1.Introduction

If there is a small pocket on the sliding bearing surface, its effects must be verified clearly for analyzing the characteristics of pocket surface bearing.

In high precision slider bearing, an interesting behavior on the pocket surface slider bearing has increased and its effects must be considered in designing and making the slider bearing. A study on a slider bearing considering the surface roughness have started in 1970s, and Chow^[1] have studied about the effects of one-dimensional random roughness on the load carrying capacity. Also, Christen^[2] have studied the lubrication characteristics considering roughness by introducing partial differential equation of Reynolds equation using normalized film thickness and pressure statistically.

Since then, Tonder analyzed this problem in finite width bearing and also Patir^[3] studied roughness effects considering shear flow factor. Teale^[4] showed that the shear flow factor is not constant. Also, White^[5] published another research about load carrying capacity, by assuming the ensemble average Reynolds equation and the roughness by Fourier series expansion. Domestically, Shin^[6,7] etc. Show the lubrication effects of bearing by considering the roughness with sine wave. However, the same characteristics are studied considering the roughness of sine wave in the study^[8]. Furthermore the lubrication effects of round pocket slider bearing are investigated in detail to see the effects of lubrication behavior to obtain appropriate lubrication condition.

In this paper, the characteristics and load carrying capacity of square pocket surface on slider bearing are discussed for thin film effect by square pocket slider bearing. For the numerical analysis, the finite difference method of central difference scheme is used for this study. In a slider bearing with square pocket surface, several simulation parameters such as pressure and load capacity of the bearing can be acquired according to independent parameters such as the slope of the slider bearing and number of pocket on the upper slider. After all, their distribution of pressure can be displayed and be analyzed easily by using the developed program. The independent parameters such as the number of pocket and the slope of the pocket surface slider are adopted for discussing simulation parameters of pressure distribution stress and load carrying capacity of the square pocket slider. These discussed results reported in this paper should be applied to the other different shaped slider bearing with a rectangular wavy surface.

2.Numerical Analysis

2.1.Geometry of Slider Bearing Surface

Fig.1 shows the flat finite width slider bearing having square pocket surface on the upper slider. Also, three different sliders having square pocket on the upper slider is configured for its bearing analysis respectively and they are shown in Fig. 1(a)~(c). Fig. 1(a) shows one-pocket slider and Fig. 1(b) shows nine pocket surface of the slider bearing. For both case the film thickness equation can be represented as follows;(1)(2)

h = h_{0} + s (1 - x) + A : at pocket

(1)

h = h_{0} + s (1 - x) : outside of pocket

(2)

2.2.Governing Equation

The lubrication fluid is incompressible and ideal and it has no viscosity change, then the governing Reynolds equation between bearing and upper slider can be written and the pressure may be acquired from this reynolds equation. By considering that upper slider is moving to the right side against the lower bearing with steady state speed of U and the temperature of lubricant is constant, also the boundary pressure condition of slider bearing is $P_{a} = 1 a t m (1 a t m = 1.01325 bar)$ . The pressure distribution of the bearing can be obtained and the load carrying capacity can be acquired by the following equation^[8,9].(3)

W = \frac{1}{L B} \int_{0}^{L} \int_{0}^{B} p d x d y

(3)

2.3.Numerical Method

Fig. 2 shows a small element of the differential bearing surface. For the solution of Reynolds equation^[8,9], can be differentiated and the finite differential equation using the small ΔX and ΔY can be made with central difference scheme. And the non-dimensional terms of Reynolds can be written respectively as follows^[8,9];(5)

\begin{array}{l} \frac{\partial}{\partial X} (H^{3} \frac{\partial P}{\partial X}) = \\ \frac{1}{Δ X} (H_{i + \frac{1}{2}, j}^{3} \frac{P_{i + 1, j} - P_{i, j}}{Δ X} - H_{i - \frac{1}{2}, j}^{3} \frac{P_{i, j} - P_{i - 1, j}}{Δ X}) \end{array}

(4)

\begin{array}{l} \frac{B^{2}}{L^{2}} \frac{\partial}{\partial Y} (H^{3} \frac{\partial P}{\partial Y}) = \\ \frac{1}{Δ Y} (H_{i + \frac{1}{2}, j}^{3} \frac{P_{i + 1, j} - P_{i, j}}{Δ Y} - H_{i, j - \frac{1}{2}}^{3} \frac{P_{i, j} - P_{i - 1, j}}{Δ Y}) \end{array}

(5)

\frac{\partial H}{\partial X} \frac{H_{i + \frac{1}{2}, j} - H_{i - \frac{1}{2}, j}}{Δ X}

(6)

With eq. (4) ~ eq. (6), the normalized pressure P_i,j can be obtained using eq. (7).

P_{i, j} = a_{0} + a_{1} P_{i + 1, j} + a_{2} P_{i - 1, j} + a_{3} P_{i, j + 1} + a_{4} P_{i, j - 1}

(7)

Where, if coefficients, $a_{0}, a_{1}, a_{2}, a_{3} and a_{4}$ , are given, the pressure P_i,j at center point (i, j) can be represented with four close points as eq. (7).

One equation at each point can be drawn and if there are N_x and N_y points into x and y direction, N_x ×N_y numbers of simultaneous equations can be constructed and their solutions may be obtained using iterative method. The iterative calculation may be continued until the relative error decrease to some satisfied stop criterion ε and the differential equation can be solved using the successive over relaxation iterative method^[8,9].

3.Lubrication Behavior

3.1.Pressure distribution of slider bearing with multi square pockets

For numerical calculation of Reynolds equation, the boundary condition of slider bearing must be given. In numerical calculation, the pressure at the boundary of the bearing was fixed as $P = 1 (at x, y = 0 and L, M)$ , and the behavior of pressure and load carrying capacity are analyzed according to the amplitude, number of pocket and slope of the slider bearing. Fig. 3 shows the pressure distributions of the slider bearing at condition of h_f = 0.09mm, pocket number of 1. For finite bearing of L/W = 1 in Fig. 3, the pressure distribution may increase first but decrease after some slope, m = 3.4875. Also this pressure distributions may cause different load carrying capacity according to variable m.

Fig. 4 show the pressure distributions of the slider bearing at pocket numbers of 9(3×3) and its height h_f = 0.09mm. For finite bearing of L/W = 1, the pressure distribution shows similarly to 1-pocket slider. Also this pressure distribution cause different load carrying capacity according to slider slope.

Fig. 5 show the pressure distributions of the slider bearing at pocket numbers of 25(5×5) with same height of h_f = 0.09mm. The pressure distribution may increase first but decrease after some slope. Also it is a little bit different than that of 1-pocket or 9-pockets slider according to its slope.

Fig. 6, Fig. 7 and Fig. 8 show the 2-dimensional pressure distributions in detail at the center section of finite width slider bearing according to the slope and different pocket number for above same condition. They are the pressure distributions for the film thickness ratio, $m (h_{1} / h_{0})$ ranging from 1.6219 to 3.487511. The maximum mean pressure appears around the film thickness ratio of m = 2.25. If film thickness ratio is larger than 2.25, the mean pressure decreases and load carrying capacity may also decrease too. However, the slope is the most effective parameter that can affect to the load carrying capacity. In general, the mean pressure is proportional to load carrying capacity. So the maximum load carrying capacity may appear around the film thickness(m) ranging from 2 to 2.5 depending to pocket height.

4.Load carrying capacity according to pocket number shape and slope

Fig. 9 show the load carrying capacity of finite width slider bearing according to the slope s, for L=50mm, L/W = 1 and h₀ = 0.09mm. The load carrying capacity of the slider bearing is maximum for film thickness ratio, m(h₁/h_o) of 2.5, which is the same slope of about 0.0003 (s=(h₁-h_o)/L). In a film thickness ratio(m) larger than 2.5, the pressure distribution is decreasing and the total load carrying capacity may also decreases in Fig. 9.

Even if there is a difference of pocket shape and number, these pocket numbers cannot change the load carrying capacity. The number of pocket makes only the number of perturbation in pressure distribution. In perturbation property in the pressure distribution, it has a part of negative and positive pressure simultaneously along the mean pressure. As a slope of slider increases, its negative pressure effect may decreases. However, as a slider speed increases, its negative pressure effect appear clearly. So, the number does not increase the load capacity. However, appropriate slope is an important factor to increase the load carrying capacity. Too many pocket numbers does not reduce the load carrying capacity and show the same load carrying capacity.

5.Conclusion

1. The slider bearing with different square pocket of film thickness ratio(m) about 2.5 shows the maximum load carrying capacity. So, this slope is the most appropriate value to keep the maximum load carrying capacity. By increasing height of a pocket, its effect shows negligible in improving load carrying capacity.
2. The bearing having a pocket more than one cannot increase the load carrying capacity. So the bearing with too many pocket of low height shows the same effects as no-pocket slider bearing. As the height of pocket decreases, the pressure distribution shows the same shape as that of slider bearing with no-pocket.
3. As the sliding slope increases, the position of maximum pressure moves to exit position along the direction of bearing length and moves to center position along the direction of bearing width.

Figure

Fig. 1.Slider bearing with square pocket slider

Fig. 2.Discretization of slider bearing

Fig. 3.Pressure distribution of 1-pocket slider bearing according to its slope for hf = 0.09mm

Fig. 4.Pressure distribution of 9-pocket slider bearing according to its slope for hf = 0.09mm

Fig. 5.Pressure distribution of 25-pocket slider bearing according to its slope for hf = 0.09mm

Fig. 6.Pressure distribution of center line of 1-pocket slider according to its slope for hf = 0.09mm

Fig. 7.Pressure distribution of the center line of 9- pocket slider according to its slope for hf = 0.09mm

Fig. 8.Pressure distribution of the center line of 25- pocket slider according to its slope for hf = 0.09mm

Fig. 9.Load capacity of slider bearing with square pocket surface according to its slope at hf = 0.09mm

Table

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