Issue 
Mechanics & Industry
Volume 22, 2021



Article Number  1  
Number of page(s)  11  
DOI  https://doi.org/10.1051/meca/2020099  
Published online  08 March 2021 
Regular Article
An optimization research on groove textures of a journal bearing using particle swarm optimization algorithm
College of Power and Energy Engineering, Harbin Engineering University, Harbin, PR China
^{*} email: ssdpz041@outlook.com
Received:
14
September
2020
Accepted:
18
December
2020
This study aims to optimize the distributions of groove textures in a journal bearing to reduce its friction coefficient. Firstly, A lubrication model of a groove textured journal bearing is established, and the finite difference and overrelaxation iterative methods are used to numerically solve the model. Then, the friction coefficient is adopted as the fitness function and the groove lengths are optimized by particle swarm optimization (PSO) algorithm to evolve the optimal distributions. Furthermore, the effects of eccentricity ratios and rotary speeds on optimal distributions of groove textures are also discussed. The numerical results show the optimal distributions of groove textures are like trapeziums under different eccentricity ratios and rotary speeds, and the trapeziums become slenderer with increasing of eccentricity ratios. It is also found that the reductions of friction coefficients by optimal groove textures are more significant under lower eccentricity ratios. Briefly, this study may provide guidance on surface texture design to improve the tribological performance of journal bearings.
Key words: Journal bearing / groove textures / friction coefficient / PSO algorithm
© X. Zhang et al., Hosted by EDP Sciences 2021
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
As a feasible way to conserve oil and debris on contact surfaces, surface textures have been researched over past decades and widely used in many applications [1], such as vibration cutting tools [2], mechanical seals [3], gas face seals [4], slipper bearing [5], thrust bearing [6], piston rings [7], and medical devices [8,9]. These researches showed the applying of surface textures can reduce wear and increase load carrying capacity, which is helpful to prolong service lives of mechanical components, save energy and protect environment.
The journal bearing with surface textures also aroused general concerns of researchers, and the presetting of texture distributions is commonly used in their studies. In order to maximize load carrying capacity and minimize friction torque, Shinde and Pawar [10] adopted Taguchi and grey relational analysis methods to design groove textures with considering groove location, width, gap, height, and numbers. Their optimal solution can increase load carrying capacity by 51.01% and reduce friction torque by 9.84%. Kango et al. [11] compared the bearing performance between grooves and spherical textures under given eccentricity ratios. Their results showed when compared with spherical textures, grooves can show maximum reductions in bearing performance parameters. Manser et al. [12] showed the bearing performance can be positively affected by partial textures, and the optimal locations are depending on the working conditions and geometry parameters. Yu et al. [13] and Lin et al. [14] showed the load carrying capacity can be increased when textures are in rising part of pressure field, and vice versa. However, some studies have led to conflicting conclusions. TalaIghil et al. [15,16] indicated the extra hydrodynamic lift in bearing can be generated when textures are in declining part of pressure field. Shinde and Pawar [17] showed among three partial grooving distributions (90°–180°, 90°–270°, 90°–360°), the first distribution can maximize pressure increase and the last distribution can minimize friction loss. Their study indicated the optimal grooving distribution may depend on the optimization goal.
The above literatures have made important contributions on texture researches, but the texture distributions are preset in their researches, which probably miss the global optimal design. To resolve this issue, some optimization algorithms are adopted by researchers, such as GA (genetic algorithm) [18–20] and neural network [21]. To design an optimal bushing profile for a journal bearing, Pang et al. [18] conducted a multiobjectives optimization (minimum friction power loss and minimum leakage flowrate) by NSGAII (modified nondominant sort in genetic algorithm). Their results showed the optimal profile can be obtained at the profile of order n = 2, where n is the order of Fourier series and can be determined by the gradual method for increasing Fourier series orders. To obtain minimum friction coefficients of a journal bearing and bearing slider, Zhang et al. [19,20] adopted GA to optimize the coverage area of circular dimples. In their researches, bearing surface is divided into certain numbers of grids and the dimple in each grid center existing or not is marked with 1 or 0, respectively. Then GA is used to evolve the solution process and the final optimal coverage area is like a semielliptical shape. Sinanoğlu et al. [21] experimentally and theoretically researched the influences of shaft surface textures on film pressure and consequently on load by proposed neural network. The shaft surfaces included two types: trapezoidal and saw surfaces. Their results showed the shaft with trapezoidal surface has larger load carrying capacity than the shaft with saw surface.
The above literatures have showed the genetic algorithm and neural network can be successfully employed in surface texture design for journal bearings, while the PSO algorithm may provide a more convenient way for this issue. Particle swarm optimization (PSO), presented by Eberhart and Kennedy [22,23] in 1995, is an optimization method motivated by behaviors of bird flocking/roosting. In view of this algorithm, the individual members establish a social network and can benefit from previous experiences and discoveries of the other members. PSO algorithm is easier to implement because the swarm are updated only by updating the particle velocity and position vectors, which shows this approach has great potentials for use in the designs for air foil bearing [24], rolling element bearing [25] and magnetorheological (MR) bearing [26].
Although PSO has been used in some previous researches, few scholars adopted this algorithm to optimize surface textures for a journal bearing. The novelty of this study is to optimize the distributions of groove textures in a journal bearing to reduce its friction coefficient by PSO algorithm. The effects of eccentricity ratios and rotary speeds on optimal distributions of groove textures are also discussed. Overall, the optimization idea in this study may be helpful for journal bearings to improve their tribological performance.
2 Description of a journal bearing with groove textures
Generally, textures located in convergent area is more favorable for improving tribological performance [19]. In this study, the convergent area is covered with groove textures, as illustrated in Figure 1. A detailed description of groove textures is illustrated in Figure 2. The detailed parameters of a journal bearing and groove textures are given in Tables 1 and 2. Note the oil is assumed as isoviscous incompressible fluid.
As Table 2 shows, θ_{s}, θ_{e}, L_{c}, L_{g} and d_{g} are fixed, while L_{a} is variable from 0 to 34 mm. Note the groove length 0 mm means there is no groove and groove length 34 mm means the longest groove is slightly shorter than bearing width. The groove numbers N_{g} is$${N}_{g}=\left[\frac{R\left({\theta}_{e}{\theta}_{s}\right)\times \left(\pi /180\right)}{\left({L}_{c}+{L}_{g}\right)}\right]\text{+1}$$(1)
According to the above geometric parameters, N_{g} = 13.
Fig. 1 A textured journal bearing. 
Fig. 2 Groove textures in bearing inner surface. 
Detailed parameters of a journal bearing.
Geometric parameters of groove textures.
3 Lubrication model
3.1 Film thickness
As illustrated in Figure 1, film thickness h can be obtained by equation (2) $$h=c\left[1+\u03f5\mathrm{cos}\left(\theta \varphi \right)\right]+{\delta}_{tex}$$(2)where c is the radial clearance, ε the eccentricity ratio (ε = e/c, e eccentricity), ϕ the attitude angle, δ_{tex} the clearance added by groove textures.
3.2 Reynolds equation
The Reynolds equation under steady operating conditions is shown below [19]$$\frac{\partial}{\partial x}\left(\frac{{h}^{3}}{12\mu}\cdot \frac{\partial p}{\partial x}\right)+\frac{\partial}{\partial y}\left(\frac{{h}^{3}}{12\mu}\cdot \frac{\partial p}{\partial y}\right)=\frac{U}{2}\frac{\partial h}{\partial x}$$(3)
where U is the relative velocity, µ the oil viscosity, p the film pressure. For the journal bearing, U = Rω, R is the journal radius, ω its angular velocity. The employed Reynolds equation can be obtained by replacing the variable x with Rθ in equation (3), as expressed in equation (4) $$\frac{1}{{R}^{2}}\frac{\partial}{\partial \theta}\left(\frac{{h}^{3}}{\mu}\cdot \frac{\partial p}{\partial \theta}\right)+\frac{\partial}{\partial y}\left(\frac{{h}^{3}}{\mu}\cdot \frac{\partial p}{\partial y}\right)=6\omega \frac{\partial h}{\partial \theta}$$(4)
3.3 Load carrying capacity
Integrate film pressure p over whole the computation domain, then load carrying capacity F_{b} is obtained, as expressed in equations (5) and (6) $$\{\begin{array}{l}{F}_{x}=+{\int}_{0}^{B}{\int}_{0}^{2\pi}pR\mathrm{sin}\theta d\theta dy\\ {F}_{z}={\int}_{0}^{B}{\int}_{0}^{2\pi}pR\mathrm{cos}\theta d\theta dy\end{array}$$(5) $${F}_{b}=\sqrt{{{F}_{x}}^{2}+{{F}_{z}}^{2}}$$(6)
3.4 Friction coefficient
The friction f_{b} arising from the shearing effect of oil can be calculated by equation (7) $${f}_{b}={\int}_{0}^{B}{\int}_{0}^{2\pi}\left(\frac{\mu U}{h}+\frac{h}{2R}\frac{\partial p}{\partial \theta}\right)Rd\theta dy$$(7)
Then the friction coefficient µ_{f} can be calculated by equation (8) $${\mu}_{f}={{f}_{b}/F}_{b}$$(8)
3.5 Numerical validation
In numerical solution, the bearing inner surface is divided into rectangular grids, and nodes numbers in circumferential and axial directions are n_{θ} and n_{y}, respectively. The equation (4) is discretized by finite difference method, and the pressure is solved by overrelaxation iterative method. The rupture area of oil film is determined by Reynolds boundary conditions, and the pressure in oil hole and both ends of bearing are zero. The pressure convergence criteria ε_{p} is 10^{−5}, and the solution process will be terminated if ΔP_{i}/P_{i} <ε_{p} is satisfied, then the friction coefficient can be calculated by equations (5)–(8). The programming platform is Intel Xeon E52630 at 2.3 GHz with 64GB RAM.
It is necessary to verify the model before subsequent analysis. The Sommerfeld number S are calculated based on the researched journal bearing, and the results are compared with Pinkus and Sternlicht [27], as illustrated in Table 3. It can be seen the Sommerfeld numbers calculated by this study agree well with literature's results, which shows good accuracy of this model.
4 Particle swarm optimization algorithm
General researches consider the grooves have an equal length for convenience. In the authors' opinion, grooves with unequal lengths may bring more benefits. In this study, groove lengths are optimized by PSO algorithm. Figure 3 shows the computational process of PSO, which includes following steps:
(1) Initialize a particle swarm with 30 particles to be randomly generated. The swarm size Np = 30 is selected from 20, 30, 50 and 70, which can balance the contradiction between computing time and population diversity. The dimension of each particle is N_{g}, and the particle elements are random groove lengths.
(2) Calculate the particle fitness according to initial swarm. The friction coefficient µ_{f} is adopted as the fitness function.
(3) Search the local best pbest and global best gbest.
(4) Perform the iterative calculation. This step is the key of PSO algorithm, which can be subdivided into following substeps:
(4a) Update the particle velocity vector v_{id} and position vector x_{id} $${{v}_{id}}^{T\text{+1}}={\omega}_{i}{{v}_{id}}^{T}+{c}_{1}{r}_{1}\left(pbes{{t}_{i}}^{T}{{x}_{id}}^{T}\right)+{c}_{2}{r}_{2}\left(gbes{t}^{T}{{x}_{id}}^{T}\right)$$(9) $${{x}_{id}}^{T+1}={{x}_{id}}^{T}+{{v}_{id}}^{T+1}$$(10)
where i is the particle number, i = 1, 2, 3, …, 30; d the N_{g} dimensions, d = 1, 2, 3, …, N_{g}; ω_{i} the inertia weight. According to Shi and Eberhart [28], PSO has the best chance to find global optimum if ω_{i} satisfies 0.9 ≤ ω_{i} ≤ 1.2, here ω_{i} = 0.9; c_{1} and c_{2} the learning factors, c_{1} = c_{2} = 2 [28]; r_{1} and r_{2} the random numbers between 0 and 1 [28]; pbesti^{T} the local best; gbest^{T} the global best; T the index of generations; T_{max} the maximum index of generations.
It is necessary to check if the new velocity and position vectors satisfy boundary conditions:$$\{\begin{array}{l}v\left(v>{v}_{\mathrm{max}}\right)={v}_{\mathrm{max}}\\ v\left(v<{v}_{\mathrm{min}}\right)={v}_{\mathrm{min}}\end{array}$$(11) $$\{\begin{array}{l}x\left(x>{x}_{\mathrm{max}}\right)={x}_{\mathrm{max}}\\ x\left(x<{x}_{\mathrm{min}}\right)={x}_{\mathrm{min}}\end{array}$$(12)
where v_{max} = 4, v_{min} = −4. The velocity range [−4, 4] is determined through a multitude of trials, which can help the particles avoid flying past good solution areas or trapping into local optimal solution [29]. x_{max} and x_{min} are the maximum and minimum of groove lengths, x_{max} = 34 mm, and x_{min} = 0 mm.
(4b) Update the particle fitness according to new swarm.
(4c) Update the local best pbest and global best gbest.
(4d) Exit the loop if T is greater than T_{max} = 100; Otherwise, return to (4a) and repeat (4a)–(4d).
(5) Output the final optimal solution: gbest and µf _{best}.
Fig. 3 Computational process of PSO algorithm. 
5 Results and discussions
The mesh refinement is firstly performed based on smooth bearing. The friction coefficients µ_{f} under ε = 0.2, 0.4, ϕ = 0 of different mesh schemes n_{θ}×n_{y} are shown in Figure 4, which shows µ_{f} tends to stabilize gradually with denser grids. Considering the accuracy and solving speed, 551 × 177 mesh is adopted. In this scheme, the single grid is approximately to a square with 0.2 mm side length.
Fig. 4 Friction coefficients µ_{f} of different mesh schemes under (a) ε = 0.2, (b) ε = 0.4. 
5.1 Optimal distributions of groove textures under fixed eccentricity ratio
Based on Section 4, the changes of µ_{f}_{best} under ε = 0.1, ϕ = 0 is illustrated in Figure 5. With increasing of generations from 1 to 30, the µ_{f}_{best} shows a decreasing trend, which indicates the swarm is in continuing evolution. When the generation is beyond 30, µ_{f} _{best} maintains a constant value 0.0607, which indicates the swarm has been in a stable state. The progressive changes of distributions under ε = 0.1, ϕ = 0 are shown in Figure 6. It starts from a distribution with random groove lengths, and then gradually evolves to the final optimal distribution, whose shape is like a trapezium.
To further explain the optimal distribution of groove textures, the distribution of groove textures with an equal length 34 mm is given as reference, and the groove number in reference is same with it in the optimal one, as illustrated in Figure 7.
Figure 8 shows the pressure distributions of smooth, optimal textured and reference textured bearings under ε = 0.1, ϕ = 0, and Figure 9 illustrates the comparisons of load carrying capacity, friction, and friction coefficients between three bearings. It can be seen the optimal and reference textured bearings generate greater film pressures than smooth bearing due to local hydrodynamic pressures generated by grooves, which yield larger load carrying capacities. Moreover, the optimal textured bearing also generates greater pressure and larger load carrying capacity than referenced textured bearing. It can be explained that for the reference textured bearing, grooves with an equal length can destroy the pressure generations at upper and lower boundaries, that is, grooves located at upper and lower boundaries can suppress the pressure generation here. In contrast, gradually shortened grooves in optimal textured bearing reduce this pressure suppression, which yield greater pressure and load carrying capacity. Meanwhile, the frictions among three bearings are almost same. Hence, based on equation (8), the optimal textured bearing has a minimum friction coefficient, then followed by the reference textured and smooth bearings, respectively.
Fig. 5 The µ_{f} _{best} varying with the generations under ε = 0.1, ϕ = 0. 
Fig. 6 The progressive changes of distributions under ε = 0.1, ϕ = 0. 
Fig. 7 The optimal and reference distributions of groove textures. 
Fig. 8 Pressure distributions under ε = 0.1, ϕ = 0 of (a) smooth bearing, (b) optimal textured bearing, (c) reference textured bearing. 
Fig. 9 Comparisons of (a) load carrying capacity, (b) friction, (c) friction coefficient, between smooth, optimal textured, and reference textured bearings under ε = 0.1, ϕ = 0. 
5.2 Effects of eccentricity ratios on optimal distributions of groove textures
In Section 5.1, the eccentricity ratio ε is 0.1. In this section, the effects of eccentricity ratios (ε = 0.1–0.5) on optimal distributions of groove textures are discussed and shown in Figure 10. It can be seen this optimal distribution is still like a trapezium with increasing of ε, but it becomes slenderer. The reduction of friction coefficient between smooth and optimal textured bearings δµ_{f} is defined in equation (13) $$\delta {\mu}_{f}=\frac{{\mu}_{f}{\mu}_{f}}{{\mu}_{f}}\times 100\%$$(13)
The friction coefficients µ_{f} and reductions δµ_{f} are also given. As illustrated in Figure 11, with increasing of ε, δµ_{f} gradually decreases from approximately 30% to 8%, which indicates the reductions of friction coefficients by optimal groove textures are more significant under lower eccentricity ratios.
Fig. 10 The optimal groove textures under ε = 0.1–0.5, ϕ = 0, N = 2000 rpm. 
Fig. 11 µ_{f} and δµ_{f} under ε = 0.1–0.5, ϕ = 0, N = 2000 rpm. 
5.3 Effects of rotary speeds on optimal distributions of groove textures
In Sections 5.1 and 5.2, the rotary speed is 2000 rpm. In this section, the effects of rotary speeds on optimal distributions of groove textures and δµ_{f} are discussed. The speeds 500, 1000, 1500, 2500 and 3000 rpm are considered. As the results of these cases are similar, only the 1000 and 3000 rpm cases are given, as illustrated in Figures 12–15. It can be seen the results in Figures 12–15 are similar with these in Figures 10 and 11, which indicates the optimal distributions of groove textures and trends of δµ_{f} are consistent under different rotary speeds.
Fig. 12 The optimal groove textures under ε = 0.1–0.5, ϕ = 0, N = 1000 rpm. 
Fig. 13 µ_{f} and δµ_{f} under ε = 0.1–0.5, ϕ = 0, N = 1000 rpm. 
Fig. 14 The optimal groove textures under ε = 0.1–0.5, ϕ = 0, N = 3000 rpm. 
Fig. 15 µ_{f} and δµ_{f} under ε = 0.1–0.5, ϕ = 0, N = 3000 rpm. 
6 Conclusions
This study focuses on optimizing the distributions of groove textures in a journal bearing to reduce its friction coefficient using PSO algorithm. Some conclusions are summarized below:

For the researched journal bearing, the optimal distribution of groove textures is like a trapezium. This distribution can reduce the pressure suppression caused by grooves located at upper and lower boundaries, which yield greater film pressure and load carrying capacity. This is the main reason to reduce the friction coefficient.

With increasing of eccentricity ratios (ε = 0.1–0.5), the optimal distributions of groove textures become slenderer, and the reductions of friction coefficients by optimal groove textures are more significant under lower eccentricity ratios.

The optimal distributions and reductions of friction coefficients are similar under different rotary speeds, which indicate the conclusions of this optimization have certain universality.
In future work, the authors will research the optimal groove textures for the journal bearing by choosing other optimization variables (such as groove depth and width) or other optimization goal (such as temperature rise), and the corresponding experiments will be performed to verify these studies.
Nomenclature
θ_{s} : Start position of textured region
θ_{e} : End position of textured region
L_{a} : Groove length in axial direction
L_{c} : Groove width in circumferential direction
L_{g} : Groove gap in circumferential direction
δ_{tex} : Variation clearance caused by groove textures
ω : Angular velocity of journal
F_{b} : Load carrying capacity
ε_{p} : Allowable precision for the solution of pressure
v_{id} : Particle velocity vector
x_{id} : Particle position vector
c_{1}, c_{2} : Learning factors
r_{1}, r_{2} : Random numbers between 0 and 1
x_{min} : Minimum groove length
x_{max} : Maximum groove length
T_{max} : Maximum index of generations
µ_{f} _{best} : The minimum µ_{f} in particle swarm during each generation
µ_{f} _{smooth} : Friction coefficient of smooth bearing
µ_{f} _{optimal} : Friction coefficient of optimal textured bearing
δµ_{f} : The reduction of friction coefficients between smooth and optimal textured bearings
Acknowledgments
This study is supported by National Natural Science Foundation of China (Grant No. 51809057).
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Cite this article as: X. Zhang, C. Liu, B. Zhao, An optimization research on groove textures of a journal bearing using particle swarm optimization algorithm, Mechanics & Industry 22, 1 (2021)
All Tables
All Figures
Fig. 1 A textured journal bearing. 

In the text 
Fig. 2 Groove textures in bearing inner surface. 

In the text 
Fig. 3 Computational process of PSO algorithm. 

In the text 
Fig. 4 Friction coefficients µ_{f} of different mesh schemes under (a) ε = 0.2, (b) ε = 0.4. 

In the text 
Fig. 5 The µ_{f} _{best} varying with the generations under ε = 0.1, ϕ = 0. 

In the text 
Fig. 6 The progressive changes of distributions under ε = 0.1, ϕ = 0. 

In the text 
Fig. 7 The optimal and reference distributions of groove textures. 

In the text 
Fig. 8 Pressure distributions under ε = 0.1, ϕ = 0 of (a) smooth bearing, (b) optimal textured bearing, (c) reference textured bearing. 

In the text 
Fig. 9 Comparisons of (a) load carrying capacity, (b) friction, (c) friction coefficient, between smooth, optimal textured, and reference textured bearings under ε = 0.1, ϕ = 0. 

In the text 
Fig. 10 The optimal groove textures under ε = 0.1–0.5, ϕ = 0, N = 2000 rpm. 

In the text 
Fig. 11 µ_{f} and δµ_{f} under ε = 0.1–0.5, ϕ = 0, N = 2000 rpm. 

In the text 
Fig. 12 The optimal groove textures under ε = 0.1–0.5, ϕ = 0, N = 1000 rpm. 

In the text 
Fig. 13 µ_{f} and δµ_{f} under ε = 0.1–0.5, ϕ = 0, N = 1000 rpm. 

In the text 
Fig. 14 The optimal groove textures under ε = 0.1–0.5, ϕ = 0, N = 3000 rpm. 

In the text 
Fig. 15 µ_{f} and δµ_{f} under ε = 0.1–0.5, ϕ = 0, N = 3000 rpm. 

In the text 
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