Effect of Surface Texture Depth with Various Geometric Shapes on the Steady-State Performance and Dynamic Stability of Noncircular Lobed Journal Bearings

Abstract

The widespread use of journal bearings in rotating machinery has intensified the need to optimize their operational performance. A key determinant of bearing efficiency lies in the lubricant film thickness between the rotor and bearing surface. Recent studies demonstrate that strategically engineered surface textures can effectively modulate film thickness and enhance both static and dynamic characteristics of bearing. This investigation specifically examines how cubic, cylindrical, and semi-ellipsoidal texture geometries at varying depths influence the performance of noncircular two-lobe bearings. Through finite element analysis, the study evaluates critical performance parameters including load capacity, rotor attitude angle, critical mass threshold, and whirl frequency ratio to establish texture–depth relationships with system stability and operational efficiency. The analysis reveals that texturing the maximum pressure zone in lobe 2 significantly enhances bearing performance, with semi-ellipsoidal, cylindrical and cubic geometries, respectively. Also, the results demonstrate that texture geometry and depth significantly influence two-lobe bearing performance. Optimal enhancements in load capacity, whirl frequency reduction, and critical rotor mass occur at texture depths below the bearings clearance space width, with performance peaking before gradually declining as depth increases further. Notably, texture application in low-pressure or cavitation regions generally yields diminished or even counterproductive effects. The findings demonstrate that cubic textures provide optimal bearing performance across all depth ranges, with cylindrical and semi-ellipsoidal geometries ranking second and third, respectively, in comparative analysis.

1. Introduction

Friction and wear represent critical factors in assessing mechanical systems’ tribological behavior, with surface characteristics serving as a key determinant of these properties. Recent research has increasingly focused on surface engineering techniques, particularly texturing approaches, as a promising strategy for enhancing lubrication performance and wear resistance.

Surface texturing offers significant tribological benefits across various mechanical components, including bearings, liners, piston assemblies, seals, and drilling tools, by optimizing friction characteristics, reducing wear rates, and improving lubrication efficiency. The pioneering work of Etsion and Burstein [1] on hemispherical surface texturing demonstrated its potential to significantly enhance mechanical seal performance through improved lubrication and reduced friction. This breakthrough spurred global research efforts, leading to extensive theoretical and experimental investigations into surface texturing techniques and their tribological benefits. An investigation by Ausas et al. [2] examined how cavitation modeling influences the lubrication analysis of micro-textured journal bearings. The study compared two approaches—the conventional Reynolds model and the Elrod–Adams P-θ model—in evaluating bearing performance under uncertain operating conditions. While both models yielded nearly identical predictions for load capacity and frictional torque, the Reynolds formulation demonstrated superior numerical stability, making it the preferred choice for such analyses. Furthermore, it was indicated that when journal bearings with a micro-textured inner surface are considered, the Reynolds model significantly leads to errors in estimating several variables, such as frictional torque. Therefore, when analyzing this type of micro-textured bearing, only the mass-conserving model developed by Elrod and Adams provides accurate results. Building upon previous research, Sinanoglu [3] systematically investigated how groove-like surface textures affect journal bearing performance, examining critical parameters including pattern geometry, size, and orientation. The study combined neural network modeling with experimental validation, demonstrating that textured surfaces significantly enhance load carrying capacity compared to smooth surfaces, with important implications for various tribological applications. Subsequently, Kango and Sharma [4] expanded this research by analyzing textured journal bearings with non-Newtonian lubricants using a power-law model. Their numerical simulations revealed that while shear-thickening fluids generally increase both load capacity and friction, specific texture patterns—particularly fully positive wave textures—optimize performance characteristics. Further advancing this field, Rao et al. [5] conducted comprehensive analyses of partially textured surfaces in both journal and thrust bearing configurations. The findings demonstrate that parallel groove textures on bearing surfaces can simultaneously enhance load carrying capacity while reducing friction in the lubricating film for both bearing types examined. Chan et al. [6] performed an elasto-hydrodynamic analysis of textured journal bearings, evaluating how different groove designs affect load capacity. By solving the Reynolds equation with a parallel iterative RBSOR method while accounting for bearing deformations, they demonstrated that flash-like grooves enhance load capacity more effectively than rectangular grooves, particularly at higher eccentricity ratios. Building on this work, Lin et al. [7] examined how large-scale texturing influences bearing performance, with particular attention to cavitation effects. Their findings demonstrated that the effectiveness of surface texturing is highly dependent on its spatial distribution, with textures positioned in pressure-enhancing regions significantly boosting load capacity, while those in cavitation-prone or pressure-reducing areas yield either minimal improvement or potentially adverse effects. Together, these studies highlight the importance of both texture geometry and strategic placement for optimizing bearing performance.

Yu et al. [8] examined the performance of partially textured journal bearings with grease lubrication using CFD analysis. Their investigation revealed that strategically positioned surface textures enhance load capacity while influencing friction characteristics, with texture geometry playing a critical role in determining these effects. Subsequently, Gu et al. [9] investigated textured bearing stability with complex lubricants, developing a comprehensive lubrication model incorporating piezo-viscous and shear-thinning effects. Their findings highlighted that partial grooving in specific bearing regions effectively reduces friction, particularly at lower eccentricity ratios, while emphasizing the significant impact of shear-thinning behavior on system stability. Senator and Rao [10] comprehensively reviewed the effects of partial surface texturing on journal bearings lubricated with Newtonian fluids. Their analysis confirmed that controlled surface texturing can simultaneously reduce friction coefficients and enhance load carrying capacity, though these benefits diminish at higher eccentricity ratios. Building on these findings, Sharma et al. [11] conducted numerical optimization of partially textured journal bearings using finite element analysis. Their work specifically investigated triangular textures in pressure-enhancement regions, systematically evaluating the influence of texture density and depth on bearing performance under iso-viscous, Newtonian lubrication conditions. The simulation results demonstrate that strategically positioned textures in the pressure-enhancing region of the bearing surface significantly improve steady-state performance. Optimal performance is achieved with shallow textures and low eccentricity ratios, yielding a maximum performance improvement factor of 2.198—quantified as the ratio of enhanced load capacity to reduced friction coefficient.

Recent studies have increasingly focused on dimensional uncertainty effects in surface textures for mechanical systems, particularly journal bearings. Research demonstrates that manufacturing variations in texture patterns significantly influence tribological performance, necessitating meticulous analysis of their effects. Dimensional uncertainties in surface texture design and production can lead to substantial deviations from optimal system performance parameters [12,13].

The dynamic stability analysis of mechanical systems, particularly journal bearing supports under critical operating conditions (e.g., heavy, oscillatory, or impact loads, and high-speed operations that induce motion instabilities), has long been a focal point in mechanical system condition monitoring research [14,15]. Investigators have employed various analytical models to examine perturbed motions of system components, such as rotors in journal bearings under critical operational states [16,17,18]. Both linear and nonlinear approximation models have been historically significant in tribological research for analyzing bearing system performance. In linear dynamic stability analysis of journal bearing systems, the rotor center’s perturbed motion within the bearing clearance space is typically constrained to limit-cycle perturbations around the static equilibrium point. This assumption facilitates the development of an equivalent vibrational model for a rotor supported by a lubricant film, characterized through equivalent stiffness and damping coefficients. Conversely, nonlinear modeling approaches—increasingly prevalent in recent studies—enable analysis under extreme conditions such as sudden load shocks or significant speed variations [19,20]. These methods involve time-step tracking of component positions through coupled solutions of motion equations and governing equations, followed by vibration behavior evaluation using modal analysis techniques (e.g., power spectrum analysis, bifurcation diagram and Poincaré map) over defined intervals [21,22]. The primary objective of these dynamic analyses is to quantitatively assess parameter effects on system performance and to optimize their design and adjustment, thereby preventing surface contact and component wear caused by rotor instabilities within bearing clearance space under actual operating conditions. The combined application of signal monitoring and processing techniques for precise frequency response characterization with dynamic stability analysis methods of mechanical systems, particularly journal bearings, has advanced significantly in recent years, as these signal processing approaches enable more accurate vibrational response analysis by filtering out erroneous data and random noise [23,24].

Sharma et al. [25] investigated the enhancement of static and dynamic performance in hydrodynamic circular journal bearings through triangular surface texturing. Their study examined variations in texture depth, quantity, and spatial distribution to evaluate their effects on steady-state behavior and dynamic properties, including lubricant film stiffness and damping coefficients, with comparisons made to conventional smooth bearings. At a constant eccentricity ratio of 0.6, results demonstrated that minimal texture depth optimizes threshold stability speed and stiffness, while maximum damping occurs at greater depths. Ramos et al. [26] explored the application of textured journal bearings to minimize viscous lubricant losses and enhance energy efficiency in rotating machinery. Their investigation focused on circular journal bearings with spherical textures, demonstrating that reduced shear adhesion forces could significantly decrease operational heat generation. This thermal reduction improves bearing performance and overall machine efficiency. Also, their numerical results confirmed that textured bearings effectively diminish shear adhesive forces in rotating systems. Han and Fu [27] subsequently examined how micro-grooves affect the lubrication performance of misaligned circular journal bearings with a rotating shaft. Their findings revealed that groove angular positioning critically influences bearing performance under misalignment conditions. Both load-bearing capability and friction force were found to increase proportionally with greater misalignment angles and higher aspect ratios. Tomar and Sharma [28] conducted a study examining the influence of spherical surface textures on the dynamic behavior of hole-entry hybrid spherical bearings. Their findings revealed that incorporating spherical micro-cavities on the bearing’s inner surface enhances rotor stability. Additionally, fully textured bearings exhibited a notable increase in fluid film stiffness coefficients, whereas damping coefficients demonstrated a declining trend. Later, Rao et al. [29] investigated partially textured circular journal bearings lubricated using micropolar and power-law lubricants. Their results demonstrated an improvement in load carrying capacity alongside a reduction in frictional forces, highlighting the benefits of surface texturing under non-Newtonian lubrication conditions.

Manser et al. [30] conducted a computational analysis examining the static performance of hydrodynamic circular journal bearings featuring cylindrical micro-textures and lubricated with micropolar fluids, while incorporating mass conservation principles. Their study employed the finite difference approximations to solve the modified Reynolds equation, integrating Elrod’s mass-preserving scheme. The results demonstrated that texturing the converging region of the bearing clearance space enhances load capacity while reducing the friction coefficient. However, a fully textured bearing surface was found to adversely affect performance metrics. Awasthi and Maan [31] conducted a novel investigation into how surface texturing affects the tribological behavior of hydrodynamic journal bearings operating under turbulent lubrication conditions. Their findings demonstrate that strategically positioned surface textures can significantly improve bearing performance by increasing load capacity, reducing friction coefficients, and expanding the dynamic stability range. Building upon this research, Nie et al. [32] performed a computational fluid dynamics (CFD) analysis to examine how surface texturing influences journal bearing load capacity. Their study revealed that texture placement critically affects hydrodynamic pressure profile and load-bearing capability, with optimal performance achieved when textures are located in the converging wedge gap region. Mishra and Aggarwal [33] conducted a comprehensive review of textured journal bearing performance, concluding that surface texturing consistently enhances tribological properties by simultaneously reducing friction and increasing load capacity. Singh and Kango [34] investigated the thermohydrodynamic behavior of partially slip-textured slider bearings under varying sliding speeds. Their comparative analysis of different texture geometries showed that square-textured slip bearings outperformed other configurations, delivering superior results in key performance metrics including average pressure, friction force, and lubricant temperature distribution. Wang et al. [35] conducted a comprehensive theoretical and experimental investigation of friction behavior in textured journal bearings. Their study developed a modified Reynolds equation incorporating longitudinal surface roughness to analyze both load capacity and frictional properties. They demonstrated that surface roughness effects become particularly significant when the dimple depth approaches the magnitude of the roughness profile. This finding highlights the critical interplay between surface texture geometry and inherent roughness in determining bearing performance characteristics. Wang et al. [36] conducted a CFD study examining how cylindrical surface textures influence journal bearing performance by their internal flow. This analysis revealed that positioning textures downstream of maximum pressure zones achieves an optimal balance between friction reduction and load capacity preservation. The research identified two competing mechanisms, including beneficial micro-hydrodynamic pressure generation and detrimental high-pressure zone disruption, affecting the load capacity of the bearing. Wang et al. [37] investigated the antifriction mechanisms of surface texture, revealing an optimal relationship between texture depth and elliptical bearings performance. Their research demonstrated that bearing capacity initially improves but subsequently declines as micro-texture depth increases, with peak performance occurring at a dimensionless texture depth of 0.03 where friction reaches its minimum while load capacity maximizes. Profito et al. [38] demonstrated that surface texturing enhances journal bearing performance through dual friction-reduction mechanisms. Their comprehensive analysis of laser-textured bearing surfaces revealed that microscopic cavitation effects concurrently lower both lubricant shear stress and effective viscosity, yielding significantly reduced friction coefficients compared to conventional smooth bearings. Gu et al. [39] developed an innovative optimization methodology for designing surface textures in journal bearings. Their study employed a multi-objective grey wolf algorithm to simultaneously optimize multiple texture parameters, creating tailored solutions for varying operational requirements. The research demonstrated that strategically designed textures enhance bearing performance by increasing minimum oil film thickness while minimizing asperity contact. Notably, their adaptive scale texture design showed exceptional performance across diverse operating conditions, achieving substantial hydrodynamic improvements and friction reduction. Recently, Kumar and Singh [40] conducted a CFD investigation examining how micro-textured surfaces affect hydrodynamic journal bearing performance under turbulent flow conditions. Their study evaluated four dimple geometries using water lubrication, comparing results against conventional non-textured bearings. The research systematically analyzed critical design parameters including dimple arrangement, dimensions, and distribution patterns, demonstrating their significant influence on pressure distribution and load capacity. Findings revealed that optimal texture geometry substantially enhance bearing performance characteristics compared to smooth surfaces.

A comprehensive literature review reveals a significant research gap regarding the steady-state and transient evaluation of textured noncircular lobed journal bearings. To address this, the current investigation systematically evaluates how three distinct texture geometries (cubic, cylindrical, and semi-ellipsoidal) influence both steady-state performance and stability in two-lobe hydrodynamic bearings. The study employs finite element discretization to solve the governing equations, enabling detailed examination of multiple performance parameters, including pressure distribution, load carrying capacity, attitude angle, whirl frequency ratio, and critical rotor mass at instability threshold across various texture configurations. This study further examines how variations in texture depth influence two-lobe bearing performance when positioned within the high-pressure region of the lubricant film.

2. Theory

Surface textures represent intentionally engineered topographical features with precisely controlled geometric parameters, standing in stark contrast to the random surface roughness that emerges unpredictably from manufacturing processes. These designed microstructures exhibit repeatable patterns of cavities and protrusions—including but not limited to cubic depressions, cylindrical pillars, spherical dimples, and complex hybrid configurations—each carefully dimensioned and spatially organized to achieve specific functional outcomes. In contrast to the stochastic nature of surface roughness, textured surfaces are precision-engineered to enhance the tribological performance of dynamic interfaces—as exemplified by the tread patterns of automotive tires that improve road adhesion and the dimpled topography of golf balls that strategically reduce aerodynamic drag [25,33,41].

Textured surfaces demonstrate beneficial effects in lubrication applications, including mechanical seals, thrust bearing pads, and journal bearing inner surfaces. These surfaces are primarily employed in rotating systems to improve both static and dynamic performance parameters of bearing supports. Micro-textured surfaces (with dimensions ranging approximately from 1 to 10 μm) can control lubrication properties in mechanical seals and fluid film bearings, modifying load capacity, frictional torque, and hydrodynamic stiffness and damping parameters. Consequently, proper selection of texture geometry and placement significantly improves bearing system performance. These micro-textures function as lubrication reservoirs during startup, capture particulate contaminants, and protect contacting surfaces while improving hydrodynamic efficiency. Contemporary fabrication methods enable complex 3D surface architectures through deposition, energy beam processing, chemical etching, and mechanical texturing techniques [26,42].

Engineered surface features on bearing shell interiors designed to improve system performance typically divide into two principal forms. Protruding microstructures or positive textures with carefully specified dimensions and shapes are manufactured on the inner bearing surface while corresponding indentations or negative textures with precise depth characteristics are produced within the same surface area [33,42]. Figure 1 illustrates examples of positive and negative textures with different geometric shapes.

Surface textures created on journal bearing shells, due to variations in the thickness of the lubricating oil film trapped in the bearing clearance space, can contribute to improving the static performance parameters and dynamic stability of the rotor-bearing systems. In Figure 2, a textured two-lobe journal bearing is shown.

3. Governing Equations

This portion presents the fundamental equations describing fluid film lubrication in geometrically-modified multi-lobe journal bearings. The mathematical framework facilitates the examination of the lubricant pressure variations and quantification of operational performance metrics of the bearing, specifically the load carrying capacity, attitude angle, and dynamic response indicators, including the critical mass and whirling frequency of the rotor.

3.1. Reynolds Equation

The nondimensional Reynolds equation describing fluid film lubrication in journal bearings, considering isothermal operation, constant-density lubricants, and perfect rotor-shell alignment, takes the following form [43]:

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\theta }}\left[{\overline{h}}_n^3{\displaystyle \frac{\mathsf{\partial }\overline{P}}{\mathsf{\partial }\theta }}\right]+{\left({\displaystyle \frac{D}{L}}\right)}^2{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\overline{z}}}\left[{\overline{h}}_n^3{\displaystyle \frac{\mathsf{\partial }\overline{P}}{\mathsf{\partial }\overline{z}}}\right]=6\mu {\displaystyle \frac{\mathsf{\partial }{\overline{h}}_n}{\mathsf{\partial }\theta }}+12\mu {\displaystyle \frac{\mathsf{\partial }{\overline{h}}_n}{\mathsf{\partial }\tau }}$$

(1)

The dimensionless quantities appearing in Equation (1) are characterized as:

$$\theta ={\displaystyle \frac{x}{R}}, \overline{z}={\displaystyle \frac{z}{L/2}}, \overline{P}={\displaystyle \frac{C_m^2}{{\mu }_{bf}\omega R^2}}P , {\overline{h}}_n={\displaystyle \frac{h_n}{C_m}}, u=R\omega , \mathsf{\tau }=t\omega$$

(2)

The lubricating film thickness distribution throughout the engineered lobed journal bearing with surface textures is given by the subsequent relation [44].

$${\overline{h}}_n={\overline{h}}_{n-smooth}+\mathsf{\Delta }\overline{h}(\theta ,z)$$

(3)

Equation (3) incorporates ${\overline{h}}_{n-smooth}$ as the lubricant film dimension separating the rotating shaft and the bearing shell for an untextured circular bearing at all circumferential positions, mathematically described by Equation (4). Furthermore, $\mathsf{\Delta }\overline{h}(\theta ,z)$ quantifies the depth of engineered surface textures—including cubic, cylindrical, and semi-ellipsoidal geometries—patterned on the bearing’s interior surface [14].

$${\overline{h}}_n={\displaystyle \frac{h_n}{C_m}}=\left({\displaystyle \frac{1}{\delta }}-{\displaystyle \frac{X_j}{C_m}}\cos \theta -{\displaystyle \frac{Y_j}{C_m}}\sin \theta +\left({\displaystyle \frac{1}{\delta }}-1\right)\cos \left(\theta -{\theta }_0^n\right)\right)$$

(4)

Within Equation (4), symbol n denotes the lobe count of the two-lobe journal bearing, while variables $X_j$ and $Y_j$ specify the journal center location in static equilibrium conditions relative to the reference coordinate system. Angle ${\theta }_0^n$ corresponds to the lobe’s centerline orientation with respect to the X-axis. The noncircularity coefficient δ, characterizing the bearing geometry, is determined as the ratio between the smallest radial gap (C_m) and the maximum conventional radial clearance (C) under concentric shaft-bearing conditions.

3.2. The Geometric Shape of the Utilized Textures

This investigation examines how different texture geometries—including cubic, cylindrical, and semi-ellipsoidal forms—with controlled depth variations influence system behavior when selectively applied to specific regions of a noncircular two-lobe bearing’s interior surface (Figure 3). The analysis focuses on both steady-state operational characteristics and dynamic stability metrics of the rotor-bearing set.

As illustrated in Figure 4, the parameters $r_x$, $r_y$ and $r_z$ represent the radius in the circumferential direction, depth, and longitudinal radius of cylindrical and semi-ellipsoidal dimples along the coordinate axes x, y, and z, respectively. Similarly, the spatial dimensions, ${2a}_x$, $a_y$, ${2a}_z$ correspond to the length, height, and width of the cubic textures, respectively.

In the cylindrical and semi-ellipsoidal textures shown in Figure 4, $r_x=r_z=r$, and the equations defining cubic, cylindrical, and semi-ellipsoidal geometries are as follows:

$$\left|x-x_c\right|=a_x \mathrm{a}\mathrm{n}\mathrm{d} \left|z-z_c\right|=a_z$$

(5)

$${\left(x-x_c\right)}^2+{\left(z-z_c\right)}^2=r^2$$

(6)

$${\left({\displaystyle \frac{x-x_c}{r_x}}\right)}^2+{\left({\displaystyle \frac{y-y_c}{r_y}}\right)}^2+{\left({\displaystyle \frac{z-z_c}{r_z}}\right)}^2=1$$

(7)

where $O_c$ is the center of each texture in local coordinates $\left(x_c,y_c,z_c\right)$, located on the surface of the bearing without the texture, i.e., at $y_c=0$. Additionally, the depth of each texture, as shown in Figure 3, is equal to:

$$\mathsf{\Delta }h=r_y \mathit{for Cylindrical and Semi-ellipsoidal Textures}$$

(8)

$$\mathsf{\Delta }h=a_y \mathit{for Cubic Texture}$$

(9)

In the top view plan shown in Figure 5, the method used to discretize the bearing surface around each texture is illustrated in both the longitudinal and circumferential directions. The parameters ${\mathrm{l}}_{\mathrm{x}}$ and ${\mathrm{l}}_{\mathrm{z}}$ represent the distances between the textures in the circumferential and longitudinal directions, respectively. Furthermore, the parameters $∆\mathrm{x}$ and $∆\mathrm{z}$ denote extremely small distances between the texture edge and the asymptotic auxiliary line. These distances are essential for precise simulations of the texture walls when meshing the bearing surface along the $\mathrm{X}$ and $\mathrm{Z}$ coordinate axes.

3.3. Dynamic Behavior Analysis of Rotor Using a Linear Model

According to the simulation of the displacement of the rotor center in the form of limit cycle oscillatory disturbances as shown in Figure 6, the components of the rotor center motion in the direction of the coordinate axes can be expressed as follows [17]:

$${\overline{X}}^{\prime }=Re\left(|{\overline{X}}^{\prime }|e^{j\gamma \tau }\right) , {\overline{Y}}^{\prime }=Re\left(|{\overline{Y}}^{\prime }|e^{j\gamma \tau }\right)$$

(10)

Within this analytical framework, ${\overline{X}}^{\prime }$ and ${\overline{Y}}^{\prime }$ characterize the rotor center’s oscillatory displacements along coordinate directions referenced to its equilibrium position, with $|{\overline{X}}^{\prime }|$ and $|{\overline{Y}}^{\prime }|$ quantifying their respective magnitudes. The operator Re extracts the real component of complex expressions, where j signifies the imaginary unit ($\sqrt{-1}$). The parameter γ, known as the whirl frequency ratio, describes the relationship between the angular velocity of the perturbation motion of the shaft center around the static equilibrium point $\left({\omega }_P\right)$ and the rotor’s spin velocity (ω). The dynamic pressure fields $\left({{\overline{P}}^{\prime }}_X,{{\overline{P}}^{\prime }}_Y\right)$ across the solution domain are obtained through Equation (11), derived by applying partial differentiation to the nondimensional Reynolds equation (Equation 1) with respect to ${\overline{X}}^{\prime }$ and ${\overline{Y}}^{\prime }$, followed by evaluation at null perturbation conditions.

$${\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\theta }}\left[{\overline{h}}_0^3{\displaystyle \frac{\mathsf{\partial }{\overline{P}}_I^{\prime }}{\mathsf{\partial }\theta }}\right]+{\left({\displaystyle \frac{D}{L}}\right)}^2{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\overline{z}}}\left[{\overline{h}}_0^3{\displaystyle \frac{\mathsf{\partial }{\overline{P}}_I^{\prime }}{\mathsf{\partial }\overline{z}}}\right]={\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\theta }}\left\{\left[3{\overline{h}}_0^2{\displaystyle \frac{\mathsf{\partial }{\overline{P}}_0}{\mathsf{\partial }\theta }}-6{\mu }_{rel}\right]f\left(\theta \right)\right\}+{\left({\displaystyle \frac{D}{L}}\right)}^2{\displaystyle \frac{\mathsf{\partial }}{\mathsf{\partial }\overline{z}}}\left[3{\overline{h}}_0^{2}f\left(\theta \right){\displaystyle \frac{\mathsf{\partial }{\overline{P}}_0}{\mathsf{\partial }\overline{z}}}\right]-12j\gamma {\mu }_{rel}f\left(\theta \right)$$

(11)

The parameters ${{\overline{P}}^{\prime }}_I$ and $f\left(\theta \right)$ in the above equation are:

$${{\overline{P}}^{\prime }}_I={\overline{P}}_I+j\gamma {\overline{P}}_{\dot{I}} , I=X,Y$$

(12)

$$f\left(\theta \right)=\left\{\begin{array}{c}cos\theta for {{\overline{P}}^{\prime }}_X\\ sin\theta for {{\overline{P}}^{\prime }}_X\end{array}\right.$$

(13)

In the present study, the finite element numerical method (FEM) based on the Galerkin framework with the isoparametric rectangular elements is used to solve the governing lubrication equation of two-lobe textured journal bearings [14].

3.4. Static Performance Characteristics of the Lobed Bearing

The steady state (static) performance characteristic parameters of the investigated journal bearing can be extracted by determining the position of the static equilibrium point of the rotor center and the steady state pressure distribution created in the lubricating oil film.

3.4.1. Load Carrying Capacity

The load components that can be carried by the bearing along the coordinate axes can be calculated from Equation (14).

$$\left[\begin{array}{c}{\overline{W}}_{X0}\\ {\overline{W}}_{Y0}\end{array}\right]=\sum _{n=1}^N\left[\begin{array}{c}{\overline{W}}_{X0}^n\\ {\overline{W}}_{Y0}^n\end{array}\right]=\sum _{n=1}^N{\mathsf{\int }}_{{\theta }_1^n}^{{\theta }_2^n}{\mathsf{\int }}_{-1}^{+1}{\overline{P}}_{0n}\left[\begin{array}{c}cos\theta \\ sin\theta \end{array}\right]d\overline{z}d\theta$$

(14)

The overall load carrying capacity of the bearing is determined as follows.

$${\overline{W}}_0={\displaystyle \frac{2C_m^2}{{\mu }_{bf}\omega R^{3}L}}{W}_0=\sqrt{{\overline{W}}_{0x}^2+{\overline{W}}_{0y}^2}$$

(15)

3.4.2. Attitude Angle

The attitude angle in two-lobe noncircular journal bearings can be determined using the following equation, based on the static equilibrium position of the rotor center, to ensure the vertical resultant load is achieved.

$${\phi }_0=Arc\mathit{tan}\left(-{\displaystyle \frac{X_{J0}}{Y_{J0}}}\right)$$

(16)

3.5. Dynamic Stability of Bearings Based on the Linear Model

According to Figure 7, the dynamic stability of noncircular journal bearings in the linear model is primarily determined by the equivalent stiffness and damping coefficients. These coefficients, which characterize the lubricating film, are derived from solving the governing equations.

Equation (17) describes the calculation of dynamic force components generated by the lubricant film due to the perturbation behavior of the rotor center

$$\left[\begin{array}{c}{\overline{F}}_X\\ {\overline{F}}_Y\end{array}\right]={\displaystyle \frac{2C_m^2}{{\mu }_{bf}\omega R^{3}L}}\left[\begin{array}{c}{F}_X\\ F_Y\end{array}\right]=\sum _{n=1}^N{\int }_{-1}^{+1}{\int }_{{\theta }_1^n}^{{\theta }_2^n}{\overline{P}}_n\left[\begin{array}{c}cos\theta \\ sin\theta \end{array}\right] d\theta d\overline{z}$$

(17)

The vector representation of fluid film forces in a dynamic state can be expressed as the sum of steady-state equilibrium forces and unstable disturbance forces, as follows:

$$\left[\begin{array}{c}{\overline{F}}_X\\ {\overline{F}}_Y\end{array}\right]=\left[\begin{array}{c}{\overline{F}}_{X0}\\ {\overline{F}}_{Y0}\end{array}\right]+\left[\begin{array}{c}\mathsf{\Delta }{\overline{F}}_X\\ \mathsf{\Delta }{\overline{F}}_Y\end{array}\right]$$

(18)

In Equation (18), $\left({\mathsf{\Delta }\overline{\mathrm{F}}}_{\mathrm{X}},{\mathsf{\Delta }\overline{\mathrm{F}}}_{\mathrm{Y}}\right)$ represent the components of the unsteady disturbance force affecting the rotor in the dynamic state along the coordinate axes. In the linear dynamic analysis model, these perturbed forces $\left({\mathsf{\Delta }\overline{\mathrm{F}}}_{\mathrm{X}},{\mathsf{\Delta }\overline{\mathrm{F}}}_{\mathrm{Y}}\right)$ are considered as the linear functions of the dynamic components of rotor displacement $\left({\overline{\mathrm{X}}}^{\prime },{\overline{\mathrm{Y}}}^{\prime }\right)$ and velocity $\left({\dot{\overline{\mathrm{X}}}}^{\prime },{\dot{\overline{\mathrm{Y}}}}^{\prime }\right)$.

$$\left[\begin{array}{c}\mathsf{\Delta }{\overline{F}}_X\\ \mathsf{\Delta }{\overline{F}}_X\end{array}\right]=-\left[\begin{array}{cc}{\overline{S}}_{XX}& {\overline{S}}_{XY}\\ {\overline{S}}_{YX}& {\overline{S}}_{YY}\end{array}\right]\left\{\begin{array}{c}{\overline{X}}^{\prime }\\ {\overline{Y}}^{\prime }\end{array}\right\}-\left[\begin{array}{cc}{\overline{B}}_{XX}& {\overline{B}}_{XY}\\ {\overline{B}}_{YX}& {\overline{B}}_{YY}\end{array}\right]\left\{\begin{array}{c}\dot{{\overline{X}}^{\prime }}\\ \dot{{\overline{Y}}^{\prime }}\end{array}\right\}$$

(19)

In the above equation, the parameters ${\overline{S}}_{ij}$ and ${\overline{B}}_{ij}$ where $i, j=X, Y$ represent the equivalent stiffness and damping coefficients of the lubricating film in the linear dynamic model. By referencing Equations (14), (17), and (19) and separating the variables, the dimensionless forms of the equivalent stiffness and damping components of the lubricating fluid film are derived based on the following relationships.

$$\left[\begin{array}{cc}{\overline{S}}_{XX}& {\overline{S}}_{XY}\\ {\overline{S}}_{YX}& {\overline{S}}_{YY}\end{array}\right]=\sum _{n=1}^N{\int }_{-1}^{+1}{\int }_{{\theta }_1^n}^{{\theta }_2^n}\left[\begin{array}{c}{\overline{P}}_X\\ {\overline{P}}_Y\end{array}\right] \left\{\begin{array}{cc}cos\theta & sin\theta \end{array}\right\} d\theta d\overline{z}$$

(20)

$$\left[\begin{array}{cc}{\overline{B}}_{XX}& {\overline{B}}_{XY}\\ {\overline{B}}_{YX}& {\overline{B}}_{YY}\end{array}\right]=\sum _{n=1}^N{\int }_{-1}^{+1}{\int }_{{\theta }_1^n}^{{\theta }_2^n}\left[\begin{array}{c}{\overline{P}}_{\dot{X}}\\ {\overline{P}}_{\dot{Y}}\end{array}\right] \left\{\begin{array}{cc}cos\theta & sin\theta \end{array}\right\} d\theta d\overline{z}$$

(21)

The equations describing the dynamic displacement of the rotor center supported by hydrodynamic two-lobe journal bearings during periodic rotational movement around the static equilibrium point along the coordinate axes $\left(X,Y\right)$ are expressed as follows:

$${\overline{M}}_J\left\{\begin{array}{c}{\ddot{\overline{X}}}^{\prime }\\ {\ddot{\overline{Y}}}^{\prime }\end{array}\right\}+\left[\begin{array}{cc}{\overline{B}}_{XX}& {\overline{B}}_{XY}\\ {\overline{B}}_{YX}& {\overline{B}}_{YY}\end{array}\right]\left\{\begin{array}{c}\dot{{\overline{X}}^{\prime }}\\ \dot{{\overline{Y}}^{\prime }}\end{array}\right\}+\left[\begin{array}{cc}{\overline{S}}_{XX}& {\overline{S}}_{XY}\\ {\overline{S}}_{YX}& {\overline{S}}_{YY}\end{array}\right]\left\{\begin{array}{c}{\overline{X}}^{\prime }\\ {\overline{Y}}^{\prime }\end{array}\right\}=0$$

(22)

In Equation (22), the parameter ${\overline{M}}_J=M_J\left(2C_m^3\omega /\mathrm{\mu }{R}^{3}L\right)$ represents the dimensionless mass of the rotor. By substituting the values and derivatives of the parameters ${\overline{X}}^{\prime }$ and ${\overline{Y}}^{\prime }$ from Equation (10) into Equation (22), the resulting expression can be derived as follows:

$$\left[\begin{array}{cc}-{\overline{M}}_J{\gamma }^2+{\overline{S}}_{XX}+i\gamma {\overline{B}}_{XX}& {\overline{S}}_{XY}+i\gamma {\overline{B}}_{XY}\\ {\overline{S}}_{YX}+i\gamma {\overline{B}}_{YX}& -{\overline{M}}_J{\gamma }^2+{\overline{S}}_{YY}+i\gamma {\overline{B}}_{YY}\end{array}\right]\left\{\begin{array}{c}{\overline{X}}^{\prime }\\ {\overline{Y}}^{\prime }\end{array}\right\}=0$$

(23)

A non-trivial solution of the Equation (23) is possible if the coefficient matrix is singular. Consequently, the characteristic equation of the linear dynamic model, under the assumption of limit cycle disturbances affecting the displacement of the rotor center at the instability threshold, is expressed as:

$$\left(-{\overline{M}}_J{\gamma }^2+{\overline{S}}_{XX}+i\gamma {\overline{B}}_{XX}\right)\left(-{\overline{M}}_J{\gamma }^2+{\overline{S}}_{YY}+i\gamma {\overline{B}}_{YY}\right)-\left({\overline{S}}_{XY}+i\gamma {\overline{B}}_{XY}\right)\left({\overline{S}}_{YX}+i\gamma {\overline{B}}_{YX}\right)=0$$

(24)

Separating the real and imaginary parts of Equation (24) and setting them equal to zero yields two distinct Equations, (25) and (26). By assuming an initial value for the parameter γ and calculating the dynamic coefficients corresponding to it from Equations (20) and (21), the necessary parameters for analyzing the subsequent equations are provided.

$${\overline{M}}_J{\gamma }^2={\displaystyle \frac{\left({\overline{S}}_{XX}{\overline{B}}_{YY}+{\overline{S}}_{YY}{\overline{B}}_{XX}-{\overline{S}}_{XY}{\overline{B}}_{YX}-{\overline{S}}_{YX}{\overline{B}}_{XY}\right)}{\left({\overline{B}}_{XX}+{\overline{B}}_{YY}\right)}}$$

(25)

$${\gamma }^2={\displaystyle \frac{\left[\left({\overline{S}}_{XX}-{\overline{M}}_J{\gamma }^2\right)\left({\overline{S}}_{YY}-{\overline{M}}_J{\gamma }^2\right)-{\overline{S}}_{XY}{\overline{S}}_{YX}\right]}{\left({\overline{B}}_{XX}{\overline{B}}_{YY}-{\overline{B}}_{XY}{\overline{B}}_{YX}\right)}}$$

(26)

The equations above serve as a criterion for determining the margin of dynamic instability of the system based on the dimensionless mass parameter of the rotor and the whirl frequency ratio of the rotor center $\left({\overline{M}}_J,\gamma \right)$. By determining the final value of the corrected whirl frequency ratio and completing the steps of the mentioned iteration process in the flowchart of Figure 8, the stability field of the investigated rotor-bearing system can be determined using the following equation, based on the linear dynamic model.

$${\overline{M}}_C={\displaystyle \frac{\left({\overline{S}}_{XX}{\overline{B}}_{YY}+{\overline{S}}_{YY}{\overline{B}}_{XX}-{\overline{S}}_{XY}{\overline{B}}_{YX}-{\overline{S}}_{YX}{\overline{B}}_{XY}\right)}{\left[\left({\overline{B}}_{XX}+{\overline{B}}_{YY}\right){\gamma }^2\right]}}$$

(27)

In Equation (27), ${\overline{M}}_C$ represents the dimensionless critical mass parameter of the rotor, which positions the system at the threshold of instability. By starting with an estimated value for γ, the critical mass parameter and the whirl frequency ratio at the dynamic instability threshold of the modeled rotating system can be determined through iterative calculations.

4. Results and Discussion

To validate the accuracy of the developed computational code for analyzing the hydrodynamic performance of two-lobe journal bearings with different surface textures, a comparison is conducted. The program’s output for the minimum thickness of the lubricating fluid film $\left({\mathrm{h}}_{\mathrm{m}\mathrm{i}\mathrm{n}}\right)$ and the attitude angle $\left({\mathsf{\varphi }}_0\right)$ of the rotor within the clearance space of the textured circular journal bearing is benchmarked against previously published results in reference [44].

The comparative data, presented in

Table 1. Minimum lubricant film thickness $\left({\mathrm{h}}_{min}\right)$ and attitude angle $\left({\mathsf{\varphi }}_0\right)$ in a circular journal bearing with cylindrical texture, $\mathsf{\epsilon }=0.6$, $L/D=1$, $r_x=r_z=1 \mathrm{m}\mathrm{m}$, $r_y=0.015 \mathrm{m}\mathrm{m}$.

${\mathbf{ϴ}}_1°$	${\mathbf{ϴ}}_2°$	${\mathbf{z}}_1$	${\mathbf{z}}_2$	${\begin{array}{c}{\mathbf{h}}_{\mathbf{m}\mathbf{i}\mathbf{n}}\\ \left(\mathbf{\mu }\mathbf{m}\right)\end{array}}^{\mathbf{a}}$	${\begin{array}{c}{\mathbf{h}}_{\mathbf{m}\mathbf{i}\mathbf{n}}\\ \left(\mathbf{\mu }\mathbf{m}\right)\end{array}}^{\mathbf{b}}$	${\mathbf{\phi }°}^{\mathbf{a}}$	${\mathbf{\phi }°}^{\mathbf{b}}$
0	90	0	0.5	11.75	11.63	48.5	47.7
0	45	0	0.25	11.96	11.76	50.4	51.3
175	220	0.12	0.5	12.18	12.25	49	49.3
180	225	0.2	0.5	12.11	12.21	49.6	49.5
185	230	0.12	0.5	12.19	12.15	49	48.7

a: Ref. [44]; b: present study.

, demonstrate a good agreement between the results of this study and those reported earlier. This alignment confirms the reliability and robustness of the generalized computational code used in this analysis.

Additionally,

Table 2. Effect of rotor eccentricity ratio $\left(\mathsf{\epsilon }\right)$ on attitude angle $\left({\mathsf{\varphi }}_0\right)$, stiffness $\left(\mathrm{S}\right)$, and damping $\left(\mathrm{B}\right)$ coefficients in two-lobe journal bearings with Newtonian lubricant, $\mathsf{\delta }=0.5$, $\mathrm{L}/\mathrm{D}=1$.

Bearings Characteristics	$\mathit{\epsilon }=0.2$		$\mathit{\epsilon }=0.432$
	Present Study	Ref. [45]	Present Study	Ref. [45]
$\phi$	90.41	90.37	70.93	69.69
$S_{XX}$	0.59	0.58	1.37	1.34
$S_{XY}$	−4.68	−4.79	−0.17	−0.16
$S_{YX}$	5.49	5.58	3.54	3.60
$S_{YY}$	8.78	8.93	5.22	5.36
$B_{XX}$	4.86	4.82	1.43	1.42
$B_{XY}=B_{YX}$	−4.39	−4.5	1.36	1.34
$B_{YY}$	17.82	17.99	7.81	7.67

examines the effect of rotor eccentricity on the equivalent stiffness and damping coefficients of the lubricant film in a two-lobe bearing with a Newtonian lubricant, as reported in Reference [45], and compares them with the results obtained in the present study. The comparison of the results presented in Table 2 shows a good agreement between the findings of the present study and the previously reported results, further confirming the accuracy of the developed computer program.

It is noteworthy that, in the following sections, the results presented in all diagrams are extracted for the parameters $\epsilon =0.5$, $L/D=1$ and $\delta =0.7$. The design parameters of the two-lobe noncircular journal bearing used in the present study are provided in

Table 3. The specifications of the analyzed two-lobe journal bearing.

Parameter	Definition	Number and Magnitude (Dimensionless)
$N_{xD}$	The number of textures in the circumferential direction	5
$N_{zD}$	Number of textures in the longitudinal direction	10
$R_x$	Dimensionless radius of textures in the circumferential direction	0.047619
$R_z$	Dimensionless radius of textures in the longitudinal direction	0.0238095
$\epsilon$	Eccentricity ratio	0.5
$z_1$	The starting point of the texture in the longitudinal direction	0.05
$z_2$	Termination point of the texture in the longitudinal direction	0.95

.

The analysis results examining both static and dynamic performance parameters of two-lobe bearings are presented subsequently. To facilitate direct comparison with actual operational data,

Table A1. Properties of the investigated rotor two-lobe journal bearing system.

Symbol	Variable	Value	Unit
${\rho }_f$	Density of oil lubricant	868	$\mathrm{K}\mathrm{g}/{\mathrm{m}}^3$
$\mu$	Oil film viscosity	0.065	$\mathrm{P}\mathrm{a}.\mathrm{s}$
$L$	Bearing length	0.1	$\mathrm{m}$
$R$	Journal radius	0.05	$\mathrm{m}$
$C_m$	Minimum clearance width	$120\times {10}^{-6}$	$\mathrm{m}$
$\omega$	Angular speed of rotor	3000	$\mathrm{r}\mathrm{p}\mathrm{m}$

and

Table A2. Applied numerical coefficients for obtaining the dimensional performance parameters of the analyzed two-lobe textured journal bearings.

Quantity	Unit	Parametric Coefficients	Nondimensional Form	Numerical Coefficients
$\mathrm{p}$	$\mathrm{p}\mathrm{a}$	$=\left(\mathsf{\mu }\mathsf{\omega }{\mathrm{R}}^2/{\mathrm{C}}_{\mathrm{m}}^2\right)$	$\overline{\mathrm{P}}$	3,545,200.043
${\mathrm{W}}_0$	$\mathrm{N}$	$=\left(\mathsf{\mu }\mathsf{\omega }{\mathrm{R}}^3\mathrm{L}/2{\mathrm{C}}_{\mathrm{m}}^2\right)$	$\overline{\mathrm{W}}$	8863.000108
${\mathrm{S}}_{\mathrm{i}\mathrm{j}}$	$\mathrm{N}/\mathrm{m}$	$=\left(\mathsf{\mu }\mathsf{\omega }{\mathrm{R}}^3\mathrm{L}/2{\mathrm{C}}_{\mathrm{m}}^3\right)$	${\overline{\mathrm{S}}}_{\mathrm{i}\mathrm{j}}$	73,858,334.23
${\mathrm{B}}_{\mathrm{i}\mathrm{j}}$	$\mathrm{N}.\mathrm{s}/\mathrm{m}$	$=\left(\mathsf{\mu }{\mathrm{R}}^3\mathrm{L}/2{\mathrm{C}}_{\mathrm{m}}^3\right)$	${\overline{\mathrm{B}}}_{\mathrm{i}\mathrm{j}}$	235,098.3796
${\mathrm{M}}_{\mathrm{J}}$	$\mathrm{K}\mathrm{g}$	$=\left({\mathsf{\mu }\mathrm{R}}^3\mathrm{L}/2{\mathrm{C}}_{\mathrm{m}}^3\mathsf{\omega }\right)$	${\overline{\mathrm{M}}}_{\mathrm{J}}$	748.3413846
$\mathsf{\gamma }$	$\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$	$=\left(\mathsf{\omega }\right)$	${\overline{\mathsf{\omega }}}_{\mathrm{P}}$	314.1592654

in the Appendix A detail the specifications of a reference rotor-bearing system and provide the necessary conversion coefficients for dimensional scaling of the bearing performance characteristics.

Figure 9 displays the three-dimensional profiles of both lubricant film thickness $\left({\overline{\mathrm{h}}}_n\right)$ and corresponding pressure distribution $\left(\overline{\mathrm{P}}\right)$ in a two-lobe journal bearing, incorporating cubic, cylindrical, and semi-ellipsoidal surface textures within circumferential angular sectors of 270–300° and 300–330° at a dimensionless depth of 0.5, along with comparative results for a conventional plain journal bearing. Pressure distribution comparisons clearly demonstrate that surface texturing within the bearing’s maximum pressure zone (270–300°) elevates peak pressure values compared to untextured bearings. Among the texture geometries examined, cubic patterns produce the most substantial pressure enhancement in this angular region, with cylindrical and semi-ellipsoidal variants following in descending order of effectiveness. It is clear from Figure 9 that relocating surface textures from the main high-pressure zone to adjacent 300–330° regions triggers significant pressure redistribution where moderate pressure elevation occurs in the upper lobe’s entry section alongside substantial pressure reduction in the lower lobe’s principal load-bearing zones, ultimately reversing traditional pressure gradient behavior. Given the dominant influence of pressure distribution on both static and dynamic performance metrics of the bearing, strategic texturing of the lower lobe’s high-pressure sector (270–300°) emerges as the most impactful approach for modifying the operational behavior of the evaluated two-lobe bearing systems.

Figure 10 illustrates how varying depth magnitudes of engineered surface features (cubic, cylindrical, and semi-ellipsoidal geometries) patterned across the 270$°$ $\le {\theta }_{tex}\le$ 300$°$ and 300$°$ $\le {\theta }_{tex}\le$ 330$°$ sectors of the bearing’s interior surface influence hydrodynamic pressure development along the bearing’s axial centerline. The findings reveal that texturing the high-pressure zone corresponding to 270–300° circumferential angles enhances pressure distribution within the lower bearing lobe, albeit with marginal pressure reduction in the upper lobe. As evidenced by the data plots, surface patterning in this high-load region elevates central longitudinal plane pressures beyond maximum values observed in plain bearings at specific locations. Conversely, texture application in low-pressure zones (300–330° range) predominantly induces pressure decline across lower load-bearing lobe areas, while paradoxically increasing pressures at select upper lobe points above comparable plain bearing values. This behavioral trend stems from corresponding enhancements in fluid film thickness resulting from deeper surface modifications along the bearing’s interior wall.

The plotted data demonstrate a progressive attenuation of pressure enhancement in the bearing’s lower lobe relative to plain bearings as texture depth increases within the 270–300° angular sector. Conversely, surface modifications in the 300–330° region initially generate substantial pressure reduction in the lower lobe, with pressure variation magnitudes becoming constrained beyond geometry-specific critical depth thresholds. The results of Figure 10 fundamentally indicate the existence of an optimal depth for each type of investigated textures to maximize bearing performance enhancement. Exceeding this critical depth yields diminishing returns and, in certain cases, performance deterioration. Cubic textures exhibit the shallowest optimal depth, while cylindrical and semi-ellipsoidal configurations require progressively greater depths for peak effectiveness. The analysis reveals minimal pressure variation across most texture depths on the primary lobe surface, with more pronounced reductions observed when texturing the 270–300° circumferential segment. This behavior results from the upper lobe’s auxiliary load-bearing function in two-lobe journal bearings, where the lower lobe maintains primary pressure absorption. Consequently, all critical performance metrics in such bearing systems are predominantly determined by the lower lobe’s pressure-dependent characteristics. Figure 10 further confirms that the angular region spanning 270–330° represents the core pressure zone in two-lobe journal bearings, establishing this sector as the optimal location for precision surface texturing applications.

Figure 11 demonstrates how texture depth variations (cubic, cylindrical, and semi-ellipsoidal geometries) affect the load capacity of two-lobe noncircular journal bearings. The data reveal that surface textures within the 270–300° angular region enhance bearing load capacity across all geometry types. This improvement occurs primarily at shallow penetration depths, with diminishing returns observed beyond optimal depth thresholds. Cubical textures generate the most substantial load capacity enhancement among the three geometries examined. However, relocating textures to the 300–330° sector reverses this trend, with cubic textures exhibiting the most significant load capacity reduction under these conditions.

This analysis establishes that the 270–300° angular sector represents the optimal texturing zone, whereas the 300–330° region proves disadvantageous for surface texture application on two-lobe journal bearing interiors when enhancing load capacity relative to untextured configurations. Careful examination of load capacity variations relative to texture depth modifications reveals distinct optimal depth values corresponding to peak performance points for each geometry type. The analysis shows cubic and cylindrical textures in the lower lobe’s high-pressure region achieve maximum effectiveness at shallower penetrations, while semi-ellipsoidal geometries require greater depths for optimal performance.

Figure 12 demonstrates how strategically oriented surface textures of varying geometries influence rotor attitude angle within two-lobe bearing clearances across different penetration depths. The graphical data reveal distinct behavioral patterns for textures positioned at two circumferential locations (270–300° and 300–330°), identified as strategic angular sectors for surface modification in such rotor-bearing systems. Analysis of Figure 12 indicates that texture implementation within the 270–300° region induces significant attitude angle reduction, though this effect diminishes beyond certain depth thresholds. Notably, cubic textures exhibit the most pronounced influence on angular orientation among the three geometries studied. In contrast, texture placement in the 300–330° region produces irregular attitude angle variations that lack consistent correlation with increasing penetration depth.

The variations in bearing attitude angle shown in Figure 6 reflect vertical displacement of the rotor center within the clearance space to maintain vertical alignment of the resultant load under various operating conditions. Texturing the high-pressure bearing region (270–300° angular sector) initially reduces the attitude angle up to a critical depth for all geometry types, indicating rotor equilibrium at lower positions. Beyond this threshold, further depth increases reverse this trend. According to the graphs in Figure 12, it can be seen that cubic surface textures achieve maximum attitude angle reduction at shallower penetration depths compared to cylindrical and semi-ellipsoidal configurations.

The linear dynamic analytical model yields critical rotor-bearing system parameters characterizing dynamic rotor behavior within the two-lobe bearing clearance space. This formulation treats rotor perturbation motions about static equilibrium positions as bounded limit-cycle oscillations, enabling the derivation of dynamic pressure components and the determination of rotor critical mass, equivalent lubricant film stiffness and damping coefficients, maximum permissible rotor mass, and whirl frequency limits, which collectively establish conditions for stable equilibrium recovery and sustained operational performance. Alternative parameter identification methods employ nonlinear analytical approaches, rotor center vibrational response monitoring or experimental signal analysis techniques.

Figure 13 demonstrates the influence of texture geometry and depth variation on the critical mass of the rotor within two angular sectors (270–300° and 300–330°) of two-lobe bearings. Results reveal that surface texturing in the 300–330° region fails to enhance system stability with increasing dimple depth in noncircular journal bearings. Rather, surface modifications generally degrade stability until reaching a critical penetration threshold, beyond which the destabilizing effect stabilizes.

However, when surface textures are implemented in the 270–300° angular region (Figure 13), the system’s dynamic stability initially declines relative to untextured bearings until reaching a threshold depth. Beyond this depth, stability improves temporarily before decreasing again after exceeding a second critical depth value. This behavior reveals that in two-lobe journal bearings, surface texturing within the 270–300° sector produces a characteristic stability pattern featuring sequential descending, ascending, and descending phases compared to non-textured configurations.

Furthermore, the analysis demonstrates that two-lobe journal bearings with properly positioned cubic textures offer superior dynamic stability enhancement compared to cylindrical and semi-ellipsoidal geometries when optimal texture parameters are maintained. Analysis of the results reveals that optimal dimensionless depths for maximizing rotor critical mass differ significantly by texture geometry. Cubic and cylindrical textures achieve peak effectiveness within the 0.4–0.5 dimensionless depth range, while semi-ellipsoidal configurations positioned in the 270–300° circumferential zone require substantially greater penetration depths to reach comparable performance levels.

Figure 14 presents how different surface texture geometries (cubic, cylindrical, and semi-ellipsoidal) influence the whirl frequency ratio—a critical dynamic stability parameter—across varying penetration depths in two distinct circumferential regions of a two-lobe journal bearing.

An elevated whirling frequency at constant axial rotational velocity indicates intensified orbital motion of the rotor center about its equilibrium position, demonstrating the system’s heightened propensity for dynamic instability escalation. This condition progressively leads to critical operational states characterized by surface contact phenomena and accelerated wear mechanisms.

The data in Figure 14 show reduced stability in textured rotor-bearing systems compared to non-textured configurations when surface modifications are implemented within the 300–330° circumferential range. This destabilization intensifies with greater texture penetration depths, correlating with elevated whirl frequency ratios in the bearing clearance. For textures positioned in the 270–300° sector, system stability initially decreases, then recovers with increasing texture depth. The analysis further reveals that cubic textures exert the strongest influence on stability modification, enhancing the stable operational range while amplifying instability in critical regimes, followed sequentially by cylindrical and semi-ellipsoidal geometries.

The preceding analysis establishes that strategically engineered surface textures with optimized geometry and penetration depth effectively enhance both static performance and dynamic stability in noncircular two-lobe journal bearing systems. Among the investigated configurations, cubic surface patterns demonstrate superior influence on bearing performance metrics, with cylindrical and semi-ellipsoidal geometries exhibiting progressively lesser impact.

5. Conclusions

In the present study, the effect of the depth of cubic, cylindrical, and semi-ellipsoidal textures on the static performance and dynamic stability of two-lobe noncircular hydrodynamic journal bearings in terms of load carrying capacity, attitude angle, critical mass, and the whirl frequency ratio of the rotor is investigated. To achieve this, the Reynolds equation is solved numerically via the finite element method. This approach addresses key challenges, including the finite bearing length, the lack of an analytical solution, and variations in lubricant film thickness due to surface texturing on the bearing shell. The Reynolds boundary condition is also applied to differentiate between pressure buildup and cavitation zones in the lubricant film. Furthermore, the study determines the rotor’s equilibrium position at a given eccentricity ratio while maintaining a vertical load direction, consistent with actual operating conditions. The research reveals these conclusions:

1.

Among the tested surface textures, cubic patterns demonstrated the most significant impact on both static and dynamic performance of two-lobe bearings, particularly when positioned at 270–300° in the high-pressure region of the lower lobe. Cylindrical and semi-ellipsoidal textures followed in effectiveness, respectively.

2.

The improvement trend of two-lobe bearing performance parameters—including load capacity, critical mass, and whirling frequency reduction—with increasing texture depth is not monotonic. Beyond a geometry-dependent critical depth, the enhancement rate decreases or even reverses.

3.

Cubic textures achieve maximum performance improvements in two-lobe bearings at shallower depths compared to other patterns. The optimal dimensionless depth range for cubic textures falls between 0.4 and 0.5, while cylindrical and semi-ellipsoidal textures require greater depths for peak performance.

4.

The linear dynamic model demonstrates that implementing surface textures while maintaining constant bearing design parameters effectively controls rotor disturbances and reduces surface wear/collisions by enhancing the stability through increased critical mass and reduced whirling frequency. This performance enhancement is more pronounced with cubic textures compared to cylindrical and semi-ellipsoidal patterns.

5.

Surface texturing in low-pressure bearing regions (such as the 300–330° zone of the lower lobe or cavitation areas of the lubricant film) demonstrates negligible performance enhancement and proves economically inefficient. However, computer-aided optimization of texture location, geometry, and depth can achieve peak operational performance of two-lobe bearings while simultaneously reducing both manufacturing and maintenance costs.