## 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\({}^{\circ}\) 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\({}^{\circ}\) 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.  
Conceptualization, M.Z.M. and A.R.S.; data curation, M.Z.M. and A.R.S.; formal analysis, M.Z.M. and M.R. ([PERSON]); investigation, M.Z.M. and M.R. ([PERSON]); methodology, M.Z.M.; and M.R. ([PERSON]); project administration, M.Z.M. and M.R. ([PERSON]); supervision, M.Z.M. and M.R. ([PERSON]); visualization, M.Z.M. and M.R. ([PERSON]); writing--original draft, M.Z.M. and M.R. ([PERSON]); writing--review and editing, [PERSON] and M.R. ([PERSON]). All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest.