# Performance of Tilting Pad Journal Bearings with the Same Sommerfeld Number
## 1 Introduction  
In an initial stage of bearing design, Sommerfeld number is a good approximation to size the bearing and anticipate its performance. Identical Sommerfeld numbers for different sizes of bearings indicates the identical nondimensionalized static and dynamic force performance [1]. As Sommerfeld number is simple in nature, the assumptions for deriving Sommerfeld number neglect the effect of change in temperature influencing the decrease in viscosity and heat transfer to adjacent structures (bearing, shaft). The purpose of this paper is to give light to design engineers of the limitation of Sommerfeld number by providing how the tilting pad journal bearing (TPJB) performance differswith the thermal deformation, viscous shearing, elastic deformation of the tilting pads and pivots. The following literature review presents how the TPJB characteristics are affected by the various factors. In 1960s, [PERSON] [2] introduced a method for estimating the TPJB force coefficients assuming that the journal is subjected to harmonic motion. [PERSON] et al., 1966 [3] experimentally showed that there is a large temperature variation near the surface of the bearing and maximum temperature occurs at the minimum film thickness. This paper emphasizes the importance of estimating the effective lubricant viscosity in bearing to accurately predict the bearing performance. [PERSON] et al. (1996) [4] state that a proper thermo-hydro-dynamic (THD) analysis can give a better solution to bearing analysis. They compare their prediction of temperature variation to experimental results in reference [3] and shown a good correlation. [PERSON] et al. [5] emphasize the importance of accounting the thermal deformation in the prediction which can be as large as 45% of minimum film thickness and can lead to high temperature increase up to \(\sim\)20 \({}^{\circ}\)C.  
[PERSON] and [PERSON] [6] present thermo-elastic-hydro-dynamic (TEHD) model with thermal cavitation by defining thermal boundary conditions in cavitated regions. The paper well summarizes the theory behind their theoretical model: (1) energy equation in the fluid film, (2) flexibility of the pivot, (3) thermal expansion of the shaft, and (4) upwinding technique. They assumed the heat flow from the lubricant to the bearing is negligible. Nonetheless, the prediction of temperature distribution closely matches with reference [7]. [PERSON] and [PERSON] [8] present nonlinear analysis accounting TEHD of a TPJB. The authors showed the increase of bearing temperature leads to change in dynamic response, in particular at the critical speed. All the listed papers [2; 3; 4; 5; 6; 7; 8] emphasize that, without considering thermo-hydro-dynamic analysis in prediction, the outcome will likely result in unrealistic bearing performance. [PERSON] [9] accounted for the pad flexibility of TBJB to predict the realistic curvature of the pad by using a curved beam model. The flexible pad and pivot lead to change in the bearing film clearance due to the dynamic load on the bearing, which results in altering the bearing performance. He has shown numerically that the bearing damping can be diminished due to pad flexibility. Later, [PERSON] and [PERSON] [10] utilized the Hertzian-stress model to predict the contact stiffness of the TPJB. Later, [PERSON] and [PERSON] [11] show experimentally that prediction not accounting for the pad and pivot flexibility at large loaded condition results in a huge deviation (\(\sim\)3x-9x) from the test results. While prediction considering for the pad/pivot flexibility and hot clearance provides a close match with experimental results. [PERSON] emphasizes the importance of accounting for hot-bearing clearance in the prediction. [PERSON] [12] tested three LBP TPJBs with different pad thicknesses to experimentally identify the effect of pad flexibility on bearing performance. For decreasing pad thickness, the TPJB direct damping decreases up to \(\sim\)20%; however, the reduction of direct stiffness coefficient was minor.