Static Characteristics of a Tilting Five-Pad Journal Bearing with an Asymmetric Geometry

: In this paper, static characteristics of a tilting ﬁve-pad rocker-backed journal bearing with an asymmetric geometry, i.e., di ﬀ erent clearance for each pad, are investigated. A thermo-elasto-hydrodynamic (TEHD) model considering the elasticity of the pad and pivot is used for the simulation. The pivot sti ﬀ ness of each pad obtained by experiment is also introduced in the model. The experimental tests were carried out on a tilting pad journal bearing (TPJB) with a nominal diameter of 100 mm and a length-to-diameter (L / D) ratio of 0.7 with load-between-pad (LBP) and load-on-pad (LOP) arrangements. Several analyses, including numerical simulations and experimental measurements, are implemented in order to obtain the static behaviors of the tilting-pad bearing under variations of rotational speed, amplitude and direction of applied static load, such as clearance distribution proﬁle, static eccentricity, temperature and pressure distribution. The results show that the e ﬀ ect of asymmetric geometry on the static characteristics is not negligible.


Introduction
Journal bearings are widely used to support the load in the direction normal to the rotating shaft and widely investigated in the literature, such as in the comprehensive book about hydrodynamic lubrication by Hori [1]. Among sliding bearings, this kind of bearing is the most popular one. Hydrodynamic journal bearings are frequently instrumented in rotating machines for supporting loads. Hence, it is necessary to identify the actual or expected operating conditions of the bearings in the process of machine design. These working conditions can be investigated by simulation and experiment. Hydrodynamic journal bearings can be found in steam turbines, electric motors, generators, pumps, compressors, internal combustion engines and so on, even though rolling element bearings are widely used.
Titling pad journal bearings (TPJBs) are commonly equipped in high rotational speed machinery like steam and gas turbines. This is generally thanks to two special characteristics of this bearing, namely stability at high rotational speeds and a high tolerance of misalignment. Owing to the hydrodynamic pressure, the pad can tilt and a converging zone is generated between the pad and the shaft. The pad tilting depends on the bearing operation parameters, such as bearing load and rotational speed. In comparison with plain cylindrical journal bearings, these bearings consist of more moving mechanical elements, leading to a potential chance of component damage. Additionally, they are more complicated to study. The shaft center moves along the {X, Y} axes which gives two degrees of freedom shape as a function of operating temperature was plotted. The experimental results show that the hot clearances were nearly 30% smaller than the cold ones and were inversely proportional to the pad surface temperature. The thermal effects on the performance of TPJBs and pressure distribution were investigated in [22,23].
The impacts of deflection of a pivot on the performances of journal bearings fabricated with rocker-backed and ball-socket pivots was studied in [24]. This paper presents a thermos-hydrodynamic (THD) model in which the pivot stiffness was obtained using the Hertzian contact theory to predict the static and dynamic characteristics of journal bearings. Yingze et al. [25] presented the nonlinear dynamic behaviors of TPJBs with elastic pivots. It was found that the pivot stiffness, preload factors and pre-tightening force have significant effects on the TPJB performances.
While numerous studies of TPJBs used the Hertz theory to calculate the pivot stiffness, several projects performed experiments to estimate the stiffness value. However, in these studies, all pad pivots were assumed to have similar stiffness due to the arrangement of the test rig. In this research, the load can be applied in any direction thanks to configuration of the test rig, which will be presented in a later section. Thanks to this method, the stiffness of each pad pivot can be easily obtained.
The effect of geometry on the performance of TPJBs was studied in [26,27]. A lot of projects have only focused on the LOP or LBP configurations, or both. On the contrary, research about the influence of load direction on the TPJB performances is still limited, and numerous studies are only focused on the theoretical investigation.
In this paper, static characteristics of a tilting five-pad rocker-backed TPJB with an asymmetric geometry are investigated. The thermo-elasto-hydrodynamic (TEHD) model considers the pad and pivot flexibility in which the pivot stiffness of each pad is estimated by experimental activities. Then, some experimental tests are carried out for comparison with the analyses. Results regarding the clearance distribution profile, static eccentricity as a function of rotational speed, temperature and pressure distribution as a function of rotational speed, amplitude of static load and load direction are discussed.

Bearing Modeling
The TEHD model of TPJBs was developed and validated by experimental data in [28] and was improved for the elastic pivot stiffness and thermal feature, as partially done in [29] for two-axial groove journal bearings. Figure 1 shows the scheme of a pad-shaft system in a rocker-backed TPJB, where O b and O j are the center of the bearing and the shaft, respectively. Only the displacement of point P in the radial direction η is considered for the pivot flexibility, while the tangential displacement is neglected. The pad tilts about the line contact with angle θ (see Figure 1). So, the vector of the DOF of the system is: where x s and y s are the shaft center position and θ k and η k represent the tilt angle and the radial movement of the k-th pad, respectively. It was found that the effect of the flexibility of the pivot along the η direction in the prediction of the performances of TPJBs is very important. The pivot stiffness estimated from the Hertz contact theory is always higher than that obtained by experiments, especially for rocker-backed bearings [29]. In order to improve the numerical simulation, the pivot stiffness estimated by experiment is introduced to the code. The experimental procedure for the stiffness pivot calculation of each pad has been described in detail in [29]. point P in the radial direction is considered for the pivot flexibility, while the tangential displacement is neglected. The pad tilts about the line contact with angle (see Figure 1). So, the vector of the DOF of the system is: where xs and ys are the shaft center position and and represent the tilt angle and the radial movement of the k-th pad, respectively.  The pressure distribution of each oil film is obtained by using the well-known Reynolds equation: where h and p are the thickness and pressure distribution in the fluid film, µ, ρ are the dynamic viscosity and density of the oil and x, z are the tangential and axial direction.
To solve Equation (2), the following assumptions are used: laminar flow, Newtonian fluid, negligible inertia forces, insignificant fluid compressibility, negligible slippage between the solid surface and the fluid and constant pressure in the film thickness.
The variables (U 1 , V 1 ) and (U 2 , V 2 ) present the velocity components of the shaft and the pads, respectively [1]. Note that U, V represent the velocity component along the tangential direction and radial direction, respectively.
In a steady-state condition, x s = . y s = 0, U 1 = Ω × r j and V 1 = U 2 = V 2 = 0. The energy equation of each oil film can be calculated as: The viscosity µ and density ρ of the lubricating oil are: ISO VG46 oil was used to lubricate the bearing in this research and its properties are listed in Table 1. A finite difference method was used to solve the Reynolds equation. At node (i, j) of the mesh grid, the pressure p i,j is as follows: A simple thermal model with 2D control volume [30] was applied for the estimation of the temperatures and viscosities of fluid in the bearing. An adiabatic condition was applied to the shaft surface and the average oil temperature along the radial direction was supposed to be constant. The temperature distribution in the oil film was applied to the active surface of the 3D thermal model of the pads.
The temperature T p of the center of each control volume with dimensions ∆X and ∆Z is a function of the temperatures of the four edges of the control volume T N , T E , T S , T W and the previous value of the iteration for temperature T p0 : The term E depends on the discrepancy between the "in" and "out" flows of lubricant of the control volume and is given by: The parameters S C , S P and the intensity of the viscous heating S are obtained by: (10) In this model, the thermal power is generated by the viscous shear stress and the effect of adjacent pads is taken into account by the hot oil carry-over coefficient m T , as shown in Figure 2.
Therefore, the inlet oil temperature is: where the supply temperature is T s = 40 • C and m T = 0.3. T outlet is the average oil film temperature at the pad trailing edge and is achieved by taking into account the heat flux at the trailing edge: Actuators 2020, 9, 89 6 of 23 Additionally, 3D finite element analysis is used to evaluate pad deformation because of the pressure distribution and the thermal expansion. Then, these pad deformations are introduced in the change in oil film thickness. The temperature distribution in the pad is (at the steady state): where k is the material thermal conductivity. The deformation of the pad u owing to thermal and mechanical stresses considering an isotropic material is given by: where C is the tensor of mechanical properties and E, α, ν represent Young's modulus, the thermal expansion coefficient and Poisson's ratio of the material, respectively.
The term E depends on the discrepancy between the "in" and "out" flows of lubricant of the control volume and is given by: where: The parameters SC, SP and the intensity of the viscous heating S are obtained by: In this model, the thermal power is generated by the viscous shear stress and the effect of adjacent pads is taken into account by the hot oil carry-over coefficient mT, as shown in Figure 2. Therefore, the inlet oil temperature is: where the supply temperature is = 40 ° and = 0.3. Toutlet is the average oil film temperature at the pad trailing edge and is achieved by taking into account the heat flux at the trailing edge: = . .
Additionally, 3D finite element analysis is used to evaluate pad deformation because of the pressure distribution and the thermal expansion. Then, these pad deformations are introduced in the change in oil film thickness. The temperature distribution in the pad is (at the steady state):  Readers are strongly recommended to refer to [11,29] for a detailed description of the model. Each pad consists of two different parts: a base part (steel) and an anti-friction layer (Babbit metal with a thickness of 3 mm). The properties of the two materials are listed in Table 2. In this study, a convection boundary condition with a constant coefficient of q = 50 W/(m 2 ·K) is applied on the pad surfaces contacting lubricating oil at supply temperature (T supply = 40 • C and q air = 20 W/(m 2 ·K)) in order to evaluate the distribution of the pad temperature. The constant convection coefficient q = 50 W/(m 2 ·K) was also used in some previous publications [11,29,31,32]. Actually, this value is not uniform at the pad peripheries and it changes with the conditions of the oil flow around the pad [19]. For the estimation of the pad deformation, the Dirichlet boundary condition with null displacement is assumed for the upper surface of the pad face F2 (see Figure 3), corresponding to the pivot part. The boundary conditions used in the model of the pad are listed in Table 3.
In this study, a convection boundary condition with a constant coefficient of q = 50 W/(m 2 .K) is applied on the pad surfaces contacting lubricating oil at supply temperature (Tsupply = 40 °C and qair = 20 W/(m 2 .K)) in order to evaluate the distribution of the pad temperature. The constant convection coefficient q = 50 W/(m 2 .K) was also used in some previous publications [11,29,31,32]. Actually, this value is not uniform at the pad peripheries and it changes with the conditions of the oil flow around the pad [19]. For the estimation of the pad deformation, the Dirichlet boundary condition with null displacement is assumed for the upper surface of the pad face F2 (see Figure 3), corresponding to the pivot part. The boundary conditions used in the model of the pad are listed in Table 3.

Face Boundary Condition Type
Pad lateral and bottom surfaces Convection with oil @ T supply (qoil = 50 W/(m 2 K)) Coating lateral surfaces (Babbitt layer) Convection with oil @ T supply Supplied temperature of 40 °C Active surface Given pressure and temperature distribution

Face Boundary Condition Type
Pad lateral and bottom surfaces Convection with oil @ T supply (q oil = 50 W/(m 2 ·K)) Coating lateral surfaces (Babbitt layer) Convection with oil @ T supply (q oil = 50 W/(m 2 ·K)) Fixed surface (F2) Null displacement Leading edge (F8) Supplied temperature of 40 • C Active surface Given pressure and temperature distribution As already stated, numerous studies of TPJBs assumed the nominal dimension for all pads. Unfortunately, due to an error in manufacturing, the actual dimensions of the pads are different from each other. The effect of the manufacturing/assembly error on bearing and system was highlighted in [32,33]. Based on the bearing geometry estimation methodology which was described in [28], the actual pad thickness can be obtained.
The values of pad thickness, bearing assembly clearance and preload factor of the nominal bearing and real bearing are provided in Table 4. Table 4. Pad thickness, bearing assembly clearance and preload factor of the bearing under test [30].

Pad
Pad In this model, the optimization toolbox in Matlab is applied for finding the equilibrium position of the pads and shaft. Additionally, the Partial Differential Equation Toolbox is introduced to the 3D structural mechanics and thermal models to calculate the deformation of each pad and the temperature distribution, respectively. The main program flowchart for the TEHD model in this paper is shown in Figure 4.
In this model, the optimization toolbox in Matlab is applied for finding the equilibrium position of the pads and shaft. Additionally, the Partial Differential Equation Toolbox is introduced to the 3D structural mechanics and thermal models to calculate the deformation of each pad and the temperature distribution, respectively. The main program flowchart for the TEHD model in this paper is shown in Figure 4.

Test Rig and Bearing under Test
A picture of the test rig used to perform the experimental tests is shown in Figure 5. A detailed description of this test rig was presented in [28]. Two identical five-pad TPJBs support the shaft but only one bearing (including the housing support and the bearing ring) is fully instrumented by necessary sensors, such as load cells, proximity probes, temperature probes and accelerometers. The bearing housings are structured so that the bearings can be configured in the LOP or LBP arrangement. Two orthogonal 20 kN load cells are installed in each bearing housing in the horizontal and vertical direction.
In this test rig, two hydraulic actuators are perpendicular to each other and form an angle of 45 • in an arrangement with two load cells. These actuators can apply a maximum static load of 25 kN in the middle of the shaft by means of rolling bearings (see Figure 6). Thanks to this configuration of the test rig, the direction of the static load, as well as dynamic load, can be arranged at an arbitrary angle.
Two orthogonal proximity probes are set up on each bearing support (see in Figure 7) to measure the relative displacements between the shaft and the housing. The oil inlet temperature is also controlled by a PID temperature controller during the tests.
bearing housings are structured so that the bearings can be configured in the LOP or LBP arrangement. Two orthogonal 20 kN load cells are installed in each bearing housing in the horizontal and vertical direction.
In this test rig, two hydraulic actuators are perpendicular to each other and form an angle of 45° in an arrangement with two load cells. These actuators can apply a maximum static load of 25 kN in the middle of the shaft by means of rolling bearings (see Figure 6). Thanks to this configuration of the test rig, the direction of the static load, as well as dynamic load, can be arranged at an arbitrary angle.  Two orthogonal proximity probes are set up on each bearing support (see in Figure 7) to measure the relative displacements between the shaft and the housing. The oil inlet temperature is also controlled by a PID temperature controller during the tests. and vertical direction.
In this test rig, two hydraulic actuators are perpendicular to each other and form an angle of 45° in an arrangement with two load cells. These actuators can apply a maximum static load of 25 kN in the middle of the shaft by means of rolling bearings (see Figure 6). Thanks to this configuration of the test rig, the direction of the static load, as well as dynamic load, can be arranged at an arbitrary angle.  Two orthogonal proximity probes are set up on each bearing support (see in Figure 7) to measure the relative displacements between the shaft and the housing. The oil inlet temperature is also controlled by a PID temperature controller during the tests. One temperature probe and one pressure probe are installed on each pad, as seen in Figure 8. The pressure amplitude in the middle of the pad is measured by a pressure probe through a small hole in the pad surface. The model and sensitivity of the instrumented sensors are provided in Table 5. One temperature probe and one pressure probe are installed on each pad, as seen in Figure 8. The pressure amplitude in the middle of the pad is measured by a pressure probe through a small hole in the pad surface. The model and sensitivity of the instrumented sensors are provided in Table 5. One temperature probe and one pressure probe are installed on each pad, as seen in Figure 8. The pressure amplitude in the middle of the pad is measured by a pressure probe through a small hole in the pad surface. The model and sensitivity of the instrumented sensors are provided in Table 5.    All signals from sensors installed in the test rig are acquired by means of a signal processing system ( Figure 9). The voltage for all sensors is 24VDC. It consists of a power supply, transducer signal conditioning and a PCI DAQ chassis (NI cDAQ-9178).
Actuators 2020, 9, x FOR PEER REVIEW 10 of 23 All signals from sensors installed in the test rig are acquired by means of a signal processing system ( Figure 9). The voltage for all sensors is 24VDC. It consists of a power supply, transducer signal conditioning and a PCI DAQ chassis (NI cDAQ-9178).
The pad includes the steel part of the pad base and the 3 mm thick Babbitt layer on the pad surface. A sketch of one single pad is shown in Figure 10.  The pad includes the steel part of the pad base and the 3 mm thick Babbitt layer on the pad surface. A sketch of one single pad is shown in Figure 10.  All the bearing geometric characteristics and the operating conditions are listed in Table 6. All the bearing geometric characteristics and the operating conditions are listed in Table 6.

Clearance Profile
For a TPJB, the clearance profile has a polygon shape with n p sides with respect to the number of pads installed in the bearing.
In order to obtain the clearance profile of TPJBs, a rotating force is applied on the non-rotating shaft through the deep groove precision ball bearings, so the rotor is slowly moved inside of the bearing housing. It should be noted that during the tests, the amplitude of rotating force is high enough to ensure that the shaft contacts all five pads. Please note that, during the test, oil is still supplied in order to avoid metal to metal contact. However, this force is just large enough to prevent considerable pivot deflections during the tests. This force can be generated thanks to the configuration of the test rig in which each hydraulic actuator generates a sinusoidal force and the phases of two these force signals are shifted by 270 • . Figure 11 shows the measured pentagonal clearance profiles (colored solid line) at different pad temperatures (from cold bearing at 25 • C to hot bearing at 55 • C) and the predicted nominal (black dash-dot line) of the TPJB. The temperature shown in Figure 11 is the average temperature of each pad surface acquired during the measurement. The cold-bearing clearances were plotted when the system had not worked for several hours. Hot clearances were recorded immediately (about 10 s) after performing several tests for a given rotational speed and load. Positions of the pad surface temperature sensor are already described (see Figure 8). In this test rig, pad #1 is the loaded pad, whereas pad #3 and pad #4 are the unloaded ones.

Eccentricity Measurement
It difficult to define the actual center of the bearing because of the thermal expansion of the bearing housing in which the proximity probes are installed. Therefore, steady-state shaft eccentricities should be recorded with respect to a lightly loaded equilibrium position, which corresponds to a shaft rotation of 3000 rpm and an applied static load equal to about 350 N on each bearing. The Sommerfeld number for this light load, corresponding to an eccentricity ratio of less than 0.01, is larger than 12 [34]. This position can be considered as the bearing center.
The values of eccentricity = and the eccentricity ratio ( ) can be calculated as: where (xequilibrium, yequilibrium) is the equilibrium position of the journal center and Cb is the bearing It is noticeable that the measured clearance shape of the actual bearing presents an asymmetrical pentagonal shape and is quite different from that of the nominal one, which shows a symmetrical pentagonal profile. This strange bearing profile of the test bearing has been described and explained in [28,29]. It is evident that as the temperature of inlet oil increases, the pentagonal shape tends to shrink. Since the bearing becomes hotter, the shaft, pads and bearing expand. Supposing the bearing bore remains stable, a hotter pad will get bigger, leading to a reduced profile. This trend shows a good agreement with the clearance profile changes reported in [21].
It can be detected that the clearance profiles of the test bearing of pad #4 show an unusual shape compared to those of the other pads. This consequence is probably due to the small thickness of pad #4 (15.981 mm), so at the location of pad #4, the movement of the shaft is be larger.

Eccentricity Measurement
It difficult to define the actual center of the bearing because of the thermal expansion of the bearing housing in which the proximity probes are installed. Therefore, steady-state shaft eccentricities should be recorded with respect to a lightly loaded equilibrium position, which corresponds to a shaft rotation of 3000 rpm and an applied static load equal to about 350 N on each bearing. The Sommerfeld number for this light load, corresponding to an eccentricity ratio of less than 0.01, is larger than 12 [34]. This position can be considered as the bearing center.
The values of eccentricity e = O b O j and the eccentricity ratio (Ψ ) can be calculated as: where (x equilibrium , y equilibrium ) is the equilibrium position of the journal center and C b is the bearing clearance. Three test series were run at a constant rotational frequency at, respectively, 10 Hz, 25 Hz and 40 Hz, with measurements taken using nine applied static loads, from 1 to 9 kN, in steps of 1 kN.
As shown in Figure 12, it is clear that when the rotational speed increases, the equilibrium and shaft position move up. When the speed is low, the measured eccentricity deviates slightly from the bearing vertical centerline. The largest eccentricity, which corresponds to the maximum load of 9 kN, is about 70 µm. When the rotational speed becomes higher, the deviation of the measured eccentricity from the vertical centerline grows (especially at 40 Hz), though the largest eccentricities decrease with respect to that of the test at 10 Hz. The measured eccentricities, as a function of the applied static loads, are shown in Figure 13 for three different rotational speeds. The eccentricity is almost inversely proportional to the rotational speed and practically linear to the static load.
The measured eccentricity ratios of three test series as a function of the Sommerfeld number calculated using nominal radial clearance and pad average temperature are shown in Figure 14. These Sommerfeld numbers are based on the oil viscosity corresponding to the bearing pad temperature. The measured eccentricities, as a function of the applied static loads, are shown in Figure 13 for three different rotational speeds. The eccentricity is almost inversely proportional to the rotational speed and practically linear to the static load. The measured eccentricities, as a function of the applied static loads, are shown in Figure 13 for three different rotational speeds. The eccentricity is almost inversely proportional to the rotational speed and practically linear to the static load.

Y [ m]
The measured eccentricity ratios of three test series as a function of the Sommerfeld number calculated using nominal radial clearance and pad average temperature are shown in Figure 14. These Sommerfeld numbers are based on the oil viscosity corresponding to the bearing pad temperature. On the one hand, large Sommerfeld numbers (low applied load, high speed or high lubricant viscosity) determine low journal eccentricities, small eccentricity ratios or nearly centered operation, i.e., → 0, → 90 . It means that the applied load is nearly perpendicular to the journal eccentricity vector. The measured eccentricity ratios of three test series as a function of the Sommerfeld number calculated using nominal radial clearance and pad average temperature are shown in Figure 14. These Sommerfeld numbers are based on the oil viscosity corresponding to the bearing pad temperature. On the contrary, small Sommerfeld numbers (large applied load, low speed or low lubricant viscosity) determine large journal eccentricities or large eccentricity ratios, i.e., → 1, → 0 . It means that the vector of journal eccentricity has a similar direction to the applied load direction.

Eccentricity [ m]
It is clearly seen that, in Figure 14, most of experimental results either touch or coincide with each other when the Sommerfeld number ranges from 0.1 to 0.5.

Temperature Distribution
As mentioned above, each pad of the tested bearing is instrumented with a temperature sensor located below the bearing's surface to measure the pad temperature in the middle of the pad. Figure 15 shows the predicted oil film temperatures on the middle plane of the axial length of the loaded pad from the leading edge (A) to the trailing edge (B) in the case of a LOP configuration of five pads. The measured pad subsurface temperature is also plotted for the corresponding positions. Please note that the temperature of the oil film along the radial direction is supposed to be constant. The temperature of the oil film given by the 2D model is applied on the active surfaces of the 3D model of the pad, where convective heat exchange with oil at 40 °C is assumed for all the other On the one hand, large Sommerfeld numbers (low applied load, high speed or high lubricant viscosity) determine low journal eccentricities, small eccentricity ratios or nearly centered operation, i.e., e → 0, φ → 90 0 . It means that the applied load is nearly perpendicular to the journal eccentricity vector.
On the contrary, small Sommerfeld numbers (large applied load, low speed or low lubricant viscosity) determine large journal eccentricities or large eccentricity ratios, i.e., e → 1, φ → 0 0 . It means that the vector of journal eccentricity has a similar direction to the applied load direction.
It is clearly seen that, in Figure 14, most of experimental results either touch or coincide with each other when the Sommerfeld number ranges from 0.1 to 0.5.

Temperature Distribution
As mentioned above, each pad of the tested bearing is instrumented with a temperature sensor located below the bearing's surface to measure the pad temperature in the middle of the pad. Figure 15 shows the predicted oil film temperatures on the middle plane of the axial length of the loaded pad from the leading edge (A) to the trailing edge (B) in the case of a LOP configuration of five pads. The measured pad subsurface temperature is also plotted for the corresponding positions. Please note that the temperature of the oil film along the radial direction is supposed to be constant. The temperature of the oil film given by the 2D model is applied on the active surfaces of the 3D model of the pad, where convective heat exchange with oil at 40 • C is assumed for all the other faces of the pads.  Figure 16 shows the measured temperature of each pad in the test bearing as a function of applied static load with a speed of 20 Hz and as a function of rotational speed with an applied static load of 5 kN. The static load in both cases was applied in the vertical direction, i.e., pad #1 was the loaded pad. During the test, the oil inlet temperature of each test was also recorded. Note that, because the shaft was run in the anticlockwise direction, pad #2 was loaded more than pad #5 (see Figure 2). The other two upper pads were unloaded pads. For ease of visualization, only the temperature distribution in the loaded pad is shown in Figure 15. It is possible to see that the pad temperature grows circumferentially on the middle plane from the leading edge to the trailing edge, about 4 • C for pads #1 and #4, and more or less 6 • C for the other three pads. For the nominal bearing, the predicted oil film temperature lines are identical for all pads because of the equal sizes of the pads. The difference between the nominal and the test bearings is negligible starting from the leading edge and is then quite evident at the trailing edge of the pads, about 1 • C.
Interestingly, while there is a difference between the measured metal temperature and the predicted film temperature in pads #1, #4 and #5 (about 1−2 • C), the values of the other two pads are nearly equal. For example, the measured temperature of pad #3 is 51.33 • C, whereas the prediction gives 51.36 • C for the test bearing. Figure 16 shows the measured temperature of each pad in the test bearing as a function of applied static load with a speed of 20 Hz and as a function of rotational speed with an applied static load of 5 kN. The static load in both cases was applied in the vertical direction, i.e., pad #1 was the loaded pad.
During the test, the oil inlet temperature of each test was also recorded. Note that, because the shaft was run in the anticlockwise direction, pad #2 was loaded more than pad #5 (see Figure 2). The other two upper pads were unloaded pads. Figure 15. Predicted oil film temperatures along the center line of the loaded pad and measured pad temperatures. Positions A and B present the leading and trailing edge of the loaded pad, respectively. Figure 16 shows the measured temperature of each pad in the test bearing as a function of applied static load with a speed of 20 Hz and as a function of rotational speed with an applied static load of 5 kN. The static load in both cases was applied in the vertical direction, i.e., pad #1 was the loaded pad. During the test, the oil inlet temperature of each test was also recorded. Note that, because the shaft was run in the anticlockwise direction, pad #2 was loaded more than pad #5 (see Figure 2). The other two upper pads were unloaded pads. It can be observed that the temperature of two loaded pads increases with an increase in the applied static load, ranging from 2.5 kN to 8.75 kN, in which the temperature of pad #1 was always higher than that of pad #2 by about 1-1.5 °C. The results show that there is a reduction in the temperature of pads #3, #4 and #5 in the test of 7.5 kN before an increase in the next test. This behavior It can be observed that the temperature of two loaded pads increases with an increase in the applied static load, ranging from 2.5 kN to 8.75 kN, in which the temperature of pad #1 was always higher than that of pad #2 by about 1-1.5 • C. The results show that there is a reduction in the temperature of pads #3, #4 and #5 in the test of 7.5 kN before an increase in the next test. This behavior can be easily explained thanks to the oil inlet temperature. In the test of 6.25 kN, the temperature of the oil supply was 42.1 • C, at which temperature the cooling system was activated. The decreasing oil temperature led to the reduction of these three pads' temperatures.
It is possible to conclude that the pad temperature strongly depends on the shaft speed. This value increases by about 10-15 • C when the shaft speed increases by 30 Hz. These behaviors show an agreement with data published in [10,17].
The comparison between the predicted and measured pad temperature as a function of static load and rotational speed is shown in Figure 17. The measured lines are plotted with solid lines, while a dash-dot line is used for the predicted ones.
It is possible to see that the simulation estimates the oil film temperature on the pad surface to be about 4 • C lower than the measured temperature. However, the code correctly estimates the variation trend of oil film temperature in the considered range. For example, while the oil film temperature of pads #1 and #2 increases linearly by about 3 • C when the applied static load increases from 2.5 kN to 8.75 kN, the other three pads show a fluctuation of temperature at the pad surface.
In the case of a variation of shaft speed with a given static load, the model underestimates the film temperature when the pad is unloaded or rotational speed is high. For instance, at 40 Hz, the measured temperature of pad #1 is higher by about 7 • C than that of the simulation. However, this value is about 2.4 • C for pad #4 at the same speed.
It is possible to conclude that the pad temperature strongly depends on the shaft speed. This value increases by about 10-15 °C when the shaft speed increases by 30 Hz. These behaviors show an agreement with data published in [10,17].
The comparison between the predicted and measured pad temperature as a function of static load and rotational speed is shown in Figure 17. The measured lines are plotted with solid lines, while a dash-dot line is used for the predicted ones. It is possible to see that the simulation estimates the oil film temperature on the pad surface to be about 4 °C lower than the measured temperature. However, the code correctly estimates the variation trend of oil film temperature in the considered range. For example, while the oil film temperature of pads #1 and #2 increases linearly by about 3 °C when the applied static load increases from 2.5 kN to 8.75 kN, the other three pads show a fluctuation of temperature at the pad surface.
In the case of a variation of shaft speed with a given static load, the model underestimates the film temperature when the pad is unloaded or rotational speed is high. For instance, at 40 Hz, the measured temperature of pad #1 is higher by about 7 °C than that of the simulation. However, this value is about 2.4 °C for pad #4 at the same speed. Figure 18 and Figure 19 show temperature distribution in all pads and only in pad #1 obtained by the 3D thermal model of the pads, respectively, in two cases: as a function of shaft speed with a given static load of 5 kN and as a function of applied static load with a constant speed of 25 Hz. It is possible to highlight a slight temperature gradient in the radial direction. The main temperature gradient is obtained in the circumferential direction. Note that the temperature distribution on the pad surface is also the predicted temperature distribution in the oil film along the circumferential direction.  Figures 18 and 19 show temperature distribution in all pads and only in pad #1 obtained by the 3D thermal model of the pads, respectively, in two cases: as a function of shaft speed with a given static load of 5 kN and as a function of applied static load with a constant speed of 25 Hz. It is possible to highlight a slight temperature gradient in the radial direction. The main temperature gradient is obtained in the circumferential direction. Note that the temperature distribution on the pad surface is also the predicted temperature distribution in the oil film along the circumferential direction.   In general, it can be observed that the maximum oil film temperature (the temperature of the pad surface) is reached on pad #1, which is the loaded pad. Due to the anticlockwise direction, the average temperature on pad #2, which is the pad after the loaded one (in the direction of tangential speed), is slightly lower than that of pad #1. Conversely, the minimum temperature is reached on pad #5, that is, the pad after the unloaded pads #3 and #4.
Interestingly, due to the asymmetric geometry, the predicted oil film temperature of the trailing edge of pad #2 is nearly equal to that of pad #1 or even higher in the case of a small static load, i.e., 2.5 kN. This effect can be easily explained due to the large thickness (16.015 mm), and pad #2 is still close to the rotating shaft although the shaft moves toward the middle of pad #1 (see Figure 20). For example, at a static load of 2.5 kN, the maximum oil film temperature of pad #2 is 51.4 °C, while this temperature is 50.9 °C for pad #1. In general, it can be observed that the maximum oil film temperature (the temperature of the pad surface) is reached on pad #1, which is the loaded pad. Due to the anticlockwise direction, the average temperature on pad #2, which is the pad after the loaded one (in the direction of tangential speed), is slightly lower than that of pad #1. Conversely, the minimum temperature is reached on pad #5, that is, the pad after the unloaded pads #3 and #4.
Interestingly, due to the asymmetric geometry, the predicted oil film temperature of the trailing edge of pad #2 is nearly equal to that of pad #1 or even higher in the case of a small static load, i.e., 2.5 kN. This effect can be easily explained due to the large thickness (16.015 mm), and pad #2 is still close to the rotating shaft although the shaft moves toward the middle of pad #1 (see Figure 20). For example, at a static load of 2.5 kN, the maximum oil film temperature of pad #2 is 51.4 • C, while this temperature is 50.9 • C for pad #1.

Pressure Distribution
The oil film pressure distribution is a very essential feature of TPJBs. The effect of the direction of the applied load on the pressure distribution results in a deviation in the performance of the TPJBs. Figure 21 shows the simulated oil pressure distributions in the middle plane for test bearing and the nominal bearing as a function of load direction. For the sake of brevity, only LOP configuration on each pad was considered. In this simulation, the shaft was run at a speed of 20 Hz and the applied static load was 5 kN. Due to the identical geometry of the pads in the nominal bearing, a pressure with a maximum value of nearly 30 bar is distributed mostly on the loaded pads.
It should be noted that, for the test bearing, when pad #1 is loaded, its average pressure distribution is almost the same as that of pad #2 and pad #5. The same behavior also happens when the applied static load is at the center of pad #4. On the contrary, if the static load is applied in the middle of pad #2 (−18°), pad #3 (54°) or pad #5 (198°), the oil film pressure with a maximum pressure

Pressure Distribution
The oil film pressure distribution is a very essential feature of TPJBs. The effect of the direction of the applied load on the pressure distribution results in a deviation in the performance of the TPJBs. Figure 21 shows the simulated oil pressure distributions in the middle plane for test bearing and the nominal bearing as a function of load direction. For the sake of brevity, only LOP configuration on each pad was considered. In this simulation, the shaft was run at a speed of 20 Hz and the applied static load was 5 kN. Due to the identical geometry of the pads in the nominal bearing, a pressure with a maximum value of nearly 30 bar is distributed mostly on the loaded pads.   It should be noted that, for the test bearing, when pad #1 is loaded, its average pressure distribution is almost the same as that of pad #2 and pad #5. The same behavior also happens when the applied static load is at the center of pad #4. On the contrary, if the static load is applied in the middle of pad #2 (−18 • ), pad #3 (54 • ) or pad #5 (198 • ), the oil film pressure with a maximum pressure of nearly 38 bar distributes mainly across these three pads.
This phenomenon can be explained based on the pads' thickness. Note that the thickness values of pad #1 (15.994 mm) and pad #4 (15.981 mm) are smaller than those of the nominal one (16 mm), whereas the other three pads show a contrary tendency. For example, if the direction of the applied static load is vertical, the shaft should move down to the center of pad #1. Nevertheless, due to the smaller thickness of pad #1, its average oil film thickness is still larger than that of the two consecutive pads (pads #2 and #5). In this case, the minimum thickness of the oil film of pad #1, pad #2 and pad #5 is about 37.6 µm, 32.5 µm and 31.2 µm, respectively. It makes the pressure distribution on pad #1 smaller than that of the two other pads. It also clear that, due to having the largest thickness value (16.018 mm), pad #5 will have a maximum pressure (nearly 38 bar in Figure 21) when pad #5 is loaded. Figures 22 and 23 show the contour plots of oil film pressure distribution on all pads in both real and nominal bearings as a function of load direction. For the sake of simplicity, only the LOP configuration of each pad is considered. The red rectangle on each contour indicates the loaded pad.

Conclusions
This paper investigated the static characteristics of a five-pad rocker-backed TPJB with an asymmetric geometry. A TEHD model used for predictions takes into account the elasticity of the pads and pivots. This model uses the stiffness value of each pivot obtained from experiments. The bearing under test has a diameter of 100 mm and length of 70 mm. The bearing housing is

Conclusions
This paper investigated the static characteristics of a five-pad rocker-backed TPJB with an asymmetric geometry. A TEHD model used for predictions takes into account the elasticity of the pads and pivots. This model uses the stiffness value of each pivot obtained from experiments. The From these two plots, it is easily to observe the influence of the asymmetric geometry on oil pressure distribution.

Conclusions
This paper investigated the static characteristics of a five-pad rocker-backed TPJB with an asymmetric geometry. A TEHD model used for predictions takes into account the elasticity of the pads and pivots. This model uses the stiffness value of each pivot obtained from experiments. The bearing under test has a diameter of 100 mm and length of 70 mm. The bearing housing is manufactured to support the bearing under LBP and LOP arrangements. Some conclusions can be figured out from the results as follows: (1) The clearance profile of TPJBs has a polygon shape with the number of sides equal to the number of pads. The measured clearance profile of the test bearing is a strange pentagon, while the nominal one shows a regular pentagonal profile. As the bearing becomes hotter, the shaft, pads and bearing will expand, resulting in a reduction of the clearance profile. (2) It is more evident that when rotational speed increases, the equilibrium and shaft position will move up. When speed is low, the measured eccentricity deviates slightly from the bearing vertical centerline. The largest eccentricity corresponding to a maximum load of 9 kN and shaft speed of 10 Hz is about 70 µm. (3) The experiments show that the eccentricity is almost inversely proportional to the rotational speed. Additionally, most of the eccentricity ratios of all tests either touch or coincide with each other when the Sommerfeld number ranges from 0.1 to 0.5. (4) The measured data show that the pad temperature increases with an increase in the applied static load in which the temperature of pad #1 is always higher than that of pad #2, by nearly 1-1.5 • C. Additionally, it can be stated that the rotational speed has a considerable effect on the pad temperature. (5) As a function of load direction, on the one hand, the TEHD model underestimates the film temperature for pads #1, #4 and #5. On the other hand, the numerical results show a good agreement with measurement data for pad #2 and pad #3. The code predicts that the pad temperature grows circumferentially on the middle plane from the leading edge to the trailing edge, about 4 • C for pads #1 and #4, and more or less 6 • C for other three pads in the tested bearing. (6) The effect of the asymmetric geometry on the oil film pressure distribution is not negligible.
Owing to the identical geometry of pads in the nominal bearing, the oil film pressure with a maximum value of nearly 30 bar is distributed mostly on the loaded pads.