Efficient Flexible Multibody Models for Tilting Pad Journal Bearings

Abstract

Tilting pad journal bearings (TPJBs) are key components in modern rotating machines. They require accurate modeling to describe their behavior, and we considered the thermo-elasto-hydrodynamic approach. Thermo-elasto-hydrodynamic (TEHD) models are a powerful tool for machine design and analysis, but they have to assure a good compromise between accuracy and numerical efficiency. Current multi-physics TEHD models of TPJBs are very accurate; however, their numerical efficiency is still far from being satisfying for industrial applications. To partially fill this gap, we propose an innovative modeling approach based on efficient flexible multibody techniques, such as the Floating Frame of Reference Formulation (FFRF) and some extensions that can be easily coupled with different fluid dynamic models to describe the whole TPJB. The model was tested on a real TPJB geometry and validated through experimental tests. The results are encouraging and show the effectiveness of the proposed approach. Finally, the model proposed can be applied in different engineering fields, such as Oil & Gas and aeronautics, where TPJBs are typically used, where it is fundamental to always reach better performance with low energy consumption and where it is particularly important to have an efficient and accurate model to simulate long period times with brief simulation times.

1. Introduction

The success of tilting pad journal bearings (TPJBs) is particularly due to the comprehension of the oil behavior in the thickness between the rotor and the pads of the bearing. Currently, the most important technological limit of the turbomachinery is represented by the maximum rotor peripheral speed: to obtain more power is fundamental to increasing the rotor speed and with a higher rate to subsequently increase the flow rate of the maximum rotor peripheral speed.

In this context, the proposed work presents an efficient flexible multibody model to predict the behavior of the bearings and of the film lubrication, while paying attention to the oil film and pad modeling. The knowledge regarding these complex phenomena is essential because they influence the dynamic behavior of the rotor and its hydrodynamic lift, the bearing temperature and wear and thus the reliability of the whole machine. The work presented builds on papers that explained the rotor model and the interaction of the bearings with the rotor [1].

The novelty of the model is represented by the integration of thermal, elastic and dynamic phenomena in the same model, with a low simulation time compared to the fully three-dimensional bearing models present in the literature. Indeed, in the model proposed, the 3D discretization is used only where the interaction of the physics phenomena listed above is more complex and impacts the rotor behavior, and thus it is used in the bearings. This allows the best compromise between a sufficiently physical detailed model and a reasonable time simulation model.

2. State of the Art

In the literature, there are many studies regarding TPJBs. The simpler TPJB models consider the lubricant temperature effects on the whole system in a lumped parameter model. Those models offer considerable advantages from the computational point of view; however, they are not able to investigate all the physical interactions that involve the bearings. Indeed, most of the studies found in the literature analyzed a smaller number of physical phenomena coupled together or performed a complete coupled analysis considering simpler bearings (e.g., journal bearings). This was the approach of Knight and Barrett [2], who developed thermal analysis thermo-hydrodynamics (THD) of a Tilting Pad Journal Bearing, based on the solution of the classical energy equation, that was able to estimate the lubricant viscosity through the average value of the temperature calculated in the fluid film.

Further development is represented by the completely three-dimensional models, which analyze many of the physical phenomena involved in the tilting pad journal bearings. These models couple the classic thermo-hydrodynamic (THD) models performed on the fluid films with the elastic and thermal behaviors of the solid elements of the system, thereby, generating thermo-elasto-hydrodynamic (TEHD) models of the bearings [3,4]. Some authors, such as Ettles [5] and Brockwell [6], proposed TEHD models with a different method to study the lubrication problem: the generalized Reynolds equation was used for the calculation of the pressure distribution in the fluid film.

Another interesting model was proposed by Costantinescu [7], in which the fluid film was studied inside the bearings considering the fluid motion as the sum of an average speed that this assumes and a fluctuation component. Other authors investigated several aspects related to tilting pad journal bearings, such as Brugier and Pascal [8] who developed a TEHD model to estimate the dynamic coefficients of TPJBs: in this method, a small perturbation applies to the rotor in correspondence of the bearings needed to evaluate the bearing response and, consequently, extract the respective damping $\left[C\right]$ and stiffness $\left[K\right]$ matrices. A further model was proposed by Kim [9], in which the coefficients of the damping and stiffness matrices were evaluated considering both heat transfer and elastic deformation.

Furthermore, some authors introduced the analysis of other elements of the rotor-bearing system. Kirk and Balbahadur [10] developed a ThermoElasto model to analyze the onset of thermal instability of the rotor. Kirk and Reedy [11] considered the bearing deformability to obtain satisfactory results. Later, Monmousseau and Fillon [12] developed a TEHD model considering possible flexibility of the pads pivots. Suh and Palazzolo [13] performed transient tests analyzing the thermo-hydrodynamic behavior of the lubricant coupled with the rotor FEM model, considering the possibility for the cylindrical pivot of the pad to be deformed. An important contribution due to this work is the analysis of the distortion generated on the pads and on the rotor by the heat load generated by the fluid film within the bearing.

Other important papers are present in the current state of the art regarding the estimation of the pad leading-edge film temperature in tilting pad journal bearings: Yang and Palazzolo [14,15] developed a complex and detailed model with a 3D hybrid between pad (HBP) model, computational fluid dynamics (CFD) and machine learning (ML) to predict the radial and axial temperature distributions at the pad leading edges. The model proposed has the aim to identify all the physical parameters that influence the Morton effect. The differences with the proposed model are represented by the more complex rotor model (3D FEM) and the fluid-dynamic phenomena in the oil film described with CFD formulation: their results are accurate; however, the time simulation is higher compared to the proposed model.

To conclude, all these models offer great accuracy; however, their resolution times and their computational weight represent a major limitation, while the low time consumption is one of the most important aspects of a rotor-bearing model, independently of the type of bearing used [16].

3. Model Structure

All the model components were developed in COMSOL Multiphysics^®4:4 to simplify the analyses of the interactions between the various elements. Figure 1 shows the inputs and outputs of the various sub-models. In particular, to obtain the best compromise between the result accuracy and time simulation consumption, the following modeling choices were taken into account:

• Rotor: geometry 3D FEM discretization of the part in contact with the pads, beam modeling of the other parts to simplify the model.
• Pads: geometry 3D FEM discretization.
• Lubrication: the Reynolds formulation is used to describe the laminar hydrodynamic mechanism of lubrication (2D representation of the oil film lubricant).
• Supply plant: a lumped parameter model is used.

Figure 1. Structure of the developed model.

Figure 1. Structure of the developed model.

The ith oil film model provides the forces and the moments needed by the rotor model to determine its motion. The moments $M_{tilt,i}$ and $M_{pitch,i}$ determine the pad motion. The flow rates $Q_{in}$, $Q_{out}$ and $Q_{bord}$ are the inputs for the ith lubricant supply plant model and the lubricant temperature $T_f$, which has to be provided to the rotor and pad models to evaluate their thermal deformation. The ith lubricant supply plant model maintains a balance between the flow rates above mentioned and generates, as output, the pressure $p_{pozz,i}$ to be used as a boundary condition on the leading edge of the ith pad and on the trailing edge of the i-1th pad.

The ith pad model, subjected to the thermal and structural loads mentioned above, provides the rigid displacements, velocities, pad deformation and temperature, to be supplied as a boundary condition to the oil film model. The rotor model, on which it is possible to apply an overhung forcing due to the imbalance, receives, as inputs, the loads generated by the bearings and the bearings temperature and provides, as output, the temperature $T_A$ and the displacement $q_A$ of the rotor, which is provided to the oil film model and used for the direct calculation of the fluid film thickness.

4. Fluid Dynamical Aspects: Oil Film Modeling

The most important part of the tilting pad journal bearing model is the FEM model of the oil film interposed between the tilting pads and the rotor. The fluid dynamical equation solution allows calculating all the quantities needed to know the behavior of the other components. Lubrication is a fluid dynamical phenomenon, and it can be represented through the Navier–Stokes equations.

Given the geometrical and physical peculiarities of a lubrication problem, the Navier–Stokes equations can be simplified by obtaining the Reynolds equation for Newtonian fluids, characterized by a viscosity that depends only on temperature and pressure.

Due to the low value of the bearing clearance with respect to the bearing dimensions (see

Table 1. Main characteristics and operating conditions of the test rig and of its TPJBs.

Rotor
Total mass of the rotor	450 [kg]
Rotor length	2.5 [m]
Young’s modulus	210 [GPa]
Poisson’s ratio	0.31
Specific heat capacity	475 [J/kgK]
Thermal conductivity	44.5 [W/mK]
Thermal expansion	$12.3$·${10}^{-6}$ [1/K]
Rotational velocity of the rotor ${\omega }_a$	$8000÷20,000$ [rpm]
Bearings
Bearing radius	70 [mm]
Pad thickness	19 [mm]
Bearing axial length	56 [mm]
Radial clearance of the bearing	0.124 [mm]
Radial clearance of the pad	0.07 [mm]
Pad angle	56.3°
Pivot offset	50%
Type of lubricant	ISO VG 46

) and the oil viscosity, it is supposed that the lubricant regime in the oil film thickness is predominantly a laminar regime.

Through the CAD tool implemented in COMSOL Multiphysics^®4:4 it is possible to draw the geometric surface where the Reynolds equation must be solved (see Figure 2).

In order to solve this equation, it is necessary to implement, in the model, some mathematical functions representative of the rotor fraction peripheral speed and the oil film thickness. Typically, to efficiently complete this calculation, all the relative motions between the two moving solids are assigned to only one of them: the solid wall (i.e., the pad surface) is assumed to have zero velocity. To develop the oil film thickness formulation, all the relative motions are assigned to channel base.

COMSOL Multiphysics^®4:4 includes a vast lubricant library, useful for different kinds of analyses. The considered lubricant is denominated ISO VG 46 by ISO standards and its fluid properties are proposed by the solver through polynomial equations as functions of the fluid temperature and pressure. The considered properties are four:

• The absolute viscosity with the expression for $\psi$:

  $$\mu ={\mu }_0{10}^{\psi };$$

  (1)

  $$\psi =b_{p1}(p-p_{ref})+b_{t1}(T-T_{ref})+b_{t2}{(T-T_{ref})}^2.$$

  (2)
• The specific volume:

  $$\begin{array}{c}\hfill v_s=\frac{1}{\rho }=v_{s0}[1+a_{p1}(p-p_{ref})+a_{p2}{(p-p_{ref})}^2+a_{t1}(T-T_{ref})+\\ \hfill +a_{t2}{(T-T_{ref})}^2+a_{tp}(p-p_{ref})(T-T_{ref})].\end{array}$$

  (3)
• The specific heat:

  $$\begin{array}{c}\hfill C_p=C_{p0}[1+C_{t1}(T-T_{ref})+C_{t2}{(T-T_{ref})}^2+\\ \hfill +C_{p1}(p-p_{ref})+C_{pt}(p-p_{ref})(T-T_{ref})].\end{array}$$

  (4)
• The thermal conductivity:

  $${\lambda }_p={\lambda }_{p0}[1+d_{t1}(T-T_{ref})+d_{t2}{(T-T_{ref})}^2].$$

  (5)

In all the previous equations, $T_{ref}$ is the reference temperature (20 °C), while T is the fluid temperature. Moreover, $p_{ref}$ is the reference pressure (100,000 Pa) and p is the fluid pressure. For each of Equation (3) to Equation (5) reported above, each parameter is proposed by the solver COMSOL Multiphysics^®4:4, depending on the lubricant chosen.

Nevertheless, in the literature, different and most accurate formulations of dynamic viscosity and density are proposed, and thus the authors implemented the following formulations for the dynamic viscosity and the density of the oil lubricant instead the previous ones proposed by the solver:

$$\mu \left(T\right)={\mu }_0{e}^{-\beta (T-T_0)};$$

(6)

$$\rho \left(T\right)={\rho }_0[1-ϵ(T-T_0)];$$

(7)

where T is the absolute temperature, ${\mu }_0$ and ${\rho }_0$ are respectively the viscosity and the density at the reference temperature $T_0$, $\beta$ is the thermal expansion coefficient and $ϵ$ is the volume expansion coefficient.

To solve the fluid dynamical problem, the input of the oil film model are the positions and velocities of pads and rotor fraction and the pressure level on the pad edges. All these parameters are exchanged continuously during the simulation of the model and they are provided by the ODE components of the model. The fluid dynamical problem provides to the other models its outputs, in particular, the quantities calculated from the solution of the pressure field are the lubricant flow rates and the forces and moments exerted on the rotor and pad (see Figure 3).

5. Fluid Dynamical Aspects: Supply Sump Modeling

In the TPJBs, the balance between the flow rates from the hydraulic network of the power plant and to the external environment affects the pressure level on the pad edges. This fluid cavity, with the duct elements connected to it, was modeled according to a lumped parameters approach.

To formulate the characteristic equations of the lumped elements, the flow in a duct with a constant cross-section must be considered. A viscous flow generally involves many phenomena that break its uniformity; the lumped parameters modeling approach is based on a one-dimensional representation of the system and assumes the fluid properties to be constant in the duct cross-section.

Generally, the lumped models are expressed as a combination of capacitive, inertial and resistive elements that are explained below:

• Capacitive element: the capacitance of fluid is due to the intrinsic compressibility of real fluids. It reveals itself as a mass accumulation or release, related to the mass flow that crosses the boundaries of the considered control volume. The capacitive element stores energy and contributes to weakening vibrations and load oscillations within the circuit, thus avoiding sudden pressure variations in the fluid dynamical field.
• Inertial element: the inertia of a fluid is a physical characteristic related to its mass and velocity, so it is linked to the fluid motion and kinetic energy. The behavior of the inertial element can be analyzed from the momentum equation, derived for fluid from Newton’s Second Law. The inertial element contributes to avoiding sudden flow rate variations.
• Resistive element: the resistance of fluid is due to the presence of viscous effects and it generates an energy dissipation caused above all by friction actions. This energy dissipation involves a pressure drop for the flow that is directly related to the volumetric flow rate.

From the model point of the view, the hypothesis considered is that the sump has a behavior similar to the capacitive element: the balance between the flow rates entering and exiting it, with the lubricant bulk modulus, affects the pressure level inside the sump. This pressure level is a boundary condition for the oil films. The flow rates involved in the balance are calculated according to the resistive element formulation: the supply flow rate is computed considering the pressure drop between the supply pressure and the sump pressure, while the leakage flow rate is computed considering the drop between the sump pressure and the environment pressure (see Figure 4).

6. Fluid Dynamical Aspects: Pad Sub-Model

The pads have two degrees of freedom: $\gamma$ is the principal tilt angle, defined with respect to an axis parallel to the geometrical symmetry axis of the machine, while $\chi$ is the secondary tilt angle (i.e., the pitch angle), defined with respect to a circumferentially tangential axis (see Figure 5).

Considering the moments due to the pressure field inside the oil film, it is then possible to formulate the pad equations of motion as follows:

$$\left\{\begin{array}{c}{J}_{p,pad}\ddot{\gamma }=M_{tilt}\hfill \\ J_{p,pad}\ddot{\chi }=M_{pitch}.\hfill \end{array}\right.$$

(8)

7. Thermo-Elasto-Hydrodynamic Aspects: Oil Film Modeling

Regarding the oil film modeling, the novelty is represented by the analyses of the thermal effects inside the film. The thermal effects strongly influence the dynamic behavior of the rotor, due to the viscosity dependence related to the temperature. By the Navier–Stokes equations, in order to develop a reliable model of the hydrodynamic bearing, it is fundamental to take into account the influence that the temperature has on the lubricant properties (see Section 4).

Considering the velocity field obtained through the fluid dynamic analysis of the lubricant, the temperature $T_f$ inside the lubricant can be obtained using the energy equation:

$$\rho C_p\frac{\partial T_f}{\partial t}+\nabla (-\lambda \nabla T_f)+\beta \nabla T_f=f_{visc};$$

(9)

where $C_p$ is the thermal capacity of the lubricant at constant pressure, $\lambda$ is the thermal conductivity, $\beta$ is a vector defined by the multiplication of the term $\rho C_p$ for the three components of velocity, u, v and w; while $f_{visc}$ is the viscous dissipation due to the lubricant motion.

Moreover, the boundary conditions can be expressed as follows:

$$f\left(n\right)=\left\{\begin{array}{cc}{T}_f=T^i\hfill & \mathrm{on} S_3\hfill \\ n(-\lambda \nabla T_f)=0\hfill & \mathrm{on} S_1,S_2,S_4\hfill \end{array}\right.;$$

(10)

where $S_1$, $S_2$, $S_3$ and $S_4$ are respectively the leading edge, the trailing edge and the two side edges of the control volume (see Figure 6) and $T^i$ is the temperature of the lubricant supply plant preceding the ith pad.

8. Thermo-Elasto-Hydrodynamic Aspects: Supply Sump Modeling

In the lubricant supply plant model is represented the modeling of the thermal effects. The model proposed consists of an energy equation that takes into account the lubricant flow rate of the previous pad and the flow rate supplied from the outside. The model result is the lubricant temperature needed as a boundary condition on the leading edge of the next pad. These topics were already discussed in the literature [17,18]; however, in this work, the authors pay more attention not to the mixing phenomena between the oil flow rates with different temperatures (minor from the supply plant and higher from the previous pad) but to the oil temperature resultant, which affect the temperature of the rotor and of the pads and their mutual interaction.

As represented in Figure 6, within the ith lubricant supply plant, the lubricant flow rate from the outside $Q_{orif}$ and that coming from the previous pad $Q_{in}$ are at different temperatures ($T_s$ and $T_{in}$), so the resultant lubricant flow has an intermediate temperature, resulting from the convective and conduction heat transfer within the sump.

The energy balance that determines the lubricant temperature that comes out from the lubricant supply plant $T_{out}$ used as a boundary condition in the energy equation of the oil film model (Equation (9)), can be expressed as follows:

$$T_{out}=\frac{Q_{in}}{Q_{out}}{T}_{in}+\frac{Q_{orif}}{Q_{out}}{T}_s;$$

(11)

where the temperature at which thermal equilibrium is reached, is calculated by assigning weights for the two temperatures, $T_s$ and $T_{in}$, according to the two flow rate values, $Q_{orif}$ and $Q_{in}$, compared to the total flow rate that coming out from the lubricant supply plant $Q_{out}$ (see Figure 6).

The lubricant supply plant model is crucial in order to obtain an accurate model in particular for the part concerning the thermal exchange. This model, considering the mixing that occurs between the two lubricant flow rates, allows the optimal computing of the temperature field that develops in the fluid and in the solid components of the system. Without an appropriate consideration of these phenomena, it would not be possible to obtain reliable results in terms of thermal instability.

Regarding the pressure calculation inside the lubricant supply plant, this has less effect on the rotor dynamic results; its importance lies instead on the possibility of accurately calculating the lubricant flow rate required for a correct operation of the Tilting Pad Journal Bearing.

9. Thermo-Elasto-Hydrodynamic Aspects: Pad Modeling

Concerning the pad, the pad model is a 3D model that simulates the dynamical, structural and thermal behavior of the tilting pads. The model analyzes the rigid rotational motions that each pad describes around its pivot following the displacement of the rotor, but it also simulates both the elastic and thermal deformations of the pads and the thermal field developed within them, due to the presence of the fluid films. The inputs are the loads generated by the fluid films ($M_{z,pad}$ and $M_{x,pad}$), the temperature ($T_f$) and the pressure (p) of the lubricant. The outputs are the position, velocity and temperature of the pads (respectively $q_{pad}$, ${\dot{q}}_{pad}$ and $T_{pad}$).

9.1. Fem Modeling of the Pads

The elements used for the pads FEM modeling are the BRICK elements since there is not a negligible dimension compared to the others. The BRICK elements are able to represent a 3D stress state, they have eight nodes, and each node has three degrees of freedom (one for each translation).

9.2. Thermal Modeling of the Pads

In the developed model, the heat exchange inside the pads is driven by the fluid movement, which, due to the rotation of the rotor, adheres to the journal surface and is dragged in the proximity of the pads, with a consequent temperature increase. The thermal energy generated by this phenomenon is then transferred inside the pad. The study of heat transfer phenomena in solid components is simulated through the energy equation, which is written below:

$$\rho C_p\frac{\partial T}{\partial t}+\rho C_p\mathbf{u}\nabla T=\nabla (k\nabla T\underline{)}+Q,$$

(12)

where $\mathbf{u}$ is the lubricant velocity field and Q is the heat source.

Concerning the boundary conditions, forced convection, due to the fluid motion, was imposed on the pad surface in contact with the oil films (Figure 7).

Moreover, the heat exchange between pad and lubricant inside the oil film is expressed by the following equation:

$$\mathbf{n}(-\lambda \nabla T_{pad})=h_{film}(T_f-T_{pad}),$$

(13)

where $\mathbf{n}$ is the normal vector at the pad surface, $h_{film}$ is the convective heat transfer coefficient between the pad and the fluid film, $T_f$ and $T_{pad}$ are respectively the fluid film and pad temperatures.

The remaining pad surfaces shown in Figure 8 are also interested in convective phenomena with the supplied lubricant, which is at a temperature $T_s$, generally different from the pads one. For these surfaces, the heat transfer coefficient is not calculated through an empirical correlation but, since the fluid is stationary, a value of h equal to 50 W/m${}^2$K was chosen. Even in this case, the heat exchange that interests such surfaces is defined by the relationship:

$$\mathbf{n}(-\lambda \nabla T_{pad})=h_{pad}(T_s-T_{pad}),$$

(14)

where $h_{pad}$ is the convective heat transfer coefficient between the pad and the lubricant within which it is immersed, $T_s$ and $T_{pad}$ are, respectively, the supply and pad temperatures.

Finally, due to the reduced lubricant quantity that comes in contact with the pivot, heat exchange is considered equal to zero in the spherical surface of the pivot (adiabatic areas, see Figure 8).

9.3. Elastic Modeling of the Pads

The pressure field developed in the fluid film during the bearing operation acts both on the rotor and on the pads and the temperature field that develops within the pads contributes to their deformation, and thus it is fundamental to determine the elastic behavior of the pad. The pads are considered made of homogeneous, isotropic and linear elastic material. Replacing in the equilibrium Equations the relationship between stress and deformation and the relationship between deformations and displacements, it is possible to obtain the Navier equations for the elastic problem:

$$-(\lambda +\mu )\nabla (\nabla \mathbf{q})-\mu {\nabla }^2\mathbf{q}=\mathbf{F}$$

(15)

where $\mathbf{q}$ is the displacement field, $\mathbf{F}$ is the volume forces vector, $\mu$ is the Lame’s first parameter (equal to G) and $\lambda$ is the Lame’s second parameter. In addition to the elastic deformation due to the pressure field, the model takes also into account the thermal deformation due to the pad temperature distribution. These are fundamental because they modify the pad geometry and thus the oil film geometry:

$$ϵ=D^{-1}\sigma +{\alpha }_T\Delta T$$

(16)

where D is the elastic stiffness matrix, $\sigma$ is the stress tensor, ${\alpha }_T$ is the thermal expansion tensor and $\Delta T$ is the delta temperature, which generates the pad deformation $ϵ$ due to the thermal behavior.

Finally, the spherical pivot was considered rigid in the model. Thus, the equations that describe the structural behavior of the pads are defined as follows:

$${\mathbf{K}}_{pad}{\mathbf{q}}_{pad}={\mathbf{f}}_{pad}$$

(17)

where ${\mathbf{K}}_{pad}$ and ${\mathbf{q}}_{pad}$ are respectively the stiffness matrix and the vector of the pad displacements.

10. Thermo-Elasto-Hydrodynamic Aspects: Rotor Modeling

The rotor model is a 3D model composed of BEAM elements connected to two solid parts (see Figure 9). The two cylindrical parts are in the rotor fractions where there are the bearings and they allow to analyze the influence that the bearings have on the rotor thermo-structural behavior. The rotor model provides both the rotor FEM modeling and the modeling of deformation and temperature field created due to the lubricant motion inside the bearings. The inputs are forces and moments generated by the bearings, the bearing temperature field and possible external loads. The outputs are the temperature and the displacement of the rotor (needed to calculate the fluid film thickness).

11. Experimental Data: Thermo-Elasto-Hydrodynamic Characteristics

The model proposed was developed on the basis of technical data relating to a test rig specially built to study the possible onset of thermal instability in a centrifugal compressor supported by two TPJBs (Figure 10). The test rig represents a scale model of a real machine and was developed to evaluate the rotor behavior both in the start-up and shutdown phases and in some steady operating conditions, in order to highlight the onset or absence of thermal instability. The TEHD model proposed was validated through a comparison with experimental data sets supplied by General Electric Oil & Gas.

In Table 1 are the bearing characteristics and the main functioning conditions of the test rig. The bearings that support the rotor test rig are tilting pad journal bearings, with five pads, set in the Load On Pivot (LOP) configuration (load acting on the spherical pivot), whose technical data are in Table 1. The rotor is driven by an electric motor connected to the Drive End side of the rotor through a flexible coupling.

In correspondence of the Non-Drive End bearing, i.e., the side on which the unbalance mass is placed, the test rig is equipped with eight thermocouples positioned directly on the rotor surface and connected with an external telemetry device (Figure 11); thermocouples rotate with the rotor and then allow to measure the temperature profile that during the operating phase is spread on the rotor.

In addition to the thermocouples, there are also resistance thermometers on the pad placed at the bearings midspan: the resistance thermometers are able to measure the fluid film temperature inside the bearings. The lubricant used for the tests (

Table 2. Main characteristics of the lubricant ISO VG 46.

ISO VG 46
Density at 15 °C	0.856 [kg/L]
Kinematic viscosity at 40 °C	43.8 [mm${}^2$/s]
kinematic viscosity at 100 °C	7 [mm${}^2$/s]
Viscosity index	115
Pour point	−15 [°C]
Flash point	232 [°C]

) is referred to as ISO VG 46 by the ISO standards, and this was supplied to the system at a temperature of about 40 °C.

The preliminary phase of the tests is presented in Figure 12 with,

• the rotor start-up to the operating velocity, immediately followed by a drastic reduction of the velocity, to identifying the critical speeds of the rotor;
• the rotor is subjected to a continuous series of velocity increments (each lasting five minutes), to detect the possible onset of thermal instability phenomena; and
• finally, the test proceeds with the machine shutting down.

Figure 12. Experimental procedure to evaluate the critical speeds and the onset of the thermal instability.

Figure 12. Experimental procedure to evaluate the critical speeds and the onset of the thermal instability.

For the validation of the proposed model, the complete testing procedure was repeated with three different configurations. In the brackets below the distance between the Non-Drive End of the rotor and the center of mass of the disk was added:

• The end of the rotor without additional disks, W3 configuration (215 [mm]).
• The end of the rotor equipped with a disc of 10 kg, W2 configuration (249 [mm]).
• The end of the rotor equipped with a disc of 20 kg, W1 configuration (301 [mm]).

The model validation was made through a comparison of the following experimental data:

• Rotor vibrations acquired on the bearings midspan plane and on the corresponding plane to the Non-Drive End bearing.
• Rotor temperature of the Non-Drive End bearing, to evaluate the onset of the thermal instability.
• Pads temperature of the Non-Drive End bearing to evaluate the onset of the thermal instability.

12. Models Numerical Validation

In this section, we explain the comparison between the experimental and numerical values of the pressure (p), temperature ($T_f$) and oil film thickness (h): these variables are measured in correspondence of a rotor rotational velocity equal to 8000 rpm and with an applied load equal to half of the rotor mass (LOP configuration) together with an unbalanced force.

The comparisons are highlighted considering experimental data and numerical results obtained with the TEHD model and from previous HD and THD releases of the model, which do not consider the heat transfer phenomena and elastic deformation present in the bearings and in the rotor. All the analyzed variables are measured circumferentially on the fluid film and their value is those recorded at the last step of the performed transient analysis, i.e., equal to 240 s.

As anticipated in Section 4, the lubrication regime is intended to be a laminar regime with hydrodynamic lubrication, due to the bearing radial clearance (see Figure 13) compare to the bearing pad roughness (≃0.4 $\mathsf{\mu }$m). Moreover, these results are in agreement with those found in the literature [19].

12.1. Thermo-Elasto-Hydrodynamic Characteristics

In Figure 14, the values of the oil film thickness measured on the Non-Drive End bearing is minimum in correspondence of the most loaded pads and on each pad, the thickness value decreases from the leading edge to the trailing edge. The TEHD model shows minimum values of h compared to the $\mathrm{HD}$ model, this is due to a decrease of the bearing radial clearance at high rotational velocities.

Figure 15 shows the pressure values measured on the Non-Drive End bearing: it is possible to highlight how the higher levels of pressure are detected where the oil film thickness on the pads is lower. The highest pressure values are obtained with the TEHD model: this is due to the fact that if the solid components are flexible, a part of the pressure generated within the oil film contributes to the pad’s deformation while the remaining part contributes to the rotor lift.

In Figure 16, the temperature values measured on the Non-Drive End bearing are reported: the highest temperature values are obtained in correspondence of the most loaded pads. Furthermore, on each pad, there is an increasing temperature profile from the leading edge to the trailing edge: this is due to the viscous dissipation within the lubricant, which tends to raise the temperature and is maximum in correspondence of the minimum oil film thickness, the lubricant velocity reach the maximum value.

For the pressure and temperature fields, it is possible to highlight how the numerical results obtained with the proposed TEHD model are in good agreement with the experimental data. Figure 15 and Figure 16, allow to highlight how the introduction of further physical phenomena in the analysis improved the accuracy of the TEHD model with respect to the previously developed HD and THD models, in which a lower number of physical phenomena was considered.

Figure 16 shows how the temperature values obtained with the THD model are quite different from the experimental results, in particular in correspondence of the pads trailing edges. This is due to the fact that in the trailing edges the temperature and the thermal deformations are greater and the hypothesis of deformation equal to zero on the pads (such as that made in the THD model) leads to excessively distorting the numerical results.

12.2. Models Numerical Validation: Rotor and Pad Models

Figure 17 shows a three-dimensional representation of the temperature field in the rotor fraction enclosed within the Non-Drive End bearing, with a rotational velocity equal to 8000 rpm and the relative pads temperature. From the color scale, it is possible to highlight how, due to the rotor rotation, the temperature field on the rotor surface is characterized by the presence of cold and a hot zone, consistently with the result found in the literature [19]: the temperature on the journal has a sinusoidal profile due to the rotor motion.

Due to the unbalanced load, the rotor performs a synchronous orbit while rotating around its axis: this leads to a condition in which a rotor part is always in contact with a higher oil film thickness zone, while the opposite side is always in contact with a lower oil film thickness zone; then, the second part will be more heated than the other, thus, generating a sinusoidal trend of the rotor temperature. This effect can trigger thermal instability phenomena, but in the considered operating conditions the temperature gap between the two opposite rotor sides is not high enough to generate this type of behavior.

The temperature gradient is calculated according to the thermal load found in correspondence to the considered rotor fraction and applied directly to the BEAM elements used for the rotor modeling (as described in Section 10). Since the rotor rotates around its z-axis and the temperature distribution has a sinusoidal trend, the temperature gradient applied on the BEAM elements is represented by a rotating vector; this type of coupling is at the base of the onset of the thermal instability phenomenon known as the Morton effect [20,21].

13. Model Performances

The model is based on different types of finite element discretization:

• 35 2D QUAD elements for each oil film model.
• 4200 3D BRICK elements for each pad.
• 41 3D BEAM elements for the rotor model.
• 21,600 3D BRICK elements for the rotor fractions enclosed in the two bearings.

In terms of degrees of freedom, the model accounts for:

• 1,527,032 degrees of freedom for the parts listed above.
• 40 degrees of freedom related to the implementation of ordinary differential equations that model the pad’s motion and the dynamics of the lubricant supply plant.

The 2D elements for the oil film discretization allow avoiding the huge computational efforts required by a fully 3D CFD model, thus, preserving the chance to obtain a geometric and kinematic 3D analysis. Although substantially longer in comparison to the lumped parameters models, the simulations are less time-consuming compared to the fully 3D bearing models. The simulations were performed using a machine with the characteristics shown in

Table 3. Machine features and calculation time.

CPU	Intel CORE i7
clock frequency	2.30 GHz
RAM memory	16 Gb
machine time [min]/time to be simulated [s]	15.3 [min]/[s]

: a simulation of 3 s requires about 46 min of computing time.

14. Conclusions and Future Works

In this work, an innovative Tilting Pad Journal Bearing model was developed and experimentally validated. The proposed model allows analysis simultaneously of the thermo-elasto-hydrodynamic (TEHD) characteristics of the system and is suitable to be used in the industrial field. The main objective of the proposed modeling approach is to completely model the rotor system, providing results as accurate as those obtainable with a fully-3D CFD model (good accuracy) but with computational times comparable to those of lumped parameter models (good numerical efficiency).

Many future developments could begin from this work:

• the introduction of a basement model for the rotor, in order to include the dynamical coupling between the motions of rotor and structure;
• the pivot flexibility can be introduced in the model since this influences the total stiffness and damping characteristics of the bearing [22];
• from a fluid dynamical point of view, a further step is represented by the use of a turbulence model for the oil film in the classic Reynolds equation, in order to analyze the behavior of a machine in the widest operating range [23]; and
• the temperature variation along the oil film thickness to investigate thermal instability effects.

In the SIR program (Scientific Independence of Young Researchers) project, a complete rotordynamic test rig (rotor supported by two tilting pad journal bearings) is under investigation in laboratories of the University of Florence. The aim of the cited test rig is to allow further upgrades of the proposed model considering the previous several points, particularly in terms of simulation time reduction and for high bearing peripheral speeds.