Modelling of LEG tilting pad journal bearings with active lubrication

Author’s Accepted Manuscript Modelling of LEG Tilting Pad Journal Bearings with Active Lubrication Alejandro Cerda Varela, Asier Bengoechea García, Ilmar Ferreira Santos www.elsevier.com/locate/jtri

PII: DOI: Reference:

S0301-679X(16)30462-5 http://dx.doi.org/10.1016/j.triboint.2016.11.033 JTRI4473

To appear in: Tribiology International Received date: 12 August 2016 Revised date: 16 November 2016 Accepted date: 19 November 2016 Cite this article as: Alejandro Cerda Varela, Asier Bengoechea García and Ilmar Ferreira Santos, Modelling of LEG Tilting Pad Journal Bearings with Active L u b r i c a t i o n , Tribiology International, http://dx.doi.org/10.1016/j.triboint.2016.11.033 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Modelling of LEG Tilting Pad Journal Bearings with Active Lubrication Alejandro Cerda Varela Escuela de Ingenier´ıa Mec´anica Pontificia Universidad Cat´olica de Valpara´ıso Los Carrera 01567, Quilpu´e, Chile Email: [email protected]

Asier Bengoechea Garc´ıa Escuela de Ingenier´ıa Mec´anica Pontificia Universidad Cat´olica de Valpara´ıso Los Carrera 01567, Quilpu´e, Chile Email: [email protected]

Ilmar Ferreira Santos Department of Mechanical Engineering Technical University of Denmark 2800 Kgs. Lyngby, Denmark E-mail: [email protected]

Abstract This work constitutes the first step in a research effort aimed at studying the feasibility of introducing an active lubrication concept in tilting pad journal bearings (TPJBs) that feature the leading edge groove (LEG) lubrication system. The modification of the oil flow into each pad supply groove by means of servovalves renders the bearing active. This article focus on obtaining and validating a numerical model capable of simulating the studied system, in terms of its steady state and dynamics characteristics. The developed model is then used to simulate a simple system, in order to portray the feasibility of affecting its static and dynamic properties by introducing this novel active bearing design. Keywords: modelling, tilting pad bearing, leading edge groove, active 1. INTRODUCTION In order to provide machines with a higher degree of reliability and adaptability to different operational conditions, a “mechatronic” approach towards their design and operation is desirable. This approach consists of the synergetic coupling of traditional mechanical elements with sensors, processing units and actuation elements derived from electronic or electromechanical engineering [1]. Considering the current demands over the industry of the energy sector, it could be desirable to introduce this approach in the design and operation of its turbomachinery. Among the many different machine elements, bearings correspond to a suitable target for implementing the mechatronic approach. Some key parameters of the dynamic behavior of a rotating Preprint submitted to Tribology International

November 24, 2016

shaft, such as critical speeds, unbalance response and stability limit, are heavily influenced by the stiffness and damping characteristics of the lateral supporting bearings [2, 3]. A mechatronic bearing [4, 5] is able to vary such characteristics as a function of an electrical control signal obtained in open or closed loop configuration. Consequently, the dynamic behavior of the whole rotating machine can be modified on the go, according to the operation and production requirements. Furthermore, the mechatronic bearing is capable of exerting controllable forces over the rotor, which gives the possibility to compensate the dynamic forces induced by an unexpected failure or abrupt process change, or to perform parameter identification procedures in situ. Among the different oil bearing designs, the tilting pad journal bearing is widely used for high load and high speed applications due to its superior stability characteristics [6].A continuous research effort has been carried out in order to advance the understanding of its physical behavior, aiming at an improved predictability of its static, thermal and dynamic characteristics. Some of the most recent publications on the subject [7, 8] include sophisticated 3D ETHD models, able to represent with high degree of accuracy this machine element. Other recent work couple explicitly a tilting pad bearing model with equations related to the oil supply system and the rotor dynamics to study their interactions [9]. Furthermore, there are some published works dealing with the effect of installing compliant liners over the bearing dynamics [10], and other ones dealing with the effect of manufacturing errors over the bearing behaviour [11]. Several publications deal with the conversion of the tilting pad bearing into a mechatronic element, aiming at expanding even further its versatility, by rendering it adaptable to sudden operational changes. The fact that its design is composed of moving parts (namely, the pads) also makes it particularly suitable to introduce some actuation elements. Among the alternatives studied to this end, one can find linear and rotational actuators acting over the bearing pads [12, 13, 14] and pads with embedded magnetic actuators [15].The adjustment of the bearing gap to introduce controllable characteristics has also been studied and applied to an industrial steam turbine by [16]. The usage of hydraulic systems to render TPJBs controllable has also been studied, by means of hydraulic chambers in the back of the pads [17, 18], and by the injection of pressurized oil into the bearing clearance through injection nozzles located radially across the bearing pads [19]. The later approach, referred to as the active lubrication system, resorts to a servovalve to control the pressurized oil flow towards the injection nozzle in order to render the bearing controllable. An extensive series of studies has dealt with the improvement of the mathematical modelling of such active bearing design [20, 21, 22], as well as with its experimental application to control the dynamics of a rotor system [23, 24, 25, 26] and to perform parameter identificacion procedures in situ [27, 28]. This series of theoretical and experimental studies have proved the soundness of the basic concept behind the active lubrication, namely, the modification of the fluid film velocity field via controllable pressurized oil injection in order to render the bearing active. Most of the work related to actively lubricated TPJBs has resorted to a geometry where the injection nozzle is installed through the pad pivot point, due to its advantages from a construction point of view. Nevertheless, other configurations have been investigated. Bearing pads featuring an array of injection nozzles symmetrically distributed with respect to the pivot point were studied in [29, 30]. In [31, 32], the injection nozzle was located displaced from the pivot point, in a position closer to the pad leading edge. The theoretical results showed that by injecting oil at a constant pressure in that position it was possible to improve the stability margin of a flexible rotor mounted on such bearings. In [33], it was proved by theoretical means that a pad configuration where the injection nozzle is displaced from the pivot line offers advantages in terms of an increase of the controllability of the actively lubricated TPJB, when compared to a 2

centrally located nozzle configuration. It means that the “displaced” nozzle configuration offers more “output” controllable force for the same control input signal and associated hydraulic system, a consequence of the increase of the resulting moment of the pressurized oil injection over the pads tilting motion. So far, a circular injection nozzle has been the geometry of choice for the investigations concerning the active lubrication concept. It seems feasible to apply the same concept using other geometrical configurations, if they provide advantages compared to the original geometry. In this line, several passive lubrication systems are currently being used in industrial TPJBs. Among them, the leading edge groove (LEG) system is widely used, which employs a deep groove close to the pad leading edge to provide the passive lubrication supply. This bearing design exhibits advantages in terms of a reduction of the required oil supply flow and the bearing power loss, as it has been proven by some authors [34, 35]. From the dynamic point of view, investigations related to LEG TPJBs have demonstrated that a modification of the pressure field around the groove can introduce significant variations of the bearing dynamic behavior. In [36], the authors investigated instability issues experienced by tilting pad bearings when operating under light load conditions. By introducing some geometrical modifications over the pad leading edge groove, the authors managed to solve the instability issue. They also showed that the modification had a significant impact over the equivalent stiffness and damping of the bearing. Considering this background, it seems feasible to obtain a controllable TPJB design based on the active lubrication concept, where the oil injection is performed closer to the pad leading edge. Moreover, the leading edge groove oil supply system, which is already used in industrial TPJBs due its advantages in terms of reduction of oil supply and bearing losses, could be employed to perform the active oil injection into the bearing clearance. By doing this, the introduction of controllable characteristics into standard TPJBs that feature the leading edge groove system could be straightforward, as it only implies a modification of the oil supply loop consisting of the introduction of a controller, a servovalve and a high pressure oil supply pump. This article explores, by theoretical means, the introduction of the active lubrication concept into LEG tilting pad bearings. A mathematical model is developed, which links the oil flow and pressure in the oil supply system (servovalve, oil feedline and leading edge groove), with the fluid flow and pressure within the bearing clearance, the thermal effects in the lubricant film and the pad surface motion, due to tilting action, elastic deflections and thermal growth. The modelling of the hydraulic system follows in general lines the method employed in previous publications related to actively lubricated bearings [19], although the LEG geometry implies a different mathematical model for the coupling of injection flow with bearing pressure field. The validation of such simplified model is undertaken by means of benchmark CFD calculations. The elastothermohydrodynamic model for the TPJB is validated using experimental data available in the literature [37, 38]. Finally, the Active LEG TPJB model is applied to a simple system (bearing supporting a rigid disk), in order to study the implications of the introduction of controllable characteristics into this bearing design. 2. MATHEMATICAL MODEL: ACTIVE LEG TPJB By introducing the active lubrication concept into the LEG lubrication scheme, one aims at transforming the bearing into a mechatronic actuator. An electric input signal is sent to a servovalve in order to regulate the flow at its port. The flow is supplied to the LEG cavity, resulting in a modification of the fluid velocity and pressure within the groove, which in turn will modify the 3

Figure 1: Schematics of the hydraulic system associated with the Actively Lubricated LEG TPJB.

Figure 2: Reference frame used for analyzing the LEG TPJB fluid film domain

bearing clearance pressure field. As a result of the increase of pressure within the groove and the bearing clearance, a modification of the servovalve port flow is obtained. The mathematical model for the Actively Lubricated LEG must be able to portray this two-way coupling between flow and pressure, while keeping a simplified and efficient formulation. Such simplified formulation is desirable when coupling this model with the model of the rotor system where the active vibration control must take place. In order to achieve this objective, the current work expands the modelling approach followed in [39], accounting for the different geometry of the injection area, namely a circular injector radially located across the bearing pad in [39] versus a deep groove in the current work. The modelling of the hydraulic system (servovalve and oil feedline) is based on a lumped parameter approach, whereas the bearing clearance pressure field is obtained via the FEM solution of the Reynolds equation in the circumferential and axial direction. The main difference of this work with the earlier ones arise in the way in which the injection flow is related to the groove and bearing clearance pressure fields, as it will be presented here. 2.1. Oil Flow and Pressure within the Hydraulic System and the Bearing Clearance For the sake of simplicity, consider a system composed of a journal supported vertically by a single tilting pad with a leading edge groove, associated to a single servovalve and pressurized oil feedline, see Fig.1. The mathematical model is stated as follows: 4

Figure 3: Nomenclature for the LEG flow and pressure modelling

1. Servovalve spool driven flow q xV as a function of an input control signal uV [40, 41]: q¨ xV + 2ξV ωV q˙ xV + ω2V q xV = ω2V RV uV

(1)

This second order differential equation portrays the frequency dependent relationship between input control signal uV and servovalve spool driven flow q xV . For the Active LEG TPJB, one must ensure that oil flow is reaching the LEG at all times to maintain the required passive lubrication. Consequently, even though a servovalve usually presents two output ports, only one of them is connected with the LEG, which implies that each pad must be associated to a single servovalve. Furthermore, the control signal uV fed into the servovalve must be always positive, to ensure that the oil supply flow is not interrupted. 2. Servovalve port flow qV , modelled by means of a first order linearization [42], that considers leakage flow q∗V , spool driven flow q xV and flow - servovalve port pressure p port relationship via K pq coefficient: qV = q∗V + q xV − K pq p port

(2)

3. Oil feedline flow, modelled by considering a Hagen Pouiseuille flow profile driven by the pressure difference between the servovalve port pressure p port and the LEG cavity pressure pLEG : p port − pLEG =

8μL pipe πR4pipe

qV

Where L pipe and R pipe are the length and radius of the feedline, respectively. 5

(3)

4. For simplicity, it is considered that oil pressure value within the LEG cavity does not present spatial variation, setting it to a single value equal to pLEG . 5. At the LEG cavity, mass conservation must hold. Consequently, it must be ensured that the flow entering the cavity through the feeding line qV is equal to the summatory of the flows leaving or entering through the bearing clearance on the LEG trailing, leading and side edge. Using a similar approach to [43], one assumes that the Reynolds model holds for the fluid flow within the bearing clearance, in particular at the LEG edges. Referring to Fig.2 and Fig.3, by imposing axial simmetry for the system (no flow crosses through the pad axial center line), the mass conservation requirement can be stated as: qV + qlead − qtrail − q side = 0

(4)

Where the flows through the leading edge qlead , trailing edge qtrail and side edge q side of the supply groove are modelled by:  qlead = 

LEG Laxial

0

qtrail =

q side

h3 ∂p ΩRh − 2 12μ ∂ xˆ

 dˆz lead

 h3 ∂p ΩRh − dˆz 2 12μ ∂ xˆ trail   LLEG  3 trail h ∂p = d xˆ LEG 12μ ∂ˆz side Llead

LEG Laxial

0

 

(5)

Where h is the oil film thickness function, Ω is the rotor rotational speed and R its radius. Eqn. (5) introduces the simplificatory assumption that the flow pattern fully develops in the interface between the LEG cavity and the bearing clearance. The integrals corresponding to the Couette and Pouiseuille flows in Eqn. (5) are numerically evaluated for the finite element grid nodes located in the leading, trailing and side edge respectively. Since the bearing clearance pressure field p is not known a priori, and the oil film thickness is a function of the rotor position xr , and pad modal coordinates b, the numerical evaluation via FEM of the integrals in Eqn. (5) yields an expression of the following structure: [W (xr , b)] {p} − qV = qcouette (xr , b)

(6)

6. The oil film pressure field within the bearing clearance is modelled by means of the Reynolds Equation for laminar and incompressible flow in circumferential and axial direction:     ∂ h3 ∂p ∂ h3 ∂p ΩR ∂h ∂h (7) + = + ∂ xˆ 12μ ∂ xˆ ∂ˆz 12μ ∂ˆz 2 ∂ xˆ ∂t The oil film thickness function h depends on the journal position xr and the pad dynamics, represented as a structural finite element model reduced to the modal coordinates b. By applying FEM to Eqn.(7) , one obtains the discretized set of equations for the pressure field:   [A (xr , b)] {p} = {r xr , b, x˙ r , b˙ } (8) Among the nodes of the {p} field, one finds the nodes lying on top of the LEG cavity, as well as the ones sitting on the LEG leading, trailing and side edge plead , ptrail , p side . 6

7. For the numerical solution of the Reynolds equation, Eqn. (8), a suitable set of boundary conditions must be enforced. The static pressure is set to zero at the pad external boundaries. For the FEM nodes located on top of the LEG cavity and on its edges, the pressure is set to be constant and equal to the LEG cavity pressure pLEG . At the LEG trailing edge, the Rayleigh step effect can increase ptrail above the pLEG value. In [44], Constantinescu and Galetuse obtained approximated integral momentum equations for analyzing the contribution of fluid inertia effects in bearing lubricant flows. For the case of an infinitely long step bearing, by neglecting the contribution of viscous effects at the step, they stated that the pressure variation at the discontinuity from clearance h1 to h2 can be modelled by: 2 ρ (ΩR)2 dh d p αρQ step dh = − β d xˆ h d xˆ h3 d xˆ α = 1.2 for laminar flow β = 0.133 for laminar flow

(9)

Which yields that the local pressure increment at the LEG trailing edge can be modelled as: ⎛ ⎞   ρ 2 ⎜⎜⎜ 1 1 ⎟⎟⎟ h1 2 ⎜ ⎟ Δp step = α Q step ⎝ 2 − 2 ⎠ − βρ (ΩR) ln (10) 2 h2 h2 h1 In order to apply Eqn.(10) to the situation under analysis, some assumptions are introduced. Firstly, the axial variation of pressure on the LEG trailing edge is neglected. This assumption is based on CFD benchmarking results, as it will be shown in a following section of this article. Consequently, it is possible to apply here Eqn.(10), originally deviced for an infinitely long bearing , i.e. a bearing with negligible axial pressure gradient. Based on this reasoning, the pressure at the trailing edge is assumed to be constant, and it is modelled as: ptrail = pLEG + Δp step

(11)

Consequently, the pressure at the calculation nodes located on top of the groove trailing edge is imposed as a boundary condition. Hence, at these calculation nodes the Reynolds equation is not being solved, which eliminates the possible inconsistency associated to using a finite length bearing model (Reynolds) with an infinite length model (Constantinescu and Galetuse). Secondly, the flow trough the step Q step is assumed to be equal to the superposition of the Couette and Pouiseuille flow at the groove trailing edge, see Eqn.(5). Thirdly, the oil film thickness before the step h1 is set as a fitting parameter, based on the results delivered by benchmarking CFD calculations, while the h2 is set to the actual bearing clearance at the trailing edge. By solving in a coupled manner Eqn.(1,2,3,6,8), it is possible to determine simultaneously p,pLEG , p port ,qV as function of an electric input signal uV fed into the servovalve. Consequently, it is feasible to determine the resulting forces over the rotor as a function of this electrical signal. 2.2. Thermal and Flexibility Effects The importance of including thermal effects as well as pad and pivot flexibility for an accurate modelling of TPJB has been widely discussed in the literature. The current model introduces flexibility effects by means of the FEM solution of the following model: 7

1. The system matrix A and the source term r related to the Reynolds equation solution are a function of the rotor position xr , and the pad displacements reduced to the modal coordinates b, that define the oil film thickness function h. Considering a rigid rotor, and arranging the pad mode shapes vectors in a modal matrix V, their dynamics can be modeled by [45, 21]: [Mr ] {¨xr } = {fr (p)} + {fext } ¨ + [V]T [Kb ] [V] {b} = [V]T {fb (p)} [V]T [Mb ] [V] {b}

(12)

Where the pads mass matrix Mb and stiffness matrix Kb are obtained by applying the FEM method, considering a rigid pivot and a linear elastic two dimensional pad model in the circumferential and radial direction. 2. The pivot flexibility effect is considered by inserting an additional pad mode shape vector to the pad modal matrix V, in a similar way to [7]. Its modal mass is set to the pad mass, whereas its modal flexibility is set to the pivot stiffness, determined by experimental measurement or using the analytical relations determined by [46]. The bearing thermal model is defined in the following manner: 1. It is assumed that the oil temperature inside the LEG cavity does not present spatial variation, setting it to be equal to the oil supply temperature. This value is applied as a boundary condition for calculating the oil and pad temperature field. 2. The oil film T oil and pad T pad temperature field are calculated using a two dimensional FEM model, formulated in the circumferential and radial direction at the pad axial center line. Temperature variation in the bearing axial direction is assumed negligible. The calculation is performed by the coupled solution of the oil film energy equation and the Fourier heat conduction model for the pad material, stated in a similar way to the model used in [47]. The oil film energy equation includes convective and diffusive effects, plus the source term due to viscous heating. The heat transfer between oil and pad surface is included explicitly in the model formulation, whereas the heat transfer towards the environment is neglected. The constitutive equations are:  2   2 ∂u ∂T oil ∂ T oil ∂2 T oil = μ + ρC + u + Soil koil p boundary ∂ xˆ ∂ xˆ ∂ xˆ2 ∂ˆy2  2  ∂ T pad ∂2 T pad pad k pad + = Sboundary ∂ xˆ2 ∂ˆy2 ∂T oil (13) Soil boundary boundary = koil ∂ˆy ∂T pad pad Sboundary = k pad boundary ∂ˆy pad Soil boundary = Sboundary

Where koil and k pad are the thermal conductivity of oil and pad material, C p is the oil thermal capacity, and u is the oil velocity field in the xˆ direction. 3. The calculated pad temperature field is used to predict the resulting pad thermal growth. The resulting strain is calculated by applying to the pad flexibility model a linear thermal expansion coefficient α and the temperature field, in a similar way to [47]. = αΔT pad 8

(14)

4. The calculated oil film temperature field is used to update its viscosity. For doing so, an exponential viscosity - temperature dependency law is used. μ (T ) = μ∗ eν(T −T

∗)

(15)

2.3. Active LEG TPJB Dynamic Coefficients

The equations that govern the dynamics of the studied system are: [Mr ] {¨xr } = {fr (p)} + {fext } ¨ + [V]T [Kb ] [V] {b} = [V]T {fb (p)} [V]T [Mb ] [V] {b}   [A (xr , b)] {p} = {r xr , b, x˙ r , b˙ } [W (xr , b)] {p} − qV = qcouette (xr , b) 8μL pipe p port − pLEG = qV πR4pipe

(16)

qV = q∗V + q xV − K pq p port q¨ xV + 2ξV ωV q˙ xV + ω2V q xV = ω2V RV uV Considering the state of the system as defined by the variables s = {xr , b, p, qV , pLEG , p port , q xV }, it is possible to obtain a first order linearization of the constitutive equations following the method exposed in [27, 48], yielding: [Mr ] {δ¨xr } − {G} = 0 ¨ + [V]T [Kb ] [V] {δb} − {F} = 0 [V]T [Mb ] [V] {δb}  ∗ ˙ A {δp} + {H}δpLEG + [B] {δb} + [C] {δb} + [D] {δxr } + [E] {δ˙xr } = 0  ∗ W {δp} + [J] {δxr } + [L] {δb} − δqV = 0 8μL pipe δp port − δpLEG = δqV πR4pipe δqV = δq xV − K pq δp port δq¨ xV + 2ξV ωV δq˙ xV + ω2V δq xV = ω2V RV δuV 9

(17)

Where ∗ denotes the magnitudes or matrices calculated for the system equilibrium position, δ denotes the linearized magnitudes, and: ∂ [A∗ ] ∗ ∂{r} p − ∂b ∂b ∂{r} [C] = − ∂b˙ ∂ [A∗ ] ∗ ∂{r} [D] = p − ∂xr ∂xr ∂{r} [E] = − ∂˙xr ∂fb {F} = [V]T { }T {δp} ∂p ∂fr {G} = [V]T { }T {δp} ∂p ∂{p} {H} = ∂pLEG ∂ [W∗ ] ∗ ∂qcouette [J] = p − ∂xr ∂xr ∂ [W∗ ] ∗ ∂qcouette [L] = p − ∂b ∂b [B] =

(18)

The perturbed pressure field δp can be expressed as a function of the rotor, pad movements, as well as variations of the pressure in the groove pLEG by means of:   ˙ {δp} = − A∗ −1 ({H}δpLEG + [B] {δb} + [C] {δb} (19) + [D] {δxr } + [E] {δ˙xr }) By using Eqn.(19), Eqn.(17) can be reduced to the following form: [MALEG ] {δ¨s} + [DALEG ] {δ˙s} + [KALEG ] {δs} = {δfALEG } {δs} = {δxr , δb, δqV , δpLEG , δp port , δq xV }T {δfALEG } = {0, 0, 0, 0, 0, δuV }

(20)

T

The control electrical signal δuV can be fixed to a constant value, or it can be set as function of the system states, for instance, the rotor position δxr , which yields a control loop scheme. For instance, a PD controller can be set up in the following form: δuV = {GP }{δxr } + {GD }{δ˙xr }

(21)

Which yields: [MALEG ] {δ¨s} + [DALEG (GD )] {δ˙s} + [KALEG (GP )] {δs} = 0 {δs} = {δxr , δb, δqV , δpLEG , δp port , δq xV }T

(22)

In Eqn.(22), the system dynamics have become a function of the controller proportional and derivative gains, rendering the system controllable. By performing a standard dynamic condensation, it becomes possible to condense away the variables related to the hydraulic system and 10

the pad movements. The obtained reduced system includes only the degrees of freedom related to the rotor as follows: [Mr ] {¨xr } + [Dreduced ] {˙xr } + [Kreduced ] {xr } = 0

(23)

The reduced set of dynamic coefficients Dreduced , Kreduced exhibit frequency dependency, due to contribution in terms of dynamics coming from the pads and the hydraulic system. 2.4. Numerical Implementation The mathematical model for the Actively Lubricated LEG TPJB is implemented in a FORTRAN 95 program. The finite element method is used for solving the constitutive equations related to the oil film pressure field, the thermal model and the pad flexibilility. The FEM implementation is heavily based on the one presented in [21], hence further details about it can be found there. The finite element model for the oil film pressure field is stated in circumferential and axial direction, whereas the thermal and pad flexibility model are stated in the radial and circumferential direction. In all cases, quadrilateral second order elements are implemented. The Galerkin method is used to obtain the weak form of the partial differential equations, with the exception of the oil film energy equation, where the Streamline Upwinded Petrov-Galerkin Method [49] is implemented. The program calculates the bearing equilibrium state using the Newton-Raphson method, using the following scheme: 1. The initial state of the system is imposed by means of an initial guess, in terms of the oil and pad temperature fields (set to be constant and equal to the oil supply temperature), and the rotor position xr and the pads modal coordinates b. Both x˙ r , b˙ are set to zero. Furthermore, the control signal sent to the servovalves uV is set to the desired value. 2. The flow and pressure within the hydraulic system and the bearing clearance are obtained by solving the following system of equations, under the already discussed boundary conditions: [A (xr , b)] {p} = {r (xr , b)} [W (xr , b)] {p} − qV = qcouette (xr , b) 8μL pipe p port − pLEG = qV πR4pipe

(24)

qV = q∗V + q xV − K pq p port q xV = RV uV 3. The linearized stiffness matrix KALEG at the current system state s is calculated and then it is used to solve for the increment of the state δs: [KALEG ] {δs} = {fr (p) + fext , [V]T {fb (p)}, 0, 0, 0, 0}T {δs} = {δxr , δb, δqV , δpLEG , δp port , δq xV }T

(25)

4. The system state is updated and set to s + δs. 5. Steps 2, 3 and 4 are repeated until convergence is achieved, i.e. the system has reached static equilibrium. 11

6. The thermal model, Eqn.(13), is solved for the calculated system equilibrium position, and the oil film viscosity is updated by means of Eqn.(15). 7. If the temperature field exhibits convergence with respect to the previously set value, then the system has reached thermal equilibrium and the calculation is stopped. Otherwise, the procedure gets back to step 2. 8. Once the system has reached steady state (static and thermal equilibria), the full set of dynamic coefficients is calculated as shown in Eqn.(20). Calculation time is highly dependent on the initial guess for the system state. Specifically, most of the calculation time is devoted to solve the system of equations in step 2 and 3. This is due to the structure of the matrix, which is non symmetric and presents off diagonal terms, that prevents the usage of solvers optimized for banded matrices. Currently, a frontal solver is implemented, but the authors are working on incorporating into the calculation code existing multithreaded solvers for non symmetric sparse linear systems. 3. MODEL BENCHMARKING The numerical model for the Active LEG TPJB is essentially composed of two parts: the leading edge groove fluid domain model and the elastothermohydrodynamic model for the tilting pad bearing clearance. A validation process is undertaken for both components, based on benchmark numerical and experimental results. 3.1. LEG Model: Validation against benchmark CFD results The objective is to validate the simplified model that couples the oil feedline flow qV with the pressure value within the leading edge groove pLEG and within the bearing clearance p. Namely, this corresponds to the coupled of solution of Eqn.(1,2,3,6,8). To this end, the results from a CFD model implemented in COMSOL are used as benchmark. The simulated geometry is depicted in Fig.4. It corresponds to a simplified geometry consisting of an inlet oil feeding line, a leading edge groove, and two flat surfaces. The gap between the surfaces is 150 μm at the pivot line. The upper face is rotated around the pivot line with an angle of 10−3 rad, in order to conform a converging wedge. The lower surface is fixed, whereas a tangential speed ΩR is imposed in the upper one. Only half of the geometry is modelled, hence a simmetry boundary condition is enforced. At the open end of the oil feedline, an input flow qV is imposed. At the open ends of the converging wedge, the pressure is set to zero. The analysis is run considering incompressible laminar flow. No thermal effects or flexibility effects are considered, since their modelling is validated in the next section. Inspired by the analysis presented in [50], a parametric study is carried out, aimed at evaluating the correspondence between benchmark CFD results and the simplified model for several operational conditions. Since the pressure generation mechanism for the studied bearing comes both from the hydrodynamic as well as the oil injection effect, the study includes cases where each effect is stronger or weaker than the other one. Furthermore, the effect of the groove depth is also included, as it could entail a variation of the strength of fluid inertia effects within the injection area. For the sake of brevity, only the most representative cases are included in this article. Three non dimensional parameters are varied, namely: Reynolds number at bearing clearance, Reynolds number at the oil feeding line, and the aspect ratio AR of the leading edge groove, 12

Table 1:

Cases for the parametric study to validate the simplified active LEG model Case # 1 2 3 4 5 6 7 8

Reh 25 25 250 250 25 25 250 250

Re pipe 300 600 300 600 300 600 300 600

AR 1.5 1.5 1.5 1.5 0.5 0.5 0.5 0.5

defined as: ρΩRh μ 2ρqV Re pipe = μπR pipe GD AR = 10[mm] Reh =

(26)

The simulated cases are listed in Tab.1. The mesh used for the CFD benchmark model is portrayed in Fig.4. The discretization used consists of third order elements for the pressure field, and second order elements for the fluid velocity field. A convergence study for different operational conditions was performed in order to validate the discretization used. Fig.5 shows that the selected mesh delivers the same results for oil pressure field, when compared to denser meshes. 3.1.1. Velocity field results Previous investigations on the active lubrication concept focused on studying a geometry where the oil injection into the bearing clearance is performed directly by a circular nozzle [19]. The modelling approach followed in that case resorted to an assumed velocity profile (fully developed Hagen Pouseuille flow profile) for the oil injected into the bearing clearance, in order to link the hydraulic system with the bearing pressure field. Consequently, the question arises whether a similar approach could be followed for modelling the Actively Lubricated LEG bearing. Fig.6 provide some insight into this issue. It can be seen that the oil velocity field in the vertical direction at the interface between feedline and LEG closely resembles the Hagen Pouseuille profile, although it tends to become perturbed when the Reynolds number at the bearing clearance is increased, due to the stronger recirculation effect entailed within the groove. Such perturbation of the velocity profile takes place even for the “deeper” groove geometry simulated. On the other hand, when looking at the vertical velocity profile at the LEG upper boundary, it can be seen that the alternative of adjusting a simplified injection velocity profile to the whole LEG area is not possible. This confirms that the “mass conservation” approach followed in this work is the correct one for modelling the active LEG bearing. The velocity profile becomes quite complex, due to the superposition of the oil injection effect on top of oil feedline, a strong recirculation pattern 13

Figure 4: Geometry and mesh used for the benchmark CFD model (all dimensions in milimeters): the parameters varied for the study are the journal tangential velocity ΩR, the injection flow qV , and the groove depth GD

around it, a “negative” (downwards from the shaft surface) velocity profile in the proximities of the LEG trailing edge due to the pressure build up in that area, and a “positive” (towards the shaft surface) velocity profile closer to the LEG side edge, due to the lower pressure taking place within the bearing clearance in that area. 14

·106 2.5

Mesh #1 Mesh #2 (4 times number of nodes) Mesh #3 (16 times number of nodes)

Pressure [Pa]

2

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1

Pad length [non dim]

Figure 5: Convergence study for the CFD benchmark model: pressure profile at the pad centerline for Mesh 1 (used for the study) and two other denser meshes

3.1.2. Pressure field results The previous results portray the difficulties to obtain a simplified model for the velocity field within the LEG. However, from the perspective of the design of the controllable bearing, the interest is set in modelling with sufficient accuracy its input/output relationships. In this case, it refers to the relation between input oil flow and the resulting bearing pressure field, with the consequent loads over rotor and pads. Fig.7 depicts the comparison between the benchmark CFD results and the simplified model for the bearing pressure field. It can be seen that the simplified model is able to resemble with sufficient accuracy the benchmark result. Main features of the obtained pressure field is negligible variation of the pressure field within the LEG extents, and a relevant contribution from Rayleigh step effect at the LEG trailing edge in terms of a local pressure increment. From the CFD results, it can be seen that pressure gradient in the LEG axial direction is to a large extent negligible, which enables to use the simplified model from [44] to model the Rayleigh step effect in terms of pressure increment. A more detailed comparison between CFD and simplified model for the pressure field results is portrayed in Fig.8. Good resemblance between the two implemented models is observed in all cases tested. It can be seen that the simplified model for the Rayleigh step is able to track such effect with good accuracy. In order to obtain these results, the oil film thickness before the LEG trailing edge was set equal to 10 times the bearing clearance at the LEG edge. Furthermore, it can be seen that the modification of the LEG aspect ratio does not entail a significant modification of the pressure field, at least within the order of magnitude of aspect ratios tested. 15

Figure 6: Oil velocity field (m/s) in the vertical yˆ direction obtained via CFD (A) LEG bottom AR = 1.5,Reh = 250,Re pipe = 600 (B) LEG bottom AR = 0.5,Reh = 25,Re pipe = 300 (C) LEG upper boundary (interface with bearing clearance) AR = 1.5,Reh = 250,Re pipe = 600 (D) LEG upper boundary (interface with bearing clearance) AR = 0.5,Reh = 25,Re pipe = 300

Figure 7: CFD (upper half) vs. Simplified Model (lower half) : Pressure profile in Pascals at the pad surface for two cases (A) AR = 1.5,Reh = 250,Re pipe = 600, (B) AR = 0.5,Reh = 25,Re pipe = 300

3.2. TPJB ETHD Model: Validation against experimental data Having verified the model for the LEG in terms of the relationship between flow and pressure, the next step is to verify the implemented elastothermohydrodynamic TPJB theoretical model. There is a large number of publications dealing with the experimental characterization of TPJBs. Here, one of the most recent publications has been selected. Gaines and Childs [37] provided an experimental study on the effect of pad flexibility over the TPJB static and dynamic behavior. Their investigation used a three-pad, load between pad bear16

Figure 8: CFD vs. Simplified Model: Pressure profile at the pad centerline for different Reynolds numbers at the injection pipe and bearing clearance, and two different aspect ratios of the leading edge groove

ing configuration, with three pad sets featuring different radial thickness (8.5 mm to 11.5 mm). No LEG is featured in the tested bearing, hence this feature is turned off for these simulations. Since the focus here is to validate the ETHD model of the TPJB, the LEG presence or absence becomes irrelevant. The only difference it introduces from the modelling point of view is a different boundary conditions for the thermal model, since the absence of the LEG entails the introduction of oil mixture equations to determine the oil temperature between pads. The applied unit loads ranged from 172 kPa to 1724 kPa. Some additional experimental results arising from this study were included by San Andres and Li [38] in an article aimed at validating a TPJB numerical model. The experimental results obtained by the aforementioned authors with the pad thickness equal to 11.5 mm are used here as benchmark. Fig.9 depicts the comparison between the model and the experimental results regarding rotor eccentricity and variation of pad surface temperature (difference between maximum and minimum surface temperature). Better coherence between the model and the experiment is observed for the results obtained at the lower rotational speed (6000 17

Table 2:

Simulated TPJB geometrical and hydraulic system parameters

Parameter # Rotor Diameter Number of Pads Load Configuration Bearing Diameter Radial Pad Clearance Radial Bearing Clearance Preload Pad Axial Length LEG Axial Length Pad Thickness Pad Arc Angle LEG Arc Angle Pivot Offset Lubricant Lubricant Inlet Temperature Servovalve cut-off frequency ωV Servovalve leakage flow q∗V Servovalve flow pressure coeff. K pq Servovalve flow voltage coeff. RV Servovalve damping ratio ξV Length Oil Feedline L pipe Radius Oil Feedline R pipe

Value 101.59 [mm] 3 Load between pads 101.74 [mm] 0.1016 [mm] 0.0762 [mm] 0.25 60.96 [mm] 50 [mm] 11.5 [mm] 90 [degrees] 50 - 65 [degrees] 50% ISO VG 46 49 [oC] 200 Hz 2e-6 m3 /s 1.13e-12 m3 /(sPa) 1.5e-4 m3 /(sV) 0.6085 1m 5e-3 m

RPM). The causes for the divergence at higher rotational speed (12000 RPM) are not clear, but it was also observed in the results provided by [38]. The authors provided a discussion stating that keeping a constant oil supply flow rate for the whole testing program could be a factor contributing to some divergences observed between their theoretical model and the same experimental results. Fig.10 and 11 depicts the comparison between the model prediction and the experimental data concerning the tested bearing impedance for two different rotational speeds and an applied unit load of 1724 kPa. It can be seen that the model is able to predict in a good manner both the magnitude of the impedance, as well as its frequency dependency. Critical for obtaining such good coherence was the fact that the authors of [37] thoroughly reported the pivot stiffness of the tested bearing. Slightly better correlation is observed for the 6000 RPM results, a consequence of the accurate prediction of the system static and thermal equilibrium observed before in Fig.9. 4. SIMULATION: RIGID DISK SUPPORTED BY ACTIVE LEG TPJB The model for the Active LEG TPJB is used here to study the effect of the introduced active characteristics over the bearing performance. To do so, the TPJB defined in Tab.2 is simulated here again. This bearing is simulated both without LEG in the pads (in the following labelled as ’No LEG’ case) to provide a reference line, and also with the active LEG features turned on. 18

The bearing applied unit load is exerted in the ’between pads’ direction, and it is set to a constant value of 689 kPa for all studied cases. The supported element corresponds to a rigid disk with only two degrees of freedom (vertical and horizontal movement). The modelling of the coupling between the bearing and a flexible rotor dynamics will be left for a later step in this research effort. Concerning the definition of the servovalve control signals, two cases are studied: open loop and close loop implementation 4.1. Active LEG TPJB in open loop configuration In this case, the control signal governing each servovalve is imposed as a fixed value. The simulated combinations of control signals for each one of the three servovalves that compose the tested active bearing are listed in Tab.3. Case 1 corresponds to the base case, with all the servovalves providing a minimum flow to the LEGs, in order to ensure lubrication supply at all times. Case 2 and 3 entail equal opening of all the servovalves, with a higher value of supply flow and resulting LEG pressure. Case 4 to 7 correspond to the opening of the servovalves aiming at moving the rotor in four different directions (up, down, right and left, respectively). Fig.12 depicts the modification of the oil film pressure, thickness and temperature at the bearing upper pad center line for different cases of the open loop implementation. The ’No LEG case’ provided for comparison exhibits a more symmetric, lower magnitude pressure profile yielding lower tilting angles, when compared to the active LEG cases. The maximum oil temperature achieved in the ’No LEG’ case is higher than in the active bearing configuration. This can be attributed to the lack of shear stress heat generation on top of the LEG surface for the active case. When increasing the oil flow reaching all of the LEGs (case 3), an increment of both the LEG pressure and the overall oil film pressure are observed. In order to reach load equilibrium, the pad is forced to increase its tilting angle, with the resulting increment of the maximum oil temperature. Variations of pressure, clearance and temperature are also observed when the rotor is forced towards the upper pad via an increase of LEG inlet flow at the lower pads (case 4), although in such case the variation of the upper pad tilting angle is smaller. This can be related to negligible variation of the LEG pressure and its resulting moment around the pad pivot point. These results indicate that it becomes possible to alter the system equilibrium position by altering the flow and pressure at the LEG cavities. Fig.13 portrays the modification of the rotor equilibrium position for the different cases of the open loop implementation. The span of rotor movement that can be imposed by the active bearing becomes narrower for higher rotational speed, due to the increase of the relative strength of the hydrodynamic pressure generation when compared to the oil injection effect. The cases where all three servovalves receive similar signals exhibit weak modification of the rotor position due to the opposition of the generated forces, although they do entail variation of the pad tilting angles as it was shown in Fig.12. These results entail that it could be possible to implement integral controllers using the Active LEG TPJB, aiming at modifying the rotor equilibrium position when required. Being the Active TPJB a non linear system, the modification of its equilibrium position should imply a modification of its equivalent stiffness and damping characteristics, with potential benefits from the system dynamics perspective. Since the implemented model for the active bearing also delivers the equivalent dynamic coefficients, it is possible to perform a study in that direction. Fig.14 depicts the amplitude plot of the frequency response function for different operational conditions of the Active LEG bearing, including also the ’No LEG’ case as baseline. The ’No LEG’ case exhibits lower equivalent stiffness and lower damping ratio, when compared to the rest of the cases. As it was seen before in Fig.12, significant differences exist in the pressure profile, oil film temperature and thickness obtained for the ’No LEG’ and the active LEG 19

Table 3:

Servovalve control signals imposed for the Active LEG bearing operating in open loop

regime Case 1 2 3 4 5 6 7

Servo 1 uV (Upper pad) 0.1 V 0.5 V 1.0 V 0.1 V 1.0 V 0.1 V 0.1 V

Servo 2 uV (Left pad) 0.1 V 0.5 V 1.0 V 1.0 V 0.1 V 1.0 V 0.1 V

Servo 3 uV (Right Pad) 0.1 V 0.5 V 1.0 V 1.0 V 0.1 V 0.1 V 1.0 V

cases. The higher tilting angle observed for the Active LEG cases implies that the oil film should become stiffer than for the ’No LEG’ case. In the FRF, the increment in the equivalent stiffness when introducing the LEG can be clearly observed. Furthermore, the average oil temperature for the Active LEG cases was lower than for the ’No LEG’ case. This fact can explain the increase of the system equivalent damping ratio observed in the FRFs for the Active LEG cases. When comparing the different Active LEG cases, it can be seen that the modification of the system equilibrium position observed before in Fig.12 and Fig.13 imply variations of the equivalent damping ratio and stiffness characteristics of the simulated system. The modification becomes slightly stronger for lower rotational speed, following the trend observed before for the induced rotor displacement. In general, the system tends to become stiffer when the LEG injection is engaged, which can be related to the increase of the pads tilting angle. Regarding damping ratio, only a modest variation of such parameter can be observed when different signals are fed into the LEG governing servovalves. 4.2. Active LEG TPJB in closed loop configuration The effect of introducing a closed loop control strategy into the configuration of the Active LEG TPJB is demonstrated here. It must be stressed that these results are only provided for illustrative purposes, since the control rule introduced is the simplest one available, and no optimization procedure has been undertaken to optimize its gains. The implemented controller is built on top of the set of open loop signals considered in case 5. Namely, the control signal fed into servovalve 2 and 3 (lower pads) is zero, whereas servovalve 1 (upper pad) is controlled by: uV = 1.0 + G P y + G D y˙

(27)

The control signal is built so that there is always a positive signal reaching servovalve 1, in order to prevent oil starvation at the bearing upper pad. The gains used for the simulation were G P = 2.5e3[V/m] and G D = 50[V s/m]. Fig.15 depicts the FRF amplitudes obtained when this controller is implemented, and compare them to some open loop cases. It can be seen that the introduction of the controller has almost no influence over the disk dynamic behavior in the horizontal direction. This is explained by the control rule (only vertical displacements are considered) and the position of the pad affected by servovalve 1 (upper pad), which should exert predominantly vertical forces over the rotor. On the other hand, the FRF in the vertical direction 20

is affected by reducing the amplitude of its resonant peak. Fig.16 depicts the stability map of the studied system, for different configurations. It can appreciated that the introduction of the LEG injection in open loop already implies an increase of the stability margin of the system, when compared to the ’No LEG’ case. By introducing the control rule in combination with case 5, such stability margin can be further enhanced. In the studied case, these results can be explained by the modification suffered by the damping characteristics and stiffness asymmetry when introduction the Active LEG. Although the results portrayed here are obtained for a specific controller and bearing geometry, they suggest the possibility of combining “passive” and “active” pads within the same bearing. This strategy could simplify even further the introduction of the active characteristics into LEG TPJB, since only the “active” ones would require a servovalve within their hydraulic supply system. Of course, this implementation would reduce the degree of controllability of the supported rotor, but under certain circumstances it could be enough to improve the system behavior. 5. CONCLUSION AND FUTURE ASPECTS This work constitutes the first step of a research effort aimed at studying the feasibility of introducing the active lubrication concept in tilting pad journal bearings (TPJBs) that feature the leading edge groove (LEG) lubrication system. At this stage, the focus was set on obtaining a theoretical model capable of simulating the studied system, in terms of its steady state and dynamic characteristics, considering the coupling between the supply hydraulic system and the elastothermohydrodynamic effects taking place in the bearing clearance. From the modelling point of view, the relationship between oil supply flow and pressure within the LEG cavity and bearing clearance can be accurately predicted by means of a simplified mathematical model based on the mass conservation principle. To improve its accuracy, it becomes relevant to include the Rayleigh step pressure build up effect at LEG trailing edge, in this case by means of a tuned analytical model. Within the range of parameters tested, the LEG depth exhibited a weak influence over the obtained results. Considering the theoretical results presented in this article, it seems feasible to modify the steady state and dynamics characteristics of LEG TPJB by means of the active lubrication concept. The implemented active characteristics are able to modify the system equilibrium position depending on the control signals, which yields a modification of the equivalent stiffness and damping characteristics. Such modifications can be obtained both with signals obtained in open loop configuration, as well as closed loop by means of adequate control rules. The next steps of this research effort will include theoretical and experimental investigations, aiming at simulating the system in a configuration closer to a real application (flexible rotor supported by the Active LEG TPJBs), and to validate experimentally the mathematical model developed in this article, considering the steady state and dynamic characteristics of this novel active bearing design. 6. ACKNOWLEDGMENTS The authors of this article would like to acknowledge CONICYT Chile (Project FONDECYT Iniciacion No. 11150112) for its financial support for this research project. 1. Preumont, A.. Mechatronics: Dynamics of Electromechanical and Piezoelectric Systems; vol. 136. Springer Science & Business Media; 2006.

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2. Vance, J.M.. Rotordynamics of Turbomachinery. John Wiley & Sons; 1988. 3. Childs, D.W.. Turbomachinery Rotordynamics: phenomena, modeling, and analysis. John Wiley & Sons; 1993. 4. Santos, I.F.. Mechatronics Applied to Machine Elements with Focus on Active Control of Bearing, Shaft and Blade Dynamics. Technical University of Denmark, Department of Mechanical Engineering; 2010. 5. Santos, I.F.. Trends in Controllable Oil Film Bearings. IUTAM Bookseries: Symposium on Emerging Trends in Rotor Dynamics 2011;1011:185–199. 6. Lanes, R., Flack, R., Lewis, D.. Experiments on the stability and response of a flexible rotor in three types of journal bearings. Tribol Trans 1982;25(3):289–298. 7. Suh, J., Palazzolo, A.. Three-dimensional dynamic model of TEHD tilting-pad journal bearing part I: Theoretical modeling. ASME J Tribol 2015;137(4):041703. 8. Suh, J., Choi, Y.S.. Pivot design and angular misalignment effects on tilting pad journal bearing characteristics: Four pads for load on pad configuration. Tribol Int 2016;. 9. Conti, R., Frilli, A., Galardi, E., Meli, E., Nocciolini, D., Pugi, L., Rindi, A., Rossin, S.. An efficient quasi-3d rotordynamic and fluid dynamic model of tilting pad journal bearing. Tribol Int 2016;103:449–464. 10. Cha, M., Glavatskih, S.. Nonlinear dynamic behaviour of vertical and horizontal rotors in compliant liner tilting pad journal bearings: Some design considerations. Tribol Int 2015;82:142–152. 11. Dang, P.V., Chatterton, S., Pennacchi, P., Vania, A.. Effect of the load direction on non-nominal five-pad tilting-pad journal bearings. Tribol Int 2016;98:197–211. 12. Deckler, D.C., Veillette, R.J., Braun, M.J., Choy, F.K.. Simulation and control of an active tilting-pad journal bearing. Tribol Trans 2004;47(3):440–458. 13. Wu, A., De Queiroz, M.. A New Active Tilting-Pad Bearing: Non Linear Modeling and Feedback Control. Tribol Trans 2010;53(5):755–763. 14. Wu, A., Cai, Z., De Queiroz, M.S.. Model-based control of active tilting-pad bearings. IEEE/ASME Trans Mechatronics 2007;12(6):689–695. 15. Viveros, H.P., Nicoletti, R.. Lateral vibration attenuation of shafts supported by tilting-pad journal bearing with embedded electromagnetic actuators. ASME J Eng Gas Turbines Power 2014;136(4):042503. 16. Chasalevris, A., Dohnal, F.. Improving stability and operation of turbine rotors using adjustable journal bearings. Trib Int 2016;104:369–382. 17. Santos, I.F.. Design and evaluation of two types of active tilting pad journal bearings. The Active Control of Vibration 1994;:79–87. 18. Santos, I.F.. On the adjusting of the dynamic coefficients of tilting-pad journal bearings. Tribol Trans 1995;38(3):700–706. 19. Santos, I.F., Russo, F.. Tilting-pad journal bearings with electronic radial oil injection. ASME J Tribol 1998;120(3):583–594. 20. Santos, I.F., Nicoletti, R.. THD Analysis in Tilting-Pad Journal Bearings using Multiple Orifice Hybrid Lubrication. ASME J Tribol 1999;121:892–900. 21. Haugaard, A.M., Santos, I.F.. Multi-orifice active tilting-pad journal bearings: Harnessing of synergetic coupling effects. Tribol Int 2010;43(8):1374–1391. 22. Cerda, A., Nielsen, B.B., Santos, I.F.. Steady state characteristics of a tilting pad journal bearing with controllable lubrication: Comparison between theoretical and experimental results. Tribol Int 2013;58(1):85–97. 23. Nicoletti, R., Santos, I.F.. Linear and Non-Linear Control Techniques Applied to Actively Lubricated Journal Bearings. JSV 2003;260(5):927–947. 24. Nicoletti, R., Santos, I.F.. Control System Design for Flexible Rotors Supported by Actively Lubricated Bearings. ASME J Vibr Acoust 2008;14(3):347–374. 25. Salazar, J.G., Santos, I.F.. Feedback-controlled lubrication for reducing the lateral vibration of flexible rotors supported by tilting-pad journal bearings. Proc Inst Mech Eng, Part J: J Eng Tribol 2015;229(10):1264–1275. 26. Salazar, J.G., Santos, I.F.. Exploring integral controllers in actively-lubricated tilting-pad journal bearings. Proc Inst Mech Eng, Part J: J Eng Tribol 2015;:1350650115570697. 27. Cerda, A., Santos, I.F.. Tilting-Pad Journal Bearings with Active Lubrication Applied as Calibrated Shakers: Theory and Experiment. ASME J Vibr Acoust 2014;136(6). 28. Santos, I.F., Svendsen, P.. Non-invasive parameter identification in rotor dynamics via fluid film bearings: Linking active lubrication and operational modal analysis. In: ASME Turbo Expo 2016: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers; 2016:GT2016–57618. 29. Santos, I.F., Nicoletti, R.. Influence of Orifice Distribution on the Thermal and Static Properties of Hybridly Lubricated Bearings. International Journal of Solids and Structures 2001;38:2069–2081. 30. Santos, I.F., Scalabrin, A.. Control System Design for Active Lubrication with Theoretical and Experimental Examples. ASME J Eng Gas Turbines Power 2003;125:75–80. 31. Cerda, A., Santos, I.F.. Stability Analysis of an Industrial Gas Compressor supported by Tilting-Pad Journal Bearings under Different Lubrication Regimes. ASME J Eng Gas Turbines Power 2012;134(2).

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32. Cerda, A., Santos, I.F.. Performance Improvement of Tilting-Pad Journal Bearings by means of Controllable Lubrication. Mechanics & Industry 2012;13:17–32. 33. Salazar, J.G., Santos, I.F.. On the controllability and observability of actively-lubricated journal bearings with pads with different nozzle-pivot configurations. ASME J Tribol 2016;. 34. Dmochowski, W.K.S.A., Brockwell K.and DeCamillo, S., Mikula, A.. A study of the thermal characteristics of the leading edge groove and conventional tilting pad journal bearings. ASME J Tribol 1993;115(2):219–226. 35. Bang, K.B., Kim, J.H., Cho, Y.J.. Comparison of power loss and pad temperature for leading edge groove tilting pad journal bearings and conventional tilting pad journal bearings. Tribol Int 2010;43(8):1287–1293. 36. Edney, S., DeCamillo, S.M.. Testing, analysis and CFD modelling of a profiled leading edge groove tilting pad bearing. ASME International Gas Turbine and Aeroengine Congress and Exhibition 1998;. 37. Gaines, J.E., Childs, D.W.. The impact of pad flexibility on the rotordynamic coefficients of tilting-pad journal bearings. In: ASME Turbo Expo 2015: Turbine Technical Conference and Exposition. American Society of Mechanical Engineers; 2015:V07AT31A002–V07AT31A002. 38. San Andr´es, L., Li, Y.. Effect of pad flexibility on the performance of tilting pad journal bearingsbenchmarking a predictive model. ASME J Eng Gas Turbines Power 2015;137(12):122503. 39. Santos, I.F., Russo, F.. Tilting-Pad Journal Bearings with Electronic Radial Oil Injection. ASME J Tribol 1998;120(3):583–594. 40. Edelmann, F.. High-Response Servovalves and their applications. Hydraulik und Pneumatik 1986;(30):1–6. 41. Thayer, W.. Transfer Functions for MOOG Servovalves. MOOG Technical Bulletin 1965;103:1–11. 42. Merritt, H.. Hydraulic Control Systems. Wiley and Sons, Inc.; 1967. 43. Santos, I.F., Watanabe, F.Y.. Compensation of cross-coupling stiffness and increase of direct damping in multirecess journal bearings using active hybrid lubrication: Part I theory. ASME J Tribol 2004;126(1):146–155. 44. Constantinescu, V.N., Galetuse, S.. On the possibilities of improving the accuracy of the evaluation of inertia forces in laminar and turbulent films. ASME J Lubr Tech 1974;96(1):69–77. 45. Kim, J., Palazzolo, A., Gadangi, R.. Dynamic characteristics of TEHD tilt pad journal bearing simulation including multiple mode pad flexibility model. ASME J Vibr Acoust 1995;117(1):123–135. 46. Kirk, R., Reedy, S.. Evaluation of pivot stiffness for typical tilting-pad journal bearing designs. ASME J Vibr Acoust 1988;110(2):165–171. 47. Fillon, M., Bligoud, J.C., Frene, J.. Experimental study of tilting-pad journal bearings,comparison with theoretical thermoelastohydrodynamic results. ASME J Tribol 1992;114(3):579–587. 48. Cerda, A., Santos, I.F.. Dynamic coefficients of a tilting pad with active lubrication: Comparison between theoretical and experimental results. ASME J Tribol 2015;137(3):031704. 49. Brooks, A.N., Hughes, T.J.. Streamline upwind/petrov-galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier-stokes equations. Computer methods in applied mechanics and engineering 1982;32(1):199–259. 50. Horvat, F., Braun, M.. Comparative experimental and numerical analysis of flow and pressure fields inside deep and shallow pockets for a hydrostatic bearing. Tribol Trans 2011;54(4):548–567.

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A b

DALEG Dreduced δi ξV FRF (ω) fext fr fb GP GD h ( xˆ, zˆ) K pq Kb KALEG Kreduced LEG L pipe Mr Mb MALEG μ p ( xˆ, zˆ, t) p p port pLEG qlead qtrail q side qV q∗V q xV r R

System matrix for Reynolds equation discretized solution Pad modal coordinates vector Linearized global damping matrix Synchronously reduced damping matrix Linearized i-variable Servovalve damping ratio Frequency response function External force over the journal,[N] Pressure field resultant force over the journal,[N] Pressure field resultant force over the bearing pad mode Controller Proportional Gain Controller Derivative Gain Oil film thickness,[m]  3 m Servovalve flow pressure coefficient, sPa Bearing pad stiffness matrix Linearized global stiffness matrix Synchronously reduced stiffness matrix Leading Edge Groove Oil feedline length,[m] Journal mass matrix Bearing pad mass matrix Linearized global mass matrix Oil dynamic viscosity,[sPa] Oil film pressure field,[Pa] Discretized oil film pressure field,[Pa] Servovalve port pressure,[Pa] Pad leading edge groove pressure,[Pa]  3 Groove leading edge flow, ms  3 Groove trailing edge flow, ms  3 Groove side edge flow, ms  3 Servovalve flow, ms  3 Servovalve leakage flow, ms  3 Servovalve spool driven flow, ms Right Hand Side vector for Reynolds equation discretized solution Journal Radius,[m]

24

R pipe RV ρ t T PJB s s∗ uV V W xˆ, yˆ , zˆ xV xr yr ωV ω

  Oil Feedline Radius, m2  3 Servovalve flow voltage coefficient, msV  kg  Oil density, m3 Time s Tilting Pad Journal Bearing Global system state vector Global system state vector in static equilibrium Servovalve control signal,[V] Pad modal matrix Mass conservation equation, coefficients vector Oil film curvilinear reference frame Servovalve spool position,[m] Journal position vector,[m] Journal vertical position,[m]   Servovalve cut-off frequency, rad s   Frequency, rad s

25

26

Figure 9: TPJB model benchmarking against experimental data [37, 38]: rotor eccentricity and variation of pad surface temperature for different applied unit loads (#1 172 kPa, #2 345 kPa, #3 689 kPa, #4 1034 kPa, #5 1724 kPa

27

Figure 10: TPJB model benchmarking against experimental data [37, 38]: bearing impedance obtained with an applied unit load of 1724 kPa and a rotational speed of 6000 RPM.

28

Figure 11: TPJB model benchmarking against experimental data [37, 38]: bearing impedance obtained with an applied unit load of 1724 kPa and a rotational speed of 12000 RPM.

29

Figure 12: Active LEG TPJB: Modification of the oil film pressure, thickness and temperature at the bearing upper pad, for 6000 RPM and different servovalve control signals in open loop (see Tab.3)

Rotor Eccentricity 6000 RPM

Rotor Eccentricity 12000 RPM

0

0 Case 4

Case 4 −0.1

−0.2 Case 3 Case 2 Case 1

−0.3

No LEG −0.4 Case 5

−0.5 −0.6

−0.4

−0.2

Case 7

Case 6

Case 6

Case 7

Eccentricity Y [non-dim]

Eccentricity Y [non-dim]

−0.1

0

Case 1,2,3

−0.2

No LEG −0.3 Case 5 −0.4 −0.5

0.2

−0.6

0.4

Eccentricity X [non-dim]

−0.4

−0.2

0

0.2

0.4

Eccentricity X [non-dim]

Figure 13: Active LEG TPJB: Rotor eccentricity for different configurations of the servovalve control signals in open loop (see Tab.3)

30

Figure 14: Active LEG TPJB: FRF Amplitude for two different rotational speeds and different configurations of the servovalve control signals in open loop (see Tab.3)

31

Figure 15: Active LEG TPJB: FRF Amplitude when implementing a simple proportional derivative controller

32

Stability Map 100 No LEG Case 3 Case 5 + PD Control

Real Part Eigenvalues

0

−100 −200 −300 −400 −500

0.9

1

1.1

1.2

1.3

Rotational speed [RPM]

1.4

1.5 ·104

Figure 16: Active LEG TPJB: Stability map of the simulated system, for different configurations of the servovalve control signals

33