Comparative analysis on the performance envelope of a three-pad and four-pad active journal bearing

Abstract

Controllable fluid film bearings with the ability to actively modify the operational conditions of rotor bearing systems possess high demand in the field of rotating machinery. Controllable or active bearing geometries can effectively enhance the performance envelope and life of machine support systems. Alternate forms of an active bearing pad geometry with unique adjustability features having the ability to modify the rotor bearing performance are presented in this study. The adjustability mechanism incorporated in the active pad geometry allows the multiple-bearing pads to translate radially and undergo tilted motions on a real-time basis. These pad adjustments are simulated by incorporating the radial and tilt adjustable functions into the developed theoretical model. In this study, a major focus is on a three-pad active bearing geometry and will further compare its steady state and dynamic performance behavior with a four-pad active bearing model. Effect of varying pad positions, pad angles and number of pads on the steady state and dynamic characteristics are analyzed in this study. For various combinations of operating parameters, pads adjusted to negative radial and negative tilt position were found to provide a notable improvement in the steady state and dynamic performance parameters. Identifying the optimal pad adjustments will further help to develop an active bearing with a feedback control mechanism operated on a real-time basis.

Keywords:

• – Active bearing
• radial adjustments
• tilt adjustments
• film thickness
• static characteristics
• dynamic coefficients

1. Introduction

Active journal bearing geometries find extensive applications in rotating machines and exhibit great potential in reducing the bearing susceptibility to oil whirl. The adjustability concept in journal bearings is devised to suppress the lateral vibrations generated in a rotor bearing system by modifying the bearing stiffness and damping properties. Krodkiewski et al. (Krodkiewski et al., 1997) mathematically modelled an active journal bearing consisting of a hydraulic damper and a deformable flexible sleeve. The sleeve was modelled to get activated by regulating the hydraulic chamber pressures which thereby influences the film thickness and dynamical behavior of rotor system. Krodkiewski and Sun (Krodkiewski & Sun, 1998) further developed the simulation model by considering the active bearing geometry in a multi rotor system. The journal equilibrium positions, critical speeds, threshold of instability and whirl motions of the journal were simulated for a set of operating parameters. Later, Sun et al. (Sun et al., 1998) formulated a feedback-controlled algorithm to minimize the vibrations generated in a rotor system operated with the developed active journal bearing.

A journal bearing embedded with movable pads was also proposed by Chasalevris and Dohnal (Chasalevris & Dohnal, 2012), to efficiently control the bearing dynamic coefficients and thereby minimize the rotor vibrations at resonance. The operating principle was based on introducing an additional fluid film layer in the bearing clearances by providing controlled variations to the bearing pad. By utilizing the self-controllable bearing, a 60–70% reduction in vibration amplitudes was observed for simulations in large-size rotor systems (Chasalevris & Dohnal, 2014). A working prototype of the proposed bearing model was further developed to be incorporated into an experimental setup. The dynamic responses of the system were measured in Chasalevris and Dohnal (Chasalevris & Dohnal, 2015), which highlighted its efficiency in vibration quenching under resonance conditions. Chasalevris and Dohnal (Chasalevris & Dohnal, 2018) further modelled different variable forms of partial bearings with controllable bearing parts to periodically vary the dynamic behavior of lubricant film. An improvement in stability regions was further attained by considering parametric excitation of bearing dynamic parameters at certain operational frequencies (Chasalevris & Dohnal, 2016; Chasalevris A, Dohnal F. Improving stability and operation of turbine rotors using adjustable journal bearings. Tribology International, 2016).

Due to their controllable properties, these active journal bearings can also effectively improve the dynamic stability of a rotor bearing system. Parkins and Martin (Parkins & Martin, 1998) invented two novel journal bearings with unique adjustable features to provide a controlled variation to bearing pads during operation. Martin (Martin, 1999, 2002) focused his studies particularly on an inversely oriented movable journal bearing geometry. The movable bearing segments were modelled using finite element technique, in which the displacements were provided to the nodes adjacent to the location of tapered pins. Analysis was further extended by Martin and Parkins (Martin & Parkins, 2002) to simulate the influence of varying load magnitudes and orientations on different bearing parameters under symmetrical and asymmetrical pad displacement conditions. For an increase in segment adjustments, a notable improvement in the bearing dynamic coefficients was particularly observed during operation. Martin and Parkins (Martin & Parkins, 2001) later performed experimental tests on a conventional type of adjustable bearing, which exhibited a notable decrease in the size of journal orbits. A measurement method was proposed by Martin (Martin, 2004a, 2004b) for recording the rotor displacement coefficients for subsequent changes in rotor positions and incremental load values.

In the early years, studies were also carried out by various researchers in investigating the performance behavior of partial arc journal bearings under different operating conditions. Recently, extensive studies were focused on modifying the geometry of partial journal bearings to control the bearing operation through pre-defined pad adjustments. Shenoy (Shenoy, 2008) modelled a 120° partial arc bearing with adjustable features to translate the pad in radial and tilt directions. Initial studies conducted by Shenoy and Pai (Shenoy & Pai, 2009) were focused on investigating the static performance of the variable bearing geometry subjected to varied L/D ratios, journal eccentricities and pad adjustment positions. For different pad adjustments and flow conditions, the dynamic performance of the proposed bearing was further evaluated using a linearized solution technique. In the analysis, the stability margins of the rotor system were determined by Shenoy and Pai (Shenoy & Pai, 2010), by employing the computed system dynamic coefficients. From the results, pad displaced in inward directions was found to produce better steady state and dynamic performances at higher journal eccentricities. Pai and Parkins (Pai & Parkins, 2018) later carried out detailed experimental studies on a conventional form of variable bearing geometry with multiple movable elements and experimentally measured the static performance envelope.

In view of controlling the vibrations in rotating machinery, Santos (Santos, 1994, 1995) designed and evaluated two different types of active tilting pad bearing geometries. In one of the proposed journal bearing configuration, the tilted bearing pads were mounted on a hydraulically operated chamber system consisting of two control valves. This control mechanism altered the dynamic properties of the bearing and in turn helped to attenuate the rotor vibration amplitudes crossing critical speeds. A further improvement in damping and bearing stability was also possible by directly influencing the kinematics of oil film through the activation of control valves. This active lubrication principle was further theoretically investigated by Santos and Russo (Santos & Russo, 1998) to analyse the feasibility of modifying the hydrodynamic forces using the electronic lubricant injection technique. In such actively lubricated tilted pad bearings, Nicoletti and Santos (Nicoletti & Santos, 2003) employed some linear and non-linear controller systems to alter the damping coefficients and rotor equilibrium positions. Simulations were carried out to analyse the control system efficiency and to determine the unbalanced and frequency responses of the rotor bearing system. Satisfactory results were obtained from the unbalance response analysis, where all the employed active control systems were able to regulate the rotor vibrations and eliminate the whirl instabilities under different operational conditions. The proposed active oil injection technique was then experimentally investigated by Santos and Scalabrin (Santos & Scalabrin, 2003), using a relatively light rotor operated under different rotational speeds and bearing parameters. Wu and Queiroz (Wu & De Queiroz, 2010) proposed an alternate approach to actively operate the tilting pad bearing by controlling the tilt mechanism. In the non-linear simulation, the angular velocities of tilting pads were modelled as control inputs to modify the journal center trajectories as per the requisite conditions.

Varela and Santos (Varela & Santos, 2012) further studied the controllable mechanism of oil injection influencing the thermal as well as dynamic behaviour of a tilting pad bearing model. The actively lubricated tilting pad bearing had proven to be useful in generating controllable forces in a wide range of operating frequencies. Due to this feature, studies were also conducted by Varela and Santos (Varela & Santos, 2014) to analyse the possibility of employing such bearing models as calibrated shakers for parameter identification in rotating machineries. Varela and Santos (Varela & Santos, 2015) later validated the developed mathematical model with the results measured from an experimental test rig employed with an actively operated single pad bearing configuration. Varela et al. (Varela AC, García AB, Santos IF. Modelling of LEG tilting pad journal bearings with active lubrication. Tribology International, 2017) also introduced the established active lubrication concept into a tilting pad bearing design featuring a groove lubrication mechanism at the leading edges of the pad. This proposed design was able to modify the rotor equilibrium positions for signals received from both open and closed loop configurations. Bearing fault diagnosis and performance analysis can be evaluated using multi-objective optimization techniques and an artificial neural network approach. Zhao et al. (Zhao et al., 2022) utilized a continuous wavelet transform technique featuring vibration amplitude spectrum image extraction and conversion for bearing fault diagnosis. Gauss Convolutional Deep Belief Network approach is further considered to classify the representative features under varying signal to noise ratios and operational conditions. The proposed method is validated by comparing it with actual bearing data and satisfactory results were obtained. To further enhance the precision of hyperspectral image classification and to reduce the dimensions of its spectral features, an advanced extraction technique incorporating local binary pattern is presented in Chen et al. (2021). For optimizing the operational parameters of kernel extreme learning machine, gray wolf optimization algorithm was applied for image analysis. Further to improve the scale selection and to compute the signal characteristics, a scale-based mathematical morphology spectrum entropy technique was presented by Yao et al. (Yao et al., 2022). Optimization techniques including multi-objective particle swarm and ant colony hybrid optimization approach, support vector machine regression models can be utilized to identify the greatest influence of different bearing performance parameters (Deng et al., 2022; Onyekwena et al., 2022; Zhou et al., 2022).

An adjustable fluid film bearing geometry capable enough to maintain optimum bearing performance during operation is presented in this study. The proposed bearing configuration can be developed as an electronically operated active journal bearing setup with embedded control system. Such active bearing systems can modify the journal position and provide improved bearing stability through automatic adjustments of multiple pads in real-time conditions. The present theoretical study focuses on identifying the optimum pad adjustment positions for three-pad and four-pad bearing configuration under varying journal eccentricity ratios. These pad adjustment data will be helpful in developing the source code for the control system to be integrated in the experimental bearing setup. The development of experimental setup is an ongoing study, and the data presented in the current paper contribute to identifying the appropriate pad adjustments based on journal center location. The active bearing system has the provision to incorporate industrial internet of things (IIoT) to record and monitor journal positions and provide controlled symmetric and asymmetric position inputs to multiple pads. The present study does not consider asymmetric pad adjustments in radial and tilt directions. In a real journal bearing, multiple pads will be automatically adjusted with symmetric and asymmetric pad adjustments based on the positioning of journal center. Geometrical configuration of the proposed three-pad and four-pad active journal bearings is presented in Figure 1. The design configuration of proposed active bearing model involves an adjustability mechanism which allows the multiple bearing pads to translate radially and tilt in real time. Detailed steady state and dynamic analysis is performed to measure the variation in the performance envelope of three-pad active journal bearings. For varying pad positions, variation in static performance parameters recorded for the three-pad active bearing model is further compared with the published results of four-pad bearing geometry given in Girish and Pai (2019). From the present simulation, enhanced performance behavior was observed by translating the multiple adjustable pads in negative radial and negative tilt directions.

Figure 1. Geometrical design configuration of active journal bearings. (a) three-pad active bearing (b) four-pad active bearing.

2. Analysis

By considering the lubricant film pressure variation, the Reynolds equation can be formulated as

(1) $\frac{\mathrm{\partial }}{\mathrm{\partial }\mathit{x}}\left(\frac{h^3}{12}\frac{\mathrm{\partial p}}{\mathrm{\partial x}}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial z}}\left(\frac{h^3}{12}\frac{\mathrm{\partial p}}{\mathrm{\partial z}}\right)=\frac{1}{2}\mathit{\eta }\mathit{U}\frac{\mathrm{\partial h}}{\mathrm{\partial x}}+\mathit{\eta }\frac{\mathrm{\partial h}}{\mathrm{\partial t}}$(1)

For static analysis, the Reynolds equation will be simplified and non-dimensionalized by neglecting the time dependent term

(2) $\frac{\mathrm{\partial }}{\mathrm{\partial \theta }}\left({\left(\bar{h}\right)}^3\frac{\mathrm{\partial }\bar{p}}{\mathrm{\partial \theta }}\right)+\frac{1}{4}{\left(\frac{D}{L}\right)}^2{\left(\bar{h}\right)}^3\frac{{\mathrm{\partial }}^2\bar{p}}{\mathrm{\partial }{\bar{z}}^2}=\mathrm{\Lambda }\frac{\mathrm{\partial }\bar{h}}{\mathrm{\partial }\mathit{\theta }}$(2)

where bearing number $\mathrm{\Lambda }=\frac{6\mathit{\eta \omega }}{p_s{\left(\frac{C}{R}\right)}^2}(3)$

3. Film thickness equation formulation method

(4) $\bar{h}=\left(\frac{h}{C}\right)=\left(\frac{Z_L-\mathit{R}}{\mathit{C}}\right)+\mathit{\epsilon }cos\left(\mathit{\theta }{ }^{\mathrm{\prime }}\right)$(4)

where

(5) $Z_L=\sqrt{{\left(R_p+C_p\right)}^2+{\left(L_M\right)}^2-\left(2\left(R_p+C_p\right)\ast L_M\right)cos\left(\mathit{\alpha }+\mathit{Y}\right)}$(5)

where $\mathit{\alpha }$ = pad angle

$\mathit{\theta }{ }^{\mathrm{\prime }}=\left(\mathit{\theta }-\mathit{\psi }-\frac{\alpha }{2}\right)$ for load-on-pad configuration (6)

The model description of a three-pad active journal bearing and formulation technique applied to develop the modified film thickness relation is detailed in Figure 2. Equal positional adjustments in radial and tilt manner will be provided to the three bearing pads. Controlled position inputs will affect the radial clearances and vary the lubricant film profile generated. Radial and tilt adjustment parameters considered in the derivation technique will be incorporated into the developed film thickness equation. Reduction in clearance spaces is observed for negative R_adj, whereas an additional fluid film zone is developed for positive R_adj. Pad flexibility is provided at one corner of the pad, by tilting it to a certain angle in positive and negative directions. Under tilt pad adjustments, the pad center attains a tilted position which varies depending upon the inward and outward tilt angles. The distance of tilting pad relative to the common bearing center $\left(B_C\right)$ is a variable factor considered across the pad and is determined using Eq. (5)(5) $Z_L=\sqrt{{\left(R_p+C_p\right)}^2+{\left(L_M\right)}^2-\left(2\left(R_p+C_p\right)\ast L_M\right)cos\left(\mathit{\alpha }+\mathit{Y}\right)}$(5) . Non-dimensionalized film thickness relation formulated by considering tilted and radially displaced pad positions are defined in Eq. (4)(4) $\bar{h}=\left(\frac{h}{C}\right)=\left(\frac{Z_L-\mathit{R}}{\mathit{C}}\right)+\mathit{\epsilon }cos\left(\mathit{\theta }{ }^{\mathrm{\prime }}\right)$(4) .

Figure 2. Trigonometric approach applied on a three-pad active bearing to derive the modified film thickness relation.

4. Boundary conditions

Film pressures at the edges of the finite difference grid are assumed to be zero and cavitation is taken into account in the analysis:

(7) $\bar{p}=0;\mathit{at}\bar{z}=0\mathrm{\&}1$(7)

(8) $\bar{p}=0;\mathit{at}\mathit{\theta }=0\mathrm{\&}1$(8)

(9) $\bar{p}=\frac{\mathrm{\partial }\bar{p}}{\mathrm{\partial }\mathit{\theta }}=0\mathit{at}\mathit{\theta }={\theta }_m$(9)

5. Static characteristics

5.1. Load capacity

The load capacity can be computed for a three-pad bearing using

(10) ${\bar{W}}_{\mathit{r}}=\left(\frac{W_r}{\mathit{LD}{p}_s}\right)=-{\int }_0^1{\int }_0^{2\pi }\bar{p}cos\left(\mathit{\theta }{ }^{\mathrm{\prime }}-\mathit{\psi }\right)\mathit{d\theta }\mathit{d}\bar{z}$(10)

(11) ${\bar{W}}_{\mathit{t}}=\left(\frac{W_t}{\mathit{LD}{p}_s}\right)={\int }_0^1{\int }_0^{2\pi }\bar{p}sin\left(\mathit{\theta }{ }^{\mathrm{\prime }}-\mathit{\psi }\right)\mathit{d\theta }\mathit{d}\bar{z}$(11)

Resultant reaction force can be calculated as

(12) $\bar{W}=\left(\frac{W}{LD{p}_s}\right)=\sqrt{{{\bar{W}}_{\mathit{r}}}^2+{{\bar{W}}_{\mathit{t}}}^2}$(12)

5.2. Attitude angle

(13) $\mathit{\varphi }={tan}^{-1}\left(\frac{{\bar{W}}_{\mathit{t}}}{{\bar{W}}_{\mathit{r}}}\right)$(13)

5.3. Side leakage flow

(14) $\mathit{Q}=-{\int }_0^{2\pi }\frac{h^3}{12\mathit{\eta }}{\left.\frac{\mathrm{\partial p}}{\mathrm{\partial z}}\right|}_{\mathit{z}=0}\mathit{R}\mathit{d\theta }-{\int }_0^{2\pi }\frac{h^3}{12\mathit{\eta }}{\left.\frac{\mathrm{\partial p}}{\mathrm{\partial z}}\right|}_{\mathit{z}=\mathit{L}}\mathit{R}\mathit{d\theta }$(14)

In non-dimensional form

(15) $\bar{Q}=\left(\frac{2\mathit{Q\eta L}}{C^3{p}_s\mathit{D}}\right)=-{\int }_0^{2\pi }\frac{{\left(\bar{h}\right)}^3}{12}{\left.\frac{\mathrm{\partial }\bar{p}}{\mathrm{\partial }\bar{z}}\right|}_{\bar{z}=0}\mathit{d\theta }-{\int }_0^{2\pi }\frac{{\left(\bar{h}\right)}^3}{12}{\left.\frac{\mathrm{\partial }\bar{p}}{\mathrm{\partial }\bar{z}}\right|}_{\bar{z}=1}\mathit{d\theta }$(15)

5.4. Friction variable

Friction force can be computed from

(16) $\mathit{F}={\int }_0^L{\int }_0^{2\pi }\mathit{\tau Rd\theta dz}$(16)

(17) $\mathit{F}={\int }_0^L{\int }_0^{2\pi }\frac{h}{2}\frac{\mathrm{\partial p}}{\mathrm{\partial \theta }}\mathit{d\theta dz}+{\int }_0^L{\int }_0^{2\pi }\frac{\mathit{R\eta U}}{h}\mathit{d\theta dz}$(17)

In non-dimensional form

(18) $\bar{F}=\left(\frac{F}{LC{p}_s}\right)={\int }_0^1{\int }_0^{2\pi }\left(\frac{\bar{h}}{2}\frac{\mathrm{\partial }\bar{p}}{\mathrm{\partial \theta }}+\frac{\mathrm{\Lambda }}{6}\frac{1}{\bar{h}}\right)\mathit{d\theta d}\bar{z}$(18)

Friction variable

(19) $\bar{\mu }=\mathit{\mu }\left(\frac{R}{C}\right)=\frac{\bar{F}}{\bar{W}}$(19)

5.5. Sommerfeld number

(20) $\mathit{S}=\frac{\mathit{\eta }{\mathit{N}}^{\prime }}{\mathit{P}}{\left[\frac{R}{C}\right]}^2$(20)

6. Dynamic analysis

6.1. Linear perturbation method

Under dynamic conditions, a whirl motion of the journal center is often noted with respect to its equilibrium location. This small amplitude whirling motion is primarily caused due to the dynamic forces generated from the stiffness and damping behavior of oil film. The self-excited vibrations generated due to fluid induced instabilities can affect the overall dynamic performance of a rotor-bearing system. Hence, it is important to design the journal bearings to operate within a range of stable rotational speeds. Stiffness and damping coefficients of the bearing primarily influence the natural frequencies located near the rotor operational speeds and also limits the forced vibrations generated to a permissible level.

In this study, a linear perturbation method is adopted to compute the direct and cross-coupled dynamic coefficients. Small linear perturbations in X and Y directions are applied to the journal center during simulation. During operation, when low-magnitude unbalanced dynamic forces acts on the journal, the equilibrium location of journal will undergo displacements in horizontal and vertical directions along with associated whirl velocities. The resulting forces can be arranged in the following matrix form:

(21) $\left\{\begin{array}{c}{W}_x\\ W_y\end{array}\right\}=\left\{\begin{array}{c}{\left(W_x\right)}_0\\ 0\end{array}\right\}+\left[\begin{array}{cc}{K}_{\mathit{xx}}& K_{\mathit{xy}}\\ K_{\mathit{yx}}& K_{\mathit{yy}}\end{array}\right]\left\{\begin{array}{c}\mathrm{\Delta x}\\ \mathrm{\Delta y}\end{array}\right\}+\left[\begin{array}{cc}{C}_{\mathit{xx}}& C_{\mathit{xy}}\\ C_{\mathit{yx}}& C_{\mathit{yy}}\end{array}\right]\left\{\begin{array}{c}\mathrm{\Delta }\dot{x}\\ \mathrm{\Delta }\dot{y}\end{array}\right\}$(21)

The above equation contains principal and cross-coupled stiffness and damping. Applying Taylor expansion of the film pressure profile,

(22) $\mathit{p}={\left(p\right)}_0+{\left(\frac{\mathrm{\partial p}}{\mathrm{\partial x}}\right)}_0\mathrm{\Delta x}+{\left(\frac{\mathrm{\partial p}}{\mathrm{\partial y}}\right)}_0\mathrm{\Delta y}+{\left(\frac{\mathrm{\partial p}}{\mathrm{\partial }\dot{x}}\right)}_0\mathrm{\Delta }\dot{x}+{\left(\frac{\mathrm{\partial }\mathit{p}}{\mathrm{\partial }\dot{y}}\right)}_0\mathrm{\Delta }\dot{y}$(22)

Simplification

(23) ${\left(p\right)}_0=p_0;{\left(\frac{\mathrm{\partial }\mathit{p}}{\mathrm{\partial x}}\right)}_0=p_x;{\left(\frac{\mathrm{\partial p}}{\mathrm{\partial y}}\right)}_0=p_y;{\left(\frac{\mathrm{\partial p}}{\mathrm{\partial }\dot{x}}\right)}_0=p_{\dot{x}};{\left(\frac{\mathrm{\partial p}}{\mathrm{\partial }\dot{y}}\right)}_0=p_{\dot{y}}$(23)

The reaction force components can be determined through integration:

(24) $\left\{\begin{array}{c}{W}_x\\ W_y\end{array}\right\}={\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}\left(p_0+p_x\mathrm{\Delta x}+p_y\mathrm{\Delta y}+p_{\dot{x}}\mathrm{\Delta }\dot{x}+p_{\dot{y}}\mathrm{\Delta }\dot{y}\right)\left\{\begin{array}{c}cos\mathit{\theta }{ }^{\mathrm{\prime }}\\ sin\mathit{\theta }{ }^{\mathrm{\prime }}\end{array}\right\}\mathit{Rd\theta dz}$(24)

(25) $\left\{\begin{array}{c}{\left(W_x\right)}_0\\ \left(0\right)\end{array}\right\}=\left\{\begin{array}{c}{\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_0\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\\ {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{0}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\end{array}\right\}$(25)

(26) $\left\{\begin{array}{cc}{K}_{\mathit{xx}}& K_{\mathit{xy}}\\ K_{\mathit{yx}}& K_{\mathit{yy}}\end{array}\right\}=\left\{\begin{array}{cc}{\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_x\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_y\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\\ {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{x}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{y}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\end{array}\right\}$(26)

(27) $\left\{\begin{array}{cc}{C}_{\mathit{xx}}& C_{\mathit{xy}}\\ C_{\mathit{yx}}& C_{\mathit{yy}}\end{array}\right\}=\left\{\begin{array}{cc}{\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{x}}\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{y}}\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\\ {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{x}}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{y}}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\end{array}\right\}$(27)

Principal dynamic coefficients define the dynamic force variation in a particular direction as a result of journal center displacement in the same direction. Whereas cross-coupled dynamic coefficients give the relation of dynamic force variation in a particular direction as a result of journal center displacement in the opposite direction. Eq. (26)(26) $\left\{\begin{array}{cc}{K}_{\mathit{xx}}& K_{\mathit{xy}}\\ K_{\mathit{yx}}& K_{\mathit{yy}}\end{array}\right\}=\left\{\begin{array}{cc}{\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_x\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_y\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\\ {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{x}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{y}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\end{array}\right\}$(26) and (27(27) $\left\{\begin{array}{cc}{C}_{\mathit{xx}}& C_{\mathit{xy}}\\ C_{\mathit{yx}}& C_{\mathit{yy}}\end{array}\right\}=\left\{\begin{array}{cc}{\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{x}}\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{y}}\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\\ {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{x}}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{y}}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\end{array}\right\}$(27) ) will be utilized to compute the stiffness and damping coefficients by considering the perturbed film pressures such as $p_x,p_y,p_{\dot{x}}$ and $p_{\dot{y}}$

6.2. Theoretical equations for dynamic analysis

For dynamic analysis, the film thickness equation will be modified by considering the journal center displacements:

(28) $\mathit{h}=h_0+\mathrm{\Delta x}cos\mathit{\theta }{ }^{\mathrm{\prime }}+\mathrm{\Delta }\mathit{y}sin\mathit{\theta }{ }^{\mathrm{\prime }}$(28)

After differentiation, we get

(29) $\frac{\mathrm{\partial }\mathit{h}}{\mathrm{\partial t}}=\mathrm{\Delta }\dot{x}cos\mathit{\theta }{ }^{\mathrm{\prime }}+\mathrm{\Delta }\dot{y}sin\mathit{\theta }{ }^{\mathrm{\prime }}$(29)

Substituting Eq. (22)(22) $\mathit{p}={\left(p\right)}_0+{\left(\frac{\mathrm{\partial p}}{\mathrm{\partial x}}\right)}_0\mathrm{\Delta x}+{\left(\frac{\mathrm{\partial p}}{\mathrm{\partial y}}\right)}_0\mathrm{\Delta y}+{\left(\frac{\mathrm{\partial p}}{\mathrm{\partial }\dot{x}}\right)}_0\mathrm{\Delta }\dot{x}+{\left(\frac{\mathrm{\partial }\mathit{p}}{\mathrm{\partial }\dot{y}}\right)}_0\mathrm{\Delta }\dot{y}$(22) and Eq. (28)(28) $\mathit{h}=h_0+\mathrm{\Delta x}cos\mathit{\theta }{ }^{\mathrm{\prime }}+\mathrm{\Delta }\mathit{y}sin\mathit{\theta }{ }^{\mathrm{\prime }}$(28) in Eq. (1(1) $\frac{\mathrm{\partial }}{\mathrm{\partial }\mathit{x}}\left(\frac{h^3}{12}\frac{\mathrm{\partial p}}{\mathrm{\partial x}}\right)+\frac{\mathrm{\partial }}{\mathrm{\partial z}}\left(\frac{h^3}{12}\frac{\mathrm{\partial p}}{\mathrm{\partial z}}\right)=\frac{1}{2}\mathit{\eta }\mathit{U}\frac{\mathrm{\partial h}}{\mathrm{\partial x}}+\mathit{\eta }\frac{\mathrm{\partial h}}{\mathrm{\partial t}}$(1) ), we get

$\frac{1}{R^2}\frac{\mathrm{\partial }}{\mathrm{\partial }\mathit{\theta }}\left\{\frac{{\left(h_0+\mathrm{\Delta x}cos\mathit{\theta }{ }^{\mathrm{\prime }}+\mathrm{\Delta }\mathit{y}sin\mathit{\theta }{ }^{\mathrm{\prime }}\right)}^3}{12}\frac{\mathrm{\partial }}{\mathrm{\partial }\mathit{\theta }}\left(p_0+p_x\mathrm{\Delta x}+p_y\mathrm{\Delta y}+p_{\dot{x}}\mathrm{\Delta }\dot{x}+p_{\dot{y}}\mathrm{\Delta }\dot{y}\right)\right\}$

$+\left(\frac{{\left(h_0+\mathrm{\Delta }\mathit{x}cos\mathit{\theta }{ }^{\mathrm{\prime }}+\mathrm{\Delta }\mathit{y}sin\mathit{\theta }{ }^{\mathrm{\prime }}\right)}^3}{12}\right)\left\{\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{z}^2}\left(p_0+p_x\mathrm{\Delta }\mathit{x}+p_y\mathrm{\Delta y}+p_{\dot{x}}\mathrm{\Delta }\dot{x}+p_{\dot{y}}\mathrm{\Delta }\dot{y}\right)\right\}$

(30) $=\frac{1}{2}\mathit{\eta \omega }\frac{\mathrm{\partial }}{\mathrm{\partial \theta }}\left(h_0+\mathrm{\Delta x}cos\mathit{\theta }{ }^{\mathrm{\prime }}+\mathrm{\Delta }\mathit{y}sin\mathit{\theta }{ }^{\mathrm{\prime }}\right)+\mathit{\eta }\left(\mathrm{\Delta }\dot{x}cos\mathit{\theta }{ }^{\mathrm{\prime }}+\mathrm{\Delta }\dot{y}sin\mathit{\theta }{ }^{\mathrm{\prime }}\right)$(30)

After non-dimensionalization, we get

(31) $\frac{\mathrm{\partial }}{\mathrm{\partial }\mathit{\theta }}\left(\frac{{{\bar{h}}_0}^3}{12}\frac{\mathrm{\partial }{\bar{p}}_0}{\mathrm{\partial \theta }}\right)+\frac{1}{4}{\left(\frac{D}{L}\right)}^2\frac{{{\bar{h}}_0}^3}{12}\left(\frac{{\mathrm{\partial }}^2{\bar{p}}_0}{\mathrm{\partial }{\bar{z}}^2}\right)=\frac{1}{2}\frac{\mathrm{\partial }{\bar{h}}_0}{\mathrm{\partial }\mathit{\theta }}$(31)

Perturbation pressure ${\bar{p}}_{\mathit{x}}$

(32) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial \theta }}\left(\frac{{{\bar{h}}_0}^3}{12}\frac{\mathrm{\partial }{\bar{p}}_{\dot{x}}}{\mathrm{\partial \theta }}\right)+\frac{1}{4}{\left(\frac{D}{L}\right)}^2\frac{{{\bar{h}}_0}^3}{12}\left(\frac{{\mathrm{\partial }}^2{\bar{p}}_{\dot{x}}}{\mathrm{\partial }{\bar{z}}^2}\right)\\ & =\frac{1}{c}\left[\frac{-\frac{1}{2}\left(sin\mathit{\theta }{ }^{\mathrm{\prime }}+\frac{3cos\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}{{\bar{h}}_0}\frac{\mathrm{\partial }{\bar{h}}_0}{\mathrm{\partial }\mathit{\theta }}\right)}{+\frac{3{\bar{h}}_0}{12}\frac{\mathrm{\partial }{\bar{p}}_0}{\mathrm{\partial \theta }}\left({\bar{h}}_{0}sin\mathit{\theta }{ }^{\mathrm{\prime }}+cos\mathit{\theta }{ }^{\mathrm{\prime }}\frac{\mathrm{\partial }{\bar{h}}_0}{\mathrm{\partial }\mathit{\theta }}\right)}\right]\end{array}$(32)

Perturbation pressure ${\bar{p}}_{\mathit{y}}$

(33) $\begin{array}{rl}& \frac{\mathrm{\partial }}{\mathrm{\partial \theta }}\left(\frac{{{\bar{h}}_0}^3}{12}\frac{\mathrm{\partial }{\bar{p}}_{\dot{x}}}{\mathrm{\partial \theta }}\right)+\frac{1}{4}{\left(\frac{D}{L}\right)}^2\frac{{{\bar{h}}_0}^3}{12}\left(\frac{{\mathrm{\partial }}^2{\bar{p}}_{\dot{x}}}{\mathrm{\partial }{\bar{z}}^2}\right)\\ & =\frac{1}{C}\left[\frac{\frac{1}{2}\left(cos\mathit{\theta }{ }^{\mathrm{\prime }}-\frac{3sin\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}{{\bar{h}}_0}\frac{\mathrm{\partial }{\bar{h}}_0}{\mathrm{\partial }\mathit{\theta }}\right)}{-\frac{3{\bar{h}}_0}{12}\frac{\mathrm{\partial }{\bar{p}}_0}{\mathrm{\partial \theta }}\left({\bar{h}}_{0}cos\mathit{\theta }{ }^{\mathrm{\prime }}-sin\mathit{\theta }{ }^{\mathrm{\prime }}\frac{\mathrm{\partial }{\bar{h}}_0}{\mathrm{\partial }\mathit{\theta }}\right)}\right]\\ & \end{array}$(33)

Perturbation pressure ${\bar{p}}_{\dot{x}}$

(34) $\frac{\mathrm{\partial }}{\mathrm{\partial \theta }}\left(\frac{{{\bar{h}}_0}^3}{12}\frac{\mathrm{\partial }{\bar{p}}_{\dot{x}}}{\mathrm{\partial \theta }}\right)+\frac{1}{4}{\left(\frac{D}{L}\right)}^2\frac{{{\bar{h}}_0}^3}{12}\left(\frac{{\mathrm{\partial }}^2{\bar{p}}_{\dot{x}}}{\mathrm{\partial }{\bar{z}}^2}\right)=\left(\frac{1}{\mathit{C\omega }}\right)cos\mathit{\theta }{ }^{\mathrm{\prime }}$(34)

Perturbation pressure ${\bar{p}}_{\dot{y}}$

(35) $\frac{\mathrm{\partial }}{\mathrm{\partial }\mathit{\theta }}\left(\frac{{{\bar{h}}_0}^3}{12}\frac{\mathrm{\partial }{\bar{p}}_{\dot{y}}}{\mathrm{\partial \theta }}\right)+\frac{1}{4}{\left(\frac{D}{L}\right)}^2\frac{{{\bar{h}}_0}^3}{12}\left(\frac{{\mathrm{\partial }}^2{\bar{p}}_{\dot{y}}}{\mathrm{\partial }{\bar{z}}^2}\right)=\left(\frac{1}{\mathit{C\omega }}\right)sin\mathit{\theta }{ }^{\mathrm{\prime }}$(35)

7. Computational method

Unlike a conventional bearing, the position of angular coordinate $\mathit{\theta }{ }^{\mathrm{\prime }}$ must be known in advance for an adjustable bearing. Therefore, an attitude angle $\left(\mathit{\psi }\right)$ value is assumed as an initial condition in the developed computational algorithm. The computational procedure followed in the present steady state analysis is illustrated in Figure 3. Necessary design parameters mentioned in Table 1, such as pad angle, journal radius, clearance and L/D ratio are defined in the computational code as initial input parameters. The critical part of the computational algorithm is to provide position inputs to the multiple pads in radial and tilt directions. Modified fluid film relation defined in Eq. (4)(4) $\bar{h}=\left(\frac{h}{C}\right)=\left(\frac{Z_L-\mathit{R}}{\mathit{C}}\right)+\mathit{\epsilon }cos\left(\mathit{\theta }{ }^{\mathrm{\prime }}\right)$(4) is utilized to provide film thickness variation under varied journal eccentricities by incorporating the different radial adjustment (R_adj) and tilt angle $\left(\mathit{\delta }\right)$ parameters. Non-dimensionalized Reynolds equation and fluid film domain are discretized using finite difference approximation technique. To discretize the partial derivatives in Eq. (2(2) $\frac{\mathrm{\partial }}{\mathrm{\partial \theta }}\left({\left(\bar{h}\right)}^3\frac{\mathrm{\partial }\bar{p}}{\mathrm{\partial \theta }}\right)+\frac{1}{4}{\left(\frac{D}{L}\right)}^2{\left(\bar{h}\right)}^3\frac{{\mathrm{\partial }}^2\bar{p}}{\mathrm{\partial }{\bar{z}}^2}=\mathrm{\Lambda }\frac{\mathrm{\partial }\bar{h}}{\mathrm{\partial }\mathit{\theta }}$(2) ), central finite difference method is applied. Reynolds boundary conditions are applied to the finite difference grid to solve the discretized mathematical relations. After discretization of Reynolds equation, the resulting simultaneous equations are solved by Gauss-Seidel iterative method with successive over relaxation scheme (SOR). The pressure convergence criterion of the present computational analysis is given in Eq. (36)(36) $\left|\frac{{\left({{\bar{p}}_0}_{\mathit{i},\mathit{\hspace{0.17em}}\mathit{j}}\right)}_{\mathit{present}}-{\left({{\bar{p}}_0}_{\mathit{i},\mathit{\hspace{0.17em}}\mathit{j}}\right)}_{\mathit{old}}}{{\left({{\bar{p}}_0}_{\mathit{i},\mathit{\hspace{0.17em}}\mathit{j}}\right)}_{\mathit{present}}}\right|\le 0.001$(36) . To reduce the complexity and iteration time of the computational analysis, a pressure convergence limit and attitude angle convergence limit is set as 10⁻⁴.

Figure 3. Solution methodology to compute the static and dynamic characteristics of the three-pad active bearing.

(36) $\left|\frac{{\left({{\bar{p}}_0}_{\mathit{i},\mathit{\hspace{0.17em}}\mathit{j}}\right)}_{\mathit{present}}-{\left({{\bar{p}}_0}_{\mathit{i},\mathit{\hspace{0.17em}}\mathit{j}}\right)}_{\mathit{old}}}{{\left({{\bar{p}}_0}_{\mathit{i},\mathit{\hspace{0.17em}}\mathit{j}}\right)}_{\mathit{present}}}\right|\le 0.001$(36)

Table 1. Design data for the multi-pad adjustable bearing (Pai and Parkins (Pai & Parkins, 2018))

Pad Angle	60 degrees
L/D ratio	0.53
Journal radius	23.75 mm
Radial clearance	42.5 µm

Based on the calculated load components, the attitude angle $\left(\mathit{\varphi }\right)$ will be computed using Eq. (13)(13) $\mathit{\varphi }={tan}^{-1}\left(\frac{{\bar{W}}_{\mathit{t}}}{{\bar{W}}_{\mathit{r}}}\right)$(13) . Attitude angle ($\mathit{\psi })$ will be modified in every iteration by comparing the assumed $\mathit{\psi }$ value with the calculated value $\mathit{\varphi }$. Both pressure convergence and attitude angle convergence criterion need to be satisfied to achieve the converged pressure values and load carrying capacity for further computation of static performance characteristics. Detailed analysis is further carried out to compute the frictional force, friction variable and side leakage from the multiple pads of the adjustable bearing. After computing the static characteristics, perturbation pressures are calculated from Eq. (31)(31) $\frac{\mathrm{\partial }}{\mathrm{\partial }\mathit{\theta }}\left(\frac{{{\bar{h}}_0}^3}{12}\frac{\mathrm{\partial }{\bar{p}}_0}{\mathrm{\partial \theta }}\right)+\frac{1}{4}{\left(\frac{D}{L}\right)}^2\frac{{{\bar{h}}_0}^3}{12}\left(\frac{{\mathrm{\partial }}^2{\bar{p}}_0}{\mathrm{\partial }{\bar{z}}^2}\right)=\frac{1}{2}\frac{\mathrm{\partial }{\bar{h}}_0}{\mathrm{\partial }\mathit{\theta }}$(31) to Eq. (35)(35) $\frac{\mathrm{\partial }}{\mathrm{\partial }\mathit{\theta }}\left(\frac{{{\bar{h}}_0}^3}{12}\frac{\mathrm{\partial }{\bar{p}}_{\dot{y}}}{\mathrm{\partial \theta }}\right)+\frac{1}{4}{\left(\frac{D}{L}\right)}^2\frac{{{\bar{h}}_0}^3}{12}\left(\frac{{\mathrm{\partial }}^2{\bar{p}}_{\dot{y}}}{\mathrm{\partial }{\bar{z}}^2}\right)=\left(\frac{1}{\mathit{C\omega }}\right)sin\mathit{\theta }{ }^{\mathrm{\prime }}$(35) . By utilizing the computed perturbation pressures, stiffness and damping coefficients are further calculated using Eq. (26)(26) $\left\{\begin{array}{cc}{K}_{\mathit{xx}}& K_{\mathit{xy}}\\ K_{\mathit{yx}}& K_{\mathit{yy}}\end{array}\right\}=\left\{\begin{array}{cc}{\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_x\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_y\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\\ {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{x}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{y}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\end{array}\right\}$(26) and (27(27) $\left\{\begin{array}{cc}{C}_{\mathit{xx}}& C_{\mathit{xy}}\\ C_{\mathit{yx}}& C_{\mathit{yy}}\end{array}\right\}=\left\{\begin{array}{cc}{\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{x}}\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{y}}\cos \mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\\ {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{x}}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}& {\int }_z^{\hspace{0.17em}}{\int }_{\mathit{\theta }{\mathit{ }}^{\mathrm{\prime }}}^{\mathit{\hspace{0.17em}}}{p}_{\dot{y}}sin\mathit{\theta }{ }^{\mathrm{\prime }}\mathit{R}\mathit{d\theta }\mathit{dz}\end{array}\right\}$(27) ). An optimum value for all the considered set of performance parameters were obtained with less computational efforts using the developed algorithm.

8. Validation

The solution method developed in the present study for an active journal bearing is validated by applying the same to a single pad active journal bearing geometry. A single pad bearing having a pad angle of 120° and L/D ratio = 1 is considered in this study. Film thickness equation given in Eq. (4(4) $\bar{h}=\left(\frac{h}{C}\right)=\left(\frac{Z_L-\mathit{R}}{\mathit{C}}\right)+\mathit{\epsilon }cos\left(\mathit{\theta }{ }^{\mathrm{\prime }}\right)$(4) ), developed to consider the radial and tilt adjustments in an active journal bearing is applied to the single pad active bearing geometry. For varying pad adjustments, variation in dimensionless load capacity and friction variable recorded for different operating eccentricity ratios of single pad bearing is computed for validating the computational methodology employed in this study. This variation in the recorded values is further compared with the simulated values of Shenoy and Pai (Shenoy & Pai, 2009). By comparing Figures 4 (a) and (b), similar variation trend is noted for different controlled position inputs of the bearing pad, which validate the solution methodology adopted in this study.

Figure 4. Non-dimensional load and friction variable variation recorded for a single pad active bearing geometry (a) Shenoy and Pai (Shenoy & Pai, 2009; b) present simulation.

9. Results and discussions

Variation in the static and dynamic performance envelope of proposed active fluid film bearing models under the effect of different pad positions, pad angle and number of pads is analysed in the present study. Results obtained from the comparative analysis of three-pad and four-pad active bearing operated under symmetrical pad displacement conditions are shown in Figures 5–12. Figure 5 presents the lubricant film thickness variation recorded for a three-pad active bearing. A notable reduction in film thickness is observed for the three-pad bearing model under negatively adjusted pad positions than compared with other adjustments. This gradual reduction in lubricant film profile under negative pad displacement conditions is primarily caused due to the inward displacement of bearing pads, both radially and in tilted manner. These inward radial and tilt displacement towards the common bearing center (B_c) effectively reduces the pad clearances and thereby influence the lubricant film thickness. Whereas positively adjusted pad positions in radial and tilt direction result in the increase of bearing clearances caused by the displacement of bearing pads away from the common bearing center (B_c). For a three-pad active bearing model, convergent film region is developed over the bearing pad 2 due to the self weight and equilibrium positions of the journal. For varying pad positions, minimum film thickness region is developed in the convergent zone of the pad 2. In these regions, the hydrodynamic pressures generated will be maximum and contribute to higher load capacity and improve overall bearing performance. The convergent fluid film region developed will be varied depending upon the load directions. Under load between pad conditions, the convergent region can get distributed over two bearing pads. However, such variation will negatively influence both hydrodynamic film pressures generated and the bearing performance.

Figure 5. Lubricant film profile developed for a three-pad active bearing model for varying pad positions.

Figure 6. Variation in ɛ-φ characteristics of a three-pad and four-pad active bearing model for varying pad positions.

Figure 7. Variation in non-dimensional load of a three-pad and four-pad active bearing model for varying pad positions.

Figure 8. Variation in friction variable of a three-pad and four-pad active bearing model for varying pad positions.

Figure 9. Variation in side leakage of a three-pad and four-pad active bearing model for varying pad positions.

Figure 10. Variation in Sommerfeld Number of a three-pad and four-pad active bearing model for varying pad positions.

Figure 11. Variation of stiffness coefficients for a three-pad and four-pad active bearing model for varying pad positions (a) ${\bar{K}}_{\mathit{xx}}$ (b) ${\bar{K}}_{\mathit{yx}}$ (c) ${\bar{K}}_{\mathit{xy}}$ (d) ${\bar{K}}_{\mathit{yy}}$.

Figure 12. Variation of damping coefficients for a three-pad and four-pad active bearing model for varying pad positions (a) ${\bar{C}}_{\mathit{xx}}$ (b) ${\bar{C}}_{\mathit{yx}}$ (c ${\bar{C}}_{\mathit{xy}}$) (d) ${\bar{C}}_{\mathit{yy}}$.

Figure 6 presents the variation in the attitude angle recorded for three-pad and four-pad active bearing geometries subjected to different pad adjustments. A gradual reducing trend in the ɛ-φ characteristic curve is observed for a wide variation in journal eccentricities. In comparison, for both three-pad and four-pad active bearing models, the journal equilibrium positions recorded for increasing eccentricity ratios are found to be near the bearing vertical for pads subjected to negative radial and tilt motion. This recorded variation of ɛ-φ characteristics indicates that the bearing model is highly stable for negative pad adjustments and at large eccentricity ratios. By comparing the ɛ-φ variation of both active bearing models, lower attitude angle is developed for three-pad active bearing model at different operating eccentricity ratios. The reduction in number of pads and increased pad angle is found to have significant effect on the ɛ-φ characteristics of the adjustable journal bearing. Increased film pressures generated under negative pad displaced positions will help to modify the journal equilibrium positions closer to the bearing vertical. This modification of journal equilibrium positions will help to enhance the dynamic stability of the active rotor bearing system. In comparison with the ɛ-φ characteristics, improved bearing stability can be obtained for three-pad active bearing model than compared with four-pad bearing geometry.

Figure 7 illustrates the varying dimensionless load recorded for three-pad and four-pad active bearing geometries. With the increase in operating eccentricity ratios, an incremental variation in dimensionless load is recorded for both bearing models. For varying eccentricity ratios, higher dimensionless load is obtained for negatively adjusted pad positions than compared with positive or zero adjustment positions. This increase in load capacity is the result of higher lubricant film pressures generated in the convergent fluid film zone of the active bearing pad geometry under negative pad adjustments. The resulting reduction in clearance spaces and minimum film thickness regions developed under negative pad adjustments directly influence and result in higher lubricant film pressures. The tilting motion of bearing pads plays a major role in increasing the load capacity of the proposed bearing models than conventional fixed profile journal bearings. In comparison, a sizeable increase in load carrying capacity is noted for a three-pad active bearing model than utilizing a four-pad configuration subjected to varying pad adjustments. From the results, it is observed that reducing the number of pads with increased pad angle up to 60 degree each can provide higher load carrying capacity under different operating conditions.

The variation in friction variable recorded for three-pad and four-pad active bearing geometry operated under different pad displaced positions and pad angles is presented in Figure 8. A reduction in friction variable is observed for both bearing pad geometries subjected to zero and extreme pad adjustments towards and away from the common bearing center (B_c). Based on the displaced pad positions, lowest friction variable values are noted near the maximum possible journal eccentricities. Higher fluid film pressures developed under negative pad adjusted positions influence the decreasing trend of friction variable for both three-pad and four-pad active bearing models. At higher journal eccentricities, a notable increase in friction variable is obtained for positively adjusted pad position. This variation is mainly caused due to the resulting increase of pad clearances from the displacement of bearing pads away from the common bearing center (B_c). By comparing the friction variable trend recorded for both bearing models, reduced friction variable is obtained for the three-pad active bearing model with pad angle of $\mathit{\alpha }={60}^0$ for all possible pad translations. From the results, it is noted that an increased pad angle with lower number of pads is found to increase the convergent fluid film zone, which thereby results in the increase of lubricant film pressures and effectively reduce the friction variable.

The comparative variation of side leakage generated in a three-pad and four-pad bearing model is detailed in Figure 9. In comparison, less side leakage is noted for the four-pad active bearing geometry with pads having zero and positive adjustments than noted in the three-pad geometry. Increased clearance spaces developed due to the positive displacement of bearing pads causes higher leakage flow from the edges of the bearing pads. Since the bearing pads are applied with symmetrical pad adjustments, the variation in side leakage is found to be uniform across the bearing pads. However, for asymmetrical pad adjustments, the variation in side leakage differs from one another depending upon the positioning of pads in either inward or outward directions. At lower eccentricity ratios, due to generation of higher pressure gradients in the convergent fluid film zone, a further increase in the side leakage is noted for both bearing geometries. Variation in pad angle and number of pads is found to have minimal influence over the side leakage. Significant variation in side leakage can be expected from asymmetrical pad adjustment conditions. In Figure 10, influence of pad angle, number of pads and pad adjusted positions on Sommerfeld number is presented. From the results, higher Sommerfeld number is generated for negatively adjusted pad positions of both three-pad and four-pad active bearing models. In comparison, a notable reduction in Sommerfeld number is attained for three-pad active bearing geometry than measured in a four-pad bearing model under varied pad translation conditions.

Under controlled variation of pad positions for a three-pad and four-pad bearing configuration, a notable deviation observed for direct and cross stiffness coefficients is represented in Figure 11. In Figure 11(a), a gradual increasing trend of direct stiffness ${\bar{K}}_{\mathit{xx}}$ is observed for varying journal eccentricities. Increase in number of pads is found to have minimal impact on direct stiffness ${\bar{K}}_{\mathit{xx}}$ at higher journal eccentricities. At lower eccentricity ratios, higher direct stiffness values are generated for three-pad bearing configurations than compared with four-pad model. Further rise in direct stiffness values are noted for three-pad bearing configuration under extreme negative adjustment conditions. This variation is mainly caused due to the influence of reduced pad clearances under negative or inward pad displacements. The resulting higher fluid film pressures developed under negative pad adjustments plays a major role influencing the rise in direct stiffness at varying eccentricity ratios. From Figure 11(b), influence of pad adjustments, pad angle and number of pads on cross stiffness ${\bar{K}}_{\mathit{yx}}$ can be interpreted in detail. Different combination of pad adjustments has different range of operating eccentricity ratios. Due to the effect of reduced pad clearances under negative adjustment conditions, maximum operating eccentricity ratio range is limited from $\mathit{\epsilon }$ = 0.1 to 0.6. Similarly, positive pad adjustments have large operating eccentricity ratio range from $\mathit{\epsilon }$ = 0.1 to 1.27. Operating eccentricity ratios above $\mathit{\epsilon }$ = 1 is obtained due to the increase in pad clearances under outward adjustment of pad positions. In comparison, maximum cross stiffness ${\bar{K}}_{\mathit{yx}}$ is obtained for negative pad adjustment conditions irrespective of the number of pads and pad angle. At lower eccentricity ratios, four-pad bearing configuration generates higher cross stiffness ${\bar{K}}_{\mathit{yx}}$ for all the applied pad adjustments. However, minimal variation in cross stiffness ${\bar{K}}_{\mathit{yx}}$ values are noted at higher journal eccentricities for both three-pad and four-pad bearing configuration. In Figure 11(c), the variation in cross stiffness value ${\bar{K}}_{\mathit{xy}}$ is found to have notable reduction at lower journal eccentricities and a further increase is observed at maximum operating eccentricity ratios for varied pad adjustments. Higher cross stiffness value ${\bar{K}}_{\mathit{xy}}$ is observed for the active bearing configuration with 3 number of pads. Increase in pad angle and lower number of bearing pads is found to have higher influence on the variation of cross stiffness value ${\bar{K}}_{\mathit{xy}}$. The resulting increase in pad clearance and lower film pressures developed under positive pad adjustments has minimal effect on the cross-stiffness coefficients. In Figure 11(d), variation in direct stiffness ${\bar{K}}_{\mathit{yy}}$ is noted for both three-pad and four-pad active bearing models. For three-pad bearing model, an incremental variation in direct stiffness ${\bar{K}}_{\mathit{yy}}$ is noted for increasing eccentricity ratios. However, for four-pad bearing model, an initial decreasing trend in direct stiffness ${\bar{K}}_{\mathit{yy}}$ is observed at smaller journal eccentricities. In comparison, for both bearing models, lower direct stiffness ${\bar{K}}_{\mathit{yy}}$ values are obtained for pads under extreme positive adjustment conditions. This is mainly caused due to the lower film pressures developed in the convergent fluid film region of bearing pad. The increase in direct and cross stiffness coefficients at larger journal eccentricities is directly influenced by the peak dynamic pressures developed in the smaller clearance spaces.

Figure 12 presents the detailed variation of direct and cross-coupled damping coefficients for a three-pad and four-pad bearing model. In Figure 12(a), a similar variation in direct damping coefficient ${\bar{C}}_{\mathit{xx}}$ is noted for both the active bearing models under increasing eccentricity ratios. For different pad adjustment positions, an initial reduction in direct damping coefficient ${\bar{C}}_{\mathit{xx}}$ followed by a gradual increase at maximum operating eccentricities is noted. In comparison, a small improvement in direct damping is obtained for three-pad active bearing model for applied pad positions. At lower journal eccentricities, increased pad angle in three-pad bearing model is found to have low influence on the direct damping coefficient ${\bar{C}}_{\mathit{xx}}$. Negative tilt angles applied to the leading edges of the bearing pad result in the increase of direct damping at higher eccentricity ratios. In Figure 12(c) and (d), variation in cross damping coefficient ${\bar{C}}_{\mathit{yx}}$ and ${\bar{C}}_{\mathit{xy}}$ are noted for varied pad adjusted positions. For both three-pad and four-pad bearing models, variation in cross damping ${\bar{C}}_{\mathit{yx}}$ and ${\bar{C}}_{\mathit{xy}}$ are found to be similar with increasing operating eccentricity ratios. For four-pad bearing model, an incremental variation in cross damping is noted for all the applied pad positions. Whereas an initial decrease in cross damping is noted for three-pad bearing model with increased pad angles. This variation is mainly influenced due to the presence of increased minimum film thickness region and higher film pressures generated in the three-pad bearing model. Number of adjustable pads, pad angle and inward adjustment of bearing pads are found to have maximum influence on the cross-damping coefficients. In Figure 12(d), a decreasing variation in direct damping ${\bar{C}}_{\mathit{yy}}$ is noted for different pad positions and increase in eccentricity ratios. At maximum operating eccentricities, higher direct damping ${\bar{C}}_{\mathit{yy}}$ values are obtained for four-pad bearing model with pads adjusted in negative directions. Lower pad angles and higher number of pads are found to have greater influence on the direct damping coefficient ${\bar{C}}_{\mathit{yy}}$. Due to the improvement in stiffness and damping coefficients, the proposed bearing model with three-pad and four-pad bearing configurations is less susceptible to whirl motion at larger journal eccentricities for varied pad displaced positions.

10. Conclusions

The steady state and dynamic performance envelope of a three-pad active bearing model is analysed in this study. For comparative analysis of three-pad and four-pad active bearing models, the position of the multiple bearing pads with controllable mechanism are varied and fixed with load acting directly on it. Radial and tilt adjustments of the bearing pads are applied by translating the pads towards and away from the common bearing center. Both the controllable parameters are incorporated in theoretical simulation using the developed film thickness relation. A notable variation in the steady state and dynamic performance parameters is obtained by translating the bearing pads with radial and tilt motions to negative and positive directions.

• In a three-pad bearing model, reduction in clearances and lower film thickness values are generated for the pads displaced with radial and tilt motion in inward directions.

• Negative pad adjustments result in higher load carrying capacity due to the larger pressure gradients generated in the increased convergent region of three-pad active bearing model.

• Increase in lubricant film pressures has a positive impact on the ɛ-φ characteristics, by modifying the journal equilibrium positions and thereby providing improved bearing stability.

• For both bearing models, enhanced steady state performance was noticed for bearing pads adjusted with negative radial and negative tilt motions.

• Number of pads and variation in pad angles have shown a nominal influence on the direct and cross-coupled stiffness and damping coefficients. However, varied pad adjustments is found to have major influence over the dynamic characteristics.

• Due to the significant improvement in dynamic characteristics under negative pad adjustments, the proposed bearing models will be less susceptible to whirl motion at larger journal eccentricities.

• Based on the comparative simulation, improved performance envelope in an active bearing model can be attained by considering three bearing pads with loads positioned to act directly on it.

Acknowledgements

I would like to thank Prof. Dr Raghuvir Pai, Professor, Manipal Institute of Technology, MAHE, Manipal for the supervision in the work carried out

Disclosure statement

No potential conflict of interest was reported by the author(s).

Table

Nomenclature
C	Bearing pad clearance (m)
R	Journal Radius (m)
e	Journal eccentricity (m), $\mathit{\epsilon }=\left(\mathit{e}/\mathit{C}\right)$
F	friction force (N), $\bar{F}=\left(\mathit{F}/\mathit{LC}{p}_s\right)$
h	Fluid film thickness (m), $\bar{h}=\left(\mathit{h}/\mathit{C}\right)$
L	Length of bearing (m)
N’	Journal speed (rps)
p	steady state pressures (N/m^2), $\bar{p}=\mathit{p}/p_s$
$p_s$	Lubricant supply pressures (N/m^2)
Q	flow leak (m^3/s), $\bar{Q}=\left(2\mathit{Q\eta L}/C^3{p}_s\mathit{D}\right)$
Radj	radial adjustments (m)
t	time (s)
$\mathit{W}$	Journal load (N), $\bar{W}=\left(\mathit{W}/\mathit{LD}{p}_s\right)$
$W_r$	radial load component (N), $\overline{W_r}=\left(W_r/\mathit{LD}{p}_s\right)$
$W_t$	Transverse load component (N), $\overline{W_t}=\left(W_t/\mathit{LD}{p}_s\right)$
U	velocity of journal (m/s), $\mathit{U}=\mathit{\omega R}$
x	circumferential coordinates (m), $\mathit{x}=\mathit{R\theta }$
z	axial coordinates (m), $\mathit{z}=\bar{z}\mathit{L}$
$\mathit{\alpha }$	Bearing pad angle (rad)
$\mathit{\delta }$	tilted angle of pads (rad)
$\mathit{\eta }$	lubricant viscosity (Ns/m^2)
$\mathit{\theta }{ }^{\mathrm{\prime }},\mathit{\theta }$	angular coordinate (rad), $\mathit{\theta }{ }^{\mathrm{\prime }}=\left(\mathit{\theta }-\frac{\alpha }{2}\right)$
$\mathit{\tau }$	Shearing force (N/m^2)
$\mathrm{\Lambda }$	bearing number, $\mathrm{\Lambda }=6\mathit{\eta \omega }/\left[p_s{\left(\mathit{C}/R\right)}^2\right]$
$\mathit{\mu }$	coefficient of friction
$\bar{\mu }$	friction variable, $\bar{\mu }=\mathit{\mu }\left(\mathit{R}/\mathit{C}\right)$
	attitude angle (rad)
$\mathit{\psi }$	assumed attitude angle (rad)
S	Sommerfeld number, $\mathit{S}=\left(\mathit{\eta N}/\mathit{P}\right)\left(\mathit{R}/\mathit{C}\right){ }^2$
$\mathit{\omega }$	journal angular velocity (rad/s)

Additional information

Funding

The authors have no funding to report.