Electro-rheological fluid lubricated textured multi-lobe hole-entry hybrid journal bearing system

Abstract

Lubrication of fluid film bearings with smart lubricants provides better dynamic behavior. The behavior of these smart lubricants is described by Bingham model of non-Newtonian fluid flow. In the recent years, surface texturing has open up enormous possibilities for enhancing the performance of tribo-components and is being successfully applied in fluid film bearing applications. This article deals with the theoretical investigation pertaining to the influence of electro-rheological fluid lubrication with textured bearing on the performance of multi-lobe (three-lobe) hole-entry hybrid journal bearing. For numerical analysis, the continuous Bingham model has been used to derive the modified Reynold’s equation and solved using the finite element method numerical technique in order to get the static and dynamic characteristics of the electro-rheological lubricated textured three-lobe hybrid journal bearing. The results obtained from the present numerical model are compared with the published results of previous investigations. In this work, the effects of electric field and Sommerfeld number on the performance of multi-lobe (three-lobe) textured/non-textured journal bearings have been comprehensively studied. It has been observed that electro-rheological fluid lubrication in the three-lobe textured journal bearing significantly improves the performance characteristics vis-à-vis non-textured circular journal bearing lubricated with Newtonian lubricant.

Keywords

Three-lobe hole-entry hybrid journal bearing electro-rheological lubricant surface texturing finite element formulation

Introduction

In spite of development of efficient bearing procedures, the journal bearings do fail in the practical application with serious consequences and designers all over the world focus their efforts to devise better bearing. The tribologists and lubrication scientist introduced the electro-rheological (ER) and magneto-rheological (MR) fluids in lubrication problems to develop the smart tribo-components (Urreta et al., 2010; Winslow, 1947). An ER fluid usually consists of dielectric particles, which are dispersed into base oil (Nikolakopoulos and Papadopoulos, 1997). When an electric field is applied on these fluids, the dispersed particles are fibrillated in a complex micro-structure column and change their rheological behavior rapidly (in few milliseconds) according to the field change. Once the applied electric field is removed, the original flow state can be recovered immediately. Since the applied field can be generated from sensors, this reversible change of state in ER fluids may occur in response to external environment variations, for this reason, it is also generally called “Smart Fluid.” The invention of ER fluids is credited to Winslow (1947) due to his patent in 1947. In the recent years, some studies have been reported on ER fluid lubricated journal bearing (smart journal bearing) dealing with different issues. Kollias and Dimarogonas (1994) developed the ER fluid lubricated partial journal bearing. They computed the hydrodynamic pressure and observed good agreement with theoretical Bingham model of Wada et al. (1973). Later, Nikolakopoulos and Papadopoulos (1998) solved Reynolds equation using the finite element method (FEM) numerical technique to compute the dynamic behavior of the ER fluid lubricated journal bearing against the electric field. They used the Bingham plastic model to describe the flow behavior of ER lubricant. They observed that the use of ER fluid may act as smart fluid on journal bearing to appreciatively control the vibration of journal bearing in terms of stability parameter. Zhun and Ke-Qin (2002) presented the performance analysis of ER fluid lubricated journal bearing with special properties of ER fluid. The velocity and pressure distribution profiles were computed in their study. They reported that the load-carrying capacity of journal bearing increases with an increase in the value of applied electric field. Recently, Peng and Zhu (2005, 2006) applied the computational fluid dynamic (CFD) technique to study the influence of ER fluids on the performance of hydrodynamic journal bearing. They considered the finite length journal bearing and imposed the electric field along a contractive portion of journal bearing. They found that the ER lubricant has a significant impact on the performance of fluid film journal bearings.

In the recent times, in a quart to improve the performance of tribo-contacts, the use of textured surfaces has emerged out as an important area. The surface texturing offers great versatility vis-à-vis the improvement of tribological performance, that is, dimple acting as micro-reservoir for lubricant, increasing the fluid film thickness between contact zones consequences reducing the frictional coefficient, providing an additional micro-hydrodynamic lift effect to increase the load-carrying capacity, and dimples trap wear particles to minimize further abrasion (Qiu and Raeymaekers, 2015; Tala-Ighil et al., 2007). Due to these attractive features of textured surfaces, the bearing designers have focused their research effort to investigate the influence of surface texturing on the journal bearing during past decades. In 1966, Hamilton et al. (1966) developed the idea of textured surfaces by providing the micro-irregularities on surface of hydrodynamic journal bearing. They observed that these micro-irregularities act as micro-hydrodynamic bearing, improving the performance of hydrodynamic journal bearing. Furthermore, Brizmer et al. (2003) and Etsion et al. (2004) investigated the influence of dimple surfaces on the performance of hydrodynamic oil bearings. They found that the optimal features of textured surfaces like shape, pattern, dimension, and density of dimples pattern are essential to enhance the bearing performance. Tala-Ighil et al. (2011) used a finite difference numerical technique to analyze the influence of cylindrical shape dimple on the hydrodynamic journal bearing performance. They showed that the presence of textured surfaces increases the local fluid film thickness and decreases the frictional torque. Lu and Khonsari (2007) experimentally examined the effect of fully and partially textured patterns on the performance of journal bearing. They also considered the various shapes and sizes of dimples in their analysis. They found that the fully textured pattern gives better performance than that of partially textured pattern in mixed as well as hydrodynamic lubrication regime. Furthermore, they stated that the dimple parameters like shape, sizes, density, orientations, and so on are crucial for efficient textured journal bearing design. Recently, Tala-Ighil and Fillon (2015) reported the influence of surface texturing on the values of static parameters of thermo-hydrodynamic journal bearing. It was observed that large dimple size is more effective to enhance the static characteristics like minimum fluid film thickness and friction coefficient than that of small dimple size. They also suggested that the influence of dimple surfaces strongly depends on the geometrical features of dimples and operating condition of journal bearing.

It is a well-known fact that circular journal bearings face the problem of fluid-induced instability operating at high speeds. Fluid-induced instability produces the large amplitude self-excited vibration in terms of whirl phenomena (Li et al., 1980). Furthermore, the bearing designers were successfully able to control fluid-induced instability problem by changing the geometry of journal bearing to non-circular profile. Non-circular bearings are usually characterized by their number of lobes, aspect ratio, and offset factor (Lund and Thomsen, 1978). A three-lobe journal bearing is the one of simplest configurations of non-circular journal bearing. It consists of different centers of curvature for each of the lobes, which provides difference from the conventional cylindrical fluid film journal bearing. Pinkus (1959) and Pinkus and Lynn (1956) were the first to develop the idea of three-lobe journal bearing and provided the design data for power loss and stability parameter using numerical methods. Three-lobe journal bearings were experimentally tested for the influence of load speed and oil inlet pressure on the bearing load-carrying capacity by Saad (1979). Ten Napel and Bosma (1980) optimized the three-lobe journal bearing with respect to minimum film thickness and friction. Also, stability and lubricant flowcharts were presented in their study. They found that the three-lobe profile journal bearing gives higher value of stability and stiffness characteristics compared to cylindrical journal bearings.

The available literature survey indicates that the studies related to multi-lobe hole-entry hybrid journal bearing systems are rather very limited. Also, it has been observed from the literature that most of the studies with respect to the use of surface texturing and ER lubricant, respectively, mainly concerned with hydrodynamic journal bearings, whereas few studies exist for hydrostatic/hybrid journal bearing systems. Therefore, more work is required in the area of textured hybrid journal bearings to provide the needful information for the design process. The contribution of this article is to numerically study the three-lobe hole-entry hybrid journal bearing having spherical dimples lubricated with ER lubricant. In this work, the bearing characteristics parameters are shown as a function of applied electric field and Sommerfeld number. Furthermore, the combined influence of surface texturing and ER lubricant on the dynamic behavior of the bearing is numerically performed in terms of linear trajectories for better understanding on the stability of bearing.

Mathematical model

General layouts of textured/non-textured three-lobe hole-entry hybrid journal bearings are illustrated in Figure 1(a) to (c). The general layouts also incorporate a subassembly demonstrating practical application of electric field in above-mentioned bearings. A strong electric field across fluid film domain is applied using direct current (DC) power supply (0–1200 V). Furthermore, the local Reynolds number for the flow in the clearance between surfaces is measured to be 0.0036, it means the inertia forces are negligible in comparison to the viscous forces. Therefore, laminar flow condition is used in the present analysis. Hence, the non-dimensional modified average Reynolds equation governing the laminar flow of ER lubricant through the clearance of journal and textured bearing considering the variable viscosity is expressed as (Khatri and Sharma, 2016; Sharma and Yadav, 2014)

\begin{array}{l} \frac{\partial}{\partial α} ({\bar{h}}^{3} {\bar{F}}_{2} \frac{\partial \bar{p}}{\partial α}) + \frac{\partial}{\partial β} ({\bar{h}}^{3} {\bar{F}}_{2} \frac{\partial \bar{p}}{\partial β}) \\ = Ω [\frac{\partial}{\partial α} {(1 - \frac{{\bar{F}}_{1}}{{\bar{F}}_{0}}) \bar{h}}] + \frac{\partial \bar{h}}{\partial \bar{t}} \end{array}

(1)

where ${\bar{F}}_{0},$ ${\bar{F}}_{1}$ , and ${\bar{F}}_{2}$ are the fluid film viscosity integrals and obtained by the following expressions

{\bar{F}}_{0} = \int_{0}^{1} \frac{1}{\bar{μ}} d \bar{z;} {\bar{F}}_{1} = \int_{0}^{1} \frac{\bar{z}}{\bar{μ}} d \bar{z}; {\bar{F}}_{2} = \int_{0}^{1} \frac{\bar{z}}{\bar{μ}} (\bar{z} - \frac{{\bar{F}}_{1}}{{\bar{F}}_{0}}) d \bar{z}

Fluid film thickness

The non-dimensional value of nominal fluid film thickness of non-textured three-lobe hole-entry hybrid journal bearing is simply related to the journal center coordinates $({\bar{X}}_{J}, {\bar{Z}}_{J})$ and lobe center coordinates $({\bar{X}}_{L}^{i}, {\bar{Z}}_{L}^{i})$ and is expressed by (Crosby and Chetti, 2009; Kushare and Sharma, 2013)

\bar{h} = \frac{1}{δ} - ({\bar{X}}_{J} - {\bar{X}}_{L}^{i}) \cos α - ({\bar{Z}}_{J} - {\bar{Z}}_{L}^{i}) \sin α

(2)

where ${\bar{X}}_{J}$ and ${\bar{Z}}_{J}$ are the journal center coordinates and are expressed as ${\bar{X}}_{J} = ε \sin ϕ$ and ${\bar{Z}}_{J} = - ε \cos ϕ$ , $δ$ is the offset factor, and ${\bar{X}}_{L}^{i}$ and ${\bar{Z}}_{L}^{i}$ are the coordinates of lobe center and are expressed as ${\bar{X}}_{L}^{i} = (c - c_{m}) \cos α_{o}^{i}$ and ${\bar{Z}}_{L}^{i} = (c - c_{m}) \sin α_{o}^{i}$ .

For the case of textured (spherical dimple) three-lobe journal bearing, the nominal fluid film thickness in non-dimensional form is given by (Khatri and Sharma, 2016; Sharma and Yadav, 2014)

\bar{h} = \frac{1}{δ} - ({\bar{X}}_{J} - {\bar{X}}_{L}^{i}) \cos α - ({\bar{Z}}_{J} - {\bar{Z}}_{L}^{i}) \sin α + {\bar{h}}_{sd}

(3)

where

\begin{matrix} {\bar{h}}_{s d} = {[{(\frac{{\bar{h}}_{p}}{2} + \frac{{\bar{δ}}^{2}}{2 {\bar{h}}_{p}})}^{2} - {\bar{δ}}^{2} ({\bar{x}}_{l}^{2} + {\bar{z}}_{l}^{2})]}^{1 / 2} - [(\frac{{\bar{δ}}^{2}}{2 {\bar{h}}_{p}} - \frac{{\bar{h}}_{p}}{2})] & at r^{'} < r_{p} \\ [1] & at r^{'} > r_{p} \end{matrix}

where $\bar{δ} = r_{p} / c; \bar{x} = x / r_{p}; \bar{z} = z / r_{p}; r' = \sqrt{{\bar{x}}_{l}^{2} + {\bar{z}}_{l}^{2}}$ .

ER fluid model

In this study, the flow behavior of an ER lubricant is demonstrated by continuous Bingham model, which is proposed by Dorier and Tichy (1992). In the presence of electric field in ER lubricant, the particles oriented themselves along the electric field line. This makes the fluid more viscous, consequently transferring from a liquid to a semi-solid state. The dimensional form of continuous Bingham model may be expressed as

μ (\overset{\cdot}{γ}) = μ_{0} + \frac{2 τ_{0}}{π \overset{\cdot}{γ}} \tan^{- 1} (\frac{\overset{\cdot}{γ}}{{\overset{\cdot}{γ}}_{0}})

(4)

where $τ_{0}$ represents the yield stress parameter and is expressed as (Nikolakopoulos and Papadopoulos, 1998; Peng and Zhu, 2006)

τ_{0} (E) = A E^{B}

(5)

In equation (5), $A$ and $B$ are the ER fluid parameter and ER fluid field parameter, respectively. Furthermore, the electric field is determined by the following relation (Nikolakopoulos and Papadopoulos, 1998; Peng and Zhu, 2006)

E = \frac{V}{h}

(6)

where $V$ represents the imposed voltage in volts and $h$ is the nominal fluid film thickness.

The continuous Bingham model may be expressed in non-dimensional form by introducing the following parameters

{\bar{μ}}_{0} = \frac{μ_{0}}{μ_{r}}; {\bar{τ}}_{0} = τ_{0} (\frac{R_{J}}{c p_{s}}); \bar{\overset{\cdot}{γ}} = \overset{\cdot}{γ} (\frac{μ_{r} R_{J}}{c p_{s}}); and E = \frac{V}{\bar{h}}

(7)

Hence, non-dimensional continuous Bingham model is given below

{\bar{μ}}_{a} = \bar{μ} (\bar{\overset{\cdot}{γ}}) = {\bar{μ}}_{0} + \frac{2 {\bar{τ}}_{0}}{π \bar{\overset{\cdot}{γ}}} \tan^{- 1} (\frac{\bar{\overset{\cdot}{γ}}}{{\bar{\overset{\cdot}{γ}}}_{0}})

(8)

where ${\bar{μ}}_{a}$ is the apparent viscosity of ER lubricant and $\bar{\overset{\cdot}{γ}}$ is the shear strain rate of ER lubricant. The shear strain rate $(\bar{\overset{\cdot}{γ}})$ is written in non-dimensional form as (Khatri and Sharma, 2016; Kushare and Sharma, 2013)

\bar{\dot{γ}} = {[{(\frac{\bar{h}}{\bar{μ}} \frac{\partial \bar{p}}{\partial α} (\bar{z} \frac{{\bar{F}}_{1}}{{\bar{F}}_{0}}) + \frac{Ω}{\bar{μ} \bar{h} {\bar{F}}_{0}})}^{2} + {(\frac{\bar{h}}{\bar{μ}} \frac{\partial \bar{p}}{\partial β} (\bar{z} - \frac{{\bar{F}}_{1}}{{\bar{F}}_{0}}))}^{2}]}^{1 / 2}

(9)

In equation (8), when ${\bar{\overset{\cdot}{γ}}}_{0}$ approaches to infinity $(\infty)$ and $V$ to zero $(0)$ , the lubricant behaves like Newtonian lubricant.

FEM approach

In the present analysis, domain of an ER lubricant flow is discretized using four-noded bilinear isoperimetric elements. The unknown fluid film pressure is considered to be distributed linearly within four-noded quadrilateral element and is approximated by (Khatri and Sharma, 2016; Kushare and Sharma, 2013)

\bar{p} = \sum_{i = 1}^{n_{l}^{e}} {\bar{p}}_{j} N_{j}

(10)

Here, $N_{j}$ is elemental nodal shape function and $n_{l}^{e}$ is nodes per isoperimetric element.

Using Galerkin’s orthogonality conditions, the weak formulation of modified average Reynolds equation (1) can be expressed as

\begin{matrix} \int \int_{A_{e}} [\frac{\partial}{\partial α} ({\bar{h}}^{3} {\bar{F}}_{2} \frac{\partial \bar{p}}{\partial α}) + \frac{\partial}{\partial β} ({\bar{h}}^{3} {\bar{F}}_{2} \frac{\partial \bar{p}}{\partial β}) \\ - Ω [\frac{\partial}{\partial α} {(1 - \frac{{\bar{F}}_{1}}{{\bar{F}}_{0}}) \bar{h}}] - \frac{\partial \bar{h}}{\partial \bar{t}}] N_{i} d α d β = 0 \end{matrix}

(11)

where $A_{e}$ is the area of element and $N_{i}$ is the Langrangian interpolation functions.

After FEM formulation and simplification of equation (11), the elemental system of equation for obtaining nodal fluid film pressure vector is written as follows (Khatri and Sharma, 2016; Sharma and Yadav, 2014)

{[\bar{F}]}^{e} {\bar{p}}^{e} = {\bar{Q}}^{e} + Ω {{\bar{R}}_{H}}^{e} + {\bar{\overset{\cdot}{X}}}_{J} {{\bar{R}}_{Xj}}^{e} + {\bar{\overset{\cdot}{Z}}}_{J} {{\bar{R}}_{Zj}}^{e}

(12a)

The eth element equation in the above matrices are expressed as

\begin{matrix} {\bar{F}}_{ij}^{e} = \int_{A} \int_{e} {\bar{h}}^{3} [{\bar{F}}_{2} \frac{\partial N_{i}}{\partial α} \frac{\partial N_{j}}{\partial α} + {\bar{F}}_{2} \frac{\partial N_{i}}{\partial β} \frac{\partial N_{j}}{\partial β}] d α d β \end{matrix}; Fluidity matrix

(12b)

\begin{matrix} {\bar{Q}}_{i}^{e} = \int_{Γ^{e}} {({\bar{h}}^{3} {\bar{F}}_{2} \frac{\partial \bar{p}}{\partial α} - Ω \bar{h} (1 - \frac{{\bar{F}}_{1}}{{\bar{F}}_{0}})) n_{α} + ({\bar{h}}^{3} {\bar{F}}_{2} \frac{\partial \bar{p}}{\partial β}) n_{β}} \\ N_{i} d {\bar{Γ}}^{e}; Nodal flow matrix \end{matrix}

(12c)

{\bar{R}}_{H_{i}}^{e} = \int_{A} \int_{e} {(1 - \frac{{\bar{F}}_{1}}{{\bar{F}}_{0}}) \bar{h}} \frac{\partial N_{i}}{\partial α} d α d β; Hydrodynamic squeeze matrix

(12d)

\begin{matrix} {\bar{R}}_{X_{Ji}}^{e} = \int_{A} \int_{e} N_{i} \cos α d α d β; Journal center velocity \\ matrix along X_{J} direction \end{matrix}

(12e)

{\bar{R}}_{Z_{J i}}^{e} = \int_{A} \int_{e} N_{i} \sin α d α d β; Journal center velocity matrix along Z_{J} direction

(12f)

where the notations $n_{α}$ and $n_{β}$ are directional cosines along respective directions and $i, j = 1, 2, 3, 4$ are numbers of nodes.

Capillary restrictor flow equation

The capillary restrictor is a narrow diameter tube introduced between the main lubricant supply line and holes for controlling the lubricant flow. The flow of ER lubricant through the capillary restrictor in non-dimensional form is expressed as follows (Khatri and Sharma, 2016; Rowe et al., 1982)

{\bar{Q}}_{R} = {\bar{C}}_{s 2} (1 - {\bar{p}}_{c})

(13)

where ${\bar{C}}_{s 2} = 1 / 2 (π r_{cp}^{4} / 8 c^{3} l_{cp});$ ${\bar{C}}_{s 2}$ is the restrictor design parameter.

Boundary conditions

The following boundary conditions are used on the performance analysis of three-lobe hole-entry hybrid journal bearing (Kushare and Sharma, 2013; Rowe et al., 1982; Yoshimoto et al., 1988):

At the external boundary, the nodal fluid film pressure is zero

\bar{p} |_{β = \pm 1} = 0.00

The nodal fluid film pressure for the nodes appearing on the hole boundary are equal.

The total lubricant input flow in the bearing is equal to total lubricant flow through the restrictor $(i . e ., \bar{Q} = {\bar{Q}}_{R})$ .

$\bar{p} = \partial \bar{p} / \partial α = 0.00$ is assigned at trailing edge of the positive region. It is known as Reynolds Boundary Condition (RBC). In the RBC, the nodal fluid film pressure will assign zero at the portion where cavitation occurs.

Bearing performance characteristics

After establishing the nodal pressure distribution in lubricant flow field, the static as well as dynamic performance characteristics are to be simulated. Estimation of the behavior of non-textured/textured circular/three-lobe hole-entry hybrid journal bearing lubricated with ER lubricant has been accomplished with the help of the following mathematical relations.

Static performance characteristics

After computing the nodal fluid film pressure from the modified Reynolds equation, the static performance characteristics are obtained at steady-state condition $({\bar{\overset{\cdot}{X}}}_{J} = {\bar{\overset{\cdot}{Z}}}_{J} = 0)$ using the following relations.

Fluid film reaction

The fluid film nodal pressure obtained from the modified Reynolds equation (1) is integrated to obtain the fluid film reactions along and perpendicular to the line of centers as follows (Khatri and Sharma, 2016; Peng and Zhu, 2005)

{\bar{F}}_{x} = - \int_{- λ}^{λ} \int_{0}^{2 π} \bar{p} \cos α d α d β

(14a)

{\bar{F}}_{z} = - \int_{- λ}^{λ} \int_{0}^{2 π} \bar{p} \sin α d α d β

(14b)

The resulting radial fluid film reaction components is given by

\bar{F} = {[{\bar{F}}^{2}_{x} + {\bar{F}}^{2}_{z}]}^{1 / 2}

(14c)

Minimum fluid film thickness

After computing the nodal fluid film pressure and equilibrium journal center coordinates $({\bar{X}}_{J}, {\bar{Z}}_{J})$ , the minimum fluid film thickness is computed using the following relation

{\bar{h}}_{min} = minimum of (\bar{h})_{n}

(15)

where $n = 1, 2, 3, \dots, n_{l};$ $n_{l}$ represents the number of nodes along the flow field.

Attitude angle

The attitude angle is defined as the angle between the line of action of external load, ${\bar{W}}_{o}$ , and line of center and expressed for different equilibrium positions of journal center as follows

For ${\bar{X}}_{J} \geq 0, {\bar{Z}}_{J} > 0$

ϕ = \frac{π}{2} + \tan^{- 1} [\frac{{\bar{Z}}_{J}}{{\bar{X}}_{J}}]

(16a)

For ${\bar{X}}_{J} \leq 0, {\bar{Z}}_{J} > 0$

ϕ = π + \tan^{- 1} [\frac{{\bar{X}}_{J}}{{\bar{Z}}_{J}}]

(16b)

For ${\bar{X}}_{J} < 0, {\bar{Z}}_{J} < 0$

ϕ = 3 π / 2 + \tan^{- 1} [\frac{{\bar{Z}}_{J}}{{\bar{X}}_{J}}]

(16c)

For ${\bar{X}}_{J} > 0, {\bar{Z}}_{J} < 0$

ϕ = 2 π + \tan^{- 1} [\frac{{\bar{X}}_{J}}{{\bar{Z}}_{J}}]

(16d)

Frictional torque

For a hole-entry hybrid journal bearing lubricated with ER lubricant, the non-dimensional value of frictional torque is expressed as follows (Crosby and Chetti, 2009)

{\bar{T}}_{f} = \int_{- λ}^{λ} \int_{0}^{2 π} (\frac{\bar{h}}{2} \frac{\partial \bar{p}}{\partial α} + \frac{\bar{μ} Ω}{\bar{h}}) d α d β

(17)

Dynamic performance characteristics

The dynamic performance characteristics are determined when the journal center fluctuated around its equilibrium position $({\bar{\overset{\cdot}{X}}}_{J} \neq 0, {\bar{\overset{\cdot}{Z}}}_{J} \neq 0)$ .

Fluid film stiffness coefficients

The fluid film stiffness coefficients are obtained from the partial derivative of fluid film reaction with respect to journal center displacement $({\bar{q}}_{j} = {\bar{X}}_{J} or {\bar{Z}}_{J})$ and are expressed as (Crosby and Chetti, 2009; Sharma and Yadav, 2014)

{\bar{S}}_{xx} = - \frac{\partial}{\partial {\bar{X}}_{J}} \int_{- λ}^{λ} \int_{0}^{2 π} [\bar{p} \cos α d α d β]

(18a)

{\bar{S}}_{xz} = - \frac{\partial}{\partial {\bar{Z}}_{J}} \int_{- λ}^{λ} \int_{0}^{2 π} [\bar{p} \cos α d α d β]

(18b)

{\bar{S}}_{zx} = - \frac{\partial}{\partial {\bar{X}}_{J}} \int_{- λ}^{λ} \int_{0}^{2 π} [\bar{p} \sin α d α d β]

(18c)

{\bar{S}}_{zz} = - \frac{\partial}{\partial {\bar{Z}}_{J}} \int_{- λ}^{λ} \int_{0}^{2 π} [\bar{p} \sin α d α d β]

(18d)

Fluid film damping coefficients

The fluid film damping coefficients are obtained from the partial derivative of fluid film reaction with respect to journal center velocity $({\bar{\overset{\cdot}{q}}}_{j} = {\bar{\overset{\cdot}{X}}}_{J} or {\bar{\overset{\cdot}{Z}}}_{J})$ and are expressed as (Crosby and Chetti, 2009; Sharma and Yadav, 2014)

{\bar{C}}_{xx} = - \frac{\partial}{\partial {\bar{\overset{\cdot}{X}}}_{J}} \int_{- λ}^{λ} \int_{0}^{2 π} [\bar{p} \cos α d α d β]

(19a)

{\bar{C}}_{xz} = - \frac{\partial}{\partial {\bar{\overset{\cdot}{Z}}}_{J}} \int_{- λ}^{λ} \int_{0}^{2 π} [\bar{p} \cos α d α d β]

(19b)

{\bar{C}}_{zx} = - \frac{\partial}{\partial {\bar{\overset{\cdot}{X}}}_{J}} \int_{- λ}^{λ} \int_{0}^{2 π} [\bar{p} \sin α d α d β]

(19c)

{\bar{C}}_{zz} = - \frac{\partial}{\partial {\bar{\overset{\cdot}{Z}}}_{J}} \int_{- λ}^{λ} \int_{0}^{2 π} [\bar{p} \sin α d α d β]

(19d)

Linearized equations of motion and stability margins

The linearized equation of disturbed motion of the journal center which represents lateral $({\bar{X}}_{J}, {\bar{Z}}_{J})$ motion is obtained by equating the inertia force components to the out of balance fluid film force and moment components. In the compact form, the linearized equation of motion can be expressed as below in matrix form (Crosby and Chetti, 2009; Khatri and Sharma, 2016)

[\begin{matrix} {\bar{M}}_{J} & 0 \\ 0 & {\bar{M}}_{J} \end{matrix}] {\begin{matrix} {\bar{\overset{\cdot\cdot}{X}}}_{J} \\ {\bar{\overset{\cdot\cdot}{Z}}}_{J} \end{matrix}} + [\begin{matrix} {\bar{C}}_{xx} & {\bar{C}}_{xz} \\ {\bar{C}}_{zx} & {\bar{C}}_{zz} \end{matrix}] {\begin{matrix} {\bar{\overset{\cdot}{X}}}_{J} \\ {\bar{\overset{\cdot}{Z}}}_{J} \end{matrix}} + [\begin{matrix} {\bar{S}}_{xx} & {\bar{S}}_{xz} \\ {\bar{S}}_{zx} & {\bar{S}}_{zz} \end{matrix}] {\begin{matrix} {\bar{X}}_{J} \\ {\bar{Z}}_{J} \end{matrix}} = {\begin{matrix} 0 \\ 0 \end{matrix}}

(20)

Furthermore, the system stability margin in terms of critical mass $({\bar{M}}_{c})$ for the lateral motion is obtained by satisfying the necessary and sufficient conditions of Routh’s criteria. The non-dimensional critical mass $({\bar{M}}_{c})$ of journal is written as (Khatri and Sharma, 2016; Li et al., 1980)

{\bar{M}}_{c} = \frac{{\bar{G}}_{1}}{{\bar{G}}_{2} - {\bar{G}}_{3}}

(21)

where {\bar{G}}_{1} = [{\bar{C}}_{xx} {\bar{C}}_{zz} - {\bar{C}}_{zx} {\bar{C}}_{xz}]; {\bar{G}}_{2} = \frac{[{\bar{S}}_{xx} {\bar{S}}_{zz} - {\bar{S}}_{zx} {\bar{S}}_{xz}] [{\bar{C}}_{xx} + {\bar{C}}_{zz}]}{[{\bar{S}}_{xx} {\bar{C}}_{zz} + {\bar{S}}_{zz} {\bar{C}}_{xx} - {\bar{S}}_{xz} {\bar{C}}_{zx} - {\bar{S}}_{zx} {\bar{C}}_{xz}]}; {\bar{G}}_{3} = \frac{[{\bar{S}}_{xx} {\bar{C}}_{xx} + {\bar{S}}_{xz} {\bar{C}}_{xz} + {\bar{S}}_{zx} {\bar{C}}_{zx} + {\bar{S}}_{zz} {\bar{C}}_{zz}]}{[{\bar{C}}_{xx} + {\bar{C}}_{zz}]}

The stability threshold speed margin ${\bar{ω}}_{th}$ of journal can be evaluated using the following relation (Li et al., 1980)

{\bar{ω}}_{th} = {[\frac{{\bar{M}}_{c}}{{\bar{F}}_{o}}]}^{1 / 2}

(22)

where ${\bar{F}}_{o}$ is resultant fluid film reaction for steady-state condition $(\partial \bar{h} / \partial \bar{t} = 0)$ .

The system becomes stable when the stability threshold speed is greater than the operating journal speed $(i . e ., Ω < {\bar{ω}}_{th})$ .

Solution scheme

The solution algorithm adopted for studying the combined influence of ER lubricant and textured surfaces on the performance of three-lobe hole-entry hybrid journal bearing is given in the flowchart as shown in Figure 2. In this study, a MATLAB source code based on the FEM numerical technique is used to solve the modified Reynolds equation (1) coupled with restrictor flow equation (13). The overall numerical procedure comprises computing the static and dynamic performance characteristics are as follows:

The lubricant flow domain is discretized using four-noded iso-parametric quadrilateral elements and bearing operating and geometric parameters have been taken as input.

The values of fluid film thickness $(\bar{h})$ for non-textured and textured circular/three-lobe journal bearings are initialized from equations (2) and (3), respectively, for given value of bearing load $({\bar{W}}_{o})$ .

The dynamic yield stress $(τ_{o})$ and electric field $(E)$ are calculated at all Gaussian points using equations (5) and (6), respectively. Then, the apparent viscosity $({\bar{μ}}_{a})$ is obtained at all Gaussian points using equation (8) with the help of shear strain rate $(\bar{\overset{\cdot}{γ}})$ of ER lubricant (equation (9)). Furthermore, the values of ${\bar{F}}_{0},$ ${\bar{F}}_{1}$ , and ${\bar{F}}_{2}$ corresponding to the apparent viscosity are computed across the fluid film thickness by numerical integration using Simpson’s rule.

The modified Reynolds equation (1) reduces to the assembled global matrix equation (12a) using Galerkin’s orthogonality technique.

RBCs are applied to global fluidity matrices.

Global system of equation (12a) is solved using the Gauss–Siedel iterative numerical technique to compute the nodal fluid film pressure.

Using the nodal fluid film pressure obtained in step 6, the journal center equilibrium position is established for a chosen value of external load $({\bar{W}}_{o})$ .

The iteration steps 2 to 7 are repeated until the following appropriate convergence criteria are achieved

| \frac{{[{(Δ {\bar{X}}_{J}^{i})}^{2} - {(Δ {\bar{Z}}_{J}^{i})}^{2}]}^{1 / 2}}{{[{({\bar{X}}_{J}^{i})}^{2} - {({\bar{Z}}_{J}^{i})}^{2}]}^{1 / 2}} | \times 100 \leq 10^{- 3}

where $({\bar{X}}_{J}^{i}, {\bar{Z}}_{J}^{i}) = i th$ journal center coordinates position and $(Δ {\bar{X}}_{J}^{i}, Δ {\bar{Z}}_{J}^{i})$ = increment in $ith$ journal center coordinates position.

Finally, the converged pressure is obtained using the equilibrium value of journal center $({\bar{X}}_{J}, {\bar{Z}}_{J})$ . Based on the computed value of nodal fluid film pressure, the performance characteristics of textured three-lobe hole-entry hybrid journal bearing are simulated using the mathematical relations given in the previous analysis section.

Numerical model validation

The computer program used in this study for obtaining the nodal fluid film pressure and the bearing performance characteristic has been developed in MATLAB. FEM is used as solution technique for evaluating the nodal fluid film pressure, which requires various steps discussed in sections “Mathematical model” and “Solution scheme.” As the computer code is developed from the elementary stage, it is essential to test the validity of program thoroughly so as to rely on the accuracy of simulated results. The fluid film reaction of plain cylindrical hole-entry journal bearing at hydrostatic mode is computed from current developed MATLAB code and validated with the published results of Yoshimoto et al. (1988). Figure 3(a) indicates that the computed results from the present numerical model closely follow the results of Yoshimoto et al. (1988). The performance parameters of three-lobe journal bearing are simulated in hydrodynamic mode and validated with the available results of Lund and Thomsen (1978) as shown in Table 1. In addition, to validate the developed numerical model for the textured journal bearing, the pressure distribution of textured hydrodynamic journal bearing is computed. Figure 3(b) shows the good agreement with the results of Brizmer and Kligerman (2012). The results computed from the current developed numerical model for ER lubricant is consistent with the reported results of Peng and Zhu (2006) and illustrated in Figure 3(c). The simulated results of current numerical model appear to be in good concordance with published results, indicating the accuracy of the present numerical model.

Table 1.

Comparison of performance characteristic of three-lobe hydrodynamic journal bearing.

$[L / D = 1, Ω = 1, δ = 0.5]$ , Hydrodynamic journal bearing
Performance characteristics	${\bar{W}}_{o} = 0.89$		${\bar{W}}_{o} = 1.3$
	Lund and Thomsen (1978)	This work	Lund and Thomsen (1978)	This work
Eccentricity ratio, $ε$	0.144	0.141	0.192	0.189
Lubricant flow rate, $\bar{Q}$	0.271	0.283	0.285	0.297
Fluid film stiffness, ${\bar{S}}_{11} / {\bar{W}}_{o}$	3.49	3.34	2.77	2.65
Fluid film stiffness, ${\bar{S}}_{12} / {\bar{W}}_{o}$	5.92	5.82	4.34	4.28
Fluid film stiffness, ${\bar{S}}_{21} / {\bar{W}}_{o}$	−5.85	−5.97	−4.75	−4.86
Fluid film stiffness, ${\bar{S}}_{22} / {\bar{W}}_{o}$	6.85	6.69	6.27	6.18
Fluid film damping, ${\bar{C}}_{11} / {\bar{W}}_{o}$	9.93	9.98	7.12	7.20
Fluid film damping, $C_{12} / {\bar{W}}_{o}$	−0.37	−0.375	−0.02	−0.0203
Fluid film damping, $C_{21} / {\bar{W}}_{o}$	−0.37	−0.373	−0.02	−0.0201
Fluid film damping, ${\bar{C}}_{22} / {\bar{W}}_{o}$	14.74	15.14	12.07	12.45

Furthermore, the flow of lubricant in textured hydrodynamic fluid bearing is often subjected to cavitation. The aforementioned studies used for validation purposes have utilized Reynolds boundary condition (RBC) which is not mass conserving in cavitation zone. An additional study is performed to compare the results obtained using RBC and Jakobsson–Floberg–Olsson (JFO) boundary condition in textured hybrid journal bearing. The algorithm used to implement JFO boundary condition is taken from published reference studies (Elrod, 1981; Qiu and Khonsari, 2009). The results presented in Table 2 indicate a negligible deviation between the two studies, that is, JFO boundary condition and RBC. This is due to the fact that in hybrid hole-entry journal bearings, the pressurized lubricant is supplied via an external pump. Hence, lubricant pressure within the fluid film domain is sufficiently high in hydrostatic/hybrid journal bearing as compared to hydrodynamic bearing. Another reason could be due to the presence of many supply holes provided in circumferential and radial directions; thus, a net positive pressure is maintained throughout the entire fluid film domain. The results from JFO boundary conditions are bound to be more accurate as it is mass conserving over entire fluid flow domain. However, implementation of JFO requires higher computational resources vis-à-vis RBC. Hence, further simulation in this study is carried out employing RBC.

Table 2.

Comparison of Reynolds boundary condition with JFO boundary condition in terms of minimum fluid film thickness $({\bar{h}}_{min})$ .

$λ = 1, {\bar{a}}_{b} = 0.25, Ω = 1, {\bar{h}}_{p} = 1.8, {\bar{C}}_{s 2} = 0.0873, B = 2.185, V = 1200 V$
	By Reynolds boundary condition	By JFO boundary condition	By Reynolds boundary condition	By JFO boundary condition	By Reynolds boundary condition	By JFO boundary condition	By Reynolds boundary condition	By JFO boundary condition
${\bar{W}}_{o}$	CWTWN	CWTWN	CTWN	CTWN	CWTWER	CWTWER	CTWER	CTWER
1	0.84903	0.84913	0.83189	0.83196	0.88850	0.88859	0.87173	0.87182
1.1	0.83432	0.83443	0.81557	0.81564	0.87755	0.87764	0.85909	0.85919
1.2	0.81973	0.81985	0.79938	0.79946	0.86665	0.86675	0.84651	0.84662
1.3	0.80526	0.80539	0.78334	0.78343	0.85581	0.85592	0.83399	0.83410
1.4	0.79093	0.79106	0.76745	0.76754	0.84503	0.84515	0.82153	0.82165
1.5	0.77673	0.77687	0.75172	0.75182	0.83432	0.83445	0.80914	0.80927
1.6	0.76268	0.76283	0.73615	0.73626	0.82369	0.82382	0.79681	0.79695

JFO: Jakobsson–Floberg–Olsson; CWTWN: circular non-textured journal bearing operating with Newtonian lubricant; CTWN: circular textured journal bearing operating with Newtonian lubricant; CWTWER: circular non-textured journal bearing operating with ER lubricant; CTWER: circular textured journal bearing operating with ER lubricant.

Results and discussion

For computing the performance characteristics of three-lobe textured hole-entry hybrid journal bearing, the grid size 49 × 20 has been adopted in this study which is based on the grid convergence studies by varying the elements along circumferential as well as axial dimension of journal bearing. Furthermore, the geometric and operating input values used in this work have been chosen from the available reported literature (Brizmer and Kligerman, 2012; Lund and Thomsen, 1978; Peng and Zhu, 2005, 2006; Rowe et al., 1982; Sharma and Yadav, 2014; Yoshimoto et al., 1988) and indicated in Table 3. In addition, for optimization of electric field on the present numerical model, the variation of minimum fluid film thickness $({\bar{h}}_{min})$ and fluid film stiffness coefficient $({\bar{S}}_{22})$ , respectively, with respect to external applied electric voltage $(V)$ are illustrated in Figure 4(a) and (b). From Figure 4(a) and (b), it has been observed that influence of electric field on bearing performance operating with ER lubricant is dependent on the numerical value of voltage applied across fluid film. Based on simulated results, it can be said that the influence of electric field on the performance of bearing is observed to be more significant for value of electric field line between voltages $V = 400 and 1200 V$ . Thus, the numerical value of electric field $(E = 23.90 kV / mm)$ corresponding to potential difference $(V = 1200 V)$ is observed to be suitable optimization value of electric field. Hence, in this study, numerical simulated results are obtained for three values of electric field, that is, $E = 0.0 kV / mm$ (Newtonian lubricant), $E = 11.95 kV / mm$ (corresponding $V = 600 V$ ) and $E = 23.90 kV / mm$ (ER lubricant). Based on the above chosen grid size, geometric and operating input values, the static and dynamic performance parameters of three-lobe textured journal bearing are presented in non-dimensional form as shown in Figures 5 to 11.

Table 3.

Geometric and operating non-dimensional input parameters for three-lobe hole-entry hybrid journal bearing (Brizmer and Kligerman, 2012; Lund and Thomsen, 1978; Peng and Zhu, 2005, 2006; Rowe et al., 1982; Sharma and Yadav, 2014; Yoshimoto et al., 1988).

Parameters	Value/range
Non-dimensional
Bearing aspect ratio $(λ)$	1
Clearance ratio $(\bar{c})$	0.001
Land width ratio $({\bar{a}}_{b})$	0.25
Number of holes per row $(N)$	12
Number of rows $(N_{r})$	2
Speed parameter $(Ω)$	1.0
Sommerfeld number $({\bar{S}}_{o})$	0.3978–0.6366
Offset factor $(δ)$	1, 1.1
Restrictor design parameter $({\bar{C}}_{s 2})$	0.0873
Dimple depth $({\bar{h}}_{p})$	1.8
Dimple radius $(\bar{δ})$	100
ER fluid parameter $(A)$	6.619 × 10⁻¹³ Pa (m/V)^B
ER fluid field parameter $(B)$	2.18
Applied voltage $(V)$	0, 600, 1200 V
Compensating element	Capillary
Geometric shape of microdimple	Spherical
Dimensional
Radius of journal $(R_{J})$	50 mm
Bearing length $(L)$	100 mm
Radial clearance $(c)$	0.05020 mm
Applied electric field $(E)$	0, 11.95, 23.90 kV/mm
Shaft speed $(N)$	2500 r/min
ER lubricant density $(ρ)$	894 kg/m³
ER lubricant viscosity $(μ)$	0.0816 N s/m²

ER: electro-rheological.

Fluid film pressure distribution

The nodal fluid film pressure distribution contours are illustrated in Figure 5 for Newtonian and ER fluid lubricated textured/non-textured circular/three-lobe hole-entry hybrid journal bearings. From these pressure contours, the influence of textured surfaces with ER fluid lubrication on the fluid film pressure distribution along the axial as well as circumferential direction of corresponding bearings has been easily investigated . Figure 5 indicated that the three-lobe profile journal bearing operates with higher value of maximum fluid film pressure $({\bar{p}}_{max})$ in comparison to circular journal bearing. It may be the reason that the three-lobe $(δ = 1.1)$ profile in journal bearing gives more converging and diverging portion for hydrodynamic squeeze action which help to improve the load-carrying capacity of journal bearing. Furthermore, from Figure 5, it can be seen that the ER fluid lubrication in journal bearing provides enhanced value of ${\bar{p}}_{max}$ as compared to Newtonian fluid lubrication. Because of the presence of electric field, the suspended particles orient themselves along the electric field line, this makes the fluid more viscous and provides more film thickness to support the external load. Furthermore, it has been observed that the use of surface texturing increases the value of ${\bar{p}}_{max}$ for ER fluid lubrication and reduces for Newtonian lubrication. For a selected value of Sommerfeld number ${\bar{S}}_{o} = 0.636$ according to the external applied load ${\bar{W}}_{o} = 1.0$ , the Newtonian fluid lubrication in textured circular journal bearing reduces the value of ${\bar{p}}_{max}$ by order of 2.02% with respect to Newtonian fluid lubrication. However, the ER fluid $(E = 23.90 kV / mm)$ lubrication in three-lobe $(δ = 1.1)$ textured journal bearing gets enhanced value of ${\bar{p}}_{max}$ in order of 22.15% in comparison to Newtonian fluid lubrication in non-textured circular journal bearing. Moreover, in the ER fluid lubricated textured three-lobe journal bearing, the higher pressure is more uniformly distributed over a larger land area vis-à-vis Newtonian fluid lubricated non-textured circular journal bearing. Thus, the ER fluid lubricated textured three-lobe journal bearing gives higher load-carrying capacity than that of Newtonian fluid lubricated non-textured circular journal bearing.

Minimum fluid film thickness

Figure 6 shows the combined influence of ER fluid lubrication and textured surface on the minimum fluid film thickness $({\bar{h}}_{min})$ at different values of Sommerfeld number $({\bar{S}}_{o})$ . Figure 6 shows that the textured journal bearing reduces the value of ${\bar{h}}_{min}$ as compared to non-textured journal bearing. The lower value of minimum fluid film thickness may cause wear and scuffing problem in bearing surface. Furthermore, it has been observed that the value of minimum fluid film thickness increases for three-lobe $(δ = 1.1)$ journal bearing than that of circular journal bearing. This is due to the fact that the use of three-lobe $(δ = 1.1)$ profile bearing gives more convergence gap at the bottom of the bearing. As result of this, the minimum fluid film thickness is increased by adding extra lubricant in the clearance gap. Furthermore, the value of ${\bar{h}}_{min}$ increases with an increase in the value of applied electric field $(kV / mm)$ in the ER lubricant. For a given value of Sommerfeld number ${\bar{S}}_{o} = 0.489$ according to the external applied load ${\bar{W}}_{o} = 1.3$ , the textured circular journal bearing gives reduction in the value of ${\bar{h}}_{min}$ by an amount of 2.72% in comparison to non-textured journal bearing for Newtonian lubricant. The ER fluid lubrication $(E = 23.90 kV / mm)$ in textured three-lobe $(δ = 1.1)$ journal bearing enhances the value of ${\bar{h}}_{min}$ in order of 4.35% as compared to Newtonian fluid lubricated non-textured circular journal bearing. Thus, we can say that the ER fluid lubricated textured three-lobe hole-entry hybrid journal bearing can sustain higher load-carrying capacity than that of Newtonian fluid lubricated circular non-textured journal bearing.

Attitude angle

Figure 7 depicts the variation of attitude angle $(ϕ)$ against the Sommerfeld number $({\bar{S}}_{o})$ under the influence of ER lubricant for textured and non-textured three-lobe hole-entry hybrid journal bearing configurations. In general, it may be observed that the value of attitude angle $(ϕ)$ is not much affected by variation of non-dimensional external load. From Figure 7, it has been noticed that the use of ER fluid lubrication increases the value of $ϕ$ vis-à-vis Newtonian fluid lubrication for both circular and three-lobe journal bearings. The similar pattern has also been observed in the study reported by Nikolakopoulos and Papadopoulos (1998) for the value of attitude angle in the case of plain journal bearing lubricated with ER fluids. However, the three-lobe $(δ = 1.1)$ textured journal bearing operates with lower attitude angle $(ϕ)$ in comparison to non-textured circular journal bearing. Hence, the combination of three-lobe $(δ = 1.1)$ profile and textured surface bearing is beneficial from the viewpoint of stability. For a case of hole-entry circular hybrid journal bearing, the ER fluid lubrication $(E = 23.90 kV / mm)$ increases the value of $ϕ$ by the order of 5.65%–6.06% as compared to Newtonian fluid lubrication. The ER fluid lubricated $(E = 23.90 kV / mm)$ textured three-lobe $(δ = 1.1)$ journal bearing reduces the value of $ϕ$ in order of 13.08%–13.44% in comparison to Newtonian fluid lubricated non-textured circular journal bearing.

Frictional torque

The variation of frictional torque $({\bar{T}}_{f})$ under the combined influence of ER fluid with applied electric field $(E)$ , offset factor $(δ)$ , and textured surface on three-lobe journal bearing is illustrated in Figure 8. In general, increasing value of Sommerfeld number $({\bar{S}}_{o})$ is observed to increase the value of frictional torque $({\bar{T}}_{f})$ . From Figure 8, it can be seen that the three-lobe $(δ = 1.1)$ journal bearing is operating with higher value of frictional torque $({\bar{T}}_{f})$ in comparison to circular journal bearing. Based on simulated results, it is clearly visible that the value of ${\bar{T}}_{f}$ increases with the increase in the value of applied electric field $(E)$ . This is due to the increase in the apparent viscosity of ER fluid, consequently increasing the frictional torque according to equation (17). To minimize the operating temperature of journal bearing, the value of frictional torque should be minimum. Furthermore, it can be seen that the presence of textured surface on the bearing surfaces significantly minimizes the value of ${\bar{T}}_{f}$ for both circular and three-lobe journal bearing. This is because of the micro-texture surface storing extra lubricant and flow to the contact zone directly. As seen in Figure 8, the three-lobe $(δ = 1.1)$ non-textured journal bearing increases the values of ${\bar{T}}_{f}$ in order of 1.48%–1.58% with respect to circular non-textured journal bearing for Newtonian fluid. However, the ER fluid $(E = 23.90 kV / mm)$ lubricated textured three-lobe $(δ = 1.1)$ journal bearing minimizes the value of ${\bar{T}}_{f}$ by an amount of 3.22%–3.83% over a Newtonian lubricant lubricated non-textured circular journal bearing.

Rotor fluid film stiffness coefficients

Figure 9(a) and (b) shows the combined influence of applied electric field in ER fluid, surface texturing, and non-circular profile configuration on the values of fluid film stiffness coefficients $({\bar{S}}_{11}, {\bar{S}}_{22})$ of hole-entry journal bearing. As seen from Figure 9(a) and (b), it has been noticed that the ER fluid lubrication in the three-lobe $(δ = 1.1)$ profile of bearing improves the values of ${\bar{S}}_{11}$ and ${\bar{S}}_{22}$ with respect to the Newtonian fluid lubrication in the circular journal bearing. This is because the ER fluid has a yield stress proportional to the square of applied electric field. Furthermore, it may be noticed that the textured journal bearing significantly enhances the values of ${\bar{S}}_{11}$ and ${\bar{S}}_{22}$ for both Newtonian and ER lubricant. It is interesting to note that the influence of ER fluid on the values of ${\bar{S}}_{11}$ and ${\bar{S}}_{22}$ is high in textured journal bearing than that of non-textured journal bearing. For the given value of Sommerfeld number, ${\bar{S}}_{o} = 0.489$ , the ER fluid lubricated $(E = 23.90 kV / mm)$ non-textured three-lobe $(δ = 1.1)$ journal bearing enhanced the values of ${\bar{S}}_{11}$ and ${\bar{S}}_{22}$ by amount of 9.30% and 11.75% respectively, in comparison to Newtonian fluid lubricated circular journal bearing. However, the ER fluid lubrication $(E = 23.90 kV / mm)$ in textured three-lobe $(δ = 1.1)$ journal bearing gets improvement in the values of ${\bar{S}}_{11}$ and ${\bar{S}}_{22}$ by 136.80% and 66.56% respectively, as compared to Newtonian fluid lubrication in circular non-textured journal bearing.

Rotor fluid film damping coefficients

Figure 10(a) and (b) illustrates the variation of fluid film damping coefficients $({\bar{C}}_{11}, {\bar{C}}_{22})$ against Sommerfeld number $({\bar{S}}_{o})$ for the ER fluid lubricated textured/non-textured circular/three-lobe journal bearing. From Figure 10(a) and (b), it can be seen that the use of dimple surfaces on the bearing surface reduces the values of ${\bar{C}}_{11}$ and ${\bar{C}}_{22}$ in comparison to non-textured surface journal bearing configuration. In order to damp out the vibration of journal under dynamic loading, the value of damping coefficients $({\bar{C}}_{11}, {\bar{C}}_{22})$ needs to be larger. Furthermore, the use of ER fluid lubrication in three-lobe $(δ = 1.1)$ profile journal bearing may compensate the reduction in the values of ${\bar{C}}_{11}$ and ${\bar{C}}_{22}$ caused by textured journal bearing. As seen in Figure 10(a) and (b), the textured circular hole-entry journal bearing reduces the values of ${\bar{C}}_{11}$ and ${\bar{C}}_{22}$ by order of 22.36%–23.51% and 21.03%–21.42%, respectively, relative to circular non-textured journal bearing for Newtonian fluid. However, the ER fluid lubricated $(E = 23.90 kV / mm)$ textured three-lobe $(δ = 1.1)$ journal bearing improved the values of ${\bar{C}}_{11}$ and ${\bar{C}}_{22}$ by an amount of 27.15%–30.72% and 31.86%–32.56% with respect to Newtonian fluid lubricated circular non-textured journal bearing.

Stability threshold speed margin and linear journal center motion trajectories

The stability of rotating system is an important parameter to design of stable system. Generally, the stability threshold speed $({\bar{ω}}_{th})$ of journal bearing is a function of the bearing dynamic coefficients. Due to the combined influence of ER fluid lubrication, surface texturing, and non-circular profile of bearing, the values of stiffness and damping coefficients get altered. Thus, it is expected that the value of ${\bar{ω}}_{th}$ also gets affected. Figure 11(a) shows the variation of ${\bar{ω}}_{th}$ in terms of Sommerfeld number $({\bar{S}}_{o})$ for circular/three-lobe textured/non-textured journal bearings. Figure 11(a) shows that the textured journal bearing gets significant improvement in the value of ${\bar{ω}}_{th}$ for both circular and three-lobe journal bearing, respectively, relative to non-textured journal bearing. Furthermore, the value of ${\bar{ω}}_{th}$ increases with the increase in the value of applied electric field $(E)$ in ER fluid. In addition, it can be seen that the three-lobe $(δ = 1.1)$ profile journal bearing gives improved value of ${\bar{ω}}_{th}$ than that of circular journal bearing. The numerically simulated results in Figure 11(a) indicate that the percentage (%) increment in the value of ${\bar{ω}}_{th}$ is order of 5.77%–6.13% due to the use of ER fluid lubrication $(E = 23.90 kV / mm)$ in non-textured circular journal bearing instead of Newtonian fluid lubrication. Whereas the use of surface texturing in ER fluid lubricated $(E = 23.90 kV / mm)$ three-lobe $(δ = 1.1)$ journal bearing provides significant enhancement in the value of ${\bar{ω}}_{th}$ by order of 28.61%–32.72% in comparison to Newtonian fluid lubricated non-textured circular journal bearing.

Furthermore, the linear journal center motion trajectories for circular/two-lobe textured/non-textured journal bearings are drawn in Figure 11(b) to (d). From Figure 11(b) to (d), it may be clearly noticed that the textured journal bearing gives small locus of journal center and settles down quickly to equilibrium position of journal center in comparison to non-textured journal bearing. In addition, the ER fluid lubricated three-lobe $(δ = 1.1)$ journal bearing also provided the small radius of whirl orbit to settle down the journal center to equilibrium position after some disturbances owing to unbalanced fluid film forces. Smaller whirl orbit size indicates that journal center fluctuates around its equilibrium position with lower amplitude of vibration consequently providing the higher stability. Therefore, it may be said that the ER fluid lubricated textured three-lobe journal bearing gives significantly stable operation under given operating conditions. As seen in trajectory plots, it can also be observed that the condition ${\bar{M}}_{J} = 0.9 {\bar{M}}_{C}$ provides better stability than that of other combinations, that is, ${\bar{M}}_{J} = 1.1 {\bar{M}}_{C}$ and ${\bar{M}}_{J} = {\bar{M}}_{C}$ . Hence, for a stable operation of journal bearing system, designers may take proper care about selection of suitable bearing geometry, surface texture geometry, and use of lubricant.

Comparative study

A comparative study between textured and non-textured circular/three-lobe journal bearings lubricated with ER and Newtonian lubricant, respectively, has also been carried out and is shown in Table 3 in terms of percentage variation (%). The percentage increase in bearing performance parameters $({\bar{h}}_{min}, ϕ, {\bar{T}}_{f}, {\bar{S}}_{11}, {\bar{S}}_{22}, {\bar{C}}_{11}, {\bar{C}}_{22}, {\bar{ω}}_{th})$ has been evaluated with respect to base bearing, that is, non-textured circular journal bearing operated with Newtonian fluid and expressed

% Change (Performance Parameters) = \frac{(Performance Parameters - {Performance Parameters |}_{base bearing})}{({Performance Parameters |}_{base bearing})} \times 100

Table 4 indicates that the use of ER fluid lubrication with textured and non-circular profile improved the performance characteristics of hole-entry hybrid journal bearing. For example, it can be seen that the use of ER fluid lubrication in three-lobe $(δ = 1.1)$ journal bearing gives enhancement in the value of ${\bar{S}}_{22}$ by an amount of 66.56% in comparison to Newtonian fluid lubrication in non-textured circular journal bearing. Hence, it may be concluded that the combination of ER fluid lubrication and three-lobe $(δ = 1.1)$ textured hole-entry journal bearing is more beneficial from the viewpoint of efficient bearing performance.

Table 4.

Percentage change in the values of static and dynamic characteristic of three-lobe hole-entry hybrid journal bearing.

$[λ = 1, {\bar{a}}_{b} = 0.25, Ω = 1, {\bar{C}}_{s 2} = 0.0873, {\bar{S}}_{o} = 0.4896, B = 2.185]$
BPC	Circular non-textured bearing $(δ = 1.0, {\bar{h}}_{p} = 0)$			Circular textured bearing $(δ = 1.0, {\bar{h}}_{p} = 1.8)$			Three-lobe non-textured bearing $(δ = 1.1, {\bar{h}}_{p} = 0)$			Three-lobe textured bearing $(δ = 1.1, {\bar{h}}_{p} = 1.8)$
	$E = 0$	$E = 11.95$	$E = 23.90$	$E = 0$	$E = 11.95$	$E = 23.90$	$E = 0$	$E = 11.95$	$E = 23.90$	$E = 0$	$E = 11.95$	$E = 23.90$
${\bar{h}}_{\min}$	0.8052	0.8222	0.8558	0.7833	0.8006	0.8339	0.8122	0.8288	0.8616	0.7916	0.8083	0.8402
% Change	+0.000	+2.133	+6.276	−2.722	−0.576	+3.567	+0.862	+2.932	+7.000	−1.687	+0.388	+4.347
$ϕ \circ$	71.784	72.554	76.011	61.443	62.743	64.138	69.990	70.450	74.405	58.432	59.839	62.288
% Change	+0.000	+1.072	+5.888	−14.406	−12.595	−10.653	−2.500	−1.859	+3.650	−18.600	−16.641	−13.230
${\bar{S}}_{11}$	1.4390	1.4698	15.242	2.7540	2.9093	3.2474	1.4802	1.5085	1.5730	2.8902	3.0528	3.4077
% Change	+0.000	+2.138	+5.922	+91.400	+102.169	+125.660	+2.860	+4.831	+9.308	+100.841	+112.139	+136.804
${\bar{S}}_{22}$	2.1187	2.1840	2.2930	2.9442	3.0685	3.3614	2.2271	2.2702	2.3678	3.1109	3.2390	3.5292
% Change	+0.000	+3.115	+8.223	+38.960	+44.825	+58.652	+5.116	+7.152	+11.755	+46.826	+52.875	+66.568
${\bar{C}}_{11}$	13.4168	15.6184	22.7455	10.3493	11.8919	16.6979	13.9060	16.2386	23.7892	10.6809	12.2981	17.3240
% Change	+0.000	+16.409	+69.529	−22.862	−11.365	+24.455	+3.646	+21.031	+77.308	−20.391	−8.337	+29.122
${\bar{C}}_{22}$	12.7437	14.8267	21.6443	10.0355	11.5428	16.2402	13.2089	15.4080	22.6216	10.3715	11.9495	16.8520
% Change	+0.000	+16.345	+69.843	−21.250	−9.423	+27.437	+3.650	+20.906	+77.511	−18.614	−6.232	+32.237
${\bar{ω}}_{th}$	2.3683	2.4137	2.4837	2.8205	2.8934	3.0563	2.3910	2.4358	2.5056	2.8555	2.9287	3.0898
% Change	+0.000	+1.915	+4.873	+19.091	+22.172	+29.048	+0.957	+2.850	+5.795	+20.569	+23.663	+30.465
${\bar{T}}_{f}$	12.9280	12.9517	12.9814	12.1773	12.2134	12.2837	13.1274	13.1483	13.1709	12.3607	12.4003	12.4747
% Change	+0.000	+0.1836	+0.413	−5.807	−5.527	−4.984	+1.542	+1.703	+1.878	−4.388	−4.082	−3.506

BPC: bearing performance characteristics, Newtonian lubricant $(V = 0)$ , electro-rheological lubricant $(V = 600, 1200)$ .

% Change = (BPC – BPC|_{Base bearing}) × 100/BPC|_{Base bearing}.

Base bearing: smooth circular journal bearing lubricated with Newtonian lubricant.

Conclusion

The influence of ER effects, caused by the applied electric field, on the multi-lobe (three-lobe) textured hole-entry journal bearings is quite significant. On the basis of the result discussed, the following salient conclusions can be drawn:

The textured hole-entry journal bearing reduces the value of minimum fluid film thickness $({\bar{h}}_{min})$ relative to non-textured journal bearing. However, the use of ER fluid lubrication $(E = 23.90 kV / mm)$ and three-lobe $(δ = 1.1)$ profile in textured hole-entry hybrid journal bearing gives higher values of ${\bar{h}}_{min}$ in order of 4.35% over the Newtonian fluid lubricated circular non-textured journal bearing. It means the ER fluid lubricated smart three-lobe $(δ = 1.1)$ textured journal bearing can sustain higher value of external load than that of non-textured circular journal bearing lubricated with Newtonian fluid.

In case of hole-entry hybrid journal bearing configuration, the use of ER fluid lubrication increases the attitude angle $(ϕ)$ . Furthermore, the use of textured surface on the surface of three-lobe $(δ = 1.1)$ profile bearing reduces the value of attitude angle $(ϕ)$ vis-à-vis non-textured circular journal bearing. Hence, for minimizing the attitude angle, the ER fluid lubricated smart three-lobe $(δ = 1.1)$ textured hole-entry journal bearing is beneficial than the Newtonian fluid lubricated circular non-textured journal bearing.

The ER fluid lubricated three-lobe $(δ = 1.1)$ profile hole-entry hybrid journal bearing increases the value of frictional torque $({\bar{T}}_{f})$ in comparison to Newtonian fluid lubricated circular journal bearing. However, the presence of microdimple surfaces reduces the value of ${\bar{T}}_{f}$ significantly for all types of bearing configurations. Hence, from the viewpoint of frictional power losses (minimization of bearing temperature), the ER fluid lubricated smart three-lobe $(δ = 1.1)$ textured hole-entry journal bearing is beneficial than the Newtonian fluid lubricated circular non-textured journal bearing.

In a case of hole-entry hybrid journal bearing system, the presence of surface texturing reduces the values of fluid film damping coefficients $({\bar{C}}_{11}, {\bar{C}}_{22})$ . In addition, the use of ER fluid lubrication in a three-lobe $(δ = 1.1)$ journal bearing provides enhancement in the values of ${\bar{C}}_{11}$ and ${\bar{C}}_{22}$ . So to damp out the oscillations of journal bearing, the combination of textured three-lobe profile and ER fluid lubrication is to provide better performance vis-à-vis Newtonian fluid lubricated circular non-textured journal bearing.

For a circular/three-lobe hole-entry journal bearing system, the use of surface texturing significantly enhances the values of fluid film stiffness coefficients $({\bar{S}}_{11}, {\bar{S}}_{22})$ and stability threshold speed $({\bar{ω}}_{th})$ with respect to non-textured surface journal bearing. The influence of ER lubricant is significantly high on the values of ${\bar{S}}_{11}$ , ${\bar{S}}_{22}$ , and ${\bar{ω}}_{th}$ for textured journal bearing in comparison to non-textured journal bearing.

Furthermore, it may be noticed that the combination of three-lobe $(δ = 1.1)$ profile, ER fluid lubrication, and surface texturing generates the small radius of whirl orbit (linear trajectory) to settle down the journal center to equilibrium position after some disturbances owing to unbalanced fluid film forces. Therefore, it may be pointed out that the ER fluid $(E = 23.90 kV / mm)$ lubricated smart three-lobe $(δ = 1.1)$ textured journal bearing is highly beneficial from stability point of view.

Footnotes

Appendix 1 Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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