The lubrication characteristics of magnetized journal bearings affected by magnetic fluids considering the elasticity of its liner

Abstract

In order to investigate the lubrication characteristics of journal bearings with the elasticity of liner affected by magnetic fluids, a theoretic computation model was developed based on magnetic fluids supporting force parameter ( $γ$ ) and couple-stress factor ( $L^{*}$ ). The results indicate that maximum dimensionless pressure peak ( $P_{max}$ ) of oil-film increases with increase in magnetic fluids supporting force parameter ( $γ$ ), couple-stress factor ( $L^{*}$ ) and with decrease in elastic deformation coefficient (Ce) of the liner. The loading capacity ( $S$ ) increases with increase in $γ$ , $L^{*}$ and with decrease in Ce particularly in the high eccentricity ratio (ε); The attitude angle ( $ψ$ ) of S with different $L^{*}$ and $γ$ decreases in a small range of ε at first, and then increases, it increases with the decrease of $Ce$ and $γ$ , and with the increase of $L^{*}$ , theses phenomena are more obvious for $γ$ at low ε and for $L^{*}$ and $Ce$ at moderate and high ε; the modified coefficient of friction ( $C_{f} (R / C)$ ) decreases with increase in $γ$ , $L^{*}$ and with decrease in Ce; the side leakage (Q) decreases with increase in $γ$ , with decrease in $L^{*}$ and with increase in Ce particularly in the low (ε). Moreover, the increase of elastic deformation coefficient (Ce) weaken the impacts of $γ$ , $L^{*}$ on the increase of $P_{max}$ , $ψ$ , $S$ , and the decrease of $C_{f} (R / C)$ , Q.

Keywords

Magnetized journal bearings magnetic fluids supporting force couple-stress fluids squeeze dynamic action elastohydrodynamic effect

Introduction

Carrying the larger loading capacity of the magnetized journal bearings with a squeeze dynamic action such as in the internal combustion engine, the measure unit of oil-film thickness is usually expressed in micron, for the maximum oil-film pressure that is often bigger than the value of 10 MPa and the stiffness of the journal bearing (bearing block) is smaller due to its structure design limit.^1,2 According to this analysis, the effects of the elastic deformation of working surface have a great importance in the pressure distribution of the oil film and its loading capacity. For example, some papers^3,4 have reported that loading capacity of rigid journal bearing is bigger than that of journal bearing considering its materials elastic deformation. Meanwhile, the elastic deformation of the liner has also been considered about its lubrication influence on journal bearings,⁵ authors mainly investigated that the elasticity effect of the liner reduces with increasing loading capacity and increases with increasing coefficient of friction. Usov⁶ analyzed that a plane elastohydrodynamic problem for a radial sliding bearing with a thin liner under reciprocating motion under a constant load is considered, and he found that the film thickness and pressure distributions of journal bearing at various moments in time depend on the elastic deformation and the eccentricity motion. Patel et al.⁷ found that the dimensionless pressure is greater when the width-diameter ratio of the bearing is slightly smaller. In addition, the increase of eccentricity expands the liquid film convergence area between the journal and the bearing, thus increasing the pressure again. Some studies^8,9 have also explored the effects of texture and surface roughness on the lubrication and other properties of journal bearings. However, these papers above neglect the coupling effect among the relative characteristics parameters such as the squeeze dynamic action, couple stresses, the elastic deformation, and the extent of the elastic deformation about the impact on the couple stresses and other relative factors on the oil-film lubrication characteristics. The paper mainly shows that the lubrication characteristics of journal bearing with the elastic deformation of the liner and with a squeeze dynamic action based on magnetic fluids supporting force and its couple stress. Moreover, the elasticity extent of the liner on the effects of other characteristics parameters of oil-film lubrication characteristics is also discussed.

There are some papers discovered that magnetic fluids are non-Newtonian fluids^10,11 with the solid magnetic nano-materials are analogical to the couple-stress fluids, which show the analogical mechanical properties. Up to now, few researches were mentioned about a lubricant with the magnetic couple-stress fluids under the applied magnetic field. Das¹² showed the maximum loading capacity of journal bearings increase with an increase in magnetic fluids supporting force parameters and couple-stress factors by using a non-compressible lubricant of electrical-magnetic, couple-stress fluids under the applied uniform magnetic field; Nada and Osman¹³ analyzed the impact of magnetic fluids lubricant with considering it as non-Newtonian fluids, and pointed out that the increasing with the solid particle size concerning couple-stress factors can enhance the oil-film pressure of the bearing and its loading capacity and decrease the modified coefficient of friction. Moreover, the increasing of the applied magnetic field factors can also enhance the oil-film pressure of the bearing and its loading capacity. Hu and Xu¹⁴ found that the lubrication characteristics of journal bearing with magnetic fluids supporting force and its couple stress can be obviously improved better than that of journal bearing with no magnetic fluids supporting force and its couple stress. However, the researches above have tended to mainly focus on the lubrication characteristics of journal bearing without the elasticity of the liner and the coupling effects among these characteristics parameters. However, the presence of the liner is helpful to protect the working life of journal bearing.^5,6 Therefore, the study of magnetized journal bearing with the elastic deformation of the liner and its elasticity extent on the lubrication characteristics are necessarily discussed with the influence of squeeze oil-film action in axial taken into account together in the paper, which is few mentioned in other research on the journal bearing affected by magnetic fluids. Mokhiamer et al.¹⁵ assumes that without considering the effect between own force and couple stress, the expressions of load, friction, and pressure center are deduced. The results show that the couple-stress factor have a significant impact on the bearing capacity, pressure distribution and friction coefficient in the journal bearing lubricated by stress-couple fluid.

The modified Reynolds formulas on the lubrication characteristics of journal bearing affected by magnetic fluids have been derived on the basis of Stokes micro-continuum theory,¹⁶ considering the couple action between the rotating action of journal bearing and its oil-film squeeze dynamic action based on Hu and Xu’s theoretical research.¹⁴ On the basis of this, the impact of the magnetic supporting oil-film pressure due to the symmetric gradient magnetic field in axial and its lubrication characteristics have been analyzed. The combined effect above among the elastic deformation coefficient (Ce) of the liner, the squeeze dynamic action parameter (q) of journal bearing, magnetic fluids supporting force parameter ( $γ$ ) and couple-stress factor (L*) have been also considered together in the paper. This is beneficial to use as a reference for the future magnetic fluid bearings with the elastic or rigid liner for its lubrication characteristics.

Theory

Modeling analysis

The physical model of journal bearings with the elastic or rigid liner affected by Magnetic fluids and considering its squeeze dynamic action are stated in Figure 1.

Figure 1.

Journal bearings with the elastic/rigid liner affected by magnetic fluids and the axial symmetric gradient configuration of its magnetic field (arrows present the strength of the magnetic gradient field and its direction).

Considering the application of hydrodynamic lubrication, the magnetic fluids’ velocities $u$

and $w$ in the bearings can be obtained in the directions of x and z are¹⁴:

u = U \frac{y}{h} + \frac{1}{2 μ} (\frac{\partial p}{\partial x} - \frac{\partial F_{m}}{\partial x}) {y (y - h) + 2 l^{2} [1 - \frac{\cosh (\frac{2 y - h}{2 l})}{\cosh (\frac{h}{2 l})}]}

(1a)

w = \frac{1}{2 μ} (\frac{\partial p}{\partial z} - \frac{\partial F_{m}}{\partial z}) {y (y - h) + 2 l^{2} [1 - \frac{\cosh (\frac{2 y - h}{2 l})}{\cosh (\frac{h}{2 l})}]}

(1b)

Where $l = \sqrt{η / μ}$ , p is the oil-film pressure, $μ$ and $η$ are the classic shear viscosity of couple-stress fluids and its material constant respectively. Due to the elastic deformation of the liner,⁵ $h = c + e \cos θ$ should be written as $h = c + e \cos θ + δ$ , where $δ = \frac{pt}{E} (1 - k^{2})$ , $t$ is Bearing liner thickness, and $k$ is the Poisson’s ratio.

Introducing the velocity components formulas of $u$ and $w$ into the continuum theory function and then integrating with respect to y, the modified generalized Reynolds formula of magnetized journal bearings with squeeze dynamic action and with the elastic deformation of the liner based on magnetic fluids supporting force and its couple stress with the axial magnetic field configuration¹⁴ can be written as follows.

\begin{matrix} \frac{\partial}{\partial x} (g (h, l) \frac{\partial p}{\partial x}) + \frac{\partial}{\partial z} (g (h, l) \frac{\partial p}{\partial z}) = \\ 6 μ U \frac{\partial (h)}{\partial x} + 12 μ \frac{\partial (h)}{\partial t} + \frac{\partial}{\partial z} (g (h, l) \frac{\partial F_{m}}{\partial z}) \end{matrix}

(2)

Here $g (h, l) = h^{3} - 12 l^{2} (h - 2 l \tanh (\frac{h}{2 l}))$ . As $l$ equals to zero, $g (h, l)$ is written as $h^{3}$ . Therefore, the formula (2) is changeable and used to the case of magnetic Newtonian fluids.

For obtaining dimensionless form, rewrite $ω^{*} = ω - 2 (\frac{d ϕ}{dt})$ and introduce the following variables into the formula of (2):

\begin{matrix} θ = {\frac{x}{R}}_{b}, Z = \frac{z}{B}, λ = \frac{B}{2 R_{b}}, H = \frac{h}{c}, \\ L = \frac{l}{c}, P = \frac{p {(c / R_{b})}^{2}}{6 μ ω^{*}}, q = \frac{2}{ω^{*}} \frac{d ε}{dt}, H_{m} = \frac{h_{m}}{h_{m 0}} \end{matrix}

(3)

Thus, formula (2) considering the elastic deformation of the liner becomes a dimensionless form:

\begin{matrix} \frac{\partial}{\partial θ} (G (H, L) \frac{\partial P}{\partial θ}) + {(\frac{1}{2 λ})}^{2} \frac{\partial}{\partial Z} (G (H, L) \frac{\partial P}{\partial Z}) = \\ - ε \sin θ + q \cos θ - C_{e} \frac{\partial P}{\partial θ} \\ + \frac{1}{6} γ \frac{\partial}{\partial Z} (G (H, L) H_{m} \frac{\partial H_{m}}{\partial Z}) \end{matrix}

(4)

Here $H = 1 + ε \cos θ + C_{E} P$ , $C_{E} = \frac{6 μ UR ω^{*} t (1 - ν^{2})}{c^{3} E}$ is the elastic deformation coefficient, $ω$ is journal angular velocity, $ω^{*}$ is the effective angular velocity, $R_{b}$ is the radius of bearing, $q$ is the squeeze dynamic action parameter, $γ = \frac{μ_{0} X_{m} {(h_{m 0})}^{2} c^{2}}{μ ω B^{2}}$ is the magnetic fluids supporting force parameter and $h_{m 0} = h_{mc}$ , $G (H, L) = H^{3} - 12 L^{2} H + 24 L^{3} \tanh (\frac{H}{2 L})$ .

The oil-film pressure with boundary conditions can be expressed as:

\begin{matrix} {P (θ, Z) |}_{Z = \pm 1} = 0, {\frac{\partial P}{\partial Z} (θ, Z) |}_{Z = 0} = 0 \\ {P (θ, Z) |}_{θ = 0, 2 π} = 0, P (θ^{*}) = \frac{\partial P}{\partial θ} (θ^{*}) = 0 \end{matrix}

(5)

Where $θ^{*}$ represents the position of the film rupture or the beginning of the cavitation region.

Characteristics of the bearing

Loading capacity

By integrating the oil-film pressure on the magnetic supporting journal bearing, the loading components can be acquired by

\begin{matrix} S_{p} = 2 \int_{0}^{2 π} \int_{0}^{0.5} P \cos θ d θ dZ, \\ S_{V} = 2 \int_{0}^{2 π} \int_{0}^{0.5} P \sin θ d θ dZ \end{matrix}

(6)

Thus, the dimensionless magnetic loading capacity ( $S$ ) and its attitude angle ( $ψ$ ) can be written as:

S = \sqrt{{S_{p}}^{2} + {S_{V}}^{2}},

ψ = \tan^{- 1} (\frac{S_{V}}{S_{p}})

(7)

Force of friction and its coefficient

The shear stress¹⁷ on the moving journal surface is:

τ = {μ \frac{\partial u}{\partial y} |}_{y = h} - {η \frac{\partial^{3} u}{\partial y^{3}} |}_{y = h}

(8)

From equation (8a):

τ = μ \frac{U}{h} + \frac{h}{2} (\frac{\partial p}{\partial x} - \frac{\partial F_{m}}{\partial x})

(9)

Thereafter, the force of friction is developed by integrating the formula (9):

F = 2 \int_{0}^{0.5} \int_{0}^{2 π} (\frac{1}{H} (1 + \frac{c \overset{\cdot}{ε} \sin θ - e \overset{\cdot}{φ} \cos θ}{ω R_{b}}) + 3 H \frac{\partial P}{\partial θ}) d θ dZ

(10)

The modified coefficient of friction can be written as follows.

C_{f} = \frac{F}{S} \frac{c}{R_{b}}

(11)

Side leakage

By integrating formula (1b), the side leakage flow of the journal bearing across its end segment can be acquired as follows.

q_{E} = 2 \int_{0}^{2 π} {- \frac{g (h, l)}{12 μ} (\frac{\partial p}{\partial z} - \frac{\partial F_{m}}{\partial z}) |}_{z = - \frac{B}{2}} R_{b} d θ

(12)

Thereafter, the dimensional side leakage is obtained as:

Q = 2 \int_{0}^{2 π} {- G (H, L) \cdot (\frac{1}{4 λ^{2}} \frac{\partial P}{\partial Z} - \frac{1}{6} γ H_{m} \frac{\partial H_{m}}{\partial Z}) |}_{Z = 0.5} d θ

(13)

Where Q = \frac{2 q_{E}}{B R_{b} c ω}

Method of solution

According to the finite difference scheme (seen as in Figure 2), the formula (4) can be solved numerically. The domain of solution is divided into the circumferential direction with $m - 1$ intervals and the length direction with $n - 1$ intervals. Thus, the intersections with the dividing lines resulted in producing a mesh size of $m \times n$ points. The modified Reynolds formula (4) by using the central difference method for derivatives can be written as:

a_{1} P_{i, j} - a_{2} P_{i + 1, j} - a_{3} P_{i - 1, j} - a_{4} P_{i, j + 1} - a_{5} P_{i, j - 1} = f_{i}

(14)

Figure 2.

Difference meshed grid.

Where

\begin{matrix} a_{1} = \frac{G_{i + (1 / 2), j} + G_{i - (1 / 2), j}}{Δ θ^{2}} + \frac{G_{i, j + (1 / 2)} + G_{i, j - (1 / 2)}}{4 λ^{2} Δ Z^{2}} \\ a_{2} = \frac{G_{i + (1 / 2), j}}{Δ θ^{2}}, a_{3} = \frac{G_{i - (1 / 2), j}}{Δ θ^{2}} \\ a_{4} = \frac{G_{i, j + (1 / 2)}}{4 λ^{2} Δ Z^{2}}, a_{4} = \frac{G_{i, j - (1 / 2)}}{4 λ^{2} Δ Z^{2}} \\ f_{i, j} = ε \sin θ_{i} - q \cos θ_{i} + \frac{1}{2} C_{e} \frac{P_{i + 1, j} - P_{i - 1, j}}{Δ θ} \\ - \frac{1}{6} γ G_{i, j} (\frac{\partial^{2} H_{m}}{\partial Z^{2}} + {(\frac{\partial H_{m}}{\partial Z})}^{2}) \\ - \frac{1}{6} γ G_{i, j + (1 / 2)} (\frac{1}{2 Δ Z} H_{m} \frac{\partial H_{m}}{\partial Z}) \\ + \frac{1}{6} γ G_{i, j - (1 / 2)} (\frac{1}{2 Δ Z} H_{m} \frac{\partial H_{m}}{\partial Z}) \end{matrix}

(15)

$Δ θ$ in circumferential and $Δ Z$ in axial are mesh size in sequence. The difference of formula (14) is calculated and the profile of the oil-film pressure can be acquired by Gauss-Seidel methods with under relaxation. The iterative process with the formula (16) can be stopped once the difference value (t) of the successive iteration is not more than the error precision value of 10⁻⁷.

t = \frac{\sum_{i = 1}^{m} \sum_{j = 1}^{n} | P_{i, j}^{- k} - P_{i, j}^{- (k - 1)} |}{\sum_{i = 1}^{m} \sum_{j = 1}^{n} | P_{i, j}^{- k} |} \leq 10^{- 7}

(16)

Results and discussion

According to the solutions of the formulas above, the effects of magnetic fluids supporting force and couple stress on the lubrication characteristics of magnetized journal bearing with the elastic deformation of liner can be analyzed. The magnetic fluids supporting force parameter ( $γ$ ), couple-stress factor (L*), the elastic deformation coefficient (Ce), structure parameters, and working conditions of the bearings are presented in the Table 1 are derived from actual data and used as follows.

Table 1.

Parameters.

Parameters	Range taken value
Bearing length, B	2.8 × 10⁻² m
Bearing radius, $R_{b}$	1.4 × 10⁻² m
Radial clearance, c	2.8 × 10⁻⁴ m
Ratio of width to diameter of bearing, $λ$	1.0
Journal speed, n	2500 RPM
Derivative of eccentricity ratio, $\overset{\cdot}{ε}$	0.1(q = 1.0,0.1,0.5), 0(q = 0.0)
Derivative of attitude angle, $\overset{\cdot}{φ}$	130.8(q = 1.0), 130.7(q = 0.5), 129.9(q = 0.1)
Magnetic fluids supporting force parameter, $γ$ ^18,19	0.0, 0.05, 0.1, and 0.2
Couple-stress factor L*^20–22	0.0, 0.2, 0.4
Elastic deformation coefficient Ce	0.0, 0.3, 0.6
Magnetic fluid viscosity, $μ$	1.1 × 10⁻⁴ Pa · s
Free space permeability, $μ_{0}$ ^8,23	4π × 10⁻⁷ H/m
Susceptibility of magnetic fluids, $X_{m}$	2
Magnetic field strength, $h_{mc}$	2 × 10³ A/m
Magnetic field distribution parameter, β¹⁴	0.8

Figure 3 exhibits that the maximum dimensionless pressure peak ( $P_{max}$ ) increases with increase in γ. Under constant conditions of q = 0.1, L* = 0.2, Ce = 0.3, ε = 0.1, λ = 1.0 and β = 0.8, it is evident that $P_{max}$ for magnetized journal bearings obviously increase with increase in magnetic fluids supporting force (γ) by about 5.5%, 16.7%, and 38.1% for $γ$ = 0.05, $γ$ = 0.1 and $γ$ = 0.2 in comparison with magnetized journal bearings without magnetic fluids supporting force (γ = 0.0) respectively. This phenomenon is mainly due to the magnetic fluids supporting force generated by the viscosity increase of magnetic fluids under the applied magnetic field, which increases $P_{max}$ . The larger the magnetic fluids supporting force parameter, the bigger maximum dimensionless pressure peak.

Figure 3.

Variation of oil-film pressure configuration with various γ (Calculating constants: $γ$ = 0.0→0.2, q = 0.1, L* = 0.2, Ce = 0.3, and ε = 0.1): (a) γ = 0.0, (b) γ = 0.05, (c) γ = 0.1, and (d) γ = 0.2.

Figure 4 depicts that maximum dimensionless pressure peak ( $P_{max}$ ) obviously increases with increase in L*. Under constant conditions of $γ$ = 0.0, 0.2, q = 0.1, Ce = 0.3 and ε = 0.1, it is evident that $P_{max}$ for magnetized journal bearing obviously increase with increase in couple stress (L* ≠ 0.0) by about 36.7% and 90.2% for L* = 0.2 and L* = 0.4 in comparison with magnetized journal bearing with no couple stress (L* = 0.0) respectively. It can be inferred that the additive with polymer long chain can improve the lubricating oil-film pressure of magnetized journal bearing, so the bearing capacity increases and the friction coefficient decreases. The longer the chain length of the additive in lubricant, the greater the influence of couple stress.

Figure 4.

Variation of dimensionless centerline pressure (P) at the direction of circumference with various L* (Calculating constants: $γ$ = 0.0, 0.2, q = 0.1, Ce = 0.3, and ε = 0.1).

Figure 5 demonstrates that maximum dimensionless pressure peak ( $P_{max}$ ) obviously decreases with increase in Ce. Under constant conditions of $γ$ = 0.0, 0.2, q = 0.1, L* = 0.2 and ε = 0.1, it is evident that $P_{max}$ for magnetized journal bearing obviously decreases with increase in the elastic liner(Ce ≠ 0.0) by about 8.5% and 15.8% for Ce = 0.3 and Ce = 0.6 in comparison with magnetic journal bearing with the rigid liner (Ce = 0.0) respectively. Under load conditions, the elastic deformation of journal bearing will affect the clearance space geometry of bearing, which makes its performance different from that of rigid bearing.

Figure 5.

Variation of dimensionless centerline pressure (P) at the direction of circumference with various Ce (Calculating constants: $γ$ = 0.0, 0.2, q = 0.1, L* = 0.2 and ε = 0.1).

In this paper, based on the rigid liner (Ce = 0.0) and the elastic liner (Ce = 0.6), the effects of different magnetic fluids supporting force parameter $γ$ and couple-stress factor L* on the lubrication characteristics (S, $Ψ$ , C_f (R/c), and Q) of journal bearing affected by magnetic fluids are studied as follows.

Figure 6(a) shows that dimensionless loading capacity (S) of magnetized journal bearing affected by magnetic fluids and its supporting force ( $γ$ ≠ 0.0) slightly increases with increase in $γ$ . This is because the magnetic particles in the magnetic fluid are magnetized and gathered under the action of the external magnetic field, and the static magnetic force is generated in the magnetic fluid, which provides the support force for the journal bearing. Therefore, the larger the parameter of the magnetic fluid support force, that is, the greater the support force provided by the static magnetic force, the greater the maximum dimensionless bearing capacity. Figure 6(b) exhibits that S of journal bearing affected by magnetic fluids with couple stress (L* ≠ 0.0) clearly increase in comparison with that of journal bearing affected by magnetic fluids with no couple stress (L* = 0.0) with increase in L* particularly in the high ε. When the couple-stress factor increases, the increase of magnetic fluids supporting force parameter will make the increase of dimensionless loading capacity more obvious. This phenomenon is mainly caused by the enhancement of the molecular chain-length size effect of the additive in the magnetic fluids lubricant.

Figure 6.

Variation of dimensionless loading capacity (S) with eccentricity ratio (ε) for various $γ$ and L* (Calculating constants: L* = 0.0, 0.2 and ε = 0.1→0.8): (a) The impact of $γ$ at q = 0.1, L* = 0.2 and (b) The impact of L* at q = 0.1, $γ$ = 0.2.

Figure 6 also illustrates that S of journal bearing affected by magnetic fluids with rigid liner (Ce = 0.0) is clearly higher than those of journal bearing affected by magnetic fluids with the elastic liner (Ce = 0.6) especially in the high ε, which can be inferred that journal bearings affected by magnetic fluids with the elastic liner (Ce = 0.6) weakens the increase of S.

Figure 7 shows that attitude angle ( $ψ$ ) firstly decreases and then increases with increase in ε for various $γ$ and L*, which is due to the existence of q = 0.1,¹⁴ and it can be also seen from equations (4) and (7).

Figure 7.

Variation of attitude angle ( $ψ$ ) with eccentricity ratio (ε) for various $γ$ and L* (Calculating constants: Ce = 0.0, 0.6 and ε = 0.1→0.8): (a) The impact of γ at q = 0.1, L* = 0.2 and (b) The impact of L* at q = 0.1, γ = 0.2.

Figure 7(a) illustrates that magnetized journal bearing affected by magnetic fluids with supporting force has a lower ψ in comparison with that of journal bearing affected by magnetic fluids with no supporting force with increase in $γ$ especially in the low ε, which mainly results from the narrower cavitation zone caused by the applied magnetic field.¹⁴ Figure 7(b) exhibits that ψ of magnetized journal bearing affected by magnetic fluids with couple stress(L* ≠ 0.0) obviously increase than that of magnetized journal bearing affected by magnetic fluids with no couple stress (L* = 0.0) by increasing L* particularly in the moderate and high ε. It can be inferred that journal bearing with the elastic liner (Ce = 0.6) affected by magnetic fluids can strengthen the great stability of journal bearing particularly in the moderate and high ε.

Figure 8(a) illustrates that modified coefficient of friction (Cf (R/c)) of magnetized journal bearing affected by magnetic fluids with supporting force ( $γ$ ≠ 0.0) obviously decreases in comparison with magnetized journal bearing affected by magnetic fluids without supporting force ( $γ$ = 0.0) with increase in $γ$ particularly in the low ε. The decrease in Cf (R/c) above caused by the increase in oil-film loading capacity results from magnetic fluids supporting force and meanwhile almost constant value of force of friction with regard to the formula (11).

Figure 8.

Variation of the modified coefficient of friction C_f (R/c) with eccentricity ratio ε for various $γ$ and L* (Calculating constants: Ce = 0.0, 0.6 and ε = 0.1→0.8): (a) The impact of $γ$ at q = 0.1, L* = 0.2 and (b) The impact of L* at q = 0.1, $γ$ = 0.2.

Figure 8(b) indicates that C_f (R/c) of magnetized journal bearing affected by magnetic fluids with couple stress (L* ≠ 0.0) decrease significantly than that of magnetized journal bearing affected by magnetic fluids with no couple stress (L* = 0.0) with increase in L* particularly in the low ε. The results above are mainly due to the increase of ε and L* leading to a significant increase of effective loading capacity, which leads to the decrease of C_f (R/c). Furthermore, the increase of $γ$ also reduces the friction coefficient to a certain extent.

Figure 8 also exhibits that there exists a lower C_f (R/c) of magnetized journal bearing affected by magnetic fluids with the rigid liner (Ce = 0.0) in contrast to magnetized journal bearing affected by magnetic fluids with the elastic liner (Ce = 0.6) particularly in the low ε.

Figure 9 shows that dimensionless side leakage flow (Q) increases with increase in ε for various $γ$ and L*. Figure 9(a) illustrates that magnetized journal bearing affected by magnetic fluids with supporting force ( $γ$ ≠ 0.0) exists a lower Q in contrast to those of magnetized journal bearing affected by magnetic fluids without supporting force ( $γ$ = 0.0) with increase in $γ$ increase particularly in the high ε, which caused by the sealing effect of magnetized journal bearing affected by magnetic fluids, and there has higher Q due to the enhancement of dynamic pressure effect and its extrusion effect when $γ$ = 0.

Figure 9.

Variation of dimensionless side leakage flow (Q) with eccentricity ratio (ε) for various $γ$ and L* (Calculating constants: Ce = 0.0, 0.6 and ε = 0.1→0.8): (a) The impact of $γ$ at q = 0.1, L* = 0.2 and (b) The impact of L* at q = 0.1, $γ$ = 0.2.

Figure 9(b) indicates that magnetized journal bearing affected by magnetic fluids with couple stress (L* ≠ 0.0) exists a lower Q than those of magnetized journal bearing affected by magnetic fluids without couple stress (L* = 0.0) with increase in L* particularly in the high ε, which caused by hydrodynamic effect increasing with increase in L*. In addition, Figure 9 also exhibits that there exists a lower Q of magnetized journal bearing with the elastic liner (Ce = 0.6) affected by magnetic fluids in contrast to magnetized journal bearing with the rigid liner (Ce = 0.0) affected by magnetic fluids particularly in the high ε.

The magnetized magnetic fluid journal bearing can meanwhile achieve lubrication and certain sealing effect. Continuous and stable lubrication can be effectively gotten through a small amount of magnetic fluid medium to reduce the side leakage flow.

Conclusion

The modified Reynolds formula of journal bearing with the elastic deformation of liner is derived by taking into account the presence of magnetic fluids supporting force and couple stress due to the existence of applied magnetic field and magnetic fluids as non-Newtonian lubricants filled with micro-additives. The following results are shown as.

This study reveals that maximum dimensionless pressure peak ( $P_{max}$ ) of journal bearing affected by magnetic fluids increases with increase in magnetic fluids supporting force parameter ( $γ$ ), couple-stress factor (L*), with decrease in elastic deformation coefficient (Ce) of the liner. $P_{max}$ for journal bearing affected by magnetic fluids with supporting force ( $γ$ ≠ 0.0) is larger as compared to that for journal bearing affected by magnetic fluids without supporting force ( $γ$ = 0.0) by increasing about 5.5%, 16.7%, and 38.1% at $γ$ = 0.05, $γ$ = 0.1, and $γ$ = 0.2 in sequence; $P_{max}$ for journal bearing affected by magnetic fluids with couple stress (L* ≠ 0.0) is clearly larger as compared to that for magnetic journal bearing without couple stress (L* = 0.0) by increasing about 36.7% and 90.2% at L* = 0.2 and L* = 0.4 in sequence; $P_{max}$ for magnetized journal bearing with the elastic liner(Ce ≠ 0.0) is larger as compared to that for magnetic journal bearing with the rigid liner (Ce = 0.0) by increasing about 8.5% and 15.8% at Ce = 0.3 and Ce = 0.6 in sequence.

The loading capacity (S) increases with increase in $γ$ , L* and with decrease in Ce; the attitude angle ( $ψ$ ) increase with decrease in $γ$ , with increase in L* and with decrease in Ce; C_f(R/c) decreases with increase in $γ$ , L* and with decrease in Ce; Q decreases with increase in $γ$ , with decrease in L* and with increase in Ce.

The existence of elastic deformation coefficient (Ce) weaken the effects of γ, L* on the increase of $P_{max}$ , $ψ$ , S and the decrease of C_f (R/c), Q.

It can be gotten that the lubrication characteristics of journal bearing with the rigid liner affected by magnetic fluids considering the effects of $γ$ , L* is better as compared to that of journal bearing with the elastic liner affected by magnetic fluids considering the impacts of $γ$ , L*.

Footnotes

Appendix

Handling Editor: Chenhui Liang

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was financially supported by Science and Technology Project Founded by the Education Department of JiangXi Province in China (ID: GJJ180960); National Natural Science Foundation of PR China for the financial support (ID: 51765044); The State Scholarship Found Project of China Scholarship Council in 2019(ID: 201908360023); Natural Science Foundation of JiangXi Province in China (ID: 20212BAB204038, 20161BAB216105).

ORCID iD

Chunxia Xu

References

Khatir

Bouchetara

Djafri

, et al. Influence of balancing of internal combustion engines on the operating conditions of hydrodynamic bearings. J Mech Sci Technol 2017; 31: 4579–4588.

Shao

Liu

, et al. Analysis of thermoelastohydro- dynamic characteristics of journal misaligned engine main bearings. Chin J Mech Eng 2015; 28: 511–520.

Liu

Zhang

Bader

, et al. Scale and contact geometry effects on friction in thermal EHL: twin-disc versus ball-on-disc. Tribol Int 2021; 154: 106694.

Albahrani

Philippon

Vergne

, et al. A review of in situ methodologies for studying elastohydrodynamic lubrication. Proc Inst Mech Eng J: J Eng Tribol 2016; 230: 86–110.

Chetti

Zouggar

. Steady-state performance of a circular journal bearing lubricated with a non-Newtonian fluid considering the elastic deformation of the liner. Proc Inst Mech Eng J J Eng Tribol 2019; 233: 1389–1396.

Usov

. Elastohydrodynamic problem for journal sliding bearing under reciprocating motion. J Friction Wear 2016; 37: 204–212.

Patel

Vakharia

Deheri

. A study on the performance of a magnetic-fluid-based hydrodynamic short journal bearing. ISRN Mech Eng 2012; 2012: 603460.

Hsu

Chen

Chiang

, et al. Lubrication performance of short journal bearings considering the effects of surface roughness and magnetic field. Tribol Int 2013; 61: 169–175.

Galda

Sep

Olszewski

, et al. Experimental investigation into surface texture effect on journal bearings performance. Tribol Int 2019; 136: 372–384.

10.

Urreta

Aguirre

Kuzhir

, et al. Actively lubricated hybrid journal bearings based on magnetic fluids for high-precision spindles of machine tools. J Intell Mater Syst Struct 2019; 30: 2257–2271.

11.

Nada

Abdel-Jaber

Abdo

. Thermal effects on hydrodynamic journal bearings lubricated by magnetic fluids with couple stresses. In: 2nd International conference on energy engineering, Egypt, 27–29 December 2010.

12.

Das

. A study of optimum load-bearing capacity for slider bearings lubricated with couple stress fluids in magnetic field. Tribol Int 1998; 31: 393–400.

13.

Nada

Osman

. Static performance of finite hydrodynamic journal bearings lubricated by magnetic fluids with couple stresses. Tribol Lett 2007; 27: 261–268.

14.

. Influence of magnetic fluids’ cohesion force and squeeze dynamic effect on the lubrication performance of journal bearing. Adv Mech Eng 2017; 9: 1–13.

15.

Mokhiamer

Crosby

El-Gamal

. A study of a journal bearing lubricated by fluids with couple stress considering the elasticity of the liner. Wear 1999; 224: 194–201.

16.

Laouadi

Lahmar

Bou-Saïd

, et al. Analysis of couple-stress effects in gas foil bearings using the V. K. Stokes micro-continuum theory. Lubrication Sci 2018; 30: 401–439.

17.

Lin

, et al. Squeeze film characteristics of parallel circular disks lubricated by ferrofluids with non-Newtonian couple stresses. Tribol Int 2013; 61: 56–61.

18.

Shah

Bhat

. Ferrofluid squeeze film in a long journal bearing. Tribol Int 2004; 37: 441–446.

19.

Osman

Nada

Safar

. Effect of using current-carrying-wire models in the design of hydrodynamic journal bearings lubricated with ferrofluid. Tribol Lett 2001; 11: 61–70.

20.

Lin

Chu

, et al. Combined effects of piezo-viscous dependency and non-Newtonian couple stresses in wide parallel-plate squeeze-film characteristics. Tribol Int 2011; 44: 1598–1602.

21.

Naduvinamani

Hanumagowda

Siddangouda

. Effect of surface roughness on magneto-hydrodynamic couple-stress squeeze film lubrication between circular stepped plates. Lubrication Sci 2012; 24: 61–74.

22.

Wang

Zhu

Wen

. On the performance of dynamically loaded journal bearings lubricated with couple stress fluids. Tribol Int 2002; 35: 185–191.

23.

Rao

Rani

Nagarajan

, et al. Analysis of porous layered journal bearing lubricated with ferrofluid. Appl Mech Mater 2016; 819: 474–478.