Performance and characterization of dynamically-loaded engine bearings with provision for misalignment

Accepted Manuscript Performance and characterization of dynamically-loaded engine bearings with provision for misalignment J.Y. Jang, M.M. Khonsari PII:

S0301-679X(18)30484-5

DOI:

10.1016/j.triboint.2018.10.003

Reference:

JTRI 5422

To appear in:

Tribology International

Received Date: 28 August 2018 Revised Date:

26 September 2018

Accepted Date: 5 October 2018

Please cite this article as: Jang JY, Khonsari MM, Performance and characterization of dynamicallyloaded engine bearings with provision for misalignment, Tribology International (2018), doi: https:// doi.org/10.1016/j.triboint.2018.10.003. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Performance and characterization of dynamically-loaded engine bearings with provision for misalignment

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J. Y. Jang and M. M. Khonsari ∗ Department of Mechanical and Industrial Engineering 3283 Patrick Taylor Hall Louisiana State University Baton Rouge, LA 70803 USA

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Abstract The influence of misalignment on a dynamically-loaded engine bearing is investigated using a mass-conserving cavitation algorithm. Whereas in statically loaded bearings misalignment can be conveniently characterized using two parameters known as the degree of misalignment DM and the misalignment angle α, an alternative approach is needed when the load changes direction. In this paper, the complication is treated by representing the severity of misalignment using two deflection angles θx and θy since these deflection angles remain constant in an engine bearing during operation. Effects of the variation of load, rotating speed, oil viscosity, supply pressure, journal mass and deflection angles are presented.

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Keywords: Misalignment; dynamic-loaded engine bearings; mass-conservative cavitation

∗

Corresponding author ([email protected]), V. 225.578.9192

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1. Introduction The assumption of the perfectly aligned bearing is an idealization that is hard to achieve in practice due to a multiplicity of factors ranging from imperfect manufacturing to installation errors to external elements that come about during the bearing’s lifetime. Simply put, if the shaft and the bushing axes do not remain parallel, then the protective film thickness at one end of the bearing will suffer and if it falls below a certain threshold, then surface-to-surface contact becomes imminent with looming failure. It is thus no surprise that rich volumes of tribology literature are devoted to the understanding the nature of misalignment and its effect on bearing performance. The historical viewpoint with a comprehensive review of the subject matter and ample references is given in Jang and Khonsari [1].

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Misalignment greatly complicates the analytical formulation and performance predictions even in the case of a hydrodynamically-lubricated, plain journal bearing where the load is constant in both magnitude and direction. In this paper, we turn our attention to dynamically-loaded engine bearings where the analysis is far more complex. In contrast to the stationary-loaded bearings where the minimum film thickness remains invariant and stationary, the behavior of an engine bearing is time-dependent since the direction of the load—and consequently the location minimum film thickness—changes with time. This is particularly crucial in marine diesel engines with large crankshafts with series of crank webs and pins along their lengths. Misalignment of such long crankshafts can critically cause excessive wear and serious damage to the engine with the very costly repair.

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It is appropriate to begin with some of the pertinent publications on this subject notable among which is the work of Goenka [2] who in 1984 developed a finite element program to describe the transient behavior of a misaligned journal bearing under the dynamic loading. The program had the capability of treating full or partial arc bearing with a different form of oil supply arrangements such as a hole or oil-feed grooves. Later, in 2002, Lahmar et al. [3] analyzed the effect of static misalignment on the dynamic behavior of the main crankshaft bearing using mobility method originally developed by Booker [4]. They showed that locus of the shaft center and the associated minimum film thickness is altered even at low values of misalignment parameters. Further, they illustrated that a circumferential groove may deteriorate the performance by causing metal-to-metal contact. Elrod [5] in 1981 developed a so-called massconservative algorithm that automatically accounts for film rupture due to cavitation by introducing a switch function. In 1989, Vijayaraghavan and Keith [6] modified the cavitation algorithm using a differencing procedure. They took into account the film rupture and reformation. Bouyer and Fillon [7] experimentally investigated effects of misalignment on the performances of the hydrodynamic journal bearings. Sun et al. [8] studied the misalignment caused by shaft deformation and their results are validated by experiments. They showed that the misalignment makes the obvious increment in the maximum pressure and the reduction in the minimum film thickness. More recently, Boedo [9] investigated a dynamically-loaded misaligned bearing with a hybrid mobility method. These researchers [2-3,9] take into account the influence of cavitation with film rupture. However, their models do not satisfy the mass conservation. Nikolakopoulos et al. [10] used the Pareto analysis to optimize the geometric configuration of a misaligned journal bearing. They showed that a misaligned sleeve bearing can take advantage of a limited amount of wear. Mishra [11] investigated the effect of nonNewtonian behavior of lubricant on the performance of misaligned journal bearings using the

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power law model including thermal effects. Khonsari and Booser [12] in 2017 provided the causes and effects of the misalignment in bearings from the practical design point of view.

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The traditional Swift-Stieber boundary condition generally suffices for the steadily-loaded, aligned bearings. However, Swift-Stieber boundary condition does not satisfy the mass conserving near the cavitation. For the dynamically loaded misaligned bearings, this boundary condition is not valid anymore and the mass conservative cavitation algorithm is necessary. In this paper, we formulate and treat the problem of engine bearing misalignment taking into account the influence of cavitation (fluid rupture as well as reformation) using the cavitation algorithm developed by Elrod and Adams [13]. An extensive set of simulations are presented to investigate how a misaligned engine bearing is influenced by the variation of load, rotating speed, oil viscosity, supply pressure, journal mass and deflection angles.

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2. Theory Figure 1 shows the geometric configuration of a misaligned engine bearing and the appropriate nomenclature involved.

Y Y

WY

θ

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Y

MY

Front-end

Rear-end

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Oil Film

h(θ,z)

O C

C2

θy

C1

X

φo Front-end

α Z

Rear-end

MX

O

C1 C

C2

B

X WX

A

Journal Center Line L/2

L/2

Bushing Mid-plane

e Projection of Journal 3

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Figure 1. Configuration of an engine bearing with misaligned journal

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2.1 Film thickness expression The geometry of the film thickness in a bearing with a misaligned shaft can be expressed using the following relationship with provision for both axial and circumferential variations [14-15].  z 1 (1) h = C + eo cos(θ − φo ) + e′ −  cos(θ − α − φo )  L 2 where eo is the eccentricity in the mid-plane of the bearing, and φo represents the attitude angle between the line of centers and the Y-axis. The parameter e′ is the length of the projection of the misaligned journal on the mid-plane and L denotes the bearing length. The film thickness in a perfectly aligned bearing is a special case of Eq. (1) with e′ = 0 , and the moments MX and MY shown in Figure 1 disappear. The parameter α represents the misalignment angle between the line of centers and C1C2 , i.e. the line of the projected front and rear centers on the mid-plane.

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We define the following dimensionless parameters: y u h e x z R v , z= u= , v= , h = , εo = o , x = θ = , y = C C R L C uS h( x ) uS w u t η C2P C 2M , P= , η = , t = S and M = (2) ηi RuS R ηi R 2 L2uS ηi uS Using which the dimensionless film thickness becomes 1  (3) h = 1 + ε o cos(θ − φo ) + ε ′ z −  cos(θ − α − φo ) 2  ′ , where DM In Eq. (3), ε ′ represents the misalignment eccentricity ratio defined as ε ′ = DM ε max ′ is the is often referred to as the degree of the misalignment that ranges from 0 to 1. ε max maximum possible ε ′ determined from the following expression [16].

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w=

(

′ = 2 1 − ε o2 sin 2 α − ε o cos α ε max

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(4) The severity of the misalignment is determined by two independent parameters. If the direction of applied load remains invariant—as in the case of a stationary-loaded bearing—then one possibility for characterizing misalignment is to simply specify DM and α. However, in the case of dynamically loaded bearings these parameters vary with time and cannot be easily specified. The crankshaft assembly in large diesel engines and compressors consists of several crank-pins, crank-webs, and main journals. After some period of constant use, the crankshaft often becomes deflected, but the deflection generally remains constant during operation due to its heavy weight. In this situation, it is more suitable to describe the severity of misalignment via the deflection angles, θx and θy, in the horizontal and vertical planes, respectively. Two deflection angles θx and θy qualify as set of independent parameters to identify the misalignment of the crankshaft [17]. The bearing is considered to be perfectly aligned if the two deflection angles are both nil. The side view of the deflection angle θy is shown in Figure 1. The deflection angle θx can be shown in a similar way from the top view. In Figure 1, the deflection angles are positive when the center of the front-end is located at the right and top of the center of the rear-end. The lengths

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A and B are geometrically related the deflection angle: A = L tan θ x and B = L tan θ y . Therefore, the angle of misalignment can be computed from the deflection angles as follows:  tan θ x   − φo for θ y < 0 α = Tan −1  −  θ tan y    tan θ x   − φo + π for θ y > 0 α = Tan −1  − (5)  tan θ y  

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2.2 Mass-Conservative Reynolds equation The so-called generalized Reynolds equation that governs the pressure distribution is: ∂   F  ∂ ∂  ∂  ∂P  ∂P  (6) ρF2   +  ρF2   =  ρus  h − 1  + (ρh )  ∂x  F0  ∂t  ∂x  ∂z   ∂z  ∂x   where h 1 h y h 1  F  (7) F0 = ∫ dy , F1 = ∫ dy and F2 = ∫   y 2 − 1 y  dy 0 η 0 η 0 F0  η  Further modification of the equation is due to the mass-conservative cavitation algorithm of Elrod [5]. For this purpose, a parameter known as the fractional film content, Θ = ρ / ρ c , was

introduced with the corresponding pressure P = Pc + β ln Θ , so that the Reynolds equation is reformulated so that the unknown variable is Θ instead of P :  ∂  ∂ ∂  F  ∂ g β F2 ∂Θ  +  g β F2 ∂Θ  = uS Θ  h − 1   + ( Θh ) (8) ∂x  ∂z  ∂z  ∂x ∂x  F t ∂ 0   

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where β is the lubricant’s bulk modulus, ρc is the lubricant at the cavitation pressure Pc and us is the shaft speed. The parameter g(Θ) is a binary parameter referred to as the cavitation switch function defined as g = 1 when Θ ≥ 1 (9) g = 0 when Θ < 1 Equation (8) governs the pressure distribution in both in the full-film and cavitated region and the Elrod’s solution algorithm automatically satisfies the film rupture-and-reformation. The mathematical approach and the numerical scheme for the mass-conserve cavitation algorithm were successfully applied to the journal bearings under periodic loading by Vijayaraghavan and Brewe [18]. However, this algorithm is sensitive and often causes numerically instability when the switch function oscillates between 0 and 1 from one iteration to the next near the cavitation boundary [19]. Elrod and Adams [13] introduced the following so-called p-Θ formulation that reduces the numerical instability since the switch function is removed from the generalized Reynolds equation:  ∂   ∂P  ∂   ∂P  ∂  F1  ∂   + (Θh ) (10) F + F = u Θ h −      2 2 s  ∂x   ∂x  ∂z   ∂z  ∂ x  F ∂t 0  

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In the fluid-film region Θ=1. In the cavitated region, the pressure terms vanish —since the pressure remains constant— and Θ represents the fractional-film content. The dimensionless form of equation (10) is: ∂ ∂  3  ∂ P  1 ∂  3  ∂ P  ∂ (11) UΘ h + Θh h F 2  ∂ θ  + 2 h F 2   = ∂θ  ∂t   4Λ ∂ z   ∂ z  ∂θ  F1  L  . The where the aspect ratio and dimensionless speed are defined as Λ = and U = 1 −  2R h F 0   dimensionless viscosity functions are defined as: 1y 11 1 1  2 F 1  (12) F 0 = ∫ d y , F 1 = ∫ d y and F 2 = ∫   y − y d y 0η 0η 0 η F 0    Note that if the viscosity remains constant, i.e. isoviscous, η = 1 and the viscosity functions are simplified as follows. 1 1 (13) F 0 = 1 , F 1 = and F 2 = 2 12

)

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Discretization scheme The shear flow term (source term) in the full-film region is centrally differenced. In the cavitated cavitation region, the Reynolds equation changes from elliptic to parabolic partial differential equation and upwind finite differencing is used for the source term. In the discretized form, the Reynolds equation (11) reads

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The P Equation

n n n  Sin+1 / 2, k + Sin−1 / 2, k  Sin−1 / 2, k  n 1 Si , k +1 / 2 + Si , k −1 / 2  n  Si +1 / 2, k  n P − + 2 P  Pi, j +   2  i −1, k 2 2  i +1, k 4Λ (∆θ )2 ∆z   (∆θ )   (∆θ )  

 1 Sn  n  1 Sn  n +  2 i , k −1 /22  P i , k −1 +  2 i , k +1 2/ 2  P i , k +1  4Λ ∆ z   4Λ ∆ z  The Θ Equation

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n

( )

( ) (U Θh) − (U Θh) = n

n

i,k

i −1, k

∆θ

n

( )

(Θh) − (Θh) + n

n −1

i,k

i,k

(14)

∆t

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U i , j 1  U i−1, j  n n n−1 1 +  Θh i , j =    Θh i−1, j +   Θh i , j  ∆t   ∆ x ∆t   ∆x 

Sn  n Sn  n  1 Sn  n  1 Sn  n +  i −1 / 22,j  P i−1, j +  i +1 / 22,j  P i+1, j +  2 i , j −1 /22  P i , j −1 +  2 i , j +1 /22  P i , j +1  ∆ x   ∆ x   4Λ ∆ z   4Λ ∆ z 

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( )

( )

( )

(15)

n n  Sin+1 / 2, j + Sin−1 / 2, j 1 Si , j +1 / 2 + Si , j −1 / 2  n − + 2  Pi, j 2 2 4Λ ∆x ∆z  

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where n is the current and n-1 represents the previous time step. Parameter S is taken the average 3

between

two

adjacent

3

Si ±1/ 2,k =

i.e.

and

2

3

h i ,k ±1 F2 i ,k ±1 + h i ,k F2 i ,k 2

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Si ,k ±1/ 2 =

nodes,

3

h i ±1,k F2 i±1,k + h i ,k F2 i ,k

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The numerical solution procedure is as follows: (i) Recall the fractional film content and the pressure computed at the previous iteration. (ii) If Θ=1 or P> Pc, then compute the new pressure from equation (14). (iii) If P ≥ Pc , then set Θ=1. If P < Pc , then set P=Pc. (iv) If Θ < 1 or P ≤ Pc , then solve equation (15) for Θ. (v) Go to step (ii) for the next node.

2.3 Force balance The equations of motion are:

eɺtX − eɺtX− ∆t =0 ∆t (16) t t − ∆t ɺ ɺ e − e FYAppl + FYoil (etX , eYt , eɺtX , eɺYt ) − M J Y Y = 0 ∆t Appl Appl where MJ is the mass of journal. FX and FY are the applied loads in the x- and y directions.

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FXAppl + FXoil (etX , eYt , eɺtX , eɺYt ) − M J

FXoil and FYoil represent the reaction forces. These forces are functions of the translational journal

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velocity ( eɺtX & eɺYt ), instantaneous journal position ( etX & eYt ), and journal rotation. The reaction forces are evaluated by integrating the film pressure over the appropriate area. The last terms are the inertia forces of the journal. Typically, the inertia forces are relatively small compared to the other terms except when the load changes its direction.

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The equations of motion have four unknowns and therefore, two more equations are required. They are: etX = etX−∆t + eɺtX ∆t (17) eYt = eYt −∆t + eɺYt ∆t Inserting equation (17) into (16), the dimensionless form of the equations of force balance become Appl oil MJ t f X = F X + F X (εɺ Xt , εɺYt ) − εɺ X − εɺ Xt −∆ t = 0 ∆t (18) Appl oil M J f Y = F Y + F Y (εɺ Xt , εɺYt ) − εɺYt − εɺYt −∆ t = 0 ∆t u C 3M C 2F where the dimensionless mass is M J = S 4 J and the dimensionless force is F = . ηi R L ηi R 2 Lu S

(

)

(

)

( ) ( ) − (εɺ ) i

Equation (18) is solved iteratively by calculating the Jacobian matrix. Let ∆ εɺXt = εɺXt

( ) ( ) − (εɺ ) at the i iteration where (εɺ ) determined. ∆ (εɺ ) and ∆ (εɺ ) can be computed from i

and ∆ εɺYt = εɺYt

t Y

cr

t X

i

i

th

t Y

t X cr

( )

and εɺYt

cr

cr

t X

i

are the unknown roots to be

i

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∂ fX ∂ fX  i  ∂ εɺ t  f εɺ t i , εɺ t i  ɺYt  ∆ εɺ Xt  ∂ ε X X X Y   (19)  i  = − i t t v i ∂ ∂ f f Y   ∆ εɺ  Y    ɺ ɺ f ε , ( ε ) Y  Y   Y X  t   ∂ εɺ t  X ∂ εɺY  Solving the above equation by Gaussian elimination and updating results at (i+1)th iteration as follows.  εɺ t i +1   εɺ t i  ∆ εɺ t i  (20)  X i +1  =  X i  +  X i   εɺYt   εɺYt   ∆ εɺYt  The above process is repeated until both errors of estimated εɺX and εɺY are reduced below the tolerance. This method is very efficient, and the solution converges in less than ten iterations for most time steps. Table 1 shows that errors of εɺX and εɺY reduce quickly below the tolerance for the simulation of Asus et al. [20] at three loading during operating cycle, i.e., the beginning of the load cycle and two load peaks.

( ) ( )

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Table 2. Number of iterations for convergence of εɺX and εɺY Number of At beginning At first load peak At second load peak Iterations Error for εɺX Error for εɺY Error for εɺX Error for εɺY Error for εɺX Error for εɺY 0.1161 ×10-1 0.5767 ×10-3 0.2481 ×10-5 0.1332 ×10-6 0.1820 ×10-8 -

0.9527 0.2151 ×102 0.9516 0.5427 0.3696 0.6718 ×10-1 0.6512 ×10-2 0.2329 ×10-4 0.1313 ×10-5 0.4258 ×10-7

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0.3701×10-1 0.3471 ×10-2 0.1397 ×10-3 0.2699 ×10-5 0.4748 ×10-7 -

0.1706 0.2876 0.1959 0.9385 ×10-1 0.4004 ×10-1 0.9804 ×10-2 0.1033 ×10-2 0.2782 ×10-4 0.6987 ×10-6 0.1952 ×10-7

0.2139 0.3336 ×10-1 0.5126 ×10-2 0.1655 ×10-3 0.2869 ×10-5 0.5900 ×10-7 -

0.1366 0.1391 ×10-1 0.2099 ×10-2 0.7288 ×10-4 0.1417 ×10-5 0.2859 ×10-7 -

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2.4 Bearing performance parameters The instantaneous film loads and the bearing attitude angle are given by L 2π R  η R 2UL  1 2π FXoil = ∫ ∫ P sin θ dxdz =  i 2   ∫ ∫ P sin θ d xd z   0 0   C  0 0 oil Y

F

= −∫

L

0

∫

2π R

0

 ηi R 2UL   1 2π  P cos θ dxdz =    − ∫0 ∫0 P cos θ d xd z  2 C  

(F ) + (F ) = Tan ( F / F )

F oil =

ϕo

oil 2 X

−1

oil X

oil 2 Y

(21)

(22)

oil Y

The lubricant leakage flowrate is the sum of the flowrate exiting the bearing axially from the front end and the rear end.

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π ∫ (w

(

)

)

 1 2π h w − h w d xd y  (23) − w dxdy = − CRu S z =1 z =0  ∫0 ∫0  0 0 z =0 z =1 where w is the axial flow velocity that can be determined by solving the continuity equation. The expression for friction force is  L 2πR du η u RL  1 2π η du F =∫ ∫ η dxdz = i S d xd z  (24) ∫ ∫ 0 0 dy y =0 C  0 0 h d y y =0    As shown in Figure 1, the components of the moments are evaluated the following equations L 2πR  η u R 2 L2  1 2π  1  L  M X = ∫ ∫ PR z −  cos θ dθdz = i S 2 P z −  cos θ d xd z   ∫ ∫ 0 0 2 C 2  0 0   (25) 2 2 L 2πR  η i u S R L  1 2π  1  L  M Y = ∫ ∫ PR z −  sin θ dθdz =  ∫0 ∫0 P z − 2  sin θ d xd z  0 0 2 C2      The total moment and its direction angle are: h

2 R

M = M X2 + M Y2

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Qleak = ∫

(26)

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φM = Tan −1 (− M X / M Y ) Initial Conditions Advance Time

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Applied Load at Given Time

Film Thickness, h (EQ 1)

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Reynolds Equation, P & Θ (EQs 14 & 15)

Iterations for Jacobian Matrix

Jacobian Matrix

( ) & εɺ (t ) at ith iteration (EQ 19)

εɺ Xi t

i Y

Force Balance, fX & fY (EQ 18)

fX=0 & fY=0

No

Iterations for Force Balance

Yes Compute Bearing Parameters (EQs 21-26) n=n+1 End 9

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Figure 2. Computational flow chart

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2.5 Computational procedure The method of finite differences is employed to treat the Reynolds equation. The Reynolds equation is solved using the successive overrelaxation method. The overall solution scheme is as follows and illustrated in Figure 2. First, with given or guessed φo, εo, θx, θy, the film thickness is computed, and the applied load is determined at the current time. Second, the Jacobian matrix is computed to obtain the solutions of εɺ X and εɺY . For this purpose, the Reynolds equation and the bearing load are solved iteratively. When the force balance is satisfied with computed εɺ X and εɺY , the program proceeds to the next time step. The flow chart of the computational procedure is shown in Figure 2. The number of mesh points is 81 in the circumferential direction, 41 in the axial direction, and 21 across the film. The time step is 0.0001 second, and the error tolerance is 10-6 for both the pressure and the moving speeds. The error EΨ between two successive iterations is computed using 1 N Ψ now − Ψ old (27) EΨ = ∑ N i=1 Ψ now where N is the total number of nodes and Ψ represents the parameters to be solved.

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3. Results and discussion We begin by providing a set of validation results with comparison to existing literature, available for perfectly aligned bearings. Then we focus on misaligned journal bearing results. The first category of simulations pertains to a bearing with two load pulses representing the power strokes. These are shown in Section 3.2 for the aligned bearings and Sections 3.3-3.5 for the misaligned bearings. Table 2 shows the input data for two load pulses. The results of the engine bearing simulations subject to the load cycle of 720  are presented in Section 3.6. Table 2 shows the input data for load cycle of 720  . Unless specified otherwise, these data are used in all the simulations.

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Table 2. Input Data Shaft Radius, R 12.5 mm Bearing width, L 7.854 mm Clearance, C 20 µm Shaft Speed, N 4000 rpm Supply Pressure, Ps 347 kPa Ambient Pressure, Pa 0 kPa Cavitation Pressure, Pc 0 kPa 0.0075 Pa.s Viscosity, ηi Journal mass, MJ 3.21 Kg

3.1 Validation: Comparison with Ausas’ results for transient journal bearings

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To test the validation of the model, we begin by comparing the predictions with published results. Asus et al. in 2009 [20] developed an algorithm for treating the dynamics of lubrication problems based on the Elrod–Adams formulation of the Reynolds equation. The journal bearing is operating under the transient loads of typical automotive engines, which is essentially characterized by two impulses due to the power strokes. The applied load components during the load cycle are shown in the inset of Figure 3. The operating conditions are the same as those listed in Table 2. The oil is supplied from the circumferential groove ion the middle of the bearing.

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Fig. 3 shows the comparison of the eccentricity ratios between the present results and those of Ausas et al. during three load cycles (0.045 seconds). To compare, our results are transformed into coordinates used in Ausas’ results, i.e. X=-εy and Y=εx. The maximum dimensionless load is 0.01, which corresponds to 285.4 kN. Figure 3 shows that the present results are in good agreement with Ausas’ results. A very small discrepancy occurs when the gradient of the eccentricity ratios are near zero. One of the possible reasons is the bigger time step used in our simulations.

Figure 3. Locus of shaft center comparing to Ausas et al. results [20]

3.2 Benchmark solutions: Aligned dynamically loaded journal bearings Figure 4 shows the applied load variations in both x and y directions during the load cycle (360 degrees), where the time for one load cycle is tP=0.015 s. Using the time for one load cycle t p = 60 / N , the applied load during one load cycle used in the simulations [20] is given by Wx = 843.35e Wy = −2853.78e

(

(

−400 t / t p −0.5

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−400 t / t p −0.25 2

)2

− 2726.33e

(

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− 400 t / t p −0.5 2

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Figure 4 shows two peak loads at t = 0.00375 s (tP/4) and t = 0.0075 s (tP/2) during the load cycle. At the first peak, the load is 2.85 kN and the direction is downward for the 1x load. At the second peak, the load is 2.85 kN for the 1x load and the direction is θ=287.19o, i.e. 17.19o deviation from the first peak. To investigate how a drastic change in the applied load affects the performance, two more applied load at one-half, ½ x, and twice the original load, 2x, are also simulated. Figure 4 shows that changing the magnitude of the impulse load to (Wx/2, Wy/2) or (2Wx, 2Wy) does not alter the trend of the results. In what follows, we will examine the locus of the shaft center and the bearing performance parameters. Note that in simulations the weight of the journal is not considered as the applied load since the weight is much smaller than the applied load.

Figure 4. Various applied loads during the load cycle

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Shaft locus Figure 5 shows the movement of the bearing journal center within the clearance. It also shows the positions of the journal center at the beginning of the cycle as well as the positions of the maximum and minimum journal translating speed at 1x load. At the beginning of the cycle, the load is not applied to the journal, and the journal begins to move as a result of the reaction force due to the generated pressure. For the case of 1x load, at roughly t=0.002s the load is applied to the journal in the downward direction and the journal changes its direction. During the downward movement, the journal reaches to its maximum translating speed at t=0.003s where eɺmax = 55 mm/s. After the first peak load, again the journal changes its moving direction in the counterclockwise since the applied load decreases. It is shown that when the applied load is due to the journal weight only, the shaft travels in a circular path in the counterclockwise until it reaches to its steady-state position. After the second peak load, the journal changes its direction to downward direction again but quickly resumes its circular path. The position of the minimum translating speed ( eɺmin = 0.41 mm/s) is almost matching with the second load peak. In Figure 5, it is also shown that when the journal weight is considered without the consideration of the applied load, the radius of the trajectory is reducing with time and the journal rotates counterclockwise as

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it approaches to the position where the steady-state is satisfied. This is labeled “Journal weight Only” in Figure 5. The maximum eccentricity ratio εmax=0.891 at ½x load, εmax=0.931 at 1x load, and εmax=0.957 at 2x load, which implies that much higher load is required to reach the higher eccentricity ratio than the lower eccentricity ratio. At 2x load, the minimum film thickness is hmin=0.86 µm, which is corresponding to εmax=0.957. The roughness (r.m.s) of the bearing surface is generally σ=0.2-1.3 µm and the surface-to-surface contact generally occurs within the gap of 3σ. With a very smooth bearing (σ=0.2), the bearing will still operate since hmin=0.86 µm at 2x load is larger than 3σ. However, with a larger bearing surface roughness, there will be the surface-to-surface contact leading to the bearing failure. Therefore, the bearing surface finishing is very important in bearing design. If the thermal effect is considered, hmin would be reduced further due to the viscosity drop and, therefore, operating the bearing at 2x load is in danger even with a very smooth bearing (σ=0.2).

Figure 5. Movement of aligned journal center at various loads

Performance parameters Figure 6 shows the variations of the maximum pressure, friction force, eccentricity ratios and film thickness during the load cycle corresponding to Figure 5. The maximum hydrodynamic pressure increases with increasing the applied load, and pressure peaks form at the same time as the two applied load peaks without any delay. Note that the magnitude of the maximum pressure is more than doubled as the applied load is doubled.

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The results in Figure 6(b) show that the friction force generally increases with the applied load. However, the location of the two peaks in the friction force is delayed and does not occur at the same instant that load peaks are imposed. The lowest friction force is just before the first load peak where the eccentricity ratio is low and the moving speed is high. The smallest friction force is corresponding to the largest value of the minimum film thickness in Figure 6(d).

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The eccentricity ratios are also shown in Figure 6(c) as a function of time. Figure 6(c) shows two peaks in the eccentricity ratio εy just after the two peak loads. The magnitude of eccentricity ratios increases with the increasing the applied load. The difference in the magnitude of the eccentricity ratios due to the applied loads is more pronounced at high eccentricity ratios.

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Figure 6(d) shows the minimum film thickness at various loads during the cycle. The variation of the minimum film thickness is influenced by the location of the shaft and the direction of the load at a given time. The figure is important in determining the susceptibility of bearing to surface-to-surface contact when the film is at its lowest value. The minimum film thickness is at its highest values just before the first load peak and drops to it smallest value just after the second load peak at 0.0076 s.

Figure 6. Variations of aligned bearing parameters during load cycle at various loads Influence of journal mass, speed, supply pressure and inlet viscosity Figure 7 shows the movement of the bearing journal center within the clearance when subjected to different loads, speeds, supply pressure, and lubricant viscosities. Figure 7(a) shows the results of the variation of the journal mass. The effect of the journal mass on the trajectory of the journal

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center is clearly shown between the times for the maximum moving speed and the first load peak. At a high eccentricity ratio, the effect of journal mass is negligibly small. The same result was also reported in 1983 by Martin [21]. Figure 7(b) shows the results for the variation of the rotating speed. At a higher speed, the eccentricity ratio is lower since a hydrodynamic pressure generated is greater and the shaft runs on a thicker film. Note that the time for one load cycle varies with the rotating speed. The time for one load cycle is 0.03 s at 2000 rpm, and 0.015 s at 4000 rpm, respectively. Figure 7(c) shows the results for the variation of the supply pressure. The effect of the supply pressure on the trajectory of the journal center appears only at a low eccentricity ratio. Figure 7(d) shows how the oil viscosity influences the shaft locus. Similar to the rotating speed, the eccentricity ratio is less pronounced when the oil viscosity is large. As expected, a low viscosity oil generates a thinner film and much higher eccentric ratio results to generation the necessary pressure to support the load.

Figure 7. Movement of journal center in a dynamically loaded bearing at various operating conditions

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3.3 Effect of applied load in misaligned journal bearings

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Shaft locus In these simulations, the deflection angles θx and θy are chosen to investigate the effects of misalignment. Equation (28) is also used in the simulations for the misaligned journal bearings. Figures 8 show the movement of the misaligned bearing journal center in the mid-plane as well as on both front and rear ends when the deflection angle in the y-direction is θy=-0.0003 rad and the deflection angle in the x-direction is θx=0 rad. Figures 8 also show the movement of the aligned journal center in dotted lines for the comparison. Figure 8(a) shows that the locus of the misaligned journal center in the mid-plane is very close to the locus of the aligned journal center. The difference is clearer when the eccentricity ratio is higher and the applied load is larger. Similar trends are observed by Lahmar et al. [3] and Boedo [9] using non mass-conserving cavitation algorithms. Figure 8(b) shows the movement of the misaligned bearing journal center on the front end. It is shown that under the 2x load, the eccentricity ratio on the front end is εf=0.989 at 278.7o (near the first load peak) and εf=0.996 at 296.4o (near the second load peak), where the possibility of the metal-to-metal contact is very high. Therefore, it is important to consider the surface roughness to predict the performances of the misaligned journal bearing. Figure 8(c) shows the movement of the misaligned bearing journal center on the rear end, where the film thickness is large enough and safe from the metal-to-metal contact. Referring to Figure 8(d) the minimum film thickness occurs on the front end in the fourth quadrant and on the rear end in the first quadrant and indicative of damage due to metal-to-metal contact, particularly in the high eccentricity region of the 2x load.

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Figure 8. Movement of misaligned journal center at mid-plane at various loads (θx=0 and θy=0.0003 rad)

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Figure 9 shows the variations of the maximum pressure, moment, degree of misalignment DM and misalignment angle α at fixed θx=0 rad and θy=-0.0003 rad, which is corresponding to Figure 8. Similar to the aligned case, the maximum pressure in Figure 9(a) increases with increasing the applied load. Two pressure peaks also appear at the same time as two applied load peaks without any delay. Comparing to Figure 6, the maximum pressure for the misaligned journal is similar to that for the aligned journal when the load is small (low misalignment). However, it is much higher when the deflection angle is larger (high misalignment) since the local film thickness of the misaligned bearing is smaller than that of the aligned bearing. This is clearly visible in Figure 9(c), which shows the effect of the degree of misalignment. The second peak of the maximum pressure occurs at the front end where the film thickness is minimum. Figure 9(b) shows the moment generated due to the misalignment. The moment increases with increasing the load leading to a greater misalignment. The degree of misalignment increases when the eccentricity ratio increases since the deflection angles are fixed. The degree of misalignment also shows two peaks near the load peaks where the eccentricity ratio is high. 17

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Figure 9(d) shows the variation of the misalignment angle during the load cycle. The misalignment angle does not change significantly with the change of the applied load since θx rad and θy are fixed (see equation 5). It is shown that DM and α are varying significantly during operation at fixed θx rad and θy. It is worth noting again that in stationary loaded bearings one can specify DM and α for the severity of the misalignment since they remain constant, but in dynamically loaded bearings DM and α vary with time and, therefore, it would be important to formulate the problem in terms of θx rad and θy.

Figure 9. Variations of misaligned bearing parameters during load cycle at various loads (θx=0 and θy=-0.0003 rad) 3.4 Effect of deflection angle θy Figure 10 shows the variations of the maximum pressure, moment, degree of misalignment and misalignment angle at various deflection angles in the y-direction, θy. The deflection angle in the x-direction is fixed as θx=0 in these simulations, which implies that the crankshaft is deflecting in the vertical direction only. Figure 10(a) shows the locus of the journal center during the load cycle. The locus for small deflection angle (θy=0.0001 rad) does not show any difference in the

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journal locus in comparison with the locus for the aligned bearing. At higher angle (θy=0.0005 rad), the journal trajectory is influenced due to the much larger degree of misalignment. Note that the minimum film thickness, which occurs at one end of the journal, is generally much smaller at θy=0.0005 rad. In Figure 10(a), it is shown that at θy=0.0005 rad, the eccentricity ratio on the front end is εf=0.981 at 279.8o and εf=0.996 at 298.3o, where the metal-to-metal contact could occur. The eccentricity ratio on the rear end, εr, is smaller than εf since the deflection angle is negative.

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Figure 10(b) shows that the moment increases with increasing load. It is shown that two peaks in the moments occur at two load peaks. The moment increases with increasing the misalignment, while it is nil for the aligned bearings. Figure 10(c) shows how the degree of misalignment parameter DM changes with time. At a large deflection angle θy=0.0005 rad, the maximum degree of misalignment is DM=0.95 after the second load peak. Under this condition, metal-tometal contact is very likely with subsequent wear leading to eventual bearing failure. Figure 10(d) reveals that when θx=0, misalignment angles are not sensitive to the deflection angle in the y direction since the misalignment angles are only a function of the attitude angle when (see equation 5). The attitude angles are almost the same from the Figure 10(a).

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Figure 10. Variations of misaligned bearing parameters at various θy (θx=0)

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3.5 Effect of deflection angle θx Figure 11 shows the variations of the locus of the journal center, moment, degree of misalignment and misalignment angle at various deflection angle in the x-direction, θx with a fixed value of θy=-0.0001 rad. Figure 11(a) shows the locus of the journal center during the load cycle. The variation of the deflection angle in the x-direction influences the locus when the load is small. Figure 11(a) also shows the locus corresponding to the minimum film thickness at θx=0.0005, where its value is larger than that at θx=-0.0005. The maximum value of the eccentricity ratio εf=0.986 occurs on the front end at 301.3o, where the possibility of the metalto-metal contact is higher. Figure 11(b) shows the components of the moment near the second peak are positive when θx=-0.0005 rad since the maximum pressure is generated on the front end, while they are negative when θx=0.0005 rad or θx=0 since the maximum pressure is generated on the rear end. Near the second peak, the moments at θx=0.0005 rad are larger than those at θx=-0.0005 rad since its corresponding degree of misalignment in Figure 11(c) is large, which implies the smaller film thickness and the higher pressure. The maximum degree of misalignment for θx=0.0005 rad is DM=0.84 which is much lower than the case for θy=-0.0005

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rad in Figure 10 since the load direction is mostly close to the y-direction. Figure 11(d) shows the variation of the misalignment angles during the load cycle. From Figure 10(a), the influence of θx on the locus of journal center is very small which implies the difference of the attitude angle φo between different θx is very small. This can be explained by referring to Equation (5) that shows a relationship between α and φo through θx and θy. The first term of the RHS has a 79o difference between each other. Therefore, the misalignment angles are roughly parallel to each other during the load cycle.

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Figure 11. Variations of misaligned bearing parameters at various θx (θy=-0.0001 rad) 4 Conclusions A model is developed to investigate the influence of misalignment on the performance of a dynamically loaded journal bearing. The model takes into account the lubricant’s film rupture and reformation by adopting the mass-conserving cavitation algorithm by Elrod and Adams. Newton’s method is used to find the translating speeds εɺ X and εɺY at each time step via the force balance between the applied load, inertia and reaction force by the hydrodynamic pressure generated.

In general, for characterizing misalignment two parameters are needed. The set frequently used is DM and α. However, this set of parameters are ideal for the treatment of misalignment in a 21

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statically loaded bearing, where the load direction is invariant with time and the position of the minimum film thickness remains stationary. In the case of dynamically-loaded engine bearings, the deflection angles θx and θy are more convenient since θx and θy remain constant in the engine bearings while DM and α vary during operations. Appropriate relations for treatment of dynamically-loaded bearings using the deflection angles are utilized for engine performance predictions, and a series of parametric simulations are presented to gain insight particularly to assess the susceptibility of bearing to damage during load variation (0.5x, 1x, and 2x).

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The locus of the misaligned bearing in the mid-plane is very similar to that for the aligned bearing. However, DM and α are varying significantly during the load cycle since DM is sensitive when the eccentricity ratio is high. The moment and the degree of misalignment increase with increasing |θy|. If θy<0, the moment is negative. If θx=0, the change of θy does not influence the misalignment angle α. At a fixed θy, the moment and the degree of misalignment at |θx|>0 are relatively larger than those at θx>0. At θx>0, the moment is positive. The plots of misaligned angles at each θx during the load cycle are parallel to each other. When the load is downwards, the locus of journal center is located in the right half of the bearing since the journal is rotating in the counterclockwise. Therefore, the positive θx increases the degree of misalignment higher than the negative θx, and the chance of the surface-to-surface contact increases.

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It is shown that if the rotation is counterclockwise, the direction of the translating journal movement is likely to be counterclockwise without the applied load. When the load is applied, the locus of the journal center changes depending upon the magnitude and the direction of the applied load. At a high eccentricity ratio, a small reduction of the film thickness generates a large reaction force. Therefore, upon imposing a large load, the eccentricity ratio increases and the movement of journal center becomes slow. At a low eccentricity ratio, a smaller load increases the eccentricity ratio easily and the speed of the journal center is fast. The peak maximum pressure occurs at the same time as the peak of the applied load. The simulation results predict that some of the bearing performance parameters such as the friction force exhibit a delay in their peak value corresponding to a large input peak load. This can be attributed to the fact that friction coefficient is strongly related to the film thickness and the film thickness is a function of several factors such as the load and its direction, the journal location, etc. The journal mass and the supply pressure does not significantly influence the locus of journal center at a high eccentricity ratio since the difference of forces due to the change of the journal mass and the supply pressure is very small in comparison to the applied load at a high eccentricity ratio. The change of the applied load, the rotating speed and the viscosity influence the locus of journal center at any eccentricity ratio. These parameters are the key operating parameters in the design stage to prevent the surface-to-surface contact. The minimum film thickness occurs at one of bearing ends when the bearing is misaligned. When the misalignment is severe (large deflection angles), the degree of misalignment easily reaches near one, and the minimum film thickness could be less than three times of the standard deviation (r.m.s) where the metal-to-metal contact takes place. Therefore, it is required that the surface roughness of the bearing should be taken into consideration when the misalignment is investigated. Also, important to investigate is the thermal effects since it, too, can play a major role in reducing the film thickness and promoting metal to metal contacts. These considerations

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add additional degrees of complexity into the formulation and numerical solutions but are necessary to guard against bearing damage. 5 Acknowledgements

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This research was funded in part by Naval Surface Warfare Center (NSWC), Contract #N6449817P5391. The data, analysis and interpretation of the results are the responsibility of the authors and do not pertain to the contract. The authors greatly acknowledge support and encouragement and helpful suggestions of Dr. Gregory Anderson at NSWC.

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6 References [1] Jang, J. Y. and Khonsari, M. M., 2015, “On the Characteristics of Misaligned Journal Bearings,” Lubricants, 3, 3010057, pp. 27-53.

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[2] Goenka, P. K. 1984, “Dynamically Loaded Journal Bearings: Finite Element Method Analysis,” ASME J. Tribol., 106, pp. 429–437. [3] Lahmar, M., Frihi, D. and Nicolas, D., 2002, “The Effect of Misalignment on Performance Characteristics of Engine Main Crankshaft Bearings,” European J. Mech. A/Solids, 21, pp. 703– 714. [4] Booker, J. F., 1971, “Dynamically Loaded Journal Bearings: Numerical Application of the Mobility Method,” ASME J. Tribol., 93, pp. 168-174.

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[5] Elrod, H. G., 1981, “A Cavitation Algorithm,” ASME J. Lubrication Technology, 103, pp. 350-354.

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[6] Vijayaraghavan, D. and Keith, T. G., 1989, “Effect of Cavitation on the Performance of a Grooved Misaligned Journal Bearing,” Wear, 134, pp. 377-397. [7] Bouyer, J., and Fillon, M., 2002, “An Experimental Analysis of Misalignment Effects on hydrodynamic Plain Journal Bearing Performances,” ASME J. Tribol., 124, pp. 313–319

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[8] Sun, J., Gui, C. L., and Li, Z. Y., 2005, “Influence of Journal Misalignment Caused by Shaft Deformation Under Rotational Load on Performance of Journal Bearing,” Proc. Inst. Mech. Eng., Part J: J. Eng. Tribol., 219, pp. 275–283. [9] Boedo, S., 2013, “A Hybrid Mobility Solution Approach for Dynamically Loaded Misalignment Journal Bearings,” ASME J. Tribol., 135, 024501, pp. 1-5. [10] Nikolakopoulos, P., Papadopoulos, C. and Kaiktsis, L., 2011, “Elastohydrodynamic Analysis and Pareto Optimization of Intact, Worn and Misaligned Journal Bearings,” Meccanica, 46, pp. 577-588.

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[11] Mashra, P. C., 2013, “Mathematical Modeling of Stability in Rough Elliptic Bore Misaligned Journal Bearing Considering Thermal and non-Newtonian Effects,” Appl. Math. Model., 37, pp. 5596-5912.

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[12] Khonsari, M. M. and Booser, E. R., 2017, Applied Tribology: Bearing Design and Lubrication, 3rd ed., John Wiley and Sons, West Sussex, UK. [13] Elrod, H. G. and Adams, M., 1974, “A Computer Program for Cavitation and Starvation Problems,” In Proceedings of the 1st Leeds-Lyon Symposium on Cavitation and Related Phenomena in Lubrication, Leeds, England, Sept., pp. 37–41.

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[14] Jang, J. Y. and Khonsari, M. M., 2010, “On the Behavior of Misaligned Journal Bearings Based on Mass-Conservative Thermohydrodynamic Analysis,” ASME J. Tribol., 132, 011702.

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[15] Jang, J. Y. and Khonsari, M. M., 2015, “On the Characteristics of Misaligned Journal Bearings,” Lubricants, 3, pp. 27-53. [16] Vijayaraghavan, D. and Keith, T. G., 1989, “Effect of Cavitation on the performance of a Grooved Misaligned Journal Bearing,” Wear, 134, pp. 377-397. [17] Jang, J. Y. and Khonsari, M. M., 2016, “On the Relationship between Journal Misalignment and Web Deflection in Crankshafts,” J. Eng. Gas Turbines Power, 138, 122501, pp. 1-10.

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[18] Vijayaraghavan, D. and Brewe, D. E., 1992, “ Frequency Effects on the Stability of a Journal Bearing for Periodic Loading,” J Tribol., 114, pp. 107-115. [19] Fesanghary, M. and Khonsari, M. M., 2011, “A Modification of the Switch Function in the Elrod Cavitation Algorithm,” ASME J. Tribol., 2011, 133, 024501, pp. 1-4.

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[20] Ausas, R. F., Jai, M. and Buscaglia, G. C., 2009, “A Mass-Conserving Algorithm for Dynamical Lubrication Problems with Cavitation,” ASME J. Tribol., 131, 031702, pp. 1-7.

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[21] Martin, F. A., 1983, “Developments in Engine Bearing Design,” Tribol. Int., 16, pp. 147164.

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Nomenclature C Bearing radial clearance, m DM Bearing degree of the misalignment Journal eccentricity at the mid-plane, m eo Projected length between centers of the front and rear ends, m e′ F Bearing frictional force, N g Elrod switch function h Lubricant film thickness, m L Width of the bearing, m M Moment due to misalignment, Nm P Fluid pressure, Pa Pa Ambient Pressure, Pa Cavitation pressure, Pa Pc Ps Lubricant supply pressure, Pa Qleak Lubricant leakage flow-rate, m3/s R Journal radius, m Journal rotational speed, m/s us w Axial flow velocity, m/s W Applied load, N x,y,z Coordinate system, m α Angle between the axis Y ′ and the line from C1 to C2 β Bulk modulus of the Lubricant, Pa εo Eccentricity ratio at the mid-plane ε o = eo / C ′ Maximum possible value of ε ′ ε max

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Θ

Attitude angle at the mid-plane Misalignment angle Lubricant viscosity, Pa.s Coefficient of temperature-viscosity relationship Circumferential angle θ=R/x Deflection angles of the journal Density of the lubricant, Kg/m3 Density of the lubricant at the cavitation, Kg/m3 Fractional film content Θ = ρ / ρc Aspect ratio of the bearing Λ = L / D

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φo φM η λ θ θx, θy ρ ρc

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Highlights

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Misalignment affects the performance of dynamically-loaded engine bearings Severity of misalignment can be characterized using two deflection angles Governing equations with provision for mass-conservative cavitation are derived Large defection angles are indicative of metal-to-metal contact and looming failure

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• • • •