Fluid dynamic bearings: Modelling of elastic deformations Текст научной статьи по специальности «Физика»

Journal of Siberian Federal University. Engineering & Technologies 3 (2015 8) 270-282

УДК 628.822

Fluid Dynamic Bearings: Modelling of Elastic Deformations

Viktor A. Ivanova*, Nikolai V. Erkaevab and Daniel Langmayrc

aSiberian Federal University 79 Svobodny, Krasnoyarsk, 660041, Russia bInstitute of Computational Modelling SB RAS 50/44 Akademgorodok, Krasnoyarsk, 660036, Russia

cANSYS Germany GmbH Staudenfeldweg 12, Otterfing, 83624, Germany

Received 02.03.2015, received in revised form 27.03.2015, accepted 07.04.2015

This article deals with a new approach for calculation of self-consistent pressure distribution and surface deflection for a lubricated journal bearing. This approach is based on the numerical solution of the 2-D Reynolds' equation for the lubrication layer, numerical calculation of the surface deformations by the 3-D ANSYS package and Fourier series expansion for the compliance matrix. A simple analytical approximation is found for the obtained compliance matrix, which can be used for heavy loaded journal bearings. The compliance matrix is implemented into the iterative procedure for calculation of self-consistent pressure distribution and surface deflection in the contact zone. Results of calculations are presented for the particular journal bearing.

Keywords: journal bearing, elastic hydrodynamics, compliance matrix.

© Siberian Federal University. All rights reserved Corresponding author E-mail address: Vintextrim@yandex.ru

*

Гидродинамический подшипник скольжения: моделирование упругих деформаций

В.А. Иванов3, Н.В. Еркаевяб, Д. Лангмайр в

аСибирский федеральный университет Россия, 660041, Красноярск, пр. Свободный, 79 бИнститут вычислительного моделирования СО РАН Россия, 660036, Красноярск, Академгородок, 50/44 вГерманский филиал фирмы АНСИС Германия, 83624, Оттерфинг, Штауденфельдвег, 12

В статье рассмотрен новый подход к вычислению самосогласованного распределения давления и деформации поверхности для цилиндрического подшипника скольжения. Предлагаемый метод основан на численном решении 2-мерного уравнения Рейнольдса для смазочного слоя, вычислении деформации поверхности с помощью 3-мерного пакета АНСИС с использованием разложения Фурье для вычисления матрицы податливости. Найдена простая аналитическая аппроксимация для матрицы податливости, которая может применяться для расчета тяжело нагруженных подшипников скольжения. Найденная матрица податливости используется в итеративной процедуре для расчета самосогласованного распределения давления и прогиба поверхности в зоне контакта. Представлены результаты расчета конкретного подшипника скольжения.

Ключевые слова: подшипник скольжения, упругогидродинамический контакт, матрица податливости.

1. Introduction

Hydrodynamic lubrication theory is a commonly used tool for calculating and designing fluid dynamic journal bearings, which are important parts of various mechanisms and vehicles. There exist many publications devoted to this subject [1 - 4]. The role of elastic deformations becomes very important for heavy loaded journal bearings. Therefore elastic effects were incorporated into the lubrication theory, and the so called elastic hydrodynamic approach was developed, which considers both lubricant flow and surface deformations caused by enhanced pressure in the lubricant film [5]. In this approach, a key problem is to find a relationship between lubricant pressure distributions and surface deformations. The main constructive elements of a fluid dynamic bearing are shaft (journal), lubricant film, sleeve and housing. Usually housing material is more rigid compared to that of a sleeve. This is the reason why deformations of the housing are often neglected, and the sleeve deformations are used to be taken into account [6]. In such simplified case, the deformations of the thin sleeve constrained by the rigid housing are dependent on a small parameter, which is a ratio of the sleeve thickness to the curvature radius. As shown in [7], in the first order approximation with respect to the small parameter, the sleeve deformation is proportional to the local pressure in the lubricant film. The coefficient of proportionality is called as "sleeve compliance". This approximation is related to the hypothesis of Winkler. In case of cylindrical symmetry, the compliance coefficient can be determined from the analytical solution [8].

Generally, it is necessary to take into account not only sleeve deformations, but also housing deformations, because housing has a finite rigidity. In such a case, calculations of surface deformations

have to be performed self-consistently with calculations of pressure distributions along the whole lubrication layer. For this purpose we have to determine a generalized equation for relationship between the local surface deformations and the lubricant pressure distributions, taking into account different material properties of the sleeve and housing. The main goal of this work is to elaborate a method for deriving such an equation, and to apply it for self-consistent solution of the elastic hydrodynamic problem.

2. Satement of problem

To describe our approach we consider the journal bearing with steel shaft and bronze sleeve shown in Fig. 1. Here ra is the angular speed of the shaft, 9 is the azimuthal angle counted clockwise off the maximal clearance, and n is the eccentricity of the shaft, R0 is the journal radius, R is the internal radius of the sleeve, R2 and R3 are the internal and external radii of the steel housing, L is the length of the bearing. We assume that the external boundary of the housing is fixed. This means that deformations vanish at the external boundary of rhe housm^ Tlae sheft and sleeve surfaces are aesumed fo be separated by a fhin film of a liquid lubrhaoi, so called the lubrication layer. We also set zero boundary eondition for the pressure ad she edgss of Ihe biasing.

The pressure distaibution in the lfbsrcation tayer is determlned by tine conventional Reynolds' equation [2]

d h h3 dP 1 d h h3 dP^ + -

1

Rd ^cp 2 12(0, Sep J dy ( 12( dy

h = R1- R0 + n cos(cp)+S(P),

1 dhu R0 dp

P> 0;

dhu

= 0,

P < 0.

(1) (2) (3)

Here h is the thicknesr of rye lubrication layer, i ir the visco ssi^tys coefficient, u is the mean velociiy of thte boundary surfaces, y is the coordinate aloni0 the axis of the journal bearing, 9 is the

P- ->1

x- / / /

\ \ \ \ \' \\

i t£ i t£ DC

nl

sc 1

\\\\P \\\\\ W

/// /

y

Fig. 1. Geometrical scheme of the journal bearing: 1 is the shaft, 2 is the bronze sleeve, 3 is the steel housing

azimutlinl angle. § is the radial deflection of the sin eve surface , which is dependent on the pres sure tn the lubrication layer.

For computational convenience we intsoduce normalized paramsterr as follows:

„ ~ 6lR=2

P = P , , h = Hd, d = R,-R0

d (4)

y=Riyi, d = C0Rj/2, T| =

where P iid tlo(3 dtmensionless ^rcsisEsi^iree^ j ii t^lt^e dimensionltes coordinaOe i^lomngrr "^l^i;i:'ot;£itio:n.:ol axis, and H ts tej dimenskinkas thonkne rs oi tie luOoinatiof laynr. WMi oetct^^ii^^iLeiaittilons (4) wf transform the Reynoiao' equntinn to tlie timenoioniens form ^ncH add a relexation poyametee a

dep

( 3 ~ 3D A

<3q>

+JL\H>—

HI dy

dH dP

--——, (5)

Sep dt

# = lHT|coh(cp)+s(p), P >0; (6)

|9 = 0, 0. (7)

Theoe are myny publicaeiono demoted do numericml methads for innegration (if the Reynolds' equation [f - r2n. I]f our caoe, eumerical solution of Eq. (5) ie obiained be a roldxahion mrthod based on ad implicr snlie me witli rlie finite difforeece opproximations [1 3]:

Ap

p)—-j * a))+4j/2J—

A Hf —H pn+u 2 _ pn

i i i „n ----„n \ n j— — i—1 H a _p_-j

(Pn _ 2 Pn Pn ^ -

^ /P' J+r'iJ—l) iAcp Af (8)

A . - . , Pn+V2 — P" A

pJ I pn+l - — ,1 \

i pn+l __ 2 P"+1 + P n+1 \ — n j_i -— Pj ( Pn —OP" 4- P" ^

"VriJ+1 2 ■2\riJ+1 Zri) ~rГг•j—l)•

Ay2V'1 ''J ''J-1' At A/

Here the lower indices i,j enumerate tlti^ grid points, and the upper index n enumesates eime steps. Eqs. (3) £nr<s solved by a tri-diagonal matrix orgorithm. A stationary pressure distribution is obtained as a result oatime relaxation of the numerical solution.

3. Results of calculations. 2-d ANSYS model

For illustrating our method, we take the aollowing input parameters of the journal bearing: Ri = 0.0:3 m, R2 = 0.0t5 m^ R3 = 0.1 m, d =e 0.000l:a m, == ]Pa, Ee2 =t ^.(rin i^^1 »t1 = 0.31,

m2 = 0.334, (i = 0.02244 Pa/s, tf = 31-4.;i6> , whete i?i and E2 ^re the elasticity moduluses of ^h^e steel housing and bronze sleeve, mi and md ^^^ itlies Poisson c;o)T^s5]p(3ndi^l5 ijo tlte materials of

the housing find sleeve, reepectivelyi "Thhiie lenJa^ll and diameter of the journal bearing are assumed to be equal too each other.

Fig. 2 shows pressures in the lubrication layer as functions of the azimuthal angle. These pressures are obtained from the numerical solution of equation (5) for journal eccentricities if = 0,9 and n =0,8. Further, we used these pressure distributions for calculating the elastic deformations of the sleeve and housing. For this purpose we applied the ANSYS package based on the finite element numerical method [14]. Fig. 3 shows mesh spacing of the 2-D calculation domain with 5 elements across the bronze sleeve, 9 elements across the steel housing, and 100 elements along the azimuthal direction. The previously calculated pressures are applied to the grid points at the internal boundary of the sleeve.

Performing 2-D ANSYS calculations for the given pressure distribution along the lubrication layer (Fig. 2), we obtained results presented in Fig. 4. Here the sleeve and housing deformations are

2 3

0 [rad]

Fig. 2. Distribution of the pressure along the lubrication layer for two eccentricities: 0.9 (curve 1) and 0.8 (curve 2)

Fig. 3. Mesh spacing of the 2-D ANSYS model

2 1.8 1 .6 1.4

1.2 1 1

25

0.8 0.6 2.4 81.2 8

Fig. 4. Deformations of the sleeve (curve 1) and housing (curve 2)

<b [rad]

shown by the solid and dashed lines, respectively. These numerical results can be compared to the approximate analytical formula [6] implying a proportional relationship between deformation of a thin sleeve and the corresponding local hydrodynamic pressure in the lubrication film

8 = DP, D (1 + m)(l ). (9)

E2 l1 - m2 )

This formula implie s proportionality of the pres sure and deflection, and thus it neglects nonlocal influence of the pressure.

Moie general and realistic formuladescribing dependence of rlastic deformations on pressure distributions nan be written in tire integral form:

2n

S(cp)={ P(q')K (cp-qj'W, (10)

0

where Sand P are the surface deflection and ptesrurt aa functions of tine azimuthal angle, ^(9-9') is the kernel huncrion, which determines a compliance matrix. This Cunction does not depend on the paessure distribution, but it depends on the geomerric characteristics of the journal bearing. It can be deCermined based on the numerical resuhs deacribed above. For this purpose we apply Fourier series expansion:

(p) = [[ (fccs(>kq3)+jitik sin(fcp)]. (11)

k=0

- 2e5 -

Multiplying equation (11) o n the harmonic functions and integrating over the angle fromO to n, we obtain a linear algebraic systemof equations for the unknown Fourier coefficients Mk and Nk.

¿n 271

Mk Jf 3S<p')sin(m(p')£h(p' +ovj =

0 0 1 2 n

= — f 8(p')sia(=p')-=p', k--

(12 )

: =1, 2, 3.

o

0 n

Mk j P((')cos(k(')d('-Nk j>((')sin('

0 0 e 2oi

= — J 8((p ')cos (-p ' ) • <2cf> ' , k-n 0

Mp J-Pfp')"' = — ^J"(è(cp')p0cp'

:=1, 2, 3...

i13)

(14)

Using ^lie; known distribution of pressure in the luboication eayer aisd surface deformations calculated by ANSYS we find the solution of linear equations (12, 13)

M

n Xii+iv

Y =}YkCk-X dy

2 X) + Y,2

(=1,2,3...

Mn =-

Dn

2%Yn

(15)

(16)

where

Xk = jxp((p^)sin(ncip')• oOcp', Yk = j>(cp')c(((ncp')-dp',

0 0

2n 2n

Ck = J8(p') sin(kcp')• Pp', D' = J8(p') co s(kp') • dcp '.

(17)

Here, the pressure and surface deflection as functions of the angle were determined by spline approximation of the grid values tf and cf . Substituting Fouoier coefficients Md end Nt (15, 16) in Eq. (11) we find

2D(cp -cp') =

2%Y,, k

2=1 X2 + Y2

1 k=n

yf

tanl,c^rs;Irft sm(c(q)-cp')^.

(18)

Xf+Ya

Here, the accucacy of the result is dependent on a fhoice of the upper limis of summing n. In particular, a value of n=10 i s quipe sufficient for good accuracy of the approximation. Rogarding formula (r8), it iis worth noeing about an influence of a small scaled noise, wdich usudly appeats in the input data obtained from ihe numerical calculations. In order to nemove tffects (elf such noise, ft is necessary "to apply some kind of tegularization. A general the ory of re gularizations for integral is quations is described in monographs [15), 16]. To supprese the noisy oscillations, we introduce multiplication

factors gk = l/t + eif) fo:r "tnli«^ FEFou^in^it (15). Here e is a smkll parameter wliicii litis strong

influeemce; on damping of the noisy oscillations. Witli the regularization facto) equations (15) are modified as follows

Mk= * XYM2

$ g_s YjCj - XjDj - n XX d2

C =1, 2, 3.

(199)

Substituting tlie modified EFouir:^e;:^ coefficients (19) into formuHa (It) one can obtain tlle smootlie d kernel function:

k (_pr) = 3

2 nY0 n

-fg^^ -+Y) " «» (((cp-cpr))-

(=l

1 krn

nhr

7lt=n 1X1 +Ik

(20)

Increase of tlie parameter e leads tottronger nampingof tine nocse, and id also causes a tlight decrease of" the ketndl function peak. Fig. 5 shows the kernel function (solid line) corresponding to e=0.000L In tlle same figure, the daslied lme is tile analytical approximation given by simple formula

Kan(<=) = f.2K« where a = 5, a = 1.7, (3 = 1.4,

f

f+ acp"

(21)

) =

((2 - Rj(1 + m1)=1 - 2m1) (R3 -R2- (1 + m2)(1- 2m2)

1 - m,

1 - m2

(22)

Tle obtained functions (21) and (22) determine the complifnfe matrix for any distribution of the pressuie an tft ludricstion layei Cor a given geometric characteristics of lhe founnal bearing. For

1.2 1

0.8 0.6 0.4 0.2 0

-3-2-1 0 12 3

<> [rad]

Fig. 5. Kernel function obtained from numerical solution (curve 1) is compared to the analytical approximation (curve 2) given by formula (21)

<|> [rad]

Fig. 6. Curve 1 shows the sleeve deflection obtained from formula (10); and curve 2 is the sleeve deflection calculated by the ANSYS package

testing formula (10) , we calculated deformations of the bearing surface related to another pressure distribution (Fig. 2b) corresponding to larger eccentricity ^=0.8. These calculations were performed with the ANSYS package. Results of the calculations are shown in Fig. 6. One can see in this figure that formula (10) with K given by (20) yields result (solid line), which is rather close to that of the direct ANSYS calculation (dashed line). A small difference (about 3 %) can be related to the effects of numerical approximation of the code.

4. 3-d ANSYS calculation

Next, we analyze a difference between the kernel functions determined for different cross sections of the journal bearing. For this study we use 3-D ANSYS calculations, taking the same parameters of the journal bearing as described above. To determine the deformation caused by the pressure distribution we apply the 3-D ANSYS package with the mesh spacing shown in Fig. 7. Using the finite difference scheme described above, we calculate the hydrodynamic pressure in the lubrication layer as a function of two coordinates in case of journal bearing with finite length. Fig. 8 shows the pressure profiles for different cross sections (y = const) of the bearing. Using results of ANSYS calculations we find deformations of the bearing surface in each cross section. Then we compare the 3-D ANSYS solutions for the central cross section with the 2-D ANSYS solution. Both solutions are presented in Fig. 9. As one can see in Fig. 9, the bearing surface deformation is larger in 2-D case than that in 3-D case. The difference is about 28 %. Then we consider the behavior of the kernel functions which determine the

<|> [rad]

Fig. 8. Azimuthal distributions of the pressure corresponding to the different cross sections of the journal bearing. Curve 1 is for the cross sections close to the edges of the bearing: y=0.02666 m. Curves 2, 3, 4, and 5 correspond to the cross sections y=0.01999 m, y=0.01333 m, y=0.00666 m and y = 0, respectively

compliance matrix for different cross sections. To obtain these functions we used the described above Fourier method, which was applied for different cross sections of the bearing. Finally we find the kernel functions shown in Fig. 10. This figure indicates clearly that the kernel functions are rather close to each other for all cross sections (besides the cross sections near by the edges). Therefore one can use the compliance matrix determined for the central cross section. The kernel function corresponding to 2-D case is about 28 % larger than that for 3-D case when the length of the bearing is of the same order as its diameter.

Fig. 9. Deformations of the bearing surface for the different cross sections (3-D model) and also for the 2-D model. Curves 1, 2, 3, 4 and 5 correspond to tlie surface deflections obtained from tlie 3-D ANSYS model for the cross sections y= 0.03 m, y=0.02333 m, y=0.01666 m, y= 0.t0999 m and y=0.00333 m, respectively; curve 6 is related to the 2-D ANSYS model

0> [Rad]

Fig. 10. Kernel functions based on the 3-D and 2-D ANSYS models respectively. Curves 1, 2, 3, 4 correspond to He cross sections y=0.02333 m, y=0.0l666 m, y= 0.00999 m and y=0.00333 m, respectively. Curve 5 is He 2-D kernel function

55. Self-consistent solution

We apply the obtained kernel function for self-consistent calculation of the lubrication pressure. The iterative procedure is as follows:

d_

dp

(h «)3

dP

H+i)

dp

d_

dy

(H (" 03

dP

(n+i)

dy

dH(n) (P (n+i)- P(n)) -+ -'-,

dp At

(23)

=S(n) = l-ncoe(9) + Kjj5Pn)(9pP(q>-cp')9, P(n) >0;

0

^ = 0, P (c) ho, dp

woliere A = 6K0nRKaid3.

This c^^mensioi^leess; paeameter A characterizes the role of the elastic surface deformations. Numerical iterations (223) converge to self consistent distributions; of the pressure and surface deflection. Those are presented in Fig. il for the central cross section (y=0), for three different eccentricities oh the shaft: if = 0.8, 0.9, 0.h5. Sn chse n = 0.95, the pressure maximum is affected quite strongly by the elastic surface deformation. In thir care, the surface deformation leads to coeresponding decrease of the peessure maximum in a factor of 1.5.

6. Conclusion

An effective approach is proposed which allows one to determine the compliance matrix on the base of preliminary caOculation ot the pressure in the lubrication layer (without elestic defocmations) and an ANSYS cflculation sali" the surface dffotmfiion. Even fat a longSournal bearing, one has to apply 33-D ANSYS model, rather then 2-D one, foe thn calculation of the rurfacn deformations. This is because the 22-DD ANSYS model overestimates the surface f eflectioh substantially. The compliance mctecx is deteamined fry thee; kternei Dunction which ir expressed via Fourier series expansion. In order to suppress the noisy oscillations, we modify thr Fouriee coefficients by multiplying them on regularization factors, which cause smoothing of the kernel function. The compliance matrices obtained for different cross sections of the journal bearing are very similar to each other. Therefore -i is tufficient todetermine tfe compliance matrix focthe central cross section only. The compliance matrix depends on constructive parametessof lhe jonrnal bearing, but it does not depend on a specific psessure di stribution. ThereSoce odtained once, the comptiance matrix canine; used ieeratively together with lhe Reynelds equation for serf coneiseeut simulatious of the dynamlral regimes of thr eournal b facing. The proprrsed iterative peocedure convergfs rather qutckly and thus it is very

Fig. 11. Self consistent pressure and surface deflection profiles corresponding to the central cross section (y =0) for three eccentricities: 0.95 (curve 1), 0.9 (curve 2) and 0.8 (curve 3). Here 5 is normalized to d = R1-R0

efficient for obtaining a self-consistent surface deformations and pressure distribution in leavy loaded journal bearings.

This work was supported by the Russian Foundation for Basic Research (project 15-0500879).

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