Radial Bearings with Porous Elements - Core

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ScienceDirect Procedia Engineering 150 (2016) 559 – 570

International Conference on Industrial Engineering, ICIE 2016

Radial Bearings with Porous Elements M.A. Mukutadze* Rostov State Transport Univercity (RSTU), 2 Rostovskogo Strelkovogo Polka Narodnogo Opolcheniya sq., Rostov-on-Don, 344038, Russia

Abstract This paper presents the analytical model of the radial bearing of the finite length with the two-layer porous insert obtained on the basis of the dimensionless equations of movement of a viscous incompressible lubricant in an operating clearance and in porous layers of a bearing sleeve and continuity equations as well. We are analyzing the case when permeability of porous layers on the border of section of a two-layer insert accepts one value. The lubricant moves in a circumferential direction through a bore in a bearing body and the subsequent filtration occurs through insert pores. Permeability anisotropy of porous layers is taken into consideration in radial and circumferential directions. As a result of the given task, we have obtained the field of speeds and pressure in porous layers and in the lubricant layer. Analytical dependences for components of a vector of supporting force and the friction moment have been found, and we have also defined the loading factor and friction factor. Moreover, in our calculations we have revealed and used the parameter characterizing specificity of lubrication feeding in a circumferential direction. It is proved that the rational mode of operation is reached by a design of a two-layer porous insert given when the resistance factor increases more intensively and from the parameter ȥ, and from a relative eccentricity İ in comparison with the bearing having a single-layer insert. Hence, anisotropy of permeability of porous layers in a circumferential and radial direction has a defined possibility of practical application of this new model. © by Elsevier Ltd. by ThisElsevier is an open © 2016 2016Published The Authors. Published Ltd.access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016. Peer-review under responsibility of the organizing committee of ICIE 2016

Keywords: hydrodynamics; radial bearing; two-layer insert; anisotropy of permeability of porous layers in radial and circumferential directions.

* Corresponding author. Tel.: +7-928-270-27-43. E-mail address: [email protected]

1877-7058 © 2016 Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICIE 2016

doi:10.1016/j.proeng.2016.07.041

560

M.A. Mukutadze / Procedia Engineering 150 (2016) 559 – 570

1. Introduction Modern machines widely use the antifrictional porous materials obtained by methods of powder metallurgy, and also the porous coverings put by a gas-thermal dusting. Powder materials possess higher oil absorption and damper ability, than compact ones [1-5]. Porous bearing materials are made by methods of powder metallurgy. The standard technology in a bearing bushing provides 25-30 % of pores: blind, one-side open and transverse. Soaked in vacuum by a heated liquid lubricant, powder bearings absorb up to 5-6 % of oil weight. Porous bearings operate usually under relatively small load in a self-lubricating mode, as a result of oil allocation under the exposure of thermoexpansion from heat generated by friction. Hence, in this case the boundary friction is realized, and only occasionally and for short periods of time - hydrodynamic. Unfortunately, modern researchers pay insufficient attention to the specified phenomena, there are practically no papers devoted to working out of mathematical models, describing the physics of this phenomena. There are very few papers which only with certain assumptions attempt to examine the performance of porous plain bearers [6-9]. These statements testify that nowadays practice of application of porous journal bearings in a mode of hydrodynamic lubrication outruns theoretical researches. Hence, continuous and accurate research in the field of the porous bearings, greased with modern lubricants, is claimed by practice. In papers [10, 11] it is shown, that porosity of powder details essentially reduces their strengthening parameters. High contact pressure in high-pressure bearings, deforming a porous surface, reduce a filtration of a lubricating fluid and possibility of its feeding through pores. In its turn, a compact bearing bushing, maintaining considerable loadings, demands special devices for feeding of a lubricating fluid. This problem has found the solution in working out of compound plain bearers in which the lubricating fluid can move through a porous part, or this porous part will be so-called depot of lubricant. The further development of this idea resulted in necessity of reception of models for compound bearings with multilayered porous inserts. A number of papers are devoted to analytical consideration of performance of radial plain bearers of finite length with constructive elements or coverings from porous pseudo-alloys [12-15], including stability of liquid mode of friction. The analysis of papers [16-18] devoted to calculation of the similar bearings operating under the pressure of feed shows that permeability anisotropy of porous layers, simultaneously in radial and circular directions, is not considered. Anisotropy consideration only in a circumferential direction does not allow presenting factor of permeability in the form of the continuous function, which is necessary for calculation and depending on radial and circumferential co-ordinates. We have made an attempt to feel this want for problems in nonlinear statement. 2. Problem statement Let us consider the established current of viscous incompressible liquid in the clearance of the radial bearing of final length with a compound two-layer porous insert. We consider the bearing bushing to be motionless, and the shaft to be rotating with angular speed . The lubrication material is fed into the operating clearance of the bearing in an axial direction under the pressure of feeding. Let us assume that in two sections the lubrication pressure is set as follows:

pz

0, r r1

pH ,

pz

l , r r1

pK

(1)

Here PH and PK are the set values. Let us place the beginning of cylindrical system of co-ordinates (r, ș, z) on the bearing shaft in its left end (fig. 1).

561


Fig. 1. Analytical model

Then, the equations of shaft and bearing outline can be written down as follows:

c1 ; r c0 : r

b, c2 : r

b h1 ; r

a 1 H , H

b h2 ; h

h1 h2

1 2

H cos T H 2 sin 2 T

(2)

where h is thickness of porous layer; a is shaft radius; b is bearing bushing radius, İ=e/a , ɟ is eccentricity, ɇ is thickness of lubricating layer. Permeability of insert porous layers can be set in such a way that at the interface it takes on equal value: c1 ; r k1'

b, c2 : r Ae

O1 ln

r b h1

b h1 ; r k T ,

b h2 ; h k2'

Ae

O2 ln

h1 h2 r b h1

k T

(3)

Here, A is a set constant; dimensionless parameters Ȝ1>0 and Ȝ2>0 characterize distribution of porous layers permeability in a radial direction. Function k T by analogy to the law of change of the form of a lubricant film can be set in a kind, k the same order İ=Ș*İ*.

A 'cos T , where A ' A H* 1 . Also it is supposed, that İ* and İ are small parameters of

2.1. The initial equations and boundary conditions Let us start with the dimensionless equations of movement of viscous incompressible liquid in lubricating layer and section of porous inserts, and also in continuity equations [19, 10]. ª wX X wXT wX X 2 º Re «Xr r T Xz z T » wz r wT r ¼ ¬ wr

1

1 D

2

wp w 2Xr 1 wXr 1 w 2Xr w 2Xr Xr2 2 wXT 2 2 2 wr wr 2 r wr r 2 wT2 wz r r wT

562


ª wX X wXT wX X X º Xz T r T » Re «Xr T T r r r ¼ w wT wz ¬

wX º ª wX X wX z Re «Xr z T Xz z » r wT wz ¼ ¬ wr

wXr 1 wXT wX z Xr r wr r wT wz

1

1 D

2

1

1 D

2

1 wp w 2XT 1 wXT 1 w 2XT w 2XT XT2 2 wXr 2 2 2 r wT wr 2 r wr r 2 wT 2 r r wT wz

wp w 2X z 1 wX z 1 w 2X z w 2X z 2 wz wr 2 r wr r 2 wT 2 wz

0

ª w 2 Ɏ 1 wɎ 1 w 2 Ɏ w 2 Ɏ º wk1 wɎ 1 wk1 wɎ k1 r , T « 2 » r w r r 2 wT 2 w z2 ¼ w r w r r wT wT ¬wr

0

ª w 2 F 1 wF 1 w 2 F w 2 F º wk2 wF 1 wk2 wF k2 r , T « 2 » r w r r 2 w T2 w z 2 ¼ w r w r r w T w T ¬wr

0

(4)

Here dimensionless sizes are as follows: ȣr , ȣz , ȣș are components of speed vector; pare hydrodynamic pressure in lubricating layer; Ɏ and F are accordingly hydrodynamic pressure in porous layers;k1 and k2 are permeability of porous layers are connected with dimensionless, Dimensional sizes r , z , X , X , X , P, Ɏ, F and k , k r

T

1

z

2

F by ratio r, z, X r , X T , X z , P , Ɏ,

r

P : ab P, 2

P

A k1 , k2'

a z, k1'

b r, z

P : ab Ɏ,

Ɏ

b a

ba

A k2 , Xr F

: a Xr , XT

P : ab F ba

: a XT (5)

Further the sign ~ for dimensionless variables is skipped. The system of equations (4) is solved under the following boundary conditions: 1. On the outline ɫ1 the continuity of pressures (P = Ɏ) is carried out, and ȣr is a component of speed vector, which is defined by Darcy's law; the other components are equal to zero. 2. On border of section of porous layers F , k1

Ɏ 3. F

r bh

wɎ wr

k2

wF wr

pg ,

where Pg is law of lubrication feeding. 4. On the outline ɫ1 pz

0

pH , p*

pz

J

pK p*

5. On the shaft surface the following conditions are satisfied:

563


§ wX · ¨ r ¸ D H ! H sin T © wr ¹r D

Xr (D D H ) Xr

r D

XT (D D H ) XT

r D

w (D D H )

wr

D

§ wX · ¨ T ¸ © wr ¹r

§ ww · ¨ ¸ © wr ¹r

DH ! 1 D

DH ! 0 D

6.

wF wr

0.

(6)

r E2

2.2. Symptom-free task solution The solution of system (4) satisfying to the above-stated boundary conditions, can be searched for as follows:

p

b P(r ,T ) ) az ,

b R1 (r ,T ) F az ,

vr

u (r , T ) vT ,

R1

) 0 (r ) H )1 (r ,T ) , R2

u

u0 (r ) H u1 (r , T ) ! , v

v ( r , T ) vz ,

b R2 (r ,T ) az ,

w (r , T ), F0 (r ) H F1 (r , T ) P ,

P0 (r ) H P1 (r , T )

v0 (r ) H v1 (r ,T ) ! , w

,

w0 (r ) H w1 (r , T ) !

(7)

Substituting (7) in (4) and in boundary conditions, we will come to the following system of the differential equations and boundary conditions. We will have ª d u v2 º Re «u0 0 0 » r ¼ ¬ dr

1

1 D

2

dp0 d 2 u0 1 d u0 u0 dr d r2 r d r r2

d 2 v0 1 d v0 v0 d r2 r d r r2

ª dv u v º Re «u0 0 0 0 » r ¼ ¬ dr

d 2 w0 1 d w0 d r2 r d r

ª d w0 º Re «u0 w02 » d r ¬ ¼

d u0 u w0 0 dr r

d 2) 0 1 d) 0 d r2 r d r

0,

a

1 D

2

O1 d) 0 r dr

ª du du v du 1º Re «u0 1 u1 0 0 1 2v0 v1 » T d r d r r d r¼ ¬

,

d 2 F0 1 dF0 d r2 r d r 1

1 D

2

O2 dF0 r dr

,

dp1 d 2 u1 1 d u1 1 d 2 u1 u1 2 w v1 d r d r 2 r d r r 2 d T 2 r 2 r 2 wT

(8)

564


ª dv d v v d v u v u0 v1 º Re «u0 1 u1 0 0 1 1 0 » d r dr r dT r ¬ ¼

ª d w1 dw v d w1 º Re «u0 u1 0 0 » dr r dT ¼ ¬ dr d u1 1 d v1 u1 d r r dT r

d 2 F1 1 dF1 1 d 2 F1 d r 2 r d r r 2 dT 2

1 D

2

d p1 d 2 v1 1 d v1 1 d 2 v1 v1 2 w u1 , dT d r 2 r d r r 2 d T 2 r 2 r 2 wT

d 2 w1 1 d w1 1 d 2 w1 d r 2 r d r r 2 dT 2

d 2)1 1 d)1 1 d 2)1 d r 2 r d r r 2 dT 2

0

1

O1 d)1 r dr

O2 dF1

(9)

r dr

Boundary conditions can be written as follows:

u0 D

0 v0 D 1 w0 D , ,

u0 1

\ 1 D ) 0' 1 v0 1 ,

0

,

0 w0 1 ,

\ 1 D a,

) 0 1 P0 (1) 0 , ) 0 E1 F0 E1 , ) 0' E1 F0' E1 , F0' E 2 0. u1 D ,T

sin T

w1 1, T

0

v1 1,T

\ 1 D

F1' E 2

0 , )1 E1 ,T

,

v1 D , T

u1 1,T

v0' D D cos T

\ 1 D

, w)1 wT

,

w\ wr

,

w1 D , T

h1 b

,

E2 1

h b

r 1

P1 1,T ) 2 1,T

F1 E1 ,T , )1' E1 , T

,\

Ab

b a

e 3

0, F1' E1 , T ,

O1 ln

b b h1

In terms of boundary conditions the solution of system (12)

du X wu 1º ª wu Re «u0 1 u1 0 0 1 2X0X1 » dr r wT r¼ ¬ wr

,

\ 1 D ) 0' 1 K *cos T ,

Here:

E1 1

w0' D D cos T

(10)

(11)

565


1

1 D

2

wp1 w 2 u1 1 wu1 1 w 2 u1 u1 2 wX1 , wr wr 2 r wr r 2 wT 2 r 2 r 2 wT

dX X wX u X u X º ª wX Re «u0 1 u1 0 0 1 1 0 0 1 » w r dr r wT r ¬ ¼

1

1 D

2

1 wp1 w 2X1 1 wX1 1 w 2X1 X1 2 wu1 , r wT wr 2 r wr r 2 wT 2 r 2 r 2 wT

w 2 w1 1 w w1 1 w 2 w1 , w r 2 r w r r 2 wT 2

ª ww w w X w w1 º Re «u0 1 u1 0 0 » w wr r r wT ¼ ¬ w u1 1 wX1 u1 w r r wT r

0,

w 2 Ɏ1 1 w Ɏ1 1 w 2 Ɏ1 w r 2 r w r r 2 wT 2

O1 w Ɏ1 r wr

,

(12)

for the first approximation can be searched for as follows:

, Q1

u1

u11 r cos T u12 r sin T

w1

w11 r cos T w12 r sin T

Ɏ1

Ɏ11 r cos T Ɏ12 r sin T

,

Q 11 r cos T Q 12 r sin T

P11 r cos T P12 r sin T

P1 ,

, ,

F11 r cos T F12 r sin T

F1

.

Substituting (13) in (9) and (11), we will have

1

1 D

2

1 2 2 u11cc u11c 2 u11 2 Q 12 , r r r Q 2Q ª º p11c Re «u0 u11c u0c u11 0 u12 0 Q 11 » r r ¬ ¼

1 2 2 u12cc u12c 2 u12 2 Q 11 r r r

Q 11cc Q 11c

1 r

2 2 Q 2 u12 2 11 r r

1 r

2 2 Q 12 2 u11 r2 r

Q 12cc Q 12c

1 1 w11'' w11' 2 w11 r r

Q 2Q ª º P12c Re «u0 u12c u0c u12 0 u11 0 Q 12 » , r r ¬ ¼

1

1 D

2

1

1 D

2

Q Q u 1 ª º P12 Re «u0Q 11c u0cQ 11 0 Q 12 0 Q 11 0 u11 » , r r r r ¬ ¼

1

1 D

2

u Q Q 1 ª º P11 Re «u0Q 12c u12c Q 0 0 Q 11 0 Q 12 0 u12 » , r r r r ¬ ¼

Q ª º Re «u0 w11' w0' u11 0 w12 » , r ¬ ¼

(13)

566


1 1 w12cc w12c 2 w12 r r u 1 u11'' Q 12 11 r r

Q ª º Re «u0 w12c w0c u12 0 w11 » , r ¬ ¼

u 1 0 , u12'' Q 11 12 r r

1 ' 1 '' Ɏ11 Ɏ11 2 Ɏ11 r r

u11 (D )

0, u12 (D )

0,

1 ' 1 ' '' , Ɏ12 Ɏ12 2 Ɏ12 Ɏ11 r r r

O1

1, Q 11 (D )

O2 ' Ɏ12 , r

(14)

Q 0c (D )D ,

Q 12 (D ) 0, w11 (D ) w0c (D )D , w12 (D ) 0, u11 (1)

c (1) (1 D )u0c (1)K * , u12 (1) \ (1 D )Ɏ11

Q11 (1)

\ (1 D )Ɏ12 (1), Q12 (1)

w11 (1)

0, w12 (1)

0, P11 (1)

0 , Ɏ12 1

Ɏ11 1 F11' E1

\ (1 D )Ɏ11 (1),

0, P12 (1)

0 , F11 E1

' Ɏ11 E1 , F12 ' E1

c (1), \ 1 D Ɏ12

0,

Ɏ11 ( E1 ) , F12 E1

' Ɏ12 E1 , F11c E2

Ɏ12 ( E1 ) ,

0, F12c E 2

0.

(15)

Replacing derivative summands in expressions (8), (10), (14) and (15) by finite-difference approximations, we will obtain a system of the algebraic equations which is solved by Gauss-Seidel method. Having defined a velocity field and pressure field in a lubricating layer for components of a vector of constraining force Rx and Ry, and also the friction moment, we will obtain the following expressions:

Rx

M ɬɪ

HSJP a 2 b: l (b a)2

p11 (D ),

Ry

HSJ P a 2b: l (b a)2

p12 (D ),

X0 (D ) º a3 P:JS l ª «X0c (D ) a » . b ¬ ¼

(16)

The basic running qualities of the examined bearing are load factor Ȣ, resistance factor to rotation ȟ friction factor f, and parameter ઼ defined by formulas:

9

N (1 D )2 , N 2laP:

Rx2 Ry2 , [

M (1 D ) f , 2la 2 P: 1 D

[ ., a 9

1 pk pn p*

3. Findings of the research

The findings of the numerical analysis shown on fig. 2-7 allow making a number of conclusions

(17)


Fig. 2. Dependence of resistance factor on permeability parameter ȥ: given Ȝ1 = Ȝ2 = 0 and ‫ = *ܭ‬0 : 1 – Ȗ = 0,5; 2 – Ȗ = 1; 3 – Ȗ = 2; given Ȝ1 = Ȝ2 = 0,1 and ‫ = *ܭ‬0,01 : 1ૼ – Ȗ = 0,5; 2ૼ – Ȗ = 1

Fig. 3. Dependence of resistance factor on eccentricity with various permeability option values ȥ: given Ȝ1 = Ȝ2 = 0 and ‫ = *ܭ‬0 : 1 – ȥ = 10–2; 3 – ȥ = 1; 4–10 –1; given Ȝ1 = Ȝ2 = 0,1 : 2 – ȥ = 10–2; ‫ = *ܭ‬0,01

567

568


Fig. 4. Dependence of load factor from permeability parameter ȥ at0 various option values of covering thickness ȕ, dimensionless p pa bearing length Ȗ and relative eccentricity ߝǁ: a1 = 1, b1 = 1, Ș* = 1, g = 0.8; 1 – ߝǁ =0; Ȗ = 0.5; Re = 0; 2-Ȗ = 1, ߝǁ = 0.2, Re = 0.9

Fig. 5. Dependence of load factor from permeability parameter ȥ: Ȝ1 = Ȝ2 = 0, Į=0.998, ‫݌‬௚଴ െ ‫݌‬ҧ௔ = 0.5, ȕ2 = 1,6; 1 – Ȗ = 0.5; 2 – Ȗ = 1; ––– ȕ1 = 1,1; – – ȕ1 = 1,2; –– –– ȕ1 = 1,4

Fig. 6. Dependence of friction factor on permeability parameter ȥ: Ȝ1 = Ȝ2 = 0, Į=0.998, p g0 pa = 0.5, ȕ2 = 1,6; 1 – Ȗ = 2; 2 – Ȗ = 0.5; ––– ȕ1 = 1,1; – – ȕ1 = 1,2; –– –– ȕ1 = 1,4


p 0 pa Fig. 7. Dependence of resistance factor on permeability parameter ȥ: Ȝ1 = Ȝ2 = 0, Į=0.998, g = 0.5, ȕ2 = 1,6; 1 – Ȗ = 2; 2 – Ȗ = 1; 3 – Ȗ = 0.5; ––– ȕ1 = 1,1; – – ȕ1 = 1,2; –– –– ȕ1 = 1,4

4. Conclusions x Given permeability value H H i.e. when permeability of a porous layer varies according to the same law, as well as the form of a lubricant film, the bearing possesses higher bearing capacity; x We can observe two maxima of a bearing. The first is at permeability value ȥ, close to 10-1, the second is at relative width of the porous insert Ȗİ1 and ȥĬ10-1. x The account, at calculation of inertia forces, shows, that bearing capacity of the bearing can really increase by 1015 %. x The carried out calculations have allowed us to establish, that reduction of permeability of porous inserts in an axial direction will allow providing to the bearing sealing properties, and in circumferential direction providing increased bearing capacity; x Given Ȝ1= Ȝ2=0 and Ȗı1 with increase in parameter of permeability ȥ, value of load factor decreases, and its sharpest drop is observed at ȥĬ10-3. Given Ȗę[1;2] with increasing of permeability ȥ, the friction factor increases, but remains practically constant in all range. x Considering value Ȗ=0.5 and parameter of insert width ȕę[1.1;1.2], the strongly pronounced minimum of friction factor is observed at permeability parameter value ȥĬ1. x Calculations show that with Ȝ1ĮȜ2 i.e. in case of the two-layer porous bearing with values Ȝ2/Ȝ11 the bearing possesses higher bearing capacity than with Ȝ2/Ȝ1>1 and h2/h1