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Magnetohydrodynamic contact between two elastic rollers
285
Wear, 65 (1981) 285 - 293 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
MAGNETOHYDRODYNAMIC ROLLERS*
CONTACT
BETWEEN
TWO ELASTIC
S. PYTKO and K. WIERZCHOLSKI Stanislaw Staszic Technical University of Mining and Metallurgy, Institute of Basic Problems of Machine Construction, ul. Mickiewicza 30, 30-059 Krakow (Poland) (Received December 14, 1979)
Summary A simultaneous estimation of the elastohydrodynamic equations for hydromagnetic flow in the contact gap of mating rollers is presented. The increments of pressure are determined in the contact gap for a non-classical lubricant in a magnetic field.
1. Introduction Electromagnetohydrodynamic contact occurs in roller bearings, ball bearings, gear teeth, gear trains etc. The electric and magnetic fields are produced by an external electrical circuit, an external magnetic field or induction. A theoretical analysis of the problem of stationary contact between two mating surfaces of .infinite length in the presence of magnetic and electric fields is presented in this paper. In this case the working surfaces are deformed in the presence of magnetic and electric fields (Fig. 1).
Fig. 1. Contact between two mating surfaces in a magnetic field. *Paper presented at the 4th International Conference on Tribology, Paisley, September 10 - 15,1979.
286
2. Basic equations The analysis of the contact problem is performed using the following equations [l] : the equation of motion in the lubricant -n,i + g(ui,jj + $ uj,ji) + eijkJ$h + p,Ei + V,j(ui,j + uj,i) - $Q.iUh,/z = PUjUi~
(1)
j
the equation of continuity (P”i),i
=
(2)
O
the equation of conservation of energy P”j(CuT), j + PP”j(Ppl ), j =
J2 (gTi),i
+
’
+
(3)
7
Maxwell’s equations VxE=O
(4)
V-B=0
(5)
VxH=J
(6)
the continuity equation for the current V-J=0
(7)
Ohm’s equation J=o(E+
uX B)
(8)
and the equations of elasticity for a non-homogeneous isotropic body /JUi,
jj
+
+
(A +
/J)Uj,
ji
+
eijkU*EjBk
+
/J, j(Ui, j + Uj,i)
+
h,i”s,s
+
3dK*T),i
+
Pe*Ei = 0
for i, .j, s = 1,2,3. In eqns. (1) - (9), Uiare the components of the lubricant velocity u, Ui the components of the displacement, Bi the components of the applied magnetic field B = Hpf , Ji the components of the current density vector J, Ei the components of the electric field vector E and Hi the components of the magnetic field vector H. The following symbols are also used : pf is the magnetic permeability, pe* the space charge in the elastic body, X and ~1the Lame coefficients for the elastic body, K* the modulus of compression, (or the thermal coefficient of linear expansion, eijk the Levi-Civita tensor, ‘dethe thermal conductivity of the lubricant, p the density of the lubricant, (I* the electrical conductivity of the elastic body, u the electrical conductivity of. the lubricant, n the dynamic coefficient of viscosity, p the pressure, T the temperature, c, the specific heat and pe the space charge in the lubricant.
287
3. Assumptions The following assumptions are made. (1) The lubricant is an incompressible newtonian fluid. (2) The fluid motion is non-isothermal. (3) The dynamic viscosity coefficient of the lubricant depends on the pressure. (4) The elastic isotropic homogeneous working surfaces are deformed. (5) The magnetic permeability, the thermal and electrical conductivities and the specific heat are constant scalar quantities. (6) The Lorentz forces are the only body forces acting on the fluid. (7) No slip occurs between the surfaces of the cylinders. (8) There is no lubricant flow in the x3 direction. (9) The induced magnetic field is small compared with the applied magnetic field. (10) There is no component Rx3 of the applied magnetic field in the x3 direction; the components Rx1 and B.+ are functions of xp only. (11) The components of the electric field vector are independent of x3, the vertical component EX2 is negligibly small and the horizontal component Exl is independent of x2. (12) The elastic body is homogeneous and isothermal. (13) The space charge pe* m ’ the elastic body and the space charge pe in the lubricant are negligibly small. It follows from the first Maxwell equation (eqn. (4)) and from assumption (11) that the component Exl vanishes and the component Ex3 is independent of both x2 and x. Let us assume that Ex3 = 19VBo = constant
where 0 < 0 < 2, 0 = Ex3/E,-, and V = 2nR202; o2 is the angular roller velocity and R2 the radius of the roller. From Ohm’s law (eqn. (8)) and assumption (8) it follows that the current density components Jx, and Jx, vanish and the component J.+ is a function of x2 and x1. Equation (7) is satisfied for this current density vector. It follows from eqns. (5) and (6) that the applied magnetic field is constant across the film and obeys Go = e/R2 = 1O-4 where E is the minimum
thickness
of the contact
gap.
4. Dimensionless analysis of the problem of stationary contact between two homogeneous elastic cylinders in a magnetic field Let us assume that the lubricant dynamic viscosity pressure p according to the Barrus formula [2,3] :
depends
17= qOeap where CYis the pressure effect coefficient
on the (10)
of the lubricant
viscosity.
288
The dimensionless h = 1+ ix”
height of the gap is
+ $QU,
fll) y=o where pE = (~cR~/e and for the elastic body x = x~/R~~~ is the dimensionless horizontal coordinate, y = x2/R1p0 is the dimensionless vertical coordinate, UX= ul/Rlpo is the dimensionless horizontal displacement and U, = @RI y. is the dimensionless vertical displacement. The dimensionless horizontal and vertical coordinates in the contact gap have the form x = x,/R,s,Q and y = x2/e, and it is assumed that hl = czh v1 = 2nR20zv, v2 = 2n (&,{l + c)}~‘~c+R~v, P
(12)
=PoPl
T =T,+T,*l; J
= JaJzl
where u2 and u1 are the vertical and horizontal components of the lubricant velocity vector, u2 and u1 are the vertical and horizontal components of the surface displacement vector, w1 and w2 are the components of angular velocity, To* = EC Pr To is the characteristic temperature, $0
,1/z
( 1
cpo=c l+c
Pr is the Prandtl number, ambient temperature,
EC is the Eckert number,
J/e = e/R2, To is the
277w2rlo po
=
~o{~o(l
+ ,)}I’2
is the characteristic pressure and c = R2/RI. When eqns. (12) are substituted in systems (1) - (9), assumptions (1) (13) are taken into account and terms of order $J = low4 are neglected, we obtain
Wb)
+ u, 2 + H2(0 - u,)~ =Gz v, 3 ay
(13d)
289
h+pao
a2u, + -=-__-_& a2u, ay2 ax2
1-1
c, + -=--a2u, ay2
ax
x+1.1
ax2
I-1
ao
ay
* aux +au,
o=
ax
ay
where
=
(14)
0 (10)
is the Hartmann number, Gz = Pr Re $ = 0 (1) is the Graetz number, Re =
277o2R,vo 770
is the Reynolds number, O
0
*BoEo*Qo
(15)
is a dimensionless number and
O
2nw27loff
$o{J/o(l + c)F2
(16)
Eo* is the constant z component of the electric field in the elastic body and p is the shear modulus.
5. Height of the contact gap We now consider the problem of deformation of the contact gap under a pressure p. The problem is solved for the deformation of a flat elastic layer bounded to an absolutely rigid base. This layer is loaded by a pressure distribution. The system of equations, eqns. (13e) and (13f), is solved neglecting the term Se because 0 < Se Q 1. It is assumed that the normal stresses on the upper surface of the elastic layer are equal to the hydrodynamic pressure p with the opposite sign and the tangential stresses are ignored. Neither vertical nor horizontal displacement occurs on the bonded lower surface of the elastic layer. The dimensionless vertical displacements are of the form u, (x,y = 0) =
lk@,,x)
(17)
290
where h
2(1 - v2)
=
ITE
G{f(x,
(18)
PO
_ x)I = r (smh b - b/(3 - 4~)) cos @f (x1 -x)} o b{cosh b ++(l + b2)/(3--4v)+$ -2~)
db (20)
where f is the ratio of the contact width to double the thickness of the elastic superficial layer, 0.10 < f G 4.00,O < b < 00, G2 G x1 G $r, E is Young’s modulus, @r is related to the outlet, G2 to the inlet of the oil film and v is Poisson’s ratio. Substitution of eqn. (17) into eqn. (11) gives the dimensionless height of the contact gap.
6. The analytical solution of the magnetohydrodynamic contact between two cooperating surfaces
problem of the
The following boundary conditions [ 31 are applied to the system of equations, eqns. (13a) - (13d): u, (x,y = 0) = Cl
(2la)
u,(x,y
= h) = 1
(21b)
u, (x,y = 0) = 0
(21c)
uy(x,y=h)=
f
Wd)
T,(x,y
= 0) = 1
We)
T,(x,y
= h) = 1
Pl(G2
= x) = Pl(@l
= x1 = 0
ah
~(x=Gl)=o
Gw (2W
where cl = olRI/w2 R2 is the ratio of the velocities of the rollers. For the boundary conditions (21a) and (21b) we obtain from eqns. (13a) and (13b) u, =2 Cl
+
where
(1 --Cl)
sinh{U(y - h)} + sinh(2Uy) cosh(2Uh)
+ Cl
(22)
291
(231 The integral form of the continuity
equation
gives on integration dh s”uXdy + u&r,y = tt) - u,(x,y = 0) - &xu,(x,y =hf=O 0 When conditions (21b) - (21d) are applied, eqn. (25) implies ;
a
h
&h&=0
(26)
0
Substituting eqn. (22) into eqn. (26) gives tanhU __cI u
--
--‘I
+fc+l)U
tanh u
2cU = h
(271
where c is an arbitrary constant. Substituting eqn. (27) into eqn. (22) gives 0, =
(cl - l)k. tanh U f 2U(cU -elk) hftanh U - U) sinh(2Uy)
+ tf.
--%I
sinh(2U~~
sinh(Uy) sinh(U(y -h)) cosh( Uy)
+
+c ’
Substituting eqn. (13a) into eqn. (13d) and using the condition 0 < Gz 4 1 gives
a=q+
-
a2 Jexp@&)-----u,s 2
a=u,+(0
- 0 exp(Kp,) -
=0 (299 ay* ay2 Substituting eqn. (22) into eqn. (20) and integrating using conditions (21e) and (21f) gives the temperature distribution in the contact gap: ay=
7. Pressure distribution The pressure distribution p1 is obtained from eqns. (27) and (23) which can be expressed in implicit form as
1 2 -H”
dp, dx
-8
and U= iHexp(-
$$&)[I+
iX2t
m Csinhb-b(3-44v)}cos(bf(x,
-x)} (32)
The arbitrary constant C, the second arbitrary constant from one integration and the constant outlet ~lr are obtained from conditions (21g). 8. Numerical analysis A numerical analysis of the pressure distribution in the contact gap was made for the following parameters: K = l/Z (because the dynamic coefficient of viscosity depends on pressure); ipE= 100; k = l/1000; f = 1; v = l/3; cl = -l/2; cpa= -1; 8 = 1 (because the component E, of the electric field is of P,
=
P/P,
t
Fig. 2. Dimensionless distribution of the pressure in the contact gap between two mating surfaces for Hartmann numbers H of 1, 2 and 3.
293
the order of 27rRzwpB0). Values of 1,2 and 3 were assumed for the Hartmann number H. The pressure ordinates obtained from eqn. (31) increase with increasing H (Fig. 2).
9. Conclusions If a magnetic field (H > 0) is assumed in the contact gap between two mating rollers, the dimensionless pressure distribution increases relative to the pressure distribution for H = 0 as is the case for a classical lubricant. The pressure distribution increases with increasing Hartmann number as shown in Fig. 2.
References 1 W. F. Hughes and F. J. Young, Electromagnetodynamics of Fluids, Wiley, London, 1966. 2 S. Pytko and K. Wierzcholski, Druckverteilung im Quergleitlager fiir nichtstationaren &durchfluss mit Beriicksichtigung des Temperatureinflusses beim Wlrmeaustausch und der Spaltdeformation, ht. Tribology Congr. Eurotrib., Diisseldorf, October 1977. 3 K. Wierzcholski, The effect of the forces of inertia and of variable lubricant viscosity on the hydromagnetic non-isothermal lubricant flow in the gap of a-journal bearing, Exploitation Problems of Machines, 13 (4)(36) (1978) 403 - 430 (in Polish).
Wear, 65 (1981) 285 - 293 0 Elsevier Sequoia S.A., Lausanne - Printed in the Netherlands
MAGNETOHYDRODYNAMIC ROLLERS*
CONTACT
BETWEEN
TWO ELASTIC
S. PYTKO and K. WIERZCHOLSKI Stanislaw Staszic Technical University of Mining and Metallurgy, Institute of Basic Problems of Machine Construction, ul. Mickiewicza 30, 30-059 Krakow (Poland) (Received December 14, 1979)
Summary A simultaneous estimation of the elastohydrodynamic equations for hydromagnetic flow in the contact gap of mating rollers is presented. The increments of pressure are determined in the contact gap for a non-classical lubricant in a magnetic field.
1. Introduction Electromagnetohydrodynamic contact occurs in roller bearings, ball bearings, gear teeth, gear trains etc. The electric and magnetic fields are produced by an external electrical circuit, an external magnetic field or induction. A theoretical analysis of the problem of stationary contact between two mating surfaces of .infinite length in the presence of magnetic and electric fields is presented in this paper. In this case the working surfaces are deformed in the presence of magnetic and electric fields (Fig. 1).
Fig. 1. Contact between two mating surfaces in a magnetic field. *Paper presented at the 4th International Conference on Tribology, Paisley, September 10 - 15,1979.
286
2. Basic equations The analysis of the contact problem is performed using the following equations [l] : the equation of motion in the lubricant -n,i + g(ui,jj + $ uj,ji) + eijkJ$h + p,Ei + V,j(ui,j + uj,i) - $Q.iUh,/z = PUjUi~
(1)
j
the equation of continuity (P”i),i
=
(2)
O
the equation of conservation of energy P”j(CuT), j + PP”j(Ppl ), j =
J2 (gTi),i
+
’
+
(3)
7
Maxwell’s equations VxE=O
(4)
V-B=0
(5)
VxH=J
(6)
the continuity equation for the current V-J=0
(7)
Ohm’s equation J=o(E+
uX B)
(8)
and the equations of elasticity for a non-homogeneous isotropic body /JUi,
jj
+
+
(A +
/J)Uj,
ji
+
eijkU*EjBk
+
/J, j(Ui, j + Uj,i)
+
h,i”s,s
+
3dK*T),i
+
Pe*Ei = 0
for i, .j, s = 1,2,3. In eqns. (1) - (9), Uiare the components of the lubricant velocity u, Ui the components of the displacement, Bi the components of the applied magnetic field B = Hpf , Ji the components of the current density vector J, Ei the components of the electric field vector E and Hi the components of the magnetic field vector H. The following symbols are also used : pf is the magnetic permeability, pe* the space charge in the elastic body, X and ~1the Lame coefficients for the elastic body, K* the modulus of compression, (or the thermal coefficient of linear expansion, eijk the Levi-Civita tensor, ‘dethe thermal conductivity of the lubricant, p the density of the lubricant, (I* the electrical conductivity of the elastic body, u the electrical conductivity of. the lubricant, n the dynamic coefficient of viscosity, p the pressure, T the temperature, c, the specific heat and pe the space charge in the lubricant.
287
3. Assumptions The following assumptions are made. (1) The lubricant is an incompressible newtonian fluid. (2) The fluid motion is non-isothermal. (3) The dynamic viscosity coefficient of the lubricant depends on the pressure. (4) The elastic isotropic homogeneous working surfaces are deformed. (5) The magnetic permeability, the thermal and electrical conductivities and the specific heat are constant scalar quantities. (6) The Lorentz forces are the only body forces acting on the fluid. (7) No slip occurs between the surfaces of the cylinders. (8) There is no lubricant flow in the x3 direction. (9) The induced magnetic field is small compared with the applied magnetic field. (10) There is no component Rx3 of the applied magnetic field in the x3 direction; the components Rx1 and B.+ are functions of xp only. (11) The components of the electric field vector are independent of x3, the vertical component EX2 is negligibly small and the horizontal component Exl is independent of x2. (12) The elastic body is homogeneous and isothermal. (13) The space charge pe* m ’ the elastic body and the space charge pe in the lubricant are negligibly small. It follows from the first Maxwell equation (eqn. (4)) and from assumption (11) that the component Exl vanishes and the component Ex3 is independent of both x2 and x. Let us assume that Ex3 = 19VBo = constant
where 0 < 0 < 2, 0 = Ex3/E,-, and V = 2nR202; o2 is the angular roller velocity and R2 the radius of the roller. From Ohm’s law (eqn. (8)) and assumption (8) it follows that the current density components Jx, and Jx, vanish and the component J.+ is a function of x2 and x1. Equation (7) is satisfied for this current density vector. It follows from eqns. (5) and (6) that the applied magnetic field is constant across the film and obeys Go = e/R2 = 1O-4 where E is the minimum
thickness
of the contact
gap.
4. Dimensionless analysis of the problem of stationary contact between two homogeneous elastic cylinders in a magnetic field Let us assume that the lubricant dynamic viscosity pressure p according to the Barrus formula [2,3] :
depends
17= qOeap where CYis the pressure effect coefficient
on the (10)
of the lubricant
viscosity.
288
The dimensionless h = 1+ ix”
height of the gap is
+ $QU,
fll) y=o where pE = (~cR~/e and for the elastic body x = x~/R~~~ is the dimensionless horizontal coordinate, y = x2/R1p0 is the dimensionless vertical coordinate, UX= ul/Rlpo is the dimensionless horizontal displacement and U, = @RI y. is the dimensionless vertical displacement. The dimensionless horizontal and vertical coordinates in the contact gap have the form x = x,/R,s,Q and y = x2/e, and it is assumed that hl = czh v1 = 2nR20zv, v2 = 2n (&,{l + c)}~‘~c+R~v, P
(12)
=PoPl
T =T,+T,*l; J
= JaJzl
where u2 and u1 are the vertical and horizontal components of the lubricant velocity vector, u2 and u1 are the vertical and horizontal components of the surface displacement vector, w1 and w2 are the components of angular velocity, To* = EC Pr To is the characteristic temperature, $0
,1/z
( 1
cpo=c l+c
Pr is the Prandtl number, ambient temperature,
EC is the Eckert number,
J/e = e/R2, To is the
277w2rlo po
=
~o{~o(l
+ ,)}I’2
is the characteristic pressure and c = R2/RI. When eqns. (12) are substituted in systems (1) - (9), assumptions (1) (13) are taken into account and terms of order $J = low4 are neglected, we obtain
Wb)
+ u, 2 + H2(0 - u,)~ =Gz v, 3 ay
(13d)
289
h+pao
a2u, + -=-__-_& a2u, ay2 ax2
1-1
c, + -=--a2u, ay2
ax
x+1.1
ax2
I-1
ao
ay
* aux +au,
o=
ax
ay
where
=
(14)
0 (10)
is the Hartmann number, Gz = Pr Re $ = 0 (1) is the Graetz number, Re =
277o2R,vo 770
is the Reynolds number, O
0
*BoEo*Qo
(15)
is a dimensionless number and
O
2nw27loff
$o{J/o(l + c)F2
(16)
Eo* is the constant z component of the electric field in the elastic body and p is the shear modulus.
5. Height of the contact gap We now consider the problem of deformation of the contact gap under a pressure p. The problem is solved for the deformation of a flat elastic layer bounded to an absolutely rigid base. This layer is loaded by a pressure distribution. The system of equations, eqns. (13e) and (13f), is solved neglecting the term Se because 0 < Se Q 1. It is assumed that the normal stresses on the upper surface of the elastic layer are equal to the hydrodynamic pressure p with the opposite sign and the tangential stresses are ignored. Neither vertical nor horizontal displacement occurs on the bonded lower surface of the elastic layer. The dimensionless vertical displacements are of the form u, (x,y = 0) =
lk@,,x)
(17)
290
where h
2(1 - v2)
=
ITE
G{f(x,
(18)
PO
_ x)I = r (smh b - b/(3 - 4~)) cos @f (x1 -x)} o b{cosh b ++(l + b2)/(3--4v)+$ -2~)
db (20)
where f is the ratio of the contact width to double the thickness of the elastic superficial layer, 0.10 < f G 4.00,O < b < 00, G2 G x1 G $r, E is Young’s modulus, @r is related to the outlet, G2 to the inlet of the oil film and v is Poisson’s ratio. Substitution of eqn. (17) into eqn. (11) gives the dimensionless height of the contact gap.
6. The analytical solution of the magnetohydrodynamic contact between two cooperating surfaces
problem of the
The following boundary conditions [ 31 are applied to the system of equations, eqns. (13a) - (13d): u, (x,y = 0) = Cl
(2la)
u,(x,y
= h) = 1
(21b)
u, (x,y = 0) = 0
(21c)
uy(x,y=h)=
f
Wd)
T,(x,y
= 0) = 1
We)
T,(x,y
= h) = 1
Pl(G2
= x) = Pl(@l
= x1 = 0
ah
~(x=Gl)=o
Gw (2W
where cl = olRI/w2 R2 is the ratio of the velocities of the rollers. For the boundary conditions (21a) and (21b) we obtain from eqns. (13a) and (13b) u, =2 Cl
+
where
(1 --Cl)
sinh{U(y - h)} + sinh(2Uy) cosh(2Uh)
+ Cl
(22)
291
(231 The integral form of the continuity
equation
gives on integration dh s”uXdy + u&r,y = tt) - u,(x,y = 0) - &xu,(x,y =hf=O 0 When conditions (21b) - (21d) are applied, eqn. (25) implies ;
a
h
&h&=0
(26)
0
Substituting eqn. (22) into eqn. (26) gives tanhU __cI u
--
--‘I
+fc+l)U
tanh u
2cU = h
(271
where c is an arbitrary constant. Substituting eqn. (27) into eqn. (22) gives 0, =
(cl - l)k. tanh U f 2U(cU -elk) hftanh U - U) sinh(2Uy)
+ tf.
--%I
sinh(2U~~
sinh(Uy) sinh(U(y -h)) cosh( Uy)
+
+c ’
Substituting eqn. (13a) into eqn. (13d) and using the condition 0 < Gz 4 1 gives
a=q+
-
a2 Jexp@&)-----u,s 2
a=u,+(0
- 0 exp(Kp,) -
=0 (299 ay* ay2 Substituting eqn. (22) into eqn. (20) and integrating using conditions (21e) and (21f) gives the temperature distribution in the contact gap: ay=
7. Pressure distribution The pressure distribution p1 is obtained from eqns. (27) and (23) which can be expressed in implicit form as
1 2 -H”
dp, dx
-8
and U= iHexp(-
$$&)[I+
iX2t
m Csinhb-b(3-44v)}cos(bf(x,
-x)} (32)
The arbitrary constant C, the second arbitrary constant from one integration and the constant outlet ~lr are obtained from conditions (21g). 8. Numerical analysis A numerical analysis of the pressure distribution in the contact gap was made for the following parameters: K = l/Z (because the dynamic coefficient of viscosity depends on pressure); ipE= 100; k = l/1000; f = 1; v = l/3; cl = -l/2; cpa= -1; 8 = 1 (because the component E, of the electric field is of P,
=
P/P,
t
Fig. 2. Dimensionless distribution of the pressure in the contact gap between two mating surfaces for Hartmann numbers H of 1, 2 and 3.
293
the order of 27rRzwpB0). Values of 1,2 and 3 were assumed for the Hartmann number H. The pressure ordinates obtained from eqn. (31) increase with increasing H (Fig. 2).
9. Conclusions If a magnetic field (H > 0) is assumed in the contact gap between two mating rollers, the dimensionless pressure distribution increases relative to the pressure distribution for H = 0 as is the case for a classical lubricant. The pressure distribution increases with increasing Hartmann number as shown in Fig. 2.
References 1 W. F. Hughes and F. J. Young, Electromagnetodynamics of Fluids, Wiley, London, 1966. 2 S. Pytko and K. Wierzcholski, Druckverteilung im Quergleitlager fiir nichtstationaren &durchfluss mit Beriicksichtigung des Temperatureinflusses beim Wlrmeaustausch und der Spaltdeformation, ht. Tribology Congr. Eurotrib., Diisseldorf, October 1977. 3 K. Wierzcholski, The effect of the forces of inertia and of variable lubricant viscosity on the hydromagnetic non-isothermal lubricant flow in the gap of a-journal bearing, Exploitation Problems of Machines, 13 (4)(36) (1978) 403 - 430 (in Polish).