CC BY
45
MSC 80M20, 65N06
DOI: 10.14529/ m mp140404
A SIMULATION OF THE THERMAL STATE OF HEAVILY LOADED TRIBO-UNITS AND ITS EVALUATION
Yu. Rozhdestvensky, South Ural State University, Chelyabinsk, Russian Federation, [email protected],
E. Zadorozhnaya, South Ural State University, Chelyabinsk, Russian Federation, [email protected]
The thermal state of the elements of heavily loaded tribo-units is among the most important parameters affecting their performance. The temperature of the lubricating layer of bearings determines to a large extent their load-bearing capacity. The heat balance equation used to estimate the temperature of friction pairs fails to yield the temperature fields and the regions of their maximum values. This fact makes our problem important. We examine a mathematical model and a method for calculating the thermal state and thermohydrodynamic characteristics of heavily loaded sliding bearings, taking into account the non-Newtonian properties of the fluid as well as the heat exchange processes between the lubricating layer and the surrounding moving surfaces of tribo-units. To solve the energy equation, we propose to use finite difference approximation methods. To create the difference analogs of the energy equations for some structural elements and thin lubricant layers, we use the Pismen-Reckford scheme of implicit alternating directions. We present the calculated hydromechanical characteristics of the connecting rod bearing of a heat engine. We obtain three-dimensional distributions of temperature in the lubricant.
The results show that, if we allow for convective heat transfer in the radial direction, the processes of heat exchange between the lubricating layer and the surrounding moving surfaces enable us to determine more accurately the mean lubricant temperature and the thermal stress of a tribo-unit as a whole. Our method can be used to assess the performance and efficiency of heavily loaded tribo-units of piston and rotary machines.
Keywords: bearing; generalized energy equation; partial differential equations; boundary value problems.
Introduction
The problem of improving the reliability and durability of friction units in machines has always been a major challenge of modern engineering. The rising power of thermal machines and the rising demands for durability and fuel efficiency result in the increased loading of tribo-units (TU). In this regard, it is important to choose methods for solving concrete problems, as well as to create methods for physical and mathematical modeling of friction and the wear of friction units.
Modern mechanical engineering is based on advanced technologies. Innovative computational methods and computer technologies allow engineers and researchers to simulate and calculate TU for different machines and consider a number of design, operation, and other parameters affecting the performance of friction units. However, the thermal processes occurring in a heavily loaded bearing are of great importance for understanding the performance of friction units. Usually they are treated on the basis of a solution to the generalized energy equation (heat transfer equation) for a thin layer
of viscous incompressible fluid between two arbitrarily moving surfaces. This equation involves both convective heat transfer in the lubricant and heat transfer by conduction.
Thus, our research aims at developing models and algorithms to solve the problems of TU dynamics taking into account the temperature distribution in the lubricating layer, the non-Xewtonian properties of the lubricant, and the geometry of friction surfaces bounding the thin lubricating layer.
1. Solution
To solve the stated problem, consider a circular cylindrical radial bearing, where the bearing (bushing) 1 and the journal 2 rotate round the axes OiZ^ with i = 1, 2 passing through their centers Oi with absolute angular velocities ui. The lubricating layer of non-Xewtonian fluid is bounded by the surfaces of the bearing and the journal (see Fig. 1). Here x = rp with r ~ r1 ~ r2, where r1 = r1(p,t) and r2 = r2(p,t) are the radii of bushing (the inner surface) and the journal; p is the angular coordinate measured from the axis O1X1 rigidly fastened to the bushing. The axes Ox and Oz of the coordinate system Oxyz, in which we consider the processes in the lubricating layer, lie in the plane of the reference surface, and the axis Oy is perpendicular to it. The external force F (t) depending on time t acts on the pin in the plane Oxy in the central cross section z = 0.
l
Fig. 1. Coordinate systems for solving the heat, problem
To determine the field of hydrodynamic pressures in the thin lubricating layer, use the generalized Reynolds equation [1 3]. For a non-Xewtonian fluid we can write it as
д_
дф
^ (*■ - Э +к [ir+2 (*• - Э р
п 0 - 0] + rn (Ph)
д_
дф
Ф Фъ) . {лЛ
д , ^
where p and z are the angular and axial coordinates; p = p/p0 is the dimensionless density; p0 is the density of the Newtonian fluid; p = (p — pa) ip2/p0u0, ^ = h0/r2, z = z/r2 with —a < z < ^^d t = u0t are the dimensionless hydrodynamic pressure, relative radial clearance, width coordinate of the bearing, and time; h0 is the radial adjusting clearance; a = B/D is the relative width of the bearing; p0 is the characteristic viscosity of the lubricant; pa is atmospheric pressure; B, D = 2r2, and r2 are respectively the width, diameter, and radius of the pin; u21 = (u2 — ^i) /u0 is the dimensionless angular velocity of the pin. The dimensionless thickness h of the film and its derivative dh/dt are defined as
h = 1 — x cos (p — 8), dh/dt = —x cos (p — 8) — x8 cos (p — 8),
where x is the relative eccentricity and 8 is the angle of relative position of the center line, furthermore,
fh
tk = yk/p*dy,k = 0, 1, 2, J 0
where p* is the dimensionless non-Newtonian viscosity of the lubricant, which is a function
y
the lubricating layer.
Use a multigrid method to integrate (1) with the Swift-Stieber boundary conditions, taking into account the presence of sources of lubrication (holes or grooves) on the friction surfaces [4]:
p(p, z = ±a) = 0, p(p, z) = p(p + 2n, z), p(p, z) > 0 on (p, z) E Qs
p(p,z) = ps, S = 1, 2...,S*, where the region stands for a source of lubricant, in which the pressure is constant and
p s S*
the dependence of viscosity on the shear rate and pressure as [5, 6]:
p1 • exp (ft • p) , I2 < 104 c-1,
p • I(n-1)/2 • exp (ft • p) , 104 < I2 < 106 c-1, (2)
p2 • exp (ft • p) , I2 > 106 c-1.
The parameter n characterizes the degree of non-Newtonian behavior and ft is the pressure-viscosity coefficient of the lubricant, which is a function of temperature.
According to the model (2) in region 1 (with I2 < 104 c-1) the lubricant behaves as a Newtonian fluid with viscosity p1 (Tp,p). In region 2 (with 104 < I2 < 106 c-1) viscosity decreases following a power law. In region 3 (with I2 > 106 c-1) the lubricant is considered as a Newtonian fluid with viscosity p2 (Tp ,p).
Let us express in a dimensionless form the equations for the velocities of a volume element of the lubricant and their derivatives:
Vx = &U* + (by — t toy) hn+1 . fU = (— bty) h"+1. adt
to v to ) dp V to ) adz
Vy
h^
1 dh (_ x dt.
xy
1 dq.
zy
dVx
dy
1 p*
[<q
^ + hn+l( y -
V <qo)
a dp dp
ry „-.k
ykdy
ky
k = 0,1, 2,
*
lh i p
(3)
dp
V dy
1 • h~(qq -t) t
Write down the flux of the lubricant across the sections of unit length along the x and z coordinates as _
yy yy
qxy = Vxdy; qzy = Vzdy.
J 0 J 0
The generalized energy equation (heat transfer equation) for the lubricating layer of viscous incompressible fluid, accounting for small thermal conductivity along the x and z axes, is [6-81
dT
pco^r + pco dt
(
dT
dT
dT
Vx^~ + Vy— + Vz^- - Ao
dx dy dz
)
d2T dy2
Ds,
(4)
where p, c0, and A0 are the density, specific heat, and thermal conductivity of the lubricant (taken to be constants); Vx, Vy, and Vz are the com ponents of the velocity vector of a volume
Ds = p*
sv*)2 + (dv^ 2
dy ) V dy
is a dissipative function.
Introduce the following notation: T = T/T0 is the dimensionless temperature at a point of the lubricating layer; T0 is characteristic temperature; y = y/h, h = h/h0, ^ = h0/r, Vx = Vx/u0r, Vy = Vy fu0r, Vz = Vzju0r, p = p/p0, where h0 is the characteristic thickness of the film for the central position of the pin, Pe = p0pc0u0Al/A0 is the Peclet number, kT = p0pc0T0^2/(u0p0). In this case, we can write the energy equation for the lubricating film of the bearing with non-Newtonian lubricant in dimensionless form as
dT - dT q dT - dT 1 1
= -Vx--D— - vz— +— •
dt dp dy dq Pe h2
d2T Dq
where
D = -1
h
t dh + d (h-xy ) + hd-zy y dq dp dt
(5)
(6)
This differential equation is linear with respect to derivatives. The coefficients of the convective terms depend on p, y , z, and t. The equation is parabolic in time, and so we impose initial and boundary conditions.
Determine the distribution Ti(pi, R2, t) of temperature in the bushing, where R2 is the radial coordinate (see Fig. 1), by solving the equation for transient heat flow, which in cylindrical coordinates and dimensionless variables becomes
dT -
dt
— = ai
(
d2T i 1 dT i 1 d2T -- + == —- +---
dR2 R- dR- r2 dP
-2-
(7)
Here Ri = Ri/r and Ti = Ti/T0; furthermore, ai = Ai/ (cip\r2u0) is the dimensionless coefficient of heat transfer from the bushing to the environment, with ri ~ r2 ~ r, while pi5 c^d Ai are the density, specific heat, and thermal conductivity of the material of the bushing.
Introduce coordinate system Ox1y1 z1 and dimensionless variables
y1 = (R1 — n)/(r3 — n) = (R1 — 1) /(r3 — 1), r3 = r3/r,
where r1 and r3 are the radii of the inner and outer surfaces of the bushing. Then (7) becomes
Similarly, determine the distribution T2(p2, R2,t) of temperature in the pin by solving the equation for transient heat flow, which in cylindrical coordinates and dimensionless variables becomes
dT2 (d2T2 1 dT2 1 d2T
- (иг.dhduA (q\
dt dR2 R2 dR2 + r2 дф2) ■ w
Here R2 = R2/r and T2 = T2/T0; furthermore, a2 = \2/ (c2p2r2u0) is the dimensionless coefficient of heat transfer from the pin to the environment, while p2, c2, and A2 are the density, specific heat, and thermal conductivity of the material of the pin.
Ox2y2z2
y2 = (R2 — r4) /(r2 — r4) = (R2 — r4) /(1 — r4), T4 = ^/r,
r4
dT2 = _ ( 1 d2T2+_1_dT2d2T2 )
dt (1 — Z4)2 dy22 + [T4 + (1 — T,)y2] (1 — r 4) dZ2 + b + (1 — T4)V2]2 dp2) ' K
Let us state the boundary conditions to integrate the heat subproblem (5), (8), and (10). Since the temperatures of the lubricant and the bushing are periodic in the circumferential direction, we have
T (p, y, t) = T (p + 2n, z, t), T 1(p, R1,t) = T1(p + 2n, Rut).
On the outer surface of the bushing assume the free convection hypothesis
dT 1
dR
ir_ / Vi V
air'TA ^ - Tc
\i \ 11 x-"
c
On the common surface of the lubricant and the bushing impose the continuity condition for the heat flux (the coupling condition)
dT 1
dyi
f u dTi = - (Г3 - 1) T^T
V1=0
\1кф dy
y=0
On the surfaces of the lubricant shared with the sleeve and spike impose the equal temperature condition
T(p, y = 0,t)= Z(p, V1 = 0,t — tc), T(p, y =1,t) = T2(t — tc).
x=r3
2. Finite-Difference Approximation for the Equations of the Thermal Subproblem
Express (5), (8), and (10) in the dimensionless form:
d 2T
dT = RdT_ + RdT_ + K _
dt 1 dp 2 dy 3 dy2
dT i d2T i k •
dt dT 2
+ Kg— + K7
dy 1
dT 1 dyi
dt
- = K8 ^ + K9 ^ + K10
c3y2
2
dV.
2
+ K4,
d2 T1
dp2 '
d 2T 2
dp2
(11) (12) (13)
Here we put
T = T
1 Ds
K1(p,y,V = -Vx, K2(p,y,V = -D,K3(p) = Peh,K4(p) = kT -hn+1,
K5 = a1 K8 = a2■
1
(V3 - 1)2 1
, Kg(v1) = , K9(y2) =
a1
((r 3 - 1)) [1 + (r3 - 1)V1] a2
(1 - r4) [r4 + (1 - r4)y2Y
Kj(y1) =
a1
K10 (y22)
[1 + (r3 - 1)V1\
a2
[r4 + (1 - r4)V2]2
For generality, we also replaced y ,y1,y2 ^ ^d T, T1,T2 ^ T.
Express the system of equations for the two-dimensional distribution of temperature in the lubricating layer, bushing, and pin in the operator form
T-t + A • Lv(T) + B • Ly(T) = C • LVV(T) + D • Lyy(T) + E
(14)
where
T " -K1 ' ' -K2 ' 0 K3 K4
T = T1 , A = 0 , B = -Kg , C = K7 , D = K5 , E = 0
T2 0 -Kg K10 K8 0
dT
dT
dT
d2T
d2T
Tt = ai,Lv(T) = dp,Ly (T) = dV,Lw(T) = dp,Lyy (T) dy2 •
For the hnite difference approximation, denote by lv = 2n, lz = ^^d ly = 1 the
p z y
coordinates (pi,Zj,yk,tn) of the nodes, where
pi = (i - 1)AV (i =1,2, ...N, Av = lv/(N - 1)) ,
Zj = -1 + (j - 1)Az (j = 1, 2, ...M, Az = lz/(M - 1)); yk = (k - 1)Ay (k =1, 2....K, Ay = ly/(K - 1)); tn = nAt (n = 0,1, 2,...).
The step and the number of mesh elements depend on the direction: (N +1) = 96, (M + 1) = 25, and K = 20. At each time step we introduce arrays with elements. The one-dimensional arrays are
- ( dh) (dh) * _ * (dp*) _ (dp*) (dp*) _ (dp*) h- UJi, UJt= pij=19,{dp)i- UJ^, U Adtj,
(K)i, (K4)i, (K6)k, (K)k, (K)i, (K10)k, (Q1)i = (Q1)ij=19, (Q2)i = (Q2)ij=19. The two-dimensional arrays are
pi,k = pi,j=19,k, (Vx).,k = (Vx).^=19kk , (Vz)i,k = (Vz).^=19kk ,
idVA =i idVA =f дУЛ
\dy Jik \dyj
i ,j=19, к \dy J i к \dy Д
, pi, k = pi, j=19,k,
i,j=19,k \ wy / i,k \ wy / i,j=19,k (qxV)i,k = (qxV)i,j=19,k , (qzV)i,k = (qzV)i,j=19,k , (D S) i,k = (D S) i,j=19,k , (D) i,k = (D) i,j=19,k .
To construct the difference analogs of (14), we apply the Pismen-Reckford method [9-12] with a two-step difference scheme. Let us discretize (14), using for all derivatives the simplest central differences
dT = Ti+1 — Ti-1 d2T = Ti+1 — 2Ti + Ti-1 dp 2A^ , dp2 A* .
Firstly, consider the discretization scheme of the heat equation (8) for the bushing.
Divide the time step into two half-steps (Figure 2):
p
n+1/2
K5 • Lyy(T)n,k + К6к • Ly(T)nk + K7к • LW(T)"¡V
Ггп+1/2 _ rpn 1 i,k 1 i,k
A/2
y
Tjtn+1 _ Tn+1/2
iJk_ijk_ — К. . T i^T\n+1 , ът. T (rr\n+1 , и.. . T (rp\n+1/2
At/2
K5 • Lyy (T)n+1 + К к • Ly (T)n+1 + K7к • LVV(TU
Tn+1/2 _ tn Tn _ 2Tn + Tn
1 i,k 1 i,k 1i,k+1 21 i,k + 1i,k-1 .
~ K5---+
At/2 Ay
tn tn rrin+1/2 C)tn+1/2 \ tn+1 /2
. r^ Ti,k+1 — Ti,k-1 . Ti+1,k — 21i,k + 1 i-1,k
+Кбк-2A-+ K7 k-Аф-'
Tn+1 _ Tn+1/2 TFrn+1 _ 2rT'n+1 + TFrn+1
1 i,k — 1 i,k = k Ti,k+1 — 21 i,k + 1 i,k-1 +
a/2 =K5 Ay +
Tn+1 _ тп+1 T'n+1/2 _ 21^+1/2 + Tn+1/2
. 1 i,k+1 Ti,k-1 Tr 1 i+1,k 21 i,k + 1 i-1,k
+K6 к-2A- + K7 к-A2-■
2Ay Аф
(15)
(16)
Express (15), (16) as
a^i-li2 + b1Tn+l/2 + c^'i+^j2 = d1Tik-1 + e1Tnk + f1 ^k+n (17)
Fig. 2. Scheme of calculation by the implicit method of alternating directions. Arrows indicate the directions the scheme is implicit in
aT— + bT+ c2T^ - dT—2 + e2T^ + f2T^, (18)
where
K7k . J 1 , Klk ) Klk ( K5 K6k ) ( 1 кг
f K K6k K Кб k , J 1 , K ) к Кб и K7k
fl + 2лу ,a2 - - Щ + 2лу ,b2-2{â + âJ ,c2 - - Щ - ,d2 --щ '
2( 1 + K7A K7k
e2-2U + w ,f2
А* Аф J ' Аф'
Similarly discretize the energy equation (5). Divide the time step into two halves: p
rpU+l/2 _ rpn T i,k Ti,k
K3 i • Lyy(T)n,k + Kni,k • Ly(T)n,k + Kni,k • LV(T)n+k1 2 + K4i.
А_/2 i ) i,k 1 "2 i,k ^y
Step 2 (by y)
Tn+l _ Tn+l/2
iJk-^^ - Ki • Lyy(T)n+1 + K^i,k • Ly(T)n+kl + Knlhk • LV(T)n+1/2 + Ki
А*/2
or
Tn+1/2 _ Tn Tn _ 2Tn + Tn Tn _ Tn
Ti,k Ti,k Tr Ti.k+1 2Ti,k + Ti,k—1 T^n Ti,k+1 Ti,k-1 .
~ K3 i-1Г2--+ K2 i,k-^--+
А*/2 А2 2 ik 2ау
Tn+1/2 Tn+1/
rn Ti+1,k — Ti-1,k
~2А
n+1/2 rpn+1/2 (19)
1 T^n i+1,k i—1,k . jr
+K1 ikk-^--+ K4 h
ф
гтт+1 _тп+1/2 Тп+1 _ 2Tn+1 + Тп+1 Тп+1 _T'n+1
1 i,k — Ti,k _ u Ti,k+1 — 21 i,k + Ti,k-1 , 1 i,k+1 — 1 i,k-1
A/2 =ь A +K2 ik 2Ay +
Tjn+1/2 _ Tn+1/2 i+1,k i-1,k +K1 i,k-2A--+ K ^ i'
(20)
Express (19, 20) as
al^f + Ъ1+/2 + csT+f = dl— + в3Щ + НЦк+1 + 9з, (21) а^к-1 + b4Ti,kl + c4Ti,k+1 = d4T'i-+i,k2 + e4 Ti,кf + hT'i++i,k2 + 94, (22)
where
К, к , 2 , Кз i К™ k il К
4i, k , 2 , K3 i ^2 i, k 0 / 1 K3 i
a6 = —:—, Ъ6 = -т-, c3 = — a6, d6 = —тт;---:—, e6 = 2 —---—
3 2Аф At ' 6 Ay 2Ay \ At Ay
f±_ K61)
\At Ay)
(— + K2L [ VAt + Ay)
, _ K3 i . K2i, k _ „ _ K3 i . K2i, k , _ p. f 1 . K3 i\ _ K3 i Kni,k
f3 = A+W, 93 = K4^a4 = — A, 64 IA + Ay) ,c4 = — A—"2A7=
Kn 2 Kn
K i,k 2 K i,k d4 = — , e4 = At 4 = , 94 = K4i.
The discretization of the heat equation (10) for the pin follows the same pattern.
p
n+ / 2 n Ti,k — 1 i,k
At/2 y
K8 • Lyy (T )lk + K9 k • Ly (T )lk + K1 0 k • LVV(T )nt~
Tn+1 _ t™+1 /2
ik A_2k = К • Lyy(T)n+ 1 + Кk • Ly(T)n+ 1 + K10k • LVV(T)n+ 1 /2,
Tn+1/2 _ Tn Tn _ 2Tn 1 Tn
1 i,k_Ti,k = к 1 i,k+1 21 i,k + Ti,k-1 .
a/2 = 8 Ay +
tn tn rjin+1/2 c\tn+1/2 tn+1 /2
. 1 i,k+1 — Ti,k-1 . 1 i+1,k — 21 i,k + Ti-1,k
+K9k—2A— + K10 k-A-'
T'n+1 _ Tjn+1/2 Tjn+1 _ 2Tjn+1 I Tjn+1
1 i,k 1 i,k jr Ti,k+1 21 i,k + 1 i,k-1 .
— K8-
At/2 Ay
Tn+1 _ Tn+1 T'n+1/2 _ 2Tn+1/2 + Tn+1/2
+K9 k-2Ay-+ K10 k-Av-.
Express the latter equations as
^T^+k + 65Tin+1/2 + ^'i+ij2 = d5Ti,k-1 + e5Ti,k + f5 Tnk+1, (23)
a6Tn+-11 + b6Tn+1 + c6Tn+ = d6 T^l2 + e6T//2 + f!/2, (24)
where
K
a5 = --
i0k
A2
v
2 ( ^ + Kik \
\Ay + A2J
K
C5 = --
i0k
A2
v
d5
( K K9k\
\2A2 2Ay)
e5
( _ Ki \ f = Ks + Kk,
V Ay A2yJ ,f5 A2 2Ay,a6
Ks , K9 k ,
- A '&6
2 ( - + KS )
[Ay + A2J
c6
Kl Kk ri
K
i0k
A22 y
2Ay
Av
e6
2(± + Km)
\Ay + AV )
f6
K
i0k
AV
Write down the groups of equations (17) and (18), (21) and (22), (23) and (24) at all internal nodes of the grid for all t > 0. This splitting reduces the problem to systems of algebraic equations with tridiagonal matrices. At step 1 we solve the system for each row
k
i
We begin by integrating the energy equation with the boundary conditions reflecting the frequency of temperature change in the circumferential direction and the equality of temperatures on the surfaces of the lubricating layer shared with the journal (the Dirichlet conditions). Their difference analogs are
rqn 1i-i,k
rqn, Tn+i/2 _ Tn+i/2 T'n+i _ Tn+i qn _ T(t — Tc) Tn _ T
Ti,k, Ti-i,k = T i,k , Ti-i,k = T i,k , Ti,i = Ti i,i , Ti,k =
=,(t-tc)
Applying the sweep method first in the circumferential direction, we write the recurrence
T n+i/2
£i+i,j Ti,k i + Vi+i,k ■
Using the formulas of the left sweep, find the coefficients £i+1 and nm for all i and k, then perform the sweep in the radial direction.
The experience of many researchers has shown that implicit schemes of the method of alternating directions allow for big time intervals and speed up calculations in general. Douglas [10] successfully implemented a three-step scheme to integrate this kind of equation and showed that the scheme of the second order of accuracy O(A2, A2x, A2y, AD is certainly stable.
b
5
2
3. The Results
The first results concern the dynamics of the connecting rod bearing of an internal combustion engine of type 13/15.
During the calculation, at each point in time we obtain the three-dimensional
y
coordinate is shown in Figure 3, where for each element of the system "journal - lubricating layer - bushing" we chose 20 mesh elements.
The solution rests on a finite difference approximation. We neglected journal tilting. Solving the equation for hydrodynamic pressure, we allow the viscosity of the lubricant to depend on the second invariant of the shear rate and the resulting temperature distribution. Calculations show that the maximal temperature of the lubricating layer is found in the region of the largest hydrodynamic pressures.
Fig. 3. The distribution of temperature along the y coordinate
The results show that the heat exchange processes between the lubricating layer and the surrounding surfaces with convective heat transfer in the radial direction (see Fig. 3) reduce the mean temperature of the lubricant by 5 to 7 degrees. These results agree with the data of [13,14].
Conclusions
(1) We presented a model and an algorithm for calculating the thermal hydrodynamic characteristics of heavily loaded bearings. They enable us to account for the processes of heat exchange between the lubricating layer with the properties of a non-Xewtonian liquid and the elements of tribo-units.
(2) We proposed a solution algorithm based on a two-step difference approximation and an implicit scheme of the method of alternating directions, using which we can significantly increase the time intervals and decrease the running time of the calculation.
(3) The results show that in the design of heavily loaded tribo-units the models and algorithms we developed enable us to account for the rheological properties of the lubricant, the temperature distribution in the lubricating layer, and the thermal state of tribo-units as a whole.
Acknowledgements. This work was supported by a grant of the Ministry of Education and Science of the Russian Federation for applied research, code 2014-14-579-0109. The unique identifier for Applied Scientific R.esearch (project) is RFMEFI57714X0102. Agreement no. 14.577.21.0102.
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Received September 25, 2014
УДК 51-74; 532.5: 532.135; 621.822 DOI: 10.14529/mmpl40404
МОДЕЛИРОВАНИЕ И ОЦЕНКА ТЕПЛОВОГО СОСТОЯНИЯ СЛОЖНОНАГРУЖЕННЫХ ТРИБОСОПРЯЖЕНИЙ
Ю.В. Рождественский, Е.А. Задорожная
Тепловое состояние элементов сложнонагруженных трибосопряжений является одним из наиболее значимых параметров, влияющих на их работоспособность. Температура смазочного слоя подшипников скольжения во многом определяет его несущую способность. Использование уравнения теплового баланса для оценки температуры трибосопряжений не позволяет найти поля температур и зоны их максимальных значений. Этим определяется актуальность задачи. В статье рассмотрена математическая модель и методика расчета теплового состояния и термогидродинамических характеристик сложнонагруженных опор скольжения. При этом учитываются неньютоновские свойства жидкости, процессы теплообмена между смазочным слоем и окружающими его подвижными поверхностями трибосопряжения. Для решения уравнения энергии предложено использовать конечно-разностные аппроксимации. При построении разностных аналогов уравнений энергии для отдельных элементов конструкции и тонкого смазочного слоя был применен неявный метод переменных направлений Писмена-Рекфорда. Приведены результаты расчета гидромеханических характеристик шатунного подшипника теплового двигателя. В процессе расчета были получены трехмерные распределения температуры в смазочном материале.
Результаты показали, что при учете конвективного переноса тепла в радиальном направлении, процессы теплообмена между смазочным слоем и окружающими его подвижными поверхностями дают возможность более точно определить среднеин-тегральную температуру смазочного материала и теплонапряженность сопряжения в целом. Разработанная методика может быть использована при оценке характеристик и работоспособности сложнонагруженных трибосопряжения поршневых и роторных машин различного назначения.
Ключевые слова: опора жидкостного трения; обобщенное уравнение энергии; уравнения с частными производными; краевые задачи.
Литература
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Юрий Владимирович Рождественский, доктор технических наук, профессор, кафедра «Автомобильный транспорт и сервис автомобилей>, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Елена Анатольевна Задорожная, доктор технических наук, доцент, кафедра «Автомобильный транспорт и сервис автомобилей >, Южно-Уральский государственный университет (г. Челябинск, Российская Федерация), [email protected].
Поступила в редакцию 25 сентября 2014 г.
