analysis of the transient behavior of rubbing components
TRANSCRIPT
ANALYSIS OF THE TRANSIENT
BEHAVIOR OF RUBBING COMPONENTS
Mongi Ben Quezdou
R. L. Mullen
Department of Civil EngineeringCASE WESTERN RESERVE UNIVERSITY
Cleveland, Ohio 44106
Final Report NASA Contract NAG3-369Principal Investigator: R. L. Mullen
NASA Technical Off icersR. C. HendricksG. E. McDonald
( (NASA-CR-176546) A N A L Y S I S OF THE T R A N S I E N TBEHAVIOR OF R U B E I K G C C M P O N E N 1 S Final Eeport(Case Western Reserve Univ . ) 51 p
I HC A04/HF A01 CSC1 11GG3/27
N86-19464
(Judas05516
TABLE OF CONTENTS
Abstract i
List of Symbols ii
Introduction 1
Literature Review 3
Formulation of the Partial Differential
Equations 7
Finite Element Formulation 10
Problems and Solutions 16
Conclusions 30
References 31
Appendix I Shape Functions 33
Appendix II Stiffness Matrix 34
Appendix III Input Data Form 37
Publications Resulting from Grant 46
ANALYSIS OF THE TRANSIENT
BEHAVIOR OF RUBBING COMPONENTS
ABSTRACT
Finite element equations are developed for studying
deformations and temperatures resulting from frictional
heating in slid'ing system. The formulation is done for
l inear s teady s ta te motion in two d imens ions . The
equat ions include the e f f e c t of the ve loc i ty on the
moving components. This gives spurious oscillations in
their solutions by Galerkin finite element methods. A
method called "streamline upwind scheme" is used to try
to dea l with this d e f i c i e n c y . The f in i t e element
program is then used to invest igate the f r i c t i o n of
heating in gas path seal.
LIST OF SYMBOLS
C Specific heat at constant pressure
E Young's modulus
F Body force
H Hilbert spaces
k Thermal conductivityA
k Artificial conductivity
N Shape function
q Heat flux
T Absolute temperature
U Displacement
V Velocity
W Weighting function
a Thermal expansion coefficient
X Lame's constant
jj Shear modulus
p Density
£,r) Natural coordinates
v Poisson's ratio
a) Angular velocity
ii
Chapter 1
INTRODUCTION
The first law of thermodynamics expresses the energy balance
during mechanical and thermal process. In the analysis of rubbing
problem, the loss of mechanical energy (frictional energy) is
transformed in its largest percentage to thermal energy. During
high speed sliding, contact patches are formed. An analytical
treatment of stresses (or displacement) and temperature distribution
near the contact patches is necessary. A transient finite element
heat conduction analysis has shown (ref. 14) that within a very short
time after establishment of the contact zone the temperature
distribution approached a steady state relative to a stationary
observer. The length of time required to reach this quasi-steady
state is so short that it may be concluded that within the contact
patches a steady state temperature distribution occurs. Therefore
it is not necessary to do a transient temperature analysis. A finite
element formulation will be done for linear steady state. The
formulation will be given in a form that could be expanded to
inelastic, non-linear problems.
The advantage of the Finite Element Method is that it is
possible to model finite geometry of complex shapes or different
material properties. Both temperature and stress analysis could be
done by similar modeling. Indeed, a thermomechanical analysis could
be carried out using one element grid and two linked finite element
programs. The major difficulty in applying a Finite Element Method
is that the convection operators are nonsymmetric. For instance the
Galerkin Finite Element Method is successful when applied to linear
symmetric operators, but these methods usually give spurious
oscillations in their solutions when applied to convection dominated
problems. A "streamline upwind scheme" (ref. 10) is used to deal
with this problem by adding an artificial conductivity in a manner
which stabilize the solution without destroying the physics of the
problem.
In this work, a review of literature about the principal
subjects is given, followed by a formulation of the weak form for the
heat transfer equation and the thermoelasticity equation. Then a
finite element formulation is developped for both thermal and
thermoelastic equation for a two dimension solid. The resulting
finite element program, which gives the displacement and the
temperature distribution, is first compared to an analytical solution
such as a semi-infinite plane under a heat flux (ref. 1). The
program is then used to a problem of rubbing contact at high
velocities in a gas path seal.
Chapter 2
LITERATURE REVIEW
The heat transfer theory started with Fourier's law of heat
conduction:
i A dTq = -k A -j—^ dx
When the body is moving with a given velocity, a convection
term is added to the equation (ref. 13 & 17). The use of Galerkin
Finite Element Method to solve the heat problems for a moving body
give rise to spurious oscillations. These oscillations can be
removed in this case by severe mesh refinement which undermines the
practical utility of the methods (ref. 8, 9 and 10).
New schemes were developped trying to deal with this deficiency.
The first scheme appeared by Roache in ref.6 as a classical upwind
difference scheme. It has been noted that the Galerkin Finite
Element Method produces central difference type approximations to the
advection (conduction) term. In finite difference theory, the
adverse behavior of central differences in these circumstances has
long been noted. But this method was considered as inaccurate.
Heinrich proposed a new scheme (ref. 8). The Finite Element
Method is applied using weighted residual formulation with
bilinear quadrilateral element shape functions, and non-symmetric
weighting functions which are different from the shape
functions, and depend on parameters which allow the amount of
"upwinding" to be controlled. An increase in accuracy could be
obtained by varying these parameters from element to element. This
method is now known as Petro-Galerkin method. But the two
dimensionnal quadrilaterals proposed by ref. 8 distord the diffusion
(conduction) term when upwinding is applied. It seems very difficult
to find an upwinding function that does not disturb the diffusion
(conduction) operator, yet upwinds the advection (convection) term.
Hansen and Von Flotow (ref. 16) noted that it might be better to
apply upwind weighting to the advection (conduction) term only and
central to the remainder of the equation.
Another simpler technique proposed by Hushes (ref. 9) called
quadrature upwinding was based on moving the integration points in
the Galerkin Finite Element Method. But he came later with Brook
(ref. 10) to propose a new multi-dimensionnal upwind scheme. The
method was applied successfully to one dimension, and then
generalized for two dimensions. This method, called "streamline
upwind scheme", is applied to the advection-diffusion equation, and
then to Navier-Stokes equations.
The thermoelasticity equation is derived from the principles of
thermodynamics. A simple formulation of the equation is presented
in ref. 2 & 3 as:
uUitjj + (X+y) Ujfji - (3U2u) aTfi + Fi = 0
Together with the heat transfer equation, the thermoelasticity
equation (two equations in dimension two) leads to the determination
of the displacement and temperature fields in a thermomechanical
problem such as the problem of rubbing contact at high sliding
velocities.
The thermal analysis of bodies in sliding contact has attracted
the interest of many investigators because of its importance in many
situations in which friction occurs: bearing, seals, brakes,
clutches,... Various methods have been proposed, but none has proven
universally acceptable. Many surface temperature analyses have been
based on heat source methods (ref. 1), in which the solution for
temperature distribution due to a point source on a surface is used
to develop the solution for a distributed heat flux within a contact
patch on the surface of an infinite half space. The difficulties
involved with application of heat source techniques to bodies of
finite dimensions led to the development of integral transform
technique presented by Ling (ref. A). Although this method have
been successfully applied to a number of problems with different
geometries, its limitations to simple shapes and its mathematical
sophistication have kept this from being widely used by engineers.
Kennedy tried to solve the problem of rubbing contact at high
sliding velocities by using the Finite Element Method. He applied
the problem to two examples: aircraft disk brakes and gas path seals
in turbine engine (ref. 11). The first examples was presented
before in the study of transient temperature in disk brakes in
ref. 5, considered as one of the first documention that use Finite
Element Method in such problems. He retreated the second example
experimentally and analytically in ref. 1A & 15. In ref. 11,
Kennedy used one element grid and two linked finite element programs
to make a thermomechanical analysis of the contact.
Chapter 3
FORMULATION OF THE PARTIAL
DIFFERENTIAL EQUATION
3-1. Heat transfer:
Let k. . , the thermal conductivity, be constant, let p, the
density, and C , the specific heat be constant. Let q be the heat
flux. In a steady state, the heat transfer partial differential
equation for a moving body with a constant velocity V. is:
k.. T . . - p C V. T . + q = 0ij ,ij P i ti s
The first term represents the conduction heat transfer. The
second term represents the convection heat transfer. The problem
defined in the equation above could be given after applying
Galerkin's method, as:
Find T GH2 for all W G H° scuh that:
[ W kij T,ij - * P Cp V. Tf . + W q ] df> = 0
Where T is assumed to satisfy the essential boundary
conditions. An integration by parts allows us to rewrite the problem
as:
Find T e H1 for all W G H1 such that:
f [ W k. ..T . + W P C V. T . •] oft = [ WqJ0 'J *J •! P i ,1 J Jo
8
Here homogenous natural boundary conditions have been assumed
in those sections of the boundary where T is not specified.
3-2. Thermoelasticity:
The constants X and y are respectively Lame's constant and shear
modulus. They are related to Young's modulus E and Poisson's ratio
v by:
v E EX = ; y =
(1 + v) (1 - 2 v) 2 (1 + v)
The Navier's equation, with temperature changes, in terms of
displacement U is given by:
y lh .. + (X + y) U. ... - (3X + 2 y) a T);. + F4 = 0
Where a is the coeficient of the thermal expansion and F. is the body
force.
Applying the Galerkin method to the equation above within an
element results in the following formulation:
Find U e H2, T 6 H1 for all W G H° such that:
f [ W y IL .. + W (X + y) U. .± - W (3X + 2y ) a T ±
+ W Fi ] d°. = 0
After an integration by parts, we write the formulation as:
Find U G H1, T € H1 for all W e H such that:
f [ W . P U. . + W . (X + y) U. . + W (3X + 2y ) a T,i ]Jo »3 1»3 »x J»J
= I W F.Jo i
'Rdo.
This equation, called the weak form, could be obtained if the
law of conservation of energy is used.
Chapter 4
FINITE ELEMENT FORMULATION
4-1. Heat transfer:
4-1-1. Streamline upwind scheme:
The heat transfer equation has a convection operator which has
been shown to carry spurious oscillations in finite element solution
(ref. 10). The conventional approach to mitigate these oscillations
is to introduce an artificial diffusion (conduction) term in the
heat transfer equation. This method is called "streamline upwind
scheme".
The weak form becomes:
f [ W (k + k ) T + W p C V T ] dfi = f W q dQJQ »x J-J -"-J »J P i ti Jjj
A
If the artificial conductivity k. . is correctly chosen, no
oscillation will occur in the Galerkin Finite Element formulation.
In ref. 10, the following technique is presented:
A A A A
Assume k. . = k u. u .
A ^where u. = = with M u l l 2 = u. u. and u. = D C V.i ,,..,, l l ~ l l 1 1 i ^ p i
u.i
uA
k is a scalar artificial conductivity.
Assume that the coordinates are chosen such that locally xl-
direction is aligned with streamlines and the x2-direction is
perpendicular. Then the artificial conductivity matrix in this
10
11
coordinate system is:
k = k
In case of bilinear quadrilateral, in two dimension, k is chosen
as:
f. A
endr|define the location of the quadratic point and are given by
= coth a --
* 1n = coth a --
where J(2k)
are the element length (Fig. 1)
Figure 1. Typical four-node quadrilateral finite ele-ment geometry.
12
u and k are evaluated at the origin of the element in£-n
coordinates, a- could be expressed by:
PC V. h.p a 1
2 k
4-1-2. Heat transfer finite element formulation:
The weak form is know given by:
[ w§i (k. . + k. .) T> . + w p cp v. Tfl idn
If the body is divided into a number of finite elements, the
weighting function W and the temperature distribution T within a
discrete element may be approximated by:
W » W N
Where N-, is the shape (interpolation) function of the element.
W, and T~ are constant at the nodes. The majuscule subscripts I
indicate the node number.
Substituting these approximations into the weak form we get:
L [ wi »i.i (ku + v TJ NJJ + wi NI ( P CP v
TJ NJ.I ' dnit
= [ Wj Nj q dfiJfi
which may be simplified as follows:
f [NI§1 (k;n
=f N TJo -1
q
13
In the matrix notation, these equations has the following form:
[ K ] [ T ] = [ Q ]
dflwhere [ K ] = [ NT (k + k ) N. -f N. < P C V ) N. ]J -1*1 aJ *J J»J -1 P * J»J
is the stiffeness matrix.
[ T ] = Tt
[ Q ] =[ N, q dfiJo i
These equations are the finite element equations which will be
formed, assembled, and solved for the temperature distribution
through the body. Because of non-symmetry of the second term
in the stiffeness matrix [k], its presence require the use of
solution routines different from those used in most finite element
programs.
To compare the solution obtained with the solution without theA
use of the upwind scheme, we can just set k. . = 0. The same finite
element equations can also be used for both stationary and moving
components of a sliding system by setting V = 0 for elements in the
stationary body.
4-2. Thermoelasticity:
The weak form of the thermoelasticity equation was found to be:
•t w T dn
The weighting function W, the displacement distribution U, and
the temperature distribution T, within a discrete element may be
approximated by:
W = W, N,
U
where the majuscule subscripts indicate the node number and N are
the shape function defining the type of element. For some elements,
the shape functions are given in the appendix. WT, UT and T, are
constants at the nodes.
After substituting these approximations into the weak form, we
get:
+ Wj Nj (3A+ 2V ) a Tj Nj ± ] dJi =J Wj Nj Fj
which may be simplified as follows:
f [ NT (3A+ 2U) a NT .] dfi )TT = f NT FTJ 1 J,i J J i i
These equations can be written in the matrix notation as:
[ KE ] [ U ] + [ K£T ) [ T ] = [ F ]
15
where [ K^ ] = [ NT . V N . . + NT . ( * -I- V ) N . ] dft is the^ '^ ^
elastic stiffeness matrix.
[ U ] = U
[ K™ ] = [ NT (3 A + 2U ) a jo . ] dfi is the coupledJ Q J. J »1
stiffeness matrix.
[ T ] = T
[ F ] = f N F dJ 1 -1 fl
Based on the equations above and the heat transfer finite
element equations, defined in the previous section, a finite element
program has been written. This program, which will be used in a
thermomechanical analysis, will give the displacement and the
temperature distribution through the body. It should be noted that
one element grid can be used. With this program simpler problems
could be treated. For example:
-Heat transfer without thermal stresses problem (an example of
this problem is given later): Set no forces and a = 0.
-Elasticity problem without temperature change: Set no flux.
Some basic routines and a typical input data are given in the
appendix.
Chapter 5
PROBLEMS AND SOLUTIONS
5-1. Semi-infinite solid under heat friction:
5-1-1. Description of the problem:
The semi-infinite source strip is defined by: xe[-b,b];
ye[0,°°]; in the plane z = 0. The heat is applied at the
rate Q per unit time per unit area over the strip. The
surrounding media moves across it with velocity V in the
direction of the x-axis. This problem was treated by
Carslaw and Jaeger (Ref. 1) using the heat source method.
K-2b
Fig.2-Seroi-infinite solid under friction.
16
17
5-1-2. Finite element method:
The problem defined above is treated by the finite
element program. The elements used in the mesh are quadri-
lateral (linear or quadratic). The heat flux is applied
uniformly distributed on a width of 2b at the nodes. Four
meshes were used, which are:
- mesh 1: 200 elements, 231 nodes, four-node quadrilateral
element, with upwinding (Fig. 3).
- mesh 2: 200 elements, 231 nodes, four-node quadrilateral
element, without upwinding.
- mesh 3: 300 elements, 981 nodes, eight-node quadrilateral
element, with upwinding.
- mesh 4: 460 elements, 1493 nodes, eight-node quadrilateral
element, with upwinding.
5-1-3. Results:
Carslaw and Jaeger (Ref. 1) gave the temperature as the
following integral:
71 kV J x-B " "° ̂ T "• ' dU
where Ko(x) is the modified Bessel function of the second
kind of order zero and X, Z, B are dimensionless quantities
introduced as:
x.a.z.fe.B.a, with K . i-£.1*. f̂t. ZK \J \J_
P
Some values of the surface temperature of the solid
are shown in Fig. 4 for B = 10. The curve represents
18
EL
|5
_t-J_ I CL
IT?'
Q.-0
n> .J3^ '&. -FJfl.3-Linear quadrilateral mesh for a "semi-infinite solid under
heat friction".
19
I
u
2oum
§
IW3/A11JL
T3 II0) CQ
3 80 -H01 0)
01
C m-4 0)I -O
•^ -He ~»o> inu
o u•H
Si<0 0)
e A0) (N
o -a«2 '»V) o;*
20
(7iTkV/2KQ) vs. (x/b) given by Ref. 1. The results by finite
element method, for each mesh, is plotted on the same
representation.
5-1-4. Discussion:
When the quadratic elements (8 nodes) are used the curve
(TI TkV/2KQ) vs. (x/b) approaches the one given by the heat
source method (Ref. 1). When a smaller number of linear
elements (A nodes) is used the curve is less accurate but
has the same pattern. The oscillations in the solution are
minimized when the upwinding is used. But it is still
necessary to make a mesh refinement in order to get accurate
results.
5-2. Gas path seals:
5-2-1. Description of the problem:
The gas path seals are used in the turbine engines of
modern aircraft to prevent the axial flow of working fluid
(air) around rotating engine components. Reduction of the
clearances between rotating and stationary component of these
seals can decrease the consumption of the specific fuel and
increase the effeciency of the engine. However, such
reduction in clearance may result in occasional rubbing
between the rotating and the stationary seal components as
engine deflection occurs. These rubs, which occurs at very
high sliding speeds, can cause high surface temperatures,
excessive wear of the seal components, and possible damage
21
to the engine. The development of gas path seal designs
have been retarted by an incomplete understanding of the
temperatures, stresses, and deformations which occur during
high speed seal rubs (Ref. 11).
An attempt to get the temperature and deformation in this
sliding system is done using the finite element analysis.
A model has been developed to simulate the rubbing contact
which occurs in gas path seals between the rotating knife
edge and a stationary seal segment. The outer gas path seal,
assumed to have a circular section with 5 layers (Fig. 5 & 6),
is rotating at 20,000 r.p.m.. The friction force is taken as
100 lb.. The material properties are given in Table 1. On
the external surface the temperature is taken to be 70°F and
the displacement is zero. On the internal surface the flux,
considered concentrated at the lower point is taken to be:
q = N P V = 100 x (0.3) x (20,000 x 27tx A.85)
= 1.8284 x 107 BTU/min in2
where u is the friction coefficient.
5-2-2. Finite element method:
The thermomechanical program developed before is used
to the rubbing contract problem. A finite element mesh
based on quadratic quadrilateral (8 nodes) elements, shown in
Fig. 7, is used in the development of the model.
5-2-3. Results:
3-1. Temperature:
22
. Siaior
Fiq.5_Hiqh pressure lurbine ouler qas path sea!.
i
Fig.6-Cross section of outer gas path seal (units mm)
Table 1
Material properties
Layer
Int. radius (in.)
Ext. radius (in.)
Material 100 YSZ 85% YSZ 70% YSZ 40% YSZ MAR-M-15% CoCr 30% CoCr 60% CoCr 50gAl Y Al Y Al Y
Young's modulus 2.00 2.00 8.00 17.75 15.60(Ib/in2)xl06
1
A. 85
A. 91
2
A. 91
A.9A
3
A.9A
A. 97
A
A. 97
5.00
5
5.00
5.11
Poisson's ratio
Coef. of expansion(in/in°F)x!0~6
Thermal cond. .(BTU/min in°F)x!0
Density(lb/in3)
Specific heat-
0.25 .
A. 83
8.11
0.155
0.161
0.26
7.70
16.30
0.180
0.161
0.27
8.38
20.95
0.205
0.161
0.28
9.52
25.52
0.25A
0.158
0.30
12.20
A22.98
0.320
0.155o(BTU/lb"F)xlO
23
T=70°F, U=0.
u=20,000 rpm.
Fig.?- Finite element mesh for gas path seal friction problem.
25
The finite element results give the temperature at any
node. A plot of the temperature on the internal surface vs.
the angle near the contact is given in Fig. 8. A peak of
temperature occurs at the location of the friction contact.
3-2. Deformation:
The finite element results give the displacement in
x-direction and in y-direction of any node. The curve of the
deformation in x-direction vs. the angle for the interior
surface is given in Fig. 9. The displacement is positive for
the nodes at the right of the friction contact and negative
for the nodes at its left. The curve of the deformation in
y-direction vs. the angle is given in Fig. 10. This curve
presents a peak at the friction contact.
5-2-4. Discussion:
At the location of the friction contact, the temperature
and the deformation are increased rapidly. The temperature
might reach the melting point of the material. The
temperature peak was predicted by Harsher (Ref. 12) and
confirmed by Kennedy (Ref. 11). The latter conformed also
that the maximum amount of deformation occurs on the contact
surface, with magnitudes decreasing very rapidly in a
direction normal to the surface.
The oscillations that appear clearly in the x-direction
displacement (midside node) are due to the presence of the
oscillations in the temperature distribution. When the
26
scale
270
Fig.B-Internal temperature distribution under friction nearcontact.
27
£!')
Fig.9-Displa(.ement in x-direction near contact.
28
scalp;
250 270 290Fig.lO-Displacement in y-direction near contact.
upwjnding term is increased, the magnitude of temperature
and displacement decrease but the oscillations don't vanish.
Chapter 6
CONCLUSIONS
A finite element program has been developed which can
efficiently and accurately predict temperature and deformation
distribution in a thermomechanical problem such as sliding systems.
The method includes velocity effects with the use of the streamline
upwind scheme which eliminates spurious oscillations and is therefore
very useful in cases involving high speed sliding. The comparison
with some results that used heat source method shows the accuracy
of the method. A special application to gas path seal problem shows
the existence of a peak in the temperature and deformation near the
contact, indicating a possibility of melting and excessive wear
which might provoke an early thermomechanical failure. Further
studies could investigate the importance of thermal conductivity of
the moving and stationary component on decreasing surface
temperature and deformation.
30
REFERENCES
1. H.S. Carslaw & J.C. Jaeger: "Conduction of heat in solids". 2ndedition. Clarendon Press, Oxford, (1959).
2. B.A. Boley & J.H. Weiner: "Theory of thermal stresses". Wiley Inc,(1960).
3. A.D. Kovalenko: "Thermoelasticity: Basic theory and application".Translated from Russian by D.B. Macvean. Wolters-Noordhoffpublishing Groningen. Netherlands. (1969)
4. F.F. Ling: "Surface mechanics". Wiley, N.Y., (1973).
5. F.E. Kennedy & F.F. Ling: "A thermal, thermoelastic, and wearsimulation of a high-energy sliding contact problem". Jour. Lubr.Tech., Vol. 97, 497-508, (jul. 1974).
6. P.J. Roache: "Computationnal fluid dynamics". Hermosa publishers.Alburquerque, (1976).
7. I. Christie, D.F. Griffiths, A.R. Mitchell, and O.C. Zienkiewicz:"Finite element methods for second order differential equationwith significant first derivatives". Int. Jour. Num. Meth. Engng.,Vol. 10, 1389-1396, (1976).
8. J.C. Heinrich and O.C. Zienkiewicz: "Quadratic finite elementschsemes for two-dimensionnal convective-transport problems". Int.Jour. Num. Meth. Engng., Vol. 12, 1359-1365, (1978).
9. T.J.R. Hughes: "A simple scheme for developping 'upwind' finiteelement". Int. Jour. Num. Meth. Engng. Vol. 12. 1359-1365. (1978).
10.T.J.R. Hughes and A. Brooks: "A multi-dimensionnal upwind schemewith no crosswind diffusion". California institue of technology,Pasedena. CA.
11.F.E. Kennedy: "Thermomechanical phenomena in high speed rubbing".Wear, 59. 149-163, (1980).
12.W.D. Marscher: "A phenomenological model of abradable wear inhigh performance turbomachinery". Wear, 59. 191-211. (1980).
13.J.P. Holman: "Heat transfer". Chap 1 & 5. Me Craw-Hill Inc.(1980).
31
32
14.F.E. Kennedy: "Surface temperature in slinding systems. A finiteelement analysis." Jour. Lubr. Tech., Vol. 103, 90-96.(Jan. 1981).
15.F.E. Kennedy: "Single pass rub phenomena : Analysis and experi-ment". Jour. Lubr. Tech., Vol. 104, pp582. (Oct. 1982)
16.J.S. Hansen & A.H. Von Flotow:"Finite element operators: In-expensive evaluation of upwind schemes". Int. Jour. Num. Meth.Engng., Vol. 18, 77-88. (1982).
17.H.Wolf:"Heat transfer". Harper & row publishers, N.Y., (1983).
18.R.L. Mullen & R.C. Hendricks: "Finite element formulation fortransient heat treat problems". NASA. Technical Memorandum 83070,(Mar. 1983).
Appendix I
SHAPE FUNCTIONS NI
1-Linear quadrilateral element:
N = id -€) (i -n)(-1,1)
N2 = i (i (i -n)
N3 = i (i +€) (1 + n)
-
3(1,1)
2-Quadratic quadrilateral element:
d+C)(i-n
N7 = i d-62 )d+n )
N8 = i d-n2 )(!-€)
(-1,1)
5
-f3(l,l)
33
ORIGINAL PAGE ISOF POOR
Appendix II
STIFFENESS MATRIX ROUTINE
0001COO.'COC-3occ<ooosCCOtO O C 7ooc-tCOOS'00 JOco:i?*!?(•'.•KC-t'liO O i iC'017O C J £001P
OOl'O0021002:-P022?E*et".C*002?
'«Co5'l00300031oo?r00?3C034C03S
S03<C37002tOC7-V0040OC<100420043sm00440047w»0050I»OM
S°oll0054005tm*«0||
oo«:00<300t4OOti00610067006660690070
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35
ORIGINAL PAGE ISOF POOR QUALITY
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Appendix III
INPUT DATA FORMAT
Columns Format Descriptiontype v
\ 1-80 20A4 Title to be printed at beginning ofoutput.
2 1-5 15 (Number of elements in this problem.5-10 15 Number of nodes in this problem.
3 1-10 G10.4 Young's modulus for material type 1.11-20 G10.4 Yount's modulus for material type 2.
51-60 G10.4 Young's modulus for material type 6.
1-10 G10.4 Poisson's ratio for material type 1.11-20 G10.4 Poisson's ratio for material type 2.
51-60 G10.4 Poisson's ratio for material type 6.
1-10 G10.4 Expansion coefficient for materialtype 1.
11-20 G10.4 Expansion coefficient for materialtype 2.
51-60 G10.4 Expansion coefficient for materialtype 6.
1-10 G10.4 Conductivity in x-direction formaterial type 1.
11-20 G10.4 Conductivity in x-direction formaterial type 2.
51-60 G10.4 Conductivity in x-direction for
37
38
material typo 6.
1-10 G10.4 Conductivity in y-direction formaterial type 1.
11-20 G10.4 Conductivity in y-direction formaterial type 2.
51-60 G10.A Conductivity in y-direction formaterial type 6.
8 1-10 G10.4 Density for material type 1.11-20 G10.4 Density for material type 2.
51-60 G10.4 Density for material type 2.
1-10 G10.A Specific heat for material type 1.11-20 G10.4 Specific heat for material type 2.
51-60 G10.A Specific heat for material type 6.
10 1-10 G10.A Velocity in x-direction for materialtype 1.
11-20 G10.4 Velocity in x-direction for materialtype 2.
51-60 G10.4 Velocity in x-direction for materialtype 6.
11 1-10 G10.4 Velocity in y-direction for materialtype 1.
11-21 G10.4 Velocity in y-direction for materialtype 2.
51-60 G10.4 Velocity in y-direction for materialtype 6.
12 1-10 G10.4 Element height for material type 1.11-21 G10.4 Element he ight for mater ial type 2.
39
51-60 G10.4 Element heigh for material type 6.
13 1-5 15 Node number9-9 II Displacement fixity in x-direction at
this node= 0 applied traction= 1 applied displacement
10-10 II Displacement fixity in y-direction atthis node.
11-11 II Temperature fixity at this node= 0 given flux=1 no temperature
12-21 G10.4 x coordinate of node.22-31 G10.4 y coordinate of node32-41 G10.4 Force at node in x-direction42-51 G10.4 Force at node in y-direction52-61 G10.4 Flux or temperature at node
14 1-5 15 Element number6-10 15 Node number for first node on element11-15 15 Node number for second node on element
41-50 15 Node number for eigth node on element57-57 II Material type for element64-64 II Element type
Data cards can be omitted for nodes equispaced between node N.
and N. as long as the data for nodes N. and N. are included.J J
ORIGINAL PAGE fSOC POOR QUALITY
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PUBLICATIONS
NASA Grant NAG3-369
R. L. Mullen and R. C. Hendricks, "A 3-Body Approach to FrictionContact Modelling", to be presented 22nd Annual Meeting ofthe Society of Engineering Sciences, Pennsylvania StateUniversity, University Park, October, 1985.
R. C. Hendricks, M. J. Braun, R. L. Wheeler III and R. L. Mullen"Two-Phase Plows With Ambient Pressure above the. Therrao-dynamic Critical Pressure", Bently Roter Dynamics ResearchSymposium of Instability in Rotating Machinery, CarsonCity NV, 1985.
R. C. Hendricks, M. J. Braun ,R. L. Mullen, R. E. Bucham andW. A. Diamond, "Analysis of Experimental Shaft Seal Datafor High-Perromance Turbomachines As for Space ShuttleMain Engines", Proceedings Workshop on Heat and MassTransfer in Rotating Systems, 1985.
M. J . 'Braun, R. L. Mullen, Andre Prekwas and R. C. Hendricks,"Finite Difference Solution for a Generalized ReynoldsEquation with Homogeneous Two-Phase Flow", ProceedingsWorkshop on Heat and Mass Transfer in Rotating Systems,ASME,1985.
_R._L._Mul.len, Andre Prekwas. M. J. Braun and R. C. Hendricks,"Finite Element and Finite Difference Methods for aReynolds Equation using a Power Law Fluid", Workshopon Heat and Mass Transfer in Rotating Systems, ASME.1985.
R.L. Mullen, R.C. Hendricks, G. McDonald, "Finite ElementAnalysis of Residual Stresses in Plasma-Sprayed Ceramics",9th Annual Conference on Composites and Advanced CeramicMaterials, 1985 (Also in press, Ceramic Engineering andScience Proceedings) 1985.
Robert L. Mullen, M.J. Braun and R.C. Hendricks "Finite ElementSolutions for the Stiffness and Damping of A Three-Dimensional Journal Bearing Using a Non-Newtonian Fluid"Proceedings of the 10th Canadian Congress of AppliedMechanics Volume 2 pp. E-23,London, Ontario, 1985.
M . J . Braun, R .L . M u l l e n , R . C . Hendr icks , Robert L . Wheeler ,"Fluid Flow and Heat Transfer in Annuli with Axial PressureGradient a Non-Isothermal Rotating Inner Cylinder and anInsulated Outer Cylinder", to be presented at the 1985 ASMENational Heat Transfer Conference, 1985.
R. L. Mullen, "Quasi Eulerian Formulation of Thermal ElasticContact Problem Involving Moving Loads". PresentedWinter Annual Meeting ASME, 1984.
47
M. J. Braun, M. L. Adams, and R. L. Mullen, "Analysis of a TwoRow Hydrostatic Journal Bearing with Variable PropertiesInertia Ef fec t s and Surface Roughness". To be publishedIsrael J. Of Technology and presented at 18th Israel Con-ference on Mechanical Engineering, June 27, 1984 Haifa .
J. Padovan, B. Chung, M. J. Braun, R. L. Mullen "An ImprovedPlastic Flow Thermomechanical Algorithm and Heat-TransferAnalysis for Plasma-Sprayed Ceramics" , 8th AnnualConference on Composites and Advance Ceramic Materials,American Ceramic Society, 1984. To be published CeramicEngineering & Science Proceedings. ;
R. L. Mullen, M. J. Braun, and R. C. Hendricks, "Finite ElementFormulation of a Reynolds Type Equation for a Power LawFluid", presented, Fifth International Symposium onFinite Element Methods in Flow Problems, Austin Texas,1984.
M. J. Braun, R. C. Hendricks, and R. L. Mullen."Studies of TwoPhase Flow in Hydrostatic Journal Bearings", 7th AnnualEnergy Sources Conference, Cavitation and Multi-PhaseFlow,FED-9 ASME, pp. 61-65 , 1984.
M. J. Braun, R. L. Mullen, and R. C. Hendricks,"A ParametricStudy of Pressure and Temperature in a Narrow Gap Between aStationary Outer Cylinder and an Inner Rotating Mis-alignedShaft", presented National Heat Transfer Conference, 1984.
M. J. Braun, R. L. Mul l en , and R. C. Hendricks , "A ThreeDimensional Numerical Method with Applications toTribological Problems", Proceedings of the 10th Leeds-LyonSymposium on Tribology, 1983.
R. L. Mullen and R. C. Hendricks, "Finite Element Formulationfor Transient Heat Treat Problems", ASME-JSME ThermalEngineering Joint Conference Proceedings, Volume 3, 1983,pp. 367-374.(also NASA Technical Memorandum 83070).