JOURNAL OF MECHANICAL ENGINEERING( VOL.9.nO.1 ) is annual Journal published in the field of space technolgies and related sciences by Space Society of Mechanical Engineers, (SSME). The Journal Editorial Board is chaired by A R Srinivas, Scientist in Indian Space Organization, (ISRO)

1. ISSN 2249-0531
JOURNAL OF
MECHANICAL
ENGINEERING VOL. 9, NO. 1, 2011
A Publication of Space Society of Mechanical Engineers
SPACE SOCIETY OF MECHANICAL ENGINEERS
SPACE APPLICATIONS CENTRE, ISRO
AHMEDABAD 380 015

2. SPACE SOCIETY OF MECHANICAL ENGINEERS
Reg. No: GUJ-5390
EXECUTIVE COMMITTEE, SSME JME EDITORIAL BOARD
2010-2012 2010-2012
President : Sh. A C Mathur Chairman : Sh. A R Srinivas
acmathur@sac.isro.gov.in arsrinivas@sac.isro.gov.in
Phone: 079-2691 2106 Phone : 079-2691 5284/97
Vice President: Sh. C. P. Dewan
dewanchirag@sac.isro.gov.in Member Secretary: Sh. Hemant Arora
Phone : 079-2691 4310/27 hemant_arora@sac.isro.gov.in
Phone : 079-2691 5289
Secretary : Sh. S. G. Vaishnav
sgv@sac.isro.gov.in
Phone: 079-2691 2112
Members : Sh. Nitin Sharma
Sh. B. Satyanarayana
Jt. Secretary : Sh. Sammir Sakhare Sh. Gaurav Sharma
sammir@sac.isro.gov.in
Phone: 079-2691 2144 Sh. Anurag Verma
Sh. Durai C. Raju
Treasurer : Sh. Vimal M. Shah
vmshah@sac.isro.gov.in Sh. Rahul Dev
Phone: 079-2691 4505 Sh. Bhavik Shah
Sh. Ravi Prakash
Members : Sh. G. Gupta
Sh. S. R. Joshi
Sh. N. J. Bhatt JME Advisory Review Board
Sh. Bindav Pandya • Dr. Bishak Bhattacharya
(Asso. Prof., Dept. of Mech. Engg., IIT Kanpur)
Sh. R. G. Nipane
Sh. Dinesh Nolakha • Dr. S. C. Jain
(Prof. Dept. of Mech. Engg., IIT Roorkey)
Sh. Amit Agarwal
Sh. Deepak Yadav • Dr. P. Seshu
(Prof. Dept. of Mech. Engg., IIT Bombay)
• Dr. P. S. Nair
(Ex. Dy. Director, ISAC Banglore)
Ex. Officio Members: Dr. PVBAS Sarma • Mr. Bhanu Pant
(Ex. President)
(Sci./Engr. ‘SG’, VSSC Trivandrum)
Dr. B.S.Munjal
(Ex. Vice President) • Sh. D. J. Dave
Sh. Ulkesh. B. Desai (Ex. Prof. & Head, Applied Mechanics, GCET
(Ex. Secretary) College of Engg., Vallabh Vidhyanagar)
Address for correspondence Address for correspondence
The Secretary, The Chairman, JME
Space Society of Mechanical Engineers Space Society of Mechanical Engineers
Building No. 21, Room no. 12 Building No. 52, Room no. 84
Space Application Centre-ISRO, Space Application Centre-ISRO,
Jodhpur Tekra PO, Jodhpur Tekra PO,
Ahmedabad-380015, Gujarat, India Ahmedabad-380015, Gujarat, India
Ph. No. +9179-26912112 Ph. No. +9179-26915284/89/97
Title Design: Simulations from this issue of the Journal and Galileo IOV satellite launch from Soyuz Rocket, Credits: European Space Agency/ Science Photo Library

3. From the Editor’s Desk
“Simulation" the life blood of Scientists, Engineers and Academia.
A. R. Srinivas, Chairman, JME, arsrinivasu@yahoo.com
Way back in early eighties, when I was in my high school, my science teacher gave me a target of
developing a model to display how atoms/molecules move in matter. After a lot of scratching of
heads and scribbling with sketches on papers, I ultimately could convey my idea to a carpenter
who developed a wooden box with a glass on one end which is pasted with used photographic
protection black rolls to protect the light from passing through except from the elliptic orbit cut in
the rolls. I further managed to have a point source of light on the back side of the box with a steel
L-bent shaft which carried a wooden frame with a plus sign cut out and pasted with color
translucent paper. When I rotated the shaft with the lamp lit, one could see an illuminated dot
following the orbit thus resembling electrons moving around nucleus. Though my target was met
and appreciated by the teachers and friends at that time, now a sneak peek into you my past
leaves me intimidating.
Those were the days when visual and virtual reality was not known to many of us. I was unable to
Editorial
comprehend the power and need of having what was later referred as “Simulation”. Now 25
years since then I did take yet another look at the metamorphosis, simulation industry underwent
particularly in scientific and engineering worlds. Simulation was the name of the game which has
been incubated, nurtured and transformed over the years into a Multi Billion Dollar Industry.
Now one can virtually simulate anything, from ‘melting of metal, solidification process’, to ‘solar
system simulation’. These tools have now the life blood to scientists, engineers, academia etc.
Thanks to efforts of many researchers and their contributions in computational sciences which led
to spectrum of technologies in computerized behavioral simulations. The cover page of this
journal illustrates proofs of heights the simulation industry has reached.
This issue, Volume 9 of the JME also orients on simulation efforts done by many of our authors
around the country. Though use of Finite Elements and Difference based numerical methods has
almost become inevitable in many of our day to day engineering and scientific problems, an in-
depth analysis of effective utilization of these methods and functions never can see the dead end.
In similar simulative efforts here, Paul Murugan presented coupled FEA for Peizo electric
composites, Vishal has simplified hydrodynamic gas bearing simulation, Naimesh contributed on
Shock simulation, Krunal on simulation & optimization of mirrors, Mavani came out with
simulation of thermo-elastic cases, Veeresha, Rajesha and Durai performed simulations for space
subsystems like reaction wheels, antenna mechanisms and x-ray polarimeters. Munish and
Meena have performed experimental simulations to establish the basis for facts. Hope these
contributions would add additional feathers to Simulation and help you directly or by deriving
analogues.
I now, take the privilege to announce the launch of JME Vol.9, No.1, 2011 on behalf of the
editorial board. I congratulate all the authors and thank all the advertisers for their respective
contributions. The Volume 10 of the JME is in editing stage, you can still join with your
contributions on real-life systems design and complex turnkey project implementations challenges
you participate.
While our continuous endeavors are for improving the quality, frequency and standard of the
Journal it would not be possible without your (authors/readers) suggestions/comments/
appreciations. Do write to us….!.

4. JME Vol. 9 No. 1, 2011
CONTENTS
1. COUPLED FIELD FINITE ELEMENT ANALYSIS (CFFEA) OF PIEZO
ELECTRIC COMPOSITE
Paul Murugan J, Thomas Kurian, Srinivasn V
.........1-10
2. SIMULATION OF HYDRODYNAMIC GAS BEARING STATIC
CHARACTERISTICS USING MATLAB
Vishal Ahlawat, S. K. Verma, K. D. Gupta
.........11-16
3. EXPERIMENTAL SET-UP FOR ELECTRONIC CIRCUIT COOLING USING
SHARP EDGED WAVY PLATE
Munish Gupta, Prithpal Singh
.........17-22
4. 'SRS' SIMULATION ON SPACE BORNE OPTO-MECHANICAL PAYLOAD
Naimesh Patel , A.P.Vora, S.R.Joshi , C.P.Dewan , D.Subrahmanyam
.........23-28
5. DESIGN OPTIMIZATION OF A MIRROR SEGMENT FOR PRIMARY
SEGMENTED MIRROR
Krunal Shah, Vijay Chaudhary, Hemant Arora, A.R.Srinivas
.........29-34
6. THERMO-ELASTIC ANALYSIS OF SPACEBORNE ELECTRONIC PACKAGE
Hiren H. Mavani, Satish B.Sharma, Anup P. Vora, C. P. Dewan, D Subrahmanayam
........35-40
7. THERMAL MANAGEMENT OF REACTION WHEELS OF ASTROSAT
SPACECRAFT
D. R. Veeresha, B. Thrinatha Reddy, Venkata Narayana, Randhir Rai, Venkata Raghavendra, S. G.
Barve
.........41-47
8. THERMAL CONTROL SYSTEM FOR KA BAND ANTENNA
S. Rajesha Kumar*, Chaitanya B. S., AnujSoral, R. Varaprasad, S. G. Barve
.........48-54
9. DESIGN AND OPTIMIZATION OF X-RAY POLARIMETER DETECTOR
USING FINITE ELEMENT METHOD
R. Duraichelvan, A. R. Srinivas, C. M. Ateequlla, Biswajit Paul, P.V. Rishin, Ramanath Cowsik
.........55-64
10. CHALLENGES IN TEMPERATURE CYCLING TESTS OF SPACE
HARDWARE
B L Meena, J J Mistry .........65-70

5. JME Vol. 9 No. 1, 2011 JME09011101, COUPLED FIELD FINITE.....
COUPLED FIELD FINITE ELEMENT ANALYSIS (CFFEA) OF
PIEZO ELECTRIC COMPOSITE
Paul Murugan J*, Thomas Kurian, Srinivasn V
PRSO Entity,Vikram Sarabhai Space Centre,ISRO,Trivandrum, India
*
email: paulmurugan1iitm@gmail.com
strains since piezoelectric
1. Introduction ABSTRACT materials are capacitive in nature
and cannot measure continuous
Coupled field analysis can be used Simulation of smart structures is very static stresses. While static stress
for solving thermal-mechanical, similar to conventional structures except will cause an initial output, this
electrical–mechanical or the modeling has to take care of signal will slowly decay based on
electromagnetic-electro-mechanical additional complexities arising due to the piezoelectric material and time
problems using finite element the material properties. These are constant of the attached
methods (FEM).There is an reflected in constitutive laws in the form electronics. Ray et al [1] presented
increasing awareness of the of electro-mechanical coupling. From an analytical solution for static
benefits to be derived from the the modeling point of view, these analysis of simply supported
development and exploitation of complexities would lead to additional rectangular plate type smart
smart materials and structures in matrices in Finite Element method structure composed of composite
applications ranging from (FEM). Coupled field analysis through laminates coupled with actuators
hydrospace to aerospace. With the FEM gives the confidence before doing and sensors. Exact solutions are
ability to respond autonomously to the expensive experimental test on piezo obtained for cylindrical bending of
changes in their environment, electric system. Though many literature simply supported laminated plates
smart systems can offer a are available in the piezo composite, developed by [2,3,4]. A
simplified approach to the control this paper gives the basic ideas of segmented piezoelectric actuator
of various material and system Electrical-mechanical coupling in terms is simulated by applying an
characteristics such as light of analytical expressions with unique electric potential only over a part
transmission, viscosity, strain, notations, stiffness matrix for coupled of a distributed piezoelectric
noise and vibration etc. depending field element , Finite Element Modeling actuator [5,6,7]. Suresh et al [8,9]
on the smart materials used. Smart approach of piezo composite with developed a simple analytical
structures use actuators and sensors different substrate material and solution to study the flexural
at milli and micro-scales to achieve convergence results through mesh behavior of the smart panels
a certain goal. However, at present density. Finite element analysis of Piezo subjected to electro mechanical
time, the relevant control problems bimorph using coupled field element in loads. Most of these three-
in this area are poorly understood. ANSYS Multiphysics has been carried dimensional analytical solutions
There is a need for improved out and these results were compared are applicable only when the
sensing and actuation both at the with analytical solution. edges are simply supported and
material and systems level. Keywords: piezo, poling, Electro- subjected to specific types of
Research on smart structures is Mechanical, FEM, Coupling electric boundary conditions.
interdisciplinary because it Analytical material models
involves materials, structural currently used to evaluate the
mechanics, electronics, signal processing, response of metallic parts in ANSYS Multi Physics
communication and control. The goal of multi- Finite element package are adequate to characterize the
disciplinary research is to develop techniques to piezo electric materials compared to the shape memory
design, control, analyze, and visualize optimal or near alloys. In this paper analytical equation derived for
optimal smart and adaptive structures using newly finding the displacements and strains for piezo
developed smart materials. Piezo electric materials are composite bimorph beams and also the effect substrate
widely considered for smart structure design due to the material is studied. Finite element analysis using
fact that they are light weight and compact, relatively coupled field element in ANSYS has been carried out
inexpensive, and exhibit moderately linear field strain and these results were compared with analytical
relations at low drive levels. By bonding piezoelectric solution. The present study will be helpful for
actuators to structures, desired localized strains can be understanding the behavior of piezo electric system
induced by applying appropriate voltages to the which can be used for various applications.
actuators. Piezoelectric materials bonded to structures
can also be used as dynamic strain sensors. It should be
noted that piezoelectric sensors measure only dynamic
1

6. JME Vol. 9 No. 1, 2011 JME09011101, COUPLED FIELD FINITE.....
2. Poling Direction is loaded, it generates an electric field. In other words,
the above constitutive law demonstrates electro
The dipoles are oriented with respect to one another mechanical
through a process called poling. Poling direction is
indicated by ‘P’ in Fig.2. Poling requires that the
piezoelectric material be heated up above its Curie
temperature and then placed in a strong electric field
(typically, 2KV/mm).Heating the material allows the
dipoles to rotate freely, since the material is softer at Charge
flow
higher temperatures [10]. The electric field produces
an alignment of the dipoles along the direction of the
electric field. Quickly reducing the temperature and
removing the electric field produces a material whose Closed
electric dipoles are oriented in the same direction. This
direction is referred to as the poling direction of the
material.To increase or produce the electric field,
voltage needs to be increased or thickness of piezo Open
should be decreased. That is why piezo-electric
materials are of small thickness and in the range of 0.5
to 1mm by cosndiering the fabrication feasibility. The
poling direction always should be in thickness
direction or 3-direction to produce higher electric filed
with smaller input voltage. i.e if poling in thickness
direction, then electric field, E = V / t. If the poling in
Fig. 1:Typical diagram for stress-strain tests on a
width direction, then E = V/W. The width of square or
piezoelectric material to study the effect of boundary
rectangular piezo always higher than the thickness. So conditions (Reference:10)
higher voltage is to be applied to produce the more
electric field and it leads higher cost for making coupling, which is exploited for variety of structural
electrical setup. applications, such as vibration control, noise control,
shape control and structural health monitoring. The
3. Constitutive Equations electro mechanical coupling in the material is
The basic properties of a piezoelectric material are represented by the off diagonal terms of Eqn (3). A
expressed mathematically as a relationship between larger off-diagonal term will result in a material that
two mechanical variables, stress and strain, and two produces more strain for applied electric field and
electrical variables, electric field and electric more electric displacement for an applied mechanical
displacement. The direct and converse piezoelectric stress. For these reasons, the piezoelectric strain
effects are written as the set of linear equations. coefficient is an important parameter for comparing the
The constitutive equation in compact form relative strength of different types of piezoelectric
material. In the limit as d approaches zero, we are left
= C σ + d E ( Converse piezo -Actuation)
ε (1)
with a material that exhibits very little
electromechanical coupling. Eqn.(3) has expressed
= d σ + κ E (Direct Piezo – sensor )
D (2) with stress and electric field as the independent
variables and strain and electric displacement as the
These constitutive relationships would exist in a dependent variables. The above equation can be
material that was either purely elastic or purely inverted to write expression with stress and field as the
dielectric. The first part of Eqn. (1) represents the dependent variables and strain and the electric
displacement as the independent variables. Taking the
In matrix form, inverse of the 2x2 matrix produces the expression
1 C − d −1 K c 2  ε 
−1
σ   
ε  C d  σ    = 2     (4)
 E  1 − K c −d K c
−1 −1
 =  D
2
  (3)  k 
 D  d k   E  Where K c is piezo electric coupling coefficient,
stresses developed due to mechanical load, while the d
second part of the same equation gives the stresses due
Kc = . (5)
Ck
to voltage input. From Eqn. (1) and eqn. (2) it is clear
An important property of the piezoelectric coupling
that the structure will be stressed due to the application
coefficient is that it is always positive and bounded
of the electric field, even in the absence of mechanical
between 0 and 1. The bounds on the coupling
load [2]. Alternatively when the mechanical structure
2

7. JME Vol. 9 No. 1, 2011 JME09011101, COUPLED FIELD FINITE.....
coefficient are related to the energy conversion
properties in the piezo electric material and the bounds
={ε } { C E σ (1 − K c 2 ) (11)
0 and 1 represent the fact that only a fraction of the The fact that eqn (11) was derived assuming an open
energy is converted between mechanical and electrical circuit (D=0), now the relationship between the short
circuit and open circuit mechanical compliance can be
domains. The K c quantifies the electro mechanical
written as
energy conversion.
=CD { C E (1 − K c 2 ) (12)
3.1 Effect of Mechanical and Electrical An analogous relationship exists for specifying
Boundary conditions electrical quantities such as the dielectric permittivity.
The relationship between electrical displacement and
Electromechanical coupling in piezoelectric device
applied field changes depending on mechanical
gives rise to the fact that the properties of the material
boundary conditions. A stress free (σ = 0) condition is
are also a function of the mechanical and electrical
achieved by applying a field without mechanical
boundary conditions. Consider a piezoelectric cube in
constraints placed at the boundary of the piezoelectric
which the mechanical compliance C is being measured
material, whereas a strain free (ε = 0) condition is
by applying a known stress and measuring the induced
achieved by clamping both ends of the material such
strain. An important parameter here is the electrical
that there is zero motion. Performing an analysis
boundary conditions that exist between the opposing
similar to the one presented for the electrical boundary
faces. Assume for a short circuit condition in which the
condition, the expression can be arrived as
faces of the piezoelectric cube are connected directly
as shown in Fig.1.This electrical boundary condition = κ σ (1 − K c 2 )
κE (13)
results is zero field across the faces of the material but It is to be noted that, the piezoelectric strain coefficient
does not allow charge to flow form the positive (d) is independent of the mechanical or electrical
terminal to negative terminal. Substitute the condition boundary condition. It is observed from Eqn (12) and
E =0 in to Equation (3) results in the expression the (13) that the a material with coupling coefficient of 0.5
removal of second column. By performing the would be able to change the mechanical compliance by
experiment [10] when the electrical terminal is open 25%.
such that no charge flows between the faces of the
material. In this D =0 and the constitutive relationship z
P
in Eqn(4) reduces to E
σ=
1
1 − Kc2
{C −1}{ε } (6) Piezo
tp/2
Substrate
ts
1
K c 2 {ε }
x
E=
d {1 − K c }
2
(7) V
Piezo
tp/2
E
Inverting Eqn (6), we see that P
{ε } = {Cσ  Short circuit (8) Fig. 2 : Bimorph Piezo composite Actuator
= { Cσ (1 − K c 2 ) Open circuit
{ε } (9) 3.2 Material Properties
The result demonstrates that [10] the mechanical The PZT (Lead –Zirconate-titanate) or Pb (Zr,Ti)O3
compliance changes when the electrical boundary is the most commonly employed class of piezo ceramic
conditions is changed. The fact that K c 2 > 0 indicates for smart material application. These compounds are
that the mechanical compliance decreases when the comprised of PbTi1-x O3 and PbZrxO3 with x chosen
electrical boundary condition is changed from a short to optimize electro mechanical coupling [13]. The
circuit to open circuit condition. For this reason it is input material properties for PZT-5H piezo electric
improper to refer to the mechanical compliance ceramic is: PZT-5H has orthotropic material properties
without specifying the electrical boundary condition. It and it has 5 independent elastic constants.
is convenient to adopt a superscript to denote the Density = 7500 Kg/ m3
boundary condition associated with the measurement 1. Compliance matrix (C) (m2/N): size : 6x6
of a particular mechanical or electrical property. The C11 = C22 = 7.93 x 10-12 ; C12= C21 = 1.26 x10-11
superscript E or D denotes constant electric field and C13 = C31 = 1.18 x 10-11 ; C33 = 8.55 x10-12
constant electrical displacement respectively for C44 = C55 = C66 =4.35 x10-11
mechanical property. Rewriting the above equations 2. Dielectric or Permittivity matrix (κ) (Farad/meter):
using this notation produces size: 3x3
κ11= κ22 = 1.505 x 10-8
{ε } = {C D σ (10) κ33 = 1.301 x 10-8
3. Piezo-electric strain coefficient matrix or piezo-
3

8. JME Vol. 9 No. 1, 2011 JME09011101, COUPLED FIELD FINITE.....
electric constants (d)  t p 3 t p 2t s t p t s 2   t p2 t p ts 
(Coulomb/N): size : 3x6
Y1E wp K 
 24
+
8
+=
8 
 ∫ σ 1 z dydz + Y1E wp d13 

+
4 
 E3
  y,z  8 
d31 = d32 = 274 x 10-12 wp ts 3 (17)
d15 = d24 = 741 x 10-12 Ys K
12
= ∫ σ 1 z dydz
d33 = 591 x 10-12 y,z
Adding the results from the three domains together
3.3 Piezo composite bimorph bending yields
Actuator  t p 3 t p 2t s t p t s 2  wp ts 3  t p2 t p ts 
Y1E wp K 
 12
+
4
+
4 
 + Ys K =
12 ∫ σ 1 z dydz + Y1E wp d13 

+
2 
 E3
  y,z  4 
A piezoelectric bimorph is composed of two (18)
piezoelectric layers joined together with opposite
polarities. Piezoelectric bimorph are widely used for The integration of the stress component on the right
actuation and sensing. In the actuation mode, on the hand side of the expression in Eqn (18) is the
application of an electric field across the beam momentum resultant from the externally applied loads.
thickness, one layer contracts while the other expands. If this moment resultant is zero, we can solve the
This results in the bending of the entire structure and curvature as a function of
end deflection. In the sensing mode, the bimorph is
t 2 t t 
used to measure an external load by monitoring the Y1E  p + p s  d13 E3
 4 2 
piezoelectrically induced electrode voltages. Piezo
K=   (19)
composite bimorph consists of two piezo layer bonded
t 3
t t t t 
2 2
t 3 
on the top and bottom of the substrate. In this section Y1E  p + p s + s p  + Ys  s 
 12 4 
the analytical derivation of piezo electric composite for  4   12 
bending actuator is explained. A non dimensional expression for the curvature of the
composite beam due to piezo electric actuation is
3.3.1 Derivation of Analytical Equation for piezo
obtained by dividing the numerator and denominator
composite
by the inertia per unit width and substitute τ = ts / tp,
Assume Euler Bernoulli’s beam for deriving the strain  3τ 2
equation for piezo layer bonded with substrate [10] i.e
 ts   + 3τ 
 2 
composite piezo bimorph actuator shown in Fig.2 K = (20)
x-axis= 1- direction (length)  2d13 E3  τ 3  Ys  + 3τ 2 + 3τ + 1
y-axis = 2-direction (width)  E
 Y1 
z-axis = 3-direction (thickness- poling direction)
The relationship between the strain and the curvature is
From Eqn. (6), the curvature K can be obtained. Now
ε1(z) = K z (14)
we can get the strain at the interface between the piezo
Where, K = curvature = -d2u3(x)/dx2
layer and substrate
z = distance from neutral axis
Under the assumption that the field is in the poling
 ts  t  (21)
direction in the top layer and opposite to the poling ε1  at=
z = Kz K  s 
 =
direction in the bottom layer, we can write the  2  2 
constitutive equations for the piezo composite as
Strain at the outer surface (i.e it is the normal strain in
1-direction due to electric field in 3-direction) of the
1/Y1 σ1(z) + d13 E3 , ≤ z ≤ ( ts + t p )
E ts 1
2 2 piezo composite can be obtained as
ε1(z) = 1/Ys σ1(z) , −
ts t
≤z ≤ s (15) 
ε1  at z =
 ts t p  
+ =
( ts + t p ) (22)
2 2  K
  2 2  2
1/Y1E σ1(z) - d13 E3, − 1 ( ts + t p ) ≤ z ≤ − ts
2 2
Substitute Equn (14) in to the constitutive relations 3.3.2 Stiffness matrix for coupled field element
and rewriting, we obtain From Eqn (1) and (2), we can write the constitutive
Y1 (= σ 1 ( z ) + Y1 d13 E3
E
Kz ) E
Ys ( Kz ) = σ 1 ( z ) relation as
σ Sε − d T E
= and D dε − κ ε E
= (23)
Y1 (= σ 1 ( z ) − Y1 d13 E3
E
Kz ) E
(16)
Where S= stiffness matrix
Weak form from principle of Virtual work can be
The equilibrium expressions for the moment are written as [11]
obtained by multiplying above equation by z and
integrating over the domain in y and z. The results is
4

9. JME Vol. 9 No. 1, 2011 JME09011101, COUPLED FIELD FINITE.....
translational dof per node (8x3=24) and one voltage
or potential dof per node (8x1= 8). i.e
(24)
Essential Boundary conditions: u = u0 on su ;
φ = φ0 on sφ (25)
Natural boundary conditions: σ .n = t0 on st ;
D.n = − q0 on sq (26)
Nodal dof: displacement and potential at each node:
{u} = [ Nu ]{un } ; {φ} =  Nφ  {φn }
  (27)
Strain-displacement relation; {ε } = [ Bu ]{un } (28)
Electric field – potential or voltage relation;
{ E} = −  Bφ  {φn }
  (29)
Eqation (23) can be rewritten as
σ =S ( ε − ε s ) ε − d T E
D = d (ε − ε s ) − κ σ E + P s (30)
The unit conversion is expressed as above.
Substitute Eqn (28) and (29) in Equn (23) and Equn
3.3.3 Effect of substrate material
(24) we get respectively,
(31) The substrate material in piezo composite plays
σ = − d T E =[ Bu ]{un } + d T  Bφ  {φn }
Sε S   important role to control the bending strain. The
D =ε − κ ε E = [ Bu ]{un } − κ ε  Bφ  {φn }
d d  
stiffness ratio is (Ys/Yp) and thickness ratio is τ =
ts/tp). For same value of substrate thick but with diff
piezo thick, the lowest stiffness ratio gives higher
∫ [ B ]δ {u }(σ )dv − ∫  Bφ  δ {φ }( D )dv
u n  n (32)
curvature which gives more bending strain (Fig.3). At
large values of τ,we note that the induced strain is
= ∫ [ N ] δ {u }( t )ds + ∫  Nφ  δ {φ }( q )ds
  u n n small due to the fact that the substrate layer is much
thicker than the piezo layer. At small value of τ, the
δ {un } and
induced strain at the interface becomes very small due
Substitute (31) in (32) and rearranging to the small ts and the interface is becoming very
δ {φn } terms separately in two eqns and neglect close to the neutral axis of composite bimorph (Fig. 4).
higher order terms
4.0 Coupled Field Finite Element
∫ [ B ]  S  [ B ]{u }dv + ∫ [ B ] {d }  Bφ  {φ }dv = ∫ [ N ]{t}ds (33)
T E T
u   u n   u n u
Analysis
− ∫  Bφ  [ d ] [ B ]{u }dv + ∫  Bφ  [κ ]  Bφ  {φ }dv = ∫  Nφ  {q}ds
T T T T
  u  n     n
There are two type of analysis has been carried out.
In matrix form,  [ kuu ]  kuφ   un   F 
   (34) The 8-noded coupled field (solid5) element is
 = 
  kφu   kφφ   φn  Q  considered in both analyses. The model description
   
and the results are explained in this section.
Where
[ Kuu ] = ∫ [ Bu ] {S E } [ Bu ]dv
T
4.1 Piezo single layer subjected to
 K uφ  = ∫ [ Bu ] {d }  Bφ dv
T
(35)
    Voltage
 Kφu  = − ∫  Bφ  {d } [ Bu ]dv
T T
    First, the actuator effect of piezo ceramic single layer
by the application of voltage is studied. This analysis
 Kφφ  = ∫  Bφ  {κ }  Bφ dv
T
      was useful to validate the polarization axis, material
properties and electrical and mechanical boundary
For 8-noded coupled field brick element (Solid5 in conditions. The mechanical boundary conditions ( At
ANSYS) the size of the matrix is expressed with 3 y-axis, Ux = Uz=0 and at x- axis, Uy = Uz =0)
imposed on the PZT to avoid the rigid body motion i.e
5

10. JME Vol. 9 No. 1, 2011 JME09011101, COUPLED FIELD FINITE.....
PZT in the hung condition with exposing electric field of a bimorph actuator are chosen such that the electric
or voltage boundary condition. The dimensions of the field is in the same direction as the poling direction in
piezo electric plate are: Length of the piezo in x- one of the layers, whereas in the second layer the
direction (L) = 76 mm ; width(W) in y-direction = 26 electric field is in the direction opposite the poling
mm, thickness in z- direction (h) = 1mm; Z or direction.
thickness direction is the poling direction. Voltage
Table -2 : Comparison between the finite element and
boundary
results analytical solution
conditions are: zero voltage is applied at the nodes at
location z=0 and 100 volts is applied at the nodes of FE
location of z = 1 mm. The static analysis for the PZT- Displacement results Analytical %
5H actuator with considering the material properties (m) (Ansys) x10-6 Error
mentioned in section- 3.2 has been carried out. The x10-6
displacement in X-direction (Refer Fig 5) and their Ux (length
corresponding strains are obtained from FE results. 2.11 2.08
direction) 1.44
Mesh convergence study (Table- 1) was carried out by
reducing the size of the elements without altering the Uy (width
0.699 0.712
existing nodes location. It is converged for the aspect direction) 1.96
ratio (AR) of one. Table- 2 shows the displacement Uz (thickness
comparison between the analytical and FE results and 0.052 0.0591 0.33
direction)
the error within acceptable limits. The analytical
solution is obtained using [12, 13]. 1. Displacement in The brass aluminum, magnesium and Titanium used as
length direction, Ux = (d31xVxL)/h = 274x10-12 x 100 substrate material. Application of electric field to both
x 0.076 / 0.001= 2.11 x 10-6 m 2.Displacement in width layers produces extension in one of the layers and
direction, Uy = (d32xVxW)/h = 274x10-12 x 100 x contraction in the other. The net result is a bending of
0.026 / 0.001=0.699 x 10-6 m ( d32 = d31) 3. the material. Assuming a perfect bond between the
Displacement in thickness direction, Uy = (d33xVxh)/h inactive layer and piezo layers and assuming that the
= 591x10-12 x 100 x 0.001 / 0.001 =-0.052 x 10-6 m piezo layers are symmetric about the neutral axis of the
Composite, the bending will result in the deformed
Table-1: Mesh convergence table shape. Fig.6 shows the Finite element model with
voltage boundary conditions. The converged mesh is
Mesh Displacements (m) considered in this analysis also. The length and width of
density( the substrate and piezo layers are same and the
LxW) Ux Uy Uz thickness of the substrate is 0.002m and thickness of
for FE ( x10-6) ( x10-6) ( x10-6) each piezo layer is 0.001m. The voltage boundary
analysis conditions are: zero volts are applied in the nodes at z=
4x1 1.76 0.617 0.05 - (ts/2+tp/2) and 100 volts at z = - ts/2 and ts/2 (interface
between substrate and piezo) and again zero volts are
8x2 1.83 0.634 0.044 applied at z= ts/2+tp/2. Fig.7 shows the normal strain
16x4 1.95 0.670 0.045 distribution due to bending at outer surface of the piezo
composite for brass substrate. It is observed that the
32x8 2.02 0.691 0.0453
strain is more for magnesium substrate due to lower
64x16 2.05 0.702 0.044 stiffness compared to others. Fig.8 shows the normal
128x32 2.07 0.708 0.038 strains plot across thickness for aluminium substrate
which shows the typical behavior of the beam due to
76x26 bending. Convergence study also carried out by
(AR=1) 2.11 0.699 0.059 changing the mesh density across the substrate
thickness and the converged results only is tabulated.
Fig. 9 shows the strain comparison in X- direction
4. 2. Finite Element analysis of Bimorph across thickness (Neutral axis to –Z side). The strain at
piezo composite bending Actuator outer surface of the piezo composite is obtained using
Equn (22). Table-3 shows the comparison of the results
Composite piezoelectric device is useful for between the finite element solution and analytical
extensional actuation and the primary use of 31- solution and both the results were in good agreement.
multilayer piezoelectric actuators is as a bending
device. As discussed in section 3.3.1, a three layer 4.3 Piezo layer with laminated
device in which the piezo electric layers are fixed to
composites
the outer surfaces of an inactive substrate is typically
called a bimorph actuator. The electrical connections Beam like structural components made of composite
6

11. JME Vol. 9 No. 1, 2011 JME09011101, COUPLED FIELD FINITE.....
materials are being increasingly used in engineering causes the substrate to undergo deflection in opposite
applications. Laminated composite structures consist of direction to that caused by mechanical load alone. This
several layers of different fiber-reinforced laminae observation shows that the deflection can be reduced
bonded together to obtain desired structural properties. by applying an appropriate electrical potential on the
The desired structural properties are achieved by actuator. The stress variation across thickness of
varying the lamina thickness, lamina material substrate also decreased when the voltage is applied to
properties, and stacking sequence. With the availability the actuator (Refer Fig.14). This study will be useful in
of functional materials and feasibility of embedding or active shape control and vibration control for
bonding them to composite structures, new smart aerospace structures.
structural concepts have emerged for potentially high-
performance structural applications [14].The powerful
electro-mechanical coupling attribute of piezoelectric 5.0 Conclusion
materials enables these materials with laminated
composites to act as effective actuators [15]. Here the PZT widely used in many applications due to it is high
carbon fiber (T300) with epoxy resin and also graphite dielectric and piezoelectric strength, moderate cost and
with epoxy is used as substrate to study the behavior of broad range of operating temperatures. Electro
piezo layers. The dimensions for piezo and voltage mechanical behavior of these piezo electric patches
with different metallic substrate is studied through
boundary conditions are same as mentioned above analytical and coupled field analysis. Laminated
section except the following orthotropic material composites with piezo layers for simply supported
substrate. Material Properties for T300: E11= 130 GPa; condition also studied only through finite element
E22 =10 Gpa; G12 =5Gpa; µ12 = 0.35; Material analysis without analytical solution. This study gives
Properties for Graphite: E11= 172.4 GPa; E22 =6.89 the confidence of using piezo patches in the actual
Gpa; G12 =3.45Gpa; µ12 = 0.25; The thickness of structures for many applications. The future work is to
lamina (fiber+matrix) is 0.125 mm and thickness of study the delamination of piezo layers at the interface.
laminate (four layers with different angle) substrate is
0.5 mm. Layered solid [16] elements (solid46) are used Table -3: Strain comparison between the FE result and
analytical solution for different substrate
for modeling composite. The strains are more for T300
substrate due to lower modulus compared to graphite
strain at strain at
substrate. Table-4 shows the strain comparison for two
k outer outer (μs) %
different substrate which is bonded with piezo electric Substrate
(curvature) (μs)-FEM- surface- error
layer. The strains for graphite fiber composite substrate
(Ansys ) Analytical
with 0.5 mm thickness are lower than metallic
substrate of 2mm thickness. It is due to high specific 1. Brass 0.014069 28.8 28.14 2.2
stiffness of the fiber composite. 2. Aluminium 0.015317 29.9 30.63 2.4
3.
4.3.1 Piezo electric material with layered 0.016097 31.3 32.19 2.8
Magnesium
composite for simply supported condition
4. Titanium 0.014107 28.9 28.22 2.4
FEM of a simply supported plate type smart structure
is considered for static analysis, to determine the
displacement and stress for mechanical and electrical Table-4: Finite Element result for composite substrate
loads. The smart structure considered in this study is a with different ply sequence
composite substrate with 0/90/90/0 stacking sequence.
The piezoelectric material (PZT) is bonded on the top
Sl. Total strain
and bottom of the substrate. Top layer is taken as Ply Total strain
actuator. Actuator voltage is applied at the interface. N (µs)
(µs) for
100V is considered. The strains and stresses for sequence for Graphite +
0 T300+Epoxy
mechanical load only are shown Fig. 10 and Fig.11. Epoxy
Strain is continuous across thickness at the substrate- 1 0/45/45/0 -37.1 -27.6
actuator interface. Stress is discontinuous at the
2 0/90/90/0 -36.9 -27.5
substrate-actuator interface due to different Young’s
Modulus. Fig.12 shows the stresses at centre of the 3 0/30/30/0 -37.1 -27.5
substrate (L/2, W/2) across thickness (Z = - ts/2 to Z =
ts/2) for various loadings. It is observed from the 4 45/-45/-45/45 -36.8 -28.7
Fig.13 that the maximum transverse displacement of 5 0/45/60/90 -37.9 -29.2
the substrate decreases when 100 V is applied at
actuator interface along with uniform pressure load.
The actuator when applied with electrical potential
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12. JME Vol. 9 No. 1, 2011 JME09011101, COUPLED FIELD FINITE.....
0.6
0.5
kts/(2*d13*E3)
0.4
Ys/Yp=1.6
0.3 Ys/Yp=1.06
Ys/Yp=3.15l
0.2
0.1
0
0 1 0.5 0.75 2 4 6
ts/tp
Fig. 3:Curvature versus stiffness ratio
Voltage Vs Strains for (ts/tp=0.5)-Al substrate
250.00
225.00
200.00
175.00
150.00
strain
strain at outer
125.00
100.00
75.00
strain at interface Fig. 7: Normal strain (bending) at outer surface of piezo
50.00
25.00
composite for brass substrate
0.00
100 200 400 600 800
voltage (V)
Fig. 4:Strain at outer and interface of smart structure
Fig. 8: strains across thickness (-Z to + Z) in Aluminium
substrate
Fig. 5: Displacement (in ‘m’) in length or x-direction Strain in X-dir between FEM Vs Analytical(Al substrate)
3.50E-05
3.00E-05
strain in x-direction(m/m)
2.50E-05
2.00E-05
FEM
1.50E-05
Analytical
1.00E-05
5.00E-06
0.00E+00
0 0.0005 0.001 0.0015 0.002 0.0025
-5.00E-06
Distance from Neutral axis to (-)ve Z side
Fig. 9: Strain comparison in X-direction across thickness
for Al substrate (Neutral axis to –Z side)
Fig. 6: Finite Element model with boundary conditions
8

13. JME Vol. 9 No. 1, 2011 JME09011101, COUPLED FIELD FINITE.....
Fig. 10: strains across thickness (-Z to + Z) in composite Fig. 14: Stress variation across width at outer surface (+
plate Z side)
Acknowledgement
Authors wish to thank Prof. M.S Sivakumar and Dr.
Arockia Rajan, Applied Mechanics Dept. of IIT
Madras for their course work on Smart Materials and
Structures and their motivation. Authors wish to
acknowledge K. Pradeep, Scientist ‘SF; and Dr. R.
Suresh, Scientist ‘SF, VSSC, ISRO for thir valuable
suggestions.
Symbols
ts thickness of substrate
Fig. 11: stresses across thickness (-Z to + Z) in tp thickness of piezo layer
composite plate Y1 E Young’s modulus of piezo-layer
Ys Young’s modulus of substrate
stress across thickness of substrate for various loading conditions C Complinace matrix
κ Dielectric or Permittivity matrix
1.70E+07
d Piezo-electric strain coefficient
1.00E+07 matrix or piezo-electric constants
D Electric displacement or Electric flux
stresess (N/m2)
P=1 and V=0-Sx
3.00E+06
P=1 and V=100-Sx
P=1 and V=0-Sy density
-4.00E+06 0 0.0001 0.0002 0.0003 0.0004 0.0005
P=1 and V=100-Sy K curvature
-1.10E+07 z distance from neutral axis
-1.80E+07
Distance across thickness of substrate(m ) from Z=-ts/2 E Electric field
ε
to Z=ts/2
Strain
σ Stress
Fig. 12: Stress across thickness of substrate for various
Loadings Kc Piezoelectric coupling coefficient
Transverse displacement across thickness of substrate for various CD Compliance at constant Electric
loadings
displacement
-4.74E-05
0 0.0001 0.0002 0.0003 0.0004 0.0005 CE Compliance at constant Electric field
displacement(m)
S Stiffness matrix or Elasticity matrix
Transverse
-4.77E-05
P=1and V=0
-4.80E-05
P=1and V=100
References
-4.83E-05 [1] M.C.Ray, R. Bhattacharya, B. Samantha, “Exact
Distance across thickness of substrate(m) from Z=- solutions for static analysis of intelligent structures’,
ts/2 to Z=ts/2
AIAA J.vol.31, No.9, sep 1993.
[2] Heyliger, P., Brooks, S., 1996, “Exact Solutions
Fig. 13: Transverse displacement across thickness
of substrate for various loadings For Laminated Piezoelectric Plates In Cylindrical
Bending”, Journal of Applied Mechanics, Vol. 63, pp.
9

14. JME Vol. 9 No. 1, 2011 JME09011101, COUPLED FIELD FINITE.....
[3] Vel, S. S., Batra, R C., 2001, “Exact Solution in the academic year 2002. He has
forRectangular Sandwich Plates With Embedded completed his Masters degree
Piezoelectric Shear Actuators”, AIM Journal, Vol. 39, (M.Tech) in Machine Design
pp. 1363-1373. from the Indian Institute of
[4] Batra, R. C., Liang, X. Q., Yang, J. S., Technology (IIT- Madras). He
1996b,“Shape Control of Vibrating Simply Supported was the second topper in
Rectangular Plates”, AIAA Journal, Vol. 34, pp. 1 16- Machine Design stream in the
1 22 academic year 2010. He has
[5] Brooks, S., Heyliger, P., 1994, “Static Behavior of presented 12 technical papers in National and
Piezoelectric Laminates With Distributed and Patched International conferences. He has worked as Research
Actuators, Journal of Intelligent Material Systems and Fellow in Air Frame composite Stress Analysis group
Structures”, Vol. 5, pp. 635-646. in Aeronautical Development Agency (ADA),
[6] Batra, R. C., Liang, X. Q., Yang, J. S., 1996a, Bangalore. He is working as Scentist/Engineer in
“The Vibration of a Simply Supported Rectangular VSSC/ISRO from 2004 onwards in Solid Motors
Elastic Plate Due to Piezoelectric Actuators”, Group in the areas of design, analysis, realisation and
International Journal of Solids and Structures, Vol. 33, testing of solid motor hardware. His areas of interest
pp. 1597- 1 6 18. include Finite element methods and Analysis (FEM &
[7] Batra, R. C., Liang, X. Q., Yang, J. S., 1996b, FEA), Fracture Mechanics, Advanced Solid
“Shape Control of Vibrating Simply Supported Mechanics, Vibrations, Composites and Smart
Rectangular Plates”, AIAA Journal, Vol. 34, pp. 1 16- Structures.
1 22 Thomas Kurian is working in VSSC, ISRO since
[8] Suresh R, Gajbir singh, G. Venkatesawara rao,
Nov., 1992. He passed B.Tech from University of
“An analytical solution for the flexural response of
Kerala securing first rank in
intelligent composites and sandwich panels, Acta
Mechanical Engg. discipline. He
Mechanica 152, 81-93 (2001).
holds M.E (Mech Engg.) degree
[9] Suresh R, Gajbir singh, G. Venkatesawara rao,
from IISc Bangalore. He has got
“An investigation of flexural behavior of smart
experience in the areas of design,
composite panel subjected to elector mechanical
structural analysis, fabrication and
loads”, Proceedings of the international conference on
acceptance testing of Solid Rocket
smart materials, structures and systems, July 1999.
Motor hardware, pressure vessel
[10] Donald J. Leo, ‘Engineering analysis of smart
systems and fixtures. He has contributed significantly
material systems’, John Wiley & sons Inc, 2007.
towards the design and realization of Solid Rocket
[11] Smith R.C, Smart material systems: Model
Motor hardware related to PSLV, GSLV, LVM3, RLV-
development, Siam Publications, 2004.
TD and ATVP Projects. He is currently working as the
[12] Indira Priyadharshini, ‘Buckling control of thin
Deputy Head of Hardware Design and Realization
plates using PZT actuators, MS Thesis report, IIT
Division in Solid Motors Group, VSSC.
Madras, 2007.
[13] Vijay K Varadan,, Vinoy K J, Gopalakrishnan,
Smart material systems and MEMS: Design and Srinivasan V holds Bachelors degree in Chemical
Development methodologies, John Wiley &sons Engineering from NIT, trichy, University of Madras.
Ltd,,2006. He has completed his Masters degree (M.E.,) in
[14] S.J. Lee, J.N. Reddy, F. Rostam-Abadi, Nonlinear Chemical Engg. with distinction
finite element analysis of laminated composite shells from IISc Bangalore . He has held
with actuating layers, J of many positions like Dy. Project
Finite Elements in Analysis and Design 43 (2006) 1 – Diretor, Associate Project Director
21 and Project Director, S200 Project.
[15] Jinquan Cheng, Farid Taheri, A smart single-lap Concurrently he was Group
adhesive joint integrated with partially distributed Director, Solid Motors Group,
piezoelectric patches, International Journal of Solids VSSC. Currently he is Dy.
and Structures 43 (2006) 1079–1092. Director, Propulsion, Propellants & Space Ordinace
[16] ANSYS-10.0 help Manual (PRSO) Entity, VSSC. He received ISRO merit award-
2007 and Team excellence award-2009 for his
contribution towards design and development of large
About Authors solid boosters. He was awarded with the Eminent
Chemical Engineer-2011 award by IICHE, Kochi.
Shri Paul Murugan J holds Bachelors degree in
Mechanical Engineering from Madurai Kamaraj
University (MKU). He was an outstanding student and
over all topper in Mechanical Engineering department
10

15. JME Vol. 9 No. 1, 2011 JME09011102, SIMULATION OF HYDRODYNAMIC ...
SIMULATION OF HYDRODYNAMIC GAS BEARING
STATIC CHARACTERISTICS USING MATLAB
Vishal Ahlawat1*, S. K. Verma2, and K. D. Gupta2
1
Department of Mechanical Engineering, U.I.E.T., K. U. Kurukshetra, India
2
Departmentof Mechanical Engineering,D.C.R.U.S.T.,Murthal, India
*
email: mail2vishal1986@yahoo.co.in
1. Introduction of numerical analysis to the
ABSTRACT problems encountered in
Miniaturization in a lot of research hydrodynamic lubrication. A
domains has led to a demand for In this paper, the Reynolds equation is popular numerical technique, the
small-scale systems running at solved to determine the pressure ‘finite difference method’ is
high operational speeds. Gas distribution for hydrodynamic gas- introduced and its application to
bearings operate with the pressure lubricated journal bearings in which the hydrodynamic lubrication is
generated inside lubricating film. the compressibility effect must be taken demonstrated. Piekos and Breuer
Because gas bearings have such into account. The non-linear Reynolds [6] explored the effect of axially
characteristics as low friction, high equation for self-acting gas-lubricated varying clearance on
precision, and low pollution, they journals bearing is linearized through microfabricated gas journal
have been successfully used in appropriate approximation and a bearings. Taper and Bow types of
many commercial applications, modified Reynolds equation is derived clearance variation commonly
such as navigation systems, and solved by means of the finite observed in etched bearings. Peng
computer disk drives, high- difference method (FDM). The gas film [7] presented the development of
precision instruments and sensors, pressure distribution of a self-acting mathematical models and
dental drills, machine tools and gas-lubricated journal bearing is numerical schemes for simulating
turbo compressors [11]. The attained and the load capacity is the hydrodynamic pressure and
theory of lubrication had not been calculated. A computer code using temperature rise of compliant foil
extended to gas films. Except for MATLAB has been developed to find bearings lubricated by a thin gas
an approximate solution for an out the static performance film in between its compliant
infinitely long journal bearing, characteristics for gas-lubricated bearing surface and the rotating
neither mathematical solutions nor journal bearings. The results of the shaft. Lo et al. [8] presented a
appropriate experimental study show that the present solution is detailed theoretical analysis of
techniques had been developed. in better agreement with experimental bearing performance in which the
Now, with the theory verified and data which validates the formulation gas flow within the bearing is
some sophisticated mathematical and the computer program so initially expressed in the form of
tools at hand, along with electronic developed. simplified dimensionless Navier
equipment to aid experimentation, Stokes equations. Vleugels et al.
fundamental understanding and [9] presented an overview of the
optimization of gas bearings is Keywords: Self-acting gas journal
total rotor dynamic modelling
becoming possible. Gross [1] bearings, Reynolds equation, Finite
process of a micro-turbine rotor
found adequate mathematical tools difference method (FDM)
supported on aerostatic bearings.
and experimental facilities in a A both accurate and efficient
survey for verifying the theory of gas lubrication. He modelling technique is outlined to obtain static and
also found that the gas lubricants are used in three dynamic air bearing properties. Yang et al. [10] studied
basic bearing types: self- acting, externally pressurized, the non-linear stability of finite length self-acting gas
and squeeze-film. Chandra et al. [2] presented an journal bearings by solving a time-dependent Reynolds
exhaustive design data for the static and dynamic equation using finite difference method. Zhang et al.
characteristics of centrally loaded partial arc gas- [11] calculated the gas film pressure distribution of a
lubricated journal bearings. Florin Dimofte [3] self-acting (hydrodynamic) gas-lubricated journal
adopted, modified, and applied an alternating direction bearing and the load capacity is calculated. Taking a
implicit (ADI) method to the Reynolds equation for small pressure change in the gas film of self-acting
thin gas fluid films. An efficient code is developed to gas-lubricated journal bearings into account, the
predict both the steady-state and dynamic performance corresponding nonlinear Reynolds equation is
of an aerodynamic journal bearing. Faria and Andres linearized through appropriate approximation and a
[4] presented a numerical study of high-speed modified Reynolds equation is derived and solved by
hydrodynamic gas bearing performance using both the means of the finite difference method (FDM).
finite difference and finite element methods.
Stachowiak and Batchelor [5] described the application
11