ISSN 2249-0531JOURNAL OFMECHANICALENGINEERING             VOL. 9, NO. 1, 2011A Publication of Space Society of Mechanical ...
SPACE SOCIETY OF MECHANICAL ENGINEERS  Reg. No: GUJ-5390           EXECUTIVE COMMITTEE, SSME                              ...
From the Editor’s Desk              “Simulation" the life blood of Scientists, Engineers and Academia.              A. R. ...
JME Vol. 9 No. 1, 2011                                       CONTENTS1. COUPLED FIELD FINITE ELEMENT ANALYSIS (CFFEA) OF P...
JME Vol. 9 No. 1, 2011                                                        JME09011101, COUPLED FIELD FINITE.....COUPLE...
JME Vol. 9 No. 1, 2011                                                     JME09011101, COUPLED FIELD FINITE.....2. Poling...
JME Vol. 9 No. 1, 2011                                                           JME09011101, COUPLED FIELD FINITE.....   ...
JME Vol. 9 No. 1, 2011                                                                            JME09011101, COUPLED FIE...
JME Vol. 9 No. 1, 2011                                                                                                 JME...
JME Vol. 9 No. 1, 2011                                                     JME09011101, COUPLED FIELD FINITE.....PZT in th...
JME Vol. 9 No. 1, 2011                                                       JME09011101, COUPLED FIELD FINITE.....materia...
JME Vol. 9 No. 1, 2011                                                                                                    ...
JME Vol. 9 No. 1, 2011                                                                                              JME090...
JME Vol. 9 No. 1, 2011                                                   JME09011101, COUPLED FIELD FINITE.....[3] Vel, S....
JME Vol. 9 No. 1, 2011                                             JME09011102, SIMULATION OF HYDRODYNAMIC ...      SIMULA...
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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)

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JOURNAL OF MECHANICAL ENGINEERING, VOL.9 NO.1

  1. 1. ISSN 2249-0531JOURNAL OFMECHANICALENGINEERING VOL. 9, NO. 1, 2011A Publication of Space Society of Mechanical EngineersSPACE SOCIETY OF MECHANICAL ENGINEERS SPACE APPLICATIONS CENTRE, ISRO AHMEDABAD 380 015
  2. 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. +9179-26915284/89/97Title Design: Simulations from this issue of the Journal and Galileo IOV satellite launch from Soyuz Rocket, Credits: European Space Agency/ Science Photo Library
  3. 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 ofdeveloping a model to display how atoms/molecules move in matter. After a lot of scratching ofheads and scribbling with sketches on papers, I ultimately could convey my idea to a carpenterwho developed a wooden box with a glass on one end which is pasted with used photographicprotection black rolls to protect the light from passing through except from the elliptic orbit cut inthe rolls. I further managed to have a point source of light on the back side of the box with a steelL-bent shaft which carried a wooden frame with a plus sign cut out and pasted with colortranslucent paper. When I rotated the shaft with the lamp lit, one could see an illuminated dotfollowing the orbit thus resembling electrons moving around nucleus. Though my target was metand appreciated by the teachers and friends at that time, now a sneak peek into you my pastleaves me intimidating.Those were the days when visual and virtual reality was not known to many of us. I was unable to Editorialcomprehend the power and need of having what was later referred as “Simulation”. Now 25years since then I did take yet another look at the metamorphosis, simulation industry underwentparticularly in scientific and engineering worlds. Simulation was the name of the game which hasbeen 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 ‘solarsystem 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 ledto spectrum of technologies in computerized behavioral simulations. The cover page of thisjournal 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 authorsaround the country. Though use of Finite Elements and Difference based numerical methods hasalmost 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 electriccomposites, Vishal has simplified hydrodynamic gas bearing simulation, Naimesh contributed onShock simulation, Krunal on simulation & optimization of mirrors, Mavani came out withsimulation of thermo-elastic cases, Veeresha, Rajesha and Durai performed simulations for spacesubsystems like reaction wheels, antenna mechanisms and x-ray polarimeters. Munish andMeena have performed experimental simulations to establish the basis for facts. Hope thesecontributions would add additional feathers to Simulation and help you directly or by derivinganalogues. I now, take the privilege to announce the launch of JME Vol.9, No.1, 2011 on behalf of theeditorial board. I congratulate all the authors and thank all the advertisers for their respectivecontributions. The Volume 10 of the JME is in editing stage, you can still join with yourcontributions on real-life systems design and complex turnkey project implementations challengesyou participate.While our continuous endeavors are for improving the quality, frequency and standard of theJournal it would not be possible without your (authors/readers) suggestions/comments/appreciations. Do write to us….!.
  4. 4. JME Vol. 9 No. 1, 2011 CONTENTS1. COUPLED FIELD FINITE ELEMENT ANALYSIS (CFFEA) OF PIEZO ELECTRIC COMPOSITE Paul Murugan J, Thomas Kurian, Srinivasn V .........1-102. SIMULATION OF HYDRODYNAMIC GAS BEARING STATIC CHARACTERISTICS USING MATLAB Vishal Ahlawat, S. K. Verma, K. D. Gupta .........11-163. EXPERIMENTAL SET-UP FOR ELECTRONIC CIRCUIT COOLING USING SHARP EDGED WAVY PLATE Munish Gupta, Prithpal Singh .........17-224. SRS SIMULATION ON SPACE BORNE OPTO-MECHANICAL PAYLOAD Naimesh Patel , A.P.Vora, S.R.Joshi , C.P.Dewan , D.Subrahmanyam .........23-285. DESIGN OPTIMIZATION OF A MIRROR SEGMENT FOR PRIMARY SEGMENTED MIRROR Krunal Shah, Vijay Chaudhary, Hemant Arora, A.R.Srinivas .........29-346. THERMO-ELASTIC ANALYSIS OF SPACEBORNE ELECTRONIC PACKAGE Hiren H. Mavani, Satish B.Sharma, Anup P. Vora, C. P. Dewan, D Subrahmanayam ........35-407. THERMAL MANAGEMENT OF REACTION WHEELS OF ASTROSAT SPACECRAFT D. R. Veeresha, B. Thrinatha Reddy, Venkata Narayana, Randhir Rai, Venkata Raghavendra, S. G. Barve .........41-478. THERMAL CONTROL SYSTEM FOR KA BAND ANTENNA S. Rajesha Kumar*, Chaitanya B. S., AnujSoral, R. Varaprasad, S. G. Barve .........48-549. 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 .........55-6410. CHALLENGES IN TEMPERATURE CYCLING TESTS OF SPACE HARDWARE B L Meena, J J Mistry .........65-70
  5. 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 piezoelectric1. Introduction ABSTRACT materials are capacitive in nature and cannot measure continuousCoupled field analysis can be used Simulation of smart structures is very static stresses. While static stressfor solving thermal-mechanical, similar to conventional structures except will cause an initial output, thiselectrical–mechanical or the modeling has to take care of signal will slowly decay based onelectromagnetic-electro-mechanical additional complexities arising due to the piezoelectric material and timeproblems using finite element the material properties. These are constant of the attachedmethods (FEM).There is an reflected in constitutive laws in the form electronics. Ray et al [1] presentedincreasing awareness of the of electro-mechanical coupling. From an analytical solution for staticbenefits to be derived from the the modeling point of view, these analysis of simply supporteddevelopment and exploitation of complexities would lead to additional rectangular plate type smartsmart materials and structures in matrices in Finite Element method structure composed of compositeapplications ranging from (FEM). Coupled field analysis through laminates coupled with actuatorshydrospace to aerospace. With the FEM gives the confidence before doing and sensors. Exact solutions areability to respond autonomously to the expensive experimental test on piezo obtained for cylindrical bending ofchanges in their environment, electric system. Though many literature simply supported laminated platessmart systems can offer a are available in the piezo composite, developed by [2,3,4]. Asimplified approach to the control this paper gives the basic ideas of segmented piezoelectric actuatorof various material and system Electrical-mechanical coupling in terms is simulated by applying ancharacteristics such as light of analytical expressions with unique electric potential only over a parttransmission, viscosity, strain, notations, stiffness matrix for coupled of a distributed piezoelectricnoise 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 analyticalstructures use actuators and sensors different substrate material and solution to study the flexuralat milli and micro-scales to achieve convergence results through mesh behavior of the smart panelsa certain goal. However, at present density. Finite element analysis of Piezo subjected to electro mechanicaltime, 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 solutionsThere is a need for improved out and these results were compared are applicable only when thesensing and actuation both at the with analytical solution. edges are simply supported andmaterial and systems level. Keywords: piezo, poling, Electro- subjected to specific types ofResearch on smart structures is Mechanical, FEM, Coupling electric boundary conditions.interdisciplinary because it Analytical material modelsinvolves materials, structural currently used to evaluate themechanics, electronics, signal processing, response of metallic parts in ANSYS Multi Physicscommunication and control. The goal of multi- Finite element package are adequate to characterize thedisciplinary research is to develop techniques to piezo electric materials compared to the shape memorydesign, control, analyze, and visualize optimal or near alloys. In this paper analytical equation derived foroptimal smart and adaptive structures using newly finding the displacements and strains for piezodeveloped smart materials. Piezo electric materials are composite bimorph beams and also the effect substratewidely considered for smart structure design due to the material is studied. Finite element analysis usingfact that they are light weight and compact, relatively coupled field element in ANSYS has been carried outinexpensive, and exhibit moderately linear field strain and these results were compared with analyticalrelations at low drive levels. By bonding piezoelectric solution. The present study will be helpful foractuators to structures, desired localized strains can be understanding the behavior of piezo electric systeminduced by applying appropriate voltages to the which can be used for various applications.actuators. Piezoelectric materials bonded to structurescan also be used as dynamic strain sensors. It should benoted that piezoelectric sensors measure only dynamic 1
  6. 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 electroThe dipoles are oriented with respect to one another mechanicalthrough a process called poling. Poling direction isindicated by ‘P’ in Fig.2. Poling requires that thepiezoelectric material be heated up above its Curietemperature and then placed in a strong electric field(typically, 2KV/mm).Heating the material allows thedipoles to rotate freely, since the material is softer at Charge flowhigher temperatures [10]. The electric field producesan alignment of the dipoles along the direction of theelectric field. Quickly reducing the temperature andremoving the electric field produces a material whose Closedelectric dipoles are oriented in the same direction. Thisdirection is referred to as the poling direction of thematerial.To increase or produce the electric field,voltage needs to be increased or thickness of piezo Openshould be decreased. That is why piezo-electricmaterials are of small thickness and in the range of 0.5to 1mm by cosndiering the fabrication feasibility. Thepoling direction always should be in thicknessdirection or 3-direction to produce higher electric filedwith smaller input voltage. i.e if poling in thicknessdirection, then electric field, E = V / t. If the poling in Fig. 1:Typical diagram for stress-strain tests on awidth direction, then E = V/W. The width of square or piezoelectric material to study the effect of boundaryrectangular piezo always higher than the thickness. So conditions (Reference:10)higher voltage is to be applied to produce the moreelectric field and it leads higher cost for making coupling, which is exploited for variety of structuralelectrical setup. applications, such as vibration control, noise control, shape control and structural health monitoring. The3. Constitutive Equations electro mechanical coupling in the material isThe basic properties of a piezoelectric material are represented by the off diagonal terms of Eqn (3). Aexpressed mathematically as a relationship between larger off-diagonal term will result in a material thattwo mechanical variables, stress and strain, and two produces more strain for applied electric field andelectrical variables, electric field and electric more electric displacement for an applied mechanicaldisplacement. The direct and converse piezoelectric stress. For these reasons, the piezoelectric straineffects are written as the set of linear equations. coefficient is an important parameter for comparing theThe 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 theThese constitutive relationships would exist in a dependent variables. The above equation can bematerial that was either purely elastic or purely inverted to write expression with stress and field as thedielectric. The first part of Eqn. (1) represents the dependent variables and strain and the electric displacement as the independent variables. Taking theIn 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 dsecond part of the same equation gives the stresses due Kc = . (5) Ckto voltage input. From Eqn. (1) and eqn. (2) it is clear An important property of the piezoelectric couplingthat the structure will be stressed due to the application coefficient is that it is always positive and boundedof the electric field, even in the absence of mechanical between 0 and 1. The bounds on the couplingload [2]. Alternatively when the mechanical structure 2
  7. 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. 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 dydzd33 = 591 x 10-12 y,z Adding the results from the three domains together3.3 Piezo composite bimorph bending yieldsActuator  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 oppositepolarities. Piezoelectric bimorph are widely used for The integration of the stress component on the rightactuation and sensing. In the actuation mode, on the hand side of the expression in Eqn (18) is theapplication 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 theThis results in the bending of the entire structure and curvature as a function ofend 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 is3.3.1 Derivation of Analytical Equation for piezo obtained by dividing the numerator and denominatorcomposite 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 + 1y-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. Nowε1(z) = K z (14) we can get the strain at the interface between the piezoWhere, K = curvature = -d2u3(x)/dx2 layer and substratez = distance from neutral axisUnder 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 asε1(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 2Substitute Equn (14) in to the constitutive relations 3.3.2 Stiffness matrix for coupled field elementand rewriting, we obtain From Eqn (1) and (2), we can write the constitutiveY1 (= 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 beThe equilibrium expressions for the moment are written as [11]obtained by multiplying above equation by z andintegrating over the domain in y and z. The results is 4
  9. 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. 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 electricor voltage boundary condition. The dimensions of the field is in the same direction as the poling direction inpiezo electric plate are: Length of the piezo in x- one of the layers, whereas in the second layer thedirection (L) = 76 mm ; width(W) in y-direction = 26 electric field is in the direction opposite the polingmm, thickness in z- direction (h) = 1mm; Z or direction.thickness direction is the poling direction. Voltage Table -2 : Comparison between the finite element andboundary results analytical solutionconditions are: zero voltage is applied at the nodes atlocation z=0 and 100 volts is applied at the nodes of FElocation of z = 1 mm. The static analysis for the PZT- Displacement results Analytical %5H actuator with considering the material properties (m) (Ansys) x10-6 Errormentioned in section- 3.2 has been carried out. The x10-6displacement in X-direction (Refer Fig 5) and their Ux (lengthcorresponding strains are obtained from FE results. 2.11 2.08 direction) 1.44Mesh convergence study (Table- 1) was carried out byreducing the size of the elements without altering the Uy (width 0.699 0.712existing nodes location. It is converged for the aspect direction) 1.96ratio (AR) of one. Table- 2 shows the displacement Uz (thicknesscomparison between the analytical and FE results and 0.052 0.0591 0.33 direction)the error within acceptable limits. The analyticalsolution is obtained using [12, 13]. 1. Displacement in The brass aluminum, magnesium and Titanium used aslength direction, Ux = (d31xVxL)/h = 274x10-12 x 100 substrate material. Application of electric field to bothx 0.076 / 0.001= 2.11 x 10-6 m 2.Displacement in width layers produces extension in one of the layers anddirection, Uy = (d32xVxW)/h = 274x10-12 x 100 x contraction in the other. The net result is a bending of0.026 / 0.001=0.699 x 10-6 m ( d32 = d31) 3. the material. Assuming a perfect bond between theDisplacement 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- direction4. 2. Finite Element analysis of Bimorph across thickness (Neutral axis to –Z side). The strain atpiezo composite bending Actuator outer surface of the piezo composite is obtained using Equn (22). Table-3 shows the comparison of the resultsComposite piezoelectric device is useful for between the finite element solution and analyticalextensional actuation and the primary use of 31- solution and both the results were in good agreement.multilayer piezoelectric actuators is as a bendingdevice. As discussed in section 3.3.1, a three layer 4.3 Piezo layer with laminateddevice in which the piezo electric layers are fixed to compositesthe outer surfaces of an inactive substrate is typicallycalled a bimorph actuator. The electrical connections Beam like structural components made of composite 6
  11. 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 oppositeapplications. Laminated composite structures consist of direction to that caused by mechanical load alone. Thisseveral layers of different fiber-reinforced laminae observation shows that the deflection can be reducedbonded together to obtain desired structural properties. by applying an appropriate electrical potential on theThe desired structural properties are achieved by actuator. The stress variation across thickness ofvarying the lamina thickness, lamina material substrate also decreased when the voltage is applied toproperties, and stacking sequence. With the availability the actuator (Refer Fig.14). This study will be useful inof functional materials and feasibility of embedding or active shape control and vibration control forbonding them to composite structures, new smart aerospace structures.structural concepts have emerged for potentially high-performance structural applications [14].The powerfulelectro-mechanical coupling attribute of piezoelectric 5.0 Conclusionmaterials enables these materials with laminatedcomposites to act as effective actuators [15]. Here the PZT widely used in many applications due to it is highcarbon fiber (T300) with epoxy resin and also graphite dielectric and piezoelectric strength, moderate cost andwith epoxy is used as substrate to study the behavior of broad range of operating temperatures. Electropiezo layers. The dimensions for piezo and voltage mechanical behavior of these piezo electric patches with different metallic substrate is studied throughboundary conditions are same as mentioned above analytical and coupled field analysis. Laminatedsection except the following orthotropic material composites with piezo layers for simply supportedsubstrate. Material Properties for T300: E11= 130 GPa; condition also studied only through finite elementE22 =10 Gpa; G12 =5Gpa; µ12 = 0.35; Material analysis without analytical solution. This study givesProperties for Graphite: E11= 172.4 GPa; E22 =6.89 the confidence of using piezo patches in the actualGpa; G12 =3.45Gpa; µ12 = 0.25; The thickness of structures for many applications. The future work is tolamina (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 is0.5 mm. Layered solid [16] elements (solid46) are used Table -3: Strain comparison between the FE result and analytical solution for different substratefor modeling composite. The strains are more for T300substrate due to lower modulus compared to graphite strain at strain atsubstrate. 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- errorlayer. The strains for graphite fiber composite substrate (Ansys ) Analyticalwith 0.5 mm thickness are lower than metallicsubstrate of 2mm thickness. It is due to high specific 1. Brass 0.014069 28.8 28.14 2.2stiffness 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 2.4FEM of a simply supported plate type smart structureis considered for static analysis, to determine thedisplacement and stress for mechanical and electrical Table-4: Finite Element result for composite substrateloads. The smart structure considered in this study is a with different ply sequencecomposite substrate with 0/90/90/0 stacking sequence.The piezoelectric material (PZT) is bonded on the top Sl. Total strainand bottom of the substrate. Top layer is taken as Ply Total strainactuator. Actuator voltage is applied at the interface. N (µs) (µs) for100V is considered. The strains and stresses for sequence for Graphite + 0 T300+Epoxymechanical load only are shown Fig. 10 and Fig.11. EpoxyStrain is continuous across thickness at the substrate- 1 0/45/45/0 -37.1 -27.6actuator interface. Stress is discontinuous at the 2 0/90/90/0 -36.9 -27.5substrate-actuator interface due to different Young’sModulus. Fig.12 shows the stresses at centre of the 3 0/30/30/0 -37.1 -27.5substrate (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 -28.7Fig.13 that the maximum transverse displacement of 5 0/45/60/90 -37.9 -29.2the substrate decreases when 100 V is applied atactuator interface along with uniform pressure load.The actuator when applied with electrical potential 7
  12. 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 substrateFig. 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. 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 thicknessof substrate for various loadings For Laminated Piezoelectric Plates In Cylindrical Bending”, Journal of Applied Mechanics, Vol. 63, pp. 9
  14. 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 hasforRectangular Sandwich Plates With Embedded completed his Masters degreePiezoelectric Shear Actuators”, AIM Journal, Vol. 39, (M.Tech) in Machine Designpp. 1363-1373. from the Indian Institute of[4] Batra, R. C., Liang, X. Q., Yang, J. S., Technology (IIT- Madras). He1996b,“Shape Control of Vibrating Simply Supported was the second topper inRectangular Plates”, AIAA Journal, Vol. 34, pp. 1 16- Machine Design stream in the1 22 academic year 2010. He has[5] Brooks, S., Heyliger, P., 1994, “Static Behavior of presented 12 technical papers in National andPiezoelectric Laminates With Distributed and Patched International conferences. He has worked as ResearchActuators, Journal of Intelligent Material Systems and Fellow in Air Frame composite Stress Analysis groupStructures”, 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 MotorsElastic Plate Due to Piezoelectric Actuators”, Group in the areas of design, analysis, realisation andInternational Journal of Solids and Structures, Vol. 33, testing of solid motor hardware. His areas of interestpp. 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 SmartRectangular 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 inintelligent composites and sandwich panels, Acta Mechanical Engg. discipline. HeMechanica 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 andloads”, Proceedings of the international conference on acceptance testing of Solid Rocketsmart materials, structures and systems, July 1999. Motor hardware, pressure vessel [10] Donald J. Leo, ‘Engineering analysis of smart systems and fixtures. He has contributed significantlymaterial 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 Realizationplates 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 ChemicalDevelopment 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 distinctionfinite element analysis of laminated composite shells from IISc Bangalore . He has heldwith actuating layers, J of many positions like Dy. ProjectFinite Elements in Analysis and Design 43 (2006) 1 – Diretor, Associate Project Director21 and Project Director, S200 Project. [15] Jinquan Cheng, Farid Taheri, A smart single-lap Concurrently he was Groupadhesive 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 largeAbout Authors solid boosters. He was awarded with the Eminent Chemical Engineer-2011 award by IICHE, Kochi.Shri Paul Murugan J holds Bachelors degree inMechanical Engineering from Madurai KamarajUniversity (MKU). He was an outstanding student andover all topper in Mechanical Engineering department 10
  15. 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.in1. Introduction of numerical analysis to the ABSTRACT problems encountered inMiniaturization in a lot of research hydrodynamic lubrication. Adomains has led to a demand for In this paper, the Reynolds equation is popular numerical technique, thesmall-scale systems running at solved to determine the pressure ‘finite difference method’ ishigh operational speeds. Gas distribution for hydrodynamic gas- introduced and its application tobearings operate with the pressure lubricated journal bearings in which the hydrodynamic lubrication isgenerated inside lubricating film. the compressibility effect must be taken demonstrated. Piekos and BreuerBecause gas bearings have such into account. The non-linear Reynolds [6] explored the effect of axiallycharacteristics as low friction, high equation for self-acting gas-lubricated varying clearance onprecision, and low pollution, they journals bearing is linearized through microfabricated gas journalhave been successfully used in appropriate approximation and a bearings. Taper and Bow types ofmany commercial applications, modified Reynolds equation is derived clearance variation commonlysuch as navigation systems, and solved by means of the finite observed in etched bearings. Pengcomputer disk drives, high- difference method (FDM). The gas film [7] presented the development ofprecision instruments and sensors, pressure distribution of a self-acting mathematical models anddental drills, machine tools and gas-lubricated journal bearing is numerical schemes for simulatingturbo compressors [11]. The attained and the load capacity is the hydrodynamic pressure andtheory of lubrication had not been calculated. A computer code using temperature rise of compliant foilextended to gas films. Except for MATLAB has been developed to find bearings lubricated by a thin gasan approximate solution for an out the static performance film in between its compliantinfinitely long journal bearing, characteristics for gas-lubricated bearing surface and the rotatingneither mathematical solutions nor journal bearings. The results of the shaft. Lo et al. [8] presented aappropriate experimental study show that the present solution is detailed theoretical analysis oftechniques had been developed. in better agreement with experimental bearing performance in which theNow, with the theory verified and data which validates the formulation gas flow within the bearing issome sophisticated mathematical and the computer program so initially expressed in the form oftools at hand, along with electronic developed. simplified dimensionless Navierequipment to aid experimentation, Stokes equations. Vleugels et al.fundamental understanding and [9] presented an overview of theoptimization of gas bearings is Keywords: Self-acting gas journal total rotor dynamic modellingbecoming possible. Gross [1] bearings, Reynolds equation, Finite process of a micro-turbine rotorfound adequate mathematical tools difference method (FDM) supported on aerostatic bearings.and experimental facilities in a A both accurate and efficientsurvey for verifying the theory of gas lubrication. He modelling technique is outlined to obtain static andalso found that the gas lubricants are used in three dynamic air bearing properties. Yang et al. [10] studiedbasic bearing types: self- acting, externally pressurized, the non-linear stability of finite length self-acting gasand squeeze-film. Chandra et al. [2] presented an journal bearings by solving a time-dependent Reynoldsexhaustive 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 alubricated journal bearings. Florin Dimofte [3] self-acting (hydrodynamic) gas-lubricated journaladopted, modified, and applied an alternating direction bearing and the load capacity is calculated. Taking aimplicit (ADI) method to the Reynolds equation for small pressure change in the gas film of self-actingthin gas fluid films. An efficient code is developed to gas-lubricated journal bearings into account, thepredict both the steady-state and dynamic performance corresponding nonlinear Reynolds equation isof 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 byhydrodynamic 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

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