Modeling and Simulation of the Electrochemical Machining (ECM) Material Removal Process for the Manufacture of Aero Engine Components

Size: px
Start display at page:

Download "Modeling and Simulation of the Electrochemical Machining (ECM) Material Removal Process for the Manufacture of Aero Engine Components"

Transcription

1 Aville online t Proedi CIRP 8 (2013 ) th CIRP Conferene on Modeling of Mhining Opertions (CIRP CMMO) Modeling nd Simultion of the Eletrohemil Mhining (ECM) Mteril Removl Proess for the Mnufture of Aero Engine Components Astrt F. Kloke, M. Zeis, *, S. Hrst, A. Klink, D. Veselov, M. Bumgärtner Lortory formhine Tools nd Prodution Engineering, Steinhstrße 19, Ahen 52074, Germny Leistritz Turomshinen Tehnik GmH, Mrkgrfenstrsse 29-39, Nuremerg 90459, Germny * Corresponding uthor. Tel.: ; fx: ; E-mil ddress: m.zeis@wzl.rwth-hen.de. In order to inrese the effiieny of jet engines hrd to mhine nikel-sed nd titnium-sed lloys re in ommon use for ero engine omponents suh s ldes nd lisks (lde integrted disks). Here Eletrohemil Mhining (ECM) is promising lterntive to milling opertions. Due to lk of pproprite proess modeling pilities eforehnd still knowledge sed nd ost intensive thode design proess is pssed through. Therefore this pper presents multi-physil pproh for modeling the ECM mteril removl proess y oupling ll relevnt onservtion equtions. The resulting simultion model is vlidted y the exmple of ompressor lde. Finlly new pproh for n inverted thode design proess is introdued nd disussed The Authors. Pulished y y Elsevier Elsevier B.V. B.V. Open ess under CC BY-NC-ND liense. Seletion nd nd/or peer-review peer-review under responsiility under responsiility of The Interntionl of The Interntionl Sientifi Committee Sientifi of Committee the 14th CIRP of Conferene the 14th CIRP on Modeling Conferene of Mhining on Opertions Modeling of in Mhining the person of Opertions" the Conferene in the Chir person Prof. of Lu the Conferene Settineri Chir Prof. Lu Settineri Keywords: Eletrohemil Mhining (ECM); simultion; ero engine omponents; inverse prolem 1. Introdution To hieve weight redution nd inresed therml effiieny of jet engines, hrd to mhine lloys suh s Ti-6Al-4V nd Inonel 718 re in ommon use for the mnufture of ero engine omponents. Espeilly the milling proess of lisks mde of Ni-sed lloys rehes its tehnologil nd eonomil limit. Here Eletrohemil Mhining (ECM) is ost-effetive lterntive. In ECM high mteril removl rtes n e relized without developing ny white lyer or het ffeted zone. Additionlly vi ECM it is possile to hieve finished surfes qulities during rough mhining opertions, whih elimintes the need for further tretment like ost-intensive finish milling steps or polishing opertions [1, 2, 3]. But due to ost intensive tool developing proesses nd rther high investment osts for the mhine tools, ECM is mostly used in produtions with lrge th sizes. Min reson for high tool osts is n only knowledge-sed, itertive thode designing proess. After test run the produed workpiee hs to e mesured nd the differene etween trget nd tul geometry due to lolly hnged onditions of eletrolysis is sutrted from the thode nd so forth. On the other hnd the theoreti kground of the eletrohemil mteril removl proess with ll its different physil spets is very well known, ut up to now it hs never een sueeded to omine ll different spets together into one suffiiently preise model to simulte the proess for the mnufture of ero engine omponents. This pper presents new pproh for modeling the ECM mteril removl proess y oupling the onservtion equtions for fluid flow, eletri fields, eletrohemil surfe retions, ioni trnsport nd het trnsfer. Bsed on initilly introdued governing equtions, possile oupling strtegies re presented nd disussed. Afterwrds, the erodynmi ross-setion of ompressor lde is lulted in two different wys The Authors. Pulished y Elsevier B.V. Open ess under CC BY-NC-ND liense. Seletion nd peer-review under responsiility of The Interntionl Sientifi Committee of the 14th CIRP Conferene on Modeling of Mhining Opertions in the person of the Conferene Chir Prof. Lu Settineri doi: /j.proir

2 266 F. Kloke et l. / Proedi CIRP 8 ( 2013 ) Besides the simultion of the lssil ECM proess with the im to lulte the lde ontour y using predetermined thode geometries new inverted simultion strtegy is presented. This new pproh uses the lde trget geometry in order to lulte thode shpe y inverting the eletri field. 2. Theoretil Bkround ECM is mnufturing tehnology where mny different physil disiplines nd spets need to e onsidered (p. figure 1). In order to simulte this mteril removl tehnology esides the modeling of the eletromgneti field the eletrolyte flow hs to e nd the ioni trnsport. So in this hpter the governing equtions for omplete desription of the eletrohemil mteril removl proess re nmed nd possile physil ouplings re disussed. Eletri Field urrent density equls zero euse only n externl voltge is pplying. The eletri displement field is defined s:. (4) Herein is the eletril field onstnt nd the mteril dependent reltive permittivity. Altogether urrent density n e written s: 2.2. Eletrolyte. (5) Flow fields of Newtonin fluids n ompletely e desried y the Nvier-Stokes equtions [6]. Equivlent to the ontinuity eqution of hrge the onservtion of mss n e written s:. (6) Here the veloity vetor. ed to ontinuum the onservtion of momentum results to:. (7) Surfe Retions Fluid mehnis Fig. 1. Physil Couplings in ECM [4] 2.1. Eletri Field Het Trnsfer Geometry Struture The eletri field forms the sis for eletrohemil equtions [5] nd the ontinuity eqution of hrge whih sttes tht lol eletri hrge density hnges y the divergene of the vetor of urrent density :. (1) If mgneti fields re negleted, the eletril field is defined y the negtive grdient of eletri voltge :. (2) With the speifi eletri ondutne, the eletri displement field nd the externl vetor of urrent density the vetor of urrent density results to:. (3) In se of eletrohemil mhining the externl Within eqution (7) is n elertion vetor, the pressure nd the tensor of sher stresses. In ECM used y the eletrolysis of wter hydrogen gs is produed fluid hd to e desried s two phse flow whih in generl n e solved y the Nvier-Stokes equtions s well. Cused y high flow veloities nd shrp inlet ngles of the thodes, flow fields in ECM proesses for ero engine omponents re mostly turulent. In order to hieve dequte simultion times the Nvier-Stokes equtions re not omputed diretly. Hene turulene hs to e modeled seprtely e. g. y the k- -model [7] Eletrode Surfe Retion To model the ECM proess the mthemtil desription of surfe retions is of entrl importne. Bsilly urrent density n e written s the differene of node (index A) nd thode (index C) retions:. (8) Herein is Frd represent the retion veloities. These retion veloities n e desried with the help of the trnsition stte theory nd re defined s [8]:. (9)

3 F. Kloke et l. / Proedi CIRP 8 ( 2013 ) dependent frequeny ftor is proportionl to the rtio ) onstnt [9]:. (10) Differene of free enthlpy (Gis free energy) is expressed y the sum of the enthlpy differene in the energeti ground stte nd the Glvni potentil.. (11) Eqution (11) here denotes the mximum potentil differene whih does not our in relity so tht pssge ftor with is pplied. If the nodi urrent density equls the thodi urrent density, potentil differene equls the equilirium rest potentil. This interreltionship leds to the exhnge urrent density :, (12). (13) If n externl potentil is pplied to the eletrohemil ell, the potentil within the ell hnges from equilirium rest potentil to nd the over-voltge :. (14) Hene the new potentil differene is:. (15) Summrized nd with the sustitution the lol urrent density n e lulted y the so-lled Butler-Volmer eqution: 2.4. Het Trnsfer. (16) In order to model n ECM-proess the het trnsfer is divided into het trnsfer in solids nd het trnsfer in fluids under negleted therml rdition. Bsed on the first trnsfer in solids n e lulted y [6]:. (17) Herein is the speifi het pity, the isotropi therml ondutivity nd speifi het soure. In omintion with the onservtion of mss nd momentum the onservtion of energy for fluid n e written s [6]: 2.5. Ioni Trnsport Mehnisms In generl it is distinguished etween the three types of ioni trnsport mehnisms diffusion, migrtion nd onvetion [8]. Diffusion sys tht moleulr prtile flux is proportionl to the grdient of molr onentrtion :. (19) the onservtion of mss for eh element:. (20) With the help of the Stokes-Einstein eqution in whih is the dynmi visosity of the fluid nd n e lulted y:. (21) Migrtion Ioni migrtion desries the movement of ions in n eletri field. Positive hrged ions move through the eletri field in the diretion of the negtive pole. Assumed tht the ions reh sttionry vlue of migrtion veloity, ioni moility n e lulted y the Nernst-Einstein eqution. Therein denotes the universl gs onstnt:. (22) Convetion Finlly dissolved prtiles n e moved y onvetion. This veloity is severl orders of mgnitude higher thn diffusion or migrtion nd n e desried y the Nvier-Stokes equtions (p. hpter 2.2). But the wll ner oundry lyer is n exeption used y the no-slip ondition. In summry, ll ioni trnsport mehnisms n e expressed for one speies (index ) to eqution (23). Therein denotes hemil retion term: 2.6. Frdy s Lw (23) where denotes the inner energy., (18) Besides the omplete desription sed on onservtion equtions the eletrohemil mhining lw [4]. For multiphse mterils nd due to the ft tht

4 268 F. Kloke et l. / Proedi CIRP 8 ( 2013 ) not eh individul retion is removl effetive,. (24) In eqution (24) represents the effetive mteril removl rte, the dissolving veloity, the speifi mteril removl rte, the urrent effiieny, the speifi weight of the lloying ddition, the molr mss nd the eletrohemil vlene. Due to the ft tht eh element n e ionized differently strong, eletrohemil vlenes in eqution (24) re hrd to forest so tht the djustment ftor indites the proility for eh individul retion tking ple. But these proilities re not known extly euse they re on their prt funtion of temperture, ph-vlue nd so forth. For this reson nd due to the ft, tht the effetive mteril removl rte n e determined very extly in nlogil experiments, vlues of re often used s sis for further simultions [10] Physil Coupling First of ll the lol urrent densities out of the Butler-Volmer eqution (16) hve to e oupled with eh elementry retion re known, the lol, nlytil dissolving veloities n e lulted y:. (25) Here denotes the stoihiometri oeffiient of the redued speies. Up to now no generl nlytil expression for the ondutne of strong eletrolytes s they re used in ECM exists [11]. For this reson nd due to the ft tht this interreltionship is experimentlly well proven, here liner pproh (slope ftor ) is used to model s funtion of temperture:. (26) Similr to eqution (26) the eletril ondutivity of two-phse flows n e lulted y semiphenomenologil pprohes, e. g. of Bruggemnn et l. [12]. Generlly in ECM het is generted y Joule heting nd eletrohemil retions. Provided tht nerly ll eletril energy is onverted into het [13], Joule heting of the eletrolyte n e lulted in. (27) Therein is the differene in potentil etween node nd thode nd the lol gp width. Due to their low speifi resistnes, Joule heting of used solids is negleted. Het generted y surfe retions n e expressed y the produt of lol urrent density generted y Butler-Volmer retion nd over-voltge : Simultion step 1 Simultion step 3 Simultion step 3. (28) Finlly ll equtions of this hpter re introdued into one multi-physil model 3. Modeling nd Simultion 3.1. Simultion Softwre For modeling nd simultion the ommeril FEsoftwre COMSOL Multiphysis hs een used. Due to the modulr design of this softwre tool it is possile to omine different physil phenomen together into one simultion model. COMSOL hs lredy proven to e suessful in severl different eletrohemil mhining opertions like jet-ecm [14] or in the mnufture of shver ps [15] Coupling Strtegies In order to solve the Butler-Volmer eqution for eh elementry retion the exhnge urrent density nd the pssge ftor hs to e known. While in most ses tkes the vlue 0.5, vlues for unfortuntely only exist for smll numer of individul retions [16]. Thus, urrently it is not possile to set up omplete nlytil model euse mny prmeters re not existing nd hve to e found in future experimentl studies. experimentl dt of the effetive mteril removl rte [10]. In figure 2 simultion steps nd possile oupling strtegies re summrized y their nmes within COMSOL Multiphysis nd prtiulr physil effets they tke into ount. Fluid Flow Turulent Flow Het Trnsfer Het Trnsfer in Fluids AC/DC Eletri Currents AC/DC Eletri Currents Mthemtis Moving Mesh Fig. 2. Simultion steps nd possile oupling strtegies

5 F. Kloke et l. / Proedi CIRP 8 ( 2013 ) Initil nd Boundry Conditions Tle 1 summrizes typil initil nd oundry onditions of ECM proesses for the mnufture of ero engine omponents. Finl simultion exmples re sed on these vlues. Aero engine omponents suh s ldes nd lisks re mnuftured in severl steps so tht feed rte nd voltge re vried in predefined rnge. Tle 1. Typil initil nd oundry onditions Initil nd oundry ondition Symol nd physil unit Vlue Inflow temperture T i / K Inflow ondutivity of the eletrolyte 0 / (ms/m) 55 Slope of the ondutivity 2.8 Feed rte v f / (mm/min) Effetive mteril removl rte V eff 1.51 Voltge U / V 7-20 Surrounding temperture T 0 / K Results nd Disussion Bsed on rel ompressor lde geometry finlly simultion with the three simultion steps nd ouplings desried ove ws mde. Besides the trget geometry thode geometry whih hs proven itself in prtie ws given in order to vlidte the simultion model. Figure 4 shows the results of the simultion. It is mpped exellent y the simultion in omprison to trget geometry. Geometril devitions of less thn 150 μm in triling nd leding edge n primrily e explined y inuries of the predetermined inflow ondutivity of the eletrolyte. Another reson whih is reognizle generted y COMSOL. So shrp triling edge is uilt whih does not orrespond with the trget geometry in this re. Cthode geometry Simulted geometry Mximum devition < 10 μm In this hpter the results of simultion using the model with the initil nd oundry onditions desried ove re presented for rel ompressor lde pplition. Bsed on the lde trget geometry itself nd thode geometry whih hs proven to e suessful in prtie the simultion model is vlidted. In seond step new inverted pproh is presented where thode geometry is omputed Clssi Simultion Fig. 4. Clssi ECM proess simultion 4.2. Alterntive Cthode Clultion 1 mm Figure 4 shows shemtilly the ECM proess of the mnufture of ompressor or turine lde. Tool thodes re moved with onstnt feed rte towrds the lde (node) nd due to lol onditions of eletrolysis the lde is formed. Cthode Inlet Feed diretion Leding edge Outlet Due to the ft tht even with the help of working simultion model the thode design proess would still e itertive now new pproh is presented. By inverting the eletri field it is virtully possile to ompute thode geometry y using the trget lde geometry s node. ECM proess is inverted so tht lter tool geometry forms itself due to lol onditions of eletrolysis nd lde trget geometry. Figure 5 shows the results of this inverted pproh for thode design y ompring lulted geometry (lk line) with the one lredy proven itself in prtie (lue line). For the flow surfes nd the triling edge good results n e stted nd onerning the leding Triling edge Feed diretion Fig. 3. Priniple of ECM lde mnufture Cthode Anode slight differene etween trget nd lulted ontour. is of essentil

6 270 F. Kloke et l. / Proedi CIRP 8 ( 2013 ) importne for suessful model of the eletrohemil removl proess. At the moment is only impreisely determined t the inflow oundry nd hs to e improved in order to reh higher simultion ury. It is impossile tht thodes ut themselves like in of this lultion is to prove how ner the simultion n mp relity. Furthermore the inverse prolem nnot e solved unique so tht severl solutions re possile. Existing thode geometry Workpiee geometry Simulted geometry Mximum devition < 80 μm Fig. 5. Alterntive thode design simultion 5. Summry nd Outlook All neessry equtions to desrie the ECM proess nlytilly nd how they re onneted to eh other hve een presented nd disussed. With experimentl results for the effetive mteril removl rte ording een uilt up tking into ount fluid flow, eletri field nd het trnsfer. In order to vlidte the simultion model rel ompressor lde hs een simulted in one erodynmi ross-setion in two different wys. Besides the simultion of the lssil ECM proess with the im to lulte the lde ontour y using predetermined thode geometries new inverted simultion strtegy ws presented. This new pproh uses the lde trget geometry in order to lulte the thode shpe y inverting the eletri field. Both simultions showed good results ompred to relity. To hieve even etter results, the effets of the hydrogen evolution t the thode hve to e inorported in the existing model. It is neessry to model the eletrolyte either s two-phse flow or to use semi phenomenologil pproh whih onsiders the gs phse influene to the ondutivity of the eletrolyte, e. g. of Bruggemnn et l. In next step 3D-model is going to e uilt nd y the here presented inverted pproh thode is eing lulted nd 1 mm mnuftured. Afterwrds lde is going to e mhined with this tool nd mesured s well s ompred to trget geometry. Aknowledgements This work hs prtilly een funded y the Germn Federl Lnd NRW within the projet EF 2037 von Nikelsis-Turinenshufeln für 700 Grd Referenes [1] Rjurkr K.P., Levy G., Mlshe A., Sundrm M.M., MGeough A., Hu X., Resnik R., De Silv A., 2006, Miro nd Nno Mhining y Eletro-Physil nd Chemil Proesses, CIRP Annls - Mnufturing Tehnology, 55: [2] Kloke F., Zeis M., Klink A., Veselov D., 2012, Tehnologil nd Eonomil Comprison of Roughing Strtegies vi Milling, EDM nd ECM for Titnium- nd Nikel-sed Blisks, Pro. CIRP, 2: [3] MGeough J.A., 1974, Priniples of Eletrohemil Mhining, Chpmn nd Hll. [4] Kloke F., Zeis M., Klink A., 2012, Tehnologil nd eonomil pilities of mnufturing titnium- nd nikel-sed lloys vi Eletrohemil Mhining (ECM), Key Engineering Mterils, : [5] Tipler P., Mos G., 2004, Physik für Wissenshftler und Ingenieure, Spektrum Akdemisher Verlg. [6] Shröder W., 2004, Fluidmehnik, Ahener Beiträge zur Strömungsmehnik, Wissenshftsverlg Minz in Ahen. [7] Wilox D., 1994, Turulene Modeling for CFD, DCW Industries, In. [8] Forker W., 1989, Elektrohemishe Kinetik, Akd. Berlin Verlg. [9] Steinfeld J., Frniso J., Hse W., 1998, Chemil Kinetis nd Dynmis - Seond Edition, Prentie Hll. [10] Kloke F., Zeis M., Klink A., Veselov D., 2013, Experimentl Reserh on the Eletrohemil Mhining of Modern Titnium- nd Nikel-sed Alloys for Aero Engine Components, ISEM 2013, in press. [11] Atkins P.W., de Pul J., 2006, Physiklishe Chemie, Wiley-VCH. [12] Kozk J., 1998, Mthemtil models for omputer simultion of eletrohemil mhining proesses, Journl of Mterils Proessing Tehnology, 76/1-3: [13] Kueth H., 1965, Der Aildungsvorgng zwishen Werkzeugelektrode und Werkstük eim Elektrohemishen Senken, Disserttion RWTH Ahen. [14] Hkert M., 2009, Entwiklung und Simultion eines Verfhrens zum elektrohemishen Atrgen von Mikrogeometrien mit geshlossenem elektrolytishen Freistrhl, Disserttion TU Chmnitz. [15] Vn Tijum R., Pjk P.T., 2008, Simultion of Prodution Proess using the Multiphysis pproh: The Eletrohemil mhining Proess, COMSOL Conferene. [16] Holze R., 2007, Eletrohemistry, Springer.

8 THREE PHASE A.C. CIRCUITS

8 THREE PHASE A.C. CIRCUITS 8 THREE PHSE.. IRUITS The signls in hpter 7 were sinusoidl lternting voltges nd urrents of the so-lled single se type. n emf of suh type n e esily generted y rotting single loop of ondutor (or single winding),

More information

Cyclic voltammetry simulation at microelectrode arrays with COMSOL Multiphysics Ò

Cyclic voltammetry simulation at microelectrode arrays with COMSOL Multiphysics Ò J Appl Eletrohem (009) 39:9 63 DOI 0.007/s0800-009-9797- ORIGINAL PAPER Cyli voltmmetry simultion t miroeletrode rrys with COMSOL Multiphysis Ò Alessndro Lvhi Æ U. Brdi Æ C. Borri Æ S. Cporli Æ A. Fossti

More information

Modelling the Electrolyte Flow in a Full-scale Copper Electrorefining Tankhouse Cell

Modelling the Electrolyte Flow in a Full-scale Copper Electrorefining Tankhouse Cell Modelling the Eletrolyte Flow in Full-sle Copper Eletrorefining Tnkhouse Cell Andres Kemminger, Andres Ludwig Montnuniversitet Leoben Deprtment Metllurgy, Chir of Simultion nd Modelling of Metllurgil Proesses

More information

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and

Lecture 27: Diffusion of Ions: Part 2: coupled diffusion of cations and Leture 7: iffusion of Ions: Prt : oupled diffusion of tions nd nions s desried y Nernst-Plnk Eqution Tody s topis Continue to understnd the fundmentl kinetis prmeters of diffusion of ions within n eletrilly

More information

NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE

NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE NEW CIRCUITS OF HIGH-VOLTAGE PULSE GENERATORS WITH INDUCTIVE-CAPACITIVE ENERGY STORAGE V.S. Gordeev, G.A. Myskov Russin Federl Nuler Center All-Russi Sientifi Reserh Institute of Experimentl Physis (RFNC-VNIIEF)

More information

Table of Content. c 1 / 5

Table of Content. c 1 / 5 Tehnil Informtion - t nd t Temperture for Controlger 03-2018 en Tble of Content Introdution....................................................................... 2 Definitions for t nd t..............................................................

More information

THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL

THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL THE INFLUENCE OF MODEL RESOLUTION ON AN EXPRESSION OF THE ATMOSPHERIC BOUNDARY LAYER IN A SINGLE-COLUMN MODEL P3.1 Kot Iwmur*, Hiroto Kitgw Jpn Meteorologil Ageny 1. INTRODUCTION Jpn Meteorologil Ageny

More information

A Study on the Properties of Rational Triangles

A Study on the Properties of Rational Triangles Interntionl Journl of Mthemtis Reserh. ISSN 0976-5840 Volume 6, Numer (04), pp. 8-9 Interntionl Reserh Pulition House http://www.irphouse.om Study on the Properties of Rtionl Tringles M. Q. lm, M.R. Hssn

More information

Review Topic 14: Relationships between two numerical variables

Review Topic 14: Relationships between two numerical variables Review Topi 14: Reltionships etween two numeril vriles Multiple hoie 1. Whih of the following stterplots est demonstrtes line of est fit? A B C D E 2. The regression line eqution for the following grph

More information

Project 6: Minigoals Towards Simplifying and Rewriting Expressions

Project 6: Minigoals Towards Simplifying and Rewriting Expressions MAT 51 Wldis Projet 6: Minigols Towrds Simplifying nd Rewriting Expressions The distriutive property nd like terms You hve proly lerned in previous lsses out dding like terms ut one prolem with the wy

More information

1.3 SCALARS AND VECTORS

1.3 SCALARS AND VECTORS Bridge Course Phy I PUC 24 1.3 SCLRS ND VECTORS Introdution: Physis is the study of nturl phenomen. The study of ny nturl phenomenon involves mesurements. For exmple, the distne etween the plnet erth nd

More information

(h+ ) = 0, (3.1) s = s 0, (3.2)

(h+ ) = 0, (3.1) s = s 0, (3.2) Chpter 3 Nozzle Flow Qusistedy idel gs flow in pipes For the lrge vlues of the Reynolds number typilly found in nozzles, the flow is idel. For stedy opertion with negligible body fores the energy nd momentum

More information

Unit 4. Combinational Circuits

Unit 4. Combinational Circuits Unit 4. Comintionl Ciruits Digitl Eletroni Ciruits (Ciruitos Eletrónios Digitles) E.T.S.I. Informáti Universidd de Sevill 5/10/2012 Jorge Jun 2010, 2011, 2012 You re free to opy, distriute

More information

Polyphase Systems. Objectives 23.1 INTRODUCTION

Polyphase Systems. Objectives 23.1 INTRODUCTION Polyphse Systems 23 Ojetives eome fmilir with the opertion of threephse genertor nd the mgnitude nd phse reltionship etween the three phse voltges. e le to lulte the voltges nd urrents for three-phse Y-onneted

More information

Chapter Gauss Quadrature Rule of Integration

Chapter Gauss Quadrature Rule of Integration Chpter 7. Guss Qudrture Rule o Integrtion Ater reding this hpter, you should e le to:. derive the Guss qudrture method or integrtion nd e le to use it to solve prolems, nd. use Guss qudrture method to

More information

Polyphase Systems 22.1 INTRODUCTION

Polyphase Systems 22.1 INTRODUCTION 22 Polyphse Systems 22.1 INTRODUTION n genertor designed to develop single sinusoidl voltge for eh rottion of the shft (rotor) is referred to s single-phse genertor. If the numer of oils on the rotor is

More information

Electromagnetism Notes, NYU Spring 2018

Electromagnetism Notes, NYU Spring 2018 Eletromgnetism Notes, NYU Spring 208 April 2, 208 Ation formultion of EM. Free field desription Let us first onsider the free EM field, i.e. in the bsene of ny hrges or urrents. To tret this s mehnil system

More information

SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS

SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS SECOND HARMONIC GENERATION OF Bi 4 Ti 3 O 12 FILMS IN-SITU PROBING OF DOMAIN POLING IN Bi 4 Ti 3 O 12 THIN FILMS BY OPTICAL SECOND HARMONIC GENERATION YANIV BARAD, VENKATRAMAN GOPALAN Mterils Reserh Lortory

More information

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point

Numbers and indices. 1.1 Fractions. GCSE C Example 1. Handy hint. Key point GCSE C Emple 7 Work out 9 Give your nswer in its simplest form Numers n inies Reiprote mens invert or turn upsie own The reiprol of is 9 9 Mke sure you only invert the frtion you re iviing y 7 You multiply

More information

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

1 PYTHAGORAS THEOREM 1. Given a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. 1 PYTHAGORAS THEOREM 1 1 Pythgors Theorem In this setion we will present geometri proof of the fmous theorem of Pythgors. Given right ngled tringle, the squre of the hypotenuse is equl to the sum of the

More information

Some Aspects of Non-Orthogonal Stagnation-Point Flow towards a Stretching Surface

Some Aspects of Non-Orthogonal Stagnation-Point Flow towards a Stretching Surface Engineering, 00,, 705-709 doi:0.436/eng.00.909 Published Online September 00 (http://www.sirp.org/journl/eng) Some Aspets of Non-Orthogonl Stgntion-Point Flow towrds Strething Surfe Abstrt Mothr Rez, Andi

More information

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R

Thermodynamics. Question 1. Question 2. Question 3 3/10/2010. Practice Questions PV TR PV T R /10/010 Question 1 1 mole of idel gs is rought to finl stte F y one of three proesses tht hve different initil sttes s shown in the figure. Wht is true for the temperture hnge etween initil nd finl sttes?

More information

ANALYSIS AND MODELLING OF RAINFALL EVENTS

ANALYSIS AND MODELLING OF RAINFALL EVENTS Proeedings of the 14 th Interntionl Conferene on Environmentl Siene nd Tehnology Athens, Greee, 3-5 Septemer 215 ANALYSIS AND MODELLING OF RAINFALL EVENTS IOANNIDIS K., KARAGRIGORIOU A. nd LEKKAS D.F.

More information

Journal of Chemical and Pharmaceutical Research, 2013, 5(12): Research Article

Journal of Chemical and Pharmaceutical Research, 2013, 5(12): Research Article Avilble online www.jopr.om Journl of Chemil nd Phrmeutil Reserh, 2013, 5(12):1283-1288 Reserh Artile ISSN : 0975-7384 CODEN(USA) : JCPRC5 Study on osion resistne of zin lloy oting of mehnil plting by eletrohemil

More information

U Q W The First Law of Thermodynamics. Efficiency. Closed cycle steam power plant. First page of S. Carnot s paper. Sadi Carnot ( )

U Q W The First Law of Thermodynamics. Efficiency. Closed cycle steam power plant. First page of S. Carnot s paper. Sadi Carnot ( ) 0-9-0 he First Lw of hermoynmis Effiieny When severl lterntive proesses involving het n work re ville to hnge system from n initil stte hrterize y given vlues of the mrosopi prmeters (pressure p i, temperture

More information

Thermal energy 2 U Q W. 23 April The First Law of Thermodynamics. Or, if we want to obtain external work: The trick of using steam

Thermal energy 2 U Q W. 23 April The First Law of Thermodynamics. Or, if we want to obtain external work: The trick of using steam April 08 Therml energy Soures of het Trnsport of het How to use het The First Lw of Thermoynmis U W Or, if we wnt to otin externl work: U W 009 vrije Universiteit msterm Close yle stem power plnt The trik

More information

University of Sioux Falls. MAT204/205 Calculus I/II

University of Sioux Falls. MAT204/205 Calculus I/II University of Sioux Flls MAT204/205 Clulus I/II Conepts ddressed: Clulus Textook: Thoms Clulus, 11 th ed., Weir, Hss, Giordno 1. Use stndrd differentition nd integrtion tehniques. Differentition tehniques

More information

Engr354: Digital Logic Circuits

Engr354: Digital Logic Circuits Engr354: Digitl Logi Ciruits Chpter 4: Logi Optimiztion Curtis Nelson Logi Optimiztion In hpter 4 you will lern out: Synthesis of logi funtions; Anlysis of logi iruits; Tehniques for deriving minimum-ost

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 H 3 CH3 C. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR

6.3.2 Spectroscopy. N Goalby chemrevise.org 1 NO 2 CH 3. CH 3 C a. NMR spectroscopy. Different types of NMR 6.. Spetrosopy NMR spetrosopy Different types of NMR NMR spetrosopy involves intertion of mterils with the lowenergy rdiowve region of the eletromgneti spetrum NMR spetrosopy is the sme tehnology s tht

More information

Introduction to Olympiad Inequalities

Introduction to Olympiad Inequalities Introdution to Olympid Inequlities Edutionl Studies Progrm HSSP Msshusetts Institute of Tehnology Snj Simonovikj Spring 207 Contents Wrm up nd Am-Gm inequlity 2. Elementry inequlities......................

More information

Appendix C Partial discharges. 1. Relationship Between Measured and Actual Discharge Quantities

Appendix C Partial discharges. 1. Relationship Between Measured and Actual Discharge Quantities Appendi Prtil dishrges. Reltionship Between Mesured nd Atul Dishrge Quntities A dishrging smple my e simply represented y the euilent iruit in Figure. The pplied lternting oltge V is inresed until the

More information

NON-DETERMINISTIC FSA

NON-DETERMINISTIC FSA Tw o types of non-determinism: NON-DETERMINISTIC FS () Multiple strt-sttes; strt-sttes S Q. The lnguge L(M) ={x:x tkes M from some strt-stte to some finl-stte nd ll of x is proessed}. The string x = is

More information

Chapter 4 State-Space Planning

Chapter 4 State-Space Planning Leture slides for Automted Plnning: Theory nd Prtie Chpter 4 Stte-Spe Plnning Dn S. Nu CMSC 722, AI Plnning University of Mrylnd, Spring 2008 1 Motivtion Nerly ll plnning proedures re serh proedures Different

More information

Section 4.4. Green s Theorem

Section 4.4. Green s Theorem The Clulus of Funtions of Severl Vriles Setion 4.4 Green s Theorem Green s theorem is n exmple from fmily of theorems whih onnet line integrls (nd their higher-dimensionl nlogues) with the definite integrls

More information

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix

Matrices SCHOOL OF ENGINEERING & BUILT ENVIRONMENT. Mathematics (c) 1. Definition of a Matrix tries Definition of tri mtri is regulr rry of numers enlosed inside rkets SCHOOL OF ENGINEERING & UIL ENVIRONEN Emple he following re ll mtries: ), ) 9, themtis ), d) tries Definition of tri Size of tri

More information

ERT 316: REACTION ENGINEERING CHAPTER 3 RATE LAWS & STOICHIOMETRY

ERT 316: REACTION ENGINEERING CHAPTER 3 RATE LAWS & STOICHIOMETRY ER 316: REIO EGIEERIG HER 3 RE LWS & SOIHIOMERY 1 OULIE R 1: Rte Lws Reltive Rtes of Retion Retion Orer & Rte Lw Retion Rte onstnt, k R 2: Stoihiometry th System Stoihiometri le low System Stoihiometri

More information

Chemical Equilibrium

Chemical Equilibrium Chpter 16 Questions 5, 7, 31, 33, 35, 43, 71 Chemil Equilibrium Exmples of Equilibrium Wter n exist simultneously in the gs nd liquid phse. The vpor pressure of H O t given temperture is property ssoited

More information

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e

Green s Theorem. (2x e y ) da. (2x e y ) dx dy. x 2 xe y. (1 e y ) dy. y=1. = y e y. y=0. = 2 e Green s Theorem. Let be the boundry of the unit squre, y, oriented ounterlokwise, nd let F be the vetor field F, y e y +, 2 y. Find F d r. Solution. Let s write P, y e y + nd Q, y 2 y, so tht F P, Q. Let

More information

Dorf, R.C., Wan, Z. T- Equivalent Networks The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000

Dorf, R.C., Wan, Z. T- Equivalent Networks The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 orf, R.C., Wn,. T- Equivlent Networks The Eletril Engineering Hndook Ed. Rihrd C. orf Bo Rton: CRC Press LLC, 000 9 T P Equivlent Networks hen Wn University of Cliforni, vis Rihrd C. orf University of

More information

EE 330/330L Energy Systems (Spring 2012) Laboratory 1 Three-Phase Loads

EE 330/330L Energy Systems (Spring 2012) Laboratory 1 Three-Phase Loads ee330_spring2012_l_01_3phse_lods.do 1/5 EE 330/330L Energy Systems (Spring 2012) Lortory 1 ThreePhse Lods Introdution/Bkground In this lortory, you will mesure nd study the voltges, urrents, impednes,

More information

Lecture Notes No. 10

Lecture Notes No. 10 2.6 System Identifition, Estimtion, nd Lerning Leture otes o. Mrh 3, 26 6 Model Struture of Liner ime Invrint Systems 6. Model Struture In representing dynmil system, the first step is to find n pproprite

More information

Learning Partially Observable Markov Models from First Passage Times

Learning Partially Observable Markov Models from First Passage Times Lerning Prtilly Oservle Mrkov s from First Pssge s Jérôme Cllut nd Pierre Dupont Europen Conferene on Mhine Lerning (ECML) 8 Septemer 7 Outline. FPT in models nd sequenes. Prtilly Oservle Mrkov s (POMMs).

More information

College of engineering/ Babylon University, Babylon, Iraq

College of engineering/ Babylon University, Babylon, Iraq Experimentl Investigtion of Three Phse Flow (Liquid-Gs-Solid) in Horizontl Pipe Riydh S. Al-Turihi Deprtment of Mehnil Engineering Astrt: -The study of three phse flow in horizontl nd vertil pipe re importnt

More information

Estimation of Sequence Components using Magnitude Information

Estimation of Sequence Components using Magnitude Information 6th NATIONAL POWER SYSTEMS CONFERENCE, 5th-7th DECEMBER, 2 47 Estimtion of Sequene Components using Mgnitude Informtion P.S. Ngendr ro nd Ssikirn Veknuru Deprtment of Eletril Engineering Indin Institute

More information

Symmetrical Components 1

Symmetrical Components 1 Symmetril Components. Introdution These notes should e red together with Setion. of your text. When performing stedy-stte nlysis of high voltge trnsmission systems, we mke use of the per-phse equivlent

More information

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points.

a) Read over steps (1)- (4) below and sketch the path of the cycle on a P V plot on the graph below. Label all appropriate points. Prole 3: Crnot Cyle of n Idel Gs In this prole, the strting pressure P nd volue of n idel gs in stte, re given he rtio R = / > of the volues of the sttes nd is given Finlly onstnt γ = 5/3 is given You

More information

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals

AP Calculus BC Chapter 8: Integration Techniques, L Hopital s Rule and Improper Integrals AP Clulus BC Chpter 8: Integrtion Tehniques, L Hopitl s Rule nd Improper Integrls 8. Bsi Integrtion Rules In this setion we will review vrious integrtion strtegies. Strtegies: I. Seprte the integrnd into

More information

Chapter 19: The Second Law of Thermodynamics

Chapter 19: The Second Law of Thermodynamics hpter 9: he Seon Lw of hermoynmis Diretions of thermoynmi proesses Irreversile n reversile proesses hermoynmi proesses tht our in nture re ll irreversile proesses whih proee spontneously in one iretion

More information

Generalization of 2-Corner Frequency Source Models Used in SMSIM

Generalization of 2-Corner Frequency Source Models Used in SMSIM Generliztion o 2-Corner Frequeny Soure Models Used in SMSIM Dvid M. Boore 26 Mrh 213, orreted Figure 1 nd 2 legends on 5 April 213, dditionl smll orretions on 29 My 213 Mny o the soure spetr models ville

More information

Lecture Summaries for Multivariable Integral Calculus M52B

Lecture Summaries for Multivariable Integral Calculus M52B These leture summries my lso be viewed online by liking the L ion t the top right of ny leture sreen. Leture Summries for Multivrible Integrl Clulus M52B Chpter nd setion numbers refer to the 6th edition.

More information

6.5 Improper integrals

6.5 Improper integrals Eerpt from "Clulus" 3 AoPS In. www.rtofprolemsolving.om 6.5. IMPROPER INTEGRALS 6.5 Improper integrls As we ve seen, we use the definite integrl R f to ompute the re of the region under the grph of y =

More information

SECTION A STUDENT MATERIAL. Part 1. What and Why.?

SECTION A STUDENT MATERIAL. Part 1. What and Why.? SECTION A STUDENT MATERIAL Prt Wht nd Wh.? Student Mteril Prt Prolem n > 0 n > 0 Is the onverse true? Prolem If n is even then n is even. If n is even then n is even. Wht nd Wh? Eploring Pure Mths Are

More information

Chapter 8 Roots and Radicals

Chapter 8 Roots and Radicals Chpter 8 Roots nd Rdils 7 ROOTS AND RADICALS 8 Figure 8. Grphene is n inredily strong nd flexile mteril mde from ron. It n lso ondut eletriity. Notie the hexgonl grid pttern. (redit: AlexnderAIUS / Wikimedi

More information

#A42 INTEGERS 11 (2011) ON THE CONDITIONED BINOMIAL COEFFICIENTS

#A42 INTEGERS 11 (2011) ON THE CONDITIONED BINOMIAL COEFFICIENTS #A42 INTEGERS 11 (2011 ON THE CONDITIONED BINOMIAL COEFFICIENTS Liqun To Shool of Mthemtil Sienes, Luoyng Norml University, Luoyng, Chin lqto@lynuedun Reeived: 12/24/10, Revised: 5/11/11, Aepted: 5/16/11,

More information

TOPIC: LINEAR ALGEBRA MATRICES

TOPIC: LINEAR ALGEBRA MATRICES Interntionl Blurete LECTUE NOTES for FUTHE MATHEMATICS Dr TOPIC: LINEA ALGEBA MATICES. DEFINITION OF A MATIX MATIX OPEATIONS.. THE DETEMINANT deta THE INVESE A -... SYSTEMS OF LINEA EQUATIONS. 8. THE AUGMENTED

More information

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b

where the box contains a finite number of gates from the given collection. Examples of gates that are commonly used are the following: a b CS 294-2 9/11/04 Quntum Ciruit Model, Solovy-Kitev Theorem, BQP Fll 2004 Leture 4 1 Quntum Ciruit Model 1.1 Clssil Ciruits - Universl Gte Sets A lssil iruit implements multi-output oolen funtion f : {0,1}

More information

THE PYTHAGOREAN THEOREM

THE PYTHAGOREAN THEOREM THE PYTHAGOREAN THEOREM The Pythgoren Theorem is one of the most well-known nd widely used theorems in mthemtis. We will first look t n informl investigtion of the Pythgoren Theorem, nd then pply this

More information

A Matlab/Simulink Model of a Langevin s Ultrasonic Power Transducers

A Matlab/Simulink Model of a Langevin s Ultrasonic Power Transducers Mtl/Simulink Model of Lngevin s Ultrsoni Power Trnsduers Igor Jovnović, Ugleš Jovnović nd Drgn Mnčić strt Ultrsoni sndwih trnsduer, lso known s Lngevin s trnsduer, is hlf-wve resonnt struture tht osilltes

More information

Lecture 1 - Introduction and Basic Facts about PDEs

Lecture 1 - Introduction and Basic Facts about PDEs * 18.15 - Introdution to PDEs, Fll 004 Prof. Gigliol Stffilni Leture 1 - Introdution nd Bsi Fts bout PDEs The Content of the Course Definition of Prtil Differentil Eqution (PDE) Liner PDEs VVVVVVVVVVVVVVVVVVVV

More information

1 This question is about mean bond enthalpies and their use in the calculation of enthalpy changes.

1 This question is about mean bond enthalpies and their use in the calculation of enthalpy changes. 1 This question is out men ond enthlpies nd their use in the lultion of enthlpy hnges. Define men ond enthlpy s pplied to hlorine. Explin why the enthlpy of tomistion of hlorine is extly hlf the men ond

More information

THE ANALYSIS AND CALCULATION OF ELECTROMAGNETIC FIELD AROUND OVERHEAD POWER LINE HongWang Yang

THE ANALYSIS AND CALCULATION OF ELECTROMAGNETIC FIELD AROUND OVERHEAD POWER LINE HongWang Yang 5th Interntionl Conferene on Advned Mterils nd Computer Siene (ICAMCS 6) THE ANALYSIS AN CALCULATION OF ELECTROMAGNETIC FIEL AROUN OVERHEA POWER LINE HongWng Yng eprtment of eletril engineering, North

More information

Dense Coding, Teleportation, No Cloning

Dense Coding, Teleportation, No Cloning qitd352 Dense Coding, Teleporttion, No Cloning Roert B. Griffiths Version of 8 Ferury 2012 Referenes: NLQI = R. B. Griffiths, Nture nd lotion of quntum informtion Phys. Rev. A 66 (2002) 012311; http://rxiv.org/rhive/qunt-ph/0203058

More information

Research Article Comparative Studies of Different Switching Patterns for Direct and Indirect Space Vector Modulated Matrix Converter

Research Article Comparative Studies of Different Switching Patterns for Direct and Indirect Space Vector Modulated Matrix Converter dvnes in Power Eletronis Volume, rtile ID 854, 8 pges doi:.55//854 Reserh rtile omprtive Studies of Different Swithing Ptterns for Diret nd Indiret Spe Vetor Modulted Mtrix onverter min Shnpour, Ssn Gholmi,

More information

ANALYSIS OF CFD HEAT TRANSFER OF VACUUM FREEZE-DRYING SHELF

ANALYSIS OF CFD HEAT TRANSFER OF VACUUM FREEZE-DRYING SHELF HEFAT202 9 th Interntionl Conferene on Het Trnsfer, Fluid Mehnis nd Thermodynmis 6 8 July 202 Mlt ANALYSIS OF CFD HEAT TRANSFER OF VACUUM FREEZE-DRYING SHELF Hong-Ping Cheng *, Shin-min Tsi 2 * Professor,

More information

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1)

Math 32B Discussion Session Week 8 Notes February 28 and March 2, f(b) f(a) = f (t)dt (1) Green s Theorem Mth 3B isussion Session Week 8 Notes Februry 8 nd Mrh, 7 Very shortly fter you lerned how to integrte single-vrible funtions, you lerned the Fundmentl Theorem of lulus the wy most integrtion

More information

Chemical Equilibrium. Problem Set: Chapter 16 questions 25, 27, 33, 35, 43, 71

Chemical Equilibrium. Problem Set: Chapter 16 questions 25, 27, 33, 35, 43, 71 Chemil Equilibrium roblem Set: Chpter 16 questions 5, 7, 33, 35, 43, 71 Exmples of Equilibrium Wter n exists simultneously in the gs nd liquid phse. The vpor pressure of H O t given temperture is property

More information

If only one fertilizer x is used, the dependence of yield z(x) on x first was given by Mitscherlich (1909) in form of the differential equation

If only one fertilizer x is used, the dependence of yield z(x) on x first was given by Mitscherlich (1909) in form of the differential equation Mitsherlih s Lw: Generliztion with severl Fertilizers Hns Shneeerger Institute of Sttistis, University of Erlngen-Nürnerg, Germny 00, 5 th August Astrt: It is shown, tht the rop-yield z in dependene on

More information

Supporting Information

Supporting Information tom-thik Interlyer Mde of VD-Grown Grphene Film on Seprtor for dvned thium-sulfur tteries Zhenzhen Du 1, hengkun Guo 2, njun Wng 3, jun Hu 1, Song Jin 1, Timing Zhng 1, Honghng Jin 1, Zhiki Qi 1, Sen Xin

More information

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets.

Reference : Croft & Davison, Chapter 12, Blocks 1,2. A matrix ti is a rectangular array or block of numbers usually enclosed in brackets. I MATRIX ALGEBRA INTRODUCTION TO MATRICES Referene : Croft & Dvison, Chpter, Blos, A mtri ti is retngulr rr or lo of numers usull enlosed in rets. A m n mtri hs m rows nd n olumns. Mtri Alger Pge If the

More information

A Mathematical Model for Unemployment-Taking an Action without Delay

A Mathematical Model for Unemployment-Taking an Action without Delay Advnes in Dynmil Systems nd Applitions. ISSN 973-53 Volume Number (7) pp. -8 Reserh Indi Publitions http://www.ripublition.om A Mthemtil Model for Unemployment-Tking n Ation without Dely Gulbnu Pthn Diretorte

More information

On the Scale factor of the Universe and Redshift.

On the Scale factor of the Universe and Redshift. On the Sle ftor of the Universe nd Redshift. J. M. unter. john@grvity.uk.om ABSTRACT It is proposed tht there hs been longstnding misunderstnding of the reltionship between sle ftor of the universe nd

More information

1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light.

1 This diagram represents the energy change that occurs when a d electron in a transition metal ion is excited by visible light. 1 This igrm represents the energy hnge tht ours when eletron in trnsition metl ion is exite y visile light. Give the eqution tht reltes the energy hnge ΔE to the Plnk onstnt, h, n the frequeny, v, of the

More information

VIBRATION ANALYSIS OF AN ISOLATED MASS WITH SIX DEGREES OF FREEDOM Revision G

VIBRATION ANALYSIS OF AN ISOLATED MASS WITH SIX DEGREES OF FREEDOM Revision G B Tom Irvine Emil: tom@virtiondt.om Jnur 8, 3 VIBRATION ANALYSIS OF AN ISOLATED MASS WITH SIX DEGREES OF FREEDOM Revision G Introdution An vionis omponent m e mounted with isoltor grommets, whih t s soft

More information

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1. 1 [(y ) 2 + yy + y 2 ] dx,

MATH34032: Green s Functions, Integral Equations and the Calculus of Variations 1. 1 [(y ) 2 + yy + y 2 ] dx, MATH3403: Green s Funtions, Integrl Equtions nd the Clulus of Vritions 1 Exmples 5 Qu.1 Show tht the extreml funtion of the funtionl I[y] = 1 0 [(y ) + yy + y ] dx, where y(0) = 0 nd y(1) = 1, is y(x)

More information

Finite Element Simulation on Frictional and Brittle Preseismic fault slip

Finite Element Simulation on Frictional and Brittle Preseismic fault slip Finite Element Simultion on Fritionl nd Brittle Preseismi fult slip Zhishen Wu (1) Yun Go (1) Yutk Murkmi (2) (1) Deprtment of Urn & Civil Engineering. Irki University, Jpn (e-mil: zswu@ip.irki..jp; goyun@hs.irki..jp,

More information

VISIBLE AND INFRARED ABSORPTION SPECTRA OF COVERING MATERIALS FOR SOLAR COLLECTORS

VISIBLE AND INFRARED ABSORPTION SPECTRA OF COVERING MATERIALS FOR SOLAR COLLECTORS AGRICULTURAL ENGINEERING VISIBLE AND INFRARED ABSORPTION SPECTRA OF COVERING MATERIALS FOR SOLAR COLLECTORS Ltvi University of Agriulture E-mil: ilze.pelee@llu.lv Astrt Use of solr energy inreses every

More information

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version

A Lower Bound for the Length of a Partial Transversal in a Latin Square, Revised Version A Lower Bound for the Length of Prtil Trnsversl in Ltin Squre, Revised Version Pooy Htmi nd Peter W. Shor Deprtment of Mthemtil Sienes, Shrif University of Tehnology, P.O.Bo 11365-9415, Tehrn, Irn Deprtment

More information

AC/DC/AC Converters: Two-Level and Multilevel VSI

AC/DC/AC Converters: Two-Level and Multilevel VSI Sortes Ersmus Visit A/D/A onerters: Two-Leel nd Multileel VSI Josep Pou Antoni Aris Pge 1 Sortes Ersmus Visit Outline 1. Two-Leel Inerter 2. Multileel Inerters - sde H-Bridge Inerter - Flying-pitor Inerter

More information

Distributed Generation Placement in Unbalanced Distribution System with Seasonal Load Variation

Distributed Generation Placement in Unbalanced Distribution System with Seasonal Load Variation Distriuted Genertion Plement in Unlned Distriution System with Sesonl Lod Vrition Rvi Tej Bhimrsetti Dept. of Eletril Engg., NT Kurukshetr Kurukshetr, ndi svrtej@gmil.om Ashwni Kumr, Memer, EEE Dept. of

More information

H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3.

H 4 H 8 N 2. Example 1 A compound is found to have an accurate relative formula mass of It is thought to be either CH 3. . Spetrosopy Mss spetrosopy igh resolution mss spetrometry n e used to determine the moleulr formul of ompound from the urte mss of the moleulr ion For exmple, the following moleulr formuls ll hve rough

More information

The Stirling Engine: The Heat Engine

The Stirling Engine: The Heat Engine Memoril University of Newfounln Deprtment of Physis n Physil Oenogrphy Physis 2053 Lortory he Stirling Engine: he Het Engine Do not ttempt to operte the engine without supervision. Introution Het engines

More information

f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums

f (x)dx = f(b) f(a). a b f (x)dx is the limit of sums Green s Theorem If f is funtion of one vrible x with derivtive f x) or df dx to the Fundmentl Theorem of lulus, nd [, b] is given intervl then, ording This is not trivil result, onsidering tht b b f x)dx

More information

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem.

Lesson 2: The Pythagorean Theorem and Similar Triangles. A Brief Review of the Pythagorean Theorem. 27 Lesson 2: The Pythgoren Theorem nd Similr Tringles A Brief Review of the Pythgoren Theorem. Rell tht n ngle whih mesures 90º is lled right ngle. If one of the ngles of tringle is right ngle, then we

More information

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS

The University of Nottingham SCHOOL OF COMPUTER SCIENCE A LEVEL 2 MODULE, SPRING SEMESTER MACHINES AND THEIR LANGUAGES ANSWERS The University of ottinghm SCHOOL OF COMPUTR SCIC A LVL 2 MODUL, SPRIG SMSTR 2015 2016 MACHIS AD THIR LAGUAGS ASWRS Time llowed TWO hours Cndidtes my omplete the front over of their nswer ook nd sign their

More information

Line Integrals and Entire Functions

Line Integrals and Entire Functions Line Integrls nd Entire Funtions Defining n Integrl for omplex Vlued Funtions In the following setions, our min gol is to show tht every entire funtion n be represented s n everywhere onvergent power series

More information

Comparing the Pre-image and Image of a Dilation

Comparing the Pre-image and Image of a Dilation hpter Summry Key Terms Postultes nd Theorems similr tringles (.1) inluded ngle (.2) inluded side (.2) geometri men (.) indiret mesurement (.6) ngle-ngle Similrity Theorem (.2) Side-Side-Side Similrity

More information

The study of dual integral equations with generalized Legendre functions

The study of dual integral equations with generalized Legendre functions J. Mth. Anl. Appl. 34 (5) 75 733 www.elsevier.om/lote/jm The study of dul integrl equtions with generlized Legendre funtions B.M. Singh, J. Rokne,R.S.Dhliwl Deprtment of Mthemtis, The University of Clgry,

More information

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES

PAIR OF LINEAR EQUATIONS IN TWO VARIABLES PAIR OF LINEAR EQUATIONS IN TWO VARIABLES. Two liner equtions in the sme two vriles re lled pir of liner equtions in two vriles. The most generl form of pir of liner equtions is x + y + 0 x + y + 0 where,,,,,,

More information

Power System Representation and Equations. A one-line diagram of a simple power system

Power System Representation and Equations. A one-line diagram of a simple power system Power ystem epresenttion nd Equtions Lod B Lod A Bus Bus A oneline digrm of simple power system Oil or liquid iruit reker otting mhine Twowinding power trnsformer Wye onnetion, neutrl ground PerPhse, Per

More information

Something found at a salad bar

Something found at a salad bar Nme PP Something found t sld r 4.7 Notes RIGHT TRINGLE hs extly one right ngle. To solve right tringle, you n use things like SOH-H-TO nd the Pythgoren Theorem. n OLIQUE TRINGLE hs no right ngles. To solve

More information

Nondeterministic Automata vs Deterministic Automata

Nondeterministic Automata vs Deterministic Automata Nondeterministi Automt vs Deterministi Automt We lerned tht NFA is onvenient model for showing the reltionships mong regulr grmmrs, FA, nd regulr expressions, nd designing them. However, we know tht n

More information

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx

for all x in [a,b], then the area of the region bounded by the graphs of f and g and the vertical lines x = a and x = b is b [ ( ) ( )] A= f x g x dx Applitions of Integrtion Are of Region Between Two Curves Ojetive: Fin the re of region etween two urves using integrtion. Fin the re of region etween interseting urves using integrtion. Desrie integrtion

More information

Particle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 3 : Interaction by Particle Exchange and QED. Recap

Particle Physics. Michaelmas Term 2011 Prof Mark Thomson. Handout 3 : Interaction by Particle Exchange and QED. Recap Prtile Physis Mihelms Term 2011 Prof Mrk Thomson g X g X g g Hnout 3 : Intertion y Prtile Exhnge n QED Prof. M.A. Thomson Mihelms 2011 101 Rep Working towrs proper lultion of ey n sttering proesses lnitilly

More information

Magnetically Coupled Coil

Magnetically Coupled Coil Mgnetilly Coupled Ciruits Overview Mutul Indutne Energy in Coupled Coils Liner Trnsformers Idel Trnsformers Portlnd Stte University ECE 22 Mgnetilly Coupled Ciruits Ver..3 Mgnetilly Coupled Coil i v L

More information

Elecrochemical Behaviour of Zinc on Copper and on Vitreous. Carbon Electrodes. The Influence of Gluconate

Elecrochemical Behaviour of Zinc on Copper and on Vitreous. Carbon Electrodes. The Influence of Gluconate Portuglie Eletrohimi At 21 (2003) 179-189 PORTUGALIAE ELECTROCHIMICA ACTA Elerohemil Behviour of Zin on Copper nd on Vitreous Cron Eletrodes. The Influene of Gluonte J. Torrent-Burgués *, E. Guus Group

More information

Environmental Science

Environmental Science ISSN : 0974-74 Volume 8 Issue Environmentl Siene An Indin Journl Current Reserh Pper ESAIJ, 8(), 03 [436-44] Theoretil nd experimentl study of rdon seprtion from ter by bubbling system Hossein Noorinezhd,

More information

INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable

INTEGRATION. 1 Integrals of Complex Valued functions of a REAL variable INTEGRATION NOTE: These notes re supposed to supplement Chpter 4 of the online textbook. 1 Integrls of Complex Vlued funtions of REAL vrible If I is n intervl in R (for exmple I = [, b] or I = (, b)) nd

More information

21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae

21.1 Using Formulae Construct and Use Simple Formulae Revision of Negative Numbers Substitution into Formulae MEP Jmi: STRAND G UNIT 1 Formule: Student Tet Contents STRAND G: Alger Unit 1 Formule Student Tet Contents Setion 1.1 Using Formule 1. Construt nd Use Simple Formule 1.3 Revision of Negtive Numers 1.4

More information

Modeling and Simulation of Permanent Magnet Brushless Motor Drives using Simulink

Modeling and Simulation of Permanent Magnet Brushless Motor Drives using Simulink INDIAN INSTITUTE OF TECHNOLOGY, KHARAGPUR 72102, DECEMBER 27-29, 2002 25 Modeling nd Simultion of Permnent Mgnet Brushless Motor Dries using Simulink Mukesh Kumr, Bhim Singh nd B.P.Singh Astrt: Permnent

More information