CIVL 8/ D Boundary Value Problems - Quadrilateral Elements (Q4) 1/8
|
|
- Blake Reed
- 6 years ago
- Views:
Transcription
1 CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 1/8 ISOPARAMERIC ELEMENS h inar rianguar mn and h iinar rcanguar mn hav vra imporan diadvanag. 1. Boh mn ar una o accura rprn curvd oundari, and. h provid poor approimaion for an drivd varia. B uiizing highr-ordr mn ioparamric mn h dficinci of h inar mn ar ovrcom. ISOPARAMERIC ELEMENS h rm ioparamric man ha an qua numr of paramr ar ud o rprn h gomr and h dpndn varia. In ohr word, h am hap funcion ud o inrpoa h dpndn varia ar ud o rprn h gomr. A uparamric mn i on whr h gomr i dfind fwr paramr han ud o inrpoa h ouion. A uprparamric mn i on whr h gomr i dfind mor paramr han ud o inrpoa h ouion. ISOPARAMERIC ELEMENS h rm ioparamric man ha an qua numr of paramr ar ud o rprn h gomr and h dpndn varia. u un i i 6-nod rianguar mn (6) N N i i i i 8-nod quadriara mn (Q8) h four-nod quadriara mn i vr imiar o h four-nod rcanguar mn. Boh provid iinar inrpoaion ovr h mn for h gnra ouion and a inar approimaion of h fir drivaiv. Howvr, h quadriara mn i no rricd o rcanguar hap; in fac i ma rprn an numr of rciinar hap. h gnra approimaion ovr a quadriara mn i aumd o : u, un, un, un, un, u NuN,,,, h mna coordina ar and. h inrpoaion funcion N i (, ) ar idnica o h rcanguar mn inrpoaion funcion. N, N, N, N,
2 CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) /8 h ranformaion from goa coordina (, ) o mna or parna coordina (, ) i accompihd uing h foowing coordina mapping:,,,,, N N N N,,,,, N N N N N N N N h inrpoaion funcion for a quadriara mn ar rfrrd a hap funcion inc h prion dfin h hap of h mn. L ook a how h in and poin in h parn mn (, ) ar mappd or ranformd o h goa m (, ).,, * (,) 1 1,, * (,) 1 1 1, 1, (,) 1 a 1 1 1, 1, (,) 1 a 1 1 In gnra raigh in in h - coordina m ar mappd ino raigh in in h quadriara mn. For amp, poin a and ar mappd o poin a* and * in h quadriara. h raigh in = 1 in h parn mn, conncing nod and, ma mappd ino h quadriara a:,, * (,) 1 1,, * (,) 1 1 1, 1, (,) 1 a 1 1 1, 1, (,) 1 a 1 1 In ordr o undrand h naur of h ranformaion, h chain ru i ud o form h diffrnia raionhip: In mari noaion h drivaiv ma wrin a: J whr J J h drminan of h Jacoian mari, J, i a of h invriii of h ranformaion = (, ) and = (, ). Whn J i poiiv vrwhr in a rgion, h ranformaion ma invrd o drmin = (,) and = (, ). hi man ha for a givn poin (, ) in h quadriara hr i a uniqu corrponding poin (, ) in h parn mn.
3 CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) /8 h mn of h Jacoian mari ma compud a: hrfor, J ha h form: J j1 j j j whr h conan j i ar funcion of h coordina of h nod (funcion of h gomr of h mn). h conan j i acua zro o ha J i a inar funcion of and. h drminan J gomrica rprn h raionhip wn h diffrnia ara da(, ) = d d and h diffrnia ara da(, ). J da, da, J d d PROBLEM #1 - Show ha h conan j dfining h drminan of h Jacoian J i zro. Eamp - Conidr h quadriara mn givn ow. 1, 0, 10,10 0, 11 h mn of h Jacoian mari ma cacuad a: Eamp - Conidr h quadriara mn givn ow. 1, 0, 10,10 0, 11 h mn of h Jacoian mari ma cacuad a: Eamp - h drminan i: J 8 Conidr h ocaion of nod hr a = 19 and =, hn h ranformaion i givn : J 8
4 CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) /8 Eamp - Ovr h inrva -1 1 and -1 1 h drminan of h Jacoian, J, i poiiv, indicaing ha hr ar no prom wih h mapping from - o -. h ara of h quadriara ma compud a: 98 A A d d J d d d d Eamp - h parn mn in h - m and h hap of h quadriara mn ar hown in h figur ow: 1, 0 10,10 * 19, (,) (,) 0, 11 1 a Eamp - Conidr h ocaion of nod hr a = 16 and = 1, hn h ranformaion i givn : J J Eamp - Ovr h inrva -1 1 and -1 1 h drminan of h Jacoian, J, i no awa poiiv. In fac, J i ngaiv for ( + ) 5/. For poin (, ) who vau aif ( + ) 5/, h mapping oca hm ouid of h quadriara. h prom i aociad wih h fac h rgion i conv aong h id dfind nod,, and. 9 7 J Eamp - h parn mn in h - m and h hap of h quadriara mn ar hown in h figur ow: 1, 0 10,10 (,) 16, 1 (,) 0, 11 1 a h mna maric for h Poion prom ar: NN NN k da f f da N A A a NN d h Nh d
5 CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 5/8 h dvopmn of oh h k and f rm i idnica o ha prnd for rcanguar mn: k JJ d d f Nf J d d whr JJ = (J 1 J 1 + J J ) J. h vau of h mari ma compud a: Rca, J 1 and J ar h fir and cond row of h invr of h Jacoian mari. 1 1 J J1 J J J J h mn of h invr of h Jacoian ar: J For a gnra quadriara mn, h mari JJ i a funcion of and wih h Jacoian J in h dnominaor. h ruing prion for JJ i vr difficu o vaua ac and h ingraion ar uua don numrica. hrfor, h rm k and f ma ca in h foowing form: G, dd whr G(,) i a compicad funcion of h varia and. In princip, i ma poi o vaua h f rm, howvr, numrica ingraion i pica mor pracica. For h k rm, h apparanc of h Jacoian J in h ingrand gnra indica h u of numrica quadraur. I i foruna ha h imi of h ingra conform o h forma ud in Gau-Lgndr quadraur. Sinc h ingraion ar in oh h and -dircion, on approach i o app Gauian quadraur para in ach dircion. hrfor, h gnra prion for k and f an N-rm Gauian quadraur i: k JJ d d ww JJ f Nf Jdd i n i n 1 j1 1 j1 whr h ucrip ij rfr o h vauaion of h quani a i and j. n n i j ij ij ij,,,, ww N f J i j i j i j i j ij Conidr h ingra a and h : a NN d h Nh d whr h ingraion i aong a oundar gmn of h mn. Sinc, h ingraion i compud aong a ing id of h quadriara mn, h quadriara hap funcion rduc o inar funcion.
6 CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 6/8 h ru for hi ingraion ar idnica o prion oaind for inar rianguar and rcanguar mn. j a h k h k 0 i Nk N k k k Nk hn k k hn d N h N N N N d Nk 1 N Evauaing h ingra, h a and h rm rduc o: j a k h k 0 i k k 1 k k h Nk 1 N h h k h 6 hk h h ruing a mna iffn mari and h 1 h mna oad vcor conriu o h goa m quaion if h mn ha a id a par of h oundar. h goa m quaion ar compod from h foowing ummaion: K k a F f h G G G G G G h ruing m quaion ar, in mari form, givn a: Ku F G G G Eamp - Conidr h quadriara mn givn ow. 1, 0, 10,10 0, 11 Cacua h vau of h iffn mari k for h aov mn whn = 19 and = 1. k JJ d d f Nf J d d 1 1 Eamp - Rca JJ = (J 1 J 1 + J J ) J. h vau of h mari ma compud a: whr J 1 and J ar h fir and cond row of h invr of h Jacoian mari J J J 1 J J1 J Eamp - h mn of h invr of h Jacoian ar:
7 CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 7/8 Eamp - h cacuaion ar prformd compuing h vau of ach componn of,, J 1, J, and J maric a ach Gau poin, prforming h mari muipicaion, and fina umming h ru for a h Gau poin. h u of numrica quadraur a hi poin in h formuaion ma gnra om rror. Howvr, if nough Gau poin ar ud h numrica ru ar qui accura whn compard o h ac vau for h ingraion. Eamp - A wo-dimniona quadraur uing N = i acua Gau poin and uing N = i 16 quadraur poin. 1 N = 1 Gau poin 1 N = 1 N = Eamp - hi fac i vidn in h foowing cacuaion uing Gauian quadraur wih N = 1,,, and. N k mmric Eac k mmric Eamp - hi fac i vidn in h foowing cacuaion uing Gauian quadraur wih N = 1,,, and. N Eac k k mmric mmric Eamp - hi fac i vidn in h foowing cacuaion uing Gauian quadraur wih N = 1,,, and. N Eac k k mmric mmric Eamp - hi fac i vidn in h foowing cacuaion uing Gauian quadraur wih N = 1,,, and. N Eac k k mmric mmric
8 CIVL 8/7111 -D Boundar Vau Prom - Quadriara Emn (Q) 8/8 Eamp - h ru for N = 1 ar no vr accura. Sinc on on Gau poin i ud in h quadraur, hi ru i no unpcd. h vau for k uing Gauian quadraur wih N = and indica ha h accurac of h vauaion incra a h numr of Gau poin incra. Eamp - Drivd Varia - h drivd varia for h Poion or Lapac quaion ar h paria drivaiv u/ and u/. h rm ar compud a funcion of poiion in an mn uing h hap funcion and h noda vau. h ru uing N = ar ac o fiv or i dcima pac. Eamp - Drivd Varia - h paria drivaiv ma comind o giv h norma or dirciona drivaiv: nin jn u u n un u u n n u u J u u J1 u 1 u J u u J Eamp - Drivd Varia - Rca J 1 and J ar h fir and cond row of J -1, rpciv. Rca ha u / i: u [ Nu] u u u [ Nu] hrfor: u u J1 u J u h dirciona or norma drivaiv i: u nj n J u n 1 End of Quadriara mn (Q)
Elementary Differential Equations and Boundary Value Problems
Elmnar Diffrnial Equaions and Boundar Valu Problms Boc. & DiPrima 9 h Ediion Chapr : Firs Ordr Diffrnial Equaions 00600 คณ ตศาสตร ว ศวกรรม สาขาว ชาว ศวกรรมคอมพ วเตอร ป การศ กษา /55 ผศ.ดร.อร ญญา ผศ.ดร.สมศ
More informationDEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018
DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Aoc. Prof. Dr. Burak Kllci Spring 08 OUTLINE Th Laplac Tranform Rgion of convrgnc for Laplac ranform Invr Laplac ranform Gomric valuaion
More information2. The Laplace Transform
Th aac Tranorm Inroucion Th aac ranorm i a unamna an vry uu oo or uying many nginring robm To in h aac ranorm w conir a comx variab σ, whr σ i h ra ar an i h imaginary ar or ix vau o σ an w viw a a oin
More informationCIVL 8/ D Boundary Value Problems - Triangular Elements (T6) 1/8
CIVL 8/7 -D Boundar Valu Problm - rangular Elmn () /8 SI-ODE RIAGULAR ELEMES () A quadracall nrpolad rangular lmn dfnd b nod, hr a h vrc and hr a h mddl a ach d. h mddl nod, dpndng on locaon, ma dfn a
More informationBoyce/DiPrima 9 th ed, Ch 2.1: Linear Equations; Method of Integrating Factors
Boc/DiPrima 9 h d, Ch.: Linar Equaions; Mhod of Ingraing Facors Elmnar Diffrnial Equaions and Boundar Valu Problms, 9 h diion, b William E. Boc and Richard C. DiPrima, 009 b John Wil & Sons, Inc. A linar
More informationTransfer function and the Laplace transformation
Lab No PH-35 Porland Sa Univriy A. La Roa Tranfr funcion and h Laplac ranformaion. INTRODUTION. THE LAPLAE TRANSFORMATION L 3. TRANSFER FUNTIONS 4. ELETRIAL SYSTEMS Analyi of h hr baic paiv lmn R, and
More informationChapter 12 Introduction To The Laplace Transform
Chapr Inroducion To Th aplac Tranorm Diniion o h aplac Tranorm - Th Sp & Impul uncion aplac Tranorm o pciic uncion 5 Opraional Tranorm Applying h aplac Tranorm 7 Invr Tranorm o Raional uncion 8 Pol and
More informationMidterm exam 2, April 7, 2009 (solutions)
Univrsiy of Pnnsylvania Dparmn of Mahmaics Mah 26 Honors Calculus II Spring Smsr 29 Prof Grassi, TA Ashr Aul Midrm xam 2, April 7, 29 (soluions) 1 Wri a basis for h spac of pairs (u, v) of smooh funcions
More informationBoyce/DiPrima 9 th ed, Ch 7.8: Repeated Eigenvalues
Boy/DiPrima 9 h d Ch 7.8: Rpad Eignvalus Elmnary Diffrnial Equaions and Boundary Valu Problms 9 h diion by William E. Boy and Rihard C. DiPrima 9 by John Wily & Sons In. W onsidr again a homognous sysm
More informationWave Equation (2 Week)
Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris
More informationChap.3 Laplace Transform
Chap. aplac Tranorm Tranorm: An opraion ha ranorm a uncion ino anohr uncion i Dirniaion ranorm: ii x: d dx x x Ingraion ranorm: x: x dx x c Now, conidr a dind ingral k, d,ha ranorm ino a uncion o variabl
More informationS.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]
S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:
More informationUNIT #5 EXPONENTIAL AND LOGARITHMIC FUNCTIONS
Answr Ky Nam: Da: UNIT # EXPONENTIAL AND LOGARITHMIC FUNCTIONS Par I Qusions. Th prssion is quivaln o () () 6 6 6. Th ponnial funcion y 6 could rwrin as y () y y 6 () y y (). Th prssion a is quivaln o
More informationLecture 26: Leapers and Creepers
Lcur 6: Lapr and Crpr Scrib: Grain Jon (and Marin Z. Bazan) Dparmn of Economic, MIT May, 005 Inroducion Thi lcur conidr h analyi of h non-parabl CTRW in which h diribuion of p iz and im bwn p ar dpndn.
More informationLaPlace Transform in Circuit Analysis
LaPlac Tranform in Circui Analyi Obciv: Calcula h Laplac ranform of common funcion uing h dfiniion and h Laplac ranform abl Laplac-ranform a circui, including componn wih non-zro iniial condiion. Analyz
More informationEXERCISE - 01 CHECK YOUR GRASP
DIFFERENTIAL EQUATION EXERCISE - CHECK YOUR GRASP 7. m hn D() m m, D () m m. hn givn D () m m D D D + m m m m m m + m m m m + ( m ) (m ) (m ) (m + ) m,, Hnc numbr of valus of mn will b. n ( ) + c sinc
More informationChapter 5 The Laplace Transform. x(t) input y(t) output Dynamic System
EE 422G No: Chapr 5 Inrucor: Chung Chapr 5 Th Laplac Tranform 5- Inroducion () Sym analyi inpu oupu Dynamic Sym Linar Dynamic ym: A procor which proc h inpu ignal o produc h oupu dy ( n) ( n dy ( n) +
More informationInstructors Solution for Assignment 3 Chapter 3: Time Domain Analysis of LTIC Systems
Inrucor Soluion for Aignmn Chapr : Tim Domain Anali of LTIC Sm Problm i a 8 x x wih x u,, an Zro-inpu rpon of h m: Th characriic quaion of h LTIC m i i 8, which ha roo a ± j Th zro-inpu rpon i givn b zi
More information4.i. 4 Numerical Model Development 4.1
4i Chapr 4 4 Numrica Mod Dvopmn 41 Asrac 41 Résumé 41 41 Govrning quaions 4 4 Souion sragy: h iraiv mhod 45 43 Numrica mhod: h fini-voum approximaion 48 431 Grid arrangmn 48 43 Compuaion of h surfac ara
More informationLecture 1: Numerical Integration The Trapezoidal and Simpson s Rule
Lcur : Numrical ngraion Th Trapzoidal and Simpson s Rul A problm Th probabiliy of a normally disribud (man µ and sandard dviaion σ ) vn occurring bwn h valus a and b is B A P( a x b) d () π whr a µ b -
More informationREPETITION before the exam PART 2, Transform Methods. Laplace transforms: τ dτ. L1. Derive the formulas : L2. Find the Laplace transform F(s) if.
Tranform Mhod and Calculu of Svral Variabl H7, p Lcurr: Armin Halilovic KTH, Campu Haning E-mail: armin@dkh, wwwdkh/armin REPETITION bfor h am PART, Tranform Mhod Laplac ranform: L Driv h formula : a L[
More informationCHAPTER. Linear Systems of Differential Equations. 6.1 Theory of Linear DE Systems. ! Nullcline Sketching. Equilibrium (unstable) at (0, 0)
CHATER 6 inar Sysms of Diffrnial Equaions 6 Thory of inar DE Sysms! ullclin Skching = y = y y υ -nullclin Equilibrium (unsabl) a (, ) h nullclin y = υ nullclin = h-nullclin (S figur) = + y y = y Equilibrium
More informationCIVL 8/ D Boundary Value Problems - Rectangular Elements 1/7
CIVL / -D Boundr Vlu Prolms - Rctngulr Elmnts / RECANGULAR ELEMENS - In som pplictions, it m mor dsirl to us n lmntl rprsnttion of th domin tht hs four sids, ithr rctngulr or qudriltrl in shp. Considr
More informationCopyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.
Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin
More informationLogistic equation of Human population growth (generalization to the case of reactive environment).
Logisic quaion of Human populaion growh gnralizaion o h cas of raciv nvironmn. Srg V. Ershkov Insiu for Tim aur Exploraions M.V. Lomonosov's Moscow Sa Univrsi Lninski gor - Moscow 999 ussia -mail: srgj-rshkov@andx.ru
More informationFinal Exam : Solutions
Comp : Algorihm and Daa Srucur Final Exam : Soluion. Rcuriv Algorihm. (a) To bgin ind h mdian o {x, x,... x n }. Sinc vry numbr xcp on in h inrval [0, n] appar xacly onc in h li, w hav ha h mdian mu b
More informationPWM-Scheme and Current ripple of Switching Power Amplifiers
axon oor PWM-Sch and Currn rippl of Swiching Powr Aplifir Abrac In hi work currn rippl caud by wiching powr aplifir i analyd for h convnional PWM (pulwidh odulaion) ch and hr-lvl PWM-ch. Siplifid odl for
More informationSpring 2006 Process Dynamics, Operations, and Control Lesson 2: Mathematics Review
Spring 6 Procss Dynamics, Opraions, and Conrol.45 Lsson : Mahmaics Rviw. conx and dircion Imagin a sysm ha varis in im; w migh plo is oupu vs. im. A plo migh imply an quaion, and h quaion is usually an
More information1 Finite Automata and Regular Expressions
1 Fini Auom nd Rgulr Exprion Moivion: Givn prn (rgulr xprion) for ring rching, w migh wn o convr i ino drminiic fini uomon or nondrminiic fini uomon o mk ring rching mor fficin; drminiic uomon only h o
More informationLet s look again at the first order linear differential equation we are attempting to solve, in its standard form:
Th Ingraing Facor Mhod In h prvious xampls of simpl firs ordr ODEs, w found h soluions by algbraically spara h dpndn variabl- and h indpndn variabl- rms, and wri h wo sids of a givn quaion as drivaivs,
More informationAR(1) Process. The first-order autoregressive process, AR(1) is. where e t is WN(0, σ 2 )
AR() Procss Th firs-ordr auorgrssiv procss, AR() is whr is WN(0, σ ) Condiional Man and Varianc of AR() Condiional man: Condiional varianc: ) ( ) ( Ω Ω E E ) var( ) ) ( var( ) var( σ Ω Ω Ω Ω E Auocovarianc
More informationTransmission Line Theory
Tranmiion in Thory Dr. M.A.Moawa nroducion: n an cronic ym h divry of powr rquir h conncion of wo wir bwn h ourc and h oad. A ow frqunci powr i conidrd o b divrd o h oad hrough h wir. n h microwav frquncy
More informationwhere: u: input y: output x: state vector A, B, C, D are const matrices
Sa pac modl: linar: y or in om : Sa q : f, u Oupu q : y h, u u Du F Gu y H Ju whr: u: inpu y: oupu : a vcor,,, D ar con maric Eampl " $ & ' " $ & 'u y " & * * * * [ ],, D H D I " $ " & $ ' " & $ ' " &
More informationThe transition:transversion rate ratio vs. the T-ratio.
PhyloMah Lcur 8 by Dan Vandrpool March, 00 opics of Discussion ransiion:ransvrsion ra raio Kappa vs. ransiion:ransvrsion raio raio alculaing h xpcd numbr of subsiuions using marix algbra Why h nral im
More information14.02 Principles of Macroeconomics Fall 2005 Quiz 3 Solutions
4.0 rincipl of Macroconomic Fall 005 Quiz 3 Soluion Shor Quion (30/00 poin la a whhr h following amn ar TRUE or FALSE wih a hor xplanaion (3 or 4 lin. Each quion coun 5/00 poin.. An incra in ax oday alway
More informationSOLUTIONS. 1. Consider two continuous random variables X and Y with joint p.d.f. f ( x, y ) = = = 15. Stepanov Dalpiaz
STAT UIUC Pracic Problms #7 SOLUTIONS Spanov Dalpiaz Th following ar a numbr of pracic problms ha ma b hlpful for compling h homwor, and will lil b vr usful for suding for ams.. Considr wo coninuous random
More informationfiziks Institute for NET/JRF, GATE, IIT JAM, JEST, TIFR and GRE in PHYSICAL SCIENCES MATEMATICAL PHYSICS SOLUTIONS are
MTEMTICL PHYSICS SOLUTIONS GTE- Q. Considr an ani-symmric nsor P ij wih indics i and j running from o 5. Th numbr of indpndn componns of h nsor is 9 6 ns: Soluion: Th numbr of indpndn componns of h nsor
More informationCSE 245: Computer Aided Circuit Simulation and Verification
CSE 45: Compur Aidd Circui Simulaion and Vrificaion Fall 4, Sp 8 Lcur : Dynamic Linar Sysm Oulin Tim Domain Analysis Sa Equaions RLC Nwork Analysis by Taylor Expansion Impuls Rspons in im domain Frquncy
More informationDE Dr. M. Sakalli
DE-0 Dr. M. Sakalli DE 55 M. Sakalli a n n 0 a Lh.: an Linar g Equaions Hr if g 0 homognous non-homognous ohrwis driving b a forc. You know h quaions blow alrad. A linar firs ordr ODE has h gnral form
More informationOn the Speed of Heat Wave. Mihály Makai
On h Spd of Ha Wa Mihály Maai maai@ra.bm.hu Conns Formulaion of h problm: infini spd? Local hrmal qulibrium (LTE hypohsis Balanc quaion Phnomnological balanc Spd of ha wa Applicaion in plasma ranspor 1.
More informationApplied Statistics and Probability for Engineers, 6 th edition October 17, 2016
Applid Saisics and robabiliy for Enginrs, 6 h diion Ocobr 7, 6 CHATER Scion - -. a d. 679.. b. d. 88 c d d d. 987 d. 98 f d.. Thn, = ln. =. g d.. Thn, = ln.9 =.. -7. a., by symmry. b.. d...6. 7.. c...
More informationLecture 2: Current in RC circuit D.K.Pandey
Lcur 2: urrn in circui harging of apacior hrough Rsisr L us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R and a ky K in sris. Whn h ky K is swichd on, h charging
More informationLecture 4: Laplace Transforms
Lur 4: Lapla Transforms Lapla and rlad ransformaions an b usd o solv diffrnial quaion and o rdu priodi nois in signals and imags. Basially, hy onvr h drivaiv opraions ino mulipliaion, diffrnial quaions
More informationMa/CS 6a Class 15: Flows and Bipartite Graphs
//206 Ma/CS 6a Cla : Flow and Bipari Graph By Adam Shffr Rmindr: Flow Nwork A flow nwork i a digraph G = V, E, oghr wih a ourc vrx V, a ink vrx V, and a capaciy funcion c: E N. Capaciy Sourc 7 a b c d
More information15th European Signal Processing Conference (EUSIPCO 2007), Poznan, Poland, September 3-7, 2007, copyright by EURASIP
5h Eropan Signa Proing Confrn (EUSIPCO 7 Poznan Poand Spmbr -7 7 opyrigh by EURASIP COMPARISO OF HE COHERE A HE COMPOE MEHOS FOR ESIMAIG HE CHARACERISICS OF HE PERIOICALLY CORRELAE RAOM PROCESSES Ihor
More informationNARAYANA I I T / P M T A C A D E M Y. C o m m o n P r a c t i c e T e s t 1 6 XII STD BATCHES [CF] Date: PHYSIS HEMISTRY MTHEMTIS
. (D). (A). (D). (D) 5. (B) 6. (A) 7. (A) 8. (A) 9. (B). (A). (D). (B). (B). (C) 5. (D) NARAYANA I I T / P M T A C A D E M Y C o m m o n P r a c t i c T s t 6 XII STD BATCHES [CF] Dat: 8.8.6 ANSWER PHYSIS
More informationMath 266, Practice Midterm Exam 2
Mh 66, Prcic Midrm Exm Nm: Ground Rul. Clculor i NOT llowd.. Show your work for vry problm unl ohrwi d (pril crdi r vilbl). 3. You my u on 4-by-6 indx crd, boh id. 4. Th bl of Lplc rnform i vilbl h l pg.
More informationOn the Derivatives of Bessel and Modified Bessel Functions with Respect to the Order and the Argument
Inrnaional Rsarch Journal of Applid Basic Scincs 03 Aailabl onlin a wwwirjabscom ISSN 5-838X / Vol 4 (): 47-433 Scinc Eplorr Publicaions On h Driais of Bssl Modifid Bssl Funcions wih Rspc o h Ordr h Argumn
More informationwhereby we can express the phase by any one of the formulas cos ( 3 whereby we can express the phase by any one of the formulas
Third In-Class Exam Soluions Mah 6, Profssor David Lvrmor Tusday, 5 April 07 [0] Th vrical displacmn of an unforcd mass on a spring is givn by h 5 3 cos 3 sin a [] Is his sysm undampd, undr dampd, criically
More information3(8 ) (8 x x ) 3x x (8 )
Scion - CHATER -. a d.. b. d.86 c d 8 d d.9997 f g 6. d. d. Thn, = ln. =. =.. d Thn, = ln.9 =.7 8 -. a d.6 6 6 6 6 8 8 8 b 9 d 6 6 6 8 c d.8 6 6 6 6 8 8 7 7 d 6 d.6 6 6 6 6 6 6 8 u u u u du.9 6 6 6 6 6
More information2.1. Differential Equations and Solutions #3, 4, 17, 20, 24, 35
MATH 5 PS # Summr 00.. Diffrnial Equaions and Soluions PS.# Show ha ()C #, 4, 7, 0, 4, 5 ( / ) is a gnral soluion of h diffrnial quaion. Us a compur or calculaor o skch h soluions for h givn valus of h
More informationA THREE COMPARTMENT MATHEMATICAL MODEL OF LIVER
A THREE COPARTENT ATHEATICAL ODEL OF LIVER V. An N. Ch. Paabhi Ramacharyulu Faculy of ahmaics, R D collgs, Hanamonda, Warangal, India Dparmn of ahmaics, Naional Insiu of Tchnology, Warangal, India E-ail:
More informationInstitute of Actuaries of India
Insiu of Acuaris of India ubjc CT3 Probabiliy and Mahmaical aisics Novmbr Examinaions INDICATIVE OLUTION Pag of IAI CT3 Novmbr ol. a sampl man = 35 sampl sandard dviaion = 36.6 b for = uppr bound = 35+*36.6
More informationXV Exponential and Logarithmic Functions
MATHEMATICS 0-0-RE Dirnial Calculus Marin Huard Winr 08 XV Eponnial and Logarihmic Funcions. Skch h graph o h givn uncions and sa h domain and rang. d) ) ) log. Whn Sarah was born, hr parns placd $000
More informationH is equal to the surface current J S
Chapr 6 Rflcion and Transmission of Wavs 6.1 Boundary Condiions A h boundary of wo diffrn mdium, lcromagnic fild hav o saisfy physical condiion, which is drmind by Maxwll s quaion. This is h boundary condiion
More informationChapter 3: Fourier Representation of Signals and LTI Systems. Chih-Wei Liu
Chapr 3: Fourir Rprsnaion of Signals and LTI Sysms Chih-Wi Liu Oulin Inroducion Complx Sinusoids and Frquncy Rspons Fourir Rprsnaions for Four Classs of Signals Discr-im Priodic Signals Fourir Sris Coninuous-im
More informationPoisson process Markov process
E2200 Quuing hory and lraffic 2nd lcur oion proc Markov proc Vikoria Fodor KTH Laboraory for Communicaion nwork, School of Elcrical Enginring 1 Cour oulin Sochaic proc bhind quuing hory L2-L3 oion proc
More informationMidterm Examination (100 pts)
Econ 509 Spring 2012 S.L. Parn Midrm Examinaion (100 ps) Par I. 30 poins 1. Dfin h Law of Diminishing Rurns (5 ps.) Incrasing on inpu, call i inpu x, holding all ohr inpus fixd, on vnuall runs ino h siuaion
More information[ ] 1+ lim G( s) 1+ s + s G s s G s Kacc SYSTEM PERFORMANCE. Since. Lecture 10: Steady-state Errors. Steady-state Errors. Then
SYSTEM PERFORMANCE Lctur 0: Stady-tat Error Stady-tat Error Lctur 0: Stady-tat Error Dr.alyana Vluvolu Stady-tat rror can b found by applying th final valu thorm and i givn by lim ( t) lim E ( ) t 0 providd
More information4.1 The Uniform Distribution Def n: A c.r.v. X has a continuous uniform distribution on [a, b] when its pdf is = 1 a x b
4. Th Uniform Disribuion Df n: A c.r.v. has a coninuous uniform disribuion on [a, b] whn is pdf is f x a x b b a Also, b + a b a µ E and V Ex4. Suppos, h lvl of unblivabiliy a any poin in a Transformrs
More informationSME 3033 FINITE ELEMENT METHOD. Bending of Prismatic Beams (Initial notes designed by Dr. Nazri Kamsah)
Bnding of Prismatic Bams (Initia nots dsignd by Dr. Nazri Kamsah) St I-bams usd in a roof construction. 5- Gnra Loading Conditions For our anaysis, w wi considr thr typs of oading, as iustratd bow. Not:
More informationDecline Curves. Exponential decline (constant fractional decline) Harmonic decline, and Hyperbolic decline.
Dlin Curvs Dlin Curvs ha lo flow ra vs. im ar h mos ommon ools for forasing roduion and monioring wll rforman in h fild. Ths urvs uikly show by grahi mans whih wlls or filds ar roduing as xd or undr roduing.
More information3.4 Repeated Roots; Reduction of Order
3.4 Rpd Roos; Rducion of Ordr Rcll our nd ordr linr homognous ODE b c 0 whr, b nd c r consns. Assuming n xponnil soluion lds o chrcrisic quion: r r br c 0 Qudric formul or fcoring ilds wo soluions, r &
More informationHeat flow in composite rods an old problem reconsidered
Ha flow in copoi ro an ol probl rconir. Kranjc a Dparn of Phyic an chnology Faculy of Eucaion Univriy of jubljana Karljva ploca 6 jubljana Slovnia an J. Prnlj Faculy of Civil an Goic Enginring Univriy
More information( ) ( ) + = ( ) + ( )
Mah 0 Homwork S 6 Soluions 0 oins. ( ps I ll lav i o you vrify ha h omplimnary soluion is : y ( os( sin ( Th guss for h pariular soluion and is drivaivs ar, +. ( os( sin ( ( os( ( sin ( Y ( D 6B os( +
More informationModule 1-2: LTI Systems. Prof. Ali M. Niknejad
Modu -: LTI Sysms Prof. Ai M. Niknad Dparmn of EECS Univrsiy of Caifornia, Brky EE 5 Fa 6 Prof. A. M. Niknad LTI Dfiniion Sysm is inar sudid horoughy in 6AB: Sysm is im invarian: Thr is no cock or im rfrnc
More informationReview Lecture 5. The source-free R-C/R-L circuit Step response of an RC/RL circuit. The time constant = RC The final capacitor voltage v( )
Rviw Lcur 5 Firs-ordr circui Th sourc-fr R-C/R-L circui Sp rspons of an RC/RL circui v( ) v( ) [ v( 0) v( )] 0 Th i consan = RC Th final capacior volag v() Th iniial capacior volag v( 0 ) Volag/currn-division
More informationControl System Engineering (EE301T) Assignment: 2
Conrol Sysm Enginring (EE0T) Assignmn: PART-A (Tim Domain Analysis: Transin Rspons Analysis). Oain h rspons of a uniy fdack sysm whos opn-loop ransfr funcion is (s) s ( s 4) for a uni sp inpu and also
More informationCPSC 211 Data Structures & Implementations (c) Texas A&M University [ 259] B-Trees
CPSC 211 Daa Srucurs & Implmnaions (c) Txas A&M Univrsiy [ 259] B-Trs Th AVL r and rd-black r allowd som variaion in h lnghs of h diffrn roo-o-laf pahs. An alrnaiv ida is o mak sur ha all roo-o-laf pahs
More information1. Inverse Matrix 4[(3 7) (02)] 1[(0 7) (3 2)] Recall that the inverse of A is equal to:
Rfrncs Brnank, B. and I. Mihov (1998). Masuring monary policy, Quarrly Journal of Economics CXIII, 315-34. Blanchard, O. R. Proi (00). An mpirical characrizaion of h dynamic ffcs of changs in govrnmn spnding
More informationLecture 1: Growth and decay of current in RL circuit. Growth of current in LR Circuit. D.K.Pandey
cur : Growh and dcay of currn in circui Growh of currn in Circui us considr an inducor of slf inducanc is conncd o a DC sourc of.m.f. E hrough a rsisr of rsisanc and a ky K in sris. Whn h ky K is swichd
More informationTHE INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES *
Iranian Journal of Scinc & Tchnology, Tranacion A, Vol, No A Prind in h Ilamic Rpublic of Iran, 009 Shiraz Univriy THE INVOLUTE-EVOLUTE OFFSETS OF RULED SURFACES E KASAP, S YUCE AND N KURUOGLU Ondokuz
More informationFourier. Continuous time. Review. with period T, x t. Inverse Fourier F Transform. x t. Transform. j t
Coninuous im ourir rnsform Rviw. or coninuous-im priodic signl x h ourir sris rprsnion is x x j, j 2 d wih priod, ourir rnsform Wh bou priodic signls? W willl considr n priodic signl s priodic signl wih
More informationFIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Introduction and Linear Systems
FIRST-ORDER SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS I: Inroducion and Linar Sysms David Lvrmor Dparmn of Mahmaics Univrsiy of Maryland 9 Dcmbr 0 Bcaus h prsnaion of his marial in lcur will diffr from
More informationLaplace Transforms recap for ccts
Lalac Tranform rca for cc Wha h big ida?. Loo a iniial condiion ron of cc du o caacior volag and inducor currn a im Mh or nodal analyi wih -domain imdanc rianc or admianc conducanc Soluion of ODE drivn
More informationPhys463.nb Conductivity. Another equivalent definition of the Fermi velocity is
39 Anohr quival dfiniion of h Fri vlociy is pf vf (6.4) If h rgy is a quadraic funcion of k H k L, hs wo dfiniions ar idical. If is NOT a quadraic funcion of k (which could happ as will b discussd in h
More informationWhy Laplace transforms?
MAE4 Linar ircui Why Lalac ranform? Firordr R cc v v v KVL S R inananou for ach Subiu lmn rlaion v S Ordinary diffrnial quaion in rm of caacior volag Lalac ranform Solv Invr LT V u, v Ri, i A R V A _ v
More informationMath 3301 Homework Set 6 Solutions 10 Points. = +. The guess for the particular P ( ) ( ) ( ) ( ) ( ) ( ) ( ) cos 2 t : 4D= 2
Mah 0 Homwork S 6 Soluions 0 oins. ( ps) I ll lav i o you o vrify ha y os sin = +. Th guss for h pariular soluion and is drivaivs is blow. Noi ha w ndd o add s ono h las wo rms sin hos ar xaly h omplimnary
More informationPartial Fraction Expansion
Paial Facion Expanion Whn ying o find h inv Laplac anfom o inv z anfom i i hlpfl o b abl o bak a complicad aio of wo polynomial ino fom ha a on h Laplac Tanfom o z anfom abl. W will illa h ing Laplac anfom.
More informationPart I: Short Answer [50 points] For each of the following, give a short answer (2-3 sentences, or a formula). [5 points each]
Soluions o Midrm Exam Nam: Paricl Physics Fall 0 Ocobr 6 0 Par I: Shor Answr [50 poins] For ach of h following giv a shor answr (- snncs or a formula) [5 poins ach] Explain qualiaivly (a) how w acclra
More informationPart 3 System Identification
2.6 Sy Idnificaion, Eiaion, and Larning Lcur o o. 5 Apri 2, 26 Par 3 Sy Idnificaion Prpci of Sy Idnificaion Tory u Tru Proc S y Exprin Dign Daa S Z { u, y } Conincy Mod S arg inv θ θ ˆ M θ ~ θ? Ky Quion:
More informationChapter 10. The singular integral Introducing S(n) and J(n)
Chaptr Th singular intgral Our aim in this chaptr is to rplac th functions S (n) and J (n) by mor convnint xprssions; ths will b calld th singular sris S(n) and th singular intgral J(n). This will b don
More informationCOHORT MBA. Exponential function. MATH review (part2) by Lucian Mitroiu. The LOG and EXP functions. Properties: e e. lim.
MTH rviw part b Lucian Mitroiu Th LOG and EXP functions Th ponntial function p : R, dfind as Proprtis: lim > lim p Eponntial function Y 8 6 - -8-6 - - X Th natural logarithm function ln in US- log: function
More informationDiscussion 06 Solutions
STAT Discussion Soluions Spring 8. Th wigh of fish in La Paradis follows a normal disribuion wih man of 8. lbs and sandard dviaion of. lbs. a) Wha proporion of fish ar bwn 9 lbs and lbs? æ 9-8. - 8. P
More informationNumerical Simulation for the 2-D Heat Equation with Derivative Boundary Conditions
IOSR Joural of Applid Chmisr IOSR-JAC -ISSN: 78-576.Volum 9 Issu 8 Vr. I Aug. 6 PP 4-8 www.iosrjourals.org Numrical Simulaio for h - Ha Equaio wih rivaiv Boudar Codiios Ima. I. Gorial parm of Mahmaics
More informationy z P 3 P T P1 P 2. Werner Purgathofer. b a
Einführung in Viual Compuing Einführung in Viual Compuing 86.822 in co T P 3 P co in T P P 2 co in Geomeric Tranformaion Geomeric Tranformaion W P h f Werner Purgahofer b a Tranformaion in he Rendering
More informationCharging of capacitor through inductor and resistor
cur 4&: R circui harging of capacior hrough inducor and rsisor us considr a capacior of capacianc is conncd o a D sourc of.m.f. E hrough a rsisr of rsisanc R, an inducor of inducanc and a y K in sris.
More informationLecture 4: Parsing. Administrivia
Adminitrivia Lctur 4: Paring If you do not hav a group, pla pot a rqut on Piazzza ( th Form projct tam... itm. B ur to updat your pot if you find on. W will aign orphan to group randomly in a fw day. Programming
More information16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 3: Ideal Nozzle Fluid Mechanics
6.5, Rok ropulsion rof. nul rinz-snhz Lur 3: Idl Nozzl luid hnis Idl Nozzl low wih No Sprion (-D) - Qusi -D (slndr) pproximion - Idl gs ssumd ( ) mu + Opimum xpnsion: - or lss, >, ould driv mor forwrd
More informationSource code. where each α ij is a terminal or nonterminal symbol. We say that. α 1 α m 1 Bα m+1 α n α 1 α m 1 β 1 β p α m+1 α n
Adminitrivia Lctur : Paring If you do not hav a group, pla pot a rqut on Piazzza ( th Form projct tam... itm. B ur to updat your pot if you find on. W will aign orphan to group randomly in a fw day. Programming
More informationRevisiting what you have learned in Advanced Mathematical Analysis
Fourir sris Rvisiing wh you hv lrnd in Advncd Mhmicl Anlysis L f x b priodic funcion of priod nd is ingrbl ovr priod. f x cn b rprsnd by rigonomric sris, f x n cos nx bn sin nx n cos x b sin x cosx b whr
More informationEE 434 Lecture 22. Bipolar Device Models
EE 434 Lcur 22 Bipolar Dvic Modls Quiz 14 Th collcor currn of a BJT was masurd o b 20mA and h bas currn masurd o b 0.1mA. Wha is h fficincy of injcion of lcrons coming from h mir o h collcor? 1 And h numbr
More informationwith Dirichlet boundary conditions on the rectangle Ω = [0, 1] [0, 2]. Here,
Numrical Eampl In thi final chaptr, w tart b illutrating om known rult in th thor and thn procd to giv a fw novl ampl. All ampl conidr th quation F(u) = u f(u) = g, (-) with Dirichlt boundar condition
More informationANALYSIS OF LAMINATED CONICAL SHELL STRUCTURES USING HIGHER ORDER MODELS. and J. Herskovits c
ANALYI OF LAMINATED CONICAL HELL TUCTUE UING HIGHE ODE MODEL I. F. Pino Corria a Criovão M. Moa oar b* Caro A. Moa oar b an. Hrovi c a ENIDH - Dparamno Máquina Maríima Paço Arco 78-57 Oira Poruga. b IDMEC/IT
More informationAN INTRODUCTION TO FOURIER ANALYSIS PROF. VEDAT TAVSANOĞLU
A IRODUCIO O FOURIER AALYSIS PROF. VEDA AVSAOĞLU 994 A IRODUCIO O FOURIER AALYSIS ABLE OF COES. HE FOURIER SERIES ---------------------------------------------------------------------3.. Priodic Funcions-----------------------------------------------------------------------3..
More informationFourier Series and Parseval s Relation Çağatay Candan Dec. 22, 2013
Fourir Sris nd Prsvl s Rlion Çğy Cndn Dc., 3 W sudy h m problm EE 3 M, Fll3- in som dil o illusr som conncions bwn Fourir sris, Prsvl s rlion nd RMS vlus. Q. ps h signl sin is h inpu o hlf-wv rcifir circui
More informationVoltage v(z) ~ E(z)D. We can actually get to this wave behavior by using circuit theory, w/o going into details of the EM fields!
Considr a pair of wirs idal wirs ngh >, say, infinily long olag along a cabl can vary! D olag v( E(D W can acually g o his wav bhavior by using circui hory, w/o going ino dails of h EM filds! Thr
More information5. An object moving along an x-coordinate axis with its scale measured in meters has a velocity of 6t
AP CALCULUS FINAL UNIT WORKSHEETS ACCELERATION, VELOCTIY AND POSITION In problms -, drmin h posiion funcion, (), from h givn informaion.. v (), () = 5. v ()5, () = b g. a (), v() =, () = -. a (), v() =
More informationERROR ANALYSIS A.J. Pintar and D. Caspary Department of Chemical Engineering Michigan Technological University Houghton, MI September, 2012
ERROR AALYSIS AJ Pinar and D Caspary Dparmn of Chmical Enginring Michigan Tchnological Univrsiy Houghon, MI 4993 Spmbr, 0 OVERVIEW Exprimnaion involvs h masurmn of raw daa in h laboraory or fild I is assumd
More informationEAcos θ, where θ is the angle between the electric field and
8.4. Modl: Th lctric flux flows out of a closd surfac around a rgion of spac containing a nt positiv charg and into a closd surfac surrounding a nt ngativ charg. Visualiz: Plas rfr to Figur EX8.4. Lt A
More information