First looking at the scalar potential term, suppose that the displacement is given by u = φ. If one can find a scalar φ such that u = φ. u x.
|
|
- Jared Simon
- 6 years ago
- Views:
Transcription
1 7.4 Eastodynams 7.4. Propagaton of Wavs n East Sods Whn a strss wav travs throgh a matra, t ass matra parts to dspa by. It an b shown that any vtor an b wrttn n th form φ + ra (7.4. whr φ s a saar potnta and a s a vtor. hs two trms n th dspamnt fd an b xamnd sparaty. h most gnra dspamnt fd an b obtand by addng both sotons togthr. Irrotatona Wavs Frst oong at th saar potnta trm, sppos that th dspamnt s gvn by φ. If on an fnd a saar φ sh that φ, thn t foows that r 0, or r / / / ( hs ah of th trms nsd th brats s zro. Bt ths trms rprsnt rotatons of matra parts (s Eqns...0. For xamp, as stratd n Fg. 7.4., ω (7.4. x ω ω x Fgr 7.4.: a rotaton from th Hmhotz thory
2 hs r 0 an b ntrprtd as no rotaton of matra parts. A sma mnt of matra an st ndrgo norma and shar stran, bt th mnt w not rotat as a rgd body n spa. ang th dspamnt fd φ, wrtng t n ndx notaton, j φ / j, and sbstttng nto Navr s qatons, ads to th thr-dmnsona wav qaton: t, λ + μ ρ ρ E( ν ( + ν ( ν (7.4.4 hs dspamnt fd ths orrsponds to strss wavs travng at spd, asng matra to stran bt not to rotat. hs rrotatona wavs ar aso ad wavs of dataton. Eqvomna Wavs Consdr now th dspamnt fd ra. If on an fnd a vtor a sh that ra, thn t foows that 0, or + + ε + ε + ε ΔV V (7.4.5 hs th ondton that th dspamnt fd b dvrgn-fr mps that thr s no vom hang. hr an b norma strans ony so ong as thr sm s zro. ang ε / and sbstttng nto Navr s qatons thn ads mmdaty to μ ρ (7.4.6 t or th thr-dmnsona wav qaton: t, μ ρ E ρ ( + ν (7.4.7 hs dspamnt fd ths orrsponds to strss wavs travng at spd, asng matra to shar. hs qvomna wavs ar aso ad shar wavs or wavs of dstorton. In smmary, whn an vnt sh as an xposon ors, two dffrnt typs of wav mrg, rrotatona wavs whh rst n rrotatona dspamnt fds, and qvomna wavs whh rst n qvomna dspamnts. hs wavs trav at dffrnt spds.
3 7.4. Pan Wavs At a sffnt dstan from any nta dstrban, a strss wav w trav n a pan. It an b assmd that a matra parts w dspa thr para to th drton of wav propagaton (ongtdna wavs or prpndar to ths drton (transvrs wavs. t th wav trav n th x drton. Irrotatona (p / ongtdna Pan Wavs Consdr parts whh dspa n th drton of wav propagaton aordng to ( x, t. hs s an rrotatona wav sn r 0, and th strss wav s govrnd by th on-dmnsona wav qaton x t (7.4.8 hs ongtdna pan wavs ar aso ad p-wavs. x omprsson / rarfaton wavfront Fgr 7.4.: a ongtdna wav Eqvomna (s / transvrs / shar Pan Wavs Consdr parts whh dspa aordng to ( x, t. hs s an qvomna wav sn 0, and th strss wav s govrnd by th on-dmnsona wav qaton x t (7.4.9 p stands for prmary 4
4 hs transvrs/shar wavs ar aso ad s-wavs. σ N σ S 0 0 x Fgr 7.4.: a transvrs wav 7.4. Vbraton Anayss A vbraton anayss an b arrd ot n xaty th sam way as n Chaptr, ony th wav spds n th D wav qatons and ar now dffrnt from th D spd E / ρ. h partar sotons, ford vbraton and rsonan thory of Chaptr an agan b appd hr. h anayss hr s approprat for thn pats nfnty wd n th x, x drtons, Fg h fgr shows ongtdna vbraton, bt on an aso hav transvrs vbraton whr th parts dspa prpndar to th x axs. x Fgr 7.4.4: strth vbraton of a pat Wavs at Bondars Pan wavs xst n nbondd ast ontna. In a fnt body, a pan wav w b rftd whn t hts a fr srfa. In ths as, on nds to sov Navr s qatons s stands for sondary 5
5 sbjt to th bondary ondtons of zro norma and shar strss at th fr srfa. Wavs of both typs w n gnra b rftd for any sng typ of ndnt wav. Smary, whn a wav mts an ntrfa btwn two dffrnt matras, thr w b rfton and rfraton. h bondary ondtons ar that th dspamnts ar ontnos and th norma and shar strsss ar ontnos, Fg E E ( ( (, ν, ρ ( ( (, ν, ρ ( x ( ( ( x, y y σ σ ( N ( ( ( N, σ S σ S Fgr 7.4.5: rfton and rfraton of a wav at an ntrfa Wavs at Bondars h wavs dsssd ths far ar body wavs. Whn a fr srfa xsts, for xamp th srfa of th arth, anothr typ of wav moton s possb; ths ar th Raygh wavs and trav aong th srfa vry mh watr wavs. It an b shown that th spd of Raygh wavs s btwn 90% and 95% of, dpndng on th va of Posson s rato. Smar typs of wavs an propagat aong th ntrfa btwn two dffrnt matras Probms. Consdr th moton π sn ( x t, 0, 0, What ar th strans n th matra? What ar th orrspondng strsss? What s th vom hang n th matra? What s th nam (or nams gvn to th typ of wav whh ass ths nd of moton?. Consdr th moton π 0, sn ( x t, 0, What ar th strans n th matra? What ar th orrspondng strsss? What s th vom hang n th matra? What s th nam (or nams gvn to th typ of wav whh ass ths nd of moton?. Drv an xprsson for th rato / n trms of th matra s Posson s rato ony. Whh s th fastr, th ongtdna or transvrs wav? 6
6 4. Show that th moton π 0, 0, os s qvomna. ( px os ( x t 5. Consdr th moton [ sn β ( x t + α sn β ( x + t ], 0, 0 ( what nd of ast strss wav dos ths nvov? (Sth th pan of th wav and ts drton of propagaton. ( what ar th strans and strsss. ( s th qatons of moton to dtrmn th wav spd. Is t what yo xptd? (v Sppos that th pan x 0 s a fr srfa. Dtrmn α. (v Sppos aso that x h s a fr srfa. Dtrmn β. 6. Consdr a pat wth ft fa ( 0 α Ωt and th rght fa ( x x sbjtd to a ford dspamnt sn fr. ( fnd th thnss-strth vbraton of th pat. What ar th natra frqns? ( Whn dos rsonan or? 7. Consdr a pat wth ft fa ( 0 th rght fa ( x x sbjtd to a traton t α os Ωt and fxd, as shown n th fgr bow. ( fnd th thnss-shar vbraton of th pat. What ar th natra frqns? ( Whn dos rsonan or?, x fxd x σ [not: assm a dspamnt ( x, t ; as wth transvrs wavs, ths w satsfy th -d wav qaton wth bng th transvrs wav spd. Us th traton to obtan an xprsson for th shar strss σ ovr th ft hand fa. Whn appyng th strss bondary ondton, yo w nd th stran-dspamnt xprsson and strss-stran aw, 7
7 ε +, ε σ μ 8. Consdr th as of α ( osωt + sn Ωt ovr th ft fa ( 0 rght fa ( x x wth th fxd. Drv an xprsson for th partar soton and show that t rprsnts rar moton of th parts n th x x pan. [hnt: vaat th partar sotons for and sparaty and thn show that + for som r (ndpndnt of tm] r 8
The Hyperelastic material is examined in this section.
4. Hyprlastcty h Hyprlastc matral s xad n ths scton. 4..1 Consttutv Equatons h rat of chang of ntrnal nrgy W pr unt rfrnc volum s gvn by th strss powr, whch can b xprssd n a numbr of dffrnt ways (s 3.7.6):
More information1.9 Cartesian Tensors
Scton.9.9 Crtsn nsors s th th ctor, hghr ordr) tnsor s mthmtc obct hch rprsnts mny physc phnomn nd hch xsts ndpndnty of ny coordnt systm. In ht foos, Crtsn coordnt systm s sd to dscrb tnsors..9. Crtsn
More informationAPPENDIX H CONSTANT VOLTAGE BEHIND TRANSIENT REACTANCE GENERATOR MODEL
APPNDIX H CONSAN VOAG BHIND RANSIN RACANC GNRAOR MOD h mprov two gnrator mo uss th constant votag bhn transnt ractanc gnrator mo. hs mo gnors magntc sancy; assums th opratng ractanc of th gnrator s th
More informationJones vector & matrices
Jons vctor & matrcs PY3 Colást na hollscol Corcagh, Ér Unvrst Collg Cork, Irland Dpartmnt of Phscs Matr tratmnt of polarzaton Consdr a lght ra wth an nstantanous -vctor as shown k, t ˆ k, t ˆ k t, o o
More informationACOUSTIC WAVE EQUATION. Contents INTRODUCTION BULK MODULUS AND LAMÉ S PARAMETERS
ACOUSTIC WAE EQUATION Contnts INTRODUCTION BULK MODULUS AND LAMÉ S PARAMETERS INTRODUCTION As w try to vsualz th arth ssmcally w mak crtan physcal smplfcatons that mak t asr to mak and xplan our obsrvatons.
More information9.5 Complex variables
9.5 Cmpl varabls. Cnsdr th funtn u v f( ) whr ( ) ( ), f( ), fr ths funtn tw statmnts ar as fllws: Statmnt : f( ) satsf Cauh mann quatn at th rgn. Statmnt : f ( ) ds nt st Th rrt statmnt ar (A) nl (B)
More informationLecture 14. Relic neutrinos Temperature at neutrino decoupling and today Effective degeneracy factor Neutrino mass limits Saha equation
Lctur Rlc nutrnos mpratur at nutrno dcoupln and today Effctv dnracy factor Nutrno mass lmts Saha quaton Physcal Cosmoloy Lnt 005 Rlc Nutrnos Nutrnos ar wakly ntractn partcls (lptons),,,,,,, typcal ractons
More informationGrand Canonical Ensemble
Th nsmbl of systms mmrsd n a partcl-hat rsrvor at constant tmpratur T, prssur P, and chmcal potntal. Consdr an nsmbl of M dntcal systms (M =,, 3,...M).. Thy ar mutually sharng th total numbr of partcls
More informationA FINITE ELEMENT MODEL FOR COMPOSITE BEAMS WITH PIEZOELECTRIC LAYERS USING A SINUS MODEL
A FINIE ELEMEN MODEL FOR COMPOSIE BEAMS WIH PIEZOELECRIC LAYERS USING A SINUS MODEL S.B. Bhsht-Aval * M. Lzgy-Nazargah ** Dpartmnt of Cvl Engnrng Khajh Nasr oos Unvrsty of hnology (KNU) hran, Iran ABSRAC
More informationA Note on Estimability in Linear Models
Intrnatonal Journal of Statstcs and Applcatons 2014, 4(4): 212-216 DOI: 10.5923/j.statstcs.20140404.06 A Not on Estmablty n Lnar Modls S. O. Adymo 1,*, F. N. Nwob 2 1 Dpartmnt of Mathmatcs and Statstcs,
More informationHeisenberg Model. Sayed Mohammad Mahdi Sadrnezhaad. Supervisor: Prof. Abdollah Langari
snbrg Modl Sad Mohammad Mahd Sadrnhaad Survsor: Prof. bdollah Langar bstract: n ths rsarch w tr to calculat analtcall gnvalus and gnvctors of fnt chan wth ½-sn artcls snbrg modl. W drov gnfuctons for closd
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 informationte Finance (4th Edition), July 2017.
Appndx Chaptr. Tchncal Background Gnral Mathmatcal and Statstcal Background Fndng a bas: 3 2 = 9 3 = 9 1 /2 x a = b x = b 1/a A powr of 1 / 2 s also quvalnt to th squar root opraton. Fndng an xponnt: 3
More informationCHAPTER 33: PARTICLE PHYSICS
Collg Physcs Studnt s Manual Chaptr 33 CHAPTER 33: PARTICLE PHYSICS 33. THE FOUR BASIC FORCES 4. (a) Fnd th rato of th strngths of th wak and lctromagntc forcs undr ordnary crcumstancs. (b) What dos that
More informationFINITE ELEMENT ANALYSIS OF
FINIT LMNT NLYSIS OF D MODL PROBLM WITH SINGL VRIBL Fnt lmnt modl dvlopmnt of lnr D modl dffrntl qton nvolvng sngl dpndnt nknown govrnng qtons F modl dvlopmnt wk form. JN Rddy Modlqn D - GOVRNING TION
More information4.8 Huffman Codes. Wordle. Encoding Text. Encoding Text. Prefix Codes. Encoding Text
2/26/2 Word A word a word coag. A word contrctd ot of on of th ntrctor ar: 4.8 Hffan Cod word contrctd ng th java at at word.nt word a randozd grdy agorth to ov th ackng rob Encodng Txt Q. Gvn a txt that
More informationCOMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP
ISAHP 00, Bal, Indonsa, August -9, 00 COMPLEX NUMBER PAIRWISE COMPARISON AND COMPLEX NUMBER AHP Chkako MIYAKE, Kkch OHSAWA, Masahro KITO, and Masaak SHINOHARA Dpartmnt of Mathmatcal Informaton Engnrng
More informationEconomics 600: August, 2007 Dynamic Part: Problem Set 5. Problems on Differential Equations and Continuous Time Optimization
THE UNIVERSITY OF MARYLAND COLLEGE PARK, MARYLAND Economcs 600: August, 007 Dynamc Part: Problm St 5 Problms on Dffrntal Equatons and Contnuous Tm Optmzaton Quston Solv th followng two dffrntal quatons.
More informationZero Point Energy: Thermodynamic Equilibrium and Planck Radiation Law
Gaug Institut Journa Vo. No 4, Novmbr 005, Zro Point Enrgy: Thrmodynamic Equiibrium and Panck Radiation Law Novmbr, 005 vick@adnc.com Abstract: In a rcnt papr, w provd that Panck s radiation aw with zro
More informationChapter 6 Student Lecture Notes 6-1
Chaptr 6 Studnt Lctur Nots 6-1 Chaptr Goals QM353: Busnss Statstcs Chaptr 6 Goodnss-of-Ft Tsts and Contngncy Analyss Aftr compltng ths chaptr, you should b abl to: Us th ch-squar goodnss-of-ft tst to dtrmn
More information3.4 Properties of the Stress Tensor
cto.4.4 Proprts of th trss sor.4. trss rasformato Lt th compots of th Cauchy strss tsor a coordat systm wth bas vctors b. h compots a scod coordat systm wth bas vctors j,, ar gv by th tsor trasformato
More informationBasic Electrical Engineering for Welding [ ] --- Introduction ---
Basc Elctrcal Engnrng for Wldng [] --- Introducton --- akayosh OHJI Profssor Ertus, Osaka Unrsty Dr. of Engnrng VIUAL WELD CO.,LD t-ohj@alc.co.jp OK 15 Ex. Basc A.C. crcut h fgurs n A-group show thr typcal
More informationClausius-Clapeyron Equation
ausius-apyron Equation 22000 p (mb) Liquid Soid 03 6. Vapor 0 00 374 (º) oud drops first form whn th aporization quiibrium point is rachd (i.., th air parc bcoms saturatd) Hr w dop an quation that dscribs
More information10/7/14. Mixture Models. Comp 135 Introduction to Machine Learning and Data Mining. Maximum likelihood estimation. Mixture of Normals in 1D
Comp 35 Introducton to Machn Larnng and Data Mnng Fall 204 rofssor: Ron Khardon Mxtur Modls Motvatd by soft k-mans w dvlopd a gnratv modl for clustrng. Assum thr ar k clustrs Clustrs ar not rqurd to hav
More informationthe output is Thus, the output lags in phase by θ( ωo) radians Rewriting the above equation we get
Th output y[ of a frquncy-sctiv LTI iscrt-tim systm with a frquncy rspons H ( xhibits som ay rativ to th input caus by th nonro phas rspons θ( ω arg{ H ( } of th systm For an input A cos( ωo n + φ, < n
More informationFundamentals of Continuum Mechanics. Seoul National University Graphics & Media Lab
Fndmntls of Contnm Mchncs Sol Ntonl Unvrsty Grphcs & Md Lb Th Rodmp of Contnm Mchncs Strss Trnsformton Strn Trnsformton Strss Tnsor Strn T + T ++ T Strss-Strn Rltonshp Strn Enrgy FEM Formlton Lt s Stdy
More informationStatic/Dynamic Deformation with Finite Element Method. Graphics & Media Lab Seoul National University
Statc/Dynamc Dormaton wth Fnt Elmnt Mthod Graphcs & Mda Lab Sol Natonal Unvrsty Statc/Dynamc Dormaton Statc dormaton Dynamc dormaton ndormd shap ntrnal + = nrta = trnal dormd shap statc qlbrm dynamc qlbrm
More informationLecture 08 Multiple View Geometry 2. Prof. Dr. Davide Scaramuzza
Lctr 8 Mltpl V Gomtry Prof. Dr. Dad Scaramzza sdad@f.zh.ch Cors opcs Prncpls of mag formaton Imag fltrng Fatr dtcton Mlt- gomtry 3D Rconstrcton Rcognton Mltpl V Gomtry San Marco sqar, Vnc 4,79 mags, 4,55,57
More informationHiroaki Matsueda (Sendai National College of Tech.)
Jun 2 26@YITP 4-das confrnc: oograph & Quantum Informaton Snapshot ntrop: An atrnatv hoographc ntangmnt ntrop roa atsuda Snda Natona Cog of Tch. Purpos of ths wor Purpos: Stratg: Entangmnt hoograph RG
More informationLECTURE 21 Mohr s Method for Calculation of General Displacements. 1 The Reciprocal Theorem
V. DEMENKO MECHANICS OF MATERIALS 05 LECTURE Mohr s Method for Cacuaton of Genera Dspacements The Recproca Theorem The recproca theorem s one of the genera theorems of strength of materas. It foows drect
More informationMath 656 March 10, 2011 Midterm Examination Solutions
Math 656 March 0, 0 Mdtrm Eamnaton Soltons (4pts Dr th prsson for snh (arcsnh sng th dfnton of snh w n trms of ponntals, and s t to fnd all als of snh (. Plot ths als as ponts n th compl plan. Mak sr or
More information3. Stress-strain relationships of a composite layer
OM PO I O U P U N I V I Y O F W N ompostes ourse 8-9 Unversty of wente ng. &ech... tress-stran reatonshps of a composte ayer - Laurent Warnet & emo Aerman.. tress-stran reatonshps of a composte ayer Introducton
More informationdt d Chapter 30: 1-Faraday s Law of induction (induced EMF) Chapter 30: 1-Faraday s Law of induction (induced Electromotive Force)
Chaptr 3: 1-Faraday s aw of induction (inducd ctromotiv Forc) Variab (incrasing) Constant Variab (dcrasing) whn a magnt is movd nar a wir oop of ara A, currnt fows through that wir without any battris!
More information:2;$-$(01*%<*=,-./-*=0;"%/;"-*
!"#$%'()%"*#%*+,-./-*+01.2(.*3+456789*!"#$%"'()'*+,-."/0.%+1'23"45'46'7.89:89'/' ;8-,"$4351415,8:+#9' Dr. Ptr T. Gallaghr Astrphyscs Rsarch Grup Trnty Cllg Dubln :2;$-$(01*%
More information(1) Then we could wave our hands over this and it would become:
MAT* K285 Spring 28 Anthony Bnoit 4/17/28 Wk 12: Laplac Tranform Rading: Kohlr & Johnon, Chaptr 5 to p. 35 HW: 5.1: 3, 7, 1*, 19 5.2: 1, 5*, 13*, 19, 45* 5.3: 1, 11*, 19 * Pla writ-up th problm natly and
More informationEinstein Summation Convention
Ensten Suaton Conventon Ths s a ethod to wrte equaton nvovng severa suatons n a uncuttered for Exape:. δ where δ or 0 Suaton runs over to snce we are denson No ndces appear ore than two tes n the equaton
More informationReview - Probabilistic Classification
Mmoral Unvrsty of wfoundland Pattrn Rcognton Lctur 8 May 5, 6 http://www.ngr.mun.ca/~charlsr Offc Hours: Tusdays Thursdays 8:3-9:3 PM E- (untl furthr notc) Gvn lablld sampls { ɛc,,,..., } {. Estmat Rvw
More informationGuo, James C.Y. (1998). "Overland Flow on a Pervious Surface," IWRA International J. of Water, Vol 23, No 2, June.
Guo, Jams C.Y. (006). Knmatc Wav Unt Hyrograph for Storm Watr Prctons, Vol 3, No. 4, ASCE J. of Irrgaton an Dranag Engnrng, July/August. Guo, Jams C.Y. (998). "Ovrlan Flow on a Prvous Surfac," IWRA Intrnatonal
More informationLecture 3: Phasor notation, Transfer Functions. Context
EECS 5 Fall 4, ctur 3 ctur 3: Phasor notaton, Transfr Functons EECS 5 Fall 3, ctur 3 Contxt In th last lctur, w dscussd: how to convrt a lnar crcut nto a st of dffrntal quatons, How to convrt th st of
More informationa 1and x is any real number.
Calcls Nots Eponnts an Logarithms Eponntial Fnction: Has th form y a, whr a 0, a an is any ral nmbr. Graph y, Graph y ln y y Th Natral Bas (Elr s nmbr): An irrational nmbr, symboliz by th lttr, appars
More informationNAME: ANSWER KEY DATE: PERIOD. DIRECTIONS: MULTIPLE CHOICE. Choose the letter of the correct answer.
R A T T L E R S S L U G S NAME: ANSWER KEY DATE: PERIOD PREAP PHYSICS REIEW TWO KINEMATICS / GRAPHING FORM A DIRECTIONS: MULTIPLE CHOICE. Chs h r f h rr answr. Us h fgur bw answr qusns 1 and 2. 0 10 20
More information2. Background Material
S. Blair Sptmbr 3, 003 4. Background Matrial Th rst of this cours dals with th gnration, modulation, propagation, and ction of optical radiation. As such, bic background in lctromagntics and optics nds
More informationThe van der Waals interaction 1 D. E. Soper 2 University of Oregon 20 April 2012
Th van dr Waals intraction D. E. Sopr 2 Univrsity of Orgon 20 pril 202 Th van dr Waals intraction is discussd in Chaptr 5 of J. J. Sakurai, Modrn Quantum Mchanics. Hr I tak a look at it in a littl mor
More informationFolding of Regular CW-Complexes
Ald Mathmatcal Scncs, Vol. 6,, no. 83, 437-446 Foldng of Rgular CW-Comlxs E. M. El-Kholy and S N. Daoud,3. Dartmnt of Mathmatcs, Faculty of Scnc Tanta Unvrsty,Tanta,Egyt. Dartmnt of Mathmatcs, Faculty
More informationPhy213: General Physics III 4/10/2008 Chapter 22 Worksheet 1. d = 0.1 m
hy3: Gnral hyscs III 4/0/008 haptr Worksht lctrc Flds: onsdr a fxd pont charg of 0 µ (q ) q = 0 µ d = 0 a What s th agntud and drcton of th lctrc fld at a pont, a dstanc of 0? q = = 8x0 ˆ o d ˆ 6 N ( )
More informationName: Jeffy We have already seen the document object and written HTML directly into it.
C3E3 -- Vsua Programmng Envronmnts Programmng n Javacrpt -- cont Mscanous Matra -- mor on objcts Usr-Dfn J objcts us a functon to crat th cass tmpat thn us th constructor, nw, to crat th objct nstanc s
More informationFirst derivative analysis
Robrto s Nots on Dirntial Calculus Chaptr 8: Graphical analysis Sction First drivativ analysis What you nd to know alrady: How to us drivativs to idntiy th critical valus o a unction and its trm points
More informationChemistry 342 Spring, The Hydrogen Atom.
Th Hyrogn Ato. Th quation. Th first quation w want to sov is φ This quation is of faiiar for; rca that for th fr partic, w ha ψ x for which th soution is Sinc k ψ ψ(x) a cos kx a / k sin kx ± ix cos x
More informationVertical Sound Waves
Vral Sond Wavs On an drv h formla for hs avs by onsdrn drly h vral omonn of momnm qaon hrmodynam qaon and h onny qaon from 5 and hn follon h rrbaon mhod and assmn h snsodal solons. Effvly h frs ro and
More information16.512, Rocket Propulsion Prof. Manuel Martinez-Sanchez Lecture 6: Heat Conduction: Thermal Stresses
16.512, okt Proulon Prof. Manul Martnz-Sanhz Ltur 6: Hat Conduton: Thrmal Str Efft of Sold or Lqud Partl n Nozzl Flow An u n hhly alumnzd old rokt motor. 3 2Al + O 2 Al 2 O 2 3 m.. 2072 C, b.. 2980 C In
More informationMath 34A. Final Review
Math A Final Rviw 1) Us th graph of y10 to find approimat valus: a) 50 0. b) y (0.65) solution for part a) first writ an quation: 50 0. now tak th logarithm of both sids: log() log(50 0. ) pand th right
More informationMath 656 Midterm Examination March 27, 2015 Prof. Victor Matveev
Math 656 Mdtrm Examnatn March 7, 05 Prf. Vctr Matvv ) (4pts) Fnd all vals f n plar r artsan frm, and plt thm as pnts n th cmplx plan: (a) Snc n-th rt has xactly n vals, thr wll b xactly =6 vals, lyng n
More informationIntroduction to the quantum theory of matter and Schrödinger s equation
Introduction to th quantum thory of mattr and Schrödingr s quation Th quantum thory of mattr assums that mattr has two naturs: a particl natur and a wa natur. Th particl natur is dscribd by classical physics
More informationParametric Investigation of Foundation on Layered Soil under Vertical Vibration
IOSR Jornal of Engnrng IOSRJEN ISSN : 5-3 ISSN p: 78-879 Vol. 4 Iss 7 Jly. 4 V3 PP 3- www.osrn.org Paramtr Invstgaton of Fonaton on Layr Sol nr Vrtal Vbraton S. N. Swar P. K. Prahan ** B. P. Mshra ***
More informationOptimal Ordering Policy in a Two-Level Supply Chain with Budget Constraint
Optmal Ordrng Polcy n a Two-Lvl Supply Chan wth Budgt Constrant Rasoul aj Alrza aj Babak aj ABSTRACT Ths papr consdrs a two- lvl supply chan whch consst of a vndor and svral rtalrs. Unsatsfd dmands n rtalrs
More informationPHYS ,Fall 05, Term Exam #1, Oct., 12, 2005
PHYS1444-,Fall 5, Trm Exam #1, Oct., 1, 5 Nam: Kys 1. circular ring of charg of raius an a total charg Q lis in th x-y plan with its cntr at th origin. small positiv tst charg q is plac at th origin. What
More informationFREE VIBRATION ANALYSIS OF FUNCTIONALLY GRADED BEAMS
Journal of Appl Mathatcs an Coputatonal Mchancs, (), 9- FREE VIBRATION ANAYSIS OF FNCTIONAY GRADED BEAMS Stansław Kukla, Jowta Rychlwska Insttut of Mathatcs, Czstochowa nvrsty of Tchnology Czstochowa,
More informationMinimum Spanning Trees
Mnmum Spnnng Trs Spnnng Tr A tr (.., connctd, cyclc grph) whch contns ll th vrtcs of th grph Mnmum Spnnng Tr Spnnng tr wth th mnmum sum of wghts 1 1 Spnnng forst If grph s not connctd, thn thr s spnnng
More informationAdvances in the study of intrinsic rotation with flux tube gyrokinetics
Adans n th study o ntrns rotaton wth lux tub gyroknts F.I. Parra and M. arns Unrsty o Oxord Wolgang Paul Insttut, Vnna, Aprl 0 Introduton In th absn o obous momntum nput (apart rom th dg), tokamak plasmas
More informationEquil. Properties of Reacting Gas Mixtures. So far have looked at Statistical Mechanics results for a single (pure) perfect gas
Shool of roa Engnrng Equl. Prort of Ratng Ga Mxtur So far hav lookd at Stattal Mhan rult for a ngl (ur) rft ga hown how to gt ga rort (,, h, v,,, ) from artton funton () For nonratng rft ga mxtur, gt mxtur
More informationLECTURE 5 Guassian Wave Packet
LECTURE 5 Guassian Wav Pact 1.5 Eampl f a guassian shap fr dscribing a wav pact Elctrn Pact ψ Guassian Assumptin Apprimatin ψ As w hav sn in QM th wav functin is ftn rprsntd as a Furir transfrm r sris.
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationME311 Machine Design
ME311 Machin Dsign Lctur 4: Strss Concntrations; Static Failur W Dornfld 8Sp017 Fairfild Univrsit School of Enginring Strss Concntration W saw that in a curvd bam, th strss was distortd from th uniform
More informationPhysics 256: Lecture 2. Physics
Physcs 56: Lctur Intro to Quantum Physcs Agnda for Today Complx Numbrs Intrfrnc of lght Intrfrnc Two slt ntrfrnc Dffracton Sngl slt dffracton Physcs 01: Lctur 1, Pg 1 Constructv Intrfrnc Ths wll occur
More informationWhy switching? Modulation. Switching amp. Losses. Converter topology. i d. Continuous amplifiers have low efficiency. Antag : u i
Modlaton Indtral Elctrcal Engnrng and Atomaton Lnd nvrty, Swdn Why wtchng? Contno amplfr hav low ffcncy a b Contno wtch pt ( t ) = pn( t) = ( a b) Antag : ( a b) = Pn = Pt η = = = Pn Swtchng amp. Lo Convrtr
More informationMP IN BLOCK QUASI-INCOHERENT DICTIONARIES
CHOOL O ENGINEERING - TI IGNAL PROCEING INTITUTE Lornzo Potta and Prr Vandrghynst CH-1015 LAUANNE Tlphon: 4121 6932601 Tlfax: 4121 6937600 -mal: lornzo.potta@pfl.ch ÉCOLE POLYTECHNIQUE ÉDÉRALE DE LAUANNE
More informationJunction Tree Algorithm 1. David Barber
Juntion Tr Algorithm 1 David Barbr Univrsity Collg London 1 Ths slids aompany th book Baysian Rasoning and Mahin Larning. Th book and dmos an b downloadd from www.s.ul.a.uk/staff/d.barbr/brml. Fdbak and
More informationSeptember 27, Introduction to Ordinary Differential Equations. ME 501A Seminar in Engineering Analysis Page 1. Outline
Introucton to Ornar Dffrntal Equatons Sptmbr 7, 7 Introucton to Ornar Dffrntal Equatons Larr artto Mchancal Engnrng AB Smnar n Engnrng Analss Sptmbr 7, 7 Outln Rvw numrcal solutons Bascs of ffrntal quatons
More informationANALYTICITY THEOREM FOR FRACTIONAL LAPLACE TRANSFORM
Sc. Rs. hm. ommn.: (3, 0, 77-8 ISSN 77-669 ANALYTIITY THEOREM FOR FRATIONAL LAPLAE TRANSFORM P. R. DESHMUH * and A. S. GUDADHE a Prof. Ram Mgh Insttt of Tchnology & Rsarch, Badnra, AMRAVATI (M.S. INDIA
More information2F1120 Spektrala transformer för Media Solutions to Steiglitz, Chapter 1
F110 Spktrala transformr för Mdia Solutions to Stiglitz, Chaptr 1 Prfac This documnt contains solutions to slctd problms from Kn Stiglitz s book: A Digital Signal Procssing Primr publishd by Addison-Wsly.
More informationCHAPTER 24 HYPERBOLIC FUNCTIONS
EXERCISE 00 Pag 5 CHAPTER HYPERBOLIC FUNCTIONS. Evaluat corrct to significant figurs: (a) sh 0.6 (b) sh.8 0.686, corrct to significant figurs (a) sh 0.6 0.6 0.6 ( ) Altrnativly, using a scintific calculator,
More informationAdvanced Queueing Theory. M/G/1 Queueing Systems
Advand Quung Thory Ths slds ar rad by Dr. Yh Huang of Gorg Mason Unvrsy. Sudns rgsrd n Dr. Huang's ourss a GMU an ma a sngl mahn-radabl opy and prn a sngl opy of ah sld for hr own rfrn, so long as ah sld
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by D. Klain Vrsion 207.0.05 Corrctions and commnts ar wlcom. Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix A A k I + A + k!
More informationSouthern Taiwan University
Chaptr Ordinar Diffrntial Equations of th First Ordr and First Dgr Gnral form:., d +, d 0.a. f,.b I. Sparabl Diffrntial quations Form: d + d 0 C d d E 9 + 4 0 Solution: 9d + 4d 0 9 + 4 C E + d Solution:
More informationComplex Numbers Practice 0708 & SP 1. The complex number z is defined by
IB Math Hgh Leel: Complex Nmbers Practce 0708 & SP Complex Nmbers Practce 0708 & SP. The complex nmber z s defned by π π π π z = sn sn. 6 6 Ale - Desert Academy (a) Express z n the form re, where r and
More informationTitle: Vibrational structure of electronic transition
Titl: Vibrational structur of lctronic transition Pag- Th band spctrum sn in th Ultra-Violt (UV) and visibl (VIS) rgions of th lctromagntic spctrum can not intrprtd as vibrational and rotational spctrum
More informationThe Matrix Exponential
Th Matrix Exponntial (with xrciss) by Dan Klain Vrsion 28928 Corrctions and commnts ar wlcom Th Matrix Exponntial For ach n n complx matrix A, dfin th xponntial of A to b th matrix () A A k I + A + k!
More informationA Propagating Wave Packet Group Velocity Dispersion
Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to
More informationA general Kirchhoff approximation for echo simulation in ultrasonic NDT
A genera rchhoff approxmaton for echo smaton n trasonc NDT V. Dorva, S. Chaton, B. L, M. Darmon, S. Mahat (CEA, LIST) CEA, LIST, F-99 Gf-sr-Yvette, France Pan A genera rchhoff approxmaton for echo smaton
More informationECE 2210 / 00 Phasor Examples
EE 0 / 00 Phasor Exampls. Add th sinusoidal voltags v ( t ) 4.5. cos( t 30. and v ( t ) 3.. cos( t 5. v ( t) using phasor notation, draw a phasor diagram of th thr phasors, thn convrt back to tim domain
More informationElectrochemical Equilibrium Electromotive Force. Relation between chemical and electric driving forces
C465/865, 26-3, Lctur 7, 2 th Sp., 26 lctrochmcal qulbrum lctromotv Forc Rlaton btwn chmcal and lctrc drvng forcs lctrochmcal systm at constant T and p: consdr G Consdr lctrochmcal racton (nvolvng transfr
More informationBackground: We have discussed the PIB, HO, and the energy of the RR model. In this chapter, the H-atom, and atomic orbitals.
Chaptr 7 Th Hydrogn Atom Background: W hav discussd th PIB HO and th nrgy of th RR modl. In this chaptr th H-atom and atomic orbitals. * A singl particl moving undr a cntral forc adoptd from Scott Kirby
More informationEpistemic Foundations of Game Theory. Lecture 1
Royal Nthrlands cadmy of rts and Scncs (KNW) Mastr Class mstrdam, Fbruary 8th, 2007 Epstmc Foundatons of Gam Thory Lctur Gacomo onanno (http://www.con.ucdavs.du/faculty/bonanno/) QUESTION: What stratgs
More informationThat is, we start with a general matrix: And end with a simpler matrix:
DIAGON ALIZATION OF THE STR ESS TEN SOR INTRO DUCTIO N By th us of Cauchy s thorm w ar abl to rduc th numbr of strss componnts in th strss tnsor to only nin valus. An additional simplification of th strss
More informationThe gravitational field energy density for symmetrical and asymmetrical systems
Th ravtatonal ld nry dnsty or symmtral and asymmtral systms Roald Sosnovsy Thnal Unvrsty 1941 St. Ptrsbur Russa E-mal:rosov@yandx Abstrat. Th rlatvst thory o ravtaton has th onsdrabl dults by dsrpton o
More informationShortest Paths in Graphs. Paths in graphs. Shortest paths CS 445. Alon Efrat Slides courtesy of Erik Demaine and Carola Wenk
S 445 Shortst Paths n Graphs lon frat Sls courtsy of rk man an arola Wnk Paths n raphs onsr a raph G = (V, ) wth -wht functon w : R. Th wht of path p = v v v k s fn to xampl: k = w ( p) = w( v, v + ).
More informationLECTURE 6 TRANSFORMATION OF RANDOM VARIABLES
LECTURE 6 TRANSFORMATION OF RANDOM VARIABLES TRANSFORMATION OF FUNCTION OF A RANDOM VARIABLE UNIVARIATE TRANSFORMATIONS TRANSFORMATION OF RANDOM VARIABLES If s a rv wth cdf F th Y=g s also a rv. If w wrt
More informationSoft k-means Clustering. Comp 135 Machine Learning Computer Science Tufts University. Mixture Models. Mixture of Normals in 1D
Comp 35 Machn Larnng Computr Scnc Tufts Unvrsty Fall 207 Ron Khardon Th EM Algorthm Mxtur Modls Sm-Suprvsd Larnng Soft k-mans Clustrng ck k clustr cntrs : Assocat xampls wth cntrs p,j ~~ smlarty b/w cntr
More informationBifurcation Theory. , a stationary point, depends on the value of α. At certain values
Dnamic Macroconomic Thor Prof. Thomas Lux Bifurcation Thor Bifurcation: qualitativ chang in th natur of th solution occurs if a paramtr passs through a critical point bifurcation or branch valu. Local
More informationLucas Test is based on Euler s theorem which states that if n is any integer and a is coprime to n, then a φ(n) 1modn.
Modul 10 Addtonal Topcs 10.1 Lctur 1 Prambl: Dtrmnng whthr a gvn ntgr s prm or compost s known as prmalty tstng. Thr ar prmalty tsts whch mrly tll us whthr a gvn ntgr s prm or not, wthout gvng us th factors
More informationConsider a system of 2 simultaneous first order linear equations
Soluon of sysms of frs ordr lnar quaons onsdr a sysm of smulanous frs ordr lnar quaons a b c d I has h alrna mar-vcor rprsnaon a b c d Or, n shorhand A, f A s alrady known from con W know ha h abov sysm
More informationGive the letter that represents an atom (6) (b) Atoms of A and D combine to form a compound containing covalent bonds.
1 Th diagram shows th lctronic configurations of six diffrnt atoms. A B C D E F (a) You may us th Priodic Tabl on pag 2 to hlp you answr this qustion. Answr ach part by writing on of th lttrs A, B, C,
More informationSuperposition. Thinning
Suprposition STAT253/317 Wintr 213 Lctur 11 Yibi Huang Fbruary 1, 213 5.3 Th Poisson Procsss 5.4 Gnralizations of th Poisson Procsss Th sum of two indpndnt Poisson procsss with rspctiv rats λ 1 and λ 2,
More informationNewtonian noise mitigation by using mini-sogros
Nwtonan nos mtgaton by usng mn-sogos Ho Jung Pa /8/6 Sn SOGO s a vry snstv gravty stran gaug on may b ab to mpoy sad- and rmov t down SOGOs n pa of a arg array of ssmomtrs to drty masur Nwtonan nos NN
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 401 Digital Signal Procssing Prof. Mark Fowlr Dtails of th ot St #19 Rading Assignmnt: Sct. 7.1.2, 7.1.3, & 7.2 of Proakis & Manolakis Dfinition of th So Givn signal data points x[n] for n = 0,, -1
More informationSummary: Solving a Homogeneous System of Two Linear First Order Equations in Two Unknowns
Summary: Solvng a Homognous Sysm of Two Lnar Frs Ordr Equaons n Two Unknowns Gvn: A Frs fnd h wo gnvalus, r, and hr rspcv corrspondng gnvcors, k, of h coffcn mar A Dpndng on h gnvalus and gnvcors, h gnral
More information425. Calculation of stresses in the coating of a vibrating beam
45. CALCULAION OF SRESSES IN HE COAING OF A VIBRAING BEAM. 45. Calulaton of stresses n the oatng of a vbratng beam M. Ragulsks,a, V. Kravčenken,b, K. Plkauskas,, R. Maskelunas,a, L. Zubavčus,b, P. Paškevčus,d
More informationStability of an Exciton bound to an Ionized Donor in Quantum Dots
Stablty of an Exton bound to an Ionzd Donor n Quantum Dots by S. Baskoutas 1*), W. Shommrs ), A. F. Trzs 3), V. Kapakls 4), M. Rth 5), C. Polts 4,6) 1) Matrals Sn Dpartmnt, Unvrsty of Patras, 6500 Patras,
More informationMultiple-Choice Test Runge-Kutta 4 th Order Method Ordinary Differential Equations COMPLETE SOLUTION SET
Multpl-Co Tst Rung-Kutta t Ordr Mtod Ordnar Drntal Equatons COMPLETE SOLUTION SET. To solv t ordnar drntal quaton sn ( Rung-Kutta t ordr mtod ou nd to rwrt t quaton as (A sn ( (B ( sn ( (C os ( (D sn (
More informationLie Groups HW7. Wang Shuai. November 2015
Li roups HW7 Wang Shuai Novmbr 015 1 Lt (π, V b a complx rprsntation of a compact group, show that V has an invariant non-dgnratd Hrmitian form. For any givn Hrmitian form on V, (for xampl (u, v = i u
More information