ME 501A Seminar in Engineering Analysis Page 1
|
|
|
- Gregory Sharp
- 5 years ago
- Views:
Transcription
1 Seod ad igher Order Liear Differeial Equaios Oober 9, 7 Seod ad igher Order Liear Differeial Equaios Larr areo Mehaial Egieerig 5 Seiar i Egieerig alsis Oober 9, 7 Oulie Reiew las lass ad hoewor ppl aerial fro las lass o ehaial ibraios igher order equaios wih osa oeffiies oogeous ad ohoogeous soluios Exisee ad uiqueess of soluios for higher order equaios Reiew Udeeried oeffiies Used for osa oeffiie equaio + a + b = r( Soluio is = P +, where is soluio of + a + b = Posulae a soluio for P followig guidelies o ex wo hars Plug soluio io ODE ad sole for uow oeffiies Oerall oeffiies of lie ers o boh sides of ODE us aish Table of Trial P Soluios or hese r( r( = e ax r( = x r( = si r( = os r( = e ax si r( = e ax os Sar wih his P P = e ax P = a + a x + + a x P = si + os P = e ax ( si + os ) Speial Rules If he righ-had-side, r( osiss of ore ha oe er fro he preious able, use a P ha oais all he orrespodig P ers or r( = os bx + e, use P = E si bx + os bx + Ge If r( is proporioal o a soluio for he hoogeous equaio, use P equal o x ies he P show i he able or a double roo, ulipl able P b x 5 Reiew Paraeer Variaio Wa o sole liear equaio d d p( q( r( ae o sole hoogeous equaio o ge wo (LI) soluios ad Defie W fro hese wo soluios d d W 6 ME 5 Seiar i Egieerig alsis Page
2 Seod ad igher Order Liear Differeial Equaios Oober 9, 7 Reiew Paraeer Variaio II Defie u( ad ( suh ha P = u +, where u ad are foud fro he followig iegrals r( r( u W ( W ( r( r( P u W ( W ( x ) Ge = + P ad ealuae osas i soluio fro iiial odiios Nohoogeous Suar Udeeried oeffiies is sipler approah bu is liied osa oeffiie equaios Liied se of fuios Variaio of paraeers is ore oplex, bu hadles ore ases I reali, here are o geeral ehods o ge hoogeous soluio o liear, seod-order ODE wihou osa oeffiies 7 8 igher Order Equaios Geeral h order liear equaio d d d p ( p( p( r( Treae siilar o seod order Loo a hoogeous soluio firs obie wih pariular soluio Mus osider ODE wih osa oeffiies o ge a geeral resuls This is siilar o seod order 9 igher Order Equaios II Loo a geeral h order differeial equaio wih osa oeffiies d d d a a a r( id fro hoogeous ODE d a d d a a x e oogeous soluio igher Order Equaios III oogeous soluio I hoogeous soluio, he alues of are soluios o he equaio + a a a + a = oplex soluios our as oplex ojugaes giig sies ad osies or double roos = DR, we odif soluio o gie ( + + x ) e (DR)x e x igher Order Equaios IV or ohoogeous equaios we a fid he oal soluio = + P P a be foud b udeeried oeffiies or ariaio of paraeers Use sae proess for ehod of udeeried oeffiies Variaio of paraeers is ore oplex sie i ioles soluio of siulaeous equaios for ew soluios ME 5 Seiar i Egieerig alsis Page
3 Seod ad igher Order Liear Differeial Equaios Oober 9, 7 igher Order Equaios V There are liearl-idepede soluios o a liear, hoogeous h order ODE The liearl-idepede soluios for a basis for all soluios Use sae proess for ehod of udeeried oeffiies Variaio of paraeers is ore oplex sie i ioles soluio of siulaeous equaios for ew soluios Exisee ad Uiqueess Geeral liear, hoogeous, h order ODEs hae a uique soluio oer a < x < b if all he p ( are oiuous here d p d d ( p ( p( The proposed soluios ( o he hoogeous ODE are liearl idepede if he Wrosia (see ex har) is ozero Wrosi Deeria Wrosia, W, for h order ODE (wih oaio ha () deoes d / ) () () W ( ) () () ( ) () () ( ) () () ( ) 5 ppliaio: Sruural Meber elasi bea wih a applied load, f(, per ui legh, i he direio (oral o he bea) ea is be uder his load edig oe, M( is gie b seodorder ODE: d M/ = f( ial defleio is d / M = EI d / where E is Youg s odulus ad I is oe of ieria 6 Sruural Meber ODE obie d M/ = f( ad M = EI d / o ge EId / = f( SI uis for hese quaiies are eers for x ad, N/ for E, for I, N/ for f(, ad N for M diesios for h order deriaie are diesios of ueraor diided b (deoiaor diesios) ae a oal of four boudar odiios a x = ad x = L Equaio has separable soluio 7 d f ( EI d EI d EI d EI Solig he Equaio d EI f ( f ( x f ( x x f ( x x x ppl boudar odiios o fid osas of iegraio 8 ME 5 Seiar i Egieerig alsis Page
4 Seod ad igher Order Liear Differeial Equaios Oober 9, 7 ME 5 Seiar i Egieerig alsis Page 9 ppliaio: ored Vibraios Las wee we showed soluios for free ibraios of sprig-ass-daper sse ODE was d /d + d/d + = Iposed fore gies ohoogeous ODE d /d + d/d + = f() osider exaple where f() = os Udeeried oeffiie rial soluio is P = si + os ored Vibraios II Deriaies of P = si + os P = os - si P = - si - os Subsiue io ODE: d /d + d/d + = os [- si - os ] + [ os - si ] + [ si + os ] = os ored Vibraios III Rearrage o olle sies ad osies [- si - os ] + [ os - si ] + [ si + os ] = os [- + ] si + [- + + ] os = os Equae oeffiies of sie ad osie ers o boh sides of he equaio ored Vibraios IV [- + ] si + [- + + ] os = os (- + ) = (sie ers) (- + ) = (osies) raer s rule soluio gies Defie = / i Soluio Udaped ase, = P = si + os = os ro las wee, = si + Dos = E os( + ) Loo a iiial odiios
5 Seod ad igher Order Liear Differeial Equaios Oober 9, 7 ME 5 Seiar i Egieerig alsis Page 5 5 Udaped ase II D P os os si Iiial odiios () = ad () = = D + //( ) = os os si os os si 6 Udaped ase III / is a fuio of ad he followig hree (diesioless) paraeers: /, / ad / os os si 7 Udaped ase IV Sar wih soluio below ad oer sie ad osie ers o a osie er os os si a os os 8 Udaped ase V opue diesioless ad a a a os os 9 Udaped ase VI / is a fuio of, /, /, ad / a os os Zero Iiial odiios Wihou forig ( = ), whe = =, he soluio is = for all orig gies a ozero soluio Sar wih geeral soluio for = D P os os si = () = gies = = () = gies D = - //( ) os os
6 Seod ad igher Order Liear Differeial Equaios Oober 9, 7 Zero Iiial odiios II Rearrage soluio for = = = os os os os Plo / ersus wih / as a paraeer / Udaped ored Osiallaios = = Raio paraeer is raio of forig freque o aural freque / raio =. raio =.5 raio =.95 raio =.9 Resoae odiio urre equaio for / has seeral ers wih / i deoiaor Soluio is o alid whe = If =, r( = os is proporioal o hoogeous equaio soluio ae o ge ew pariular soluio Use udeeried oeffiies approah sarig wih P = [ si + os ] = Resoae Soluio II Reeber = = (/) / here Deriaies of P = [ si + os ] P = [ os - si ] + si + os P = [ - si - os ] + os - si Subsiue io ODE for = ad = : d P /d + P = d P /d + P = os = Resoae Soluio III [- si - os ] + [ os - si ] + [ si + os ] = os fer aellaios we hae [ os - si ] = os This gies = ad = / P = [ /] si Pariular soluio ireases wihou boud as ireases Exaie Dapig Daped osillaios wihou exeral fore deried (hoogeous equaio) soluios las wee Three ases: uderdapig, riial dapig, ad uderdapig ll ases show goes o zero as ireases Loo a pariular soluio ol o show effe of fored osillaios Effes oe fro apliude of osillaios 5 6 ME 5 Seiar i Egieerig alsis Page 6
7 Seod ad igher Order Liear Differeial Equaios Oober 9, 7 ME 5 Seiar i Egieerig alsis Page 7 7 Geeral ase for oer P = si + D os = E os( -) o exaie apliude + D = E ad = a - (/D) ppl his o wrie P = E os( -) os si P a a 8 id ha Maxiizes d d 9 pliude of P ersus Maxiu apliude equaio o alid if / > = / Loo a behaior of P = os( - ) b exaiig ersus Wrie diesioless equaio for, whih has diesios of legh pliude of P ersus II Diesioless apliude depeds o / ad / = / Preious resul: ( / ) ax = - / Diesioless Dapig pliude / / Diesioless pliud * =. * =. * =.5 * =.5 * =. * = diesioless dapig = / s / beoes saller, loaio of / for axiu beoes oe ad axiu apliude beoes / Suar Geeral soluios for ODEs wih order for osa oeffiies ol Soluios are series of e x ers where are soluios of algebrai equaio Speial ases: double ad oplex roos Ge geeral soluio as = + P Use ehod of udeeried oeffiies (sipler ha ariaio of paraeers) o fid P
ME203 Section 4.1 Forced Vibration Response of Linear System Nov 4, 2002 (1) kx c x& m mg
ME3 Setio 4.1 Fored Vibratio Respose of Liear Syste Nov 4, Whe a liear ehaial syste is exited by a exteral fore, its respose will deped o the for of the exitatio fore F(t) ad the aout of dapig whih is
Energy Density / Energy Flux / Total Energy in 1D. Key Mathematics: density, flux, and the continuity equation.
ecure Phys 375 Eergy Desiy / Eergy Flu / oal Eergy i D Overview ad Moivaio: Fro your sudy of waves i iroducory physics you should be aware ha waves ca raspor eergy fro oe place o aoher cosider he geeraio
Chapter 10. Laser Oscillation : Gain and Threshold
Chaper 0. aser Osillaio : Gai ad hreshold Deailed desripio of laser osillaio 0. Gai Cosider a quasi-moohromai plae wave of frequey propaai i he + direio ; u A he rae a whih
FBD of SDOF Base Excitation. 2.4 Base Excitation. Particular Solution (sine term) SDOF Base Excitation (cont) F=-(-)-(-)= 2ζω ωf
.4 Base Exiaio Ipoa lass of vibaio aalysis Peveig exiaios fo passig fo a vibaig base hough is ou io a suue Vibaio isolaio Vibaios i you a Saellie opeaio Dis dives, e. FBD of SDOF Base Exiaio x() y() Syse
ODEs II, Supplement to Lectures 6 & 7: The Jordan Normal Form: Solving Autonomous, Homogeneous Linear Systems. April 2, 2003
ODEs II, Suppleme o Lecures 6 & 7: The Jorda Normal Form: Solvig Auoomous, Homogeeous Liear Sysems April 2, 23 I his oe, we describe he Jorda ormal form of a marix ad use i o solve a geeral homogeeous
Numerical Method for Ordinary Differential Equation
Numerical ehod for Ordiar Differeial Equaio. J. aro ad R. J. Lopez, Numerical Aalsis: A Pracical Approach, 3rd Ed., Wadsworh Publishig Co., Belmo, CA (99): Chap. 8.. Iiial Value Problem (IVP) d (IVP):
1. Solve by the method of undetermined coefficients and by the method of variation of parameters. (4)
7 Differeial equaios Review Solve by he mehod of udeermied coefficies ad by he mehod of variaio of parameers (4) y y = si Soluio; we firs solve he homogeeous equaio (4) y y = 4 The correspodig characerisic
A modified method for solving Delay differential equations of fractional order
IOSR Joural of Mahemais (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume, Issue 3 Ver. VII (May. - Ju. 6), PP 5- www.iosrjourals.org A modified mehod for solvig Delay differeial equaios of fraioal order
On the Existence and Uniqueness of Solutions for. Q-Fractional Boundary Value Problem
I Joural of ah Aalysis, Vol 5, 2, o 33, 69-63 O he Eisee ad Uiueess of Soluios for Q-Fraioal Boudary Value Prolem ousafa El-Shahed Deparme of ahemais, College of Eduaio Qassim Uiversiy PO Bo 377 Uizah,
Transverse Wave Motion
Trasverse Wave Moio Defiiio of Waves wave is a disurbae ha moves hrough a medium wihou givig he medium, as a whole, a permae displaeme. The geeral ame for hese waves is progressive wave. If he disurbae
State-Space Model. In general, the dynamic equations of a lumped-parameter continuous system may be represented by
Sae-Space Model I geeral, he dyaic equaio of a luped-paraeer coiuou ye ay be repreeed by x & f x, u, y g x, u, ae equaio oupu equaio where f ad g are oliear vecor-valued fucio Uig a liearized echique,
MAHALAKSHMI ENGINEERING COLLEGE TIRUCHIRAPALLI
MAHALAKSHMI EGIEERIG COLLEGE TIRUCHIRAALLI 6 QUESTIO BAK - ASWERS -SEMESTER: V MA 6 - ROBABILITY AD QUEUEIG THEORY UIT IV:QUEUEIG THEORY ART-A Quesio : AUC M / J Wha are he haraerisis of a queueig heory?
Math-303 Chapter 7 Linear systems of ODE November 16, Chapter 7. Systems of 1 st Order Linear Differential Equations.
Mah-33 Chaper 7 Liear sysems of ODE November 6, 7 Chaper 7 Sysems of s Order Liear Differeial Equaios saddle poi λ >, λ < Mah-33 Chaper 7 Liear sysems of ODE November 6, 7 Mah-33 Chaper 7 Liear sysems
Key Questions. ECE 340 Lecture 16 and 17: Diffusion of Carriers 2/28/14
/8/4 C 340 eure 6 ad 7: iffusio of Carriers Class Oulie: iffusio roesses iffusio ad rif of Carriers Thigs you should kow whe you leave Key Quesios Why do arriers use? Wha haes whe we add a eleri field
Comparison between Fourier and Corrected Fourier Series Methods
Malaysia Joural of Mahemaical Scieces 7(): 73-8 (13) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Joural homepage: hp://eispem.upm.edu.my/oural Compariso bewee Fourier ad Correced Fourier Series Mehods 1
Ideal Amplifier/Attenuator. Memoryless. where k is some real constant. Integrator. System with memory
Liear Time-Ivaria Sysems (LTI Sysems) Oulie Basic Sysem Properies Memoryless ad sysems wih memory (saic or dyamic) Causal ad o-causal sysems (Causaliy) Liear ad o-liear sysems (Lieariy) Sable ad o-sable
Electrical Engineering Department Network Lab.
Par:- Elecrical Egieerig Deparme Nework Lab. Deermiaio of differe parameers of -por eworks ad verificaio of heir ierrelaio ships. Objecive: - To deermie Y, ad ABD parameers of sigle ad cascaded wo Por
Harmonic excitation (damped)
Harmoic eciaio damped k m cos EOM: m&& c& k cos c && ζ & f cos The respose soluio ca be separaed io par;. Homogeeous soluio h. Paricular soluio p h p & ζ & && ζ & f cos Homogeeous soluio Homogeeous soluio
Problems and Solutions for Section 3.2 (3.15 through 3.25)
3-7 Problems ad Soluios for Secio 3 35 hrough 35 35 Calculae he respose of a overdamped sigle-degree-of-freedom sysem o a arbirary o-periodic exciaio Soluio: From Equaio 3: x = # F! h "! d! For a overdamped
Math 2414 Homework Set 7 Solutions 10 Points
Mah Homework Se 7 Soluios 0 Pois #. ( ps) Firs verify ha we ca use he iegral es. The erms are clearly posiive (he epoeial is always posiive ad + is posiive if >, which i is i his case). For decreasig we
BE.430 Tutorial: Linear Operator Theory and Eigenfunction Expansion
BE.43 Tuorial: Liear Operaor Theory ad Eigefucio Expasio (adaped fro Douglas Lauffeburger) 9//4 Moivaig proble I class, we ecouered parial differeial equaios describig rasie syses wih cheical diffusio.
Chemistry 1B, Fall 2016 Topics 21-22
Cheisry B, Fall 6 Topics - STRUCTURE ad DYNAMICS Cheisry B Fall 6 Cheisry B so far: STRUCTURE of aos ad olecules Topics - Cheical Kieics Cheisry B ow: DYNAMICS cheical kieics herodyaics (che C, 6B) ad
In this section we will study periodic signals in terms of their frequency f t is said to be periodic if (4.1)
Fourier Series Iroducio I his secio we will sudy periodic sigals i ers o heir requecy is said o be periodic i coe Reid ha a sigal ( ) ( ) ( ) () or every, where is a uber Fro his deiiio i ollows ha ( )
Homework 6: Forced Vibrations Due Friday April 6, 2018
EN40: Dyais ad Vibratios Hoework 6: Fored Vibratios Due Friday April 6, 018 Shool of Egieerig Brow Uiversity 1. The vibratio isolatio syste show i the figure has 0kg, k 19.8 kn / 1.59 kns / If the base
N! AND THE GAMMA FUNCTION
N! AND THE GAMMA FUNCTION Cosider he produc of he firs posiive iegers- 3 4 5 6 (-) =! Oe calls his produc he facorial ad has ha produc of he firs five iegers equals 5!=0. Direcly relaed o he discree! fucio
Calculus BC 2015 Scoring Guidelines
AP Calculus BC 5 Scorig Guidelies 5 The College Board. College Board, Advaced Placeme Program, AP, AP Ceral, ad he acor logo are regisered rademarks of he College Board. AP Ceral is he official olie home
ES 330 Electronics II Homework 03 (Fall 2017 Due Wednesday, September 20, 2017)
Pae1 Nae Soluios ES 330 Elecroics II Hoework 03 (Fall 017 ue Wedesday, Sepeber 0, 017 Proble 1 You are ive a NMOS aplifier wih drai load resisor R = 0 k. The volae (R appeari across resisor R = 1.5 vols
Lecture contents Macroscopic Electrodynamics Propagation of EM Waves in dielectrics and metals
Leure oes Marosopi lerodyamis Propagaio of M Waves i dieleris ad meals NNS 58 M Leure #4 Maxwell quaios Maxwell equaios desribig he ouplig of eleri ad magei fields D q ev B D J [SI] [CGS] D 4 B D 4 J B
S n. = n. Sum of first n terms of an A. P is
PROGREION I his secio we discuss hree impora series amely ) Arihmeic Progressio (A.P), ) Geomeric Progressio (G.P), ad 3) Harmoic Progressio (H.P) Which are very widely used i biological scieces ad humaiies.
EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D. S. Palimkar
Ieraioal Joural of Scieific ad Research Publicaios, Volue 2, Issue 7, July 22 ISSN 225-353 EXISTENCE THEORY OF RANDOM DIFFERENTIAL EQUATIONS D S Palikar Depare of Maheaics, Vasarao Naik College, Naded
λiv Av = 0 or ( λi Av ) = 0. In order for a vector v to be an eigenvector, it must be in the kernel of λi
Liear lgebra Lecure #9 Noes This week s lecure focuses o wha migh be called he srucural aalysis of liear rasformaios Wha are he irisic properies of a liear rasformaio? re here ay fixed direcios? The discussio
Online Supplement to Reactive Tabu Search in a Team-Learning Problem
Olie Suppleme o Reacive abu Search i a eam-learig Problem Yueli She School of Ieraioal Busiess Admiisraio, Shaghai Uiversiy of Fiace ad Ecoomics, Shaghai 00433, People s Republic of Chia, she.yueli@mail.shufe.edu.c
Solutions Manual 4.1. nonlinear. 4.2 The Fourier Series is: and the fundamental frequency is ω 2π
Soluios Maual. (a) (b) (c) (d) (e) (f) (g) liear oliear liear liear oliear oliear liear. The Fourier Series is: F () 5si( ) ad he fudameal frequecy is ω f ----- H z.3 Sice V rms V ad f 6Hz, he Fourier
Optimization of Rotating Machines Vibrations Limits by the Spring - Mass System Analysis
Joural of aerials Sciece ad Egieerig B 5 (7-8 (5 - doi: 765/6-6/57-8 D DAVID PUBLISHING Opimizaio of Roaig achies Vibraios Limis by he Sprig - ass Sysem Aalysis BENDJAIA Belacem sila, Algéria Absrac: The
1 Notes on Little s Law (l = λw)
Copyrigh c 26 by Karl Sigma Noes o Lile s Law (l λw) We cosider here a famous ad very useful law i queueig heory called Lile s Law, also kow as l λw, which assers ha he ime average umber of cusomers i
Time Dependent Queuing
Time Depede Queuig Mark S. Daski Deparme of IE/MS, Norhweser Uiversiy Evaso, IL 628 Sprig, 26 Oulie Will look a M/M/s sysem Numerically iegraio of Chapma- Kolmogorov equaios Iroducio o Time Depede Queue
Numerical methods for ordinary differential equations
A Maser Scieiic copuig Nuerical ehos or oriar iereial equaios Oriar iereial equaios Le be a ocio o oe variable havig successive erivaives. A h orer oriar iereial equaio ODE is a equaio o he or: A s orer
PVP Plastic Instability Pressure of Conical Shells. PVP2011 July 17-21, 2011, Baltimore, Maryland, USA. R. Adibi-Asl *
Proeedigs Proeedigs of he ASME of 11 he Pressure ASME 11 Vessels Pressure & Pipig Vessels Divisio & Pipig Coferee Divisio PVP11 July 17-1, 11, Balimore, Marylad, USA Plasi Isabiliy Pressure of Coial Shells
Processamento Digital de Sinal
Deparaeo de Elecróica e Telecouicações da Uiversidade de Aveiro Processaeo Digial de ial Processos Esocásicos uar ado Processes aioar ad ergodic Correlaio auo ad cross Fucio Covariace Fucio Esiaes of he
More on ODEs by Laplace Transforms October 30, 2017
More on OE b Laplace Tranfor Ocober, 7 More on Ordinar ifferenial Equaion wih Laplace Tranfor Larr areo Mechanical Engineering 5 Seinar in Engineering nali Ocober, 7 Ouline Review la cla efiniion of Laplace
Communication Systems Lecture 25. Dong In Kim School of Info/Comm Engineering Sungkyunkwan University
Commuiaio Sysems Leure 5 Dog I Kim Shool o Io/Comm Egieerig Sugkyukwa Uiversiy 1 Oulie Noise i Agle Modulaio Phase deviaio Large SNR Small SNR Oupu SNR PM FM Review o Agle Modulaio Geeral orm o agle modulaed
Homework 2 Solutions
Mah 308 Differenial Equaions Fall 2002 & 2. See he las page. Hoework 2 Soluions 3a). Newon s secon law of oion says ha a = F, an we know a =, so we have = F. One par of he force is graviy, g. However,
Lecture 15: Three-tank Mixing and Lead Poisoning
Lecure 15: Three-ak Miig ad Lead Poisoig Eigevalues ad eigevecors will be used o fid he soluio of a sysem for ukow fucios ha saisfy differeial equaios The ukow fucios will be wrie as a 1 colum vecor [
Review for the Midterm Exam.
Review for he iderm Exm Rememer! Gross re e re Vriles suh s,, /, p / p, r, d R re gross res 2 You should kow he disiio ewee he fesile se d he udge se, d kow how o derive hem The Fesile Se Wihou goverme
Introduction to Mechanical Vibrations and Structural Dynamics
Inroducion o Mechanical Viraions and Srucural Dynaics The one seeser schedule :. Viraion - classificaion. ree undaped single DO iraion, equaion of oion, soluion, inegraional consans, iniial condiions..
Using Linnik's Identity to Approximate the Prime Counting Function with the Logarithmic Integral
Usig Lii's Ideiy o Approimae he Prime Couig Fucio wih he Logarihmic Iegral Naha McKezie /26/2 aha@icecreambreafas.com Summary:This paper will show ha summig Lii's ideiy from 2 o ad arragig erms i a cerai
Lecture 15 First Properties of the Brownian Motion
Lecure 15: Firs Properies 1 of 8 Course: Theory of Probabiliy II Term: Sprig 2015 Isrucor: Gorda Zikovic Lecure 15 Firs Properies of he Browia Moio This lecure deals wih some of he more immediae properies
Introduction to Mechanical Vibrations
CHAPTER 1 Iroducio o Mechaical Vibraios Vibraio is he oio of a paricle or a body or syse of coeced bodies displaced fro a posiio of equilibriu. Mos vibraios are udesirable i achies ad srucures because
Digital Signal Processing. Homework 2 Solution. Due Monday 4 October Following the method on page 38, the difference equation
Digital Sigal Proessig Homework Solutio Due Moda 4 Otober 00. Problem.4 Followig the method o page, the differee equatio [] (/4[-] + (/[-] x[-] has oeffiiets a0, a -/4, a /, ad b. For these oeffiiets A(z
Amit Mehra. Indian School of Business, Hyderabad, INDIA Vijay Mookerjee
RESEARCH ARTICLE HUMAN CAPITAL DEVELOPMENT FOR PROGRAMMERS USING OPEN SOURCE SOFTWARE Ami Mehra Indian Shool of Business, Hyderabad, INDIA {Ami_Mehra@isb.edu} Vijay Mookerjee Shool of Managemen, Uniersiy
Basic Results in Functional Analysis
Preared by: F.. ewis Udaed: Suday, Augus 7, 4 Basic Resuls i Fucioal Aalysis f ( ): X Y is coiuous o X if X, (, ) z f( z) f( ) f ( ): X Y is uiformly coiuous o X if i is coiuous ad ( ) does o deed o. f
Conservation of Momentum. The purpose of this experiment is to verify the conservation of momentum in two dimensions.
Conseraion of Moenu Purose The urose of his exerien is o erify he conseraion of oenu in wo diensions. Inroducion and Theory The oenu of a body ( ) is defined as he roduc of is ass () and elociy ( ): When
(C) x 3 + y 3 = 2(x 2 + y 2 ) (D) x y 2 (A) (10)! (B) (11)! (C) (10)! + 1 (D) (11)! 1. n in the expansion of 2 (A) 15 (B) 45 (C) 55 (D) 56
Cocep rackig paper-7 (ST+BT) Q. If 60 a = ad 60 b = 5 he he value of SINGLE OPTION CORRECT a b ( b) equals (D) Time-5hrs 0mis. Q. ( + x) ( + x + x ) ( + x + x + x )... ( + x + x +... + x 00 ) whe wrie
[ m] x = 0.25cos 20 t sin 20 t m
![[ m] x = 0.25cos 20 t sin 20 t m](/thumbs/88/116457721.jpg "[ m] x = 0.25cos 20 t sin 20 t m")
. x.si ( 5 s [ ] CHAPER OSCILLAIONS x ax (.( ( 5 6. s s ( ( ( xax. 5.7 s s. x.si [] x. cos s Whe, x a x.5. s 5s.6 s x. x( x cos + si a f ( ( [ ] x.5cos +.59si. ( ( cos α β cosαcos β + siαsi β x Acos φ
Manipulations involving the signal amplitude (dependent variable).
Oulie Maipulaio of discree ime sigals: Maipulaios ivolvig he idepede variable : Shifed i ime Operaios. Foldig, reflecio or ime reversal. Time Scalig. Maipulaios ivolvig he sigal ampliude (depede variable).
Viscous Damping Summary Sheet No Damping Case: Damped behaviour depends on the relative size of ω o and b/2m 3 Cases: 1.
Viscous Daping: && + & + ω Viscous Daping Suary Shee No Daping Case: & + ω solve A ( ω + α ) Daped ehaviour depends on he relaive size of ω o and / 3 Cases:. Criical Daping Wee 5 Lecure solve sae BC s
MODERN CONTROL SYSTEMS
MODERN CONTROL SYSTEMS Lecure 9, Sae Space Repreeaio Emam Fahy Deparme of Elecrical ad Corol Egieerig email: emfmz@aa.edu hp://www.aa.edu/cv.php?dip_ui=346&er=6855 Trafer Fucio Limiaio TF = O/P I/P ZIC
ECE-314 Fall 2012 Review Questions
ECE-34 Fall 0 Review Quesios. A liear ime-ivaria sysem has he ipu-oupu characerisics show i he firs row of he diagram below. Deermie he oupu for he ipu show o he secod row of he diagram. Jusify your aswer.
Fourier transform. Continuous-time Fourier transform (CTFT) ω ω
Fourier rasform Coiuous-ime Fourier rasform (CTFT P. Deoe ( he Fourier rasform of he sigal x(. Deermie he followig values, wihou compuig (. a (0 b ( d c ( si d ( d d e iverse Fourier rasform for Re { (
Department of Mathematical and Statistical Sciences University of Alberta
MATH 4 (R) Wier 008 Iermediae Calculus I Soluios o Problem Se # Due: Friday Jauary 8, 008 Deparme of Mahemaical ad Saisical Scieces Uiversiy of Albera Quesio. [Sec.., #] Fid a formula for he geeral erm
David Randall. ( )e ikx. k = u x,t. u( x,t)e ikx dx L. x L /2. Recall that the proof of (1) and (2) involves use of the orthogonality condition.
! Revised April 21, 2010 1:27 P! 1 Fourier Series David Radall Assume ha u( x,) is real ad iegrable If he domai is periodic, wih period L, we ca express u( x,) exacly by a Fourier series expasio: ( ) =
Supplement for SADAGRAD: Strongly Adaptive Stochastic Gradient Methods"
Suppleme for SADAGRAD: Srogly Adapive Sochasic Gradie Mehods" Zaiyi Che * 1 Yi Xu * Ehog Che 1 iabao Yag 1. Proof of Proposiio 1 Proposiio 1. Le ɛ > 0 be fixed, H 0 γi, γ g, EF (w 1 ) F (w ) ɛ 0 ad ieraio
Effects of Forces Applied in the Middle Plane on Bending of Medium-Thickness Band
MATEC We of Cofereces 7 7 OI:./ maeccof/77 XXVI R-S-P Semiar 7 Theoreical Foudaio of Civil Egieerig Effecs of Forces Applied i he Middle Plae o Bedig of Medium-Thickess Bad Adre Leoev * Moscow sae uiversi
Solutions 3.2-Page 215
Solutios.-Page Problem Fid the geeral solutios i powers of of the differetial equatios. State the reurree relatios ad the guarateed radius of overgee i eah ase. ) Substitutig,, ad ito the differetial equatio
Calculus Limits. Limit of a function.. 1. One-Sided Limits...1. Infinite limits 2. Vertical Asymptotes...3. Calculating Limits Using the Limit Laws.
Limi of a fucio.. Oe-Sided..... Ifiie limis Verical Asympoes... Calculaig Usig he Limi Laws.5 The Squeeze Theorem.6 The Precise Defiiio of a Limi......7 Coiuiy.8 Iermediae Value Theorem..9 Refereces..
Thus the force is proportional but opposite to the displacement away from equilibrium.
Chaper 3 : Siple Haronic Moion Hooe s law saes ha he force (F) eered by an ideal spring is proporional o is elongaion l F= l where is he spring consan. Consider a ass hanging on a he spring. In equilibriu
Optimum design of complementary transient experiments for estimating thermal properties
Opiu desig of copleeary rasie experies for esiaig heral properies Jaes V. Beck*, Filippo de Moe, Doald E. Aos *Depare of Mechaical Egieerig, Michiga Sae Uiversiy, USA Depare of Idusrial ad Iforaio Egieerig
Let s express the absorption of radiation by dipoles as a dipole correlation function.
MIT Deparme of Chemisry 5.74, Sprig 004: Iroducory Quaum Mechaics II Isrucor: Prof. Adrei Tokmakoff p. 81 Time-Correlaio Fucio Descripio of Absorpio Lieshape Le s express he absorpio of radiaio by dipoles
K3 p K2 p Kp 0 p 2 p 3 p
Mah 80-00 Mo Ar 0 Chaer 9 Fourier Series ad alicaios o differeial equaios (ad arial differeial equaios) 9.-9. Fourier series defiiio ad covergece. The idea of Fourier series is relaed o he liear algebra
ECE 340 Lecture 19 : Steady State Carrier Injection Class Outline:
ECE 340 ecure 19 : Seady Sae Carrier Ijecio Class Oulie: iffusio ad Recombiaio Seady Sae Carrier Ijecio Thigs you should kow whe you leave Key Quesios Wha are he major mechaisms of recombiaio? How do we
Lecture 25 Outline: LTI Systems: Causality, Stability, Feedback
Lecure 5 Oulie: LTI Sye: Caualiy, Sabiliy, Feebac oucee: Reaig: 6: Lalace Trafor. 37-49.5, 53-63.5, 73; 7: 7: Feebac. -4.5, 8-7. W 8 oe, ue oay. Free -ay eeio W 9 will be oe oay, ue e Friay (o lae W) Fial
Linear System Theory
Naioal Tsig Hua Uiversiy Dearme of Power Mechaical Egieerig Mid-Term Eamiaio 3 November 11.5 Hours Liear Sysem Theory (Secio B o Secio E) [11PME 51] This aer coais eigh quesios You may aswer he quesios
STK4080/9080 Survival and event history analysis
STK48/98 Survival ad eve hisory aalysis Marigales i discree ime Cosider a sochasic process The process M is a marigale if Lecure 3: Marigales ad oher sochasic processes i discree ime (recap) where (formally
ECE 340 Lecture 15 and 16: Diffusion of Carriers Class Outline:
ECE 340 Lecure 5 ad 6: iffusio of Carriers Class Oulie: iffusio rocesses iffusio ad rif of Carriers Thigs you should kow whe you leave Key Quesios Why do carriers diffuse? Wha haes whe we add a elecric
The Structures of Fuzzifying Measure
Sesors & Trasduers Vol 7 Issue 5 May 04 pp 56-6 Sesors & Trasduers 04 by IFSA Publishig S L hp://wwwsesorsporalom The Sruures of Fuzzifyig Measure Shi Hua Luo Peg Che Qia Sheg Zhag Shool of Saisis Jiagxi
Solutions to selected problems from the midterm exam Math 222 Winter 2015
Soluios o seleced problems from he miderm eam Mah Wier 5. Derive he Maclauri series for he followig fucios. (cf. Pracice Problem 4 log( + (a L( d. Soluio: We have he Maclauri series log( + + 3 3 4 4 +...,
Fresnel Dragging Explained
Fresel Draggig Explaied 07/05/008 Decla Traill Decla@espace.e.au The Fresel Draggig Coefficie required o explai he resul of he Fizeau experime ca be easily explaied by usig he priciples of Eergy Field
Lateral torsional buckling of rectangular beams using variational iteration method
Scieific Research ad Essas Vol. 6(6), pp. 445-457, 8 March, Available olie a hp://www.academicjourals.org/sre ISSN 99-48 Academic Jourals Full egh Research Paper aeral orsioal bucklig of recagular beams
Economics 8723 Macroeconomic Theory Problem Set 2 Professor Sanjay Chugh Spring 2017
Deparme of Ecoomics The Ohio Sae Uiversiy Ecoomics 8723 Macroecoomic Theory Problem Se 2 Professor Sajay Chugh Sprig 207 Labor Icome Taxes, Nash-Bargaied Wages, ad Proporioally-Bargaied Wages. I a ecoomy
14.02 Principles of Macroeconomics Fall 2005
14.02 Priciples of Macroecoomics Fall 2005 Quiz 2 Tuesday, November 8, 2005 7:30 PM 9 PM Please, aswer he followig quesios. Wrie your aswers direcly o he quiz. You ca achieve a oal of 100 pois. There are
School of Mechanical Engineering Purdue University. ME375 Transfer Functions - 1
Trasfer Fuctio Aalysis Free & Forced Resposes Trasfer Fuctio Syste Stability ME375 Trasfer Fuctios - 1 Free & Forced Resposes Ex: Let s look at a stable first order syste: y y Ku Take LT of the I/O odel
Section 8 Convolution and Deconvolution
APPLICATIONS IN SIGNAL PROCESSING Secio 8 Covoluio ad Decovoluio This docume illusraes several echiques for carryig ou covoluio ad decovoluio i Mahcad. There are several operaors available for hese fucios:
Outline. Review Homework Problem. Review Homework Problem II. Review Dimensionless Problem. Review Convection Problem
adial diffsio eqaio Febay 4 9 Diffsio Eqaios i ylidical oodiaes ay aeo Mechaical Egieeig 5B Seia i Egieeig Aalysis Febay 4, 9 Olie eview las class Gadie ad covecio boday codiio Diffsio eqaio i adial coodiaes
Moment Generating Function
1 Mome Geeraig Fucio m h mome m m m E[ ] x f ( x) dx m h ceral mome m m m E[( ) ] ( ) ( x ) f ( x) dx Mome Geeraig Fucio For a real, M () E[ e ] e k x k e p ( x ) discree x k e f ( x) dx coiuous Example
ECE 570 Session 7 IC 752-E Computer Aided Engineering for Integrated Circuits. Transient analysis. Discuss time marching methods used in SPICE
ECE 570 Sessio 7 IC 75-E Compuer Aided Egieerig for Iegraed Circuis Trasie aalysis Discuss ime marcig meods used i SPICE. Time marcig meods. Explici ad implici iegraio meods 3. Implici meods used i circui
INSTANTANEOUS VELOCITY
INSTANTANEOUS VELOCITY I claim ha ha if acceleraion is consan, hen he elociy is a linear funcion of ime and he posiion a quadraic funcion of ime. We wan o inesigae hose claims, and a he same ime, work
CS 450: COMPUTER GRAPHICS INTRODUCTION TO MATRICES SPRING 2016 DR. MICHAEL J. REALE
CS 45: COPUTER GRAPHICS INTRODUCTION TO ATRICES SPRING 26 DR. ICHAEL J. REALE hp://www.papeleparee.e c.br/wallpapers/coigoari_2283_2824.jpg ENTER THE ATRIX ari = (p X q) 2D arra of ubers (scalars) p =
6.003: Signals and Systems
6.003: Sigals ad Sysems Lecure 8 March 2, 2010 6.003: Sigals ad Sysems Mid-erm Examiaio #1 Tomorrow, Wedesday, March 3, 7:30-9:30pm. No reciaios omorrow. Coverage: Represeaios of CT ad DT Sysems Lecures
Notes 03 largely plagiarized by %khc
1 1 Discree-Time Covoluio Noes 03 largely plagiarized by %khc Le s begi our discussio of covoluio i discree-ime, sice life is somewha easier i ha domai. We sar wih a sigal x[] ha will be he ipu io our
th m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)
![th m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)](/thumbs/80/81319353.jpg "th m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)")
1 Trasform Techiques h m m m m mome : E[ ] x f ( x) dx h m m m m ceral mome : E[( ) ] ( ) ( x) f ( x) dx A coveie wa of fidig he momes of a radom variable is he mome geeraig fucio (MGF). Oher rasform echiques
ME 321 Kinematics and Dynamics of Machines S. Lambert Winter 2002
ME 31 Kiemaic ad Dyamic o Machie S. Lamber Wier 6.. Forced Vibraio wih Dampig Coider ow he cae o orced vibraio wih dampig. Recall ha he goverig diereial equaio i: m && c& k F() ad ha we will aume ha he
Transfer Function Analysis
Trasfer Fuctio Aalysis Free & Forced Resposes Trasfer Fuctio Syste Stability ME375 Trasfer Fuctios - Free & Forced Resposes Ex: Let s s look at a stable first order syste: τ y + y = Ku Take LT of the I/O
Walter D. Pilkey. Dmitry V. Balandin. Nikolai N. Bolotnik. Sergey V. Purtsezov. Introduction. Nomenclature
Waler D. Pilkey e al Waler D. Pilkey wdp@virgiia.edu Uiversiy of Virgiia Dep. of Mechaical ad Aerospace Eg. 94-4746 Charloesville, VA. USA Diry V. Baladi baladi@pk.u.rue.ru izhy ovgorod Sae Uiversiy Dep.
SUMMATION OF INFINITE SERIES REVISITED
SUMMATION OF INFINITE SERIES REVISITED I several aricles over he las decade o his web page we have show how o sum cerai iiie series icludig he geomeric series. We wa here o eed his discussio o he geeral
Pure Math 30: Explained!
ure Mah : Explaied! www.puremah.com 6 Logarihms Lesso ar Basic Expoeial Applicaios Expoeial Growh & Decay: Siuaios followig his ype of chage ca be modeled usig he formula: (b) A = Fuure Amou A o = iial
2007 Spring VLSI Design Mid-term Exam 2:20-4:20pm, 2007/05/11
7 ri VLI esi Mid-erm xam :-4:m, 7/5/11 efieτ R, where R ad deoe he chael resisace ad he ae caaciace of a ui MO ( W / L μm 1μm ), resecively., he chael resisace of a ui PMO, is wo R P imes R. i.e., R R.
C(p, ) 13 N. Nuclear reactions generate energy create new isotopes and elements. Notation for stellar rates: p 12
Iroducio o sellar reacio raes Nuclear reacios geerae eergy creae ew isoopes ad elemes Noaio for sellar raes: p C 3 N C(p,) 3 N The heavier arge ucleus (Lab: arge) he ligher icomig projecile (Lab: beam)
Linear Time Invariant Systems
1 Liear Time Ivaria Sysems Oulie We will show ha he oupu equals he covoluio bewee he ipu ad he ui impulse respose: sysem for a discree-ime, for a coiuous-ime sysdem, y x h y x h 2 Discree Time LTI Sysems
A Two-Level Quantum Analysis of ERP Data for Mock-Interrogation Trials. Michael Schillaci Jennifer Vendemia Robert Buzan Eric Green
A Two-Level Quaum Aalysis of ERP Daa for Mock-Ierrogaio Trials Michael Schillaci Jeifer Vedemia Rober Buza Eric Gree Oulie Experimeal Paradigm 4 Low Workload; Sigle Sessio; 39 8 High Workload; Muliple
Week 8 Lecture 3: Problems 49, 50 Fourier analysis Courseware pp (don t look at French very confusing look in the Courseware instead)
Week 8 Lecure 3: Problems 49, 5 Fourier lysis Coursewre pp 6-7 (do look Frech very cofusig look i he Coursewre ised) Fourier lysis ivolves ddig wves d heir hrmoics, so i would hve urlly followed fer he