Class Notes 1: Introduction. MAE 82 Engineering Mathematics

Transcription

1 Class Noes 1: Inroducion MAE 82 Engineering Mahemaics

2 CHANGE

3 Rae of Change

4 Basic Mahemaical Models Man of he principles or laws underling he behavior of he naural world are saemens or relaions involving raes a which hings happen. Principles / Laws Relaions -> Equaions Raes -> Derivaive A differenial equaion ha describes a phsical process is ofen called a mahemaical model.

5 Air Moion - Air Flow - Wing

6

7 Air Moion Hurrian Karina

8 Air Moion Hurrian Karina

9 Waer Moion - Fluid Flow - Waves

10 Waer Moion - Fluid Flow - Waves 2004 Indian Ocean earhquake and sunami

11 Solid Flow

12 Hea Dissipaion Human Bod

13 Populaion Dnamics

14 Modeling Tpes Model Qualiaive Quaniaive (Mahemaical) Saisical Qualiaive Transfer Funcion Qualiaive Process Descripion Fuzz Logic Black Box Mechanisic Probalisic Correlaion Liner Non - Liner Lumped Parameers Disribued Parameers Transfer Funcion Time Series Time Series Neural Nework Liner Non - Linear Liner Non - Linear

15 Modeling Process Newon Euler Von Karman Ohm Kelvin Verhuls Maxell Releigh Navier Sockes Heaviside Einsein Schrodinger Real World Phenomena Model Prep Dependen/ Independen Variable Smbols Unies Principles / Law Mahemaical Represenaion e.g. Diff Eq. Daa Observaions Experimens Measuremens Refine Model Analsis Compuing Graphics Compare Model Predicion wih Daa Qualiaive Quaniaive Perdiion Mahemaical Inerface

16 Consrucing Mahemaical Models 1. Independen and dependen variables - Idenif independen and dependen variables and assign leers o represen hem. 2. Unies - Choose he unis of measure for each variable. 3. Basic Principle - Ariculae he basic principle ha underlies or governs he problem ou are invesigaing. This requires our being familiar wih he field in which he problem originaes. 4. Mahemaical Expression - Express he principle or law in he previous sep in erms of he variables idenified a he sar. This ma involve he use of inermediae variables relaed o he primar variables. 5. Unies Unificaion - Make sure each erm of our equaion has he same phsical unis. 6. Equaions / Se of Equaions - The resul ma involve one or more differenial equaions.

17 Example 1 Free Fall Paricle Modeling Formulae a differenial equaion describing moion of an objec falling in he amosphere near sea level. Variables / Unis: Independen Variable Time () [sec] Dependen Variable Veloci (v) [m/s] Basic Principle Newon s Second Law Law F m - Mass [kg] a Acceleraion [m/s 2 ] v Veloci [m/s] x posiion [m] F Force [N] ma dv m d 2 d x m 2 d

18 Example 1 Free Fall Paricle Modeling Free Bod Diagram Mahemaical Model Solve he equaion and find v=v() ha saisfied

19 Example 1 Free Fall Paricle Formulae a differenial equaion describing moion of an objec falling in he amosphere near sea level. Variables: ime veloci v Newon s 2 nd Law: F = ma = m(dv/d) ne force Force of gravi: F = mg downward force Force of air resisance: F = v upward force Then Taking g = 9.8 m/sec 2 m = 10 kg = 2 kg/sec we obain m dv d dv d mg v v

20 Example 1 Free Fall Paricle Direcion Field v v Using differenial equaion and able plo slopes (esimaes) on axes below. The resuling graph is called a direcion field. (Noe ha values of v do no depend on.) v v'

21 Example 1 Free Fall Paricle Direcion Field v v When graphing direcion fields be sure o use an appropriae window in order o displa all equilibrium soluions and relevan soluion behavior.

22 Example 1 Free Fall Paricle Equilibrium Soluion Arrows give angen lines o soluion curves and indicae where soln is increasing & decreasing (and b how much). Horizonal soluion curves are called equilibrium soluions. Use he graph below o solve for equilibrium soluion and hen deermine analicall b seing v' = 0. v v Se v 0 : v v v

23 Example 2 Mice & Owls Populaions: Mice (pra) / Owls (Predaors) Mice (pra) - A mouse populaion reproduces a a rae proporional o he curren populaion wih a rae consan equal o 0.5 mice/monh Wrie a differenial equaion describing mouse populaion assuming no owls presen. Time [monh] Populaion p() Growh Rae (Rae Consan) r = 0.5 [1/monh] Owls (Predaor) - When owls are presen he ea he mice. Suppose ha he owls ea 15 per da (average). Wrie a differenial equaion describing mouse populaion in he presence of owls. (Assume ha here are 30 das in a monh.)

24 Example 2 Mice & Owls Direcional Field Soluion curve behavior and equilibrium soluion p 0.5 p 450

25 Example 1 & 2 Direcional Field Soluion Behavior v 0.2v 9.8 p 0.5 p 450

26 dv d Soluion of he General Equaion Free Fall v g m Soluion Converge Araced b Equilibrium Limiaions: Free Fall No Obsracion d d a b Equilirium p v b / a k / r mg Mice/Owls dp d rp k Soluion Diverge Repelled b Equilibrium Limiaions: Realisic wihin limied ime

27 Soluion of he General Equaion dv d dp d v g m rp k a -b a b Differenial Equaion d a b d Iniial Condiion ( 0) 0 If:

28 Soluion of he General Equaion Inegrae boh sides From Calculus Take he exponenial of boh sides

29 Soluion of he General Equaion a ce ~ b / a c~ e c Where (Consan) Noe: when c ~ 0 b / a Equilibrium Iniial Condiion ( 0) 0 A 0 c~ b / a c~ 0 b / a a b / ae b / a a 0 0 For General Soluion ce ~ a b / a Inegral Curve Each inegral curve is associaed wih a paricular value of c ~

30 Soluion of he General Equaion

31 Soluion of he General Equaion Saisfing an iniial condiion Idenif he inegral curve ha passes hrough he given Iniial Condiion (I.C.) / 2 /5 p ce 900 v ce 49

32 Linearizaion of a Non Linear ODE Example

33 Linearizaion of a Non Linear ODE Example

34 Linearizaion of a Non Linear ODE Example

35 Classificaion of Differenial Equaions Ordinar differenial equaions (ODE). When he unknown funcion depends on a single independen variable onl ordinar derivaives appear in he equaion. The equaions discussed in he preceding wo secions are ordinar differenial equaions. For example dv dp v 0.5 p 450 d d d a b d 2 d Q( ) dq( ) 1 L R Q( ) E( ) 2 d d C

36 Classificaion of Differenial Equaions Parial Differenial Equaions (PDE) When he unknown funcion depends on several independen variables parial derivaives appear in he equaion. For example a Phsical consan u independen variable depends on x (wave equaion) ) ( ) ( (hea equaion) ) ( ) ( x u x x u a x u x x u

37 Classificaion of Differenial Equaions Ssem of Differenial Equaions One Unknown One equaion Two or more unknown Ssem of equaions For example predaor-pre equaions have he form x() Pre () - Predaor du / d dv / d a x x c u where x() and () are he respecive populaions of pre and predaor species. The consans a c depend on he paricular species being sudied. Ssems of equaions are discussed in Chaper 7.

38 Classificaion of Differenial Equaions Block Diagram Differenial Equaions Linear Differenial Equaions Non Linear Differenial Equaions Parial Linear Differenial Equaions Ordinar Linear Differenial Equaions Variable Coefficiens Consan Coefficiens

39 Order of Classificaion The order of a differenial equaion is he order of he highes derivaive ha appears in he equaion. Examples: We will be suding differenial equaions for which he highes derivaive can be isolaed: d d 1 e 4 2 d d u u sin xx ( n ) ( ) f ( n 1)

40 Linear & Non Linear ODE An ordinar differenial equaion is linear if F is linear in he variables Thus he general linear ODE has he form 0 ) ( n F ) ( n ) ( ) ( ) ( ) ( 1) ( 1 ) ( 0 g a a a n n n

41 u u u uu u d d d d e xx xx cos ) sin( sin Linear & Non Linear ODE Examples

42 Soluion o Differenial Equaions A soluion () o an ordinar differenial equaion saisfies he equaion: Example: Verif he following soluions of he ODE sin 2 ) ( cos ) ( sin ) ( ; ) ( ) ( ) ( n n f 1) ( ) ( ) ( n n f

43 Soluion o Differenial Equaions Three imporan quesions in he sud of differenial equaions: Is here a soluion? (Exisence) If here is a soluion is i unique? (Uniqueness) If here is a soluion how do we find i? (Analical Soluion Numerical Approximaion ec)

44 Soluion o Differenial Equaions A soluion of he ordinar differenial equaion on he inerval is a f ( ( n) ( n1) funcion such ha ( n) ( ) (n ) ( f ( ( ) ( ) ) exis and saisf n1) ( )) for ever in α β Unlesssaedoherwisewe assumeha he funcionf is a real value funcionandwe are ineresedinobainingreal values soluionsof φ()

45 Soluion o Differenial Equaions 1) Is here a soluion?(exisence) - I is answeredb heoremssaing ha under cerain resricions on he funcion f in ( f( ( n ) n 1) alwas hasa soluion Reasons A. If a problemhasno soluionwe wouldprefer o know ha fac before invesing ime and effor ina vain aemp o solve o problem B. Diff eq.of phsicalmodel- somehingis wrong wih he formulaion - Check he validi of he mahemaicalmodel )

46 2) If 3)If of - Soluion o Differenial Equaions here isa soluionisi unique?(uniqueness) A. B.If - Even One soluion can be sure ha compleelsolved he problem here ismore ha more here is a soluionhow do we find i? (Analical soluionnumericalapproximaionec.) ma be ha hough we ma know ha he soluionisno he usual.elemenar funcions - polnomial rigonomeric exponeniallogarihmic Commonsiuaionfor one soluionconinue o search andhperbolic funcions mosof we have a soluionexissi expressible in erm he diff. eq.