More on ODEs by Laplace Transforms October 30, 2017
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1 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 ranfor Geing a ranfor b inegraion Finding ranfor and invere ranfor fro able and heore pplicaion o differenial equaion Exaple of applicaion o e of hoogenou and nonhoogeneou equaion Review Tranfor efiniion Tranfor fro a funcion of ie, f, o a funcion in a coplex pace, F, where i a coplex variable The ranfor of a funcion, i wrien a F = L f where L denoe he Laplace ranfor ue for L in oe equaion Laplace ranfor defined a he following inegral L [ f ] e f d F Siple Laplace Tranfor f F f F n n!/ n+ e a in x x+/ x+ a e a / a e a co a in / + a co / + ddiional ranfor inh / - in pp 6-67/8-5 of Krezig 9 h / h coh / - ediion Review Tranfor Properie L [af + bf ] = al [f ] + bl [f ] Fir hifing heore If L[f] = F hen L[e a f] = F a Exaple: L[co] = / + o L[e a co] = a/[ a + ] erivaive ranfor where L[f] = F L[df/d] = F f L[d f/d ] = F f f Siilar reul for higher derivaive 5 Solving ifferenial Equaion Tranfor all er in he differenial equaion o ge an algebraic equaion For a differenial equaion in we ge he ranfor = L [] Siilar noaion for oher ranfored funcion in he equaion R = L [r] Solve he algebraic equaion for Obain he invere ranfor for fro able o ge Manipulaion ofen required o ge fro equaion o ranfor in able 6 ME 5 Seinar in Engineering nali Page
2 More on OE b Laplace Tranfor Ocober, 7 Invere Tranforaion Ue ranfor able Ma need parial fracion approach Ue fir hifing heore dicued la Wedneda New ehod o be dicued onigh Ue econd hifing heore Second hifing heore depend on definiion of Heavide uni funcion and irac dela funcion 7 Srange Funcion The Heavide uni funcion, u a i defined o be for < a and for a Repreen ep fro zero o one a x = a Laplace ranfor i e -a / ela funcion, x a i defined uch ha for an vanihingl all, x a = excep for < x a < and he inegral a a x a dx L a x a e 8 Second Shifing Theore pplie o e -a F, where F i nown ranfor of a funcion f Invere ranfor i f a u a where u a i he uni ep funcion For e -a / +, we have e -a F wih F = / + for f = co Thu e -a / + i he Laplace ranfor for co[ a] u a Review Parial Fracion Mehod o conver fracion wih everal facor in denoinaor ino u of individual facor in denoinaor Exaple i F = /+a+b Wrie /+a+b = /+a + /+b Mulipl b +a+b and equae coefficien of lie power of = + b + + a + = for er and = b + a for er 9 Review Parial Fracion II + = for er and = b + a for er Solving for and give = - = /b a Reul: /+a+b = /[b a + a] /[b a + b] So f = [e -a e -b ]/b a Thi acuall ache a able enr Follow ae baic proce for ore coplex fracion Special rule for repeaed facor and coplex facor Review Parial Fracion Rule Repeaed fracion for repeaed facor n n n n n a a a a a oplex facor + i + + i i i Pure iaginar facor i coplex facor wih = i i ME 5 Seinar in Engineering nali Page
3 More on OE b Laplace Tranfor Ocober, 7 ME 5 Seinar in Engineering nali Page Oher pplicaion We can appl hi o a e of equaion for i Tranfor all equaion fro i o i Solve iulaneou algebraic equaion for each i Ge invere ranfor for i Soeie ipler o ge oe i fro differenial equaion afer olving one equaion uing ranfor Se of Equaion Loo a e of wo equaion fro pring-a e olved previoul Have equaion for and Wrie and for L[ ] and L[ ] Equaion fro Ocober lecure d d d d 5 Se of Equaion II Rearrange o how wo iulaneou algebraic equaion in and Gau eliinaion give equaion Eqn Eqn Eqn Eqn 6 Se of Equaion III Mulipl equaion b + + / 7 Se of Equaion IV 8 Se of Equaion V The facor for i he ae a he characeriic equaion obained in Ocober lecure
4 More on OE b Laplace Tranfor Ocober, 7 ME 5 Seinar in Engineering nali Page 9 Se of Equaion VI If all and all are he ae he equaion becoe Se of Equaion VII The er ulipling can be facored a follow So he equaion becoe Se of Equaion VIII Manipulae righ ide of equaion o cobine lie power of Se of Equaion IX Ue parial fracion for ; have wo pure iaginar facor Mulipl b denoinaor Expand and equae lie power of Se of Equaion X Se of Equaion XI The equaion ranfor i he u of ranfor for ine and coine f F f F in co in co
5 More on OE b Laplace Tranfor Ocober, 7 Se of Equaion XII Iniial condiion fro Ocober lecure = a, = a, = = a a Subiue,,, and value ino 5 Se of Equaion XIII a a a a a Soluion wih = a, = = = i co co in in aco 6 aco Se of Equaion XIV Ge fro original differenial equaion afer eing all and o be equal d d d d aco aco and ae a in Ocober noe 7 Oher pplicaion Laplace ranfor are ued o analze differenial equaion for conrol e efine e funcion or ranfer funcion a L[inpu] / L[oupu] for a ingle inpu Ue hi funcion o analze repone o variou inpu eerine abili of conrol e: will a diurbance dap ou? 8 Laplace Tranfor Suar Ue able o ge ranfor fro o and vice vera ifferenial equaion in f and i derivaive becoe algebraic equaion in Solve for and rearrange o ge er ha ou find in ranfor able Ue ranfor able o ge fro Tranfor ehod incorporae nonhoogenou er and iniial condiion 9 Group Exercie For group of - people Ue Laplace ranfor o olve he differenial equaion 9 = e - wih = and = ME 5 Seinar in Engineering nali Page 5
6 More on OE b Laplace Tranfor Ocober, 7 Soluion o Group Exercie Solve 9 = e - wih = and = b Laplace ranfor Tranfor differenial equaion: 9 = / + Subiue iniial condiion and olve reul for 9 = / + 9 = + / + Soluion o Group Exercie II 9 = + / Ue parial fracion for la er 9 9 Se u of lie power o zero Soluion o Group Exercie III 9 er: er: er: 9 equaion give = Subiuing = ino equaion give + + = or = = / Subiue = and = ino equaion o ge = 9 Soluion o Group Exercie IV Fro = / and = : = / Fro = / and = -: = -/ 9 9 Fro ranfor able inh hec Soluion for OE Plug oluion ino original differenial equaion: 9 = e - inh coh 6 6inh 9 8 hec oundar ondiion oundar condiion: = ; = inh inh coh 6 9 6inh inh 5 coh ME 5 Seinar in Engineering nali Page 6
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