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
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
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
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
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
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 / 5 5 5-5 - -5 - -5 Udaped ored Osiallaios = = 5 5 75 5 5 75 5 5 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
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 /.5.5.5.5.5 5 5.5.5.5.5 / 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