UNIT VIII INVERSE LAPLACE TRANSFORMS. is called as the inverse Laplace transform of f and is written as ). Here

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1 UNIT VIII INVERSE APACE TRANSFORMS Sppo } { h i clld h ivr plc rorm o d i wri } {. Hr do h ivr plc rorm. Th ivr plc rorm giv blow ollow oc rom h rl o plc rorm, did rlir. i co 6 ih 7 coh 8...,,! 9! b b b i b b co b b b ih b b coh Pril rcio: b B A b b c b B A b c b c B A c b

2 Exmpl: Fid h ivr plc rorm o. 8 i. co i co. i co i co.!!!! 6

3 .!. 6 i 6. 8 i co i co 7.

4 [ co ½ i ]. 8.!! 9. 6 i c / 6 / coh / ih / 6 / coh / ih /

5 . Fid h ivr plc rorm o c B A C B A Pig,, d rpcivl w g, 9, 9 C B A Fid h ivr plc rorm o c B A C B A Pig,, d rpcivl w g 7,, 6 C B A 9 9. Fid c B A 6 C B A

6 Pig,, d rpcivl w g,, C B A Fid., i i i ih i i Th Fid h Ivr.T o h ollowig cio: Evlio o } { F B Hviid hiig horm w hv i } { F h } { F H } { H F Fid h Ivr plc Trorm o:

7 . Fid W hv,, Eqio bcom... Fid π π π π π. W hv π π co, π π π i Hc qio bcom i / / co π π i / i π π / i π W hll ir id } {

8 ., i g w! i g U 6!! Nx w hv [ ] 6 Th. Hr F {F} { F } H H.. π. Hr F {F} i π { π F } π H π i π H π i π H π

9 i π H π i H π Evlio o F & d d F. I { } F F Th d F d { F } d I { } F h d { } { F } d d { F } d { F } Fid h ivr plc rorm o h ollowig cio:. F {F} d.

10 . F i d F d co i co co. Fid h ivr plc rorm o F i d F co i d d i co i d i co co Th

11 . i x x W kow h i x x!! i 6 i 6 i 6! i d i 6 Evl:. {log log. log Th log {} F d d Now { F} [ log log ] d d d { F} d co co co. 6. log b W h orml df d

12 b d d log log log b d d b b b 7. log Sol: log log h { log d d { [ ] { } log log d d log log coh coh coh

13 coh log 8. Evl W kow h i d d hh w Now d d i i i 9. Prov h ih i h { } F { } d d F d d Coidr { } F d d

14 { } F d d [ ] i i i i ih Exrci: log. 6. co

15 Covolio Diiio: Th covolio o wo cio d g ll dod * g i dod i h orm o igrl ollow. * g g d Propr: * g g * Th i o h h covolio oprio * i commiv. Covolio horm I [ ] d [ g ] g [. g ] g d h Proo: W hll how h g d. g W hv.h.s b h diiio g d g d g dd g dd W hll chg h ordr o igrio i rpc o hi dobl igrl. Exiig rgio: o o O chgig h ordr: o Horizol rip Vricl rip Vricl rip o Horizol rip O chgig h ordr o igrio, bcom g dd Now, l p v whr i ixd d dv

16 I, v ; I, v d hc bcom v v g v dvd d. v. g R.H.S Hc w hv provd h g d. g Th [. g ] g d Thi prov h covolio horm. Vriicio o h covolio horm Workig procdr or problm v g v dv W d o vri h horm i rpc o h wo giv cio d g. W id [ ] d g [ g ]. W vl * g g d. W id [ * g ].. I [ * g ]. g h w c cocld h h horm i vriid. Ex. Vri covolio horm or h pir o h cio i d g Sol: * g g d i d i, g i d i co * g i co i co

17 * g ] [ Alo. g Th [ * g ]. g. Th horm i vriid. Ex. Vri covolio horm or h pir o h cio co d g cob Sol. co, g cob b * g * g g d [co bb co cob b d cob b ] d i b b i bbb * g b b i i b i i b b b { } { } i i b b b b b b * g i i b b b i bi b, b b b [ * g ]. b b b.., b i [ * g ]. b b b

18 Alo. g b Th [ * g ]. g. Th horm i vriid. Tp Compio o h ivr rorm b ig covolio horm Workig procdr or problm. Th giv cio i xprd h prodc o wo cio d g. W id [ ] d [ g ] g. W ppl h covolio horm i o o h orm: [. g ] g d. W vl h covolio igrl o obi h rqird ivr. Ex. Uig covolio horm obi h ivr plc rorm o h cio Sol: ; g Tkig ivr, ; g W hv covolio horm, [. g ] g d i. d co co i

19 co Th Ex. Uig covolio horm obi h ivr plc rorm o h cio Sol: g ; g ; g ; i g ; co Now b pplig covolio horm w hv, d i co d co i d ] i [i d ] i [i co ] [ i co co i

20 i Th Ex. Uig covolio horm obi h ivr plc rorm o h cio Sol:. ; g g co ; i Now b pplig covolio horm w hv, d co i d i i d i i co i co co i i Tp plc rorm o h covolio igrl d olio o igrl qio Workig procdr or problm. plc rorm o h covolio igrl b ig h rl

21 d g g d g. i h ppropri orm.. Giv qio or ivolvig h covolio igrl w ir k plc rorm o boh id. W vl h covolio igrl d impli o obi ] [ cio o. Tkig ivr w obi Ex: Fid rom h qio o d Sol: Tkig plc rorm o boh id w hv, [ ] [ ] o d o d. g [ ] g g Hc qio bcom. or B kig ivr w hv, [ ] Th Ex: Solv h igrl qio i o d Sol: Tkig plc rorm o boh id w hv,

22 [ ] [ ] i o d. g g i g or i g whr Hc qio bcom or. or., i [ ] Now Th

23 APPICATION OF APACE TRANSFORMS: SOUTION OF DIFFERENTIA EQUATIONS USING APACE TRANSFORMS: Th plc rorm mhod o olvig diril qio giv priclr olio wiho h ci o ir idig h grl olio d h vlig h rbirr co. Thi mhod i pcill l or olvig lir diril qio wih co coici. Workig Procdr Workig Procdr o olv lir diril qio wih co coici b rorm mhod. plc rorm h o b k o boh id o h diril qio, ig h orml o driviv d h giv iiil codiio. All h rm wih giv ig r rpod o righ. Divid b h coici o [ ], gig [ ] Rolv hi cio o Rol hi cio o io pril rcio Tk h ivr plc rorm o boh id. kow cio o. Th w g cio o which i h rqird olio iig h giv codiio. No: W h ollowig h orml o driviv i vr xmpl. [ ] { } { } { }

24 Ex: Solv b h mhod o plc rorm h qio., giv d d d d Sol: Th giv diril qio i Tkig plc rorm o boh id } { } { } { } { } { ] [ Tkig Ivr plc rorm o boh id } { Ex: Solv b h mhod o plc rorm h qio, giv h 8 9 π d d Sol: Giv diril qio i

25 9 8 Tkig plc rorm o boh id { } 9{ } 8{ } 8 9 Tk k d 9 8 k 8 9 k 9 Tkig Ivr plc rorm o boh id { } 8 k k 9 k i i π π wh π π k π i i k π 9 π i i π i i h rqird olio d Ex: d d d d Solv giv,, x dx dx dx dx dx b plc rorm mhod. Sol: Giv diril qio i

26 x x x x Tkig plc rorm o boh id } { } { } { } { x x x x } { ] [ ] [ Tkig Ivr plc rorm o boh id } { C B A i C B A Pig S,, i i w obi A/, B d C / } { x x x x

27 SIMUTANEOUS DIFFRENIA EQUATIONS Egirig xprim ivolvig wo or mor dpd vribl d ol o idpd vribl rl i imlo diril qio which c b olvd b pplig plc rorm. Ex: Solv h imlo qio. dx d i ; x co d d giv h x,. Sol: Th giv qio r x i ; x co Tkig h plc rorm o boh id x x x Uig h giv iiil codiio w obi, x x x x Mliplig b d h brc rom x x Tkig h ivr plc rorm o boh id { x } x coh Th bi x i qio Tkig h ivr plc rorm o boh id

28 { } i ih x coh, i ih Ex: Solv h qio. dx d dx d x ; x d d d d Sbjc o h codiio x, Sol: Giv qio r x x x x Tkig h plc rorm o boh id x x x x x [ ] x Uig h giv iiil codiio w obi, x x x x x x Mliplig b d b x x Sbrcig, [ ] x

29 x x Tkig h ivr plc rorm o boh id } { x x i co x i

30 SOME APPICATION TO ENGINEERING PROBEMS I hi cio, w illr h o plc rorm b coidrig om xmpl cocd wih h vibrio o rig, dlcio o bm d RC circi. Vibrio o rig: lic rig b chd o ppor d hgig dowwrd. bod o m m b chd o h lowr d o lic rig. Wh h prig d h bod r r i h qilibrim poiio E, ppo h wigh m i plld dowwrd d rld, h h rig vibr i h vricl dircio rl d hr i ric d o h mdim oppoig h movm rlig i dmpd vibrio. I do h dowwrd diplcm o h bod o m rom h qilibrim poiio im, h diril qio or hi modl i md d c d dx k d d whr i h vloci, i h cclrio, c i h dmpig coici, k d d i h prig modl d i h drivig orc. I b h dic b which h bod m i plld dow h h qilibrim poiio bor rlig i, d v b h vloci wih which h bod m i rld. d Th d v d Th r clld iiil codiio or vibrio. Th qio d oghr orm iiil vl problm. Th diplcm c b obid b olvig dr h codiio. I h mdim do o c ric o h vibrio, h w hv dmpd vibrio d i ch c, c Thror bcom md d k I c> h w hv roc dmpd vibrio Th drivig orc xr orc h h vibrio r clld orcd vibrio ohrwi, h r clld impl hrmoic vibrio moio. Dlcio o bm: Coidr horizol iorm bm ppord boh h d, which i lighl b dlc b dowwrd vricl lod w rom horizol poiio. x do h horizol dic o poi P o h bm rom h o d O o h bm. Th h dlcio h poi P dic x rom o d o h bm xpricd d o bdig ii h diril qio

31 d EI w dx Whr E i h modl o lici d I i mom o iri. E d I r ppod o b co. Th co EI i clld lxrl rigidi o h bm. RC Circi: A impl lcric coiig o idcc hr ric R Ohm d cpcic C rd cocd i ri i clld RCcirci. I co lcromoiv orc m E vol i pplid o RC circi h h crr I mpr i h circi im i giv b h diril qio, which dpd po h Kircho lw i., h lgbric m o h volg droprod clod circi i ql o h rl lccromoiv orc i h circi ig h ollowig rlio. dq i d or q id Volg drop cro ric R Ri Volg drop cro di idcc d Volg drop cro q cpcic C C di q Ri E d C Thror lo c b wri d q dq q R E d d C From w c id q Diri w.r. w g d i di dq R E d d C d d i di or R i E d d C From w c id i. Th diril corrpodig o R circi i giv b di Ri E d

32 Ex: Sprig c xdd cm wh. kg m i chd o i. I i pdd vricll rom ppor d io vibrio b pig i dow cm d impoig vloci cm/c vricll pwrd. Fid h diplcm rom i qilibrim poiio im. Sol: Wh m. kg grm. B cm, kig g98cm/c w hv mg X 98 k, d/cm. b Eqio o moio i d x k x d m d x, i., x d d x 9x d Tkig plc rorm o boh id. x x x 9x Iiill, h m dph o cm blow h qilibrim poiio x. Alo h iiil vloci w cm/c. vricll pwrd x 9 x x 9 x 9 x co7 i Ex: A bm i impl ppord i d x d i clmpd h ohr d xl. I crri lod xl/. Fid h rlig dlcio poi. Sol: Th diril qio o dlcio i d x w δ x l / d EI Tkig h plc rorm, w g w { } { δ x l / } EI

33 w l EI / Uig h giv codiio d mig c, c, bcom, c c w EI l / Tkig ivr rorm w g cx cx, < x < l 6 / d w l cx c x x, l / < x < l 6E Uig h codiio, l, l,, bcom, c l cl 6 9wl 8E d c cl 6 9wl E 9wl 8w c, c 6E 8E Sbiig h vl o C d C i w g h dlcio poi. Ex: A vol E i pplid o circi o Idcc d Ric R, how b plc rorm mhod h crr im i R [ ] r E / Sol: Rqird diril qio i di Ri E d Sic i i giv h E E, qio bcom di d Ri E

34 Sic h volg i pplid or >, w ppo h i i.., i Tkig plc rorm o boh h id, w hv R i i i E. Pig i, w g R E i. E i. R E i. R / R i E. R R / Tkig ivr rorm w g i E R R /.[ ] whr > Ex: A pricl i movig wih dmpd moio ccordig o h lw d x dx 6 x. I h iiil poiio o h pricl i x d h iiil d d pd i, id h diplcm o h pricl im ig plc rorm. Sol: Th giv qio i x 6x x Iiil codiio r x, x Th x co 7 /.i Ex: Th diril qio o iorml lodd bm wih impl ppor i d wx EI l x dx Whr EI d w r co. Solv b plc rorm mhod giv h. Ex: A R circi crri m o volg E E i w whr E d w r co. Fid h crr I i h circi, i iiill hr i o crr i h circi.

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S.

Advanced Engineering Mathematics, K.A. Stroud, Dexter J. Booth Engineering Mathematics, H.K. Dass Higher Engineering Mathematics, Dr. B.S. Rfrc: (i) (ii) (iii) Advcd Egirig Mhmic, K.A. Sroud, Dxr J. Booh Egirig Mhmic, H.K. D Highr Egirig Mhmic, Dr. B.S. Grwl Th mhod of m Thi coi of h followig xm wih h giv coribuio o h ol. () Mid-rm xm : 3%