LAPLACE TRANSFORMS AND THEIR APPLICATIONS

Transcription

1 APACE TRANSFORMS AND THEIR APPICATIONS. INTRODUCTION Thi ubjc w nuncid fir by Englih Enginr Olivr Hviid (85 95) from oprionl mhod whil udying om lcricl nginring problm. Howvr, Hviid` rmn w no vry ymic nd lckd rigour which lr on ndd o nd rcpiuld by Bromwich nd Cron. plc rnform coniu n imporn ool in olving linr ordinry nd pril diffrnil quion wih conn cofficin undr uibl iniil nd boundry condiion wih fir finding h gnrl oluion nd hn vluing from i h rbirry conn. plc rnform whn pplid o ny ingl or ym of linr ordinry diffrnil quion, convr i ino mr lgbric mnipulion. In c of pril diffrnil quion involving wo indpndn vribl, lplc rnform i pplid o on of h vribl nd h ruling diffrnil quion in h cond vribl i hn olvd by h uul mhod of ordinry diffrnil quion. Thrfr, invr plc rnform of h ruling quion giv h oluion of h givn p.d.. Anohr imporn pplicion of plc Trnform i in finding h oluion of Mhmicl Modl of phyicl problm whr in h righ hnd of h diffrnil quion involv driving forc which i ihr diconinuou or c for hor im only. Dfiniion: f() b funcion dfind for ll. Thn h ingrl, { } f fd if xi, i clld plc Trnform of f(). i prmr, my b rl or complx numbr. Clrly {f()} bing funcion of i brifly wrin fif ().. { ()} f (). Hr h ymbol which rnform f() ino f () i clld plc Trnform Opror. Thn f() i clld invr plc rnform of f () or imply invr rnform of fi ().. { f ()} 755. No: Thr r wo yp of lplc rnform. Th bov form of ingrl i known on idd or unilrl rnform. Howvr, if h rnform i dfind f { } fd, whr i complx vribl, clld wo idd or bilrl lplc of f(), providd h ingrl xi. If in h fir yp rnform, vribl i complx numbr wih rul h

2 756 Enginring Mhmic hrough Applicion lplc rnform i dfind ovr porion of complx pln. If {f()} xi for rl nd hn {f()} xi in hlf of h complx pln in which R > (Fig..). Th rnform f () i n nlyic funcion wih propri: (i) f() viz. h ncry condiion for f () o b rnform. R. (ii). f A, if h originl funcion h limi. f() A Imgn. o Fig..: Rl xi. EXISTENCE CONDITIONS Th lplc rnform do no xi for ll funcion. If i xi, i i uniquly drmind. For xinc of lplc, h givn funcion h o b coninuou on vry fini inrvl nd of xponnil ordr i.. if hr xi poiiv conn M nd uch h f () M for ll. Th funcion f() i om im rmd objc funcion dfind for ll nd f () i rmd h ruln img funcion. Hr h prmr hould b ufficinly lrg o mk h ingrl convrgn. In h bov dicuion, h condiion for xinc of f () i ufficin bu no ncry, which prcily mn h if h bov condiion r ifid, h lplc rnform of f() mu xi. Bu if h condiion r no ifid, h lplc rnform my or my no xi. For.g. in c of f (), f (), prcily mn f () i no picwi coninuou on vry fini inrvl in h rng. Howvr, f() i ingrbl from o ny poiiv vlu, y. Alo f () M for ll > wih M nd. Thu, f() π, > xi vn if i no picwi coninuou in h rng.. EXISTENCE THEOREM ON APACE TRANSFORM If f() i funcion which i picwi* coninuou on vry fini inrvl in h rng nd ify f () M for ll nd for om poiiv conn nd M mn, f() i of xponnil ordr, hn h lplc rnform of f() i.. f() d xi.

3 plc Trnform nd hir Applicion 757 { } () () () Proof: W hv f () f d f d+ f d () Hr f () d xi inc f() i picwi coninuou on vry fini inrvl. Now () () f d f d M d, inc f () M ( ) ( ) Md M ; > ( ) () ( ) Bu M cn b md mll w pl by mking ( ) ufficinly lrg. Thu, from (), w conclud h {f()} xi for ll >. *No: A funcion i id o b picwi (cionlly) coninuou on clod inrvl [, b], if hi clod inrvl cn b dividd ino fini numbr of ubinrvl in ch of which f() i coninuou nd h fini lf hnd nd righ hnd limi. A funcion i id o b of xponnil ordr ( > ), if hr xi fini poiiv conn nd M uch h f() M or f() M for ll..4 TRANSFORMS OF EEMENTARY FUNCTIONS By dirc u of dfiniion, w find h lplc rnform of om of h impl funcion:. (), ( > ). ( > ). n n!, whr Γ ( n+ ) n,,,, ohrwi n + 4. (in ), + 5. (co ), + 6. (inh ), 7. (coh ), Proof:. () d, > n + ( > ) ( > ) ( > ) ( > )

4 758 Enginring Mhmic hrough Applicion ( ) d d, > ( ) ( ) ο. ( ) Rmrk: Hr h condiion > i ncry o nur h convrgnc of h ingrl inc ohrwi divrgn for.. n n n p p dp d on king p, d dp p n p ( n+ ) Γ ( n+ ) p dp p dp, n+ n+ n+ if n > nd > in in in co d, > ο co co co in d, > ο d d d 6. ( inh inh ) ( + ) + > + + d d d 7. ( coh coh ) ( + + ).,. + > +.5 PROPERTIES OF APACE TRANSFORMS Sr. No. Propry f(), funcion f, () lplc rnform. inriy f() + bg() ch() f() + bg () ch (). Chng of cl f() f, >. Iniil vlu f () f() 4. Finl vlu f () f() 5. Fir hifing f() f ( ) 6. Scond hifing f( )u( ) f()

5 plc Trnform nd hir Applicion 759 d 7. Driviv f () d d d f () n d n d f () f() f(), > f f() f'(), > n n n n f() f() f () f (), > 8. Ingrl fudu f ( du) f du n f (), > f () () f n, > for ny poiiv ingr n. d 9. Muliplicion by f() f () d f() ( ) n f() d fs d n n d f n d. Diviion by f () f () f () n fd (), providd h ingrl xi. f ()( d ), providd h ngrl xi. f ()( d ) n, providd h ingrl xi. Convoluion f()*g(), fg () () Thorm whr f * g fug ( udu ),. Priodic Funcion f( + T) f() T f() d T

6 76 Enginring Mhmic hrough Applicion Proof of Som of h propri r kn in ubqun dicuion. P inriy Propry: f(), g(), h() b ny funcion who lplc rnform xi, hn for ny conn, b, c nd k, w hv (i) [f() + bg() hf()] [f()] + b[g()] c[h()] (ii) [kf()] k[f()]. Proof: (i) (ii) HS f() + b g() ch() d. [ ] f() d + b g() d c h() d f () + b g () c h() Thi rul cn ily b gnrlizd. HS kf() kf() d k f() d kf () In viw of bov dicuion, plc opror i linr in nur. P Fir Shif Propry: [KUK, ] If r f() f() hn f() f ( ) ( ) p f () f () d fd () fd () fp, whr p Thu, if w know h rnform of f(), w cn wri h rnform of f() imply rplcing by ( ) in f () f () f () Trnlion long xi f ( ) Fig.. Applicion of hi propry giv ri o following impl rul:. ( ), ()

7 plc Trnform nd hir Applicion 76 n.. ( in b) 4. ( co) n!, n + ( ) b, + b 5. ( inh b) 6. ( cohb) + b b b, b whr in ch c >.,, n! n n + in( b) b + b + b ( cob) b b ( inhb) b ( cohb) P If Chng of Scl: f() f() hn f ( ) f dp f( ) f( ) d f( p) on puing p, d p dp ( ) p f() p dp f. No: Chnging o, h rul chng o f f( ) P 4 Chng of Scl wih Shifing: b b f f f f If () (), hn ( ) b b ( b) f( ) f( ) d f( ) d on king p, d dp b f() p dp f b p Exmpl : Find h lplc rnform of h following: (i) in co (ii) in (iii) in (iv) in (iv) coh inb (v) in inb

8 76 Enginring Mhmic hrough Applicion Soluion: in co in 5 in in 5 in (i) 5, > ( 5) (ii) ( ) (in ) in in6, Uing inθinθ in θ 4 4 (iii) ( in ) ( in ) ( in6) , > ! 5! 5 7 Γ Γ Γ 5 + 7! 5! 5 π π π π + + π 4 4! 4! 4 co (iv) ( in ) which comprbl o ( f ) f( ) co () co + 4 whr ( in ) ( + ) ( ) ( + + 5) ( + )( + ) ( + )

9 plc Trnform nd hir Applicion 76 ( + )( + + 5) + (v) ( coh in ) in ( in ) + ( in ) in, + > ; hn by hif propry, ( coh in ) + ( ) + ( + ) + Bu { + + } + {( + ) } { } {( ) } ( + ) coh whn coh,. > Alrnly: Finding h imginry pr of i { } i i i i (vi) i i ( in in b) inb ( inb) ( inb) b in b, + b >, hn by hif propry, Bu b b ( in inb) i ( i) + b ( + i) + b b i i+ b + i+ b b i { + ( b )} i + ( b ) + i { } ({ + ( b )} + i) ({ + ( b )} i) { + ( )} b i b 4i b 4i i 4 { + b + ( b )} + 4

10 764 Enginring Mhmic hrough Applicion b + b + b + 4 b ( + ( b+ ) ) + ( b ) in inb co b co b, co(a + B) Alrnly: ( ) ( + ) uing ina inb co(a B) Exmpl : Find h plc rnform of (i) (iii)*, < < f (), >, whn () T < < T f, whn > T (ii) { in (), < <π f, >π [Mdr, 5] (iv) ( π ) co, > f (), π < π *[Krl Tch. 5, NIT Kurukhr, ] (v) f () + +, o [NIT Kurukhr, ] Soluion: (i) For funcion f() dfind for ll, lplc rnform i dfind : () () (), f f f d Providd h hi ingrl xi. ( ) ( ) ( + ) d & od d ( ) ( ) i ny prmr, my hv rl or complx vlu. (ii) { in (), < < π f, >π By dfiniion, ( f() ) f() d I( y) π π I ind+ d ind π π π (. co) ( )( co) d

11 plc Trnform nd hir Applicion 765 π π π ( + ) ( in ) (. ) ind ο π π I ( + ) ind (Sinc vlu of h xprion in i zro boh h limi) I ( π + ) I or I( + ) ( + π ) π + I f() + Hnc h rul. (iii) Hr, whn < < T f () T, whn > T T T f () fd () fd () + fd () d+ d T T T T d + T ο o T T T + T T T ( ) T T T + T π π co, > (iv) Hr f () π, < T T T ( ) ( ) π π π T (Ingrion by pr) Hnc h rul π f () fd () fd () fd () co + d Now ingr by pr king co ( π/) nd funcion nd funcion. π π π I in ( ) in d π π in d π (Sinc h vlu of h xprion π in i zro boh h limi). π π I co ( ) co d π π

12 766 Enginring Mhmic hrough Applicion π I d π + co co π π π I I or I( + ) I f() π ( + ) Hnc h rul. (v) Hr f () + +, Clrly f() for for > Sy, if, if, if, i.. for ll vlu of vrying from o, vlu of + + i lwy. ikwi for >, i i. Hnc, f() f() d d+ d + d ( ) + ( ) + d ( ) P 5. Trnform of Funcion Muliplid by n If f() i funcion of cl A nd if f() f(), hn n d n n d f ( ()) f () nd f ( ()) f () n d d Hnc h rul.

13 plc Trnform nd hir Applicion 767 By dfiniion of lplc of f(), w hv f() d f() () d Diffrniing () wih rpc o, d f() d f() () d d By ibniz rul of diffrniion undr ingrl ign, () rduc o d f() d f() d d ( ) f() d f() or d d f() d f() d Thi prov h horm for n (i.. fir rul) Now um h horm i ru for n m (y), o h () giv m m m d ( f() ) d ( ) f() m d d hn m+ m m d m+ d f() d f() d Agin by ibniz rul, (5) rduc o m+ m m+ m d ( ) m+ d f () d f () d f () () (4) (5) m or m+ + m+ d m+ d f() d f() (6) Thi clrly how h if h horm i ru for n m, i i ru for n m + i.. for n +, n +, o on. Hnc h horm i ru for ll poiiv ingr vlu n. No: A funcion which i picwi or cionlly coninuou on vry fini inrvl in h rng i of xponnil ordr ( i... f() ) i known funcion of cl A. Exmpl : Find plc of (i)* (in ) (ii) co (iii) ** inhco [* KUK, ; ** NIT Kurukhr, 5] Soluion: n n! nd d n ( f ()) f () + d d 4 4 in. d (i) ( in ) ( co ) ( co) A { } ( + )

14 768 Enginring Mhmic hrough Applicion d co f (), whr f () co, > d + (ii) ( co ) ( + ). ( + ) ( + ) d d d d + d d co co 4 (iii) ( + inh co ) ( ) ( + ) ( ) ( co + co) 4 ( ) ( ) (uing fir hif propry, rplc by ) { } { } { }{ } P 6 Trnform of Funcion Dividd By f () If f f hn f d, providd h ingrl xi. Proof: W hv () () () f f fd Ingring h bov quion wih rpc o bwn h limi nd, w g f d f d d f d d (On umpion h h chng of ordr of ingrion xi). f () f () d d Sinc lim, > o x () () () o. f () fd ()

15 plc Trnform nd hir Applicion 769 f () Rmrk: Sinc corrpond o h ingrion of lplc rnform of f() wih rpc o bwn f () h limi nd, hrfor h rpd pplicion of h rul giv f()( d) n n providd for 4444 nim f () poiiv ingr n, lim x n xi. co Exmpl 4: Find h lplc rnform of. [Omni, ] Soluion: Hr, Agin co d, inc + (), co log ( + ) log log + log + log log +, + log + A, whn + + f () log d log d co d + log d + d, ingrion by pr + + log + d + + log + + d log + + n π log+ + n co log+ in Exmpl 5: Prov h n rnform of co xi? nd hnc find in. Do h lplc

16 77 Enginring Mhmic hrough Applicion f () in f () in, l l nd in f ()( y) o o + Soluion: For in π f d d n n co n + Now in in n n (Uing chng of cl propry, co f,( y), hn + Agin, f( ) f, > ) ( log ). log log co d l + Which do no xi l log ( + ) i infini. Hnc co do no xi. Exmpl 6: Uing plc Trnform, how h in d π Soluion: W know h [in( )], > + π d n n co + in () Howvr by dfiniion, From () nd (), in d co in in d () On king nd in h bov rlion, w g P 7 plc Trnform of Ingrl of f() in d π co f() u du f() whr f() f()

17 plc Trnform nd hir Applicion 77 φ () f( u) du nd φ () hn φ '() f() W know h, [ ] ( ) i.. [ ] φ ' φ φ() φ () φ(), in c φ () On puing h vlu of φ() nd φ (), w g i.. [ φ () ] ( φ () ) f() u du f() f() o Rmrk: If h bov prpoiion hold, hn h invr rnform i.. f( u) du f or fudu f lo hold. P 8 plc Trnform of Driviv f() b coninuou funcion for ll nd b of xponnil ordr, nd f () if lo b pic wi coninuou for ll nd b of xponnil ordr, hn [(f ()] (f()) f(). ( f () ) f () d () ( ). () lim () () + () o f f d f f f Now, uming f() o b uch h f() (Whn hi condiion i ifid, f() i id o b of xponnil ordr ). f () f() f() Thrfor, w conclud h (f ()) xi nd Rmrk: By ucciv pplicion of h bov rul, w obin h lplc rnform of n f () follow: fn nf nf() nf () fn(), if f (), f(),, fn () r coninuou, fn() i picwi coninuou for ll nd if f (), f (),, fn() ll r of xponnil ordr, whn. Exmpl 7: Find (co) nd dduc from i (in) [KUK, ] Soluion: By df n., ( co ) i i + d, co ( i) ( + i) d+ d ( i ) ( + i) + ( i) + i o o i + i

18 77 Enginring Mhmic hrough Applicion + ( i) ( + i), i i () + Now o dduc (in ) uing (co ), wri, in cod () Tk lplc on boh id, ( in ) cod () f () By formul, f() d, whr f() f() + ( in ) + + d in co f d Alrnly: Wri, Tk lplc on boh id, Now by h formul, (4) in f' (5) f f () f(), whr f + nd f() (co ) Uing bov, (5) bcom in Hnc h rul. (6) Exmpl 8: Find in nd hnc dduc co. [KUK, ] d co Soluion: in f, hn f (in ) d or π f f f f Bu (), whr ( in ) 4 co π.. f() in co π 4 4

19 plc Trnform nd Thir Applicion 77 Exmpl 9: Evlu in d Soluion: (), whr () (in) in f d f + π d n n co + in co ( ), by fir hif propry Now in f () co ( ) d P 9 Evluion of Ingrl by plc Trnform Exmpl : Evlu (i) Soluion: in d (ii) ind OR Find lplc of (i) in (ii) in [MDU, ; KUK,, 4, 5, 6; VTU, 7] (i) Th Ingrl ind i comprbl o f() d wih nd f() in. Now (in ) f, > + d d in f d d + + Hnc h ingrl ( + ) ( + ). d 5 in ( in ), whn. (ii) Th ingrl ind i comprbl o Hnc, f() in Now (in ) f, ( > ) + f() d

20 774 Enginring Mhmic hrough Applicion d + d ( ) d ( ) + + ( + ) d d d 4 ( in ) ( ) 4 Hnc h ingrl 4 ind in whn 4 ( + ) Exmpl : Evlu d [KUK, 5] Soluion: Wri d d comprbl o d d () g() f, whr f(), g() f() d, whn log log + d + d f () uing f d Alrnly: log log + + log + log (log), whn d d d f () f () d d (dfn. By d fn. whn f() ) for for d d log log log log for for for for for for + ( + log) + log log, (On rforming) ASSIGNMENT Find plc Trnform of following:. (i) cohco (ii) *cohin

21 plc Trnform nd Thir Applicion 775 (iii) in hco (iv) inhin (v) in coh + co inh (vi) in coh co inh *[NIT Kurukhr; Dlhi, ]. in (i) in co (ii). 4. (i), < < f (), < < 7, > (ii) in π π, > f () π, < (i) in [Ripur 5] (ii) inh (iii) co (iv) coh 5. (i) in + co (ii) inh + coh (iii) in co (iv) inh coh 6. (i) co in (ii) coh + inh 7. *(i) co cob co (ii) *[VTU 6; INTU 5; UPTU 5] in in (iii) [WBTU, 5] **(iv) [KUK,*-4, **6] 8. Find h lplc rnform in nd hnc dduc (co). uing rnform of diffrniion. 9. Find h lplc rnform of coh nd hnc dduc (inh). uing h rnform of driviv.. Find h lplc rnform of h following: *(i) cod (ii). Show h (i) co d, 5 (iii) * (ii) co cob d in log 5, d 4 *[Punjbi Univ., ] co cob b d log *[KUK, 6, ]

22 776 Enginring Mhmic hrough Applicion.6 INVERSE APACE TRANSFORMS To find h invr lplc rnform of givn funcion, w ry o rcogniz h givn funcion ihr in h givn form comprbl o om ndrd xprion who invr funcion i known ndrd funcion or pli h givn funcion of ino numbr of xprion in (wih h hlp of pril frcion) comprbl gin o om ndrd funcion of of which invr funcion r known. u li fir invr lplc rnform of om ndrd lmnry funcion nd ocid funcion. n ( n)! n in + inh ( ) + b b in b in ( + ) co ( + ) n ( ) n! + n co coh ( ) + b co b inh ( ) + coh ( ) (in co ) ( + ) (in co ) ( + ) + (co in ) ( + ) inh.in coh.in (inh coh ) ( ) (inh coh ) ( ) + (coh inh ) ( ) coh.co 4 4 inh.co

23 plc Trnform nd Thir Applicion 777 (coh in inh co ) (coh in inh co ) J ( ) J ( ) + I( ) ( ) I ( ) Invr rnform buld bov will b frqunly ud rdy rcknr, priculrly for olving diffrnil quion uing lplc rnform. Wihou going ino dil of drivion of h, w my dicu om of hm undr: + ( + ) ( + ) ( + ) ( + ) ( ) ( + ) ( + ) ( + ) in co co in ikwi, + ( ) ( ) ( ) inh coh + coh + inh A lrdy d, in dducing h rul, w nd o mk u of pril frcion. Som of h propri of invr lplc rnform nd Convoluion Thorm r kn up in ubqun dicuion. m m m Tip for Pril Frcion: In ny gnrl xprion n n b + b + + b A Corrponding o non-rpd linr fcor ( ), wri ( ) A A A Corrponding o non-rpd linr fcor ( ), wri r r A + B Corrponding o non rpd qudric fcor ( + + b), wri ( + + b) Corrponding o rpd qudric fcor ( + + b), wri n

24 778 Enginring Mhmic hrough Applicion A + B A + B A r + Br r ( + + b) ( + + b) ( + + b) Thn drmin unknown conn A, B, C, A, A, A, by quing h cofficin of qul powr of on boh id. Exmpl : Find h invr plc rnform of (i), ( + ) n (ii) ( A + B) whr A, B nd r conn nd n i poiiv ingr. Soluion: n, (i) (ii) n n n, uing fir hif propry. ( + ) n! B B n A A n n n n n n ( A B) A B A + A n! + A A + B A + B Exmpl : Find h invr lplc rnform of nd C + D + E C + D + E Soluion: A + B A B (i) + C + D + E C D E C D E C C C C A + B, whr F D nd G E C + F+ G C + F+ G C C A B + C C F F F F S+ + G S+ + G 4 4 (By compling qur) () C I: Whn F G >, hn h xprion ( + F + G) k h form ( + ) + b, whr 4 F F nd 4, hn b G A B A B + + C D E C + + ( + ) + b C ( + ) + b ( + ) B C A + C + + b + + b.

25 plc Trnform nd Thir Applicion 779 ( + ) C A B C b + + b + + b A B C + b b + + C + b A B inb cob in b. C + b C b A co B b A + inb C bc () C II: whn F G <, hn 4 A B A B + + C + D + E C ( + ) b C ( + ) b ( + ) B C A + C + b + b ( + ) B C A C + + b + b + b A B C + b b C b A B inhb cohb inh b. C + b C b A + B A B A cohb + inh b C + D + E C bc () C III: whn F G, hn 4 A + B A B C D E C F F C F F + G+ + G 4 4 A B + C ( + ) C ( + )

26 78 Enginring Mhmic hrough Applicion ( + ) B ( + ) C ( + ) A + C A B C + + ( + ) C ( + ) A B. C + C A B ( ) + C C A + B A B + C + D + E C C ( ) + Exmpl 4: Find invr lplc of ( + ) ( ) + A B C D Soluion: Wri ( + ) ( ) + ( + ) ( ) ( ) Implying ( + ) A( + )( ) + B( ) + C( + ) ( ) + D( + ) () A( + )( + ) + B( + ) + C( )( ) + D( ) A( + ) + B( + ) + C( + 4) + D( ) ( + ) (A + C) + (B + C + D) + ( A B + 4D) + (A + B 4C + 4D) () From (), if ; hn ( + ) D( + ) implying D () if ; hn ( 4 + ) B( ) implying B (4) Furhr on quing cofficin of nd conn rm on boh id of quion (), w g, A + C nd A + B 4C + 4D (5) Thu on puing D nd B ; quion (5) rul in A C ( + ) ( ) ( + )

27 plc Trnform nd Thir Applicion 78 Exmpl 5: Find Soluion: Wri ( ) ( + ) ( + + )( + ) A+ B C+ D Thu, or (A + B)( + + ) + (C + D)( + ) (A + C) + (A + B C + D) + (A + B + C D) + (B + D) On compring cofficin of qul powr of on boh id, All h quion oghr giv () For, A + C For, A + B C + D For, A + B + C D Conn, B+ D 4 B, D By I hif propry in in in inh in

28 78 Enginring Mhmic hrough Applicion Exmpl 6: Find invr plc rnform of 4 4 [NIT Jlndhr, 7] A+ B C+ D Soluion: wri ( )( + ) ( ) ( + ) Implying (A + B)( + ) + (C + D)( ) (A + C) + (B + D) + ( A C) + (B D) On compring cofficin of qul powr of,, nd conn on boh id w g, A+ C B+ D All oghr implying A C, B D. ( A C) ( B D) 4 4 coh co + + [ + ] Exmpl 7: Find invr rnform of (i), 4 4 (ii) , (iv) + 4, 4 4 (v)** + 4 ( ) 4 4 (iii)* + 4, 4 4 [*KUK,, ; **NIT Kurukhr, ] Soluion: (i) Wri, ( + ) () ( + )( + + ) A B C D o h By pril frcion, A C, B D 8 4 Now ( + ) + ( + ) + ( ) + ( ) (By fir hif propry)

29 plc Trnform nd Thir Applicion 78 8 co in co in + + ( + ) ( ) in + co 4 in coh co inh 4 (ii) By pril frcion 4 ( ) + ( + ) ( ) in in in in inh 4 (iii) , (By pril frcion) ( ) + ( + ) co + in co + in 4

30 784 Enginring Mhmic hrough Applicion + co + in co inh + in coh (By pril frcion) (iv) + 4 (v) Hr co co coh co + ( ) A + B C + D On rolving pril frcion w g A C nd B D co co. co inh co..7 SOME IMPORTANT PROPERTIES ON INVERSE APACE TRANSFORMS: If f() f(), hn (i) ( f ) f ( ), (ii) () d f f (), providd f() d (iii) f () f() u du, d (iv) f() f(), d f () (v) f() d (vi) () (), whr () f f F F > <,

31 plc Trnform nd Thir Applicion 785 Exmpl 8: Find h invr lplc of (i) ( + ) ( ), (ii) co (iii) + Soluion: (i) Wri ( + ) d d f () + + d 4 5 d ( 4 5) ( + + ) () Now on uing propry; if d f() f() hn f() f() d Hr f () in ( ) ( ) + ( + ) + + f() in ( ) (ii) W know h 4 x x co + + uing, cox 4 +! 4!! 4! co +! 4! + 5! 4! 4 + (!) (4!) (iii) A in, hrfor inudu co + ( + ) implying ( + ) u du ( co ) ( in ) nd ( ) + ( + ) u inu du co

32 786 Enginring Mhmic hrough Applicion Exmpl 9: Find invr lplc rnform of.log + [NIT Kurukhr ] d d d f () log log + log d d + d Soluion: Hr, φ() Tking invr lplc on boh id, log( ) log( + + ) + + { log ( + ) log ( ) } + + φ, whr φ () { log( + ) log } () f.() () coh () d d d d Agin, φ φ log log( + ) inh + inh or φ () Uing rul () in (), w g () inh inh coh f () coh or f () Exmpl : Find invr lplc rnform of (i) ( + ), (ii) ( + ) (SVTU 7) (iii)* ( + ) [Mdr ; *KUK, Jun 4] Soluion: ( + ) ( + ) () i comprbl o f () d d i comprbl o φ + d d Now ( + )

33 plc Trnform nd Thir Applicion 787 φ φ in (), whr () + ( ) ( + ) Implying ( + ) in, which prov rul (ii)...() For (i), f ( ) ( ) + + fudu uin udu, uing () cou cou u du co in (in co ) + For (ii), Alrnly f () fd () or f () d d + + ( + ) ( + ) Tking invr, For (iii) ( ) ( + + ) f () in in or f comprbl o f d f() f(), whr f() f() nd f() ( + ) d Hr Implying in f(), rul( ii) ( + ) d in + in co ( + ) d.8 CONVOUTION THEOREM Convoluion: f() nd g() b wo funcion of cl A, hn h convoluion of h wo funcion f() nd g() dnod by f g i dfind :

34 788 Enginring Mhmic hrough Applicion f g f() u g( u) du,whrf g i lo known fling of f() nd g(). Th convoluion f g i (i) Commuiv i.. f g g f (ii) Aociiv i.. (f g) h f (g h) (iii) Diribuiv wih rpc o ddiion i.. f (g + h) f g + f h Smn: If Proof: f() f() nd g() g(), hn f() g() f( u)( g u) du ( f g) ( f g) d f()( u g u) dud () f( u) g( u) dud () (Exprion () clrly how h h ingrion i I kn long h vricl rip PQ from P o Q, P ring on h curv u nd Q ring on h curv u nd finlly h rip PQ lid bwn o o covr h full dod rgion) On chnging h ordr of ingrion, w g u ( f ( g) fug ( ud ) ) du () u ( ( u + u) f() u g( u) d) du u u ( u) f() u ( g( u) d) du u u p f() u g() p dpdu, (on puing u p o h d dp) Whnc h dird rul. f( u) g() du f() u du g() f() g() (Uing df n. of lplc) u u Exmpl : Uing Convoluion horm, find (i) ( + ) (ii)* 4 4 ( + ) *[KUK,,, NIT Kurukhr, 5] u u u Q P Fig:. Soluion: (i) Wri,, ( + ) ( + )( + ) whr in co ( + ) nd

35 plc Trnform nd Thir Applicion 789 in ( + ) in ( u) (co u) du + (ii) co u (in co u co in u ) du co co udu in inu du in co ( + co u) du in udu in in co cou + ( in )( co) in co + + co in f() g() f() g() ( )( ) ( ) whr, f () f () coh nd g () g () co + u u du I 4 4 ( ) coh co, Ingr by pr, king cohu I funcion nd co ( u) cond funcion, { } in ( u) in ( u) I coh u. inh du in ( u cou + inh u in ( u) du co( u) co ( u) ( in ) + inh u. coh u. du

36 79 Enginring Mhmic hrough Applicion in inh u co ( u) I + I implying in inh I + or I (in + inh ) Hnc [in inh ] u u + Alrnly, my k cohu in ingrl I. Exmpl : Uing convoluion horm, find Soluion: Wri χ [ KUK, 4] ( + 4) comprbl o fg wih in f () nd f () ; g () nd + + co( u) in( u) du ( 4) + co( u) ( co ) 4 4 Alrnly: ; co f () nd g () o h uco( u) du ( co ) ( 4) + 4 ASSIGNMENT Find h invr lplc rnform of. (i) (iv). ( ) A + B + ( + ), ( C + D + E) (ii) (v)* ( )( 4) ( + ) ( + ) whr A, B, C, D, r conn., (iii), *[NIT Kurukhr, 5]

37 plc Trnform nd Thir Applicion 79. (i), + 4+ (ii) + 7, (iii) (i)* + ( ), (ii) + ( + 6+ ), (iii)** [*Mdr, ; PTU, 5; **KUK, 5] 5. (i)* + log ( + ) (ii) co or n (iii)** n [*VTU, 4; **Bomby, 5] (iv)* ( + ) log ( + b ) (v) log ( + ) ( + )( + ) (vi)* log [*Ann, UP Tch ] ( + ) (vii)* log ( ) (viii) co ( + ) (ix)** log b + + [*Mdr, 5; **VTU, 7] 6. (i)* + b (ii) ( + ) + (iii)** [*Mdr, 5; **KUK, 4] 7. (i) + (ii) ( +) (iii)* ( + ) [*NIT Kurukhr, ] 8. Show h 5 in + (! ) ( 5! ) 9. U Convoluion horm o find invr of h following: (i) ( + )( + b) (ii) ( + ) (iii) + (iv) ( + ) (v)* ( + )( + 9) (vi)** ( + )( + 4) (vi)*** ( + )( + b ) [KUK, *4, 5-6; **4-5 *** JNTU, 5, SVTU, 7; KUK, ]

38 79 Enginring Mhmic hrough Applicion No: Mo of h bov problm cn b hndld ihr by pril frcion or convoluion horm. Howvr, for y hndling of problm 4 & 5, pply propry d f() f(); d in problm 7 & 8, pply propry f () f( u) du. d f() f(); d in problm (6), pply propry.9 APPICATION TO DIFFERENTIA EQUATIONS Exmpl : Uing lplc rnform, olv (D 4 K 4 )y whr y(), y () y () y () [NIT Kurukhr, 4] Soluion: Tking lplc rnform of ch individul rm of h givn quion, (D 4 y) K 4 (y) Implying 4 4 y y() y () y () y () Ky i.. 4 K 4 y() implying Thn by pril frcion, y () ( K)( + K)( + K ) A B C+ D + + () K + K + K K + K + K () implying A( + K)( + K ) + B( K)( + K ) + (C + D)( K ) On compring cofficin of qul powr of on boh id (A + B + C) (i) (A B) K + D (ii) (A + B C)K (iii) (AK BK D)K (iv) Subrcing (iii) from (i), w g C Subrcing (iv) from (ii), w g D Wih vlu of C nd D, quion (i) nd (iv) bcom A+ B A B implying A nd B 4 4 Puing h vlu of A, B, C nd D in quion () or mor prcily in (), w g y () K 4 + K + K () Tking invr rnform of ch individul rm on boh id,

39 plc Trnform nd Thir Applicion 79 K K K K + y + + cok + cok cohk + cok 4 4 dy dy dy Exmpl 4: Solv + y dx dx dx whn dy d y y, x dx dx Soluion: Tking lplc rnform on boh id of h givn quion y y y y + y y y y y y () () () () () () () () () () () Mking u of h givn condiion, y + y y y i.. ( ) y ( 4 5) () y () () 5 y () + (By pril frcion) (4) + + Tking invr lplc on boh id of quion (4), w g 5 yx () which i h dird oluion. x x x + (5) Exmpl 5: Solv h quion (D + )x co ; x Dx [KUK, ] Soluion: Tking lplc on boh id, d () () + d 4 Uing h givn condiion ( + ) x + 4 i x x x x x () By pril frcion, ( 4) ( + 4) ( + ) ( 4) A + B C + D E + F + + ( + ) ( + ) ( + ) ( + 4) 4 ( + 4) or ( 4) (A + B)( + 4) + (C + D)( + )( + 4) + (E + F)( + ) On compring cofficin of qul powr of on boh id, w g ()

40 794 Enginring Mhmic hrough Applicion 5, A+ C...( i) A C E 4, B+ D...( ii) 5 B, 8A + 5C + E...( iii) 9 5, F + 8B+ 5D...( iv) D 9, 6A+ 4C + E...( v) 8, 6B+ 4D+ F 4...( vi) F () Thu, x () ( + ) ( + ) ( + ) Tking invr lplc rnform on boh id, x () in+ in+ inco 9 8. ( + ) () uing in co whr in hi c x() in + in + ( in co) 5 4 x () in+ in co. Hnc h rul. 9 9 dx Exmpl 6: Solv + 9x co d if x () nd x(π/). [UPTch ; Ripur 4] Soluion: Tking lplc on boh id of h givn quion, x() x() x () + 9 x() + 4 i.. (). 9 (), king (), y + 4 x + x x + x ( 9) x () or x () (By pril frcion) () Tking invr lplc rnform on boh id x [ co co] + co + in () 5

41 plc Trnform nd Thir Applicion 795 A π, + + or () 5 5 x co + 4co + 4in 5 [ ] dy in, d + +. Exmpl 7: Solv y yd y Soluion: Tking lplc on boh id, y () ( y() y() ) + y() + +, uing f() d On uing h givn condiion, + ( + ) ( + ) A B C+ D + + ( + ) ( + ) ( + ) ( + ) ( + ) + + y or y By pril frcion, i.. ( + ) A( + )( + ) + B( + ) + (C + D)( + ) On compring h cofficin of qul powr of nd olving for A, B, C, D. f () A, B, C, D y () Tking invr, Exmpl 8: Solv y () + + in y () + y( u)coudu if y() 4. Soluion: Tking lplc of ch individul rm on boh id of h givn quion, y ( ()) () + y ( u)coudu n y() y() + y co ; uing df. f g f( u)( g u) du or y() 4 + y() ; ( f g) f() g() +

42 796 Enginring Mhmic hrough Applicion ( 4 + )( + ) implying 4 5 y() + 4 or y() Tking invr, y () Exmpl 9: Show h h oluion of h quion y () φ () + yu g ( u) du cn b rprnd y, whr φ ( φ ) nd g ( g ) olv φ() g y () uy ( u) du. Soluion: plc rnform of h givn ingrl quion rul in y () () φ + yug ( udu ) φ () + ( y g) φ () + yg () () On implificion, g y φ or φ() y () g Implying, () y () φ g or y () φ () φ() ψ(), whr ψ g () g φ() u ψ( u) du Now, for hnc pr y () ( y) y () + y or y or y or + + Tking invr, y ( co).. Hnc, Exmpl : Solv h quion dx dx + + x, x(), x (). d d [NIT Kurukhr, 6] Soluion: Tking lplc of ch individul rm on boh id of h givn quion, dx dx + + ( x) d d ()

43 plc Trnform nd Thir Applicion 797 d d () () '() () () () d + d ( x x x ) ( x x ) ( x ) d uing formul, f ( ) f d d d x x + x, (Tking x(), nd x () (y)) d d dx dx x + x + d d dx dx ( + ) + x + d d x + On ingring, + ( + ) C log x log logc x () + Tking invr lplc rnform, x() CJ () (I i ndrd lplc of Bl Funcion) () On uing givn condiion, x() ; CJ () or C J (). Hnc x() J () Exmpl : Solv h quion dy dy x + + xy ; dx dx y(), y'(). [NIT Kurukhr, ] Soluion: Tking invr lplc on boh id of h givn quion dy dy x + + xy, () dx dx d d W g, y y() y () + ( y y() ) ( y) d d d dy ( y ) ( y ) d + d, uing h condiion y(), y () dy implying + y d + On ingrion, w g or dy + d y ( + ) + ( + ) C log y log logc i.. y + Tking Invr lplc on boh id of quion (), w g y(x) CJ (x), Whnc h rul. (Sinc + () i ndrd rnform of Bl Funcion J (x)

44 798 Enginring Mhmic hrough Applicion dx dx Exmpl : Solv h quion + x, x(), x () d d Soluion: Tking plc Trnform of n ch individul rm of h givn quion, dx dx + () x () d d () d x() x() x () + x() x() + x() d d x () + x () + x () d dx x () + + x+ x d dx + ( + ) x + + d dx + + x + + d () Now hi quion cn b rd ibniz linr diffrnil quion in x () nd. + Hr, P, Q d + log log Pd IF.. () Now for oluion, x().. IF. (..) IF Qd+ c (4) + + d+ c d d + + d + c d d d c d d d d + + d+ c d + + d + c x () ( + ) + c or c x + + (5) Tking Invr plc Trnform on boh id, w g x() + + c.f(), whr F() i om funcion of. (6) On uing boundry condiion,

45 plc Trnform nd Thir Applicion 799 Whn x(), x() +. + c.f() i.. c.f() (7) whn x (), c.f () (8) (7) & (8) oghr imply c Whnc x() +. Exmpl : y + y + y in, y() y'() [Punjb Univ., ; NIT Kurukhr, 5] Soluion: (y ) + (y ) + (y) (in ), ( ) d d y () y() y() + y () () y y () d d d d y y + y() + y y() y d d d ( + ) y y() d +, d ( + ) y +, y() d + d y () + d + ( + ) Tking Invr plc on boh id of (), y in + (in co ), uing (in co ) ( + ) in co y () Alrnly: y () d dn d + ( + ) ( + ) Now for ( ) d I, k nθ o h d c θ + c θ + θ θ θ Implying co in I dθ θ θ θ + ( + θ) co d d n 4 n nθ I +, inθ n θ Puing (4) ino (), + + () () () () n n + y y () n + y () ( n ) ( + ) (4)

46 8 Enginring Mhmic hrough Applicion () (n ) co y (5) d g () n (6) d + Tking Invr plc on boh id, Now l g() (n ) o h g() in or () in g (7) in co Puing (7) in (5), f () in co f () Exmpl 4: A volg E i pplid o circui of inducnc, nd Rinc R. Show by h rnformion mhod h currn ny im i R E R. [VTU, ] Soluion: Equion govrning h currn flow in -R circui i or di di R E + Ri E or + i d d Tking lplc on boh id. R E i() i() + i() ( + ) E E i () R ( + ) R R R + ( + ) + Tking invr lplc rnform on boh id, R E i () R Exmpl 5: U lplc rnform mhod o obin h chrg ny inn of cpcior which i dichrgd in R C circui, fr h wich i clod if R.5 ohm, lf inducnc Hnry, cpcinc, C frd, nd h cpcior h iniilly chrg of coulomb. Iniilly h wich i opn nd hrfor, no currn i flowing. Soluion: Th dird quion in C R circui i dq dq q + R + E. () d d C Hr givn,, R 9/4, C, q(), Wih bov vlu, quion () bcom dq q (), d E

47 plc Trnform nd Thir Applicion 8 dq 9 dq q + + () d 4 d Tking lplc on boh id of h bov quion 9 q q() q () + q q() + q i q() 4 4 i.. ( + ) 4 9 q () ( ) 7 ( 4+ ) ( + ) q () 8 7 ( + 4) + Tking invr lplc rnform on boh id, q () (By pril frcion) () Exmpl 6: Alrning volg in() i pplid o circui wih n inducnc 5 mh (millihnry), Cpcinc µf (micro frd) nd rinc Ω (ohm). Find h currn i ny im cond if h iniil currn i nd chrg q r zro. Soluion: Equion of currn flow in R C circui i dq dq q dq, whr + R + E i () d d c d Hr w r givn R ohm 5 Hnry C 6 Frd E in() Wih bov vlu, quion () bcom dq dq q in 6 d d dq dq q 4in d d Tking plc on boh id, or 4 q q q + q q + q () () () q or q () 4 ( ) ( + )( + ) ( + ) ( + ) Tking Invr plc Trnform, () () q ( ) in (4)

48 8 Enginring Mhmic hrough Applicion dq Alo i ( ) co ( ) d (5) Exmpl 7: Obin h quion for h forcd ocillion of m m chd o h lowr nd of n lic pring who uppr nd i fixd nd who iffn i k, whn h driving forc i F in. Solv hi quion (Uing h plc Trnform) whn k/m, givn h iniilly vlociy nd diplcmn (from quilibrium poiion) r zro. Soluion: Th problm li undr forcd ocillion wihou dmping. (If h poin of uppor of h pring i lo vibring wih om xrnl priodic forc, hn h ruling moion i clld forcd ocillory moion, ohrwi h moion i rmd forcd ocillion wihou dmping) Tking h xrnl priodic forc o b F in, h quion of moion i dx m mg k( + x) + F in () d whr, x i h lngh of h rchd porion of h pring (diplcmn) fr im, i h longion producd in h pring by h m m, k i h roring forc pr uni rch of h pring du o liciy; i ny rbirry conn nd p i ny clr. A B x P F in k ( + x) mg Fig..4 In hi priculr problm, nion mg k, hrfor quion (), bcom dx m kx+ F in or dx F + k x in d d m m dx F + nx in d m Tking lplc on boh id of (), w g F x () x() x () + nx () m + ()

49 plc Trnform nd Thir Applicion 8 k F + x() m m +, uing x(), x () F F k x (), µ m + + k m ( )( n ) m ( + ) + m A + B C + D Now by pril frcion, + ( + )( + n ) ( + ) ( + n ) ( A + B)( + n ) + ( C + D)( + ) ( A+ C) + ( B+ D) + ( n A+ C) + ( n B+ D) On olving for A, B, C, D; w g A, B, C, D n n ( ) () Hnc x () F mn n ( ) ( + ) ( + ) Tking plc Invr on boh id, F x n n mn () in in, ( ) F k k nin in n, whr n nd mn n m m ASSIGNMENT Solv h following inr Diffrnil quion: dx. x in w, x() d + [KUK, ]. dy dy +, whr (), dx dx wy y A B x. y y + y 4 +, whn y() nd y () [Andhr, ; NIT Kurukhr, 8] 4. (D + n )x in(n + α), x Dx 5. y + y y y, givn y() y () nd y () 6 6. (D D + D )y, givn y(), y (), y () 7. dx dx + x d d wih x, x [KUK, 4] 8. y ir () + y + y in, whn y() y () y () y ()

50 84 Enginring Mhmic hrough Applicion 9. dx dy + + 5y in, d dx whr y(), y () [KUK,, 4; Bomby, 5; PTU, 5]. y + y + y co, givn h y(). y + ( )y y, whn y(), y () [Punjbi Univ. ]. u f () co + f ( udu ).: APPICATIONS TO SIMUTANEOUS INEAR EQUATIONS WITH CONSTANT COEFFICIENTS Exmpl 8: Solv h imulnou quion dx dy y, + x in, givn x(), y() [Dlhi, ] d d Soluion: Tking plc Trnform of h givn imulnou quion, [ x() x() ] y() or x() y() or x y () nd y() y() + x() or + On olving () nd () for x () nd y (), x () + ( )( + ) ( + ) x + y + () y () + nd ( )( + ) () ( + ) Tking Invr plc Trnform on boh id of quion () nd (4), w g x ( + ) ( in co) ( in co) (4) ( + in+ co co ) (5)

51 nd y + ( + ) + + in ( + in co) ( in in co ) (6) x ( + in + co co) Hnc, ( in in y co ) Exmpl 9: Th coordin (x, y) of pricl moving long pln curv ny im dy dx r givn by + x in, y co, ( > ). If, x nd y, how by d d uing rnform h h pricl mov long h curv 4x + 4xy + 5y 4. [UPTch, ] Soluion: On king plc on boh id of h givn imulnou quion, y y() + x or x + y, uing (y() ) () Similrly [ x x() ] + y or x + y +, uing (x() ) () For olving x nd y, muliply hroughou () by, () by nd k h diffrnc of h wo, y + 4 or y in () Furhr x + or x ( in + co) Equion (4) impli, x in + co i.. x + y co, uing () (x + y) 4( in ) or 4x + y + 4xy 4 4y, uing () 4x + 4xy + 5y 4 Hnc h dird pricl (x, y) mov long h curv 4x + 4xy +5y 4 Exmpl 4: Solv dy dx d y (4) dx + + in, + co wih h iniil condiion d d d d dx dy x, ; y, whn. d d Soluion: On pplying plc rnform on ch individul rm of h wo givn quion

52 86 Enginring Mhmic hrough Applicion dx dy + + in d d dx dy + co d d nd On implificion of () nd (), + x+ y + nd x y + + Adding () nd (4), w g ( ) ( y ) x + + ( + ) + + x ( y ) () () () (4) x ( ) or x ( + ) Uing (5) in (4), w g y ( + ) ( + ) ( + ) ( + ) ( + ) or y + ( + ) Tking invr of (5) nd (6), x co nd y ( + in ). (5) (6) ASSIGNMENT 4 Solv h following imulnou quion dx. + y in, dy + x co givn x, y whn. d dx [UPTch, 4; Krl, 5]. dx dy dx dy + + x, y ; givn x, y whn. d d d d dx. + 5x y, dy + x+ y ; bing givn h x y d d 4. Th currn i nd i in mh r givn by h diffrnil quion:

53 plc Trnform nd Thir Applicion 87 di di wi co p, + wi in p. Find h currn i nd i by plc rnform d d if i i. 5. Smll Ocillion of crin ym wih wo dgr of frdom r givn by h } quion D x + x y whr x y nd Dx Dx Dy x 5y whn. + + Dy Find x nd y whn.. APACE TRANSFORMS OF SOME SPECIA FUNCTIONS Sr.No. Funcion Nm f() f (). Hviid' Uni Sp Funcion u ( ) {, <, >. Dirc Dl funcion, δ ο > ( ) f() d. Priodic Funcion f () f ( + T) T inu 4. Sin Ingrl Si () du u 5. Coin Ingrl C i () 6. Error Funcion cou du u T n log ( + ) u rf du π + u 7. Complmnry Error Fun. rfc du π ( ) 8. Exponnil Fun. of Ordr Zro () { } u du log ( + ) u Bl Funcion of Ordr Zro J() A fw of h bov funcion r kn up in h ubqun dicuion.

54 88 Enginring Mhmic hrough Applicion. Uni Sp Funcion (Hviid Uni Funcion): In nginring, mny im w com cro uch frcion of which invr lplc i ihr vry difficul or cn no b obind by h formul dicud o fr. To ovrcom uch problm, Uni p funcion (Hviid Uni Funcion) h bn inroducd. Th uni p funcion i dfind follow: u ( ) for < U() u ( ) for, whr i lwy poiiv. From h grph, i i pprn h funcion i zro for ll vlu of up o nd fr which i i uniy. In phyicl problm, i im vribl. u, < A priculr c,, Obrvion: Gnrlly h uni p funcion in mchnicl nginring com ino picur forc uddnly pplid o mchin or mchin componn, whr in lcricl nginring i mnif n lcromoiv forc of bry in circui. Trnform of Uni Sp Funcion: { u( ) } u( ) d d + d nd in priculr, whn, { u ( ) } Scond Shifing Thorm (-hifing): fu (), whn < On h bi of h u( ), w hd h funcion, f (), whn, nd h funcion f( ) rprn h grph of f() diplcd hrough dinc o h righ i.. u () h jump yp of diconinuiy. f() f(), hn f( ) u ( ) f() If { } { ( ) ( )} { ( ) ( )} f u f u d od + f( ) d ( x + ) f( x) dx (on king x) x f( x) dx f Fig..5 Fig..6 Th cond hif horm i lo known hifing i.. i rplcd by in f(). f () f () f( ) u( ) +

55 plc Trnform nd Thir Applicion 89 Exmpl 4: Find plc Trnform of (i) f() ( ) u( ) (ii) f() { u( )} (iii) f() in u( ) (iv) for < < f () { for < < Soluion: (i) Givn f() ( ) u( ) () On compring h givn funcion wih f( )u( ), W h in hi c, f() nd f () ( f ()) Conqunly by cond hif horm, f u f { ( ) ( )} () (ii) Hr in hi c king f (), f () nd hn by cond hif horm + u u f () ( { }) u( ) + + o Fig..7 Exmpl 4: Find h invr lplc rnform of h following: (i)*, / + π > (ii)** (iii)***, w + π ( + b) > (iv) (v) ( )( ) ( + ) [*KUK, ; **NIT Kurukhr, ***KUK 5; VTU, ] Soluion: (i) f (), hn f () cohw w Now by cond hifing horm, { } { } coh w( ) u ( ) w (ii) For f (), f () coπ +π f ( ) f ( ) u ( )

56 8 Enginring Mhmic hrough Applicion π f (), f () inπ +π Alo by cond hifing horm { } f ( ) f ( ) u ( ), { } +π +π +π +π π + coπ u + in π( ) u coπ u + in ( π+ π) u co π u (in π) u (iii) Givn () F + + b b + b b b, (By pril frcion) + b + b b b Furhr b,,, + b Alo by cond hifing horm, f () f ( ) u ( ), ( b) b + b b b (iv) b ( ) u ( ) u ( ) + ( ) u ( ) b b b b ( ) { + b( ) } u ( ) b () F ( + ) u u [By cond hifing horm]

57 plc Trnform nd Thir Applicion 8 ( ) u (v) Hr F () f (), whr ( + ) ( ) ( ) u ( + ) f () ( + ) Exmpl 4: Find h invr lplc rnform of ( ). Soluion: ( ) ( ) ( ) Comprbl o f(), whr f() inchc. + u ( ) + u ( ) + u ( ) + Thu, h funcion cn b wrin f (), <, < f (), < Somim i i clld irc funcion. O Fig..8 Exmpl 44: In n lcricl circui wih.m.f. E(), rinc R nd inducnc, h currn i build up h r givn by di + Ri E(). If h wich i conncd d nd diconncd, find h currn i ny inn., for Soluion: Givn i nd E < < E () o, for > On king lplc of ch rm of h givn quion, w g i () i() + Ri () Ed () ( nd R r conn) ()

58 8 Enginring Mhmic hrough Applicion () + R i E d + d E ( + R) i E o i () E R R ( + ) ( + ) Now king invr lplc on boh id, w g i() E E ( + R) ( + R) By pril Frcion, Equion () bcom, + R R R+ R { } i () E E R R+ R R R+ R () () (4) Now E E i () R R R R + + R ( ) E E i () F () (5) R R F() f() f( ) u ( ), (By cond hifing propry) R R ( ) ( ). u ( ) u ( ) u ( ) R + On puing (6) ino (5), w g h vlu of h currn R R E ( ( ) ) (6) E i () u ( ) (7) R R R E Now, for < <, () ( i ) R Qu ( ), < <, for >, E R E R ( ) E R R i () R R R

59 plc Trnform nd Thir Applicion 8 II Uni Impul Funcion (Dirc Dl Funion) Impul i conidrd forc of vry high mgniud pplid for ju n inn nd h funcion rprning h impul i clld Dirc Dl Funcion. Mhmiclly, i i h limiing form of h funcion δ ( ), ( ) + δ, ohrwi, I i pprn from h figur.9, h, h high of Fig..9 h rip incr indfinily nd h widh dcr in uch wy h h r of h rip i lwy uniy. Th bov fc cn noionlly b d δ( ), for uch h o, for δ( ) d for Obrvion: In mchnic, h impul of forc f() ovr im inrvl, y ( + ) i dfind o b h ingrl of f() from o ( + ). Th nlog of n lcric circui i h ingrl of.m.f. pplid o h circui, ingrd from o ( + ). Of priculr prcicl inr i h c of vry hor ε wih limi ε i.. impul of forc cing for n inn. Trnform Of Uni Impul Funcion: [ ] + + δ( ) δ( ) d od + d + od + + ( + ) d + Tking imi o, [ ( ) δ ] o In priculr, if o, [δ()]. Filring of uni Impul Funcion: If f() i coninuou nd ingrbl for ll, hn Rlion Bwn u ( ) nd δ( ) f () δ( d ) f u u u o [ ( )] [ ( )] ( ) [ δ( )] Exmpl 45: Th diffrnil quion govrning h flow of currn i() in n R circui i givn by di Ri + Eδ() d O +

60 84 Enginring Mhmic hrough Applicion whr E δ() i h impul volg nd i(). Find h currn i(), >. Soluion: On king lplc of ch individul rm in h givn quion, Ri() + i() i() o E. E implying ( R+ )() i E or i() R + R On king invr, lplc, E i () i h rquird xprion for h currn i. Exmpl 46: An impulion volg Eδ() i pplid o n C R circui wih zro iniil condiion. If i() b h currn ny ubqun im, find h limi of i(). Soluion: Th govrning quion for n C R circui i di + Ri + id Eδ(), whr i() whn d C o Tking plc of ch individul rm on boh id of h quion, R i () E i() i() + i() +, whr i() C R E Implying, + + i() R E C or + + i() C E E i () R C b E ( + ) E + i (), ( + ) + b + + b + + b On invrion, E i () cob inb b Tking limi, R whr nd + b C E i. Though, iniilly currn i pplid, y lrg currn will dvlop innnouly du o impul pplid nd i i E. ASSIGNMENT 5. Skch h grph of h following funcion nd xpr hm i rm of uni p funcion. Hnc find hir lplc rnform: (i), for < < π f (), for >π

61 plc Trnform nd Thir Applicion 85 for < < (ii) f (), for > [Hin: f() u ()]. Find h lplc rnform of h following:, > T f () co ω +φ, < < T (i) (ii) k, for < < b, whr > f () for o < < nd > b (iii), < < h f () h, > h [Hin: Expibl f() h [ u ()]] h, < < (iv) f () > [Hin: f() [ u ()] ] (v) () {, < < π f in, >π [Hin: f() in u π ()] (vi) { in (), < π f,. Find h invr rnform of π (i)* + (ii) >π [ ] + + [ ] Hin : f in uπ uπ in in uπ (iii) + b π (iv) + 4 *[KUK, 4] f (), < < 4. Solv y + 4y + y f(), whr, > ; givn h y() y () Hin: f u, y ( ), y f( ) u ( + )( + ) 5. A bm h i nd clmpd x nd x l. A concnrd lod W c vriclly downwrd h poin x l. Find h ruling dfcion. 4 dy ω Hin: Th diffrnil quion nd boundry condiion r l x δ 4 dx EI nd y() y () y() l y () l 6. A cnilvr bm i clmpd h nd x nd i fr h nd x l. I crri uniform lod pr uni lngh from x o x l/. Clcul h dflcion y ny poin. 4 dy ω() x Hin: Th diffrnil quion nd boundry condiion, 4 dx EI l w, < x < ( < x < l) ; ω ( x) nd y() y () y () y (), x > l

62 86 Enginring Mhmic hrough Applicion III plc Trnform of Priodic Funcion If f() b priodic funcion of priod, T(> ) i.. f( + T) f(), hn T f() Proof: By dfiniion, f() d T ( ) f () fd () fd () + fd () + fd () + Subiuing u, u + T, u + T, in h ucciv ingrl, hn T T T u ( u+ T) ( u+ T) f() f( u) du + f( u + T) du + f( u + T) du + Sinc f(u) f(u + T) f(u + T), T T T u T u T u T T T u ( ) f( u) du T u f() u du T f [ fudu + fudu + fudu + ( ()) f ( ) T f() d T ( ) (On chnging h dummy vribl u o ) Obrvion: Hnc h lplc rnform of priodic funcion f() wih priod T i xpribl in h form of ingrl T f() d ovr on priod of f() ind of ovr h mi-infini inrvl. Exmpl 47: Find h lplc rnform of h ringulr wv of priod givn by f (), < <, < < } [Mdr, ; UP Tch, ; VTU, ; KUK, 4; NIT Jlndhr, 6] Or Find h lplc rnform of h ringulr wv funcion f() who grph i hown blow. Soluion: Clrly h grph of f() i righ lin from h origin wih lop in h inrvl o f(), o Howvr in h inrvl, h grph i righ lin who lop i -.

63 plc Trnform nd Thir Applicion 87 f() ( )( ), Thu, h funcion i priodic on, wih priod f() 4 Fig.. f [ ] fd + f() d ( ) d o ( ) + + o ( ) ( ) / / / ( ) / / / / / / ( + ) nh Exmpl 48: Find h lplc rnform of h full wv rcifir f() E inω, o < < π/ω; hving priod π/ω. Soluion: π/ w f() π Einωd ω

64 88 Enginring Mhmic hrough Applicion E π ω ω ω +ω ω ( in co ) π/ w / E π ω π in co ( π ) πω ω ω +ω ω +ω ω π / ω E ω π ω+ +ω +ω ω E ω ( π / ω ) π / ω +ω + Eω + ( +ω) π / ω π / ω Eω coh π x x x + + coh x ( +ω) ω Q x x x Th grph for h funcion f() which i full-in wv rcifir cn b drwn blow: Y y πω / πω / πω / πω / Fig.. X Sinc in w π + inω for ll. ω Exmpl 49: Find h lplc rnform of h qur wv (or mndr) funcion of priod dfind : f() whn < < /, whn / < < [UP Tch, 4] Soluion: [ f] () f() d o + / d d o /

65 plc Trnform nd Thir Applicion 89 / o / / / + + ( ) / / ( ) ( ) ( ) / /4 /4 / /4 /4 nh No: I my b dfind lik, f() k, whn < < / k, whn / < < Y k X o / 5 k X Fig.. Exmpl 5: Find h plc Trnform of h priodic funcion (w ooh wv), f( + T) f(); f () k for < < T T Soluion: T T T T f [ ] fd k d T T k T T T ( ) d k d T (By pr) k T T ( ) T o T o

66 8 Enginring Mhmic hrough Applicion T T k T + T T ( ) T k T T + T ( ) k k T T T ( ) T ( ) No: I my rul o priculr c: whn k, ft T f () T T T T Fig.. Exmpl 5: Find h plc rnform of h funcion (Hlf wv rcifir) π in ω, for < < f () ω π π, for < < ω ω [Mdr, ; KUK, 5] Soluion: f [ ] fd T T π ω π/ ω f() d π f () h priod, T Q ω / / π ω π ω π inω d + o d π/ ω ω π/ ω π in ωd, o ω x, + b Uing x inbxdx ( inbx bcobx)

67 plc Trnform nd Thir Applicion 8 ( inω ωcoω ) π +ω ω π ω ω +ω π ( ) +ω ω π +ω ω + w π +ω ω ω +ω ( ) π ω π/ ω πω / πω / π ω 4π ω 5π ω Fig..4 No: Clrly h funcion f() inω rprn h hlf in wv rcifir wih priod π ω. Th grph of f() in h π π inrnl < < i in funcion which vrnih nd nd f() for h r hlf from ω ω π π o ω ω. Exmpl 5: A priodic qur wv funcion f(), in rm of uni p funcion, i wrin f() k [u () u () + u () u () + ]. Show h h plc rnform of f() i k givn by [ f] nh. Soluion: f() k[u () u () + u () u () + ] f() k [u () u () + u () u () + ] k + + k + + k ( + ) k +

68 8 Enginring Mhmic hrough Applicion k + + k k + + k nh f () T T T T Fig..5 ASSIGNMENT 6 Find h plc rnform of h following priodic funcion: π. f () in ω, whn f () f ( + T); T ω [Hin: f() rprn full in wv rcifir. Ex. 48]. Sw ooh wv funcion of priod T givn by f () +, < T T, < b. Non Symmric qur wv of priod T, givn by f (), b < T 4. Sirc funcion dfind by f() kn, nt < <(n + )T; whr n,,,

69 plc Trnform nd Thir Applicion 8 ANSWERS Aignmn. (i) , (ii) ( + ) , (iii) ( ) , (iv) (i) + 4, (v), , (ii) ( ). (i) ( ) ( 5 ) (vi) (ii) 4, π, > + 4. (i) ( + ), (ii) ( + ), (iii) ( ), > (iv) + ( ), 5. (i) ( + ), (ii) ( + ), (iii) ( ), > (iv) ( ), > 6. (i) ( + ), (ii) ( ), 7. (i) b log + + / (ii) 4 log + 8. (iii) co, + + (iv) co ( + ) 9., +. (i) + + (ii) b log + + Aignmn. (i) inh, co in (ii) ( + )

70 84 Enginring Mhmic hrough Applicion (iii) / co in +, (iv) (v) 5 5 in A B b b B A Ab inb bc. ( ) co + ( + ), (ii) 4, (iii) 4 +. (i) ( 6co 7in) 4. (i) in, (ii) in, (iii) 5. (i) (iv) 6. (i) co + b (vii) ( co) b co (ii), (v) 7. (i) (i) b b, (viii) in, +, (iii) in, (ix) inh inh in, (vi) ( coh) co cob (ii) ( ), (iii) ( 4 + ) 4 (ii) ( ) + (iii) ( in ) in co, (ii) ( ), (iii) ( + ) + ( ) 64 co co (iv) ( in ), (v) 8 ( + 8), (vi) ( ) (vii) ( in binb) ( b ), Aignmn. ω in co x ω ω ω B + +. y Acoω x + inωx ω + ω + ω. y ( ) ( ) x [inncoα nco( n +α)] n

71 plc Trnform nd Thir Applicion y + 6. y x + 8. y [( )in co ] 8 9. y (in + in ). y + in. y. f() co + in Aignmn x +. y + in x ( + 6) + ( + ), 7 7 y ( ) ( ) x in in, x y in in, y x 6 y 5 i p+ω ω + p i p+ω ω p ( in in ) ( co co ) 6 Aignmn 5 π (i) ( u ) ( π ) + u, + (ii) k b ω + φ (ii) ( ). (i) [() u u ( T)co( ) (iv) ( ) ( ) π (v), + (vi) (iii) (), u [ h ] h + + π (i) inu π () (ii) ( ) u () (iii) cob( ) u () (iv) ( u ) y () f () f ( u ) (), whr f () + 6 ωx ( l5x) l, < x < 8 EI yx () ω x ( l 5x) ω l x l +, < x < l 8 EI 6EI π in π

72 86 Enginring Mhmic hrough Applicion ωl l 4 ω ω 4 ω 6. yx () x x + x ( xl ) u( xl ) 6EI EI 4EI 4EI Aignmn 6.. Eω co π h + w ω T b ( + ) T ( ). 4. T k T T ( ) T coh