LECTURE NOTES OF ENGINEERING MATHEMATICS III Su Cod: MAT) Vtusoutio.i COURSE CONTENT ) Numric Aysis ) Fourir Sris ) Fourir Trsforms & Z-trsforms ) Prti Diffrti Equtios 5) Lir Agr 6) Ccuus of Vritios Tt Book: Highr Egirig Mthmtics y Dr. B.S.Grw 6th Editio ) Kh Puishrs,Nw Dhi Rfrc Book: Advcd Egirig Mthmtics y E. Kryszig 8th Editio ) Joh Wiy & Sos, INC. Nw York DEFINITIONS : FOURIER SERIES Dr.A.T.Eswr Profssor d Hd Dprtmt of Mthmtics P.E.S.Cog of Egirig Mdy -57 A fuctio y = f) is sid to v, if f-) = f). Th grph of th v fuctio is wys symmtric out th y-is. A fuctio y=f) is sid to odd, if f-) = - f). Th grph of th odd fuctio is wys symmtric out th origi. Vtusoutio.i For mp, th fuctio f) = i [-,] is v s f-) = = f) d th fuctio f) = i [-,] is odd s f-) = - = -f). Th grphs of ths fuctios r show ow : Vtusoutio.i
Vtusoutio.i Grph of f) = Grph of f) = Not tht th grph of f) = is symmtric out th y-is d th grph of f) = is symmtric out th origi.. If f) is v d g) is odd, th h) = f) g) is odd h) = f) f) is v h) = g) g) is v For mp,. h) = cos is v, sic oth d cos r v fuctios. h) = si is v, sic d si r odd fuctios. h) = si is odd, sic is v d si is odd.. If f) is v, th. If f) is odd, th f ) d f ) d Vtusoutio.i For mp, d f ) d cos d cosd, s cos is v si d, s si is odd Vtusoutio.i
PERIODIC FUNCTIONS :- A priodic fuctio hs sic shp which Vtusoutio.i is rptd ovr d ovr gi. Th fudmt rg is th tim or somtims distc) ovr which th sic shp is dfid. Th gth of th fudmt rg is cd th priod. A gr priodic fuctio f) of priod T stisfis th coditio f+t) = f) Hr f) is r-vud fuctio d T is positiv r umr. As cosquc, it foows tht f) = f+t) = f+t) = f+t) =.. = f+t) Thus, f) = f+t), =,,,.. Th fuctio f) = si is priodic of priod sic Si+ ) = si, =,,,.. Th grph of th fuctio is show ow : Not tht th grph of th fuctio tw d is th sm s tht tw d d so o. It my vrifid tht ir comitio of priodic fuctios is so priodic. Vtusoutio.i FOURIER SERIES A Fourir sris of priodic fuctio cosists of sum of si d cosi trms. Sis d cosis r th most fudmt priodic fuctios. Th Fourir sris is md ftr th Frch Mthmtici d Physicist Jcqus Fourir 768 8). Fourir sris hs its ppictio i proms prtiig to Ht coductio, coustics, tc. Th sujct mttr my dividd ito th foowig su topics. Vtusoutio.i
FOURIER SERIES Vtusoutio.i Sris with ritrry priod FORMULA FOR FOURIER SERIES Cosidr r-vud fuctio f) which oys th foowig coditios cd Diricht s coditios :. f) is dfid i itrv,+), d f+) = f) so tht f) is priodic fuctio of priod.. f) is cotiuous or hs oy fiit umr of discotiuitis i th itrv,+).. f) hs o or oy fiit umr of mim or miim i th itrv,+). Aso, t Hf-rg sris Comp sris Hrmoic Aysis f ) d f )cos f )si d, d, Vtusoutio.i ),,,... ),,,... ) Th, th ifiit sris cos si ) is cd th Fourir sris of f) i th itrv,+). Aso, th r umrs,,,., d,,. r cd th Fourir cofficits of f). Th formu ), ) d ) r cd Eur s formu. It c provd tht th sum of th sris ) is f) if f) is cotiuous t. Thus w hv f) = cos si. 5) Suppos f) is discotiuous t, th th sum of th sris ) woud f ) f ) whr f + ) d f - ) r th vus of Vtusoutio.i f) immdity to th right d to th ft of f) rspctivy.
Prticur Css Cs i) Suppos =. Th f) is dfid ovr Vtusoutio.i th itrv,). Formu ), ), ) rduc to f ) d f )cos f )si d, d,,,... 6) Th th right-hd sid of 5) is th Fourir psio of f) ovr th itrv,). If w st =, th f) is dfid ovr th itrv, ). Formu 6) rduc to Aso, i this cs, 5) coms Cs ii) = f ) d f )cosd, =,,.. 7) f )sid =,,.. f) = cos si Suppos =-. Th f) is dfid ovr th itrv -, ). Formu ), ) ) rduc to f )cos d Vtusoutio.i f ) d 8) =,, 9) f )si d, Th th right-hd sid of 5) is th Fourir psio of f) ovr th itrv -, ). =,, If w st =, th f) is dfid ovr th itrv -, ). Formu 9) rduc to = f ) d Vtusoutio.i
f ) cosd, =,,.. ) Vtusoutio.i f ) sid =,,.. Puttig = PARTIAL SUMS i 5), w gt f) = cos si Th Fourir sris givs th ct vu of th fuctio. It uss ifiit umr of trms which is impossi to ccut. Howvr, w c fid th sum through th prti sum S N dfid s foows : N SN ) cos si whr N tks positiv itgr vus. I prticur, th prti sums for N=, r S ) cos si S ) cos si cos si If w drw th grphs of prti sums d compr ths with th grph of th origi fuctio f), it my vrifid tht S N ) pproimts f) for som rg N. Som usfu rsuts :. Th foowig ru cd Broui s grizd ru of itgrtio y prts is usfu i vutig th Fourir cofficits. ' '' uvd uv u v u... Hr u, u v,.. r th succssiv drivtivs of u d Vtusoutio.i v vd, v vd,... W iustrt th ru, through th foowig mps : cos si cos sid d 6 8 6 6. Th foowig itgrs r so usfu : Vtusoutio.i
cosd sid cos si Vtusoutio.i si cos. If is itgr, th si =, cos = -), si =, cos = Emps. Oti th Fourir psio of W hv, f) = i - < < f ) d ) d = f )cosd Hr w us itgrtio y prts, so tht si cos ) ) cosd cos si Vtusoutio.i ) )sid Usig th vus of, d i th Fourir psio ) w gt, f ) cos si Vtusoutio.i
) f ) si Vtusoutio.i This is th rquird Fourir psio of th giv fuctio.. Oti th Fourir psio of f)= - i th itrv -, ). Dduc tht Hr, cosch = d ) sih ) sih cosd sid cos si = si cos ) sih = Thus, sih sih f) = For =, =, th sris rducs to sih sih f)= = cos sih si Vtusoutio.i ) ) ) or = sih sih ) Vtusoutio.i
sih or = Thus, cosch ) Vtusoutio.i ) This is th dsird dductio.. Oti th Fourir psio of f) = ovr th itrv -, ). Dduc tht... 6 Th fuctio f) is v. Hc = f ) d= or = d = = Itgrtig y prts, w gt ) f ) cosd f ) d f )cosd, sic f)cos is v cosd si cos si Vtusoutio.i Aso, f )sid sic f)si is odd. Thus Vtusoutio.i
f ) 6 ) cos Hc,... 6. Oti th Fourir psio of, f ), Dduc tht 8 5 Th grph of f) is show ow. Hr, = f ) d=... Hr OA rprsts th i f)=, AB rprsts th i f)= -) d AC rprsts th i =. Not tht th grph is symmtric out th i AC, which i tur is pr to y-is. Hc th fuctio f) is v fuctio. f ) d Vtusoutio.i = d f )cosd cosd Vtusoutio.i f )cosd sic f)cos is v. Vtusoutio.i
si cos = Vtusoutio.i ) Aso, f )sid, sic f)si is odd Thus th Fourir sris of f) is For =, w gt or Thus, f ) ) cos f ) ) cos cos ) ) 8 ) or... 8 5 This is th sris s rquird. 5. Oti th Fourir psio of, f) =, Dduc tht 8 5 Hr,... Vtusoutio.i d d ) cosd cosd Vtusoutio.i
sid sid Vtusoutio.i ) Fourir sris is ) f) = ) cos si Not tht th poit = is poit of discotiuity of f). Hr f + ) =, f - )=- t =. Hc [ f ) f )] Th Fourir psio of f) t = coms [ ) ] Simpifyig w gt, or 8 [ 5 )... 6. Oti th Fourir sris of f) = - ovr th itrv -,). Th giv fuctio is v, s f-) = f). Aso priod of f) is --)= Hr = = ] f ) d= f ) d ) d Vtusoutio.i = Itgrtig y prts, w gt f )cos f )cos ) d ) d ) cos ) d s f) cos ) is v si ) Vtusoutio.i cos ) ) si )
) Vtusoutio.i f )si ) d =, sic f)si ) is odd. = Th Fourir sris of f) is f) = 7. Oti th Fourir psio of i f) = i Dduc tht 8 ) cos ) 5... Th priod of f) is Aso f-) = f). Hc f) is v Aso, / Vtusoutio.i = / / / / / / / / f ) d d si / f )cos d / f )cos d / ) f ) d Vtusoutio.i cos /
f )si Vtusoutio.i d Thus f) = puttig =, w gt f) = ) ) 8 or =... 5 Thus,... 8 5 NOTE cos Hr vrify th vidity of Fourir psio through prti sums y cosidrig mp. W rc tht th Fourir psio of f) = ovr -, ) is ) cos f ) Lt us dfi N ) cos SN ) Th prti sums corrspodig to N =,,..6 S ) S ) r cos cos cos S6 ) cos cos cos cos cos5 cos5 9 5 9 Th grphs of S, S, S 6 gist th grph of f) = r pottd idividuy d show ow : Vtusoutio.i Vtusoutio.i
Vtusoutio.i O compriso, w fid tht th grph of f) = coicids with tht of S 6 ). This vrifis th vidity of Fourir psio for th fuctio cosidrd. Ercis Chck for th vidity of Fourir psio through prti sums og with rvt grphs for othr mps so. HALF-RANGE FOURIER SERIES Th Fourir psio of th priodic fuctio f) of priod my coti oth si d cosi trms. My tim it is rquird to oti th Fourir psio of f) i th itrv,) which is rgrdd s hf itrv. Th dfiitio c tdd to th othr hf i such mr tht th fuctio coms v or odd. This wi rsut i cosi sris or si sris oy. Si sris : Suppos f) = ) is giv i th itrv,). Th w dfi f) = - -) i -,). Hc f) coms odd fuctio i -, ). Th Fourir sris th is Vtusoutio.i f ) si ) whr f )si d Th sris ) is cd hf-rg si sris ovr,). Puttig = i ), w oti th hf-rg Vtusoutio.i si sris of f) ovr, ) giv y
f ) si Vtusoutio.i Cosi sris : Lt us dfi f ) f )sid i,)... giv i -,)..i ordr to mk th fuctio v. Th th Fourir sris of f) is giv y f ) cos ) whr, f ) d f )cos Th sris ) is cd hf-rg cosi sris ovr,) Puttig = Emps : i ), w gt d Vtusoutio.i. Epd f) = -) s hf-rg si sris ovr th itrv, ). W hv, f ) whr Itgrtig y prts, w gt f )sid )sid ) f ) d ) cos f )cosd,,,.. Vtusoutio.i
Th si sris of f) is f ). Oti th cosi sris of Hr d ) cos si Vtusoutio.i ) si, f ) ovr, ), cosd ) d )cosd Prformig itgrtio y prts d simpifyig, w gt ) cos 8,,6,,... Thus, th Fourir cosi sris is cos cos6 cos f) =... 5 cos Vtusoutio.i. Oti th hf-rg cosi sris of f) = c- i <<c Hr c c ) d c c )cos d c c Itgrtig y prts d simpifyig w gt, c c Vtusoutio.i )
c ) Th cosi sris is giv y Vtusoutio.i Ercics: f) = c c ) cos c Oti th Fourir sris of th foowig fuctios ovr th spcifid itrvs :. f) = ovr -, ). f) = + ovr -, ). f) = ovr, ). f) = ovr -, ) ; Dduc tht... 5 5. f) = ovr -, ) ; Dduc tht... 8, 6. f) = ovr -, ), Dduc tht... 8, 7. f) =,, ovr -, ) Dduc tht 5... 8. f) = si ovr ; Dduc tht 9. f) =,, ovr -, ). f) = -) ovr,). f) = ovr -,). f) =, ), Vtusoutio.i Vtusoutio.i
Oti th hf-rg si sris of th foowig fuctios ovr th spcifid itrvs :. f) = cos ovr, ). f) = si ovr, ) 5. f) = - ovr, ) Vtusoutio.i Oti th hf-rg cosi sris of th foowig fuctios ovr th spcifid itrvs : 6. f) = ovr, ) 7. f) = si ovr, ) 8. f) = -) ovr,) 9. f) = k, k ), HARMONIC ANALYSIS Th Fourir sris of kow fuctio f) i giv itrv my foud y fidig th Fourir cofficits. Th mthod dscrid cot mpoyd wh f) is ot kow picity, ut dfid through th vus of th fuctio t som quidistt poits. I such cs, th itgrs i Eur s formu cot vutd. Hrmoic ysis is th procss of fidig th Fourir cofficits umricy. To driv th rvt formu for Fourir cofficits i Hrmoic ysis, w mpoy th foowig rsut : Th m vu of cotiuous fuctio f) ovr th itrv,) dotd y [f)] is dfid s f ) f ) d. Th Fourir cofficits dfid through Eur s formu, ), ), ) my rdfid s f ) d f )cos Vtusoutio.i f )si [ f )] d d f )cos f )si Usig ths i 5), w oti th Fourir sris of f). Th trm cos+ si is cd th first hrmoic or fudmt hrmoic, th trm cos+ si is cd th scod hrmoic d so o. Th mpitud of th first hrmoic is hrmoic is d so o. Vtusoutio.i d tht of scod
Emps. Fid th first two hrmoics of th Fourir sris of f) giv th foowig t : Vtusoutio.i Vtusoutio.i 5 f)...9.7.5.. Not tht th vus of y = f) r sprd ovr th itrv d f) = f ) =.. Hc th fuctio is priodic d so w omit th st vu f ) =. W prpr th foowig t to comput th first two hrmoics. y = f) cos cos si si ycos W hv Puttig, =,, w gt f )cos f )si [ ycos] [ ycos] [ y cos] [ ysi] ycos.) 6 6 ycos.) Vtusoutio.i 6 6 s th gth of itrv= = or =.67. ycos ysi. ysi 6..5 -.5.866.866.7 -.7...9 -.5 -.5.866 -.866 -.95 -.95.65 -.65 8.7 - -.7.7.5 -.5 -.5 -.866.866 -.75 -.75.99.99..5 -.5 -.866 -.866.6 -.6 -.9 -.9 Tot -. -..76 -.7
ysi [ ysi ].9 6 Vtusoutio.i ysi [ ysi].577 6 Th first two hrmoics r cos+ si d cos+ si. Tht is -.67cos +.9 si) d -.cos.577si). Eprss y s Fourir sris upto th third hrmoic giv th foowig vus : 5 y 8 5 7 6 Th vus of y t =,,,,,5 r giv d hc th itrv of shoud gth of th itrv = 6- = 6, so tht = 6 or =. Th Fourir sris upto th third hrmoic is or y cos si cos si cos < 6. Th si y cos si cos si cos si Put, th y cos si cos si cos si ) W prpr th foowig t usig th giv vus : = y ycos ycos ycos ysi ysi ysi 6 8 - -8 6.98 6.98 Vtusoutio.i 5-7.5-7.5 5.99 -.99 8 7-7 7-7 6 - - 6-5.96 5.96 5 - - -.7 -.7 Tot -8.5 Vtusoutio.i -.5 8.99 -.598
Usig ths i ), w gt [ f )] [ Vtusoutio.i y y] ) 6 [ y cos ] 6 8.5).8 [ ysi ].99) 6. [ y cos ] 6.5).5 [ ysi ] 6.598).866 [ y cos ] 8) 6.667 [ ysi ] y 7,8cos.)si.5cos.866si This is th rquird Fourir sris upto th third hrmoic. Vtusoutio.i.667cos. Th foowig t givs th vritios of priodic currt A ovr priod T : tscs) T/6 T/ T/ T/ 5T/6 T A mp).98..5. -.88 -.5.98 Show tht thr is costt prt of.75mp. i th currt A d oti th mpitud of th first hrmoic. Not tht th vus of A t t= d t=t r th sm. Hc At) is priodic fuctio of priod T. Lt us dot t. W hv T [ A] Acos t [ Acos ] ) T Asi t [ Asi ] T W prpr th foowig t: Vtusoutio.i
t t T A cos si Acos Asi Vtusoutio.i.98.98 T/6 6..5.866.65.58 T/.5 -.5.866 -.55.99 T/ 8. - -. T/ -.88 -.5 -.866..76 5T/6 -.5.5 -.866 -.5.65 Tot.5..7 Usig th vus of th t i ), w gt A.5 6.5 Acos. 6.7 Asi.7 6.6 Th Fourir psio upto th first hrmoic is A cos t T si t t.75.7cos.6si T T Th prssio shows tht A hs costt prt.75 i it. Aso th mpitud of th first hrmoic is t T Vtusoutio.i =.77. Vtusoutio.i
ASSIGNMENT :. Th dispcmt y of prt of mchism is tutd with corrspodig gur movmt of th crk. Eprss y s Vtusoutio.i Fourir sris upto th third hrmoic. 6 9 5 8 7 y.8...6.5..6.5..5.76.. Oti th Fourir sris of y upto th scod hrmoic usig th foowig t : 5 9 5 8 5 7 5 6 y..8.. -.5.8.. Oti th costt trm d th cofficits of th first si d cosi trms i th Fourir psio of y s giv i th foowig t : 5 y 9 8 8 6. Fid th Fourir sris of y upto th scod hrmoic from th foowig t : 6 8 Y 9. 8.. 7.8 7.5. 9. 5. Oti th first thr cofficits i th Fourir cosi sris for y, whr y is giv i th foowig t : 5 y 8 5 7 6 Vtusoutio.i 6. Th turig momt T is giv for sris of vus of th crk g = 75. Vtusoutio.i
6 9 5 8 Vtusoutio.i T 5 897 785 599 66 Oti th first four trms i sris of sis to rprst T d ccut T t = 75. Vtusoutio.i Vtusoutio.i