FACULTY OF NATURAL SCIENCES CONSTANTINE THE PHILOSOPHER UNIVERSITY IN NITRA ACTA MATHEMATICA 7 FOURIER TRANSFORM AN ITS APPLICATION ARINA STACHOVÁ ABSTRACT By ans of Fourir sris can b dscribd various xapls of wav oion, such as h sound, or wav of h arhquak I can b usd in any rsarch or work, such as h daa analysis afr an arhquak or digiizing usic Gnralizaion of Fourir sris, which allows for so applicaions or appropria xprssion is h Fourir ingral Fourir ransfor basd on a Fourir ingral in h coplx for Fourir ransfor is an iporan ool in a nubr of scinific filds Is advanags, disadvanags and sublis hav bn xaind any is by dozns of ahaicians, physiciss and nginrs In his conribuion w ry o suariz iporan aspcs of his ransfor and discuss variy of is uss in conporary scinc wih phasis on donsraing conncions o dynaic inracions in h vhicl-roadway sys KEY WORS: Fourir ransfor, i sris, frquncy rprsnaion CLASSIFICATION: I55, I85, M55 Rcivd April 4; rcivd in rvisd for May 4; accpd 3 May 4 Inroducion Epirical asurns in various doains conoical, chnical, or ohr ar ofn urnd ino i sris Basd on his i is possibl o prfor analysis, which in urn allows us o br undrsand h dynaics of h facors involvd To his nd, w us h Fourir ransfor Bing discovrd a h urn of h 9 h cnury, h hory of h Fourir ransfor is currnly usd in signal procssing such as in iag sharpning, nois filring, c For us h rlvan applicaion is in h hory of dynaic inracion in h vhicl-roadway sys Th basis of h Fourir ransfor is h so-calld Fourir apping, i, h ransforaion of on funcion o anohr, fro propris of which w can obain inforaion abou h original funcion Fourir ransfor xprsss a i-dpndn signal using haronic signals, i, h sin and cosin funcions, in gnral funcions of coplx xponnials I is usd o ransfor signals fro h i doain o h frquncy doain A signal can b coninuous or discr S Figur for a dpicion of h corrspondnc bwn h i-basd and frquncy-basd rprsnaion Figur : Apliud frquncy diagra 55
ARINA STACHOVÁ Fourir ingral A gnralizaion of h Fourir sris ha pris in so applicaions a or appropria xprssion of a non-priodical funcion dfind alos vrywhr in R is h Fourir ingral Thor : L f: R R b a funcion ha a) is pic-wis coninuous on R along wih is drivaiv f ', b) is absoluly ingrabl on R, i, f d convrgs Thn all R saisfy ~ ~ f d f s cos ss d, whr f li f s li f s () s s No : Fro h clai of Thor, i follows ha h valus of h doubl ingral on h righ-hand sid of (Eq ) is qual o f() for ach R in which f is coninuous and is qual o h arihic an of h lf- and righ- liis of his funcion in ach poin of disconinuiy, providd h condiions a) and b) of his clai hold No : Using h fac ha R, s R and ω ; w hav cos ω( s) = = cos ω cos ω s + sin ω sin ω s, w can rwri () in h for ~ f a cos b sin d, () whr a(ω) = f s cos, b(ω) = f s sin, ω ; (3) finiion : Th righ-hand sid of (Eq ) is calld h doubl Fourir ingral of f: R R Th righ-hand sid of (Eq ) is h singl Fourir ingral of f No 3: I is no difficul o s ha h singl Fourir ingral (Eq ) is a gnralizaion of h doubl Fourir ingral and h funcions a: ; R, b: ; R dfind by (Eq 3) ar a gnralizaion of h sandard Fourir cofficins of a priodic funcion I is clar ha if f: R R is an vn funcion, hn b(ω) =, ~ a f s cos and f a cos d = f s cos s cos ds d f s Siilarly, if f is odd, w hav: a(ω) =, b(ω) = sin ~ f b sin d = d f s sin s sin ds and w hav No 4: Using h wll-known Eulr s forula for xponnial and gonioric i i i i in h singl Fourir ingral funcions: cos ω =, sin ω = i ib a ib ~ i i (Eq ), w obain R: a f d aib aib By ling = c(ω), c for ω ;, w hav for all R: f c d ~ i = c( ω) = whr c(ω) = [a(ω) ib(ω)] = f s i, (4) for all ω R (5) finiion : Th righ-hand sid of (4), whr c: R C is dfind by (5) is calld h Fourir ingral of f: R R in a coplx for 56
FOURIER TRANSFORM AN ITS APPLICATION 3 Fourir ransfor finiion 3: L f: R R along wih is drivaiv f ' b pic-wis coninuous on R, and l f b absoluly ingrabl on R Thn w call f h sourc of Fourir ransfor L h s of such funcions f: R R b dnod by F Thn h funcion F: iω F(f()), whr F(f()) = F(iω) = f i d whr ω, (6) ; is calld h Fourir iag of f and h apping F fro h s of funcions F dfind by (Eq 6) is calld h forward Fourir ransfor Thor : If f F, hn hr xiss a Fourir iag F(iω) of f dfind by (Eq6), ~ i which saisfis i f F d, (7) ~ whr f li f s li f s for all s s R finiion 4: Th apping F ( F ) dfind by (7) is calld h invrs Fourir ransfor, i, F ~ i (F(iω)) = i f F d, R No 5: Th Fourir iag F(iω) is in h chnical liraur ofn calld h spcral characrisic of f Is agniud F(ω) = F(iω) is h apliud characrisic of f, funcion α(ω) = Arg F(iω), ω ; is calld h phas characrisic of f and h funcion P(ω) = F(iω) powr characrisic (powr spcru) of f Hnc, for all ω R F(iω) = F(ω) iα(ω) = A(ω) ib(ω), whr A(ω) = f cos d, B(ω) = f sin d Fro his i follows ha F(ω) = B A, α(ω) = arcan [B(ω)/A(ω)], which ans ha h apliud funcion F(ω) is an vn funcion and h phas funcion α(ω) is an odd funcion of h indpndn variabl (frquncy) ω 4 Us of Fourir ransfor in solving rprsnaiv probls Rcall ha according o h Eulr s forula w can wri iω = cos ω + i sin ω a Exapl : Find h Fourir iag of h funcion f(): R R, f() =, a R + Soluion: Fro quaions (Eq 4) and (Eq 5) i follows ha a i s ai s c(ω) = a s i s ai s ai s ds ds ds a i a i ~ i =, i, f f d a i a i a a 57
ARINA STACHOVÁ Figur : Coparing a graph of a funcion wih is Fourir iag a a Thrfor h Fourir iag of is F( ) = a a Fourir ransfor has a wid variy of uss; w hav alrady shown so of h for illusraion Fourir ransfor is also usd o solv diffrnial quaions Th ky ida is ha h Fourir ransfor ransfors h opraion of aking drivaivs ino uliplicaion of h iag by h indpndn variabl If w prfor h Fourir ransfor using all indpndn variabls w obain as iag a soluion of h quaion wih no drivaivs Whn w solv i, i suffics o find h Fourir priag which usually is h os difficul par Unforunaly, i can also happn ha h soluion has no priag Thn his hod dos no work Howvr, w ay prfor h Fourir ransfor using only so indpndn variabls This yilds a diffrnial quaion wih fwr variabls and wih parars, which igh b asir o solv han h original quaion; nonhlss h ulia difficuly ay sill b in finding h priag Exapl : Using h Fourir ransfor find a soluion of h diffrnial quaion saisfying h following condiions: a) y'() + k y () = a, whr k R + {}, b R, R, li y = li y =, b) y''() + 3y'() + y () = y y =, R, = li li No: Fourir ransfor can also b usd for solving ordinary linar diffrnial quaions wih consan cofficins assuing ha h soluion of such quaion along wih is drivaivs of ordr up o h ordr of h quaion has propris fro finiion 3 Soluion: L y, y', y'' F and wri F(y()) = Y(iω) Thn F(y'()) = iωy(iω), F(y''()) = ω Y(iω) a) Sinc F( a i a i ) = a, w hav y() = Y i d = d a k Thus y() = for ;, y() = a k for ; k k b) Sinc F( i ) =, i follows ha y() = Y i d = i d Hnc y() = for ;, y() =, and y() = for ; 6 6 3 Exapl 3: So siuaions rquir spcifying h probl using a diagra Hr h subjc of analysis is h so-calld quarr odl of a vhicl shown in Figur 3 This copuaional odl rprsns on half of on axl of a vhicl Unvnnss of h road surfac is h ain sourc of kinaic xciaion of h vhicl Th vhicl s rspons o his xciaion can b found nurically in boh h i and frquncy doain In h i doain w ar ainly inrsd in i voluion of conac forcs and in h frquncy doain in h powr spcral dnsiis of powr forcs in rlaion o h powr spcral dnsiis of h unvnnss of h road [] As an xapl, w us nurical characrisics of h vhicl Tara odl T48 Wigh parars of h odl: Rigidiy consans of h coupling: a 93 kg 455 kg - k 43 76,5 N k 75 3, N - 58
FOURIER TRANSFORM AN ITS APPLICATION - aping cofficins: b 9 64, kg s No: Inhrn par of h procss of solving h probl is h forulaion of siplifid odls of h vhicl, hir ahaical dscripion, and drinaion of h vhicl s rspons in h i doain Copuaional odls of vhicls can hav varid coplxiy dpnding on h naur of probl o b solvd Ofnis h so-calld quarr- of half odls ar usd; hs odls odl oion and ffcs of a quarr or half of h vhicl Nowadays, howvr, i is no uncoon o us spaial odls of vhicls Figur 3: Quarr odl of a vhicl Th law of consrvaion of chanical nrgy is a spcial cas of h consrvaion of nrgy law, which applis o all yps of nrgy In h cas of dissipaiv forcs such as fricional forcs, par of h chanical nrgy is convrd o ha, bu h oal aoun of nrgy rains h sa Soluion: Applying a gnral procdur [] o h odl fro Figur 3, w obain quaions of oion of h odld vhicl Wih ha w also obain xprssions dscribing inracion forcs a h poin of conac of h vhicl s axl wih h road surfac r ' ' k r r b r ' r ' r '' kr r kr h b r ' r ' br ' h' (8) Using h principl of qual acion and racion, w driv h following: F FRV G kr h b r' h' Fs Fdyn, i, F s = G and F dyn () = k r h b r ' h' (9) W rarrang h quaions (8) as follows: r '' br ' br ' kr kr r '' br ' br ' br ' bh' kr kr kr kh F dyn () = b r ' bh' kr kh () Funcion f() and is i drivaiv will b hn ransford in his way: f a F, f ' for f(± ) = o i F, f ' r a o ' for f ' (± ) = f (± )= o F Th coplx Fourir ransfor of (Eq ) afr rarranging has h following for: r ω b ω k r i b ω k i b ω k r ω i b ω i b ω k k i b ω k i Fdyn r i b ω k+ i b ω k Th firs wo quaions of (Eq ) can b wrin as a r PS () or in h arix a a r PS for as a a r PS () A soluion is h found using h Crar s rul, i, r r (3) whr = a a a a, = PS a PS a, = a PS PS a, 59
ARINA STACHOVÁ If w hn considr ha in h Fourir ransfor h parar rprsns h rad angular frquncy in, hn h cofficins a ij in (Eq ) hav h following for: k a k k b s a i b, k i b a, PS = (k ) +, PS = + i, k a i b, i b i b Th xprssion (Eq3) is calculad nurically for chosn valus of ω in h slcd frquncy band In his soluion w ignor h daping of h ir, i b = [kg s - ] Th soluion hus applis o h siplifid odl shown in Fig 3 Sinc b =, w hav k r 5 Conclusion Why do w us ransforaions? For various rasons, for insanc: Transforaions allow ransforing a coplicad probl o a ponially siplr on Th probl can b hn solvd in h ransfor doain Using h invrs ransfor w obain soluions in h original doain Fourir ransfor is appropria for priodical signals - I allows uniquly ransforing a signal fro/o i rprsnaion f() o/fro frquncy rprsnaion F(iω) - I allows analyzing h frquncy conn (spcru) of a signal (for insanc in noninvasiv hods arial diagnosics or agnic rsonanc) Th basis of vry xprinal scinc is asurn, sinc i is h only ool o quaniaivly dscrib propris of ral-world physical procsss Soluion of dynaical probls can b ralizd boh in h i and h frquncy doain Boh fors hav hir advanags, copln on anohr and rprsn wo diffrn facs of h sa physical phnonon Rfrncs [] Mlcr, J, Lajčáková, G Aplicaion of progra sys Malab for h soluion of srucural dynaic probls (in Slovak) Zilina: ZU v Zilin EIS, 66 p ISBN 978-8-554-38-3 (Lcur nos) [] Mlcr, J Th us of Fourir and Laplac ransfor on h soluion of vhiclroadway inracion probls In: Civil and nvironnal nginring, Vol 8, No,, ISSN 336-5835 (scinific chnical journal) [3] Moravčík, J Mahaics 5, Ingral ransfors (in Slovak), Zilina, ZU v Zilin EIS,, 9 p ISBN 8-7-776-5 (Lcur nos) Auhor s Addrss RNr arina Sachová, Ph parn of ahaics, Faculy of Huaniis, Univrsiy of Zilina in Zilina, Univrzina, SK- 6 Zilina; -ail: darinasachova@fhvunizask Acknowldgn This work was producd as par of h projc VEGA SR /59/ F dyn 6