No 6 Frqucy Rpo Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada
Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada. alyical Exprio for Frqucy rpo iv ha a liar im-ivaria ym i rprd by a rafr fucio of. L h ym ipu b a iuoid, i.., r i, whr i h ampliud ad i h frqucy of h ipu. Th Laplac raform of r i giv by R Thu, o ha h ym oupu C R C or C Prformig a parial fracio xpaio o C yild fracio rm from Parial C whr [ ] S C [ ] S C C R
Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada 3 Pla o ha ad - ar complx umbr. By uig Eulr formula, h wo complx umbr ca b rprd, rpcivly, by uig h followig xpoial form whr i h magiud of, ad i h agl of. Pla o ha i h ady a, h rpo of h porio from will dcay o 0; o h ym rpo will hav oly h porio from h ipu, i.., h fir wo rm of h abov parial-fracio xpaio, i.., C Takig h ivr Laplac raformaio, o ha c Subiuig h xprio of ad foud prviouly, o ha c or c Rcallig i, o ha c i
From h abov quaio, i ca b ha h ady-a rpo of h ym i alo iuoidal. I ha h am frqucy a h iuoidal ipu, bu i ha diffr ampliud ad pha agl from h ipu. Th diffrc ar fucio of frqucy. I paricular, h ampliud i amplifid by, which i h magiud of ; h pha agl i hifd by, which i h agl of. Frqucy rpo of a ym i dfid a h frqucy-dpd rlaio i boh ampliud ad pha agl diffrc bw iuoidal ipu ad h rula ady-a iuoidal oupu. Thu, h frqucy rpo of a ym who rafr fucio i ca b obaid by imply rplacig i wih, i..,. Paricularly, h magiud of i calld h magiud frqucy rpo; ad h agl of i calld h pha frqucy rpo. Exampl Fid h aalyical xprio for h magiud frqucy rpo ad h pha frqucy rpo for h ym wih a rafr fucio of. iv ha h ym ipu i r i, wha i h xprio for h ym oupu? Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada 4
. Ploig Frqucy Rpo Frqucy rpo of a ym ca b plod i vral way; wo of hm ar bod plo ad polar plo. Bod plo. Th magiud ad pha ar plod, rpcivly, a fucio of frqucy a h frqucy vari from 0 o ifiiy. I h magiud plo, h magiud rpo i plod a magiud i dcibl db v. log, whr magiud i a ui of db 0 log. I h pha plo, h pha rpo i plod a pha v. log. Polar plo. Boh magiud ad pha ar plod i h complx pla, whr h diac from h origi i h magiud ad h agl from h ral axi o ha h courclolckwi dircio i poiiv i h pha agl. 3. Skchig Bod Plo Ca : Magiud Frqucy Rpo: Pha Frqucy Rpo: M 0log 0 80 0 if > 0 if < 0 Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada 5
Ca : B Magiud Frqucy Rpo: Pha Frqucy Rpo: M 0 log B a B Ca 3: B Magiud Frqucy Rpo: Pha Frqucy Rpo: M 0 log B a B Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada 6
Ca 4: Magiud Frqucy Rpo: Pha Frqucy Rpo: M 0 log 0 a 90 o Ca 5: Magiud Frqucy Rpo: Pha Frqucy Rpo: M 0 log 0 0 a 90 o Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada 7
So far, w hav kow how o kch h bod plo for h fiv baic ca. I raliy, h rafr fucio of a ym could b much mor complx ha ay of h fiv ca. To kch i body plo, h procdur i a follow. Sp : Facor h rafr fucio io om baic ca. For xampl, giv ha 0 00, i ca b dcompod io 0 00 00 500 00 Sp : Skch h Bod Plo for ach of h baic ca. i.., 500,, 3, 4 i h abov xampl. 00 Sp 3: Sum up h idividual Bod Plo o g h Bod Plo of h complx ym. Exampl 50 Skchig h Bod Plo for h ym wih a rafr fucio of. 0 Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada 8
Exampl.5 0 Skchig h Bod Plo for h ym wih a rafr fucio of. Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada 9
Dparm of Mchaical Egirig, Uivriy Of Sakachwa, 57 Campu Driv, Sakaoo, S S7N 59, Caada 0 Ca 6: Sym wih Complx Pol Coidr a cod-ordr ym wih a rafr fucio of ζ Dividig h umraor ad domiaor by yild ζ S W ar wih ζ. I hi ca, h rafr fucio ca b rwri a If.3 0 < ζ, h bod plo ca b approximad by uig h o wih ζ. if 3 < 0. ζ, hi approximaio i oo iaccura ad w hav o u compur.