How delay equations arise in Engineering? Gábor Stépán Department of Applied Mechanics Budapest University of Technology and Economics
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1 How y quos rs Egrg? Gábor Sépá Dprm of App Ms Bups Ursy of Toogy Eooms Cos Aswr: Dy quos rs Egrg by o of bos by formo sysm of oro - Lr sby bfuros summry - M oo bros - Smmyg ws of rus moorys - Bg um robo - Robo poso for oro Sby of r RFDEs of DoF sysms Dy m sysms u rs: M B K Auoomous sysms: B B K K Tr souo: A AR Crrs roos: R < = sby D M D-urs: R R D S ImD [ R r: S r r sby sg S B K Emps w DoF = w w D s R m R os S S for S r R 4 r sg S r o R Sby r ẋ / Dy osors w bro frqus / / sby r
2 Dy osor w mpg mpg: b = b / / / / / / s w w 6 No-uoomous r RFDEs Tm-pro sysms: Tr souo: H s f mso rm rrs fuo rrs roos R < = sby < = for rrs muprs of fu op T K B M K T K B T B p s os T T B A p T p T T y Mu quo Hrmo b H s rm os b B A B A C C b T y Mu sby rs b= Sru-Iz 98 ε= Hsu-B 966 ε= os b
3 Sby r of y Mu os b Sm-srzo mo rouo Isprgr Sépá T ppromg DDE s o-uoomous m / [ / m / [ m m Irouo o SDM y osor m Dy osor sby r by SDM m / K os K s m / K s K os m m y m m y Ay o m I A m sby Fu srzo - omprso Dsrzo so wr m rs sow org Irouo o SDM Mu quo os [ s os 3
4 Sm-srzo gr s η η T η M m M m S ~ m m Isprgr Sp: I J of Numr Mos Egrg Emps s o y Mu os4 os4 s 6 os Nor RFDEs Egrg S&sp usb pro moo Sby yss of sy-ss s foow by bfuro yss Hopf bfuro sf- bros Suprr s: sy o o bros by owg r sby bour Subr s: usb pro souos m m om of ro for sr sy-s bour o b pr by r sby yss 4
5 Usb m y gos bro Cr ~ g frquy m oo bro Cr s mos obsur of probms fg ms probby o rus or formu b s w w ury gu ms g mmum us sps possb wou proug r Tyor 97 Effy of ug Spf mou of mr u w r m D V w wr w p w p ss Ω ~ ug sp Effy of ug Spf mou of mr u w r m D V w wr w p w p ss Ω ~ ug sp surf quy Mog rgr ff M mo τ m pro of rouo Mm mo F m 5
6 6 Cug for ¾ ru for or ug for Cug off 4 3/ w w F 4 / 4 3 w w F w 3 3 F F Lr yss sby Dmsoss m Dmsoss p w Dmsoss ug sp ~ ~ ~ ~ ~ ~ ~ w w m w ~ ~ ~ m m Sby r of urg Br sby proprs pr ow g ug sps! ~ w r r Sor rgr ff p m p Wg fuos / p q 5 D q Wg fuos Drg ow sp bros ry ow frqus! D q s p
7 Drg Bfuros of urg Bog Subr Hopf bfuro: usb bros rou sb ug w ~ r r M surf Usb qus-pro bro Co-mso Hopf bfuro: D=76 [mm] τ =75 [s] f f 53 88[Hz] f f 53 35[Hz] 5 Sf-rrup ug Hg-sp mg Prmry rrup ug Low umbr of gs Low mmrso Hgy rrup 7
8 Mog g-sp mg Two yms: - fr-fg - ug w rgr ff Nor sr mp of HS mg Lr sby: r rrs muprs A s os w~ r m 3; r b os s F m Sby r of H-S mg Subr fp bfuro Ss of pro oubg or fp bfuro? Bfuro grm os Amo of sb pro oubg 8
9 9 Bo pro-s usb b Sruur of os rso mr Smmy A M Ayr I Gorg quos & mmory ff Trg w souo of PDE: [ ] [ o q q q q I A q / 3 L L L V A I L V / 3
10 Sby r of smmy mo Nor bros wou y w y??? 3 Bg 6 g 6 Q Q = - o oro m g 6 = s usb Q = P + D PD oro 6 6 D P mg m m = s symp sb D > P > mg 3 Q = P + D w rf y 6 6 6g D P m m Sby r & r rf y r sby 3g 3[m] [s] f 5[Hz] 4 Lbyr um bg org Bo g gur oy sgs r
11 Rom osos of robo bg smpg m A s Aurs Wor Lws Crro 899 quzo rou-off Smpg y of g oro Dgy oro puum 6g u [ 6 u D P m Sby of g oro smpg u A s A s 6 6 P D m m 6g s I A pfor 3 Hopf Sby of g oro rou-off o g or o oro for 6 D P u m B g g s 6 P D m s B s I B = = > = 3 =
12 D roo -os mp Drop msos rs w b P b A pur mm ppro p > p < q y py q y souo w = y s o -os mp = p b = q p /p > < b < sm s: + = rg s: + = b Mro-os mp rg s b sm s Typ gy oro ms us pry by y 4 Robo poso oro Equo of moo m Q Csg Q D P Poso rror: C / P Sby P D W smpg y msoss m T / P D T T [ m m Sby of g poso oro T p T [ : T T T T T T [ z : z Az A p 3 I A p p 3 Sby r R 3 p p 4 4 p Sby oos: p > H > = Hopf Mmum g: m Pm 4 Mmum poso rror m 4C / m Sf- bro frquy: < f < f smpg /6
13 5 Robo for oro my y P y F ycsg y Equbrum: y F / For rror: F C / P Sby P Bu w smpg y F y [ Q P y Dmsoss prmrs: / f / f P s Sby r of for oro Vbro frquy: < f < f s / Mmum g: P m 5 Mmum for rror: F m /3C Qus-pro oso DoF mo m sory sprum sby r 3
14 6 Hum-robo for oro og oupu oro sg ro Rom y msurms Comb pur y smpg Mog um-robo for oro Sby of rbo robo m [ P for rror: F msoss g: p P / Dm s ssum o b 8 smpg R prmrs us suy T umo mpg of um pm or sous ffs p o sbz os prmrs 4
15 7 Sorg 8 Hum-um for oro Ffows Cmbrg 997 Couso How os y rs Egrg? By s-ps o PDERFDE By formo g oro T you for your o! 5
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