Elecroecicl Syse Dyics, eergy Coersio, d Elecroecicl Alogies Modelig of Dyic Syses Modelig of dyic syses y e doe i seerl wys: Use e sdrd equio of oio Newo s Lw for ecicl syses Use circuis eores O s lw d Kircoff s lws: KCL d KVL Aoer pproc uilizes e oio of eergy o odel e dyic syse Lgrge odel
Meicl Modelig d Syse Dyics Newoi Mecics: Trsliol Moio Te equios of oio of ecicl syses c e foud usig Newo s secod lw of oio F is e ecor su of ll forces pplied o e ody; is e ecor of ccelerio of e ody wi respec o ieril referece fre; d is e ss of e ody To pply Newo s lw, e free-ody digr i e coordie syse used sould e sudied F
Force : Fcoulo Trsliol Moio i Elecroecicl Syses Cosiderio of fricio is esseil for udersdig e operio of elecroecicl syses Fricio is ery cople olier peoeo d is ery difficul o odel fricio Te clssicl Coulo fricio is rerdig friciol force for rsliol oio or orque for roiol oio cges is sig wi e reersl of e direcio of oio, d e pliude of e friciol force or orque re cos Viscous fricio is rerdig force or orque is lier fucio of lier or gulr elociy 3
Newoi Mecics: Trsliol Moio For oe-diesiol roiol syses, Newo s secod lw of oio is epressed s e followig equio M is e su of ll oes ou e ceer of ss of ody N; J is e oe of ieril ou is ceer of ss kg/ ; d α is e gulr ccelerio of e ody rd/s M jα 4
Te Lgrge Equios of Moio Aloug Newo s lws of oio for e fudel foudio for e sudy of ecicl syses, ey c e srigforwrdly used o derie e dyics of elecroecicl oio deices ecuse elecrogeic d circuiry rsies eior us e cosidered Tis es, e circui dyics us e icorpored o fid ugeed odels Tis c e perfored y iegrig orsiol-ecicl dyics d sesor/cuor circuiry equios, wic c e deried usig Kircoff s lws Lgrge cocep llows oe o iegre e dyics of ecicl d elecricl copoes I eploys e sclr cocep rer e ecor cocep used i Newo s lw of oio o lyze uc wider rge of syses F Wi Lgrge dyics, focus is o e eire syse rer idiidul copoes Γ, D, Π re e ol kieic, dissipio, d poeil eergies of e syse q i d Q i re e geerlized coordies d e geerlized pplied forces ipu d d dγ d q i dγ dqi dd d q i dπ dq i Q i 5
Elecricl d Mecicl Couerprs Eergy Mecicl Elecricl Kieic Mss / Ieri 05 / 05 jω Iducor 05 Li Poeil Griy: g Sprig: 05 k Cpcior 05 C Dissipie Dper / Fricio 05 B Resisor Ri 6
Meicl Model for Siple Pedulu Te kieic eergy of e pedulu o is : Γ Te poeil eergy is : Π g gl cosθ lθ y 0 T, θ l θ g y g cosθ 7
Elecricl Coersio Ipu Elecricl Eergy Oupu Mecicl Eergy Couplig Elecrogeic Field Irreersile Eergy Coersio Eergy Losses Eergy Trsfer i Elecroecicl Syses For roiol oio, of curre d gulr displcee, is : T Were W c e elecrogeic orque, ψdi; wereψ is e flu i e i, θ s fucio dw c i, θ dθ 8
Elecroecicl Alogies Fro Newo s lw or usig Lgrge equios of oios, e secodorder differeil equios of rsliol-dyics d orsioldyics re foud s j d d d θ d B B d d k s F dθ ksθ T d Trsliol dyics Torsiol dyics 9
For series RLC circui, fid e crcerisic equio d defie e lyicl reliosips ewee e crcerisic roos d circuiry preers d i R di d i d L d LC L d R s s 0 L LC Te crcerisic roos re s R L R L LC s R L R L LC 0
Resisce, R o Appied olge Curre i i Ri R i R
Iducce, L H Appied olge Curre i i i L L di 0 d d L
Cpcice, C F Appied olge Curre i i d C d i C d 0 i C 3
4 Trsliol Dper, B N-sec F d F B d d B B F F B B F F 0 Lier posiio /sec Lier elociy i Newo force Appied B
5 Trsliol Sprig, k N F s s s s d k F d df k d d F k k F F 0 Lier posiio /sec Lier elociy i Newo force Appied
6 Roiol Dper, B N--sec/rd F θ d T B d d B B T T B B T T 0 rd Agulr displcee rd/sec Agulr elociy N - Appied orque θ θ ω ω ω θ ω ω B
7 Roiol Sprig, k s N--sec/rd F θ s s s d k T d dt k d d T k B T T 0 rd Agulr displcee rd/sec Agulr elociy N - orque Appied ω θ ω θ θ θ ω ω k s
Mss Grouded, kg Appied orque T N - Lier elociy /sec Lier posiio F 0 d d F d d d F 8
Mss Grouded, kg Appied orque T N - Agulr elociy ω rd/sec Agulr displcee θ rd θ ω T ω J J dω d θ J d d 0 T d F 9
Sedy-Se Alysis Se: Te se of dyic syse is e slles se of riles clled se riles so e kowledge of ese riles 0, ogeer wi e kowledge of e ipu for 0, deeries e eior of e syse for y ie 0 Se Vriles: Te se riles of dyic syse re e riles kig up e slles se of riles deerie e se of e dyic syse Se Vecor: If se riles re eeded o descrie e eior of gie syse, e e se riles c e cosidered e copoes of ecor Suc ecor is clled se ecor Se Spce: Te -diesiol spce wose coordies es cosis of e is, is,, is, were,,, re se riles, is clled se spce Se-Spce Equios: I se-spce lysis we re cocered wi ree ypes of riles re ioled i e odelig of dyic syse: ipu riles, oupu riles, d se riles 0
Se Vriles of Dyic Syse 0 iiil codiio u Ipu Dyic Syse Se y Oupu Te se riles descrie e fuure respose of syse, gie e prese se, e eciio ipus, d e equios descriig e dyics
Elecricl Eple: A RLC Circui i c ξ C / Li / 0 C ; d d d c L i L C u i L c is e ol iiil eergy of e ework USE KCL e jucio 0 u C i C i L C L R
3 Te Se Differeil Equio Equio Bu Se Differeil A ẋ Du Oupu Equio C y u u d d u u u u u u Se Vecor D :direc rsissio ri C :Oupu ri; B :ipu ri A :Se ri;
4 Te Oupu Equio Equio Bu Se Differeil A ẋ Du Oupu Equio H y y y y y y y y H D :direc rsissio ri H :Oupu ri; B :ipu ri A :Se ri;
Eple : Cosider e gie series RLC circui Derie e differeil equios p e circuiry dyics dc C i d di L d dc i d C di d L c Ri Ri c V R i C L 5
6 Eple : Usig e se-spce cocep, fid e se-spce odel d lyze e rsie dyics of e series RLC circui Bu A L i L R L C d di d d d d d d d d i Ri L d di i C d d c c c c 0-0 e corol is Tese re e ses ;
Coiue wi Vlues Assue R o, L 0 H, d C 05 F, fid e followig coefficies Te iiil codiios re ssued o e c 0 c0 5 V; d I 0 i 0 5 A Le e olge cross e cpcior e e oupu; y c Te oupu equio will e Te epded oupu equio i y 0 A -0 0-0 0 0 0 d B 0 5 5 y i c [ 0] H ; H [ 0] y 0 i c [ 0] [ ] H Du Te circui respose depeds o e lue of 7