a dt a dt a dt dt If 1, then the poles in the transfer function are complex conjugates. Let s look at f t H t f s / s. So, for a 2 nd order system:

Undrdamd Sysms Undrdamd Sysms nd Ordr Sysms Ouu modld wih a nd ordr ODE: d y dy a a1 a0 y b f If a 0 0, hn: whr: a d y a1 dy b d y dy y f y f a a a 0 0 0 is h naural riod of oscillaion. is h daming facor. is h sady sa gain. For diaion ariabls, whr s s 1 ys f s Gs y 0 f 0 0, h Lalac ransform will b: y s f s s s 1 Dynamic Rsons of Undrdamd nd Ordr Sysm If 1, hn h ols in h ransfr funcion ar comlx conjugas. L s look a f H f s / s. So, for a nd ordr sysm: rsons o a uni s chang. y s Inring: s s 1 s 1 s ys s s s 1 s 1 ys s 1 s s s 1 y s s 1 s John Jchura (jjchura@mins.du) - 1 - Coyrigh 017 Aril 3, 017

Undrdamd Sysms / 1 / 1 y 1 cos sin 1 Th sin & cosin rms can b combind ino a singl sin rm wih a has angl offs. Rmmbr: cos qsin r sin whr: r q and an q So: 1 r 1 1 1 an 1 Dfining: 1 / hn: y 1 sin 1 John Jchura (jjchura@mins.du) - - Coyrigh 017 Aril 3, 017

Undrdamd Sysms A C y/ Ris Tim Rsons Tim Figur 1. Rsons Cur for Undrdamd Sysm Thr ar sral rms dfind o dscrib h characrisics of an undrdamd sysm s rsons. Ths ar also shown in h Figur 1: Ulima Valu. This is h alu ha h rsons sls down o a ry larg ims. This can asily b rmind as: / lim lim y y 1 sin 1 Priod of oscillaion. Th im bwn crossing of h ulima alu. This will also b h im bwn h aks and allys. Sinc h frquncy of oscillaion is: 1 hn h riod of oscillaion is: 1 T f 1 John Jchura (jjchura@mins.du) - 3 - Coyrigh 017 Aril 3, 017

Undrdamd Sysms Ris im. Th im i aks h rsons o firs g o h ulima alu. This can b asily rmind as: ris / y ris 1 sinris 1 ris / 1 sin ris 1 1 ris / ris 1 sin 0 sin 0 ris ris 1 1 ris an 1 Orshoo. A masur of how far h rsons xcds h ulima alu. On h figur, his is A /. Th formula gin for his in h x is: Orshoo x 1 This is calculad by finding h im a which h firs maximum occurs and hn finding h corrsonding alu in h rsons cur. Th 1 s driai of h rsons cur is: dy d / sin 1 dy / / sin 1 cos / dy sin cos 1 Th sin & cosin rms can b combind ino a singl sin rm o gi: / 1 sin an dy 1 John Jchura (jjchura@mins.du) - 4 - Coyrigh 017 Aril 3, 017

Undrdamd Sysms / dy 1 1 1 1 1 sin an an 1 / dy 1 sin So, h im a which h 1 s maximum occurs is a: dy / sin 0 sin 0 1 Thn: / ymax y 1 sin 1 1 ymax 1 sinx 1 1 1 1 y max 1 sin an x 1 1 1 y max 1 1 x 1 1 y max 1 x Finally, Orshoo 1 John Jchura (jjchura@mins.du) - 5 - Coyrigh 017 Aril 3, 017 y max y 1 y 1 x x 1 Dcay Raio. Th raio of h orshoo of wo succssi aks, C/ A. Th x gis his xrssion as: Dcay Raio Orshoo x 1

Undrdamd Sysms W know from h driaion for h orshoo ha h aks and allys will occur whn: sin 0 0,,,, n, Th aks will occur a h odd mulils:, 3,, n 1, So, h n-h ak will ha a rsons alu of: n1 / yak, n yn 1 sinn 1 1 1 n 1 yak, n 1 sin x 1 1 n 1 y ak, n 1 sin x 1 1 1 1 1 n 1 y ak, n 1 sin an x 1 1 y ak, n 1 x n 1 1 No ha h allys will occur a h n mulils:, 4,, n, so h n-h ally will ha a rsons alu of: n / yally, n 1 sinn 1 1 n yally, n 1 sinx 1 y ally, n 1 x Now, h dcay raio, R, will b: n 1 John Jchura (jjchura@mins.du) - 6 - Coyrigh 017 Aril 3, 017

Undrdamd Sysms y R n 1 x y 1 1 x 1 ak, n1 yak, n y n x R x n 1 1 n 1 1 n 1 n 1 R x 1 1 R x 1 y/a Rsons Tim Figur. Occurrnc of Rsons Tim in an Undrdamd Sysm John Jchura (jjchura@mins.du) - 7 - Coyrigh 017 Aril 3, 017

Undrdamd Sysms Rsons im. Th im i aks h rsons o say wihin 5% of h ulima alu. Figur shows ha his is a much mor comlicad alu o figur ou, sinc h rsons cur may cross or h hrshold alus many ims bfor i final sls down wihin h rang. Bu w can brack h alu by finding h 1 s ak and h 1 s ally ha rmains wihin h olranc. If a ak is h 1 s xrmum o say wihin h olranc, hn h rsons im will b bwn his 1 s ak wihin h olranc and h las ally ou of h olranc. Similarly, if a ally is h 1 s xrmum o say wihin h olranc, hn h rsons im will b bwn his 1 s ally wihin h olranc and h las ak ou of h olranc. L us dno h olranc as, whr by radiion 0.05. Thn h rsons im will b bfor h lows alu of n which saisfis: y y y ak,n yak, n 1 y n 1 1 x 1 1 n 1 1 x 1 1 x n 1 1 n 1 ln 1 1 ln 1 n min n Th rsons im will also b bfor h lows alu of n which saisfis: y y y ally, n y ally, n 1 y n 1 x 1 1 x n 1 John Jchura (jjchura@mins.du) - 8 - Coyrigh 017 Aril 3, 017

Undrdamd Sysms n ln 1 ln 1 n min n So, if n n, hn h rsons im will b bwn h aks rrsnd by n and n 1, or ims (n 1) / and ( n 1) /. Howr, if n n, hn h rsons im will b bwn h aks rrsnd by n and n 1, or ims n / and (n 3) /. For xaml, Figur 1 was gnrad using 0.15. So: ln 1 3.1 n 4 1 ln 1 3.6 n 4 So, h rsons im is found from whr h rsons cur crosss h lowr limi bwn h 3 rd ally and h 4 h ak. On furhr hing o no is ha h rsons can b bfor h ris im if h 1 s ak dos no xcd h olranc for h rsons im. This will han if says largr hn a hrshold alu of: 1 ln 1 1 1 ln 1/ For h radiional olranc, his hrshold alu is 0.690. John Jchura (jjchura@mins.du) - 9 - Coyrigh 017 Aril 3, 017