Iroucio ME 30 Mecharoics II Laboraory Lab 6: Seco-Orer Dyamic Respose Seco orer iffereial equaios approimae he yamic respose of may sysems. I his lab you will moel a alumium bar as a seco orer Mass-Sprig-Damper sysem. Seco-Orer Sysems This laboraory eercise focuses o a seco-orer mass-sprig-amper sysem forme by a alumium bar fie wih srai gages. The siffess of he sysem is provie by he maerial properies of he alumium (Youg s moulus), a he ampig is provie by he air rag, a eergy issipaio i he alumium iself. The geeralize characerisic equaio ha escribes every seco-orer sysem is y+ y + y f( ) ζ () where y is he sysem oupu, ζ is he imesioless ampig raio or ampig coefficie, is he uampe aural frequecy i raias per seco, a f() is some forcig fucio. However, for his lab, he equaio of moio ha escribes he mass-sprig-amper sysem of he alumium bar is give by he followig iffereial equaio: c k y+ y + y 0 m m y+ ζ y + y The soluio o Equaio is someimes referre o as he free respose of he sysem because he sysem respose is o ecie by a eeral force (f()0). There are hree iffere soluios o he free respose: uerampe, overampe, a criically ampe soluios. Each epes o he value of ζ. If ζ <, he he sysem is sai o be uerampe. For he case of he alumium bar use i his lab, he ampig forces ha issipae he vibraioal eergy of he sysem are very small. Therefore we assume ha he sysem is uerampe. The soluio o a geeralize uerampe sysem respose wih a o-zero iiial coiio is show below i Figure. () Figure : Geeralize free respose of a uerampe seco-orer sysem o o-zero iiial coiios. of 5 Revise: 3//004
Deermiig a ζ Eperimeally Oe meho for eperimeally eermiig he parameers i Equaio comes from solvig he iffereial equaio a performig some mahemaical maipulaio. Sice he sysem is uforce oly he homogeeous respose is eee. The homogeeous soluio o Equaio is summarize i Equaios hrough 4. ζ ζ y () ye 0 cos ( ) + si( ) y ( φ) 0 ζ e cos () φ a ζ (3) ζ (4) where y 0 is he iiial isplaceme of he e of he alumium bar a y() is he isplaceme of he e of he bar as a fucio of ime. This soluio shows ha a ecayig epoeial scales a siusoial oscillaio, similar o he respose i Figure. The ime i akes o reach he firs peak ca be fou by akig he ime erivaive of y h a seig he resul equal o zero. ζ 0cos( ) y y 0si ( ) 0 φ + y φ e ζ (5) Sice he epoeial erm heoreically ever reaches zero, he ime o he firs peak ca be eermie hrough reucio usig rigoomeric ieiies a algebra. This proceure yiels he equaio for he ime o peak, p, i erms of a ζ. p (6) The ime o peak is measure i secos. If he sysem is sufficiely uerampe (ζ < 0.), he he erm uer he raical i Equaio 6 approaches uiy. Therefore we ca esimae he aural frequecy,, usig he followig relaioship. (7) p Whe a sysem is his uerampe he sysem respose ecays very slowly a i ofe akes a log ime before he oscillaios sele o a fial or seay-sae value. Geerally, he selig ime is efie as he ime i akes for he sysem o eer a regio arou he sysem s fial value wihou leavig. This boue regio is ypically efie as a perceage of he seay-sae value. For his eperime, he selig ime will be efie o be whe he sysem respose seles o wihi % of is seay-sae value. The selig ime ca be approimae aalyically as he ime i akes he ecayig epoeial of Equaio o reach 0.0: e ζ s 0.0 (8) of 5 Revise: 3//004
Solvig Equaio 8 for he selig ime, s, yiels he followig relaioship. s l(0.0) 4 ζ ζ (9) So, ow ha we have epressios for he ime o peak a he selig ime, we ca calculae w a z usig Equaios 7 a 9 by eamiig he sysem respose a recorig s a p. Log Decreme Techique Aoher way o calculae a ζ is o use he log ecreme echique. This echique is eplois he perioiciy of Equaio o eermie a ζ. The ime bewee successive peaks i he free respose of he sysem is eermie by he perio,, of he cosie fucio i Equaio. (8) The ampliue of he firs peak, which occurs a p, is efie as. Subsiuig p a io Equaio yiels he followig equaio. ζ p y0e cos( p φ ) (9) The equaio epressig he ampliue of he e peak is: ζ( p+ ) y0e cos( ( p + ) φ ) (0) Subseque epressios for ampliue peaks follow his paer accorig o he Equaio below: ( ( ) ) ζ( p+ ) + y0e cos p + φ, for,,3... () The raio of he iiial peak o each successive peak is efie by he followig epressio + ζ e, for,,3... () Takig he aural logarihm of boh sies of Equaio a subsiuig Equaio 8 i for yiels he followig relaioship. ζ l ζ, for,,3... + (3) Oce agai we assume ha for small ampig raios he erm i he raical approaches uiy, resulig i he followig: δ ( ) ζ l, for,,3... + This resul shows ha here is a liear relaioship bewee a δ() ha is liearly relae o he ampig raio of he sysem. The ampig raio ca be eermie hrough liear regressio give values for a δ(). The aural frequecy,, ca he be eermie by combiig Equaios a 8: resulig i he followig equaio. (4) (5) 3 of 5 Revise: 3//004
Eperime. Secure a alumium bar o he workbech usig a large C-clamp.. Desig a buil a amplifier circui like he oe i Figure of he Dyamomeer I lab. The overall gai shoul be arou 000. Recor he gai below. k. 3. Deermie he relaioship bewee ip eflecio a volage oupu of your amplifier circui. a. Displace he ip of he bar a kow isace a measure he oupu volage from your circui. b. Recor he values below. mm V ou V c. Divie by V ou o fi he coversio from volage o isplaceme. Recor he resul below: /V ou K mm/v 4. Coec he oupu from your amplifier circui o ACH0 of he DAQ ermial block a coec he EXTREF o he commo grou of your amplifier circui. 5. Ope NDORDER.ee from he CVI foler o he eskop. 6. Se he samplig rae a umber of samples o he appropriae value, eflec he alumium bar o creae a iiial isplaceme i he bar, click he START buo, a quickly release he e of he alumium bar. 7. Eamie he plo of he aa a repea Sep 6 uil you capure eough aa o capure he mome whe a % selig ime is reache; save he aa se wih a.a file eesio. 8. Ope he ime a volage aa usig Malab or Ecel a plo he aa. 9. Calculae he aural frequecy a ampig raio of he sysem respose usig Equaios 7 a 9; recor he values below: ζ ra/s 0. Deermie ζ a usig he log ecreme echique. ζ ra/s 4 of 5 Revise: 3//004
Quesios. Is a seco-orer approimaio sufficie o moel his sysem? Why or why o?. Compare he resuls from Seps 9 a 0. I your opiio which echique is more accurae? Why? 3. Deermie he seco-orer pole locaios for he sysem base o he resuls from eiher Sep 9 or 0. 4. Briefly escribe a leas wo oher seco-orer sysems. 5 of 5 Revise: 3//004