Mechatronics II Laboratory Exercise 5 Second Order Response

Transcription

1 Mechatroics II Laboratory Exercise 5 Seco Orer Respose Theoretical Backgrou Seco orer ifferetial equatios approximate the yamic respose of may systems. The respose of a geeric seco orer system ca be see i Figure at the bottom of the page. I this lab you will moel a alumium bar as a seco orer Mass- Sprig-Damper system. The alumium bar itself provies the mass, stiffess, a viscous ampig for your moel. The characteristic ODE that escribes this system is of the form, y + ζω y + ω y = () where y is the system output, ζ is terme the ampig ratio or ampig coefficiet (a imesioless quatity), a ω is the uampe atural frequecy i ra/s. If ζ <, the the system is uerampe. This is the situatio for the alumium bars that will be use for this experimet. Oe metho for experimetally etermiig the parameters i Equatio () comes from solvig the ifferetial equatio a performig some mathematical maipulatio. Sice the system is uforce (thus the o the right-ha sie of equatio ()), oly the homogeeous respose ees to be solve: ζωt ζ yh = ye cos( ω ζ t) + si ( ω ζ t) () ζωt ζ = ye cos( ω t ), where ta a φ φ = ω = ω ζ Output (uits) Time (s) Figure : Time respose of a uerampe seco-orer system

2 where y is the iitial coitio of the output variable y. Ispectio of this solutio reveals a combiatio of a ecayig expoetial a siusoial oscillatio, similar to the respose i figure. The techiques presete here are base o measurig the amplitue of the peaks as well as whe they occur. The time it takes to reach the first peak ca be fou by takig the time erivative of y h a settig the result equal to zero, like so: ζω ω ζωt y = y cos( ω t φ ) + y si( ω t φ ) e = (3) ζ Sice the expoetial term theoretically ever reaches zero, the time to the first peak ca be etermie through reuctio usig trigoometric ietities a algebra, resultig i t p = ω (4) =, ω where t p is the time to peak i secos. If the system is sufficietly uerampe (ζ<.), the the term uer the raical i equatio (4) approaches uity, allowig the atural frequecy term to be solve accorig to the relatio: ω (5) t p For such a uerampe system, its takes some time for the respose to settle to its fial value. Geerally, the settlig time is efie as the time it takes for the system to eter a give boue regio about the system s fial value without leavig. This boue regio is typically efie as a percetage of the fial value. For this experimet, the settlig time will be efie to be whe the respose settles to % of its fial value (that is, % of the ifferece betwee the iitial a fial values). The settlig time ca be approximate aalytically as the time it takes the ecayig expoetial to reach.: ts e ζω =. (6) which ca be solve for the settlig time as follows: l(.) t s = ζω (7) 4 ζω for sufficietly uerampe ( ζ ) systems. Usig the preceig equatios, the parameters ζ a ω ca be etermie. The seco metho you will use to characterize the system is the log ecremet techique. The time betwee successive peaks i the respose is etermie to be t = (8) ω which is equivaletly the perio of the fuctio si(ω - φ). The amplitue of the first peak which occurs at t p is efie as x. Substitute ito equatio (), this becomes

3 ζωt p x = ye cos( ωtp φ) (9) The amplitue of the ext peak has the magitue ζω( tp+ t ) x = ye cos( ω ( tp + t) φ) () Subsequet amplitues follow the patter ζω( tp+ t ) x + = ye cos( ω ( tp + t) φ) () where =,, 3, The ratio of two amplitues is x ζω t = e () x+ Takig the atural logarithm of both sies yiels x ζ l = ζωt = (3) x + As above, for small ampig ratios, the term i the raical approaches uity, resultig i the relatio x ( ) ( ) δ = ζ, where δ = l (4) x + If δ versus is plotte, a straight lie through the ata shoul result, whose slope is ζ. The atural frequecy ω is the fou from the ampe atural frequecy ω usig equatios (8) a (). Experimet. Secure a alumium bar to the workbech usig a large C-clamp.. Buil a amplifier circuit. Verify that your circuit works a experimetally etermie the gai of your amplifier usig the bech potetiometer a a voltmeter. k =. 3. Calibrate your system by isplacig the tip a measurig the output voltage from your circuit. You must covert the output voltage to the eflectio of the tip of the sow-alumium bar for the post lab report. isplacemet =, V out =. 4. Coect the output from your amplifier circuit to AI_CH of the DAQ termial block. Make sure to referece a commo grou. 5. Ru the NDORDER program from the CVI foler o the esktop. 6. Set the samplig rate a umber of samples to the appropriate value a give the alumium bar a step iput by eflectig a releasig it while ruig the program. Be sure you hit the START butto before you release the alumium bar. Esure the alumium bar is sufficietly catilevere. You may ee to physically hol ow the back of the alumium bar securely to keep it from liftig off the bech. 7. Save your ata whe you get a respose that ecays sufficietly by the e of the ata set. Esure a % settlig time is reache. Use Matlab or Excel to plot the ata. 3

4 Questios. Determie ζ a ω usig equatios (5) a (7). ζ =, ω =.. Determie ζ a ω usig the log ecremet techique (show your calculatios here, but plot your ata separately). ζ =, ω =. 3. Compare the results from questios a : 4

5 4. Is a seco-orer approximatio sufficiet to moel this system? Why or why ot? 5. Determie the seco-orer pole locatios for the system base o the results from either questio or. 6. Briefly escribe at least two other seco-orer systems. 5