Time Domain Techniques

Transcription

1 ME365 FALL 008 Overview O ID Techiques 10/06/008 Four techiques are used to idetiy the system parameters ω ad ζ or the d order system studied i this experimet. These techiques all ito two categories: (1) Time domai techiques ad () Frequecy domai techiques. Time Domai Techiques Both the time domai techiques require a step iput to be give to the system. This is doe i this lab by usig a square wave rom the uctio geerator as iput. The square wave ca be thought o as a series o step iputs. The output is recorded usig SYSTEMID.VI. 1. Percetage Overshoot Method Iput: Step Iput Output: As show i Figure 1: Figure 1: Step respose o a secod order system or percetage overshoot calculatios Numbers are or illustrative purposes oly. y 0 is ot equal to 0 i this experimet. 1

2 ME365 FALL The aalytical expressio or the above step respose (y(t)) is give by ( ζω t) exp yt () = y0 + ( y y0) 1.si( ω ) dt+ φ 1 1 ζ where φ ta =. A versio o this equatio with y 0 =0 is available i equatio ζ 16 i page 6-1 o the course packet. - This ca be algebraically maipulated to give: y max y y y 0 = exp ζπ where y max is the amplitude o the irst peak i the oscillatios, y is the steady state value o the output ad y 0 is the iitial value o the output. All these are marked i Figure 1. The above equatio ca be rearraged to solve or ζ. - I the time period o the oscillatios is T d, the the damped atural requecy is give π by ω =. From the damped atural requecy ad dampig ratio, the udamped d T d atural requecy ca be calculated usig ω = ωd. Limitatios: For this method to work, the output must show oscillatios. This oly happes whe the system is uderdamped i.e. oly i 1 ζ <.. Log Decremet Method Iput: Step Iput Output: As show i Figure - Deote the irst peak i the output as y i (which is the same as y max or the percetage overshoot method). The peaks ater the irst peak are labeled i order as y i+1, y i+,, y i+,

3 ME365 FALL yi y - Deie = l where reers to the th peak ater the irst peak. Hece i yi+ y =, use y i+ = y i+ i the deomiator. With this deiitio, it ca be show that: ζ = π The above equatio ca be arraged to solve or ζ. - The, ollowig the same procedure as i the percetage overshoot method, the udamped atural requecy ω ca be evaluated. Figure : Step respose o a secod order system or log decremet calculatios Numbers are or illustrative purposes oly. y 0 is ot equal to 0 i this experimet. Limitatios: Same as or the percetage overshoot method. Frequecy Domai Techiques Both the requecy domai techiques require the geeratio o the Bode plot or the system. I this lab, this is geerated ad recorded usig BODE.VI. 3

4 ME365 FALL Hal-Power Method The Bode plot must be coverted rom log log scale to a liear scale or both the x ad y axis. The magitude plot i the Bode plot, plotted o a liear scale is show i Figure 3. Figure 3: FRF magitude plot i liear scale. Numbers are or illustrative purposes oly. - I order to evaluate ω, reer to the phase plot o the Bode plot. The udamped atural requecy, ω, is the requecy at which the phase φ = -90 o. Alterately, a Lissajous igure ca be used i the lab to evaluate ω. - Let the peak value o the requecy respose uctio be deoted M max. This peak occurs at a requecy called the resoat atural requecy, deoted by ω r. M ma x - Calculate = 0.707M ma x. Now id the two requecies, ω 1 ad ω, at which the ω ω1 FRF attais this magitude. The, the dampig ratio ca be calculated usig ζ = ω 4

5 ME365 FALL 008 Limitatios: For this method to work, the FRF magitude plot must show a peak. This 1 oly happes whe ζ <... Slope o Phase method - Covert the phase plot o the Bode plot rom log-log scale to liear scale i both phase ad requecy. This is show i Figure 4. - Express phase i radias ad requecy i radias/secod. - Evaluate the udamped atural requecy, ω, i the same maer as or the halpower method. 1 1 dφ dφ - The dampig ratio is evaluated usig ζ = ω dω where ω= ω dω is the ω = ω slope o the phase plot at ω = ω or at φ = -π/ radias. Limitatios:Noe Figure 4: FRF phase vs. requecy i liear scale Numbers are or illustrative purposes oly. 5