Measurements in Fluid Mechanics 058:180:001 (ME:5180:0001) Time & Location: 2:30P - 3:20P MWF 218 MLH Office Hours: 4:00P – 5:00P MWF 223B-5 HL Instructor: Lichuan Gui

22 Dynamic response of first-order system Step response Assume y(t)/K is the measurement of x(t), measurement error: K – static sensitivity  – time constant

33 Dynamic response of first-order system Frequency response As , B/A  0, and  -  /2. Thus a first-order system acts like a low-pass filter. K – static sensitivity  – time constant

44 Dynamic response of second-order system Step response - Damping ratio  determines response - Critically damped & overdamped system output increases monotonically towards static level i.e. high  n expected for desired output - output of underdamped system oscillates about the static level with diminishing amplitude. i.e. high  n expected for desired output - Lightly damped system (  <<1) are subjected to large-amplitude oscillation that persist over a long time and obscure a measurement. i.e. should be aviod  – damping ratio  n – undamped natural frequency

55 Dynamic response of second-order system - Critically damped & overdamped systems act like low-pass filters and have diminishing output amplitudes - Undamped systems have infinite output amplitude when  =  n - Underdamped systems with have no resonant peak - Underdamped systems with present a peak at resonant frequency.  – damping ratio  n – undamped natural frequency Frequency response

6 Lecture 6. Measurement uncertainty

Probability density function: f(x) Mean value:  7 Measurement errors Physical property: x True value: x true Measured value: x i, i=1,2, ,N Absolut error:  x i =x i -x true Relative error:  x i /x true Standard deviation:  Bias (fixed or system) error:  =  - x true  Precision (random) error:  i =x i -  ii Normal (Gaussian) distribution

8 Accuracy and uncertainty Accurate measurement: |  |<< | x true | and  << | x true | Inaccurate measurement: - Precise but biased, i.e.  << | x true | but |  | relatively large - Unbiased but imprecise, i.e. |  | << | x true | but  relatively large - Biased and imprecise, i.e. both |  | and  relatively large Measurement uncertainty: Bias limit b signifies that the experimenter is 95% confident that |  |<b Precision limit p signifies that, for every single repeat measurement x i, the mean value  would fall within the interval [ x i -p,x i +p ] 95% of the time. b k – individual bias limits of totally K sources

9 Uncertainty of derived properties Derived property: Uncertainty: Bias limit: Precision limit: Precision limit of mean value of repeat number N: Propagation of uncertainty:

10 Homework - Questions and Problems: 2 and 5 on page 53 - Read textbook on page Due on 09/07 (Q5 is optional)