1. 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 lichuan-gui@uiowa.edu http://lcgui.net

2. 2 Lecture 5. Dynamic response of measuring systems

3. 3 Models of dynamic response Dynamic measuring system - at least one of inputs is time dependent Description of dynamic response - differential equation that contains time derivatives. - Linear dynamic response: linear differential equation Simple dynamic response - Non-linear dynamic response: non-linear differential equation Zero-order systems K – static sensitivity - approximated by single, linear, ordinary differential equation with constant coefficients x – input y – output t – time constant coefficients: a i, i=1,2, ,n ; b j, j=1,2, ,m - example of zero-order systems: electric resistor - time independent

4. 4 Models of dynamic response First-order systems Second-order systems K – static sensitivity  – time constant - example of first-order systems: thermometer K – static sensitivity  – damping ratio  n – undamped natural frequency  =0: undamped second-order system  =1: critically damped second-order system 0<  <1: underdamped second-order system  >1: overdamped second-order system Damped natural frequency (for 0<  <1): - example of second-order systems: liquid manometer

5. 5 Type of input Unit-step (or Heaviside) function Unit-impulse (or Dirac’s delta) function - A relative fast change of the input from one constant level to another. - A sudden, impulsive application of different value of input, lasting only briefly before it returns to the original level for continuous function f(x):

6. 6 Type of input Unit-slope ramp function Periodic function - A gradual change of the input, starting from a constant level persisting monotonically. - Function f(t) with period T so that f(t)=f(t+nT) T - Can be decomposed in Fourier series

7. 7 Dynamic response of first-order system Step response t/  1234  x/A 37%13.5%5%1.8%

8. 8 Dynamic response of first-order system Impulse response t/  1234 -  x/A 37%13.5%5%1.8% Ramp response t/  1234 -  (  x/A-  ) 37%13.5%5%1.8%

9. 9 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.

10. 10 Dynamic response of second-order system Step response - Damping ratio  determines response - Critically damped & overdamped system output increases monotonically towards static level - output of underdamped system oscillates about the static level with diminishing amplitude. - Lightly damped system (  <<1) are subjected to large-amplitude oscillation that persist over a long time and obscure a measurement.

11. 11 Dynamic response of second-order system Impulse response Ramp response - Critically damped & overdamped system output increases monotonically towards static level - underdamped system oscillates with diminishing amplitude. - undamped system with large-amplitude oscillation

12. 12 Dynamic response of second-order system Frequency response - 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.

13. 13 Dynamic response of higher-order and non-linear system Dynamic analysis by use of Laplace transform - Laplace transform of time-dependent property f(t) : - Inverse Laplace transform: - Differentiation property of Laplace transform: Experimental determination of dynamic response - square-wave test: input switched periodically from one level to another - frequency test: sinusoidal input of constant amplitude and varying frequency Direct dynamic calibration suggested when measuring system exposed to time-dependent inputs

14. 14 Distortion, loading and cross-talk Flow distortion - caused by instrument inserted in flow Loading of measuring system - measuring component extracts significant power from flow Instrument cross-talk - output of one measuring component acts as undesired input to the other

15. 15 Homework - Questions and Problems: 10 on page 43 - Read textbook 2.3-2.4 on page 31-41 - Due on 09/02