Measurements in Fluid Mechanics 058:180 (ME:5180) Time & Location: 2:30P - 3:20P MWF 3315 SC Office Hours: 4:00P – 5:00P MWF 223B-5 HL Instructor: Lichuan Gui Phone: (Lab), (Cell)

2 Lecture 5. Dynamic response of measuring systems

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 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 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):

T 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) - Can be decomposed in Fourier series

77 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 t/  1234  x/A 37%13.5%5%1.8%

8 Dynamic response of first-order system Impulse response t/   x/A 37%13.5%5%1.8% Ramp response t/   (  x/A-  ) 37%13.5%5%1.8% K – static sensitivity  – time constant

99 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

10 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 avoided  – damping ratio  n – undamped natural frequency

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  – damping ratio  n – undamped natural frequency

12 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. Frequency response  – damping ratio  n – undamped natural frequency

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 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 Homework - Questions and Problems: 10 on page 43 - Read textbook on page Due on 09/05