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

[email protected]

http://lcgui.net

Lecture 5. Dynamic response of measuring systems

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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 -

Non-linear dynamic response

: non-linear differential equation

Simple dynamic response

- 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

Zero-order systems

K

– static sensitivity - time independent - example of zero-order systems: electric resistor 3

Models of dynamic response

First-order systems

K

– static sensitivity  – time constant - example of first-order systems: thermometer

Second-order systems

K

– static sensitivity 

n

 – damping ratio – undamped natural frequency  =0: undamped second-order system 0<  <1: underdamped second-order system  =1: critically damped second-order system  >1: overdamped second-order system Damped natural frequency (for 0<  <1): - example of second-order systems: liquid manometer 4

Type of input

Unit-step (or Heaviside) function

- A relative fast change of the input from one constant level to another.

Unit-impulse (or Dirac’s delta) function

for continuous function f(x):

- A sudden, impulsive application of different value of input, lasting only briefly before it returns to the original level 5

Type of input

Unit-slope ramp function

- A gradual change of the input, starting from a constant level persisting monotonically.

Periodic function

- Function

f(t)

with period

T

so that

f(t)=f(t+nT)

- Can be decomposed in Fourier series

T

6

Dynamic response of first-order system

Step response

𝑥 𝑡 = 𝐴𝑈 𝑡 𝜏 𝑑𝑦 𝑑𝑡 + 𝑦 = 𝐾𝐴𝑈 𝑡 = 𝐾𝐴 for t ≥ 0 𝑦 𝑡 𝐾𝐴 = 1 − 𝑒 −𝑡/𝜏 ∆𝑥 𝑡 𝐴 = 1 − 𝑦 𝑡 𝐾𝐴 = 𝑒 −𝑡/𝜏

t/

 

x/A

1

37%

2

13.5%

3

5%

4

1.8% 7

Dynamic response of first-order system

Impulse response

𝑥 𝑡 = 𝐴𝛿 𝑡 ,

t/



-



x/A

𝜏 𝑑𝑦 𝑑𝑡 + 𝑦 = 𝐾𝐴𝛿 𝑡 ,

1

37% 𝑦 𝑡 𝐾𝐴 = 1 𝑒 −𝑡/𝜏 𝜏 ,

2

13.5%

3

5% ∆𝑥 𝑡 𝐴 = − 1 𝜏 𝑒 −𝑡/𝜏

4

1.8%

Ramp response

𝑥 𝑡 = 𝐴𝑟 𝑡 , 𝜏 𝑑𝑦 𝑑𝑡 + 𝑦 = 𝐾𝐴𝑟 𝑡 , 𝑦 𝑡 𝐾𝐴 = 𝑒 −𝑡/𝜏 + 𝑡 − 𝜏 ,

t/



-

 ( 

x/A-

 )

1

37%

2

13.5%

3

5% ∆𝑥 𝑡 𝐴 = −𝜏𝑒 − 𝑡 𝜏 + 𝜏

4

1.8% 8

Dynamic response of first-order system

Frequency response

𝐵 𝐾𝐴 = 1 𝜔 2 𝜏 2 + 1 𝑥 𝑡 = 𝐴sin 𝜔𝑡 𝜑 = −arctan 𝜔𝜏 𝑦 𝑡 = 𝐵sin 𝜔𝑡 − 𝜃 As  , B/A  0, and   /2. Thus a first-order system acts like a low-pass filter. 9

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. 10

Dynamic response of second-order system

Impulse response

- undamped system with large-amplitude oscillation - underdamped system oscillates with diminishing amplitude.

- Critically damped & overdamped system output increases monotonically towards static level

Ramp response

11

Dynamic response of second-order system

Frequency response

𝑥 𝑡 = 𝐴sin 𝜔𝑡 𝑦 𝑡 = 𝐵sin 𝜔𝑡 − 𝜃 𝐵 𝐾𝐴 = 1 1 − 𝜔/𝜔 𝑛 2 2 + 4𝜁 2 𝜔 2 /𝜔 𝑛 2 𝜑 = −arctan 2𝜁𝜔/𝜔 𝑛 1 − 𝜔/𝜔 𝑛 2 - 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 0 < 𝜁 < 2/2 present a peak at resonant frequency.

𝜔 𝑟 = 𝜔 𝑛 1 − 2𝜁 2 - Underdamped systems with have no resonant peak 𝜁 > 2/2 12

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

Direct dynamic calibration suggested when measuring system exposed to time-dependent inputs - square-wave test: input switched periodically from one level to another - frequency test: sinusoidal input of constant amplitude and varying frequency 13

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 14

Homework

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

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