Measurements in Fluid Mechanics058:180:001 (ME:5180:0001)

Time & Location: 2:30P - 3:20P MWF 218 MLH

Office Hours: 4:00P – 5:00P MWF 223B-5 HL

Instructor: Lichuan Guilichuan-gui@uiowa.edu

http://lcgui.net

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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

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: ai , i=1,2,,n ; bj , j=1,2,,m

- example of zero-order systems: electric resistor

- time independent

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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

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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):

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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

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Dynamic response of first-order system

Step response

𝑥 (𝑡 )=𝐴𝑈 (𝑡 )

𝜏𝑑𝑦𝑑𝑡

+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/ 1 2 3 4

x/A 37% 13.5% 5% 1.8%

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Dynamic response of first-order system

Impulse response

𝑥 (𝑡 )=𝐴𝛿 (𝑡 ) , 𝜏𝑑𝑦𝑑𝑡

+𝑦=𝐾𝐴𝛿 (𝑡 ) , 𝑦 (𝑡 )𝐾𝐴

= 1𝜏𝑒−𝑡 /𝜏 ,

∆ 𝑥 (𝑡 )𝐴

=−1𝜏𝑒− 𝑡 /𝜏

t/ 1 2 3 4

-x/A 37% 13.5% 5% 1.8%

Ramp response

𝑥 (𝑡 )=𝐴𝑟 (𝑡 ) , 𝜏𝑑𝑦𝑑𝑡

+𝑦=𝐾𝐴𝑟 (𝑡 ) ,𝑦 (𝑡 )𝐾𝐴

=𝑒−𝑡 /𝜏+𝑡−𝜏 ,∆ 𝑥 (𝑡 )𝐴

=−𝜏 𝑒− 𝑡𝜏+𝜏

t/ 1 2 3 4

-(x/A-) 37% 13.5% 5% 1.8%

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Dynamic response of first-order system

Frequency response 𝑥 (𝑡 )=𝐴 sin (𝜔𝑡 ) 𝑦 (𝑡 )=𝐵 sin (𝜔 𝑡−𝜃 )𝐵𝐾𝐴

=1

√𝜔2𝜏2+1𝜑=−arctan (𝜔𝜏 )

As , B/A 0, and -/2. Thus a first-order system acts like a low-pass filter.

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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.

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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

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Dynamic response of second-order system

Frequency response

𝑥 (𝑡 )=𝐴 sin (𝜔𝑡 ) 𝑦 (𝑡 )=𝐵 sin (𝜔 𝑡−𝜃 )

𝐵𝐾𝐴

=1

√ [ 1− (𝜔 /𝜔𝑛)2 ]2+4 𝜁 2𝜔2/𝜔𝑛

2𝜑=−arctan ( 2𝜁𝜔/𝜔𝑛

1− (𝜔/𝜔𝑛)2 )

- Critically damped & overdamped systems act like low-pass filters and have diminishing output amplitudes

- Undamped systems have infinite output amplitude when =n

𝜁 >√2/2- Underdamped systems with have no resonant peak

𝜔𝑟=𝜔𝑛√1− 2𝜁 2

- Underdamped systems with present a peak at resonant frequency.

0<𝜁 <√2/2

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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

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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

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Homework

- Questions and Problems: 10 on page 43

- Read textbook 2.3-2.4 on page 31-41

- Due on 09/02