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
lichuan-gui@uiowa.edu
Phone: 319-384-0594 (Lab), 319-400-5985 (Cell)
http://lcgui.net
Lecture 5. Dynamic response of measuring systems
2
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: ai , i=1,2,,n ; bj , 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
 – damping ratio
n – 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)
T
- Can be decomposed in Fourier series
6
Dynamic response of first-order system
K – static sensitivity
 – time constant
Step response
𝑥 𝑡 = 𝐴𝑈 𝑡
𝜏
𝑑𝑦
+ 𝑦 = 𝐾𝐴𝑈 𝑡 = 𝐾𝐴 for t ≥ 0
𝑑𝑡
𝑦 𝑡
= 1 − 𝑒 −𝑡/𝜏
𝐾𝐴
𝜏
𝑑𝑦
= 𝐾𝐴𝑒 −𝑡/𝜏
𝑑𝑡
𝑦 𝑡
=𝑥 𝑡
𝐾
𝑦 𝑡 = 𝐾𝑥 𝑡 − 𝐾𝐴𝑒 −𝑡/𝜏
𝑤ℎ𝑒𝑛 𝑡 → ∞
Assume y(t)/K is the measurement of x(t),
measurement error:
∆𝑥 𝑡
= 𝑒 −𝑡/𝜏
𝐴
∆𝑥 𝑡 =
𝑦 𝑡
− 𝑥 𝑡 = 𝐴𝑒 −𝑡/𝜏
𝐾
t/
1
2
3
4
x/A
37%
13.5%
5%
1.8%
7
Dynamic response of first-order system
K – static sensitivity
 – time constant
Impulse response
𝜏
𝑥 𝑡 = 𝐴𝛿 𝑡 ,
𝑑𝑦
+ 𝑦 = 𝐾𝐴𝛿 𝑡 ,
𝑑𝑡
𝑦 𝑡
1
= 𝑒 −𝑡/𝜏 ,
𝐾𝐴
𝜏
∆𝑥 𝑡
1
= − 𝑒 −𝑡/𝜏
𝐴
𝜏
t/
1
2
3
4
-x/A
37%
13.5%
5%
1.8%
Ramp response
𝑥 𝑡 = 𝐴𝑟 𝑡 ,
𝜏
𝑑𝑦
+ 𝑦 = 𝐾𝐴𝑟 𝑡 ,
𝑑𝑡
𝑦 𝑡
= 𝑒 −𝑡/𝜏 + 𝑡 − 𝜏 ,
𝐾𝐴
𝑡
∆𝑥 𝑡
= −𝜏𝑒 −𝜏 + 𝜏
𝐴
t/
1
2
3
4
-(x/A-)
37%
13.5%
5%
1.8%
8
Dynamic response of first-order system
K – static sensitivity
Frequency response
𝐵
1
=
𝐾𝐴
𝜔2𝜏 2 + 1
 – time constant
𝑥 𝑡 = 𝐴sin 𝜔𝑡
𝑦 𝑡 = 𝐵sin 𝜔𝑡 − 𝜑
𝜑 = −arctan 𝜔𝜏
As  , B/A  0, and  -/2. Thus a first-order system acts like a low-pass filter.
9
Dynamic response of second-order system
n – undamped natural frequency
 – damping ratio
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
10
Dynamic response of second-order system
n – undamped natural frequency
 – damping ratio
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
n – undamped natural frequency
𝑥 𝑡 = 𝐴sin 𝜔𝑡
Frequency response
𝐵
=
𝐾𝐴
1
1 − 𝜔/𝜔𝑛
2 2
+ 4𝜁 2 𝜔 2 /𝜔𝑛 2
 – damping ratio
𝑦 𝑡 = 𝐵sin 𝜔𝑡 − 𝜑
𝜑 = −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
- Underdamped systems with 0 < 𝜁 < 2/2
present a peak at resonant frequency.
𝜔𝑟 = 𝜔𝑛 1 − 2𝜁 2
- Underdamped systems with 𝜁 > 2/2
have no resonant peak
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 09/05
15