Lab 7 — Frequency response computations
Agenda
Time
Item
5 min
Lab introduction
10 min
Computing the frequency response of a system
20 min
Plotting the frequency response of a system
125 min
The big picture:
Lab activity 1 – Frequency response of an active band-pass filter
Lab activity 2 – Frequency response of a two-degree-of-freedom oscillator
Practice performing frequency response calculations with a single input frequency.
Practice generating a Bode plot for a given transfer function.
Model an active band-pass filter, obtain a transfer function, perform frequency
response computations, and plot a Bode plot.
Model a two-degree-of-freedom oscillator, obtain a transfer function, perform
frequency response computations, and plot a Bode plot.
Submit a memo presenting your results.
ES 205
Lab 7
Page 1 of 8
Computing the frequency response of a system
The big picture of frequency response analysis:
Important frequency response quantities:
•
Input amplitude:
•
Input frequency:
•
Transfer function:
()
•
Frequency response function (FRF):
(i) = ()|
•
FRF magnitude:
|(i)|
•
FRF phase:
∠(i)
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Lab 7
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Practice Problem 1
() =
5(10 + 50)
,
(2 + 5)( + 20 + 30)
withinput() = 3 sin 2.
Given:
A transfer function
Taks:
Compute the FRF magnitude |(i)| and FRF phase ∠(i).
Create an m-file and record the following lines of code:
w = 2;
s = i*w;
%
%
Input frequency from f(t)
Set s = i*w
Giw = (10*s + 50)/(s^2 + 20*s + 30)
%
FRF
magGiw = abs(Giw)
%
FRF magnitude
phaseGiw = angle(Giw)*180/pi
%
FRF phase in degrees
Results:
(i) =
_______________________________
|(i)| =
_______________________________
∠(i) =
_______________________________
The steady-state response "## () to the given input is
"## () = __________________________________________________________________
ES 205
Lab 7
Page 3 of 8
Plotting the frequency response of a system
The Bode plot:
Plot the FRF magnitude and phase for a range of frequencies:
50
Magnitude (dB)
45
40
35
30
25
20
10
1
10
0
1
10
Frequency (rad/s)
10
2
10
3
40
Phase (deg)
20
0
20
40
60
80
10
1
10
0
1
10
Frequency (rad/s)
•
Frequency is represented on a logarithmic scale in rad/s.
•
Magnitude is plotted on a linear scale in decibels (dB):
10
2
10
3
$%& = 20 log*+(|(i)|)
•
ES 205
Phase is plotted on a linear scale in degrees.
Lab 7
Page 4 of 8
Practice Problem 2
() =
5(10 + 50)
.
(2 + 5)( + 20 + 30)
Given:
A transfer function
Task:
Plot the Bode plot over the frequency range 0.1 rad/s to 1000 rad/s.
In your m-file, record and complete the following lines of code:
a
b
n
w
s
=
=
=
=
=
___;
___;
___;
logspace(a,b,n)';
i*w;
%
%
%
%
%
10^a = 0.1 rad/s
10^b = 1000 rad/s
Number of frequency points
Define the frequency range
Set s = i*w
Giw = _____________;
magGiw = abs(Giw);
MdB = 20*log10(magGiw);
phaseGiw = unwrap(angle(Giw))*180/pi;
%
%
%
%
FRF – use ‘.’ operators
FRF magnitude
Magnitude in dB
FRF phase in degrees
figure(1)
set(gcf, 'color', 'w')
subplot(211)
semilogx(w,MdB)
grid on
xlabel('Frequency (rad/s)')
ylabel('Magnitude (dB)')
%
%
Create a subplot figure
Bode magnitude plot
%
Bode phase plot
subplot(212)
semilogx(w,phaseGiw)
grid on
xlabel('Frequency (rad/s)')
ylabel('Phase (deg)')
Print your Bode plot.
Mark on the graph the points that correspond to the magnitude and phase you calculated in Practice
Problem 1.
ES 205
Lab 7
Page 5 of 8
Lab activity 1 – Frequency response of an active band-pass filter
As a team, do the following:
1.
Model the active band-pass filter shown below.
2.
Use Maple to obtain the transfer function relating the input voltage ,- to the output voltage ,./0 .
Use the following parameter values:
•
1* = 1 = 10kΩ
•
14 = 15 = 1kΩ
•
6* = 10μF
•
6 = 1μF
3.
Let ,- () = 5 cos 100 V. Calculate the FRF (i), the FRF magnitude |(i)|, the magnitude
$%& in dB, and the FRF phase ∠(i) at the input frequency.
4.
Write an expression for the steady-state output voltage, ,./0,## ().
5.
Plot the Bode plot over the frequency range 0.1 rad/s to 1000 rad/s.
ES 205
Lab 7
Page 6 of 8
Lab activity 2 – Frequency response of a two-degree-of-freedom oscillator
As a team, do the following:
1.
Model the two-degree-of-freedom oscillator shown below.
2.
Use Maple to obtain the transfer function relating the input force () to the block displacement
"*. Use the following parameter values:
•
:* = 2 kg
•
: = 1 kg
•
;* = ; = 0.5 N-s/m
•
<* = 10 N/m
•
< = 20 N/m
3.
Let () = 2 sin 4 N. Calculate the FRF (i), the FRF magnitude |(i)|, the magnitude $%& in
dB, and the FRF phase ∠(i) at the input frequency.
4.
Write an expression for the steady-state block displacement, "*,## ().
5.
Plot the Bode plot over the frequency range 0.1 rad/s to 100 rad/s.
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Lab 7
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Submit a memo detailing your results
Download the Lab 7 memo template from the course website.
As a team, complete the Lab 7 memo:
•
Complete all tables. Include units.
•
Paste in all requested plots. Do NOT use the Snipping Tool.
•
Provide descriptive captions for all tables and figures.
•
Type in the transfer function for each system analyzed. Use Equation Editor.
•
No additional text is required in your memo.
•
Attach printouts of your Maple worksheet(s) and MATLAB m-file(s).
Submit your Lab 7 memo by the end of the lab period.
ES 205
Lab 7
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