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
Lab activity 1 – Frequency response of an active band-pass filter
125 min
Lab activity 2 – Frequency response of a two-degree-of-freedom oscillator
The big picture: 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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Practice Problem 1
5(10 + 50)
Given: A transfer function () = , with input ( ) = 3 sin 2 .
(2 + 5)( + 20 + 30)
Taks: Compute the FRF magnitude |(i)| and FRF phase ∠(i).
Create an m-file and record the following lines of code:
w = 2; % Input frequency from f(t)
s = i*w; % 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
"## ( ) = __________________________________________________________________
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Plotting the frequency response of a system
The Bode plot:
Plot the FRF magnitude and phase for a range of frequencies:
50
45
Magnitude (dB)
40
35
30
25
20 1 0 1 2 3
10 10 10 10 10
Frequency (rad/s)
40
20
0
Phase (deg)
20
40
60
80
1 0 1 2 3
10 10 10 10 10
Frequency (rad/s)
• Frequency is represented on a logarithmic scale in rad/s.
• Magnitude is plotted on a linear scale in decibels (dB):
$%& = 20 log*+(|(i)|)
• Phase is plotted on a linear scale in degrees.
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Practice Problem 2
5(10 + 50)
Given: A transfer function () = .
(2 + 5)( + 20 + 30)
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 = ___; % 10^a = 0.1 rad/s
b = ___; % 10^b = 1000 rad/s
n = ___; % Number of frequency points
w = logspace(a,b,n)'; % Define the frequency range
s = i*w; % Set s = i*w
Giw = _____________; % FRF – use ‘.’ operators
magGiw = abs(Giw); % FRF magnitude
MdB = 20*log10(magGiw); % Magnitude in dB
phaseGiw = unwrap(angle(Giw))*180/pi; % FRF phase in degrees
figure(1)
set(gcf, 'color', 'w')
subplot(211) % Create a subplot figure
semilogx(w,MdB) % Bode magnitude plot
grid on
xlabel('Frequency (rad/s)')
ylabel('Magnitude (dB)')
subplot(212)
semilogx(w,phaseGiw) % Bode phase plot
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 = 10 kΩ
• 14 = 15 = 1 kΩ
• 6* = 10 μF
• 6 = 1 μF
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.
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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.
ES 205 Lab 7 Page 7 of 8
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 Page 8 of 8
