Transcribed Text
Part 1 Explanation
involves the following:
A Mechanical
 Derive the transfer function for a mechanical system.
 Reference the transfer function to the generalized second order transfer function.
 Use the transfer function in Matlab to plot the system’s transient response (for a step input)
under different damping conditions. Use the Matlab results to determine system damping, and
other transient characteristics.
 Plot the system poles in the complex plan. Use the poles plot to infer some conclusions
 Plot the frequency response under different damping conditions. Use the response to infer the
following: natural frequency, cutoff frequency, rolloff rate, type of response
B Electrical
 Write the state equations for a second order electrical system
 use the derived state equations to:
o determine the system’s eigenvalues
o predict the system’s generalized transient response due to a step input
o Use Matlab to plot the step  transient response of the system for different states of the
system
o Plot the systems response for a sinusoidal input
o make concluding remarks about the response in relation to system poles.
C Simulation
 Simulate the electrical circuit and investigate:
o the transient response for a step input
o show the response of different states due to the step input
o compare this result to that obtained with Matlab
o investigate the output (response) for nonzero initial conditions and compete to Matlab
results
o investigate the response (output) for a sinusoidal input and compare to Matlab Results.
D Discussion and Conclusions
o Discuss results
o explain certain phenomenon of system response
o demonstrate understanding of the material
Deliverables:
All deliverables must be complete, well organized, have professional appearance, show all work, provide
commented and organized Matlab Code, provide Matlab results and plots (label correctly), discuss
results.
MUST be well organized and very easy to follow.
For each part of the sections above:
1 give the problem statement
2 provide schematics/drawings
3 show all theoretical work  well presented
4 discussion of results thus far
5 provide compete and well documented Matlab code anytime used
6 provide results of software (Matlab or Simulink). It must be well presented ( don’t just show results
and leave to the reader to figure out)
7 correctly labeled and titles figures.
8 draw conclusions or state comments that demonstrate your understanding
Solutions must be complete, clear, professionally presented, well organized
This is individual work. All Matlab code must be placed in the corresponding dropbox for this exam
on D2L
Part 2
Problem No.1 (Mechanical System)
For the massspringdamper system shown, determine the expression for the motion, x(t) and
plot it. K= 9 N/m, m=3kg, f = 28N and the damping constant has the following three
possibilities:
� = #
���� (�) 12
���� (�) 1
���� (� ) 10.39232 �. �/� . Assume zero initial conditions{�(0) = 0 �, �̇(0) = 0 �/�
� is the displacement and �̇ is the velocity.
Hint: Refer to Chapter 3 and to lectures of 4/17 through 4/27
Do not substitute values for m, K, B, or f yet.
Step 1 Write the differential equation for the displacement, y(t)
�̈(�) = − ?
? �̇(�) − ?
?
�(�) + ?
? �(�)
Step 2 With zero initial conditions and using your result in Step 1, determine the transfer function G(s).
Ensure it is written in the standardized form (as was done in class.)
�(�) = C(D)
E(D) = ?
Step 3 In this step, you will substitute the values of m and K as given above and for each case of B
you are to:
 calculate the poles and zeros of the transfer function
 predict the step response of the system
 calculate the damping ratio, the natural and damping frequencies
 complete the table
Using your result of G(s) found in Step 2, calculate and complete the table below. Must Show work
for case (b)
Table1: Second order Translational System’s Data
B
(N.s/m) ��(rad/sec) ��(rad/sec) � poles zeros Expected Step Response
classification
Case( a)
12
Case( b)
1
Case( c)
10.3923
Work for Case (b):
Step 4 Use Matlab (Submit the complete code in the designated dropbox on D2L)
Write a complete and well organized and documented Matlab code as follows:
Name this code: Pr1Stp4.m
 First, make a data section for the system constants m, K, B, and A. Note A is the magnitude of
the input (in our case f = 28N, thus A = 28.)
you may want to be more creative here and make B a vector of three quantities. Your code
loops the number of quantities in vector B. And in each iteration, it picks the proper B case
based on iteration number (index).
However, you can run the program separately each time by providing a new value of B based on
the case under test.
 define the numerator and denominator vectors
**Repeat all of the below for each value of B. This is indicated on the next page.
**
Case (a) B =
num = [ ] ?
den = [ ]?
create and display the transfer function
G= tf(num,den) notice: no semicolon, thus G will be displayed
 pause the code here so the user can observe the transfer function. Of course, instruct the user
to “ hit any key to continue …”
 determine the poles and zeros of the transfer function. (There are several commands that one
may use including commands that map the poles and zeros onto the complex plane. for
example: pzmap(G) followed by the grid command
will only use the commands:
pole(G)
zero(G)
 pause the code here so the user can observe the transfer function. Of course, instruct the user
to “ hit any key to continue …”
 write the poles and zeros down as you will need to enter them into Table 2
poles are:
zeros are:
 Determine the step response due to f = 28N
step(A*G)
 title and label the axis properly. Also include a grid
 of course do the pause steps to allow user to complete and paste the figure. Or you may use
the figure handle as we did in class, this way all figures can be accessed.
 copy and paste the figure into Word (Place below this line )
 do not close the figure yet
 right click on the figure and show different transient characteristics (rise time, overshoot, steady
state value, and settling time)
 copy the figure with the data and paste into Word. Data on this figure will need to be placed in
Table 2 later. (Place the figure below this line)
Frequency Response
 Use the command:
bode(G) to plot the frequency response
 in the magnitude plot, move the cursor to the lowest possible frequency (close to 1Hz) and
write down the corresponding gain (this is called the Low Frequency Gain.)
Ao(dB) = ? (write this value as it will be needed for Table 2 later)
 move the cursor until you reach a gain that is 3dB below Ao(dB). Record the frequency at that
point. This is the cutoff frequency
wc = ? Write down this value as it will be needed later for Table 2
 For case(b) only, move the cursor to the maximum value of the magnitude frequency response,
the frequency at this point corresponds to the natural frequency. wn Record this frequency and
provide the Matlab figure. ( Insert the figure below this line)

 �K = = ?
Case (b) B =
Repeat the steps of case (a) above **
Case (c ) B =
Repeat the steps of Case (a) above **
Matlab summary results. Insert your results in the table below.
Transient response characterization refers to the response type: overdamped, underdamped,
or critically damped.
Table 2: Matlab Summary Results for the 2nd Order Mechanical system
Case( a)
B = 12 (N.s/m)
Case( b)
B = 1 (N.s/m)
Case( c)
B = 10.3923 (N.s/m)
poles
zeros
tr (rise time) (sec.)
ts (settling time) (sec.)
% Overshoot
steady state value (m)
A0(dB)
�L (rad/sec.)
�K (rad/sec.)
Transient Response
Characterization
Problem No. 2 (Electrical System)
Given the electrical system shown. Assume zero initial energy stored in the dynamic elements L and C.
A LaPlace Domain
Step 1 Derive the transfer function G(s)
�(�) = �L(�)
�OK(�)
Make sure the transfer function is in the standard form per the class lecture
Step 2 From the equation, determine the following in terms of the circuit elements R,L and C:
The natural frequency: �K
The damping Ration �
Step 3 Given your equation in Step 1, Write a code using Matlab to determine the step response of the
voltage across the capacitor for the following cases of R,L and C . Also, from the step response, complete
Table 3 for each of the cases. Ensure to plot properly labeled and titled step response figures for each of
the cases. Also provide the well organized and commented code in the designated dropbox. Name this
code as: Pr2A_St3.m
Additionally, in your code, using the equations of Step 2, calculate �K and � for each case of the
component values.
Cases:
R = 8 ohms, L =0.2H,
case a) C = 0.04F
Case b) C =0.0125F
Case c) C = 0.0004F
Table 3: Second Order Electrical System Data
Case (a) Case (b) Case (c)
��
�
steady state value
tr (rise time) (sec.)
ts (settling time) (sec.)
% Overshoot
Step Response Type
Insert the transient response figures below
B State Equations
The goal is to write the state space model for the same electric circuit (shown below for convenience) in
terms of E, R, L and C (assume zero initial energy is stored in the system). Let the states be as follows:
�R = �L ��� �V = �X
Where the output y is :
case (i) y = vc
case (ii) y= iL
case (iii) the output is a vector of two quantities: � = Y
�L
�X
Z
case (iv) the output is a vector of three quantities: � = [
�L
�X
�\
]
Procedure:
Step 1 Write the KVL differential equation. Refer to Chapter 3 and to the class notes on a series RLC
circuit.
Step 2 Convert the differential equation to state equations. Show work
^
�̇
R
�̇
V
_ = Y ZY Z + Y Z �
Step 3 For case (i) where y = vc, complete the below
� = [ ]Y Z + [ ]�
Step 4 For case (ii) where y = iL, complete the below
� = [ ]Y Z + [ ]�
Step 5 For case (iii) where y is a vector of two quantities: � = Y
�L
�X
Z, complete the below
� = Y
�L
�X
Z = Y ZY Z + Y Z �
Step 6 for case (iv) where y is a vector of three quantities: � = [
�L
�X
�\
]. write the output equations below
Step 8 Matlab Simulations (See hints below) name this code: Pr2B_St8.m
Matlab® Requirements: (see some Matlab hints at the end of this document) Just like with Part A of this
problem, we will have three cases of the values R, L and C.
Cases:
R = 8 ohms, L =0.2H,
case a) C = 0.04F
Case b) C =0.0125F
Case c) C = 0.0004F
Do the following for each of the component cases
Step 1
Using the state equations, plot the step response of the system for each case of the outputs above
(as noted in steps 3,4,5, and 6.) Make sure to label and title each figure clearly. Include the plots in
this document. Include the Matlab code (submit into the designated dropbox.)
place the figures here
Step 2 Sinusoidal Input (Name this code Pr2B_sim.m and place the code in the designated dropbox.)
plot the output vc  this is case (i) for each of the situations below
A) E(t) = 10sin(2 π*100 t) u(t) and zero initial conditions
B) E(t) = 10sin(2 π*100 t) u(t) and iL(0) = 100mA, vc(0) = 1 volt.
Some Matlab® hints:
To create a state space model (like a struct)
Model = ss(A,B,C,D)
To find the step response of the above model (given zero initial conditions)
step(Model)
What if the input is a sine wave, E(t) = 20sin(2 π*100 t) Then:
(BE CAREFUL here, you already included the magnitude 20 in the system matrices created above. Thus,
either change this E magnitude of 20 to 1 in the original system equations or change the 20 to 1 in the
below.)
Need to create a time vector:
t = 0:0.0001: 0.2;
u= 20*sin(2*pi*100*t);
y=lsim(Model,u,t)
If the nonzero initial conditions:
Then, add a vector for the initial conditions:
X0=[1; 0.1] ; % This means vc(0) =1, iL(0) = 100mA
Then
t = 0:0.0001: 0.2;
u= 20*sin(2*pi*100*t);
y=lsim(Model,u,t,X0)
in Matlab® command window, type help lsim
Problem No. 3 (Multisim Simulation)
Use Multisim to simulate the series RLC circuit as follows:
time, t goes from 0 to 0.2 seconds.
Step 1 Simulate the RLC circuit and plot the transient response of yc , vL and iL . Assume zero initial
conditions. In your transient simulations, ensure to set initial conditions to 0. Note, each of the plots is
done by a new simulation.
Step 2 Let the initial conditions be iL(0) = 100mA, vc(0) = 1 volt. simulate and plot the transient response
of yc , vL and iL . In your transient simulations, ensure to set initial conditions to User defined. Also, set
the initial conditions on the circuit in Multisim.
Problem No.4 (Conclusions)
Provide a well written and concise explanations of the results of your Matlab and Multisim simulations.
 discuss the overall process of each
 discuss whether the results agreed or not
 Also, discuss and explain each of the following. Give scientific explanation (short and to the point)
o natural frequency of a mechanical system,
o damping frequency of a system
o damping ratio
§ type of transient response if the damping ratio is greater than 1
§ type of transient response if the damping ratio is equal to 1
§ type of transient response if the damping ratio is greater than zero but less than
1
§ type of transient response if damping ratio is equal to zero
§ type of response if damping ratio is negative.
o General and good description of the pole locations in a second order system for each of
the following cases of damping ratio, � .
case 1: � > 1
case 2: � = 1
case 3: 0 < � < 1
case 4: � = 0
case5: � < 0
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