ENS3553: SIGNALS & SYSTEMS
Due 5:00 p.m. Friday, 2nd November 2018
This assignment is worth 10% of the unit mark.
This assignment is divided into 3 parts. Total 80 marks
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Analysing a system
A vehicle suspension system can be modelled by the block diagram shown in Figure 1 below:
Figure 1: Block diagram of vehicle suspension system
In this block diagram, the variation in the road surface height as the vehicle moves is the
input to the system. The tyre is modelled by the spring and dashpot (damping) system with
spring constant and damping coefficient respectively and this results in the
displacement of the wheel (), represented by the mass . The wheel’s displacement acts
as an input to the suspension system, modelled by the spring and dashpot with spring
constant and damping coefficient respectively and this results in the displacement (
of the body, represented by the mass . When the car is at rest, it is taken that = 0, =
= 0. (Note: is normally a quarter of the vehicle mass since it is assumed the weight
is distributed evenly between the 4 wheels.
This system is composed of two mass-spring-damper systems ‘stacked’ one on top of the
other. We shall first consider the behaviour of a single sub-system and then later attempt to
combine these to find the overall system behaviour.
Consider the simple mass-spring damper system shown in Figure 2 below:
Figure 2: A single mass-spring-damper system
In Figure 2:
x is the position of input body/surface, with its rest position given by x = 0.
The mass m represents the mass
The height of mass m above its reference level is called y. The reference level is chosen
such that when system is at rest, y = 0.
Section 1: Mathematical Analysis of System (25 marks)
1. Draw a free-body diagram showing all the forces acting on the mass m shown in Figure 2.
2. From the earlier description, diagrams and the laws of Physics, show that the motion of
the system in Figure 2 can be described by the LCCDE (linear constant-coefficient
differential equation) below:
3. Using the Laplace transform of the equation above, find an expression for , the
system transfer function.
The mass-spring-damper system is a damped second order system. It is common to express
the homogenous second order DE for such a damped system as
= 0 2
where is the damping ratio and is the undamped natural (resonant) frequency.
4. From equations (1) and (2), determine expressions for (the damping ratio) and (the
natural frequency) in terms of the parameters m, k and C
5. Determine the characteristic equation and eigenvalues (characteristic values) for this
system based on equation (2) above.
6. From the answer to part 5, determine the natural response of the system for the following
a. = 0
b. 0 < < 1
c. = 1
d. > 1
Consider a suspension system with the following parameters:
= 340 kg
= 21,000 N/m
7. Determine (in rad/s) for this suspension system and the corresponding value for
8. Calculate the required value of in order to achieve = 1
Note: Complete and clear working is required for all answers for this section.
Section 2: System analysis using Matlab (30 Marks)
In this section, the system responses should be analysed using Matlab. Refer to the document
“A Brief MATLAB Guide” in order to understand how to represent LTI systems in Matlab, and
hence how to determine impulse response, step response and frequency response of
systems. Students are advised to refer to the help function within Matlab as well as online
Matlab documentation for more details.
MATLAB is installed in the engineering computer labs.
Using the commands given in the Guide, analyse the response of the suspension system using
the and parameters given in Section 1 and value calculated in question 8:
9. Plot the impulse response and step response of the system (for 2 seconds duration and
time ‘step size’ of 1 millisecond) using the impulse and step functions. Include all plots
(properly labelled) in your submission.
10. Determine the frequency response from 0 to 200 rad/s using the freqs command. Plot the
magnitude and phase response over this frequency range. Hint: Use frequency ‘step size’
of 0.1 rad/s.
Hint 1: You can plot all 4 graphs in one go using a 2 x 2 matrix of plots using subplot(22n),
where n determines which of the 4 subplots gets used.
Hint 2: In order to clearly see variations over a range of frequencies, it is best to use a log
scale for the frequency and magnitude (phase would still be displayed using linear scale).
The functions loglog (for magnitude) and semilogx (for phase) can be used instead of plot.
11. Determine the magnitude response at . Determine the frequency of the -3dB point
(where magnitude = 1⁄√2). Hint: Use the ‘data cursor’ tool on the plot of the magnitude
response. It shows the x and y values of the plot as you move along the curve.
12. Discuss the response of the system. Why do the impulse and step responses have that
particular shape? How well will this system fulfil its purpose of a vehicle suspension?
Note: The function of a suspension system is to ‘filter out’ the effect of bumps, potholes
and other such vibrations, but allow the vehicle to ‘follow the road’ as the height of the
road surface varies.
13. Repeat the analysis above (steps 9 – 11) for the following damping ratios
a. = 0.5
b. = 0.7
c. = 1.5
Hint 3: It would be more efficient to put all the necessary commands into a script file
(a .m file) so you can edit the parameters and then run all the commands at once.
14. Based on the results of the Matlab analysis above, which of the 4 values of damping ratio
would be best for application as a suspension system. Justify your selection.
Section 3: Including the wheel & tyre response in the analysis (25 marks)
The tyre and wheel assembly can also be modelled as a mass-spring-damper system (refer to
The parameters for a wheel and tyre system are as follows:
= 45 kg
= 192,000 N/m
= 100 Ns/m
15. Based on the parameters above, determine the natural frequency of this system
16. Find the impulse response, step response and frequency response of the system (same as
in steps 9 – 10).
17. Do the impulse and step responses look like what you would expect from a vehicle wheel?
Discuss any differences and reasons for them. (Hint: Refer to Tutorial 7 Question 7
18. The frequency response of a ‘cascade’ system, such as this suspension and wheel
combined system, can normally be worked out by combining the frequency response of
the individual subsystems. Combine the frequency responses obtained in Sections 2 and
3 to find a predicted overall system frequency response (plots of combined magnitude
and phase responses).
19. Discuss the overall frequency response found using this method and its validity.
20. Analysis of such suspension systems sometimes ignore the effect of the tyre’s ‘spring and
damper’ effect. Discuss the impact of this based on your previous results.
ASSIGNMENT 1 MARKING SCHEME
Section 1: Mathematical Analysis (25 marks)
Free body diagram 3
Deriving DE formula 4
System transfer function 2
Expressions for damping coefficient and natural frequency 2
Characteristic equation and eigenvalues 4
Natural response (4 cases) 8
Calculation of and 2
Section 2: Matlab System Analysis (30 marks)
Impulse and step response, frequency plot, and specific magnitude values 12
Description of responses and relating it to the system’s desired function 3
Analysis of various damping ratios 12
Selection and justification of preferred damping ratio 3
Section 3: Including the wheel & tyre response in the analysis (25 marks)
Calculation of 1
Impulse and step response, frequency plot, and specific magnitude values 8
Discussion on time response 4
Combined frequency response 5
Discussion on shape and validity of frequency response 4
Discussion on validity of ignoring tyre effects 3
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