ENS3553: SIGNALS & SYSTEMS

ASSIGNMENT

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

You will need to reference all external sources of information.

IMPORTANT: Submission Instructions

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answers to the stated problems.

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The use of materials drawn from other sources without appropriate acknowledgement is

plagiarism and consequences for such actions will apply. Refer to Academic Misconduct

section at the end of the assignment.

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 and

= 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:

+

+

=

+

1

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

+ 2

+

= 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

cases:

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

(in Hz).

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

Figure 1).

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

solutions).

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)

Description 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

TOTAL 25

Section 2: Matlab System Analysis (30 marks)

Description 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

TOTAL 30

Section 3: Including the wheel & tyre response in the analysis (25 marks)

Description 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

TOTAL 25

Academic Misconduct

Edith Cowan University has firm rules governing academic misconduct and there are substantial

penalties that can be applied to students who are found in breach of these rules. Academic

misconduct includes, but is not limited to:

plagiarism;

unauthorised collaboration;

cheating in examinations;

theft of other students’ work.

Additionally, any material submitted for assessment purposes must be work that has not been

submitted previously, by any person, for any other unit at ECU or elsewhere.

The ECU rules and policies governing all academic activities, including misconduct, can be accessed

through the ECU website.

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