For details please see the attached document
Part 1: Analysing a simple system (20 marks)
Consider a “unicycle” shown below, which consists simply of the chassis connected to the
wheels through a spring:
In this figure:
is rolling on flat, level ground, h = 0.
chosen so that when the car is in level motion, y = 0.
y-direction, depending on the length of the spring and the spring constant k,
according to Hooke’s law of elasticity.
Note: At equilibrium the spring is slightly compressed from its natural length due to the
weight of the vehicle, making its length equal to L0. Then the total force acting on M – gravity
plus spring force – is zero, and M neither rises nor falls.
Section 1A: Mathematical Analysis
1. From the description and the laws of Physics, show that the motion of the unicycle can be
described by the LCCDE (linear constant-coefficient differential equation) below:
2. Determine the characteristic equation and eigenvalues for this system
3. Work out the frequency response of this system. Do you foresee any problems with this
(a) Simplified suspension system
(b) The driving frequency is determined by the spacing of the
bumps and the speed of the vehicle
Section 1B: Analysis using Matlab
In this section the system responses should be analysed using Matlab. Refer to the attached
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. MATLAB is installed in the computer labs.
Using the commands given in the Guide, analyse the response of the system using the
M = 200 kg
k = 1800 N/m
?? = 0.25 m
4. Determine the frequency response from 0.1 Hz to 20 Hz using the freqs command. Plot
the magnitude and phase response over this frequency range.
5. Plot the impulse response and step response of the system (for 5 seconds duration) using
the impulse and step functions. Include all plots (properly labelled) in your submission.
6. Discuss the response of the system. Does the frequency response match what was
calculated in Part 1A? How does the response correspond to the systems physical
parameters? Would this setup perform the required function as it is?
Note: The function of the system is to minimise the up and down motion of the vehicle (while
still following contours of road).
Part 2: Improving the system (35 marks)
To improve the system, what is done is to add another component to the suspension system,
as shown below:
The new component is called (somewhat misleadingly) a shock absorber. It is the same
as a dashpot: a viscous-damping device consisting of a plunger moving through a
viscous fluid. It adds an additional frictional force proportional to velocity and to a
constant c describing the drag, determined by the size and shape of the shock
absorber. This frictional force resists motion, i.e., acts in the opposite direction,
slowing the motion and absorbing energy.
Improved suspension system with “shock absorber”.
7. Work out the LCCDE for the system including the “shock absorber”. Show all working.
8. Use Matlab to plot the frequency response (magnitude and phase), impulse response
and step response of the system for the following values of c:
a. C = 200
b. C = 400
c. C = 1800
Hint 1: 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 one go.
Hint 2: You can plot all 4 graphs at one go using a 2 x 2 matrix of plots using
subplot(22n), where n determines which of the 4 subplots gets used.
9. From the plots determined above, describe the behaviour of the system as the value
of c is changed. How has adding the “shock absorber” helped the system? What are
the advantages and disadvantages of the different c values in terms of the primary
function of the system?
10. Use Matlab to determine the optimal value of c in terms of the function of the system
(in your opinion). Show all 4 response plots for this chosen value of c, and justify your
selection (discuss and give reasoning) based on these plots.
Part 3: Investigation of State Space Analysis Techniques (25 marks)
In this section the task is to research the state space analysis approach to signals and systems
design and investigation and to produce a concise (~1000 – 1500 words) description of what
it is and how it could be applied.
You should explain what the state space analysis approach is (the final chapter in the textbook
is a good starting point), its advantages and disadvantages, and what types of problems it is
appropriate for. You should identify in the literature particular classes of problems that state
space analysis techniques have been applied to and the outcomes.