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signal and systems (Analyzing a simple system), computer science homework help

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:

  • h is the height of the wheel; the origin of the coordinates is such that when the wheel
  • is rolling on flat, level ground, h = 0.

  • The mass M represents the weight of the unicycle.
  • The height of mass M above its reference level is called y. The reference level is
  • chosen so that when the car is in level motion, y = 0.

  • Under these conditions the length of the spring is L0. The spring exerts forces in the
  • 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:

    ??????

    ??? +

    ?

    ???? =

    ?

    ?ℎ???

    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

    response?

    (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

    following parameters:

    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.

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