Characteristic Equation and its Roots - MATLAB - Engineering Assignment Help

Assignment Task:

Produce a typeset report on the assignment below.  Discussion is important.  Be clear and give your qualitative insights supported by quantitative results and plots.  Be creative.  More complex systems and deeper insight earn more points.  This is an open-ended assignment. You may explore topics that we did not cover in class or are not listed here if you feel comfortable and/or want to earn even more.
Choose a mechanical, electrical, thermal, fluid, or coupled (e.g. electromechanical) system composed of linear elements and with a  single or multiple inputs. Be creative, but the system must be physically plausible.   The system must contain at least one element that yields an overall mathematical model to be a second-order linear differential equation (e.g. a mechanical damper, an electrical inductor, etc).
1.  Sketch your system.
2.  Derive the mathematical model for your system.  Show all steps.
3.  Compute the Laplace transform of the model.
4.  Compute the Transfer Function for each input.
5.  Determine the free, forced, steady-state, and transient components of the response.
6.  Choose reasonable values for the model parameters while varying the damping ratio; ζ. Show solutions for the response of the system for the undamped,  underdamped, critically damped, and overdamped cases.  For the underdamped case choose parameters such that ζ ≤ 0.707 (this is important for resonant behavior in later parts of this project).  Plot these solutions using MATLAB.
7.  Discuss the characteristic equation and its roots.  For example, comments like these show your understanding of the response.   Play with numbers in MATLAB to gain more intuition.
(a)  Unstable behavior occurs if any root lies to the right of the imaginary axis.
(b)  The response oscillates only when a root has a nonzero imaginary part.
(c)  The greater the imaginary part, the higher the frequency of the oscillation.
(d)  The farther to the left the root lies,  the faster the response due to that root decays.
You may use additional plots to show the effects the roots have on the response and make qualitative statements.
8.  For the underdamped case you have selected, determine important response parameters like ωn, ωd, ζ, τ, Mp, tp, tr, etc.  Derive an expression for them and calculate the value for the constants you chose to plot with.
9.  Discuss the behavior of your system to a harmonic input of the form f (t) =sin(ωt). Solve for the response using frequency domain techniques (i.e.G(jω)).

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