Consider a RLC $($resister$-$inductor$-$capacitor$)$ circuit with all impedance elements connected in series that are powered by a$$

sinusoidal voltage source $($recall that you already analyzed an AC RL circuit in Assessment$2).$ In this RLC circuit the inductor and$$

resister have fixed constant values however the capacitor has a variable capacitance. The variable capacitance is obtained by$$

constructing the capacitance from two parallel metal upper and lower plates separated by a small distance where the parallel plates

lie in a horizontal orientation in a constant gravitational field separated by a known specified dielectric such as a fluid such as air or$$

oil. A wire connects the positive end of the voltage source to the upper plate of a known mass which is suspended overhead to a$$

ceiling with a spring and a damper that are connected in parallel. Another wire is connected to the lower plate which maintains a$$

constant distance from the overhead ceiling with a rigid support that supports the lower plate such that this wire is in turn connected$$

to the LR portion of the circuit which in turn is connected to the negative end of the voltage source. The combination of the RLC$$

circuit which forms an electrical system for a band pass filter to receive radio signals from a nearby telecommunications cellular$$

tower, and the mass$-$spring$-$damper which forms a mechanical system, together constitutes an integrated mechatronics system.$$

The mechatronic system has parameters such that the equivalent natural frequency of vibration of the mechanical system differs$$

from the equivalent resonance frequency of the electronic system, such that the system exhibits a complex overall system response

based on the coupling interactive effect of the governing differential equation of the mechanical system and the governing$$

differential equation of the electronic system.$$

Investigate appropriate parameters for such a physical system and substantiate the chosen parameters with underlying engineering$$

judgement and reasons, then construct a mathematical model of the entire mechatronic system, and use any appropriate software$$

package $($e$.$g$.$ Scilab$/$Xcos$,$ Matlab$/$Simulink or Mathematica$/$Wolfram Systems Modeler etc.$)$ to numerically simulate the$$

mechatronic system response in the time domain and$/$or frequency domain. Document your investigation, findings, and reflections$$

in a type written engineering report inclusive of all derivations, calculations, graphical plots, and computer codes in appropriate

appendices to demonstrate your level of knowledge and competence in mechatronics.$$

For additional marks: utilize your model to investigate and analyze how the mechatronic system response of the capacitor plates$$

could potentially be used for a position and$/$or speed sensor$/$transducer and perform an engineering analysis to determine how

feasible such a mechatronic system may be in practical applications.$$