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EE 350 Control Systems Assignments

Assignment 4: State space equation and it's solution

4.16 Consider automobile damping system as depicted in fig below.

Fig 4.5 Damping system of an automobile

The data for the system are:
a) Mass of automobile $(m_{1})=375kg$ and mass of wheel ($m_2$)=30kg
b) $k_{1}=1500N/m$ and spring constant of tyre $k_{2}=6500N/m$
c) Damping constant C=0, 375, 750, 1125
d) $x_{1}\rightarrow$ Displacement of automobile body from equillibrium ,m; $x_{2}=\dot{x}_{1}$
$x_{3}\rightarrow$ displacement of the wheel from equillibrium position ,m; $x_{4}=\dot{x}_{3}$
e) $v\rightarrow$ velocity of car=9, 18, 27, 36m/sec
The model of the system is given by $\begin{bmatrix} \dot{x}_{1} \\ \dot{x}_{2} \\ \dot{x}_{3} \\ \dot{x}_{4} \end{bmatrix}=\begin{bmatrix} 0 & 1 &0 & 0 \\ -k_{1}/m_{1} & -c/m_{1} & k_{1}/m_{1} & c/m_{1} \\ 0 & 0& 0 & 1 \\ k_{1}/m_{2} & c/m_{2} & -(k_{1}+k_{2})/m_{2} & -c/m_{2} \\ \end{bmatrix}\begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{bmatrix}+\begin{bmatrix} 0 \\ 0 \\ 0 \\ k_{2}/m_{2} \end{bmatrix}u(t)$ where $u(t)=(1/6)sin(2vt\pi/20)$. Describe the profile of the road way. Now
a) Assumeing $x(0)=\begin{bmatrix} 0 & 0 & 0 & 0 \end{bmatrix}^T$, plot $x_{1},x_{2},x_{3},x_{4}$ for all combination of C and v
b) Double and half the $k_{1}$ and plot $x_{1},x_{2},x_{3},x_{4}$
c) Double and half the $k_{2}$ and plot $x_{1},x_{2},x_{3},x_{4}$
d) Compare all the result and comment on your results.