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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.

    Automobile Damping

    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.


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