(Solved): solve number 3 Figure 1: Cart with simple pendulum. This mechanisin has one translational degrce of ...

Figure 1: Cart with simple pendulum. This mechanisin has one translational degrce of frecdom arid one rotational degree of freedom. The system physically consists of a cart $C$ with mass $m_1$ and center of mass $c$ rolls along ia as shown in Figure 1. The distance from $w$ to $c$ along $i\mathrm{A}$ is $q.\mathrm{A}$ simple pendulum $\mathcal{L}$ with length $f$ is connected to the cart at $c_1$, and the particle $y$ mountod on the end of the pendulum has mass m2. As shown in the Figure, an external force ${\vec{f}}_{\text{ext }}=f_{\text{ext }}{\hat{l}}_A$ is applied to the cart. The cquation of motion of the cart and the pendulum is $\left[\begin{array}{cc}{m}_1+m_2& m_2\ell \cos \theta \\ m_2\ell \cos \theta & m_2{\ell }^2\end{array}\right]\left[\begin{array}{l}q\\ \theta \end{array}\right]+\left[\begin{array}{c}-m_2\ell {\dot{\theta }}^2\sin \theta \\ m_2\ell g\sin \theta \end{array}\right]=\left[\begin{array}{c}{f}_{\mathrm{cxt}}\\ 0\end{array}\right].$ \begin{tabular}{|c|c|} \hline Parameter & Value \\ \hline Mass of the cart, $m_1$ & $2\mathrm{kg}$ \\ \hline Mass of the pendulum, $m_2$ & $1.\mathrm{kg}$ \\ \hline Length of the pendulum, $\ell$ & $30\mathrm{cm}$ \\ \hline Acceleration due to gravity, $g$ & $10\mathrm{m}/{\mathrm{s}}^2$ \\ \hline \end{tabular} 3. Lineariazation. Lincarize the nonlincar system (1) about all cquilibriums you found in the previous step. At each equilibrium, classify the linearized system as asymptotically stable, semi-stable, Lyapunov stable, or unstable.