91Ó°ÊÓ

Consider a door that opens to only one side. A spring mechanism closes the door automatically. The state of the door at a given time $t$is determined by the angular velocity $\mathrm{θ}\left(t\right)$and the angular velocity $\mathrm{Ó\neg }\left(t\right)=\frac{d\mathrm{θ}\left(t\right)}{dt}$.

When the door is moving freely, its movement is subject to the following differential equations.

$$\begin{gathered} \frac{\mathbf{d}\mathbf{θ}}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathit{Ó\neg } \\ \frac{\mathbf{d}\mathbf{Ó\neg }}{\mathbf{d}\mathbf{t}}\mathbf{=}\mathbf{-}\mathbf{2}\mathit{θ}\mathbf{-}\mathbf{3}\mathit{Ó\neg } \end{gathered}$$

Matrix A for the system is:

$A=\left[\begin{array}{cc}0& 1\\ -2& -3\end{array}\right]$

Eigen values are given by the characteristic equation $\left|A-\mathrm{λ}I\right|=0$

Hence, the values of Eigen values are $\mathrm{λ}=-1,-2$

The corresponding eigen vectors are:

The phase portrait is as follows:

Let$E_{{\mathrm{λ}}_1},E_{{\mathrm{λ}}_2}$are Eigen spaces of${\mathrm{λ}}_1$androle="math" localid="1664894365518" ${\mathrm{λ}}_2$. Since both eigenvalues are negative.

Hence, the solution is:

$\stackrel{→}{x}\left(t\right)=C_1{e}^{{\mathrm{λ}}_{1}t}\stackrel{→}{v_1}+C_2{e}^{{\mathrm{λ}}_{2}t}\stackrel{→}{v_2}$

Here, the values are$\stackrel{→}{x}=\left[\begin{array}{c}\mathrm{θ}\\ \mathrm{Ó\neg }\end{array}\right]$,$\stackrel{→}{v_1}=\left[\begin{array}{c}1\\ -1\end{array}\right]$corresponding torole="math" localid="1664894343401" ${\mathrm{λ}}_1=-1$and $\stackrel{→}{v_2}=\left[\begin{array}{c}1\\ -2\end{array}\right]$corresponding to${\mathrm{λ}}_2=-2$.

As $t→∞,\stackrel{→}{x}\left(t\right)→0$solution converges to origin.

Also, $C_1$and $C_2$are arbitrary constants.

Dominant terms is $C_2{e}^{{\mathrm{λ}}_{2}t}\stackrel{→}{v_2}$as ${\mathrm{λ}}_2$is larger in magnitude, therefore distant trajectories are parallel to $E_{{\mathrm{λ}}_2}$.

All trajectories converge to origin.

Along the trajectory 1, the movement of the door slow down continuously until it reaches to the closed state as along this trajectory $w$decreases continuously.

Door slams if the trajectory 2 is followed that is initially if :

role="math" localid="1664894281284" $\frac{\mathrm{Ó\neg }\left(0\right)}{\mathrm{θ}\left(0\right)}<{\mathrm{λ}}_2=-2$

From figure it observe that for condition $\mathrm{Ó\neg }<0$door reaches$\mathrm{θ}<0$ .

Hence, all trajectories converge to origin and it observe that for condition$\mathrm{Ó\neg }<0$ door reaches$\mathrm{θ}<0$ .