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docs/Chapters/Examples/EXAMPLE-3.md

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+# Example 3
+
+## Double Mass Cart
+
+![double mass cart](./../../Images/Examples/Example-3/double-cart.png)
+
+### Formulas
+
+- Resulting forces for cart 1:\
+$
+m_1 \ddot{p}_1 = k_2(p_2 - p_1) + b_2( \dot{p}_2 - \dot{p}_1) - 
+    k_1 p_1 - b_1 \dot{p}_1
+$
+
+- Resulting forces for cart 2:\
+$
+m_2 \ddot{p}_2 = F - k_2(p_2 - p_1) - b_2( \dot{p}_2 - \dot{p}_1) 
+$ 
+
+### Reasoning
+We now have 2 different accelerations. The highest order of derivatives is 2 for 
+2 variables, hence we need 4 variables in the `state`:
+
+$$
+x = \begin{bmatrix}
+x_1 = p_1\\
+x_2 = p_2\\
+x_3 = \dot{p}_1\\
+x_4 = \dot{p}_2
+\end{bmatrix}
+
+\dot{x} = \begin{bmatrix}
+\dot{x}_1 = \dot{p}_1 = x_3 \\
+\dot{x}_2 = \dot{p}_2 = x_4\\
+\dot{x}_3 = \ddot{p}_1 = 
+    \frac{1}{m_1} \left[ k_2(x_2 - x_1) + b_2( x_4 - x_3) - 
+    k_1 x_1 - b_1 x_3 \right]\\
+\dot{x}_4 = \ddot{p}_2 = 
+    \frac{1}{m_2} \left[ F - k_2(x_2 - x_1) - b_2( x_4 - x_3) \right]\\
+\end{bmatrix}
+$$
+
+Let's write our $S(A, B, C, D)$:
+$$
+A = \begin{bmatrix}
+0 & 0 & 1 & 0 \\
+0 & 0 & 0 & 1 \\
+% 3rd row
+- \frac{k_2 - k_1}{m_1} & 
+\frac{k_2}{m_1}  & 
+-\frac{b_2 + b_1}{m_1} & 
+\frac{b_2}{m_1} \\
+% 4th row
+\frac{k_2}{m_12} & 
+- \frac{k_2}{m_2} & 
+\frac{b_2}{m_2} & 
+- \frac{b_2}{m_2} \\
+\end{bmatrix}
+
+B = \begin{bmatrix}
+0 \\ 
+0 \\ 0 \\ 1
+\end{bmatrix}
+
+C = \begin{bmatrix}
+1 & 0 & 0 & 0 \\
+0 & 1 & 0 & 0
+\end{bmatrix}
+
+D = \begin{bmatrix}
+0
+\end{bmatrix}
+$$
+
+
+## Suspended Mass
+
+
+> [!NOTE]
+> For those of you the followed CNS course, refer to professor 
+> PDF for this excercise, as it has some unclear initial conditions
+>
+> However, in the formulas section, I'll take straight up his own
+
+![suspended mass](./../../Images/Examples/Example-3/suspended-mass.png)
+
+### Formulas
+
+- Resulting forces for mass:\
+$
+m \ddot{p} = -k(p - r) -b(\dot{p} - \dot{r})
+$
+
+### Reasoning
+
+
+$$
+x = \begin{bmatrix}
+x_1 = p \\
+x_2 = \dot{x}_1
+\end{bmatrix}
+
+\dot{x} = \begin{bmatrix}
+\dot{x}_1 = x_2 \\
+\dot{x}_2 = \frac{1}{m} \left[-k(x_1 - r) -b(x_2 - \dot{r}) \right]
+\end{bmatrix}
+$$
+
+<!-- TODO: Correct here looking from book -->
+> [!WARNING]
+> Info here are wrong 
+
+Let's write our $S(A, B, C, D)$:
+$$
+A = \begin{bmatrix}
+0 & 1\\
+-\frac{k}{m} & - \frac{b}{m} 
+\end{bmatrix}
+
+B = \begin{bmatrix}
+0 \\ 
+\frac{k + sb}{m}
+\end{bmatrix}
+
+C = \begin{bmatrix}
+1 & 0 
+\end{bmatrix}
+
+D = \begin{bmatrix}
+0 & 0
+\end{bmatrix}
+$$