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