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Line 82: |
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\begin{bmatrix} |
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\begin{bmatrix} |
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0 & 1 & 0 & 0 \\ |
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0 & 1 & 0 & 0 \\ |
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0 & 0 & 0 & 0 \\
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-\frac{k_1}{m_1} & -\frac{2b_1}{m_1} & \frac{k_1}{m_1} & 0 \\ |
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0 & 0 & 0 & 1 \\ |
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0 & 0 & 0 & 1 \\ |
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0 & 0 & 0 & 0 \\ |
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0 & 0 & -\frac{k_2}{m_2} & -\frac{2b_2}{m_2}\\ |
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\end{bmatrix} |
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\end{bmatrix} |
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\begin{bmatrix} |
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\begin{bmatrix} |
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L_1 \\
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l_1 \\ |
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L_2 \\
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l_2 \\ |
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\end{bmatrix} |
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\end{bmatrix} |
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</math> |
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</math> |
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We don't need to include gravity here if we allow are initial conditions for the spring be zero with gravity accounted for. |
Revision as of 12:57, 1 December 2009
Problem Statement
For the below system set up a set of state variable equations, and then solve using Laplace transformations. Assume all motion takes place in the vertical directions.
Initial Values
For the upper mass:
And for the lower mass:
Find the Force Equations
First we need to sum forces in the y-direction for each block.
For mass 1:
For mass 2:
For the cases above
and
where l is the unstretched length of the spring and x is the displacement of the spring.
So if we put the equations above into the correct form we have:
and
State Space Equation
The general form for the state equation is as shown below:
Where denotes a matrix and denotes a vector.
If we let , , , and be the state variables, then
We don't need to include gravity here if we allow are initial conditions for the spring be zero with gravity accounted for.