Coupled Oscillator: Coupled Mass-Spring System with Damping: Difference between revisions
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<math> \ddot{x}_2= - \frac{2b_2}{m_2} \dot{x}_2 - \frac{k_2}{m_2}l_2 - \frac{k_2}{m_2}x_2 + g </math> |
<math> \ddot{x}_2= - \frac{2b_2}{m_2} \dot{x}_2 - \frac{k_2}{m_2}l_2 - \frac{k_2}{m_2}x_2 + g </math> |
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==State Space Equation== |
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The general form for the state equation is as shown below: |
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<math> \underline{\dot{x}}(t) = \widehat{A} \, \underline{x}(t) + \widehat{C} \, \underline{u}(t) </math> |
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Where <math>\widehat{M}</math> denotes a matrix and <math>\underline{v}</math> denotes a vector. |
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If we let <math>x_1 \frac{}{}</math>, <math>\dot{x}_1 \frac{}{}</math>, <math>x_2 \frac{}{}</math>, and <math>\dot{x_2} \frac{}{}</math> be the state variables, then |
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<math> |
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\begin{bmatrix} |
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\dot{x}_1 \\ |
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\ddot{x}_1 \\ |
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\dot{x}_2 \\ |
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\ddot{x}_2 |
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\end{bmatrix} |
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= |
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</math> |
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<math> |
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\begin{bmatrix} |
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0 & 0 & 0 & 0 \\ |
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0 & 0 & 0 & 0 \\ |
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0 & 0 & 0 & 0 \\ |
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0 & 0 & 0 & 0 \\ |
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\end{bmatrix} |
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\begin{bmatrix} |
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x_1 \\ |
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\dot{x}_1 \\ |
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x_2 \\ |
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\dot{x}_2 |
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\end{bmatrix} |
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+ |
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\begin{bmatrix} |
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0 & 0 & 0 & 0 \\ |
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-1 & \frac{k_1}{m_1} & \frac{-k_2}{m_1} & \frac{1}{m_1} \\ |
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0 & 0 & 0 & 0 \\ |
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-1 & 0 & \frac{k_2}{m_2} & 0 |
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\end{bmatrix} |
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\begin{bmatrix} |
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g \\ |
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L_1 \\ |
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L_2 \\ |
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F |
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\end{bmatrix} |
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</math> |
Revision as of 12:30, 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