Coupled Oscillator: Coupled Mass-Spring System with Damping: Difference between revisions
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<math> \ddot{x}_1= - \frac{2b_1}{m_1} \dot{x}_1 - \frac{k_1}{m_1} |
<math> \ddot{x}_1= - \frac{2b_1}{m_1} \dot{x}_1 - (\frac{k_1}{m_1}+ \frac{k_2}{m_2})x_1 - \frac{k_2}{m_2}x_2 + g </math> |
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and |
and |
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\begin{bmatrix} |
\begin{bmatrix} |
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0 & 1 & 0 & 0 \\ |
0 & 1 & 0 & 0 \\ |
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-\frac{k_1}{m_1} & -\frac{2b_1}{m_1} & \frac{k_1}{m_1} & 0 \\ |
-(\frac{k_1}{m_1}+\frac{k_2}{m_2}) & -\frac{2b_1}{m_1} & \frac{k_1}{m_1} & 0 \\ |
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0 & 0 & 0 & 1 \\ |
0 & 0 & 0 & 1 \\ |
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\frac{k_2}{m_2} & 0 & -\frac{k_2}{m_2} & -\frac{2b_2}{m_2}\\ |
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\end{bmatrix} |
\end{bmatrix} |
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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. |
We don't need to include gravity here if we allow are initial conditions for the spring to be zero with gravity accounted for. |
Revision as of 11:33, 3 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:
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 to be zero with gravity accounted for.