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 |
<math> \ddot{x}_1= - \frac{2b_1}{m_1} \dot{x}_1 - \frac{k_1}{m_1}x_1 + \frac{k_2}{m_2}x_2 1 \frac{k_2}{m_2}x_1 + g </math> |
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and |
and |
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<math> \ddot{x}_2= - \frac{2b_2}{m_2} \dot{x}_2 - \frac{k_2}{m_2} |
<math> \ddot{x}_2= - \frac{2b_2}{m_2} \dot{x}_2 - \frac{k_2}{m_2}x_2 + \frac{k_2}{m_2}x_1 + g </math> |
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==State Space Equation== |
==State Space Equation== |
Revision as of 11:27, 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 be zero with gravity accounted for.