Coupled Oscillator: Coupled Mass-Spring System with Damping: Difference between revisions
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For mass 1: |
For mass 1: |
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<math> + \uparrow \sum F_{y_1} = m_1 \ddot{x}_1 \Rightarrow\ m_1 \ddot{x}_1=-2b_1\dot{x}_1- |
<math> + \uparrow \sum F_{y_1} = m_1 \ddot{x}_1 \Rightarrow\ m_1 \ddot{x}_1=-2b_1\dot{x}_1-k_1x_1+k_2(x_2-x_1)+m_1g</math> |
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For mass 2: |
For mass 2: |
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<math> + \uparrow \sum F_{y_2} = m_2 \ddot{x}_2 \Rightarrow\ m_2 \ddot{x}_2=-2b_2\dot{x}_2- |
<math> + \uparrow \sum F_{y_2} = m_2 \ddot{x}_2 \Rightarrow\ m_2 \ddot{x}_2=-2b_2\dot{x}_2-k_2(x_2-x_1)+m_2g</math> |
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For the cases above |
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<math> s_1=(l_1+x_1)\, </math> and <math> s_2=(l_2+x_2)\, </math> |
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where l is the unstretched length of the spring and x is the displacement of the spring. |
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Revision as of 11:25, 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.