Coupled Oscillator: Coupled Mass-Spring System with Damping: Difference between revisions
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<math> \underline{\dot{x}}(t) = \widehat{A} \, \underline{x}(t) + \widehat{ |
<math> \underline{\dot{x}}(t) = \widehat{A} \, \underline{x}(t) + \widehat{B} \, \underline{u}(t) </math> |
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\begin{bmatrix} |
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Revision as of 11:36, 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.
Solution
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.