Coupled Oscillator: Coupled Mass-Spring System with Input: Difference between revisions
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<math>k_2 = 80000 \frac{N}{m}</math> |
<math>k_2 = 80000 \frac{N}{m}</math> |
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Let the initial conditions be zero for the time being. |
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==Force Equations== |
==Force Equations== |
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[[Image:BP_FBD2-2.jpg|right|FBD for m1]] |
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[[Image:BP_FBD1-2.jpg|right|FBD for m1]] |
[[Image:BP_FBD1-2.jpg|right|FBD for m1]] |
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Sum of the forces in the x direction yields |
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[[Image:BP_FBD2-2.jpg|right|FBD for m2]] |
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For <math>m_1 \frac{}{}</math> |
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<math> |
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+ \uparrow \sum F_{x_2} = m_2 \ddot{x}_2 |
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</math> |
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<math> |
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\Rightarrow\ m_2 \ddot{x}_2=F_{s_2} - m_2 g |
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</math> |
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Since |
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<math> |
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F_{s_2} = k_2(L_2+x_1-x_2) |
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</math> |
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<math> |
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\Rightarrow\ \ddot{x}_2=-g + \frac{k_2}{m_2} \, L_2 + \frac{k_2}{m_2} \, x_1 - \frac{k_2}{m_2} \, x_2 |
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</math> |
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And for <math>m_1 \frac{}{}</math> |
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<math> |
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+ \uparrow \sum F_{x_1} = m_1 \ddot{x}_1 |
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</math> |
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<math> |
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\Rightarrow\ m_1 \ddot{x}_1=F_{s_1} + F - m_1 g - F_{s_2} |
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</math> |
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Since |
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<math> |
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F_{s_1} = k_1(L_1-x_1) |
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</math> |
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Where |
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<math>F \frac{}{} = F(t)</math> is the input force |
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<math> |
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\Rightarrow\ \ddot{x}_1 = - g +\frac{1}{m_1} \, F + \frac{k_1}{m_1} \, L_1 - \frac{k_1}{m_1} \, x_1 |
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-\frac{k_2}{m_1} \, L_2 - \frac{k_2}{m_1} \, x_1 + \frac{k_2}{m_1} \, x_2 |
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</math> |
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==State Space Equation== |
==State Space Equation== |
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The general form of the state equation is |
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<math> |
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\underline{\dot{x}}(t) = \widehat{A} \, \underline{x}(t) + \widehat{C} \, \underline{u}(t) |
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</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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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 & 1 & 0 & 0 \\ |
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\frac{-k_1}{m_1}-\frac{k_2}{m_1} & 0 & \frac{k_2}{m_1} & 0 \\ |
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0 & 0 & 0 & 1 \\ |
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\frac{k_2}{m_2} & 0 & \frac{-k_2}{m_2} & 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> |
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Latest revision as of 16:25, 13 December 2009
Setup State Space Equation
Problem Statement
Find an input function such that the lower mass, , is stationary in the steady state. Find the equation of motion for the upper mass, .
The use of one spring between the masses is just a simplification of a multi-spring system, so the possibility of being off-kilter is neglected and just the vertical forces are considered.
Initial Conditions and Values
Force Equations
Sum of the forces in the x direction yields
For
Since
And for
Since
Where is the input force
State Space Equation
The general form of the state equation is
Where denotes a matrix and denotes a vector.
Let , , , and be the state variables, then