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Line 192: |
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: <math> |
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: <math> |
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x |
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x = c_1</math><math>\begin{bmatrix}0.0520 \\-0.1609 \\-0.3031\\0.9378\end{bmatrix}\,</math><math>e^{-3.0937}+ c_2</math> |
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= |
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<math>\begin{bmatrix}0.4176i \\-0.8928\\- 0.0716i\\0.1532\end{bmatrix}\,</math><math>e^{2.1380i}+ c_3</math><math>\begin{bmatrix}- 0.4176i \\-0.8928\\0.0716i\\0.1532\end{bmatrix}\,</math><math>e^{- 2.1380i}+ c_4</math><math>\begin{bmatrix}-0.0520 \\-0.1609\\0.3031\\0.9378\end{bmatrix}\, |
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c_1 |
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</math><math>e^{3.0937}\,</math> |
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\begin{bmatrix} |
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-0.2 \mathbf{i} \\ |
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0.4 \\ |
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0.4 \mathbf{i} \\ |
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-0.8 \\ |
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\end{bmatrix} |
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e^{2 \mathbf{i}} |
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+ |
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c_2 |
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\begin{bmatrix} |
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0.2 \mathbf{i} \\ |
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0.4 \\ |
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-0.4 \mathbf{i} \\ |
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-0.8 \\ |
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\end{bmatrix} |
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e^{-2 \mathbf{i}} |
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+ |
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c_3 |
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\begin{bmatrix} |
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-0.29277 \mathbf{i} \\ |
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0.33806 \\ |
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0.58554 \mathbf{i} \\ |
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-0.67621 \\ |
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\end{bmatrix} |
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e^{1.1547 \mathbf{i}} |
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+ |
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c_4 |
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\begin{bmatrix} |
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0.29277 \mathbf{i} \\ |
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0.33806 \\ |
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0.58554 \mathbf{i} \\ |
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0.67621 \\ |
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\end{bmatrix} |
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e^{-1.1547 \mathbf{i}} |
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</math> |
Revision as of 23:00, 11 December 2009
By Jimmy Apablaza
This problem is described in Page 321-322, Section 7.6 of the A first Course in Differential Equations textbook, 8ED (ISBN 0-534-41878-3).
Figure 1. Coupled Pendulum.
Problem Statement
Consider the double-pendulum system consisting of a pendulum attached to another pendulum shown in Figure 1.
Assumptions:
- the system oscillates vertically under the influence of gravity.
- the mass of both rod are neligible
- no dumpung forces act on the system
- positive direction to the right.
The system of differential equations describing the motion is nonlinear
In order to linearize these equations, we assume that the displacements and are small enough so that and . Thus,
Solution
Since our concern is about the motion functions, we will assign the masses and , the rod lenghts and , and gravitational force constants to different variables as follows,
Hence,
Solving for and we obtain,
Therefore,
State Space
Let's plug some numbers. Knowing , and assuming that , , and , the constants defined previously become,
Hence, the state space matrix is,
Eigenvalues
The eigenvalues are obtained from 's identity matrix,
According to my TI-89, the eigenvalues are,
and the eigenvectors,
Standard Equation
Now, we plug the eigenvalues and eigenvectors to produce the standar equation,