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</math> |
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</math> |
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Let's plugs some numbers. Knowing <math>g (32)</math>, and assuming that <math>m_1=3</math>, <math>m_2=1</math>, and <math>l_1=l_2=16</math>, the state space matrix is, |
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=== Laplace Transform === |
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=== Laplace Transform === |
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s & 1 & 0 & 0 \\ |
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s & 1 & 0 & 0 \\ |
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& & & \\ |
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& & & \\ |
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\dfrac{-CD}{AD+B^2} & s & \dfrac{-BE}{AD+B^2} & 0 \\ |
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\dfrac{-8}{5} & s & \dfrac{-8}{5} & 0 \\ |
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& & & \\ |
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& & & \\ |
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0 & 0 & s & 1 \\ |
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0 & 0 & s & 1 \\ |
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& & & \\ |
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& & & \\ |
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\dfrac{BC}{AD+B^2} & 0 & \dfrac{-AE}{AD+B^2} & s \\ |
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\dfrac{8}{5} & 0 & \dfrac{-8}{5} & s \\ |
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\end{bmatrix} |
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\end{bmatrix} |
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</math> |
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</math> |
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Hence, the inverse matrix is, |
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Hence, the inverse matrix (thanks TI-89!) is, |
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: <math> |
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: <math> |
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s & 1 & 0 & 0 \\ |
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s & 1 & 0 & 0 \\ |
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& & & \\ |
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& & & \\ |
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\dfrac{-CD}{AD+B^2} & s & \dfrac{-BE}{AD+B^2} & 0 \\ |
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\dfrac{-8}{5} & s & \dfrac{-8}{5} & 0 \\ |
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& & & \\ |
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& & & \\ |
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0 & 0 & s & 1 \\ |
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0 & 0 & s & 1 \\ |
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& & & \\ |
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& & & \\ |
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\dfrac{BC}{AD+B^2} & 0 & \dfrac{-AE}{AD+B^2} & s \\ |
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\dfrac{8}{5} & 0 & \dfrac{-8}{5} & s \\ |
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\end{bmatrix} |
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\end{bmatrix} |
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Revision as of 12:39, 9 December 2009
By Jimmy Apablaza
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
Plugging the constants yields,
Let's plugs some numbers. Knowing , and assuming that , , and , the state space matrix is,
Laplace Transform
First, we determine the eigenvalues of the matrix,
Hence, the inverse matrix (thanks TI-89!) is,