Coupled Oscillator: Double Pendulum: Difference between revisions

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</math>
</math>


=== Matrix Exponential ==
=== Matrix Exponential ===
Let's consider the space state equation, The matrix exponential is defined as,

:<math>\dot{\bar{z}}=\bold{\hat{A}}\bar{z}=\bold{TAT^{-1}}\bar{z}</math>


Now we can use the equation for a transfer function to help us solve through the use of matrix exponentials.
Now we can use the equation for a transfer function to help us solve through the use of matrix exponentials.
:<math>\bar{z}=\bold{T}\bar{x}\,</math>
:<math>\bar{z}=\bold{T}\bar{x}\,</math>
Line 243: Line 248:
Combining the two equations we then get
Combining the two equations we then get
:<math>\bold{T^{-1}}\dot{\bar{z}}=\bold{AT^{-1}}\bar{z}</math>
:<math>\bold{T^{-1}}\dot{\bar{z}}=\bold{AT^{-1}}\bar{z}</math>

Multiplying both sides of the equation on the left by '''T''' we get
:<math>\dot{\bar{z}}=\bold{TAT^{-1}}\bar{z}</math>
where
:<math>\bold{\hat{A}}=</math>

Revision as of 16:08, 12 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,

Matrix Exponential

Let's consider the space state equation, The matrix exponential is defined as,


Now we can use the equation for a transfer function to help us solve through the use of matrix exponentials.

This can be rearranged by multiplying T-inverse to the left side of the equations.

Now we can bring in the standard form of a state space equation

Combining the two equations we then get

Multiplying both sides of the equation on the left by T we get

where