Coupled Oscillator: Double Pendulum
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).
Consider the double-pendulum system consisting of a pendulum attached to another pendulum shown in Figure 1.
- the system oscillates vertically under the influence of gravity.
- the mass of both rod are negligible
- no damping 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,
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,
Solving for and we obtain,
Let's plug some numbers. It's known that . In addition, we assume that , , and , so the constants defined previously become,
Hence, the state space matrix is,
Eigenvalues & Eigenvectors
The eigenvalues and eigenvectors are easily obtained with the help of a TI-89 calculator. First, we consider the 's identity matrix,
Once we define the matrix, the eigenvalues are determined by using the eigVi() function,
On the other hand, we use the eigVc() function to find the eigenvectors,
Now, we plug the eigenvalues and eigenvectors to produce the standar equation,
The matrix exponential is,
Again, we can resort to the TI-89 calculator. As it is mentioned above, the matrix exponential is obtained by typing eigVc(a)^-1*a*eigVc(a), where a is the matrix. Thus,
So, the exponential matrix becomes,
so, the matrix exponential can be solved using matrix multiplication.