# 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).

## Contents |

# 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 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,

## 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. 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,

### Standard Equation

Now, we plug the eigenvalues and eigenvectors to produce the standar equation,

### Matrix Exponential

The matrix exponential is,

where

- ,

and

- ,

so

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

*is the matrix. Thus,*

**a**So, the exponential matrix becomes,

where

Hence,

so, the matrix exponential can be solved using matrix multiplication.