1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Systems with Constant Coefficients 391

0 1 0 0

u' 1 U1

• (~ + ~)^0

k

0

u~ m U2

(9)

u' 3 0 0 0 1 U3

u' k

-(~ + ~)

4 0 0 U4

m

Exercise 6. Use EIGEN or your computer software to find the general solu-

tion of system (9) form= 25 g, £ = 50 cm, k = 400 g/s^2 , and g = 980 cm/s^2.

A Double Pendulum A double pendulum consists of a bob of mass m 1

attached to a fixed support by a rod of length £ 1 and a second bob of mass

m2 attached to the first bob by a rod of length £2 as shown in F igure 9.6.

I

~

I Y2

vert 1. ca^11

1

Figure 9.6 A Double Pendulum

Let y 1 and y2 denote the displacement from the vertical of the rods of length
£ 1 and £ 2 respectively. Assuming the double pendulum oscillates in a vertical
p lane and neglecting the mass of the rods and any damping forces, it can
be shown that the displacements, Y1 and Y2, satisfy the following system of
differential equations

(10)

where g is the constant of gravitational acceleration.