1550078481-Ordinary_Differential_Equations__Roberts_

390 Ordinary Differential Equations

9.2 Pendulum Systems

Couple d P endulums A coupled pendulum system consists of a pair of
identical pendulums with bobs of mass m and rods of length f coupled by a
spring with spring constant k as shown in Figure 9.5. Let Y1 and Y2 denote
the displacement of the pendulums from vertical (positive displacement is to
the right). Neglecting damping and assuming the displacements y 1 and y 2 are
small , we showed in section 8.3 that the displacements satisfy the foll owing
system of second-order differential equations

my~ = -mgyi - k(y1 - Y2)

g

II -mgy2 k( )

my2 = g - Y2 - Y1 ·

mg

vertical

Figure 9.5 Coupled Pendulums

mg

Also in section 8.3, we showed by letting u 1 = y 1 , u 2 = y~, u 3 = y 2 , and u 4 =

Y2 that the second-order system above can be rewritten in matrix notation as
the following first-order system of differential equations.