1550078481-Ordinary_Differential_Equations__Roberts_

Linear Systems of First-Order Differential Equations 363

8.3 Linear Systems with Constant Coefficients

Numerous physical phenomena can be modelled by systems of first-order

linear differential equations with constant coefficients. We will show how to

model coupled pendulums by such a system.

Coupled Pendulums A pair of identical pendulums with bobs of mass m

and rods of length e are coupled by a spring with spring constant k as shown

in Figure 8.5.

vertical

Figure 8.5 Coupled Pendulums

Let y 1 and Y2 denote the displacement of each pendulum from the vertical
(positive displacement is to the right). When the pendulums are in their

equilibrium positions (y 1 = Y2 = 0) the spring is horizontal and not stretched

nor compressed. Suppose at some time t, the positive horizontal displacement
of the bobs are y 1 and Y2, so that the spring is stretched by the amount Y2 -y1
and the spring exerts a force of k(y1 -y2). Assuming the motion is undamped
and the displacement of the pendulum bobs is small so that the restoring force
due to the weight mg is mgy/C, then the second order system of equations
satisfied by the displacements of the coupled pendulums is

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

e

-mgy2

my~ = e -k(y2 - Y1).