1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Equations with Constant Coefficients 299

System at

Equilibrium

System at

Timet

Figure 6.8 A Coupled Spring-Mass System

Exercise 2. For m 1 = .4 kg, m2 = .25 kg, k 1 = 7 kg-m^2 /s^2 , k 2 =

6 kg-m^2 /s^2 , and k3 = 9 kg-m^2 /s^2 find the solution to the initial value problem

consisting of the system of differential equations (32) and the initial conditions

Yi (0) = -.6 m, y~ (0) = .45 m/s, y2(0) = .3 m, and y~(O) = -.37 m/s. (HINT:

Equation (32a) is the same as equation (28a). Eliminate Y2 and y~ from
equation (32b) and obtain a fourth order linear differential equation in y 1.
Use POLYRTS or your computer software to find the roots of the associated
auxiliary equation. Write the general solution y 1 , then find y 2 and satisfy the
initial conditions.)

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 m 2
attached to the first bob by a rod of length £2 as shown in Figure 6.9. 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 plane
and neglecting the mass of the rods and any damping forces, it can be shown
that the displacements, y 1 and y 2 , satisfy the following system of differential
equations