1550078481-Ordinary_Differential_Equations__Roberts_

388 Ordinary Differential Equations

Exercise 1. Write the real general solution for the coupled spring-mass

system (3) for

a. m 1 = 5 g, m 2 = 10 g, k 1 = 10 g/s and k2 = 10 g/s.

b. m1 = 5 g, m2 = 10 g, k1 = 5 g/s and k2 = 10 g/s.

Exercise 2. Find the solution of the initial value problem consisting of the

differential system of exercise 1. a. subj ect to the initial conditions u1 (0) =

3 cm and u2(0) = u3(0) = u4(0) = 0.

Exercise 3. Let u1 = Y1, u2 = y~, u3 = Y2, and U4 = Y~·

a. Write system (6) as an equivalent system of four first-order differential
equations.

b. Write the first-order system which is equivalent to (6) in matrix-vector
notation.

c. Use EIGEN or your computer software to find the general solution to the

homogeneous system of part b. for m 1 = m 2 = 10 g, k 1 = 5 g/ s^2 , k2 = 10 g/ s^2

and d1 = d2 = 15 g/s.

System 2 A second coupled spring-mass system which consists of two
masses, m 1 and m2, connected to two fixed supports by three springs which
have spring constants k 1 , k 2 , and k3 is shown in Figure 9.4. Neglecting the
effects of damping, the system of differential equations which describes the
displacements y 1 and Y2 of masses m 1 and m2, respectively, from their equi-
librium positions is

(7)

When damping is assumed to be present in the coupled spring-mass system
shown in Figure 9.4, the equations of motion satisfy the following second-order
linear system

(8)

where d 1 and d2 are the damping constants.