1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Systems with Constant Coefficients 389

System at

Equilibrium

System at

Timet

.. I

I

Figure 9.4 A Coupled Spring-Mass System

Exercise 4. Let u 1 = Y1, u2 = y~, u3 = Y2 and U4 = y~.

a. Write the system (7) as an equivalent system of four first-order differen-
t ial equ ations.

b. Write the first-order system of equations which is equivalent to (7) 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 = m2 = 2 g, k 1 = k3 = 4 g/s^2 , and
k2 = 6 g/s^2.

Exercise 5. a. Write system (8) as an equivalent system of first-order equa-
t ions in matrix-vector notation.

b. Use EIGEN or your computer software to find the general solution of
the resulting system for the following cases.

(i) m1 = m2 = 2 g, k1 = k3 = 10 g/s^2 , k2 = 3 g/s^2 , and d1 = d2 = 12 g/s.

(ii) m 1 = m2 = 1 g, k1 = k3 = 3 g/s^2 , k2 = 1 g/s^2 , and d1 = d2 = 4 g/s.

(iii) m 1 =m2=1 g, k 1 = k3 = 2 g/s^2 , k2 = 1 g/s^2 , and d1=d2=2 g/s.