1550078481-Ordinary_Differential_Equations__Roberts_

384 Ordinary Differential Equations

For i = 1, 2 let Yi represent the vertical displacement of mass mi from its

equilibrium position. We will assign downward displacement from equilibrium
to be positive and upward displacement from equilibrium to be negative. Ap-
plying Newton's second law of motion and assuming no damping is present,
it can be shown that the equations of motion for this coupled spring-mass
system satisfy the following linear system of differential equations

(1)

m2y~ = -k2(Y2 - Y1).

Dividing the first equation in system (1) by m 1 , dividing the second equation
of system (1) by m2, and letting u1 = Y1, u2 = y~, U3 = Y2, and U4 = y~, we
see we may rewrite (1) as the following equiva lent first-order system

U~ = U2

u~ = -------'---- k1u1 + k2(u3 - u1) --'-

m1

-(k1 + k2)u1 k2U3

------'--+ --

m1 m1

(2)

Next, using matrix-vector notation, we may rewrite (2) as

0 1 0 0

-(k1 + k2)

0

k2

0

m1 m1

(3) u'= u.

0 0 0 1

k2

0

-k 2

0

m2 m2

EXAMPLE 1 Finding the Real General Solution of a Coupled

Spring-Mass System

Write the real general solution of system (3) for m 1
ki = 18 g/s^2 , and k 2 = 3 g/s^2.

3 g, m2 5 g,