1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Linear Systems with Constant Coefficients 387

where {3 = 2.66316 and "( = .712450. A graph of the position of the mass

m1, u1(x), and the position of the mass m 2 , u3(x), on the interval [O, 30]

is displayed in Figure 9.3. Notice the interesting, oscillatory nature of both

functions.

(^2) u
1
5
Figure 9.3 A Graph of the Positions of the Masses
of a Coupled Spring-Mass System
x
0
When damping is assumed to be present in the coupled spring-mass system
shown in Figure 9.1, the equations of motion satisfy the following second-order
linear system
(6)
where, as before, y 1 and Y2 are the displacements of the masses from the
equilibrium positions, m 1 and m 2 are the masses, and k 1 and k 2 are the
respective spring constants. The positive constants d 1 and d 2 a re due to the
d amping forces and are called the damping constants.