1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 489

(ii) Y1(0) = 2.1, Y2(0) = 0, y3(0) = .05, and y4(0) = 0.

(The spring pendulum is set into motion by pulling the mass down

.1 decimeter from its equilibrium position and moving it to the right so

that e = .05 radians and then releasing the mass.)

For both sets of initial conditions display a graph of the length of the spring

as a function of time-y 1 (t) = L(t); display a graph of the angle the spring

makes with the vertical as a function of time-y 3 (t) = B(t); display a phase-

plane graph of Y2 versus Y1; and display a phase-plane graph of y 4 versus y 3.

Is the motion of the spring pendulum periodic?

10.8 Duffing's Equation

(Nonlinear Spring-Mass Systems)

In chapter 6 we saw that the equation of motion for a mass on a spring

which satisfied Hooke's law (a linear spring) is

(1) my"+ ky = 0

where m is the mass and k > 0 is the spring constant. The restoring force

for linear springs is ky. Springs which do not obey Hooke's law are called

nonlinear springs. The restoring force for hard nonlinear springs is ky+py^3

where p > 0 while the restoring force for soft nonlinear springs is ky - py^3.

Thus, the equation of motion for a nonlinear spring is

(2) my" + ky + py^3 = 0

where m and k are positive constants and p -j. 0. Adding an external force of
the form a sinwt to drive the spring-mass system, we obtain the equation

(3) my" + ky + py^3 = a sin wt.

Dividing by m, letting K = kjm, P = p/m and A= a/m, we obtain Duffing's

equation for the motion of a nonlinear spring with a periodic forcing function

(4) y" +Ky+ Py^3 = Asinwt.

Including a damping term which is proportional to the velocity of the mass,

yields

(5) y" +Cy'+ Ky+ Py^3 =A sin wt

where the damping constant is C 2'. 0.