1550078481-Ordinary_Differential_Equations__Roberts_

480 Ordinary Differential Equations

b. Use computer software to so lve system (8) on the interval [O, 1] for

S 1 = 100 , S2 = 50, S3 = 1000 , S4 = 1300 , a1 = .1, a2 = b1 = .05,

bz = b3 = c = d 1 = d 2 =. 01 , and r 1 = rz = r3 = r 4 =. 02 for the ini-

tial conditions Y1(0) = 1, Y2(0) = y3(0) = y4(0) = 0. Display Y1, Y2, y3, and

y4 on a single graph.

10. 7 Pendulums

In t his section , we will determine crit ical points for several kinds of pendu-

lums and pendulum systems and study their b ehavior by examining phase-

plane graphs. Since electrical and other mechanical systems give rise to similar

systems of differential equations, the results which we obtain here for p endu-

lums will apply to those electrical and mechanical systems also.

Simple Pendulum A simple pendulum consists of a rigid, straight rod of

negligible mass and length f with a bob of mass m attached at one end. The

other end of the rod is attached to a fixed support, S, so that t he pendulum

is free to move in a vertical plane. Let y denote the angle (in radia ns) which

the rod makes with the vert ical extending downward from S- a n equilibrium

position of the system. We arbitrarily choose y to b e positive if the rod is to

the right of the downward vertical and negative if the rod is to t he left of the

downward vertical as shown in Figure 10.21.

s

I

I

I

I

Vertical 1

I

y

bob of

massm

Figure 10.21 Simple Pendulum

We will assume the only forces acting on the pendulum are the force of
gravity, a force due to air resistance which is proportional to the a ngular
velocity of t he bob, and an external force, f(t), acting on the pendulum
system. Under these assumptions it can b e shown by applying Newton's
second law of motion that the position of the pendulum satisfies the initial
value problem

(1) my" + cy' + k sin y = f(t); y(O) = co, y' (0) = C1