1550078481-Ordinary_Differential_Equations__Roberts_

484 Ordinary Differential Equations

and if we assume there is no forcing function (f(t) = 0), then the differential
equation (1) for the motion of a simple pendulum is

II I mg , 0

my + cy + T sm y =.

Dividing this equation by m, choosing£= g in magnitude, and setting C =
c/m, this equation becomes

Letting Y1 = y and Y2
first-order equations

(7)

y^11 +Cy'+ siny = 0.

y', we obtain the following equivalent system of

y~ = - siny1 - Cyz.

Simple Pendulum with Constant Forcing Function

If we assume the forcing function f(t) = a, a constant, then from (1) the
equation of motion for the pendulum is

II I mg •

my + cy + T sm y = a.

Dividing this equation by m, setting£= gin magnitude, and letting C = c/m

and A = a/m, we can write this equation as

y^11 + Cy' + sin y = A

or as the equivalent first-order system

(8)

y~ = -siny 1 - Cy2 +A

where y 1 = y and Y2 = y'.

Variable Length Pendulum A variable length pendulum consists of a
mass m attached to one end of an inextensible string (a string which does
not stretch). The string is pulled over some support, S, such as a thin rod.
The length of the pendulum, £(t), can be varied with time by pulling on the
string. A diagram of a variable length pendulum is shown in Figure 10.24.
As before we let y denote the angle the pendulum makes with the downward
vertica l from the support S. The equation of motion for this pendulum is

(9) m£(t)y^11 + (2m£'(t) + c£(t))y' + mgsiny = f(t)

where m is the mass of the bob, C(t) is the length of the pendulum, c is
the damping constant, g is the gravitational constant, and f(t) is the forcing