1550078481-Ordinary_Differential_Equations__Roberts_

486 Ordinary Differential Equations

and f, is the length of the pendulum. The x-axis and y-axis form a rectangular

coordinate system on the floor beneath the suspended pendulum. The origin

of the coordinate system lies directly below the equilibrium position of the

pendulum.

Spring Pendulum A spring pendulum consists of a spring of natural

length Lo suspended by one end from a fixed support S. A bob of m ass mis

attached to the other end of the spring. We assume the spring is stiff enough

to remain straight and free to move in a vertical plane. Let L(t) be the length

of the spring at time t and let B(t) be the angle (in radians) the spring makes

with the downward vertical from S. See Figure 10.25.

s I I I I I I I

Vertical 1

I

bob of

mass m

Figure 10. 25 Spring Pendulum

If the spring obeys Hooke's law with spring constant k, then applying New-

ton's second law of motion along and p erpendicular to the spring it can be

shown that L and e simultaneously satisfy the two second-order differential

equations

L"-L(B')^2 -gcose+ k(L - Lo) =0

(12) m

Le"+ 2L'e' + gsine = o.

Letting Y1 = L(t), Y2 = L' (t), y3 = e(t), and y4 = e' (t), we can rewrite
system (12) as the following system of four first-order differential equations

(13)

, 2 k(y1 - Lo)

Y2 = Y1Y4 + gcosy 3 - ----

m

-2y2y4 - g sin y3

y~=-------

Y1