1550078481-Ordinary_Differential_Equations__Roberts_

282 Ordinary Differential Equations

6.1.1.1 Free Undamped Motion

Furthermore, when there is no damping force acting on the system- that is ,

when b = 0 the system executes free undamped motion which is also called

simple harmonic motion. This type of motion occurs if the pendulum or

spring-mass system operates in a vacuum or if all resistance, R, is removed

from the RLC series circuit making the circuit an LC series circuit. Later you

may encounter simple harmonic oscillators (motion) in the study of quantum

mechanics or in the study of the vibration of strings, membranes, and beams.

When f(t) = 0 and b = 0, r1 = -/d]O.,i = wi and r2 = --/d]O.,i = -wi, so

the general solution of ay" + dy = 0 is

(14) y =Ai sin wt+ A 2 cos wt

where Ai and A 2 are arbitrary constants. We would like to show that equa-

tion (14) may also be written in the equivalent form

(15) y = A sin (wt + ¢)

where A and ¢ are arbitrary constants. Applying the trigonometric formula

for the sine of the sum of two angles, wt and¢, to equation (15), we see

( 16) y = A sin (wt + ¢) = A cos ¢sin wt + A sin ¢ cos wt.

Equating coefficients ofsinwt and cos wt in equations (14) and (16), we obtain

the following two relationships between the constants Ai and A 2 of (14) and

A and ¢ of (16)

(17a)

(17b)

Ai= A cos¢

A2 =A sin¢.

Squaring both equations (17a) and (17b), adding the results, and solving for

A, we find

(18) A= VAi +A~.

Substituting (18) into (17a) and (17b) and solving for cos¢ and sin¢, we see

that ¢ must simultaneously satisfy

Ai

cos¢ = --;:===

JAi +A~

and

The reason for wanting to write the general solution (14) in the form (15)

is because it is easier to understand the physical significance of the arbitrary

constants A and¢ of (15). The constant A is the amplitude of oscillation

and -A::::; y(t) ::::; A for all t. The constant¢ is the phase angle. The phase