Page 135 - Schaum's Outline of Differential Equations

118 SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS [CHAP. 14

motions, while forced damped systems (assuming the external force to be sinusoidal) yield both transient and
steady-state motions.
Free undamped motion defined by Eq. (14.11) with a 1 = 0 and/(?) = 0 always has solutions of the form

which defines simple harmonic motion. Here c 1, c 2, and ft) are constants with ft) often referred to as circular
frequency. The natural frequency j'is

and it represents the number of complete oscillations per time unit undertaken by the solution. The period of
the system of the time required to complete one oscillation is

Equation (14.12) has the alternate form

where the amplitude the phase angle (j) = arctan (c 2lcj), and k is zero when c 1 is positive and unity
when c 1 is negative.

Solved Problems

14.1. A steel ball weighing 128 Ib is suspended from a spring, whereupon the spring is stretched 2 ft from its
natural length. The ball is started in motion with no initial velocity by displacing it 6 in above the equi-
librium position. Assuming no air resistance, find (a) an expression for the position of the ball at any
time t, and (b) the position of the ball att=nl 12 sec.
(a) The equation of motion is governed by Eq. (14.1). There is no externally applied force, so (t) = 0, and no
F
2
resistance from the surrounding medium, so a = 0. The motion is free and undamped. Here g = 32 ft/sec ,
m = 128/32 = 4 slugs, and it follows from Example 14.1 that k = 64 Ib/ft. Equation (14.1) becomes x + 16x = 0.
The roots of its characteristic equation are X = ±4i, so its solution is

At t = 0, the position of the ball is x 0 = - j ft (the minus sign is required because the ball is initially displaced
above the equilibrium position, which is in the negative direction). Applying this initial condition to (_/), we
find that

so (1) becomes

The initial velocity is given as v 0 = 0 ft/sec. Differentiating (2), we obtain

v(t) = x(i) = 2sin4t + 4c 2 cos4f