# Elementary differential equations 7th edition - Boyce W.E

ISBN 0-471-31999-6

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? 29. The position of a certain spring-mass system satisfies the initial value problem

u" + 4 u' + 2u = 0, u(0) = 0, u'(0) = 2.

(a) Find the solution of this initial value problem.

(b) Plot u versus t and u' versus t on the same axes.

(c) Plot u' versus u in the phase plane (see Problem 28). Identify several corresponding points on the curves in parts (b) and (c). What is the direction of motion on the phase plot as t increases?

30. In the absence of damping the motion of a spring-mass system satisfies the initial value problem

mu" + ku = 0, u(0) = a, u'(0) = b.

(a) Show that the kinetic energy initially imparted to the mass is mb2/2 and that the

potential energy initially stored in the spring is ka2/2, so that initially the total energy in the system is (ka2 + mld)/2.

(b) Solve the given initial value problem.

(c) Using the solution in part (b), determine the total energy in the system at any time t. Your result should confirm the principle of conservation of energy for this system.

31. Suppose that a mass m slides without friction on a horizontal surface. The mass is attached to a spring with spring constant k, as shown in Figure 3.8.10, and is also subject to viscous air resistance with coefficient y. Show that the displacement u(t) of the mass from its equilibrium position satisfies Eq. (21). How does the derivation of the equation of motion in this case differ from the derivation given in the text?

200

Chapter 3. Second Order Linear Equations

1A

FIGURE 3.8.10 A spring-mass system.

? 32. In the spring-mass system of Problem 31, suppose that the spring force is not given by Hooke’s law but instead satisfies the relation

Fs = —(ku + e u3),

where k > 0 and e is small but may be of either sign. The spring is called a hardening spring if e > 0 and a softening spring if e < 0. Why are these terms appropriate?

(a) Show that the displacement u(t) of the mass from its equilibrium position satisfies the differential equation

mu" + y u' + ku + eu3 = 0.

Suppose that the initial conditions are

u(0) = 0, u (0) = 1.

In the remainder of this problem assume that m = 1, k = 1, and y = 0.

(b) Find u(t) when e = 0 and also determine the amplitude and period of the motion.

(c) Let e = 0.1. Plot (a numerical approximation to) the solution. Does the motion appear to be periodic? Estimate the amplitude and period.

(d) Repeat part (c) for e = 0.2 and e = 0.3.

(e) Plot your estimated values of the amplitude A and the period T versus e. Describe the way in which A and T, respectively, depend on e.

(f) Repeat parts (c), (d), and (e) for negative values of e.

3.9 Forced Vibrations

Consider now the case in which a periodic external force, say F0 cos mt with m > 0, is

applied to a spring-mass system. Then the equation of motion is

mu" + y u'+ ku = F0cos mt. (1)

First suppose that there is no damping; then Eq. (1) reduces to

mu" + ku = F0 cos mt. (2)

If

m0

= V k/ m = m, then the general solution of Eq. (2) is

F0

u = c1 cos m0t + c2sin m0t +----------2-----^ cos mt. (3)

m (m0 — m )

The constants c1 and c2 are determined by the initial conditions. The resulting motion

is, in general, the sum of two periodic motions of different frequencies (m0 and m) and

amplitudes. There are two particularly interesting cases.

3.9 Forced Vibrations

201

Beats. Suppose that the mass is initially at rest, so that u(0) = 0 and u'(0) = 0. Then it turns out that the constants c1 and c2 in Eq. (3) are given by

F0

ci =--------77-------a ’ c2 = 0 (4)

m(m0 — m )

and the solution of Eq. (0) is

F0

u =--------2 T (cos Mt — cos m0t). (5)

m (m0 — m )

This is the sum of two periodic functions of different periods but the same amplitude. Making use of the trigonometric identities for cos( A ± B) with A = (m0 + M)t/0 and

B = (m0 — m)t/0, we can write Eq. (5) in the form

• (m0 + m) t

sin -------------. (6)

u =

2 Fo . (m0 — rn)t

sin -------------------

m(a>2 — a>2)

If |m0 — m| is small, then m0 + m is much greater than |m0 — m|. Consequently, sin(M0 + M)t/0 is a rapidly oscillating function compared to sin(M0 — m)t/0. Thus the motion is a rapid oscillation with frequency (m0 + m)/0, but with a slowly varying sinusoidal amplitude

0F0 . (M0 — M)t

0 T sin Z .

m(mQ — m ) 0

This type of motion, possessing aperiodic variation of amplitude, exhibits what is called a beat. Such a phenomenon occurs in acoustics when two tuning forks of nearly equal frequency are sounded simultaneously. In this case the periodic variation of amplitude is quite apparent to the unaided ear. In electronics the variation of the amplitude with

FIGURE 3.9.1 A beat; solution of u" + u = 0.5cos0.8f, u(0) = 0, u'(0) = 0; u = 2.77778 sin0.1 f sin0.9f.

202

Chapter 3. Second Order Linear Equations

time is called amplitude modulation. The graph of u as given by Eq. (6) in a typical case is shown in Figure 3.9.1.

Resonance. As a second example, consider the case m = m0; that is, the frequency of the forcing function is the same as the natural frequency of the system. Then the nonhomogeneous term F0 cos mt is a solution of the homogeneous equation. In this case the solution of Eq. (2) is

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