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u(0) = 0, il (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)
= V k/ m = m, then the general solution of Eq. (2) is
u = cx cos m0t + c2sin m0t +----------2-----^ cos mt. (3)
m (m0 — m )
The constants q 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
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
ci =-------^ ’ c2 = 0 (4)
m(M0 — m )
and the solution of Eq. (2) is
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/2 and
B = (m0 — M)t/2, we can write Eq. (5) in the form
• (m0 + m) t
2 F0 . («0 — M)t
m(M°° — m2)
If |m0 — m| is small, then m0 + m is much greater than |«0 — m|. Consequently, sin(«0 + M)t/2 is a rapidly oscillating function compared to sin(«0 — M)t/2. Thus the motion is a rapid oscillation with frequency (m0 + m)/2, but with a slowly varying sinusoidal amplitude
2F0 . («0 — M)t
m(mQ — m ) 2
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 = 0.5 cos 0.8t, u(0) = 0, u'(0) = 0; u = 2.77778 sin0.1t sin0.9t.
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
u = c1 cos m0t + c2 sin m0t + -----1 sin m0t. (7)
2 m m
Because of the term t sin m01, the solution (7) predicts that the motion will become unbounded as t regardless of the values of c1 and c2; see Figure 3.9.2 for a typical example. Of course, in reality unbounded oscillations do not occur. As soon as u becomes large, the mathematical model on which Eq. (1) is based is no longer valid, since the assumption that the spring force depends linearly on the displacement requires that u be small. If damping is included in the model, the predicted motion remains bounded; however, the response to the input function F0 cos mt may be quite large if the damping is small and m is close to m0. This phenomenon is known as resonance.