This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: CHAPTER RESONANCE First of all, it is necessary to note that each pendulum has its own time of
vibration, so limited and fixed in advance that it is impossible to move it in any
other period than its own unique natural one. Take in hand any string you like,
to which a weight is attached, and try the best you can to increase or diminish
the frequency of its vibrations; this will be a mere waste of effort. On the other
hand, we confer motion on any pendulum, though heavy and at rest, by merely
blowing on it. This motion may be quite large if we repeat our puffs; yet it will
take place only in accord with the time appropriate to its oscillations. If at the
first puff we shall have moved it half an inch from the vertical, by adding the
second when, returned toward us, it would commence its second vibration, we
confer a new motion on it; and thus successively with more puffs given at the
right time (not when the pendulum is going toward us, for thus we should
impede the motion and not assist it), and continuing with many impulses, we
shall confer on it impetus such that much greater force than a breath would be
needed to stop it. Galileo Galilei, Two New Sciences (1638) 21.1 FORCED OSCILLATIONS Galileo was not the only famous member of the Galilei family; his father Vincenzo was
an accomplished and articulate musician. Understandably, Vincenzo was interested in
how sound was produced. In 1589 he published his work on the relationship of the lengths
and tensions of strings to the tones they produced. This study may have been the first
experimentally derived law ever to have been discovered to replace a rival law. In the 399 400 RESONANCE sixteenth century, music was considered a branch of mathematics. The Pythagorean idea
that harmonious tones are produced by strings whose lengths are in deﬁnite ratios dom
inated music theory. Vincenzo argued that the complex sounds of musical instruments
had to be determined by ear, rather than by mathematics alone. His ideas instilled a keen
ear and insatiable curiosity into the eldest of his seven children, Galileo. Not only did Galileo uncover the factors that determine the frequency of a pendulum,
but he also understood the phenomenon of resonance. As the lines opening this chapter
reveal, he noted that the swings of a pendulum can be made increasingly large by repeated,
timed applications of a small force, like a puff of air. This method of making a pendulum
swing is an example of forced oscillations  vibrations induced by an external driving
force. Galileo further realized that if the frequency of the external driving force exactly
matches the natural frequency (no of the system, a spectacular effect takes place: the
amplitude of the vibrations becomes exceedingly large. When a vibrating system is driven
by a periodic force at the natural frequency of the system, we say that resonance occurs.
Galileo knew that the phenomenon of resonance lay at the heart of the sounding boards
of his father’s clavichords and violins, and even of the power of a singer’s voice. Sometimes resonance can cause an oscillating system literally to break apart. Tele
vision commercials and movies have capitalized on this dramatic effect by showing wine
glasses shattering when a singer hits a certain note. Is this impressive effect possible, or
is it just Hollywood trickery? Before we can answer, we need first to understand how and why resonance occurs.
Perhaps the most familiar example of forced oscillations is a child’s swing. Everyone
knows how to push a swing to make it oscillate with a large amplitude. If you want a
child to swing high, you push in step with the motion of the swing, as Fig. 21. 1a illustrates.
By applying a small force at the same point of each swing, you are timing your pushes
to the natural frequency of the swing, which is a type of pendulum. The oscillations
become larger and larger because you are adding energy to the system with each push.
If, however, the pushes are not in step with the motion, as in Fig. 21.1b, the driving
force opposes the motion and can cause the amplitude to diminish. (a) Figure 21.1 Forced oscillations of a swing by a force (a) in phase
and (b) out of phase with the natural frequency. The repeated application of a small force can create largeamplitude vibrations if
(a) the force is in step (or in phase) with the oscillating system, and (b) the driving force
repeats with the same frequency as the natural frequency of the system. Under these
conditions, resonance occurs. 21.2 DESCRIBING RESONANCE 401 Although the driving force may be small, the results of resonance may be spectacularly
large. In Fig. 21.2 a tuning fork is shown attached to a sounding box, which amplifies
the sound of the tuning fork. When an identical tuning fork is placed on a box nearby
and struck, the first begins to ring with what are called sympathetic vibrations. Here’s
why: When a series of sound waves from the second tuning fork impinges on the ﬁrst,
each compression of the air gives the fork a tiny push. Since these pushes occur at the
natural frequency of the tuning fork (remember that the forks are identical), they suc
cessively increase the amplitude of the vibration. The result is striking when you consider
how weak a disturbance sound is: a soft sound like the tuning fork causes a change in
air pressure of about one part in 10“, yet that is enough to cause the second tuning fork to vibrate. ﬁéﬁﬂMlHﬂ Figure 21.2 An example of resonance: a vibrating tuning fork causes
an identical tuning fork to vibrate. Questions 1. Suppose you have a marble which is free to roll inside a shallow bowl. Describe
how by horizontal motion of the bowl alone, you can cause the marble to roll over the edge of the bowl. 2. Similar to a pendulum, a ship has a natural frequency corresponding to a rocking
motion of the entire ship. What happens if the frequency of ocean waves matches
the natural frequency of a ship? 3. Your arm is a type of pendulum, which according to its shape and length has a
natural frequency of swinging. What happens if you are walking with a stride that
matches the natural frequency of your arms? Try to observe this by watching peo—
ple walking. 21.2 DESCRIBING RESONANCE Now that we know the conditions for resonance, let’s describe the phenomenon mathe—
matically. For a start, remember the simple harmonic oscillator equation, d2 _ k
FM) — r;x(t). . (20.1)
which describes the motion of a mass m under the inﬂuence of a spring force F = — kx. The natural frequency of the oscillations is speciﬁed by (00 = \/k/m. (20.2) 402 RESONANCE If the oscillator is disturbed, it vibrates at its natural frequency. In addition, remember
that the harmonic oscillator is a potent model for all sorts of complex oscillatory systems.
Each system will have a set of natural frequencies. (Sometimes for brevity we call (no
the frequency rather than the angular frequency. This should not cause any confusion.) Example 1 As you are emptying a jug of water, does the frequency of the gurgles increase, decrease,
or remain the same? In other words, does the sound change from a deep bass to a higher
pitch or vice versa? First we note that as the liquid runs out, the size of the air space inside the jug
becomes larger. The air in the jug will have a resonant frequency at which it will oscillate.
For a mass—andspring system, the square of natural frequency is inversely proportional
to the mass; the more massive the system, the greater its inertia, and the more sluggishly
it vibrates. Similarly, the natural frequency of the air inside the jug depends on the mass
of the air. Therefore, as the space becomes larger and increases in mass, the natural
frequency decreases, because it is more difficult to accelerate the larger mass. All this
means that you hear the pitch becoming deeper. Try observing this effect. Can you predict
what you’ll hear when a jug is being filled? Instead of allowing the system simply to oscillate at its natural frequency, suppose
we push it back and forth with a force that oscillates with a frequency on. We can describe
the driving force by an oscillating mathematical function, F(t) = F0 sin out, where F0 is just a constant. With this new additional force, Newton’s second law for the
system is dzx
m— = —k.x + F sin tot.
dt2 0
After dividing by m we can write this as
dzx _ 2 .
E — —oo0x + a0 Sln out, (21.1) where we have set a0 = Fo/m, a measure of the size of the driving force. The form of
this equation is very general; it can apply not only to a mass on a spring, but to any
harmonic oscillator whose natural frequency is (no and which is subject to a forcing
function proportional to sin wt. Common sense tells us that if we continue to push on an oscillator with some frequency
to, it will eventually oscillate at that frequency, so it is reasonable to expect that a function
of the form x(t) = A sin wt (21.2) 21.2 DESCRIBING RESONANCE 403 will be a solution of Eq. (21.1). We now show this is so by substituting x = A sin (or
into (21.1) and determining A to satisfy the equation.
The second derivative of x(t) is proportional to x(t) itself: you can verify this through
differentiation:
d_2x
dt2
Substituting this result into Eq. (21.1), which we write as
dzx it; + (1)8): = (to sin (oz, (21.1) = —u)2x. we obtain —w2A sin wt + (95A sin (or = no sin wt
or (—(x)2A + ng — a0)sin wt = 0. The only way this can be true for all times is if —(()2A + (08A — a0 = 0. Let’s now pause and recall for a moment what we are looking for. We want to
understand how and when resonance occurs. More dramatically, we want to know if it
is possible to break a wine glass by singing the right note. The wine glass is represented
by our basic equation, Eq. (21.1), with (no its natural frequency (which you hear if you
tap the wine glass), and x(t) the distortion of the shape of the wine glass. The singer’s
(live or taped) voice causes the air to push the glass with a driving force F 0 sin wt, leading
to a disturbance of the glass, A sin wt. The size of the resulting disturbance is proportional
to A; if A becomes too large, the glass will shatter. From our last equation we already
know what the amplitude of the disturbance will be: A = —a0 (21.3) (pg—(1)2 provided that u) 75 (90. Let’s interpret this equation for A. The wine glass rings with a definite frequency
(no. If the sound waves striking the glass are from a bass note, that is, a low frequency
0), which is much less than (no, u) < (00, then we can ignore (92 compared to (1)3 and Eq.
(21.3) gives us A z Clo/(1)8. For sound waves a0 is very small, consequently A is also very small. This means that
the glass vibrates only slightly at the bass note frequency Q). In other words, nothing spectacular happens. On the other hand, if the sound comes from a soprano, it has a high frequency,
u) > (.00. This time we ignore (1)3 compared to (92 in (21.3) and the resulting amplitude
of the glass is A z —aO/a)2. 404 RESONANCE This effect is even smaller than that of a bass note because a) is larger than (no. The
minus sign tells us that the glass vibrates exactly opposite to the way it is being pushed
by the sound waves. But if the impinging sound waves have precisely the frequency (no then something
startling happens. According to Eq. (21.3), no matter how small a0 is, as a) approaches
(no, the resulting amplitude of the system becomes arbitrarily large, blows up, and so
does the system. We have resonance. In real life, a glass is seldom shattered by the voice of a singer. Instead an audio
generator, a device which produces pure tones, tuned precisely to the natural frequency
of the glass is needed. And the natural frequency of the glass must be determined by a
microphone held close to the glass which detects at what frequency of the audio generator
the glass vibrates the most. The difficulty in breaking glasses is useful, because sound
of every possible frequency, although at low intensities, is always in the air. There
wouldn’t be a glass left in the world if they really broke easily. Our analysis, however, does not explain why it is actually difficult to break a wine
glass. The reason is that we’ve oversimplified the situation to expose the essential phe
nomenon. In describing the forces acting on the harmonic oscillator, we ignored friction,
viscosity, and so on. These forces are always present for any oscillator in the real world,
and they are often proportional to the velocity dx/dt. In this case the differential equation
of motion takes the form dzx dx 2 .
E + a; + wox = a0 s1n wt, (21.4) where a is a positive constant due to friction or viscosity. A detailed analysis of this
equation (which is not difficult but somewhat lengthy and so will be omitted) reveals
that if a2 < 4mg the solution x(t) consists of two parts, a purely sinusoidal term plus a
damped sinusoidal term. The damped term has a damping factor e_“”2 which decreases
to zero very rapidly as t—> 00; the only visible part of the motion is the purely sinusoidal
part, which can be written as x(t) = A sin(wt — or), (21.5)
where 0: = arctan [aw/(036 — (02)] is the phase angle and A is the amplitude, given by do A = ——————— . V ((93 — (1)2)2 + (am)2
If there is no resistive force, the friction coefficient a is 0 and the formula for the amplitude
becomes A = ao/Iwg — 002],
in agreement with the result derived earlier in Eq. (21.3). If the friction coefficient a is
small but nonzero, then the amplitude A exhibits a resonance peak of finite height nearly equal to ao/(awo) if u) is near to too. The larger the friction coefﬁcient (1 becomes, the
lower the peak height, as illustrated in Fig. 21.3. 212 DESCRIBING RESONANCE 405 W Example 2
Why do frictional forces prevent the amplitude of a driven oscillator from becoming inﬁnite at resonance?
Let’s ﬁrst recall what we know about simple harmonic oscillators when friction is present, without any driving force. In Chapter 20 we argued that the effect of friction
was to cause the amplitude (but not the frequency) of the oscillations to diminish. The
reason is that the oscillator loses energy in the form of heat. Now if the oscillator is
forced to oscillate, friction not only causes it to lose energy with each oscillation but also
to lag behind the driving force. Recall that we found that for a frequency much greater
than the natural frequency, the amplitude is negative, indicating that the oscillator vibrates
in a way opposite to the driving force. A similar effect is caused by friction and depends
upon the amount of friction. As a result, the oscillator can never be completely in step
with the driving force and moreover it keeps on losing energy. Consequently, the am—
plitude can never become infinite, only very large if friction is small. very low
/ friction low friction 0 0.5 1.0 1.5 2.0 2.5
w/wo Figure 21.3 Resonance curves (A vs. (ll/(.00) for different amounts of
frictional force acting on an oscillator. In the idealized situation where there is no friction or viscosity, so a = 0 in Eq.
(21.4), the damping factor e_‘"/2 which we neglected in Eq. (21.5) is not there to carry
the second term to 0. In this case, (1 = 0 and the solution consists of two purely sinusoidal tCI'IIISI x(t) = C sin (wot — B) + 750—2 sin (Jot (21.6)
(1) (.00— 406 RESONANCE provided that (1)2 75 (93. Again, the amplitude of the second term is large if m is near to
the natural frequency 000. But if the friction coefficient a = 0 and m2 = 035 the solution
takes yet a different form. In this case it becomes x(t) = C sin (wot — B) — ﬂ: cos (not. (21.7)
2000
The first term is purely sinusoidal, as above, but the second term oscillates with increasing
amplitude as t increases because of the presence of the factor I multiplying the cosine.
In this type of resonance the amplitude of the oscillation becomes arbitrarily large as 1
increases. An example of a solution x(t) with C = 0 is shown in Fig. 21.4. Figure 21.4 Resonant displacement of an oscillator in the absence of
friction. Breaking wine glasses is just one minor albeit dramatic example of the phenomenon
of resonance. Many other things in everyday life exhibit resonance. Most cars have
something that starts to rattle at certain motor speeds. This means that there is a natural
harmonic oscillator somewhere (usually you can never quite pinpoint it) whose oscillation
has the frequency of that motor speed. When the motor speed reaches that particular
value, vibrations set in and the object begins to rattle. Another example is the rattling of
windows when an airplane ﬂies overhead. The windows have a natural frequency that is
excited by the sound of the airplane engines. Similarly, the sounding board of a piano, which is a piece of wood with many natural
frequencies, resonates when a vibrating string is attached to it. Resonance also occurs in
the cavity of a violin; and air inside can have large oscillations for certain frequencies. Sometimes the effects of resonance can be ominous. In an earthquake, seismic waves
are sent out from the epicenter in a range of frequencies, mostly at low frequencies
compared to audible sound, known as infrasound. What happens if a structure has a
resonance at one of those frequencies? Buildings between 5 and 40 stories high are
typically resonant at earthquake frequencies. In an earthquake, these structures can literally
come apart as a result of resonance. Architects try to minimize the resonant response of
buildings by increasing friction in the joints. 21.3 SWINGING AND SINGING WIRES IN THE WIND 407 Questions 4. (a) 5. Find the resonant frequencies for the following systems: (a) k = 2000 N/m, (b) L = 0.35 m,
m = 2.5 kg; m = 0.20 kg. k =2000 N/m (b) L =0.35 m m = 2.5 kg m = 0.20 kg I
\ t._...____.__ / If pendulum E is given a slight push, which pendulum(s) will also begin to oscil—
late? Explain why. 6. 10. 11. Every child knows that by blowing across an empty Coke bottle a sound can be
produced; the frequency of the sound is the natural frequency of the air inside the
bottle. Explain what happens to the frequency produced when the bottle is par
tially filled with liquid. A cabinet next to a refrigerator contains pots and pans which make a vibrational
noise when the refrigerator motor runs. What is the source of this sound? How
can it be eliminated? A pendulum is forced to oscillate at a frequency 0) = $00, that is, at one—quarter
of its natural frequency. Compare the amplitude of the oscillations to those which occur at 00 = £000. In 1831 a bridge collapsed near Manchester, England, when soldiers marched
across it in step. Ever since then, soldiers break step while marching across a bridge. Why? Use the solution x(t) = A sin out, where A is given by Eq. (21.3), to find the total
energy of a forced harmonic oscillator. What happens to the energy at resonance? When the base of a vibrating tuning fork is touched to a table, the sound is ampli
fied. Explain why. 408 RESONANCE 21.3 SWINGING AND SINGING WIRES IN THE WIND One fascinating example of resonance is provided by telephone wires singing in the wind.
Imagine a taut wire suspended in the wind. The air ﬂow around a cross section of the
wire is illustrated in Fig. 21 .5a. This smooth ﬂow of air around the wire becomes unstable
if the wind speed is great enough. The wind tries to move around the wire and prevent
a vacuum from forming. If the speed is too high, the wind can’t achieve this in a smooth
ﬂow, and instead it forms an eddy on both sides, as in Fig. 21.5b. The eddies start to
occur behind the wire, and after a short time these vortices begin peeling off on alternate
sides and running downstream in the wake of the wire, as shown in Fig. 21.5c. This
complicated, yet very stable, ﬂow pattern was first explained by the aerodynamicist
Theodore von Karman. The full pattern with a line of vortices in opposite directions is
called the von Karman Vortex Street; Fig. 21.6 is an actual picture of the pattern. (a) (b) m
\m/—‘ (C)
Figure 21.5 Formation of vortices whi...
View
Full Document
 Fall '07
 Goodstein

Click to edit the document details