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which the frictional force 1s proportional to the speed with which the object moves.
An example of such friction is the friction for slow motion of an object in oil or
a thick liquid. “PThere is no force when ï§ is just standing still, but the faster 1t
moves the faster the oil has to go past the object, and the greater is the resistance.
So we shall assume that there is, in addition to the terms in (23.2), another
term, a resistance force proportional to the velocity: = —cdz/di. It will be
convenient, in our mathematical analysis, to write the constant é as rm times +
to simplify the equation a little. Phis is Just the same trick we use with k when
we replace it by ma, just to simplify the algebra. Thus our equation will be
m(dÊ+/dt?) + e(daz/dt) + kz = F (23.6)
or, writing e = my and k = mổ and dividing out the mass rm,
(d2+/d1?) + +(dz/dÐ) + + = F/m. (23.6a)
Now we have the equation in the most convenient form to solve. lÍ y is very
small, that represents very little friction; if +y is very large, there is a tremendous
amount of friction. How do we solve this new linear diferential equation? Suppose
that the driving force is equal to #ọ cos (¿# + A); we could put this into (23.6a)
and try to solve it, but we shall instead solve it by our new method. Thus we
write ` as the real part of #'e”““f and z as the real part of êe”“!, and substitute
these into Eq. (23.6a). It is not even necessary to do the actual substituting, for
we can see by inspection that the equation would become
[(¿œ)22 -+ +(#ø)ê + äöẬ£]e**t = (Ê/m)e**t, (23.7)
[As a matter of fact, IÝ we tried to solve Eq. (23.6a) by our old straightforward
way, we would really appreciate the magic of the “complex” method.] IÝ we divide
by e?”“f on both sides, then we can obtain the response £ to the given force /) it
â@= ÊJ/m(uŸ — 0 + iu). (23.8)
Thus again ê is given by Ê tỉmes a certain faetor. There is no technical name
for this factor, no particular letter for it, but we may call ít for discussion
DUTPOSGS:
th = ———cc
m(u8 — 2 + i2)
--- Trang 411 ---
â= ÊR. (23.9)
(Although the letters and œọ are in very common use, this f‡ has no particular
name.) This factor ## can either be written as p-L ¿g, Or as a certain magnitude ø
times e', If it is written as a certain magnitude times e'”, let us see what it
means. NÑow /' = Fụe!^, and the actual force ` is the real part of Fụel2e'“t, that
1s, Fo cos (J@£ + A). Next, Bq. (23.9) tells us that £ is equal to #'?. So, writing
R= pể?® as another name for Ï, we get
@—= RÊ = øe?Fụef2 = pFucf+A),
Einally, going even further back, we see that the physical , which is the real part
of the complex £e”“f, is equal to the real part of øFuef®+A)e/“f, But ø and Fụ
are real, and the real part of e#+^Ã**®) is simply cos (w# + A +9). Thus
# = pFọ cos (u‡ + A +8). (23.10)
'This tells us that the amplitude of the response is the magnitude of the force †
multipHed by a certain magnifcation factor, /ø; this gives us the “amount” of
oscillation. It also tells us, however, that #ø is not oscillating in phase with the
force, which has the phase A, but is shifted by an extra amount Ø. 'Therefore ø
and Ø represent the size of the response and the phase shift of the response.
Now let us work out what ø is. IÝ we have a complex number, the square of
the magnitude is equal to the number times its complex conJugate; thus
ø”= :
mˆ2(uậ — . + 0)(„8 — 2 — iu) (23.11)
—_ m3[(w2 — œ8)? + +42]
In addition, the phase angle Ø is easy to find, for if we write
1/R= 1/pe'” = (1/p)e"”” = m(ưậ — Ÿ + i20),
we see that
tan 0 = —+w/(uä — œ). (23.12)
]t is minus because tan(—0) = — tan/. A negative value for Ø results for all œ,
and this corresponds to the displacement zø lagging the force #'.
--- Trang 412 ---
(Q9 ứ)
Fig. 23-2. Plot of øˆ versus ứ.
0°
~90° (0 œ
—180°4~~~~~~~~~~~~~~~~~~~~~~—~~~~-—~~~---====
Fig. 23-3. Plot of Ø versus ứ.
Figure 23-2 shows how øŸ varies as a funetion of frequency (øZ is physically
more interesting than ø, because øŸ is proportional to the square of the amplitude,
or more or less to the energw that is developed in the oscillator by the force). We
see that iŸ + is very small, then 1/(œđ — @”)Ÿ is the most important term, and
the response tries to go up toward infũnity when œ equals œạ. Now the “infnity”
is not actually infũnite because if œ = œọ, then 1/22 is still there. The phase
shift varies as shown In Fig. 23-3.
In certain circumstances we get a slightly diferent formula than (23.8), also
called a “resonance” formula, and one might think that it represents a dierent
phenomenon, but it does not. The reason is that IŸ + is very small the most
interesting part oŸ the curve is near œ = œọ, and we may replace (23.8) by
an approximate formula which is very accurate iŸ + is small and œ is near œ0.
Since „8 — 2 = (œg — œ)(œ0 + œ@), l œ is near œọ this is nearly the same as
2(o(o — 0) and +0 is nearly the same as +œo. Dsing these in (23.8), we see that
tu — 0Ÿ + 20 200(0 — @ + 2/2), so that
â Ê/2mœg(œạ =@+ 11/9) lÍ +<øg and @®úg. (23.13)
--- Trang 413 ---
It is easy to fnd the corresponding formula for øŸ. It is
g2 % 1/4mŠ8|(ao — ø)Š + 42/4.
W©e shall leave it to the student to show the following: if we call the maximum
height of the curve of øŸ vs. œ one unit, and we ask for the width Aø of the curve,
at one half the maximum height, the full width at half the maximum height of
the curve is A¿ = +, supposing that + is small. The resonance is sharper and
sharper as the frictional efects are made smaller and smaller.
As another measure of the width, some people use a quantity @Q which is
defined as Q = œg/+. The narrower the resonance, the higher the Q: = 1000
means a resonance whose width is only 1000th of the frequency scale. The @Q of
the resonance curve shown in Fig. 23-2 is ð.
'The importance of the resonance phenomenon is that it occurs in many other
circumstances, and so the rest of this chapter will describe some of these other
circumstances.
23-3 Electrical resonance
'The simplest and broadest technical applications of resonanece are in electricity.
In the electrical world there are a number of obJects which can be connected to