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Файл:A General Course of Physics. Mechanics. Textbook
.pdf
91
We will represent both oscillations with the help of amplitude
1
À
2
À
1
À
2
À
À
)cos(2
1221
2
2
2
12
AAAAA
2211
2211
cosαAcosαA
sinαAsinαA
OC
AC
tgα
А
О
x
t+
x = Acos (t
A
Fig. 42
Fig. 43
x
x1
x2
1
2
О
x
1
A
2
A
А B C
vectors
and
, and draw the resultant vector А by the rule of vector ad-
dition. In Fig. 43, it can be clearly seen that the projection of this vector onto
the axis x equals the sum total of the added vectors’ projections:
x = x1 + x2
The vectors
and
are rotating at an identical velocity .
Consequently, the angle between them remains constant at all times and
equals 2–
1
will have a constant magnitude
and, obviously, will be rotating at the same angular velocity. Thus, the
resultant motion will be a harmonic motion at the frequency , amplitude
А and initial phase :
x = Acos(t + )
We will determine the amplitude А and initial phase using the
graph (Fig. 43).
;
∙
The resultant oscillation amplitude А depends on the phase dif-
ference 2–1 of the summed oscillations:
92
1)cos(1
12
.
2121
AAAAA
21
AÀÀ
1
À
)()(
1212
t
)α(α)tω(ωcosA2AAAA
121221
2
2
2
1
2
ttAxxx )cos(cos
21
ttAttAttA
cos)('cos)
2
cos2(
2
cos
2
2
cos2
Therefore,
.
If 2–1 = 2k(k = 0,1,2…), then А = А1 + А2 and the oscillations
are intensified. In case 2–1 = (2k + 1) (k = 0,1,2…), then
,
the oscillations decrease.
6.5. Beats
When added oscillations have the same direction but differ in
frequency, the resultant oscillation is not harmonic:
x1 = А1 соs(1t + 1); x2=А2 соs(2t + 2).
The phase difference will be
.
Then the amplitude of the resultant oscillation can be expressed
in the following way:
.
It means that the amplitude will change with time and with the
frequency 2–1.
A particular case of oscillation addition with different frequencies is beats, i.e. when oscillations occurring after the addition of two oscillations with the same direction differ slightly in frequency.
We will receive the equation for beats and, for simplicity, we will
assume that the amplitudes and initial phases of the oscillations added are
equal А1 = А2, 1 = 2 = 0:
x1 = Асоst; x2 = Асоs( + ), <<;
.
93
It follows from the equation that the resulting oscillation can be
))2cos((2)( tAtA
2T
2
b
T
roughly considered as harmonic motion with pulsing amplitude
.
In Fig. 44, there is a graph of beats, where
is a period
of oscillations, and
is a period of beats (a period of ampli-
tude variation). But strictly speaking, it is not harmonic motion, but amplitude-modulated oscillation.
An important task of the oscillation theory is harmonic analysis,
i.e. representation of complex modulated oscillations as a number of simple harmonic oscillations. Fourier, a French scientist, showed that any
complex harmonic oscillations can be represented as a sum of harmonic
oscillations with integer multiples:
94
The frequency is called the fundamental frequency, the frequencies 2, 3,… are overtones, or harmonic partials (2nd harmonic, 3
harmonic).
Representation of oscillations as a sum of harmonic oscillations
is called spectral decomposition, while a spectrum contains information
about harmonic frequency and amplitude, which are inherent components
of the oscillation.
Fig. 45 gives a view of a spectrum of a non-harmonic oscillation.
The closer the oscillation graph to the sinusoid, the lower the harmonic
amplitude and the smaller the number of harmonics. If the oscillation
graph has little resemblance with a sinusoid, then the amplitudes of some
harmonics can be comparable with the amplitude of the fundamental frequency, and the number of harmonics can be high.
The spectrum of the harmonic signal (Fig. 46, a) consists of one
frequency only (Fig. 46, b). The spectrum of the non-periodic process
(Fig. 47, a) is continuous, i.e. consists of an endless number of harmonics
(Fig. 47, b).
Beats are used for radio-signals reception, signal measurements
and comparison of their frequencies and musical instrument tuning.
6.6. Addition of Mutually Orthogonal Oscillations. Lissajous Figures
Assume that a point particle is simultaneously involved in two
mutually orthogonal oscillations with equal frequencies:
d
95
x = А1cos (t + 1); y = А2cos (t + 2).
;
1
2
A
A
x
y
x
A
A
y
1
2
1
2
A
A
tg
)cos()cos(
2
2
2
1
22
tAtAAyxS
2
2
2
1
AAA
Fig. 48
x
y
S
The trajectory of the resulting oscillation will depend on the
phase difference of the added oscillations. Let us examine a few particular cases.
Case One. The phase difference is
2–1 = 0; 2 = 1 = .
The oscillation equations here are the following:
х = А1cos (t + 1);
y = А2cos (t + 2);
;
;
.
Thus, the resulting motion will also be harmonic with the same
frequency and with the same initial phase as the added oscillations. The
motion is executed along the straight line S, which makes up the angle
with the axis х (Fig. 48). The resulting oscillation amplitude is
.
The oscillations producing a
trajectory in the form of a straight line
as a result of a point particle motion are
called linearly polarized oscillations.
Case Two. The phase difference
is
2–1 = .
Assume that 2 = 1 + . In this
case, we will write the oscillations in
the following way:
96
х = А1cos (t + 1),
x
A
A
y
1
2
2
)(cos
1
22
1
2
tÀõ
)(sin
1
22
2
2
tAy
1
2
2
22
1
2
AyAx
212
212
Fig. 49
x
y
S
y = А2cos (t + 1 + ) =
= –А2соs (t + 1);
.
The resulting motion will also
be executed along a straight line which
crosses the second and fourth quarters
(Fig. 49).
Case Three. The phase difference is
2–1 =
.
Assume that 2 = 1 + 2. Then,
х = А1cos (t + 1);
y = А2cos (t + 2) = А2cos (t +
+ 1 + 2) = –А2sin (t + 1);
;
;
.
In this case, the resulting motion will be executed along an ellipse (Fig. 50). If the phase difference is
clockwise, whereas if the phase difference is
, the motion is
, the motion
is counterclockwise. The oscillations producing a trajectory in the form of
an ellipse as a result of a point particle motion are called elliptically pola-
rized oscillations.
If А1 = А2 the ellipse becomes a circle (Fig. 51), the oscillations become circularly polarized.
To sum up, two mutually orthogonal oscillations with the same
amplitude and phase difference 2–1=/2, in total produce uniform
motion along the circumference with the radius A and the angular velocity .
97
)(sin)cos(2
12
2
12
21
2
2
2
2
1
2
AA
xy
A
y
A
x
21
yx
2
Fig. 50
x
y
A
1
A
2
Fig. 51
x
y
A
1
A
2
The uniform circular motion can be decomposed back into two
mutually orthogonal oscillations.
If phase differences have different values, they result in arbitrarily oriented ellipses.
The formula for the resulting trajectory has the following form:
.
The cases with the corresponding phase differences described
above result from this formula.
If mutually orthogonal waves oscillate at different frequencies,
added trajectories result in more complex forms called Lissajous figures.
Their shape depends on the correlation between frequencies and phase
difference of the oscillations added.
Now we will consider particular cases.
Case One. Frequencies differ twofold, the phase difference is zero:
x = A1cost; y = A2cos2t;
; = 0.
The resulting motion follows the parabolic curve (Fig. 52).
Case Two. Frequencies differ twofold, the phase difference is
:
98
x = A1cost; y = A2cos (2t + 2);
21
yx
xrrF
v
fr
kxF
Fig. 53
x y A1 A
2
Fig. 52
y
A1 A
2
; =2
The resulting motion follows the pattern given in Fig. 53.
Provided that the frequency of one of the oscillations is known,
the shape of Lissajous figures can help to determine the frequency of the
other oscillation and the phase difference.
6.7. Damped Oscillations
As a result of resistance to motion, the oscillating system is constantly giving up energy to the medium. Energy is proportional to the
amplitude squared, so if energy decreases, the amplitude diminishes as
well, i.e. the oscillations are damped.
Let us consider a case when a point particle executes rectilinear
oscillations in a viscous medium. Then, resistance will depend on the
viscous friction in the medium. If the velocity is low, the friction force is
proportional to the velocity v:
,
where r is the resistance coefficient.
Apart from this force, an elastic force is also exerted:
.
is as follows:
Thus, the equation of motion (according to Newton’s second law)
99
fr
FFma
;
xrkxxm
0 kxxrxm
0 x
m
k
x
m
r
x
2;
2
0
mrmk
02
2
0
xxx
)cos(0teAx
t
)cos( ttA
t
AtA
β
0
e
22
0
Fig. 54
x
t
;
.
Let us introduce the nota-
tion:
viz. is the attenuation coefficient ( > 0), 0 is the undamped
frequency.
Therefore,
. (70)
Equation (70) is a differential equation of damped oscillations.
The general solution (70) we will be written as follows:
, (71)
;
,
specifically,
is the damped oscillations amplitude.
As it can be seen from formula (71), the amplitude of oscillations
is decreasing with time; that is determined with the help of the attenuation coefficient .
The value
is a cyclic frequency of the damped os-
cillations; 0 is a cyclic oscillation frequency of the same system in the
absence of friction. The damped oscillations (Fig. 54) are non-periodic
oscillations because displacement, velocity and acceleration values of
every other period are different. For this reason, the period T can be considered as the time interval which the system goes through to return to the
equilibrium position.
The attenuation rate is characterized by several quantities – the
attenuation coefficient , damping logarithmic decrement , relaxation
time and mechanical quality factor Q.
100
The logarithmic ratio of two successive amplitude values, de-
Te
eA
eA
A
A
T
Tt
t
Tt
t
lnlnln
)(
0
0
ee
eA
eA
A
A
t
t
t
t
)(
0
0
TQ
22
0
2
Ò
0
β2mr
layed from each other by the time interval equal to the T period, is called
damping logarithmic decrement:
; =T.
The time interval over which the amplitude decreases e-fold is
relaxation time:
.
From this equation we will get that = 1, i.e. the attenuation
coefficient = 1/ is a physical quantity, reciprocal to the relaxation
time.
In actual practice, it is essential that oscillations should attenuate slowly when possible. The quantity which characterizes this parameter is called the Q-factor. By definition, the Q-factor is
.
When the attenuation coefficient is increased, the oscillation period extends:
;
if
; T = ;
; r = 2m.
If = 0 rk = 2m0 is the critical resistance, the process itself is
called critical. If r rk , the motion becomes non-periodic (Fig. 55).
When the motion is non-periodic, the entire mechanical energy of
the oscillating system is consumed by friction resistance (during an oscillating motion, the system, when returning to the equilibrium position, has
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