Let the point simultaneously participate in two harmonic oscillations of the same period, directed along one straight line.

The addition of oscillations will be carried out by the method of vector diagrams (Fig. 2.2). Let the oscillations be given by the equations

and

(2.2.1)

Set aside from the point O a vector at an angle φ 1 to the reference line and a vector at an angle φ 2 . Both vectors rotate counterclockwise with the same angular velocity ω, so their phase difference does not depend on time (). Such vibrations are called coherent.

We know that the total projection of a vector is equal to the sum of the projections onto the same axis. Therefore, the resulting oscillation can be represented by an amplitude vector rotating around the point O with the same angular velocity ω as , and . The resulting oscillation must also be harmonic with frequency ω:

.

By the rule of vector addition, we find the total amplitude:

The resulting amplitude is found by the formula

Thus, the body, participating in two harmonic oscillations of the same direction and the same frequency, also performs a harmonic oscillation in the same direction and with the same frequency as the summed oscillations.

From (2.2.2) it follows that the amplitude BUT the resulting oscillation depends on the difference in the initial phases . Possible values BUT lie in the range (the amplitude cannot be negative).

Let's consider some simple cases.

1. Phase difference is zero or even numberπ, that is, where . Then and

,

(2.2.4)

since , i.e. resulting oscillation amplitude BUT is equal to the sum of the amplitudes of the added oscillations (oscillations in-phase) (Fig. 2.3).

2. The phase difference is an odd numberπ , i.e , where . Then . From here

.

(2.2.5)

On fig. 2.4 shows the amplitude of the resulting oscillation BUT, equal to the difference in the amplitudes of the added oscillations (oscillations in out of phase).

3. The phase difference changes in time in an arbitrary way:

(2.2.6)

From equation (2.2.6) it follows that and will change in accordance with the value of . Therefore, when adding incoherent oscillations, it makes no sense to talk about adding amplitudes, but in some cases quite definite patterns are observed. For practice, of particular interest is the case when two added oscillations of the same direction differ little in frequency. As a result of the addition of these oscillations, oscillations with a periodically changing amplitude are obtained.

Periodic changes in the oscillation amplitude arising from the addition of two harmonic oscillations with close frequencies, are called beats . Strictly speaking, these are no longer harmonic oscillations.

Let the amplitudes of the added oscillations be equal to BUT, and the frequencies are equal to ω and , and . We choose the reference point so that the initial phases of both oscillations are equal to zero:

We add these expressions, neglecting , since .

The nature of dependence (2.2.8) is shown in Fig. 2.5, where solid thick lines give a graph of the resulting oscillation, and their envelopes - a graph of slowly changing amplitude according to equation (2.2.7).

Determination of the tone frequency (sound of a certain height) of beats between the reference and measured vibrations is the most widely used method in practice for comparing the measured value with the reference. The beat method is used for tuning musical instruments, hearing analysis, etc.

In general, oscillations of a species are called modulated . Special cases: amplitude modulation and phase or frequency modulation. beat is the simplest form of modulated oscillations.

Any complex periodic oscillations can be represented as a superposition of simultaneously occurring harmonic oscillations with different amplitudes, initial phases, and also frequencies that are multiples of the cyclic frequency ω:

.

The representation of a periodic function in this form is associated with the concept harmonic analysis of a complex periodic oscillation, or Fourier expansion(that is, the representation of complex modulated oscillations as a series (sum) of simple harmonic oscillations). The terms of the Fourier series, which determine harmonic oscillations with frequencies ω, 2ω, 3ω, ..., are called first(or main), second, third etc. harmonics complex periodic oscillation.

Along with the translational and rotational motions of bodies in mechanics, oscillatory motions are also of considerable interest. Mechanical vibrations called the movements of bodies that repeat exactly (or approximately) at regular intervals. The law of motion of an oscillating body is given by some periodic function of time x = f (t). The graphic representation of this function gives a visual representation of the course of the oscillatory process in time.

Examples of simple oscillatory systems are a load on a spring or a mathematical pendulum (Fig. 2.1.1).

Mechanical oscillations, like oscillatory processes of any other physical nature, can be free and forced. Free vibrations are made under the influence internal forces system after the system has been brought out of equilibrium. The oscillations of a weight on a spring or the oscillations of a pendulum are free oscillations. vibrations under the action external periodically changing forces are called forced .

The simplest type of oscillatory process are simple harmonic vibrations , which are described by the equation

x = x m cos (ω t + φ 0).

Here x- displacement of the body from the equilibrium position, x m - oscillation amplitude, i.e. the maximum displacement from the equilibrium position, ω - cyclic or circular frequency hesitation, t- time. The value under the cosine sign φ = ω t+ φ 0 is called phase harmonic process. At t= 0 φ = φ 0 , so φ 0 is called initial phase. The minimum time interval after which the movement of the body is repeated is called period of oscillation T. The physical quantity reciprocal to the period of oscillation is called oscillation frequency:

Oscillation frequency f shows how many vibrations are made in 1 s. Frequency unit - hertz(Hz). Oscillation frequency f is related to the cyclic frequency ω and the oscillation period T ratios:

On fig. 2.1.2 shows the positions of the body at regular intervals with harmonic vibrations. Such a picture can be obtained experimentally by illuminating an oscillating body with short periodic flashes of light ( stroboscopic lighting). The arrows represent the velocity vectors of the body at different points in time.

Rice. 2.1.3 illustrates the changes that occur on the graph of a harmonic process if either the amplitude of the oscillations changes x m , or period T(or frequency f), or the initial phase φ 0 .

When the body oscillates along a straight line (axis OX) the velocity vector is always directed along this straight line. Velocity υ = υ x body movement is determined by the expression

In mathematics, the procedure for finding the limit of the ratio at Δ t→ 0 is called the calculation of the derivative of the function x (t) by time t and denoted as or as x"(t) or finally as . For the harmonic law of motion Calculation of the derivative leads to the following result:

The appearance of the term + π / 2 in the cosine argument means a change in the initial phase. Maximum modulo values ​​of velocity υ = ω x m are achieved at those moments of time when the body passes through the equilibrium positions ( x= 0). Acceleration is defined in a similar way a = ax bodies with harmonic vibrations:

hence the acceleration a is equal to the derivative of the function υ ( t) by time t, or the second derivative of the function x (t). The calculations give:

The minus sign in this expression means that the acceleration a (t) always has the opposite sign of the offset x (t), and, therefore, according to Newton's second law, the force that causes the body to perform harmonic oscillations is always directed towards the equilibrium position ( x = 0).

a) The body participates in two harmonic oscillations with the same circular frequenciesw , but with different amplitudes and initial phases.

The equation of these oscillations will be written as follows:

x 1 \u003d a 1 cos (wt + j 1)

x 2 \u003d a 2 cos (wt + j 2),

where x 1 and x 2- offsets; a 1 and a 2- amplitudes; w- circular frequency of both oscillations; j1 and j2- initial phases of oscillations.

Let's add these fluctuations using a vector diagram. Let us represent both oscillations as amplitude vectors. To do this, from an arbitrary point O lying on the axis X, we set aside two vectors 1 and 2, respectively, at the angles j1 and j2 to this axis (Fig. 2).

The projections of these vectors onto the axis X will be equal to the displacements x 1 and x 2 according to expression (2). When both vectors rotate counterclockwise with angular velocity w projections of their ends onto the axis X will make harmonic vibrations. Since both vectors rotate with the same angular velocity w, then the angle between them j=j 1 -j 2 remains constant. Adding both vectors 1 and 2 according to the parallelogram rule, we get the resulting vector . As can be seen from Fig. 2, the projection of this vector onto the axis X is equal to the sum of the projections of the terms of the vectors x \u003d x 1 + x 2. On the other side: x \u003d a cos (wt + j o).

Consequently, the vector rotates with the same angular velocity as vectors 1 and 2 and performs a harmonic oscillation that occurs along the same straight line as the terms of the oscillations, and with a frequency equal to the frequency of the original oscillations. Here j o - the initial phase of the resulting oscillation.

As can be seen from Fig. 2, to determine the amplitude of the resulting oscillation, you can use the cosine theorem, according to which we have:

a 2 \u003d a 1 2 + a 2 2 - 2a 1 a 2 cos

a \u003d a 1 2 + a 2 2 + 2a 1 a 2 cos (j 2 - j 1)(3)

It can be seen from expression (3) that the amplitude of the resulting oscillation depends on the difference in the initial phases ( j 2 - j 1) terms of oscillations. If the initial phases are equal ( j 2 = j 1), then formula (3) shows that the amplitude a is equal to the sum a 1 and a 2. If the phase difference ( j 2 - j 1) is equal to ±180 o (i.e., both oscillations are in antiphase), then the amplitude of the resulting oscillation is equal to the absolute value of the difference in the amplitudes of the oscillation terms : a = |a 1 - a 2 |.

b) The body participates in two oscillations with the same amplitudes, initial phases equal to zero, and different frequencies.

The equations for these oscillations will look like:

x 1 \u003d a sinw 1 t,

x 2 \u003d a sinw 2 t.

In doing so, it is assumed that w 1 little different in size from w 2. Adding these expressions, we get:

x \u003d x 1 + x 2 \u003d 2a cos[(w 1 -w 2)/2]t+sin[(w 1 +w 2)/2]t=

=2а cos[(w 1 -w 2)/2]t sin wt (4)

The resulting motion is a complex oscillation called beats(Fig. 3) Since the value w1-w2 small compared to the size w1+w2, then this motion can be considered as a harmonic oscillation with a frequency equal to half the sum of the frequencies of the added oscillations w=(w1+w2)/2, and variable amplitude.

It follows from (4) that the amplitude of the resulting oscillation changes according to the periodic cosine law. A full cycle of changing the values ​​of the cosine function occurs when the argument changes by 360 0 , while the function passes values ​​from +1 to -1. The state of the system that beats at the instants of time corresponding to the specified values ​​of the cosine function in formula (4) does not differ in any way. In other words, beat cycles occur with a frequency corresponding to a change in the cosine argument in formula (4) by 180 0 . So the period T a amplitude changes during beats (beat period) is determined from the condition:

T a \u003d 2p / (w 1 - w 2).

Given that w=2pn, we get:

T a \u003d 2 p / 2 p (n 1 - n 2) \u003d 1 / (n 1 - n 2). (5)

The frequency of change in the amplitude of the resulting oscillation is equal to the difference in the frequencies of the added oscillations:

n=1/T a =n 1 -n 2 .

Addition of harmonic oscillations of one direction.

beats

Consider an oscillatory system with one degree of freedom, the state of which is determined by the dependence of some quantity on time. Let the oscillation in this system be the sum of two harmonic oscillations with the same frequency but different amplitudes and initial phases, i.e.

Since the "shift" of the oscillatory system from the equilibrium position occurs along one single "direction", in this case one speaks of the addition of harmonic oscillations of one direction. On the vector diagram, the added oscillations will be displayed as two vectors and , rotated relative to each other by an angle (Fig. 6.1). Since the frequencies of the added oscillations are the same, their mutual position will remain unchanged at any time, and the resulting oscillation will be represented by a vector equal to the sum of the vectors and . Adding the vectors according to the parallelogram rule and using the cosine theorem, we get

. (6.3)

Thus, when adding two harmonic oscillations of the same direction with the same frequencies, a harmonic oscillation of the same frequency is obtained, the amplitude and initial phase of which are determined by the expressions(6.2), (6.3).

Two harmonic oscillations that occur at the same frequency and have a constant phase difference are called coherent. Consequently, when adding coherent oscillations, a harmonic oscillation of the same frequency is obtained, the amplitude and initial phase of which are determined by the amplitudes and initial phases of the added oscillations.

If the added oscillations have different frequencies and , but the same amplitudes , then, using the expression known from trigonometry for the sum of the cosines of two angles, we obtain

It can be seen from the resulting expression that the resulting oscillation is not harmonic.

Let the frequencies of the added oscillations be close to each other so that and . This case is called beating two frequencies.

Denoting , and , can be written

. (6.5)

It follows from expression (6.5) that the resulting oscillation can be represented as a harmonic oscillation with a certain average frequency , whose amplitude slowly (with a frequency ) changes in time. Time called beat period, a – beat frequency. The beat graph is shown in Figure 6.2. Beats occur when simultaneous sounding of two tuning forks of the same key. They can be observed using an oscilloscope when adding the harmonic oscillations of two generators tuned to the same frequency. In both cases, the frequencies of the oscillation sources will differ slightly, resulting in beats.

Since the oscillations occur at different frequencies, the phase difference of the added oscillations changes with time, therefore, the oscillations are not coherent. The change in time of the amplitude of the resulting oscillations is a characteristic consequence of the incoherence of the added oscillations.

The addition of oscillations is very often observed in electrical circuits and, in particular, in radio communication devices. In some cases, this is done purposefully in order to obtain a signal with specified parameters. So, for example, in a heterodyne receiver, the received signal is added (mixed) with the local oscillator signal in order to obtain an intermediate frequency oscillation as a result of subsequent processing. In other cases, the addition of oscillations occurs spontaneously when some kind of interference is received at the input of the device, in addition to the useful signal. In fact, the whole variety of the form of electrical signals is the result of the addition of two or more harmonic oscillations.

The same body can simultaneously participate in two or more movements. A simple example is the movement of a ball thrown at an angle to the horizontal. We can assume that the ball participates in two independent mutually perpendicular movements: uniform horizontally and equally variable vertically. One and the same body (material point) can participate in two (or more) movements of an oscillatory type.

Under addition of vibrations understand the definition of the law of the resulting oscillation, if the oscillatory system simultaneously participates in several oscillatory processes. There are two limiting cases - the addition of oscillations of one direction and the addition of mutually perpendicular oscillations.

2.1. Addition of harmonic oscillations of one direction

1. Addition of two oscillations of the same direction(codirectional vibrations)

can be done using the vector diagram method (Figure 9) instead of adding the two equations.

Figure 2.1 shows the amplitude vectors BUT 1(t) and BUT 2 (t) summed oscillations at an arbitrary time t, when the phases of these oscillations are respectively equal and . The addition of oscillations is reduced to the definition . Let us use the fact that in the vector diagram the sum of the projections of the added vectors is equal to the projection of the vector sum of these vectors.

The resulting oscillation corresponds on the vector diagram to the amplitude vector and phase .

Figure 2.1 - Addition of co-directional oscillations.

Vector magnitude BUT(t) can be found using the cosine theorem:

The phase of the resulting oscillation is given by the formula:

.

If the frequencies of the added oscillations ω 1 and ω 2 are not equal, then both the phase φ(t) and the amplitude BUT(t) The resulting fluctuation will change over time. Added vibrations are called incoherent in this case.

2. Two harmonic oscillations x 1 and x 2 are called coherent, if their phase difference does not depend on time:

But since , then to fulfill the condition of coherence of these two oscillations, their cyclic frequencies must be equal.

The amplitude of the resulting oscillation obtained by adding codirectional oscillations with equal frequencies (coherent oscillations) is equal to:

The initial phase of the resulting oscillation can be easily found by projecting the vectors BUT 1 and BUT 2 on the coordinate axes OX and OY (see Figure 9):

.

So, the resulting oscillation obtained by adding two harmonic co-directional oscillations with equal frequencies is also a harmonic oscillation.

3. We investigate the dependence of the resulting oscillation amplitude on the difference between the initial phases of the added oscillations.

If , where n is any non-negative integer

(n = 0, 1, 2…), then minimum. The added vibrations at the moment of addition were in out of phase. At , the resulting amplitude is zero.

If a , then , i.e. the resulting amplitude will be maximum. At the moment of addition, the added oscillations were in one phase, i.e. were in phase. If the amplitudes of the added oscillations are the same , then .

4. Addition of codirectional vibrations with unequal but close frequencies.

The frequencies of the added oscillations are not equal, but the frequency difference both ω 1 and ω 2 are much smaller. The condition for the closeness of the added frequencies is written by the relations .

An example of the addition of co-directional oscillations with close frequencies is the movement of a horizontal spring pendulum, the spring stiffness of which is slightly different k 1 and k 2 .

Let the amplitudes of the added oscillations be the same , and the initial phases are equal to zero. Then the equations of the added oscillations have the form:

, .

The resulting oscillation is described by the equation:

The resulting oscillation equation depends on the product of two harmonic functions: one with a frequency , the other - with a frequency , where ω is close to the frequencies of the added oscillations (ω 1 or ω 2). The resulting oscillation can be viewed as harmonic oscillation with a harmonic-changing amplitude. This oscillatory process is called beats. Strictly speaking, the resulting oscillation is generally not a harmonic oscillation.

The absolute value of the cosine is taken because the amplitude is a positive value. The nature of the dependence x res. for beats is shown in Figure 2.2.

Figure 2.2 - The dependence of the displacement on time during beats.

The beat amplitude changes slowly with frequency . The absolute value of the cosine repeats, if its argument changes by π, then the value of the resulting amplitude will repeat after a time interval τ b, called beat period(See Figure 12). The value of the beat period can be determined from the following relationship:

The value is the beat period.

Value is the period of the resulting oscillation (Figure 2.4).

2.2. Addition of mutually perpendicular oscillations

1. A model that can demonstrate the addition of mutually perpendicular vibrations is shown in Figure 2.3. A pendulum (material point of mass m) can oscillate along the OX and OY axes under the action of two elastic forces directed mutually perpendicular.

Figure 2.3

The summed oscillations have the form:

Oscillation frequencies are defined as , , where , are spring stiffness coefficients.

2. Consider the case of adding two mutually perpendicular vibrations with the same frequencies , which corresponds to the condition (the same springs). Then the equations of the added oscillations will take the form:

When a point participates in two movements simultaneously, its trajectory can be different and quite complex. The equation for the trajectory of the resulting oscillations on the OXY plane when two mutually perpendicular ones with equal frequencies are added can be determined by excluding the time t from the initial equations for x and y:

The type of trajectory is determined by the difference in the initial phases of the added oscillations, which depend on the initial conditions (see § 1.1.2). Consider the possible options.

and if , where n = 0, 1, 2…, i.e. the summed oscillations are in-phase, then the equation of the trajectory will take the form:

(Figure 2.3 a).

Figure 2.3.a	Figure 2.3 b

b) If (n = 0, 1, 2…), i.e. the summed oscillations are in antiphase, then the trajectory equation is written as follows:

(Figure 2.3b).

In both cases (a, b), the resulting movement of the point will oscillate along a straight line passing through the point O. The frequency of the resulting oscillation is equal to the frequency of the added oscillations ω 0 , the amplitude is determined by the ratio.