Along with progressive and rotational movements In the mechanics of bodies, oscillatory motions are also of significant interest. Mechanical vibrations are movements of bodies that repeat exactly (or approximately) at equal intervals of time. The law of motion of a body oscillating is specified using a certain periodic function time x = f (t). Graphic image this function gives visual representation about the course of the oscillatory process over time.

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

Mechanical vibrations, like the oscillatory processes of any other physical nature, can be free And forced. Free vibrations are committed under the influence internal forces system after the system has been brought out of equilibrium. Oscillations of a weight on a spring or oscillations of a pendulum are free oscillations. Vibrations occurring under the influence 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 mcos(ω t + φ 0).

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

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

In Fig. 2.1.2 shows the positions of the body at equal intervals of time during harmonic vibrations. Such a picture can be obtained experimentally by illuminating an oscillating body with short periodic flashes of light ( strobe lighting). The arrows represent the velocity vectors of the body at different times.

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

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

In mathematics, the procedure for finding the limit of a ratio at Δ t→ 0 is called calculating the derivative of the function x (t) by time t and is denoted as or as x"(t) or, finally, like . For the harmonic law of motion, calculating 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 absolute values ​​of speed υ = ω x m are achieved at those moments in time when the body passes through equilibrium positions ( x= 0). Acceleration is determined in a similar way a = ax bodies during 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). Calculations give:

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

The same body can simultaneously participate in two or more movements. A simple example is the motion 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 uniformly variable vertically. Same body ( material point) can participate in two (or more) oscillatory movements.

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

2.1. Addition of harmonic vibrations of one direction

1. Addition of two oscillations of the same direction(co-directional oscillations)

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

Figure 2.1 shows the amplitude vectors A 1(t) and A 2 (t) added oscillations at an arbitrary moment of time t, when the phases of these oscillations are respectively equal And . The addition of oscillations comes down to the definition . Let's take advantage of the fact that in a vector diagram the sum of the projections of the vectors being added is equal to the projection vector sum these vectors.

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

Figure 2.1 – Addition of co-directional oscillations.

Vector magnitude A(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 A(t) The resulting fluctuations will change over time. Added oscillations are called incoherent in this case.

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

But since, in order 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 is easy to find if you project the vectors A 1 and A 2 on coordinate axes OX and OU (see Figure 9):

.

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

3. Let us study the dependence of the amplitude of the resulting oscillation on the difference in the initial phases of the added oscillations.

If , where n is any non-negative integer

(n = 0, 1, 2…), then minimum. The added oscillations at the moment of addition were in antiphase. When the resulting amplitude is zero.

If , That , 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 , That .

4. Addition of co-directional oscillations with unequal but similar frequencies.

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

An example of the addition of co-directional oscillations with similar 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 frequency , the other – with frequency , where ω is close to the frequencies of the added oscillations (ω 1 or ω 2). The resulting oscillation can be considered as harmonic oscillation with amplitude varying according to a harmonic law. Such oscillatory process called beats. Strictly speaking, the resulting fluctuation in general case is not a harmonic oscillation.

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

Figure 2.2 – Dependence of displacement on time during beating.

The amplitude of the beats changes slowly with frequency. The absolute value of the cosine is repeated if its argument changes by π, which means that the value of the resulting amplitude will be repeated 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 beating period.

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

2.2. Addition of mutually perpendicular vibrations

1. A model on which the addition of mutually perpendicular oscillations can be demonstrated is presented in Figure 2.3. A pendulum (a material point of mass m) can oscillate along the OX and OU axes under the action of two elastic forces directed mutually perpendicularly.

Figure 2.3

The folded oscillations have the form:

The oscillation frequencies are defined as , , where , are the spring stiffness coefficients.

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

When a point is involved 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 adding two mutually perpendicular ones with equal frequencies can be determined by excluding time t from the original 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 initial conditions(see § 1.1.2). Let's consider the possible options.

and if , where n = 0, 1, 2…, i.e. the added oscillations are in phase, then the trajectory equation 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 added 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 be an oscillation along a straight line passing through 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 relation.

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 in the following way:

x 1 = a 1 cos(wt + j 1)

x 2 = a 2 cos(wt + j 2),

Where x 1 And x 2- displacement; a 1 And a 2- amplitudes; w- circular frequency of both vibrations; j 1 And j 2- initial phases of oscillations.

Let's add these oscillations using a vector diagram. Let us represent both oscillations as amplitude vectors. For this from arbitrary point Oh, lying on the axis X, let us plot two vectors 1 and 2, respectively, at angles j 1 And j 2 to this axis (Fig. 2).

Projections of these vectors onto the axis X will be equal to the offsets x 1 And x 2 according to expression (2). When both vectors are rotated counterclockwise with angular velocity w projections of their ends onto the axis X will perform harmonic oscillations. 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 obtain the resulting vector . As can be seen from Fig. 2, the projection of this vector onto the axis X equal to the sum of the projections of the terms of the vectors x=x 1 +x 2. On the other side: x=a·cos(wt+j o).

Consequently, the vector rotates with the same angular velocity as vectors 1 and 2 and performs a harmonic oscillation, occurring along the same straight line as the components 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 = a 1 2 + a 2 2 - 2a 1 a 2 cos

a = a 1 2 + a 2 2 + 2a 1 a 2 cos(j 2 - j 1)(3)

From expression (3) it is clear that the amplitude of the resulting oscillation depends on the difference in the initial phases ( j 2 - j 1) components of vibrations. If the initial phases are equal ( j 2 =j 1), then from formula (3) it is clear that the amplitude A 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 absolute value difference in amplitudes of vibration 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 = а·sinw 1 t,

x 2 = a·sinw 2 t.

It is assumed that w 1 differs little in size from w 2. Adding these expressions, we get:

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

=2a cos[(w 1 -w 2)/2]t sin wt (4)

The resulting movement is a complex oscillation called beats(Fig. 3) Since the value w 1 -w 2 small compared to size w 1 + w 2, then this movement can be considered as a harmonic oscillation with a frequency equal to half the sum of the frequencies of the added oscillations w=(w 1 +w 2)/2, and variable amplitude.

From (4) it follows that the amplitude of the resulting oscillation varies according to periodic law cosine. Full cycle changes in the values ​​of the cosine function occur when the argument changes by 360 0, and the function passes through values ​​from +1 to -1. The state of the system beating at times corresponding to specified values The cosine functions in formula (4) are no different. In other words, beat cycles occur with a periodicity corresponding to a change in the cosine argument in formula (4) by 180 0. Thus, the period T a changes in amplitude during beats (beat period) is determined from the condition:

T a = 2p/(w 1 - w 2).

Considering that w=2pn, we get:

T a = 2 p /2 p (n 1 - n 2) = 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 harmonic vibrations one direction.

Beats

Let us consider an oscillatory system with one degree of freedom, the state of which is determined by the dependence of a certain 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 "offset" oscillatory system from the equilibrium position occurs along one single “direction”, then in this case they speak of the addition of harmonic oscillations of one direction. On the vector diagram, the added oscillations will be depicted in the form of 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 relative position will remain unchanged at any time, and the resulting oscillation will be represented by a vector, equal to the amount vectors and . Adding 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

From the resulting expression it is clear 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 beat of two frequencies.

Having designated , And , we can write

. (6.5)

From expression (6.5) it follows that the resulting oscillation can be represented as a harmonic oscillation with a certain average frequency, the amplitude of which changes slowly (with frequency) over time. Time called beat period, A – beat frequency. The beat graph is shown in Figure 6.2. Beating occurs when the simultaneous sounding of two tuning forks of the same tonality. They can be observed using an oscilloscope by adding the harmonic oscillations of two generators tuned to the same frequency. In both cases, the frequencies of the vibration sources will be slightly different, resulting in beats.

Since oscillations occur with different frequencies, then the phase difference of the added oscillations changes over 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 the specified parameters. 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, in addition to the useful signal, some kind of interference arrives at the input of the device. In fact, the entire variety of electrical signal shapes is the result of the addition of two or more harmonic vibrations.

Addition of vibrations

Addition of two harmonic oscillations with the same amplitudes and frequencies

Let's look at an example sound waves, when two sources create waves with the same amplitudes A and frequencies?. We will install a sensitive membrane at a distance from the sources. When the wave “travels” the distance from the source to the membrane, the membrane will come to oscillatory motion. The effect of each wave on the membrane can be described by the following relationships using oscillatory functions:

x1(t) = A cos(?t + ?1),

x2(t) = A cos(?t + ?2).

x(t) = x1 (t) + x2 (t) = A (1.27)

The expression in parentheses can be written differently using trigonometric function sums of cosines:

In order to simplify function (1.28), we introduce new quantities A0 and?0 that satisfy the condition:

A0 = ?0 = (1.29)

Substituting expressions (1.29) into function (1.28), we obtain

Thus, the sum of harmonic vibrations with the same frequencies? is there a harmonic oscillation of the same frequency?. In this case, the amplitude of the total oscillation A0 and the initial phase?0 are determined by relations (1.29).

Addition of two harmonic oscillations with the same frequency, but different amplitude and initial phase

Now consider the same situation, changing the oscillation amplitudes in function (1.26). For the function x1 (t) we replace the amplitude A with A1, and for the function x2 (t) A with A2. Then functions (1.26) will be written in the following form

x1 (t) = A1 cos(?t + ?1), x2 (t) = A2 cos (?t + ?2); (1.31)

Let us find the sum of harmonic functions (1.31)

x= x1 (t) + x2 (t) = A1 cos(?t + ?1) + A2 cos (?t + ?2) (1.32)

Expression (1.32) can be written differently, using the trigonometric sum cosine function:

x(t) = (A1cos(?1) + A2cos(?2)) cos(?t) - (A1sin(?1) + A2sin(?2)) sin(?t) (1.33)

In order to simplify function (1.33), we introduce new quantities A0 and?0 that satisfy the condition:

Let us square each equation of system (1.34) and add the resulting equations. Then we get the following relation for the number A0:

Let's consider expression (1.35). Let us prove that the quantity under the root cannot be negative. Since cos(?1 - ?2) ? -1, which means that this is the only quantity that can affect the sign of the number under the root (A12 > 0, A22 > 0 and 2A1A2 > 0 (from the definition of amplitude)). Let's consider the critical case (cosine is equal to minus one). Under the root is the formula for the square of the difference, which is always a positive quantity. If we begin to gradually increase the cosine, then the term containing the cosine will also begin to increase, then the value under the root will not change its sign.

Now let’s calculate the relationship for the quantity?0 by dividing the second equation of system (1.34) by the first and calculating the arctangent:

Now let’s substitute the values ​​from system (1.34) into function (1.33)

x = A0(cos(?0) cos?t - sin(?0) sin?t) (1.37)

Transforming the expression in parentheses using the cosine sum formula, we get:

x(t) = A0 cos(?t + ?0) (1.38)

And again it turned out that the sum of two harmonic functions of the form (1.31) is also harmonic function the same type. More precisely, the addition of two harmonic vibrations with the same frequencies? is also a harmonic oscillation with the same frequency?. In this case, the amplitude of the resulting oscillation is determined by relation (1.35), and the initial phase - by relation (1.36).