Oscillatory processes are an important element of modern science and technology, therefore, their study has always been given attention as one of the “eternal” problems. The task of any knowledge is not a simple curiosity, but its use in everyday life. And for this, new technical systems and mechanisms exist and appear every day. They are in motion, show their essence, performing some kind of work, or, being motionless, retain the potential under certain conditions to go into a state of motion. And what is movement? Without delving into the jungle, we will accept the simplest interpretation: a change in the position of a material body relative to any coordinate system, which is conventionally considered motionless.

Among the huge number of possible variants of motion, of particular interest is oscillatory, which differs in that the system repeats the change in its coordinates (or physical quantities) at regular intervals - cycles. Such fluctuations are called periodic or cyclical. Among them, a separate class is distinguished in which characteristic features (speed, acceleration, position in space, etc.) change in time according to a harmonic law, i.e. having a sinusoidal appearance. A remarkable property of harmonic vibrations is that their combination represents any other options, incl. and inharmonic. A very important concept in physics is the "phase of oscillations", which means fixing the position of an oscillating body at a certain moment in time. The phase is measured in angular units - radians, rather arbitrarily, just as a convenient technique for explaining periodic processes. In other words, the phase determines the value of the current state of the oscillatory system. It cannot be otherwise - after all, the phase of the oscillations is the argument of the function that describes these oscillations. The true value of the phase for a character can mean coordinates, speed and other physical parameters that vary according to a harmonic law, but they have a common time dependence.

Demonstrating oscillations is not at all difficult - for this you need a simple mechanical system - a thread of length r, and a "material point" suspended from it - a weight. Let's fix the thread in the center of the rectangular coordinate system and make our "pendulum" spin. Let us assume that he willingly does this with an angular velocity w. Then, in time t, the angle of rotation of the load will be φ = wt. Additionally, this expression should take into account the initial phase of oscillations in the form of an angle φ0 - the position of the system before the start of movement. So, the total angle of rotation, phase, is calculated from the relation φ = wt + φ0. Then the expression for the harmonic function, and this is the projection of the coordinate of the load on the X-axis, can be written:

x = A * cos (wt + φ0), where A is the vibration amplitude, in our case equal to r - the radius of the thread.

Similarly, the same projection onto the Y-axis will be written as follows:

y = A * sin (wt + φ0).

It should be understood that the phase of the oscillation does not mean in this case a measure of rotation "angle", but an angular measure of time, which expresses time in units of angle. During this time, the load rotates through a certain angle, which can be unambiguously determined based on the fact that for a cyclic oscillation w = 2 * π / Т, where Т is the oscillation period. Therefore, if one period corresponds to a rotation of 2π radians, then part of the period, time, can be proportionally expressed by the angle as a fraction of the total rotation of 2π.

Oscillations do not exist by themselves - sounds, light, vibration are always a superposition, an overlap, of a large number of vibrations from different sources. Undoubtedly, the result of superposition of two or more oscillations is influenced by their parameters, incl. and the phase of the oscillations. The formula for the total oscillation, as a rule, is inharmonic, and it can have a very complex form, but this only makes it more interesting. As mentioned above, any non-harmonic vibration can be represented as a large number of harmonic ones with different amplitudes, frequencies and phases. In mathematics, such an operation is called "function expansion in a series" and is widely used in calculations, for example, the strength of structures and structures. The basis of such calculations is the study of harmonic oscillations taking into account all parameters, including the phase.

Another characteristic of harmonic vibrations is the vibration phase.

As we already know, for a given vibration amplitude, at any moment in time we can determine the coordinate of the body. It will be uniquely specified by the argument of the trigonometric function φ = ω0 * t. The quantity φ, which stands under the sign of the trigonometric function, called the phase of oscillation.

For phase, the units are radians. The phase unambiguously determines not only the ted coordinate at any given time, but also the speed or acceleration. Therefore, it is believed that the phase of the oscillations determines the state of the oscillatory system at any moment of time.

Of course, provided that the vibration amplitude is set. Two oscillations with the same frequency and period of oscillation can differ from each other in phases.

• φ = ω0 * t = 2 * pi * t / T.

If we express the time t in the number of periods that have passed from the beginning of the oscillations, then any value of the time t corresponds to the value of the phase, expressed in radians. For example, if we take the time t = T / 4, then this value will correspond to the value of the phase pi / 2.

Thus, we can plot the dependence of the coordinate not on time, but on the phase, and we will get exactly the same dependence. The following figure shows such a graph.

Initial phase of oscillation

When describing the coordinates of the oscillatory motion, we used the sine and cosine functions. For the cosine, we wrote the following formula:

• x = Xm * cos (ω0 * t).

But we can describe the same trajectory with the help of sine. In this case, we need to shift the argument by pi / 2, that is, the difference between sine and cosine is pi / 2 or a quarter of the period.

• x = Xm * sin (ω0 * t + pi / 2).

The pi / 2 value is called the initial phase of the oscillation. The initial phase of the oscillation is the position of the body at the initial moment of time t = 0. In order to force the pendulum to oscillate, we must remove it from the equilibrium position. We can do this in two ways:

• Take him aside and let him go.
• Hit him.

In the first case, we immediately change the coordinate of the body, that is, at the initial moment of time, the coordinate will be equal to the value of the amplitude. To describe such an oscillation, it is more convenient to use the cosine function and the form

• x = Xm * cos (ω0 * t),

or the formula

• x = Xm * sin (ω0 * t + & phi),

where φ is the initial phase of the oscillation.

If we hit the body, then at the initial moment of time its coordinate is equal to zero, and in this case it is more convenient to use the form:

• x = Xm * sin (ω0 * t).

Two oscillations that differ only in the initial phase are called phase-shifted.

For example, for vibrations described by the following formulas:

• x = Xm * sin (ω0 * t),
• x = Xm * sin (ω0 * t + pi / 2),

the phase shift is pi / 2.

Phase displacement is also sometimes called phase difference.

But since the turns are shifted in space, then the EMF induced in them will not reach the amplitude and zero values ​​at the same time.

At the initial moment of time, the EMF of the loop will be:

In these expressions, the angles are called phase , or phase ... Angles and are called initial phase ... The phase angle determines the value of the EMF at any moment in time, and the initial phase determines the value of the EMF at the initial moment of time.

The difference between the initial phases of two sinusoidal quantities of the same frequency and amplitude is called phase angle

Dividing the phase angle by the angular frequency, we get the time elapsed since the beginning of the period:

Graphical representation of sinusoidal values

U = (U 2 a + (U L - U c) 2)

Thus, due to the presence of the phase angle, the voltage U is always less than the algebraic sum U a + U L + U C. The difference U L - U C = U p is called reactive voltage component.

Consider how the current and voltage change in a series AC circuit.

Impedance and phase angle. If we substitute in the formula (71) the values ​​U a = IR; U L = lL and U C = I / (C), then we will have: U = ((IR) 2 + 2), whence we obtain the Ohm's law formula for a series alternating current circuit:

I = U / ((R 2 + 2)) = U / Z (72)

where Z = (R 2 + 2) = (R 2 + (X L - X c) 2)

The quantity Z is called circuit impedance, it is measured in ohms. The difference L - l / (C) is called circuit reactance and denoted by the letter X. Therefore, the total resistance of the circuit

Z = (R 2 + X 2)

The relationship between the active, reactive and impedances of an alternating current circuit can also be obtained by the Pythagorean theorem from the triangle of resistances (Fig. 193). The triangle of resistances A'B'S 'can be obtained from the voltage triangle ABC (see Fig. 192, b), if we divide all its sides by current I.

The phase angle is determined by the ratio between the individual resistances included in the circuit. From triangle А'В'С (see fig. 193) we have:

sin? = X / Z; cos? = R / Z; tg? = X / R

For example, if the resistance R is significantly greater than the reactance X, the angle is relatively small. If there is a large inductive or large capacitive resistance in the circuit, then the phase angle increases and approaches 90 °. Wherein, if the inductive reactance is greater than the capacitive one, the voltage is ahead of the current i by an angle; if the capacitive resistance is greater than the inductive one, then the voltage lags behind the current i by an angle.

Ideal inductor, real coil and capacitor in AC circuit.

A real coil, in contrast to an ideal one, has not only inductance, but also an active resistance, therefore, when an alternating current flows in it, it is accompanied not only by a change in energy in a magnetic field, but also by the conversion of electrical energy into another form. In particular, in the coil wire, electrical energy is converted into heat in accordance with the Lenz-Joule law.

It was previously found that in an alternating current circuit, the process of converting electrical energy into another form is characterized by active power of the circuit P , and the change in energy in a magnetic field is reactive power Q .

In a real coil, both processes take place, i.e., its active and reactive powers are different from zero. Therefore, one real coil in the equivalent circuit must be represented by active and reactive elements.

Fluctuations are called movements or processes that are characterized by a certain repetition in time. Oscillations are widespread in the surrounding world and can be of a very different nature. It can be mechanical (pendulum), electromagnetic (oscillatory circuit) and other types of oscillations. Free, or own vibrations are called vibrations that occur in a system left to itself, after it has been brought out of equilibrium by an external influence. An example is the vibrations of a ball suspended from a thread. Harmonic vibrations such oscillations are called in which the oscillating quantity changes from time to time according to the law sinus or cosine . Harmonic Equation looks like:, where A - vibration amplitude (the value of the greatest deviation of the system from the equilibrium position); - circular (cyclic) frequency. Periodically changing cosine argument - called phase of oscillation ... The oscillation phase determines the displacement of the oscillating quantity from the equilibrium position at a given time t. The constant φ is the phase value at the time t = 0 and is called the initial phase of the oscillation .. This period of time T is called the period of harmonic oscillations. The period of harmonic oscillations is : T = 2π /. Mathematical pendulum- an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. Period of small natural oscillations of a mathematical pendulum of length L motionlessly suspended in a homogeneous gravity field with the acceleration of gravity g is equal to

and does not depend on the amplitude of oscillations and the mass of the pendulum. Physical pendulum- An oscillator, which is a solid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of action of the forces and does not pass through the center of mass of this body.

24. Electromagnetic vibrations. Oscillatory circuit. Thomson's formula.

Electromagnetic vibrations- these are fluctuations in electric and magnetic fields, which are accompanied by periodic changes in charge, current and voltage. The simplest system where free electromagnetic oscillations can arise and exist is an oscillatory circuit. Oscillatory circuit- this is a circuit consisting of an inductor and a capacitor (Fig. 29, a). If the capacitor is charged and closed to the coil, then a current will flow through the coil (Fig. 29, b). When the capacitor is discharged, the current in the circuit will not stop due to self-induction in the coil. The induction current, in accordance with Lenz's rule, will have the same direction and will recharge the capacitor (Fig. 29, c). The process will be repeated (Fig. 29, d) by analogy with the oscillations of pendulums. Thus, electromagnetic oscillations will occur in the oscillatory circuit due to the transformation of the energy of the electric field of the capacitor () into the energy of the magnetic field of the coil with current (), and vice versa. The period of electromagnetic oscillations in an ideal oscillatory circuit depends on the inductance of the coil and the capacitance of the capacitor and is found by the Thomson formula. Frequency and period are inversely related.

When studying this section, it should be borne in mind that hesitation of different physical nature are described from a unified mathematical standpoint. Here it is necessary to clearly understand such concepts as harmonic oscillation, phase, phase difference, amplitude, frequency, oscillation period.

It should be borne in mind that in any real oscillatory system there are resistances of the medium, i.e. oscillations will be damped. To characterize the damping of oscillations, the damping coefficient and the logarithmic damping decrement are introduced.

If vibrations are performed under the influence of an external, periodically changing force, then such vibrations are called forced. They will be continuous. The amplitude of the forced vibrations depends on the frequency of the driving force. When the frequency of forced vibrations approaches the frequency of natural vibrations, the amplitude of the forced vibrations increases sharply. This phenomenon is called resonance.

Moving on to the study of electromagnetic waves, one must clearly understand thatelectromagnetic waveis an electromagnetic field propagating in space. The simplest system that emits electromagnetic waves is an electric dipole. If the dipole performs harmonic oscillations, then it emits a monochromatic wave.

Formula table: oscillations and waves