2 oscillatory movement. Vibrations and waves

There are different types of oscillations in physics, characterized by certain parameters. Consider their main differences, classification according to various factors.

Basic definitions

Oscillation is understood as a process in which, at regular intervals, the main characteristics of the movement have the same values.

Such oscillations are called periodic, in which the values ​​of the basic quantities are repeated at regular intervals (period of oscillations).

Varieties of oscillatory processes

Let us consider the main types of oscillations that exist in fundamental physics.

Free vibrations are those that occur in a system that is not subjected to external variable influences after the initial shock.

As an example free vibrations is a mathematical pendulum.

those kinds mechanical vibrations, which arise in the system under the action of an external variable force.

Features of the classification

By physical nature distinguish the following types of oscillatory movements:

• mechanical;
• thermal;
• electromagnetic;
• mixed.

According to the option of interaction with the environment

Types of vibrations by interaction with environment distinguish several groups.

Forced oscillations appear in the system under the action of an external periodic action. As examples of this type of oscillation, we can consider the movement of hands, leaves on trees.

For forced harmonic oscillations, a resonance may appear, in which at equal values the frequency of the external action and the oscillator with a sharp increase in amplitude.

Own vibrations in the system under the influence of internal forces after it is taken out of equilibrium. The simplest variant of free vibrations is the movement of a load that is suspended on a thread or attached to a spring.

Self-oscillations are called types in which the system has a certain margin potential energy going to make oscillations. hallmark their is the fact that the amplitude is characterized by the properties of the system itself, and not by the initial conditions.

For random oscillations, the external load has a random value.

Basic parameters of oscillatory movements

All types of oscillations have certain characteristics, which should be mentioned separately.

Amplitude is the maximum deviation from the equilibrium position, the deviation of a fluctuating value, it is measured in meters.

The period is the time of one complete oscillation, after which the characteristics of the system are repeated, calculated in seconds.

The frequency is determined by the number of oscillations per unit of time, it is inversely proportional to the period of oscillation.

The oscillation phase characterizes the state of the system.

Characteristic of harmonic vibrations

Such types of oscillations occur according to the law of cosine or sine. Fourier managed to establish that any periodic oscillation can be represented as a sum of harmonic changes by expanding a certain function in

As an example, consider a pendulum having a certain period and cyclic frequency.

What characterizes these types of oscillations? Physics considers an idealized system, which consists of a material point, which is suspended on a weightless inextensible thread, oscillates under the influence of gravity.

Such types of vibrations have a certain amount of energy, they are common in nature and technology.

With prolonged oscillatory motion, the coordinates of its center of mass change, and with alternating current, the value of current and voltage in the circuit changes.

There are different types of harmonic oscillations according to their physical nature: electromagnetic, mechanical, etc.

Shaking acts as a forced vibration vehicle, which moves on a rough road.

The main differences between forced and free vibrations

These types of electromagnetic oscillations differ in physical characteristics. The presence of medium resistance and friction forces lead to damping of free oscillations. In the case of forced oscillations, energy losses are compensated by its additional supply from an external source.

The period of a spring pendulum relates the mass of the body and the stiffness of the spring. In the case of a mathematical pendulum, it depends on the length of the thread.

With a known period, it is possible to calculate the natural frequency of the oscillatory system.

In technology and nature, there are fluctuations with different values frequencies. For example, a pendulum that swings St. Isaac's Cathedral Petersburg, has a frequency of 0.05 Hz, while for atoms it is several million megahertz.

After a certain period of time, the damping of free oscillations is observed. That is why forced oscillations are used in real practice. They are in demand in a variety of vibration machines. The vibratory hammer is a shock-vibration machine, which is intended for driving pipes, piles, and other metal structures into the ground.

Electromagnetic vibrations

Characteristics of vibration modes involves the analysis of the main physical parameters: charge, voltage, current strength. As an elementary system, which is used to observe electromagnetic oscillations, is an oscillatory circuit. It is formed by connecting a coil and a capacitor in series.

When the circuit is closed, free electromagnetic oscillations occur in it, associated with periodic changes electric charge on the capacitor and current in the coil.

They are free due to the fact that when they are performed there is no external influence, but only the energy that is stored in the circuit itself is used.

In the absence of external influence, after a certain period of time, attenuation of the electromagnetic oscillation is observed. The reason for this phenomenon will be the gradual discharge of the capacitor, as well as the resistance that the coil actually has.

That is why damped oscillations occur in a real circuit. Reducing the charge on the capacitor leads to a decrease in the energy value in comparison with its original value. Gradually, it will be released in the form of heat on the connecting wires and the coil, the capacitor will be completely discharged, and the electromagnetic oscillation will be completed.

The Significance of Fluctuations in Science and Technology

Any movements that have a certain degree of repetition are oscillations. For example, a mathematical pendulum is characterized by a systematic deviation in both directions from the original vertical position.

For a spring pendulum, one complete oscillation corresponds to its movement up and down from the initial position.

In an electrical circuit that has capacitance and inductance, there is a repetition of charge on the plates of the capacitor. What is the cause of oscillatory movements? The pendulum functions due to the fact that gravity causes it to return to its original position. In the case of a spring model, a similar function is performed by the elastic force of the spring. Passing the equilibrium position, the load has a certain speed, therefore, by inertia, it moves past the average state.

Electrical oscillations can be explained by the potential difference that exists between the plates of a charged capacitor. Even when it is completely discharged, the current does not disappear, it is recharged.

AT modern technology fluctuations are used, which differ significantly in their nature, degree of repetition, nature, as well as the "mechanism" of occurrence.

Mechanical vibrations are made by the strings of musical instruments, sea waves, and a pendulum. Chemical fluctuations associated with a change in the concentration of reactants are taken into account when conducting various interactions.

Electromagnetic oscillations make it possible to create various technical devices, for example, a telephone, ultrasonic medical devices.

Cepheid brightness fluctuations are of particular interest in astrophysics, and scientists from different countries are studying them.

Conclusion

All types of oscillations are closely related to a huge number of technical processes and physical phenomena. Great is them practical value in aircraft construction, shipbuilding, construction residential complexes, electrical engineering, radio electronics, medicine, fundamental science. An example of a typical oscillatory process in physiology is the movement of the heart muscle. Mechanical vibrations are found in organic and inorganic chemistry, meteorology, and also in many other natural sciences.

The first studies of the mathematical pendulum were carried out in the seventeenth century, and by the end of the nineteenth century, scientists were able to establish the nature of electromagnetic oscillations. The Russian scientist Alexander Popov, who is considered the "father" of radio communications, conducted his experiments precisely on the basis of the theory of electromagnetic oscillations, the results of research by Thomson, Huygens, and Rayleigh. He managed to find practical use electromagnetic waves, use them to transmit a radio signal over a long distance.

Academician P. N. Lebedev for many years conducted experiments related to the production of high-frequency electromagnetic oscillations using alternating electric fields. Through numerous experiments related to various types fluctuations, scientists managed to find areas of their optimal use in modern science and technology.

Lab #3

"Determination of the coefficient of elasticity of a spring using a spring pendulum"

UDC 531.13(07)

The laws of oscillatory motion are considered on the example of a spring pendulum. Guidelines are given for performing laboratory work to determine the coefficient hardness springs by dynamic methods. Dan analysis typical tasks on the topic “Harmonic vibrations. Addition of harmonic vibrations.

Theoretical introduction

Oscillatory motion is one of the most common motions in nature. Sound phenomena, alternating current, electromagnetic waves are associated with it. Oscillations are made by individual parts of a wide variety of machines and devices, atoms and molecules in solids, liquids and gases, heart muscles in humans and animals, etc.

hesitation called a physical process characterized by the repetition in time of the physical quantities associated with this process. The movement of a pendulum or swing, contractions of the heart muscle, alternating current are all examples of systems that oscillate.

Oscillations are considered periodic if the values ​​of physical quantities are repeated at regular intervals, called period T. The number of complete oscillations performed by the system per unit time is called frequency v. Obviously, T = 1/v. Frequency is measured in hertz (Hz). At a frequency of 1 hertz, the system makes 1 oscillation per second.

The simplest type of oscillatory motion is free harmonic vibrations. free, or own are called oscillations that occur in the system after it has been taken out of equilibrium by external forces, which in the future do not take part in the movement of the system. The presence of periodically changing external forces calls in the system forced vibrations.

Harmonic called free oscillations occurring under the action of an elastic force in the absence of friction. According to Hooke's law, at small deformations, the elastic force is directly proportional to the displacement of the body x from the equilibrium position and is directed to the equilibrium position: F ex. = - κx, where κ is the coefficient of elasticity, measured in N/m, and x is the displacement of the body from the equilibrium position.

Forces that are not elastic in nature, but similar in appearance to displacement dependence, are called quasi-elastic(lat. quasi - supposedly). Such forces also cause harmonic oscillations. For example, quasi-elastic forces act on electrons in an oscillatory circuit, causing harmonic electromagnetic oscillations. An example of a quasi-elastic force can also be the gravity component of a mathematical pendulum at small angles of deviation from the vertical.

Harmonic vibration equation. Let the body mass m attached to the end of a spring whose mass is small compared to the mass of the body. An oscillating body is called an oscillator (Latin oscillum - oscillation). Let the oscillator be able to slide freely and without friction along a horizontal guide along which we direct the coordinate axis OX (Fig. 1). The origin of coordinates will be placed at the point corresponding to the equilibrium position of the body (Fig. 1, a). Apply a horizontal force to the body F and shift it from the equilibrium position to the right to the point with coordinate X. The stretching of the spring by an external force causes the appearance of an elastic force F ynp in it. , directed to the equilibrium position (Fig. 1, b). If we now remove the external force F, then under the action of the elastic force the body acquires an acceleration a, moves to the equilibrium position, and the elastic force decreases, becoming equal to zero in the equilibrium position. Having reached the equilibrium position, however, the body does not stop in it and moves to the left due to its kinetic energy. The spring is compressed again, there is an elastic force directed to the right. When the kinetic energy of the body is converted into the potential energy of the compressed spring, the load will stop, then start moving to the right, and the process repeats.

Thus, if during non-periodic motion the body passes each point of the trajectory only once, moving in one direction, then during oscillatory motion for one complete oscillation at each point of the trajectory, except for the most extreme ones, the body happens twice: once moving in the forward direction, the other times in reverse.

Let's write Newton's second law for the oscillator: ma= Fynp. , where

F control = –κ x (1)

The “–” sign in the formula indicates that the displacement and force have opposite directions, in other words, the force acting on the load attached to the spring is proportional to its displacement from the equilibrium position and is always directed towards the equilibrium position. The coefficient of proportionality "κ" is called the coefficient of elasticity. Numerically, it is equal to the force that causes the deformation of the spring, at which its length changes by one. Sometimes it is called hardness coefficient.

Since acceleration is the second derivative of the displacement of the body, this equation can be rewritten as

, or
(2)

Equation (2) can be written as:

, (3)

where both sides of the equation are divided by the mass m and introduced the notation:

(4)

It is easy to check by substitution that the solution satisfies this equation:

x \u003d A 0 cos (ω 0 t + φ 0) , (5)

where A 0 is the amplitude or maximum displacement of the load from the equilibrium position, ω 0 is the angular or cyclic frequency, which can be expressed in terms of a period T natural vibrations by the formula
(see below).

The value φ \u003d φ 0 + ω 0 t (6), which is under the cosine sign and measured in radians, is called oscillation phase at the time t, and φ 0 - initial phase. Phase is a number that determines the magnitude and direction of displacement of the oscillating point at a given time. From (6) it is seen that

. (7)

Thus, the value of ω 0 determines the rate of phase change and is called cyclic frequency. It is associated with ordinary purity by the formula

If the phase changes by 2π radians, then, as is known from trigonometry, the cosine takes its original value, and therefore, the displacement also takes its original value X. But since time changes by one period, it turns out that

ω 0 ( t + T) + φ 0 = (ω 0 t + φ 0) + 2π

Expanding brackets and canceling like terms, we get ω 0 T= 2π or
. But since from (4)
, then we get:
. (9)

In this way, body oscillation period, suspended on a spring, as follows from formula (8), does not depend on the amplitude of oscillations, but depends on the body mass and on the coefficient of elasticity(or hardness) springs.

Differential equation harmonic vibrations:
,

Natural circular frequency oscillations, determined by the nature and parameters of the oscillating system:

–
- for a material point with a mass m, oscillating under the action of a quasi-elastic force, characterized by the coefficient of elasticity (stiffness) k;

–
-for a mathematical pendulum having a length l;

–
- for electromagnetic oscillations in a circuit with a capacitance FROM and inductance L.

IMPORTANT NOTE

These formulas are correct for small deviations from the equilibrium position.

Speed for harmonic vibration:

.

Acceleration for harmonic vibration:

total energy harmonic oscillation:

.

EXPERIMENTAL PART

Exercise 1

Determination of the dependence of the period of natural oscillations of a spring pendulum on the mass of the load

1. Hang a weight from one of the springs and bring the pendulum out of balance by about 1 - 2 cm.

2. After allowing the load to oscillate freely, measure the time interval with a stopwatch t, during which the pendulum will make n (n = 15 - 25) complete oscillations
. Find the period of the pendulum's swing by dividing the amount of time you measured by the number of swings. For greater accuracy, take measurements at least 3 times and calculate the average value of the oscillation period.

Note: Make sure that there are no lateral oscillations of the load, i.e. that the pendulum oscillations are strictly vertical.

3. Repeat measurements with other weights. Record the measurement results in a table.

4. Plot the dependence of the period of oscillation of the pendulum on the mass of the load. The graph will be simpler (straight line) if the values ​​of the mass of goods are plotted on the horizontal axis, and the values ​​of the squared period are plotted on the vertical axis.

Task 2

Determination of the coefficient of elasticity of the spring by the dynamic method

1. Suspend a weight of 100 g from one of the springs, remove it from the equilibrium position by 1 - 2 cm and, having measured the time of 15 - 20 complete oscillations, determine the period of oscillation of the pendulum with the selected load using the formula
. From the formula
calculate the coefficient of elasticity of the spring.

2. Make similar measurements with weights from 150 g to 800 g (depending on the equipment), determine the coefficient of elasticity for each case and calculate the average value of the coefficient of elasticity of the spring. Record the measurement results in a table.

Task 3. According to the results of laboratory work (tasks 1 - 3):

- find the value of the cyclic frequency of the pendulum ω 0 .

– answer the question: does the amplitude of pendulum oscillations depend on the mass of the load.

Take on the graph obtained when executing tasks 1, an arbitrary point and draw perpendiculars from it until it intersects with the axes Om and OT 2. Define values ​​for this point m and T 2 and according to the formula
calculate the value of the coefficient of elasticity of the spring.

Application

BRIEF THEORETICAL INFORMATION

BY ADDITION OF HARMONIC OSCILLATIONS

Amplitude BUT the resulting oscillation obtained by adding two oscillations with the same frequencies and amplitudes A 1 and A 2 occurring along one straight line is determined by the formula

where φ 0, 1, φ 0, 2 - initial phases.

Initial phaseφ 0 of the resulting oscillation can be found by the formula

tg
.

beats arising from the addition of two vibrations x 1 =A cos2π ν 1 t occurring along one straight line with different, but close in value, frequencies ν 1 and ν 2 are described by the formula

x= x 1 + x 2 + 2A cos π (ν 1 - ν 2) t cosπ(ν 1 +ν 2) t.

Trajectory equation point participating in two mutually perpendicular oscillations of the same frequency with amplitudes BUT 1 and BUT 2 and initial phases φ 0, 1 and φ 0, 2:

If the initial phases φ 0, 1 and φ 0, 2 oscillation components are the same, then the trajectory equation takes the form
. If the initial phases differ by π, then the trajectory equation has the form
. These are the equations of straight lines passing through the origin, in other words, in these cases, the point moves in a straight line. In other cases, the movement occurs along an ellipse. With phase difference
the axes of this ellipse are located along the axes OX and OY and the trajectory equation becomes
. Such oscillations are called elliptical. When A 1 \u003d A 2 \u003d A x 2 + y 2 \u003d A 2. This is the equation of a circle, and the vibrations are called circular. For other values ​​of frequencies and phase differences, the trajectory of the oscillating point forms curves of a bizarre shape, called Lissajous figures.

ANALYSIS OF SOME TYPICAL TASKS

ON THE SPECIFIED TOPIC

Task 1. It follows from the graph of oscillations of a material point that the modulus of speed at time t = 1/3 s is ...

The period of the harmonic oscillation shown in the figure is 2 seconds. The amplitude of this oscillation is 18 cm. Therefore, the dependence x(t) can be written as x(t) = 18sin π t. The speed is equal to the derivative of the function X(t) by time v(t) = 18π cos π t. Substituting t = (1/3) s, we get v(1/3) = 9π (cm/s).

Correct is the answer: 9 π cm/s.

Two harmonic oscillations of the same direction are added with the same periods and equal amplitudes A 0 . At the difference
the amplitude of the resulting oscillation is...

The solution is greatly simplified if the vector method for determining the amplitude and phase of the resulting oscillation is used. To do this, we represent one of the added oscillations as a horizontal vector with an amplitude BUT one . From the end of this vector we construct the second vector with amplitude BUT 2 so that it forms an angle
with the first vector. Then the length of the vector drawn from the beginning of the first vector to the end of the last one will be equal to the amplitude of the resulting oscillation, and the angle formed by the resulting vector with the first vector will determine the difference in their phases. Vector diagram corresponding task condition, shown in the figure. This immediately shows that the amplitude of the resulting oscillation in
times the amplitude of each of the summed oscillations.

Correct is the answer:
.

Point M simultaneously oscillates according to the harmonic law along the coordinate axes OH and OY with different amplitudes but the same frequencies. With a phase difference π/2, the trajectory of the point M looks like:

With the phase difference given in the condition, the trajectory equation is ellipse equation, reduced to the coordinate axes, and the semi-axes of the ellipse are equal to the corresponding vibration amplitudes (see theoretical information).

Correct is the answer: 1.

Two identically directed harmonic oscillations of the same period with amplitudes A 1 \u003d 10 cm and A 2 \u003d 6 cm are added into one oscillation with amplitude A res \u003d 14 cm. Phase difference
summed oscillations is equal to...

In this case, it is convenient to use the formula . Substituting the data from the task condition into it, we get:
.

This cosine value corresponds to
.

The correct answer is: .

Test questions

1. What oscillations are called harmonic? 2. What is the form of the graph of undamped harmonic oscillations? 3. What are the values ​​of the harmonic oscillatory process? 4. Give examples of oscillatory movements from biology and veterinary medicine. 5. Write an equation for harmonic oscillations. 6. How to get an expression for the period of the oscillatory motion of a spring pendulum?

LITERATURE

Grabovsky R. I. Course of physics. - M.: graduate School, 2008, part I, § 27-30.

Fundamentals of physics and biophysics. Zhuravlev A. I., Belanovsky A. S., Novikov V. E., Oleshkevich A. A. and others - M., Mir, 2008, ch. 2.

Trofimova T. I. Course of physics: Textbook for students. universities. - M.: MGAVMiB, 2008. - Ch. eighteen.

Trofimova T. I. Physics in tables and formulas: Proc. allowance for university students. - 2nd ed., corrected. - M.: Bustard, 2004. - 432 p.

Along with translational and rotational motion, oscillatory motion plays an important role in the macro- and microworld.

Distinguish between chaotic and periodic oscillations. Periodic oscillations are characterized by the fact that at certain equal intervals of time the oscillating system passes through the same positions. An example is a human cardiogram, which is a record of fluctuations in the electrical signals of the heart (Fig. 2.1). On the cardiogram, one can distinguish oscillation period, those. time T one complete swing. But periodicity is not an exclusive feature of oscillations, it is also possessed by rotary motion. The presence of an equilibrium position is a feature of mechanical oscillatory motion, while rotation is characterized by the so-called indifferent equilibrium (a well-balanced wheel or a gambling roulette, being spun, stops in any position with equiprobability). With mechanical vibrations in any position, except for the equilibrium position, there is a force that tends to return the oscillating system to its initial position, i.e. restoring force, always directed towards the equilibrium position. The presence of all three features distinguishes mechanical vibration from other types of motion.

Rice. 2.1.

Consider specific examples of mechanical vibrations.

We clamp one end of the steel ruler in a vice, and take the other, free, to the side and release it. Under the action of elastic forces, the ruler will return to its original position, which is the equilibrium position. Passing through this position (which is the equilibrium position), all points of the ruler (except for the clamped part) will have a certain speed and a certain amount of kinetic energy. By inertia, the oscillating part of the ruler will pass the equilibrium position and will do work against the internal elastic forces due to a decrease in kinetic energy. This will lead to an increase in the potential energy of the system. When the kinetic energy is completely exhausted, the potential energy will reach a maximum. The elastic force acting on each oscillating point will also reach a maximum and will be directed towards the equilibrium position. This is described in subsections 1.2.5 (relation (1.58)), 1.4.1, and also in 1.4.4 (see Fig. 1.31) in the language of potential curves. This will be repeated until the total mechanical energy of the system is converted into internal energy (the energy of particle oscillations solid body) and will not dissipate into the surrounding space (recall that the resistance forces are dissipative forces).

Thus, in the motion under consideration there is a repetition of states and there are forces (elasticity forces) tending to return the system to the equilibrium position. Therefore, the ruler will oscillate.

Another well-known example is the oscillation of a pendulum. The equilibrium position of the pendulum corresponds to the lowest position of its center of gravity (in this position, the potential energy due to gravity is minimal). In a deflected position, a moment of force about the axis of rotation will act on the pendulum, tending to return the pendulum to its equilibrium position. In this case, there are also all signs of oscillatory motion. It is clear that in the absence of gravity (in a state of weightlessness), the above conditions will not be met: in a state of weightlessness, there is no gravity and the restoring moment of this force. And here the pendulum, having received a push, will move in a circle, that is, it will not oscillate, but rotate.

Vibrations can be not only mechanical. So, for example, we can talk about charge fluctuations on the plates of a capacitor connected in parallel with an inductor (in an oscillatory circuit), or the electric field strength in a capacitor. Their change over time is described by the equation, like that, which determines the mechanical displacement from the equilibrium position of the pendulum. In view of the fact that the same equations can describe the oscillations of the most diverse physical quantities, it turns out to be very convenient to consider the oscillations regardless of which physical quantity fluctuates. This gives rise to a system of analogies, in particular, an electromechanical analogy. For definiteness, we will consider mechanical vibrations for the time being. Only periodic fluctuations are subject to consideration, in which the values ​​of physical quantities changing in the process of fluctuations are repeated at regular intervals.

The reciprocal of a period T oscillations (as well as the time of one complete revolution during rotation), expresses the number of complete oscillations per unit time, and is called frequency(it's just a frequency, it's measured in hertz or s -1)

(with oscillations in the same way as with rotational motion).

The angular velocity is related to the frequency v introduced by relation (2.1) by the formula

measured in rad/s or s -1 .

It is natural to begin the analysis of oscillatory processes with the simplest cases of oscillatory systems with one degree of freedom. Number of degrees of freedom is the number of independent variables needed to completely determine the position in space of all parts of a given system. If, for example, the oscillations of a pendulum (a load on a thread, etc.) are limited to a plane in which the pendulum can only move, and if the pendulum thread is inextensible, then it is sufficient to set only one angle of deviation of the thread from the vertical or only the amount of displacement from the equilibrium position - for a load oscillating along one direction on a spring to fully determine its position. In this case, we say that the system under consideration has one degree of freedom. The same pendulum, if it can occupy any position on the surface of the sphere on which the trajectory of its motion lies, has two degrees of freedom. Three-dimensional vibrations are also possible, as is the case, for example, with thermal vibrations of atoms in a crystal lattice (see subsection 10.3). To analyze the process in a real physical system, we choose its model, limiting the study in advance to a number of conditions.

• Hereinafter, the oscillation period will be denoted by the same letter as the kinetic energy - T (do not confuse!).
• Chapter 4 " Molecular physics» another definition of the number of degrees of freedom will be given.

Oscillation characteristic

Phase determines the state of the system, namely the coordinate, speed, acceleration, energy, etc.

Cyclic frequency characterizes the rate of change of the oscillation phase.

The initial state of the oscillatory system characterizes initial phase

Oscillation amplitude A is the largest displacement from the equilibrium position

Period T- this is the period of time during which the point performs one complete oscillation.

Oscillation frequency is the number of complete oscillations per unit time t.

The frequency, cyclic frequency and oscillation period are related as

Types of vibrations

Vibrations that occur in closed systems are called free or own fluctuations. Vibrations that occur under the influence of external forces are called forced. There are also self-oscillations(forced automatically).

If we consider oscillations according to changing characteristics (amplitude, frequency, period, etc.), then they can be divided into harmonic, fading, growing(as well as sawtooth, rectangular, complex).

During free vibrations in real systems, energy losses always occur. Mechanical energy is expended, for example, to perform work to overcome the forces of air resistance. Under the influence of the friction force, the oscillation amplitude decreases, and after a while the oscillations stop. It is obvious that the greater the force of resistance to movement, the faster the oscillations stop.

Forced vibrations. Resonance

Forced oscillations are undamped. Therefore, it is necessary to replenish energy losses for each period of oscillation. To do this, it is necessary to act on an oscillating body with a periodically changing force. Forced oscillations are performed with a frequency equal to the frequency of changes in the external force.

Forced vibrations

The amplitude of forced mechanical oscillations reaches the greatest value in the event that the frequency of the driving force coincides with the frequency of the oscillating system. This phenomenon is called resonance.

For example, if you periodically pull the cord in time with its own oscillations, then we will notice an increase in the amplitude of its oscillations.

If a wet finger is moved along the edge of the glass, the glass will make ringing sounds. Although not noticeable, the finger moves intermittently and transfers energy to the glass in short bursts, causing the glass to vibrate.

The walls of the glass also begin to vibrate when directed at it. sound wave with a frequency equal to its own. If the amplitude becomes very large, then the glass may even break. Due to the resonance during the singing of F.I. Chaliapin, the crystal pendants of the chandeliers trembled (resonated). The emergence of resonance can be traced in the bathroom. If you sing sounds of different frequencies softly, then resonance will occur at one of the frequencies.

AT musical instruments the role of resonators is performed by parts of their cases. A person also has his own resonator - this is the oral cavity, which amplifies the sounds made.

The phenomenon of resonance must be taken into account in practice. In some situations it can be useful, in others it can be harmful. Resonant phenomena can cause irreversible damage to various mechanical systems, such as improperly designed bridges. So, in 1905, the Egyptian bridge in St. Petersburg collapsed when an equestrian squadron passed through it, and in 1940, the Tacoma bridge in the USA collapsed.

The resonance phenomenon is used when, with the help of a small force, it is necessary to obtain a large increase in the amplitude of oscillations. For example, the heavy tongue of a large bell can be swung by a relatively small force with a frequency equal to the natural frequency of the bell.

Therefore, the generalized theory of oscillations and waves is engaged in the study of these patterns. The fundamental difference from waves: during vibrations, there is no transfer of energy, these are, so to speak, “local” transformations.

Classification

Selection different types oscillations depends on the emphasized properties of systems with oscillatory processes (oscillators).

According to the mathematical apparatus used

• Nonlinear vibrations

By frequency

Thus, periodic oscillations are defined as follows:

By physical nature

• Mechanical(sound, vibration)
• electromagnetic(light, radio waves, heat)
• mixed type- combinations of the above

By the nature of interaction with the environment

• Forced- fluctuations occurring in the system under the influence of external periodic influence. Examples: leaves on trees, raising and lowering a hand. With forced oscillations, a resonance phenomenon may occur: a sharp increase in the amplitude of oscillations when the natural frequency of the oscillator coincides with the frequency of the external influence.
• Free (or own)- these are oscillations in the system under the action of internal forces after the system is taken out of equilibrium (in real conditions, free oscillations are always damped). The simplest examples of free vibrations are the vibrations of a load attached to a spring, or a load suspended from a thread.
• Self-oscillations- oscillations in which the system has a reserve of potential energy spent on oscillations (an example of such a system is a mechanical watch). A characteristic difference self-oscillations from forced oscillations is that their amplitude is determined by the properties of the system itself, and not by the initial conditions.
• Parametric- fluctuations that occur when any parameter of the oscillatory system changes as a result of external influence.

Options

Oscillation period T (\displaystyle T\,\ !} and frequency f (\displaystyle f\,\ !}- reciprocal values;

In circular or cyclic processes, instead of the "frequency" characteristic, the concept is used circular (cyclic) frequency ω (\displaystyle \omega \,\ !} (rad/s, Hz, s −1), showing the number of oscillations per 2 π (\displaystyle 2\pi ) units of time:

• Bias- deviation of the body from the equilibrium position. Designation X, Unit of measure - meter.
• Oscillation phase- determines the displacement at any time, that is, determines the state of the oscillatory system.

Short story

Harmonic vibrations have been known since the 17th century.

The term "relaxation oscillations" was proposed in 1926 by van der Pol. The introduction of such a term was justified only by the circumstance that all such fluctuations seemed to the specified researcher to be associated with the presence of "relaxation time" - that is, with the concept that at that historical moment in the development of science seemed the most understandable and widespread. The key property of the new type of oscillations described by a number of the researchers listed above was that they differed significantly from linear ones, which manifested itself primarily as a deviation from the well-known Thomson formula. Careful historical research showed that van der Pol in 1926 was not yet aware of the fact that the physical phenomenon“relaxation oscillations” corresponds to the mathematical concept introduced by Poincaré “limit cycle”, and he understood this only after the publication of A. A. Andronov, published in 1929.

Foreign researchers recognize the fact that among Soviet scientists the students of L. I. Mandelstam gained world fame, who published in 1937 the first book in which they summarized modern information about linear and non-linear oscillations. However, Soviet scientists did not accept the term "relaxation oscillations" proposed by van der Pol. They preferred the term "discontinuous motion" used by Blondel, in part because it was intended to describe these oscillations in terms of slow and fast regimes. This approach has become mature only in the context of singular perturbation theory.» .

Brief description of the main types of oscillatory systems

Linear vibrations

An important type of oscillations are harmonic oscillations - oscillations that occur according to the law of sine or cosine. As Fourier established in 1822, any periodic oscillation can be represented as the sum of harmonic oscillations by expanding the corresponding function into