One of the most interesting topics in physics is oscillations. The study of mechanics is closely connected with them, with how bodies behave, which are affected by certain forces. So, studying oscillations, we can observe pendulums, see the dependence of the oscillation amplitude on the length of the thread on which the body hangs, on the stiffness of the spring, and the weight of the load. Despite the apparent simplicity, this topic is far from being given to everyone as easily as we would like. Therefore, we decided to collect the most well-known information about oscillations, their types and properties, and compile for you a brief summary on this topic. Perhaps it will be useful to you.

Concept definition

Before talking about such concepts as mechanical, electromagnetic, free, forced vibrations, about their nature, characteristics and types, conditions of occurrence, this concept should be defined. So, in physics, oscillation is a constantly repeating process of changing the state around one point in space. The simplest example is a pendulum. Each time it oscillates, it deviates from a certain vertical point, first in one direction, then in the other direction. Engaged in the study of the phenomenon of the theory of oscillations and waves.

Causes and conditions of occurrence

Like any other phenomenon, fluctuations occur only if certain conditions are met. Mechanical forced vibrations, as well as free vibrations, arise when the following conditions are met:

1. The presence of a force that brings the body out of a state of stable equilibrium. For example, the push of a mathematical pendulum, at which the movement begins.

2. The presence of a minimum friction force in the system. As you know, friction slows down certain physical processes. The greater the friction force, the less likely the oscillations to occur.

3. One of the forces must depend on the coordinates. That is, the body changes its position in a certain coordinate system relative to a certain point.

Types of vibrations

Having dealt with what oscillation is, we will analyze their classification. There are two most famous classifications - by physical nature and by the nature of interaction with the environment. So, according to the first sign, mechanical and electromagnetic are distinguished, and according to the second - free and forced vibrations. There are also self-oscillations, damped oscillations. But we will only talk about the first four types. Let's take a closer look at each of them, find out their features, and also give a very brief description of their main characteristics.

Mechanical

It is with mechanical that the study of oscillations in the school course of physics begins. Students begin their acquaintance with them in such a branch of physics as mechanics. Note that these physical processes occur in the environment, and we can observe them with the naked eye. With such vibrations, the body repeatedly performs the same movement, passing through a certain position in space. Examples of such oscillations are the same pendulums, the vibration of a tuning fork or a guitar string, the movement of leaves and branches on a tree, a swing.

electromagnetic

After such a concept as mechanical oscillations is firmly mastered, the study of electromagnetic oscillations begins, which are more complex in structure, since this type occurs in various electrical circuits. In this process, oscillations are observed in electric as well as magnetic fields. Despite the fact that electromagnetic oscillations have a slightly different nature of occurrence, the laws for them are the same as for mechanical ones. With electromagnetic oscillations, not only the strength of the electromagnetic field can change, but also such characteristics as the strength of the charge and current. It is also important to note that there are free and forced electromagnetic oscillations.

Free vibrations

This type of oscillation occurs under the influence of internal forces when the system is taken out of a state of stable equilibrium or rest. Free oscillations are always damped, which means that their amplitude and frequency decrease with time. A striking example of this type of rocking is the movement of a load suspended on a thread and oscillating from one side to the other; a load attached to a spring, then falling down under the action of gravity, then rising up under the action of the spring. By the way, it is precisely this kind of oscillations that is paid attention to in the study of physics. Yes, and most of the tasks are devoted just to free vibrations, and not to forced ones.

Forced

Despite the fact that this kind of process is not studied in such detail by schoolchildren, it is forced oscillations that are most often encountered in nature. A rather striking example of this physical phenomenon can be the movement of branches on trees in windy weather. Such fluctuations always occur under the influence of external factors and forces, and they arise at any moment.

Oscillation characteristics

Like any other process, oscillations have their own characteristics. There are six main parameters of the oscillatory process: amplitude, period, frequency, phase, displacement and cyclic frequency. Naturally, each of them has its own designations, as well as units of measurement. Let's analyze them in a little more detail, dwelling on a brief description. At the same time, we will not describe the formulas that are used to calculate a particular value, so as not to confuse the reader.

Bias

The first one is displacement. This characteristic shows the deviation of the body from the equilibrium point at a given time. It is measured in meters (m), the common designation is x.

Oscillation amplitude

This value denotes the greatest displacement of the body from the equilibrium point. In the presence of undamped oscillation is a constant value. It is measured in meters, the generally accepted designation is x m.

Oscillation period

Another value that denotes the time for which one complete oscillation takes place. The generally accepted designation is T, measured in seconds (s).

Frequency

The last characteristic we will talk about is the oscillation frequency. This value indicates the number of oscillations in a certain period of time. It is measured in hertz (Hz) and is denoted as ν.

Types of pendulums

So, we have analyzed forced oscillations, talked about free ones, which means that we should also mention the types of pendulums that are used to create and study free oscillations (in school conditions). There are two types - mathematical and harmonic (spring). The first is a body suspended from an inextensible thread, the size of which is equal to l (the main significant value). The second is a weight attached to a spring. Here it is important to know the mass of the load (m) and the stiffness of the spring (k).

findings

So, we figured out that there are mechanical and electromagnetic oscillations, gave their brief description, described the causes and conditions for the occurrence of these types of oscillations. We said a few words about the main characteristics of these physical phenomena. We also figured out that there are forced vibrations and free ones. Determine how they differ from each other. In addition, we said a few words about pendulums used in the study of mechanical oscillations. We hope this information was useful to you.

General oscillation characteristic

Rhythmic processes of any nature, characterized by repetition in time, are called oscillations.

Oscillation is a process characterized by the repeatability in time of the parameters that describe it. The unity of the regularities of rhythmic processes made it possible to develop a single mathematical apparatus for their description - the theory of oscillations. There are many features by which fluctuations can be classified.

By physical nature oscillating system distinguish between mechanical and electromagnetic oscillations.

The fluctuations are called periodic, if the value characterizing the state of the system is repeated at regular intervals - the period of oscillation.

Period (T) - the minimum time after which the state of the oscillatory system repeats, i.e. the time of one complete oscillation.

For such fluctuations

x(t)=x(t+T);(3. 1)

Periodic are the oscillations of the pendulum of the clock, alternating current, the beating of the heart, and the oscillations of trees under a gust of wind, foreign exchange rates are not periodic.

In addition to the period, in the case of periodic oscillations, their frequency is determined.

Frequency()those. number of oscillations per unit of time.

Frequency is the reciprocal of the oscillation period,

The frequency unit is Hertz: 1 Hz \u003d 1 s -1, the frequency corresponding to one oscillation per second. When describing periodic oscillations, one also uses cyclic frequency– number of oscillations for 2 π seconds:

With periodic oscillations, these parameters are constant, while with other oscillations they can change.

The law of oscillations - the dependence of a fluctuating quantity on time x(t)- may be different. The simplest are harmonic fluctuations (Fig. 3.1), for which the fluctuating value changes according to the sine or cosine law, which allows using one function to describe the process in time:

Here: x(t) - the value of the fluctuating value at a given time t, BUT – amplitude- the largest deviation of the oscillating value from the average value., ω - cyclic frequency, ( ωt+φ) – oscillation phase, φ - initial phase.

Many well-known oscillatory processes obey the harmonic law. including mentioned above, but most importantly, with the help of Fourier method any periodic function decomposing into harmonic components ( harmonics) with multiple frequencies:

f(t)= BUT + BUT 1 cos( t + )+ BUT cos(2t+ )+…; (3.5)

Here the main frequency is determined by the period of the process: .

Each harmonic is characterized by frequency () and amplitude ( BUT). The set of harmonics is called spectrum. The spectra of periodic oscillations are discrete (linear) (Fig. 3.1a), and not periodic continuous (Fig. 3.1b).

Rice. 3.1 Discrete (a) and continuous (b) spectra of complex vibrational

Types of vibrations

The oscillatory system has a certain energy, due to which vibrations are made. The energy depends on the amplitude and frequency of oscillations.

Oscillations are divided into the following types: free or natural, damped, forced, self-oscillations.

Free oscillations occur in a system that is once taken out of equilibrium and then left to itself. In this case, oscillations occur with own frequency (), which does not depend on their amplitude, i.e. determined by the properties of the system itself.

In real conditions, fluctuations are always fading, i.e. energy decreases over time due to its dissipation and as a result, the amplitude of oscillations decreases. Dissipation is an irreversible transition of a part of the energy of ordered processes (“order energy”) into the energy of disordered processes (“chaos energy”). Dissipation occurs in any oscillating open system.

To create undamped oscillations in real systems, a periodic external action is necessary - a periodic replenishment of the energy lost due to dissipation. Harmonic oscillations that occur due to external periodic influences (“driving force”) are called forced. Their frequency coincides with the frequency of the driving force (), and the amplitude turns out to depend on the ratio between the frequency of the force and the natural frequency of the system. The most important effect that occurs during forced oscillations is resonance– a sharp increase in amplitude when the frequency of forced oscillations approaches the natural frequency of the oscillatory system. The resonant frequency is the closer to its own, and the maximum amplitude is the greater, the less dissipation.

Self-oscillations are undamped oscillations that occur due to an energy source, the type and operation of which is determined by the oscillatory system itself. With self-oscillations, the main characteristics - amplitude, frequency - are determined by the system itself. This distinguishes these oscillations from both forced ones, in which these parameters depend on external influences, and from natural ones, in which the external influence sets the oscillation amplitude. The simplest self-oscillating system includes:

oscillatory system (with damping),

oscillation amplifier (energy source),

non-linear limiter (valve),

feedback link

With self-oscillations, nonlinearity is important for their establishment, which controls the input and output of source energy, and allows you to set oscillations of a certain amplitude. Examples of self-oscillating systems are: mechanical - pendulum clock, thermodynamic - heat engine, electromagnetic - tube generator, optical - laser (optical quantum generator). The laser scheme is shown in Fig. 4.5. Here the oscillatory system is an optically active medium that fills the optical resonator, there is an external energy source that provides the "pumping" process, a valve and feedback - a translucent mirror at the output of the optical resonator, the nonlinearity is determined by the conditions of stimulated emission.

In all self-oscillatory systems, feedback regulates the inclusion of an external source and the energy supply to the oscillatory system: as long as the energy input (contribution) is higher than the loss, self-excitation (buildup) occurs, oscillations in the system increase; when the energy loss equals the energy gain, the valve closes. The system oscillates in a stationary mode with a constant amplitude; as the loss increases, the amplitude decreases and the valve opens again, the contribution increases, the amplitude is restored, and the valve closes.

Much of physics sometimes remains incomprehensible. And it's not always that a person just read a little on this topic. Sometimes the material is given in such a way that it is simply impossible for a person who is not familiar with the basics of physics to understand it. One rather interesting section that people do not always understand the first time and are able to comprehend is periodic oscillations. Before explaining the theory of periodic oscillations, let's talk a little about the history of the discovery of this phenomenon.

Story

The theoretical foundations of periodic oscillations were known in the ancient world. People saw how the waves move evenly, how the wheels rotate, passing through the same point after a certain period of time. It is from these seemingly simple phenomena that the concept of oscillations originated.

The first evidence of the description of oscillations has not been preserved, however, it is known for certain that one of their most common types (namely, electromagnetic) was theoretically predicted by Maxwell in 1862. After 20 years, his theory was confirmed. Then he conducted a series of experiments proving the existence of electromagnetic waves and the presence of certain properties that are unique to them. As it turned out, light is also an electromagnetic wave and obeys all relevant laws. A few years before Hertz, there was a man who demonstrated to the scientific community the generation of electromagnetic waves, but due to the fact that he was not strong in theory as well as Hertz, he could not prove that the success of the experiment was explained precisely by oscillations.

We've gone off topic a bit. In the next section, we will consider the main examples of periodic oscillations that we can meet in everyday life and in nature.

Kinds

These phenomena occur everywhere and all the time. And besides the waves and rotation of the wheels already cited as an example, we can notice periodic fluctuations in our body: contractions of the heart, movement of the lungs, and so on. If you zoom in and move on to larger objects than our organs, you can see fluctuations in such a science as biology.

An example would be periodic fluctuations in the number of populations. What is the meaning of this phenomenon? In any population, there is always an increase, then a decrease. And this is due to various factors. Due to the limited space and many other factors, the population cannot grow indefinitely, therefore, with the help of natural mechanisms, nature has learned to reduce the number. At the same time, periodic fluctuations in numbers occur. The same thing happens with human society.

Now let's discuss the theory of this concept and analyze a few formulas relating to such a concept as periodic oscillations.

Theory

Periodic oscillations are a very interesting topic. But, as in any other, the further you dive - the more incomprehensible, new and complex. In this article we will not go deep, we will only briefly describe the main properties of oscillations.

The main characteristics of periodic oscillations are the period and the frequency shows how long it takes the wave to return to its original position. In fact, this is the time it takes a wave to travel the distance between its adjacent crests. There is another value that is closely related to the previous one. This is the frequency. The frequency is the inverse of the period and has the following physical meaning: it is the number of wave crests that have passed through a certain area of ​​space per unit of time. Frequency of periodic oscillations , if presented in mathematical form, has the formula: v=1/T, where T is the oscillation period.

Before moving on to the conclusion, let's talk a little about where periodic fluctuations are observed and how knowledge about them can be useful in life.

Application

Above, we have already considered the types of periodic oscillations. Even if you are guided by the list of where they meet, it is easy to understand that they surround us everywhere. emit all our electrical appliances. Moreover, phone-to-phone communication or listening to the radio would not be possible without them.

Sound waves are also vibrations. Under the influence of electrical voltage, a special membrane in any sound generator begins to vibrate, creating waves of a certain frequency. Following the membrane, air molecules begin to vibrate, which eventually reach our ear and are perceived as sound.

Conclusion

Physics is a very interesting science. And even if it seems that you kind of know everything in it that can be useful in everyday life, there is still such a thing that it would be useful to understand it better. We hope that this article has helped you understand or remember the material on the physics of vibrations. This is indeed a very important topic, the practical application of the theory from which is found everywhere today.

Introduction

Studying the phenomenon, we simultaneously get acquainted with the properties of the object and learn how to apply them in technology and in everyday life. As an example, let us turn to an oscillating filament pendulum. Any phenomenon is "usually" peeped in nature, but can be predicted theoretically, or accidentally discovered when studying another. Even Galileo drew attention to the vibrations of the chandelier in the cathedral and "there was something in this pendulum that made it stop." However, observations have a major drawback, they are passive. In order to stop depending on nature, it is necessary to build an experimental setup. Now we can reproduce the phenomenon at any time. But what is the purpose of our experiments with the same filament pendulum? Man took a lot from "our smaller brothers" and therefore one can imagine what experiments an ordinary monkey would have carried out with a thread pendulum. She would have tasted it, sniffed it, pulled the string, and lost all interest in it. Nature taught her to study the properties of objects very quickly. Edible, inedible, tasty, tasteless - this is a short list of the properties that the monkey has studied. However, the man went further. He discovered such an important property as periodicity, which can be measured. Any measurable property of an object is called a physical quantity. No mechanic in the world knows all the laws of mechanics! Is it possible to single out the main laws by means of theoretical analysis or the same experiments? Those who managed to do this forever inscribed their name in the history of science.

In my work, I would like to study the properties of physical pendulums, to determine to what extent the already studied properties can be applied in practice, in people's lives, in science, and can be used as a method for studying physical phenomena in other areas of this science.

fluctuations

Oscillations are one of the most common processes in nature and technology. High-rise buildings and high-voltage wires oscillate under the influence of the wind, the pendulum of a wound clock and a car on springs during movement, the level of the river during the year and the temperature of the human body during illness.

One has to deal with oscillatory systems not only in various machines and mechanisms, the term "pendulum" is widely used in application to systems of various nature. So, an electric pendulum is called a circuit consisting of a capacitor and an inductor, a chemical pendulum is a mixture of chemicals that enter into an oscillatory reaction, an ecological pendulum is two interacting populations of predators and prey. The same term is applied to economic systems in which oscillatory processes take place. We also know that most sources of sound are oscillatory systems, that the propagation of sound in air is possible only because the air itself is a kind of oscillatory system. Moreover, in addition to mechanical oscillatory systems, there are electromagnetic oscillatory systems in which electrical oscillations can occur, which form the basis of all radio engineering. Finally, there are a lot of mixed - electromechanical - oscillatory systems used in a wide variety of technical fields.

We see that sound is fluctuations in the density and pressure of air, radio waves are periodic changes in the strength of electric and magnetic fields, visible light is also electromagnetic vibrations, only with slightly different wavelengths and frequencies. Earthquakes - soil vibrations, ebbs and flows - changes in the level of the seas and oceans, caused by the attraction of the moon and reaching 18 meters in some areas, pulse beats - periodic contractions of the human heart muscle, etc. Change of wakefulness and sleep, work and rest, winter and summer. Even our everyday going to work and returning home falls under the definition of fluctuations, which are interpreted as processes that repeat exactly or approximately at regular intervals.

So, vibrations are mechanical, electromagnetic, chemical, thermodynamic and various others. Despite this diversity, they all have much in common and are therefore described by the same differential equations. A special section of physics - the theory of oscillations - deals with the study of the laws of these phenomena. Shipbuilders and aircraft builders, industry and transport specialists, creators of radio engineering and acoustic equipment need to know them.

Any fluctuations are characterized by amplitude - the largest deviation of a certain value from its zero value, period (T) or frequency (v). The last two quantities are interconnected by an inversely proportional relationship: T=1/v. The oscillation frequency is expressed in hertz (Hz). The unit of measurement is named after the famous German physicist Heinrich Hertz (1857-1894). 1Hz is one cycle per second. This is the rate at which the human heart beats. The word "hertz" in German means "heart". If desired, this coincidence can be seen as a kind of symbolic connection.

The first scientists who studied oscillations were Galileo Galilei (1564...1642) and Christian Huygens (1629...1692). Galileo established isochronism (independence of the period from the amplitude) of small oscillations, watching the swinging of the chandelier in the cathedral and measuring the time by the beats of the pulse on his hand. Huygens invented the first pendulum clock (1657) and in the second edition of his monograph "Pendulum Clock" (1673) investigated a number of problems associated with the movement of the pendulum, in particular, found the center of swing of a physical pendulum. A great contribution to the study of oscillations was made by many scientists: English - W. Thomson (Lord Kelvin) and J. Rayleigh, Russians - A.S. Popov and P.N. Lebedev, Soviet - A.N. Krylov, L.I. Mandelstam, N.D. Papaleksi, N.N. Bogolyubov, A.A. Andronov and others.

Periodic fluctuations

Among the various mechanical movements and oscillations that take place around us, repetitive movements are often encountered. Any uniform rotation is a repetitive movement: with each revolution, any point of a uniformly rotating body passes the same positions as during the previous revolution, and in the same sequence and at the same speeds. If we look at how the branches and trunks of trees sway in the wind, how a ship sways on the waves, how the pendulum of a clock moves, how the pistons and connecting rods of a steam engine or diesel engine move back and forth, how the needle of a sewing machine jumps up and down; if we observe the alternation of the ebb and flow of the sea, the shifting of the legs and the swinging of the arms while walking and running, the beating of the heart or the pulse, then in all these movements we will notice the same feature - the repeated repetition of the same cycle of movements.

In reality, repetition is not always and under all conditions exactly the same. In some cases, each new cycle very accurately repeats the previous one (swinging of a pendulum, movements of parts of a machine operating at a constant speed), in other cases, the difference between successive cycles can be noticeable (ebb and flow, swinging branches, movements of machine parts during its operation). start or stop). Deviations from an absolutely exact repetition are very often so small that they can be neglected and the motion can be considered as repeating quite exactly, i.e., it can be considered periodic.

Periodic is a repetitive movement in which each cycle exactly reproduces any other cycle. The duration of one cycle is called a period. The period of oscillation of a physical pendulum depends on many circumstances: on the size and shape of the body, on the distance between the center of gravity and the point of suspension, and on the distribution of body mass relative to this point.

MECHANICAL VIBRATIONS

1. Fluctuations. Characteristics of harmonic oscillations.

2. Free (natural) vibrations. Differential equation of harmonic oscillations and its solution. Harmonic oscillator.

3. Energy of harmonic oscillations.

4. Addition of identically directed harmonic oscillations. beat. Vector diagram method.

5. Addition of mutually perpendicular vibrations. Lissajous figures.

6. Damped oscillations. The differential equation of damped oscillations and its solution. Frequency of damped oscillations. Isochronous oscillations. Coefficient, decrement, logarithmic damping decrement. Quality factor of the oscillatory system.

7. Forced mechanical oscillations. Amplitude and phase of forced mechanical vibrations.

8. Mechanical resonance. The relationship between the phases of the driving force and the velocity at mechanical resonance.

9. the concept of self-oscillations.

Fluctuations. Characteristics of harmonic oscillations.

fluctuations- movement or processes that have a certain degree of repetition in time.

Harmonic (or sinusoidal) oscillations- a kind of periodic oscillations that can be replaced in the form

where a is the amplitude, is the phase, is the initial phase, is the cyclic frequency, t is the time (i.e. applied over time according to the sine or cosine law).

Amplitude (a) - the largest deviation from the average value quantity that oscillates.

Oscillation phase () is the changing argument of the function describing the oscillatory process(value t+ under the sine sign in expression (1)).

Phase characterizes the value of a changing quantity at a given time. The value at time t=0 is called initial phase ( ).

As an example, figure 27.1 shows mathematical pendulums in extreme positions with a phase difference of oscillations = 0 (27.1.a) and = (27.1b)

The phase difference of pendulum oscillations is manifested by the difference in the position of the oscillating pendulums.

Cyclic or circular frequency is the number of oscillations in 2 seconds.

Oscillation frequency(or line frequency) is the number of oscillations per unit time. The unit of frequency is the frequency of such oscillations, the period of which is equal to 1 s. This unit is called Hertz(Hz).

The time interval during which one complete oscillation takes place, and the phase of the oscillation receives an increment equal to 2, is called period of oscillation(Fig. 27.2).

The frequency is related to the

ratio T ratio-

t

		X

Dividing both sides of the equations by m

and moving to the left side

Denoting , we obtain a linear differential homogeneous equation of the second order

(2)

(linear - that is, both the value x itself and its derivative to the first degree; homogeneous - because there is no free term that does not contain x; second order - because the second derivative of x).

Equation (2) is solved by (*) substituting x = . Substituting into (2) and differentiating

.

We get the characteristic equation

This equation has imaginary roots: ( -imaginary unit).

The general solution has the form

where and are complex constants.

Substituting the roots, we get

(3)

(Comment: complex number z is a number of the form z = x + iy, where x,y are real numbers, i is an imaginary unit ( = -1). The number x is called the real part of the complex number z. The number y is called the imaginary part of z).

(*) In a shortened version, the solution can be omitted

An expression of the form can be represented as a complex number using the Euler formula

likewise

We set and in the form of complex constants = A, a = A, where A and arbitrary constants. From (3) we get

Denoting we get

Using the Euler formula

Those. we obtain the solution of the differential equation for free oscillations

where is the natural circular oscillation frequency, A is the amplitude.

The offset x is applied over time according to the law of cosine, i.e. the motion of the system under the action of the elastic force f = -kx is a harmonic oscillation.

If the quantities describing the oscillations of a certain system change periodically with time, then for such a system the term " oscillator».

Linear harmonic oscillator is called such, the movement of which is described by a linear equation.

3. Energy of harmonic oscillations. The total mechanical energy of the system shown in fig. 27.2 is equal to the sum of mechanical and potential energies.

Differentiate with respect to time the expression ( , we obtain

A sin( t + ).

Kinetic energy load (we neglect the mass of the spring) is equal to

E= .

Potential energy is expressed by a well-known formula, substituting x from (4), we obtain

total energy

the value is constant. In the process of oscillations, potential energy transforms into kinetic energy and vice versa, but each energy remains unchanged.

4. Addition of equally directed oscillations.. Usually the same body is involved in several oscillations. So, for example, the sound vibrations that we perceive when listening to an orchestra are sum of fluctuations air, caused by each of the musical instruments separately. We will assume the amplitudes of both oscillations to be the same and equal to a. To simplify the problem, we set the initial phases equal to zero. Then the beats. During this time, the phase difference changes by , i.e.

Thus the beat period