Fluctuations are short. Mechanical vibrations. Oscillatory motion parameters. Basic parameters of oscillatory movements

There are different types of oscillations in physics, characterized by certain parameters. Let's consider their main differences, classification according to different factors.

Basic definitions

By oscillation is meant a process in which, at regular intervals, the basic characteristics of movement have the same values.

Periodic fluctuations are those in which the values ​​of the basic quantities are repeated at regular intervals (period of fluctuations).

Varieties of oscillatory processes

Consider the main modes of vibration that exist in fundamental physics.

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

An example of free vibrations is a mathematical pendulum.

Those types of mechanical vibrations that arise in the system under the influence of an external variable force.

Features of the classification

According to the physical nature, the following types of oscillatory movements are distinguished:

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

According to the variant of interaction with the environment

The types of vibrations in interaction with the environment distinguish several groups.

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

For forced harmonic oscillations, the appearance of resonance is possible, in which, with equal values ​​of the frequency of the external influence and the oscillator, with a sharp increase in the amplitude.

These are natural oscillations in the system under the influence of internal forces after it is taken out of equilibrium. The simplest option for free vibrations is the movement of a weight that is suspended on a thread or attached to a spring.

Self-oscillations are the types in which the system has a certain supply of potential energy, which goes to make oscillations. Their distinctive feature is the fact that the amplitude is characterized by the properties of the system itself, and not by the initial conditions.

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

Basic parameters of oscillatory movements

All modes of vibration have certain characteristics, which should be mentioned separately.

The amplitude is called the maximum deviation from the equilibrium position, the deviation of the fluctuating value, it is measured in meters.

The period is the time of one full oscillation through 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 oscillation period.

The oscillation phase characterizes the state of the system.

Harmonic characteristic

These types of oscillations occur according to the cosine or sine law. Fourier was able 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 with a certain period and cyclic frequency.

What are these types of vibrations characterized by? Physics considers an idealized system, which consists of a material point, which is suspended on a weightless inextensible thread, vibrates under the influence of gravity.

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

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

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

Forced vibrations are the shaking of a vehicle that moves on an uneven road.

The main differences between forced and free oscillations

These types of electromagnetic waves differ in physical characteristics. The presence of resistance of the medium and the friction force lead to damping of free vibrations. In the case of forced fluctuations, energy losses are compensated for by its additional input from an external source.

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

With a known period, you can calculate the natural frequency of the oscillatory system.

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

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

Electromagnetic vibrations

The characteristic of the modes of vibration involves the analysis of the basic physical parameters: charge, voltage, current. As an elementary system, which is used to observe electromagnetic oscillations, there is an oscillatory circuit. It is formed when a coil and a capacitor are connected in series.

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

They are free due to the fact that during their execution 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 possesses.

That is why damped oscillations occur in a real circuit. A decrease in the charge on the capacitor leads to a decrease in the energy value in comparison with its original indicator. 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 end.

The importance of fluctuations in science and technology

Any movement that has a certain degree of repeatability is wobble. For example, a mathematical pendulum is characterized by a systematic deviation in both directions from the initial vertical position.

For a spring pendulum, one full swing 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 capacitor plates. What is the reason for the oscillatory movements? The pendulum functions due to the force of gravity causing 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 middle state.

Electrical fluctuations 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.

In modern technology, vibrations are used that differ significantly in their nature, degree of repeatability, character, and also the "mechanism" of their appearance.

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

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

The brightness fluctuations of Cepheids are of particular interest in astrophysics; scientists from different countries are studying them.

Conclusion

All types of vibrations are closely related to a huge number of technical processes and physical phenomena. They are of great practical importance in aircraft construction, ship building, construction of residential complexes, electrical engineering, radio electronics, medicine, and 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, as well as in many other natural science fields.

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. 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, Rayleigh. He managed to find a practical application for electromagnetic oscillations, to use them to transmit a radio signal over a long distance.

Academician P. N. Lebedev for many years carried out experiments related to obtaining high-frequency electromagnetic oscillations using alternating electric fields. Thanks to numerous experiments related to various types of vibrations, scientists have managed to find areas of their optimal use in modern science and technology.

Themes of the USE codifier: harmonic oscillations; amplitude, period, frequency, phase of oscillations; free vibrations, forced vibrations, resonance.

Fluctuations - these are changes in the state of the system that are repeated in time. The concept of vibrations covers a very wide range of phenomena.

Fluctuations in mechanical systems, or mechanical vibrations- This is a mechanical movement of a body or a system of bodies, which has a repeatability in time and occurs in the vicinity of the equilibrium position. Equilibrium position is called a state of the system in which it can remain for an arbitrarily long time, without experiencing external influences.

For example, if the pendulum is deflected and released, then oscillations will begin. The equilibrium position is the position of the pendulum in the absence of deflection. In this position, the pendulum, if not touched, can remain indefinitely. When oscillating, the pendulum passes the equilibrium position many times.

Immediately after the deflected pendulum was released, it began to move, passed the equilibrium position, reached the opposite extreme position, stopped there for a moment, moved in the opposite direction, passed the equilibrium position again and returned back. One thing happened full swing... Further, this process will be periodically repeated.

Amplitude of body vibrations is the value of its greatest deviation from the equilibrium position.

Oscillation period - this is the time of one complete oscillation. We can say that during the period the body travels a path of four amplitudes.

Oscillation frequency is the reciprocal of the period:. Frequency is measured in hertz (Hz) and indicates how many complete oscillations occur in one second.

Harmonic vibrations.

We will assume that the position of the oscillating body is determined by a single coordinate. The equilibrium position corresponds to the value. The main task of mechanics in this case is to find a function that gives the coordinate of the body at any time.

For a mathematical description of oscillations, it is natural to use periodic functions. There are many such functions, but two of them - sine and cosine - are the most important. They have many good properties and are closely related to a wide range of physical phenomena.

Since the sine and cosine functions are obtained from each other by shifting the argument by, you can limit yourself to only one of them. We will use cosine for definiteness.

Harmonic vibrations- these are oscillations at which the coordinate depends on time according to the harmonic law:

(1)

Let us find out the meaning of the quantities included in this formula.

A positive value is the largest value of the coordinate in absolute value (since the maximum value of the modulus of the cosine is equal to one), i.e., the greatest deviation from the equilibrium position. Therefore - the amplitude of the oscillations.

The cosine argument is called phase hesitation. The value equal to the phase value at is called the initial phase. The initial phase corresponds to the initial coordinate of the body:.

The quantity is called cyclic frequency... Let's find its connection with the oscillation period and frequency. One complete oscillation corresponds to a phase increment equal to radians: whence

(2)

(3)

The cyclic frequency is measured in rad / s (radians per second).

In accordance with expressions (2) and (3), we obtain two more forms of writing the harmonic law (1):

The graph of function (1), expressing the dependence of the coordinate on time with harmonic oscillations, is shown in Fig. one .

The harmonic law of the form (1) is of the most general nature. It responds, for example, to situations when two initial actions were performed simultaneously with the pendulum: they deflected it by an amount and gave it a certain initial velocity. There are two important special cases when one of these actions was not performed.

Let the pendulum be deflected, but the initial speed was not reported (released without initial speed). It is clear that in this case, therefore, one can put. We get the cosine law:

The graph of harmonic oscillations in this case is shown in Fig. 2.

Rice. 2. Cosine law

Let us now assume that the pendulum was not deflected, but the initial velocity was given to it by impact from the equilibrium position. In this case, so you can put. We get the sine law:

The oscillation graph is shown in Fig. 3.

Rice. 3. Sine law

Equation of harmonic vibrations.

Let's return to the general harmonic law (1). We differentiate this equality:

. (4)

Now we differentiate the obtained equality (4):

. (5)

Let's compare expression (1) for the coordinate and expression (5) for the acceleration projection. We see that the projection of the acceleration differs from the coordinate only by a factor:

. (6)

This ratio is called harmonic vibration equation... It can be rewritten as follows:

. (7)

From a mathematical point of view, equation (7) is differential equation... Functions (not numbers, as in ordinary algebra) serve as solutions to differential equations.
So, you can prove that:

A solution to equation (7) is any function of the form (1) with arbitrary;

No other function is a solution to this equation.

In other words, relations (6), (7) describe harmonic oscillations with a cyclic frequency and only them. Two constants are determined from the initial conditions - according to the initial values ​​of the coordinate and velocity.

Spring pendulum.

Spring pendulum is a spring-mounted weight capable of vibrating horizontally or vertically.

Let us find the period of small horizontal oscillations of the spring pendulum (Fig. 4). Oscillations will be small if the amount of deformation of the spring is much less than its size. For small deformations, we can use Hooke's law. This will lead to the fact that the vibrations are harmonic.

We neglect friction. The load has a mass, the stiffness of the spring is equal.

The coordinate corresponds to the equilibrium position in which the spring is not deformed. Consequently, the amount of deformation of the spring is equal to the modulus of the coordinate of the load.

Rice. 4. Spring pendulum

In the horizontal direction, only the spring force acts on the load. Newton's second law for cargo in projection onto the axis is:

. (8)

If (the load is shifted to the right, as in the figure), then the elastic force is directed in the opposite direction, and. On the contrary, if, then. The signs and are always opposite, so Hooke's law can be written as follows:

Then relation (8) takes the form:

We have obtained an equation of harmonic vibrations of the form (6), in which

The cyclic oscillation frequency of the spring pendulum is thus equal to:

. (9)

From here and from the ratio, we find the period of horizontal oscillations of the spring pendulum:

. (10)

If you suspend a weight on a spring, you get a spring-loaded pendulum that oscillates in the vertical direction. It can be shown that in this case, formula (10) is valid for the oscillation period.

Mathematical pendulum.

Mathematical pendulum is a small body suspended on a weightless inextensible thread (Fig. 5). A mathematical pendulum can oscillate in a vertical plane in a gravity field.

Rice. 5. Mathematical pendulum

Let us find the period of small oscillations of the mathematical pendulum. The length of the thread is. We neglect the air resistance.

Let us write Newton's second law for the pendulum:

and project it onto an axis:

If the pendulum occupies a position as in the figure (i.e.), then:

If the pendulum is on the other side of the equilibrium position (i.e.), then:

So, for any position of the pendulum, we have:

. (11)

When the pendulum is at rest in the equilibrium position, equality is satisfied. For small oscillations, when the deviations of the pendulum from the equilibrium position are small (in comparison with the length of the thread), an approximate equality is fulfilled. We will use it in formula (11):

This is an equation of harmonic vibrations of the form (6), in which

Consequently, the cyclical frequency of oscillations of a mathematical pendulum is equal to:

. (12)

Hence the period of oscillation of the mathematical pendulum:

. (13)

Please note that formula (13) does not include the mass of the load. Unlike a spring pendulum, the period of oscillation of a mathematical pendulum does not depend on its mass.

Free and forced vibrations.

They say that the system does free vibrations if it is once taken out of the equilibrium position and then left to itself. No periodic external
In this case, the system does not experience any influences, and the system does not have any internal sources of energy that support the oscillations.

The oscillations of the spring and mathematical pendulums considered above are examples of free oscillations.

The frequency with which free vibrations occur is called natural frequency oscillatory system. So, formulas (9) and (12) give the natural (cyclic) frequencies of oscillations of the spring and mathematical pendulums.

In an idealized situation in the absence of friction, free vibrations are non-damping, i.e., they have a constant amplitude and last indefinitely. Friction is always present in real oscillatory systems, therefore free oscillations gradually damp (Fig. 6).

Forced vibrations- these are vibrations made by the system under the influence of an external force that periodically changes in time (the so-called driving force).

Suppose that the natural frequency of the oscillations of the system is equal, and the driving force depends on time according to the harmonic law:

For some time, forced oscillations are established: the system makes a complex movement, which is an imposition of forced and free oscillations. Free oscillations gradually damp, and in the steady state the system performs forced oscillations, which also turn out to be harmonic. The frequency of steady-state forced oscillations coincides with the frequency
a compelling force (an external force seems to impose its frequency on the system).

The amplitude of the steady-state forced oscillations depends on the frequency of the driving force. The graph of this dependence is shown in Fig. 7.

Rice. 7. Resonance

We see that resonance occurs near the frequency - the phenomenon of an increase in the amplitude of forced oscillations. The resonant frequency is approximately equal to the natural oscillation frequency of the system:, and this equality is fulfilled the more accurately, the lower the friction in the system. In the absence of friction, the resonant frequency coincides with the natural vibration frequency, and the vibration amplitude increases to infinity at.

Time should be devoted to a short essay on oscillatory motion. But first it is necessary to answer one important question. What is meant by mechanical vibrations? They mean movement, during which the observed body repeatedly occupies the same positions in space.

Physicists distinguish between non-periodic and periodic oscillations. The former include those in which the coordinates and other characteristics of the body cannot be described using periodic functions of time. The second view is easier. Periodic fluctuations are those that can be described using periodic functions of time. But what do they mean by them? In physics, oscillations are also often understood as processes that are repeated to a certain extent in time. And the following should be said separately regarding the topic under consideration. Mechanical vibrations can be conventionally classified as follows:

1. Depending on the conditions of occurrence:
1. Forced;
2. Self-oscillations;
3. Free.
2. Depending on the change in kinetic energy over time:
1. Harmonic;
2. Sawtooth;
3. Fading.

The article will not cover all, but only some types of vibrations. Separately, it should be said about the formulas, their use and variety. In short, there are many of them. The variety in which mechanical vibrations are presented, formulas for determining their parameters pushed scientists to create separate reference books designed for certain situations. So you don't need to invent anything on your own. When creating an oscillatory system, it will be necessary to spend only half an hour or an hour to find a formula for a specific situation.

Characteristic of mechanical vibrations

To characterize mechanical vibrations, physical quantities are used that allow obtaining the necessary data. Amplitude of oscillation - the greatest deviation of the body, which oscillates from the initial value of the position. What is a period? In it, vibrations are the time it takes for the body to repeat all its movements, or in other words, it takes to complete one repetition of the movement. What is meant by frequency? It is understood as a number equal to the number of oscillations made in one unit of time. Often in home, school and university experiments, one second is taken as the frequency. Cyclic frequency is often used instead of the concept of the number of oscillations that occurred per unit of time, and implies the calculation of it required to complete one such cycle.

Harmonic mechanical vibrations

Harmonic oscillations are understood as those of them, the physical value of which, selected for the characteristic, changes over a time interval in the form of a sinusoidal curve, which is easy to display in graphic mode. When the coordinate of a material point changes, according to the harmonic law, the impulse, speed and acceleration also change according to it.

Free vibrations

When the vibration occurs in the system due to the initial energy, then it is called free. As a practical display of this type of physical process, special models are used: spring and mathematical pendulums. They allow you to work with the most common situations. As a mathematical pendulum, they take a point that oscillates and hangs on an inextensible and weightless thread. There is no such device on earth. Therefore, the closest thing to the theoretical model is a structure composed of a ball, the diameter (size) of which is much smaller than the length of the thread. It is necessary to carry out actions of a physical nature. Deflect such a ball from its starting position and release. And so any experimenter can see mechanical vibrations. The period, as well as their frequency, depend exclusively on the parameters of the system: the length of the thread of the mathematical pendulum, the stiffness of the spring, the mass of the load (important for a spring pendulum). It is because of this that free vibrations are also called natural vibrations of the system. It is quite logical. And the frequency with which everything happens is called the systemic frequency.

Energy conversion during mechanical vibrations

Potential and kinetic energies pass into one another during body movements. And the same is the other way around. When the system deviates from the initial equilibrium position by the maximum possible value, then the potential energy also reaches its maximum value, while the kinetics of the body - the minimum. Separately, it should be said about one popular misconception among people. When the equilibrium position is reached, the potential energy is at the point of its minimum (it is usually considered that here it is equal to zero), while the kinetics (and this is the momentum of the body and the speed of its movement) reaches its maximum. In practice, something else is taken into account. In real systems, there are non-potential forces, the value of which is not equal to zero. The energy of the system is wasted due to the work of the support forces, air friction, internal forces of the spring or suspension. The amplitude of body vibration gradually decreases. Such oscillations are called damped. If the friction force is too great, then the entire energy reserve can be expended already in the period of one vibration, and the body's movement will not be periodic.

Forced vibrations

Forced vibrations are understood as those that occur under the influence of an external force performing work, which changes over time. There is also another wording. Due to the external influx of energy, it is maintained in the system itself at a sufficient level for the actual vibrations to occur. To understand this, it is necessary to draw parallels with reality. An example of an object that performs this type of oscillation is a swing, on which one person sits, and the other swings him. There is one caveat. If an external force compensates for the loss of energy in the system continuously or periodically, without stopping the process of oscillations itself, then they are called continuous forced.

The following can be noted about the range. The amplitude of the forced vibrations is completely determined by the force that acts from the outside, as well as the ratio between the natural frequencies of the parties involved in the process. And here there is one interesting phenomenon. During forced oscillations, a sharp increase in amplitude can be periodically observed, which is called resonance.

Resonance

It occurs when the force that affects the system becomes very close to its vibration frequency. Another option is also possible. In the event that the frequency of the influencing force is a multiple of the oscillations of the system itself, on which it acts, resonance also occurs. How is he represented graphically? The dependence of the oscillation amplitudes of the system on the frequency of the influencing force is expressed using a resonance curve.

Self-oscillations

Self-oscillations have found their application in technology. They exist where sustained oscillations are maintained by the energy of a source that can be automatically turned on and off by the system itself. In such cases, it is possible to seriously consider the issue of assigning the self-oscillating status to the system. Why? The moment when it is necessary to supply energy for oscillation is monitored by the subsystem responsible for the feedback. Depending on the parameters of the body, it can have an effect strongly and immediately, or gradually and gradually. It can open or close the opportunity for energy to enter the general system. This is her main task. As an example of a self-oscillating system, we can recall a pendulum clock, where the energy source is a weight, and the anchor mechanism successfully copes with the role of a feedback subsystem that regulates the flow of kinetics on which mechanical oscillations depend.

Parametric vibrations

This type of oscillation defines those that occur in systems that periodically change their parameters. What can you say about them? The only thing that determines the amplitude and strength of an oscillatory system is its parameters.

Oscillations are the movement of a body, during which it repeatedly moves along the same trajectory and passes through the same points in space. Examples of vibrating objects are the pendulum of a clock, the string of a violin or piano, or the vibration of a car.

Oscillations play an important role in many physical phenomena outside the field of mechanics. For example, voltage and current in electrical circuits can fluctuate. Biological examples of vibrations include heartbeats, arterial pulse, and vocal cord sound production.

Although the physical nature of oscillating systems can differ significantly, various types of oscillations can be quantitatively characterized in a similar way. A physical quantity that changes over time during oscillatory motion is called displacement . Amplitude represents the maximum displacement of an oscillating object from the equilibrium position. Full swing, or cycle - this is a movement in which the body, brought out of the equilibrium position by a certain amplitude, returns to this position, deflects to the maximum displacement in the opposite direction and returns to its original position. Oscillation period T - the time required to carry out one complete cycle. The number of vibrations per unit of time is vibration frequency .

Simple harmonic vibration

In some bodies, when they are stretched or compressed, forces arise that counteract these processes. These forces are directly proportional to the length of tension or compression. Springs have this property. When a body suspended from a spring is tilted from its equilibrium position and then released, its motion is a simple harmonic vibration.

Consider a body with mass m suspended by a spring in equilibrium position. By displacing the body downward, the body can be wobbled. If - displacement of the body from the equilibrium position, then a force arises in the spring F(elastic force) directed in the opposite direction to the displacement. According to Hooke's law, the elastic force is proportional to the displacement F ctrl = -k S, where k- a constant that depends on the elastic properties of the spring. Force is negative because it tends to bring the body back into balance.

Acting on a body with a mass m, the elastic force gives it acceleration along the direction of displacement. According to Newton's law F = ma, where a = d 2 S / d 2 t. To simplify the subsequent reasoning, we neglect friction and viscosity in an oscillating system. In this case, the amplitude of the oscillations will not change over time.

If no external forces (even the resistance of the medium) act on the vibrating body, then the vibrations are carried out with a certain frequency. These vibrations are called free. The amplitude of such fluctuations remains constant.

In this way, m d 2 S / d 2 t = -k S(one) . Moving all terms of equality and dividing them by m, we get the equations d 2 S / d 2 t + (k / m)· S = 0 ,
and then d 2 S / d 2 t + ω 0 2· S = 0 (2), where k / m =ω 0 2

Equation (2) is differential equation of simple harmonic oscillation.
The solution to equation (2) gives two functions:
S = A sin ( ω 0 t + φ 0) (3) and S = A cos ( ω 0 t + φ 0) (4)

Thus, if a body of mass m carries out simple harmonic oscillations, the change in the displacement of this body from the point of equilibrium in time is carried out according to the law of sine or cosine.

(ω 0 t + φ 0) - phase of oscillation with an initial phase φ 0 . Phase is a property of oscillatory motion, which characterizes the amount of displacement of the body at any time. The phase is measured in radians.

The magnitude called angular, or circular, frequency... Measured in radians per second ω 0 = 2πν or ω 0 = 2 π / T (5)

The graph of the equation of simple harmonic oscillation is presented on Rice. one... Body originally displaced a distance A - amplitudes hesitation , and then released, continues to fluctuate from -A and before A per time T- oscillation period.

Fig 1.

Thus, in the course of a simple harmonic oscillation, the displacement of the body changes in time along a sinusoid or cosine. Therefore, a simple harmonic wave is often called a sine wave.

A simple harmonic vibration has the following main characteristics:

A) the moving body is alternately on both sides of the equilibrium position;
b) the body repeats its movement for a certain period of time;
c) the acceleration of the body is always proportional to the displacement and directed opposite to it;
e) graphically, this type of oscillation is described by a sinusoid.

Damped oscillation

A simple harmonic oscillation cannot continue for as long as desired with a constant amplitude. In real conditions, after a while, harmonic oscillations stop. Such harmonic oscillations in real systems are called damped oscillations ( fig. 2 ) ... The action of external forces, for example, friction and viscosity, leads to a decrease in the amplitude of oscillations with their subsequent termination. These forces reduce the vibration energy. They're called dissipative forces, since they contribute to the dissipation of the potential and kinetic energy of macroscopic bodies into the energy of the thermal motion of atoms and molecules of the body.

Fig 2.

The magnitude of the dissipative forces depends on the speed of the body. If the velocity ν is comparatively small, then the dissipative force F is directly proportional to this speed F tr = -rν = -r dS / dt (6)

Here r- constant coefficient, independent of the speed or vibration frequency. The minus sign indicates that the braking force is directed against the velocity vector.

Taking into account the action of dissipative forces, the differential equation of the harmonic damped oscillation has the form: m · d 2 S / d 2 t= -kS - r dS / dt .

Moving all the terms of the equality to one side, dividing each term by m and replacing k / m = ω 2, r / m = 2β, we get differential equation of free harmonic damped oscillations

where β is the damping coefficient characterizing the damping of oscillations per unit time.

The solution to the equation is the function S = A 0 e -βt sin (ωt + φ 0) (8)

Equation (8) shows that the amplitude of the harmonic oscillation decreases exponentially with time. The frequency of damped oscillations is determined by the equation ω = √(ω 0 2- β 2) (9)

If the oscillation cannot occur due to the large, then the system returns to its equilibrium position along an exponential path without oscillation.

Forced oscillation and resonance

If you do not impart external energy to the oscillating system, then the amplitude of the harmonic oscillation decreases with time due to dissipative effects. The periodic action of force can increase the amplitude of the vibration. Now the oscillation will not fade over time, since the lost energy is replenished during each cycle by the action of an external force. If a balance of these two energies is achieved, then the amplitude of the oscillations will remain constant. The effect depends on the ratio of the frequencies of the driving force ω and the natural vibration frequency of the system ω 0.

If the body vibrates under the action of an external periodic force with the frequency of this external force, then the vibration of the body is called forced.

The energy of an external force has the greatest effect on the vibrations of the system if the external force has a certain frequency. This frequency should be the same as the frequency of the natural vibrations of the system, which this system would perform in the absence of external forces. In this case, there is resonance- the phenomenon of a sharp increase in the amplitude of oscillations when the frequency of the driving force coincides with the frequency of natural oscillations of the system.

Mechanical waves

The propagation of vibrations from one place to another is called wave motion, or simply wave.

Mechanical waves are formed as a result of simple harmonic vibrations of the particles of the medium from their mean position. At the same time, the substance of the medium does not move from one place to another. But the particles of the medium, transferring energy to each other, are necessary for the propagation of mechanical waves.

Thus, a mechanical wave is a disturbance of the material environment, which passes this environment at a certain speed, without changing its shape.

If a stone is thrown into the water, a single wave will run from the place of disturbance of the environment. However, waves can sometimes be periodic. For example, a vibrating tuning fork produces alternating compression and depression of the air around it. These disturbances, perceived as sound, occur periodically with the frequency of the tuning fork.

There are two types of mechanical waves.

(1) Transverse wave... This type of waves is characterized by vibration of particles of the medium at right angles to the direction of wave propagation. Transverse mechanical waves can occur only in solids and on the surface of liquids.

In a transverse wave, all particles of the medium carry out a simple harmonic vibration near their mean positions. The position of the maximum upward displacement is called " peak"and the position of the maximum downward displacement is" hollow"The distance between two successive peaks or troughs is called the shear wavelength λ.

(2) Longitudinal wave... This type of waves is characterized by oscillations of medium particles along the direction of wave propagation. Longitudinal waves can propagate in liquids, gases and solids.

In a longitudinal wave, all particles of the medium also carry out a simple harmonic vibration about their mean position. In some places, the particles of the medium are closer, and in other places - farther than in the normal state.

Places where particles are close together are called regions. compression, and the places where they are far from each other are regions rarefaction... The distance between two successive compressions or expansions is called the longitudinal wavelength.

There are the following wave characteristics.

(1) Amplitude is the maximum displacement of an oscillating particle of the medium from its equilibrium position ( A).

(2) Period is the time required for a particle for one complete oscillation ( T).

(3) Frequency- the number of vibrations produced by a particle of the medium per unit of time (ν). There is an inverse relationship between the wave frequency and its period: ν = 1 / T.

(4) Phase an oscillating particle at any moment determines its position and direction of motion at the given moment. Phase is a fraction of a wavelength or time period.

(5) Speed wave is the speed of propagation in space of the peak of the wave (v).

The aggregate of particles of the medium, vibrating in the same phase, forms the wave front. From this point of view, waves are divided into two types.

(1) If the source of the wave is the point from which it propagates in all directions, then spherical wave.

(2) If the wave source is an oscillating flat surface, then a plane wave.

The displacement of particles of a plane wave can be described by the general equation for all types of wave motion: S = A sin ω (t - x / v) (10)

This means that the displacement value ( S) for each time value (t) and distance from the wave source ( x) depends on the amplitude of the oscillation ( A), angular frequency ( ω ) and wave velocity (v).

Doppler effect

The Doppler effect is a change in the frequency of a wave perceived by an observer (receiver) due to the relative movement of the wave source and the observer. If the source of the waves approaches the observer, the number of waves arriving at the observer of the waves each second exceeds that emitted by the source of the waves. If the wave source is moving away from the observer, then the number of waves emitted is greater than that arriving at the observer.

A similar effect follows if the observer moves relative to a stationary source.

An example of the Doppler effect is the change in the train horn frequency as it approaches and moves away from the observer.

The general equation for the Doppler effect is

Here ν source is the frequency of the waves emitted by the source, and ν receiver is the frequency of the waves perceived by the observer. ν 0 is the speed of waves in a stationary medium, ν receiving and ν source are the speeds of the observer and the source of the waves, respectively. The upper signs in the formula refer to the case when the source and the observer move towards each other. The lower signs refer to the case of the distance from each other of the source and the observer of the waves.

The change in the frequency of the waves due to the Doppler effect is called the Doppler frequency shift. This phenomenon is used to measure the speed of movement of various bodies, including red blood cells in blood vessels.

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- these are movements or processes that are characterized by a certain repetition in time.

Oscillation period T - the time interval during which one complete oscillation occurs.

Oscillation frequency ν - the number of complete fluctuations per unit of time. The SI unit is expressed in hertz (Hz).

The period and frequency of oscillations are related by the ratio:

Harmonic vibrations - these are oscillations in which an oscillating quantity, for example, the displacement of a load on a spring from an equilibrium position, changes according to the law of sine or cosine:

Acceleration during harmonic oscillations is always directed in the direction opposite to the displacement; maximum acceleration is equal in modulus

Spring and mathematical pendulums can be cited as examples of free vibrations. Spring loaded (harmonic ) pendulum - a weight of mass m attached to a spring of stiffness k, the other end of which is fixed motionlessly. The cyclic frequency of the load is equal to:

Self-oscillations Are continuous free oscillations supported by periodic pumping of energy from some source of external force. An example of a self-oscillating system is a mechanical watch.