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 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.

1. Fluctuations. periodic fluctuations. Harmonic vibrations.

2. Free vibrations. Undamped and damped oscillations.

3. Forced vibrations. Resonance.

4. Comparison of oscillatory processes. Energy of undamped harmonic oscillations.

5. Self-oscillations.

6. Oscillations of the human body and their registration.

7. Basic concepts and formulas.

8. Tasks.

1.1. Fluctuations. periodic fluctuations.

Harmonic vibrations

fluctuations processes that differ in varying degrees of repetition are called.

recurring processes continuously occur inside any living organism, for example: heart contractions, lung function; we shiver when we are cold; we hear and speak thanks to the vibrations of the eardrums and vocal cords; When walking, our legs make oscillatory movements. The atoms that make us vibrate. The world we live in is remarkably prone to fluctuations.

Depending on the physical nature of the repeating process, oscillations are distinguished: mechanical, electrical, etc. This lecture discusses mechanical vibrations.

Periodic fluctuations

periodic called such oscillations in which all the characteristics of the movement are repeated after a certain period of time.

For periodic oscillations, the following characteristics are used:

oscillation period T, equal to the time during which one complete oscillation takes place;

oscillation frequencyν, equal to the number of oscillations per second (ν = 1/T);

oscillation amplitude A, equal to the maximum displacement from the equilibrium position.

Harmonic vibrations

A special place among periodic fluctuations is occupied by harmonic fluctuations. Their importance is due to the following reasons. Firstly, oscillations in nature and technology often have a character very close to harmonic, and secondly, periodic processes of a different form (with a different time dependence) can be represented as a superposition of several harmonic oscillations.

Harmonic vibrations- these are oscillations in which the observed value changes in time according to the law of sine or cosine:

In mathematics, functions of this kind are called harmonic, therefore, oscillations described by such functions are also called harmonic.

The position of a body that performs an oscillatory motion is characterized by displacement about the equilibrium position. In this case, the quantities in formula (1.1) have the following meaning:

X- bias body at time t;

BUT - amplitude fluctuations equal to the maximum displacement;

ω - circular frequency oscillations (the number of oscillations made in 2 π seconds), related to the oscillation frequency by the ratio

φ = (ωt +φ 0) - phase fluctuations (at time t); φ 0 - initial phase oscillations (at t = 0).

Rice. 1.1. Plots of offset versus time for x(0) = A and x(0) = 0

1.2. Free vibrations. Undamped and damped oscillations

free or own called such oscillations that occur in a system left to itself, after it has been taken out of equilibrium.

An example is the oscillation of a ball suspended on a thread. In order to cause vibrations, you need to either push the ball, or, moving it aside, release it. When pushed, the ball is informed kinetic energy, and in case of deviation - potential.

Free oscillations are performed due to the initial energy reserve.

Free undamped vibrations

Free oscillations can be undamped only in the absence of friction force. Otherwise, the initial supply of energy will be spent on overcoming it, and the range of oscillations will decrease.

As an example, consider the vibrations of a body suspended on a weightless spring, which occur after the body is deflected downward and then released (Fig. 1.2).

Rice. 1.2. Vibrations of a body on a spring

From the side of the stretched spring, the body acts elastic force F proportional to the amount of displacement X:

The constant factor k is called spring rate and depends on its size and material. The "-" sign indicates that the elastic force is always directed in the direction opposite to the displacement direction, i.e. to the equilibrium position.

In the absence of friction, the elastic force (1.4) is the only force acting on the body. According to Newton's second law (ma = F):

After transferring all the terms to the left side and dividing by the body mass (m), we obtain a differential equation for free oscillations in the absence of friction:

The value ω 0 (1.6) turned out to be equal to the cyclic frequency. This frequency is called own.

Thus, free vibrations in the absence of friction are harmonic if, when deviating from the equilibrium position, elastic force(1.4).

Own circular frequency is the main characteristic of free harmonic oscillations. This value depends only on the properties of the oscillating system (in the case under consideration, on the mass of the body and the stiffness of the spring). In what follows, the symbol ω 0 will always be used to denote natural circular frequency(i.e., the frequency at which vibrations would occur in the absence of friction).

Amplitude of free vibrations is determined by the properties of the oscillatory system (m, k) and the energy imparted to it at the initial moment of time.

In the absence of friction, free oscillations close to harmonic ones also arise in other systems: mathematical and physical pendulums (the theory of these issues is not considered) (Fig. 1.3).

Mathematical pendulum- a small body (material point) suspended on a weightless thread (Fig. 1.3 a). If the thread is deflected from the equilibrium position by a small (up to 5°) angle α and released, then the body will oscillate with a period determined by the formula

where L is the length of the thread, g is the free fall acceleration.

Rice. 1.3. Mathematical pendulum (a), physical pendulum (b)

physical pendulum- a rigid body that oscillates under the action of gravity around a fixed horizontal axis. Figure 1.3 b schematically shows a physical pendulum in the form of a body of arbitrary shape, deviated from the equilibrium position by an angle α. The oscillation period of a physical pendulum is described by the formula

where J is the moment of inertia of the body about the axis, m is the mass, h is the distance between the center of gravity (point C) and the suspension axis (point O).

The moment of inertia is a quantity that depends on the mass of the body, its dimensions and position relative to the axis of rotation. The moment of inertia is calculated using special formulas.

Free damped vibrations

Friction forces acting in real systems significantly change the nature of motion: the energy of an oscillatory system constantly decreases, and oscillations either fade out or do not occur at all.

The resistance force is directed in the direction opposite to the movement of the body, and at not very high speeds it is proportional to the speed:

A graph of such fluctuations is shown in Fig. 1.4.

As a characteristic of the degree of attenuation, a dimensionless quantity is used, called logarithmic damping decrementλ.

Rice. 1.4. Displacement versus time for damped oscillations

Logarithmic damping decrement is equal to the natural logarithm of the ratio of the amplitude of the previous oscillation to the amplitude of the subsequent oscillation.

where i is the ordinal number of the oscillation.

It is easy to see that the logarithmic damping decrement is found by the formula

Strong attenuation. At

if the condition β ≥ ω 0 is fulfilled, the system returns to the equilibrium position without oscillating. Such a movement is called aperiodic. Figure 1.5 shows two possible ways to return to the equilibrium position during aperiodic motion.

Rice. 1.5. aperiodic motion

1.3. Forced vibrations, resonance

Free vibrations in the presence of friction forces are damped. Continuous oscillations can be created with the help of a periodic external action.

compelled such oscillations are called, during which the oscillating system is exposed to an external periodic force (it is called a driving force).

Let the driving force change according to the harmonic law

The graph of forced oscillations is shown in Fig. 1.6.

Rice. 1.6. Plot of displacement versus time for forced vibrations

It can be seen that the amplitude of forced oscillations reaches a steady value gradually. The steady forced oscillations are harmonic, and their frequency is equal to the frequency of the driving force:

The amplitude (A) of steady forced oscillations is found by the formula:

Resonance called the achievement of the maximum amplitude of forced oscillations at a certain value of the frequency of the driving force.

If condition (1.18) is not satisfied, then resonance does not arise. In this case, as the frequency of the driving force increases, the amplitude of forced oscillations decreases monotonically, tending to zero.

The graphical dependence of the amplitude A of forced oscillations on the circular frequency of the driving force at different values ​​of the damping coefficient (β 1 > β 2 > β 3) is shown in fig. 1.7. Such a set of graphs is called resonance curves.

In some cases, a strong increase in the amplitude of oscillations at resonance is dangerous for the strength of the system. There are cases when resonance led to the destruction of structures.

Rice. 1.7. Resonance curves

1.4. Comparison of oscillatory processes. Energy of undamped harmonic oscillations

Table 1.1 presents the characteristics of the considered oscillatory processes.

Table 1.1. Characteristics of free and forced vibrations

Energy of undamped harmonic oscillations

A body that performs harmonic oscillations has two types of energy: the kinetic energy of motion E k \u003d mv 2 / 2 and the potential energy E p associated with the action of an elastic force. It is known that under the action of elastic force (1.4) the potential energy of the body is determined by the formula E p = kx 2 /2. For undamped oscillations X= A cos(ωt), and the speed of the body is determined by the formula v= - A ωsin(ωt). From this, expressions are obtained for the energies of a body performing undamped oscillations:

The total energy of the system in which undamped harmonic oscillations occur is the sum of these energies and remains unchanged:

Here m is the mass of the body, ω and A are the circular frequency and amplitude of oscillations, k is the coefficient of elasticity.

1.5. Self-oscillations

There are systems that themselves regulate the periodic replenishment of lost energy and therefore can fluctuate for a long time.

Self-oscillations- undamped oscillations supported by an external source of energy, the supply of which is regulated by the oscillatory system itself.

Systems in which such oscillations occur are called self-oscillating. The amplitude and frequency of self-oscillations depend on the properties of the self-oscillating system itself. The self-oscillatory system can be represented by the following scheme:

In this case, the oscillatory system itself, through a feedback channel, affects the energy regulator, informing it about the state of the system.

Feedback called the impact of the results of any process on its course.

If such an impact leads to an increase in the intensity of the process, then the feedback is called positive. If the impact leads to a decrease in the intensity of the process, then the feedback is called negative.

In a self-oscillating system, both positive and negative feedback can be present.

An example of a self-oscillating system is a clock in which the pendulum receives shocks due to the energy of a raised weight or a twisted spring, and these shocks occur at those moments when the pendulum passes through the middle position.

Examples of biological self-oscillatory systems are such organs as the heart and lungs.

1.6. Oscillations of the human body and their registration

The analysis of oscillations created by the human body or its individual parts is widely used in medical practice.

Oscillatory movements of the human body when walking

Walking is a complex periodic locomotor process resulting from the coordinated activity of the skeletal muscles of the trunk and limbs. Analysis of the walking process provides many diagnostic features.

A characteristic feature of walking is the periodicity of the support position with one foot (single support period) or two legs (double support period). Normally, the ratio of these periods is 4:1. When walking, there is a periodic displacement of the center of mass (CM) along the vertical axis (normally by 5 cm) and deviation to the side (normally by 2.5 cm). In this case, the CM moves along a curve, which can be approximately represented by a harmonic function (Fig. 1.8).

Rice. 1.8. Vertical displacement of the CM of the human body during walking

Complex oscillatory movements while maintaining the vertical position of the body.

A person standing vertically experiences complex oscillations of the common center of mass (MCM) and center of pressure (CP) of the feet on the support plane. Based on the analysis of these fluctuations statokinesimetry- a method for assessing a person's ability to maintain an upright posture. By keeping the GCM projection within the coordinates of the boundary of the support area. This method is implemented using a stabilometric analyzer, the main part of which is a stabiloplatform, on which the subject is in a vertical position. Oscillations made by the subject's CP while maintaining a vertical posture are transmitted to the stabiloplatform and recorded by special strain gauges. The strain gauge signals are transmitted to the recording device. At the same time, it is recorded statokinesigram - the trajectory of the test subject's movement on a horizontal plane in a two-dimensional coordinate system. According to the harmonic spectrum statokinesigrams it is possible to judge the features of verticalization in the norm and with deviations from it. This method makes it possible to analyze the indicators of statokinetic stability (SCR) of a person.

Mechanical vibrations of the heart

There are various methods for studying the heart, which are based on mechanical periodic processes.

Ballistocardiography(BCG) - a method for studying the mechanical manifestations of cardiac activity, based on the registration of pulse micro-movements of the body, caused by the ejection of blood from the ventricles of the heart into large vessels. This gives rise to the phenomenon returns. The human body is placed on a special movable platform located on a massive fixed table. The platform as a result of recoil comes into a complex oscillatory motion. The dependence of the displacement of the platform with the body on time is called a ballistocardiogram (Fig. 1.9), the analysis of which allows one to judge the movement of blood and the state of cardiac activity.

Apexcardiography(AKG) - a method of graphic registration of low-frequency oscillations of the chest in the area of ​​the apex beat, caused by the work of the heart. Registration of the apexcardiogram is performed, as a rule, on a multichannel electrocardiogram.

Rice. 1.9. Recording a ballistocardiogram

graph using a piezocrystalline sensor, which is a converter of mechanical vibrations into electrical ones. Before recording on the anterior wall of the chest, the point of maximum pulsation (apex beat) is determined by palpation, in which the sensor is fixed. Based on the sensor signals, an apexcardiogram is automatically built. An amplitude analysis of the ACG is carried out - the amplitudes of the curve are compared at different phases of the work of the heart with a maximum deviation from the zero line - the EO segment, taken as 100%. Figure 1.10 shows the apexcardiogram.

Rice. 1.10. Apexcardiogram recording

Kinetocardiography(KKG) - a method of recording low-frequency vibrations of the chest wall, caused by cardiac activity. The kinetocardiogram differs from the apexcardiogram: the first records the absolute movements of the chest wall in space, the second records the fluctuations of the intercostal spaces relative to the ribs. This method determines the displacement (KKG x), the speed of movement (KKG v) as well as the acceleration (KKG a) for chest oscillations. Figure 1.11 shows a comparison of various kinetocardiograms.

Rice. 1.11. Recording kinetocardiograms of displacement (x), speed (v), acceleration (a)

Dynamocardiography(DKG) - a method for assessing the movement of the center of gravity of the chest. Dynamocardiograph allows you to register the forces acting from the human chest. To record a dynamocardiogram, the patient is positioned on the table lying on his back. Under the chest there is a perceiving device, which consists of two rigid metal plates measuring 30x30 cm, between which there are elastic elements with strain gauges mounted on them. Periodically changing in magnitude and place of application, the load acting on the receiving device is composed of three components: 1) a constant component - the mass of the chest; 2) variable - mechanical effect of respiratory movements; 3) variable - mechanical processes accompanying cardiac contraction.

Recording of dynamocardiogram is carried out while holding the breath of the subject in two directions: relative to the longitudinal and transverse axes of the receiving device. Comparison of various dynamocardiograms is shown in fig. 1.12.

Seismocardiography is based on the registration of mechanical vibrations of the human body caused by the work of the heart. In this method, using sensors installed in the region of the base of the xiphoid process, a cardiac impulse is recorded due to the mechanical activity of the heart during the period of contraction. At the same time, processes occur associated with the activity of tissue mechanoreceptors of the vascular bed, which are activated when the volume of circulating blood decreases. The seismocardiosignal forms the shape of the sternum oscillations.

Rice. 1.12. Recording of normal longitudinal (a) and transverse (b) dynamocardiograms

Vibration

The widespread introduction of various machines and mechanisms into human life increases labor productivity. However, the work of many mechanisms is associated with the occurrence of vibrations that are transmitted to a person and have a harmful effect on him.

Vibration- forced oscillations of the body, in which either the whole body oscillates as a whole, or its separate parts oscillate with different amplitudes and frequencies.

A person constantly experiences various kinds of vibrational effects in transport, at work, at home. Vibrations that have arisen in any place of the body (for example, the hand of a worker holding a jackhammer) propagate throughout the body in the form of elastic waves. These waves cause variable deformations of various types in the tissues of the body (compression, tension, shear, bending). The effect of vibrations on a person is due to many factors that characterize vibrations: frequency (frequency spectrum, fundamental frequency), amplitude, speed and acceleration of an oscillating point, energy of oscillatory processes.

Prolonged exposure to vibrations causes persistent disturbances in normal physiological functions in the body. "Vibration sickness" may occur. This disease leads to a number of serious disorders in the human body.

The influence that vibrations have on the body depends on the intensity, frequency, duration of vibrations, the place of their application and direction in relation to the body, posture, as well as on the state of the person and his individual characteristics.

Fluctuations with a frequency of 3-5 Hz cause reactions of the vestibular apparatus, vascular disorders. At frequencies of 3-15 Hz, disorders associated with resonant vibrations of individual organs (liver, stomach, head) and the body as a whole are observed. Fluctuations with frequencies of 11-45 Hz cause blurred vision, nausea, and vomiting. At frequencies exceeding 45 Hz, damage to the vessels of the brain, impaired blood circulation, etc. occur. Figure 1.13 shows the vibration frequency ranges that have a harmful effect on a person and his organ systems.

Rice. 1.13. The frequency ranges of the harmful effects of vibration on humans

At the same time, in some cases, vibrations are used in medicine. For example, using a special vibrator, the dentist prepares an amalgam. The use of high-frequency vibration devices allows drilling a hole of complex shape in the tooth.

Vibration is also used in massage. With manual massage, the massaged tissues are brought into oscillatory motion with the help of the massage therapist's hands. With hardware massage, vibrators are used, in which tips of various shapes are used to transmit oscillatory movements to the body. Vibrating devices are divided into devices for general vibration, causing shaking of the whole body (vibrating "chair", "bed", "platform", etc.), and devices for local vibration impact on individual parts of the body.

Mechanotherapy

In physiotherapy exercises (LFK), simulators are used, on which oscillatory movements of various parts of the human body are carried out. They are used in mechanotherapy - form of exercise therapy, one of the tasks of which is the implementation of dosed, rhythmically repeated physical exercises for the purpose of training or restoring mobility in the joints on pendulum-type devices. The basis of these devices is balancing (from fr. balancer- swing, balance) a pendulum, which is a two-arm lever that performs oscillatory (rocking) movements around a fixed axis.

1.7. Basic concepts and formulas

Table continuation

Table continuation

End of table

1.8. Tasks

1. Give examples of oscillatory systems in humans.

2. In an adult, the heart makes 70 contractions per minute. Determine: a) the frequency of contractions; b) the number of cuts in 50 years

Answer: a) 1.17 Hz; b) 1.84x10 9 .

3. What length must a mathematical pendulum have in order for its period of oscillation to be equal to 1 second?

4. A thin straight homogeneous rod 1 m long is suspended by its end on an axis. Determine: a) what is the period of its oscillations (small)? b) what is the length of a mathematical pendulum with the same period of oscillation?

5. A body with a mass of 1 kg oscillates according to the law x = 0.42 cos (7.40t), where t is measured in seconds, and x is measured in meters. Find: a) amplitude; b) frequency; c) total energy; d) kinetic and potential energies at x = 0.16 m.

6. Estimate the speed at which a person walks with a stride length l= 0.65 m. Leg length L = 0.8 m; the center of gravity is at a distance H = 0.5 m from the foot. For the moment of inertia of the leg relative to the hip joint, use the formula I = 0.2mL 2 .

7. How can you determine the mass of a small body aboard a space station if you have a clock, a spring, and a set of weights at your disposal?

8. The amplitude of damped oscillations decreases in 10 oscillations by 1/10 of its original value. Oscillation period T = 0.4 s. Determine the logarithmic decrement and damping factor.

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 oscillations.

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

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 its maximum value if the frequency of the driving force coincides with the frequency of the oscillatory 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 if a sound wave is directed at it 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.

In musical instruments, the role of resonators is performed by parts of their bodies. 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.