The solution of a number of issues, in particular the addition of several oscillations of the same direction (or, what is the same, the addition of several harmonic functions), is greatly facilitated and becomes clear if the oscillations are graphically depicted as vectors on a plane. The scheme obtained in this way is called a vector diagram.

Take the axis, which we denote by the letter x (Fig. 55.1). From the point O, taken on the axis, we plot a vector of length a, forming an angle a with the axis.

If we bring this vector into rotation with an angular velocity , then the projection of the end of the vector will move along the x-axis in the range from -a to +a, and the coordinate of this projection will change over time according to the law

Consequently, the projection of the end of the vector onto the axis will perform a harmonic oscillation with an amplitude equal to the length of the vector, with a circular frequency equal to the angular velocity of rotation of the vector, and with an initial phase equal to the angle formed by the vector with the axis at the initial moment of time.

From what has been said, it follows that a harmonic oscillation can be specified using a vector whose length is equal to the amplitude of the oscillation, and the direction of the vector forms an angle with the x-axis equal to the initial phase of the oscillation.

Consider the addition of two harmonic oscillations of the same direction and the same frequency. The displacement x of the oscillating body will be the sum of the displacements, which will be written as follows:

Let's represent both fluctuations with the help of vectors (fig. 55.2). Let us construct the resulting vector a according to the rules of vector addition.

It is easy to see that the projection of this vector on the x-axis is equal to the sum of the projections of the terms of the vectors:

Therefore, the vector a represents the resulting oscillation. This vector rotates with the same angular velocity as the vectors so that the resulting motion will be a harmonic oscillation with frequency amplitude a and initial phase a. It is clear from the construction that

So, the representation of harmonic oscillations by means of vectors makes it possible to reduce the addition of several oscillations to the operation of adding vectors. This technique is especially useful, for example, in optics, where light vibrations at a certain point are defined as the result of a superposition of many vibrations coming to a given point from different parts of the wave front.

Formulas (55.2) and (55.3) can, of course, be obtained by adding expressions (55.1) and performing the corresponding trigonometric transformations. But the way we have used to obtain these formulas is more simple and clear.

Let us analyze the expression (55.2) for the amplitude. If the phase difference of both oscillations is equal to zero, the amplitude of the resulting oscillation is equal to the sum of a and . If the phase difference is equal to or , i.e., both oscillations are in antiphase, then the amplitude of the resulting oscillation is equal to

If the oscillation frequencies are not the same, the vectors a and will rotate at different speeds. In this case, the resulting vector a pulsates in magnitude and rotates at a non-constant rate. Consequently, the resulting motion in this case will not be a harmonic oscillation, but some complex oscillatory process.

The addition of several oscillations of the same direction (or, what is the same, the addition of several harmonic functions) is greatly facilitated and becomes clear if the oscillations are depicted graphically as vectors on a plane.

Let's take the axis, which we will denote by "x". From the point O, taken on the axis, at an angle a equal to the initial phase of the oscillations, we plot the length vector A (Fig. 8.3). We project the vector A onto the x axis, we get x 0 =A cos a is the initial displacement of the oscillating point from the equilibrium position. We bring this vector into counterclockwise rotation with an angular velocity w 0 . The position of this vector at any time will be characterized by angles equal to:

w 0 t 1 +a; w 0 t 2 +a; w 0 t 3 +a; etc.

And the projection of this vector will move along the x-axis in the range from -A to +A. Moreover, the coordinate of this projection will change over time according to the law:

.

Therefore, the projection of the end of the vector on some arbitrary axis will perform a harmonic oscillation with an amplitude equal to the length of the vector, a circular frequency equal to the angular velocity of rotation of the vector and an initial phase equal to the angle formed by the vector with the axis at the initial moment of time.

So, a harmonic oscillation can be specified using a vector, the length of which is equal to the amplitude of the oscillation, and the direction of the vector forms an angle with the “x” axis equal to the initial phase of the oscillation.

Consider the addition of two harmonic oscillations of the same direction and the same frequency. The displacement of the oscillating body “x” will be the sum of the displacements x 1 and x 2, which will be written as follows:

Let's represent both fluctuations with the help of vectors and (Fig. 8.4) According to the rules of addition of vectors, we build the resulting vector. The projection of this vector on the X axis will be equal to the sum of the projections of the terms of the vectors: x=x 1 +x 2 . Therefore, the vector represents the resulting oscillation. This vector rotates with the same angular velocity w 0 as the vectors and , so that the resulting motion will be a harmonic oscillation c with frequency w 0 , amplitude "a" and initial phase a. It follows from the construction that

So, the representation of harmonic oscillations by means of vectors makes it possible to reduce the addition of several oscillations to the operation of adding vectors. This method is more simple and clear than the use of trigonometric transformations.

Let us analyze the expression for the amplitude. If the phase difference of both oscillations a 2 - a 1 = 0, then the amplitude of the resulting oscillation is equal to the sum ( a 2 + a one). If the phase difference a 2 - a 1 = +p or -p, i.e. oscillations are in antiphase, then the amplitude of the resulting oscillation is .

If the oscillation frequencies x 1 and x 2 are not the same, the vectors and will rotate at different speeds. In this case, the resulting vector pulsates in magnitude and rotates at a non-constant rate. Therefore, the resulting motion will be in this case not just a harmonic oscillation, but some complex oscillatory process.

Let's choose an axis. From the point O, taken on this axis, we set aside the length vector, which forms an angle with the axis. If we bring this vector into rotation with an angular velocity , then the projection of the end of the vector onto the axis will change with time according to the law . Therefore, the projection of the end of the vector onto the axis will make harmonic oscillations with an amplitude equal to the length of the vector; with a circular frequency equal to the angular velocity of rotation, and with an initial phase equal to the angle formed by the vector with the axis X at the initial time.

The vector diagram makes it possible to reduce the addition of oscillations to the geometric summation of vectors. Consider the addition of two harmonic oscillations of the same direction and the same frequency, which have the following form:

Let's represent both fluctuations with the help of vectors and (fig. 7.5). Let's build the resulting vector according to the vector addition rule. It is easy to see that the projection of this vector onto the axis is equal to the sum of the projections of the terms of the vectors. Therefore, the vector represents the resulting oscillation. This vector rotates with the same angular velocity as the vectors , so that the resulting motion will be a harmonic oscillation with frequency , amplitude and initial phase . According to the law of cosines, the square of the amplitude of the resulting oscillation will be equal to

So, the representation of harmonic oscillations by means of vectors makes it possible to reduce the addition of several oscillations to the operation of adding vectors. Formulas (7.3) and (7.4) can, of course, be obtained by adding the expressions for and analytically, but the vector diagram method is more simple and clear.

DAMPING OSCILLATIONS

In any real oscillatory system there are resistance forces, the action of which leads to a decrease in the energy of the system. If the loss of energy is not replenished by the work of external forces, the oscillations will decay. In the simplest, and at the same time the most common, case, the drag force is proportional to the speed:

,

where r is a constant value called the drag coefficient. The minus sign is due to the fact that force and speed have opposite directions; hence their projections on the axis X have different signs. The equation of Newton's second law in the presence of resistance forces has the form:

.

Using the notation , , we rewrite the equation of motion as follows:

.

This equation describes fading system oscillations. The coefficient is called the damping factor.

The experimental graph of damped oscillations at a low damping coefficient is shown in Fig. . 7.6. From fig. 7.6 it can be seen that the dependence graph looks like a cosine multiplied by some function, which decreases with time. This function is represented in the figure by dashed lines. A simple function that behaves this way is the exponential function. Therefore, the solution can be written as:

,

where is the frequency of damped oscillations.

Value x periodically passes through zero and reaches a maximum and minimum an infinite number of times. The time interval between two successive passages through zero is . Its doubled value is called period of oscillation.

The multiplier in front of a periodic function is called amplitude of damped oscillations. It decreases exponentially with time. The decay rate is determined by the value . The time after which the amplitude of the oscillations decreases by a factor is called decay time. During this time, the system oscillates. It is customary to characterize the damping of oscillations logarithmic damping decrement. The logarithmic damping decrement is the logarithm of the ratio of amplitudes at the moments of successive passages of an oscillating value through a maximum or minimum:

.

It is related to the number of oscillations by the ratio:

The value is called quality factor of the oscillatory system. The quality factor is higher, the greater the number of oscillations the system has time to complete before the amplitude decreases by a factor.

The constants and , as in the case of harmonic oscillations, can be determined from the initial conditions.

FORCED VIBRATIONS

Oscillations that occur under the influence of an external periodic force are called forced. The external force performs positive work and provides an influx of energy to the oscillatory system. It does not allow oscillations to fade, despite the action of resistance forces.

A periodic external force can vary in time according to various laws. Of particular interest is the case when an external force, changing according to a harmonic law with a frequency ω, acts on an oscillatory system capable of performing natural oscillations at a certain frequency ω 0 . For example, if you pull a load suspended on a spring with a frequency , then it will perform harmonic oscillations with the frequency of the external force, even if this frequency does not coincide with the natural frequency of the spring.

Let a periodic external force act on the system. In this case, one can obtain the following equation describing the motion of such a system:

,

(7.5)

where . With forced oscillations, the amplitude of the oscillations, and, consequently, the energy transmitted to the oscillatory system, depend on the ratio between the frequencies and , as well as on the damping coefficient .

After the beginning of the impact of an external force on the oscillatory system, some time ωt is needed to establish forced oscillations. At the initial moment, both processes are excited in the oscillatory system - forced oscillations at a frequency ω and free oscillations at a natural frequency ω 0 . But free vibrations are damped due to the inevitable presence of friction forces. Therefore, after some time, only stationary oscillations at the frequency ω of the external driving force remain in the oscillatory system. The settling time is equal in order of magnitude to the decay time ω of free oscillations in the oscillatory system. The steady forced oscillations of the load on the spring occur according to the harmonic law with a frequency equal to the frequency of the external influence. It can be shown that in the steady state the solution of equation (7.6) is written as:

,

,
.

Thus, forced oscillations are harmonic oscillations with a frequency equal to the frequency of the driving force. The amplitude of forced oscillations is proportional to the amplitude of the driving force. For a given oscillatory system (that is, a system with certain values ​​of and ), the amplitude depends on the frequency of the driving force. The forced vibrations are out of phase with the driving force. The phase shift depends on the frequency of the driving force.

RESONANCE

The dependence of the amplitude of forced oscillations on the frequency of the driving force leads to the fact that at a certain frequency determined for a given system, the oscillation amplitude reaches its maximum value. The oscillatory system is especially responsive to the action of the driving force at this frequency. This phenomenon is called resonance, and the corresponding frequency is resonant frequency. Graphically, the dependence of the amplitude x m of forced oscillations on the frequency ω of the driving force is described by a resonant curve (Fig. 7.9).

We investigate the behavior of the amplitude of forced oscillations depending on the frequency . Leaving the amplitude of the driving force unchanged, we will change its frequency. When we get static deflection under the action of a constant force:

As the frequency increases, the displacement amplitude first also increases, then passes through a maximum, and finally asymptotically tends to zero. From fig. 7.9 also shows that the smaller , the higher and to the right lies the maximum of this curve. In addition, the smaller , the stronger the amplitude near the resonance changes with frequency, the sharper the maximum.

The phenomenon of resonance can cause the destruction of bridges, buildings and other structures, if the natural frequencies of their oscillations coincide with the frequency of a periodically acting external force. The phenomenon of resonance has to be taken into account when designing machines and various kinds of structures. In no case should the natural frequency of these devices be close to the frequency of possible external influences.

Examples

In January 1905 Petersburg, the Egyptian bridge collapsed. Guilty of this were 9 passers-by, 2 cab drivers and the 3rd squadron of the Peterhof Horse Guards Regiment. The following happened. All the soldiers paced rhythmically across the bridge. The bridge began to sway from this - to oscillate. By a coincidence, the natural frequency of the bridge coincided with the step frequency of the soldiers. The rhythmic step of the formation informed the bridge of more and more portions of energy. As a result of the resonance, the bridge swayed so much that it collapsed. If there were no resonance of the natural frequency of the bridge with the step frequency of the soldiers, nothing would have happened to the bridge. Therefore, when passing soldiers on weak bridges, it is customary to give the command to “knock down the leg”.

It is said that the great tenor Enrico Caruso could cause a glass goblet to shatter by singing a note of the proper pitch at the top of his voice. In this case, the sound causes forced vibrations of the walls of the glass. At resonance, the vibrations of the walls can reach such an amplitude that the glass breaks.

Do experiments

Go to some stringed musical instrument and loudly shout "a": one of the strings will respond - it will sound. The one that is in resonance with the frequency of this sound will vibrate more strongly than the other strings - it will respond to the sound.

Stretch a thin rope horizontally. Attach a pendulum of thread and plasticine to it. Throw another similar pendulum over the rope, but with a longer thread. The length of the suspension of this pendulum can be changed by pulling the free end of the thread by hand. Bring this pendulum into oscillatory motion. In this case, the first pendulum will also begin to oscillate, but with a smaller amplitude. Without stopping the oscillations of the second pendulum, gradually reduce the length of its suspension - the amplitude of oscillations of the first pendulum will increase. In this experiment, illustrating the resonance of mechanical vibrations, the first pendulum is the receiver of vibrations excited by the second pendulum. The reason forcing the first pendulum to oscillate is the periodic vibrations of the rope with a frequency equal to the frequency of oscillations of the second pendulum. Forced oscillations of the first pendulum will have a maximum amplitude only when its natural frequency coincides with the oscillation frequency of the second pendulum.

AUTO OSCILLATIONS

Numerous and diverse are the creations of human hands, in which self-oscillations arise and are used. First of all, these are various musical instruments. Already in ancient times - horns and horns, pipes, whistles, primitive flutes. Later - violins, in which the force of friction between the bow and the string is used to excite the sound; various wind instruments; harmonies in which sound is produced by metal reeds vibrating under the influence of a constant stream of air; organs from whose tubes resonating columns of air escape through narrow slits.

Rice. 7.12

It is well known that the force of sliding friction is practically independent of speed. However, it is due to the very weak dependence of the friction force on speed that the violin string sounds. A qualitative view of the dependence of the force of friction of the bow on the string is shown in Fig. 7.12. Due to the static friction force, the string is captured by the bow and is displaced from the equilibrium position. When the elastic force exceeds the friction force, the string will break away from the bow and rush towards the equilibrium position with ever increasing speed. The speed of the string relative to the moving bow will increase, the friction force will increase and at a certain moment it will become sufficient to capture the string. Then the process will repeat again. Thus, a bow moving at a constant speed will cause undamped vibrations of the string.

In bowed string instruments, self-oscillations are supported by the friction force acting between the bow and string, and in wind instruments, blowing an air jet maintains self-oscillations of the air column in the instrument's pipe.

More than a hundred Greek and Latin documents from different times mention the singing of the famous "Memnon colossus" - a majestic sounding statue of one of the pharaohs, who ruled in the XIV century BC, installed near the Egyptian city of Luxor. The height of the statue is about 20 meters, the mass reaches a thousand tons. In the lower part of the colossus, a number of crevices and holes were found with chambers of complex shape located behind them. The Colossus of Memnon is a gigantic organ that sounds under the influence of natural air currents. The statue imitates the human voice.

Natural self-oscillations of a somewhat exotic nature are singing sands. As far back as the 14th century, the great traveler Marco Polo mentioned the "sounding shores" of the mysterious lake Lop Nor in Asia. For six centuries, singing sands have been discovered in various places on all continents. In the local population, they in most cases cause fear, are the subject of legends and legends. Jack London describes the meeting with the singing sands of the characters of the novel "Hearts of Three", who went with a guide in search of the treasures of the ancient Maya.

"When the gods laugh, beware!" shouted the old man warningly. He drew a circle in the sand with his finger, and as he drew, the sand howled and screeched; then the old man knelt down, the sand roared and trumpeted.

There are singing sands and even a whole singing sandy mountain near the Ili River in Kazakhstan. Mount Kalkan, a giant natural organ, rose almost 300 meters. People call it differently: “singing dune”, “singing mountain”. It is built of light-colored sand and, against the background of the dark spurs of the Dzungarian Alatau, the Big and Small Kalkans, it presents an extraordinary sight due to the color contrast. In the wind and even when a person descends from it, the mountain makes melodious sounds. After the rain and during the calm, the mountain is silent. Tourists like to visit the Singing Dune and, having climbed one of its three peaks, admire the opened panorama of the Ili and the Zailiysky Alatau ridge. If the mountain is silent, impatient visitors "make it sing". To do this, you need to quickly run down the slope of the mountain, sandy streams will run from under your feet, and a buzz will arise from the depths of the dune.

Many centuries have passed since the discovery of the singing sands, and a satisfactory explanation for this amazing phenomenon has not been offered. In recent years, English acousticians, as well as the Soviet scientist V.I. Arabadzhi. Arabadji suggested that the sound-emitting upper layer of sand moves under some kind of constant perturbation over the lower, harder layer, which has a wavy surface profile. Due to the forces of friction during the mutual displacement of the layers, sound is excited.

Forced vibrations are undamped vibrations. The inevitable loss of energy due to friction during forced oscillations is compensated by the supply of energy from an external source of a periodically acting force. There are systems in which undamped oscillations arise not due to periodic external influences, but as a result of the ability of such systems to regulate the flow of energy from a constant source. Such systems are called self-oscillatory, and the process of undamped oscillations in such systems is called self-oscillations. . Schematically, a self-oscillating system can be represented as an energy source, a damped oscillator, and a feedback device between the oscillating system and the source (Fig. 7.10).

As an oscillatory system, any mechanical system capable of performing its own damped oscillations (for example, a pendulum of a wall clock) can be used. The energy source can be a deformed spring or a load in a gravitational field. The feedback device is a mechanism by which the self-oscillatory system regulates the flow of energy from the source.

An example of a mechanical self-oscillating system is a clockwork with an anchor stroke (Fig. 7.11). In a clock with an anchor stroke, a running wheel with oblique teeth is rigidly fastened to a gear drum, through which a chain with a weight is thrown. At the upper end of the pendulum, an anchor is fixed with two plates of hard material bent along an arc of a circle centered on the axis of the pendulum. In wristwatches, the weight is replaced by a spring, and the pendulum is replaced by a balancer, fastened to a spiral spring. The balancer performs torsional vibrations around its axis. The oscillatory system in the clock is a pendulum or balancer, the source of energy is a weight lifted up or a wound spring. The feedback device is an anchor that allows the running wheel to turn one tooth in one half cycle. Feedback is provided by the interaction of the anchor with the running wheel. With each oscillation of the pendulum, the travel wheel tooth pushes the anchor fork in the direction of the pendulum movement, transferring to it a certain portion of energy, which compensates for the energy losses due to friction. Thus, the potential energy of the weight (or twisted spring) is gradually, in separate portions, transferred to the pendulum.

In everyday life, we, perhaps without noticing it ourselves, encounter self-oscillations more often than oscillations caused by periodic forces. Self-oscillations surround us everywhere in nature and technology: steam engines, internal combustion engines, electric bells, clocks, a sounding violin string or organ pipe, a beating heart, vocal cords when talking or singing - all these systems perform self-oscillations.

Do the experience!

Rice. 7.13

Oscillatory motion is usually studied by considering the behavior of some kind of pendulum: spring, mathematical or physical. All of them are solids. You can create a device that demonstrates the vibrations of liquid or gaseous bodies. To do this, use the idea behind the design of the water clock. Two one and a half liter plastic bottles are connected in the same way as in a water clock, fastening the lids. The cavities of the bottles are connected with a glass tube 15 centimeters long, with an inner diameter of 4-5 millimeters. The side walls of the bottles should be smooth and non-rigid, easily crushed when squeezed (see Fig. 7.13).

To start oscillations, a bottle of water is placed on top. Water from it immediately begins to flow through the tube into the lower bottle. After about a second, the jet spontaneously stops flowing and gives way to a passage in the tube for the oncoming movement of a portion of air from the lower bottle to the upper one. The order of passage of oncoming flows of water and air through the connecting tube is determined by the pressure difference in the upper and lower bottles and is automatically adjusted.

Pressure fluctuations in the system are evidenced by the behavior of the side walls of the upper bottle, which, in time with the release of water and the inlet of air, periodically squeeze and expand. Insofar as

WAVES FORMATION

How does vibration propagate? Is a medium necessary for the transmission of vibrations or can they be transmitted without it? How does sound from a sounding tuning fork reach the listener? How does a rapidly alternating current in a radio transmitter's antenna cause current to flow into a receiver's antenna? How does light from distant stars reach our eyes? To consider this kind of phenomena, it is necessary to introduce a new physical concept - a wave. Wave processes represent a general class of phenomena, despite their different nature.

The sources of waves, whether they are sea waves, waves in a string, earthquake waves or sound waves in the air, are vibrations. The process of propagation of oscillations in space is called a wave. For example, in the case of sound, the oscillatory motion is performed not only by the sound source (string, tuning fork), but also by the sound receiver - the eardrum or microphone membrane. The medium through which the wave propagates also oscillates.

The wave process is due to the presence of connections between the individual parts of the system, depending on which we have an elastic wave of one nature or another. A process taking place in any part of space causes changes in neighboring points of the system, transferring a certain amount of energy to them. From these points, the perturbation passes to those adjacent to them, and so on, spreading from point to point, that is, creating a wave.

Elastic forces acting between the elements of any solid, liquid or gaseous body lead to the appearance of elastic waves. An example of elastic waves is a wave propagating along a cord. If, by moving the hand up and down, vibrations of the end of the cord are excited, then the neighboring sections of the cord, due to the action of the elastic forces of the connection, will also begin to move, and a wave will propagate along the cord. A common property of waves is that they can propagate over long distances, and the particles of the medium oscillate only in a limited region of space. The particles of the medium in which the wave propagates are not involved by the wave in translational motion, they only oscillate around their equilibrium positions. Depending on the direction of oscillation of the particles of the medium with respect to the direction of wave propagation, longitudinal and transverse waves are distinguished. In a longitudinal wave, the particles of the medium oscillate along the direction of wave propagation; in the transverse - perpendicular to the direction of wave propagation. Elastic transverse waves can arise only in a medium with shear resistance. Therefore, in liquid and gaseous media, only longitudinal waves can occur. In a solid medium, both longitudinal and transverse waves can occur.

On fig. 8.1 shows the motion of particles during propagation in a medium of a transverse wave and the location of particles in the wave at four fixed points in time. Numbers 1, 2, etc. particles are indicated that are separated from each other by the distance traveled by the wave in a quarter of the period of oscillations performed by the particles. At the moment of time taken as zero, the wave, propagating along the axis from left to right, reached the particle 1 , as a result of which the particle began to move upward from the equilibrium position, dragging the next particles with it. After a quarter of the period, the particle 1 reaches the highest position; at the same time, the particle begins to move from the equilibrium position 2 . After another quarter of the period, the first particle will pass the equilibrium position, moving in the direction from top to bottom, the second particle will reach the extreme upper position, and the third particle will begin to move upward from the equilibrium position. At the moment of time equal to , the first particle will complete the complete oscillation and will be in the same state of motion as at the initial moment. The wave will reach the particle by the time 5 .

On fig. 8.2 shows the movement of particles during propagation in a medium of a longitudinal wave. All considerations concerning the behavior of particles in a transverse wave can also be applied to this case with the displacements up and down replaced by displacements to the right and left. From fig. 8.2 it can be seen that during the propagation of a longitudinal wave in the medium, alternating concentrations and rarefaction of particles are created, moving in the direction of wave propagation with a speed of .

Bodies that act on the medium, causing vibrations, are called wave sources. The propagation of elastic waves is not associated with the transfer of matter, but the waves transfer energy, which is provided by the wave process from the source of oscillations.

The locus of points to which perturbations reach a given moment of time is called the wave front. That is, the wave front is the surface that separates a part of the space already involved in the wave process from the area that the disturbances have not yet reached.

The locus of points oscillating in the same phases is called the wave surface. The wave surface can be drawn through any point in the space covered by the wave process. Wave surfaces can be of any shape. In the simplest cases, they have the shape of a plane or sphere. Accordingly, the wave in these cases is called flat or spherical. In a plane wave, the wave surfaces are a set of planes parallel to each other; in a spherical wave, a set of concentric spheres.

The distance over which a wave propagates in a time equal to the period of oscillation of the particles of the medium is called the wavelength. Obviously, , where is the wave propagation velocity.

On fig. 8.3, made using computer graphics, shows a model of the propagation of a transverse wave on water from a point source. Each particle performs harmonic oscillations around the equilibrium position.

Rice. 8.3. Propagation of a transverse wave from a point source of vibrations

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Vector diagram. Addition of vibrations.

The solution of a number of problems in the theory of oscillations is greatly facilitated and becomes more obvious if the oscillations are depicted graphically using the method vector diagrams. Let's choose some axis X. From a point 0 on the axis we plot the length vector , which first forms an angle with the axis (Fig. 2.14.1). If we bring this vector into rotation with an angular velocity , then the projection of the end of the vector onto the axis X will change over time according to the law

.

Therefore, the projection of the end of the vector onto the axis will perform a harmonic oscillation with an amplitude equal to the length of the vector, with a circular frequency equal to the angular velocity of rotation of the vector, and with an initial phase equal to the angle that the vector forms with the axis at the initial moment of time. The angle formed by the vector with the axis at a given moment of time determines the phase of the oscillation at that moment - .

From what has been said, it follows that a harmonic oscillation can be represented using a vector, the length of which is equal to the amplitude of the oscillation, and its direction forms an angle with some axis equal to the phase of the oscillation. This is the essence of the method of vector diagrams.

Addition of oscillations of the same direction.

Consider the addition of two harmonic oscillations, the directions of which are parallel:

. (2.14.1)

Resulting offset X will be the sum of and . It will be an oscillation with amplitude .

Let's use the method of vector diagrams (Fig. 2.14.2). in the figure, and are the phases of the resulting and added oscillations, respectively. It is easy to see what can be found by adding the vectors and . However, if the frequencies of the added oscillations are different, then the resulting amplitude changes in magnitude over time and the vector rotates at a non-constant speed, i.e. the oscillation will not be harmonic, but will represent some complex oscillatory process. In order for the resulting oscillation to be harmonic, the frequencies of the added oscillations must be the same

and the resulting oscillation occurs at the same frequency

.

It is clear from the construction that

Let us analyze the expression (2.14.2) for the amplitude of the resulting oscillation. If a the phase difference of the added oscillations is equal to zero(oscillations are in-phase), the amplitude is equal to the sum of the amplitudes of the added oscillations, i.e. has the maximum possible value . If a the phase difference is(oscillations are in antiphase), then the resulting amplitude is equal to the amplitude difference, i.e. has the smallest possible value .

Addition of mutually perpendicular oscillations.

Let the particle perform two harmonic oscillations with the same frequency: one along the direction, which we denote X, the other is in the perpendicular direction y. In this case, the particle will move along some, in the general case, a curvilinear trajectory, the shape of which depends on the phase difference of the oscillations.

We choose the origin of the time reference so that the initial phase of one oscillation is equal to zero:

. (2.14.3)

To obtain the particle trajectory equation, it is necessary to exclude from (2.14.3) t. From the first equation, a. means, . Let's rewrite the second equation

or

.

Transferring the first term from the right side of the equation to the left side, squaring the resulting equation and performing transformations, we obtain

. (2.14.4)

This equation is the equation of an ellipse whose axes are rotated relative to the axes X and y to some angle. But in some special cases simpler results are obtained.

1. The phase difference is zero. Then from (2.14.4) we get

or . (2.14.5)

This is the equation of a straight line (Fig. 2.14.3). Thus, the particle oscillates along this straight line with a frequency and amplitude equal to .

A vector diagram is a way to graphically define an oscillatory motion as a vector.

An oscillating value ξ (of any physical nature) is plotted along the horizontal axis. The vector plotted from the point 0 is equal in absolute value to the oscillation amplitude A and is directed at an angle α , equal to the initial phase of the oscillation, to the axis ξ. If we bring this vector into rotation with an angular velocity ω equal to the cyclic frequency of oscillations, then the projection of this vector onto the ξ axis gives the value of the oscillating quantity at an arbitrary moment in time.

Addition of oscillations of the same frequency and the same direction

Let there be two oscillations: we build vector diagrams and add vectors:

According to the law of cosines

As then

It is obvious (see the diagram) that the initial phase of the resulting oscillation is determined by the relation:

Addition of oscillations of close frequencies

P est, two oscillations with almost identical frequencies are added, i.e.

From trigonometry:

Applying to our case, we get:

The graph of the resulting oscillation is a beat graph, i.e. almost harmonic oscillations of the frequency ω, the amplitude of which slowly changes with the frequency Δω .

Amplitude due to the presence of the sign of the modulus (the amplitude is always > 0), the frequency with which the amplitude changes is not equal to Δω / 2, but twice as high - Δω.

Addition of mutually perpendicular oscillations

Let a small body oscillate on mutually perpendicular springs of the same stiffness. On what trajectory will this body move?

These are the trajectory equations in parametric form. To obtain an explicit relationship between the x and y coordinates, the parameter t must be excluded from the equations.

From the first equation: ,

From the second

After substitution

Let's get rid of the root:

is the equation of an ellipse

H
special cases:

27. Damped vibrations. Forced vibrations. Resonance.

Damping of free oscillations

Due to resistance, free oscillations always die out sooner or later. Let us consider the process of oscillation damping. Let us assume that the resistance force is proportional to the speed of the body. (the proportionality factor is indicated by 2mg for reasons of convenience, which will be revealed later). Let us keep in mind the case when its damping is small over the period of oscillation. Then we can assume that damping will have little effect on the frequency, but it will affect the amplitude of the oscillations. Then the equation of damped oscillations can be represented as Here A(t) represents some decreasing function that needs to be determined. We will proceed from the law of conservation and transformation of energy. The change in the energy of oscillations is equal to the average work of the resistance force over the period, i.e. We divide both sides of the equation by dt. On the right we will have dx/dt, i.e. speed v, and on the left you get the derivative of energy with respect to time. Therefore, taking into account But the average kinetic energy equal to half of the total energy. Therefore, it can be written that divide both its parts by E and multiply by dt. We get that We integrate both parts of the resulting equation: After potentiation, we get The integration constant C is found from the initial conditions. Let at t = 0 E = E0, then E0 = C. Therefore, But E~A^2. Therefore, the amplitude of damped oscillations also decreases according to the exponential law:

And so, due to the resistance, the amplitude of the oscillations decreases and they generally look as shown in Fig. 4.2. The coefficient is called the attenuation coefficient. However, it does not quite characterize the attenuation. Usually, the damping of oscillations is characterized by the damping decrement. The latter shows how many times the oscillation amplitude decreases over a time equal to the oscillation period. That is, the damping factor is defined as follows: The logarithm of the damping decrement is called the logarithmic decrement, it is obviously equal to

Forced vibrations

If the oscillatory system is subjected to the action of an external periodic force, then the so-called forced oscillations arise, which have an undamped character. Forced oscillations should be distinguished from self-oscillations. In the case of self-oscillations in the system, a special mechanism is assumed, which, in time with its own oscillations, "delivers" small portions of energy from some energy reservoir to the system. Thus, natural oscillations are maintained, which do not decay. In the case of self-oscillations, the system, as it were, pushes itself. Clocks can serve as an example of a self-oscillating system. The clock is equipped with a ratchet mechanism, with the help of which the pendulum receives small shocks (from a compressed spring) in time with its own oscillations. In the case of forced oscillations, the system is pushed by an external force. Below we dwell on this case, assuming that the resistance in the system is small and can be neglected. As a model of forced oscillations, we will mean the same body suspended on a spring, which is affected by an external periodic force (for example, a force that has an electromagnetic nature). Without taking into account the resistance, the equation of motion of such a body in the projection on the x-axis has the form: where w* is the cyclic frequency, B is the amplitude of the external force. It is known that fluctuations exist. Therefore, we will look for a particular solution of the equation in the form of a sinusoidal function We substitute the function into the equation, for which we differentiate twice with respect to time . The substitution leads to the relation

The equation turns into an identity if three conditions are met: . Then and the equation of forced oscillations can be represented as They occur with a frequency coinciding with the frequency of the external force, and their amplitude is not set arbitrarily, as in the case of free oscillations, but is set by itself. This established value depends on the ratio of the natural oscillation frequency of the system and the frequency of the external force according to the formula

H and fig. 4.3 shows a plot of the dependence of the amplitude of forced oscillations on the frequency of the external force. It can be seen that the amplitude of oscillations increases significantly as the frequency of the external force approaches the frequency of natural oscillations. The phenomenon of a sharp increase in the amplitude of forced oscillations when the natural frequency and the frequency of the external force coincide is called resonance.

At resonance, the oscillation amplitude must be infinitely large. In reality, at resonance, the amplitude of forced oscillations is always finite. This is explained by the fact that at resonance and near it, our assumption of a negligibly small resistance becomes incorrect. Even if the resistance in the system is small, then it is significant in resonance. Its presence makes the oscillation amplitude in resonance a finite value. Thus, the actual graph of the dependence of the oscillation amplitude on the frequency has the form shown in Fig. 4.4. The greater the resistance in the system, the lower the maximum amplitude at the resonance point.

As a rule, resonance in mechanical systems is an undesirable phenomenon, and its they try to avoid: they try to design mechanical structures subject to oscillations and vibrations in such a way that the natural frequency of oscillations is far from the possible values ​​of the frequencies of external influences. But in a number of devices resonance is used as a positive phenomenon. For example, the resonance of electromagnetic oscillations is widely used in radio communications, the resonance of g-rays - in precision devices.

The state of the thermodynamic system. Processes

Thermodynamic states and thermodynamic processes

When, in addition to the laws of mechanics, the application of the laws of thermodynamics is required, the system is called a thermodynamic system. The need to use this concept arises if the number of elements of the system (for example, the number of gas molecules) is very large, and the movement of its individual elements is microscopic in comparison with the movement of the system itself or its macroscopic components. In this case, thermodynamics describes macroscopic movements (changes in macroscopic states) of a thermodynamic system.

The parameters describing such movement (changes) of a thermodynamic system are usually divided into external and internal. This division is very conditional and depends on the specific task. So, for example, gas in a balloon with an elastic shell has the pressure of the surrounding air as an external parameter, and for a gas in a vessel with a rigid shell, the external parameter is the volume bounded by this shell. In a thermodynamic system, volume and pressure can vary independently of each other. For a theoretical description of their change, it is necessary to introduce at least one more parameter - temperature.

In most thermodynamic problems, three parameters are sufficient to describe the state of a thermodynamic system. In this case, changes in the system are described using three thermodynamic coordinates associated with the corresponding thermodynamic parameters.

equilibrium state- a state of thermodynamic equilibrium - such a state of a thermodynamic system is called, in which there are no flows (energy, matter, momentum, etc.), and the macroscopic parameters of the system are steady and do not change in time.

Classical thermodynamics states that an isolated thermodynamic system (left to itself) tends to a state of thermodynamic equilibrium and, after reaching it, cannot spontaneously leave it. This statement is often called zero law of thermodynamics.

Systems in a state of thermodynamic equilibrium have the following properties mi:

If two thermodynamic systems that have thermal contact are in a state of thermodynamic equilibrium, then the total thermodynamic system is also in a state of thermodynamic equilibrium.

If any thermodynamic system is in thermodynamic equilibrium with two other systems, then these two systems are in thermodynamic equilibrium with each other.

Let us consider thermodynamic systems that are in a state of thermodynamic equilibrium. The description of systems that are in a non-equilibrium state, that is, in a state where macroscopic flows take place, is dealt with by non-equilibrium thermodynamics. The transition from one thermodynamic state to another is called thermodynamic process. Below we will consider only quasi-static processes or, what is the same, quasi-equilibrium processes. The limiting case of a quasi-equilibrium process is an infinitely slow equilibrium process that consists of continuously successive states of thermodynamic equilibrium. In reality, such a process cannot take place, however, if macroscopic changes in the system occur rather slowly (over time intervals significantly exceeding the time for establishing thermodynamic equilibrium), it becomes possible to approximate the real process as quasi-static (quasi-equilibrium). This approximation makes it possible to carry out calculations with sufficiently high accuracy for a large class of practical problems. The equilibrium process is reversible, that is, one in which a return to the values ​​of the state parameters that took place at the previous moment of time should bring the thermodynamic system to the previous state without any changes in the bodies surrounding the system.

The practical application of quasi-equilibrium processes in any technical devices is ineffective. Thus, the use of a quasi-equilibrium process in a heat engine, for example, that occurs at a practically constant temperature (see the description of the Carnot cycle in the third chapter), inevitably leads to the fact that such a machine will work very slowly (in the limit - infinitely slowly) and have a very small power. Therefore, in practice, quasi-equilibrium processes in technical devices are not used. Nevertheless, since the predictions of equilibrium thermodynamics for real systems coincide with a sufficiently high accuracy with experimental data for such systems, it is widely used to calculate thermodynamic processes in various technical devices.

If during a thermodynamic process the system returns to its original state, then such a process is called circular or cyclic. Circular processes, as well as any other thermodynamic processes, can be both equilibrium (and therefore reversible) and non-equilibrium (irreversible). In a reversible circular process, after the thermodynamic system returns to its original state, no thermodynamic perturbations arise in the bodies surrounding it, and their states remain in equilibrium. In this case, the external parameters of the system, after the implementation of the cyclic process, return to their original values. In an irreversible circular process, after its completion, the surrounding bodies pass into non-equilibrium states and the external parameters of the thermodynamic system change.