The study of pendulum oscillations is carried out on the installation, the scheme of which is shown in Fig.5. The installation consists of a spring pendulum, a vibration registration system based on a piezoelectric sensor, a forced vibration excitation system, and an information processing system on a personal computer. The investigated spring pendulum consists of a steel spring with a stiffness coefficient k and pendulum body m with a permanent magnet in the center. The movement of the pendulum occurs in a liquid and at low oscillation speeds the resulting friction force can be approximated with sufficient accuracy by a linear law, i.e.

Fig.5 Block diagram of the experimental setup

To increase the resistance force when moving in a liquid, the body of the pendulum is made in the form of a washer with holes. To register the vibrations, a piezoelectric sensor is used, to which the pendulum spring is suspended. During the movement of the pendulum, the elastic force is proportional to the displacement X,
Since the EMF that occurs in the piezoelectric sensor is in turn proportional to the pressure force, the signal received from the sensor will be proportional to the displacement of the pendulum body from the equilibrium position.
Excitation of oscillations is carried out using a magnetic field. The harmonic signal generated by the PC is amplified and fed to an excitation coil located under the body of the pendulum. As a result of this coil, a magnetic field that is variable in time and non-uniform in space is formed. This field acts on a permanent magnet mounted in the body of the pendulum and creates an external periodic force. When a body moves, the driving force can be represented as a superposition of harmonic functions , and the pendulum oscillations will be a superposition of oscillations with frequencies mw. However, only the force component at the frequency w, since it is closest to the resonant frequency. Therefore, the amplitudes of the components of the oscillations of the pendulum at frequencies mw will be small. That is, in the case of an arbitrary periodic action, oscillations with a high degree of accuracy can be considered harmonic at a frequency w.
The information processing system consists of an analog-to-digital converter and a personal computer. The analog signal from the piezoelectric sensor is represented in digital form using an analog-to-digital converter and fed to a personal computer.

Computer control of the experimental setup
After turning on the computer and loading the program, the main menu appears on the monitor screen, the general view of which is shown in Fig.5. Using the cursor keys , , , , you can select one of the menu items. After pressing the button ENTER the computer starts the selected mode of operation. The simplest hints on the selected mode of operation are contained in the highlighted line at the bottom of the screen.
Consider the possible modes of operation of the program:
Statics- this menu item is used to process the results of the first exercise (see Fig. 5) After pressing the button ENTER the computer asks for the mass of the pendulum's weight. After the next button press ENTER a new picture appears on the screen with a blinking cursor. Consistently write down on the screen the mass of the load in grams and, after pressing the space bar, the magnitude of the stretching of the spring. Pressing ENTER go to a new line and again write down the mass of the load and the amount of stretching of the spring. Data editing within the last line is allowed. To do this, pressing the key backspace delete the incorrect value of the mass or tension of the spring and record the new value. To change the data in other rows, you must successively press Esc and ENTER and then iterate over the result set.
After entering the data, press the function key F2. The values ​​of the spring stiffness coefficient and the frequency of free oscillations of the pendulum calculated using the least squares method appear on the screen. After clicking on ENTER a graph of the dependence of the elastic force on the magnitude of the spring extension appears on the monitor screen. Return to the main menu occurs after pressing any key.
Experiment- this item has several sub-items (Fig. 6). Consider the features of each of them.
Frequency- in this mode, using the cursor keys, the frequency of the driving force is set. In the event that an experiment with free vibrations is being carried out, then it is necessary to set the frequency value equal to 0 .
Start- in this mode after pressing the button ENTER the program begins to record the experimental dependence of the pendulum deflection on time. In the case when the frequency of the driving force is equal to zero, a picture of damped oscillations appears on the screen. In a separate window, the values ​​of the oscillation frequency and damping constant are recorded. If the frequency of the driving force is not equal to zero, then along with the graphs of the dependences of the pendulum deflection and the driving force on time, the values ​​of the frequency of the driving force and its amplitude, as well as the measured frequency and amplitude of the pendulum oscillations, are recorded in separate windows on the screen. Pressing the key Esc you can exit to the main menu.
Save- if the result of the experiment is satisfactory, then it can be saved by pressing the corresponding menu key.
New Series- this menu item is used if there is a need to discard the data of the current experiment. After pressing the key ENTER in this mode, the results of all previous experiments are erased from the machine's memory, and a new series of measurements can be started.
After the experiment, they switch to the mode measurements. This menu item has several sub-items (Fig. 7)
Frequency response graph- this menu item is used after the end of the experiment on the study of forced oscillations. The amplitude-frequency characteristic of forced oscillations is plotted on the monitor screen.
PFC chart- In this mode, after the end of the experiment on the study of forced oscillations, the phase-frequency characteristic is built on the monitor screen.
Table- this menu item allows you to display the values ​​of the amplitude and phase of oscillations on the monitor screen depending on the frequency of the driving force. These data are rewritten in a notebook for a report on this work.
Computer menu item Output- the end of the program (see, for example, Fig. 7)

Exercise 1. Determination of the spring stiffness coefficient by the static method.

Measurements are carried out by determining the elongation of the spring under the action of loads with known masses. It is recommended to spend at least 7-10 measurements of the elongation of the spring by gradually suspending the loads and thereby changing the load from 20 before 150 d. Using the menu item of the program Statistics the results of these measurements are entered into the computer memory and the coefficient of spring stiffness is determined using the least squares method. During the exercise, it is necessary to calculate the value of the natural frequency of the pendulum

Bodies under the action of an elastic force, the potential energy of which is proportional to the square of the displacement of the body from the equilibrium position:

where k is the stiffness of the spring.

With free mechanical vibrations, the kinetic and potential energies change periodically. At the maximum deviation of the body from the equilibrium position, its velocity, and hence the kinetic energy, vanishes. In this position, the potential energy of the oscillating body reaches its maximum value. For a load on a horizontally located spring, the potential energy is the energy of elastic deformations of the spring.

When the body in its motion passes through the equilibrium position, its speed is maximum. At this moment, it has the maximum kinetic and minimum potential energy. An increase in kinetic energy occurs at the expense of a decrease in potential energy. With further movement, the potential energy begins to increase due to the decrease in kinetic energy, etc.

Thus, during harmonic oscillations, a periodic transformation of kinetic energy into potential energy and vice versa occurs.

If there is no friction in the oscillatory system, then the total mechanical energy during free vibrations remains unchanged.

For spring load:

The start of the oscillatory movement of the body is carried out using the Start button. The Stop button allows you to stop the process at any time.

Graphically shows the relationship between potential and kinetic energies during oscillations at any time. Note that in the absence of damping, the total energy of the oscillatory system remains unchanged, the potential energy reaches its maximum at the maximum deviation of the body from the equilibrium position, and the kinetic energy reaches its maximum value when the body passes through the equilibrium position.

Definition

Oscillation frequency($\nu$) is one of the parameters that characterize the fluctuations. This is the reciprocal of the fluctuation period ($T$):

\[\nu=\frac(1)(T)\left(1\right).\]

Thus, the frequency of oscillations is called a physical quantity equal to the number of repetitions of oscillations per unit of time.

\[\nu =\frac(N)(\Delta t)\left(2\right),\]

where $N$ is the number of complete oscillatory motions; $\Delta t$ - time during which these fluctuations occurred.

The cyclic oscillation frequency ($(\omega )_0$) is related to the frequency $\nu $ by the formula:

\[\nu =\frac((\omega )_0)(2\pi )\left(3\right).\]

The unit of frequency in the International System of Units (SI) is hertz or reciprocal second:

\[\left[\nu \right]=c^(-1)=Hz.\]

Spring pendulum

Definition

Spring pendulum called a system that consists of an elastic spring to which a load is attached.

Let's assume that the weight of the load is $m$, the coefficient of elasticity of the spring is $k$. The mass of the spring in such a pendulum is usually not taken into account. If we consider the horizontal movements of the load (Fig. 1), then it moves under the action of the elastic force, if the system was taken out of equilibrium and left to itself. In this case, it is often believed that friction forces can be ignored.

Oscillation equations for a spring pendulum

A spring pendulum that oscillates freely is an example of a harmonic oscillator. Let it perform oscillations along the X axis. If the oscillations are small, Hooke's law is satisfied, then we write the equation for the movement of the load as:

\[\ddot(x)+(\omega )^2_0x=0\left(4\right),\]

where $(\omega )^2_0=\frac(k)(m)$ is the cyclic frequency of oscillations of the spring pendulum. The solution of equation (4) is a sine or cosine function of the form:

where $(\omega )_0=\sqrt(\frac(k)(m))>0$ is the cyclic oscillation frequency of the spring pendulum, $A$ is the oscillation amplitude; $((\omega )_0t+\varphi)$ - oscillation phase; $\varphi $ and $(\varphi )_1$ - initial phases of oscillations.

Oscillation frequency of the spring pendulum

From formula (3) and $(\omega )_0=\sqrt(\frac(k)(m))$, it follows that the oscillation frequency of the spring pendulum is:

\[\nu =\frac(1)(2\pi )\sqrt(\frac(k)(m))\ \left(6\right).\]

Formula (6) is valid if:

• the spring in the pendulum is assumed to be weightless;
• the weight attached to the spring is a perfectly rigid body;
• there are no torsional vibrations.

Expression (6) shows that the oscillation frequency of a spring pendulum increases with a decrease in the mass of the load and an increase in the coefficient of elasticity of the spring. The oscillation frequency of a spring pendulum does not depend on the amplitude. If the oscillations are not small, the elastic force of the spring does not obey Hooke's law, then the dependence of the oscillation frequency on the amplitude appears.

Examples of problems with a solution

Example 1

Exercise. The oscillation period of the spring pendulum is $T=5\cdot (10)^(-3)c$. What is the oscillation frequency in this case? What is the cyclic frequency of this weight?

Decision. The oscillation frequency is the reciprocal of the oscillation period, therefore, to solve the problem, it is enough to use the formula:

\[\nu=\frac(1)(T)\left(1.1\right).\]

Calculate the desired frequency:

\[\nu =\frac(1)(5\cdot (10)^(-3))=200\ \left(Hz\right).\]

The cyclic frequency is related to the $\nu$ frequency as:

\[(\omega )_0=2\pi \nu \ \left(1.2\right).\]

Let's calculate the cyclic frequency:

\[(\omega )_0=2\pi \cdot 200\approx 1256\ \left(\frac(rad)(c)\right).\]

Answer.$1)\ \nu =200$ Hz. 2) $(\omega )_0=1256\ \frac(rad)(s)$

Example 2

Exercise. The mass of the load hanging on the elastic spring (Fig. 2) is increased by $\Delta m$, while the frequency decreases by $n$ times. What is the mass of the first load?

\[\nu =\frac(1)(2\pi )\sqrt(\frac(k)(m))\ \left(2.1\right).\]

For the first load, the frequency will be equal to:

\[(\nu )_1=\frac(1)(2\pi )\sqrt(\frac(k)(m))\ \left(2.2\right).\]

For the second load:

\[(\nu )_2=\frac(1)(2\pi )\sqrt(\frac(k)(m+\Delta m))\ \left(2.2\right).\]

By the condition of the problem $(\nu )_2=\frac((\nu )_1)(n)$, we find the relation $\frac((\nu )_1)((\nu )_2):\frac((\nu )_1)((\nu )_2)=\sqrt(\frac(k)(m)\cdot \frac(m+\Delta m)(k))=\sqrt(1+\frac(\Delta m)( m))=n\ \left(2.3\right).$

We obtain from equation (2.3) the desired mass of the load. To do this, we square both parts of expression (2.3) and express $m$:

Answer.$m=\frac(\Delta m)(n^2-1)$

(1.7.1)

If the ball is displaced from the equilibrium position by a distance x, then the elongation of the spring will become equal to Δl 0 + x. Then the resulting force will take the value:

Taking into account the equilibrium condition (1.7.1), we obtain:

The minus sign indicates that displacement and force are in opposite directions.

The elastic force f has the following properties:

1. It is proportional to the displacement of the ball from the equilibrium position;
2. It is always directed towards the equilibrium position.

In order to inform the system of the displacement x, it is necessary to perform work against the elastic force:

This work goes to create a reserve of potential energy of the system:

Under the action of an elastic force, the ball will move towards the equilibrium position with an ever-increasing speed. Therefore, the potential energy of the system will decrease, but the kinetic energy will increase (we neglect the mass of the spring). Having come to the equilibrium position, the ball will continue to move by inertia. This is a slow motion and will stop when the kinetic energy is completely converted into potential. Then the same process will proceed when the ball moves in the opposite direction. If there is no friction in the system, the ball will oscillate indefinitely.

The equation of Newton's second law in this case is:

Let's transform the equation like this:

Introducing the notation , we obtain a linear homogeneous differential equation of the second order:

By direct substitution, it is easy to verify that the general solution of equation (1.7.8) has the form:

where a is the amplitude and φ is the initial phase of the oscillation - constant values. Therefore, the oscillation of a spring pendulum is harmonic (Fig. 1.7.2).

Rice. 1.7.2. harmonic oscillation

Due to the periodicity of the cosine, various states of the oscillatory system are repeated after a certain period of time (oscillation period) T, during which the phase of the oscillation receives an increment of 2π. You can calculate the period using the equation:

from where follows:

The number of oscillations per unit time is called the frequency:

The unit of frequency is the frequency of such an oscillation, the period of which is 1 s. This unit is called 1 Hz.

From (1.7.11) it follows that:

Therefore, ω 0 is the number of oscillations made in 2π seconds. The value ω 0 is called the circular or cyclic frequency. Using (1.7.12) and (1.7.13), we write:

Differentiating () with respect to time, we obtain an expression for the speed of the ball:

From (1.7.15) it follows that the speed also changes according to the harmonic law and is ahead of the phase shift by ½π. Differentiating (1.7.15), we get the acceleration:

1.7.2. Mathematical pendulum

Mathematical pendulum called an idealized system consisting of an inextensible weightless thread on which a body is suspended, the entire mass of which is concentrated at one point.

The deviation of the pendulum from the equilibrium position is characterized by the angle φ formed by the thread with the vertical (Fig. 1.7.3).

Rice. 1.7.3. Mathematical pendulum

When the pendulum deviates from the equilibrium position, a torque arises that tends to return the pendulum to the equilibrium position:

Let us write an equation for the dynamics of rotational motion for the pendulum, taking into account that its moment of inertia is equal to ml 2:

This equation can be brought to the form:

Restricting ourselves to the case of small fluctuations sinφ ≈ φ and introducing the notation:

equation (1.7.19) can be represented as follows:

which coincides in form with the equation of oscillations of a spring pendulum. Therefore, its solution will be a harmonic oscillation:

From (1.7.20) it follows that the cyclic oscillation frequency of a mathematical pendulum depends on its length and free fall acceleration. Using the formula for the oscillation period () and (1.7.20), we obtain the known relation:

1.7.3. physical pendulum

A physical pendulum is a rigid body capable of oscillating around a fixed point that does not coincide with the center of inertia. In the equilibrium position, the center of inertia of the pendulum C is under the suspension point O on the same vertical (Fig. 1.7.4).

Rice. 1.7.4. physical pendulum

When the pendulum deviates from the equilibrium position by an angle φ, a torque arises that tends to return the pendulum to the equilibrium position:

where m is the mass of the pendulum, l is the distance between the suspension point and the center of inertia of the pendulum.

Let us write the equation for the dynamics of rotational motion for the pendulum, taking into account that the moment of inertia is equal to I:

For small fluctuations sinφ ≈ φ. Then, introducing the notation:

which also coincides in form with the equation of oscillations of a spring pendulum. From equations (1.7.27) and (1.7.26) it follows that with small deviations of the physical pendulum from the equilibrium position, it performs a harmonic oscillation, the frequency of which depends on the mass of the pendulum, the moment of inertia and the distance between the axis of rotation and the center of inertia. Using (1.7.26), you can calculate the oscillation period:

Comparing formulas (1.7.28) and () we get that a mathematical pendulum with a length:

will have the same oscillation period as the considered physical pendulum. The quantity (1.7.29) is called reduced length physical pendulum. Therefore, the reduced length of a physical pendulum is the length of such a mathematical pendulum, the period of oscillation of which is equal to the period of oscillation of a given physical pendulum.

A point on the straight line connecting the point of suspension with the center of inertia, which lies at a distance of the reduced length from the axis of rotation, is called swing center physical pendulum. According to Steiner's theorem, the moment of inertia of a physical pendulum is:

where I 0 is the moment of inertia about the center of inertia. Substituting (1.7.30) into (1.7.29), we get:

Therefore, the reduced length is always greater than the distance between the suspension point and the pendulum's center of inertia, so that the suspension point and the swing center lie on opposite sides of the center of inertia.

1.7.4. Energy of harmonic vibrations

During harmonic oscillation, there is a periodic mutual transformation of the kinetic energy of the oscillating body E k and the potential energy E p, due to the action of a quasi-elastic force. From these energies, the total energy E of the oscillatory system is added:

Let's write the last expression

But k \u003d mω 2, so we get the expression for the total energy of the oscillating body

Thus, the total energy of a harmonic oscillation is constant and proportional to the square of the amplitude and the square of the circular frequency of the oscillation.

1.7.5. damped vibrations .

When studying harmonic oscillations, the forces of friction and resistance that exist in real systems were not taken into account. The action of these forces significantly changes the nature of the motion, the oscillation becomes fading.

If, in addition to the quasi-elastic force, the resistance forces of the medium (friction forces) act in the system, then Newton's second law can be written as follows:

where r is the coefficient of friction, which characterizes the properties of the medium to resist movement. We substitute (1.7.34b) into (1.7.34a):

The graph of this function is shown in Fig. 1.7.5 as a solid curve 1, and a dashed line 2 shows the change in amplitude:

With very little friction, the period of damped oscillation is close to the period of undamped free oscillation (1.7.35.b)

The rate of decrease in the oscillation amplitude is determined by damping factor: the larger β, the stronger the retarding effect of the medium and the faster the amplitude decreases. In practice, the degree of attenuation is often characterized logarithmic damping decrement, meaning by this a value equal to the natural logarithm of the ratio of two successive oscillation amplitudes separated by a time interval equal to the oscillation period:

;

Therefore, the damping coefficient and the logarithmic damping decrement are related by a fairly simple relationship:

With strong damping, it can be seen from formula (1.7.37) that the oscillation period is an imaginary quantity. The movement in this case is already called aperiodic. The aperiodic motion graph is shown in Fig. 1.7.6. Undamped and damped oscillations are called own or free. They arise as a result of the initial displacement or initial velocity and occur in the absence of external influence due to the initially accumulated energy.

1.7.6. Forced vibrations. Resonance .

compelled Oscillations are called those that arise in the system with the participation of an external force that changes according to a periodic law.

Let us assume that, in addition to the quasi-elastic force and the friction force, an external driving force acts on the material point

,

where F 0 - amplitude; ω - circular frequency of oscillations of the driving force. We compose a differential equation (Newton's second law):

,

The amplitude of the forced oscillation (1.7.39) is directly proportional to the amplitude of the driving force and has a complex dependence on the attenuation coefficient of the medium and the circular frequencies of natural and forced oscillations. If ω 0 and β are given for the system, then the amplitude of forced oscillations has a maximum value at a certain specific frequency of the driving force, called resonant.

The phenomenon itself - reaching the maximum amplitude for given ω 0 and β - is called resonance.

Rice. 1.7.7. Resonance

In the absence of resistance, the amplitude of forced oscillations at resonance is infinitely large. In this case, from ω res = ω 0, i.e. resonance in a system without damping occurs when the frequency of the driving force coincides with the frequency of natural oscillations. The graphical dependence of the amplitude of forced oscillations on the circular frequency of the driving force for different values ​​of the damping coefficient is shown in Fig. 5.

Mechanical resonance can be both beneficial and detrimental. The harmful effect of resonance is mainly due to the destruction it can cause. So, in technology, taking into account different vibrations, it is necessary to foresee the possible occurrence of resonant conditions, otherwise there may be destruction and disasters. Bodies usually have several natural vibration frequencies and, accordingly, several resonant frequencies.

If the attenuation coefficient of the internal organs of a person were not large, then the resonant phenomena that arose in these organs under the influence of external vibrations or sound waves could lead to tragic consequences: rupture of organs, damage to ligaments, etc. However, such phenomena are practically not observed under moderate external influences, since the attenuation coefficient of biological systems is quite large. Nevertheless, resonant phenomena under the action of external mechanical vibrations occur in the internal organs. This, apparently, is one of the reasons for the negative impact of infrasonic oscillations and vibrations on the human body.

1.7.7. Self-oscillations

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

The undamped oscillations that exist in any system in the absence of a variable external influence are called self-oscillations, and the systems themselves self-oscillating.

The amplitude and frequency of self-oscillations depend on the properties in the self-oscillating system itself; in contrast to forced oscillations, they are not determined by external influences.

In many cases, self-oscillating systems can be represented by three main elements (Fig. 1.7.8): 1) the actual oscillating system; 2) energy source; 3) a regulator of energy supply to the actual oscillatory system. The oscillatory system through the feedback channel (Fig. 6) affects the regulator, informing the regulator about the state of this system.

A classic example of a mechanical self-oscillating system is a watch, in which a pendulum or balance is an oscillatory system, a spring or a raised weight is a source of energy, and an anchor is a regulator of the energy supply from the source to the oscillatory system.

Many biological systems (heart, lungs, etc.) are self-oscillatory. A typical example of an electromagnetic self-oscillating system is generators of self-oscillating oscillations.

1.7.8. Addition of vibrations in one direction

Consider the addition of two harmonic oscillations of the same direction and the same frequency:

x 1 \u003d a 1 cos (ω 0 t + α 1), x 2 \u003d a 2 cos (ω 0 t + α 2).

A harmonic oscillation can be specified using a vector, the length of which is equal to the amplitude of the oscillations, and the direction forms an angle with some axis equal to the initial phase of the oscillations. If this vector rotates with an angular velocity ω 0, then its projection on the selected axis will change according to the harmonic law. Based on this, we choose some axis X and represent the oscillations using the vectors a 1 and a 2 (Fig. 1.7.9).

From Figure 1.7.6 it follows that

.

Schemes in which oscillations are depicted graphically as vectors on a plane are called vector diagrams.

It follows from formula 1.7.40. That if the phase difference of both oscillations is equal to zero, the amplitude of the resulting oscillation is equal to the sum of the amplitudes of the added oscillations. If the phase difference of the added oscillations is equal to , then the amplitude of the resulting oscillation is equal to . If the frequencies of the added oscillations are not the same, then the vectors corresponding to these oscillations will rotate at different speeds. In this case, the resulting vector pulsates in magnitude and rotates at a non-constant rate. Consequently, as a result of addition, not a harmonic oscillation is obtained, but a complex oscillatory process.

1.7.9. beats

Consider the addition of two harmonic oscillations of the same direction, slightly different in frequency. Let the frequency of one of them be equal to ω , and the frequency of the second ω + ∆ω, and ∆ω<<ω. Положим, что амплитуды складываемых колебаний одинаковы и начальные фазы обоих колебаний равны нулю. Тогда уравнения колебаний запишутся следующим образом:

x 1 \u003d a cos ωt, x 2 \u003d a cos (ω + ∆ω) t.

Adding these expressions and using the formula for the sum of cosines, we get:

Oscillations (1.7.41) can be considered as a harmonic oscillation with frequency ω, the amplitude of which varies according to the law . This function is periodic with a frequency twice the frequency of the expression under the module sign, i.e. with frequency ∆ω. Thus, the frequency of amplitude pulsations, called the beat frequency, is equal to the difference in the frequencies of the added oscillations.

1.7.10. Addition of mutually perpendicular vibrations (Lissajous figures)

If a material point oscillates both along the x-axis and along the y-axis, then it will move along some curvilinear trajectory. Let the oscillation frequency be the same and the initial phase of the first oscillation be equal to zero, then we write the oscillation equations in the form:

Equation (1.7.43) is the equation of an ellipse, the axes of which are arbitrarily oriented relative to the x and y coordinate axes. The orientation of the ellipse and the size of its semiaxes depend on the amplitudes a and b and the phase difference α. Let's consider some special cases:

(m=0, ±1, ±2, …). In this case, the equation has the form

This is the equation of an ellipse, the axes of which coincide with the coordinate axes, and its semiaxes are equal to the amplitudes (Fig. 1.7.12). If the amplitudes are equal, then the ellipse becomes a circle.

Fig.1.7.12

If the frequencies of mutually perpendicular oscillations differ by a small amount ∆ω, they can be considered as oscillations of the same frequency, but with a slowly changing phase difference. In this case, the oscillation equations can be written

x=a cos ωt, y=b cos[ωt+(∆ωt+α)]

and the expression ∆ωt+α is considered as a phase difference that slowly changes with time according to a linear law. The resulting movement in this case follows a slowly changing curve, which will successively take the form corresponding to all values ​​of the phase difference from -π to +π.

If the frequencies of mutually perpendicular oscillations are not the same, then the trajectory of the resulting motion has the form of rather complex curves called Lissajous figures. Let, for example, the frequencies of the added oscillations be related as 1 : 2 and phase difference π/2. Then the oscillation equations have the form

x=a cos ωt, y=b cos.

While along the x-axis the point manages to move from one extreme position to another, along the y-axis, leaving the zero position, it manages to reach one extreme position, then another and return. The curve view is shown in fig. 1.7.13. The curve with the same frequency ratio, but the phase difference equal to zero is shown in Fig. 1.7.14. The ratio of the frequencies of the added oscillations is inverse to the ratio of the number of points of intersection of the Lissajous figures with straight lines parallel to the coordinate axes. Therefore, by the appearance of the Lissajous figures, one can determine the ratio of the frequencies of the added oscillations or an unknown frequency. If one of the frequencies is known.

Fig.1.7.13

Fig.1.7.14

The closer to unity the rational fraction expressing the ratio of vibration frequencies, the more complex the resulting Lissajous figures.

1.7.11. Wave propagation in an elastic medium

If in any place of an elastic (solid liquid or gaseous) medium vibrations of its particles are excited, then due to the interaction between particles this vibration will propagate in the medium from particle to particle with a certain speed υ. the process of propagation of vibrations in space is called wave.

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 directions of particle oscillations with respect to the direction in which the wave propagates, there are longitudinal and transverse waves. In a longitudinal wave, the particles of the medium oscillate along the propagation of the wave. In a transverse wave, the particles of the medium oscillate in directions 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, the occurrence of both longitudinal and transverse waves is possible.

On fig. 1.7.12 shows the motion of particles during propagation in a medium of a transverse wave. Numbers 1,2, etc. denote particles lagging behind each other by a distance equal to (¼ υT), i.e. by the distance traveled by the wave in a quarter of the period of oscillations made by the particles. At the time taken as zero, the wave, propagating along the axis from left to right, reached 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, particle 1 reaches the uppermost equilibrium position of particle 2. After another quarter of the period, the first part will pass the equilibrium position, moving in the direction from top to bottom, the second particle will reach the uppermost position, and the third particle will begin to move upward from the equilibrium position. At the moment of time equal to T, the first particle will complete the complete oscillation cycle and will be in the same state of motion as the starting moment. The wave by the time T, having passed the path (υT), will reach particle 5.

On Fig. 1.7.13 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.

It can be seen from the figure that during the propagation of a longitudinal wave in the medium, alternating condensations and rarefaction of particles are created (the places of condensation are circled in the figure by a dotted line), moving in the direction of wave propagation with a speed υ.

Rice. 1.7.15

Rice. 1.7.16

On fig. 1.7.15 and 1.7.16 shows oscillations of particles whose positions and equilibria lie on the axis x. In reality, not only particles oscillate along the axis x, but a collection of particles enclosed in a certain volume. Spreading from the sources of oscillations, the wave process covers more and more parts of space, the locus of points, to which oscillations reach by the time t, is called wave front(or wave front). The wave front is the surface that separates the part of space already involved in the wave process from the area in which oscillations have not yet arisen.

The locus of points oscillating in the same phase is called wave surface . The wave surface can be drawn through any point in the space covered by the wave process. Consequently, there are an infinite number of wave surfaces, while there is only one wave front at any time. The wave surfaces remain stationary (they pass through the equilibrium positions of particles oscillating in the same phase ). The wavefront is constantly moving.

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.

Rice. 1.7.17

Let a plane wave propagate along the axis x. Then all points of the sphere, positions, equilibria of which have the same coordinate x(but the difference in coordinate values y and z), oscillate in the same phase.

On Fig. 1.7.17 shows a curve that gives an offset ξ from the equilibrium position of points with different x at some point in time. This drawing should not be taken as a visible image of a wave. The figure shows a graph of functions ξ (x, t) for some fixed point in time t. Such a graph can be built for both longitudinal and transverse waves.

The distance λ, for a short wave propagates in a time equal to the period of oscillation of the particles of the medium, is called wavelength. It's obvious that

where υ is the wave speed, T is the oscillation period. The wavelength can also be defined as the distance between the nearest points of the medium, oscillating with a phase difference equal to 2π (see Fig. 1.7.14)

Replacing in relation (1.7.45) T through 1/ν (ν is the oscillation frequency), we obtain

This formula can also be arrived at from the following considerations. In one second, the wave source performs ν oscillations, generating in the medium during each oscillation one "crest" and one "trough" of the wave. By the time the source completes the ν -th oscillation, the first "ridge" will have time to go through the path υ. Consequently, ν "crests" and "troughs" of the wave must fit into the length υ.

1.7.12. Plane wave equation

The wave equation is an expression that gives the displacement of an oscillating particle as a function of its coordinates x, y, z and time t :

ξ = ξ (x, y, z; t)

(meaning the coordinates of the equilibrium position of the particle). This function must be periodic with respect to time t , and relative to the coordinates x, y, z. . Periodicity in time follows from the fact that points separated from each other at a distance λ , fluctuate in the same way.

Find the type of function ξ in the case of a plane wave, assuming that the oscillations are harmonic. To simplify, we direct the coordinate axes so that the axis x coincides with the direction of wave propagation. Then the wave surfaces will be perpendicular to the axis x and since all points of the wave surface oscillate equally, the displacement ξ will only depend on x and t:

ξ = ξ (x, t) .

Fig.1.7.18

Let oscillations of points lying in the plane x = 0 (Fig. 1.7.18), have the form

Let us find the type of oscillation of points in the plane corresponding to an arbitrary value x . To go way from the plane x=0 to this plane, the wave takes time ( υ is the speed of wave propagation). Consequently, oscillations of particles lying in the plane x , will be behind in time by τ from vibrations of particles in the plane x = 0 , i.e. will look like

So, plane wave equation(longitudinal, and transverse), propagating in the direction of the axis x , as follows:

This expression defines the relationship between time t and that place x , in which the phase has a fixed value. The resulting dx/dt value gives the speed at which the given phase value moves. Differentiating the expression (1.7.48), we obtain

The equation of a wave propagating in the direction of decreasing x :

When deriving formula (1.7.53), we assumed that the oscillation amplitude does not depend on x . For a plane wave, this is observed when the wave energy is not absorbed by the medium. When propagating in an energy-absorbing medium, the intensity of the wave gradually decreases with distance from the source of oscillations - wave attenuation is observed. Experience shows that in a homogeneous medium such damping occurs according to an exponential law:

Respectively plane wave equation, considering damping, has the following form:

(1.7.54)

(a 0 is the amplitude at the points of the plane x = 0).

10.4. The law of conservation of energy during harmonic oscillations

10.4.1. Conservation of energy at mechanical harmonic vibrations

Conservation of energy during oscillations of a mathematical pendulum

With harmonic vibrations, the total mechanical energy of the system is conserved (remains constant).

Total mechanical energy of a mathematical pendulum

E = W k + W p ,

where W k - kinetic energy, W k = = mv 2 /2; W p - potential energy, W p = mgh ; m is the weight of the cargo; g - free fall acceleration modulus; v - load speed modulus; h is the height of the load above the equilibrium position (Fig. 10.15).

With harmonic oscillations, the mathematical pendulum goes through a series of successive states, so it is advisable to consider the energy of the mathematical pendulum in three positions (see Fig. 10.15):

Rice. 10.15

1) in equilibrium position

potential energy is zero; the total energy coincides with the maximum kinetic energy:

E = W k max ;

2) in extreme position(2) the body is raised above the initial level to the maximum height h max , so the potential energy is also maximum:

W p max = m g h max ;

kinetic energy is zero; the total energy coincides with the maximum potential energy:

E = W p max ;

3) in intermediate position(3) the body has an instantaneous speed v and is raised above the initial level to a certain height h, so the total energy is the sum

E = m v 2 2 + m g h ,

where mv 2 /2 - kinetic energy; mgh - potential energy; m is the weight of the cargo; g - free fall acceleration modulus; v - load speed modulus; h is the height of the load above the equilibrium position.

With harmonic oscillations of a mathematical pendulum, the total mechanical energy is conserved:

E = const.

The values ​​of the total energy of the mathematical pendulum in its three positions are shown in Table. 10.1.

№	Position	Wp	Wk	E = W p + W k
1	Equilibrium	0	m v max 2 / 2	m v max 2 / 2
2	Extreme	mgh max	0	mgh max
3	Intermediate (instant)	mgh	mv 2 /2	mv 2 /2 + mgh

The values ​​of the total mechanical energy presented in the last column of Table. 10.1, have equal values ​​for any position of the pendulum, which is a mathematical expression:

m v max 2 2 = m g h max ;

m v max 2 2 = m v 2 2 + m g h ;

m g h max = m v 2 2 + m g h ,

where m is the weight of the cargo; g - free fall acceleration modulus; v is the module of the instantaneous speed of the load in position 3; h is the height of the load above the equilibrium position in position 3; v max - maximum load speed module in position 1; h max - the maximum lifting height of the load above the equilibrium position in position 2.

Thread deflection angle mathematical pendulum from the vertical (Fig. 10.15) is determined by the expression

cos α = l − h l = 1 − h l ,

where l is the length of the thread; h is the height of the load above the equilibrium position.

Max Angle deviations α max is determined by the maximum lifting height of the load above the equilibrium position h max:

cos α max = 1 − h max l .

Example 11. The period of small oscillations of a mathematical pendulum is 0.9 s. At what maximum angle from the vertical will the thread deviate if, passing through the equilibrium position, the ball moves with a speed equal to 1.5 m/s? There is no friction in the system.

Decision . The figure shows two positions of the mathematical pendulum:

• equilibrium position 1 (characterized by the maximum speed of the ball v max);
• extreme position 2 (characterized by the maximum lifting height of the ball h max above the equilibrium position).

The desired angle is determined by the equality

cos α max = l − h max l = 1 − h max l ,

where l is the length of the pendulum thread.

The maximum lifting height of the pendulum ball above the equilibrium position can be found from the law of conservation of total mechanical energy.

The total energy of the pendulum in the equilibrium position and in the extreme position is determined by the following formulas:

• in an equilibrium position

E 1 \u003d m v max 2 2,

where m is the mass of the pendulum ball; v max - ball speed modulus in the equilibrium position (maximum speed), v max = 1.5 m/s;

• in extreme position

E 2 \u003d mgh max,

where g is the free fall acceleration modulus; h max - the maximum height of the ball above the equilibrium position.

The law of conservation of total mechanical energy:

m v max 2 2 = m g h max .

From this we express the maximum height of the ball above the equilibrium position:

h max = v max 2 2 g .

We determine the length of the thread from the formula for the oscillation period of a mathematical pendulum

T = 2 π l g ,

those. thread length

l = T 2 g 4 π 2 .

Substitute h max and l into the expression for the cosine of the desired angle:

cos α max = 1 − 2 π 2 v max 2 g 2 T 2

and perform the calculation, taking into account the approximate equality π 2 = 10:

cos α max = 1 − 2 ⋅ 10 ⋅ (1.5) 2 10 2 ⋅ (0.9) 2 = 0.5 .

It follows that the maximum deflection angle is 60°.

Strictly speaking, at an angle of 60°, the oscillations of the ball are not small, and it is illegal to use the standard formula for the oscillation period of a mathematical pendulum.

Conservation of energy during oscillations of a spring pendulum

Total mechanical energy of a spring pendulum is the sum of kinetic energy and potential energy:

E = W k + W p ,

where W k - kinetic energy, W k = mv 2 /2; W p - potential energy, W p = k (Δx ) 2 /2; m is the weight of the cargo; v - load speed modulus; k - coefficient of rigidity (elasticity) of the spring; Δx - deformation (tension or compression) of the spring (Fig. 10.16).

In the International System of Units, the energy of a mechanical oscillatory system is measured in joules (1 J).

With harmonic oscillations, a spring pendulum goes through a series of successive states, so it is advisable to consider the energy of a spring pendulum in three positions (see Fig. 10.16):

1) in equilibrium position(1 ) the speed of the body has a maximum value v max , so the kinetic energy is also maximum:

W k max = m v max 2 2 ;

the potential energy of the spring is zero, since the spring is not deformed; the total energy coincides with the maximum kinetic energy:

E = W k max ;

2) in extreme position(2) the spring has a maximum deformation (Δx max), so the potential energy also has a maximum value:

W p max \u003d k (Δ x max) 2 2;

the kinetic energy of the body is zero; the total energy coincides with the maximum potential energy:

E = W p max ;

3) in intermediate position(3) the body has an instantaneous speed v, the spring has some deformation at this moment (Δx), so the total energy is the sum

E \u003d m v 2 2 + k (Δ x) 2 2,

where mv 2 /2 - kinetic energy; k (Δx ) 2 /2 - potential energy; m is the weight of the cargo; v - load speed modulus; k - coefficient of rigidity (elasticity) of the spring; Δx - deformation (tension or compression) of the spring.

When the weight of the spring pendulum is displaced from the equilibrium position, it is affected by restoring force, whose projection on the pendulum motion direction is determined by the formula

F x = −kx ,

where x is the displacement of the spring pendulum load from the equilibrium position, x = ∆x , ∆x is the deformation of the spring; k - coefficient of rigidity (elasticity) of the pendulum spring.

With harmonic oscillations of a spring pendulum, the total mechanical energy is conserved:

E = const.

The values ​​of the total energy of the spring pendulum in its three positions are shown in Table. 10.2.

№	Position	Wp	Wk	E = W p + W k
1	Equilibrium	0	m v max 2 / 2	m v max 2 / 2
2	Extreme	k (Δxmax) 2 /2	0	k (Δxmax) 2 /2
3	Intermediate (instant)	k (Δx) 2 /2	mv 2 /2	mv 2 /2 + k (Δx) 2 /2

The values ​​of the total mechanical energy presented in the last column of the table have equal values ​​for any position of the pendulum, which is a mathematical expression law of conservation of total mechanical energy:

m v max 2 2 = k (Δ x max) 2 2 ;

m v max 2 2 = m v 2 2 + k (Δ x) 2 2 ;

k (Δ x max) 2 2 \u003d m v 2 2 + k (Δ x) 2 2,

where m is the weight of the cargo; v is the module of the instantaneous speed of the load in position 3; Δx - deformation (tension or compression) of the spring in position 3; v max - maximum load speed module in position 1; Δx max - maximum deformation (extension or compression) of the spring in position 2 .

Example 12. A spring pendulum performs harmonic oscillations. How many times is its kinetic energy greater than the potential energy at the moment when the displacement of the body from the equilibrium position is a quarter of the amplitude?

Decision . Let's compare the two positions of the spring pendulum:

• extreme position 1 (characterized by the maximum displacement of the pendulum load from the equilibrium position x max);
• intermediate position 2 (characterized by intermediate values ​​of displacement from the equilibrium position x and speed v →).

The total energy of the pendulum in the extreme and intermediate positions is determined by the following formulas:

• in extreme position

E 1 \u003d k (Δ x max) 2 2,

where k is the coefficient of stiffness (elasticity) of the spring; ∆x max - oscillation amplitude (maximum displacement from the equilibrium position), ∆x max = A ;

• in an intermediate position

E 2 \u003d k (Δ x) 2 2 + m v 2 2,

where m is the mass of the pendulum load; ∆x - displacement of the load from the equilibrium position, ∆x = A /4.

The law of conservation of total mechanical energy for a spring pendulum has the following form:

k (Δ x max) 2 2 = k (Δ x) 2 2 + m v 2 2 .

We divide both parts of the written equality by k (∆x) 2 /2:

(Δ x max Δ x) 2 = 1 + m v 2 2 ⋅ 2 k Δ x 2 = 1 + W k W p ,

where W k is the kinetic energy of the pendulum in an intermediate position, W k = mv 2 /2; W p - potential energy of the pendulum in an intermediate position, W p = k (∆x ) 2 /2.

Let us express the desired ratio of energies from the equation:

W k W p = (Δ x max Δ x) 2 − 1

and calculate its value:

W k W p = (A A / 4) 2 − 1 = 16 − 1 = 15 .

At the indicated moment of time, the ratio of the kinetic and potential energies of the pendulum is 15.