1.21. DECAYING, FORCED OSCILLATIONS

The differential equation of damped oscillations and its solution. Attenuation coefficient. logarithmic decdamping band.Q factorbody system.aperiodic process. The differential equation of forced oscillations and its solution.Amplitude and phase of forced oscillations. The process of establishing oscillations. Resonance case.Self-oscillations.

The damping of oscillations is the gradual decrease in the amplitude of oscillations over time, due to the loss of energy by the oscillatory system.

Natural vibrations without damping is an idealization. The reasons for fading can be different. In a mechanical system, vibrations are damped by the presence of friction. When all the energy stored in the oscillating system is used up, the oscillations will stop. Therefore, the amplitude damped oscillations decreases until it becomes zero.

Damped oscillations, as well as natural ones, in systems that are different in nature, can be considered from a single point of view - common features. However, characteristics such as amplitude and period require redefinition, while others require additions and clarifications in comparison with the same characteristics for natural undamped oscillations. The general signs and concepts of damped oscillations are as follows:

The differential equation must be obtained taking into account the decrease in vibrational energy in the process of oscillations.

The oscillation equation is the solution of a differential equation.

The amplitude of damped oscillations depends on time.

The frequency and period depend on the degree of damping of the oscillations.

Phase and initial phase have the same meaning as for undamped oscillations.

Mechanical damped oscillations.

mechanical system : spring pendulum subject to friction forces.

Forces acting on the pendulum :

Elastic force., where k is the spring stiffness coefficient, х is the displacement of the pendulum from the equilibrium position.

Resistance force. Consider the resistance force proportional to the speed v of movement (such a dependence is typical for a large class of resistance forces): . The minus sign shows that the direction of the resistance force is opposite to the direction of the body's velocity. The drag coefficient r is numerically equal to the drag force that occurs at a unit speed of the body:

Law of motion spring pendulum is Newton's second law:

m a = F ex. + F resist.

Considering that and , we write Newton's second law in the form:

. (21.1)

Dividing all the terms of the equation by m, moving them all to the right side, we get differential equation damped oscillations:

Let's denote , where β – damping factor , , where ω 0 is the frequency of undamped free oscillations in the absence of energy losses in the oscillatory system.

In the new notation, the differential equation of damped oscillations has the form:

. (21.2)

This is a second order linear differential equation.

This linear differential equation is solved by a change of variables. We represent the function x, depending on the time t, in the form:

.

Let's find the first and second time derivatives of this function, given that the function z is also a function of time:

, .

Substitute the expressions in the differential equation:

We bring like terms in the equation and reduce each term by , we get the equation:

.

Let us denote the quantity .

Equation solution are the functions , .

Returning to the variable x, we obtain the formulas for the equations of damped oscillations:

Thus , equation of damped oscillations is a solution of the differential equation (21.2):

Damped oscillation frequency :

(only the real root has a physical meaning, therefore).

Period of damped oscillations :

(21.5)

The meaning that was put into the concept of a period for undamped oscillations is not suitable for damped oscillations, since the oscillatory system never returns to its original state due to the loss of vibrational energy. In the presence of friction, the oscillations are slower: .

The period of damped oscillations called the minimum time interval for which the system passes twice the equilibrium position in the same direction.

For the mechanical system of the spring pendulum we have:

, .

Amplitude of damped oscillations :

For spring pendulum.

The amplitude of damped oscillations is not a constant value, but changes with time the faster, the greater the coefficient β. Therefore, the definition for the amplitude, given earlier for undamped free oscillations, must be changed for damped oscillations.

For small attenuation amplitude of damped oscillations called the largest deviation from the equilibrium position for the period.

Graphs the offset vs. time and amplitude vs. time curves are shown in Figures 21.1 and 21.2.

Figure 21.1 - The dependence of the displacement on time for damped oscillations.

Figure 21.2 - Dependences of the amplitude on time for damped oscillations

Characteristics of damped oscillations.

1. Attenuation factor β .

The change in the amplitude of damped oscillations occurs according to the exponential law:

Let the oscillation amplitude decrease by “e” times over time τ (“e” is the base of the natural logarithm, e ≈ 2.718). Then, on the one hand, , and on the other hand, having painted the amplitudes A zat. (t) and A at. (t+τ), we have . These relations imply βτ = 1, hence .

Time interval τ , for which the amplitude decreases by “e” times, is called the relaxation time.

Attenuation factor β is a value inversely proportional to the relaxation time.

2. Logarithmic damping decrement δ - a physical quantity numerically equal to the natural logarithm of the ratio of two successive amplitudes separated in time by a period.

If the attenuation is small, i.e. the value of β is small, then the amplitude changes slightly over the period, and the logarithmic decrement can be defined as follows:

,

where A at. (t) and A at. (t + NT) - oscillation amplitudes at time e and after N periods, i.e. at time (t + NT).

3. Quality factor Q oscillatory system is a dimensionless physical quantity equal to the product of the value (2π) νа the ratio of the energy W(t) of the system at an arbitrary moment of time to the energy loss over one period of damped oscillations:

.

Since the energy is proportional to the square of the amplitude, then

For small values ​​of the logarithmic decrement δ, the quality factor of the oscillatory system is equal to

,

where N e is the number of oscillations, during which the amplitude decreases by “e” times.

So, the quality factor of a spring pendulum is -. The greater the quality factor of an oscillatory system, the less attenuation, the longer the periodic process in such a system will last. Quality factor of the oscillatory system - dimensionless quantity that characterizes the dissipation of energy in time.

4. With an increase in the coefficient β, the frequency of damped oscillations decreases, and the period increases. At ω 0 = β, the frequency of damped oscillations becomes equal to zero ω zat. = 0, and T zat. = ∞. In this case, the oscillations lose their periodic character and are called aperiodic.

At ω 0 = β, the system parameters responsible for the decrease in vibrational energy take values ​​called critical . For a spring pendulum, the condition ω 0 = β will be written as:, from where we find the value critical drag coefficient:

.

Rice. 21.3. The dependence of the amplitude of aperiodic oscillations on time

Forced vibrations.

All real oscillations are damped. In order for real oscillations to occur for a sufficiently long time, it is necessary to periodically replenish the energy of the oscillatory system by acting on it with an external periodically changing force

Consider the phenomenon of oscillations if the external (forcing) force varies with time according to the harmonic law. In this case, oscillations will arise in the systems, the nature of which, to one degree or another, will repeat the nature of the driving force. Such fluctuations are called forced .

General signs of forced mechanical oscillations.

1. Let us consider the forced mechanical oscillations of a spring pendulum, which is acted upon by an external (compelling ) periodic force . The forces that act on a pendulum, once taken out of equilibrium, develop in the oscillatory system itself. These are the elastic force and the drag force.

Law of motion (Newton's second law) is written as follows:

(21.6)

Divide both sides of the equation by m, take into account that , and get differential equation forced vibrations:

Denote ( β – damping factor ), (ω 0 is the frequency of undamped free oscillations), the force acting per unit mass. In these notations differential equation forced oscillations will take the form:

(21.7)

This is a second-order differential equation with a non-zero right side. The solution of such an equation is the sum of two solutions

.

is the general solution of a homogeneous differential equation, i.e. differential equation without the right side when it is equal to zero. We know such a solution - this is the equation of damped oscillations, written up to a constant, the value of which is determined by the initial conditions of the oscillatory system:

Where .

We discussed earlier that the solution can be written in terms of sine functions.

If we consider the process of pendulum oscillations after a sufficiently long period of time Δt after the driving force is turned on (Figure 21.2), then the damped oscillations in the system will practically stop. And then the solution of the differential equation with the right side will be the solution .

A solution is a particular solution of an inhomogeneous differential equation, i.e. equations with the right side. It is known from the theory of differential equations that with the right side changing according to the harmonic law, the solution will be a harmonic function (sin or cos) with a change frequency corresponding to the change frequency Ω of the right side:

where A ampl. – amplitude of forced oscillations, φ 0 – phase shift , those. phase difference between the phase of the driving force and the phase of forced oscillations. And amplitude A ampl. , and the phase shift φ 0 depend on the parameters of the system (β, ω 0) and on the frequency of the driving force Ω.

Forced oscillation period equals (21.9)

Schedule of forced oscillations in Figure 4.1.

Fig.21.3. Schedule of forced oscillations

The steady forced oscillations are also harmonic.

Dependences of the amplitude of forced oscillations and phase shift on the frequency of external action. Resonance.

1. Let's return to the mechanical system of a spring pendulum, which is affected by an external force that changes according to a harmonic law. For such a system, the differential equation and its solution, respectively, have the form:

, .

Let us analyze the dependence of the oscillation amplitude and phase shift on the frequency of the external driving force, for this we find the first and second derivatives of x and substitute them into the differential equation.

Let's use the vector diagram method. It can be seen from the equation that the sum of the three swings on the left side of the equation (Figure 4.1) should be equal to the swing on the right side. The vector diagram is made for an arbitrary time t. It can be determined from it.

Figure 21.4.

, (21.10)

. (21.11)

Considering the value , ,, we obtain formulas for φ 0 and A ampl. mechanical system:

,

.

2. We investigate the dependence of the amplitude of forced oscillations on the frequency of the driving force and the magnitude of the resistance force in an oscillating mechanical system, using these data we construct a graph . The results of the study are shown in Figure 21.5, they show that at a certain frequency of the driving force the amplitude of oscillations increases sharply. And this increase is the greater, the lower the attenuation coefficient β. At , the oscillation amplitude becomes infinitely large.

The phenomenon of a sharp increase in amplitude forced oscillations at a frequency of the driving force equal to is called resonance.

(21.12)

The curves in Figure 21.5 reflect the relationship and are called amplitude resonance curves .

Figure 21.5 - Graphs of the dependence of the amplitude of forced oscillations on the frequency of the driving force.

The amplitude of resonant oscillations will take the form:

Forced vibrations are undamped fluctuations. The inevitable losses of energy due to friction are 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 self-oscillations.

In a self-oscillatory system, three characteristic elements can be distinguished - an oscillatory system, an energy source and a feedback device between the oscillatory system and the source. 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 the deformation energy of the spring or the potential energy of the load in the gravitational field. The feedback device is a mechanism by which the self-oscillatory system regulates the flow of energy from the source. On fig. 21.6 shows a diagram of the interaction of various elements of a self-oscillating system.

An example of a mechanical self-oscillating system is a clockwork with anchor move (Fig. 21.7.). A running wheel with oblique teeth is rigidly fastened to a toothed drum, through which a chain with a weight is thrown. At the upper end of the pendulum, an anchor (anchor) is fixed with two plates of hard material bent along an arc of a circle centered on the axis of the pendulum. In a wristwatch, the weight is replaced by a spring, and the pendulum is replaced by a balancer - a handwheel fastened to a spiral spring.

Figure 21.7. Clock mechanism with a pendulum.

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.

Mechanical self-oscillatory systems are widespread in the life around us and in technology. Self-oscillations are made by steam engines, internal combustion engines, electric bells, strings of bowed musical instruments, air columns in the pipes of wind instruments, vocal cords when talking or singing, etc.

In real oscillatory systems, in addition to quasi-elastic forces, there are forces of resistance of the medium. The presence of friction forces leads to dissipation (dissipation) of energy and a decrease in the oscillation amplitude. By slowing down the movement, the friction forces increase the period, i.e. reduces the frequency of oscillation. Such oscillations will not be harmonic.

Oscillations with amplitude continuously decreasing in time due to energy dissipation are called fading . At sufficiently low speeds, the friction force is proportional to the speed of the body and is directed against the motion

where r is the coefficient of friction, which depends on the properties of the medium, the shape and size of the moving body. The differential equation of damped oscillations in the presence of friction forces will have the form:

or
(21)

where
- attenuation coefficient,

- natural circular frequency of free oscillations in the absence of friction forces.

The general solution of Eq. (21) in the case of low damping (
) is an:

It differs from harmonic (8) in that the oscillation amplitude:

(23)

is a decreasing function of time, and the circular frequency related to natural frequency and damping factor ratio:

. (24)

The period of damped oscillations is equal to:

. (25)

The dependence of the displacement X on t damped oscillations is shown in Fig.4.

C the degree of decrease in amplitude is determined by the attenuation coefficient .

During
amplitude (23) decreases by a factor of e ≈ 2.72. This time natural decay is called relaxation time. Therefore, the damping factor is the reciprocal of the relaxation time:

.(26)

The rate of decrease in the amplitude of oscillations is characterized by logarithmic damping decrement. Let A(t) and A(t+T) be the amplitudes of two successive oscillations corresponding to time points that differ by one period. Then the relation:

(27)

called damping decrement, which shows how many times the amplitude of oscillations decreases in a time equal to the period. The natural logarithm of this ratio is:

(28)

is called the logarithmic damping factor. Here, N e is the number of oscillations performed during the time when the amplitude decreases by a factor of e, i.e. during relaxation time.

Thus, the logarithmic damping decrement is the reciprocal of the number of oscillations, after which the oscillation amplitude decreases by a factor of e.

The rate of decrease in the energy of the oscillatory system is characterized by the quality factor Q. Quality factor of the oscillatory system- a value proportional to the ratio of the total energy E(t) of the oscillatory system to the energy (- E) lost during the period T:

(29)

The total energy of the oscillatory system at an arbitrary moment of time and for any value of X has the form:

(30)

Since the energy is proportional to the square of the amplitude, the energy of damped oscillations decreases in proportion to the value
, you can write:

. (31)

Then, according to the definition, the expression for the quality factor of the oscillatory system will look like:

Here it is taken into account that at low attenuations (1): 1st -2   ​​2.

Therefore, the quality factor is proportional to the number of oscillations N e performed by the system during the relaxation time.

The quality factor of oscillatory systems can vary greatly, for example, the quality factor of a physical pendulum is Q~ 10 2 , while the quality factor of an atom, which is also an oscillatory system, reaches Q~ 10 8 .

In conclusion, we note that when the damping coefficient β=ω 0, the period becomes infinite T =∞ (critical damping). With a further increase in β, the period T becomes imaginary, and the attenuation of the motion occurs without oscillations, as they say, aperiodically. This case of movement is shown in Fig.5. Critical damping (calming) occurs in a minimum time and is important in measuring instruments, for example, in ballistic galvanometers .

AT FORCED VASCULATION AND RESONANCE

If an elastic force F y \u003d -kX acts on a body with mass m, the friction force
and external periodic force
, then it performs forced oscillations. In this case, the differential equation of motion has the form:

where
,
- attenuation coefficient,
- natural frequency of free undamped vibrations of the body, F 0 - amplitude, ω - frequency of the periodic force.

At the initial moment of time, the work of the external force exceeds the energy that is spent on friction (Fig. 6). The energy and amplitude of the body's oscillations will increase until all the energy communicated by the external force is completely spent on overcoming friction, which is proportional to the speed. Therefore, an equilibrium is established in which the sum of kinetic and potential energy is constant. This condition characterizes the stationary state of the system.

In this state, the movement of the body will be harmonic with a frequency equal to the frequency of the external excitation, but due to the inertia of the body, its oscillations will be shifted in phase with respect to the instantaneous value of the external periodic force:

X = ACos(ωt + φ). (34)

Unlike free oscillations, the amplitude A and the phase  of forced oscillations do not depend on the initial conditions of motion, but will be determined only by the properties of the oscillating system, the amplitude and frequency of the driving force:

, (35)

. (36)

It can be seen that the amplitude and phase shift depend on the frequency of the driving force (Fig. 7, 8).

A characteristic feature of forced oscillations is the presence of resonance. Phenomenon a sharp increase in the amplitude of forced oscillations when the frequency of the driving force approaches the natural frequency of free undamped oscillations of the body ω 0 is called mechanical resonance . Vibration amplitude of the body at resonant frequency
reaches the maximum value:

(37)

Concerning the resonance curves (see Fig. 7), let us make the following remarks. If ω → 0, then all curves (see also (35)) come to the same nonzero limit value
, the so-called statistical deviation. If ω→ ∞, then all curves tend asymptotically to zero.

Under the condition of low damping (β 2 ‹‹ω 0 2), the resonant amplitude (see (37))

(37a)

Under this condition, we take the ratio of the resonant displacement to the static deviation:

from which it can be seen that the relative increase in the amplitude of oscillations at resonance is determined by the quality factor of the oscillatory system. Here, the quality factor is, in fact, the gain of the response
system and at low attenuation can reach large values.

This circumstance determines the great importance of the phenomenon of resonance in physics and technology. It is used if they want to amplify vibrations, for example, in acoustics - to enhance the sound of musical instruments, in radio engineering - to isolate the desired signal from many others that differ in frequency. If resonance can lead to an undesirable increase in oscillations, a system with a low quality factor is used.

RELATED VIBRATIONS

The second oscillatory system, elastically connected with the first, can serve as a source of external periodic force. Both oscillatory systems can act one on the other. So, for example, the case of two coupled pendulums (Fig. 9).

The system can perform both in-phase (Fig. 9b) and anti-phase (Fig. 9c) oscillations. Such oscillations are called normal type or normal mode of oscillation and are characterized by their own normal frequency. With in-phase oscillations, the displacement of the pendulums at all times X 1 \u003d X 2, and the frequency ω 1 is exactly the same as the frequency of a single pendulum
. This is because the light spring is in a free state and does not have any effect on the movement. With antiphase oscillations at all times - X 1 \u003d X 2. The frequency of such oscillations is greater than and equal to
, since the spring, which has rigidity k and carries out the connection, is always in a stretched, then in a compressed state.

L
Any state of our coupled system, including the initial displacement X (Fig. 9a), can be represented as a superposition of two normal modes:

If we set the system in motion from the initial state X 1 = 0,
, X 2 \u003d 2A,
,

then the displacements of the pendulums will be described by the expressions:

On fig. 10 shows the change in the displacement of individual pendulums over time.

The oscillation frequency of the pendulums is equal to the average frequency of two normal modes:

, (39)

and their amplitude changes according to the sine or cone law with a lower frequency equal to half the frequency difference of the normal modes:

. (40)

A slow change in amplitude with a frequency equal to half the difference between the frequencies of the normal modes is called “ beats ” two vibrations with almost the same frequency. The frequency of “beats” is equal to the difference ω 1 – ω 2 frequencies, (and not half of this difference), since the maximum amplitude 2A is reached twice in a period corresponding to the frequency

Hence, the beat period is equal to:

(41)

When the pendulums beat, energy is exchanged. However, a complete energy exchange is possible only when both masses are the same and the ratio (ω 1 + ω 2 / ω 1 -ω 2) is equal to an integer. One important point to note is that while individual pendulums can exchange energy, there is no exchange of energy between normal modes.

The presence of such oscillating systems that interact with each other and are able to transfer their energy to each other, form the basis of wave motion.

An oscillating material body placed in an elastic medium entrains and sets in oscillatory motion the particles of the medium adjacent to it. Due to the presence of elastic bonds between the particles, the vibrations propagate at a speed characteristic of a given medium throughout the entire medium.

The process of vibration propagation in an elastic medium is called wave .

There are two main types of waves: longitudinal and transverse. In longitudinal waves particles of the medium oscillate along the direction of wave propagation, and in transverse is perpendicular to the direction of wave propagation. Not every elastic medium can propagate a transverse wave. A transverse elastic wave is possible only in such media in which elastic shear deformation takes place. For example, only longitudinal elastic waves (sound) propagate in gases and liquids.

The locus of points of the medium, to which the oscillation has reached a given point in time, is called wave front . The wave front separates the part of space already involved in the wave process from the area in which oscillations have not yet arisen. Depending on the shape of the front, waves are plane, spherical, cylindrical, etc.

The equation for a plane wave propagating without loss in a homogeneous medium is:
, (42)

where ξ(X,t) is the displacement of particles of the medium with the coordinate X from the equilibrium position at time t, A is the amplitude,
- wave phase,
- circular frequency of oscillation of particles of the medium, v - speed of wave propagation.

Wavelength λ the distance between points oscillating with a phase difference of 2π is called, in other words, the wavelength is the path traveled by any phase of the wave in one period of oscillation:

phase velocity, i.e. propagation speed of this phase:

λ / T (44)

wave number is the number of wavelengths that fit on a length of 2π units:

k = ω / v = 2π / λ. (45)

Substituting these notations into (42), plane traveling monochromatic wave equation can be represented as:

(46)

Note that the wave equation (46) exhibits a double periodicity in coordinate and time. Indeed, the phases of the oscillations coincide when the coordinate changes by λ and when the time changes by a period T. Therefore, it is impossible to graphically depict a wave on a plane. The time t is often fixed and the dependence of the displacement ξ on the X coordinate is presented on the graph, i.e. instantaneous distribution of displacements of particles of the medium along the direction of wave propagation (Fig. 11). The phase difference Δφ of the oscillations of the points of the medium depends on the distance ΔX \u003d X 2 - X 1 between these points:

(47)

If the wave propagates opposite to the X direction, then the backward wave equation will be written as:

ξ (X,t) = ACos(ωt + kX). (48)

STANDING WAVES are the result of a special kind of wave interference. They are formed when two traveling waves propagate towards each other with the same frequencies and amplitudes.

The equations of two plane waves propagating along the X axis in opposite directions are:

ξ 1 \u003d ACos (ωt - kX)

ξ 2 = ACos(ωt + kX). (49)

Adding these equations using the formula of the sum of cosines and taking into account that k = 2π / λ, we obtain the standing wave equation:

. (50)

Cos ωt multiplier shows that oscillations of the same frequency ω occur at the points of the medium with amplitude
, depending on the X-coordinate of the considered point. At points in the environment where:
, (51)

the oscillation amplitude reaches a maximum value of 2A. These points are called antinodes.

From expression (51) one can find the antinode coordinates:
(52)

At the points where
(53) the oscillation amplitude vanishes. These points are called knots.

Node coordinates:
. (54)

R the distances between neighboring antinodes and neighboring nodes are the same and equal to λ/2. The distance between the node and the neighboring antinode is equal to λ / 4. When passing through the node, the multiplier
changes sign, so the phases of the oscillations on opposite sides of the node differ by π, i.e. points lying on opposite sides of the node oscillate in antiphase. Points enclosed between two neighboring nodes oscillate with different amplitudes, but with the same phases.

The distribution of nodes and antinodes in a standing wave depends on the conditions that take place at the interface between two media, from which reflection occurs. If the wave is reflected from a denser medium, then the phase of the oscillations at the place where the wave is reflected changes to the opposite, or, as they say, half of the wave is lost. Therefore, as a result of the addition of oscillations of opposite directions, the displacement at the boundary is zero, i.e. there is a node (Fig. 12). When a wave is reflected from the boundary of a less dense medium, the phase of oscillations at the place of reflection remains unchanged and oscillations with the same phases are added near the boundary - an antinode is obtained.

In a standing wave, there is no phase movement, no wave propagation, no energy transfer, which is the reason for the name of this type of wave.

A decrease in the energy of the oscillatory system leads to a gradual decrease in the amplitude of oscillations, because

In this case, they say that fluctuations are damped .

A similar situation develops in the oscillatory circuit. The real coil, which is part of the circuit, always has active resistance. When current flows through the active resistance of the coil, Joule heat will be released. In this case, the energy of the circuit will decrease, which will lead to a decrease in the amplitude of charge, voltage and current oscillations.

Our task- find out by what law the decrease in the amplitude of oscillations occurs, according to which law the oscillating value itself changes, with what frequency damped oscillations occur, how long the oscillations “fade out”.

§1 Damping of vibrations in systems with viscous friction

Consider an oscillatory system in which the force of viscous friction acts. An example of such an oscillatory system is a mathematical pendulum that oscillates in the air.

In this case, when the system is taken out of equilibrium by

the pendulum will be acted upon by two forces: a quasi-elastic force and a resistance force (viscous friction force).

Newton's second law is written as follows:

(1)

We know that at low speeds, the viscous friction force is proportional to the speed of motion:

We take into account that the velocity projection is the first derivative of the body coordinate, and the acceleration projection is the second derivative of the coordinate:

Then equation (2) will take the form:

we obtain the equation of motion in the following form:

(3)

where d is the damping coefficient, it depends on the friction coefficient r,

w 0 - cyclic frequency of ideal oscillations (in the absence of friction).

Before solving equation (3), consider the oscillatory circuit. The active resistance of the coil is connected in series with the capacitance C and the inductance L.

Let's write down Kirchhoff's second law

Let's take into account that, , .

Then Kirchhoff's second law takes the form:

Divide both sides of the equation by:

Let us introduce the notation

Finally we get

Pay attention to the mathematical identity of the differential equations (3) and (3'). There is nothing surprising. We have already shown the absolute mathematical identity of the process of oscillation of the pendulum and electromagnetic oscillations in the circuit. Obviously, the processes of damping oscillations in the circuit and in systems with viscous friction also occur in the same way.

By solving equation (3), we will get answers to all the above questions.

We know the solution to this equation

Then for the desired equation (3) we obtain the final result

It is easy to see that the charge of a capacitor in a real oscillatory circuit will change according to the law

Analysis of the result:

1 As a result of the joint action of the quasi-elastic force and the resistance force, the system maybe make an oscillating motion. For this, the condition w 0 2 - d 2 > 0 must be satisfied. In other words, the friction in the system must be small.

2 The frequency of damped oscillations w does not coincide with the oscillation frequency of the system in the absence of friction w 2 = w 0 2 - d 2< w 0 2 . Over time, the frequency of damped oscillations remains unchanged.

If the damping coefficient d is small, then the frequency of damped oscillations is close to the natural frequency w 0 .

This decrease in amplitude occurs exponentially.

4 If w 0 2 - d 2< 0, то есть трение в системе велико, то уравнение (3) имеет решение вида

(4)

where .

By direct substitution, it is easy to verify that function (4) is indeed a solution to equation (3). Obviously, the sum of two exponential functions is not a periodic function. From a physical point of view, this means that there will be no oscillations in the system. After removing the system from the equilibrium position, it will slowly return to it. Such a process is called aperiodic .

§2 How quickly do oscillations decay in systems with viscous friction?

Decrement of damping

quantity value . It can be seen that the value of d characterizes the rate of damping of the oscillations. For this reason, d is called the damping factor.

For electrical oscillations in the circuit, the attenuation coefficient depends on the parameters of the coil: the greater the active resistance of the coil, the faster the amplitude of the charge on the capacitor, voltage, and current decrease.

The function is the product of a decreasing exponential function and a harmonic function, so the function is not harmonic. But it has a certain degree of "repeatability", which consists in the fact that the maxima, minima, zeros of the function occur at regular intervals. The graph of the function is a sinusoid bounded by two exponents.

Let us find the ratio of two successive amplitudes separated by a time interval of one period. This relationship is called damping decrement

Please note that the result does not depend on whether you consider two consecutive periods - at the beginning of the oscillatory movement or after some time has passed. For each period, the amplitude of oscillations changes not the same size, but the same number of times !!

It is easy to see that for any different time intervals, the amplitude of damped oscillations decreases by the same number of times.

Relaxation time

The relaxation time is called the time during which the amplitude of damped oscillations decreases by e times:

Then .

From here it is not difficult to establish the physical meaning of the attenuation coefficient:

Thus, the damping factor is the reciprocal of the relaxation time. Let, for example, in the oscillatory circuit, the damping coefficient is equal to . This means that after a time s the oscillation amplitude will decrease by e once.

Logarithmic damping decrement

Often, the damping rate of oscillations is characterized by a logarithmic damping decrement. To do this, take the natural logarithm of the ratio of the amplitudes separated by a period of time.

Let us find out the physical meaning of the logarithmic damping decrement.

Let N be the number of oscillations performed by the system during the relaxation time, that is, the number of oscillations during which the oscillation amplitude decreases in e once. Obviously, .

It can be seen that the logarithmic damping decrement is the reciprocal of the number of oscillations, after which the amplitude decreases in e once.

Suppose, , this means that after 100 oscillations, the amplitude will decrease by e once.

Quality factor of the oscillatory system

In addition to the logarithmic damping decrement and relaxation time, the damping rate of oscillations can be characterized by such a value as quality factor of the oscillating system . Under the quality factor

It can be shown that for weakly damped oscillations

The energy of the oscillatory system at an arbitrary point in time is equal to . The energy loss over a period can be found as the difference between the energy at a point in time and the energy after a time equal to the period:

Then

The exponential function can be expanded into a series at<< 1. после подстановки получаем .

When withdrawing, we imposed a restriction<< 1, что верно только для слабо затухающих колебаний. Следовательно, область применения выражения для добротности ограничена только слабо затухающими колебаниями. Тогда как выражение применимо к любой колебательной системе.

The formulas obtained by us for the quality factor of the system do not yet say anything. Let's say the calculations give a value of quality factor Q = 10. What does this mean? How fast do vibrations decay? Is it good or bad?

It is usually conditionally considered that oscillations have practically ceased if their energy has decreased by 100 times (amplitude - by 10). Let us find out how many oscillations the system has made by this moment:

We can answer the question posed earlier: N = 8.

Which oscillatory system is better - with a large or small quality factor? The answer to this question depends on what you want to get from the oscillatory system.

If you want the system to make as many oscillations as possible before stopping, the quality factor of the system must be increased. How? Since the quality factor is determined by the parameters of the oscillatory system itself, it is necessary to choose these parameters correctly.

For example, Foucault's pendulum, installed in St. Isaac's Cathedral, was supposed to perform weakly damped oscillations. Then

The easiest way to increase the quality factor of a pendulum is to make it heavier.

In practice, inverse problems often arise: it is necessary to extinguish the oscillations that have arisen as soon as possible (for example, the oscillation of the arrow of a measuring instrument, oscillations of the car body, oscillations of the ship, etc.) Devices that allow increasing attenuation in the system are called dampers (or shock absorbers). For example, a car shock absorber in the first approximation is a cylinder filled with oil (viscous liquid), in which a piston with a number of small holes can move. The piston rod is connected to the body, and the cylinder is connected to the wheel axle. The vibrations of the body that have arisen quickly die out, since the moving piston encounters a lot of resistance on its way from the viscous fluid that fills the cylinder.

§ 3 Damping of vibrations in systems with dry friction

The damping of oscillations occurs fundamentally differently if the sliding friction force acts in the system. It is she who is the reason for the stop of the spring pendulum, which oscillates along any surface.

Suppose a spring pendulum located on a horizontal surface was brought into oscillatory motion by compressing the spring and releasing the load, that is, from the extreme position. In the process of moving a load from one extreme position to another, it is affected by the force of gravity and the reaction force of the support (vertically), the force of elasticity and the force of sliding friction (along the surface).

Note that in the process of moving from left to right, the friction force is unchanged in direction and modulus.

This allows us to assert that during the first half of the period the spring pendulum is in a constant force field.

The displacement of the equilibrium position can be calculated from the condition that the resultant is equal to zero in the equilibrium position:
It is important that during the first half of the period of oscillation of the pendulum harmonic !

When moving in the opposite direction - from right to left - the friction force will change direction, but during the entire transition it will remain constant in magnitude and direction. This situation again corresponds to the oscillations of a pendulum in a constant force field. Only now this field is different! It changed direction. Consequently, the equilibrium position when moving from right to left also changed. Now it has shifted to the right by the amount D l 0 .

Let us depict the dependence of the body coordinate on time. Since for each half of the period the movement is a harmonic oscillation, the graph will be halves of sinusoids, each of which is built relative to its equilibrium position. We will perform the operation of "sewing solutions".

Let's show how this is done with a specific example.

Let the mass of the load attached to the spring be 200 g, the spring stiffness be 20 N/m, and the coefficient of friction between the load and the table surface be 0.1. The pendulum was brought into oscillatory motion by stretching the spring by

6.5 cm.

In contrast to oscillatory systems with viscous friction, in systems with dry friction, the amplitude of oscillations decreases with time according to a linear law - for each period it decreases by two widths of the stagnation zone.

Another distinctive feature is that oscillations in systems with dry friction, even theoretically, cannot occur indefinitely. They stop as soon as the body stops in the "stagnation zone".

§4 Examples of problem solving

Problem 1 The nature of the change in the amplitude of damped oscillations in systems with viscous friction

The amplitude of the damped oscillations of the pendulum during the time t 1 = 5 min decreased by 2 times. In what time t 2 will the oscillation amplitude decrease by 8 times? After what time t 3 can we consider that the oscillations of the pendulum have stopped?

Decision:

The amplitude of oscillations in systems with viscous friction over time

decreases exponentially , where is the oscillation amplitude at the initial moment of time, is the damping factor.

1 Let's write down the law of change of amplitude two times

2 We solve equations together. Taking the logarithm of each equation, we get

We divide the second equation not the first and find the time t 2

4

After transformations, we get

Divide the last equation by equation (*)

Task 2 Period of damped oscillations in systems with viscous friction

Determine the period of damped oscillations of the system T, if the period of natural oscillations T 0 \u003d 1 s, and the logarithmic damping decrement. How many oscillations will this system make before it comes to a complete stop?

Decision:

1 The period of damped oscillations in a system with viscous friction is greater than the period of natural oscillations (in the absence of friction in the system). The frequency of damped oscillations, on the contrary, is less than the natural frequency and is equal to , where is the attenuation coefficient.

2 Express the cyclic frequency through the period. and take into account that the logarithmic damping decrement is equal to:

3 After transformations, we get .

The energy of the system is equal to the maximum potential energy of the pendulum

After transformations, we get

5 We express the attenuation coefficient in terms of a logarithmic decrement , we obtain

The number of oscillations that the system will make before stopping is equal to

Problem 3 The number of oscillations made by the pendulum until the amplitude is halved

The logarithmic damping decrement of the pendulum is equal to q = 3×10 -3 . Determine the number of complete oscillations that the pendulum must make in order for the amplitude of its oscillations to decrease by 2 times.

Decision:

3 It is easy to see that is the logarithmic damping decrement. We get

Finding the number of vibrations

Task 4 Quality factor of the oscillatory system

Determine the quality factor of the pendulum, if during the time during which 10 oscillations were made, the amplitude decreased by 2 times. How long does it take for the pendulum to stop?

Decision:

1 The amplitude of oscillations in systems with viscous friction decreases exponentially with time, where is the amplitude of oscillations at the initial moment of time, is the damping coefficient.

Since the oscillation amplitude decreases by 2 times, we obtain

2 The oscillation time can be represented as the product of the period of oscillations by their number:

Substitute the resulting time value into the expression (*)

3 It is easy to see that is the logarithmic damping decrement. We get the logarithmic damping decrement equal to

4 Quality factor of the oscillatory system

The energy of the system is equal to the maximum potential energy of the pendulum

After transformations, we get

Find the time after which the oscillations will stop .

Task 5 Vibrations of a magnet

Vasya Lisichkin, a well-known experimenter throughout the school, decided to make the magnetic figurine of his favorite literary hero Kolobok vibrate along the refrigerator wall. He attached the figurine to a spring with stiffness k = 10 N/m, stretched it by 10 cm, and let it go. How many oscillations will the Gingerbread Man make if the mass of the figurine is m = 10 g, the coefficient of friction between the figurine and the wall is μ = 0.4, and it can be torn off the wall with the force F = 0.5 N.

Decision:

1 When moving from the extreme lower to the extreme upper position, when the speed of the load is directed upwards, the sliding friction force is directed downwards and is numerically equal to . Thus, the spring pendulum is in a constant force field created by the forces of gravity and friction. In a constant force field, the pendulum shifts its equilibrium position:

where is the stretching of the spring in the new "equilibrium position".

2 When moving from the extreme upper to the extreme lower position, when the speed of the load is directed downwards, the sliding friction force is directed upwards and is numerically equal to . Thus, the spring pendulum is again in a constant force field created by the forces of gravity and friction. In a constant force field, the pendulum shifts its equilibrium position:

where is the deformation of the spring in the new "equilibrium position", the "-" sign says that in this position the spring is compressed.

3 The stagnation zone is limited by spring deformations from - 1 cm to 3 cm and is 4 cm. The middle of the stagnation zone, in which the spring deformation is 1 cm, corresponds to the position of the load in which there is no friction force. In the stagnation zone, the elastic force of the spring is less in modulus than the resultant maximum static friction force and gravity. If the pendulum stops in the stagnation zone, the oscillations stop.

4 For each period, the deformation of the spring is reduced by two widths of the stagnation zone, i.e. by 8 cm. After one oscillation, the deformation of the spring will become equal to 10 cm - 8 cm = 2 cm. This means that after one oscillation, the Kolobok figure enters the stagnation zone and its oscillations stop.

§5 Tasks for independent solution

Test "Damped vibrations"

1 Damping of vibrations is understood as ...

A) decrease in the frequency of oscillations; B) decrease in the period of oscillations;

C) decrease in the amplitude of oscillations; D) decrease in the phase of oscillations.

2 The reason for the damping of free vibrations is

A) the effect on the system of random factors that inhibit oscillations;

B) the action of a periodically changing external force;

C) the presence of a friction force in the system;

D) a gradual decrease in the quasi-elastic force, which tends to return the pendulum to the equilibrium position.

?

A) 5 cm; B) 4 cm; C) 3 cm;

D) It is not possible to give an answer, because the time is unknown.

6 Two identical pendulums, being in different viscous media, oscillate. The amplitude of these oscillations changes over time as shown in the figure. Which medium has more friction?

7 Two pendulums, being in the same environment, oscillate. The amplitude of these oscillations changes over time as shown in the figure. Which pendulum has the largest mass?

C) It is impossible to give an answer, since the scale is not set along the coordinate axes and it is impossible to perform calculations.

8 Which figure correctly shows the dependence of the coordinate of damped oscillations in a system with viscous friction on time?

A) 1; B) 2; IN 3; D) All graphs are correct.

9 Establish a correspondence between the physical quantities characterizing the damping of oscillations in systems with viscous friction, and their definition and physical meaning. Fill the table

A) This is the ratio of the amplitudes of oscillations after a time equal to the period;

B) This is the natural logarithm of the ratio of the oscillation amplitudes after a time equal to the period;

C) This is the time during which the amplitude of the oscillations decreases in e once;

G) D) E)

G) This value is the reciprocal of the number of oscillations, for which the amplitude of oscillations decreases in e once;

H) This value shows how many times the amplitude of oscillations decreases over a time equal to the period of oscillations.

10 Make a correct statement.

Goodness means...

A) the ratio of the total energy of the system E increased by a factor of 2p to the energy W dissipated over a period;

B) the ratio of amplitudes after a period of time equal to the period;

C) the number of oscillations that the system makes by the moment when the amplitude decreases by e times.

The quality factor is calculated according to the formula ...

BUT) B) C)

The quality factor of an oscillatory system depends on…

A) the energy of the system;

B) energy losses for the period;

C) parameters of the oscillatory system and friction in it.

The greater the quality factor of the oscillatory system, the ...

A) oscillations decay more slowly;

B) fluctuations decay faster.

11 A mathematical pendulum is set into oscillatory motion, deviating the suspension from the equilibrium position in the first case by 15 °, in the second - by 10 °. In which case will the pendulum make more oscillations before stopping?

A) When the hanger is deflected by 15°;

B) When the hanger is deflected by 10°;

C) In both cases, the pendulum will make the same number of oscillations.

12 Balls of the same radius are attached to two threads of the same length - aluminum and copper. The pendulums are set in oscillatory motion, deflecting them at the same angles. Which of the pendulums will make the most number of oscillations before stopping?

A) aluminum; B) Copper;

C) Both pendulums will make the same number of oscillations.

13 A spring pendulum, located on a horizontal surface, was brought into oscillation by stretching the spring by 9 cm. After making three complete oscillations, the pendulum was at a distance of 6 cm from the position of the undeformed spring. How far from the position of the undeformed spring will the pendulum be after the next three oscillations?

A) 5 cm; B) 4 cm; C) 3 cm.

damped vibrations

Damped oscillations of a spring pendulum

damped vibrations- fluctuations, the energy of which decreases with time. An infinitely continuing process of species is impossible in nature. Free oscillations of any oscillator sooner or later fade and stop. Therefore, in practice, one usually deals with damped oscillations. They are characterized by the fact that the amplitude of oscillations A is a decreasing function. Typically, damping occurs under the action of the forces of resistance of the medium, most often expressed as a linear dependence on the speed of oscillations or its square.

In acoustics: attenuation - reducing the signal level to complete inaudibility.

Damped oscillations of a spring pendulum

Let there be a system consisting of a spring (obeying Hooke's law), one end of which is rigidly fixed, and on the other there is a body of mass m. Oscillations occur in a medium where the resistance force is proportional to the velocity with a coefficient c(see viscous friction).

The roots of which are calculated by the following formula

Solutions

Depending on the value of the attenuation coefficient, the solution is divided into three possible options.

• aperiodicity

If , then there are two real roots, and the solution of the differential equation takes the form:

In this case, the oscillations decay exponentially from the very beginning.

• Aperiodicity boundary

If , the two real roots are the same, and the solution to the equation is:

In this case, there may be a temporary increase, but then an exponential decay.

• Weak attenuation

If , then the solution of the characteristic equation are two complex conjugate roots

Then the solution to the original differential equation is

Where is the natural frequency of damped oscillations.

The constants and in each of the cases are determined from the initial conditions:

see also

• Decrement of damping

Literature

Lit .: Saveliev I. V., Course of General Physics: Mechanics, 2001.

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