The reason for attenuation is that in any oscillatory system, in addition to the restoring force, there are always various types of air resistance

etc., which slow down the movement. With each swing, a part is spent on work against friction forces. Ultimately, this work consumes the entire supply of energy initially supplied to the oscillatory system.

When considering , we were dealing with ideal, strictly periodic natural oscillations. By using such a model to describe real oscillations, we deliberately allow for inaccuracy in the description. However, such a simplification is suitable due to the fact that in many oscillatory systems the damping of oscillations caused by friction is really small: the system manages to make many oscillations before they decrease noticeably.

Graphs of damped oscillations

In the presence of damping, the natural oscillation (Fig. 1) ceases to be harmonic. Moreover, a damped oscillation ceases to be a periodic process - friction affects not only the amplitude of the oscillations (that is, it causes damping), but also the duration of the swings. As friction increases, the time required for the system to complete one complete oscillation increases. The graph of damped oscillations is shown in Fig. 2.

Fig.1. Free harmonic graph

Fig.2. Damped oscillation graph

A characteristic feature of oscillatory systems is that slight friction affects the period of oscillation to a much lesser extent than the amplitude. This circumstance played a huge role in the improvement of watches. The first clock was built by the Dutch physicist and mathematician Christiaan Huygens in 1673. This year can be considered the birth date of modern clock mechanisms. The movement of pendulum clocks is little sensitive to changes due to friction, which in general depend on many factors, while the speed of the previous pendulum clocks was very dependent on friction.

In practice, there is a need to both reduce and increase the damping of oscillations. For example, when designing watch movements, they strive to reduce the damping of the oscillations of the watch balancer. To do this, the balancer axis is equipped with sharp tips, which rest on well-polished conical bearings made of hard stone (agate or ruby). On the contrary, in many measuring instruments it is very desirable that the moving part of the device is installed quickly during the measurement process, but undergoes a large number of oscillations. To increase attenuation in this case, various dampers are used - devices that increase friction and, in general, energy loss.

1.21. 3DAMPED, FORCED OSCILLATIONS

Differential equation of damped oscillations and its solution. Attenuation coefficient. Logarithmic deckdecay time.Oscillation quality factorbody system.Aperiodic process. Differential equation of forced oscillations and its solution.Amplitude and phase of forced oscillations. The process of establishing oscillations. The case of resonance.Self-oscillations.

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

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

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

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

The oscillation equation is a solution to a differential equation.

The amplitude of damped oscillations depends on time.

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

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

Mechanical damped oscillations.

Mechanical system : spring pendulum taking into account friction forces.

Forces acting on a pendulum :

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

Resistance force. Let's consider a resistance force proportional to the speed v of movement (this 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 speed of the body. The drag coefficient r is numerically equal to the drag force arising at a unit speed of body movement:

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

m a = F ex. + F resistance

Considering that both , we write Newton’s second law in the form:

. (21.1)

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

Let us denote where β – attenuation coefficient , , Where ω 0 – 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 changing variables. Let us represent the function x, depending on time t, in the form:

.

Let's find the first and second derivatives of this function with respect to time, taking into account that the function z is also a function of time:

, .

Let's substitute the expressions into the differential equation:

Let's present similar terms in the equation and reduce each term by , we get the equation:

.

Let us denote the quantity .

Solving the equation 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 to the differential equation (21.2):

Damped frequency :

(only the real root has 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 losses of oscillatory energy. In the presence of friction, vibrations are slower: .

Period of damped oscillations is the minimum period of time during which the system passes the equilibrium position twice in one direction.

For the mechanical system of a spring pendulum we have:

, .

Amplitude of damped oscillations :

For a spring pendulum.

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

For small attenuations amplitude of damped oscillations is called the largest deviation from the equilibrium position over a period.

Charts The displacement versus time and amplitude versus time plots are presented in Figures 21.1 and 21.2.

Figure 21.1 – Dependence of displacement on time for damped oscillations.

Figure 21.2 – Dependence of amplitude on time for damped oscillations

Characteristics of damped oscillations.

1. Attenuation coefficient β .

The amplitude of damped oscillations changes according to an exponential law:

Let the oscillation amplitude decrease by “e” times during time τ (“e” is the base of the natural logarithm, e ≈ 2.718). Then, on the one hand, , and on the other hand, having described the amplitudes A zat. (t) and A zat. (t+τ), we have . From these relations it follows βτ = 1, hence .

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

Attenuation coefficient β – a quantity 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 is A zat. (t) and A zat. (t+NT) – amplitudes of oscillations at time e and after N periods, i.e. at time (t + NT).

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

.

Since 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.

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

4. As the coefficient β increases, 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 on values ​​called critical . For a spring pendulum, the condition ω 0 = β will be written as follows: from where we find the quantity critical resistance coefficient:

.

Rice. 21.3. Dependence of the amplitude of aperiodic oscillations on time

Forced vibrations.

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

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

General signs of forced mechanical vibrations.

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

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

(21.6)

Let's divide both sides of the equation by m, take into account that , and get differential equation forced oscillations:

Let us denote ( β – attenuation coefficient ), (ω 0 – frequency of undamped free oscillations), force acting on a unit of mass. In these notations differential equation forced oscillations will take the form:

(21.7)

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

.

– 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 accurate to a constant, the value of which is determined by the initial conditions of the oscillatory system:

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

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

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

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

Period of forced oscillations equals (21.9)

Graph of forced vibrations in Figure 4.1.

Fig.21.3. Forced oscillation graph

Steady-state forced oscillations are also harmonic.

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

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

, .

Let's analyze the dependence of the oscillation amplitude and phase shift on the frequency of the external driving force; to do this, we will find the first and second derivatives of x and substitute them into the differential equation.

Let's use the vector diagram method. The equation shows that the sum of the three vibrations on the left side of the equation (Figure 4.1) must be equal to the vibration on the right side. The vector diagram is made for an arbitrary moment of time t. From it you can determine.

Figure 21.4.

, (21.10)

. (21.11)

Taking into account the value of , ,, we obtain formulas for φ 0 and A ampl. mechanical system:

,

.

2. We study 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 reflected in Figure 21.5, which shows that at a certain driving force frequency the amplitude of oscillations increases sharply. And this increase is greater, the lower the attenuation coefficient β. When the amplitude of oscillations becomes infinitely large.

The phenomenon of a sharp increase in amplitude forced oscillations at a driving force frequency 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 energy losses due to friction are compensated by the supply of energy from an external source of 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 supply of energy from a constant source. Such systems are called self-oscillating, and the process of undamped oscillations in such systems is self-oscillations.

In a self-oscillating system, three characteristic elements can be distinguished - an oscillatory system, an energy source, and a feedback device between the oscillatory system and the source. Any mechanical system capable of performing its own damped oscillations (for example, the pendulum of a wall clock) can be used as an oscillatory system.

The energy source can be the deformation energy of a spring or the potential energy of a load in a gravitational field. A feedback device is a mechanism by which a self-oscillating system regulates the flow of energy from a source. In Fig. Figure 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 clock mechanism with anchor progress (Fig. 21.7.). The running wheel with oblique teeth is rigidly attached to a toothed drum, through which a chain with a weight is thrown. At the upper end of the pendulum there is an anchor (anchor) with two plates of hard material, bent along a circular arc with the center on the axis of the pendulum. In hand watches, the weight is replaced by a spring, and the pendulum is replaced by a balancer - a handwheel connected to a spiral spring.

The balancer performs torsional vibrations around its axis. The oscillatory system in a clock is a pendulum or balancer. The source of energy is a raised weight or a wound spring. The device used to provide feedback is an anchor, which 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, a tooth of the running wheel pushes the anchor fork in the direction of movement of the pendulum, transferring to it a certain portion of energy, which compensates for 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-oscillating systems are widespread in life around us and in technology. Self-oscillations occur in 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.

As you study this section, please keep in mind that fluctuations of different physical nature are described from common mathematical positions. Here it is necessary to clearly understand such concepts as harmonic oscillation, phase, phase difference, amplitude, frequency, oscillation period.

It must be borne in mind that in any real oscillatory system there is resistance of the medium, i.e. the oscillations will be damped. To characterize the damping of oscillations, a damping coefficient and a logarithmic damping decrement are introduced.

If oscillations occur under the influence of an external, periodically changing force, then such oscillations are called forced. They will be undamped. The amplitude of forced oscillations depends on the frequency of the driving force. As the frequency of forced oscillations approaches the frequency of natural oscillations, the amplitude of forced oscillations increases sharply. This phenomenon is called resonance.

When moving on to the study of electromagnetic waves, you need to clearly understand thatelectromagnetic waveis an electromagnetic field propagating in space. The simplest system emitting electromagnetic waves is an electric dipole. If a dipole undergoes harmonic oscillations, then it emits a monochromatic wave.

Formula table: oscillations and waves

Damped oscillations

Damped oscillations of a spring pendulum

Damped oscillations- vibrations whose energy decreases over time. An endlessly lasting process of species is impossible in nature. Free oscillations of any oscillator sooner or later fade and stop. Therefore, in practice we usually deal with damped oscillations. They are characterized by the fact that the amplitude of oscillations A is a decreasing function. Typically, attenuation occurs under the influence of resistance forces of the medium, most often expressed as a linear dependence on the oscillation speed 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 (subject to 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 speed with a coefficient c(see viscous friction).

The roots of which are calculated using 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 to the differential equation takes the form:

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

• Aperiodicity limit

If , two real roots coincide, 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 to the characteristic equation is 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 case are determined from the initial conditions:

see also

• Decrement of attenuation

Literature

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

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Until now, we have considered harmonic oscillations that arise, as already noted, in the presence of a single force in the system - an elastic force or a quasi-elastic force. In the nature around us, strictly speaking, such fluctuations do not exist. In real systems, in addition to elastic or quasi-elastic forces, there are always other forces that differ in the nature of action from elastic forces - these are forces that arise during the interaction of bodies of the system with the environment - dissipative forces. The end result of their action is the conversion of the mechanical energy of a moving body into heat. In other words, scattering occurs or dissipation mechanical energy. The process of energy dissipation is not purely mechanical and for its description requires the use of knowledge from other branches of physics. In the framework of mechanics, we can describe this process by introducing friction or resistance forces. As a result of energy dissipation, the oscillation amplitude decreases. In this case, it is customary to say that the vibrations of a body or system of bodies dampen. Damped oscillations are no longer harmonic, since their amplitude and frequency change over time.

Oscillations that, due to energy dissipation in an oscillating system, occur with a continuously decreasing amplitude are called fading. If an oscillatory system, removed from a state of equilibrium, oscillates under the influence of only internal forces, without resistance and dissipation (dissipation) of energy, then the oscillations occurring in it are called free(or own) undamped oscillations. In real mechanical systems with energy dissipation, free oscillations are always damped. Their frequency co differs from the frequency co 0 of oscillations of the system without damping (the greater the influence of resistance forces, the greater the influence of the resistance forces.

Let us consider damped oscillations using the example of a spring pendulum. Let us limit ourselves to considering small oscillations. At low oscillation speeds, the resistance force can be taken to be proportional to the speed of oscillatory displacements

Where v = 4 - oscillation speed; G - a proportionality factor called the drag coefficient. The minus sign in expression (2.79) for the resistance force is due to the fact that it is directed in the direction opposite to the speed of movement of the oscillating body.

Knowing the expressions for the quasi-elastic force i^p = - and the resistance force F c= taking into account the combined action of these forces, we can write down the dynamic equation of motion of a body performing damped oscillations

In this equation, we replace the coefficient (3 in accordance with formula (2.49) with You], after which we divide the last equation and get

We will look for a solution to equation (2.81) as a function of time of the form

Here the constant value y is still undefined. For simplicity, the initial phase in our consideration will be assumed to be equal to zero, i.e. we can “turn on” the stopwatch when the oscillatory displacement passes through the equilibrium position (zero coordinate).

We can determine the value y by substituting into the differential equation of damped oscillations (2.81) the assumed solution (2.82), as well as the speeds obtained from it

and acceleration

Substituting (2.83) and (2.84) together with (2.82) into (2.81) gives After reducing by /1 () e": " and multiplying by “-1” we get Solving this quadratic equation for y, we have

Substituting y into (2.82), we find how the displacement depends on time during damped oscillations. Let us introduce the notation

where the symbol co denotes the angular frequency of damped oscillations and coo the angular frequency of free oscillations without damping. It can be seen that for S > 0 the frequency of damped oscillations is always less than the frequency

Thus, and, therefore, the displacement during damped oscillations can be expressed as

The choice of the “+” or “-” sign in the second exponent is arbitrary and corresponds to a phase shift of the oscillations by l. We will write down damped oscillations taking into account the choice of the “+” sign, then expression (2.90) will be

This is the desired dependence of the displacement on time. It can also be rewritten in trigonometric form (limited to the real part)

The desired amplitude dependence A(t) from time to time can be represented as

Where A(,- amplitude at time t = 0.

Constant 8, equal according to (2.88) to the ratio of the resistance coefficient G to double the mass T oscillating body is called vibration damping coefficient. Let us find out the physical meaning of this coefficient. Let us find the time t during which the amplitude of the damped oscillations will decrease by e (the base of natural logarithms e = 2.72) times. To do this, let's put

Using relation (2.93), we obtain: or

whence follows

Hence, attenuation coefficient 8 is the reciprocal of time t, after which the amplitude of damped oscillations will decrease by e times. The quantity m, which has the dimension of time, is called time constant of a damped oscillatory process.

In addition to coefficient 8, the so-called logarithmic damping decrement X, equal to the natural logarithm of the ratio of two oscillation amplitudes separated from each other by a time interval equal to the period T

The expression under the logarithm, indicated by the symbol d, called simply decrement of fluctuations (decrement of attenuation).

Using the amplitude expression (2.93), we obtain:

Let us find out the physical meaning of the logarithmic damping decrement. Let the amplitude of oscillations decrease by e times after N oscillations. The time t during which the body will complete N oscillations can be expressed through the period t = N.T. Substituting this value m into (2.97), we obtain 8NT= 1. Since 67 "= A., then NX = 1, or

Hence, logarithmic damping decrement is the reciprocal of the number of oscillations during which the amplitude of damped oscillations will decrease by e times.

In some cases, the dependence of the oscillation amplitude on time A(t) It is convenient to express it in terms of the logarithmic damping decrement A. Exponent 6 1 Expressions (2.93) can be written according to (2.99) as follows:

Then expression (2.93) takes the form

where the value is equal to the number N oscillations made by the system during time t.

Table 2.1 shows approximate values ​​(in order of magnitude) of the logarithmic damping decrements of some oscillatory systems.

Table 2.1

Values ​​of attenuation decrements of some oscillatory systems

Let us now analyze the influence of resistance forces on the oscillation frequency. When a body moves from an equilibrium position and returns to an equilibrium position, a resistance force will act on it all the time, causing it to decelerate.

This means that the same sections of the path during damped oscillations will be covered by the body in a larger time interval than during free oscillations. Period of damped oscillations T, therefore, there will be a greater period of natural free oscillations. From expression (2.89) it is clear that the difference in frequencies becomes greater, the greater the attenuation coefficient b. For large b (b > coo), damped oscillations degenerate into aperiodic (non-periodic) process, in which, depending on the initial conditions, the system returns to the equilibrium position immediately without passing through it, or before stopping it passes through the equilibrium position once (performs only one oscillation) - see Fig. 2.16.

Rice. 2.16. Damped oscillations:

In Figure 2.16, A shows a dependence graph %(t) And A(t)(at 5 > co 0 and the initial phase со, oscillations are completely impossible (this case corresponds to the imaginary value of the frequency determined from equality (2.89). The system becomes damping, and the oscillatory process becomes aperiodic (Fig. 2.16, b).

• The notation exp(x) is equivalent to e*. We will use both forms.
• In a general consideration of oscillations, the full value of the oscillation phase is given by the initial conditions, i.e. the magnitude of the displacement 4(0 and speed 4(0) at the initial moment of time (t = 0) and includes the term