Subject: Damped and forced oscillations

Attenuation coefficient.

Amplitude

and frequency of damped oscillations.

Logarithmic damping decrement.

Quality factor of the oscillatory system.

Aperiodic process.

Natural vibrations real system. Differential equation of damped oscillations.

Attenuation coefficient.

. (1)

Previously, we considered the natural vibrations of conservative (ideal) oscillatory systems. In such systems, harmonic oscillations arise, which are characterized by constant amplitude and period, and are described by the following differential equation In real oscillatory systems, there are always forces that prevent oscillations (resistance forces). For example, in mechanical systems there is always a frictional force. In this case, the oscillation energy is gradually spent on work against the friction force. Therefore, the energy and amplitude of the oscillations will decrease, and the oscillations will fade. In an electric oscillatory circuit, the oscillation energy is spent on heating the conductors. That is.

real oscillatory systems are dissipative

Natural oscillations in real systems are damped. To obtain the equation of oscillations in a real system, it is necessary to take into account the resistance force. In many cases, it can be assumed that at low rates of change in the quantity S

drag force is proportional to speed Where r

– resistance coefficient (friction coefficient during mechanical vibrations), and the minus sign indicates that the resistance force is opposite to the speed. Substituting the resistance force into formula (2), we obtain differential equation

, describing oscillations in a real system Let's move all the terms to the left side and divide by the value m

and introduce the following notation ω As before the value 0 defines natural frequency of an ideal system. β The size characterizes the dissipation of energy in the system and is called attenuation coefficient. Let's move all the terms to the left side and divide by the value From formula (5) it is clear that the attenuation coefficient can be reduced by increasing the value of the quantity Where.

at a constant value Taking into account the introduced notations, we obtain

differential equation of damped oscillations

Solution of the differential equation of damped oscillations. Amplitude and frequency of damped oscillations. the general solution to the differential equation of damped oscillations has the following form

where the quantity in front of the sine is called amplitude of damped oscillations

Frequencyω damped oscillations is defined by the following expression

From the above formula (7) it is clear that the frequency of natural oscillations of a real oscillatory system is less than the oscillation frequency of an ideal system.

G
The graph of the equation of damped oscillations is shown in the figure. The solid line shows a graph of the displacement S(t), and the dash-dot line shows the change in the amplitude of damped oscillations.

It should be borne in mind that as a result of attenuation, not all values ​​of the quantities are repeated. Therefore, strictly speaking, the concepts of frequency and period are not applicable to damped oscillations. In this case, the period is understood as the period of time after which the fluctuating values ​​take on maximum (or minimum) values.

Logarithmic damping decrement. Quality factor of the oscillatory system. Aperiodic process.

To quantitatively characterize the rate of decrease in the amplitude of damped oscillations, a logarithmic damping decrement is introduced δ .

The logarithmic attenuation decrement is the natural logarithm of the ratio of amplitudes at instants of timetAndt+ T, i.e. different for the period.

A-priory The logarithmic decrement is given by the following formula

. (8)

If instead of amplitudes in formula (8) we substitute formula (6), we obtain a formula connecting the logarithmic decrement with the damping coefficient and period

. (9)

Time interval τ , during which the amplitude of oscillations decreases by e times, it's called relaxation time. Taking this into account, we obtain that , where N is the number of oscillations during which the amplitude decreases in e once. That is the logarithmic damping decrement is inversely proportional to the number of oscillations during which the amplitude decreases byeonce. If, for example, β =0.001, this means that after 100 oscillations the amplitude will decrease by e once.

The quality factor of an oscillatory system is a dimensionless quantity θ, equal to the product of the number 2π and the energy ratioW(t) oscillations at an arbitrary moment in time and the loss of this energy over one period of damped oscillations

. (10)

Since the energy is proportional to the square of the oscillation amplitude, replacing the energies in formula (10) with the squares of the amplitudes determined by formula (6), we obtain

For minor attenuations, and . Taking this into account, for the quality factor we can write

. (12)

The relationships given here can be written for various oscillatory systems. For this purpose the size is sufficient To obtain the equation of oscillations in a real system, it is necessary to take into account the resistance force. In many cases, it can be assumed that at low rates of change in the quantity, Let's move all the terms to the left side and divide by the value, k And Where replace with appropriate values ​​characterizing specific fluctuations. For example, for electromagnetic oscillations S→ q, Let's move all the terms to the left side and divide by the value→ L, k→1/C and Where→R.

Aperiodic process.

P
With a large value of the attenuation coefficient β there is not only a rapid decrease in amplitude, but also an increase in the period of oscillations. From formula (7) it is clear that when the cyclic frequency of oscillations vanishes ( T= ∞), i.e. no oscillations occur. This means that with a large resistance, all the energy imparted to the system, by the time it returns to the equilibrium position, is spent on work against the resistance force. A system removed from an equilibrium position returns to an equilibrium position without any energy reserve. The process is said to proceed aperiodically. In this case, the time to establish equilibrium is determined by the resistance value.

The reader is invited to see for himself how the values ​​of the quantities influence Where, Let's move all the terms to the left side and divide by the value, T 1 and φ 0 on the nature of oscillations of a real oscillatory system.

To do this, you need to hover the cursor over the diagram and double-click to activate it. Then, in the window that opens, change the values ​​of the quantities shown in the colored cells. Upon completion of work with the schedule tableEXEL close with or without saving data.

Self-test questions:

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tense... Free vibrations

with decreasing amplitude are called damped.

In a mechanical oscillating system, energy losses are most often associated with friction. If it is viscous, then at low speeds v is the friction force, where r is the friction coefficient, depending on the shape and size of the body and the viscosity of the medium.

Let us write down the equation of motion of a point, which occurs under the action of two forces: F = -khx (restoring force or quasi-elastic force), and friction force,

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Period of damped oscillations

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Quality factor Q oscillator characterizes the energy loss of the oscillatory system over the period:

determined by a driving force, and the undamped oscillations arising under its action are forced.

In the simplest case, the driving force changes according to the law of sine or cosine, i.e.

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Meaning resonant amplitude

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highlight">system quality factor gets one more physical meaning: it shows how many times a force acting at a resonant frequency causes more displacement than a constant force, i.e. How many times is the resonant displacement greater than the static displacement?

Test questions and tasks

1. Write down the differential equation of mechanical damped oscillations. What physical law did you use?

2. According to what law does the amplitude of a damped oscillation change?

3. What is relaxation time?

4. What physical meaning does the logarithmic damping decrement have?

5. The amplitude of damped oscillations of a mathematical pendulum decreased by 3 times in 1 minute. Determine how many times it will decrease in 4 minutes.

6. What oscillations are called forced?

7. What is the physical meaning of the quality factor of an oscillatory system?

8. What determines the frequency of forced oscillations?

9. What is the difference between resonance in a system with high and low quality factors?

10. What mode of forced oscillations is called steady?

11. Write it down common decision differential equation of forced oscillations. What parts does it consist of?

12. What is the phenomenon of resonance? Give examples of the use of this phenomenon in nature and technology?

Physics answers (Semyonov).docx

10. Oscillatory motion. Free, forced and damped oscillations.

1) Oscillations are called free(or own), if they occur due to the initially imparted energy in the subsequent absence of external influences on the oscillatory system (the system that oscillates). Differential equation 2) Available damped oscillations– oscillations, the amplitudes of which decrease over time due to energy losses by the real oscillatory system. The simplest mechanism for reducing vibration energy is its conversion into heat due to friction in mechanical oscillatory systems, as well as ohmic losses and radiation of electromagnetic energy in electrical oscillatory systems. 3) Differential equation Oscillations arising under the influence of an external periodically varying force or an external periodically varying emf are called respectively forced mechanical AndThe simplest mechanism for reducing vibration energy is its conversion into heat due to friction in mechanical oscillatory systems, as well as ohmic losses and radiation of electromagnetic energy in electrical oscillatory systems.

forced electromagnetic oscillations 11. Addition of harmonic vibrations of the same direction and the same frequency.

An oscillating body can participate in several oscillatory processes, then it is necessary to find the resulting oscillation, in other words, the oscillations must be added.

Let's add harmonic vibrations of the same direction and the same frequency

The equation for the resulting oscillation will be In the expression amplitude A  and initial phase are given accordingly by the ratios. Thus, the body, participating in two harmonic oscillations of the same direction and the same frequency, also performs harmonic oscillation  2 -  in the same direction and with the same frequency as the added vibrations. The amplitude of the resulting oscillation depends on the phase difference (

1) folded oscillations.

The result of the addition of two harmonic oscillations of the same frequency , occurring in mutually perpendicular directions along the axes X forced mechanical u. For simplicity, we choose the origin so that the initial phase of the first oscillation is equal to zero, and write Where  - phase difference of both oscillations, In the expression amplitude forced mechanical IN - amplitudes of folded oscillations. t. The equation for the trajectory of the resulting oscillation is found by eliminating the parameter expressions

Writing the folded vibrations in the form  t and replacing cos in the second equation on Ha  t and replacing cos in the second equation , andsin we get after simple transformations ellipse equation, whose axes are oriented relative to the coordinate axesarbitrarily: Since the trajectory of the resulting vibration has the shape of an ellipse, such vibrations are called

elliptically polarized.

12. Lissajous figures Closed trajectories drawn by a point that simultaneously performs two mutually perpendicular oscillations are called Lissajous figures

.* The appearance of these curves depends on the ratio of amplitudes, frequencies and phase differences of the added oscillations.

13. Laws of ideal gases. Clapeyron-Mendeleev equation. Boyle-Mariotte Law

*: for a given mass of gas at a constant temperature, the product of gas pressure and its volume is a constant value: pV=constat T=const,m=const*:1) Gay-Lussac's laws

2) the volume of a given mass of gas at constant pressure changes linearly with temperature: V=Vo(1+t) At V=const

the pressure of a given mass of gas at a constant volume changes linearly with temperature: p=po(1+t) at V=const,m=const Dalton's law *: the pressure of a mixture of ideal gases is equal to the sum of the partial pressures 1 , *: the pressure of a mixture of ideal gases is equal to the sum of the partial pressures 2 p ,..., R n

gases included in it: The state of a certain mass of gas is determined by three thermodynamic parameters: pressure R, volume V and temperature T.

There is a certain relationship between these parameters, called the equation of state, which is generally given by the expression IN - The expression is Clapeyron's equation, in which gas constant,

different for different gases. Equation

satisfies only an ideal gas, and it is the equation of state of an ideal gas, also called the Clapeyron-Mendeleev equation. Clapeyron-Mendeleev equation for mass T

gas  = Let's move all the terms to the left side and divide by the value/ Where - M N amount of substance where / volume A = ,..., R - m

concentration of molecules (number of molecules per unit volume). Thus, from Eq. call a decrease in the amplitude of vibrations over time, caused by the loss of energy by the oscillatory system (for example, the conversion of vibration energy into heat due to friction in mechanical systems). Damping breaks the periodicity of oscillations, so they are no longer a periodic process. If the attenuation is small, then we can conditionally use the concept of oscillation period - T(in Figure 7.6 In the expression amplitude 0 – initial amplitude of oscillations).

Figure 7.6 – Characteristics of damped oscillations

Damped mechanical oscillations of a spring pendulum occur under the influence of two forces: the elastic force and the resistance force:

drag force is proportional to speed Where– resistance coefficient.

Using the equation of Newton's second law, we can obtain:

or

Divide the last equation by Let's move all the terms to the left side and divide by the value and introduce the notation or

drag force is proportional to speed β damping coefficient, then the equation takes the form

(7.20)

This expression is the differential equation of damped oscillations. The solution to this equation is

This implies the exponential nature of damped oscillations, i.e. the amplitude of oscillations decreases according to an exponential law (Figure 7.6):

(7.22)

The relative decrease in the amplitude of oscillations over a period is characterized by a damping decrement equal to

(7.23)

or logarithmic attenuation decrement:

(7.24)

Attenuation coefficient β inversely proportional to time τ during which the amplitude of oscillations decreases by e once:

those. (7.25)

The frequency of damped oscillations is always less than the frequency of natural oscillations and can be found from the expression

(7.26)

where ω 0 is the frequency of natural oscillations of the system.

Accordingly, the period of damped oscillations is equal to:

Or (7.27)

With increasing friction, the oscillation period increases, and when the period .

To obtain undamped oscillations, it is necessary to act on an additional variable external force, which would push the material point in one direction or the other and the work of which would continuously replenish the loss of energy spent on overcoming friction. This variable force is called forcingF out, and the undamped oscillations arising under its influence are forced.

If the driving force changes in accordance with the expression, then the equation of forced oscillations will take the form

(7.28)

(7.29)

where ω is the cyclic frequency of the driving force.

This differential equation of forced oscillations. Its solution can be written in the form

The equation describes a harmonic oscillation occurring with a frequency equal to the frequency of the driving force, differing in phase by φ relative to the oscillations of the force.

Amplitude of forced oscillation:

(7.30)

The phase difference between the oscillations of the force and the system is found from the expression

(7.31)

The graph of forced oscillations is shown in Figure 7.7.

Figure 7.7 – Forced oscillations

During forced oscillations, a phenomenon such as resonance can be observed. Resonance this is a sharp increase in the amplitude of oscillations of the system.

Let us determine the condition under which resonance occurs; for this we consider equation (7.30). Let us find the condition under which the amplitude takes its maximum value.

It is known from mathematics that the extremum of a function will be when the derivative is equal to zero, i.e.

The discriminant is equal to

Hence

After transformation we get

Hence – resonant frequency.

In the simplest case, resonance occurs when an external periodic force F changes with frequency ω , equal to the frequency of natural oscillations of the system ω = ω 0 .

Mechanical waves

The process of propagation of oscillations in a continuous medium, periodic in time and space, is called wave process or wave.

When a wave propagates, the particles of the medium do not move with the wave, but oscillate around their equilibrium positions. Together with the wave, only the state of oscillatory motion and its energy are transferred from particle to particle of the medium. Therefore, the main property of waves, regardless of their nature, is transfer of energy without transfer of matter.

The following types of waves are distinguished:

Elastic(or mechanical) waves are called mechanical disturbances propagating in an elastic medium. In any elastic wave, two types of motion simultaneously exist: oscillation of particles of the medium and propagation of disturbance.

A wave in which the oscillations of particles of the medium and the propagation of the wave occur in the same direction is called longitudinal, and a wave in which the particles of the medium oscillate perpendicular to the direction of propagation of the wave is called transverse.

Longitudinal waves can propagate in media in which elastic forces arise during compression and tension deformations, i.e. solid, liquid and gaseous bodies. Transverse waves can propagate in a medium in which elastic forces arise during shear deformation, i.e. V solids. Thus, only longitudinal waves arise in liquids and gases, and both longitudinal and transverse waves occur in solids.

An elastic wave is called sinusoidal(or harmonic) if the corresponding vibrations of the particles of the medium are harmonic.

The distance between nearby particles vibrating in the same phase is called wavelength λ .

The wavelength is equal to the distance over which the wave propagates in a time equal to the oscillation period:

where is the speed of wave propagation.

Since (where ν is the oscillation frequency), then

The geometric location of the points to which the oscillations reach at the moment of time t, called wave front. The geometric location of points oscillating in the same phase is called wave surface.

The oscillatory motion of a real mechanical system is always accompanied by friction, to overcome which part of the energy of the oscillatory system is consumed. Therefore, the vibration energy during the vibration process decreases, turning into heat. Since the vibration energy is proportional to the square of the amplitude, the amplitude of the vibrations gradually decreases (Fig. 53; x - displacement, t - time). When all the oscillation energy is converted into heat, the oscillation will stop (decay). This kind of oscillation is called damped.

In order for the system to perform undamped oscillations, it is necessary to replenish the loss of oscillation energy due to friction from the outside. To do this, it is necessary to influence the system with a periodically changing force

where is the amplitude (maximum) value of the force, the circular frequency of force oscillations, and time. An external force that provides undamped oscillations of the system is called a driving force, and oscillations of the system are called forced. It is obvious that forced oscillations occur with a frequency equal to the frequency of the driving force. Let us determine the amplitude of forced oscillations.

To simplify the calculation, we will neglect the friction force, assuming that only two forces act on the oscillating body: driving and restoring. Then, according to Newton’s second law,

where is the mass and acceleration of the oscillating body. But, as was shown in § 27, Then

where is the displacement of the oscillating body. According to formula (9),

where is the circular frequency of the natural oscillations of the body (i.e., oscillations caused only by the action of the restoring force). That's why

From equation (22) it follows that the amplitude of the forced oscillation

depends on the ratio of the circular frequencies of the forced and natural oscillations: when there will be In fact, due to friction, the amplitude of the forced oscillations

remains final. It reaches its maximum value when the frequency of forced oscillations is close to the frequency of natural oscillations of the system. The phenomenon of a sharp increase in the amplitude of forced oscillations at is called resonance.

Using resonance, it is possible, through a small driving force, to cause an oscillation with a large amplitude. Let us hang, for example, a pocket or wrist watch on a thread of such length that the frequency of natural oscillations of the resulting physical pendulum (Fig. 54) coincides with the oscillation frequency of the clock mechanism balancer. As a result, the clock itself will begin to oscillate, deviating from the equilibrium position by an angle of 30°.

The phenomenon of resonance occurs during vibrations of any nature (mechanical, sound, electrical, etc.). It is widely used in acoustics - to amplify sound, in radio engineering - to amplify electrical vibrations, etc.

In some cases, resonance plays a harmful role. It can cause strong vibration of structures (buildings, supports, bridges, etc.) during the operation of mechanisms installed on these structures (machine tools, motors, etc.). Therefore, when calculating structures, it is necessary to ensure a significant difference between the vibration frequencies of mechanisms and the natural vibrations of structures.

Another type of undamped oscillations is common in technology - the so-called self-oscillations, which differ from forced oscillations in that in them the energy losses of the oscillations are replenished by a constant source of energy put into action for very short periods of time (compared to the oscillation period). Moreover, this source is “turned on” at the right moments of time automatically by the oscillatory system itself. An example of a self-oscillating system is a clock pendulum. Here, the potential energy of a raised weight (or a deformed spring) is brought into play through an anchor mechanism. Another example would be a closed oscillating circuit with a vacuum tube; We will get acquainted with the action of this self-oscillating system later (see § 112).