This is a periodic oscillation, in which the coordinate, speed, acceleration, characterizing the movement, change according to the sine or cosine law. The harmonic oscillation equation establishes the dependence of the body coordinate on time

The cosine graph has a maximum value at the initial moment, and the sine graph has a zero value at the initial moment. If we begin to investigate the oscillation from the equilibrium position, then the oscillation will repeat the sinusoid. If we begin to consider the oscillation from the position of the maximum deviation, then the oscillation will describe the cosine. Or such an oscillation can be described by the sine formula with the initial phase.

Mathematical pendulum

fluctuations mathematical pendulum.
Mathematical pendulum is a material point suspended on a weightless inextensible thread (physical model).
We will consider the movement of the pendulum under the condition that the deflection angle is small, then, if we measure the angle in radians, the statement is true: .
The force of gravity and the tension of the thread act on the body. The resultant of these forces has two components: a tangential one, which changes the acceleration in magnitude, and a normal one, which changes the acceleration in direction (centripetal acceleration, the body moves in an arc).
Because the angle is small, then the tangential component is equal to the projection of gravity on the tangent to the trajectory: . Angle in radians is equal to the ratio arc length to the radius (thread length), and the arc length is approximately equal to the offset ( x ≈ s): .
Compare the resulting equation with the equation oscillatory motion. It's clear that or - cyclic frequency during oscillations of a mathematical pendulum.
Oscillation period or (Galileo's formula).	Galileo formula
The most important conclusion: the period of oscillation of a mathematical pendulum does not depend on the mass of the body!
Similar calculations can be done using the law of conservation of energy. We take into account that the potential energy of the body in the gravitational field is equal to , and the total mechanical energy is equal to the maximum potential or kinetic:
Let's write down the law of conservation of energy and take the derivative of the left and right parts of the equation: . Because the derivative of a constant value is equal to zero, then . The derivative of the sum is equal to the sum of the derivatives: and.
Therefore: , which means.

Ideal gas equation of state (Mendeleev-Clapeyron equation).
An equation of state is an equation that relates the parameters of a physical system and uniquely determines its state. In 1834 the French physicist B. Clapeyron, who worked for a long time in St. Petersburg, derived the equation of state for an ideal gas for a constant mass of gas. In 1874 D. I. Mendeleev derived an equation for an arbitrary number of molecules.
In MKT and ideal gas thermodynamics macroscopic parameters are: p, V, T, m. We know that . Hence,. Given that , we get:.
The product of constant values ​​is a constant value, therefore: - universal gas constant (universal, because it is the same for all gases).
Thus, we have: Equation of state (Mendeleev-Clapeyron equation).
Other forms of writing the equation of state of an ideal gas.
1. Equation for 1 mole of a substance. If n \u003d 1 mol, then, denoting the volume of one mole V m, we get:. For normal conditions we get:
2. Write the equation in terms of density: - Density depends on temperature and pressure!
3. Clapeyron equation. It is often necessary to investigate the situation when the state of the gas changes with its constant amount (m=const) and in the absence of chemical reactions(M=const). This means that the amount of substance n=const. Then:
This entry means that for a given mass of a given gas equality is true:
For a constant mass of an ideal gas, the ratio of the product of pressure and volume to absolute temperature in this state there is a constant value: .
gas laws.
1. Avogadro's law. Equal volumes of different gases under the same external conditions contain the same number of molecules (atoms). Condition: V 1 =V 2 =…=V n ; p 1 \u003d p 2 \u003d ... \u003d p n; T 1 \u003d T 2 \u003d ... \u003d T n
Proof: Therefore, under the same conditions (pressure, volume, temperature), the number of molecules does not depend on the nature of the gas and is the same.
2. Dalton's Law. The pressure of a mixture of gases is equal to the sum of the partial (private) pressures of each gas. Prove: p=p 1 +p 2 +…+p n Proof:
3. Pascal's law. The pressure produced on a liquid or gas is transmitted in all directions without change.

The equation of state for an ideal gas. gas laws.

Numbers of degrees of freedom: this is the number of independent variables (coordinates) that completely determine the position of the system in space. In some problems, a monatomic gas molecule (Fig. 1, a) is considered as a material point, which is given three degrees of freedom of translational motion. This does not take into account the energy of rotational motion. In mechanics, a molecule of a diatomic gas, in the first approximation, is considered to be a combination of two material points, which are rigidly connected by a non-deformable bond (Fig. 1, b). This system, in addition to three degrees of freedom of translational motion, has two more degrees of freedom of rotational motion. Rotation around the third axis passing through both atoms is meaningless. This means that a diatomic gas has five degrees of freedom ( i= 5). A triatomic (Fig. 1, c) and polyatomic nonlinear molecule has six degrees of freedom: three translational and three rotational. It is natural to assume that there is no rigid bond between atoms. Therefore, for real molecules, it is also necessary to take into account the degrees of freedom of vibrational motion.

For any number of degrees of freedom of a given molecule, the three degrees of freedom are always translational. None of the translational degrees of freedom has an advantage over the others, which means that each of them has on average the same energy equal to 1/3 of the value<ε 0 >(energy of translational motion of molecules): In statistical physics, Boltzmann's law on the uniform distribution of energy over the degrees of freedom of molecules: for a statistical system that is in a state of thermodynamic equilibrium, each translational and rotational degree of freedom has an average kinetic energy equal to kT / 2, and each vibrational degree of freedom has an average energy equal to kT. The vibrational degree has twice as much energy, because it accounts for both kinetic energy (as in the case of translational and rotational motions) and potential energy, and the average values ​​of potential and kinetic energy are the same. So the average energy of the molecule Where i- the sum of the number of translational, the number of rotational in twice the number of vibrational degrees of freedom of the molecule: i=i post + i rotation +2 i vibrations In the classical theory, molecules are considered with a rigid bond between atoms; for them i coincides with the number of degrees of freedom of the molecule. Since in an ideal gas the mutual potential energy of interaction of molecules is equal to zero (molecules do not interact with each other), then the internal energy for one mole of gas will be equal to the sum of the kinetic energies N A of molecules: (1) Internal energy for an arbitrary mass m of gas. where M - molar mass, ν - amount of substance.

Maximum speed and acceleration values

After analyzing the equations of dependence v(t) and a(t), one can guess that the maximum values ​​of speed and acceleration are taken when the trigonometric factor is equal to 1 or -1. Determined by the formula

How to get dependencies v(t) and a(t)

7. Free vibrations. Velocity, acceleration and energy of oscillatory motion. Addition of vibrations

Free vibrations(or natural vibrations) are vibrations of an oscillatory system, performed only due to the initially reported energy (potential or kinetic) in the absence of external influences.

Potential or kinetic energy can be communicated, for example, in mechanical systems through an initial displacement or an initial velocity.

Freely oscillating bodies always interact with other bodies and together with them form a system of bodies called oscillatory system.

For example, a spring, a ball, and a vertical post to which the upper end of the spring is attached (see figure below) are included in an oscillatory system. Here the ball slides freely along the string (friction forces are negligible). If you take the ball to the right and leave it to itself, it will oscillate freely around the equilibrium position (point ABOUT) due to the action of the elastic force of the spring directed towards the equilibrium position.

Another classic example of a mechanical oscillatory system is the mathematical pendulum (see figure below). In this case, the ball performs free oscillations under the action of two forces: gravity and the elastic force of the thread (the Earth also enters the oscillatory system). Their resultant is directed to the equilibrium position.

The forces acting between the bodies of an oscillatory system are called internal forces. Outside forces called the forces acting on the system from the bodies that are not included in it. From this point of view, free oscillations can be defined as oscillations in a system under the action of internal forces after the system is taken out of equilibrium.

The conditions for the occurrence of free oscillations are:

1) the emergence of a force in them that returns the system to a position of stable equilibrium after it has been taken out of this state;

2) no friction in the system.

Dynamics of free oscillations.

Vibrations of a body under the action of elastic forces. The equation of oscillatory motion of a body under the action of an elastic force F(see Fig.) can be obtained taking into account Newton's second law ( F = ma) and Hooke's law ( F control= -kx), Where m is the mass of the ball, and is the acceleration acquired by the ball under the action of the elastic force, k- coefficient of spring stiffness, X- displacement of the body from the equilibrium position (both equations are written in projection onto the horizontal axis Oh). Equating the right sides of these equations and taking into account that the acceleration A is the second derivative of the coordinate X(offsets), we get:

.

This differential equation the motion of a body oscillating under the action of an elastic force: the second derivative of the coordinate with respect to time (the acceleration of the body) is directly proportional to its coordinate, taken with the opposite sign.

Oscillations of a mathematical pendulum. To obtain the equation for the oscillation of a mathematical pendulum (figure), it is necessary to expand the force of gravity F T= mg to normal F n(directed along the thread) and tangential F τ(tangent to the trajectory of the ball - a circle) components. Normal component of gravity F n and the elastic force of the thread Fynp in total they give the pendulum a centripetal acceleration, which does not affect the magnitude of the speed, but only changes its direction, and the tangential component F τ is the force that returns the ball to its equilibrium position and causes it to oscillate. Using, as in the previous case, Newton's law for tangential acceleration ma τ = F τ and given that F τ= -mg sinα, we get:

a τ= -g sinα,

The minus sign appeared because the force and the angle of deviation from the equilibrium position α have opposite signs. For small deflection angles sinα ≈ α. In its turn, α = s/l, Where s- arc OA, I- thread length. Given that and τ= s", we finally get:

The form of the equation is similar to the equation . Only here the parameters of the system are the length of the thread and the acceleration of free fall, and not the stiffness of the spring and the mass of the ball; the role of the coordinate is played by the length of the arc (i.e., the path traveled, as in the first case).

Thus, free vibrations are described by equations of the same type (subject to the same laws) regardless of physical nature forces that cause these vibrations.

Solving equations and is a function of the form:

x = xmcos ω 0t(or x = xmsin ω 0t).

That is, the coordinate of a body that performs free oscillations changes over time according to the cosine or sine law, and, therefore, these oscillations are harmonic:

In the equation x = xmcos ω 0t(or x = xmsin ω 0t), x m- oscillation amplitude, ω 0 - own cyclic (circular) oscillation frequency.

The cyclic frequency and the period of free harmonic oscillations are determined by the properties of the system. So, for vibrations of a body attached to a spring, the following relations are true:

.

The natural frequency is the greater, the greater the stiffness of the spring or the less mass of the load, which is fully confirmed by experience.

For a mathematical pendulum, the following equalities hold:

.

This formula was first obtained and tested by the Dutch scientist Huygens(a contemporary of Newton).

The period of oscillation increases with the length of the pendulum and does not depend on its mass.

It should be especially noted that harmonic oscillations are strictly periodic (because they obey the sine or cosine law) and even for a mathematical pendulum, which is an idealization of a real (physical) pendulum, they are possible only at small oscillation angles. If the deflection angles are large, the load displacement will not be proportional to the deflection angle (the sine of the angle) and the acceleration will not be proportional to the displacement.

The speed and acceleration of a body that performs free oscillations will also perform harmonic oscillations. Taking the time derivative of the function ( x = xmcos ω 0t(or x = xmsin ω 0t)), we get the expression for the speed:

v = -v msin ω 0t = -v mx mcos (ω 0t + π/2),

Where v m= ω 0 x m- velocity amplitude.

Similarly, the expression for acceleration A we get by differentiating ( v = -v msin ω 0t = -v mx mcos (ω 0t + π/2)):

a = -a mcos ω 0t,

Where a m= ω 2 0x m- acceleration amplitude. Thus, the amplitude of the speed of harmonic oscillations is proportional to the frequency, and the acceleration amplitude is proportional to the square of the oscillation frequency.

HARMONIC OSCILLATIONS
Fluctuations in which changes in physical quantities occur according to the cosine or sine law (harmonic law), called. harmonic vibrations. For example, in the case of mechanical harmonic vibrations: In these formulas, ω is the oscillation frequency, x m is the oscillation amplitude, φ 0 and φ 0 ’ are the initial phases of the oscillation. The above formulas differ in the definition of the initial phase and at φ 0 ’ = φ 0 + π/2 completely coincide.
This is the simplest form of periodic oscillations. The specific form of the function (sine or cosine) depends on the way the system is brought out of equilibrium. If the withdrawal occurs with a push (kinetic energy is reported), then at t \u003d 0, the displacement x \u003d 0, therefore, it is more convenient to use sin function, setting φ 0 ’=0; when deviating from the equilibrium position (potential energy is reported) at t \u003d 0, the displacement x \u003d x m, therefore, it is more convenient to use cos function and φ 0 =0.
An expression under the sign cos or sin, called. oscillation phase:. The phase of the oscillation is measured in radians and determines the value of the displacement (fluctuating value) at a given time.
The oscillation amplitude depends only on the initial deviation (the initial energy imparted to the oscillating system).
Speed ​​and acceleration at harmonic vibrations.
According to the definition of speed, speed is the derivative of the coordinate with respect to time
Thus, we see that the speed during harmonic oscillatory motion also changes according to the harmonic law, but the speed fluctuations are ahead of the displacement fluctuations in phase by π/2.
The value is the maximum speed of oscillatory motion (amplitude of speed fluctuations).
Therefore, for the speed during harmonic oscillation we have: , and for the case of a zero initial phase (see graph).
According to the definition of acceleration, acceleration is the derivative of speed with respect to time: is the second derivative of the coordinate with respect to time. Then: . Acceleration during harmonic oscillatory motion also changes according to the harmonic law, but acceleration oscillations are ahead of velocity oscillations by π/2 and displacement oscillations by π (they say that oscillations occur out of phase).
Value - maximum acceleration (amplitude of acceleration fluctuations). Therefore, for acceleration we have: , and for the case of zero initial phase: (see graph).
From the analysis of the process of oscillatory motion, graphs and corresponding mathematical expressions it can be seen that when the oscillating body passes the equilibrium position (the displacement is zero), the acceleration is zero, and the speed of the body is maximum (the body passes the equilibrium position by inertia), and when the amplitude value of the displacement is reached, the speed is zero, and the acceleration is maximum in absolute value (the body changes direction of movement).
Let us compare the expressions for displacement and acceleration for harmonic oscillations: and .
You can write: - i.e. the second derivative of the displacement is directly proportional (with the opposite sign) to the displacement. Such an equation is called harmonic oscillation equation. This dependence is satisfied for any harmonic oscillation, regardless of its nature. Since we have not used the parameters of a specific oscillatory system anywhere, only the cyclic frequency can depend on them.
It is often convenient to write the equations for oscillations in the form: , where T is the oscillation period. Then, if time is expressed in fractions of a period, the calculations will be simplified. For example, if you need to find the offset after 1/8 of the period, we get: . Similarly for speed and acceleration.

It is not uncommon for a system to simultaneously participate in two or more independent oscillations. In these cases, a complex oscillatory motion is formed, which is created by superimposing (adding) vibrations to each other. Obviously, the cases of summation of oscillations can be very diverse. They depend not only on the number of added oscillations, but also on the oscillation parameters, on their frequencies, phases, amplitudes, directions. It is not possible to review all the possible variety of cases of summation of oscillations, therefore we will confine ourselves to considering only individual examples.
1. Addition of vibrations in one direction. Let us add two oscillations of the same frequency, but different phases and amplitudes.

(4.40)
When the oscillations are superimposed on each other


We introduce new parameters A and j according to the equations:

(4.42)
The system of equations (4.42) is easily solved.

(4.43)

(4.44)
Thus, for x we ​​finally obtain the equation

(4.45)
So, as a result of adding unidirectional oscillations of the same frequency, we obtain a harmonic (sinusoidal) oscillation, the amplitude and phase of which is determined by formulas (4.43) and (4.44).
Let us consider special cases in which the ratios between the phases of two summed oscillations are different:


(4.46)
Let us now add unidirectional oscillations of the same amplitude, the same phases, but different frequencies.


(4.47)
Let us consider the case when the frequencies are close to each other, i.e. w1~w2=w
Then we will approximately assume that (w1+w2)/2= w, and (w2-w1)/2 is small. The resulting oscillation equation will look like:

(4.48)
Its graph is shown in fig. 4.5 This oscillation is called a beat. It is carried out with a frequency w but its amplitude oscillates with a large period.

2. Addition of two mutually perpendicular oscillations. Let us assume that one oscillation is carried out along the x-axis, the other - along the y-axis. The resulting motion is obviously located in the xy plane.
1. Let us assume that the oscillation frequencies and phases are the same, but the amplitudes are different.

(4.49)
To find the trajectory of the resulting motion, it is necessary to exclude time from equations (4.49). To do this, it is enough to divide term by term one equation by another, as a result of which we get

(4.50)
Equation (4.50) shows that in this case, the addition of oscillations leads to oscillation along a straight line, the tangent of the slope angle of which is determined by the ratio of the amplitudes.
2. Let the phases of the added oscillations differ from each other by /2 and the equations have the form:

(4.51)
To find the trajectory of the resulting motion, excluding time, it is necessary to square the equations (4.51), first dividing them by A1 and A2, respectively, and then adding them up. The trajectory equation will take the form:

(4.52)
This is the equation of an ellipse. It can be proved that for any initial phases and any amplitudes of two added mutually perpendicular oscillations of the same frequency, the resulting oscillation will be carried out along an ellipse. Its orientation will depend on the phases and amplitudes of the added oscillations.
If the added oscillations have different frequencies, then the trajectories of the resulting motions are very diverse. Only if the oscillation frequencies in x and y are multiples of each other, closed trajectories are obtained. Such movements can be attributed to the number of periodic ones. In this case, the trajectories of movements are called Lissajous figures. Let's consider one of the Lissajous figures, which is obtained by adding oscillations with frequency ratios of 1:2, with the same amplitudes and phases at the beginning of the movement.

(4.53)
Along the y axis, oscillations occur twice as often as along the x axis. The addition of such oscillations will lead to a motion trajectory in the form of a figure eight (Fig. 4.7).

8. Damped oscillations and their parameters: decrement and oscillation coefficient, relaxation time

)Period of damped oscillations:

T = (58)

At δ << ω o vibrations do not differ from harmonic ones: T = 2π/ o.

2) Amplitude of damped oscillations is expressed by formula (119).

3) damping decrement, equal to the ratio of two successive oscillation amplitudes A(t) And A(t+T), characterizes the rate of amplitude decrease over the period:

= e d T (59)

4) Logarithmic damping decrement- natural logarithm of the ratio of the amplitudes of two successive oscillations corresponding to time points that differ by a period

q \u003d ln \u003d ln e d T \u003d dT(60)

The logarithmic damping decrement is a constant value for a given oscillatory system.

5) Relaxation time called the period of time ( t) during which the amplitude of damped oscillations decreases by a factor of e:

e d τ = e, δτ = 1,

t = 1/d, (61)

From the comparison of expressions (60) and (61) we obtain:

q= = , (62)

Where N e - the number of oscillations made during the relaxation time.

If during the time t the system makes Ν fluctuations, then t = Ν . Τ and the equation of damped oscillations can be represented as:

S \u003d A 0 e -d N T cos(w t+j)\u003d A 0 e -q N cos(w t+j).

6)Quality factor of the oscillatory system(Q) it is customary to call the quantity characterizing the energy loss in the system during the oscillation period:

Q= 2p , (63)

Where W is the total energy of the system, ∆W is the energy dissipated over the period. The less energy dissipated, the greater the quality factor of the system. Calculations show that

Q = = pNe = = . (64)

Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, the quality factor is inversely proportional to the logarithmic damping decrement. From formula (64) it follows that the quality factor is proportional to the number of oscillations N e performed by the system during the relaxation time.

7) Potential energy system at time t can be expressed in terms of potential energy W 0 at largest deviation:

W = = kA o 2 e -2 qN = W 0 e -2 qN . (65)

It is usually conditionally considered that oscillations have practically ceased if their energy has decreased by a factor of 100 (the amplitude has decreased by a factor of 10). From here you can get an expression for calculating the number of oscillations made by the system:

= e 2qN= 100, ln100 = 2 qN;

N = = . (66)

9. Forced vibrations. Resonance. aperiodic fluctuations. Self-oscillations.

In order for the system to perform undamped oscillations, it is necessary to replenish the energy losses of oscillations due to friction from the outside. To ensure that the energy of the system's oscillations does not decrease, a force is usually introduced that periodically acts on the system (we will call such a force compelling, and forced oscillations).

DEFINITION: forced called such vibrations that occur in an oscillatory system under the action of an external periodically changing force.

This force, as a rule, performs a dual role:

firstly, it shakes the system and gives it a certain amount of energy;

secondly, it periodically replenishes energy losses (energy consumption) to overcome the forces of resistance and friction.

Let the driving force change with time according to the law:

.

Let us compose an equation of motion for a system oscillating under the influence of such a force. We assume that the system is also affected by the quasi-elastic force and the drag force of the medium (which is valid under the assumption of small oscillations). Then the equation of motion of the system will look like:

Or .

By substituting , , – the natural frequency of the system oscillations, we obtain a non-homogeneous linear differential equation 2 th order:

It is known from the theory of differential equations that the general solution of an inhomogeneous equation is equal to the sum of the general solution of a homogeneous equation and the particular solution of an inhomogeneous equation.

The general solution of the homogeneous equation is known:

,

Where ; a 0 and a– arbitrary const.

.

Using a vector diagram, you can make sure that such an assumption is true, and also determine the values ​​of “ a" And " j”.

The oscillation amplitude is determined by the following expression:

.

Meaning " j”, which is the magnitude of the phase delay of the forced oscillation from the driving force that caused it, is also determined from the vector diagram and is:

.

Finally, a particular solution of the inhomogeneous equation will take the form:

(8.18)

This function, together with

(8.19)

gives a general solution to an inhomogeneous differential equation describing the behavior of a system under forced vibrations. The term (8.19) plays a significant role in the initial stage of the process, during the so-called establishment of oscillations (Fig. 8.10). In the course of time, due to the exponential factor, the role of the second term (8.19) decreases more and more, and after a sufficient time it can be neglected, keeping only the term (8.18) in the solution.

Thus, function (8.18) describes steady forced oscillations. They are harmonic oscillations with a frequency equal to the frequency of the driving force. The amplitude of forced oscillations is proportional to the amplitude of the driving force. For a given oscillatory system (defined w 0 and b) the amplitude depends on the frequency of the driving force. Forced oscillations lag behind the driving force in phase, and the amount of lag "j" also depends on the frequency of the driving force.

The dependence of the amplitude of forced oscillations on the frequency of the driving force leads to the fact that at a certain frequency determined for a given system, the oscillation amplitude reaches its maximum value. The oscillatory system is especially responsive to the action of the driving force at this frequency. This phenomenon is called resonance, and the corresponding frequency is resonant frequency.

DEFINITION: a phenomenon in which a sharp increase in the amplitude of forced oscillations is observed is called resonance.

The resonant frequency is determined from the maximum condition for the amplitude of forced oscillations:

. (8.20)

Then, substituting this value into the expression for the amplitude, we get:

. (8.21)

In the absence of medium resistance, the amplitude of oscillations at resonance would turn to infinity; the resonant frequency under the same conditions (b=0) coincides with the natural oscillation frequency.

The dependence of the amplitude of forced oscillations on the frequency of the driving force (or, what is the same, on the frequency of oscillations) can be represented graphically (Fig. 8.11). Separate curves correspond to different values ​​of “b”. The smaller “b”, the higher and to the right lies the maximum of this curve (see the expression for w res.). With very large damping, resonance is not observed - with increasing frequency, the amplitude of forced oscillations decreases monotonically (lower curve in Fig. 8.11).

The set of presented graphs corresponding to different values ​​of b is called resonance curves.

Remarks about resonance curves:

as w®0 tends, all curves come to the same nonzero value equal to . This value represents the displacement from the equilibrium position that the system receives under the action of a constant force F 0 .

as w®¥ all curves tend asymptotically to zero, since at a high frequency, the force changes its direction so quickly that the system does not have time to noticeably shift from the equilibrium position.

the smaller b, the stronger the amplitude near the resonance changes with frequency, the "sharper" the maximum.

The phenomenon of resonance is often useful, especially in acoustics and radio engineering.

Self-oscillations- undamped oscillations in a dissipative dynamic system with nonlinear feedback, supported by the energy of the constant, that is non-periodic external influence.

Self-oscillations are different from forced vibrations because the latter are caused periodical external influence and occur with the frequency of this influence, while the occurrence of self-oscillations and their frequency are determined by the internal properties of the self-oscillatory system itself.

Term self-oscillations introduced into Russian terminology by A. A. Andronov in 1928.

Examples[

Examples of self-oscillations are:

· undamped oscillations of the clock's pendulum due to the constant action of the gravity of the clockwork weight;

vibrations of a violin string under the influence of a uniformly moving bow

the occurrence of alternating current in the multivibrator circuits and in other electronic generators at a constant supply voltage;

fluctuation of the air column in the pipe of the organ, with a uniform supply of air into it. (see also Standing wave)

rotational oscillations of a brass clock gear with a steel axis suspended from a magnet and twisted (Gamazkov's experiment) (the kinetic energy of the wheel, as in a unipolar generator, is converted into the potential energy of the electric field, the potential energy of the electric field, as in a unipolar engine, is converted into the kinetic energy of the wheel etc.)

Maklakov hammer

A hammer that strikes due to the energy of alternating current with a frequency many times lower than the frequency of the current in the electric circuit.

The coil L of the oscillatory circuit is placed above the table (or other object that needs to be hit). From below, an iron tube enters into it, the lower end of which is the impact part of the hammer. The tube has a vertical slot to reduce the Foucault currents. The parameters of the oscillatory circuit are such that the natural frequency of its oscillations coincides with the frequency of the current in the circuit (for example, alternating city current, 50 hertz).

After the current is turned on and oscillations are established, a resonance of the currents of the circuit and the external circuit is observed, and the iron tube is drawn into the coil. The inductance of the coil increases, the oscillatory circuit goes out of resonance, and the amplitude of the current oscillations in the coil decreases. Therefore, the tube returns to its original position - outside the coil - under the influence of gravity. Then the current fluctuations inside the circuit begin to grow, and resonance sets in again: the tube is again drawn into the coil.

tube commits self-oscillations, that is, periodic movements up and down, and at the same time it knocks loudly on the table, like a hammer. The period of these mechanical self-oscillations is tens of times greater than the period of the alternating current supporting them.

The hammer is named after M. I. Maklakov, a lecture assistant at the Moscow Institute of Physics and Technology, who proposed and carried out such an experiment to demonstrate self-oscillations.

Mechanism of self-oscillations

Fig 1. Mechanism of self-oscillations

Self-oscillations can have a different nature: mechanical, thermal, electromagnetic, chemical. The mechanism of occurrence and maintenance of self-oscillations in different systems can be based on different laws of physics or chemistry. For an accurate quantitative description of self-oscillations of different systems, different mathematical apparatus may be required. Nevertheless, it is possible to imagine a scheme that is common to all self-oscillatory systems and qualitatively describes this mechanism (Fig. 1).

On the diagram: S- source of constant (non-periodic) impact; R- a non-linear controller that converts a constant effect into a variable (for example, intermittent in time), which “swings” oscillator V- oscillating element (elements) of the system, and oscillations of the oscillator through feedback B control the operation of the regulator R, setting phase And frequency his actions. Dissipation (energy dissipation) in a self-oscillatory system is compensated by energy entering it from a source of constant influence, due to which self-oscillations do not decay.

Rice. 2 Scheme of the ratchet mechanism of a pendulum clock

If an oscillating element of the system is capable of its own damped oscillations(so-called. harmonic dissipative oscillator), self-oscillations (with equal dissipation and energy input into the system during the period) are established at a frequency close to resonant for this oscillator, their shape becomes close to harmonic, and the amplitude, in a certain range of values, the greater, the greater the magnitude of the constant external influence.

An example of such a system is the ratchet mechanism of a pendulum clock, the diagram of which is shown in Fig. 2. On the ratchet wheel axle A(which in this system performs the function of a non-linear controller) there is a constant moment of force M transmitted through the gear from the mainspring or from the weight. When the wheel spins A its teeth impart short-term impulses of force to the pendulum P(oscillator), thanks to which its oscillations do not fade. The kinematics of the mechanism plays the role of feedback in the system, synchronizing the rotation of the wheel with the oscillations of the pendulum in such a way that during the full period of oscillation the wheel turns through an angle corresponding to one tooth.

Self-oscillating systems that do not contain harmonic oscillators are called relaxation. Oscillations in them can be very different from harmonic ones, and have a rectangular, triangular or trapezoidal shape. The amplitude and period of relaxation self-oscillations are determined by the ratio of the magnitude of the constant action and the characteristics of the inertia and dissipation of the system.

Rice. 3 Electric bell

The simplest example of relaxation self-oscillations is the operation of an electric bell, shown in Fig. 3. The source of constant (non-periodic) exposure here is an electric battery U; the role of a non-linear controller is performed by a chopper T, closing and opening the electrical circuit, as a result of which an intermittent current arises in it; oscillating elements are a magnetic field periodically induced in the core of an electromagnet E, and anchor A moving under the influence of an alternating magnetic field. The oscillations of the armature actuate the chopper, which forms the feedback.

The inertia of this system is determined by two different physical quantities: the moment of inertia of the armature A and the inductance of the electromagnet winding E. An increase in any of these parameters leads to an increase in the period of self-oscillations.

If there are several elements in the system that oscillate independently of each other and simultaneously act on a nonlinear controller or controllers (of which there may also be several), self-oscillations can take on a more complex character, for example, aperiodic, or dynamic chaos.

In nature and technology

Self-oscillations underlie many natural phenomena:

fluctuations of plant leaves under the action of a uniform air flow;

· formation of turbulent flows on riffles and rapids of rivers;

The action of regular geysers, etc.

The principle of operation of a large number of various technical devices and devices is based on self-oscillations, including:

work of all kinds of clocks, both mechanical and electric;

· sounding of all wind and string-bowed musical instruments;

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Harmonic oscillation is a phenomenon of periodic change of some quantity, in which the dependence on the argument has the character of a sine or cosine function. For example, a quantity that varies in time as follows harmonically fluctuates:

where x is the value of the changing quantity, t is time, the remaining parameters are constant: A is the amplitude of the oscillations, ω is the cyclic frequency of the oscillations, is the full phase of the oscillations, is the initial phase of the oscillations.

Generalized harmonic oscillation in differential form

(Any non-trivial solution of this differential equation is a harmonic oscillation with a cyclic frequency)

Types of vibrations

Free oscillations are performed under the action of the internal forces of the system after the system has been taken out of equilibrium. For free oscillations to be harmonic, it is necessary that the oscillatory system be linear (described by linear equations of motion), and there should be no energy dissipation in it (the latter would cause damping).

Forced oscillations are performed under the influence of an external periodic force. For them to be harmonic, it is sufficient that the oscillatory system be linear (described by linear equations of motion), and the external force itself changes over time as a harmonic oscillation (that is, that the time dependence of this force is sinusoidal).

Harmonic vibration equation

Equation (1)

gives the dependence of the fluctuating value S on time t; this is the equation of free harmonic oscillations in explicit form. However, the equation of oscillations is usually understood as a different record of this equation, in differential form. For definiteness, we take equation (1) in the form

Differentiate it twice with respect to time:

It can be seen that the following relation holds:

which is called the equation of free harmonic oscillations (in differential form). Equation (1) is a solution to differential equation (2). Since equation (2) is a second-order differential equation, two initial conditions are necessary to obtain a complete solution (that is, to determine the constants A and   included in equation (1); for example, the position and speed of an oscillatory system at t = 0.

A mathematical pendulum is an oscillator, which is a mechanical system consisting of a material point located on a weightless inextensible thread or on a weightless rod in a uniform field of gravitational forces. The period of small eigenoscillations of a mathematical pendulum of length l, motionlessly suspended in a uniform gravitational field with free fall acceleration g, is equal to

and does not depend on the amplitude and mass of the pendulum.

A physical pendulum is an oscillator, which is a rigid body that oscillates in the field of any forces about a point that is not the center of mass of this body, or a fixed axis perpendicular to the direction of the forces and not passing through the center of mass of this body.

Oscillations arising under the action of external, periodically changing forces (with a periodic supply of energy from the outside to the oscillatory system)

Energy transformation

Spring pendulum

The cyclic frequency and the oscillation period are, respectively:

A material point attached to a perfectly elastic spring

Ø plot of the potential and kinetic energy of a spring pendulum on the x-coordinate.

Ø qualitative graphs of dependences of kinetic and potential energy on time.

Ø Forced

Ø The frequency of forced oscillations is equal to the frequency of changes in the external force

Ø If Fbc changes according to the sine or cosine law, then the forced oscillations will be harmonic

Ø With self-oscillations, a periodic supply of energy from its own source inside the oscillatory system is necessary

Harmonic oscillations are oscillations in which the oscillating value changes with time according to the law of sine or cosine

the equations of harmonic oscillations (the laws of motion of points) have the form

Harmonic vibrations such oscillations are called, in which the oscillating value varies with time according to the lawsinus orcosine .
Harmonic vibration equation looks like:

,
where A - oscillation amplitude (the value of the greatest deviation of the system from the equilibrium position); -circular (cyclic) frequency. Periodically changing cosine argument - called oscillation phase . The oscillation phase determines the displacement of the oscillating quantity from the equilibrium position at a given time t. The constant φ is the value of the phase at time t = 0 and is called the initial phase of the oscillation . The value of the initial phase is determined by the choice of the reference point. The x value can take values ​​ranging from -A to +A.
The time interval T, after which certain states of the oscillatory system are repeated, called the period of oscillation . Cosine is a periodic function with a period of 2π, therefore, over a period of time T, after which the oscillation phase will receive an increment equal to 2π, the state of the system performing harmonic oscillations will repeat. This period of time T is called the period of harmonic oscillations.
The period of harmonic oscillations is : T = 2π/.
The number of oscillations per unit time is called oscillation frequency ν.
Frequency of harmonic vibrations is equal to: ν = 1/T. Frequency unit hertz(Hz) - one oscillation per second.
Circular frequency = 2π/T = 2πν gives the number of oscillations in 2π seconds.

Generalized harmonic oscillation in differential form

Graphically, harmonic oscillations can be depicted as a dependence of x on t (Fig. 1.1.A), and rotating amplitude method (vector diagram method)(Fig.1.1.B) .

The rotating amplitude method allows you to visualize all the parameters included in the equation of harmonic oscillations. Indeed, if the amplitude vector A located at an angle φ to the x-axis (see Figure 1.1. B), then its projection on the x-axis will be equal to: x = Acos(φ). The angle φ is the initial phase. If the vector A put into rotation with an angular velocity equal to the circular frequency of oscillations, then the projection of the end of the vector will move along the x-axis and take values ​​ranging from -A to +A, and the coordinate of this projection will change over time according to the law:
.
Thus, the length of the vector is equal to the amplitude of the harmonic oscillation, the direction of the vector at the initial moment forms an angle with the x-axis equal to the initial phase of the oscillation φ, and the change in the direction angle with time is equal to the phase of the harmonic oscillations. The time for which the amplitude vector makes one complete revolution is equal to the period T of harmonic oscillations. The number of revolutions of the vector per second is equal to the oscillation frequency ν.

>> Harmonic vibrations

§ 22 HARMONIC OSCILLATIONS

Knowing how the acceleration and the coordinate of an oscillating body are related, it is possible, on the basis of mathematical analysis, to find the dependence of the coordinate on time.

Acceleration is the second derivative of the coordinate with respect to time. Instant Speed point, as you know from the course of mathematics, is the derivative of the coordinate of the point with respect to time. The acceleration of a point is the derivative of its velocity with respect to time, or the second derivative of the coordinate with respect to time. Therefore, equation (3.4) can be written as follows:

where x " is the second derivative of the coordinate with respect to time. According to equation (3.11), during free oscillations, the x coordinate changes with time so that the second derivative of the coordinate with respect to time is directly proportional to the coordinate itself and is opposite in sign to it.

It is known from the course of mathematics that the second derivatives of the sine and cosine with respect to their argument are proportional to the functions themselves, taken with the opposite sign. IN mathematical analysis it is proved that no other functions have this property. All this allows us to assert with good reason that the coordinate of a body that performs free oscillations changes over time according to the law of sine or pasine. Figure 3.6 shows the change in the coordinate of a point over time according to the cosine law.

Periodic changes physical quantity depending on time, occurring according to the law of sine or cosine, are called harmonic oscillations.

Oscillation amplitude. The amplitude of harmonic oscillations is the module of the greatest displacement of the body from the equilibrium position.

The amplitude can be various meanings depending on how much we displace the body from the equilibrium position at the initial moment of time, or on what speed is reported to the body. The amplitude is determined by the initial conditions, or rather by the energy imparted to the body. But the maximum values ​​of the sine module and the cosine module are equal to one. Therefore, the solution of equation (3.11) cannot be expressed simply by sine or cosine. It should have the form of the product of the oscillation amplitude x m by a sine or cosine.

Solution of the equation describing free oscillations. We write the solution of equation (3.11) in the following form:

and the second derivative will be:

We have obtained equation (3.11). Therefore, the function (3.12) is a solution to the original equation (3.11). The solution to this equation will also be the function

According to (3.14), the graph of the dependence of the body coordinate on time is a cosine wave (see Fig. 3.6).

Period and frequency of harmonic oscillations. During vibrations, body movements are periodically repeated. The period of time T, during which the system completes one complete cycle of oscillations, is called the period of oscillations.

Knowing the period, you can determine the frequency of oscillations, that is, the number of oscillations per unit of time, for example, per second. If one oscillation occurs in time T, then the number of oscillations per second

In the International System of Units (SI), the frequency of oscillations is equal to one if one oscillation occurs per second. The unit of frequency is called hertz (abbreviated: Hz) in honor of the German physicist G. Hertz.

The number of oscillations in 2 s is:

Value - cyclic, or circular, frequency of oscillations. If in equation (3.14) time t is equal to one period, then T \u003d 2. Thus, if at time t \u003d 0 x \u003d x m, then at time t \u003d T x \u003d x m, that is, through a period of time equal to one period, the oscillations are repeated.

The frequency of free oscillations is found by the natural frequency of the oscillatory system 1.

Dependence of the frequency and period of free oscillations on the properties of the system. The natural frequency of vibrations of a body attached to a spring, according to equation (3.13), is equal to:

It is the greater, the greater the stiffness of the spring k, and the less, the greater the body mass m. This is easy to understand: a stiff spring gives the body more acceleration, changes the body's speed faster. And the more massive the body, the slower it changes speed under the influence of force. The oscillation period is:

Having a set of springs of different rigidity and bodies of different masses, it is easy to verify from experience that formulas (3.13) and (3.18) correctly describe the nature of the dependence of u T on k and m.

It is remarkable that the period of oscillation of a body on a spring and the period of oscillation of a pendulum at small deflection angles do not depend on the oscillation amplitude.

The module of the coefficient of proportionality between the acceleration t and the displacement x in equation (3.10), which describes the oscillations of the pendulum, is, as in equation (3.11), the square of the cyclic frequency. Consequently, the natural frequency of oscillations of a mathematical pendulum at small angles of deviation of the thread from the vertical depends on the length of the pendulum and the free fall acceleration:

This formula was first obtained and tested by the Dutch scientist G. Huygens, a contemporary of I. Newton. It is valid only for small angles of deflection of the thread.

1 Often in what follows, for brevity, we will refer to the cyclic frequency simply as the frequency. You can distinguish the cyclic frequency from the usual frequency by notation.

The period of oscillation increases with the length of the pendulum. It does not depend on the mass of the pendulum. This can be easily verified by experiment with various pendulums. The dependence of the oscillation period on the free fall acceleration can also be found. The smaller g, the longer the period of oscillation of the pendulum and, consequently, the slower the clock with the pendulum runs. Thus, a clock with a pendulum in the form of a weight on a rod will fall behind in a day by almost 3 s if it is lifted from the basement to the upper floor of Moscow University (height 200 m). And this is only due to the decrease in the acceleration of free fall with height.

The dependence of the period of oscillation of the pendulum on the value of g is used in practice. By measuring the period of oscillation, g can be determined very precisely. The acceleration due to gravity varies with geographic latitude. But even at a given latitude it is not the same everywhere. After all, the density earth's crust is not the same everywhere. In areas where dense rocks occur, the acceleration g is somewhat greater. This is taken into account when prospecting for minerals.

Thus, iron ore has an increased density compared to conventional rocks. Measurements of the acceleration of gravity near Kursk, carried out under the guidance of Academician A. A. Mikhailov, made it possible to clarify the location of iron ore. They were first discovered through magnetic measurements.

The properties of mechanical vibrations are used in the devices of most electronic scales. The body to be weighed is placed on a platform under which a rigid spring is installed. As a result, there are mechanical vibrations, the frequency of which is measured by the corresponding sensor. The microprocessor connected to this sensor translates the oscillation frequency into the mass of the weighed body, since this frequency depends on the mass.

The obtained formulas (3.18) and (3.20) for the oscillation period indicate that the period of harmonic oscillations depends on the parameters of the system (spring stiffness, thread length, etc.)

Myakishev G. Ya., Physics. Grade 11: textbook. for general education institutions: basic and profile. levels / G. Ya. Myakishev, B. V. Bukhovtsev, V. M. Charugin; ed. V. I. Nikolaev, N. A. Parfenteva. - 17th ed., revised. and additional - M.: Education, 2008. - 399 p.: ill.

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