Harmonic vibrations

Function Graphs f(x) = sin( x) and g(x) = cos( x) on the Cartesian plane.

harmonic oscillation- fluctuations in which a physical (or any other) quantity changes over time according to a sinusoidal or cosine law. The kinematic equation of harmonic oscillations has the form

,

where X- displacement (deviation) of the oscillating point from the equilibrium position at time t; BUT- oscillation amplitude, this is the value that determines the maximum deviation of the oscillating point from the equilibrium position; ω - cyclic frequency, a value indicating the number of complete oscillations occurring within 2π seconds - complete phase oscillations, - initial phase of oscillations.

Generalized harmonic oscillation in differential form

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

Types of vibrations

Evolution in time of displacement, velocity and acceleration in harmonic motion

• Free vibrations are made under the action of the internal forces of the system after the system has been brought out of equilibrium. To free vibrations were harmonic, it is necessary that the oscillatory system be linear (described linear equations motion), and there was no dissipation of energy (the latter would cause damping).
• Forced vibrations 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).

Application

Harmonic vibrations stand out from all other types of vibrations for the following reasons:

see also

Notes

Literature

• Physics. Elementary textbook Physics / Ed. G. S. Lansberg. - 3rd ed. - M ., 1962. - T. 3.
• Khaykin S. E. Physical foundations mechanics. - M., 1963.
• A. M. Afonin. Physical foundations of mechanics. - Ed. MSTU im. Bauman, 2006.
• Gorelik G.S. Vibrations and waves. Introduction to acoustics, radiophysics and optics. - M .: Fizmatlit, 1959. - 572 p.

Wikimedia Foundation. 2010 .

See what "Harmonic vibrations" are in other dictionaries:

Modern Encyclopedia

Harmonic vibrations- HARMONIC OSCILLATIONS, periodic changes in a physical quantity that occur according to the sine law. Graphically, harmonic oscillations are represented by a sinusoid curve. Harmonic vibrations simplest form periodic movements, characterized by ... Illustrated Encyclopedic Dictionary

Fluctuations in which a physical quantity changes over time according to the law of sine or cosine. Graphically G. to. are represented by a sinusoid or cosine curve (see fig.); they can be written in the form: x = Asin (ωt + φ) or x ... Great Soviet Encyclopedia

HARMONIC OSCILLATIONS, periodic motion, such as the movement of the PENDULUM, atomic oscillations, or oscillations in electrical circuit. A body performs undamped harmonic oscillations when it oscillates along a line, moving by the same ... ... Scientific and technical encyclopedic dictionary

Oscillations, at k ryh physical. (or any other) value changes over time according to a sinusoidal law: x=Asin(wt+j), where x is the value of the oscillating value in the given. moment of time t (for mechanical G. to., for example, displacement or speed, for ... ... Physical Encyclopedia

harmonic vibrations- Mechanical vibrations, in which the generalized coordinate and (or) the generalized speed change in proportion to the sine with an argument linearly dependent on time. [Collection of recommended terms. Issue 106. Mechanical vibrations. Academy of Sciences ... Technical Translator's Handbook

Oscillations, at k ryh physical. (or any other) quantity changes in time according to a sinusoidal law, where x is the value of the oscillating quantity at time t (for mechanical G. to., for example, displacement and speed, for electrical voltage and current) ... Physical Encyclopedia

HARMONIC OSCILLATIONS- (see), in which physical. the value changes over time according to the law of sine or cosine (for example, changes (see) and speed during oscillation (see) or changes (see) and current strength with electric G. to.) ... Great Polytechnic Encyclopedia

They are characterized by a change in the oscillating value x (for example, the deviation of the pendulum from the equilibrium position, the voltage in the alternating current circuit, etc.) in time t according to the law: x = Asin (?t + ?), where A is the amplitude of harmonic oscillations, ? corner… … Big Encyclopedic Dictionary

Harmonic vibrations- 19. Harmonic oscillations Oscillations in which the values ​​of the oscillating quantity change in time according to the law Source ... Dictionary-reference book of terms of normative and technical documentation

Periodic fluctuations, with krykh change in time physical. magnitude occurs according to the law of sine or cosine (see Fig.): s = Asin (wt + f0), where s is the deviation of the fluctuating value from its cf. (equilibrium) value, A=const amplitude, w= const circular ... Big encyclopedic polytechnic dictionary

1.18. HARMONIC OSCILLATIONS AND THEIR CHARACTERISTICS

Definition of harmonic vibrations. Characteristics of harmonic oscillations: displacement from the equilibrium position, amplitude of oscillations, phase of oscillations, frequency and period of oscillations. Velocity and acceleration of an oscillating point. Energy of the harmonic oscillator. Examples of harmonic oscillators: mathematical, spring, torsional and physical pendulums.

Acoustics, radio engineering, optics and other branches of science and technology are based on the doctrine of oscillations and waves. Big role plays the theory of vibrations in mechanics, especially in calculations for the strength of aircraft, bridges, certain types machines and nodes.

fluctuations are processes that repeat at regular intervals (however, not all repeating processes are fluctuations!). Depending on the physical nature of a repeating process, mechanical, electromagnetic, electromechanical, etc. vibrations are distinguished. During mechanical vibrations, the positions and coordinates of bodies periodically change.

Restoring force - the force under the action of which the oscillatory process occurs. This force tends to return the body or material point deviated from the rest position to its original position.

Depending on the nature of the impact on an oscillating body, free (or natural) vibrations and forced vibrations.

Depending on the nature of the impact on an oscillating system, free oscillations, forced oscillations, self-oscillations and parametric oscillations are distinguished.

free (own) oscillations are called such oscillations that occur in a system left to itself after it has been given a push, or it has been taken out of equilibrium, i.e. when only the restoring force acts on the oscillating body. An example is the vibrations of a ball suspended on a thread. In order to cause vibrations, you must either push the ball, or, moving it aside, release it. In the event that no energy dissipation occurs, free oscillations are undamped. However, real oscillatory processes are damped, because an oscillating body is affected by forces of resistance to movement (mainly friction forces).

· compelled such vibrations are called, during which the oscillating system is exposed to an external periodically changing force (for example, vibrations of a bridge that occur when people walking in step pass over it). In many cases, systems perform oscillations that can be considered harmonic.

· Self-oscillations , as well as forced oscillations, are accompanied by an impact on the oscillating system external forces, however, the moments of time when these actions are carried out are set by the oscillating system itself. That is, the system itself controls the external influence. An example of a self-oscillatory system is a clock in which the pendulum receives shocks due to the energy of a raised weight or a twisted spring, and these shocks occur at the moments of the pendulum passing through the middle position.

· Parametric oscillations are carried out with a periodic change in the parameters of the oscillating system (a person swinging on a swing periodically raises and lowers his center of gravity, thereby changing the parameters of the system). Under certain conditions, the system becomes unstable - a random deviation from the equilibrium position leads to the emergence and growth of oscillations. This phenomenon is called parametric excitation of oscillations (i.e., oscillations are excited by changing the parameters of the system), and the oscillations themselves are called parametric.

Despite the different physical nature, the oscillations are characterized by the same regularities, which are studied by general methods. An important kinematic characteristic is the form of vibrations. It is determined by the form of the function of time, which describes the change of one or another physical quantity during oscillations. The most important are those fluctuations in which the fluctuating value changes with time according to the law of sine or cosine . They're called harmonic .

Harmonic vibrations oscillations are called, in which the oscillating physical quantity changes according to the sine (or cosine) law.

This type of oscillation is especially important for the following reasons. First, oscillations in nature and technology often have a character very close to harmonic. Secondly, periodic processes of a different form (with a different time dependence) can be represented as an overlay, or superposition, of harmonic oscillations.

Harmonic oscillator equation

Harmonic oscillation is described by the periodic law:

Rice. 18.1. harmonic oscillation

W

here
- characterizes change any physical quantity during oscillations (shift of the position of the pendulum from the equilibrium position; voltage on the capacitor in oscillatory circuit etc.), A - oscillation amplitude ,
- oscillation phase , - initial phase ,
- cyclic frequency ; value
also called own oscillation frequency. This name emphasizes that this frequency is determined by the parameters of the oscillatory system. A system whose law of motion has the form (18.1) is called one-dimensional harmonic oscillator . In addition to the above quantities, the following concepts are introduced to characterize oscillations: period , i.e. time of one oscillation.

(A period of oscillation T called the smallest period of time after which the states of the oscillating system are repeated (one complete oscillation is performed) and the phase of the oscillation receives an increment 2p).

and frequencies
, which determines the number of oscillations per unit time. The unit of frequency is the frequency of such an oscillation, the period of which is 1 s. This unit is called hertz (Hz ).

Oscillation frequencyn called the reciprocal of the period of oscillation - the number of complete oscillations per unit time.

Amplitude- the maximum value of the offset or change variable in oscillatory or wave motion.

Oscillation phase- argument of a periodic function or describing a harmonic oscillatory process (ω - angular frequency, t- time, - the initial phase of oscillations, that is, the phase of oscillations at the initial moment of time t = 0).

The first and second time derivatives of a harmonically oscillating quantity also perform harmonic oscillations of the same frequency:

AT this case the equation of harmonic oscillations, written according to the law of cosine, is taken as a basis. In this case, the first of the equations (18.2) describes the law according to which the speed of the oscillating material point(body), the second equation describes the law by which the acceleration of an oscillating point (body) changes.

Amplitudes
and
equal respectively
and
. hesitation
ahead of
in phase to ; and hesitation
ahead of
on the . Values A and can be determined from given initial conditions
and
:

,
. (18.3)

Oscillator oscillation energy

P

Rice. 18.2. Spring pendulum

Let's now see what happens to vibration energy . As an example of harmonic oscillations, consider one-dimensional oscillations performed by a body of mass m Under the influence elastic strength
(for example, a spring pendulum, see fig. 18.2). Forces of a different nature than elastic, but in which the condition F = -kx is satisfied, are called quasi-elastic. Under the influence of these forces, bodies also make harmonic oscillations. Let be:

bias:
speed:
acceleration:

Those. the equation for such oscillations has the form (18.1) with natural frequency
. The quasi-elastic force is conservative . Therefore, the total energy of such harmonic oscillations must remain constant. In the process of oscillations, the transformation of kinetic energy occurs E to into a potential E P and vice versa, moreover, at the moments of the greatest deviation from the equilibrium position, the total energy is equal to the maximum value of the potential energy, and when the system passes through the equilibrium position, the total energy is equal to the maximum value of the kinetic energy. Let's find out how the kinetic and potential energy changes with time:

Kinetic energy:

Potential energy:

(18.5)

Considering that i.e. , the last expression can be written as:

Thus, the total energy of the harmonic oscillation turns out to be constant. It also follows from relations (18.4) and (18.5) that the average values ​​of the kinetic and potential energies are equal to each other and half of the total energy, since the average values
and
for the period are 0.5. Using trigonometric formulas, it can be obtained that the kinetic and potential energy change with frequency
, i.e. with a frequency twice the harmonic frequency.

Examples of a harmonic oscillator are spring pendulums, physical pendulums, mathematical pendulums, and torsional pendulums.

1. Spring pendulum- this is a load of mass m, which is suspended on an absolutely elastic spring and performs harmonic oscillations under the action of an elastic force F = -kx, where k is the stiffness of the spring. The equation of motion of the pendulum has the form or (18.8) From formula (18.8) it follows that the spring pendulum performs harmonic oscillations according to the law x \u003d Acos (ω 0 t + φ) with a cyclic frequency

(18.9) and period

(18.10) Formula (18.10) is true for elastic oscillations within the limits in which Hooke's law is fulfilled, i.e. if the mass of the spring is small compared to the mass of the body. The potential energy of a spring pendulum, using (18.9) and the potential energy formula of the previous section, is (see 18.5)

2. physical pendulum- This solid, which oscillates under the action of gravity around a fixed horizontal axis, which passes through the point O, which does not coincide with the center of mass C of the body (Fig. 1).

Fig.18.3 Physical pendulum

If the pendulum is deflected from the equilibrium position by a certain angle α, then, using the equation of dynamics of the rotational motion of a rigid body, the moment M of the restoring force (18.11) where J is the moment of inertia of the pendulum about the axis that passes through the suspension point O, l is the distance between the axis and the center of mass of the pendulum, F τ ≈ –mgsinα ≈ –mgα is the restoring force (the minus sign indicates that the directions F τ and α are always opposite; sinα ≈ α since the oscillations of the pendulum are considered small, i.e. the pendulum deviates from the equilibrium position by small angles). We write equation (18.11) as

Or Taking (18.12) we get the equation

Identical to (18.8), whose solution we find and write as:

(18.13) From formula (18.13) it follows that for small oscillations the physical pendulum performs harmonic oscillations with a cyclic frequency ω 0 and a period

(18.14) where the value L=J/(m l) - . The point O" on the continuation of the straight line OS, which is separated from the point O of the suspension of the pendulum at a distance of the reduced length L, is called swing center physical pendulum(Fig. 18.3). Applying the Steiner theorem for the moment of inertia of the axis, we find

That is, OO "is always greater than OS. The suspension point O of the pendulum and the swing center O" have interchangeability property: if the suspension point is moved to the swing center, then the old suspension point O will be the new swing center, and the oscillation period of the physical pendulum will not change.

3. Mathematical pendulum is an idealized system consisting of a material point of mass m, which is suspended on an inextensible weightless thread, and which oscillates under the action of gravity. A good approximation of a mathematical pendulum is a small, heavy ball that is suspended from a long, thin thread. Moment of inertia of a mathematical pendulum

(8) where l is the length of the pendulum.

Since a mathematical pendulum is a special case of a physical pendulum, if we assume that all of its mass is concentrated at one point - the center of mass, then, substituting (8) into (7), we find an expression for the period of small oscillations of a mathematical pendulum (18.15) Comparing formulas (18.13 ) and (18.15), we see that if the reduced length L of the physical pendulum is equal to the length l a mathematical pendulum, then the periods of oscillation of these pendulums are the same. Means, reduced length of a physical pendulum is the length of such a mathematical pendulum, in which the period of oscillation coincides with the period of oscillation of a given physical pendulum. For a mathematical pendulum (material point with mass m suspended on a weightless inextensible thread of length l in the field of gravity with free fall acceleration equal to g) at small deflection angles (not exceeding 5-10 angular degrees) from the equilibrium position natural oscillation frequency:
.

4. A body suspended on an elastic thread or other elastic element, oscillating in horizontal plane, represents torsion pendulum.

This is a mechanical oscillatory system that uses the forces of elastic deformations. On fig. 18.4 shows the angular analogue of a linear harmonic oscillator that performs torsional vibrations. A horizontally located disk hangs on an elastic thread fixed in its center of mass. When the disk rotates through an angle θ, a moment of forces arises M elastic torsion strain:

where I = IC- the moment of inertia of the disk about the axis, passing through center of gravity, ε – angular acceleration.

By analogy with the load on the spring, you can get.

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 an initial phase.

Mathematical pendulum

Oscillations of a 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 can be seen that or is a 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. Let us take into account that the potential energy of the body in the gravitational field is , and the total mechanical energy equal to the maximum potential or kinetic:
We write down the law of conservation of energy and take the derivative of the left and right parts equations: . 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 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 constant mass ideal gas the ratio of the product of pressure and volume to absolute temperature in given state is a constant value: .
gas laws.
1. Avogadro's law. AT equal volumes different gases at the same external conditions located the same number 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, when 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 diatomic gas molecule in the first approximation is considered to be a set of two material points, which are rigidly connected by a non-deformable bond (Fig. 1, b). This system except for three degrees of freedom forward movement 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 thermodynamic equilibrium, for each translational and rotational degree of freedom, there is an average kinetic energy, equal to kT/2, and for each vibrational degree of freedom - on average, an 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 ideal gas Since the mutual potential energy of interaction of molecules is 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.

Pendulum movement in hours, earthquake, alternating current in an electric circuit, the processes of radio transmission and radio reception are completely different, not bound friend with other processes. Each of them has its own special reasons, but they are united by one sign - a sign of the commonality of the nature of the change physical quantities over time. These and many other processes of different physical nature, in many cases, it turns out to be appropriate to consider as one special type physical phenomena - fluctuations.

A common feature of physical phenomena, called oscillations, is their repetition in time. With a different physical nature, many oscillations occur according to the same laws, which makes it possible to apply common methods for their description and analysis.

Harmonic vibrations. From a large number various oscillations in nature and technology, harmonic oscillations are especially common. Harmonic oscillations are those that occur according to the law of cosine or sine:

where is a value that experiences fluctuations; - time; - constant, the meaning of which will be explained later.

The maximum value of a quantity that changes according to a harmonic law is called the amplitude of oscillations. The argument of the cosine or sine for harmonic oscillations is called the phase of the oscillation

The phase of oscillation at the initial moment of time is called the initial phase. Initial phase determines the value of the quantity at the initial moment of time

The values ​​of the sine or cosine function are repeated when the function argument changes to, therefore, with harmonic oscillations, the magnitude values ​​are repeated when the oscillation phase changes to . On the other hand, during a harmonic oscillation, the quantity must take the same values ​​​​in a time interval called the oscillation period T. Therefore, the phase change on occurs

through the oscillation period T. For the case when we get:

From expression (1.2) it follows that the constant in the equation of harmonic oscillations is the number of oscillations that occur in seconds. The value is called the cyclic oscillation frequency. Using expression (1.2), equation (1.1) can be expressed in terms of frequency or period T of oscillations:

As well as in an analytical way descriptions of harmonic oscillations are widely used graphic ways their presentations.

The first way is to set a schedule of fluctuations in Cartesian system coordinates. Time I is plotted along the abscissa, and the value of the changing value is plotted along the ordinate. For harmonic oscillations, this graph is a sine or cosine wave (Fig. 1).

The second way to represent the oscillatory process is spectral. The amplitude is measured along the ordinate axis, and the frequency of harmonic oscillations is measured along the abscissa axis. The harmonic oscillatory process with frequency and amplitude is represented in this case by a vertical segment with a straight length drawn from a point with a coordinate on the abscissa axis (Fig. 2).

The third way to describe harmonic oscillations is the method vector diagrams. In this method, the following, purely formal technique is used to find at any time the value of a quantity that changes according to a harmonic law:

We choose on the plane an arbitrarily directed coordinate axis along which we will count the value of interest to us From the origin along the axis we draw a vector modulus of which is equal to the amplitude of the harmonic oscillation xm. If we now imagine that the vector rotates around the origin in a plane with a constant angular velocity c counterclockwise, then the angle a between the rotating vector and the axis at any time is determined by the expression.

Mechanical harmonic oscillation- it's straight uneven movement, at which the coordinates of an oscillating body (material point) change according to the cosine or sine law depending on time.

According to this definition, the law of coordinate change depending on time has the form:

Where wt is the value under the cosine or sine sign; w- coefficient, physical meaning which we will reveal below; A is the amplitude of mechanical harmonic oscillations.

Equations (4.1) are basic kinematic equations mechanical harmonic oscillations.

Consider next example. Let's take the Ox axis (Fig. 64). From point 0 we draw a circle with radius R = A. Let point M from position 1 begin to move around the circle at a constant speed v(or with constant angular velocity w, v = wA). After some time t, the radius will rotate through an angle f: f=wt.

With such a movement along the circumference of the point M, its projection onto the x-axis M x will move along the x-axis, the coordinate of which x will be equal to x \u003d A cos f = = A cos wt. Thus, if a material point moves along a circle of radius A, the center of which coincides with the origin, then the projection of this point onto the x-axis (and onto the y-axis) will make harmonic mechanical vibrations.

If the value wt, which is under the cosine sign, and the amplitude A are known, then x can also be determined in equation (4.1).

The value wt, which is under the cosine (or sine) sign, which uniquely determines the coordinate of the oscillating point at a given amplitude, is called oscillation phase. For a point M moving along a circle, the value w means its angular velocity. What is the physical meaning of the value w for the point M x, which performs mechanical harmonic oscillations? The coordinates of the oscillating point M x are the same at some time t and (T +1) (from the definition of the period T), i.e. A cos wt= A cos w (t + T), which means that w(t + T) - wt = 2 PI(from the periodicity property of the cosine function). Hence it follows that

Therefore, for a material point that performs harmonic mechanical oscillations, the value of w can be interpreted as the number of oscillations for a certain cycle time equal to 2l. Therefore, the value w called cyclical(or circular) frequency.

If point M starts its movement not from point 1 but from point 2, then equation (4.1) will take the form:

the value f 0 called initial phase.

We find the speed of the point M x as a derivative of the coordinate with respect to time:

We define the acceleration of a point oscillating according to the harmonic law as a derivative of the speed:

It can be seen from formula (4.4) that the speed of a point performing harmonic oscillations also changes according to the cosine law. But the velocity in phase is ahead of the coordinate by PI/2. Acceleration during harmonic oscillation changes according to the cosine law, but is ahead of the coordinate in phase by P. Equation (4.5) can be written in terms of the x coordinate:

Acceleration during harmonic oscillations is proportional to displacement c opposite sign. We multiply the right and left parts of equation (4.5) by the mass of the oscillating material point m, we obtain the following relations:

According to Newton's second law, the physical meaning of the right side of expression (4.6) is the projection of the force F x , which provides harmonic mechanical movement:

The value of F x is proportional to the displacement x and is directed opposite to it. An example of such a force is the elastic force, the magnitude of which is proportional to the deformation and directed oppositely to it (Hooke's law).

The regularity of the dependence of acceleration on displacement, which follows from equation (4.6), considered by us for mechanical harmonic oscillations, can be generalized and applied when considering oscillations of a different physical nature (for example, a change in current in an oscillatory circuit, a change in charge, voltage, induction magnetic field etc.). Therefore, equation (4.8) is called the main equation dynamics of harmonic oscillations.

Consider the movement of spring and mathematical pendulums.

Let a spring (Fig. 63), located horizontally and fixed at point 0, have a body of mass m attached at one end, which can move along the x axis without friction. Let the spring constant be equal to k. We derive the body m external force from the equilibrium position and let go. Then, along the x axis, only the elastic force will act on the body, which, according to Hooke's law, will be equal to: F ypr = -kx.

The equation of motion of this body will look like:

Comparing equations (4.6) and (4.9), we draw two conclusions:

From formulas (4.2) and (4.10) we derive the formula for the oscillation period of the load on the spring:

Mathematical pendulum is a body of mass m suspended on a long inextensible thread of negligible mass. In the equilibrium position, the force of gravity and the elastic force of the thread will act on this body. These forces will balance each other.

If the thread is deflected at an angle a from the equilibrium position, then the same forces act on the body, but they no longer balance each other, and the body begins to move along the arc under the action of the gravity component directed along the tangent to the arc and equal to mg sin a.

The equation of motion of the pendulum takes the form:

The minus sign on the right side means that the force F x = mg sin a is directed against the displacement. Harmonic oscillation will occur at small angles of deviation, i.e., under the condition a 2* sin a.

Replace sin and in equation (4.12), we obtain the following equation.