ENERGY MECHANISMS OF FREE VIBRATIONS AND RESONANCE IN ELASTIC BODIES

Annotation . The mechanisms of natural oscillations and resonance are described, considering the peculiarities of the transformation of elastic and kinetic energy in the implementation of the law of conservation of energy in local and integral volumes of the body, using the concept of mechanics based on the concepts of space, time and energy. When describing the motion in the Lagrange form, the elastic deformation energy of the particles is determined by the quadratic invariant of the tensor, whose components are the partial derivatives of Euler variables with respect to Lagrange variables. The increment of the invariant due to elastic deformation is represented as the sum of two scalars, one of which depends on the average value of the relative lengths of the edges of the particles in the form of an infinitesimal parallelepiped, the second is equal to the standard deviation of these lengths from the average value. It is shown that each of the scalars can be represented in the form of two dimensionless kinematic parameters of elastic energy, which participate in different ways in the implementation of the law of conservation of energy. One part of the elastic energy passes into kinetic energy and participates in the implementation of the law of conservation of energy for the body as a whole, considering external forces. The second part is not converted into kinetic energy but changes the deformed state of the particles in accordance with the equations of motion while maintaining the same level of the part of the elastic energy of the particles used for this. The kinematic parameters differ from the volume density of the corresponding types of energy by a factor equal to the elastic modulus, which is directly proportional to the density and heat capacity of the material and inversely proportional to the volume compression coefficient. Transverse, torsional, and longitudinal vibrations are considered free and under resonance conditions. The mechanisms of transformation of forced vibrations into their own after the termination of external influences and resonance at the superposition of free and forced vibrations with the same or similar frequency are considered. The formation of a new free wave at each cycle with an increase in the amplitude, which occurs mainly due to internal energy sources, and not external forces, is justified.


Introduction
Vibrations are among the most common processes, and no phenomenon in nature or any of the created mechanisms can do without them. They should be considered when calculating, manufacturing and operating building structures, transport systems, and technological processes in mechanical engineering [1][2][3][4][5][6][7]. The most interesting and important for practical applications include free vibrations and resonance, which are used in physics, chemistry, biology, and engineering [5], including for measuring the elastic properties of materials [6] and as an energy source [1,7].
Many papers have been devoted to the study of vibrations [3][4][5]. The analysis is usually limited to the shape, frequency, and period of vibrations. As a rule, vibrations of material points are considered; for elastic bodies, the relations of the elasticity theory are used, but without analyzing the energy state of particles in the body volume [1,5]. The Nature of free vibrations and resonance is still not fully understood. There are reasons to believe that the energy basis of free vibrations and resonance, along with the energy of external forces, are internal energy sources [8].
Free (natural) vibrations are called under the action of internal forces when the system is taken out of equilibrium. They occur at the expense of the initially reported energy from external sources without additional revenues to continue fluctuations. The resonance leads to a significant increase in the amplitude when forced and free vibrations interact in an elastic body with the same or similar frequency [1][2][3][4][5].
The purpose of the work is to describe the mechanism of natural oscillations and resonance, considering the peculiarities of the transformation of elastic and kinetic energy in the implementation of the law of conservation of energy in local and integral volumes of the body, using the concept of mechanics based on the concepts of space, time, and energy [9][10].

Fundamentals of the energy model of mechanics
The energy model of mechanics [10][11] provides a description of the motion of material particles in the form of Lagrange  (1) , , the kinematic characteristics of motion are determined, including dimensionless parameters that characterize the parts of elastic energy that play a different role in the implementation of the conservation law in different volumes of the body, in particular under free vibrations [8] and resonance [12].
The different nature of the arguments in equations (1) leads to the need to use two independent infinitesimal operators: the operator d for infinitesimal increments in time, for example xi, the operator δfor increments in space, for example, the volume of a particle 0 V  =  . When used simultaneously, the sequence of their writing can be changed d Е dЕ  = . Any movement must comply with the law of conservation of energy. As a generalized scalar function [13], the energy of a moving particle mov Е  must consider the independent invariants i  of equations (1). As shown in [10][11] In particular, the invariant v 2 on the right side of equation (3) determines the kinetic energy of a particle with a multiplier ( The increment of the energy of the external forces P when they move to dr for a particle with face dimensions ,,    for the force density ,, Considering (3) -(5) when the kinetic energy changes (4), the law of conservation of energy takes the form Because the condition (6) must be performed in any frame of reference velocity, movement (1) must satisfy the differential equation [9][10][11] , 0 , where ρ0 is the density of the material in the initial state, τpisurface density of forces on the faces of the infinitesimal parallelepiped, the normal to which in the initial state specifies the first subscript , and the direction of the voltagesecond index ( , , ) i x y z  . In essence, these are the Kirchhoff stresses for the space of Lagrange variables [10].
If we equate each bracket to zero (this assumption may lead to the loss of some possible solutions), we obtain analogs of the differential equations of motion of classical mechanics of a deformable solid [1][2][3]  The energy model of mechanics allows for a transition to a single modulus of elasticity and a new scale of average stresses, considered as the volume energy density of particles [14][15]. In this case, the concept of stresses τpi becomes redundant, since they differ from the components of the tensor (2) only by a constant factor equal to twice the elastic modulus κ τpi =2κ xi,p .
The modulus of elasticity κ is directly proportional to the density ρ0 and the heat capacity c of the material, and inversely proportional to the volume compressibility β = 3α (α is the coefficient of linear It is these physical properties that determine the behavior of materials in the region of reversible deformations [16]. Considering (8), instead of equation (7), we get where μ 2 = ρ0 /(2κ). If in (10) we equate each bracket to zero, the differential equations of motion are transformed into Poisson equations for each of the functions (1) Solutions of any types of problems based on the energy model of mechanics with a single modulus of elasticity [10][11] do not contradict those known from classical mechanics. This model leads to a significant reduction in mathematical difficulties and will be used in the future to describe the mechanism of free oscillations and resonance.
To achieve the goal set in this paper, the quadratic invariant of the matrix (2), which is equal to the sum of the squares of its elements, To achieve the goal set in this paper, particular importance the quadratic invariant of the matrix (2), which is equal to the sum of the squares of its elements, It determines the volume density of elastic energy e E  considering the initial state of the particles [10][11]

5
The energy of the particle gained (edef > 0) or lost (edef <0) due to the reduction of elastic deformation δEdef = κ edef δV0, in comparison with the initial state, determines the local kinematic parameter 2 3 def e e =  − .
The right part in equation (12) can be written in terms of the squares of the ratios of the lengths of the edges up to δl0 and after the deformation δl, initially oriented in the direction of the corresponding axes Then parameter (14) can be represented in terms of other dimensionless scalars ee and es where e is the average of the relative lengths of the edges lp of an infinitesimal parallelepiped before Considering the observations made and the initial state ( 2 3 e = ), the energy of the particle δЕdef with a volume of δV0, acquired due to deformation, can be represented as [9][10][11]  The ratio of the particle volumes after V  and before the deformation 0 V  determines the Jacobian of the transformation (1) or the cubic invariant of the tensor (2) In all the cases discussed below, an elastic rod of length L with a cross-section S0 is considered as a physical body, the ends of which are fixed in fixed arrays that do not exchange energy with the oscillating system.
For the energy justification of the resonance, the transformations of forced vibrations into free ones after the termination of the action of external forces, as well as the subsequent occurrence of the resonance when periodic external forces appear with the frequency of natural vibrations or close to it, are considered.

Transverse vibrations
Consider the energy features of transverse vibrations in accordance with the equations where ( with derivatives where q is the maximum displacement of particles along the y axis in the cross section α = L/2.
The deformation is carried out due to shifts / y y     . The quadratic invariant of the tensor determines the increment of the local energy The energy for deformation in the volume of the rod is ( ) ( ) For kinetic energy with velocity (24) after integration by volume, we get Total kinetic energy and strain energy in the volume of the oscillating rod coincides with the kinetic energy available in the system (28) At the end of the cycle (t = T = 2π/ω0), the system returns to its initial state, the elastic energy is absent, and the kinetic energy (22) takes the maximum possible value. 8 To find out the energy features of the resonance, we consider vibrations with a driving force [1][2] ( ) acting in the Central section along the length α = L/2 with an amplitude F0 and a circular frequency ω, which does not necessarily coincide with ω0. The force F0 can be determined from the integral law of conservation of energy in the form of equality of the power of the external force and the rate of change of kinetic and elastic energy in the volume of the rod.
Considering the velocity yt in the cross section α = L/2, where the force (30) is applied, we obtain the power The energy transferred to the system is converted into elastic (19) and kinetic (21) energy particles of the rod, which characterize the specific powers In the expression for the rate of change of the kinetic energy of the particle The total power integral in the rod volume at any given time will be Note that when describing the motion in the Lagrange form, it is not necessary to monitor the change in the contour of an elastic body during the oscillation, since for a Lagrangian coordinate system, it coincides with the original configuration and does not change in time. 9 Equating the powers of external (31) and internal (33) forces, we find the force F0 corresponding to the vibrations with the frequency and amplitude considered Depending on the frequency ratio, the force Similarly, when 0    the negative power of Ws < 0 will reduce the actual frequency. Only in the case that 0  =  the volume integral power is equal to 0 over the entire cycle, the sum of kinetic and elastic energy in the system remains unchanged, which corresponds to the definition of free vibrations that can continue without energy input from external sources [1][2][3][4][5].
From the point of view of resonance, the case is interesting when a cyclic force (30) acts, creating a forced oscillation with a frequency of natural vibrations 0  =  or close to it. Then the two waves will interact, forming a new wave.
In accordance with the superposition principle [11,17], to obtain the equations of joint motion, it is sufficient to replace the Lagrange variables of external (superimposed) motion with expressions for the corresponding Euler variables of internal (nested) motion. In our case, the equations for natural and forced oscillations may differ in the circular frequency ω and the amplitude q, but they are 10 equivalent in their effect on the resulting oscillation. Any of them can be considered external, the other internal.
We choose the amplitude of natural oscillations by the lower index q0 For a forced oscillation taking into account the resonance, we use the equation Replacing the variable β in the system (35) with the right part of the equations (36), for the joint motion we obtain ( , ) The integral values are equal to  The energy shares of eе2 and es2 vary in opposite phases, the sum of these shares is always 0, although each of them is comparable to the total energy of edef. In other words, the energy eе2, determined by the change in the average length of the edges of the particle in the form of an infinitesimal parallelepiped, passes into the energy es2, associated with the standard deviation of the relative lengths of the edges of the particle from their average value, and Vice versa. Such deformations do not change the energy state of the particle and the body as a whole.
The addition of velocities in accordance with the principle of superposition of motions (37) provides the necessary kinetic energy for the deformation of particles during the cycle and the fulfillment of the law of conservation of energy for the system as a whole, considering external forces.
Further development of oscillation can occur in one of the following ways: 12 1) the appearance of a new oscillation with an amplitude depending on the frequency ratio, if, after the formation of free vibrations, the external force (30) begins to act again with a frequency ω significantly different from ω0. The influence of the free wave will decrease and forced oscillations (36) with the frequency of force (30) will continue, which require power (31); 2) continuation of free oscillations with an increase in the amplitude q0+q1+…+qi after the next superposition, if the frequency of forced oscillations is close to its own and there is no energy exchange with the environment; 3) decrease the amplitude if the resonant system is used as an accumulator energy and stored in the system energy will go into the environment, including due to displacement were considered stationary supporting walls of a perfectly rigid body, in which is fixed the elastic rod.
The most dangerous is the continuation of vibrations in the conditions of resonance with the achievement and subsequent exceeding of the limit values of the stresses acting in the system, the occurrence of irreversible deformations or destruction of the system.

Torsional vibrations
During torsional vibrations, circumferential and radial movements of particles can occur. In this regard, under the condition of plane deformation in the Cartesian coordinate system, two of the three equations (11) must be considered where ω0 is the frequency of natural vibrations, θ is the angle of rotation of the rod in the cross section with the coordinate α = L/2 at t = π/(2ω0). In this case, the equations (40), as well as the boundary and initial conditions (43) are fulfilled.

Time and direction derivatives
determine the kinematic, deformation and energy characteristics of the particles and the body as a whole. Note that the obtained solution satisfies not only the system (6), but also the more general equation ( In the latter case, only the condition (44) in a simplified form should be met In accordance with equations (41) and (45), as in the case of transverse vibrations, elastic deformation is carried out due to shifts, the volume of material particles and the density of the material remain unchanged, regardless of the magnitude of the rotation angle ψ The cause of forced torsional vibrations may be the moment converted into elastic and kinetic energy of particles, for the rate of change of which (taking into As in the case of transverse vibrations, the kinetic energy of particles depends on the density of the material ρ0, so it contains a multiplier η = ω/ω0, which characterizes the ratio of the frequencies of forced and natural vibrations.
Integrating the powers by volume and using the energy identity that includes the external moment (49), as well as the internal forces (50) and (51), we find the moment The system (40) under the action of a moment (48) with an amplitude (52) corresponds to forced harmonic oscillations with a circular frequency of the external moment ω. If the external moment (48) ceases to act, for example, after the completion of the next cycle, the kinetic energy of the particles remains in the system, which causes its own vibrations.
Sum of capacities (50) and (51) in the absence of external influence, it characterizes possible transitions of elastic energy to kinetic energy and Vice versa. Stationary mode occurs when ω/ω0 = 1. If the actual frequency of vibrations is lower than the proper ω0 determined by the elastic properties of the material (46), the positive power will lead to its increase by 1 and 3 quarters of cycles. Otherwise, the oscillation frequency will decrease, and the mode will correspond to its own oscillations. Equation (53) can be considered as a mechanism for converting forced oscillations into proper ones after the external influence ceases.
This feature is confirmed by the analysis of the total elastic and kinetic energy in the rod volume, which depends on the frequency ratio and only if they are equal (η = 1) remains constant, as follows from the definition of the system's natural oscillations [1][2][3][4][5].
Resonance is possible if the free vibrations are superimposed forced with a circular frequency of natural vibrations ω0. Let's use the equations for natural oscillations and forced fluctuations x = As a result, we obtain a system of type (34), in which the angle of rotation in the joint oscillation, instead of (38), is equal to the sum of the angles of rotation of the forced and free oscillations ( ) ( ) The rationale for the energy feasibility of joint oscillations in accordance with equations (40) and (56)  and due to this, the system acquires kinetic energy ( ) Resonance from the point of view of the law of conservation of energy is possible and allows for a significant increase in the amplitude of the new phase of free oscillation and the energy parameters associated with the amplitude at the expense of internal forces [12].
Structure of local kinematic parameters of the volume energy density (15) - (16), which depend on the average value of its The analysis shows that part of the energy involved in free and combined vibrations is not associated with energy coming from external sources, is not converted into kinetic energy, but is an integral element of resonant phenomena.

Discussion and conclusions
The basis of resonance, as well as other physical phenomena, is the law of conservation of energy. The resonance involves two types of energy mentioned in the left-hand sides of equations With elastic energy, the situation is more complicated. There is reason to assume that classical mechanics with three elastic modules does not explain the phenomena associated with resonance, because it does not consider the internal energy involved in the resonance. This energy is considered by the energy model of mechanics with a single elastic modulus with a new scale of average stresses that consider the initial energy state of particles [15]. The conversion of strain (12) to elastic energy according to equation (17) ensures the implementation of the law of conservation of energy both in the volume integral form, considering the transition of part of the elastic energy to kinetic energy, and for subsystems with the transformation of deformation-related modifications of the internal elastic energy.
Analysis of the structure of the invariant (12) shows that both external and internal energy sources are used to change the volume or shape of particles in resonance. Changes in the volume of particles in the regions of tension and compression during longitudinal vibrations do not require an influx of energy from outside, just as free vibrations occur in elastic bodies without external additional energy inputs. Similarly, part of the energy for changing the shape of particles during transverse and torsional vibrations comes from internal sources, including the elastic energy of the particles themselves when the deformed state is changed equivalently.
In the mechanics of a deformable solid [10], terms are used about the energies associated with changes in the volume and shape of particles. With transverse and torsional vibrations, the volume of particles does not change, so a new terminology is proposed that clarifies the mathematical meaning of the components of deformation: parts of the energy that are associated either with changes in the average values of the particle edge lengths or with their standard deviations from the average values.
The dimensionless local kinematic parameters of the energy (15) - (16), which differ from the volume density of the corresponding types of elastic energy by a constant factor equal to the elastic modulus of the material, as in equation (13), allow us to identify the features of the energy state of particles under free vibrations and resonance. The nature of the changes during the cycle allows us to judge the possibility of their transformation into kinetic energy with the implementation of the law of conservation of energy for the body as a whole, or only to change the elastic deformation of parts of the body (due to internal sources) without using the energy of external forces. They correspond to real deformations, have a clear geometric interpretation associated with changes in the average relative length of the sides of the particles (ее) and the standard deviation of these lengths from the average value for each particle (es).
Equations (33), (53), (73) they reflect the features of the mechanism for converting forced vibrations with a frequency determined by external influences into their own after the termination of the driving force. This mechanism continues to operate with a superposition of free and forced oscillations, the frequency of which is close to its own, but does not coincide with it.
Equations (33), (53), (73) correspond to the mechanism of transformation of forced vibrations with a frequency determined by external influences into their own after the termination of the driving force. This mechanism continues to operate with a superposition of free and forced oscillations, the frequency of which is close to its own, but does not coincide with it.
Resonance is possible by of a superposition of both similar and different types of vibrations if their frequencies coincide with their own or are close to them [12]. For example, with the superposition of longitudinal and transverse vibrations at the same frequencies that coincide with the frequencies of natural vibrations, the volume densities of kinetic and elastic energy, and therefore their volume integral values, have the property of additivity, the law of conservation of energy is fulfilled, and resonance is possible.
The relations (29), (47) and (68) can be used to determine the elastic constant of a material from experimental studies with the main forms of free vibrations.