Currently, there are problems with the same and different structures related to dissipative mechanical systems, the solution of which reveals new aspects of oscillations and contributes to the creation of more effective solutions for vibration protection [1–3]. The development of accurate and stable computing technologies plays an important role in solving the problems of durability and stability arising from dynamic impacts in the fields of mechanical engineering, aircraft technology, and ground transport [4], [5]. For structures subjected to periodic external stresses, the use of dynamic vibration dampers is one of the most effective measures. There are also a number of scientific and technical developments on the creation of multi-level and network systems that provide effective dissipation in multi-mode oscillations [6], [7]. In solving problems aimed at reducing dynamic oscillations, theoretical approaches based on structural methods of mathematical modeling [8], [9] are particularly important. In this approach, the mechanical oscillation system is considered as a certain dynamic automatic control system. Thus, the structural diagram constructed for the system serves as a structural equivalent of the initial mathematical model given by differential equations [10], [11]. In the structural model of the protection system, the object can be represented as a dynamic circuit with negative feedback. This approach can lead to the emergence of second-order integrating links. Based on the feedback transfer functions, the system's damping modes are determined [12], [13]. From the point of view of the structural model, negative feedback reflects the elastic reaction of the system relative to the object of protection or conditional stiffness, which changes depending on the frequency of the impact.

A linear dissipative mechanical system is represented as a body with volume , consisting of N

parts with volumes , and is bounded by the surface In the part of the surface , in general, external loads are given, and boundary conditions are given on the surface. The contact surfaces meet the conditions of sliding (there is no friction on the contact) or rigidly fixed contact, or there is friction in the contact [14].

On some surfaces of a piecewise-inhomogeneous mechanical system, a concentrated mass L is attached using massless deformable elements (springs).

The relationship between stresses and deformations of massive deformable elements satisfies the following relations:

σ( ijn) = λ ̃nθnδji + 2μ ̃nε( ijn), (n = 1,2,3, . . . , S), (1)

Here, is the operator modulus of elasticity, which has the following form [2]:

where is an arbitrary time-dependent function;

- relaxation nucleus; is the instantaneous response modulus of elasticity of the material. Now, let's assume that the integral terms expressing the dissipative properties of the viscoelastic body in Eq. (2) are very small compared to its instantaneously acting elastic composition. Such conditions allow for the application of the "freezing method" [15]. Then, using this freezing method, the equations in Eq. (2) are replaced by approximate expressions of the following type here

The cosine and sine compositions of the Fourier image of the relaxation nucleus of the material are determined, respectively.

For modeling a viscoelastic substance, the three-parameter relaxation kernel is taken as follows:

In some special cases, elements belonging to the Lumped Distributed Mass Chain (LDMC) series can be connected to each other through N non-mass deformable connecting elements. For such elements (i.e., elements without mass or having a non-zero volume), the physical dependencies are expressed by the following equations:

where force in the i-th constituent element, - elongation of this element, - frequency of external influence.

Some deformable elements can be perfectly elastic; in this case, the nuclei of heredity, characterizing their rheological properties, are equal to zero. If the rheological nuclei of all deformable elements in the system are zero, they exhibit an absolutely elastic property.

Setting of problems in the dynamics of linear dissipative mechanical systems. For the mathematical formulation of the above-mentioned problems, the Euler-Lagrange principle is applied:

When considering natural oscillations, the Right Sides (1)-(4) are identically equal to zero. We will look for the solution in the form:

j= 1,…,N,

where complex natural frequency. The problem reduces to a complex algebraic problem with eigenvalues of the form:

with a nonlinear incoming complex parameter. The characteristic equation of Eq. (5) is solved numerically using the Muller method. As an initial approximation, a solution close to (5) is chosen for the corresponding conservative problem. In this case, the determinant of the System (5) on each iteration of the Muller method is calculated using the Gauss method, with the main element selected by rows and columns. Thus, solving the problem of natural oscillations using the Muller method does not require the disclosure of its determinant. The solution to the problem of forced oscillations of the System (5) will be sought in the form:

where is the sought complex amplitudes. The problem of stable forced oscillations reduces to a system of non-uniform algebraic equations:

whose solution is carried out by the Gauss method. The result of solving the forced oscillation problem is obtaining the Amplitude-Frequency Characteristics (AFC) of the mechanical system. If a mechanical system has one degree of freedom, then the equation of motion of the system is written as:

where is the frequency of external influence. The partial solution of Eq. (7) has the form:

When the viscous properties of deformable elements are taken into account through viscous friction, then in matrix form, relative to the matrix - column of the Integro-Differential Eq. (7) (IDU), take the form

where [M] is a positively defined matrix, the elements of which represent concentrated masses, [C] is the matrix of the damping coefficients of deformable bodies, and [K] is the matrix of the elements of the skeletal characteristics of deformable bodies (symmetric square matrix), is the viscosity matrix without mass elements. The perturbing force is denoted by the column vector -

The above matrices have a physical meaning, i.e., , and are, respectively, elements of the mass, attenuation, and stiffness matrices. All matrices are square, j is the number of rows, and k is the number of columns. The elements of this Eq. (8) are obtained from the system of differential Eq. (7).

Equation of vibration shielding systems with a finite number of possible movements in directions. The equations of motion of the mathematical model of machines and equipment can be written by the system: If active elements are introduced into the mathematical model, then the mathematical model can be written in normal form by matrix differential equations of the form:

where x is the state vector, and is an arbitrary n-dimensional square, complex matrix. If the system is complex, i.e., the system is not represented as a chain of bodies, then matrices and can be absolutely "dense."

Server communication reactions can be determined using structural diagrams equivalent to automatic control systems. This approach can also be applied to systems where mass-inertial elements are represented as solids [11].

Service communication can be implemented by mechanisms that produce various generalized forces, such as mechanical, electromechanical, hydraulic, electrohydraulic, etc. [12]. Let a linear dissipative mechanical system consist of rigid bodies connected by massless deformable elements. Then Eq. (5) reduces to a system of linear differential equations with complex rigidity:

where frequency; component of the real matrix of generalized masses; C - generalized stiffness, - Kroneker symbols; - ponderomotive forces. For natural oscillations, we seek the solution in the form:

Then, substituting the last expression, we obtain:

The characteristic Eq. (6) can be solved numerically, for example, using the Muller method. For the initial value, a solution close to the corresponding conservative problem is chosen.

For System (12), we seek the solution in the form:

where - unknown amplitudes. Then we get:

For systems with deformable bodies, Eq. (6) reduces to the system:

where - component of displacement; - density; -differentiation operators. The System (14) does not include the masses and moments of inertia of non-deformable bodies and the stiffness components of massless deformable elements. These parameters in the proposed approach are considered at the internal points of the j-th rod.

Two main parameters are proposed to express the overall degree of dissipation: the lowest damping rate of natural oscillations and the highest resonance amplitude. The boundary conditions for the System (14) are generally not written due to the extreme diversity. Only when posing problems for specific mechanical systems of equations are boundary conditions automatically obtained in the process of integration by parts of the integral terms.

In this article, all authors contributed equally, and both authors jointly developed the ideas and achieved the results through collaborative thinking.

In this article, all authors contributed equally, and both authors jointly developed the ideas and achieved the results through collaborative thinking.

Data available on request due to ethical reasons.

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