Analytical Solution Of Problem Research Articles

Analytical solution of Stefan-type problems

Abstract In this paper, free surface problems of Stefan type for the parabolic heat equation are considered. The analytical solutions of the problems are based on the method of heat polynomials and integral error function in the form of series. Convergence of the series solution is considered and proved. Both one-and two-phase Stefan-type problems are investigated. Numerical results for one-phase inverse Stefan problem are presented and discussed.

Comprehensive Recovery of Point Defect Displacement Field Function in Crystals by Computer X-ray Diffraction Microtomography

In the case of the point defect in a crystal, the inverse Radon’s problem in X-ray diffraction microtomography has been solved. As is known, the crystal-lattice defect displacement field function f(r) = h·u(r) determines phases − (±h)-structure factors incorporated into the Takagi–Taupin equations and provides the 2D image patterns by diffracted and transmitted waves propagating through a crystal (h is the diffraction vector and u(r) is the displacement field crystal-lattice-defects vector). Beyond the semi-kinematical approach for obtaining the analytical problem solution, the difference-equations-scheme of the Takagi–Taupin equations that, in turn, yield numerically controlled-accuracy problem solutions has been first applied and tested. Addressing the inverse Radon’s problem solution, the χ2-target function optimization method using the Nelder–Mead algorithm has been employed and tested in an example of recovering the Coulomb-type point defect structure in a crystal Si(111). As has been shown in the cases of the 2D noise-free fractional and integrated image patterns, based on the Takagi–Taupin solutions in the semi-kinematical and difference-scheme approaches, both procedures provide the χ2-target function global minimum, even if the starting-values of the point-defect vector P1 is chosen rather far away from the reference up to 40% in relative units. In the cases of the 2D Poisson-noise image patterns with noise levels up to 5%, the figures-of-merit values of the optimization procedures by the Nelder–Mead algorithm turn out to be high enough; the lucky trials number is 85%; and in contrast, for the statistically denoised 2D image patterns, they reach 0.1%.

Direct numerical algorithm for calculating the heat flux at an inaccessible boundary

Abstract A fast numerical algorithm for solving the Cauchy problem for elliptic equations with variable coefficients in standard calculation domains (rectangles, circles, or rings) is proposed. The algorithm is designed to calculate the heat flux at the inaccessible boundary. It is based on the separation of variables method. This approach employs a finite difference approximation and allows obtaining a solution to a discrete problem in arithmetic operations of the order of N ⁢ ln ⁡ N N\operatorname{ln}N , where 𝑁 is the number of grid points. As a rule, iterative procedures are needed to solve the Cauchy problem for elliptic equations. The currently available direct algorithms for solving the Cauchy problem have been developed only for (Laplace, Helmholtz) operators with constant coefficients and for use of analytical solutions for problems with such operators. A novel feature of the results of the present paper is that the direct algorithm can be used for an elliptic operator with variable coefficients (of a special form). It is important that in this case no analytical solution to the problem can be obtained. The algorithm significantly increases the range of problems that can be solved. It can be used to create devices for determining in real time heat fluxes on the parts of inhomogeneous constructions that cannot be measured. For example, to determine the heat flux on the inner radius of a pipe made of different materials.

METHODOLOGY FOR SUBSTANTIATING THE CALCULATION PARAMETERS OF COMPLEXLY CONSTRUCTED TERRIGENOUS RESERVOIRS IN WESTERN SIBERIA, INCLUDING THOSE WITH TRIZ, ACCORDING TO THE CORE AND GIS STUDY

The article presents an analytical solution to the direct problem of the method of spontaneous polarization of rocks in a well crossing an electrically inhomogeneous reservoir of limited thickness with a drilling mud penetration zone. The analytical solution of the problem by the employees of Schlumberger-Doll Research (M.R. Taherian, at all) for an impermeable reservoir in the absence of a penetration zone is a special case of the analytical solution of the direct problem of the spontaneous polarization (PS). The accuracy of the interpretation of the analytical and numerical solution of the inverse PS problem does not exceed the permissible accuracy of measuring the PS potential. The static potential of the ESP calculated on the basis of the analytical solution of the direct and inverse PS problem is used to construct the correlation dependence of the «core-GIS» static potential of the ESP on the volumetric humidity W, equal to the product of the porosity coefficient KP by the coefficient of residual water saturation KOV. An example of the interpretation of low-permeable, low-resistance oilsaturated reservoirs of Jurassic sediments for the operational assessment of reserves of the Yegansky, Martovsky and Zaozerny deposits of Western Siberia is given. The algorithm proposed in the article can be used for automated interpretation of the PS method and multi-probe induction logging in the corporate software of PJSC OC Rosneft.

Numerical analysis via MEF of poroelastic problems

This paper presents a fully coupled hydro-mechanical formulation, based on the finite element method in terms of the displacement and pore-water pressure, taking into account the effect of relative compressibility between the grains and the solid skeleton. Its main goal is to provide a clear formulation and reference examples to facilitate reproduction by anyone which intend to develop their own computer program to solve poroelasticity problems. The numerical model presented is verified with analytical solutions of problems in plane strain condition and others authors numerical results. The influence of material properties on the pore-water pressure dissipation process and, the performance of the time integration scheme are presented. Higher the Poisson’s coefficient and lower the grain compressibility faster is the process of dissipation highlighting the importance of taking into account the relative compressibility between the grains and the solid skeleton on the coupled analyses. The Crank Nicholson’s algorithm presents a good performance when comparing with other time integration scheme presented by other authors.

Lab-scale simulation of hydraulic fracturing: some outcomes

The paper describes the lab-scale physical simulation of hydraulic fracturing in measurement holes. The tests were carried out using cubical polymethyl methacrylate specimens with faces of 200 mm. The open-end or dead-end holes with diameter of 12 mm were made in the test specimens. In the preset intervals in the holes, the longitudinal or latitudinal (ring-wise) slots with depths up to 3 mm were precut to initiate and orient created fractures. Hydrofracturing was carried out using prototype tools for sealing a test interval in a hole (packer-to-packer spacing) and to feed it with power fluid under pressure. The model specimens were tested under different triaxial loading conditions on a special testing machine. The representative cases of the created fracture growth in the model specimens were analyzed. The physical simulation outcomes can be used in formulation and solution of analytical problems on HF stress measurement, as well as to improve procedures and equipment involved in this method of stress control in underground mines.

Pressure field in the reservoir and well, taking into account the dynamic level during the selection and subsequent shutdown

An analytical solution of the non-stationary filtration problem is presented, taking into account the change in the dynamic level of the fluid in the well during extraction with a variable flow rate, which made it possible to construct calculation formulas for the pressure field in the well when operating at a constant pump performance and subsequent shutdown. The problem contains the equation of piezoconductivity for non-stationary plane-radial filtration in a formation surrounded by an impermeable bottom and tire. It also includes an integro-differential condition connecting the pressure fields in the well and the reservoir, the condition of equality of pressures at the outlet from the reservoir to the well, the absence of disturbances at the initial time and in remote sections of the reservoir. When constructing an exact analytical solution of the problem in dimensionless variables, the Laplace — Carson integral transformation with respect to time was used. The originals of the solution are also obtained analytically and contain the Bessel functions of the zeroth and first orders. The general analytical solution for the pressure during extraction with a constant flow rate and subsequent shutdown is written using the Heaviside function. Based on the solution found, computational experiments were performed in which the reservoir parameters of the oil-saturated reservoir are varied, and pump stops are performed at various stages of well operation: relaxation, transitional and stabilization. It has been established that the main influence on the dynamics of the pressure field is exerted by the permeability of the reservoir. Reducing the displacement capacity of the reservoir by an order of magnitude leads to an increase in the amplitude of the pressure perturbation by 25-35%. However, the influence of this parameter does not exceed the contribution of permeability. The contribution of the reservoir porosity is less than 6% of the peak pressure values in the stabilization mode. The viscosity of the sampled fluid has a significant impact on the pressure field recorded in the well.

Comparison of inviscid and viscid one-dimensional models of blood flow in arteries

The paper is devoted to the comparison of one-dimensional blood flow models in application to the solution of model problems. In the viscid case, the non-Newtonian properties of blood are considered. Original one-dimensional models, based on the Carreau, Carreau–Yasuda, Cross, and Powell–Eyring rheological models, are constructed by the averaging of the Navier–Stokes equations.The approach to the analytical solution of problems in the inviscid case is proposed. The originality of the method is based on the small perturbation of the initial rest state, corresponding to zero velocity. It leads to the solution of the linear wave equations. The solutions of three problems — for the infinite, semi-infinite, and finite intervals are obtained. The examples are presented for the small parameter value ∼10−2. Analytical solutions are used for the comparison of different one-dimensional models of blood flow, where the viscosity (Newtonian and non-Newtonian) is considered. The problems for the viscid models are solved numerically by the third-order WENO scheme. As the results of the comparison of models, the effects of the viscosity and velocity profile are analyzed.From a practical viewpoint, the solutions obtained by the perturbation method can be used for the testing of programs for numerical simulations and for the comparison of different blood flow models.

Analytical Solution of Problems about the Radiative and Radiative–Conductive Stationary Heat Transfer in a Medium with an Arbitrary Dependence of the Scattering and Absorption on Frequency Boundary Conditions

We defined a method for the analytical solution of problems on stationary radiative and radiative–conductive heat transfer in a medium with an arbitrary frequency dependence of absorption and scattering near its boundary. We obtained formulas for the heat conductance of the remote surface and the thickness of the radiative–conductive relaxation of the medium. We determined characteristics of radiant heat transfer from the medium to free space such as the radiation spectrum, the radiation temperature and the medium outer boundary temperature. In addition, we solved the problem on the radiative–conductive heat transfer from one of two parallel surfaces to another with a medium between them.

Analytical solution of problem of shielding low-frequency magnetic field by thin-walled cylindrical screen in presence of cylinder

The analytical solution of boundary value problem describing the process of penetration of low-frequency magnetic field through thin-walled cylindrical screen with cylindrical inclusion is constructed by use of approximate boundary conditions. The source of the field is a thin thread of infinitely small length with an infinitely small cross-section where current circulates. Thread is located in a plane which is perpendicular to axis of cylindrical screen, in outer region with respect to a screen. Initially the potential of initial magnetic field is represented as spherical harmonic functions, then using addition theorems connecting spherical and cylindrical harmonic functions, it became as cylindrical harmonic functions superposition. Secondary potentials of magnetic field are also presented as superposition of cylindrical harmonic functions in three-dimensional space. It is shown that the solution of formulated boundary value problem is reduced to the solution of linear algebraic equations system for coefficients included in the representation of secondary fields. The influence of some aspects of the problem on the value of the screening coefficient of an external magnetic field when passing through a cylindrical copper screen in the presence of a cylindrical inclusion is studied numerically. Calculation results are presented in graphs form. Obtained results can be used to shield technical devices and biological objects against the effects of magnetic fields to provide ecological surrounding of operating electrical installations and devices.

Математична модель теплообміну в елементах цифрових технологічних пристроїв

A mathematical model of heat exchange analysis between an isotropic two-layer plate heated by a point heat source concentrated on the conjugation surfaces of layers and the environment has been developed. To do this, using the theory of generalized functions, the coefficient of thermal conductivity of the materials of the plate layers is shown as a whole for the whole system. Given this, instead of two equations of thermal conductivity for each of the plate layers and the conditions of ideal thermal contact, one equation of thermal conductivity in generalized derivatives with singular coefficients is obtained between them. To solve the boundary value problem of thermal conductivity containing this equation and boundary conditions on the boundary surfaces of the plate, the integral Fourier transform was used and as a result an analytical solution of the problem in images was obtained. An inverse integral Fourier transform was applied to this solution, which made it possible to obtain the final analytical solution of the original problem. The obtained analytical solution is presented in the form of an improper convergent integral. According to Simpson's method, numerical values of this integral are obtained with a certain accuracy for given values of layer thickness, spatial coordinates, specific power of a point heat source, thermal conductivity of structural materials of the plate and heat transfer coefficient from the boundary surfaces of the plate. The material of the first layer of the plate is copper, and the second is aluminum. Computational programs have been developed to determine the numerical values of temperature in the given structure, as well as to analyze the heat exchange between the plate and the environment due to different temperature regimes due to heating the plate by a point heat source concentrated on the conjugation surfaces. Using these programs, graphs are shown that show the behavior of curves constructed using numerical values of the temperature distribution depending on the spatial coordinates. The obtained numerical values of temperature indicate the correspondence of the developed mathematical model of heat exchange analysis between a two-layer plate with a point heat source focused on the conjugation surfaces of the layersand the environment, the real physical process.

ВЫПИСЫВАНИЕ ФОРМУЛ ВЫЧИСЛЕНИЯ СИЛ В СОЧЛЕНЕНИЯХ МАНИПУЛЯТОРОВ В СТАТИКЕ

The problem of bulkiness of mathematical models of manipulative systems of industrial robots is solved. Here we consider formulas for calculating static reactions in joints and formulas for active forces that balance the forces of gravity acting on the manipulator's bodies in its stationary state. The manipulator can be in such a state when it is before capturing the object of manipulation and releasing it, or when it is performing some assembly operations, or it is during spot welding and in slow (quasi-static) arc-welding and painting processes. Aim. The aim is to derive general recur-rence and finite formulas for calculating the reaction forces in joints and their projections to the ax-es of the coordinate system rigidly connected with the selected body. Express the formulas of force projections in terms of guiding cosines and justify their optimality in terms of the minimum of arithmetic operations. Derive general inverse recurrence formulas for writing out the guide cosines of the axes associated with the moving bodies of the coordinate system with respect to the stationary coordinate system. Research methods. The methods of research relate to vector mechanics and sys-tems analysis, and the algorithmization of calculations by reducing them to the use of recurrent formulas. Results. A systematic analysis of general formulas, in which all possible regular expres-sions are highlighted which are corresponding unambiguously to the kinematic parameters of ma-nipulators, is performed. These regular expressions are used in software for analytical modeling of manipulator, in particular, for the analytical solution of problems of statics of a manipulator. The method of analytical verification of the prescribed formulas is described. The tasks of writing out optimal formulas for calculating the projections of static reaction forces in joints have been solved. And the tasks of writing out optimal formulas for calculating active forces in progressive joints of universal manipulators with six degrees of freedom, operating in Cartesian, cylindrical, spherical and angular coordinate systems, have been solved also. Analytical verification of the derived equations of stat-ics is performed. Examples of the reuse of the derived formulas for manipulators with the same kin-ematic schemes of their subsystems. Conclusion. Expressions of the equations of statics of manipu-lators through the guide cosines of the axes of the associated coordinate systems of their bodies al-low us to write these equations through the known parameters of body orientation. The recurrent formulas for calculating directional cosines allows to use recursive functions in their software im-plementation, i.e. to increase the computational efficiency of the software.

МАТЕМАТИЧНА МОДЕЛЬ ТЕПЛООБМІНУ В ЕЛЕМЕНТАХ ЦИФРОВИХ ПРИСТРОЇВ

A mathematical model of heat exchange analysis between an isotropic two-layer plate heated ba point heat source concentrated on the conjugation surfaces of layers and the environment has been developed. To do this, using the theory of generalized functions, the coefficient of thermal conductivity of the materials of the plate layers is shown as a whole for the wholesystem.Given this, instead of two equations of thermal conductivity for each of the plate layers and the conditions of ideal thermal contact, one equation of thermal conductivity ingeneralized derivatives with singular coefficients is obtained between them. To solve the boundary value problem of thermal conductivity containing this equation and boundary conditions on the boundary surfaces of the plate, the integral Fourier transform was used and as a result an analytical solution of the problem in images was obtained. An inverse integral Fourier transform was applied to this solution, which made it possible to obtain the final analytical solution of the original problem. The obtained analytical solution is presented in the form of an improper convergent integral. According to Simpsons method, numerical values of this integral are obtained with a certain accuracy for given values of layer thickness, spatial coordinates, specific power of a point heat source, thermal conductivity of structural materials of the plate and heat transfer coefficient from the boundary surfaces of the plate. The material of the first layer of the plate is copper, and the second is aluminum. Computational programs have been developed to determine the numerical values of temperature in the given structure, as well as to analyze the heat exchange between the plate and the environment due to different temperature regimes due to heating the plate by a point heat source concentrated on the conjugation surfaces. Using these programs, graphs are shown that show the behavior of curves constructed using numerical values of the temperature distribution depending on the spatial coordinates. The obtained numerical values of temperature indicate the correspondence of the developed mathematical model of heat exchange analysis between a two-layer plate with a point heatsource focused on the conjugation surfaces of the layersand the environment, the real physical process.

Формирование профессиональной коммуникации у студентов-баскетболистов

Features of the restructuring of the communication of students involved in basketball during the transition to professional activity are revealed. We consider the disadvantages of communication at the initial stage, manifested in the absence of tolerance, the ability to defend their views and interests, the lack of development of emotional manifestations in communication. In the process of pedagogical influence at the first stage, the student approaches the development of the characteristics of his personality (I-I) and the knowledge of himself, his abilities, with the transition from (I-I) to I-Other in solving common interests and problems; at the second stage, attention is directed to the collective analytical solution of problems, intra-group communication in the I – Collective, I – Team ligaments; at the third stage, the personality-reflective nature of communication was defined as experience that can be implemented in practice. The pedagogical impact of the I-I, I-Other, I-Team (sports team) allowed us to get a number of positive changes necessary for the successful start of a professional career.

THE PRESSURE FIELD IN THE RESERVOIR AT A GIVEN WELL FLOW RATE

Due to the problems of hydrocarbon production, the study of pressure fields in natural reservoirs is the primary task of the filtration theory and special interest to the practitioners. Because of the variety of natural conditions necessary to account for in setting, the number of tasks is steadily increasing. At present, several analytical solutions of problems for homogeneous reservoirs with the simplest boundary conditions have been constructed. Studying the influence of borehole conditions on pressure fields in productive formations presents interest, as in this case, it is expressed as a boundary condition in the form of an integro-differential equation, and the corresponding class of problems is not sufficiently studied in the fundamental sections of the mathematical physics. In contrast to the existing works, this article constructs an analytical solution of the problem about the pressure field in the formation, represented by the classical equation of piezo-pipeline, for the case when the boundary condition in the form of the integro-differential equation describes the process of fluid extraction from the well with the help of a pump of the given productivity. The authors study the oil-saturated porous formation, opened by the producing well for the whole thickness. The pressure field in the reservoir is described by the classical piezoelectricity equation in the cylindrical coordinate system in the assumption of axial symmetry. The function of pressure perturbation distribution across the reservoir is determined considering the influence of the well. The relationship of pressures in the well and formation is established based on the law of mass conservation and Pascal’s law. The dimensionless criteria characterizing the filtration process and allowing to simplify the task definition are defined. It is shown, that all parameters, characterizing influence of well and formation, are grouped in two interrelated criteria, which define filtration process in such conditions. The authors have presented an exact solution of the problem in the space of Laplace — Carson integral transformation with an analytical transition to the original space. In addition, they have created a numerical conversion program, based on Den Iseger’s algorithm, and developed a finite-difference scheme for the task at hand. Having calculated the space-time distributions of pressure fields, the authors have compared the graphical pressure dependencies based on an analytical formula and a numerical program with the results of finite-difference calculations. That increases the reliability of the results obtained, which seems to be of great importance, since the existence and uniqueness theorems are not proved for this class of the problem in question. The analysis of the calculation results allowed determining the contribution of the well and reservoir parameters to the pressure field. The results show that in the relaxation mode, the parameters of the well and pump have a significant influence on the formation of the pressure field. In the stabilization mode, the contribution of its physical parameters is predominant.

Analytical solutions of the problems for equations of one-dimensional hemodynamics

The paper is devoted to the construction of an analytical solution of the initial problem for the one-dimensional system of equations of non-Newtonian hemodynamics. In the model, the blood is considered as the power-law fluid. The perturbation method for the solution is proposed. The cases of the first and second perturbations are considered. The obtained solutions can be used for the comparison of different rheological models of blood or for the testing of programs, which realize the numerical algorithms.

Thermal 3D model for non-homogeneous elements in mobile devices

A mathematical model for the analysis of heat exchange between the environment and an isotropic space layer with an alien inclusions is developed, which is heated by a heat flux centered on one of the boundary surfaces. For this purpose, using the theory of generalized functions, the coefficient of thermal conductivity of this structure is depicted as one unit for the whole system. In view of this, instead of two equations of thermal conductivity for the layer and the inclusion and conditions of perfect thermal contact on the surfaces of the junction between them, one equation of thermal conductivity was obtained in the generalized derivatives with breaking coefficients. We consider the case when the inclusion sizes are small compared to the distances from the inclusion surfaces to the boundary surfaces of the layer. In this connection, the combined thermophysical parameters were introduced and the thermal coefficients of the thermal conductivity equation were transformed into singular ones. For the solution of the boundary value problem of thermal conductivity containing this equation and boundary conditions on the boundary surfaces of the layer, an integral Fourier transform was used and, as a result, an analytical solution of the problem in the images was obtained. The inverse integral Fourier transform was applied to this solution, which made it possible to obtain the final analytical solution of the original problem. The analytical solution obtained is presented as a non-native double convergent integral. To determine the numerical values ​​of the temperature in the above design, as well as to analyze the heat exchange between the layer and the environment caused by different temperature regimes due to the heating of the inhomogeneous layer by a heat source concentrated in the area of ​​inclusion, computational programs have been developed. Using these programs, graphs are displayed showing the behavior of curves constructed using numerical values ​​of the temperature distribution depending on the spatial coordinates for different inclusion materials. The obtained numerical values ​​of temperature indicate a significant influence of the inclusion on its distribution in the design "layer-inclusion". The software also makes it possible to analyze these inhomogeneous media with respect to their heat resistance during heating. As a consequence, it becomes possible to raise and protect it from overheating, which can cause destruction not only of individual elements, but also of the whole structure.

On Incompressibility Constraint and Crack Direction in Soft Solids

Most soft materials resist volumetric changes much more than shape distortions. This experimental observation led to the introduction of the incompressibility constraint in the constitutive description of soft materials. The incompressibility constraint provides analytical solutions for problems which, otherwise, could be solved numerically only. However, in the present work, we show that the enforcement of the incompressibility constraint in the analysis of the failure of soft materials can lead to somewhat nonphysical results. We use hyperelasticity with energy limiters to describe the material failure, which starts via the violation of the condition of strong ellipticity. This mathematical condition physically means inability of the material to propagate superimposed waves because cracks nucleate perpendicular to the direction of a possible wave propagation. By enforcing the incompressibility constraint, we sort out longitudinal waves, and consequently, we can miss cracks perpendicular to longitudinal waves. In the present work, we show that such scenario, indeed, occurs in the problems of uniaxial tension and pure shear of natural rubber. We also find that the suppression of longitudinal waves via the incompressibility constraint does not affect the consideration of the material failure in equibiaxial tension and the practically relevant problem of the failure of rubber bearings under combined shear and compression.

Analysis of three-dimensional ellipsoidal inclusions in thermoelectric solids

ABSTRACT Thermoelectric (TE) materials as energy functional semiconductors can directly convert heat into electrical power. Due to the brittle nature of semiconductors, voids or inclusions are easy to be produced in the TE materials. This paper studies the three-dimensional ellipsoidal inclusion problem in thermoelectric materials. By introducing two auxiliary functions, we successfully simplify the non-linear coupled governing equations into linear un-coupled equations and the linearization is validated by two practical cases. Based on the Green's function method, the analytical solutions of this problem are derived. We find that the thermoelectric field inside the ellipsoidal inclusion is always uniform when subjected to far-field uniform loads. Furthermore, we derive the effective material properties of the matrix-inclusion system in closed-form. We find that it is possible to enhance the thermoelectric properties as well as the figure of merit by inserting specific inclusions. Moreover, among the different shapes of inclusions, the elliptical cylinder fiber that lies along the loading direction has the most significant improvements to the material properties. This paper is the first that derives the analytical solutions of three-dimensional inclusion problems in TE materials.

Analysis of the natural frequencies of oscillations of a planar truss with an arbitrary number of panels

Introduction. Analytical solutions for problems of structural mechanics are not only an alternative approach to solving problems of strength, reliability and dynamics of structures, but also the possibility for simple performance evaluations and optimization of structures. Frequency analysis of planar trusses, most often used in construction and engineering, is an important part of the study of structures. Objectives - development of a three-parameter induction algorithm for deriving the analytical dependence of the natural oscillation frequencies of the truss on the number of panels. Materials and methods. A flat, statically definable truss with one additional external link and double braces has been considered. The inertia properties of the truss are modeled by point masses located in the nodes of the lower straight truss belt. Each mass is assumed to have only one vertical degree of freedom. The stiffness of all truss rods is assumed to be the same. The task is to obtain analytical dependences of the oscillation frequencies of the proposed truss model on the number of panels. The derivation of the desired formulas is performed by the method of induction in three stages - according to the numbers of rows and columns of the compliance matrix, calculated using the Maxwell - Mohr formula and the number of panels. To find common members of the obtained sequences of coefficients, an apparatus was used to compile and solve the recurrent equations of the Maple computer mathematics system. The task of determining frequencies has been reduced to the eigenvalue problem of a bisymmetric matrix. Results. For the elements of the compliance matrix, general formulas have been found, according to which the frequency equations are compiled and solved. It is shown that in the frequency spectra of trusses with different numbers of panels there is always one common frequency (middle frequency) located in the middle of the spectrum. An expression is found for the maximum value of the average oscillation frequency as a function of the height of the truss. Conclusions. The proposed truss scheme, despite its external static indeterminacy and the lattice, which does not allow for the calculation of forces by such methods as the method of cutting nodes and the cross section method, allows analytical solutions for the natural frequencies of loads in the nodes. The obtained formulas have a rather simple form, and some general properties, such as frequency coincidences for different numbers of panels and the presence of an analytically calculated maximum of the average frequency function of the truss height, make this solution convenient for practical structural evaluations.