Introduction. Modern trends in construction, related to the optimization of weight and materials, require accurate methods for calculating the stress-strain state, particularly of beams with variable stiffness. Analytical calculation of the stressstrain state for such beams is fraught with considerable difficulties, limiting its practical application. Numerical methods, specifically the Finite Element Method (FEM), are widely used to solve these problems, where the law of stiffness change is typically approximated by a piecewise (discrete) function. This study is aimed at the development of an approach based on piecewise-linear approximation of stiffness. Linear stiffness approximation suggests an optimal balance of accuracy and computational resources. This approach provides significantly higher accuracy compared to the traditional discrete approximation with similar computational complexity, allowing for adequate modeling of both smooth stiffness gradients and its violent changes.

Materials and Methods. A first-approximation stiffness matrix for a one-dimensional beam finite element with linearly varying flexural stiffness was derived on the basis of a variational formulation of the problem. An exact stiffness matrix was obtained by direct integration of the differential equation for beam bending. In the calculation examples, an exact solution was obtained using the Maple software package. The numerical solution using FEM was implemented in the author's program written in Python.

Results. During the study, approximate and exact stiffness matrices of the beam finite element were obtained, as well as the vector of nodal reactions (loads) from distributed loads. The efficiency of the proposed approach was demonstrated by numerical examples. The results obtained by the FEM were verified using analytical calculations. Based on the performed calculations, recommendations and criteria for using the exact or approximate stiffness matrix were developed.

Discussion. Finite elements that account for linear change of stiffness along the length make it possible to increase the accuracy of the results and reduce the degree of discretization of the computational scheme by more than two times. The approximate matrix shows good convergence with a smooth change in stiffness along the length. In such cases, discrete approximation is also acceptable. The exact matrix allows for calculating cases where the stiffness within the beam changes by orders of magnitude with low error. The classical discrete approximation in this case does not ensure high accuracy of the calculation results.

Conclusion. The paper presents stiffness matrices for finite elements that account for linear change of stiffness along the length. Their derivation is performed by two methods: on the basis of a variational formulation of the problem, and by direct integration of the differential equation of bending. The resulting matrices enable more accurate stress-strain analysis of beams with variable stiffness. They have an analytical format that simplifies their integration into existing software systems. Further research will be directed towards applying the obtained matrices to the calculation of reinforced concrete beams, considering physical nonlinearity, as well as to solving problems of stability and dynamics of beams with variable stiffness.

Introduction. Briquetting and pressing of wood and other powdered materials are becoming key processes in the circular economy and recycling of wood processing waste. Accurate calculation of compaction pressure is essential for equipment selection and optimization, making the task of modeling the deformation of conglomerates both practical and economically significant. The literature addresses the mechanics of powder media, porous materials, and the modeling of elastic-plastic deformations of granular conglomerates. However, most models assume fixed mechanical characteristics or approximations that do not account for the dependence of strength and elastic properties on changing density under compression. This leaves a gap in theoretical and applied approaches to adequately calculating pressure for materials with variable density. Therefore, the objective of this work is to develop an approach for calculating the compaction pressure of a particle conglomerate as a function of the degree of elastic compression, taking into account changes in the mechanical characteristics of the medium.

Materials and Methods. In the mathematical description of the research problem, the provisions of the theory of elasticity were used. Based on the principle of superposition, the process of medium deformation was divided into a number of stages, within which the particle conglomerate received a small increment in height, and the mechanical characteristics assumed a constant value. The proposed approach for determining the compaction pressure was based on the solution to a series of inverse elastic problems in which the displacement of the upper boundary of a conglomerate of rectangular particles was specified, and the normal stress that caused this increment was sought. To account for changes in the density of the medium during deformation, the method of sequential loads was used, within each of them, the density was taken to be constant and was determined depending on the magnitude of the total compressive deformation. The Hencky strain, which has the property of additivity, was used as a measure of deformation.

Results. As part of the study, an iterative model was constructed for calculating the compaction pressure of a particle conglomerate when the mechanical characteristics change depending on the degree of elastic compression. Series of test calculations were conducted using a conglomerate of wood particles, whose Young's modulus is described by a power-law density function. At each stage of deformation, the elastic constants of the material were assumed to be constant, depending on the density of the medium. Using the equilibrium equation and the superposition principle, based on the results of solving elastic deformation problems, the compaction pressure was calculated at each loading stage, and the dependence of the compaction pressure on the magnitude of the compressive deformation and the degree of compaction was constructed.

Discussion. The obtained results of deformation of the medium taking into account the change in mechanical characteristics depending on the degree of compression showed a clearly expressed nonlinearity of the curve of dependence of the compaction pressure on the compression deformation — with an increase in pressure, both the degree of compaction of the medium and the compression deformation increase. A comparative analysis of calculations using the example of a conglomerate of wood particles under the condition of a constant density of the medium and taking into account the change in density during the deformation process revealed a significant error in estimating the compaction pressure when averaging the density or when using constant density values corresponding to the initial (undeformed) or final state.

Conclusion. The constructed iterative model allows for calculating the compaction pressure of a particle conglomerate, taking into account changes in mechanical properties under elastic compression. The proposed approach accounts for the nonlinearity of the compaction pressure dependence on the degree of compaction of the medium and can be applied to briquetting processes for wood waste.

Introduction. In industry, the process of obtaining technological vacuum using ejectors that utilize the kinetic energy of a jet of compressed air is widely used. The selection of the required ejector model, as well as their number (when creating a field of ejectors), is performed proceeding from the compliance of the ejector characteristics with the key parameters of the designed process technology. One of the most important characteristics of an ejector, significantly affecting the overall performance of the vacuum system, is the evacuation time of the graduated (calibrated) container. However, in technical literature, this parameter is not specified for the maximum vacuum depth produced by the ejector, nor for the corresponding supply pressure, but for certain, less-defined parameters, referred to as optimal by ejector manufacturers. In such cases, it is impossible to accurately estimate the actual value of an important criterion. Therefore, the objective of this work is to experimentally determine the actual value of the vacuum time of a graduated (calibrated) vessel for various types of ejectors.

Materials and Methods. Experimental studies were performed on a stand specifically designed and manufactured by the authors, which made it possible to study various parameters of vacuum ejectors. In particular, the stand provided establishing the exact time of vacuuming a measuring vessel using ejectors with a nozzle diameter from 0.1 to 4.0 mm at a supply pressure value that induced the maximum vacuum depth for each model under study. The research was carried out using the most popular vacuum ejectors of the VEB, VEBL, VED and VEDL families manufactured by Camozzi at a pre-determined, precisely set input supply pressure for each ejector size. The actual values of the vacuum time at the highest vacuum depth for each ejector were experimentally determined.

Results. It has been established that the performance of VEB, VEBL, VEDL, and VED series ejectors differs from that stated in the manufacturer's catalog. The time required to reach maximum vacuum for each ejector exceeds the manufacturer's specifications by 25–40%, which impacts the performance of the vacuum system.

Discussion. The experimental data have shown that the actual values of the vacuum time of the measuring vessel differ from the values given in the catalogs of manufacturers of ejectors. This difference is explained by the fact that when conducting appropriate tests, manufacturers are guided not by the maximum vacuum depth created by the ejector, but by the vacuum depth created by a certain “optimal” (the wording of the ejector manufacturer) value of the supply pressure. In almost all the cases considered by us, this “optimal” supply pressure produced a vacuum, whose depth differed from the maximum. In this regard, it seems advisable to adjust the value of the inlet supply pressure to attain the maximum vacuum depth for each type of ejector.

Conclusions. The results of the obtained values of the vacuum creation time in one liter of volume at the maximum depth of the vacuum produced by the ejector provide a more accurate selection of vacuum ejectors depending on the required process tasks, ensure the greatest efficiency and cost-effectiveness of automated vacuum systems. The research results can be used by all ejector manufacturers to adjust their basic catalogs and appropriate recommendations for the use of these products. Further research will be conducted to study the accuracy of the geometric shapes of the surface of the ejector channel, the purity of processing, and their production technology, which affect the passage of air flow.

Introduction. The implementation of high-precision attitude control systems of a new generation with improved technical characteristics remains a key task in precision instrumentation — this is required for the reliable operation of moving objects with a long service life. One of the promising ways is the use of sensors based on the Bryan effect (hemispherical resonator gyroscopes, HRG),), which show significant advantages in stability of characteristics under external factors. Over the past 10 years, foreign and domestic research has reached noticeable success in increasing the target parameters of HRG, however, certain improvement problems remain open. Thus, in the literature, attention is paid to reducing the errors in measuring the HRG through compensating for the impact of imperfections of the resonator, but more often these methods are applicable at stages after geometry generation. Methods for early identification of material inhomogeneities (density variation) during workpiece inspection are insufficiently developed, creating a gap in the process chain and reducing the efficiency of subsequent balancing and calibration. The objective of this study is to develop a method for identifying resonator density variations at an early stage of the process — during workpiece inspection.

Materials and Methods. An optically transparent material is considered – fused quartz glass, which is the most common material for making a HRG resonator, in particular, the KU-1 brand (foreign analogs — Corning HPFS 7980, JGS1). The identification method is based on the relationship of the optical properties of quartz glass (absorption coefficient) with the desired density distribution over the volume of the workpiece. A virtual experiment was conducted, which consisted in the formation and resolution of a system of linear algebraic equations (SLAE) based on the measurements series results of a light beam intensity passing through a workpiece. A polynomial approximation was used to describe the density distribution in order to increase the robustness of the method. The SLAE roots were obtained through finding a pseudosolution by the least square method based on the singular value decomposition.

Results. A method for identifying the density variation of quartz glass at the stage of quality control of the technological workpiece of the HRG resonator was developed. The desired density distribution of quartz glass over the volume of the workpiece was obtained, coinciding with the “true” one — the difference was no more than 5%. The sensitivity of the method to the presence of macrodefects in the volume of the workpiece (pores, bubbles, etc.) was assessed.

Discussion. The results show that the proposed method can effectively control the density variation of the workpieces and optimize the resonator production, thereby improving the efficiency of the processes and minimizing the impact of imperfections on their characteristics. Virtual experiments have demonstrated that measuring the light beam intensity passing through the workpiece allows for the accurate reconstruction of the absorption coefficient and density distribution with an accuracy of at least 0.005%. The developed system of linear algebraic equations (SLAE) makes it possible to determine these parameters by volume. The paper highlights some features related to solving uncertain SLAE. Particular attention is paid to the need to control the ratio between the number of roots and unknowns to obtain a stable solution.

Conclusion. The proposed method for identifying the density variation of quartz glass at the stage of workpiece quality control in the production of HRG resonators demonstrates high efficiency and accuracy. The presented method has high accuracy for describing the distribution function, and is also flexible in terms of obtaining the optimal dimension of the SLAE, which is directly related to the number of experiments performed. The obtained results confirm the applicability of the material optical properties for controlling the density distribution over the volume, which allows for improved control of workpieces and optimization of production processes. The required measurement accuracy, determined by the level of density variation that affects the HRG characteristics, is practically achievable, which indicates that the method can be used in the manufacturing process. This approach can be applied in future research and development of highprecision systems, which will contribute to progress in the precision instrumentation industry and improve the quality of manufactured products.

Introduction. The design of road pavements for highways is a key stage of project development, directly impacting their durability and operational costs. In recent years, in the context of increasing traffic intensity and dynamic loads, technologies for strengthening roadbeds and bases, such as geosynthetic reinforcement and stabilized layers, have become widespread, making the study on their efficiency a challenge. Literature notes the practical advantages of reinforced layers — increased load-bearing capacity and reduced deformation. However, models for energy dissipation under dynamic impacts in structures with such layers are underdeveloped. Theoretical approaches to analyzing energy dissipation, including linear-elastic and viscoelastic models and finite element methods, have been primarily applied to traditional structures. Their adaptation to reinforced and stabilized layers requires further development, as there remain gaps in the quantitative comparison of efficiency by location and rigidity of reinforcements. The objective of the presented work is to analyze the dissipation of deformation energy in the structure of road pavements with different options for the arrangement of reinforced layers, and to determine optimal design solutions that contribute to increasing the durability of road pavements. To achieve this, it is required to formalize an energy dissipation model for structures with reinforcements, conduct a comparative analysis of different locations and rigidity levels of the layers.

Materials and Methods. The research utilized a comprehensive approach to the analysis of deformation processes in layered media using road pavements as an example, involving both a calculation tool and modern experimental equipment. As a calculation tool, a mathematical model of a layered half-space in an axisymmetric formulation in a cylindrical coordinate system was used. It was based on the solution to the system of dynamic Lame equations and allowed for the construction of amplitude-time characteristics of vertical displacements and impact loading impulse, on the basis of which it was possible to construct dynamic hysteresis loops. The FWD PRIMAX 1500 shock loading unit was used as experimental equipment, which made it possible to register similar characteristics of the road pavement response under field conditions at a load equivalent to the calculated one.

Results. The study involved numerical modeling of road pavement structures traditionally used in the Russian Federation and so-called full-depth road pavements, which were composed almost entirely of materials reinforced with binders. Dynamic hysteresis loops were constructed, and a comparative analysis of the results was provided. A numerical experiment revealed that strengthening only the subgrade layer, even without installing a reinforced base layer beneath the asphalt concrete, reduced the amount of dissipated deformation energy. It was also concluded that the elastic modulus of the underlying half-space simulating the subgrade had the greatest impact on the amount of dissipated energy.

Discussion. The greatest effect, both technical and economic, can be reached by strengthening the top of the roadbed while preserving the loose layers in the base of the road structure. This solution will bring the functioning of the road surface closer to the elastic stage and at the same time reduce the risk of cracks appearing on the surface of the pavement due to an excessively rigid layer of reinforced base.

Conclusion. On the basis of the constructed dynamic hysteresis loops, it is shown that a reduction in the magnitude of deformation energy can be obtained both by installing reinforced layers of the road surface throughout its entire depth, and by locally strengthening the underlying half-space layer and an additional base layer made of sand. The numerical experiment demonstrated that the use of reinforced base layers reduced the amount of deformation energy dissipation in the pavement structure by more than 2–3 times. Qualitative agreement between the experimental results and the numerical simulation results was shown.