# Vol 27, No 4 (2021)

## Статьи

### ON THE NATURE OF ADDITIONAL SPACE AT CUTTING OF SPACES OF CUSP FORMS

#### Abstract

In the article we study a space of cusp forms by the method of cutting. This space is a direct sum of the subspace of forms divided by the fixed cusp form named the cutting function and the additional space. If the additional space is zero we have the situation of exact cutting. In common case the cutting is not exact and it is important to research the nature of the additional space. We prove that the basis of the additional space can be described by the space of cusp forms of small weight. This weight is not more than 14 and often is equal to 4. We give examples of all cutting functions for all levels. We prove the theorem about the basis of the additional space to the space of cusp forms in the space of modular forms of the same level, weight and character. We use properties of eta-products, Biagioli formula for orders in cusps and Cohen — Oesterle formula for dimensions.

**Vestnik of Samara University. Natural Science Series**. 2021;27(4):7-13

### DETERMINATION OF MECHANICAL PROPERTIES OF A MATERIAL MODELED USING THE MOLECULAR DYNAMICS METHOD

#### Abstract

The article discusses a method for determining the elastic constants of a material modeled using the molecular dynamics method in the LAMMPS package (Large-scale Atomic/Molecular Massively Parallel Simulator). The purpose of the study is to calculate the matrix of elastic constants Cij or the compliance

coefficients Sij of materials. Knowing them, elastic modulus is calculated along certain directions and shear in shear systems, as well as the Poisson’s ratio. Extreme values of mechanical constants can be calculated. In addition, linear combinations of elastic constants that have a physical meaning can be determined. FCC single crystals of copper and aluminum were selected as examples for modeling. The ELATE - Elastic tensor

analysis computational and graphical package is used to visualize elastic properties.

**Vestnik of Samara University. Natural Science Series**. 2021;27(4):14-21

### AUTOMATION OF DESIGN AND CALCULATION OF THE HYDRAULIC TURBINE DISTRIBUTOR

#### Abstract

The article describes the algorithm for calculating a distributor of hydraulic turbine, which includes three stages: calculation of the kinematic characteristics of the distributor of hydraulic turbine, strength calculation of the guide vane and calculation of the power characteristics of the drive. Software implementing the specified algorithm has been developed. The use of software in JSC TYAZHMASH (Syzran) made it possible to reduce the labor intensity and cost of designing a distributor of hydraulic turbine.

**Vestnik of Samara University. Natural Science Series**. 2021;27(4):22-29

### COMPARATIVE ANALYSIS OF STRESS STATE IN THE VICINITY OF THE CRACK-TIPS AND NOTCHES USING TRUNCATED WILLIAMS SERIES EXPANSION

#### Abstract

In the present study the asymptotic analysis of the multi-parameter Williams series expansion of the stress-strain state in the vicinity of seven different configurations is performed. The aim of the asymptotic analysis is to educate the sensitivity of different cracked and notched configurations to the number of kept terms in Williams series expansion. It is shown that all configurations with the concentrated forces possess the more sensitivity to the number of terms in the truncated Williams series expansions and, therefore, require the more terms in the series expansion. The cracked configurations with distributed loadings are less sensitive to the number of terms in Williams series expansion. As a whole, it is demonstrated that high order terms have principal role in the description of the stress-strain in the vicinity of the crack tip.

**Vestnik of Samara University. Natural Science Series**. 2021;27(4):30-67

### MOLECULAR DYNAMIC MODELING OF STRESS FIELDS IN PLATES WITH A CENTRAL CRACK MADE OF MATERIALS WITH A FACE-CENTERED CUBIC LATTICE

#### Abstract

The purpose of this study is to determine the stress-strain state at the atomistic level using molecular dynamic modeling of the structure of the material, which were selected as single crystals of copper and aluminium. A series of computational experiments were carried out on plates with a central crack in the LAMMPS open code. Also, LAMMPS conducted experiments on cubic samples from the same materials to find the tensor components of elastic modules. The visualizations obtained in the OVITO software package are shown, which show the distribution of stress tensor components over different time steps for copper and aluminum single crystals. The work presents graphs showing the dependence of the stress tensor components on the polar angle, obtained by atomistic and classical approaches. The comparison showed that the stress

fields at the nanoscale level are in good agreement with their macroscopic magnitudes, so a continuum approach can be applied at the atomistic level.

**Vestnik of Samara University. Natural Science Series**. 2021;27(4):68-82

### EVALUATION OF THE APPLICATION OF DOUBLE-EXPOSURE SPECKLE PHOTOGRAPHY WITH DIFFERENT APERTURES IN EXPERIMENTAL MECHANICS

#### Abstract

The article presents an experimental evaluation of the use of double-exposure speckle photography with circular and annular apertures in experimental mechanics using the example of determining the modulus of elasticity in tension. The description of the process of registration of speckle photography and its subsequent processing is given. Based on the results of calculations of the finite element model of the sample, the area

where the boundary conditions do not affect its tension is determined. A description is given of the procedure for processing speckle photographs to determine the modulus of elasticity and the error in its measurement with different apertures.

**Vestnik of Samara University. Natural Science Series**. 2021;27(4):83-91

### BIFURCATIONS OF STATIONARY REGIMES IN THE MODEL OF A WIND POWER PLANT

#### Abstract

The article should use a mathematical model of a wind power plant. In the chosen world, alternative energy is actively developing, as natural resources for harmful energy carriers are significantly reduced every year. In turn, the energy reserves of life situations turned out to be related to world problems. Thus, the relevance of this problem is solved in the free system of the wind section. The complexity of this work is determined by the fact that the model, which is not defined, is found completely, that is, part of the functions in the mathematical model is set

only graphically, based on experimental data. The graph of relative aerodynamic moment of forces is approximated. The ratio of stationary modes is studied for various values of external resistance of the dynamic model. Conditions are found under which a saddle-node bifurcation and a supercritical Andronov–Hopf bifurcation are observed in the system.

**Vestnik of Samara University. Natural Science Series**. 2021;27(4):92-98