Numerical-analytical Algorithm Research Articles

USE OF POLYNOMIAL APPROXIMATION OF FORCES IN THE METHOD OF CORRECTION FUNCTIONS

The paper considers the problem of polynomial approximation of the "force - time" dependencies in the elements of corroding hinged-rod structures from the point of view of the influence of the degree of the polynomial on the error in calculating their durability. A method for determining the coefficients of approximating polynomials is proposed, which is based on the use of a numerical-analytical algorithm for solving a system of differential equations describing the corrosion process. The results of numerical experiments are presented, illustrating for various constructions the dependence of the error in solving the problem on the degree of approximating polynomials.

The contact condition of viscoelastic contact problems

Three contact states in the viscoelastic contact deformation area between the rigid cylinder and the viscoelastic foundation are discussed based on various material mechanical properties between the rigid and the viscoelastic materials. On this basis, the relationship model between the node displacements and the node stresses in the boundary contact area is established, and the corresponding numerical analytical algorithms are used to simulate the viscoelastic contact deformation. Meanwhile, an example is given to simulate the contact deformation of the conveyor belt.

RESEARCH OF POLYNOMIAL APPROXIMATION OF FORCES IN ROD ELEMENTS OF CORRODING STRUCTURES

The paper considers the problem of polynomial approximation of the "force - time" depend-encies in the elements of corroding hinged-rod structures from the point of view of the in-fluence of the degree of the polynomial on the error in calculating their durability. A method for determining the coefficients of approximating polynomials is proposed, which is based on the use of a numerical-analytical algorithm for solving a system of differential equations describing the corrosion process. The results of numerical experiments are presented, illus-trating for various constructions the dependence of the error in solving the problem on the degree of approximating polynomials.

Construction of a mathematical model of lamb wave propagation in a steel pipeline with a protective outer coating

THE PURPOSE. Adapt the method of measuring the propagation of Lamb waves in thin two-layer plates to the study of the dependence of the phase velocity on the technical condition of pipelines with a protective outer coating. To perform a modification of the numerical-analytical algorithm for measuring the thickness of the coating and areas of non-adhesion of materials in thin two-layer plates for vibroacoustic diagnostics of technical pipelines. To investigate the propagation of a symmetric Lamb wave in a steel pipeline with a protective outer coating. METHODS. To solve the problem of localizing damage to technical pipelines, traditional methods of non-destructive testing based on vibroacoustic diagnostics are considered. RESULTS. Work on the construction of a mathematical model of the dependence of the propagation of the phase velocity of the Lamb wave on the thickness of the object under study has been carried out. The numerical analysis of the measurements was carried out using the example of thin two-layer segments of pipelines. The presence of changes in the thickness of the pipeline is taken as the effect of material defects on the propagation parameters of the Lamb wave mode. CONCLUSION. A numerical-analytical method for calculating the propagation of a symmetric Lamb wave mode in a thin segment is presented. The dependence of the propagation velocity of Lamb waves on the segment thickness is demonstrated. Based on the described technique, it seems possible to estimate not only the thickness of the segment and the area of non-adhesion of the layers, but also the total area of the defective area. This will allow in the future to record the relative changes in the thickness of the walls of pipelines to determine changes in the physical properties of the material or the presence of a defect.

Influence of a Nonstationary Surface Load on an Elastic Porous Half-Space

In this paper, we develop a method for solving planar nonstationary problems on the influence of surface pressure on an elastic-porous half-space. Using surface influence functions, we obtain an integral relation, which expresses the desired displacements with the surface pressure. A numericalanalytical algorithm is constructed. For calculating singular integrals, we derive special quadrature formulas based on the canonical regularization of integrands.

ОПТИМАЛЬНОЕ ПОДАВЛЕНИЕ ПОПЕРЕЧНЫХ КОЛЕБАНИЙ ВРАЩАЮЩИХСЯ УПРУГИХ СТЕРЖНЕЙ

The process of suppressing transverse vibrations of an elastic rod spinning in a horizontal plane and fixed at one of its ends is studied. It is supposed that the rod spins around the vertical axis at a constant angular velocity and performs transverse vibrations in the vertical plane, the vibrations are assumed small in amplitude. Transverse vibrations of the spinning rod are performed under external mechanical action. Lateral vibrations are described by the displacement function and considered in a rotating plane by using the classical beam model. The necessary conditions of optimality are derived and applied for suppressing elastic vibrations on a finite time-interval. The problem of optimal suppression of lateral vibrations caused by initial disturbances is formulated as a variational problem with constraints that take into account the suppressing effects on the rod. The limiting restrictions are presented in the form of inequalities. With the introduction of an additional variable, these restrictions are reduced to standard integral equality, while taking into account the energy constraints imposed on the control actions. The proposed iterative algorithm for solving the formulated problem is a numerical-analytical algorithm and consists in minimizing the quadratic quality criterion. This criterion characterizes the vibration suppression process and allows the implementation of improving variations. As a result of the operations carried out, the dependence of the vibration suppression process on the determining parameters, such as the angular velocity of rotation, the isoperimetric energy constant, and the length of the considered process of vibration suppression in time, has been clarified. An example that illustrates the implementation of the proposed algorithm and shows the effectiveness of this method for suppressing lateral vibrations is given.

Application of the method of integral equations based on the Lorentz lemma for calculation of matching transitions between rectangular waveguides with offset axes

This work presents a brief description of the method of integral equations. Integral equations are obtained on the basis of the integral form of Lorentz's lemma using a non-standard physical approach. This method allows one to calculate the characteristics of various three-dimensional irregularities of the waveguide path. As an example of such irregularity, the considered transition between straight shielded waveguides with offset axes and with different transverse sizes. The main attention is paid to the description of the developed numerical-analytical algorithm for calculating the transmission characteristics of the matching transition. A procedure for obtaining an approximate record of the field components on the surface of an irregular region is presented, which makes it possible to take into account the curvature of the wave surface of an electromagnetic wave propagating in the guiding structure. In this case, the profile of the longitudinal section of the guiding structure can be described by any analytical functions. The concept of a 4W-fold comparison waveguide is introduced. In addition, this paper presents the results of calculating the frequency dependences of the transmission characteristics of the fundamental wave of waveguide transition at displacements of the axis of the second waveguide relative to the axis of the first waveguide by different values. The indicated transmission characteristics are found for single-mode and multimode modes of operation of the waveguide transition. In order to verify the results obtained, they have been compared with the results obtained using a modern licensed computer-aided design tool CST Microwave Studio. To demonstrate the possibility of solving the problem of “functional” optimization using the developed algorithm, the results of calculating the transmission characteristics of a waveguide transition between rectangular waveguides with cosine functions describing the profile of the longitudinal section of an irregular region have been presented. In order to indirectly substantiate the equivalence of the initial integral equations underlying the constructed algorithm and the problem of diffraction of an electromagnetic wave in a waveguide transition, a test problem of calculating the transmission characteristics of a stepped transition of two rectangular regular waveguides with different sizes has been solved. The calculation results have been compared with the results obtained on the basis of the electrodynamic substantiated method of half-inversion of the operator of the original boundary value problem. In addition to all the above, in order to assess the validity of the proposed method and the numerical-analytical algorithm built on its basis, a study of the convergence of the calculated results against the approximation number has been made.

Generation of phase trajectories of experimental data

The phase plane method (PPM) is a graph-analytical approach for studying the stability and long-time behavior of systems described by differential equations. In order to apply the PPM to experimental data, it is necessary to identify these data with a first-or second-order differential equation. This work presents a numerical-analytical algorithm for obtaining phase trajectories from experimental data avoiding identification with a differential equation. For this purpose, the data are interpolated with a cubic spline. The first derivative was obtained analytically from the spline. The abscissa coordinates are the values of the spline at a given point, and the ordinate is the analytical representation of the first derivative. As an example, the phase trajectory of the dose-dependent curve of the drug tubazid is constructed by two methods-with identification of the differential equation describing the process and with the algorithm proposed in the article. The approach is implemented with Korelia software.

Methods to Reduce Material Intensity of Tail Sections of Launch Vehicles

A technique has been developed for reducing the material intensity of highly stressed tail sections of launch vehicles, taking into account strength and stability constraints as well as technological requirements. A cylindrical longitudinally and transversely ribbed waffle-grid (lattice) shell with rectangular holes is taken as the design scheme of the tail section, with its lower end being fastened at locations of support brackets, and the upper one being loaded with longitudinal compressive forces, evenly distributed along the contour, due to the action of the weight of higher-located structure elements. The optimization algorithm is based on the principle of ensuring discrete uniform strength of individual elements (substructures). The structural geometric dimensions of cross-sections of a standard tail section and the stiffness parameters of longitudinal and transverse load-bearing frames, the wall thicknesses of shell elements, the dimensions of lattice shells, etc., are selected from the requirements of stress-strength reliability: constraints on the limiting values of equivalent stresses (strength conditions), compressive stresses of the local and general buckling, and a number of design and technological requirements. The direct calculation of the tail section and the search for its variable geometric parameters are proposed to be performed using an interactive numerical-analytical (finite element method – engineering analysis) algorithm. The initial calculation of the static stress-strain state of the lattice-reinforced tail section was carried out by the finite element method, which is implemented in the NASTRAN package. To discretize the shell and its ribbing, flat finite elements were used. In the process of the finite-element numerical modeling of the tail section state, the reliability of the obtained results of calculating the equivalent stresses was analyzed by studying the convergence of the results of calculations on a series of meshes with different refinement. Results of the application of the developed technique to reduce the mass of the standard tail section of the Antares launch vehicle are presented.

Software package of mathematical modeling and structure for solving the dynamics problems of structurally-inhomogeneous shell structures

On the basis of the principle of modularity, an algorithm and software package for dynamic calculation of structurally-inhomogeneous multiply connected shell units that are the elements of hydro-technical structures are developed in the paper. The purpose of the module is to carry out certain transformations of the initial (source) information into a unique result. The development prospects of construction of various hydro-technical structures, aircraft, rocket and space machinery, shipbuilding, chemical engineering and many other branches of modern technology are characterized by a complication of design decisions in projecting and calculation of objects that present multiply connected spatial shell structures subjected to static and dynamic effects. Multiply connected structures, (such as hydro-technical structures), usually include a wide range of elastic and visco-elastic deformable elements with significantly different rheological properties in the form of blocks of plate stacks, tubular, cylindrical and prismatic shell structures of complex geometry, with attached masses, various types of reinforcement, supports, containing a large number of elastic and visco-elastic couplings with significantly different functions of heredity. Obviously, the search for exact analytical solutions to dynamic problems for structurally inhomogeneous visco-elastic shell structures is doomed to failure, the only possibility of engineering implementation is to build numerical and numerical-analytical algorithms with the subsequent use of modern computers. The solution to dynamic problems involves a numerical simulation of the stress-strain state (SSS) of the structures under consideration.

Regularization of the electrostatics problem for three spheres and an electrostatic charge

A numerical-analytical algorithm for investigation of the potential of a sphere with a circular hole, surrounded by external and internal closed ribbon spheres, is constructed. The number of ribbons on the spheres is arbitrary. The ribbons on the spheres are separated by non-conductive, infinitely thin partitions. The partitions are located in planes parallel to the shear plane of the sphere with a hole. Each ribbon has its own independent potential. An electrostatic charge is placed between the outer sphere and the sphere with a hole in the axis of the structure. The full potential must satisfy, in particular, Maxwell’s equations, taking into account the absence of magnetic fields, satisfy the boundary conditions, have the required singularity at the point where the charge is placed. To solve this problem, we first used the method of partial domains and the method of separating variables in a spherical coordinate system. In this case, for the Fourier series, we use power functions and Legendre polynomials of integer orders. From the boundary conditions, using an auxiliary system of 3 equations with 4 unknowns, a pairwise system of functional equations of the first kind with respect to the coefficients of the Fourier series is obtained. The system is not effective for solving by direct methods. The method of inversion of the Volterra integral operator and semi-inversion of the matrix operators of the Dirichlet problem for the Laplace equation are applied. The method is based on the ideas of the analytical method of the Riemann - Hilbert problem. In this case, integral representations for the Legendre polynomials are used. A system of linear algebraic equations of the second kind with a compact matrix operator in the Hilbert space l`2 is obtained. The system is effectively solvable numerically for arbitrary parameters of the problem and analytically for the limiting parameters of the problem. Particular variants of the problem are considered.

Transverse Oscillations of a Cantilever Rod of Rectangular Cross Section and Variable Thickness

A numerical-analytical algorithm for calculating (a type of accelerated convergence method) of eigenvalues and modes of transverse vibrations of a rod of variable cross-section is proposed. The algorithm is implemented as a software package in the Maple symbolic computations language. The use of the package was demonstrated by the example of solving problems of determining the eigen forms and natural frequencies of rods with different dependences of the cross section on the longitudinal distance.

Piecewise Polynomial Algorithms for the Analysis of Processes in Inhomogeneous Media

The authors propose, theoretically substantiate, and programmatically implement high-precision numerical-analytical algorithms for approximation of problems solutions in inhomogeneous media on the basis of linear polynomial operators.

Numerical Analysis of the Elastic-Plastic Boundaries in the Thermal Stresses Theory Frameworks

The present study is devoted to the problem of the thermal stresses theory for a hollow thermoelastic-plastic ball under unsteady heating. Throughout the paper the conventional elastic-plastic Prandt-Reuss model generalized on the thermal effects is used with the Tresca yield criterion linearly depending on temperature. A numerical-analytical algorithm for calculating the elastic-plastic boundaries positions and irreversible deformations is proposed. The strain-stress states during the plastic flow, repeated plastic flow and unloading are analyzed. The strain-stress state parameters like residual stresses and displacements are computed and analyzed.

Determination of Plane Stress-Strain States of the Plates on the Basis of the Three-Dimensional Theory of Elasticity

We construct a theory of plates loaded only on their sides by loads parallel and symmetric to the median surface. We use the general representation of three-dimensional stress-strain state and exactly satisfy the trivial boundary conditions on the plane surfaces of the plate. The numerical analysis of the three-dimensional stressed state of the plate is reduced to the determination of its two-dimensional state under the assumption that the normal stresses perpendicular to the median surface are negligible. The displacements and stresses are expressed via two two-dimensional harmonic functions. We obtain the homogeneous solutions and develop a numerical-analytic algorithm for the solution of the boundary-value problems posed for rectangular plates.

Estimates of near-circular orbits after a single correction: a geometrical method

Estimates of the maneuvers of active space objects are considered. We propose analytical and numerical-analytical algorithms to estimate short-term and long-term one-impulse maneuvers for the case where the initial and final orbits are determined with errors. Both coplanar and noncoplanar maneuvers are considered. Special attention is given to the velocity and reliability of the solution of the problem. The process to find the solution has a geometrical interpretation. We provide examples of estimates for maneuvers of spacecraft located at geosynchronous orbits. The results obtained by the proposed method are compared with the results obtained by the traditional approach, excluding errors of orbit determination.

Stabilization of nonlinear discrete-time dynamic control systems with a parameter and state-dependent coefficients

A numerical-analytical algorithm for designing nonlinear stabilizing regulators for the class of nonlinear discrete-time control systems is proposed that significantly reduces computational costs. The resulting regulator is suboptimal with respect to the constructed quadratic functional with state-dependent coefficients. The conditions for the stability of the closed-loop system are established, and a stability result is stated. Numerical results are presented showing that the nonlinear regulator designed is superior to the linear one with respect to both nonlinear and standard time-invariant cost functionals. An example demonstrates that the closed-loop system is uniformly asymptotically stable.

Characteristic analysis on train-induced vibration responses of rigid-frame RC viaducts

A three-dimensional (3D) numerical analysis for the train-bridge interaction (TBI) system is actively developed in this study in order to investigate the vibration characteristics of rigid-frame reinforced concrete (RC) viaducts in both vertical and lateral directions respectively induced by running high-speed trains. An analytical model of the TBI system is established, in which the high-speed train is described by multi-DOFs vibration system and the rigid-frame RC viaduct is modeled with 3D beam elements. The simulated track irregularities are taken as system excitations. The numerical analytical algorithm is established based on the coupled vibration equations of the TBI system and verified through the detailed comparative study between the computation and testing. The vibration responses of the viaducts such as accelerations, displacements, reaction forces of pier bottoms as well as their amplitudes with train speeds are calculated in detail for both vertical and lateral directions, respectively. The frequency characteristics are further clarified through Fourier spectral analysis and 1/3 octave band spectral analysis. This study is intended to provide not only a simulation approach and evaluation tool for the train-induced vibrations upon the rigid-frame RC viaducts, but also instructive information on the vibration mitigation of the high-speed railway.

Plane Problem of Magnetoelasticity for a Piezomagnetic Medium with Cracks

We solve a boundary-value problem of magnetoelasticity for a piezomagnetic plane weakened by cracks. For this purpose, we generalize the method aimed at the solution of similar problems for anisotropic media. The boundary-value problem is reduced to a matrix singular integral equation and the solution of this equation is found in the class of vector functions unbounded at the ends of the notches. The numerical solution of this equation is obtained by the method of mechanical quadratures. The proposed numerical-analytic algorithm is used to investigate the influence of magnetoelastic fields on the stress intensity factors in the vicinity of the crack tips.

Determination of IVC breakpoint for josephson junction stack. Non-periodic boundary conditions with γ = 1

We prove that, in the case of non-periodic (with γ = 1) boundary conditions, the calculation of the current-voltage characteristic (IVC) for a stack of n intrinsic Josephson junctions reduces to solving a system of [(n + 1)/2] non-linear differential equations instead of the n original ones. The current voltage characteristic V(I) has the shape of a hysteresis loop. On the back branch of the loop V(I) decreases to zero rapidly near the breakpoint Ib. We succeeded to derive an algorithm determining the approximate breakpoint location and to improve simultaneously the mixed numerical-analytical algorithm of IVC calculation for a stack of Josephson junctions developed by us before. The efficiency of the improved algorithm is shown by the calculations of IVC for stacks consisting of various numbers of intrinsic Josephson junctions.