ISSN 1991-2927

## Keyword: "stability"

 The study of the stability of aeroelastic systems by linear approximation This paper proposes an approximate method of studying the stability of solutions of nonlinear differential equations with partial derivatives, describing the dynamics of one class of aeroelastic systems. Based on the Galerkin (Krylov-Kantorovich) method, when presenting the desired deformations in the form (formula), the study of problems for differential equations with partial derivatives for w(x,t) reduces to the study of problems for nonlinear systems of ordinary differential equations for functions (formula) . It is shown that the structure of these systems allows using the known stability theorems by linear approximation. Thus, the study of the dynamic stability of aeroelastic systems is reduced to the study of uniform problems combined by a common approach to solving them. As examples, vibration stability conditions of a pipeline with liquid flowing inside it and a plate streamlined by a supersonic gas flow are obtained. Pipeline, plate, deformation, dynamics, aerohydroelasticity, stability, nonlinear model, partial differential equations.
 2020_ 3

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 N.V. Dorofeev A.V. Grecheneva E.S. Pankina R.V. Romanov
 Simulation and bifurcation analysis of geotechnical system stability under vibration The automation of control processes for the stability of geotechnical systems is a great challenge involving the development of methods for multivariate analyzing and forecasting the stability with the subject to the nonlinearity of material stiffness parameters. The aim of the study is to improve the efficiency of automated control systems for geotechnical stability by developing an approach to detect negative changes in bifurcation diagrams of vibration displacement parameters of object structures. The authors present a mathematical model of the dynamic behavior of structural elements of an object as an elementary unit of a geotechnical system that describes a response to an external vibration action. An algorithm of bifurcation analysis is presented, which allows authors to determine the initial transition stage of the object structure to an unstable state by the acceleration values of forced oscillations exceeding the model parameters. A bifurcation diagram of stability changes in the structure of object at the displacement resulting from the load increase under vibration action has been constructed. This diagram, which type of codimension-one bifurcations is merging, enabled to determine the critical load values resulting in an unstable state transition of a system due to the influence of a combination of vibration factors. The efficiency evaluation of the proposed approach was carried out by the comparison with the results of construction stability calculations obtained by the use of the dynamic coefficient. The difference between the values of the maximum object displacement without loss of the stability under vibration action, obtained by the standard calculation method and using the developed model, is 32.5%, and it is significant in the theory of structure stability. When exogenous vibration noise is used as a source of a sounding signal, the application of the developed approach in automated control systems for geotechnical stability enable to change the permissible stability thresholds of objects being exploited depending on the level and combination of influencing factors. Geotechnical system, automated monitoring, stability, nonlinear dynamics model, vibration action, bifurcation diagram.
 2020_ 3

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 Mathematical Modeling Of Nonlinear Dynamics Of the Pipeline In the work, various nonlinear mathematical models are proposed that describe pipeline vibrations. Models are presented that take into account only the transverse deformation of the pipeline, and models that take into account the longitudinal and transverse deformations. All models are described by partial differential equations for unknown strain functions. The numerical-analytical solution is based on the Bubnov-Galerkin method. Based on model equations, a numerical experiment was conducted for various parameters of a mechanical system, which showed a small difference in dynamic characteristics for different models. A functional of the Lyapunov type was constructed for one of the models, on the basis of which the analytical conditions for the dynamic stability of the pipeline were obtained. Also, mathematical models of the dynamics of the pipeline are proposed taking into account the delay of power and (or) inertial characteristics.Pipeline, deformation, dynamics, stability, nonlinear model, partial differential equations, Lyapunov functional method.
 2019_ 3

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 V. Kliachkin, I. Karpunina, M. Fedorova
 Evaluation of the Computer Temperature Regime Stability The temperature regime significantly affects the durability of the computer. Ensuring stability of the computer functioning considers the stability of the heating temperature of the main elements that should not exceed the specified values. The article discusses issues related to the timely warning about a possible violation of the temperature regime stability. In order to diagnose stability, the multivariate statistical control methods are proposed to use. Evaluation of stability is carried out with the use of two criteria: stability of the temperature average level and their dispersion. Independent parameters can be controlled with the use of standard shewhart charts. The algorithms on the basis of Hotelling statistics (for assessing stability of the middle-level temperature measurement process) and generalized variance (for evaluation of the dispersion process stability) are used for correlated parameters. The efficiency of these algorithms can be enhanced through the analysis of non-random structures on the control charts, the use of the warning border, as well as application of modifications based on the cumulative amounts or moving averages weighted exponentially. The multivariate statistical technique of the computer temperature regime control including monitoring in the context of a well-developed process for a training sample in order to separate the parameters controlled for a group of independent and correlated ones, process analysis for assessment of control characteristics and continuous monitoring of the process with the construction of Hotelling charts and generalized dispersion with identifying possible violations of the process based on the presence of significant structures and the use of a warning border. This technique is illustrated by the example of five computer temperature regime parameters control.Stability, temperature, hotelling algorithm, warning limit, generalized variance, control chart.
 2016_ 3

Sections: Mathematical modeling

Subjects: Mathematical modeling, Automated control systems, Electrical engineering and electronics.

 A. Ivanov
 Optimization of Hierarchical Control System Stability Designing of hierarchical control systems with optimal stability indices in condition of price limitation is considered in the article. The formal description of hierarchical system structure and functioning algorithm was made. The choice of optimal solution comes to the operational research on the basis of analytic dependencies between system stability indices and objects stability ones. In order to plot the analytic dependence, the authors propose to approximate the experimental data got in simulation process. The simulation model was made with the use of Monte-Carlo method. The paper proposes approximate formula models including orthogonal transformation reducing given dependencies to simple approximate forms. The possibility of direct application of scarce resources distribution search with the increasing performance of computer aids for the purpose of optimizing stability indices of the system is demonstrated. On the base of simulation, optimizing software has been developed. Also the article presents the calculation results that prove reasonability of direct application of simulation models in systems of special class.Stability, hierarchical computer-aided control systems, optimal design, simulation.
 2015_ 3

Sections: Automated control systems

Subjects: Automated control systems.

 P. Velmisov, S. Kireev
 Numerical Method for Solving a Class of Nonlinear Boundary Value Problems of Aerohydroelastity On the basis of nonlinear models proposed and a numerical method developed for solving corresponding boundary value problems in nonlinear integro-differential equations static instability (divergence) of the pipeline with the fluid flowing in it is investigated. A numerical method for solving the bifurcation problem includes the Runge-Kutta 6-th order method with the error control at each step, Newton's method for solving nonlinear equations and integration with the use of Newton-Kotesa quadrature formulas. Solving the boundary value problem is reduced to solving a Cauchy problem. The complexity of a Cauchy problem is that there is an integral term in the equation. Calculation of this term needs values of the whole integration interval. It makes the direct application of the Runge-Kutta method impossible. To solve this problem (integration) a special iterative process was developed. Numerical realization is provided with the use of a program written in Delphi 7. Bifurcation diagrams showing the dependence between maximal element bending and the inflow velocity are obtained. Moreover, the element’s forms of deflection are specified. The obtained numerical solutions were compared with analytical ones.Stability, divergence, elastic element, pipeline, nonlinear model, differential equations, boundary value problem, mathematical modeling, numerical method.
 2015_ 1

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 P. Velmisov, A. Korneev
 Mathematical Modeling in the Problem of Dynamic Stability of a Pipeline The paper presents mathematical models for a viscoelastic pipeline that is a hollow rod containing flowing the fluid (gas). The article is devoted to the problem of the dynamic stability of a pipeline. Linear and non-linear models describe partial differential equations for an unknown function (the displacement of the pipeline points from the equilibrium state). By means of Lyapunov functionals designed stability theorems are formulated and analytical stability conditions for the parameters of the mechanical system and different types of initial conditions are found. The obtained stability conditions are sufficient but not necessary. A mathematical software package is developed to solve this problem. It allows to find an approximate numerical solution of differential equation for describing pipeline vibration and to plot a stability area appropriate to both sufficient and necessary stability conditions. A numerical experiment of stability areas designing is conducted on the basis of the software package. The obtained numerical results are interpreted and compared with analytical stability conditions. The influence of the model parameters variation on the stability is researched.Mathematical modeling, viscoelastic pipeline, aerohydroelasticity, stability, functional, partial differential equations, numerical methods, galerkin method.
 2015_ 1

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 A. Ankilov, P. Velmisov
 Dynamics and Stability of Elastic Aileron of Aircraft Wing in Subsonic Streamline A mathematical model of a wing with the aileron blown by a subsonic flow of the ideal gas (fluid) is offered. It is supposed that the wing is absolutely rigid, and the aileron is elastic. Dynamics and dynamic stability of the aileron are investigated. The model is described by a related system of partial differential equations for two unknown functions: the potential of velocity of gas coming on the wing and deformations of the elastic aileron. Based on the methods of the theory of functions of complex variable, the potential of velocity is expelled from the system of equations and the solution of a problem of aerohydroelasticity is consolidated to a research of the integro-differential equation containing only an unknown function of deformation of the elastic aileron. It is supposed that thickness of the elastic aileron is a variable that results in a system of equations with variable coefficients. A study of stability is performed on basis of creation of positively certain functionality corresponding to the received partial integro-differential equation. The stability conditions imposing restrictions on velocity of the incoming flow, thickness, flexural rigidity of the aileron and on other parameters of the mechanical system are received. The solution of the specified integro-differential equation for function of deformation of an element relies on basis of the Galerkin method with carrying out a numerical experiment.Aerohydroelasticity, stability, dynamics, elastic element, wing, aileron, subsonic flow.
 2014_ 3

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 A. Ankilov, P. Velmisov, Y. Tamarova
 The Dynamical Stability of an Elastic Element of the Flow Channel A mathematical model of a device pertaining to the vibration equipment which is intended for intensification of technological processes, for example, stirring process, is offered. Operation of such devices is based on fluctuations of elastic elements blown by a flow of gas or fluid. The dynamic stability of an elastic element placed on one of the walls of the channel where a subsonic flow of gas or fluid blows (in a model of ideal compressible medium) is investigated. The model is described by a related system of partial differential equations for two unknown functions: the potential of velocity of gas or fluid and deformation of the elastic element. The problem is investigated in a linear statement corresponding to small perturbations of the flow in the channel and small deformations of the elastic element. Determination of the stability of the elastic body corresponds to a concept of stability of dynamical systems by Lyapunov. Based on building of the mixed functional, sufficient conditions of stability are obtained. Conditions impose limitations on the speed of the uniform gas flow, compressed (tensile) efforts of the element, elastic element stiffness and other parameters of the mechanical system. Examples of building stability areas for concrete parameters of the mechanical system are given.Aerohydroelasticity, stability, dynamics, channel, elastic plate, deformation, subsonic flow.
 2014_ 3

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 A. Moiseev, V. Kalnikov
 Analysis of the State of a Distributed Control System Abstract The article examines an issue about how to estimate the current state of distributed control systems with the analytical models obtained as for indices of secrecy, information awareness and stability. When estimating the secrecy of control, the emission time and the amplitude of the signal of a give-away sign are taken into account. It is suggested to calculate the information awareness of operations posts with the weapons under their control using the entropic approach. The obtained results of the mathematical expression have the parameters for the required observation accuracy, errors, acquisition range, and the information update rate included. We have developed statistical and dynamic variants for such a characteristic of a distributed control system. It is suggested to analyze the control stability along with the estimation of survivability, reliability and noise-resistance of operations posts. The average number of data transmission paths, bandwidth ratio and its assignment evenness, the operations posts’ average load and its assignment evenness are chosen as the private parameters for the control survivability. A recurrent algorithm for the appropriate graph conversion and analysis is suggested to estimate the system reliability. The control noise-resistance is represented as a quantity derivative of the noise probability and the probability of an operation post to keep its performance ability.State, distributed control systems, combat readiness, secrecy, information awareness, stability, estimation.
 2014_ 1

Sections: Automated control systems

Subjects: Automated control systems, Architecture of ship's system.

 P. Velmisov, S. Kireev
 Mathematical Modelling in Instability Problems of Elastic Structural Elements in Gas Flow On the basis of the proposed non-linear models and developed numerical method for the solution to the corresponding non-linear boundary-value problems, the static instability (divergence) of the elastic element of the design streamlined and supersonic flow of ideal gas is investigated. A numerical procedure for the bifurcation-problem solution includes the 6-th order Runge-Kutta method with the error control at the step, the Newton's method required for solving non-linear equations, and integration using Newton-Kotesa quadrature. The solution to the boundary-value problem is reduced to the Cauchy problem solution, the complexity of which is that the integral term is present in the equation. In order to calculate this integral term the values of the integrand function on the whole interval of integration are required. It makes impossible the direct application of the Runge-Kutta Method. A special iterative process was developed to solve this problem as integral evaluation. Numerical implementation is carried out by the program written in Delphi 7. Bifurcation diagrams are given that showing the maximal element dependence on incident stream velocity. Element bending-forms are defined. The comparison of obtained numerical solutions against analytical solutions is carried out. The dynamic stability of the elastic structural element in a supersonic gas flow is researched by the Galerkin’s method. The element bending dependences on time in a fixed point are obtained.Stability, divergence, elastic element, plate, supersonic flow, nonlinear model, differential equations, boundary-value problem, mathematical modelling, numerical method.
 2014_ 1

Sections: Mathematical modeling

Subjects: Mathematical modeling.

 V. Sukhov
 A Way of Modeling of Strength and Vibration-protection Systemsfor Radio Devices Optimization of vibroprotection systems for radio devices is an essential element for increase of reliability and strength. Under permanent increase in prices of similar equipment, modeling of its mechanical strength at all the design stages lets ensure compliance with reliability requirements and replace mechanical trials of devices with modeling.Radio equipment, vibroprotection, device design, modeling, strength, stability.
 2012_ 1

Sections: Mathematical modeling, calculi of approximationsandsoftware systems

Subjects: Mathematical modeling, Electrical engineering and electronics.

 M. Gilvanov
 Calculation of Eigenvalues and Eigenvectors By Jacobi Method At Variable Capacity The article deals with possible increase of rate of convergence of eigenvalue and eigenvector calculation by Jacobi method. The method is based on change of capacity of original variables in case of stable- state approximation. In general, the suggested method is more advantageous with respect to direct implementation.Iteration process, jacobi method, variable capacity, stability.
 2009_ 4

Sections: Theoretical issues of automation of command and control processes

Subjects: Mathematical modeling.

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