ISSN 1991-2927

## Author: "Petr Alexandrovich Velmisov"

 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.

 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.

 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.

 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.

 P. Velmisov, Y. Tamarova
 Mathematical Modeling of Transonic Flows The article is devoted to the development of the mathematical theory of gas flow with a speed close to the speed of sound, namely transonic gas flows, i.e. flows that contain both subsonic and supersonic area. The main problems arising in the study of such flows should be classified as non-linearity and mixed type equations describing transonic flow. Transonic gas flows taking into account the transverse perturbations are studied on the basis of nonlinear equation obtained in this paper. Some exact particular solutions of this equation are constructed and their application to solving a number of transonic aerodynamics problems are shown. In particular, a solution of polynomial form describing axisymmetric gas flow in Laval nozzles with constant acceleration and flow swirling is obtained. The unsteady flows in the channels between the rotating planes are researched. The partial solutions are shown and the examples of steady flows are constructed on their basis. The asymptotic equation describing the flows arising at the unseparated and separated flow past the body that is practically identical to the cylindrical one is derived.Aerodynamics, transonic gas flows, partial differential equations, asymptotic expansion.
 2014_ 1

Sections: Mathematical modeling

Subjects: Mathematical modeling.

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