and Computational Methods

#### 539.3 Modelling of torsional vibrations of the viscoelastic round bar rotating with the constant angular velocity

**Abdirashidov A. (Samarkand State University)**

doi: 10.18698/2309-3684-2016-1-3851

The purpose of this article is to deduce general and approximate equations for the torsional vibration of the viscoelastic round bar rotating around the symmetry axis with the constant angular velocity. Within the research we develop the algorithm allowing to define the bar deflected mode. The received approximate equations enabled to numerically solve the problem of the bar torsional vibrations. Moreover, we carry out a comparative analysis of the results obtained for exponential and weakly singular kernels of the viscoelastic operator. As a result, we estimate the rotation influence on the bar vibrations

Khudoynazarov K., Abdirashidov A., Burkutboyev . Torsional vibrations of the viscoelastic round bar rotating with the constant angular velocity. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 38-51

#### 517.9:539.3:519.6 Numerical simulation of absolutely flexible bAR motion in the air flow

**Sorokin F. D. (Bauman Moscow State Technical University/Keldysh Institute of Applied Mathematics of the Russian Academy of Scienсes), Nizametdinov F. R. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2016-1-316

The article offers the calculation algorithm of deflected mode of an absolutely flexible bar interacting with the external air flow. The algorithm is based on the replacement of the continual mechanical system by the discrete set of rectilinear finite elements and concentrated masses. The authors show differential equations of mass motion with allowance for an aerodynamic load and dissipative forces and integrate them by numerical method. That made it possible to find both the equilibrium position of the flexible bar in the flow, and the critical flow velocity which causes violent bar vibrations in case of its excess.

Sorokin F., Nizametdinov F. Numerical simulation of absolutely flexible bAR motion in the air flow. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 3-16

#### 539.3 Mathematical modeling of massive tire stationary rolling on the chassis dynamometer with regard to energy dissipation in rubber

**Belkin A. E. (Bauman Moscow State Technical University), Semenov V. K. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2016-1-1737

The article examines the problem of mathematical modeling tests of massive tire bench run with the chassis dynamometer. Conducted tests enable to define the characteristics of resistance to the tire rolling. The article contains the main stages of model building. We give a formulation for the contact problem of tire stationary free rolling on the test drum considering the energy dissipation in the rubber during cyclic deformation. We also describe a rubber viscoelastic behavior by the model Bergstrom – Boyce and ascertain its numerical parameters according to the samples tests results. The contact conditions for normal and tangential directions are formulated on basis of the penetration function. For the contact restrictions implementation we use the penalty method and obtain the numerical solution of the three-dimensional viscoelasticity problem by the finite element method. To estimate the adequacy of the built model, we compare the calculation results with the test data received for massive tire on Hasbach test equipment. For this purpose rolling resistance forces under different loads were collated. The pressure distribution in the contact area obtained from calculations and experiments by using XSENSOR Technology Corporation equipment are also juxtaposed.

Belkin A., Semenov V. Mathematical modeling of massive tire stationary rolling on the chassis dynamometer with regard to energy dissipation in rubber. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 17-37

#### 539.3 Mathematical modeling of shock loading of the meniscus liner

**Asmolovskyy N. A. (Bauman Moscow State Technical University), Baskakov V. D. (Bauman Moscow State Technical University), Boyarskaya R. V. (Bauman Moscow State Technical University), Zarubina O. V. (Bauman Moscow State Technical University), Tarasov V. A. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2016-1-5267

Mathematical modeling of the high-velocity element formation process based on finite element method is presented. The proposed approach enables considering geometrical imperfections of the explosive device. The article contains general description of the proposed mathematical model including corresponding numerical algorithms. Selection of the suitable finite element formation is performed. Practical application of the proposed method is illustrated on the example of analysis of the imperfection influence on the kinematical and geometrical parameters of the elements.

Asmolovskyy N., Baskakov V., Boyarskaya R., Zarubina O., Tarasov V. Mathematical modeling of shock loading of the meniscus liner. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 52-67

#### 629.762 Aircraft motion parameters recovery from the data of their discrete registration. Part 1. Methods without use of regularization

**Plyusnin A. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2016-1-6888

The article examines methods of aircraft motion parameters recovery from the data of their low resolution recordings in the gas-dynamic ejection experimental test. Desired conditions were satisfied by the use of Hermitian piecewise polynomial interpolation. Implementation of Tikhonov regularization provides the most flexible approach to the problem under consideration.

Plyusnin A. Aircraft motion parameters recovery from the data of their discrete registration. Part 1. Methods without use of regularization. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 68-88

#### 519.8 Models of bilateral warfare of numerous groups

**Chuev V. U. (Bauman Moscow State Technical University), Dubogray I. V. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2016-1-89104

Based on the theory of Markov processes the model of "poorly organized" battle was developed. Formulae for calculating its basic parameters at different initial numbers of the opposing sides were obtained. A comparison of the results of modeling a battle using probabilistic and deterministic models was performed. It was found that the dynamics model errors of the average are primarily affected by the balance of forces of the opposing sides in the beginning of the battle. It was shown that in case of military groups of similar forces the first-strike attack is of significant importance. When one of the warring parties at the beginning of the battle has a great advantage, the influence of first-strike attack is negligible. An increase in the influence of first-strike attack on the expected losses of a strong hand, and a reduction of its impact on the expected losses of the weaker party, as the number of groups involved in the fight increases proportionally, is also shown.

Chuev V., Dubogray I. Models of bilateral warfare of numerous groups. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 89-104

#### 539.3+519.86 A model of multidimensional deformable continuum for forecasting the dynamics of large scale array of individual data

**Dimitrienko Y. I. (Bauman Moscow State Technical University), Dimitrienko O. Y. (Bauman Moscow State Technical University)**

doi: 10.18698/2309-3684-2016-1-105122

The article considers the concept of applying the multidimensional continuum model to one of the main problems emerging in the theory of large scale data array treatment i.e. forecasting the dynamics of data cluster change. The concept is based on the model of multidimensional continua in spaces of high dimensionality (more than three) earlier developed by the authors. The model includes the integral conservation laws, which are reformulated for informational data clusters, as well as the model of motion kinematics and cluster deformation. The model of deformable multidimensional cluster is developed. The movement of the cluster in multidimensional data space includes translational and rotational motion and uniform tension-compression strain. The system of differential tensor equations describing the dynamics of the deformable multivariate cluster motion over time is formulated. A numerical algorithm for solving the system of differential equations for the ellipsoidal model of multidimensional cluster is worked out. An example of the developed model application for predicting the dynamics of economic data (data on goods purchases in a large supermarket) is considered. The results of forecasting the data on purchases of different consumer groups are shown.

Dimitrienko Y., Dimitrienko O. A model of multidimensional deformable continuum for forecasting the dynamics of large scale array of individual data. Маthematical Modeling and Coтputational Methods, 2016, №1 (9), pp. 105-122