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Том 221, № 2 (2017)
- Год: 2017
- Статей: 8
- URL: https://ogarev-online.ru/1072-3374/issue/view/14809
Article
Sessions of the Workshop of the Mathematics and Mechanics Department of Lomonosov Moscow State University, “Urgent Problems of Geometry and Mechanics” Named After V. V. Trofimov
155-160
Methods of Mathematical Modeling of the Action of a Medium on a Conical Body
Аннотация
We consider a mathematical model of a plane-parallel action of a medium on a rigid body whose surface has a part which is a circular cone. We present a complete system of equations of motion under the quasi-stationarity conditions. The dynamical part of equations of motion form an independent system that possesses an independent second-order subsystem on a two-dimensional cylinder. We obtain an infinite family of phase portraits on the phase cylinder of quasi-velocities corresponding to the presence in the system of only a nonconservative pair of forces.
161-168
On Stabilization of Potential Systems by Circular Forces
Аннотация
In this paper, we consider a linear potential system possessing an arbitrary number of negative stability coefficient and solve the problem on stabilization (until a stable state) of a nonstable potential system by circular forces. We obtain stability conditions in terms of the initial system that generalize results obtained earlier for the case of a single negative stability coefficient. We also examine the action of dissipative forces.
169-173
Eigenvalue Problem for Tensors of Even Rank and its Applications in Mechanics
Аннотация
In this paper, we consider the eigenvalue problem for a tensor of arbitrary even rank. In this connection, we state definitions and theorems related to the tensors of moduli ℂ2p(Ω) and ℝ2p(Ω), where p is an arbitrary natural number and Ω is a domain of the n-dimensional Riemannian space ℝn. We introduce the notions of minor tensors and extended minor tensors of rank (2ps) and order s, the corresponding notions of cofactor tensors and extended cofactor tensors of rank (2ps) and order (N−s), and also the cofactor tensors and extended cofactor tensors of rank 2p(N−s) and order s for rank-(2p) tensor. We present formulas for calculation of these tensors through their components and prove the Laplace theorem on the expansion of the determinant of a rank-(2p) tensor by using the minor and cofactor tensors. We also obtain formulas for the classical invariants of a rank-(2p) tensor through minor and cofactor tensors and through first invariants of degrees of a rank-(2p) tensor and the inverse formulas. A complete orthonormal system of eigentensors for a rank-(2p) tensor is constructed. Canonical representations for the specific strain energy and determining relations are obtained. A classification of anisotropic linear micropolar media with a symmetry center is proposed. Eigenvalues and eigentensors for tensors of elastic moduli for micropolar isotropic and orthotropic materials are calculated.
174-204
New Cases of Integrability of Equations of Motion of a Rigid Body in the n-Dimensional Space
205-259
Some Problems of Qualitative Analysis in the Modeling of the Motion of Rigid Bodies in Resistive Media
Аннотация
In this paper, we present a qualitative analysis of plane-parallel and spatial problems on the motion of realistic rigid bodies in a resistive medium and construct a nonlinear model of the influence of the medium on the rigid body.
260-296
On the Asymptotic Expansion of Solutions to the Goursat Problem for the Sine-Gordon Equation
Аннотация
We construct an expansion of a solution to the Goursat problem for the sine-Gordon equation with respect to a small parameter that is involved in the data on characteristics. We prove that the expansion obtained is an asymptotic series.
297-303
On Extremal Properties of Mean Values of Continuous Random Variables and Relations Between Them
Аннотация
In this paper, we introduce the notions of separating value and tantiles of continuous random variables and examine their extremal properties and extremal properties of quantiles. We obtain estimates for the differences between expected values, medians, and separating values of continuous random variables.
304-312