An existence result for a class of shape optimization problems

Given a bounded open subset of R ⁿ, we prove the existence of a minimum point for a functional F defined on the family A() of all quasiopen subsets of , under the assumption that F is decreasing with respect to set inclusion and that F is lower semicontinuous on A() with respect to a suitable topology, related to the resolvents of the Laplace operator with Dirichlet boundary condition. Applications are given to the existence of sets of prescribed volume with minimal k ^th eigenvalue (or with minimal capacity) with respect to a given elliptic operator.     

Synthesis of a controller for stabilizing the motion of a rigid body about a fixed point

A method for the approximate design of an optimal controller for stabilizing the motion of a rigid body about a fixed point is considered. It is assumed that rigid body motion is nearly the motion in the classical Lagrange case. The method is based on the common use of the Bellman dynamic programming principle and the averagingmethod. The latter is used to solve theHamilton–Jacobi–Bellman equation approximately, which permits synthesizing the controller. The proposed method for controller design can be used in many problems close to the problem of motion of the Lagrange top (the motion of a rigid body in the atmosphere, the motion of a rigid body fastened to a cable in deployment of the orbital cable system, etc.).     

On the rotational motion of a gyrostat about a fixed point with mass distribution

The perturbed rotational motion of a gyrostat about a fixed point with mass distribution near to Lagrange’s case is investigated. The gyrostat is subjected under the influence of a variable restoring moment vector, a perturbing moment vector, and a third component of a gyrostatic moment vector. It is assumed that the angular velocity of the gyrostat is sufficiently large, its direction is close to the axis of dynamic symmetry, and the perturbing moments are small as compared to the restoring ones. These assumptions permit us to introduce a small parameter. Averaged systems of the equations of motion in the first and second approximations are obtained. Also, the evolution of the precession angle up to the second approximation is determined. The graphical representations of the nutation and precession angles are presented to describe the motion at any time.     

Hybrid projection method for generalized mixed equilibrium problems,variational inequality problems,and fixed point problems in Banach spaces

A new hybrid projection iterative scheme is introduced to approximate a common element of the solution set of a generalized mixed equilibrium problem, the solution set of a variational inequality problem, and the set of fixed points of a relatively weak nonexpansive mapping in the Banach spaces. The obtained results generalize and improve the recent results announced by many other authors.     

Averaging of initial-value and boundary-value problems for one class of oscillatory impulsive systems

We give a new classification of fixed times of pulse action (uniform, functional, limiting, and quantitatively limiting). Several results obtained earlier for oscillatory systems with uniform and limiting times of pulse action are generalized to similar systems with functional times of pulse action. Namely, we obtain estimates, which are exact with respect to a small parameter ε, for the deviation of solutions and their partial derivatives for original and averaged initial-value, boundary-value, and multipoint problems. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 68–84, January–March, 2006.     

The evolution of a rotational motion of an elastic- rigid body about a fixed point in a gravitational field

The rotational motion of a rigid dynamical system including a three dimensional elastic part about a fixed point is studied. The system is under the action of a general gravitational field. The stability anlysis is carried out by expanding the displacement of the deformable part in terms of orthonormalized functions with the independent coefficients. Results are obtained easily based on Routh's equation and Andoyer variables. The system is found to be stable provided that the dissipation energy is minimum corresponds to the case of axially symmetric deformation. On the other hand, the system is unstable, when the angular momentum vector is parallel to the equatorial plane.     

Mathematical models for soil displacement under a rigid wheel

Soil–wheel interaction especially soil deformation caused by the wheel motion was investigated experimentally using a sophisticated soil bin test apparatus and an on-line measurement system for soil displacement. Based on these test results, characteristics of soil deformation were summarized focusing on the behavior and distribution of displacement increment vectors. Mathematical models were examined in order to describe the displacements of soil particles. Properties of the displacement loci are described. The magnitude of the displacement increment vector, its horizontal and vertical components are discussed, and characteristics of these distributions with respect to the relative horizontal distance from the vertical centerline of the wheel to the target point are clarified. Shapes of these distribution curves were closely similar to those of the derivatives of a Gaussian function. A distribution curve of the horizontal displacement increment had two peaks and that of the vertical one had three peaks. Based on the results, mathematical models for those displacement increments were proposed by employing a Gaussian function through multiplication of a linear function and a quadratic function. Predicted distributions and displacement loci of the models agreed with high accuracy to the measured values. The mathematical models were extended taking into consideration the wheel slip. The predicted distributions according to test conditions agreed very well to the measured results.     

On the flow between a rotating and a porous fixed disk with suction

Numerical self-similar solutions are reported for the laminar, incompressible flow between a rotating disk and a porous, fixed one with suction. Validation of the method is obtained through the numerical integration of the full Navier-Stokes equations applied to a reference radially confined geometry, and also with hot-wire measurements of the tangential velocity component. The flow structure is analysed for different values of the rotational and suction Reynolds numbers. It is shown that suction causes an important angular acceleration of the rotating core, whose velocity may thus considerably exceed that of the rotating disk. The physical reason for this unusual behavior is discussed in detail.     

On the trajectories of a mass point on a rotating surface

N. E. Zhukovskii proposed a special method [1] for determining the trajectories of amass point (particle) on a plane in the field of potential forces. In [2], this method was generalized to the case of particle motions on curvilinear surfaces. In the present paper, this method is generalized to the case in which the surface, together with the field of potential forces, rotates about a fixed axis. Several examples are considered.