Chaos and optimal control of steady-state rotation of a satellite-gyrostat on a circular orbit

The present paper is devoted to discuss both the chaos and optimal control of the steady rotations of a satellite-gyrostat on a circular orbit. In this the satellite is controlled with the help of three independent control moments that are developed by three rotors attached to the satellite principal axes of inertia and rotate with the help of motors rigidly mounted on the satellite body. The optimal controllers that asymptotically stabilize these chaotic rotations and minimize the required like-energy cost are derived as a function of the phase coordinates of the system. The asymptotic stability of the resulting nonlinear system is proved using the Liapunov technique. Numerical study and examples are introduced.     

The global stability of the steady rotations of a solid

The dynamical Euler equations describing the motion of a non-symmetrical solid about the centre of mass in the field of a constant external moment and a dissipative one are considered. It is assumed that the external moment specified with respect to axes attached to the body acts about the intermediate central axis of inertia of the body. The conditions for global asymptotic stability as well as the stability in total of steady rotations of the solid are obtained.     

The dynamics of a rigid body colliding with a rigid surface

Basic investigation techniques, algorithms, and results are presented for nonlinear oscillations and stability of steady rotations and periodic motions of a rigid body, colliding with a rigid surface, in a uniform gravity field.      

The asymptotic properties of the motions of satellites in a central field due to internal dissipation

A satellite in the form of a system of bodies that does not have the property of a gyrostat in the general case is considered. An algorithm for determining all the equilibrium configurations of the system that correspond to steady motions in a central gravitational field and an algorithm for analysing their stability are given. A method based on Routh's first theorem is used to investigate the asymptotic stability of the steady motions in the unconstrained problem. Three effects caused by internal dissipation are established in a model example: stabilization of the satellites in a neighbourhood of rotations about a normal to the orbital plane, which is codirectional with the axis of the largest moment of inertia, evolution of elliptic orbits into circular orbits, and capture of the satellites in resonant oscillatory modes of motion.     

Steady flows of Cosserat–Bingham fluids

The equations describing the steady flow of Cosserat–Bingham fluids are considered, and existence of weak solution is proved for the three‐dimensional boundary‐value problem with the use of the Lipschitz truncation argument. In contrast to the classical Bingham fluid, the micropolar Bingham fluid supports local micro‐rotations and two types of plug zones. Our approach is based on an approximation of the constitutive relation by a generalized Newtonian constitutive relation and a subsequent limiting process. Copyright © 2016 John Wiley & Sons, Ltd.     

Fundamental solutions in the theory of microfluids

This paper is concerned with the theory of non-heat-conducting microfluids. We consider a specialized microfluid continuum theory in which the micromotions consist of the intrinsic rotations and stretch. First, we establish a representation of Galerkin type for the solutions to the field equations. Then, we derive the fundamental solutions for the differential system governing the motion in the case of steady vibrations.      

Stability of steady rotations in the nonholonomic Routh problem

We have discovered a new first integral in the problem of motion of a dynamically symmetric ball, subject to gravity, on the surface of a paraboloid. Using this integral, we have obtained conditions for stability (in the Lyapunov sense) of steady rotations of the ball at the upmost, downmost and saddle point.      

Asymptotic properties of the motions of a heavy gyroscope with internal dissipation

The limiting motions of a heavy gyroscope, simulated by a system of rigid bodies, are considered when there is internal friction. The whole set of limiting motions is determined and the nature of their stability is studied in detail for cases when the carried body of the gyroscope has a) three degrees and b) one degree of freedom with respect to the supporting body. The results of an analysis of case a are extended to the motion of a gyroscope with a fluid filling. For case b, the values of the parameters are determined for which the gyroscope, apart from steady rotations, has unsteady limiting motions that are integrable motions in the special Bobylev-Steklov case.     

多夹层板壳的非线性理论及应用(Ⅰ)──基本理论

本文建立了多夹层壳体小应变状态下的中转动二阶大挠度的理论,接着进行适当的简化,获得了中转动、中小转动的一阶大挠度的理论.     

On the Stability of a Class of Permanent Rotations of a Heavy Asymmetric Gyrostat

We consider the motion of an asymmetric gyrostat under the attraction of a uniform Newtonian field. It is supposed that the center of mass lies along one of the principal axes of inertia, while a rotor spins around a different axis of inertia. For this problem, we obtain the possible permanent rotations, that is, the equilibria of the system. The Lyapunov stability of these permanent rotations is analyzed by means of the Energy–Casimir method and necessary and sufficient conditions are derived, proving that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space. The geometry of the gyrostat and the value of the gyrostatic momentum are relevant in order to get stable permanent rotations. Moreover, it seems that the necessary conditions are also sufficient, but this fact can only be proved partially.     

A class of real cocycles having an analytic coboundary modification

We define here a certain class of procedures (a.a.c.c.p.) for constructing real valued cocycles over irrational rotations. Each such procedure is realizable over a residual set of possible rotations, and we prove that each such cocycle is cohomologous to a real analytic cocycle. The procedure in Section 3 of [10] is seen to be of this type and hence not only is cohomologous toC ^∞ as is shown there, but is actually cohomologous to a real analytic cocycle. We also show that following the method of [6] a procedure can be given to obtain rank-1 Anzai skew products of mixed spectral type that are real analytic. Research supported by KBN grant 512/2/91. Research supported by KBN grant 512/2/91. Research supported by NSF grant DMS 01524351.     

关于Morley元的误差估计

§1.引言 解薄板弯曲问题的三角形Morley元是六十年代出现的一种非协调元,它的形函数是完整的二次多项式,节点参数是单元顶点上的三个函数值及三边中点上的法向导数值.由于板弯曲问题的常应变是二次多项式,所以这是一个参数最少的非协调板元.由     

QR Factorization for Linear Least-Squares Problems on a Hypercube Multiprocessor

Least squares problems occur in many branches of science. Typicallythere may be a large number of data points or observations andonly a small to moderate number of variables. On sequentialmachines these problems can be time-consuming and thereforethe use of parallel machines to solve large least-squares problemsmay well yield substantial savings. The solution of least-squaresproblems by a QR factorization using Givens rotations seemsto be particularly suitable for a parallel machine, becausethere is much choice in the order of the Givens rotations andmany Givens rotations can be carried out in parallel. In this paper, an implementation of a QR factorization on theIntel hypercube is described. Each row of the least-squaresmatrix is assigned to a processor and most of the rotationsinvolve rows within one processor in the usual case when eachprocessor receives several rows. However, it is also necessaryto carry out rotations involving rows in different processorsand we call these rotations merges. Two ways of implementingthe merges are described and they are compared on the groundsof load balance and the number of communications required. Onefeature of the implementations is that processors can continueto do Givens rotations on rows within the processor while waitingfor messages that are required for merges. There is also someflexibility in the order of the merges and this can be incorporatedinto the algorithm. For each column, the merges are carriedout according to a tree structure and the choices of trees andtheir roots are discussed. Numerical results are given to showthe usefulness and efficiency of the proposed algorithms.     

四元数的复数形式及其在6R机器人反解中的应用

把四元数应用于平面旋转的特殊情况,并把旋转角度的正弦、余弦改写为复指数形式,导出平面四元数的两个复数形式的基.利用这两个基可以把表示三维变换的四元数和对偶四元数改造为复数形式,得到复数形式四元数的4个基和复数形式对偶四元数的8个基,以及它们之间乘法运算的运算法则.通过把它们应用到空间$6R$机器人的位移反解问题,证明了Dixon结式展开后的次数为16次,而不是形式上的24次,并且得到单变量的16次方程.     

Fast Givens rotations for orthogonal similarity transformations

Summary Fast Givens rotations with half as many multiplications are proposed for orthogonal similarity transformations and a matrix notation is introduced to describe them more easily. Applications are proposed and numerical results are examined for the Jacobi method, the reduction to Hessenberg form and the QR-algorithm for Hessenberg matrices. It can be seen that in general fast Givens rotations are competitive with Householder reflexions and offer distinct advantages for sparse matrices.     

Metaheuristics approach to the aircrew rostering problem

The solution of the aircrew-scheduling problem is represented by a set of rotations developed from a given set of flight segments. Once the set of rotations to be made by aircrew members has been determined, the air carrier must solve the aircrew rostering problem that entails the monthly assignment of aircrew members to planned rotations. This paper attempts to solve the aircrew rostering problem, thus constructing personalized monthly schedules using Simulated Annealing, Genetic Algorithms, and Tabu Search techniques. The developed models are tested on numerical examples that consist of constructing schedules for pilots. Dimensions of the considered examples are characteristic of small and medium-sized airlines.     

A variable gravity model of the otolith membrane

This paper describes a finite difference model of the otolith membrane which allows the acceleration due to gravity to vary, thus simulating conditions of gravity on the lunar and planetary surfaces.The differential coefficients of the second-order system of elliptic partial differential equations governing the steady state displacements of points of the membrane are replaced by finite differences. The resulting system of difference equations is seen to be consistent with the system of differential equations and to have a truncation error of order four.A close approximation to the physical boundary of a typical otolith membrane is used and two sets of numerical experiments are carried out which which simulate rotations of the membrane on the Moon and on a number of planets.The displacements at thirty nodes of the membrane are computed by solving the linear system of sixty equations obtained by applying the difference equations to each of the thirty nodes. The numerical results obtained are seen to be in general agreement with experimental results reported in the literature.     

The instability of spiky steady states for a competing species model with cross diffusion

This paper is concerned with the stability/instability of a class of positive spiky steady states for a quasi-linear cross-diffusion system describing two-species competition. By detailed spectral analysis, it is proved that the spiky steady states for the related shadow system are linearly unstable and the spiky steady states for the original cross-diffusion system are non-linearly unstable.     

Permanent rotations of a body with a viscous filler suspended on a rod

Problems of the existence, stability, and branching of the permanent rotations of a heavy, dynamically symmetrical rigid body suspended on a rod and which has an axisymmetric ellipsoidal cavity filled with a fluid are discussed. The phenomenological model of the friction of the fluid against the cavity wall proposed by Samsonov is used. All the trivial permanent rotations of the system and the non-trivial rotations that branch off from the trivial ones are found. Their stability and branching are investigated using a modified Routh's theory. The results obtained are presented in the form of an atlas of bifurcation diagrams.     

Complete balanced Howell rotations for 8k+5 teams

Complete balanced Howell rotations are mathematical designs for highly structured paired-comparison experiments. They have been widely used in bridge tournaments, and many special cases are applicable to round-robin tournaments for all kinds of sports. In this paper we construct complete balanced Howell rotations of 8k + 5 teams, for 8k + 5 a prime power greater than 5. We also show that “table balanced” complete balanced Howell rotations for 8k + 5 teams do not exist unless 8k + 5 is the sum of two squares of integers.