Conservative methods of controlling the rotation of a gyrostat

A method for shaping the control of the rotation of a gyrostat consisting of a rigid body, within which there are three rotors rotating about non-coplanar axes rigidly connected to the body, is discussed. The state of the system is defined by the position and angular velocity of rotation of the body, as well as by the angular velocities of the rotors. Control is achieved by torques applied to the rotors. The idea behind the proposed control method is to choose the controlling torques so that the angular velocities of rotation of the rotors are linear functions of the components of the angular velocity vector of the body. The linear dependence thus specified defines a 3 × 3 matrix, that is, a “controlled inertia tensor.” This matrix, which is specified by the parameters of the control selected, does not necessarily have the properties of an inertia tensor. As a result of such a choice of controls, the equations that define the variation of the angular velocity of the body are written in a form similar to Euler's dynamical equations. The system of equations obtained is used to formulate and solve problems of controlling the angular motion of a satellite in a circular orbit. The proposed method for constructing controlling actions enables both the Lagrangian structure of the equations of motion and the fundamental symmetries of the problem to be maintained. Expressions for the torques acting on the rotors and realizing the motion of the required classes are written in explicit form.     

Stabilization and optimal decay rate for a non-homogeneous rotating body-beam with dynamic boundary controls

In this paper, we consider the boundary stabilization of a flexible beam attached to the center of a rigid disk. The disk rotates with a non-uniform angular velocity while the beam has non-homogeneous spatial coefficients. To stabilize the system, we propose a feedback law which consists of a control torque applied on the disk and either a dynamic boundary control moment or a dynamic boundary control force or both of them applied at the free end of the beam. By the frequency multiplier method, we show that no matter how non-homogeneous the beam is, and no matter how the angular velocity is varying but not exceeding a certain bound, the nonlinear closed loop system is always exponential stable. Furthermore, by the spectral analysis method, it is shown that the closed loop system with uniform angular velocity has a sequence of generalized eigenfunctions, which form a Riesz basis for the state space, and hence the spectrum-determined growth condition as well as the optimal decay rate are obtained.     

Investigation on a rotating FGPM circular disk under a coupled hygrothermal field

In this paper, the distributions of the temperature, moisture, displacement and stress of a functionally graded piezoelectric material (FGPM) circular disk rotating around its axis at a constant angular velocity under a coupled hygrothermal field are presented by a numerical method. The material properties of the FGPM circular disk are assumed to vary along the radial coordinate exponentially. First, the coupled hygrothermal field along the radius of a rotating circular disk is achieved by solving the coupled hygrothermal equations, and then the dynamic equilibrium is solved by utilizing the finite difference method. Finally, numerical results show the effects of functionally graded index, inner radius, angular speed and hygrothermal index on the hygrothermal behaviors of the FGPM circular disk. The results can be useful for the optimal design of rotating FGPM circular disks under a coupled hygrothermal field.     

Study on feet forces' distributions,energy consumption and dynamic stability measure of hexapod robot during crab walking

This paper deals with the development of a dynamical model related to crab walking of a hexapod robot to determine the feet forces' distributions, energy consumption and dynamic stability measure considering the inertial effects of the legs on the system, which has not been attempted before. Both forward and inverse kinematic analyses of the robot are carried out with an assigned fixed global frame and subsequent local frames in the trunk body and joints of each leg. Coupled multi-body dynamic model of the robot is developed based on free-body diagram approach. Optimal feet forces and corresponding joint torques on all the legs are determined based on the minimization of the sum of the squares of joint torques, using quadratic programming (QP) method. An energy consumption model is developed to determine the minimum energy required for optimal values of feet forces. To ensure dynamically stable gaits, dynamic gait stability margin (DGSM) is determined from the angular momentum of the system about the supporting edges. Computer simulations have been carried out to test the effectiveness of the developed dynamic model with crab wave gaits on a banking surface. It is observed that when the swing leg touches the ground, impact forces (sudden shoot outs) are generated and their effects are also observed on the joints of the legs. The effects of walking parameters, namely trunk body velocity, body stroke, leg offset, body height, crab angle etc. on power consumption and stability during crab motion for duty factors (DFs) like 1/2, 2/3, 3/4 have also been studied.     

Exact Relativistic Treatment of Stationary Counter-Rotating Dust Disks: Axis,Disk, and Limiting Cases

We continue to study the construction of explicit solutions of the stationary axisymmetric Einstein equations, interpretable as counterrotating disks of dust. We discuss the previously constructed class of solutions for disks with constant angular velocity and constant relative density. The metric for these space–times is given in terms of theta functions on a Riemann surface of genus two. We discuss the metric functions at the axis of symmetry and on the disk. Interesting limiting cases are the Newtonian, static, and ultrarelativistic limits (in the latter limit, the central red shift diverges).     

Flow of a Reiner‐Rivlin fluid between two infinite coaxial rotating disks

Present study deals with the steady flow and heat transfer of a non‐Newtonian Reiner‐Rivlin fluid between two coaxially rotating infinite disks. Using similarity transformations, the governing equations are reduced to a set of nonlinear, highly coupled ordinary differential equations and by means of an effective analytical method called homotopy analysis method; analytical solutions are constructed in series form. Different cases, such as, when one disk is at rest and the other is rotating with constant angular velocity, two disks rotating with different angular velocities in same as well as opposite sense, two disks rotating with same angular velocities in opposite sense, are discussed. The effects of non‐Newtonian parameter, Reynolds number, are also discussed, and results are presented graphically.     

Optimal motion of a multilink system in a resistive medium

We consider the motion in a resistive medium of a mechanical system consisting of a main body and one or two links attached to it by means of cylindrical joints. The motion is controlled through high-frequency periodic oscillations of the links. For this system, an equation of motion is deduced and the average velocity of locomotion is estimated under certain assumptions. This velocity is positive if the angular velocity of diverting the attached links is less than the angular velocity of bringing them to the axis of the body. An optimal control problem of maximizing the average velocity is formulated and solved. An example is given.     

An Invariant Measure and the Probability of a Fall in the Problem of an Inhomogeneous Disk Rolling on a Plane

This paper addresses the problem of an inhomogeneous disk rolling on a horizontal plane. This problem is considered within the framework of a nonholonomic model in which there is no slipping and no spinning at the point of contact (the projection of the angular velocity of the disk onto the normal to the plane is zero). The configuration space of the system of interest contains singular submanifolds which correspond to the fall of the disk and in which the equations of motion have a singularity. Using the theory of normal hyperbolic manifolds, it is proved that the measure of trajectories leading to the fall of the disk is zero.     

Steady flow generated by the differential rotation of a porous disk of finite thickness

When a body of fluid bounded by a porous disk of finite thickness is disturbed from a state of rigid rotation by an enhanced (or reduced) angular velocity of the disk, a few authors followed Darcys model and observed that the centrifugal pumping occurs through the entire porous layer regarded as a convection zone. The shear stress can develop only at the edge of the porous layer. We use a porous disk of high permeability that allows the fluid in the porous disk to deform in response to the changing angular velocity. Based on the Birkmans model, we solve for the steady non-linear flow and observe that there arises (i) a convection zone of nearly uniform angular velocity at the boundary (within the porous layer) and (ii) a transition zone adjacent to the convection zone which provides a smooth transition to the interior. This makes the model relevant to some astrophysical situations as described by some authors [1, 3]. The two point boundary value problem is solved subject to the boundary conditions, the far field conditions, and the matching conditions at the fluid-porous medium interface. The solution is obtained using a numerical procedure known as the method of Adjoints.Received: June 13, 2002; revised: July 7, 2003     

Steady flow generated by the differential rotation of a porous disk of finite thickness

When a body of fluid bounded by a porous disk of finite thickness is disturbed from a state of rigid rotation by an enhanced (or reduced) angular velocity of the disk, a few authors followed Darcys model and observed that the centrifugal pumping occurs through the entire porous layer regarded as a convection zone. The shear stress can develop only at the edge of the porous layer. We use a porous disk of high permeability that allows the fluid in the porous disk to deform in response to the changing angular velocity. Based on the Birkmans model, we solve for the steady non-linear flow and observe that there arises (i) a convection zone of nearly uniform angular velocity at the boundary (within the porous layer) and (ii) a transition zone adjacent to the convection zone which provides a smooth transition to the interior. This makes the model relevant to some astrophysical situations as described by some authors [1, 3]. The two point boundary value problem is solved subject to the boundary conditions, the far field conditions, and the matching conditions at the fluid-porous medium interface. The solution is obtained using a numerical procedure known as the method of Adjoints.     

Ultracentrifuge Sedimentation during the Period of Constant Acceleration

Initially, the angular velocity of an ultracentrifuge increasesat a uniform rate until a given maximum is reached, after whichit is maintained at a constant value. The concentration distributionduring the accelerating period is obtained as the solution ofa one-dimensional approximation to the diffusion equation, analogousto that previously used (Fujita & MacCosham, 1959) for thecase of constant angular velocity.     

Dynamical flexibility of damped vibratory clamped-clamped beam in transportation

This work concerns the issue of damped vibratory beam in transportation. Support conditions were defined for clamped-clamped beamlike systems. Structural loads were defined as with unitary amplitude ones up to the used dynamical flexibility definition. Geometric representation of analyzed element is a beam with symmetric cross-section constant on the whole length of the beam. The load assumed as bending one. The beam was fixed onto the rotational disk with the angular velocity treated as transportation velocity. In thesis the dynamical flexibility of analyzed beam was presented and the sample dynamical characteristics were drawn on graphical charts. In derived mathematical model mutual interactions between the damping forces, dynamical flexibilities and angular working velocity were took into consideration. Presented model can be put into use in high speed pumps and turbines as vanes. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical simulation of flow of a micropolar fluid between a porous disk and a non-porous disk

Two dimensional steady, laminar and incompressible motion of a micropolar fluid between an impermeable disk and a permeable disk is considered to investigate the influence of the Reynolds number and the micropolar structure on the flow characteristics. The main flow stream is superimposed by constant injection velocity at the porous disk. An extension of Von Karman’s similarity transformations is applied to reduce governing partial differential equations (PDEs) to a set of non-linear coupled ordinary differential equations (ODEs) in dimensionless form. An algorithm based on finite difference method is employed to solve these ODEs and Richardson’s extrapolation is used to obtain higher order accuracy. The numerical results reflect the expected physical behavior of the flow phenomenon under consideration. The study indicates that the magnitude of shear stress increases strictly and indefinitely at the impermeable disk while it decreases steadily at the permeable disk, by increasing the injection velocity. Moreover, the micropolar fluids reduce the skin friction as compared to the Newtonian fluids. The magnitude of microrotation increases with increasing the magnitude of R and the micropolar parameters. The present results are in excellent comparison with the available literature results.     

SOLUTIONS OF EULER-POISSON EQUATIONS IN R^n

In this article, the authors study the structure of the solutions for the EuierPoisson equations in a bounded domain of Rn with the given angular velocity and n is an odd number. For a ball domain and a constant angular velocity, both existence and nonexistence theorem are obtained depending on the adiabatic gas constant 7. In addition, they obtain the monotonicity of the radius of the star with both angular velocity and center density. They also prove that the radius of a rotating spherically symmetric star, with given constant angular velocity and constant entropy, is uniformly bounded independent of the central density. This is different to the case of the non-rotating star.     

双缸单作用活塞泵中活塞匀速运动的探讨

通常双缸单作用活塞泵的恒转速运行会导致管路流量波动,使之在恒流量场合较少应用.对双缸单作用活塞泵的运动建立微分方程,给出了MATLAB7的仿真结果,提出了活塞近似匀速运动的条件,并对电动机转轴的角速度的计算方法作了讨论,得出了在中、低速运行时采用变角速度控制能近似实现恒流量的结论.     

A new solution of the equilibrium equations of a system of bodies with elastic joints

Different ways of representing the elastic moments are proposed, which can be used for the finite-dimensional modelling of rod systems using a system of n axisymmetric solids, connected by elastic spherical joints. Using the example of a closed plane rod, possible states of equilibrium of the finite-dimensional model of the rod are analysed for different methods of specifying the elastic torques at the joints. The case when the rod axis has the form of a “figure of eight”, which is modelled by a system of six axisymmetric solids with a relative torsion angle that depends on the bending, is investigated in detail.     

On the Bödewadt–Rosenblat flow

Fluid motion induced by the torsional oscillations (of angular velocity bΩcosω T) of an infinite disk in contact with an incompressible viscous rotating (with angular velocity aΩ) fluid of semi-infinite extent is analysed when the amplitude parameter α( = b/a) varies from zero to infinity. Composite solutions valid over the whole of the flow regime and specific expressions for the shearing stress components at the disk and for the axial flow in the far region are obtained for low and high frequencies of torsional oscillations. Using the method of matched asymptotic expansions, we find that the region of the mean flow increases with α and reaches a maximum before settling down to the Rosenblat profile. Series expressions (for α < 1) are deduced for physical quantities of interest when the fluid in the far field and the disk are rotating with different angular velocities (in the same or in the opposite sense), which agree well with the known numerical results. (Received: April 7, 2003; revised: September 29, 2005)     

Localized minimizers of flat rotating gravitational systems

We study a two-dimensional system in solid rotation at constant angular velocity driven by a self-consistent three-dimensional gravitational field. We prove the existence of stationary solutions of such a flat system in the rotating frame as long as the angular velocity does not exceed some critical value which depends on the mass. The solutions can be seen as stationary solutions of a kinetic equation with a relaxation-time collision kernel forcing the convergence to the polytropic gas solutions, or as stationary solutions of an extremely simplified drift-diffusion model, which is derived from the kinetic equation by formally taking a diffusion limit. In both cases, the solutions are critical points of a free energy functional, and can be seen as localized minimizers in an appropriate sense.     

On the B?dewadt–Rosenblat flow

Fluid motion induced by the torsional oscillations (of angular velocity bΩcosω T) of an infinite disk in contact with an incompressible viscous rotating (with angular velocity aΩ) fluid of semi-infinite extent is analysed when the amplitude parameter α( =  b/a) varies from zero to infinity. Composite solutions valid over the whole of the flow regime and specific expressions for the shearing stress components at the disk and for the axial flow in the far region are obtained for low and high frequencies of torsional oscillations. Using the method of matched asymptotic expansions, we find that the region of the mean flow increases with α and reaches a maximum before settling down to the Rosenblat profile. Series expressions (for α < 1) are deduced for physical quantities of interest when the fluid in the far field and the disk are rotating with different angular velocities (in the same or in the opposite sense), which agree well with the known numerical results.     

On the numerical solution of the flow between a rotating and a stationary disk

The flow between two co-axial, infinite disks, one rotating with constant angular velocity and one stationary is treated in this paper. The problem is reduced to that of finding the solution of a two-point boundary value for a sixth order nonlinear ordinary differential equation and three boundary conditions at each of a finite interval. The numerical solutions are obtained by using a fourth order Runge-Kutta integration scheme in modification due to Gill and in conjunction with a modified shooting method to correct the initial guesses at one boundary. The numerical calculations for different Reynolds numbers are carried out. The results obtained by this method are compared with available results. The comparison shows excellent agreement.