On a periodic motion of a rigid body carrying a material point in the presence of impacts with a horizontal plane

A material system consisting of an outer rigid body (a shell) and an inner body (a material point) is considered. The system moves in a uniform field of gravity over a fixed absolutely smooth horizontal plane. The central ellipsoid of inertia of the shell is an ellipsoid of rotation. The material point moves according to the harmonic law along a straight-line segment rigidly attached to the shell and lying on its axis of dynamical symmetry. During its motion, the shell may collide with the plane. The coefficient of restitution for an impact is supposed to be arbitrary. The periodic motion of the shell is found when its symmetry axis is situated along a fixed vertical, and the shell rotates around this vertical with an arbitrary constant angular velocity. The conditions for existence of this periodic motion are obtained, and its linear stability is studied.     

Flocking Dynamics of the Inertial Spin Model with a Multiplicative Communication Weight

In this paper, we study a flocking dynamics of the deterministic inertial spin (IS) model. The IS model was introduced for the collective dynamics of active particles with an internal angular momentum, or spin. When the generalized moment of inertia becomes negligible compared to spin dissipation (overdamped limit) and mutual communication weight is a function of a relative distance between interacting particles, the deterministic inertial spin model formally reduces to the Cucker–Smale (CS) model with constant speed constraint whose emergent dynamics has been extensively studied in the previous literature. We present several sufficient frameworks leading to the asymptotic mono-cluster flocking, in which spins and relative velocities tend to zero asymptotically. We also provide several numerical simulations for the decoupled and coupled inertial spin models to see the effect of the C–S velocity flocking and compare them with our analytical results.

     

A Note on Vorticity

The relationship between vorticity and mean angular velocityis explored by methods that do not involve tensors so as toreveal the general relations between any component of vorticityand the mean angular velocity in a perpendicular plane and betweenvorticity and the mean angular velocity of a short line of fluidparticles, over all orientations at a point. Two results aredemonstrated concerning the relation between the moment of momentumof the fluid and suitably defined mean values of the vorticityover a sphere of finite radius. One result concerns the contentsof the sphere, the other a spherical shell.     

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.     

Hybrid coordinate approach for the modelling and simulation of multibody systems

The main goal of the present work is to provide an add-on scheme for the formulation of multibody dynamics, based on natural coordinates, in regard to ideally balanced rigid bodies with high rotational spin, e.g. gyroscopes. The underlying aim of this approach is to achieve higher numerical accuracy whenever the preferred axis of rotation coincides with the balanced main axis of the body. This will be achieved by seperating the spin of the balanced rigid body along the denoted axis as an additional angular coordinate, whereas the other rotations will be covered by a carried frame, parameterized via natural coordinates. At the same time the carried frame provides a link to the existing modelling framework in terms of natural coordinates, enabling a straightforward implementation into existing multibody systems (e.g. rotary crane [2]). (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the nonstationary motion of a viscous incompressible liquid over a rotating ball

We consider the free boundary problem for the Navier–Stokes equations governing a nonstationary motion of a layer of a viscous incompressible liquid that covers the surface of a rigid ball rotating around a fixed axis with constant angular velocity ω. The liquid is subject to the gravitation force generated by the mass of the ball. The self-gravitation forces between the liquid particles and capillary forces on the free surface are not taken into account. We consider the problem of stability of the regime of the rigid rotation of the liquid with the same angular velocity and prove that it is stable if |ω| is less than a certain constant. Bibliography: 10 titles. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 91–145.     

A new type of gravitational interaction with parity violation and its role in gravidynamics and electrodynamics

Lagrangian densities for spinor particles and photons in a gravitational field, including terms with C- and P-parity violations that remove degeneracy in spin states, are postulated. The modified Dirac and Maxwell equations are obtained. Hamiltonians for particles (bodies) possessing quantum (classical in the case of bodies) angular momentum are constructed in the semiclassical approximation. It is shown that the new interactions can produce the following effects: variation of body weight because of the partial spin ordering of electrons and nucleons, angular splitting and the relative retardation of left- and right-handed polarized waves passing near the Sun, modification of atomic spectra and of the atomic shell structure, and generation of high-power X-rays in the vicinity of neutron stars and of isotropically distributed high-energy cosmic rays in the early Universe.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 2, pp. 324–340, November, 1995.     

The vibrations of a rotating thin piezoceramic cylindrical shell

G. H. Bryan [8] discovered and studied the dynamic phenomena that arise during the vibrations of thinwalled shells in natural modes while rotating uniformly about an axis of symmetry. In the present article a more detailed mathematical analysis has been conducted for the solution of the problem of the vibrations of a ring rotating with constant angular velocity under feasible conditions of perturbation of the vibrations and realistic mechanical properties of ceramic materials. It is established that two slowly precessing waves arise during the vibration of the ring, one of which (the one discovered by Bryan) is an inverse precession wave having an angular velocity of precession less than the angular velocity of rotation of the ring, while the second is a direct precession wave having angular velocity larger than the angular velocity of rotation of the ring. Under the actual working conditions for piezoceramic resonators the amplitudes of the direct and inverse wave become comparable. Consequently the vibrations present as a modulated standing wave, that is, the phenomenon of beats in the mode of steady-state normal vibration occurs. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 7–18.     

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).     

Comparison of Numerical Simulation Models for Open Loop Flight Simulations in the Human Centrifuge

To achieve predefined Gz load profile in high G training in human centrifuge, it is necessary to determine angular velocity and acceleration of a planetary axis (centrifuge arm). Initial value problem that can not be solved in closed form was obtained. Several discretization methods for calculating angular velocity of centrifuge arm driven by DC servo motor are presented. Simulations are performed for different positive and negative values of Gz onset. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The symmetry of rotating fluid bodies

Consider an incompressible fluid body (in outer space) rotating about an axis with a given angular velocity , and which is in equilibrium relative to the potential energy of its own gravitational field and the surface energy due to surface tension. We show that such a body possesses a plane of symmetry perpendicular to the axis of rotation such that any line parallel to the axis and meeting the body cuts it in a line segment whose center lies on the plane of symmetry. This extends an earlier result of L. Lichtenstein [4].This work was done with the support of S.F.B. 72 while the author was a visitor at the University of Bonn     

Stochastic flocking dynamics of the inertial spin model with state-dependent noises

We study stochastic flocking dynamics of the inertial spin (IS) model with state-dependent noises. The IS model was considered to describe the collective behaviors of starling flocks moving with constant speed. Unlike mechanical flocking models extensively studied in the literature, this model incorporates an internal dynamic observable, namely spin (internal angular momentum) in addition to mechanical observables (position and velocity), and it describes how spin interacts with mechanical observables. In previous works, emergent dynamics of the deterministic counterparts for the IS model and its mean-field limit have been investigated under some specific setting in which network topology is multiplicatively separable. In this work, we present sufficient frameworks for stochastic flocking dynamics of the IS model, which state-dependent noises vanish at the equilibria of the deterministic IS model. The proposed frameworks are in terms of coupling strength, friction, and inertial coefficients, and our asymptotic convergence results for sample paths are given in both an almost sure and an expectation sense. We have also conducted several numerical experiments to verify our analytical results and to explore what can be studied further in future work     

Parametric resonance of a FG cylindrical thin shell with periodic rotating angular speeds in thermal environment

Parametric resonance of a functionally graded (FG) cylindrical thin shell with periodic rotating angular speeds subjected to thermal environment is studied in this paper. Taking account of the temperature-dependent properties of the shell, the dynamic equations of a rotating FG cylindrical thin shell based upon Love's thin shell theory are built by Hamilton's principle. The multiple scales method is utilized to obtain the instability boundaries of the problem with the consideration of time-varying rotating angular speeds. It is shown that only the combination instability regions exist for a rotating FG cylindrical thin shell. Moreover, some numerical examples are employed to systematically analyze the effects of constant rotating angular speed, material heterogeneity and thermal effects on vibration characteristics, instability regions and critical rotating speeds of the shell. Of great interest in the process is the combined effect of constant rotating angular speed and temperature on instability regions.     

On rotational gas heating

An exact solution of the Navier-Stokes equations for a normal state viscous heat conduct ing gas (with constant viscosity and heat conductivity) is obtained in the form of a stationary plane-parallel flow in a cylinder; the gas is heated by self-rotation at the angular velocity that monotonically increases (or decreases) along the central axis.     

3-D axisymmetric subsonic flows with nonzero swirl for the compressible Euler–Poisson system

We address the structural stability of 3-D axisymmetric subsonic flows with nonzero swirl for the steady compressible Euler–Poisson system in a cylinder supplemented with non-small boundary data. A special Helmholtz decomposition of the velocity field is introduced for 3-D axisymmetric flow with a nonzero swirl (=?angular momentum density) component. With the newly introduced decomposition, a quasilinear elliptic system of second order is derived from the elliptic modes in Euler–Poisson system for subsonic flows. Due to the nonzero swirl, the main difficulties lie in the solvability of a singular elliptic equation which concerns the angular component of the vorticity in its cylindrical representation, and in analysis of streamlines near the axis $r=0$.     

Normal modes of oscillations of a rotating membrane with a rigid central insert (an application of Heun functions)

The eigenvalues problem eigenvalues for the equation of the transverse oscillations of a homogeneous annular membrane with a rigid insert, rotating with a constant angular velocity about its central axis, is considered. Exact analytical expressions for the eigenfunctions in terms of special functions (local Heun functions), as well as normalization integrals, are found. An explicit expression for the time-invariant shape of the membrane during regular precession of its rotation axis is obtained.     

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.     

The local boundedness of the perturbed motions of a gyroscope in gimbals with dissipative and accelerating forces

The motion of an unbalanced gyroscope in gimbals in a central Newtonian field of forces is considered, taking the masses of the suspension rings into account. It is assumed that there is a moment of forces of viscous friction acting on the axis of rotation of one of the rings, and there is an accelerating (electromagnetic) moment applied to the axis of rotation axis of the other ring. The equations of motion have a partial solution such that the mean velocity of the outer ring is perpendicular to the direction from the centre of gravitation S to the stationary point O, the middle plane of the inner ring contains this direction, and the gyroscope rotates about SO with an arbitrary constant angular velocity.     

Flexural Vibrations of Shafts with Delaminations

The purpose of this theoretical work is to present a general model of laminated rotating shaft with circumferential delaminations. The shaft is treated as a thin‐walled composite cylindrical shell. The delamination of constant width is parallel to the shell reference surface and it covers the entire circumference. The edge delamination is modeled by changing the effective reduced stiffnesses of debonded parts. The stabilizing effect of external damping and destabilizing effect of internal damping are taken into account in the dynamic stability analysis. The influence of the relative delamination length and configuration on the critical angular velocity of shaft is shown.     

Resonant Flow of a Rotating Fluid Past an Obstacle: The General Case

We consider the flow of a rotating fluid past an antisymmetric obstacle placed on the axis of a cylindrical tube, for the case when the upstream flow is nearly resonant, or critical, so that the speed of a free linear long wave is nearly zero in the frame of reference of the obstacle. The perturbed flow is dominated by the resonant mode, whose amplitude satisfies a forced Korteweg—de Vries equation in this general case when the upstream flow contains radial shear andjor radially dependent angular velocity.