Uniform Gravity Field Research Articles

Stability of a bounded liquid layer on a rotating horizontal plane

The paper is devoted to the study of the stability of the relative equilibrium of a bounded liquid layer on a flat solid rotating base. A uniform gravity field is oriented perpendicular to the solid surface and presses the droplet against it. The equilibrium shape and its perturbations are axisymmetric. The free surface is simply-connected. The analysis is performed both for the case of the free contact line and for the case of the fixed one. The results obtained by these two models are compared, and the effect of the input parameters on the stability is investigated. It is established that the second model is in better accordance with empirical data. Unlike the first one, it allows the possibility of a zero height of the layer at the center at certain values of the contact angle and determines the negative effect of a low wettability of the solid substrate on the stability of the droplet. The minimum potential energy principle is used as a stability criterion. In this process all physically admissible small variations of a free surface shape are considered. An equilibrium state is supposed to be stable if and only if it corresponds to a minimum potential energy on the set of allowable virtual displacements, which is more restricted when the contact line is fixed.

Does a charge in a uniform gravity field radiate? (The Accelerating Charge Paradox)

Does a charge in a uniform gravity field radiate? (The Accelerating Charge Paradox)

To the Problem of Motion Cueing Simulation on a Robotic Stand for Aircraft Flight

Various simulators with motion cueing simulation stands, which make it possible to create an acceleration environment for the pilot that is close to a real flight, are used for training aircraft pilots. The article considers the formulation of the motion cueing simulation on a stand based on an industrial manipulator. Motion cueing simulation algorithms include two phases: motion cueing simulation phase and phase of return to the working area center. During simulation phase the stand must implement such a movement that the angular accelerations acting on the person and the overload vector acting on the center of mass of the operator completely coincide with the real ones. If it is not possible then just the directions of these vectors should coincide. During the second phase the stand end point must return to the working area center with acceleration values below the threshold, but in the fastest way. This task can be presented as a generalization of the brachistochrone problem. The article considers the problem of the material point motion in a uniform gravity field along a curve located in a vertical plane, in the presence of restrictions on the trajectory curvature. It is necessary to choose the curve shape in such a way that the descent time is minimal. The problem solution is obtained by optimal control methods, the cases of regular and singular control realization are considered, the question of its conjugation. Also, the switching number between sections of regular and singular control is studied.

Periodic Waves and Ligaments on the Surface of a Viscous Exponentially Stratified Fluid in a Uniform Gravity Field

The theory of singular perturbations in a unified formulation is used, for the first time, to study the propagation of two-dimensional periodic perturbations, including capillary and gravitational surface waves and accompanying ligaments in the 10−4<ω<103 s−1 frequency range, in a viscous continuously stratified fluid. Dispersion relations for flow constituents are given, as well as expressions for phase and group velocities for surface waves and ligaments in physically observable variables. When the wave-length reaches values of the order of the stratification scale, the liquid behaves as homogeneous. As the wave frequency approaches the buoyancy frequency, the energy transfer rate decreases: the group velocity of surface waves tends to zero, while the phase velocity tends to infinity. In limiting cases, the expressions obtained are transformed into known wave dispersion expressions for an ideal stratified or actually homogeneous fluid.

Механический аналог движений неоднородной жидкости

The article considers the homogeneous vortex motion of an inhomogeneous fluid in a stationary ellipsoidal cavity arbitrarily oriented with respect to the direction of a uniform gravity field using the Lagrange variables. Uniform vortex motion of a fluid is a motion in which the rotor speed for all particles has the same value, and depends only on time. The article shows that the equations of homogeneous vortex motion of a heavy inhomogeneous fluid are possible in an ellipsoidal cavity with a linear density distribution and proposes a geometric interpretation of fluid motion. Using the Lagrange variables, the equations of motion of an inhomogeneous fluid are obtained, coinciding with the equations of a heavy rigid body motion around a fixed point, written in a fixed coordinate system arbitrarily located relative to the direction of a homogeneous gravity field.

On stability and instability of stratified elastic solids in a gravity field

Stability and instability of the equilibrium of stratified elastic solids in a uniform gravity field are studied by means of analysis of the second variation of total (elastic and gravitational) potential energy of the solid. By stratification is meant a continuous dependence of density and incremental elastic moduli of the material on vertical coordinate. The unperturbed stress state is supposed to be hydrostatic, whereas the incremental shear stiffnesses for perturbing deformations are nonzero. Obtained analytically both the necessary condition and the sufficient condition for stability demonstrate stabilizing effect of shear stiffness. For shear stiffnesses approaching zero, each of the conditions converts into obtained earlier by the authors the necessary and sufficient stability condition for stratified compressible fluids (having zero shear stiffness).The sufficient condition for stability for given stratifications of density and bulk modulus, and arbitrary dimensions of the domain, occupied by the solid, yields semi-infinite range of values of shear stiffnesses ensuring stability of the solid for conditions under which the corresponding stratified compressible fluid would be unstable.The sufficient condition for instability (negation of the necessary condition for stability) yields such a nonzero value of shear stiffness, that for shear stiffnesses below it the solid is surely unstable. This value (the lower bound for the exact critical value) depends on horizontal dimensions of the domain occupied by the solid, and on thickness of the layer where the stability condition for corresponding stratified fluid would be violated.Possible applications of obtained theoretical results are discussed.

Analysis of the orbital stability of periodic pendulum motions of a heavy rigid body with a fixed point under the Goryachev–Chaplygin condition

In this paper the motion of a rigid body with a fixed point in a uniform gravity field is considered. It is assumed that the main moments of inertia of the body correspond to the case of Goryachev–Chaplygin, i.e., they are in the ratio of 1:1:4. In contrast to the integrable case of Goryachev–Chaplygin, there are no restrictions imposed on the position of the center of mass of the body. The problem of the orbital stability of periodic pendulum motions of a body (oscillations and rotations) is investigated. The equations of perturbed motion were obtained and the problem of orbital stability was reduced to the stability problem of the equilibrium position of a second-order linear system with 2π-periodic coefficients, the right-hand sides of which depend on two parameters. Based on the analysis of the linearized system, it has been established that the rotations are orbitally unstable for all possible values of the parameters. Moreover, a diagram of stability of pendulum oscillations has been constructed, on which the regions of orbital instability (parametric resonance) and regions of orbital stability in the linear approximation are indicated. At small values of oscillations amplitudes, a nonlinear analysis was performed. Based on the analysis of the coefficients of the normalized Hamilton function using theorems of the KAM theory, rigorous conclusions on the orbital stability of pendulum oscillations with small amplitudes were obtained.

Treating the Solid Pendulum Motion by the Large Parameter Procedure

In this paper, we consider the dynamical description of a pendulum model consists of a heavy solid connection to a nonelastic string which suspended on an elliptic path in a vertical plane. We suppose that the dimensions of the solid are large enough to the length of the suspended string, in contrast to previous works which considered that the dimensions of the body are sufficiently small to the length of the string. According to this new assumption, we define a large parameter ε and apply Lagrange’s equation to construct the equations of motion for this case in terms of this large parameter. These equations give a quasi-linear system of second order with two degrees of freedom. The obtained system will be solved in terms of the generalized coordinates θ and φ using the large parameter procedure. This procedure has an advantage over the other methods because it solves the problem in a new domain when fails all other methods for solving the problem in such a domain under these conditions. It is one of the most important applications, when we study the slow spin motion of a rigid body in a Newtonian field of force under an external moment or the rotational motion of a heavy solid in a uniform gravity field or the gyroscopic motions with a sufficiently small angular velocity component about the major or the minor axis of the ellipsoid of inertia. There are many applications of this technique in aerospace science, satellites, navigations, antennas, and solar collectors. This technique is also useful in all perturbed problems in physics and mechanics, for example, the perturbed pendulum motions and the perturbed mechanical systems. The results of this paper also are useful in moving bridges and the swings. For satisfying the validation of the obtained solutions, we consider numerical considerations by one of the numerical methods and compare the obtained analytical and numerical solutions.

Numerical study on the effect of rotation radius of geotechnical centrifuge on the dynamic behavior of liquefiable sloping ground

For a small-size geotechnical centrifuge, it is well known that a uniform gravity field, which is position-independent, cannot be achieved in a model ground due to finite lengths of rotating radius: a gravity field in the centrifuge becomes radial. However, little works have been done related to the radial gravity effect on seismic responses of the model ground. This paper presents finite element simulation results for dynamic centrifuge model tests of a liquefiable sloping ground conducted at two centrifuge facilities having different small-radial arms with the shaking direction being tangential to the axis, aiming to show the importance of considering the radial gravity effect in numerical simulation. The simulations are performed in the centrifuge model scale by using a strain space multiple mechanism model; the radial gravity field is applied to the model ground at the stage of self-weight analyses before seismic response analyses are carried out. Comparison of the simulated seismic response with the centrifuge test results demonstrates that the experimental deformation mode due to lateral spreading during shaking is simulated with higher accuracy, particularly near the side boundaries, by considering the small-radius effect (i.e., the radial gravity field instead of the uniform gravity field) in an appropriate manner.

CURVE OF DESCENT OF A MATERIAL POINT IN THE SHORTEST TIME ON A TRANSCENDENTAL SURFACE IN A UNIFORM VERTICAL GRAVITATIONAL FIELD

Background. The article deals with the original variational problem of the brachystochronous motion of a material point on a cycloidal surface between two given points in a vertically homogeneous gravitational field. The novelty and relevance of the work is explained by the choice of the transcendental surface, since earlier the motion of a material point was considered on algebraic surfaces of the second order.Objective. Find a curve on the transcendental surface, moving from one set point (starting point) to another set point (finish point) on this surface without friction a material point will make such a transition in a minimum time. The transcendental surface has a guide curve of the cycloid lying in one of the coordinate planes, and its generatrix are perpendicular to that plane.Methods. To achieve this goal, we used the classical methods of variational calculus (Euler–Lagrange equation), as well as the classical method of integrating ordinary differential equations in a closed form (Bernoulli method).Results. A time functional was constructed, using which the differential equations of the spatial brachystrochron, which lies on the transcendental surface, are analytically deduced. After integration in a closed form, algebraic equations of the spatial brachystrochron in parametric form are obtained. The results of the study are illustrated graphically: the projections of the trajectory of the brachystrochron on the coordinate planes OXY and OXZ. The slope angles of the optimal trajectory at the start point are determined. A comparative analysis of the time of action in the process of motion of a material point along two trajectories is carried out: along the obtained brachystrochron and along the alternative trajectory.Conclusions. The proposed approach allows to pre-plot such a logistic route of a material point on a given transcendental surface between two fixed points, which will provide a minimum travel time between them in a uniform vertical gravity field. In this case, an extreme trajectory will not necessarily be the shortest line on the surface that connects the two predetermined points (start and finish).

On periodic motions of a rigid body suspended on a thread in a uniform gravity field

On periodic motions of a rigid body suspended on a thread in a uniform gravity field

Stable Levitation of Superconducting Myxines of Galathea Plasma Traps

Designing magnetic systems of Galathea plasma traps on the basis of levitating superconducting magnetic coils requires searching for their stable levitating states. For this purpose, proceeding from the property of superconductors to preserve the trapped magnetic flux, the potential energy of a system of several coaxial superconducting rings (one of the rings is fixed) with given trapped magnetic fluxes is obtained as an analytic function of the coordinates of the free rings along the system axis and angular deviations of their axes from the common axis of the system in a uniform gravity field in the thin-ring approximation. Computations demonstrate that, under certain values of physical parameters (the trapped magnetic flux and the dimensions and masses of the rings), this dependence has local minima, which correspond to stable equilibrium states of the levitating rings. For high-temperature superconducting (HTSC) rings and short-circuited HTSC coils manufactured for experiments on levitation, equilibrium states are found in different cases by calculating the aforementioned dependence of the potential energy. In the case where the magnetic fluxes trapped by the rings have the same signs, the calculated equilibrium levitating states of an HTSC ring in the field of a short-circuited HTSC coil (or an HTSC ring), namely, states that are stable against vertical displacements and angular deviations of the axis from the vertical, are implemented experimentally.

Stability in the Regular Precession of an Asymmetrical Gyroscope in the Critical Case of Fourth-Order Resonance

The motion of a solid around a stationary point in a uniform gravity field is considered. The mass geometry of this body is such that it can perform regular precession around an axis inclined to a vertical (Grioli precession). The problem about the orbital stability of this precession is solved in the critical case of fourth-order resonance, when the terms to a power higher than the fourth with respect to perturbations (including the sixth-power terms) must be taken into account in the expansion of the Hamiltonian function.

A pendulum on a rotating and vibrating foundation

The motion of a pendulum is investigated in a uniform gravity field. The foundation of the pendulum is assumed to rotate around the vertical simultaneously accomplishing harmonic vibrations along it with a high frequency and small amplitude. With use of the classical perturbation theory and modern methods of analysis of nonlinear dynamical sets, the problem on the existence and stability of periodic motions of a pendulum with the period equal to that of the vibrations of the foundation is solved.

Conditions of stability and instability for a pair of arbitrarily stratified compressible fluids in an arbitrary non-uniform gravity field

The work continues and develops authors’ previous investigation of stability in the small for a two-layer system of inhomogeneous compressible fluids in the uniform gravity field. Here we present a solution of a similar problem in the case of arbitrary non-uniform potential gravity field. The equilibrium stratification of both density and elastic properties of the fluids is supposed arbitrary, as well as the shape of open on top reservoir filled by the fluids. The problem of stability of equilibrium is analyzed as the corresponding problem for the non-linearly elastic bodies, basing on the static energy criterion with regard for the boundary conditions at all parts of the boundary. The crucial element of the analysis is conversion of the quadratic functional of second variation of total potential energy of the system into a “canonical” form that enables to determine its sign. Making use of this canonical form, we obtain almost coinciding with each other necessary and sufficient conditions for stability (those being valid also for an arbitrary number of layers).

On the stability of the permanent rotations of a charged rigid body-gyrostat

We consider the motion of a charged rigid body about a fixed point carrying a rotor that is attached along one of the principal axes of the body. This motion occurs under the action of the resultant of the uniform gravity field and the homogeneous magnetic field. The equations of motion are formulated, and they are presented by means of the Hamiltonian function in the framework of the Lie–Poisson system. These equations of motion have six equilibrium solutions. The sufficient conditions for instability for these equilibria are studied by utilizing the linear approximation method, while the sufficient conditions for stability are presented by means of the energy-Casimir method. For certain configuration of the body, the regions of Lyapunov stability and instability are determined in the plane of some parameters. Furthermore, we clarify that the regions of Lyapunov stability are a portion of the regions of linear stability.

Dynamics of Maxwell’s pendulum

The stability of motion of Maxwell’s pendulum is investigated in a uniform gravity field. By means of several canonic transforms of the equations of pendulum motion and the method of the surfaces of Poincare sections, the problem is reduced to investigation of the immobile-point stability retaining the area of mapping of the plane into itself. In the space of dimensionless parameters, the stability and instability regions are singled out.

Rayleigh–Bénard stability and the validity of quasi-Boussinesq or quasi-anelastic liquid approximations

The linear stability threshold of the Rayleigh–Bénard configuration is analysed with compressible effects taken into account. It is assumed that the fluid under investigation obeys a Newtonian rheology and Fourier’s law of thermal transport with constant, uniform (dynamic) viscosity and thermal conductivity in a uniform gravity field. Top and bottom boundaries are maintained at different constant temperatures and we consider here mechanical boundary conditions of zero tangential stress and impermeable walls. Under these conditions, and with the Boussinesq approximation, Rayleigh (Phil. Mag., vol. 32 (192), 1916, pp. 529–546) first obtained analytically the critical value $27\unicode[STIX]{x03C0}^{4}/4$ for a dimensionless parameter, now known as the Rayleigh number, at the onset of convection. This paper describes the changes of the critical Rayleigh number due to the compressibility of the fluid, measured by the dimensionless dissipation parameter ${\mathcal{D}}$ and due to a finite temperature difference between the hot and cold boundaries, measured by a dimensionless temperature gradient $a$. Different equations of state are examined: ideal gas equation, Murnaghan’s model (often used to describe the interiors of solid but convective planets) and a generic equation of state with adjustable parameters, which can represent any possible equation of state. In the perspective to assess approximations often made in convective models, we also consider two variations of this stability analysis. In a so-called quasi-Boussinesq model, we consider that density perturbations are solely due to temperature perturbations. In a so-called quasi-anelastic liquid approximation model, we consider that entropy perturbations are solely due to temperature perturbations. In addition to the numerical Chebyshev-based stability analysis, an analytical approximation is obtained when temperature fluctuations are written as a combination of only two modes, one being the original symmetrical (between top and bottom) mode introduced by Rayleigh, the other one being antisymmetrical. The analytical solution allows us to show that the antisymmetrical part of the critical eigenmode increases linearly with the parameters $a$ and ${\mathcal{D}}$, while the superadiabatic critical Rayleigh number departs quadratically in $a$ and ${\mathcal{D}}$ from $27\unicode[STIX]{x03C0}^{4}/4$. For any arbitrary equation of state, the coefficients of the quadratic departure are determined analytically from the coefficients of the expansion of density up to degree three in terms of pressure and temperature.

Nonequilibrium fluctuations from a nematic under a thermal gradient and a gravity field

We use a fluctuating hydrodynamics (FH) approach to study the fluctuations of the hydrodynamic variables of a thermotropic nematic liquid crystal (NLC)in a nonequilibrium steady state (NESS). This NESS is produced by an externally imposed temperature gradient and a uniform gravity field. We calculate analytically the equilibrium and nonequilibrium seven modes of the NLC in this NESS. These modes consist of a pair of sound modes, one orientation mode of the director and two visco-heat modes formed by the coupling of the shear and thermal modes. We find that the nonequilibrium effects produced by the external gradients only affect the longitudinal modes. The analytic expressions for the visco-heat modes show explicitly how the heat and shear modes of the NLC are coupled. We show that they may become propagative, a feature that also occurs in the simple fluid and suggests the realization of new experiments. We show that in equilibrium and in the isotropic limit of the NLC, our modes reduce to well-known results in the literature. For the NESS considered we point out the differences between our our modes and those reported by other authors. We close the paper by proposing the calculation of other physical quantities that lend themselves to a more direct comparison with possible experiments for this system.

Nonequilibrium fluctuations from a nematic under a thermal gradient and a gravity field*

The fluctuating hydrodynamic treatment developed in the previous article for a nematic liquid crystal under the influence of a thermal gradient α and a uniform gravity field g , is used to calculate its Rayleigh light scattering spectrum. We find that the dissipative thermal gradient force enhances the Rayleigh-line intensity which varies as k -4 with the fluctuations of the wave number k . The Rayleigh line consists of three central Lorentzians, two of which are determined by the visco-heat modes coupling the entropy and director fluctuations, which is a pure non-equilibrium effect. The third Lorentzian is due only to director fluctuations. We find that the former peaks contain the Rayleigh wings owing to the orientational fluctuations of the aniosotropic molecules. It is also shown that the obtained spectrum reduces to the known equilibrium spectrum of a nematic and to that of a simple fluid. For the particular case in which the decay rates are diffusive, we calculate and plot the amplitudes of non-equilibrium fluctuations of the dynamic structure factor as a function of |α |2 /k 4 , and also, the intermediate function in the equilibrium and non-equilibrium states.