Rotating Reference System Research Articles

Electrodynamics in noninertial frames

The electromagnetic theory is considered in the framework of the generally covariant approach, that is applied to the analysis of electromagnetism in noninertial coordinate and frame systems. The special-relativistic formulation of Maxwell’s electrodynamics arises in the flat Minkowski spacetime when the general coordinate transformations are restricted to a class of transformations preserving the Minkowski line element. The particular attention is paid to the analysis of the electromagnetism in the noninertial rotating reference system. For the latter case, the general stationary solution of the Maxwell equations in the absence of the electric current is constructed in terms of the two scalar functions satisfying the Poisson and the biharmonic equations with an arbitrary charge density as a matter source. The classic problem of Schiff is critically revisited.

Nuclear microbomb propulsion for manned deep space exploration with return travel

For manned deep space exploration with return travel a novel nuclear microbomb propulsion concept is proposed. It assumes the presence of easily mineable natural uranium and lithium deposits on Solar System planets. The concept makes use of large negative mass densities in a rapidly rotating reference system predicted by Hund as a consequence of Einstein's general theory of relativity. It permits the ignition of the deuterium thermonuclear reaction by the implosion of Li6D salt inside a rapidly rotating natural uranium shell by an intense relativistic electron beam of over 100 MJ. This concept avoids the drawbacks of carrying along liquid deuterium-tritium and Plutonium 239 or Uranium 235 for fission spark plugs on a deep space mission.The generation of the pulsed high-energy relativistic beams can be done by solar collectors charging the high-voltage Marx generators, or by a small natural uranium reactor, requiring not much more than a small chemical factory, not a large isotope separating facility. A small chemical plant is also needed to make the high explosives used to achieve the large rotational target velocity.

Effect of the Coriolis force on the macrosegregation of aluminum in the centrifugal casting of Ti-Al alloys

Within the framework of the ESA GRADECET project, experiments of directional solidification of cylindrical Ti-Al samples were conducted in hypergravity. The experiments were performed in a centrifuge with the apparent gravity (sum of centrifugal and terrestrial gravity) aligned along the cylinder centerline. 3D numerical simulations of aluminum macrosegregation in these samples are presented. A volume-averaging solidification model is used that accounts for centrifugal and Coriolis accelerations in a non-inertial rotating reference system. We compare the melt flow pattern and the macrosegregation formation under terrestrial gravity and under centrifugation. The results show that the Coriolis acceleration, although very weak, breaks the symmetry of the thermosolutal convection, having an important impact on the final macrosegregation pattern. The macrosegregation is entirely modified in comparison with a sample solidified under terrestrial gravity conditions. Besides the aluminum segregation intensity increases with the centrifugation level.

Application of Multiple-Scales Method for the Dynamic Modelling of a Gear Coupling

Thin-walled gears, designed for aeronautical applications, have shown very rich dynamics that must be investigated in advance of the design phase. One of the signatures of their dynamics is coupling due to the meshing teeth which stand-alone gear models cannot capture. This paper aims to investigate the dynamics of thin-walled gears considering time-varying coupling due to the gear meshing. Each gear is modelled with lumped parameters according to a local rotating reference system and the coupling is modelled by a traveling meshing stiffness. The set of equations of motion is solved by the non-linear Method of Multiple-Time-Scales (MMTS). MMTS is a very powerful technique that is widely used to solve perturbation problems in many fields of mathematic and physics. In the analyzed numerical test case, the relevance of gear coupling is demonstrated as well as the capability of the MMTS to capture the fundamental features of the system dynamics. In this study the analytical methodology, which uses MMTS, allows for the calculation of the forced response of the system made of two meshing gears despite the presence of a parametric quantity, i.e., the mesh stiffness. The calculation is performed in the frequency domain using modal coordinates, which ensures a fast computation. The result is compared with time domain analysis for validation purposes.

Coriolis force-assisted inertial confinement fusion

AbstractA fundamental problem for the realization of laser fusion through the implosion of a spherical target is Kidder's E−1/6 law, where E is the energy needed for ignition, proportional to the 6th power of the ratio R/R0, where R0 and R are the initial and final implosion radii, respectively. This law implies that the ignition energy is very sensitive to the ratio R0/R, or vice versa, the ratio R0/R is very insensitive to the energy input, with R0/R limited by the Rayleigh–Taylor instability. According to still classified data of the Centurion–Halite experiment at the Nevada Test Site, ignition would require an energy of ${\rm E}\simeq 50\,{\rm MJ}$, 25 times larger than the 2 MJ laser at the National Ignition Facility (NIF) reported in the New York Times. This means that even a tenfold increase from 2 to 20 MJ would only decrease the R/R0 ratio by an insignificant factor of 10−1/6 ≃ 0.7. To overcome this problem, it is proposed that the spherical target is replaced with a hollowed-out, rapidly rotating, cm-size ferromagnetic target, accelerated by a rotating traveling magnetic wave to a rotational velocity of ~1 km/s, at the limit of its tensile strength. In a rotating reference system, the general theory of relativity predicts the occurrence of negative gravitational field masses in the center of rotation, with their source located in the Coriolis force field. The density of this negative gravitational field mass can be larger than the magnitude of the positive mass density of a neutron star. The repulsive gravitational force causes the centrifugal force. For a magnetized plasma placed in the rapidly spinning, hollowed-out target chamber, this repulsive force can be balanced by the magnetic force generated by thermomagnetic currents of the Nernst effect. Such a configuration does not suffer from the Rayleigh–Taylor instability, but becomes a small magnetohydrodynamic generator, amplifying the magnetic field to values about equal to those of the Nernst effect, axially confining the plasma. By placing the spinning target in the center of a lithium vortex, the fusion neutrons absorbed in the vortex can breed tritium, and at the same time remove heat from the target chamber to sustain the Nernst effect. A hot spot is thereby produced in the target chamber, which launches a thermonuclear burn wave into a cylindrical deuterium–tritium configuration. With the stability of a rapidly rotating target greatly increased, and the range of 10 MeV electrons in the wall of the cm-size ferromagnetic target, an intense 10 MeV relativistic electron beam drawn from a 10 MJ Marx generator should be sufficient to implode the target for thermonuclear ignition.

Influence of the rotor angle precision in control of interior permanent magnet synchronous machine drives and improvement method using sensorless estimator with Hall sensors

This document studies torque degradation of an internal permanent magnet motor due to an incorrect rotor angle measurement when low rotor angle resolution sensors are used. The performance of the control with different rotor angle resolution is also analysed to determine the torque error due to the use of an incorrect synchronous rotating reference system. To address torque degradation due to the low resolution of Hall sensors, a new method of estimating the rotor angle is proposed that incorporates the information from Hall sensors. In this way, good resolution is achieved, allowing the machine to be controlled over a wide speed range even with rated torque. This method is validated through simulations and real experiments.

Effect of angular aberration using the example of the laser location of a spacecraft in a geosynchronous orbit

A mathematical model for the laser location of a spacecraft in a geosynchronized orbit is generated in a rotating reference system. A formula for calculating the angular aberration that appears when the noninertial reference system is considered is derived. The maximum and minimum values of angular aberration are calculated for when the laser station is at the Equator.

The Foucault pendulum viewed as a spherical pendulum placed in a rotating reference system

For a pendulum of a given length, installed at any fixed latitude, the motion depends on three variables: the total mechanical energy, connected with the maximum angular distance of the bob from the vertical; the projection of the angular momentum along the vertical, connected with the shape of its orbit along the surface of a sphere, which may be described as a veering ellipse recalling the orbit of Mercury; and the angular velocity, Ω, of the rotating reference system given by that of the Earth. When the value of one of these variables is negligible, the ensuing motion has already been calculated in analytical grounds: a plummet, a Foucault pendulum, or a spherical pendulum including as a special case the so-called plane pendulum. This paper offers a Lagrangian approach in the case when Ω is the sidereal angular velocity of the Earth. To our knowledge, these are the first analytical solutions for any boundary conditions. When the boundary conditions are such that the bob passes through the vertical (i.e., the direction defined by the plummet), the behavior of the pendulum is as expected: the veering velocity of the plane equals the angular velocity of the Earth times the sine of the latitude. This state is reached when the projection of its angular momentum along the vertical vanishes. However, if that projection takes any other value, the motion is produced along a veering ellipse and the time rate of advance of the major axis is the sum of the previous result plus the spherical pendulum contribution. Our solutions include, as special cases, those of the spherical pendulum, or the ideal Foucault pendulum with finite amplitude.

Solar surface rotation: N-S asymmetry and recent speed-up

Context. The relation between solar surface rotation and sunspot activity still remains open. Sunspot activity has dramatically reduced in solar cycle 24 and several solar activity indices and flux measurements experienced unprecedentedly low levels during the last solar minimum. Aims. We aim to reveal the momentary variation of solar surface rotation, especially during the recent years of reducing solar activity. Methods. We used a dynamic, differentially rotating reference system to determine the best-fit annual values of the differential rotation parameters of active longitudes of solar X-ray flares and sunspots in 1977-2012. Results. The evolution of rotation of solar active longitudes obtained with X-ray flares and with sunspots is very similar. Both hemispheres speed up since the late 1990s, with the southern hemisphere rotating slightly faster than the north. Earlier, in 1980s, rotation in the northern hemisphere was considerably faster, but experienced a major decrease in the early 1990s. On the other hand, little change was found in the southern rotation during these decades. This led to a positive asymmetry in north-south rotation rate in the early part of the time interval studied. Conclusions. The rotation of both hemispheres has been speeding up at roughly the same rate since late 1990s, with the southern hemisphere rotating slightly faster than the north. This period coincides with the start of dramatic weakening of solar activity, as observed in sunspots and several other solar, interplanetary and geomagnetic parameters.

Short Lyapunov time: a method for identifying confined chaos

Context. The orbital instability of minor solar system bodies (asteroids, small satellites, moonlets, and particles) is frequently studied in terms of the Lyapunov characteristic exponent (LCE). Asteroids interior to Jupiter often exihibit very short Lyapunov times, TL, and very large radial variations, becoming Jupiter’s crossers and escapers. However, a few cases of asteroids with very short TL and no significant radial variation have been found. These orbits were called “confined chaos” or even “stable chaos”. This feature also appeared in the case of moonlets embedded in Saturn’s F ring and disturbed by the nearby satellites Prometheus and Pandora. Aims. We present a simple approach to estimating the contribution of the radial component of the LCE to identify trajectories in the “confined chaos” regime. Methods. To estimate the radial contribution to the maximum LCE, we considered a rotating reference system in which one of the axis was aligned with the radial direction of the reference trajectory. Measuring the distance in the phase space between the two nearby orbits then allowed us to separate the contribution of the radial component from the others. We applied the method to two different dynamical systems: (a) an asteroid around the Sun disturbed by Jupiter; (b) a moonlet of Saturn’s F-ring disturbed by the satellites Prometheus and Pandora. Results. In all cases, we found that the method of comparing the radial contribution of the LCE to the entire contribution allows us to correctly distinguish between confined chaos and escapers.

An improvement of ship wake detection based on the radon transform

The purpose of this article is to present a new process for ship wake detection in synthetic aperture radar (SAR) images. Most of detection algorithms uses the Radon transform. In fact, due to its properties, this approach is particularly suited for such a detection. The main originality of our work is the wedding between the Radon transform and a filtering method used here to interpolate the image in a rotating reference system, introduced by the Radon transform. This filtering method is the stochastic matched filtering technique, which allows to maximize the signal to noise ratio after process. Because of the amount of calculus this method implies, we used a new formulation based on the signal expansion into discrete cosine transform. Experimental results on SIR-C/X-SAR and ERS SAR images are presented and compared to those obtained using some classical approaches.

Slowing Down of Gamma Photons

By a transformation to a rotating reference system, a nuclear level mixing scheme is equivalent to a scheme with pure quadrupole splitting and a strong RF-drive between quadrupole split lines. It is shown that this scheme is similar to the one used in optics in which electromagnetically induced transparency (EIT) and low group velocity have been demonstrated. One can therefore consider the slowing down of gamma photons. However, with gamma photons one always works with single photons and the question remains whether the concept of group velocity can be applied.

The Sagnac effect: correct and incorrect explanations

Different explanations for the Sagnac effect are discussed. It is shown that this effect is a consequence of the relativistic law of velocity composition and that it can also be explained adequately within the framework of general relativity. When certain restrictions on the rotational velocity are imposed, the Sagnac effect can be attributed to the difference in the time dilation (or phase change) of material particle wave functions in the scalar (or correspondingly vector) gravitational potential of the inertial forces in a rotating reference system for counterpropagating waves. It is also shown that all the nonrelativistic interpretations of the Sagnac effect, which are unfortunately sometimes found in scientific papers, monographs and textbooks, are wrong in principle, even though the results they yield are accurate up to relativistic corrections in some special cases.

Numerical Solution of the Incompressible Navier–Stokes Equations with Coriolis Forces Based on the Discretization of the Total Time Derivative

In this paper we present a numerical formulation to solve the incompressible Navier–Stokes equations written in a rotating frame of reference. The method is based on a finite difference discretization in time and a finite element discretization in space. When the viscosity is very small, numerical oscillations may appear due both to the high Reynolds number and to the presence of the Coriolis forces. To overcome these oscillations, a special discretization in time is proposed. The idea is to discretize the total time derivative in an inertial basis rather than only the partial time derivative in the rotating reference system. After this is done, a further high-order approximation is introduced, leading to a problem posed in the rotating frame of reference and in spatial coordinates. In contrast with the straightforward discretization of the original equations, some additional terms appear that enhance the stability of the numerical scheme. In the absence of Coriolis forces, the method is a generalization of the characteristic Galerkin technique for convection-dominated flows.

The optical-mechanical analogy for stationary metrics in general relativity

The optical-mechanical analogy allows for a common description of the motion of particles in mechanics and for light in geometrical optics. In a recent series of articles in this journal, it has been shown that the optical-mechanical analogy can be extended to general relativity for the case of static metrics expressible in isotropic coordinates. In this paper, we extend the optical-mechanical analogy in general relativity to the case of stationary metrics. A variational principle for the trajectories of both photons and particles is derived which takes the form of Fermat’s principle or the principle of Maupertuis. When the (suitably defined) spatial portion of the metric is written as or restricted to an isotropic form, exact equations of motion for both massless and massive particles are obtained in the form of Newtonian mechanics, describing objects moving in a medium with a spatially varying index of refraction. Such restrictions of the metric commonly occur, for example, when orbital motion is considered in a plane perpendicular to the axis of rotation of a rotating black hole or spacetime. The Newtonian form of the equations of geodesic motion are illustrated by applications to a uniformly rotating reference system and a rotating black hole (the Kerr metric).

Rotating fields and particle-like states of electron-positron systems

It is argued that the field of a charge in a stationary orbit co-rotates with the charge about the centre of mass. Such a field has no radiation term. It is obtained by transformation from the co-rotating frame of reference; direct calculation would require that the constitutional properties of the medium be known, these being such as to curve light rays. For rotating reference system Corum's tetrad field is employed, the object of anholonomity affecting the derivation of fields from potentials. The force between the charge *e of positronium is found to be (1 -p2)'/' e2/ri2 when the charges follow with velocity o x r (= pc) a stationary circular orbit of diameter r12 (= 2r). Stationary orbits are selected by Bohr quantisation of canonical angular momentum. A quadratic equation is obtained for yp and for r12, where y = (1 -p2)-1/2. One solution gives the well-known 'atom-like' states. The other gives 'particle-like' states for which orbital motion is ultrarelativistic (yp = n/a, where a = 1/137), orbital diameters are small compared with eZ/mc2 (zd), m being the electron mass, and energies are 2amc2/n, small compared with mc2. In such a state positronium is termed a 'positronium unit'. The unit has a very long lifetime for radiative transitions, and should possess a weak magnetic moment (p = cued/2n). If it is assigned half-integral orbital angular momentum (hence imaginary parity) it might be identified with the neutrino. A system in which two units orbit around a stationary charge has the properties of the muon, and a system with one orbiting unit has the properties of the charged pion. Spin, charge, rest energy and decay mode are explained in each case. The very different reactivities also can be explained. A system, 'trionium', in which two charges of one sign orbit about a central stationary charge of the opposite sign is explored. Again ultrarela- tivistic states emerge, but the energies differ inappreciably from the rest energy mc2 of the central charge. However, if one adds further pairs of charges in orbits of increasing radius, the magnetic coupling between orbital currents provides energies of order mc2/a. It is speculated that such a system may afford a model for the proton. 1. Introduction and general philosophy In the early years of this century physicists were confronted with an empirical classification of atoms based on their chemical reactivity -the periodic table of the elements. In addition, there was some spectroscopic evidence to be explained, chiefly the spectrum of the hydrogen atom and Moseley's law for x-ray lines. The key for the unravelling of this situation was provided by Rutherford when he proposed his electrodynamical model of the atom, and further crucial steps were taken by Bohr when he introduced the concept of stable orbits and thereby found at least a first-order explanation for the hydrogen atom spectrum. One wonders how long progress might have been delayed had not these steps been taken. Today one has an empirical

The formation of reference systems in visual motion perception.

In visual perception complex movements are usually split up into components — e.g., in such a way that a moving reference system is applied, to which partial movements can be related. In the following experiment a perceptual vector analysis of this kind was investigated. It was shown that 1) a radial translatory movement (relative component) can be isolated perceptually from a rotatory movement (global component) and that 2) the precision of this decomposition of the complex movement is dependent on the strength at which the rotating reference system is anchored perceptually.

The relativistic magnetohydrodynamics of an accelerating plasma

In analog to the Irvine's tetrads of a rotating reference system, a tetrad field associated with a medium experiencing a linear acceleration is derived. This is used for a tetrad formulation of relativistic equations of a system composed of an accelerating plasma coupled to a frozen-in magnetic field. It is found that the magnetic and hydrodynamic variables are independent of the time in comoving coordinates, and it is shown that a steady flow with constant values of the unperturbed quantities is not possible. Some relativistic barometric formulas for magnetohydrodynamic systems which are not in thermodynamic equilibrium are obtained.

A rotating reference system

A rotating reference system

Electrodynamics in a rotating system of reference

The usual definition of electric and magnetic field intensities ( E and H) in a non-inertial coordinate system is not unique. A system of reference is here newly defined as a continuous, three dimensional distribution of observers, with each of which a frame of reference (orthonormal tetrad) is associated. The fields E and H in such a reference system will always be given by the same physical measurements as determine these quantities in an inertial system, once the effect of inertial (“fictitious”) forces has been subtracted out. A rotating reference system is strictly defined, and it is shown that E and H in this system are related to the corresponding quantities in an inertial system by a special Lorentz transformation with velocity ν = ω × r, where ω is the angular velocity of the rotating system as observed from an inertial system. The field equations in the rotating reference system are given. The angular velocity measured by local experiments by an observer in the rotating system is found to be Ω = ω (1 − r 2ω 2/c 2) −1 .