Principal Axes Of Inertia Research Articles

Model based estimation of inertial parameters of a rigid rotor having dynamic unbalance on Active Magnetic Bearings in presence of noise

Unbalance in rotors is known as the primary source of excitation, the state of unbalance changes over time and, therefore, industrial rotors require online or smart identification and attenuation of unbalance for smooth operation. To begin with, this paper proposes the use of Active Magnetic Bearings (AMBs) to identify the unbalance. Unbalance is defined as the location of centre of mass of the rotor and orientation of the axes of principal mass moments of inertia, both with respect to the spin-axis, assumed as the geometric axis of symmetry of rotor. Non-uniform distribution of mass makes a rotor dynamically unbalanced. The centre of mass shifts from the spin-axis and none of the principal axes of inertia is aligned with it. Dynamic behaviour of such a rotor is modelled by a set of differential equations with time varying coefficients. A rigid rotor is assumed to be supported on two AMBs with all known characteristics. A methodology based on modelling and simulation is proposed to determine the eccentricity and the orientation of the principal axes of inertia (angles of its skewness) by utilizing the force and response of the rotor recorded by each AMB. The median of simulated inertial properties are shown to be in close agreement with their actual values. The proposed methodology is also found robust enough in the presence of different types of simulated noise used to pollute signals from the bearings. Therefore, the methodology is suitable for online identification of inertial properties of a rotor.

ПОСТУПАТЕЛЬНО-ВРАЩАТЕЛЬНОЕ ДВИЖЕНИЕ НЕСТАЦИОНАРНОГО ОСЕСИММЕТРИЧНОГО ТЕЛА

A nonstationary two-body problem is considered such that one of the bodies has a spherically symmetric density distribution and is central, while the other one is a satellite with axisymmetric dynamical structure, shape, and variable oblateness. Newton’s interaction force is characterized by an approximate expression of the force function up to the second harmonic. The masses of the central body and the satellite vary isotropically at different rates and do not occur reactive forces and additional rotational moments. The nonstationary axisymmetric body have an equatorial plane of symmetry. Thus, it has three mutually perpendicular planes of symmetry. The axes of its intrinsic coordinate system coincide with the principal axes of inertia and they are directed along the intersection lines of these three mutually perpendicular planes. This position remains unchangeable during the evolution. Equations of motion of the satellite in a relative system of coordinates are considered. The translational- rotational motion of the nonstationary axisymmetric body in the gravitational field of the nonstationary ball is studied by perturbation theory methods. The equations of secular perturbations reduces to the fourth order system with one first integral. This first integral is considered and three-dimensional graphs of this first integral are plotted using the Wolfram Mathematica system.

Revealing the Structure and Internal Rotation of the Sagittarius Dwarf Spheroidal Galaxy with Gaia and Machine Learning

Abstract We present a detailed study of the internal structure and kinematics of the core of the Sagittarius dwarf spheroidal galaxy (Sgr). Using machine-learning techniques, we have combined the information provided by 3300 RR Lyrae stars, more than 2000 spectroscopically observed stars, and the Gaia second data release to derive the full phase space, i.e., 3D positions and kinematics, of more than 1.2 × 105 member stars in the core of the galaxy. Our results show that Sgr has a bar structure ∼2.5 kpc long, and that tidal tails emerge from its tips to form what it is known as the Sgr stream. The main body of the galaxy, strongly sheared by tidal forces, is a triaxial (almost prolate) ellipsoid with its longest principal axis of inertia inclined 43° ± 6° with respect to the plane of the sky and axis ratios of 1:0.67:0.60. Its external regions are expanding mainly along its longest principal axis, yet the galaxy conserves an inner core of about 500×330×300 pc that shows no net expansion and is rotating at v rot = 4.13 ± 0.16 km s−1. The internal angular momentum of the galaxy forms an angle θ = 18° ± 6° with respect to its orbital angular momentum, meaning that Sgr is in an inclined prograde orbit around the Milky Way. We compared our results with predictions from N-body models with spherical, pressure-supported progenitors and a model whose progenitor is a flattened rotating disk. Only the rotating model, based on preexisting simulations aimed at reproducing the line-of-sight velocity gradients observed in Sgr, was able to reproduce the observed properties in the core of the galaxy.

ГЕОДИНАМІКА

This study aims to derive the Earth’s temporally varying Earth’s tensor of inertia based on the dynamical ellipticity , the coefficients , from UT/CSR data. They allow to find the time-varying Earth’s mechanical and geometrical parameters during the following periods: (a) from 1976 to 2020 based on monthly and weekly solutions of the coefficient ; (b) from 1992 to 2020 based on monthly and weekly solutions of the non zero coefficients , related to the principal axes of inertia, allowing to build models their long-term variations. Differences between and , given in various systems, represent the average value , which is smaller than time variations of or , characterizing a high quality of UT/CSR solutions. Two models for the time-dependent dynamical ellipticity were constructed using long-term variations for the zonal coefficient during the past 44 and 27.5 years. The approximate formulas for the time-dependent dynamical ellipticity were provided by the additional estimation of each parameter of the Taylor series, fixing at epoch =J2000 according to the IAU2000/2006 precession-nutation theory. The potential of the time-dependent gravitational quadrupole according to Maxwell theory was used to derive the new exact formulas for the orientation of the principal axes , , via location of the two quadrupole axes. Hence, the Earth’s time-dependent mechanical and geometrical parameters, including the gravitational quadrupole, the principal axes and the principal moments of inertia were computed at each moment during the past 27.5 years from 1992 to 2020. However, their linear change in all the considered parameters is rather unclear because of their various behavior on different time-intervals including variations of a sign of the considered effects due to a jump in the time-series during the time-period 1998 – 2002. The Earth’s 3D and 1D density models were constructed based on the restricted solution of the 3D Cartesian moments inside the ellipsoid of the revolution. They were derived with conditions to conserve the time-dependent gravitational potential from zero to second degree, the dynamical ellipticity, the polar flattening, basic radial jumps of density as sampled for the PREM model, and the long-term variations in space-time mass density distribution. It is important to note that in solving the inverse problem, the time dependence in the Earth's inertia tensor arises due to changes in the Earth's density, but does not depend on changes in its shape, which is confirmed by the corresponding equations where flattening is canceled.

Analysis of regular precession conditions for asymmetrical liquid-filled rigid bodies

To describe the rotation of a “rigid mantle + liquid core” system, the Poincare–Hough–Zhukovsky equations are used. An analysis is made of the previously obtained (Ol’shanskii in Celest Mech Dyn Astron 131(12):Article number:57, 2019) conditions for regular precession of a system that does not have an axial symmetry. Upon receipt of the conditions, it is considered that the external torque can be neglected as, for example, for free-floating planetary bodies. In the case when the axis of proper rotation is one of the principal axes of inertia, the formulas for the rates of precession and proper rotation have been simplified. For a particular case, when the shape of the core differs little from spherical, it is shown that the precession and proper rotation rates differ from the rates of empty axisymmetric rigid mantle by values of the first order of smallness. The ratio of these rates differs from the ratio for the rigid mantle by the second-order value. For a system with the axis of proper rotation deviated from a principal axis of inertia, the expression of the principal moments of inertia of the mantle through the moments of inertia of the core and one arbitrary parameter is found. The formulas for finding the angular velocities and for determining the position of the axis of proper rotation relative to the mantle are written in simple parametric form. The possibility of regular precession with the axis of proper rotation, which does not coincide with any principal axes, is studied in the case when the shape of the core differs little from the spherical one. Examples of elongated non-axisymmetric systems that allow precession with the axis of proper rotation deviated from principal axis of inertia are given.

Gigayear-timescale Destruction of High-eccentricity Asteroids by Spin and Why 2006 HY51 Has Been Spared

Abstract Asteroids and other small celestial bodies have markedly prolate shapes, and the perturbative triaxial torques that are applied during pericenter passages in highly eccentric orbits trigger and sustain a state of chaotic rotation. Because the prograde spin rate around the principal axis of inertia is not bounded from above, it can accidentally reach the threshold value corresponding to rotational breakup. Previous investigations of this process were limited to integrations of ∼103 orbits because of the stiff equation of motion. We present here a fast 1D simulation method to compute the evolution of this spin rate over ∼109 orbits. We apply the method to the most eccentric solar system asteroid known, 2006 HY51 (with e = 0.9684), and find that for any reasonably expected shape parameters it can never be accelerated to breakup speed. However, primordial solar system asteroids on more eccentric orbits may have already broken up from this type of rotational fission. The method also represents a promising opportunity to investigate the long-term evolution of extremely eccentric triaxial exo-asteroids (e > 0.99), which are thought to be common in white dwarf planetary systems.

Motion and inertial parameter estimation of non-cooperative target on orbit using stereo vision

In the on-orbit servicing missions, autonomous close proximity operations require knowledge of the target’s motion and inertial parameters in order to safely interact with the target. In particular, the motion and parameter estimation of an uncooperative target is a challenging task because of the lack of some prior information about the target in unfamiliar environments. In this paper, a novel method is developed for an accurate estimation of the motion and inertial parameters of an unknown target using stereo vision measurements only. Instead of using relative position and pose as observation information, this paper chooses three non-collinear feature points on the target as estimation measurements. This treatment allows us to estimate the principal axis of inertia of the target directly, and avoid estimating the quaternion between the target measurement coordinate system and the principal axis of inertia coordinate system, which is a bilinear problem, and is difficult to obtain global convergence. Based on the relative kinematics and dynamics equations of the target, the state equation and observation equation are derived, and the Extended Kalman Filter (EKF) is designed. As a result, the developed algorithm realizes the estimation of the complete unknown information of the target, including relative position, relative velocity, attitude quaternion, angular velocity, the direction of the principal axis of inertia, inertia ratio, and the position of center of mass. Numerical simulations and experimental results verify the convergence and effectiveness of the proposed filter estimation method.

Precessing Flaring Magnetar as a Source of Repeating FRB 180916.J0158+65

Abstract Recently, the Canadian Hydrogen Intensity Mapping Experiment detected periodicity in the bursting rate of the repeating FRB 180916.J0158+65. In a popular class of models, the fast radio bursts (FRBs) are created by magnetic flares of a hyperactive magnetar driven by fast ambipolar diffusion in the core. We point out that in this scenario the magnetar is expected to precess freely with a period of weeks to months. The internal magnetic field B ∼ 1016 G deforms the star, and magnetic flares induce sudden changes in magnetic stresses. The resulting torques and displacements of the principal axes of inertia are capable of pumping a significant amplitude of precession. The anisotropy of the flaring FRB activity, combined with precession, implies a strong periodic modulation of the visible bursting rate. The ultrastrong field invoked in the magnetar model provides: (1) energy for the frequent giant flares, (2) the high rate of ambipolar diffusion, releasing the magnetic energy on the timescale ∼109 s, (3) the core temperature T ≈ 109 K, likely above the critical temperature for neutron superfluidity, (4) strong magnetospheric torques, which efficiently spin down the star, and (5) deformation with ellipticity ϵ ≳ 10−6, much greater than the rotational deformation. These conditions result in a precession with negligible viscous damping, and can explain the observed 16 day period in FRB 180916.J0158+65. The increase of precession period due to the magnetar spindown should become measurable in the near future.

Symbolic-Numerical Analysis of the Necessary Stability Conditions for the Relative Equilibria of an Orbital Gyrostat

Using the software developed on the basis of the computer algebra system Mathematica, we study the rotational motion along the circular orbit of a satellite-gyrostat in a Newtonian central field of forces. The linearized equations of a perturbed motion in the vicinity of the relative equilibrium of the system are constructed on a computer in symbolic form, and the necessary stability conditions are obtained for the equilibrium. Implementing the parametric analysis of the derived inequalities, we consider one of the cases when the vector of the gyrostatic moment of the system lies in one of the planes formed by the principal central axes of inertia. The obtained stability regions have an analytical form or a graphical representation as 2D images.

New estimation of triaxial three-layered Earth's inertia tensor and solutions of Earth rotation normal modes

Until now, the calculation of the principal inertia moment of the triaxial three-layered Earth mainly adopts the scaling method. This method assumes that the corresponding principal inertia axes of the layers coincide each other, but this is not the case. In this paper, a rigorous tensor transformation rule is adopted to calculate the principal inertia moments (PIMs) of different layers. Appling the new estimated PIMs to the triaxial three-layered Earth rotation theory with considering various couplings, the numerical calculations show that the periods of the Chandler Wobble (CW), Free Core Nutation (FCN), Free Inner Core Nutation (FICN) and Inner Core Wobble (ICW) are respectively 433.0, 430.8, 943.9 and 2735.9 mean solar days, which are well comparable with the corresponding values accepted at present in geoscience community. Better estimates of the PIMs of different layers may provide better constrains on relevant physical parameters of the Earth's interior.

Relative Position and Relative Rotation in Supramolecular Systems through the Analysis of the Principal Axes of Inertia: Ferrocene/Cucurbit[7]uril and Ferrocenyl Azide/β-Cyclodextrin Case Studies.

Parameters comprising the relative position and relative rotation of molecules can be evaluated when the principal axes of inertia of the entities in a supramolecular association are employed as reference. Such information applies to the characterization and identification of experimental and theoretical nonbonded systems. The parameters are relevant to geometric comparison (for theory and experiment) and, for instance, to monitoring structures by theoretical simulations. This work introduces a software developed to obtain such parameters through the discussion of some intriguing host–guest systems, the ferrocene/cucurbit[7]uril and ferrocenyl azide/β-cyclodextrin. The ideas within this contribution naturally apply to the study of other nonbonded associations beyond host–guest chemistry. A modified version of the software discussed herein serves to obtain user-defined spatial arrangements for two nonbonded entities. Therefore, with a given geometry, for instance, from X-ray data, the parameters can be derived, and with the parameters, from a theoretical perspective, a spatial arrangement can be obtained.

Drift of the Earth's Principal Axes of Inertia from GRACE and Satellite Laser Ranging Data.

The location of the Earth’s principal axes of inertia is a foundation for all the theories and solutions of its rotation, and thus has a broad effect on many fields, including astronomy, geodesy, and satellite-based positioning and navigation systems. That location is determined by the second-degree Stokes coefficients of the geopotential. Accurate solutions for those coefficients were limited to the stationary case for many years, but the situation improved with the accomplishment of Gravity Recovery and Climate Experiment (GRACE), and nowadays several solutions for the time-varying geopotential have been derived based on gravity and satellite laser ranging data, with time resolutions reaching one month or one week. Although those solutions are already accurate enough to compute the evolution of the Earth’s axes of inertia along more than a decade, such an analysis has never been performed. In this paper, we present the first analysis of this problem, taking advantage of previous analytical derivations to simplify the computations and the estimation of the uncertainty of solutions. The results are rather striking, since the axes of inertia do not move around some mean position fixed to a given terrestrial reference frame in this period, but drift away from their initial location in a slow but clear and not negligible manner.

Dynamics of a System of Two Connected Bodies Moving along a Circular Orbit around the Earth

Symbolic–numeric methods are used to investigate the dynamics of a system of two bodies connected by a spherical hinge. The system is assumed to move along a circular orbit under the action of gravitational torque. The equilibrium orientations of the two-body system are determined by the real roots of a system of 12 algebraic equations of the stationary motions. Attention is paid to the study of the conditions of existence of the equilibrium orientations of the system of two bodies refers to special cases when one of the principal axes of inertia of each of the two bodies coincides with either the normal of the orbital plane, the radius vector or the tangent to the orbit. Nine distinct solutions are found within an approach which uses the computer algebra method based on the algorithm for the construction of a Gröbner basis.

Application of Computer Algebra Methods to Investigation of Stationary Motions of a System of Two Connected Bodies Moving in a Circular Orbit

Computer algebra and numerical methods were used to investigate the properties of a nonlinear algebraic system determining the equilibrium orientations of a system of two bodies connected by a spherical hinge that move in a circular orbit under the action of a gravitational torque. Primary attention was given to equilibrium orientations of the two-body system in the special cases when one of the principal axes of inertia of both the first and second body coincides with the normal to the orbital plane, the radius vector, or the tangent to the orbit. To determine the equilibrium orientations of the two-body system, the set of stationary algebraic equations of motion was decomposed into nine subsystems. The system of algebraic equations was solved by applying algorithms for constructing Grobner bases. The equilibrium positions were determined by numerically analyzing the roots of the algebraic equations from the constructed Grobner basis.

Enhancement of the attitude dynamics capabilities of the spinning spacecraft using inertial morphing

ABSTRACTIn the previous works by the authors, an efficient method of control of the inversion of the spinning spacecraft was proposed. This method was prompted by the Dzhanibekov’s Effect or Tennis Racket Theorem, which are often seen by many as odd or even mysterious. For the spacecraft, initially undergoing periodic flipping motion, proposed method allows to completely stop these flips by transferring the unstable motion into the regular stable spin. Similarly, the method allows activation of the flipping motion of the spacecraft, which is initially undergoing its stable spin. In this paper, spacecraft designs, which have inertial morphing capabilities, are considered and their advantages are further investigated. For general formulation, the ability of the spacecraft to change its inertial properties, associated with all three principal axes of inertia, are assumed. For simulation of these types of spacecraft systems, extended Euler’s equations are used and peculiar dynamics of the spacecraft is illustrated with a several study cases. Complex spacecraft attitude dynamics manoeuvres, using geometric interpretation, employing angular momentum spheres and kinetic energy ellipsoids, are considered in detail. Contributions of the inertial morphing to the changes of the shape of the kinetic energy ellipsoid are demonstrated and are related to the resultant various feature manoeuvres, including inversion and re-orientation. A method of reduction of the compound rotation of the spacecraft into a single stable predominant rotation around one of the body axes was proposed. This is achieved via multi-stage morphing and employing proposed instalment into separatrices. Implementation of the morphing control capabilities are discussed. For the periodic inversion motions, calculation of the periods of the flipping motion, based on the complete elliptic integral of the first kind, is performed. Flipping periods for various combinations of inertial properties of the spacecraft are presented in a systematic way. This paper discusses strategies to the increase or reduction the flipping and/or wobbling motions. A discovered asymmetric ridge of high periods for peculiar combinations of the inertial properties is discussed in detail. Numerous examples are illustrated with animations in virtual reality, facilitating explanation of the analysis and control methodologies to a wide audience, including specialists, industry and students.

New cases of regular precession of an asymmetric liquid-filled rigid body

To describe the rotation of a rigid body with an ellipsoidal cavity filled with an ideal vorticated liquid, the Poincare–Hough–Zhukovsky equations are used. It is obtained constraints (hereinafter referred to as configuration conditions) on the mass distribution and cavity dimensions of an asymmetric liquid-filled rigid body under which the rigid body can perform the regular precession. Two possible nontrivial cases are indicated when one or two components of the direct vector of the axis of proper rotation are equal to zero. It is shown that if the axis of proper rotation coincides with one of the principal axes of inertia of the system, it suffices to fulfill one configuration condition. The ratio between the periods of proper rotation and precession is found. The regular precession of a system in which the principal moments of inertia are close to each other and the cavity is close to spherical is considered. For the case when the difference between the two equatorial moments of inertia is an order of magnitude smaller than the difference between the equatorial and polar moments, the main part of the ratio between the periods coincides with the Euler period, and the correction is due to the inequality between the equatorial moments of inertia. In the case when the axis of proper rotation is orthogonal to a principal axis and is not coincident with any other principal axis, the number of configuration conditions is two. Expressions for the rates of precession and proper rotation are obtained, and the position in the body of the axis of proper rotation is indicated. Special cases are given that allow simplification of the configuration conditions.

Application of Computer Algebra Methods for Investigation of the Stationary Motions of the System of Two Connected Bodies Moving along a Circular Orbit

Computer algebra and numeric methods are used to investigate properties of a nonlinear algebraic system that determines the equilibrium orientations for a system of two bodies connected by a spherical hinge that moves along a circular orbit under the action of gravitational torque. The main attention is paid to study the conditions of existence of the equilibrium orientations for the system of two bodies for special cases, when one of the principal axes of inertia both the first and second body coincides with the normal to the orbital plane, with radius vector or tangent to the orbit. To determine the equilibrium orientations for the system of two bodies, the system of 12 stationary algebraic equations is decomposed into 9 subsystems. The computer algebra method based on the algorithm for the construction of a Grobner basis applied to solve the stationary motion system of algebraic equations. Depending on the parameters of the problem, the number of equilibria is found by numerical analysis of the real roots of the algebraic equations from the Grobner basis constructed.

Comparison of the hindfoot axes of a multi-segment foot model to the underlying bony anatomy

Musculoskeletal models used in gait analysis require coordinate systems to be identified for the body segments of interest. It is not obvious how hindfoot (or rearfoot) axes defined by skin-mounted markers relate to the anatomy of the underlying bones. The aim of this study was to compare the marker-based axes of the hindfoot in a multi-segment foot model to the orientations of the talus and calcaneus as characterized by their principal axes of inertia. Twenty adult females with no known foot deformities had radio-opaque markers placed on their feet and ankles at the foot model marker locations. CT images of the feet were acquired as the participants lay supine with their feet in a semi-weight bearing posture. The spatial coordinates of the markers were obtained from the images and used to define the foot model axes. Segmented masks of the tali and calcanei were used to create 3D bone models, from which the principal axes of the bones were obtained. The orientations of the principal axes were either within the range of typical values reported in the imaging literature or differed in ways that could be explained by variations in how the angles were defined. The model hindfoot axis orientations relative to the principal axes of the bones had little bias but were highly variable. Consideration of coronal plane hindfoot alignment as measured clinically and radiographically suggested that the model hindfoot coordinate system represents the posterior calcaneal tuberosity, rather than the calcaneus as a whole.

Two-stage filter for inertia characteristics estimation of high-speed tumbling targets

The estimation of the inertia characteristics of a tumbling target before capturing is one of the greatest challenges in on-orbit servicing missions, especially when the time interval between consecutive measurements is large or the rotational speed of the target is very high. In this paper, a two-stage constant state filter (TCSF) is proposed to predict the attitude and estimate the inertia characteristics, including the principal inertia ratios and the principal inertia axes' directions, of a high-speed tumbling target. The man innovation of this paper is to design an Affiliate Filter to estimate the inertia tensor of the target, based on the estimates of angular velocity from an Adaptive Constant State Filter. Compared with the conventional estimation methods like the unscented Kalman filter method, the proposed TCSF shows better stability and accuracy in strongly nonlinear situations, such as that where the time interval between consecutive measurements is 10 s and the angular velocity is more than 14 deg/s. The Monte Carlo simulations have been proposed to demonstrate the performances of the proposed TCSF method.

Dual-arm coordinated capturing of an unknown tumbling target based on efficient parameters estimation

A malfunctioned satellite or other space debris is generally non-cooperative and tumbling, bringing great challenge to capture and remove it. In this paper, we propose a dual-arm coordinated capturing method based on efficient parameters estimation. Firstly, the dynamics model of a tumbling target is deduced in details. Its motion characteristics are then analyzed. Secondly, we design an efficient Hybrid Kalman Filter (HKF) by combining Extended Kalman Filter (EKF) with Unscented Kalman Filter (UKF). It effectively overcomes the shortcoming of low accuracy of EKF and long iteration time of UKF, and improves the speed and accuracy of the Kalman Filter iteration algorithm. Two movement cases of an uncontrolled target are considered: one is rotation around the principal axes of inertia; the other is rotation around arbitrary axes. Thirdly, the estimated motion and inertia parameters are used to plan the trajectories of a dual-arm space robot to capture the tumbling target. Finally, the simulation environment is created and the proposed method is verified. The simulation results show that the proposed HKF algorithm can estimate the attitude quaternion, angular velocity, and the inertia tensor (including Ixx, Iyy, Izz, Ixy, Ixz and Iyz) with higher accuracy (compared to EKF) and lower computation cost (compared to UKF); the planned trajectories of the dual-arm space robot are effectively for tumbling target capturing.