Axes Of Inertia Research Articles

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.

The Oval-like Cross-section of Femoral Neck Isthmus in Three-dimensional Morphological Analysis.

ObjectivesTo investigate the cross‐section shape of the femoral neck isthmus (FNI) in three‐dimensional reconstruction model of the femoral neck.MethodsFrom December 2009 to December 2012, computed tomography (CT) data of bilateral hip joint from 200 consecutive patients (137 males and 63 females, 69.41 ± 9.21 years old, ranged from 50–85 years old) who underwent surgical treatments for proximal femoral fracture were retrospectively reviewed. The 3D model of the proximal femur was reconstructed, and the “inertia axis” method, which was applied to measure the long and short axes of the cross‐section of the FNI, was established. The cross‐sectional area and perimeter were calculated by a formula using the length of the long and short axes and then compared with the actual measured values by the software. Correlation between the descriptive parameters of the FNI cross‐section (area, perimeter, and eccentricity) and patients' demographics (age, height, and weight) was analyzed. Stepwise linear regression analysis was used to determine the main relevant factors.ResultsThe ICC results showed excellent data reproducibility ranged from 0.989 to 0.996. There was no significant difference in the cross‐sectional area of the FNI between the actual measured values and the predicted values using the formula (732.83 ± 126.74 mm2 vs 731.62 ± 128.15 mm2, P = 0.322). The perimeter using the two methods showed narrow while significant difference (97.86 ± 8.60 mm vs 92.84 ± 8.65 mm, P < 0.001), the actual measured values were about 5 mm greater than the predicted values. The parameters (area, perimeter, and eccentricity) were significantly larger in male than female (P < 0.001). A positive correlation between the cross‐sectional area, perimeter, height, and weight was observed. The stepwise linear regression analysis showed that the regression equation of the FNI area was as follows: Y = −1083.75 + 1033.86 × HEIGHT + 1.92 × WEIGHT, R 2 = 0.489.ConclusionThe cross‐section shape of the FNI appears to be oval‐like in the 3D model, which is separated according to the inertia axis, and the findings proposed an anatomical basis for the further study of the spatial configuration of cannulated screws in the treatment of femoral neck fractures.

Characterization calibration method of strapdown inertial navigation axis in photoelectric target calibration

光电校靶采用捷联惯导系统，需将惯导轴线用光电自准直系统表征捷联，用于对设备轴线的测量。光电自准直系统所表征的轴线与惯导轴线的一致性是影响校靶精度的重要因素，为提高光电校靶系统测量精度和效率，提出了一种惯导轴线的光电表征和校准方法。该方法是在分析测量结果偏差与一致性关系的基础上，采用实验数据拟合的方式得到自准直系统光轴与惯导轴之间的角度偏差值，用于系统修正，从而实现高精度校准。通过试验，将传统光机校正法和光电校准法结合使用，可大幅度提高系统校正效率，同时得到惯导与光轴一致性精度在15″以内。试验结果表明，与传统光学平晶引出的光机校正法相比，该方法的表征准确度和校准精度更高，适用于高精度惯性测量系统。

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.

Test of Tethered Deorbiting of Space Debris

Current investigations on space tethers include their application to space debris deorbiting, specifically on the set of manoeuvres performed by a chaser tug to change the orbital parameters of a target body. Targets can be cooperative spacecraft at the end of their life or uncontrolled objects such as defunct satellites without clearly available capturing interfaces. In this latter case, a link joining tug and target may be misaligned with the target body inertia axes, influencing the attitude of both bodies; in case of rigid links, torques transmitted during tugging operations may overcome the tug attitude control system. This issue is clearly less significant in case of non-rigid connections, such as tethers; furthermore, with such connections the chaser can remain at a safe distance from the target during the whole deorbiting operation. On the other side, the initial phase of tethered space debris removal manoeuvres can be influenced by transient events, such as sudden tether tension spikes, that may cause longitudinal and lateral oscillations and, in case of resonance with the target attitude dynamics, could represent a serious issue for tug safety. In this paper it is proposed to provide the tug with a tether deployer mechanism capable to perform reel-in and reel-out, smoothing loads transmission to the target and damping oscillations. This concept is validated through a representative test campaign performed with the SPAcecRraft Testbed for Autonomous proximity operatioNs experimentS (SPARTANS) on a low friction table. A prototype of the deployer is manufactured and the deployment and rewind of a thin aluminium tape tether is proven. Test results include the verification of the tether visco-elastic characteristics with the direct measurement of spikes and oscillations and the estimation of the proposed system damping capabilities.

OPTICAL METHOD FOR MEASURING RUNOUT PARAMETERS AXIS OF THE PART RELATIVE TO THE AXIS OF ROTATION

The article is devoted to the development of an optical method and a device for measuring the runout parameters of a part relative to the axis of its rotation. A graphical diagram is provided to indicate and explain all the runout parameters of rotating part. It is shown that these parameters can be determined by the optical method using the visualization of the rotation axis and the inertia axis of the part by a collimated laser beam. The process of measuring these parameters consists of two stages. In the first stage, the inertia axis of the part is aligned with the axis of the laser beam, and in the second stage, the rotation axis of the part is aligned with the axis of the measuring camera. The measurement is based on determining the coordinates of the photoenergy center of the laser spot in the plane of the sensing array. The results of the scientific work are the developed optical method and the device based on a measuring camera, which allow to measure and reduce the runout parameters in a controlled manner, bringing them to acceptable limits by means of the known vectors of bearing adjustment movements.

The Eu:CROPIS Mass Property Campaign: Trimming a Spin-Stabilized Compact Satellite for a Long-Term Artificial Gravity Experiment

Eu:CROPIS (Euglena Combined Regenerative Organic Food Production in Space) is the first mission of DLR’s compact satellite program. The launch of Eu:CROPIS took place on December 3rd in 2018 on-board the Falcon 9 SSO-A mission. The satellite’s primary payload Eu:CROPIS features a biological experiment in the context of closed loop coupled life support systems. The Eu:CROPIS satellite mission uses spin stabilization along its Z -axis to provide defined acceleration levels for the primary and secondary payloads to simulate either a Moon or Mars gravity environment. For the payload performance, it is vital to achieve a minimum deviation between spacecraft Z -axis and the major moment of inertia (MoI) axis to minimize the offset of the envisaged acceleration levels. Specific moment of inertia ratios between the spin- and minor axes had to be maintained to allow the attitude control system to keep the satellite at a stable rotation despite environmental disturbances. This paper presents the adaptive and flexible trimming strategy applied during the flight model production, as well as the mass property measurement acceptance campaign and the respective results.

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.

МЕТОД ВІРТУАЛЬНОГО БАЗУВАННЯ ДЕТАЛЕЙ З ФОРМОЮ, НАБЛИЖЕНОЮ ДО ФОРМИ ЗАГОТОВОК

The subject matter of the article is the processes of virtual localization of near shape parts during adaptive machining. The aim is to develop an effective method for finding the starting location of a CAD model of a part with virtual localization inside a point cloud obtained by laser scanning of a workpiece. The task is to formalize the procedure for starting positioning of the part model as the first stage of the virtual localization process. The second stage for final localization proposed to use iterative algorithms with the objective function which is sensitive to the intersection of the surface parts and the workpiece. In solving the problem the starting position used tools available in today's CAD packages and 3D scanning tools. The methods used are the methods of matrix algebra, in particular, the methods for finding the main central moments of inertia of three-dimensional objects based on the tensor of inertia. The following results were obtained. When calculating the inertia tensor components is proposed to use three-dimensional scanning data of workpiece and geometrical data of part obtained from the CAD system. The result is an algorithm starting location of CAD model in the virtual localization, which in the case of blanks with oversize close to uniform can provide enough current location parts for adaptive machining tasks. It is shown that to minimize computational errors and to ensure satisfactory accuracy of localization proposed algorithm can require several iterations of the shift vector search model. Conclusions. The scientific novelty of the results obtained is as follows: in contrast to the previously used approaches, when solving the problem of virtual localization for the starting position, using the condition of coincidence of the centers of the weight of thin shells coinciding with the surfaces of the workpiece and the part, it was proposed to additionally ensure the alignment of the main central axes of inertia of these shells, which, in the case of near shape blanks, provides a positioning accuracy that may not require additional iterative procedures.

Pengembangan Proses Gerinda Silinder Permukaan Luar

The abrasive machining process is called the grinding process using a wheel grinding. The process of abrasive machining is usually in the final stages of making a product or finishing product so that it becomes essential in making products because it determines the final quality of a product. The purpose of this research is to develop the external cylindrical grinding process in column products for one of the components for drilling machines by modifying the grinding process in the lathe machine. The study was conducted using a wheel head of tools sharpening machine that is mounted on a lathe machine. The stages of this research began with the lathe process. Then without removing the product from the lathe, the external surface cylindrical grinding process is carried out so that there is no need to be set up the workpiece because the central inertia axis of the workpiece has one axis with its geometric axis. The measurement results of the external surface cylindrical grinding process are carried out using a bird-cage strategy by dividing the measurement into six segments. From each segment, the data is collected at eight measuring points. Based on the measurement results, in sample no.3 in segments B and D there is an increase in measurement results, that is Ø 100 + 0.018, but still below the maximum limit of Ø 100 + 0.020.

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 &gt; 0.99), which are thought to be common in white dwarf planetary systems.

Deformation of Lie–Poisson algebras and chirality

Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian symmetric spectra: If λ is an eigenvalue of the linearized generator, −λ and λ¯ (hence, −λ¯) are also eigenvalues—the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical Hamiltonian systems) works out differently, resulting in breaking of the Hamiltonian symmetry of spectra; time-reversal asymmetry causes chirality. This interesting phenomenon was first found in analyzing the chiral motion of the rattleback, a boat-shaped top having misaligned axes of inertia and geometry [Z. Yoshida et al., Phys. Lett. A 381, 2772–2777 (2017)]. To elucidate how chiral spectra are generated, we study the three-dimensional Lie–Poisson systems and classify the prototypes of singularities that cause symmetry breaking. The central idea is the deformation of the underlying Lie algebra; invoking Bianchi’s list of all three-dimensional Lie algebras, we show that the so-called class-B algebras, which are produced by asymmetric deformations of the simple algebra so(3), yield chiral spectra when linearized around their singularities. The theory of deformation is generalized to higher dimensions, including the infinite-dimensional Poisson manifolds relevant to fluid mechanics.

Quantum Persistent Tennis Racket Dynamics of Nanorotors.

Classical rotations of asymmetric rigid bodies are unstable around the axis of intermediate moment of inertia, causing a flipping of rotor orientation. This effect, known as the tennis racket effect, quickly averages to zero in classical ensembles since the flipping period varies significantly upon approaching the separatrix. Here, we explore the quantum rotations of rapidly spinning thermal asymmetric nanorotors and show that classically forbidden tunneling gives rise to persistent tennis racket dynamics, in stark contrast to the classical expectation. We characterize this effect, demonstrating that quantum coherent flipping dynamics can persist even in the regime where millions of angular momentum states are occupied. This persistent flipping offers a promising route for observing and exploiting quantum effects in rotational degrees of freedom for molecules and nanoparticles.

Parametric synthesis of the digital system of the course stability of a car

The problem of choosing the variable parameters of a control algorithm, implemented by a digital electronic unit of the system of the course stability of a car and providing the maximum stability margin and the maximum speed of the stabilized process during car braking is considered. It is believed that the angular velocity sensors of the car body are used as the sensitive elements of the system, whose sensitivity axes coincide in direction with the main central axes of the body inertia, and the angular deviation of the longitudinal axis of the body from a given direction is calculated by a digital electronic unit using algorithms of platformless inertial systems (PINS) in the form of a lattice function. Information about the angular velocity of a body rotation is a lattice function at the output of the Lanczos digital low-pass filter, whose input is supplied with the lattice function of the angular deviation of the body from a given direction of the car movement. The lattice control function is a weighted sum of these functions, the weight coefficients of which are variable digital parameters of the Vehicle Stability Controls that are to be selected. The perturbed motion of the continuous part of the closed system is described by ordinary differential equations, the work of the discrete part is described by relations in finite differences. By means of special transformations, the mathematical model of the perturbed motion of the closed-loop control system is represented in the form of a system of difference equations, using which the characteristic equation of the closed discrete system of the course stability of the car is constructed. Using the bilinear W-transformation, the characteristic equation is reduced to a form convenient for constructing the lines of equal degree of stability in the plane of the variable parameters of the digital electronic control unit, followed by determining the optimal values of its variable parameters, as well as assessing the effect of the selected value of the quantization period of the digital electronic unit on the area of stability of the closed digital system of the course stability of a car. Key words: system of the course stability of a car, Rodrigue Hamilton parameters, lattice function, low-frequency digital filters, variable system parameters.

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.

Three-dimensional morphological analysis of the femoral neck torsion angle\u2014an anatomical study

BackgroundThe femoral neck torsion angle (FNTA) is an important but often neglected parameter in assessments of the anatomical morphology of the femoral neck, which is often confused with the femoral neck anteversion angle (FNAA) in the current literature. Currently, the measurement methods reported in the literature all adopt the naked eye or two-dimensional (2D) visualization method, and the measurement parameters and details are not clearly defined. The objection of this research was to provide a reliable 3D method for determining the femoral neck axis, to improve the measurement method of the FNTA, and to analyze the anatomical and clinical significance of the results.MethodsComputed tomography (CT) data of 200 patients who received a lower extremity CT angiography examination were selected, and the bilateral femurs were reconstructed with three dimensional CT (3D CT). First, the 3D axis of the femoral neck was built. Second, the long axis of the cross section the femoral neck isthmus (FNI) and femoral neck basilar part (FNB) were confirmed by the “inertia axes” method, and the plane consisting of the long axis of the cross-section and the center of the femoral head was defined as the long axial plane. Third, the coronal plane of the proximal femur was determined through the long axis of the proximal femur and the femoral coronal. Finally, the FNTAs (the angles between the long axial planes and the coronal plane of the proximal femur) of FNI and FNB were measured. The size of FNTA was compared between the sexes and sides and different locations, the correlation between the parameters and age, height, and weight were evaluated.ResultsThe difference in FNTA was statistically significant between the isthmus and the basilar part (isthmus 30.58 ± 8.90° vs. basilar part 23.79 ± 3.98°; p < 0.01). Significant difference in the FNTA was observed between the sexes (males 31.99 ± 9.25° vs. females 27.49 ± 7.19°; p < 0.01). The increase in FNTA from the basilar part to the isthmus was 6.79 ± 8.06°, and the male (7.87 ± 8.57°) was greater than the female (4.44 ± 6.23°, p < 0.01). However, no significant difference in the values was observed between sides. Height exerted the greatest effect on the FNTA according to the correlation analysis (r = 0.255, p< 0.001).ConclusionsThis study found a reliable 3D method for the determination of the femoral neck axis improved the measurement method of the FTNTA and made it more accurate and repeatable. The results provided a methodological basis and theoretical support for the research and development of internal fixation device for femoral neck fracture and the spatial configuration of implants in treatment. And the optimal opening point of the femoral medullary cavity was recommended to locate at the posterior position of the top of the femoral neck cross-section during hip replacement.

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.

INVESTIGATION OF THE EVOLUTION EQUATIONS OF THE AXISYMMETRIC BODY WITH

In this article we consider mutually gravitating non-stationary two bodies: first body is «central», it is a sphere with a spherical density distribution, the second body is «satellite», which has an axisymmetric dynamic structure and form. Newtonian force interaction is characterized by an approximate expression of the force function, which takes into account the second harmonic. The differential equation of translational-rotational motion of the axisymmetric body is derived with variable mass and variable size in a relative coordinate system. The axes of the own coordinate system for nonstationary two bodies coincides with the main axes of inertia and this position remains unchanged during evolution. The mass of bodies are varied isotropically in the different rates. The problem is investigated by methods perturbation theory. The equations of secular perturbations of translational-rotational motion of satellite are deduced in the analogues osculating elements Delaunay-Andoyer. The solutions of the differential equations of the perturbed motion are obtained by the numerical method and the graphs are constructed using the Wolfram Mathematica package.