Nutation Angle Research Articles

Analysis of satellite attitude motion in a three-body tethered system during deployment via integral manifolds

This paper aims to analyze the impact of unideal end-body configurations on attitude motion during the tether deployment process in a linear three-body tethered system (LTBTS). A main challenge in this process is the resonance between the nutation and spin angles caused by the unideal end-body configurations, leading to severe nutation angle oscillations. These oscillations can result in tether entanglement with the end-bodies or even tether rupture. To address this issue, the deployment model for the LTBTS is first established using the Lagrangian equations, with the end-body attitude described by Eulerian angles. Secondly, the system involves the separation of two subsatellites from the central main satellite in opposite directions. The dynamics response of the deployment process under unideal configuration of end-bodies is investigated. Resonance phenomena in nutation and spin angles are observed due to errors in initial angles/angular velocities, offset errors of tether connection points, and unideal structural characteristics of the satellites. Thirdly, to further understand this resonance, analytical solutions for resonance are derived by transforming the nutation angle equations, and satellite attitude equations are systematically solved via integral manifold methods. Potential resonance issues are mitigated by reducing system asymmetry and minimizing initial disturbances. Finally, the effectiveness of the dynamic analysis is validated through simulations.

The elliptope and bounds in the rotational dynamics of rigid bodies

The main result of this paper establishes a new polynomial relation among several projections of the angular momentum vector on the spatial and moving frames involved in the rotational dynamics of rigid bodies. This relation is surprisingly simple to derive and provides a geometric insight into the attitude dynamics of bodies with axial symmetry, explaining the bounds for the nutation angle [Formula: see text].

A parametric study of precession driven dynamos inside a sphere

The dynamo actions of an electrically conducting fluid in a precessing sphere are investigated over a wide range of parameters by direct numerical simulation using a Galerkin spectral method. The focus of this work is to identify the most promising parameter regimes for the dynamo action and to investigate the characteristics of the magnetic field generated by precession. The influence of different nutation angles (30°,60°,90°) and different precession ratios on the ability to drive dynamo action are investigated. The optimal angle for dynamo actions is found at 90°, followed by 60° with retrograde precession. A moderate precession ratio around 0.3 is shown to be more feasible for dynamo actions. A rich set of self-sustained dynamo solutions are obtained in the parameter space we explored, including steady, periodic, quasi-periodic, and turbulent dynamos. The structure of the generated magnetic fields is analyzed by using helical wave decomposition. None of the precession driven dynamos we obtained produce a predominantly dipolar field, contrary to the convection driven dynamos. The long-time evolution of the magnetic dipole moment is investigated and different types of polarity reversals are observed.

Simultaneous rotation-nutation suppression for a tumbling satellite via a flexible rod

Effectively capturing a malfunctioning tumbling satellite by a servicing spacecraft is nowadays a worldwide challenge. Because of the potential collision between the chaser and the target spacecraft, it is risky to directly capture the tumbling satellite. An innovative approach is to use a flexible brush/rod as the end-effector of a chaser to tackle the tumbling target. However, almost all studies are confined to reducing the pure rotational motion without considering the inevitable nutation motion. To ensure safe capture, it is necessary to suppress both the nutation and rotation simultaneously to the desired low level. There are two difficulties with the simultaneous suppression problem: i) the ideal contact point between the rod and the target is difficult to pinpoint for simultaneous suppression, and ii) the solution of the rod-target dynamic system (described by nonlinear partial differential equations) is extremely time-consuming especially with the limited computing capability of the on-board computer. To conquer these difficulties, a simultaneous rotation-nutation suppression (SRNS) method based on the equal ratio principle is proposed to simultaneously reduce angular velocity and nutation angle. Besides, a highly efficient simplified rod-target model is established based on machine learning. Numerical simulations have been carried out to verify the effectiveness and efficiency of the proposed method.

Calculation of mutual inductance between arbitrarily positioned planar spiral coils for wireless power applications

Mutual inductance is one of the main parameters required to determine the power link’s performance (output voltage, efficiency) in wireless power transfer. The coils are often misaligned angularly in these applications, which affects the mutual inductance and thus the performance. Hence, an accurate calculation of mutual inductance is necessary to decide the working region of the coil. This paper presents an analytical calculation of mutual inductance between two planar spiral coils under angular misalignment conditions. By solving the Neumann integral formula, mutual inductance is derived for constant current-carrying coils, and the final mutual inductance value is calculated numerically. The influence of angular misalignment of the coil, which can be due to nutation and spin angles, on mutual inductance is studied in detail. The mutual inductance of the spiral coil is calculated for different misalignment cases. The accuracy of the calculation results is verified by comparing it with conventional formulas (mainly the Liu, the Babic formula, and the Poletkin formula) and by simulation using the finite element method. The proposed method is a more generalized and simpler one that can be used to calculate the mutual inductance of any size of coils, either spiral or circular, with any lateral and angular misalignments. Finally, a couple of spiral coils are fabricated to validate it experimentally. The comparison of the simulation and experiment results with the calculation result shows its accuracy. Thus, the proposed method can be applied to compute mutual inductance in any angularly misaligned coupling coils for the optimization of the wireless power transfer and their design.

REORIENTATION OF THE EARTH REMOTE SENSING SPACECRAFT USING ROTORS

The article describes a method of reorienting of the Earth remote sensing gyrostat satellite using a flywheel engine, while located in the same vertical plane with the observation object. In the course of the study, a geometric dependence of the nutation angle on time was derived, differential equations of the motion of the spacecraft relative to the center of mass were generated and solutions were obtained. These obtained equations allows to determine motion parameters (coordinates and speeds) depending on the inertial mass characteristics of the system, initial conditions and time, as well as control the effect of these parameters on the system. The article presents the results of conducted studies that show the performance of the developed mathematical model. Thanks to the developed model, it is possible to determine the necessary control actions to target the gyrostat satellite to the observation object with high accuracy.

Mathematical Model of Impact Projectile Flight Dynamics as an Element of its Digital Twin

This paper gives the analysis of the structure and characteristics of the mathematical model of the impact projectile’s flight dynamics. The model is designed for being used as an element of the projectile’s digital twin. The model is based on differential motion equations of a gyro-stabilized solid with an axisymmetric mass distribution. Different types of angle variables were chosen for describing aerodynamics and formulating equations of motion. Non-linear (considering the nutation angle) dependences for aerodynamic coefficients are proposed. They are created by applying proven scientific concepts and research methods in aerodynamics of axisymmetric body and by comparing with known numerical and experimental results obtained in exterior ballistics of gyro-stabilized aviation and artillery projectiles. Special aspects of initial conditions for angles and angular velocity were also studied. Since the impact projectile is considered as an axially symmetric body, its self-rotation angle is not of practical inte rest. Using algebraic manipulations, the differential equation for this angle was eliminated from the set of equations. This has made it possible to significantly reduce stiff of the remaining system of differential equations. The Dormand-Prince method is recommended as a method of numerical integration. The method of the eight-order (with seventh-order uncertainty estimate) allows getting the high accurate solution of the differential equations set under relatively small computing costs. The model allows computing the projectile trajectory under various initial conditions, including the flight with high nutation angles up to 87°—89°. As a result, there is a possibility to determine the nature of the interaction between impact projectiles and typical targets (ricochet, surface effect, after-penetration effect) within a wide range of approach angles to the target’s surface (skin) unattainable during the full-scale tests. The possibility of solving similar problems allows to recommend the designed model as an element of the impact projectile’s digital twin intended for testing its exterior ballistics on the digital (virtual) test range. All testing calculations and final modeling were made by using the "GNU Octave" computational software package.

ВПЛИВ ПОЛЯРНОГО ТА ЕКВАТОРІАЛЬНОГО ДЕМПФУЮЧИХ МОМЕНТІВ СНАРЯДА НА ДАЛЬНІСТЬ ЙОГО ПОЛЬОТУ

A relevant and important issue in the calculation of projectile flight trajectories is the definition and presentation of the equatorial and polar damping moments in the system of differential equations of spatial motion of projectiles. It is shown that the damping moments are determined by the shape and orientation of the projectile, the nature of the flow, the type of boundary layer and its interaction with shock waves, the speed, the height of the projectile and its nutation angle. To evaluate the influence of the aerodynamic coefficients of the equatorial and polar damping moments (their aerodynamic coefficients) on the flight range of the projectile, the method of differences is used, which consists in solving the system of differential equations of the spatial motion of the projectile so that changing the value of the aerodynamic coefficient results in a change in the flight range. Numerical modeling of the dependence of the flight range of the 155-mm HE Assegai M2000 projectile on the change in the aerodynamic coefficients of the equatorial and polar damping moments by 1% was carried out. It is shown that the aerodynamic coefficient of the polar damping moment creates the largest errors in the range at the maximum and minimum charges, namely, at the maximum charge the error reaches 0.012%D, at the minimum charge – 0.01%D, the effect on intermediate charges is manifested in a much smaller form, and not exceeds the value of 0.005%D. The largest error in the flight range of the projectile from the aerodynamic coefficient of the equatorial damping moment is observed at the maximum charge, the deviation reaches 0.07%D, the smallest error at the minimum charge is 0.0001%D. The obtained results make it possible to estimate the required accuracy of determining the aerodynamic coefficients of the equatorial and polar damping moments under different firing conditions of artillery systems.

Effect of Archimedes number on the dynamics of free-falling perforated disks

The dynamics of perforated disks falling freely in a large expanse of viscous fluid at rest is investigated numerically. This complex fluid–structure interaction is solved via large eddy simulation. This numerical algorithm is verified and validated with available experimental results. The influence of Archimedes number expressing the ratio between the gravity-buoyancy and viscosity effects is discussed thoroughly, including kinematics and dynamics. Two critical Archimedes numbers are identified, Arcr1≈450 and Arcr2≈950, respectively. At these two critical Archimedes numbers, both kinematic and dynamic variables change trends. In this paper, we focus on the statistics of free-falling perforated disks. With the Archimedes number Ar increasing, the average angle of attack ⟨AoA⟩ and descent velocity ⟨Uz⟩ decrease gradually, and they arrive at a fixed value finally (here, ⟨·⟩ represents a time-average result); On the contrary, the other kinetic variables change violently when Ar is around 900, for example, terminal velocity ⟨Ut⟩. Additionally, phase differences of kinematic and dynamic variables are analyzed. A constant phase difference between the nutation angle θ and normal force FN is identified, about 66°, which is independent of Ar. Vortex structures are visualized using Q-criterion, and triangular vortex is omnipresent around holes. During the descent, a helical vortex always attaches to the perforated disk outer edge. With Ar increasing, complex vortex interaction appears, for example, merging and stretching. Some unusual behaviors in the numerical results are analyzed from the perspective of wake dynamics.

VERIFICATION OF THE MATHEMATICAL FLIGHT MODEL THE PROJECTILE AS A MODIFIED MODEL OF THE MATERIAL POINT

The analysis of mathematical models of the flight of projectiles has been carried out, it is shown that the nature of their provision varies depending on the degree of reliability of the representation of the real physical process of its flight by a mathematical model, the adequate consideration of certain forces (moments) acting on the projectile, as well as the level of information about external flight conditions. In the simplest mathematical model with three degrees of freedom, the projectile is considered as a material point that moves in the atmosphere under the influence of gravity and full aerodynamic force. The dynamics of projectile flight is most completely described by the mathematical model of the movement of a solid body with six degrees of freedom. In the intermediate model with four degrees of freedom, all aerodynamic forces are taken into account, the orientation of the projectile is characterized by taking into account the angle of nutation, and the kinetic energy of the rotational movement is taken into account through the angular velocity of the projectile around its axis of symmetry. The process of restori ng the coefficients of aerodynamic forces using the material point model does not provide the necessary accuracy and adequacy of the description of the projectile flight process; of the modified material point model is much simpler (the least complex) compared to the model of the motion of a rigid body with six degrees of freedom due to the smaller number of coefficients and differential equations included in its composition. The differential equations of the modified material point model are provided - the equation of motion of the center of mass and the equation of nutational oscillations of the projectile in scalar form; an assessment of its accuracy (adequacy) was carried out on the example of the 155-mm PF projectile M2000, one of the family of projectiles of the company Denel Naschem – Assegai M200X Series 155-mm Projectiles. A comparison of the characteristics of the 155-mm PF of the M2000 projectile calculated according to the 6DoF model and the modified material point model showed their high convergence; the error does not exceed 1%. Keywords: aerodynamic forces (moments), projectile, mathematical models, modified model, aerodynamic coefficients, simulation, accuracy, flight parameters.

The motion of a washer on a vertical steel rod

A washer on a vertical steel rod may start spinning instead of simply sliding down. The forces on the washer are analyzed, and the relationship between the angular velocity of rotation and the angular velocity of precession is obtained. A relationship about the coefficient of friction of the washer is obtained at the condition of rotation without slipping. The influence of the coefficient of friction and the ratio of the inner diameter of the washer to the diameter of the rod on the sliding state was investigated experimentally, and a two-dimensional phase diagram of the rotation without slipping was obtained. The relationship between the three parameters(the nutation angle of the washer, the angular velocity of precession, and the final vertical velocity) and the ratio of the inner diameter of the washer to the diameter of the rod in the condition of rotation without slipping was further investigated. The experimental results are in good agreement with the theoretical equations.

An Improved Method for Estimating Ballistic Target Micromotion and Structural Parameters Based on ISAR Image Sequences

Micro-motion and structural parameters extraction is a critical technique for recognizing ballistic missiles. It is generally known that parameter estimation methods based on inverse synthetic aperture radar(ISAR) image sequences can effectively extract target information. However, when the signal-to-noise ratio(SNR) is low, the traditional methods are prone to cross-association problems in the nearest-neighbor association procedure, and the structural parameters obtained may only be the local optimal solution. To solve the above two problems, an improved method for estimating ballistic target micro-motion and structural parameters based on ISAR image sequences is proposed in this paper. A distance association probability revision matrix is used to address the cross-association problem, reducing the risk of association with the incorrect nearest neighbor; at the same time, the impact of noise on the frequency estimation is decreased by utilizing refinement domain search-match algorithm, which can discover a narrower estimation interval with less noise to improve SNR. The global optimization capacity for structural parameters estimation of the traditional particle swarm optimization(PSO) algorithm is enhanced by step-by-step optimization algorithm, which reduces the dimension of the high-dimensional micro-motion projection cost function to ensure optimization direction variety. Simulation results show that when SNR is -5dB, the accuracy of the precession frequency estimation is improved by 0.2Hz compared with the traditional FFT method, and the RMSE values for cone height, precession angle, and nutation angle are 0.02m, 0.2 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$°$</tex-math></inline-formula> , and 0.3 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$°$</tex-math></inline-formula> , respectively, which are better than the traditional PSO algorithm.

Projectile-Artillery-Propellant Coupled Optimization Design Method of Artillery Weapon Systems

The artillery weapon system is a complex system of great non-linearity. There are many factors and their coupling relationship affecting the firing performance of artillery, and the key factor positioning is difficult. How to improve the firing performance of artillery through optimization is a challenge. The current optimization method considers less coupling factors. To approach this challenge, a projectile-artillery-propellant coupled optimization design method is proposed, where a multidisciplinary coupling of projectiles, artillery and propellant is proposed and used to construct an optimization model, and the minimum initial disturbance is taken as the optimization objective. The simulation results show that at the moment when the projectile exist the muzzle, the optimized muzzle vibration displacement is reduced by 47.6%, the muzzle vibration speed is reduced by 33.4%, the average nutation angle is reduced by 63%, the average nutation angular speed is reduced by 27%.

Error identification and compensation of 1T2R parallel power head based on trajectory optimization and principal component analysis

PurposeThe purpose of this paper is to propose a method for selecting the position and attitude trajectory of error measurement to improve the kinematic calibration efficiency of a one translational and two rotational (1T2R) parallel power head and to improve the error compensation effect by improving the properties of the error identification matrix.Design/methodology/approachFirst, a general mapping model between the endpoint synthesis error is established and each geometric error source. Second, a model for optimizing the position and attitude trajectory of error measurement based on sensitivity analysis results is proposed, providing a basis for optimizing the error measurement trajectory of the mechanism in the working space. Finally, distance error measurement information and principal component analysis (PCA) ideas are used to construct an error identification matrix. The robustness and compensation effect of the identification algorithm were verified by simulation and through experiments.FindingsThrough sensitivity analysis, it is found that the distribution of the sensitivity coefficient of each error source in the plane of the workspace can approximately represent its distribution in the workspace, and when the end of the mechanism moves in a circle with a large nutation angle, the comprehensive influence coefficient of each sensitivity is the largest. Residual analysis shows that the robustness of the identification algorithm with the idea of PCA is improved. Through experiments, it is found that the compensation effect is improved.Originality/valueA model for optimizing the position and attitude trajectory of error measurement is proposed, which can effectively improve the error measurement efficiency of the 1T2R parallel mechanism. In addition, the PCA idea is introduced. A least-squares PCA error identification algorithm that improves the robustness of the identification algorithm by improving the property of the identification matrix is proposed, and the compensation effect is improved. This method has been verified by experiments on 1T2R parallel mechanism and can be extended to other similar parallel mechanisms.

Ballistic Wound Evaluation Based on Dye Diffusion Method

Ballistic gelatin is widely used as a kind of tissue simulants in wound ballistics. Evaluations of the wound tract of gelatin by projectiles are mainly based on fissures in the gelatin slices. For there are difficulties in building relationships between the areas and the fissures, a new method based on dye diffusion is proposed in this paper. Colored regions in the slices are obtained after penetration of a Chinese 7.62mm rifle bullet into the gelatin block. The colored regions can be divided into ‘X’, ‘I’ and branching ‘I’ shapes. In the second half of the penetration, there is a strong correlation between the areas of the regions and the energy dissipation of the bullet. However, the correlation is poor in the first half of the penetration. Branches of the colored regions mainly distributed at the heads and tails of the permanent cavities. The permanent cavities’ shapes have a great dependence on the nutation angles of the bullets. Orientation angles of the permanent cavities are determined by the precession angles of the bullets.

The effect of nutation angle on the flow inside a precessing cylinder and its dynamo action

The effect of the nutation angle on the flow inside a precessing cylinder is experimentally explored and compared with numerical simulations. The focus is laid on the typical breakdown of the directly forced m = 1 Kelvin mode for increasing precession ratio (Poincaré number) and the accompanying transition between laminar and turbulent flows. Compared to the reference case with a 90° nutation angle, prograde rotation leads to an earlier breakdown, while in the retrograde case, the forced mode continues to exist also for higher Poincaré numbers. Depending largely on the occurrence and intensity of an axisymmetric double-roll mode, a kinematic dynamo study reveals a sensitive dependence of the self-excitation condition on the nutation angle and the Poincaré number. Optimal dynamo conditions are found for 90° angle which, however, might shift to slightly retrograde precession for higher Reynolds numbers.

On a Class of Precessions of a Rigid Body with a Fixed Point under the Action of Forces of Three Homogeneous Force Fields

This paper is concerned with a special class of precessions of a rigid body having a fixed point in a force field which is a superposition of three homogeneous force fields. It is assumed that the velocity of proper rotation of the body is twice as large as its velocity of precession. The conditions for the existence of the precessions under study are written in the form of a system of algebraic equations for the parameters of the problem. Its solvability is proved for a dynamically symmetric body. It is proved that, if the ellipsoid of inertia of the body is a sphere, then the nutation angle is equal to $\arccos\frac{1}{3}$. The resulting solution of the equations of motion of the body is represented as elliptic Jacobi functions.

Effect of porosity on the kinematics of free-falling porous disks

The effects of porosity on the kinematics of porous disks are investigated experimentally. A new falling motion is identified, namely, spiral irregular motion, which is characterized by the irregular centerline and spiral motion around the centerline. Multifractal analysis is introduced to quantify the self-similarity and space-filling of irregular centerlines. Generally, the capacity dimension D0 decreases as the diameter ratio between the inner holes and the disk diameter χ increases. However, there is a deviation at χ=0.2. To explain this unordinary deviation, wake is visualized by particle image velocimetry. An oblique vortex ring with high vorticity is responsible for this anomaly. With χ increasing, the angle of attack increases nonlinearly and the distance Rp between paths and centerlines decreases. However, the nutation angle does not vary monotonically with χ, and a minimum appears at χ=0.2. The Strouhal number St and the drag coefficient Cd share the same trend with χ; hence, Cd increases monotonically with St. Both St and Cd reach a maximum at χ=0.15. These findings can be applied to improve the aerodynamic stability of disk-shaped passive fliers and give theoretical insight into parameter selection.

МОДЕЛИРОВАНИЕ УГЛОВОГО ДВИЖЕНИЯ КОСМИЧЕСКОГО АППАРАТА ПЕРЕМЕННОГО СОСТАВА С УЧЁТОМ ДЕСТАБИЛИЗИРУЮЩИХ СВОЙСТВ РЕАКТИВНОЙ СТРУИ

The work presents simulation of dynamics of a spacecraft~(SC) with variable mass. A generalized method for analyzing the phase trajectory curvature is used, being refined for the case in which the moments of forces applied to the SC depend on the angular velocity. Special cases are considered along with analyzing a spacecraft with dynamic asymmetry. Using the method, the presence of complex evolutions of angular motion is shown, in which the jet stream of the engine creates Coriolis effects (Magnus effect), destabilizing the precession cone and leading to an increase in the nutation angle.

МАТЕМАТИЧНА МОДЕЛЬ ПОЛЬОТУ СНАРЯДА ЯК МОДИФІКОВАНА МОДЕЛЬ МАТЕРІАЛЬНОЇ ТОЧКИ У ЯВНІЙ ФОРМІ

A promising direction for improving the accuracy and operational efficiency of firing artillery systems is the development of ballistic computers that implement procedures for determining settings using the solution of a system of differential equations (mathematical model) that describe the movement of a projectile in the air. The main problem in the development of mathematical models is the possibility of determining with a given accuracy the coefficients of aerodynamic forces (moments) that are in the right-hand sides of differential equations. An urgent issue in this direction is the study of the possibility of restoring the coefficients of aerodynamic forces (moments) by solving the inverse problem of external ballistics using mathematical models of projectile flight. The movement of projectiles is usually described using one of three mathematical models, which differ from each other in the main level of complexity and, accordingly, the level of adequacy to the real process of projectile movement in the air. Therefore, the complexity of recovery procedures is determined by the number of coefficients of aerodynamic forces (moments) and differential equations included in mathematical models. A modified model of a material point is presented as a mathematical model of projectile flight in which all aerodynamic forces are taken into account, the orientation of the projectile is characterized by taking into account the nutation angle, and the kinetic energy of the rotational movement is taken into account through the angular velocity of the projectile around its axis of symmetry. It is shown that the modified material point model is determined by adding an implicit ordinary differential equation - an equation that describes the nutational oscillations of the projectile, which depend on the acceleration of its flight. Procedures for converting the system of differential equations of the modified material point model to an explicit form are disclosed, which allows them to be solved on the basis of standard numerical methods.