Nutation Damper Research Articles

The Viscous and Ohmic Damping of the Earth's Free Core Nutation.

The cause for the damping of the Earth's free core nutation (FCN) and the free inner core nutation eigenmodes has been a matter of debate since the earliest reliable estimations from nutation observations were made available. Numerical studies are difficult given the extreme values of some of the parameters associated with the Earth's fluid outer core, where important energy dissipation mechanisms can take place. We present a fully 3D numerical model for the FCN capable of describing accurately viscous and Ohmic dissipation processes taking place in the bulk of the fluid core as well as in the boundary layers. We find an asymptotic regime, appropriate for Earth's parameters, where viscous and Ohmic processes contribute to the total damping, with the dissipation taking place almost exclusively in the boundary layers. By matching the observed nutational damping, we infer an enhanced effective viscosity matching and validating methods from previous studies. We suggest that turbulence caused by the Earth's precession can be a source for the enhanced viscosity.

Optimal contact control for space debris detumbling and nutation damping

Space debris detumbling technology can be an effective supplement to the direct capture capability of space robot for simplifying and completing the active debris removal mission. This paper focuses on the optimal contact control for space debris rotation eliminating and nutation damping. Firstly, the space debris attitude motion is analyzed. The space debris detumbling strategy is presented with a designed spring contactor as the detumbling tool. Then, the contact detumbling dynamic models are established, including the spacecraft dynamic models, the contact detection method and the contact force model. Furthermore, an optimal contact control method is developed in order to reduce the three-axis angular velocities of space debris and suppress the nutation at the same time. The stability of the control method is proved by using the Lyapunov argument and the condition for nutation convergence is analyzed. Finally, numerical simulations are carried out to verify the optimal contact control method proposed in this paper. The results illustrate that the control method is effective for space debris detumbling and has certain robustness to the measurement and identification errors.

Nutation damping and spin orientation control of tethered space debris

As a preliminary of an active debris removal process, the attitude motion control of the debris need to be carried out to ensure security of following operations. The tethered space system is one of the most attractive methods for active debris removal missions. However, the tether tension becomes the only input to manipulate the attitude motion of the debris, causing a highly underactuated problem. Fortunately, the fully attitude stabilization of the debris is not a prerequisite for a secure deorbit operation. As long as the nutation of the debris is not so violent that makes the tether winds, the towing process can be applied continually, which is a significant advantage of the tether method. In this paper, a switched tension control law is presented to damp the nutation of the debris. Only two constant tension values are needed in the control method. Under the assumption that the tether direction is fixed in the inertia frame, the bounded stability of the control algorithm is proved based on the Barbalat's Lemma, and an asymptotic stability can be achieved for an asymmetrical debris symmetrically captured. Numerical simulations with tether libration considered are carried out to show the capability of the control method.

Prospects of using a permanent magnetic end effector to despin and detumble an uncooperative target

Space debris, such as defunct satellites and upper stages of rockets, becomes an uncooperative target after losing its attitude control and communication ability. In addition, tumbling motion can occur due to environmental perturbations and residual angular momentum prior to the object’s end-of-mission. To minimize the collision risk during docking and capturing of the tumbling target, a non-contact method based on the eddy current effect is put forward to transmit the control torque to the tumbling target. The main idea is to induce a controllable torque on the conducting surface of the tumbling target using a rotational magnetic field generated by a Halbach rotor. The radial and axial Halbach rotors are used to damp the spinning and nutation motions of the target, respectively. The normal and tangential force are evaluated concerning the relative pose between the chaser and the target. A simplified dynamic model of the nutation damping and despinning processes is developed and the influences of the asymmetrical principal moments of inertia and transverse angular velocity are discussed. The numerical simulation results show that the designed Halbach rotor stabilized the target attitude within an acceptable time. The electromagnetic nutation damping and despinning method provides new solutions for the development of on-orbit capture technology.

INVESTIGATION OF THE STABILITY OF THE STEADY MOTIONS OF A ROTATIONAL MECHANICAL SYSTEM

The conditional stability of steady motions of a mechanical isolated system consisting of a rotating carrier body, a material point creating its static imbalance, and a passive automatic balancer consisting of two identical mathematical pendulums are investigated. The pendulums are mounted on the longitudinal axis of the supporting body and move in the plane of static imbalance. The relative motion of the pendulums is impeded by viscous drag forces. It is established that in the case when the imbalance is and the pendulums can eliminate it with a certain margin, there is one basic movement; in the absence of imbalance, there is a one–parameter family of basic movements; in the case of maximum unbalance, which pendulums can eliminate, there is one basic movement, but it generates a pseudo–family of basic motions. It is also established that conditionally asymptotically stable are separate basic motions, if they are isolated, or a family, or pseudo–families of basic motions. For pendulum, ball and liquid automatic balancers, an approximate law is obtained for the variation of large nutation angles in the case of an axisymmetric and non–axisymmetric supporting body. It is established that the rate of change of the nutation angle is significantly affected by the ratio between the axial moments of inertia of the rotating carrier and the coefficient of viscous drag. An empirical formula is proposed for estimating the residual nutation angle, which occurs when the passive automatic balancers (nutation dampers) are incorrectly installed on the spacecraft and stabilized by rotation, and an example of its application for a particular satellite is given. It is shown that incorrect installation of the auto balance on the supporting body can lead to the formation of a residual nutation angle even in the case of a "stable" supporting body. The obtained results can be used in the design of passive automatic balancers (nutation damping) (pendulum, ball and liquid) for spacecrafts stabilized by rotation

Attitude acquisition from an arbitrary tumbling state using two skewed reaction wheels

A novel technique has been designed that creates rapid nutation damping and accurate spin rate control for a spacecraft with arbitrary inertia ratio. In this approach the satellite incorporates two symmetrically inclined reaction wheels in a V configuration and stabilization is achieved by simultaneously controlling the angular velocity of the satellite and the wheels. The method furnishes gyroscopic stiffness and steers interchange of momentum between the wheels and the satellite main body. A Monte Carlo type approach is used to verify stability and it is shown that the controller provides automatically logical recovery of the desired spin for any initial state and inertia ratio. Moreover, results of single wheel simulations demonstrate the efficacy of the proposed concept.

Nutation damping of dynamically tuned gyroscopes

The paper deals with the problem of nutation damping of dynamically tuned gyroscopes (DTG). Unlike the common two-axis gyroscopes, the DTG damping torques caused by dry and viscous frictions are absent in gimbal bearings, while those due to the rotor/gaseous medium interaction are insufficient to provide the required nutation damping. A physical explanation is given to the damping torques induced in the gyroscope by the magnetoelectric torque detectors. Control coils of the latter are included in the output circuits of the electronic amplifiers. The equation is proposed for the calculation of the damping torque. The numerical estimate of the damping coefficient is given in this paper.

Dynamic Modeling and Controller Design for a Spherical Rolling Robot Equipped With a Gyro

In this paper, we derive a dynamic model of a spherical rolling robot that has a new driving mechanism equipped with a gyro and design a controller to stabilize a desired translational motion of the robot. Due to the angular momentum of the gyro, nutation motion of the internal mechanism of the robot may be caused. A feedback controller is designed to achieve the stabilization to a straight motion of the robot and nutation damping in the internal mechanism simultaneously. The asymptotic stability is proved using a Lyapunov function, and the effectiveness of the controller is verified by experiments. The controller is also successfully applied to other translational motions such as a circular motion and a meandering motion.

Attitude stability of a dual-spin spacecraft on a stationary orbit around an asteroid subjected to gravity gradient torque

The attitude stability of a dual-spin spacecraft on a stable stationary orbit around an asteroid subjected to gravity gradient torque is studied in this paper, in which the effect of the second order and degree harmonics C 20 and C 22 of the gravity field of the asteroid is involved. Under some reasonable assumptions, the linearized equations of attitude motion of the dual-spin spacecraft with a nutation damper are presented. Then the necessary and sufficient conditions of attitude stability of the dual-spin spacecraft with a nutation damper are derived. Meantime, the linearized equations of attitude motion and the conditions of attitude stability of the dual-spin spacecraft without a nutation damper are also presented. These results are generalized and encompass a series of special cases, and from which some useful conclusions on the attitude stability of the dual-spin spacecraft are obtained. Examples are given to illustrate the application of these conditions of attitude stability.

The tumbling spin state of (99942) Apophis

Our photometric observations of Asteroid (99942) Apophis from December 2012 to April 2013 revealed it to be in a state of non-principal axis rotation (tumbling). We constructed its spin and shape model and found that it is in a moderately excited Short Axis Mode (SAM) state with a ratio of the rotational kinetic energy to the basic spin state energy E/E0=1.024±0.013. (All quoted uncertainties correspond to 3σ.) The greatest and intermediate principal moments of inertia are nearly the same with I2/I3=0.965-0.015+0.009, but the smallest principal moment of inertia is substantially lower with I1/I3=0.61-0.08+0.11; the asteroid’s dynamically equivalent ellipsoid is close to a prolate ellipsoid. The precession and rotation periods are Pϕ=27.38±0.07h and Pψ=263±6h, respectively; the strongest observed lightcurve amplitude for the SAM case is in the 2nd harmonic of P1=Pϕ-1-Pψ-1-1=30.56±0.01h. The rotation is retrograde with the angular momentum vector’s ecliptic longitude and latitude of 250° and -75° (the uncertainty area is approximately an ellipse with the major and minor semiaxes of 27° and 14°, respectively). An implication of the retrograde rotation is a somewhat increased probability of the Apophis’ impact in 2068, but it is still very small with the risk level on the Palermo Scale remaining well below zero. Apophis is a member of the population of slowly tumbling asteroids. Applying the theory of asteroid nutational damping by Breiter et al. (Breiter, S., Rożek, A., Vokrouhlický, D. [2012]. Mon. Not. R. Astron. Soc. 427, 755–769), we found that slowly tumbling asteroids predominate in the spin rate–size range where their estimated damping times are greater than about 0.2Gyr. The appearance that the PA/NPA rotators transition line seems to follow a line of constant damping time may be because there are two or more asteroid spin evolution mechanisms in play, or the factor of μQ (the elastic modulus times the quality factor) is not constant but it may decrease with decreasing asteroid size, which would oppose the trend due to decreasing collisional age or excitation time.

Research of process of the elimination autobalancers of large nutation angles

This paper studies the process of eliminating large nutation angles arising from inaccurate initial rotation or imbalance of the spin-stabilized spacecraft using passive auto-balancers (pendulum, ball and liquid). An analogy in the operation of various types of passive auto-balancers (nutation dampers) (pendulum, ball and liquid) during removal of large angles of nutation was established. For pendulum, ball and liquid auto-balancers approximate law of changing large nutation angles in the case of axisymmetric and non-axisymmetric lifting body was obtained. It was found that the rate of changing the nutation angle is significantly affected by the ratio between the axial moments of iner­tia of the lifting body and the coefficient of viscous drag forces. The em­pirical formula was proposed for estimating the residual nutation angle resulted from incorrect installation of passive auto-balancers (nutation dampers) on the spin-stabilized spacecraft, as well as an example of its application for a specific Brazilian satellite (SACI-2) was given. It is shown that improper installation of the auto-balancer (nutation damper) on the lifting body can cause residual nutation angle even in the case of “steady” lifting body. The results can be used in the design of passive auto-balancers (nutation dampers) (pendulum, ball and liquid) for spacecrafts or spin-stabilized artificial satellites of the Earth.

On the stability and bifurcation analysis of dual-spin spacecraft

The dynamics of dual-spin spacecraft under effects of energy dissipation are considered in this paper, where the damper masses in the platform (P) and the rotor (R) cause energy loss in the system. The Floquet theory is employed to obtain stability charts for different relative spin rates of the subsystem R with respect to the subsystem P. Based on the general model for the system with nutation dampers on both P and R, models are presented for a system whose nutation damper exists only in P as well as a system without nutation damper. The results obtained from the Floquet theory agree with the energy sink analysis in the literature. The bifurcation analysis based on the movement of loci of the Floquet multipliers as the system passes through the flutter stability boundary indicates that the system experiences the secondary Hopf (Neimark–Sacker) bifurcation. The investigations show that for spacecraft whose nutation damper exists only in one of the subsystems, there is no need to apply Floquet theory, and the Routh–Hurwitz criteria provides necessary and sufficient conditions for stability. Furthermore, for the case that only P has damping, the Lyapunov stability criteria agree with Routh–Hurwitz criteria.

Spin-stabilized satellite magnetic attitude control scheme without initial detumbling

The angular motion of an axisymmetrical satellite equipped with an active magnetic attitude control system is considered. The dynamics of the satellite are analytically studied on the whole control loop. The control loop is as follows: preliminary reorientation along with nutation damping, spinning about the axis of symmetry, then precise reorientation of the axis of symmetry in inertial space. Reorientation starts right after separation from the launch vehicle. Active magnetic attitude control system time-response with respect to its parameters is analyzed. It is proven that low-inclined orbit forces low control system time-response. Comparison with the common control scheme shows the time-response gain. Numerical analysis of the disturbances effect is carried out and good pointing accuracy is proved.

Study of a bunch of three algorithms for magnetic control of attitude and spin rate of a spin-stabilized satellite

We consider the angular motion of an axi-symmetrical satellite equipped with an active magnetic attitude control system. Dynamics of the satellite is studied on the entire control loop, consisting of a bunch of three successively used algorithms. The control cycle includes the stages of nutation damping, spinning up the satellite about its symmetry axis, and reorienting the symmetry axis into a preset direction in the inertial space. The results are confirmed by numerical simulations.

Asymptotic study of a complete magnetic attitude control cycle providing a single-axis orientation

The angular motion of an axisymmetrical satellite equipped with the active magnetic attitude control system is examined. Attitude control system has to ensure necessary orientation of the axis of symmetry in the inertial space. It implements the following strategy: coarse reorientation of the axis of symmetry with nutation damping or “-Bdot” without initial detumbling; spinning-up about the axis of symmetry to achieve the property of a gyro; fine reorientation of the axis in the inertial space. Dynamics of the satellite is analytically studied using averaging technique on the complete control loop consisting of five algorithms. Solutions of the equations of motion are obtained in terms of quadratures for most cases or even in closed-form. The latter allowed to study the dependence of motion parameters including time-response with respect to the orbit inclination and other parameters for all algorithms.

Development of a Nutation Damper Reservoir Mechanism

Development of a Nutation Damper Reservoir Mechanism

Attitude Motion of a Spinning Spacecraft with Fuel Sloshing and Nutation Damping

We use Kane’s method to derive the equations of motion of a spinning spacecraft with three momentum wheels, a nutation damper, and a spherical pendulum. The spherical pendulum is adopted as a simple mechanical equivalent of fuel sloshing in partially filled tanks. The proposed model is an extension of existing models in the literature. We verify and validate our model for two cases: flat spin and a simple reorientation maneuver. Numerical simulations are in agreement with existing results in the literature. The results confirm the accuracy of the model for a typical spacecraft.

Attitude simulation software to support Brazilian space missions

Cost reduction on system tests with hardware in the loop during qualification phase of the onboard attitude control software imply in increasingly simulation realism. Attitude simulation software not only has to mimic the dynamical behavior of the spacecraft, but also shall be easily configurable, to adapt to different orbit, attitude pointing requirements, environment disturbances, sensors, actuators, and to run in real-time among other minor requirements. As the requirements for accuracy and stability for attitude pointing has increased in many modern missions, so do the complexity of the simulation algorithms and necessity for dynamical realism. Rigid body dynamics can be considered excessively simple to model a satellite with several reaction wheels, non-rigid structures, unbalanced solar arrays, fuel and liquid sloshing, not to mention crew motion and vehicle docking. This work presents the efforts being carried out to fulfill the requirements of the Brazilian space missions, nominally Multi Mission Platform, with a simulation environment capable to accomplish from early definition and mission analysis phase up to acceptance tests of the attitude control software. Development is being conducted in order to achieve a high degree of realism and range of applicability of the simulation software, compatible with the expected cost reduction of the simulation hardware (ordinary PCs). The kinematics and dynamic models to support AOCS (Attitude and Orbit Control Subsystem) simulation and testing with or without hardware-in-the-loop for Brazilian space missions are presented. Normally the dynamic equations are expressed in angular momentum components, due to the high complexity of these equations when derived in terms of the angular velocity. This, of course, is particularly important in satellites composed by several articulated rigid bodies as solar panels, robotic arms, space booms, deployable antennas, etc. Nevertheless, the inverse of the inertia matrix shall be calculated on the run, since it varies in time. This can be somewhat slow, mainly considering the high degree of freedom in dynamics of satellites with articulated panels and reaction wheels. In this case a more complex but fast set of equations in terms of the angular velocity is the best choice. This work describes the attitude dynamic equations expressed in angular velocities for satellites with appendages (solar panels, telescopes, directional antennas, etc.) and reaction wheels. They can be easily extended, however, to include nutation dampers and robotic arms. As usual, this formulation requires the knowledge of the acting torque in the connection joint, which is not always understood, modeled or known. So the dynamic model presented in this work uses the angular acceleration instead of the torque at the joint, which simplifies and reduces the order of the differential equations. Simulation results are presented, emphasizing the difference between the behavior of rigid body and articulated rigid bodies.

Generalized Wahba Problems for Spinning Spacecraft Attitude and Rate Determination

Two generalized versions of Wahba’s attitude determination problem have been developed for a spinning spacecraft, and a restricted version of one problem has been solved in closed-form. These problems seek to estimate both attitude and rate based solely on a time series of vector attitude observations along with a spacecraft dynamic model. Algorithms that solve these problems will be useful for spin-stabilized spacecraft that reduce complexity by omitting rate gyros. The first generalized Wahba problem presumes that the spin axis is known and that the spin rate is constant but unknown, as for a spinning spacecraft that has a nutation damper. The second generalized problem includes full rigid-body Euler dynamics, which allow for nutations, and seeks to estimate the unknown initial attitude rate vector. Both problems are recast into the K-matrix form of Wahba’s problem with K matrices that depend on the unknown rates. Restricted problems are developed that use the minimum number of vector measurements, two for the first problem and three for the second problem. The restricted first problem is solved in closed-form. The restricted second problem is shown to be observable, and it is reduced to a small system of nonlinear equations in the axially symmetric case. The possibility of deriving global solutions for these problems makes them attractive to assist or replace an extended Kaiman filter because a global solution cannot suffer from nonlinear divergence.

Simple Analysis of Prolate-Spin Attitude Stability

How can a prolate satellite spinning about its axis of minimum moment of inertia be attitude-stabilized using dual spin? To this question, we find a simple, clear answer. We replace the satellite with an equivalent frame model and describe its attitude motion as small deviations from its nominal attitude. We show that a friction-like torque applied to the spin axis reduces its nutation motion, and that a similar torque is generated by a nutation damper on a de-spun platform. Thus, we derive a simple attitude-stability theory using minimal mathematical analysis. This theory is applicable to the spin stabilization of geosynchronous communication satellite.