Nutation Mode Research Articles

Nutation Excitations in the Gyrotropic Vortex Dynamics in a Circular Magnetic Nanodot.

A significant activity is devoted to the investigation of the ultrafast spin dynamic processes, holding a great potential for science and applications. However, a challenge of the understanding of the mechanisms of underlying spin dynamics in nanomaterials at pico- and femtosecond timescales remains under discussion. In this article, we explore the gyrotropic vortex dynamics in a circular soft magnetic nanodot, highlighting the impacts given by nutations in the high-frequency part of the dot spin excitation spectrum. Using a modified Thiele equation of the vortex core motion with a nutation term, we analyze the dynamic response of the vortex to an oscillating magnetic field applied in the dot plane. It is found that nutations affect the trajectory of the vortex core. Namely, we show that the directions of the vortex core motion in the low-frequency gyrotropic mode and the high-frequency nutation mode are opposite. The resonant frequencies of gyrotropic and nutational vortex core motions reveal themselves on different scales: gigahertz for the gyrotropic motion and terahertz for the nutations. We argue that the nutations induce a dynamic vortex mass, present estimates of the nutational mass, and conduct comparison with the mass appearing due to moving vortex interactions with spin waves and Doering domain wall mass.

Control System Design for Low Power Magnetic Bearings in a Flywheel Energy Storage System

This paper presents a theoretical and experimental study on controller design for the AMBs in a small-scale flywheel energy storage system, where the main goals are to achieve low energy consumption and improved rotordynamic stability. A H-infinity optimal control synthesis procedure is defined for the permanent-magnet-biased AMB-rotor system with 4 degrees of freedom. Through the choice of design weighting functions, notch filter characteristics are incorporated within the controller to reduce AMB current components caused by rotor vibration at the synchronous frequency and higher harmonics. Experimental tests are used to validate the controller design methodology and provide comparative results on performance and efficiency. The results show that the H-infinity controller is able to achieve stable rotor levitation and reduce AMB power consumption by more than 40% (from 4.80 to 2.64 Watts) compared with the conventional PD control method. Additionally, the H-infinity controller can prevent vibrational instability of the rotor nutation mode, which is prone to occur when operating with high rotational speeds.

Modeling and mu-synthesis control of a flexible rotor stabilized by active magnetic bearings including current free control

Designing a controller for a magnetically stabilized rotor that experiences non negligible gyroscopic moments requires careful attention on the rotordynamic behavior, the design of the position and current free control as well as the actuator and sensor characteristics. Most of the controller design procedures for magnetic bearings separate the synthesis of the current free controller and the position controller using different methods. This article connects the synthesis of the current free controller and the position controller in the μ-framework. Therefore, the modeling of the rotor, the position controller design and the design of the current free control of a flexible rotor stabilized by active magnetic bearings is described. The rotor is modeled by using the finite element method. Afterwards, the flexible system is reduced to get an appropriate model size for the control design. For the optimization of the controller parameters the μ-synthesis method is applied. However, compared to the general μ-synthesis method in this article an a priori fixed control structure is utilized. This control structure splits up the tilting subsystem to a control path for the nutation mode and one for the precession mode. Thus, the proposed controller has the advantage of a low system order and that the parameter variant nature of the plant is already considered in the control structure. Additionally, the synthesis of the current free controller is added to the μ-framework. Hence, the current free controller is optimized for a plant with defined uncertainties using necessary constraints of the closed loop systems. Finally, the stabilization of the system using the proposed control method is experimentally validated on a turbomolecular pump with a magnetically levitated rotor.

High-temperature magnetization reversal in the inertial regime

Motivated by the remarkable experimental observation of all-optical femtosecond-scale magnetization reversal at relatively high temperatures, a stochastic inertial Landau-Lifshitz-Gilbert-Bloch (iLLGB) equation is written to describe nonequilibrium magnetization dynamics in ferromagnets at elevated temperatures and at short enough timescales that an inertial effect manifests. The effect of thermal agitations is described by a Fokker-Planck equation derived from the iLLGB equation including the longitudinal relaxation effect, which is solved with perturbation theory valid at elevated temperatures. Considering a uniaxially symmetric ferromagnet with uniaxial anisotropy, a thermal diffusion-driven exponential mode and alternating field-driven nutation mode of stable magnetization reversal are identified. Our theory proposes a magnetization reversal mechanism based entirely on transfer of angular momentum to the local magnetization, but which takes into account thermal fluctuations and inertial effect at the same time. The theory explains several key observations in all-optical magnetization reversal experiments; the absence of the need for a static field, the subpicosecond switching timescale, the relative roles of thermal and field effects, and the relevance of circularly polarized light. Our results have direct implications for magnetic recording devices operating close to room temperatures and in the ultrafast regime.

Spin pumping at terahertz nutation resonances

We investigate spin pumping current injected by the nutation resonances of a ferromagnet or an antiferromagnet into an adjacent metal. Comparing the dc spin pumping current between the normal precession and nutation resonances, we find that the ratio of spin pumping current at the nutation resonance to the precession resonance is more pronounced in antiferromagnets. We further show that the spin pumping current injected by the nutation resonance is opposite in sign as compared to the normal precession mode. This could offer a useful experimental signature for identifying such nutation resonances. Analyzing the nature of the nutational eigenmodes, we show that the sign change in spin current is rooted in a reversal of the precession sense for the nutation mode(s). Furthermore, the nutational modes in antiferromagnets are found to be dominated by precession of one of the two sublattices only.

Influence of intersublattice coupling on the terahertz nutation spin dynamics in antiferromagnets

Spin nutation resonance has been well-explored in one-sublattice ferromagnets. Here, we investigate the spin nutation in two-sublattice antiferromagnets as well as, for comparison, ferrimagnets with inter-and intra-sublattice nutation coupling. In particular, we derive the susceptibility of the two-sublattice magnetic system in response to an applied external magnetic field. To this end, the antiferromagnetic and ferrimagnetic (sub-THz) precession and THz nutation resonance frequencies are calculated. Our results show that the precession resonance frequencies and effective damping decrease with intra-sublattice nutation coupling, while they increase with inter -sublattice nutation in an antiferromagnet. However, we find that the THz nutation resonance frequencies decrease with both the intra-and inter-sublattice nutation couplings. For ferrimagnets, conversely, we calculate two nutation modes with distinct frequencies, unlike antiferromagnets. The exchange-like precession resonance frequency of ferrimagnets decreases with intra-sublattice nutation coupling and increases with inter-sublattice nutation coupling, like antiferromagnets, but the ferromagnetic-like precession frequency of ferrimagnets is practically invariant to the intra and inter-sublattice nutation couplings.

Nutation Mode Suppression of Magnetically Levitated Turbo-Rotor via Damping Optimization

The significant nutation will degrade the stability of high-speed magnetically levitated turbo-rotor system. This article presents a damping optimization method to suppress the nutation mode. The proposed method consists of two parts: nutation compensation and damping optimization. The nutation compensation with the cross-channel displacement signal is designed in detail to increase the damping force of nutation mode in the first part. The root locus of the closed-loop rotor system is used to analyze the effect of damping compensation. The control parameters are iterated to configure the pole of the nutation mode with the goal of making the rotor system have the optimal damping ratio at the nutation mode. Finally, simulation results show that the nutation damping ratio of the rotor system with the proposed method is greater than 0.658, and the experiment results on a magnetically levitated turbomolecular pump prototype validate that the proposed damping optimization method can effectively suppress the nutation mode and its power spectrum level is below $-$60 dB over the entire speed range.

Filter Cross-Feedback Control for Nutation Mode of Asymmetric Rotors With Gyroscopic Effects

The nutation mode is an obstacle to the stability improvement of a magnetic-bearing-supported rotor system with significant gyroscopic effects. This article explores a parameter design method of the filter cross feedback for the nutation mode suppression of asymmetric rotor systems supported by active magnetic bearings (AMBs). First, the coupled multi-input multi-output model of the asymmetric AMB rotor system is converted into an equivalent complex single-input single-output model. The equivalent transformation aims at easing the system analysis and extending the classical stability criterion to the complex coefficient frequency domain. Then, the double-frequency bode plot and the Nyquist criterion are used for analyzing the stability of the AMB rotor system and designing the parameters of the filter cross feedback exactly. Simulation results show that the designed filter cross-feedback method has a good performance for suppression of the nutation mode. Finally, the power spectrum level of the nutation component is below $-$ 60 dB after the filter cross-feedback method is implemented in the controller of a large-scale magnetically suspended turbomolecular pump. The experimental results validate the effectiveness of the proposed method.

The coupling between inertial and rotational eigenmodes in planets with liquid cores

The Earth is a rapidly rotating body. The centrifugal pull makes its shape resemble a flattened ellipsoid and Coriolis forces support waves in its fluid core, known as inertial waves. These waves can lead to global oscillations, or modes, of the fluid. Periodic variations of the Earth's rotation axis (nutations) can lead to an exchange of angular momentum between the mantle and the fluid core and excite these inertial modes. In addition to viscous torques that exist regardless of the shape of the boundaries, the small flattening of the core-mantle boundary (CMB) allows inertial modes to exert pressure torques on the mantle. These torques effectively couple the rigid-body dynamics of the Earth with the fluid dynamics of the fluid core. Here we present the first high resolution numerical model that solves simultaneously the rigid body dynamics of the mantle and the Navier-Stokes equation for the liquid core. This method takes naturally into account dissipative processes in the fluid that are ignored in current nutation models. We find that the Free Core Nutation (FCN) mode, mostly a toroidal fluid flow if the mantle has a large moment of inertia, enters into resonance with nearby modes if the mantle's moment of inertia is reduced. These mode interactions seem to be completely analogous to the ones discovered by Schmitt (2006) in a uniformly rotating ellipsoid with varying flattening.

Tracking Compensation Control for Nutation Mode of High-Speed Rotors With Strong Gyroscopic Effects

Due to the strong gyroscopic effects, the stable control for the nutation vibration of high-speed rotors is an important concern in magnetically suspended control moment gyros (MSCMGs). This paper addresses the two key challenges on the nutation mode control of AMB-rotor systems: using the complex coefficient representation method to obtain the accuracy nutation frequency, and proposing an adaptive tracking approach to ensure stable operations over the entire speed ranges. In order to improve the phase margin of the nutation mode, this paper proposes a nutation mode tracking compensation method based on the cross-feedback-control scheme. First, the analytical calculation for the nutation frequency of the high-speed rotor under the closed-loop condition is carried out. Then, the compensation control method using an adaptive tracking filter in the nutation control path is proposed to provide an adequate damping for the nutation mode. Simulation results show that the nutation mode tracking compensation provides a greater phase lead for the nutation mode compared to the conventional nutation mode filtering approach. Finally, the experimental results on a developed double-gimbal MSCMG prototype validate the effectiveness of the proposed method.

Dynamics of the Earth’s fluid core: Implementation of a Clairaut coordinate system

If the reference state of a rotating and self gravitating fluid body is one of hydrostatic equilibrium then the figure of the body is a spheroid such that a cross sectional area parallel to the equatorial plane of the body is a circle while that parallel to a meridional plane is an ellipse. The effects of the fluid body’s flattened (spheroidal) figure is small on the frequencies of the body’s short-period (shorter than a few hours in the case of the Earth) normal modes. For the long-period normal modes, however, these effects must be considered. Furthermore, the body’s wobble and nutation modes owe their existence to its ellipsoidal figure. In the conventional approach to computing these frequencies, an orthogonal coordinate system is usually considered. It is then necessary to have the knowledge of the derivatives of the material properties of the body, such as the density and Lamé parameters, in order to include the effects of the ellipticity in the dynamical equations. In the available Earth models, however, these derivatives are not well defined. In order to minimize the effects of these derivatives in the treatment of the dynamical problems we use a non-orthogonal (Clairaut) coordinate system. Using this approach, we compute the frequencies and displacement eigenfunctions for some of the inertial modes of a realistic spheroidal model of the Earth’s fluid core and compare them to the known results for an Earth model with a homogeneous and incompressible fluid core.

Simple picture for neutrino flavor transformation in supernovae

We can understand many recently discovered features of flavor evolution in dense, self-coupled supernova neutrino and antineutrino systems with a simple, physical scheme consisting of two quasistatic solutions. One solution closely resembles the conventional, adiabatic single-neutrino Mikheyev-Smirnov-Wolfenstein (MSW) mechanism, in that neutrinos and antineutrinos remain in mass eigenstates as they evolve in flavor space. The other solution is analogous to the regular precession of a gyroscopic pendulum in flavor space, and has been discussed extensively in recent works. Results of recent numerical studies are best explained with combinations of these solutions in the following general scenario: (1) Near the neutrino sphere, the MSW-like many-body solution obtains. (2) Depending on neutrino vacuum mixing parameters, luminosities, energy spectra, and the matter density profile, collective flavor transformation in the nutation mode develops and drives neutrinos away from the MSW-like evolution and toward regular precession. (3) Neutrino and antineutrino flavors roughly evolve according to the regular precession solution until neutrino densities are low. In the late stage of the precession solution, a stepwise swapping develops in the energy spectra of ${\ensuremath{\nu}}_{e}$ and ${\ensuremath{\nu}}_{\ensuremath{\mu}}/{\ensuremath{\nu}}_{\ensuremath{\tau}}$. We also discuss some subtle points regarding adiabaticity in flavor transformation in dense-neutrino systems.

Practical Design and Flight Test of a Yo-Yo Wire Boom Deployment System

Ay o-yo-type wire-boom deployment system has been developed and flight tested on a sounding rocket mission. The goal of the work has been to validate a new mechanism that rapidly deploys wire booms from a spinning spacecraft. This work takes a theoretical system design and implements it in practical hardware. The limitations inherent in practical hardware necessitated new theoretical developments. A modified stability analysis has been developed for the case of nonzero axial separation between the wire-boom base attachment points and the spacecraft’s center of mass. This modified stability analysis dictates that a stable design is impractical for many missions because very large wire-boom tip masses are needed and because the three-dimensional deployment transients are very sensitive to small asymmetries. This problem has led to the development of design criteria and analysis techniques, which permit a short-duration mission to use a slightly unstable nutation mode. These techniques have been used to design a system that has been flown on two daughter spacecraft that were part of a formation of three sounding rocket subpayloads. Each daughter spacecraft deployed four 3-m-long wire booms in under 10 s and maintained a low level of spin instability for the remaining 700 s of the mission. The nutation oscillations showed a slow exponential growth, but the coning half-angles of both spacecraft never exceeded 16 deg.

VLBI observations of free core nutations and viscosity at the top of the core

Very long baseline interferometry (VLBI) nutation measurement series, both in excess of 23 years length, from Goddard Space Flight Center (GSFC) and the United States Naval Observatory (USNO), have been analyzed for free core nutation resonances. VLBI nutation observations can only be made when the radio sources being used are visible, rendering the data sequence inherently non-equispaced. This poses the problem of the approach to be taken with unevenly spaced sampling. Both the conventional Discrete Fourier Transform (DFT) and the Fast Fourier Transform Algorithm (FFT) for its computation strictly require a fixed sampling interval. Our approach is to find the Discrete Fourier Transform of the non-equispaced record by minimizing an objective function which weights the error between the DFT representation and the measured values in inverse proportion to the square of their standard errors. The resulting conditional equations have a coefficient matrix of Toeplitz form but the recursive Levinson algorithm has been found inadequate for their solution, even when implemented in double quad precision. Instead, we employ the Singular Value Decomposition technique to solve the least squares problem of fitting the Discrete Fourier Transform to the non-equispaced VLBI nutation observations. A novel feature of our procedure is to use the Parseval relation to determine the number of singular values of the coefficient matrix to be eliminated. We report the observation for the first time of the prograde mode predicted by Jiang [Jiang, X., 1993. Wobble–nutation modes of the earth, Ph.D. thesis, York University, Toronto, Canada]. The long series of observations allow the determination of the time evolution of the free core nutations. We observe both the prograde and the retrograde modes to be in free decay. In addition to providing measures of the viscosity just below the core–mantle boundary (CMB), the free decays suggest impulsive excitations rather than continuous excitation by electromagnetic core–mantle coupling or the atmosphere. The average recovered viscosity at the top of the core is of the order 615 Pa s in contrast to the value of 8 × 1 0 − 3 Pa s found by Gans [Gans, R., 1972. Viscosity of the Earth’s core, J. Geophys. Res. 77, 360–366] from the extrapolation of laboratory measurements, and commonly used by dynamo theorists.

Modeling of nutation and precession: New nutation series for nonrigid Earth and insights into the Earth's interior

The analytical formulation of the theories of nutation and wobble reveals the combinations of basic Earth parameters that govern the nutation‐wobble response of the Earth to gravitational (tidal) forcing by heavenly bodies and makes it possible to estimate several of them through a least squares fit of the theoretical expressions to the high‐precision data now available. This paper presents the essentials of the theoretical framework, the procedure that we used for least squares estimation of basic Earth parameters through a fit of theory to nutation‐precession data derived from an up‐to‐date very long baseline interferometry data set, the results of the estimation and their geophysical interpretation, and the nutation series constructed using the estimated values of the parameters. The theoretical formulation used here differs from earlier ones in the incorporation of anelasticity and ocean tide effects into the basic structure of the dynamical equations of the theory and in the inclusion of electromagnetic couplings of the mantle and the solid inner core to the fluid outer core, though this generalization comes at the cost of making some of the system parameters complex and frequency dependent; it is also more complete, as it takes account of nonlinear terms in these equations, including effects of the time‐dependent deformations produced by zonal and sectorial tides, which had been traditionally neglected in nonrigid Earth theories. Among the geophysical results obtained from our fit are estimates for the dynamic ellipticity e of the Earth (e = 0.0032845479 with an uncertainty of 12 in the last digit), for the dynamical ellipticity ef of the fluid core (3.8% higher than its hydrostatic equilibrium value, rather than ∼5% as hitherto), and for the two complex electromagnetic coupling constants. Our best estimates for the RMS radial magnetic fields at the core mantle boundary and at the inner core boundary, based on the estimates for these coupling constants, are ~6.9 and 72 gauss, respectively, when the magnetic field configurations are restricted to certain simple classes. The field strength needed at the inner core boundary could be lower if the density of the core fluid at this boundary or the ellipticity of the solid inner core were lower than that for the Preliminary Reference Earth Model. Our estimate for the resonance frequency of the prograde free core nutation mode, with an uncertainty of ∼10%, constitutes the first firm detection of the resonance associated with this mode; the period found is ∼1025 days, double that with electromagnetic couplings ignored. (Throughout this work, “days,” referring to periods, stands for “mean solar days.”) A new nutation series (MHB2000) is constructed by direct solution of the linearized dynamical equations (with our best fit values adopted for all the estimated Earth parameters) for each forcing frequency, and adding on the contributions from the nonlinear terms and other effects not included in the linearized equations. This series gives a considerably better fit to the nutation data than any of the earlier series based on geophysical theory. In particular, the residuals in the out of phase amplitudes of the retrograde 18.6 year and annual nutations, which had long remained at ∼0.5 milliseconds of arc (mas), are now reduced to the level of the uncertainties in the observational estimates, thanks mainly to the role played by the electromagnetic couplings. The largest remaining discrepancy is that in the out of phase prograde 18.6 year nutation, of ∼72 micorseconds of arc (μas). The frequency dependence of the nutation amplitudes cannot be exactly represented through a resonance formula, nor may the resonance frequencies themselves be interpreted as the eigenfrequencies of free modes because of the presence of complex and frequency‐dependent system parameters. Nevertheless, we have constructed a new resonance formula which reproduces our nutation series accurately for almost all nutation frequencies; for the few remaining frequencies, a listing is given of the corrections to be applied in order to reproduce the exact results of the direct solution.

Modeling of nutation‐precession: Very long baseline interferometry results

Analysis of over 20 years of very long baseline interferometry data (VLBI) yields estimates of the coefficients of the nutation series with standard deviations ranging from 5 microseconds of arc (μas) for the terms with periods <400 days to 38 μas for the longest‐period terms. The largest deviations between the VLBI estimates of the amplitudes of terms in the nutation series and the theoretical values from the Mathews‐Herring‐Buffett (MHB2000) nutation series are 56 ± 38 μas (associated with two of the 18.6 year nutations). The amplitudes of nutational terms with periods <400 days deviate from the MHB2000 nutation series values at the level standard deviation. The estimated correction to the IAU‐1976 precession constant is −2.997 ± 0.008 mas yr−1 when the coefficients of the MHB2000 nutation series are held fixed and is consistent with that inferred from the MHB2000 nutation theory. The secular change in the obliquity of the ecliptic is estimated to be −0.252 ± 0.003 mas yr−1. When the coefficients of the largest‐amplitude terms in the nutation series are estimated, the precession constant correction and obliquity rate are estimated to be −2.960 ± 0.030 and −0.237 ± 0.012 mas yr−1. Significant variations in the freely excited retrograde free core nutation mode are observed over the 20 years. During this time the amplitude has decreased from ∼300 ± 50 μas in the mid‐1980s to nearly zero by the year 2000. There is evidence that the amplitude of the mode in now increasing again.

Attitude Estimation for a Flexible Spacecraft in an Unstable Spin

Anattitudereconstruction algorithm hasbeendeveloped fora e exiblesounding rocketwhosespin vectornutates unstablyaboutitsminorinertia axis.Theattitudeestimatesareneededforposte ightanalysisoftherocket’ sscience data. An additional goal of the work is to show that a e exible-body model can be used in a Kalman e lter or in a smoother. The attitude is estimated using a smoother whose dynamic model includes a main rigid body and a pair of elasticbooms. Boom e exure is the principal causeof nutation instability. Boom bending is modeled by a Markov process, but the laws of mechanics are used to determine its ine uence on the attitude dynamics. This smoother achieves a better attitude estimation accuracy than can be achieved using a rigid-body model. The peak attitude error is estimated to be 4 deg, and the principal error source seems to be the limited accuracy of the rocket’ s attitude sensors. HIS work deals with the poste ight attitude determination for a sounding rocket mission, the Cleft Accelerated Plasma Experimental Rocket (CAPER). CAPER was launched from the Andoya Rocket Range in Norway in January 1999. Its e ight lasted slightly longer than 1200 s and reached an apogee altitude of 1360 km. The goal of this mission was to study the behavior of the ionosphere during auroral events. Attitude information is needed to transform CAPER’ s measurements of electric e eld components into geodetic coordinates. An attitude accuracy of from 2 to 4 deg is considered acceptable for the mission. From an attitude determination standpoint, the important characteristics of the CAPER sounding rocket were as follows: Its attitude sensors were a vector sun sensor with a slit e eld of view, a e xed-head horizon crossing indicator (HCI), and a three-axis magnetometer that was mounted on a short boom. The rocket’ s attitude was passively spin stabilized with the nominal spin vector directed along its minor inertia axis. CAPER deployed several booms after exit from the atmosphere and after the e nal stage motor burn. The longest of these were two e exible 3-m booms that deployed perpendicular to the nominal spin axis and in opposite directions from each other. A schematic diagram of this cone guration appears in Fig. 1. Minor-axis spin stabilization is often used in sounding rocket experiments. Near the dawn of the space age, the Explorer-1 mission demonstrated that such a cone guration has an unstable nutation mode due to energy dissipation in the e exible booms. 1 In many sounding rocket experiments it is acceptable for the nutation amplitude to grow as long as the resultant coning half-angle does not become too large by the end of the e ight. There are two major challenges in doing poste ight attitude determination for CAPER. The e rst is the lack of rate-gyro data. This challenge dictates the use of an Eulerian dynamics model to propagate the attitude and rates between measurement sample times. The model becomes especially important toward the end of the e ight, when only magnetometer data are available. The attitude estimate in rotation about the magnetic e eld vector is based solely on dynamic model propagation during this phase, and model inaccuracy

Effects of dissipation and a liquid core on forced nutations in Hamiltonian theory

In a previous paper, the authors considered the free rotation of an earth model composed of a rigid mantle and a liquid core in the presence of dissipation and under the Hamiltonian formalism, obtaining analytical expressions for the free nutation modes. In this paper we treat the forced motion. Approximate analytical solutions are worked out by means of Hori's perturbation method, the free solutions obtained in the former paper playing the role of the unperturbed solutions required in the application of the method. These solutions are consistent in the sense that, with the usual terminology, the rigid body solutions and the complex transfer functions are calculated with the same parameters. Besides in-phase terms, the dissipation at the core-mantle boundary studied in this paper gives rise to out-of-phase terms. From a qualitative perspective, we discuss the issue of the resonance in this context. The presence of dissipation changes dramatically the character of the FCN wobble; that is, it is no longer a regular oscillation but a damped one. A strict resonance phenomenon cannot take place thereby, since the forcing perturbations are oscillations with a real (non-complex) frequency.

Damped Gyroscopic Modes of Spinning Tethered Space Vehicles with Flexible Booms

The OEDIPUS rocket payload cone guration comprises two spinning subpayloads, each having long e exible booms and a common long connecting tether of about 1 km in length. An analysis demonstrates the damped gyroscopicnatural modesas a means of understanding thedynamics andstability ofthesevery complex cone gurations. For OEDIPUS-A, attention is focused on calculation of nutation mode divergence that is attributable to material damping of the tether and booms. Calculated divergence rates are compared with e ight data of OEDIPUS-A. For OEDIPUS-C attention is focused on attitude and cone guration stability. It is noted to be stable in nutation and other modes at its e ight spin rate of about 0.09 Hz. However, the cone guration is noted to be susceptible to structural instability at spin rates above 0.26 Hz.

Geometry of adiabatic changes. General analysis

The concept of adiabatic change and related notions are discussed. As a representative example, the evolution of a neutron spin in a precessing magnetic field is described briefly. The general setting of the equations for Dirac's evolution coefficients is discussed from the geometric phase point of view. An exactly solvable model of nutation is exhibited and the properties of the solutions are analyzed to reveal the holonomic structure of the problem. The corresponding expression for the geometric phase differs nontrivially from the corresponding expression in the well-known case of the precessing field. In addition, this geometric phase has an imaginary part which completes the picture of spin evolution in a nutation mode. The approach proposed for nutation is used to reexamine the twisted Landau-Zener problem.