A linearization tool for use with matrix formalisms of rotational dynamics.
Rotational dynamics equations in linearized matrix form applied to satellite librations in circular orbit and gyroscope motion
Likins, P. W.↗
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Investigation of Mars Rotational Dynamics Using Earth-based Radio Tracking of Mars Landers
The development of space geodetic techniques over the past two decades has made it possible to measure the rotational dynamics of the Earth at the milliarcsecond level, improving our geophysical models of the Earth 's interior and the interactions between the solid Earth and its atmosphere. We have found that the rotational dynamics of Mars can be determined to nearly the same level of accuracy by acquiring Earth-based two-way radio tracking observations of three or more landers globally distributed on the surface of Mars. Our results indicate that the precession and long-term obliquity changes of the Mars pole direction can be determined to an angular accuracy corresponding to about 15 cm/yr at the planet's surface. In addition, periodic nutations of the pole and seasonal variations in the spin rate of the planet can be determined to 10 cm or less. Measuring the rotation of Mars at this accuracy would greatly improve the determination of the planet' s moment of inertia and would resolve the size of a planetary fluid core, providing a valuable constraint on Mars interior models. Detecting seasonal variations in the spin rate of Mars would provide global constraints on atmospheric angular momentum changes due to sublimation of the Mars CO2 polar ice caps. Finally, observation of quasisecular changes in Mars obliquity would have significant implications for understanding long-term climatic change. The key to achieving these accuracies is a globally distributed network of Mars landers with stable, phase-coherent radio transponders. By simultaneously acquiring coherent two-way carrier phase observations between a single Earth tracking station and multiple Mars landers, Earth media errors are essentially eliminated, providing an extremely sensitive measure of changes in the differential path lengths between the Earth tracking station and the Mars landers due to Mars rotation. Time variability of the instrumental phase delay through the radio transponder may represent the limiting error source for this technique. Calibration of the transponder stability to about 0.1 ns or less, over a single tracking arc of up to 12 hr, is sufficient to provide the decimeter-level determination of Mars orientation parameters quoted above. We will provide a detailed description of the multilander tracking technique and the requirements it imposes on both the lander radio system and the Earth-based ground-tracking system. This concept is currently part of the strawman science plan for the Mars Environmental Survey (MESUR) mission and complements many of the other MESUR science goals.
Edwards, C. D., Jr.↗
Rotational dynamics with geometric algebra
A new spinor formulation of rotational dynamics is developed. A general theorem is established reducing the theory of the symmetric top to that of the spherical top. The classical problems of Lagrange and Poinsot are treated in detail, along with a modern application to the theory of magnetic resonance.
Hestenes, D.↗
Direct inversion of rigid-body rotational dynamics
The global linearization (inversion) of rigid-body rotational dynamics is reviewed and representations in terms of quaternions and direction cosines are compared. Certain properties common to quaternions and direction cosines that make their use preferable to Euler angles and that simplify the inversion procedure are described. Applications of the inversion procedure for state estimation and attitude control are discussed. To avoid complexities caused by aerodynamics, an example of direct inversion for linear feedback control of spacecraft attitude is given.
Bach, Ralph↗
Direct inversion of rigid-body rotational dynamics
The global linearization (inversion) of rigid-body rotational dynamics is reviewed, and representations in terms of quaternions and direction cosines are compared. Certain properties common to quaternions and direction cosines that make their use preferable to Euler angles and that simplify the inversion procedure are described. Applications of the inversion procedure for state estimation and attitude control are discussed. To avoid complexities caused by aerodynamics, an example of direct inversion for linear feedback control of spacecraft attitude is given.
Bach, Ralph↗
The rotational dynamics of Mercury and the state of its core
Data on the rotational dynamics of Mercury are examined together with possible events that could lead to the current state of rotation. It is shown that the dynamical evolution of Mercury's spin angular momentum controlled by the dissipative processes of tidal friction and relative motion between a solid mantle and a liquid core would lead naturally to the current state of rotation of the Mercury planet. To investigate the possibility that Mercury has a molten core, an experiment is designed for the measurement of the core properties of Mercury. It is shown that it is technically feasible to measure the four parameters necessary for the determination of the existence and the extent of a Mercurian molten core, including the amplitude of the physical vibration about the resonance spin rate, the obliquity, and the gravitational harmonic coefficients.
Peale, S. J.↗
Earth Rotation Dynamics: Review and Prospects
Modem space geodetic measurement of Earth rotation variations, particularly by means of the VLBI technique, has over the years allowed studies of Earth rotation dynamics to advance in ever-increasing precision, accuracy, and temporal resolution. A review will be presented on our understanding of the geophysical and climatic causes, or "excitations", for length-of-day change, polar motion, and nutations. These excitations sources come from mass transports that constantly take place in the Earth system comprised of the atmosphere, hydrosphere, cryosphere, lithosphere, mantle, and the cores. In this sense, together with other space geodetic measurements of time-variable gravity and geocenter motion, Earth rotation variations become a remote-sensing tool for the integral of all mass transports, providing valuable information about the latter on a wide range of spatial and temporal scales. Future prospects with respect to geophysical studies with even higher accuracy and resolution will be discussed.
Chao, Benjamin F.↗
Rotational dynamics of artificial satellites Technical report, 1 Apr. 1968 - 1 Apr. 1969
Rotational dynamics of artificial satellites
Source record↗
Rotational dynamics of irregularly shaped natural satellites
The rotational histories of irregularly shaped satellites are studied and are found to differ from the standard picture of tidal evolution of satellite rotations. Prior to capture into the synchronous rotation resonance, a narrow attitude-unstable chaotic zone is entered and the satellite begins to tumble chaotically. It is noted that enhanced dissipation of energy during the chaotic-tumbling phase may effect the orbital evolution. The theory suggests that, eventually, the rotation stays close to one of the accessible attitude-stable islands long enough for the weak tidal torque to remove the satellite from the chaotic zone.
Wisdom, Jack↗
A comparative study of rapidly and slowly rotating dynamical regimes in a terrestrial general circulation model
As a preliminary step in the development of a general circulation model for general planetary use, a simplified vesion of thef GISS Model I GCM has been run at various rotation periods to investigate differences between the dynamical regimes of rapidly and slowly rotating planets. To isolate the dynamical processes, the hydrologic cycle is suppressed and the atmosphere is forced with perpetual annual mean solar heating. All other parameters except the rotation period remain fixed at their terrestrial values. Experiments were conducted for rotation periods of 2/3, 1, 2, 4, 8, 16, 64 and 256 days. The results are in qualitative agreement with similar experiments carried out previously with other GCMs and with certain aspects of one Venus GCM simulation. As rotation rate decreases, the energetics shifts from baroclinc to quasi-barotropic when the Rossby radius of deformation reaches planetary scale. The Hadley cell expands poleward and replaces eddies as the primary mode of large-scale heat transport. Associated with this is a poleward shift of the baroclinic zone and jet stream and a reduction of the equator-pole temperature contrast. Midlatitude jet strength peaks at 8 days period, as does the weak positive equatorial zonal wind which occurs at upper levels at all rotation periods. Eddy momentum transport switches from poleward to equatorward at the same period. Tropospheric mean static stability generally increases in the tropics and decreases in midlatitudes as rotation rate decreases, but the global mean static stability is independent of rotation rate. The peak in the eddy kinetic energy spectrum shifts toward lower wavenumbers, reaching wavenumber 1 at a period of 8 days. Implications of these results for the dynamics of Venus and Titan are discussed. Specifically, it is suggested that the extent of low-level convection determines whether the Gierasch mechanism contributes significantly to equatorial superrotation on these planets.
Del Genio, Anthony D.↗
Rectified Rotational Dynamics of Mobile Inclusions in Two-Dimensional Active Nematics
We investigate the dynamics of mobile inclusions embedded in 2D active nematics. The interplay between the inclusion shape, boundary-induced nematic order, and autonomous flows powers the inclusion motion. Disks and achiral gears exhibit unbiased rotational motion, but with distinct dynamics. In comparison, chiral gear-shaped inclusions exhibit long-term rectified rotation, which is correlated with dynamics and polarization of nearby +1/2 topological defects. Furthermore, the chirality of defect polarities and the active nematic texture around the inclusion correlate with the inclusion’s instantaneous rotation rate. Inclusions provide a promising tool for probing the rheological properties of active nematics and extracting ordered motion from their inherently chaotic motion.
36 MATERIALS SCIENCE↗
Seismic effects on the rotational dynamics of the earth and its gravitational field
The effects of earthquakes on the rotational motion of the earth were studied. The connection between the fault parameters and the corresponding changes in the moments and products of inertia were analytically developed. The reciprocal theorem of elasticity and Volterra's formula were applied as well as the displacement and stress fields for the second degree static response of the earth model being used. The numerical results of the investigation yield the magnitude and direction of the pole shift as well as the change in the length of the day. The changes in the second degree coefficients of the geopotential were computed. Source parameters corresponding to the Alaskan earthquake on March 28, 1964 were used to generate numerical results.
Sanchez, B. V.↗
The application of potential flow theory to the rotational dynamics of spheroids, disks, and cylinders
An example for the considered phenomenon in the area of cloud microphysics is the rotational motion of ice particles as they fall through the air. Of fundamental importance for the general problem is a knowledge of the magnitude of the torque exerted by the fluid on the translating object as well as the frequency of the rotational oscillation which results from this torque. The present investigation has the objective to obtain estimates of these quantities, and to assess the accuracy of these estimates. Torques and oscillation frequencies for spheroidal objects immersed in flow are computed for potential flow and compared to measured values. The results form a consistent picture in which the ratios of measured to potential flow values are always less than 1 but generally greater than 0.1. The reason for this is described in terms of the well-known deviations of real flow from potential flow. The reported results are of academic interest in filling a gap in the knowledge regarding the applicability of potential flow.
Weinheimer, A. J.↗
Ground based studies of the vibrational and rotational dynamics of acoustically levitated drops and shells
A substantial amount of experimental data can be gathered on the dynamics of acoustically positioned liquids in a ground-based laboratory and during short duration low-gravity parabolic flights of the KC-135. The preliminary results of a set of measurements of the static shape, of the vibrational spectrum, and the rotation equilibrium shapes of simple drops and liquid shells carried out using ultrasonic levitators working between 19 and 40 kHz is presented. The droplet diameter ranges between 1 and 5 mm, the surface tension of the liquid used varies between 25 and 70 dynes/cm, and the viscosity is changed between 1 to 1,000 cP. Of particular interest is the variation of the frequency of the fundamental mode of shape oscillation with various factors, and the effects of static drop shape deformation on the limit of stability of the axisymmetric shape of a drop in solid-body rotation.
Trinh, E. H.↗
Rotational Dynamics and Time-variable Gravity of Mars and Implications for Volatile Cycling and Atmospheric Structure
The seasonal motion of atmospheric material, its deposition and sublimation at the poles, represent a minute redistribution of the planet's mass which has been detected in the motion of the Mars Global Surveyor (MGS) spacecraft. These observations have enabled us to estimate the quantity of material involved. Additional information is contained in the original extended abstract.
Zuber, M. T.↗
Determination of the Gravity Field and Rotational Dynamics of an Asteroid by Tracking an
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Orbiting Spacecraft↗
Toward Realistic Dynamics of Rotating Orbital Debris, and Implications for Lightcurve Interpretation
Optical observations of rotating space debris near GEO contain important information on size, shape, composition, and rotational states, but these aspects are difficult to extract due to data limitations and the high number of degrees of freedom in the modeling process. For tri-axial rigid debris objects created by satellite fragmentations, the most likely initial rotation state has a large component of initial angular velocity directed along the intermediate axis of inertia, leading to large angular reorientations of the body on the timescale of the rotation period. This lends some support to the simplest possible interpretation of light curves -- that they represent sets of random orientations of the objects of study. However, effects of internal friction and solar radiation are likely to cause significant modification of rotation states within a time as short as a few orbital periods. In order to examine the rotational dynamics of debris objects under the influences of these effects, a set of seven first-order coupled equations of motion were assembled in state form: three are Euler equations describing the rates of change of the components of angular velocity in the body frame, and four describe the rates of change of the components of the unit quaternion. Quaternions are a four-dimensional extension of complex numbers that form a seamless, singularity-free representation of body orientation on S3. The Euler equations contain explicit terms describing torque from solar radiation in terms of spherical harmonics, and terms representing effects of a prescribed rate of internal friction. Numerical integrations of these equations of motion are being performed, and results will be presented. Initial tests show that internal friction without solar radiation torque leads to rotation about the maximum principal axis of inertia, as required, and solar radiation torque is expected to lead to spin-up of objects. Because the axis of maximum rotational inertia tends to be roughly coincident with the normal to the largest projected cross-sectional area, internal friction is expected to lead to reduced variation of light curve amplitudes at a given phase angle, but a large dependence of the same on phase angle. At a given phase angle, databases are generated which contain reflected intensities for comprehensive sets of equally-likely orientations, represented as unit quaternions. When projected onto three dimensions (S2) and color-coded by intensity, the set is depicted as points within a solid, semi-transparent unit sphere, within which all possible reflected intensities for an object at a given phase angle may be inspected simultaneously. Rotational sequences are represented by trajectories through the sphere. Databases are generated for each of a set of phase angles separately, forming a comprehensive dataset of reflected intensities spanning all object orientations and solar phase angles. Symmetries in the problem suggest that preferred rotation states are likely, defined relative to the object-sun direction in inertial space and relative to the maximum principal axis of inertia in the body coordinate system. Such rotation states may greatly simplify the problem of light curve interpretation by reducing the number of degrees of freedom in the problem.
Ojakangas, Gregory W.↗