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DOE OSTI2025

Dynamics of McMillan mappings I. McMillan multipoles

In this article, we consider two dynamical systems: the McMillan sextupole and octupole integrable mappings, originally proposed by Edwin McMillan. Both represent the simplest symmetric McMillan maps, characterized by a single intrinsic parameter. While these systems find numerous applications across various domains of mathematics and physics, some of their dynamical properties remain unexplored. We aim to bridge this gap by providing a comprehensive description of all stable trajectories, including the parametrization of invariant curves, Poincaré rotation numbers, and canonical action–angle variables. In the second part, we establish connections between these maps and general chaotic maps in standard form. Our investigation reveals that the McMillan sextupole and octupole serve as first-order approximations of the dynamics around the fixed point, akin to the linear map and quadratic invariant (known as the Courant–Snyder invariant in accelerator physics), which represents zeroth-order approximations (referred to as linearization). Furthermore, we propose a novel formalism for nonlinear Twiss parameters, which accounts for the dependence of rotation number on amplitude. This stands in contrast to conventional betatron phase advance used in accelerator physics, which remains independent of amplitude. Notably, in the context of accelerator physics, this new formalism demonstrates its capability in predicting dynamical aperture around low-order resonances for flat beams, a critical aspect in beam injection/extraction scenarios.

43 PARTICLE ACCELERATORS↗

DOE OSTI2024

Winds and Disk Turbulence Exert Equal Torques on Thick Magnetically Arrested Disks

Abstract The conventional accretion disk lore is that magnetized turbulence is the principal angular momentum transport process that drives accretion. However, when dynamically important large-scale magnetic fields thread an accretion disk, they can produce mass and angular momentum outflows, known as winds,that also drive accretion. Yet, the relative importance of turbulent and wind-driven angular momentum transport is still poorly understood. To probe this question, we analyze a long-duration (1.2 × 10 5 r g /c) simulation of a rapidly rotating (a= 0.9) black hole feeding from a thick (H/r∼ 0.3), adiabatic, magnetically arrested disk (MAD), whose dynamically important magnetic field regulates mass inflow and drives both uncollimated and collimated outflows (i.e., winds and jets, respectively). By carefully disentangling the various angular momentum transport processes within the system, we demonstrate the novel result that disk winds and disk turbulence both extract roughly equal amounts of angular momentum from the disk. We find cumulative angular momentum and mass accretion outflow rates of L ̇ ∝ r 0.9 and M ̇ ∝ r 0.4 , respectively. This result suggests that understanding both turbulent and laminar stresses is key to understanding the evolution of systems where geometrically thick MADs can occur, such as the hard state of X-ray binaries, low-luminosity active galactic nuclei, some tidal disruption events, and possibly gamma-ray bursts.

Astronomy & Astrophysics↗

NASA NTRSDate not supplied

Dynamics of A Vibration Isolation System Including Inertia of the Human Body

Using an exercise device in a spacecraft is liable to transmit an unacceptable amount of vibration to that vehicle. This is commonly mitigated by a Vibration Isolation System (VIS), whose dynamics must be analyzed to confirm that the oscillatory forces on the spacecraft remain within allowed range, both from a structural and microgravity perspective (see, e.g., [1]). When modeling a VIS for countermeasures devices, one common approach is to record forces and moments applied on the floor while exercising, and then drive the VIS simulation by applying these recorded loads to the exercise platform part of the VIS mechanical model. This approach misses the fact that when exercising on a moving platform, the force and moment on it will differ from that on the stationary floor due to inertial effects involving the human body. For example, standing up on a platform as it gives under the subject’s feet reduces the foot force on it, and such inertial effects are especially complex for rotational motion. In principle, one could model both the motion of the human body and dynamics of the VIS mechanism in a single combined simulation, e.g., employing a tool such as the commonly used biomechanical simulation OpenSim. Here, the joints of the human body would be driven kinematically along prescribed exercise trajectories while the dynamics engine computed the response of the VIS degrees of freedom. However, mechanism designers and biomechanics experts have their own established tools, making it very desirable to have a way of decoupling the biomechanics from the VIS modeling, simulation, and analyses. We have derived a set of equations that rigorously accomplishes this goal, and have implemented them as an interface function that provides an alternative driving mechanism for an existing force-based VIS analysis simulation. When enabled, the simulated human/VIS system dynamics is now driven by this function, instead of the recorded force methodology described above. The function is designed to accept input from a data file containing the required time-stamped human motion and inertia terms corresponding to the specific exercise in question. This data file is generated by an OpenSim plugin written for that purpose. The existing VIS analytical simulation is developed using NASA’s Trick Simulation Environment [2], as well as its MBDyn multibody dynamics [3] package. The presentation will provide a detailed overview of the mathematical formulation, assumptions, plugin implementation, software interfaces, and results for a sample set of representative exercises. The results from this work aim to better inform VIS design efforts, as well as countermeasure device/protocol designs with respect to exercise type and frequency effects on vehicle structural and microgravity restrictions.

Countermeasures↗

NASA NTRS1978

Earth rotation and network geometry optimization for very long baseline interferometers

The accuracy of direction measurements is substantially improved by Very Long Baseline Interferometry (VLBI). The technique is geometric in that it operates independently of the earth's gravity field, and dynamic in that observations of polar motion, nutation, length of day variations, and precession are also possible. Attention is given to a VLBI study of earth rotation within a framework of network geometry parameters.

Dermanis, A.↗

NASA NTRSDate not supplied

An Overset Grid Motion Driver for Resolving Entry Vehicle CFD Simulations with Large Changes in Wake Orientation

Time-accurate computational fluid dynamics (CFD) simulations using overset meshes can enable reduced computational cost compared to a rigid, monolithic mesh for blunt-body atmospheric entry vehicles. If the vehicle has a significant change in attitude, the high-density region of the overset mesh designed for resolving the subsonic wake can be kept in position as the wake orientation changes relative to the body. Current simulations generally keep the wake mesh fixed in attitude while the body rotates, assuming the wake orientation remains relatively constant. However, changes in the relative velocity over a long-duration simulation or effects of the body geometry can invalidate this assumption. This work presents a method for calculating the desired mesh orientation to resolve a blunt body wake using the atmosphere-relative velocity of the vehicle. A motion driver using these calculations is implemented in a CFD-in-the-loop flight model bridging the FUN3D flow solver and POST2 trajectory propagator. Overset meshing is performed using the Yoga extension for FUN3D. Results are presented for simulations with both overset and monolithic meshes under large changes in relative velocity (and therefore wake orientation). The results demonstrate that the motion driver successfully orients an overset wake mesh to capture the subsonic wake with sufficient resolution despite variable body rotation and body-relative freestream velocity.

Computational Fluid Dynamics↗

NASA NTRS1992

Results of a zonally truncated three-dimensional model of the Venus middle atmosphere

Although the equatorial rotational speed of the solid surface of Venus is only 4 m s(exp-1), the atmospheric rotational speed reaches a maximum of approximately 100 m s(exp-1) near the equatorial cloud top level (65 to 70 km). This phenomenon, known as superrotation, is the central dynamical problem of the Venus atmosphere. We report here the results of numerical simulations aimed at clarifying the mechanism for maintaining the equatorial cloud top rotation. Maintenance of an equatorial rotational speed maximum above the surface requires waves or eddies that systematically transport angular momentum against its zonal mean gradient. The zonally symmetric Hadley circulation is driven thermally and acts to reduce the rotational speed at the equatorial cloud top level; thus wave or eddy transport must counter this tendency as well as friction. Planetary waves arising from horizontal shear instability of the zonal flow (barotropic instability) could maintain the equatorial rotation by transporting angular momentum horizontally from midlatitudes toward the equator. Alternatively, vertically propagating waves could provide the required momentum source. The relative motion between the rotating atmosphere and the pattern of solar heating, which as a maximum where solar radiation is absorbed near the cloud tops, drives diurnal and semidiurnal thermal tides that propagate vertically away from the cloud top level. The effect of this wave propagation is to transport momentum toward the cloud top level at low latitudes and accelerate the mean zonal flow there. We employ a semispectral primitive equation model with a zonal mean flow and zonal wavenumbers 1 and 2. These waves correspond to the diurnal and semidiurnal tides, but they can also be excited by barotropic or baroclinic instability. Waves of higher wavenumbers and interactions between the waves are neglected. Symmetry about the equator is assumed, so the model applies to one hemisphere and covers the altitude range 30 to 110 km. Horizontal resolution is 1.5 deg latitude, and vertical resolution is 1.5 km. Solar and thermal infrared heating, based on Venus observations and calculations drive the model flow. Dissipation is accomplished mainly by Rayleigh friction, chosen to produce strong dissipation above 85 km in order to absorb upward propagating waves and limit extreme flow velocities there, yet to give very weak Rayleigh friction below 70 km; results in the cloud layer do not appear to be sensitive to the Rayleigh friction. The model also has weak vertical diffusion, and very weak horizontal diffusion, which has a smoothing effect on the flow only at the two grid points nearest the pole.

Newman, M.↗

NASA NTRS1989

Inertial corrections by dynamic estimation

The highlights are presented of an Engineering Memorandum, Dynamic Estimation for Floated Gradiometers. The original impetus for the work was that gradiometers, in principle, measure components of the gravity gradient tensor, plus rotation effects, similar to centrifugal and Coriolis effects in accelerometers. The problem is that the rotation effects are often quite large, compared to the gradient, and that available inertial instruments can't measure them to adequate accuracy. The paper advances the idea that, if the instruments can be floated in a package subject to very low disturbances, a dynamic estimation, based on the Euler and translational equations of motion, plus models of all the instruments, can be used to greatly strengthen the estimates of the gradient and the rotation parameters. Moreover, symmetry constraints can be imposed directly in the filter, further strengthening the solution.

Sonnabend, David↗

NASA NTRS1991

NASA low-speed centrifugal compressor for 3-D viscous code assessment and fundamental flow physics research

A low speed centrifugal compressor facility recently built by the NASA Lewis Research Center is described. The purpose of this facility is to obtain detailed flow field measurements for computational fluid dynamic code assessment and flow physics modeling in support of Army and NASA efforts to advance small gas turbine engine technology. The facility is heavily instrumented with pressure and temperature probes, both in the stationary and rotating frames of reference, and has provisions for flow visualization and laser velocimetry. The facility will accommodate rotational speeds to 2400 rpm and is rated at pressures to 1.25 atm. The initial compressor stage being tested is geometrically and dynamically representative of modern high-performance centrifugal compressor stages with the exception of Mach number levels. Preliminary experimental investigations of inlet and exit flow uniformly and measurement repeatability are presented. These results demonstrate the high quality of the data which may be expected from this facility. The significance of synergism between computational fluid dynamic analysis and experimentation throughout the development of the low speed centrifugal compressor facility is demonstrated.

Hathaway, M. D.↗

NASA NTRS1991

NASA low-speed centrifugal compressor for 3-D viscous code assessment and fundamental flow physics research

Hathaway, M. D.↗

NASA NTRS1980

Nonlinear dynamics; Proceedings of the International Conference, New York, NY, December 17-21, 1979

Papers were presented on turbulence, ergodic and integrable behavior, chaotic maps and flows, chemical and fully developed turbulence, and strange attractors. Specific attention was given to measures describing a turbulent flow, stochastization and collapse of vortex systems, a subharmonic route to turbulent convection, and weakly nonlinear turbulence in a rotating convection layer. The Korteweg-de Vries and Hill equations, plasma transport in three dimensions, a horseshoe in the dynamics of a forced beam, and the explosion of strange attractors exhibited by Duffing's equation were also considered.

Helleman, R. H. G.↗

DOE OSTI2021

Disentangling Ferroelectric Wall Dynamics and Identification of Pinning Mechanisms via Deep Learning

Field-induced domain-wall dynamics in ferroelectric materials underpins multiple applications ranging from actuators to information technology devices and necessitates a quantitative description of the associated mechanisms including giant electromechanical couplings, controlled nonlinearities, or low coercive voltages. While the advances in dynamic piezoresponse force microscopy measurements over the last two decades have rendered visualization of polarization dynamics relatively straightforward, the associated insights into the local mechanisms have been elusive. This work explores the domain dynamics in model polycrystalline materials using a workflow combining deep-learning-based segmentation of the domain structures with nonlinear dimensionality reduction using multilayer rotationally invariant autoencoders (rVAE). The former allows unambiguous identification and classification of the ferroelectric and ferroelastic domain walls. Here, the rVAE discovers the latent representations of the domain wall geometries and their dynamics, consequently providing insight into the intrinsic mechanisms of polarization switching, that can further be compared to simple physical models. The rVAE disentangles the factors affecting the pinning efficiency of ferroelectric walls, offering insights into the correlation of ferroelastic wall distribution and ferroelectric wall pinning.

36 MATERIALS SCIENCE↗

NASA NTRS1989

A decoupled recursive approach for constrained flexible multibody system dynamics

A variational-vector calculus approach is employed to derive a recursive formulation for dynamic analysis of flexible multibody systems. Kinematic relationships for adjacent flexible bodies are derived in a companion paper, using a state vector notation that represents translational and rotational components simultaneously. Cartesian generalized coordinates are assigned for all body and joint reference frames, to explicitly formulate deformation kinematics under small deformation kinematics and an efficient flexible dynamics recursive algorithm is developed. Dynamic analysis of a closed loop robot is performed to illustrate efficiency of the algorithm.

Lai, Hao-Jan↗

DOE OSTI2022

Revealing a Hidden Intermediate of Rotatory Catalysis with X-ray Crystallography and Molecular Simulations

The mechanism of rotatory catalysis in ATPhydrolyzing molecular motors remains an unresolved puzzle in biological energy transfer. Notwithstanding the wealth of available biochemical and structural information inferred from years of experiments, knowledge on how the coupling between the chemical and mechanical steps within motors enforces directional rotatory movements remains fragmentary. Even more contentious is to pinpoint the rate-limiting step of a multistep rotation process. Here, using vacuolar or V 1 -type hexameric ATPase as an exemplary rotational motor, we present a model of the complete 4-step conformational cycle involved in rotatory catalysis. First, using Xray crystallography, a new intermediate or “dwell” is identified, which enables the release of an inorganic phosphate (or P i ) after ATP hydrolysis. Using molecular dynamics simulations, this new dwell is placed in a sequence with three other crystal structures to derive a putative cyclic rotation path. Free-energy simulations are employed to estimate the rate of the hexameric protein transformations and delineate allosteric effects that allow new reactant ATP entry only after hydrolysis product exit. An analysis of transfer entropy brings to light how the side-chain-level interactions transcend into larger-scale reorganizations, highlighting the role of the ubiquitous arginine-finger residues in coupling chemical and mechanical information. An inspection of all known rates encompassing the 4-step rotation mechanism implicates the overcoming of the ADP interactions with V 1 -ATPase to be the rate-limiting step of motor action.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗

NASA NTRS1993

Orientation and movement in unusual force environments

A manned space mission to Mars might take as long as 1 year each way. Consequently, artificial gravity is being considered as a way of preventing the debilitating effects of long-duration exposure to microgravity on the human body. The present article discusses some of the problems associated with adapting to the rotation levels that might be used to generate artificial gravity. It also describes how exposure to background-force levels greater or less than the 1-G force of Earth gravity affects orientation and movement control. The primary emphasis of the article is that human movement and orientation control are dynamically adapted to the 1-G force background of Earth and that accommodation to altered force levels or to rotating environments requires a wide range of adaptive changes.

Parabolic Flight↗

NASA NTRS1984

A convective model for turbulent mixing in rotating convection zones

The effects of rotation are included in an analytical model for the convective motions in a plane-parallel layer of an ideal fluid. The turbulent stress tensor, formed by taking products and averages of the various velocity components, is calculated for an arbitrary eddy size and shape. Heuristic formulae presented for determining the size and shape of the dominant eddy then give a fully specified stress tensor. Applications for this stress tensor in problems of stellar internal dynamics, heat flow, scalar diffusion, and dynamo theory are suggested. The resultant stresses tend to produce differential rotation profiles with rapidly rotating equators and interiors. The dynamo activity associated with these convective motions tends to occur near the lower boundary of the convection zone.

Hathaway, D. H.↗

DOE OSTI2024

Bulk viscosity of the rigid rotor one-component plasma

Bulk viscosity of a plasma consisting of strongly coupled diatomic ions is computed using molecular dynamics simulations. The simulations are based on the rigid rotor one-component plasma, which is introduced as a model system that adds two degrees of molecular rotation to the traditional one-component plasma. It is characterized by two parameters: the Coulomb coupling parameter, Γ, and the bond length parameter, Ω. Results show that the long-range nature of the Coulomb potential can lead to long rotational relaxation times, which in turn yield large values for bulk viscosity. Here, the bulk-to-shear viscosity ratio is found to span from small to large values depending on the values of Γ and Ω. Although bulk viscosity is often neglected in plasma modeling, these results motivate that it can be large in molecular plasmas with rotational degrees of freedom.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1989

Solving the BM Camelopardalis puzzle

BM Camelopardalis (=12 Cam) is a chromospherically active binary star with a relatively large orbital eccentricity. Systems with large eccentricities usually rotate pseudosynchronously. However, BM Cam has been a puzzle since its observed rotation rate is virtually equal to its orbital period indicating synchronization. All available photometry data for BM Cam have been collected and analyzed. Two models of modulated ellipticity effect are proposed, one based on equilibrium tidal deformation of the primary star and the other on a dynamical tidal effect. When the starspot variability is removed from the data, the dynamical tidal model was the better approximation to the real physical situation. The analysis indicates that BM Cam is not rotating pseudosynchronously but rotating in virtual synchronism after all.

Teke, Mathias↗

NASA NTRS1985

The exact eigenfunctions and eigenvalues of a two-dimensional rigid rotor obtained using Gaussian wave packet dynamics

Exact eigenfunctions for a two-dimensional rigid rotor are obtained using Gaussian wave packet dynamics. The wave functions are obtained by propagating, without approximation, an infinite set of Gaussian wave packets that collectively have the correct periodicity, being coherent states appropriate to this rotational problem. This result leads to a numerical method for the semiclassical calculation of rovibrational, molecular eigenstates. Also, a simple, almost classical, approximation to full wave packet dynamics is shown to give exact results: this leads to an a posteriori justification of the De Leon-Heller spectral quantization method.

Reimers, J. R.↗