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NASA NTRS1976

Rigid rotators

The two-particle, steady-state Schroedinger equation is transformed to center of mass and internuclear distance vector coordinates, leading to the free particle wave equation for the kinetic energy motion of the molecule and a decoupled wave equation for a single particle of reduced mass moving in a spherical potential field. The latter describes the vibrational and rotational energy modes of the diatomic molecule. For fixed internuclear distance, this becomes the equation of rigid rotator motion. The classical partition function for the rotator is derived and compared with the quantum expression. Molecular symmetry effects are developed from the generalized Pauli principle that the steady-state wave function of any system of fundamental particles must be antisymmetric. Nuclear spin and spin quantum functions are introduced and ortho- and para-states of rotators, along with their degeneracies, are defined. Effects of nuclear spin on entropy are deduced. Next, rigid polyatomic rotators are considered and the partition function for this case is derived. The patterns of rotational energy levels for nonlinear molecules are discussed for the spherical symmetric top, for the prolate symmetric top, for the oblate symmetric top, and for the asymmetric top. Finally, the equilibrium energy and specific heat of rigid rotators are derived.

Source record↗

NASA NTRS1984

Formulation of blade-flutter spectral analyses in stationary reference frame

Analytic representations are developed for the discrete blade deflection and the continuous tip static pressure fields in a stationary reference frame. Considered are the sampling rates equal to the rotational frequency, equal to blade passing frequency, and for the pressure, equal to a multiple of the blade passing frequency. For the last two rates the expressions for determining the nodal diameters from the spectra are included. A procedure is presented for transforming the complete unsteady pressure field into a rotating frame of reference. The determination of the true flutter frequency by using two sensors is described. To illustrate their use, the developed procedures are used to interpret selected experimental results.

Kurkov, A. P.↗

DOE OSTI2024

Mechanical systems and kinematics

Kinematics is the study of how motion is achieved without reference to the forces that create the motion. This includes the basic structures of all robotic systems, such as links and joints, as well as their mathematical representations. This chapter will provide a brief overview of the types of motion used in additive manufacturing (AM) systems, as well as the mathematical equations that govern them. Rotations are explained using both Euler rotations and quaternions. Combined rotation and translations are shown using homogeneous transform matrices. The robot Jacobian, which relates joint velocities to the end effector translational velocities, is explained with an example calculation. This section will give the reader the basic mathematical foundation in kinematics that is necessary to understand the mechanical underpinnings of path planning for AM systems.

Wang, Peter↗

DOE OSTI2020

Simulating pitch angle scattering using an explicitly solvable energy-conserving algorithm

Particle distribution functions evolving under the Lorentz operator can be simulated with the Langevin equation for pitch angle scattering. This approach is frequently used in particle based Monte-Carlo simulations of plasma collisions, among others. However, most numerical treatments do not guarantee energy conservation, which may lead to unphysical artifacts such as numerical heating and spectra distortions. In this paper, we present a novel structure-preserving numerical algorithm for the Langevin equation for pitch angle scattering. Similar to the well-known Boris algorithm, the proposed numerical scheme takes advantage of the structure-preserving properties of the Cayley transform when calculating the velocity-space rotations. The resulting algorithm is explicitly solvable, while preserving the norm of velocities down to machine precision. We demonstrate that the method has the same order of numerical convergence as the traditional stochastic Euler-Maruyama method. The numerical scheme is benchmarked by simulating the pitch angle scattering of a particle beam, and comparing with the analytical solution. Benchmark results show excellent agreement with theoretical predictions, showcasing the remarkable long time accuracy of the proposed algorithm.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

DOE OSTI2020

Simulating pitch angle scattering using an explicitly solvable energy-conserving algorithm

Particle distribution functions evolving under the Lorentz operator can be simulated with the Langevin equation for pitch angle scattering. This approach is frequently used in particle based Monte-Carlo simulations of plasma collisions, among others. However, most numerical treatments do not guarantee energy conservation, which may lead to unphysical artifacts such as numerical heating and spectra distortions. We present a novel structure-preserving numerical algorithm for the Langevin equation for pitch angle scattering. Similar to the well-known Boris algorithm, the proposed numerical scheme takes advantage of the structure-preserving properties of the Cayley transform when calculating the velocity-space rotations. The resulting algorithm is explicitly solvable, while preserving the norm of velocities down to machine precision. We demonstrate that the method has the same order of numerical convergence as the traditional stochastic Euler-Maruyama method.

Zhang, Xin↗

NASA NTRS1993

Movable Cameras And Monitors For Viewing Telemanipulator

Three methods proposed to assist operator viewing telemanipulator on video monitor in control station when video image generated by movable video camera in remote workspace of telemanipulator. Monitors rotated or shifted and/or images in them transformed to adjust coordinate systems of scenes visible to operator according to motions of cameras and/or operator's preferences. Reduces operator's workload and probability of error by obviating need for mental transformations of coordinates during operation. Methods applied in outer space, undersea, in nuclear industry, in surgery, in entertainment, and in manufacturing.

Diner, Daniel B.↗

NASA NTRS1994

Solar Rotating Fourier Telescope

Proposed telescope based on absorbing Fourier-transform grids images full Sun at unprecedented resolution. Overcomes limitations of both conventional optical and pinhole cameras. Arrays of grids and detectors configured for sensitivity to selected fourier components of x-ray images.

Campbell, Jonathan↗

NASA NTRS2018

Time-Varying Entry Heating Profile Replication with a Rotating Arc Jet Test Article

A new approach to arc jet testing that replicates the time-varying conditions of atmospheric entry heating was demonstrated. The concept relies on the spatial variation of heat flux and pressure around a cylindrical test article. The spatial variation is transformed to a temporal variation at one location by rotating the test article during exposure to the arc jet stream. The test article rotation mechanism is based on a programmable stepper motor. An entry heat flux profile was mapped to the cylinders heat flux distribution to determine the rotation angle vs. time executed by the mechanism. A series of arc jet tests with instrumented cylindrical TPS test articles was conducted to prove the concept.

Aerothermodynamics↗

NASA NTRS1999

Integration of vestibular and head movement signals in the vestibular nuclei during whole-body rotation

Single-unit recordings were obtained from 107 horizontal semicircular canal-related central vestibular neurons in three alert squirrel monkeys during passive sinusoidal whole-body rotation (WBR) while the head was free to move in the yaw plane (2.3 Hz, 20 degrees /s). Most of the units were identified as secondary vestibular neurons by electrical stimulation of the ipsilateral vestibular nerve (61/80 tested). Both non-eye-movement (n = 52) and eye-movement-related (n = 55) units were studied. Unit responses recorded when the head was free to move were compared with responses recorded when the head was restrained from moving. WBR in the absence of a visual target evoked a compensatory vestibulocollic reflex (VCR) that effectively reduced the head velocity in space by an average of 33 +/- 14%. In 73 units, the compensatory head movements were sufficiently large to permit the effect of the VCR on vestibular signal processing to be assessed quantitatively. The VCR affected the rotational responses of different vestibular neurons in different ways. Approximately one-half of the units (34/73, 47%) had responses that decreased as head velocity decreased. However, the responses of many other units (24/73) showed little change. These cells had signals that were better correlated with trunk velocity than with head velocity. The remaining units had responses that were significantly larger (15/73, 21%) when the VCR produced a decrease in head velocity. Eye-movement-related units tended to have rotational responses that were correlated with head velocity. On the other hand, non-eye-movement units tended to have rotational responses that were better correlated with trunk velocity. We conclude that sensory vestibular signals are transformed from head-in-space coordinates to trunk-in-space coordinates on many secondary vestibular neurons in the vestibular nuclei by the addition of inputs related to head rotation on the trunk. This coordinate transformation is presumably important for controlling postural reflexes and constructing a central percept of body orientation and movement in space.

Non-NASA Center↗

NASA NTRS1993

Effect of rotation rate on the forces of a rotating cylinder: Simulation and control

In this paper we present numerical solutions to several optimal control problems for an unsteady viscous flow. The main thrust of this work is devoted to simulation and control of an unsteady flow generated by a circular cylinder undergoing rotary motion. By treating the rotation rate as a control variable, we can formulate two optimal control problems and use a central difference/pseudospectral transform method to numerically compute the optimal control rates. Several types of rotations are considered as potential controls, and we show that a proper synchronization of forcing frequency with the natural vortex shedding frequency can greatly influence the flow. The results here indicate that using moving boundary controls for such systems may provide a feasible mechanism for flow control.

Burns, John A.↗

NASA NTRS1976

On differential transformations between Cartesian and curvilinear (geodetic) coordinates

Differential transformations are developed between Cartesian and curvilinear orthogonal coordinates. Only matrix algebra is used for the presentation of the basic concepts. After defining the reference systems used the rotation (R), metric (H), and Jacobian (J) matrices of the transformations between cartesian and curvilinear coordinate systems are introduced. A value of R as a function of H and J is presented. Likewise an analytical expression for J(-1) as a function of H(-2) and R is obtained. Emphasis is placed on showing that differential equations are equivalent to conventional similarity transformations. Scaling methods are discussed along with ellipsoidal coordinates. Differential transformations between elipsoidal and geodetic coordinates are established.

Soler, T.↗

DOE OSTI2022

Phenomenological R -Matrix parameterization of direct, doorway, and compound nuclear reactions [Abstract]

Although formal expressions for scattering matrix accounting for direct, doorway, and compound nuclear (CN) resonant reactions have been derived several decades ago in both the transition ( T -)matrix formalism and the reactance ( K -)matrix formalism, the absence of corresponding expressions in phenomenological R -matrix formalism has limited the application of the latter to CN resonant reactions only. We remove this limitation by parameterizing direct, doorway, and CN resonant reactions in a phenomenological R -matrix scattering matrix, and provide a parameterization for a corresponding Reich-Moore approximation of eliminated capture channels. Direct reactions induce (previously neglected) mixing among the incoming or outgoing R -matrix channel wave functions, parameterized by real and orthonormal channel-rotation matrix, M , whereby the original scattering matrix U is transformed into M T UM . Any real and orthonormal matrix, M , can be equivalently expressed as e η , where η is a real and skew-symmetric 2 rotation-generating matrix that subsequently yields a more intuitive parameterization of eliminated direct capture reactions in Reich-Moore approximation. A phenomenological R -matrix parameterization of doorway reactions is inferred by equating the expression for reactance ( K -)matrix, given in terms of Brune’s alternative R -matrix parameterization, to a corresponding expression derived using Feshbach’s projection operator formalism. Assuming that all doorway states, just like CN states, are confined within spheres defined by R -matrix channel radii, a new R -matrix-like term induced by doorway states is gleaned, wherein each doorway state is parameterized by its energy, width, and the strength of its coupling to each CN state. Since a Reich-Moore approximation for retained-channel scattering matrix ought to approximate the effect of eliminated capture channels taking place via direct, doorway, or CN reactions, each of the three kinds of reactions contributing to the capture entails a corresponding Reich-Moore parameterization in a first-order approximation: direct contribution is parameterized by introducing finite diagonal elements of a retained-channel rotation-generating matrix, doorway contribution is parameterized by doorway capture widths, while CN contribution is parameterized by conventional Reich-Moore capture widths. We will present evidence of direct and doorway reactions observed in recent measurements of resolved resonance cross sections at the Gaerttner LINAC Center at Rensselaer Polytechnic Institute, and will outline a path for implementing this new R -matrix parameterization into the SAMMY nuclear data evaluation code.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1981

Fourier Transform Spectrometer observations of solar carbon monoxide. I - The fundamental and first overtone bands in the quiet sun

Measurements of the 2200/cm fundamental and 4300/cm first overtone vibration-rotation band systems of solar carbon monoxide, were obtained with the Fourier Transform Spectrometer of the McMath telescope at Kitt Peak. The overtone measurements were taken at the east, north, and west heliocentric limbs, and at disk center. Observations of the strong fundamental bands were obtained at disk center and near the north limb. The low core brightness temperatures of the strongest fundamental carbon monoxide lines near the limb, reported previously by Noyes (1972) and Hall (1974), are confirmed. The possibility that thermal inhomogeneities might be responsible for the unusual behavior of the fundamental carbon dioxide lines have been examined. The somewhat discordant behavior of the fundamental lines at disk center compared with the north limb seems to favor a limb shadowing effect. The first overtone limb equivalent widths and the best-fit thermal and microvelocity models indicate a solar carbon abundance of 0.004 (on the scale with A sub H = 1) for an oxygen-to-carbon abundance ratio of 2.

Ayres, T. R.↗

NASA NTRS1987

Phase closure with a rotational shear interferometer

A simple and efficient way is proposed for achieving phase closure in an optical telescope (to enable recovering Fourier transform phases that would otherwise be corrupted by atmospheric and instrumental errors), by means of rotational shear interferometry. In a rotational shear interferometer, one images the telescope aperture onto the interferometer and then interferes the aperture with itself in a rotated orientation. To achieve the maximum frequency content permitted by the telescope, the shear has to be 180 deg, but better dynamic range and SNR are possible for lower frequencies. Drawbacks of the proposed method compared to radio astronomy are noted, and different approaches are indicated as to how to collect and use the phase closure data. Phase closure can be realized on existing telescopes and existing interferometers with special modifications. Although not all base lines are possible, the extra constraints provided by the closure phases greatly reduce the ambiguity now existing in phaseless image reconstruction.

Ribak, Erez↗

NASA NTRS1989

Optical pattern recognition; Proceedings of the Meeting, Los Angeles, CA, Jan. 17, 18, 1989

Papers on optical pattern recognition are presented, covering topics such as the estimation of satellite pose and motion parameters using a neural net tracker, associative memory, optical implmentation of programmable neural networks, optoelectronic neural networks, dynamic autoassociative neural memory, heteroassociative memory, bilinear pattern recognition processors, optical processing of optical correlation plane data, and a synthetic discriminant function-based nonlinear optical correlator. Other topics include an interactive optical-digital image processor, geometric transformations for video compression and human teleoperator display, quasiconformal remapping for compensation of human visual field defects, hybrid vision for automated spacecraft landing, advanced symbolic and inference optical correlation filters, and a rotationally invariant holographic tracking system. Additional topics include the detection of rotational and scale-varying objects with a programmable joint transform correlator, a single spatial light modulator binary nonlinear optical correlator, optical joint transform correlation, linear phase coefficient composite filters, and binary phase-only filters.

Liu, Hua-Kuang↗

DOE OSTI2024

Generalized Quantum Signal Processing

Quantum signal processing (QSP) and quantum singular value transformation (QSVT) currently stand as the most efficient techniques for implementing functions of block-encoded matrices, a central task that lies at the heart of most prominent quantum algorithms. However, current QSP approaches face several challenges, such as the restrictions imposed on the family of achievable polynomials and the difficulty of calculating the required phase angles for specific transformations. In this paper, we present a generalized quantum signal processing (GQSP) approach, employing general SU(2) rotations as our signal-processing operators, rather than relying solely on rotations in a single basis. Our approach lifts all practical restrictions on the family of achievable transformations, with the sole remaining condition being that | P | ≤ 1 , a restriction necessary due to the unitary nature of quantum computation. Furthermore, GQSP provides a straightforward recursive formula for determining the rotation angles needed to construct the polynomials in cases where P and Q are known. In cases where only P is known, we provide an efficient optimization algorithm capable of identifying in under a minute of GPU time, a corresponding Q for polynomials of degree on the order of 10 7 . We further illustrate GQSP simplifies QSP-based strategies for Hamiltonian simulation, offer an optimal solution to the ϵ -approximate fractional query problem that requires O ( ( 1 / δ ) + log ( 1 / ϵ ) ) queries to perform where O ( 1 / δ ) is a proved lower bound, and introduces novel approaches for implementing bosonic operators. Moreover, we propose a novel framework for the implementation of normal matrices, demonstrating its applicability through synthesis of diagonal matrices, as well as the development of a new algorithm for convolution through synthesis of circulant matrices using only O ( d log N + log 2 N ) 1 and 2-qubit gates for a filter of lengths d . Published by the American Physical Society 2024

Motlagh, Danial↗

DOE OSTI2024

System Modeling and Robust Stability Region Analysis for Multi-inverters Based on VSG

As the number of inverter-interfaced distributed generators (IIDGs) increases, the decrease in the inertia of the microgrid affects the stability of microgrid voltage and current. The virtual synchronous generator (VSG) control method simulates the output characteristics of the synchronous machine to improve the inertia and damping of the microgrid. However, low-frequency oscillations can occur in systems with multi-VSGs connected in parallel, primarily when operating in island mode. Therefore, this article develops the small-signal state-space model for multi-inverters based on VSG and analyses the robust stability region (RSR) of the control parameters. Firstly, an extensible small-signal state-space model is established and simplified in the dq axis, which is a necessary preprocessing for the subsequent stability analysis. Secondly, the guardian map method based on bialternate product operation is constructed, and the RSR is established through translation mapping and rotation mapping. Therefore, the solution of RSR is transformed into the problem of identifying the Hurwitz matrix. Finally, the simulations and experiments verify that the calculated region can ensure stable operation during startup and meet the requirement to suppress low-frequency oscillations. In conclusion, the method of solving RSR proposed in this paper provides a basis for selecting controller parameters in practical engineering.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2026

ACDC (Automated Campbell Diagram Code) [SWR-26-042]

This application provides a web-based graphical user interface to generating Campbell Diagrams and visualizing mode shapes for OpenFAST turbine models. Determining the aeroelastic stability and dynamic characteristics of wind turbines is a critical step in turbine design and analysis. Historically, extracting natural frequencies and mode shapes from OpenFAST—the industry-standard whole-turbine simulation code—has been a fragmented and tedious process. It required manual model configuration, command-line linearization execution, and complex post-processing via proprietary scripts to handle rotating-frame dynamics. To address these workflow bottlenecks, we present the Automated Campbell Diagram Code (ACDC), an open-source graphical software tool developed by the National Laboratory of the Rockies (NLR) under the DOE-funded Distributed Wind Aeroelastic Modeling (dWAM) project. ACDC streamlines the end-to-end linearization and stability analysis workflow into a single, intuitive cross-platform application. The software guides users through OpenFAST model configuration, definition of operating points, and the automated execution of steady-state trim and linearization simulations. Under the hood, ACDC automates the complex mathematical post-processing steps required for rotating systems, including Multi-Blade Coordinate (MBC) transformations, eigenanalysis, and advanced modal tracking utilizing the Modal Assurance Criterion (MAC) and spectral clustering. Finally, ACDC processes these results to automatically generate Campbell diagrams and features a robust 3D visualization engine to animate full-system mode shapes. By eliminating the reliance on external post-processing environments and manual data manipulation, ACDC significantly accelerates dynamic analysis and lowers the barrier to entry for wind energy researchers and engineers.

Summerville, Brent [National Laboratory of the Roc↗