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At least 433 records · Page 24

Mass Conservation and Positivity Preservation with Ensemble-type Kalman Filter Algorithms

Maintaining conservative physical laws numerically has long been recognized as being important in the development of numerical weather prediction (NWP) models. In the broader context of data assimilation, concerted efforts to maintain conservation laws numerically and to understand the significance of doing so have begun only recently. In order to enforce physically based conservation laws of total mass and positivity in the ensemble Kalman filter, we incorporate constraints to ensure that the filter ensemble members and the ensemble mean conserve mass and remain nonnegative through measurement updates. We show that the analysis steps of ensemble transform Kalman filter (ETKF) algorithm and ensemble Kalman filter algorithm (EnKF) can conserve the mass integral, but do not preserve positivity. Further, if localization is applied or if negative values are simply set to zero, then the total mass is not conserved either. In order to ensure mass conservation, a projection matrix that corrects for localization effects is constructed. In order to maintain both mass conservation and positivity preservation through the analysis step, we construct a data assimilation algorithms based on quadratic programming and ensemble Kalman filtering. Mass and positivity are both preserved by formulating the filter update as a set of quadratic programming problems that incorporate constraints. Some simple numerical experiments indicate that this approach can have a significant positive impact on the posterior ensemble distribution, giving results that are more physically plausible both for individual ensemble members and for the ensemble mean. The results show clear improvements in both analyses and forecasts, particularly in the presence of localized features. Behavior of the algorithm is also tested in presence of model error.

EnKR↗

Joint Spectrum Access and Power Control in Air-Air Communications - A Deep Reinforcement Learning Based Approach

This paper considers the dynamic spectrum access and power control problem in a single-hop point-to-point Air-Air Communication Network (AACN). Due to spectrum scarcity, we assume the number of Aircraft-to-Aircraft (A2A) communication links is greater than that of the available channels, such that some communication links need to share the same channel, causing co-channel interference. We formulate the joint channel selection and power control optimization problem to maximize the Weighted Sum Spectral Efficiency (WSSE). A distributed and dynamic deep Q learning-based algorithm is proposed to find the optimal solution. Specifically, we design two different policies that are trained by conducting a trial-and-error scheme. Each communication link can achieve the optimal policy by exploiting the local information from its neighbors, and this distributive approach make it scalable to large networks. Finally, our experimental results demonstrate the effectiveness of the proposed solution in various AACN scenarios.

Zhe Wang↗

Joint Spectrum Access and Power Control in Air-Air Communications - A Deep Reinforcement Learning Based Approach

This paper considers the dynamic spectrum access and power control problem in a single-hop point-to-point Air-Air Communication Network (AACN). Due to spectrum scarcity, we assume the number of Aircraft-to-Aircraft (A2A) communication links is greater than that of the available channels, such that some communication links need to share the same channel, causing co-channel interference. We formulate the joint channel selection and power control optimization problem to maximize the Weighted Sum Spectral Efficiency (WSSE). A distributed and dynamic deep Q learning-based algorithm is proposed to find the optimal solution. Specifically, we design two different policies that are trained by conducting a trial-and-error scheme. Each communication link can achieve the optimal policy by exploiting the local information from its neighbors, and this distributive approach make it scalable to large networks. Finally, our experimental results demonstrate the effectiveness of the proposed solution in various AACN scenarios.

Zhe Wang↗

Correction to: Imaging Light–Induced Migration of Dislocations in Halide Perovskites with 3D Nanoscale Strain Mapping

Owing to an error in properly normalizing the reconstruction phase data into atomic displacements, the strain values that we used to calculate the root mean squared local strain, ε rms , and to calculate the fraction of the crystals more strain than 1%, f, quoted in the original paper, are roughly one order of magnitude too large. This error was only discovered recently whilst performing further analysis.

36 MATERIALS SCIENCE↗

Examining the Impacts of High-Resolution Land Surface Initialization on Model Predictions of Convection in the Southeastern U.S.

One of the most challenging weather forecast problems in the southeastern U.S. is daily summertime pulse convection. During the summer, atmospheric flow and forcing are generally weak in this region; thus, convection typically initiates in response to local forcing along sea/lake breezes, and other discontinuities often related to horizontal gradients in surface heating rates. Numerical simulations of pulse convection usually have low skill, even in local predictions at high resolution, due to the inherent chaotic nature of these precipitation systems. Forecast errors can arise from assumptions within physics parameterizations, model resolution limitations, as well as uncertainties in both the initial state of the atmosphere and land surface variables such as soil moisture and temperature. For this study, it is hypothesized that high-resolution, consistent representations of surface properties such as soil moisture and temperature, ground fluxes, and vegetation are necessary to better simulate the interactions between the land surface and atmosphere, and ultimately improve predictions of local circulations and summertime pulse convection. The NASA Short-term Prediction Research and Transition (SPORT) Center has been conducting studies to examine the impacts of high-resolution land surface initialization data generated by offline simulations of the NASA Land Informatiot~ System (LIS) on subsequent numerical forecasts using the Weather Research and Forecasting (WRF) model (Case et al. 2008, to appear in the Journal of Hydrometeorology). Case et al. presents improvements to simulated sea breezes and surface verification statistics over Florida by initializing WRF with land surface variables from an offline LIS spin-up run, conducted on the exact WRF domain and resolution. The current project extends the previous work over Florida, focusing on selected case studies of typical pulse convection over the southeastern U.S., with an emphasis on improving local short-term WRF simulations over the Mobile, AL and Miami, FL NWS county warning areas. Future efforts may involve examining the impacts of assimilating remotely-sensed soil moisture data, and/or introducing weekly greenness vegetation fraction composites (as opposed to monthly climatologies) into ol'fline NASA LIS runs. Based on positive impacts, the offline LIS runs could be transitioned into an operational mode, providing land surface initialization data to NWS forecast offices in real time.

Case, Jonathan L.↗

Probable errors in width distributions of sea ice leads measured along a transect

The degree of error expected in the measurement of widths of sea ice leads along a single transect are examined in a probabilistic sense under assumed orientation and width distributions, where both isotropic and anisotropic lead orientations are examined. Methods are developed for estimating the distribution of 'actual' widths (measured perpendicular to the local lead orientation) knowing the 'apparent' width distribution (measured along the transect), and vice versa. The distribution of errors, defined as the difference between the actual and apparent lead width, can be estimated from the two width distributions, and all moments of this distribution can be determined. The problem is illustrated with Landsat imagery and the procedure is applied to a submarine sonar transect. Results are determined for a range of geometries, and indicate the importance of orientation information if data sampled along a transect are to be used for the description of lead geometries. While the application here is to sea ice leads, the methodology can be applied to measurements of any linear feature.

Key, J.↗

Aerosol Observability and Predictability: From Research to Operations for Chemical Weather Forecasting. Lagrangian Displacement Ensembles for Aerosol Data Assimilation

A challenge common to many constituent data assimilation applications is the fact that one observes a much smaller fraction of the phase space that one wishes to estimate. For example, remotely sensed estimates of the column average concentrations are available, while one is faced with the problem of estimating 3D concentrations for initializing a prognostic model. This problem is exacerbated in the case of aerosols because the observable Aerosol Optical Depth (AOD) is not only a column integrated quantity, but it also sums over a large number of species (dust, sea-salt, carbonaceous and sulfate aerosols. An aerosol transport model when driven by high-resolution, state-of-the-art analysis of meteorological fields and realistic emissions can produce skillful forecasts even when no aerosol data is assimilated. The main task of aerosol data assimilation is to address the bias arising from inaccurate emissions, and Lagrangian misplacement of plumes induced by errors in the driving meteorological fields. As long as one decouples the meteorological and aerosol assimilation as we do here, the classic baroclinic growth of error is no longer the main order of business. We will describe an aerosol data assimilation scheme in which the analysis update step is conducted in observation space, using an adaptive maximum-likelihood scheme for estimating background errors in AOD space. This scheme includes e explicit sequential bias estimation as in Dee and da Silva. Unlikely existing aerosol data assimilation schemes we do not obtain analysis increments of the 3D concentrations by scaling the background profiles. Instead we explore the Lagrangian characteristics of the problem for generating local displacement ensembles. These high-resolution state-dependent ensembles are then used to parameterize the background errors and generate 3D aerosol increments. The algorithm has computational complexity running at a resolution of 1/4 degree, globally. We will present the result of assimilating AOD retrievals from MODIS (on both Aqua and TERRA satellites) from AERONET for validation. The impact on the GEOS-5 Aerosol Forecasting will be fully documented.

da Silva, Arlindo↗

Direct comparison of the impact of head tracking, reverberation, and individualized head-related transfer functions on the spatial perception of a virtual speech source

A study of sound localization performance was conducted using headphone-delivered virtual speech stimuli, rendered via HRTF-based acoustic auralization software and hardware, and blocked-meatus HRTF measurements. The independent variables were chosen to evaluate commonly held assumptions in the literature regarding improved localization: inclusion of head tracking, individualized HRTFs, and early and diffuse reflections. Significant effects were found for azimuth and elevation error, reversal rates, and externalization.

NASA Discipline Space Human Factors↗

NASA Tech Briefs, May 2011

Topics covered include: 1) Method to Estimate the Dissolved Air Content in Hydraulic Fluid; 2) Method for Measuring Collimator-Pointing Sensitivity to Temperature Changes; 3) High-Temperature Thermometer Using Cr-Doped GdAlO3 Broadband Luminescence; 4)Metrology Arrangement for Measuring the Positions of Mirrors of a Submillimeter Telescope; 5) On-Wafer S-Parameter Measurements in the 325-508-GHz Band; 6) Reconfigurable Microwave Phase Delay Element for Frequency Reference and Phase-Shifter Applications; 7) High-Speed Isolation Board for Flight Hardware Testing; 8) High-Throughput, Adaptive FFT Architecture for FPGA-Based Spaceborne Data Processors; 9) 3D Orbit Visualization for Earth-Observing Missions; 10) MaROS: Web Visualization of Mars Orbiting and Landed Assets; 11) RAPID: Collaborative Commanding and Monitoring of Lunar Assets; 12) Image Segmentation, Registration, Compression, and Matching; 13) Image Calibration; 14) Rapid ISS Power Availability Simulator; 15) A Method of Strengthening Composite/Metal Joints; 16) Pre-Finishing of SiC for Optical Applications; 17) Optimization of Indium Bump Morphology for Improved Flip Chip Devices; 18) Measuring Moisture Levels in Graphite Epoxy Composite Sandwich Structures; 19) Marshall Convergent Spray Formulation Improvement for High Temperatures; 20) Real-Time Deposition Monitor for Ultrathin Conductive Films; 21) Optimized Li-Ion Electrolytes Containing Triphenyl Phosphate as a Flame-Retardant Additive; 22) Radiation-Resistant Hybrid Lotus Effect for Achieving Photoelectrocatalytic Self-Cleaning Anticontamination Coatings; 23) Improved, Low-Stress Economical Submerged Pipeline; 24) Optical Fiber Array Assemblies for Space Flight on the Lunar Reconnaissance Orbiter; 25) Local Leak Detection and Health Monitoring of Pressurized Tanks; 26) Dielectric Covered Planar Antennas at Submillimeter Wavelengths for Terahertz Imaging; 27) Automated Cryocooler Monitor and Control System; 28) Broadband Achromatic Phase Shifter for a Nulling Interferometer; 29) Super Dwarf Wheat for Growth in Confined Spaces; 30) Fine Guidance Sensing for Coronagraphic Observatories; 31) Single-Antenna Temperature- and Humidity-Sounding Microwave Receiver; 32) Multi-Wavelength, Multi-Beam, and Polarization-Sensitive Laser Transmitter for Surface Mapping; 33) Optical Communications Link to Airborne Transceiver; 34) Ascent Heating Thermal Analysis on Spacecraft Adaptor Fairings; 35) Entanglement in Self-Supervised Dynamics; 36) Prioritized LT Codes; 37) Fast Image Texture Classification Using Decision Trees; 38) Constraint Embedding Technique for Multibody System Dynamics; 39) Improved Systematic Pointing Error Model for the DSN Antennas; 40) Observability and Estimation of Distributed Space Systems via Local Information-Exchange Networks; 41) More-Accurate Model of Flows in Rocket Injectors; 42) In-Orbit Instrument-Pointing Calibration Using the Moon as a Target; 43) Reliability of Ceramic Column Grid Array Interconnect Packages Under Extreme Temperatures; 44) Six Degrees-of-Freedom Ascent Control for Small-Body Touch and Go; and 45) Optical-Path-Difference Linear Mechanism for the Panchromatic Fourier Transform Spectrometer.

Source record↗

Determining Magnetic Field Directions and Omni-directional Fluxes of Particles from STPSat-6 ZPS Plasma Measurements

This report summarizes our recent data derivation efforts on STPSat-6 ZPS plasma measurements, presenting final outcomes. We begin by outlining the methodology developed to determine local magnetic field directions from measured directional intensities of ~20 keV ions, leveraging symmetry in particles’ pitch-angle distributions, and the determined field directions over a storm period are shown with errors analyzed through comparison with NOAA GOES measurements. Next, we further assess the validity of the method and quantify its overall high performance while acknowledging certain caveats. Finally, given local magnetic field directions and ZPS data, we explain how omni-directional fluxes of particles can be derived from ZPS measurements with limited directional coverage, and estimate the related errors under several theoretical scenarios. The methods developed in this study can be applied for processing and augmenting ZPS data for the next, and the insights gained here can also inform instrument design in the future.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

Estimating Sea Surface Salinity and Wind Using Combined Passive and Active L-Band Microwave Observations

Several L-band microwave radiometer and radar missions have been, or will be, operating in space for land and ocean observations. These include the NASA Aquarius mission and the Soil Moisture Active Passive (SMAP) mission, both of which use combined passive/ active L-band instruments. Aquarius s passive/active L-band microwave sensor has been designed to map the salinity field at the surface of the ocean from space. SMAP s primary objectives are for soil moisture and freeze/thaw detection, but it will operate continuously over the ocean, and hence will have significant potential for ocean surface research. In this innovation, an algorithm has been developed to retrieve simultaneously ocean surface salinity and wind from combined passive/active L-band microwave observations of sea surfaces. The algorithm takes advantage of the differing response of brightness temperatures and radar backscatter to salinity, wind speed, and direction, thus minimizing the least squares error (LSE) measure, which signifies the difference between measurements and model functions of brightness temperatures and radar backscatter. The algorithm uses the conjugate gradient method to search for the local minima of the LSE. Three LSE measures with different measurement combinations have been tested. The first LSE measure uses passive microwave data only with retrieval errors reaching 1 to 2 psu (practical salinity units) for salinity, and 1 to 2 m/s for wind speed. The second LSE measure uses both passive and active microwave data for vertical and horizontal polarizations. The addition of active microwave data significantly improves the retrieval accuracy by about a factor of five. To mitigate the impact of Faraday rotation on satellite observations, the third LSE measure uses measurement combinations invariant under the Faraday rotation. For Aquarius, the expected RMS SSS (sea surface salinity) error will be less than about 0.2 psu for low winds, and increases to 0.3 psu at 25 m/s wind speed for warm waters (25 C). To achieve the required 0.2 psu accuracy, the impact of sea surface roughness (e.g. wind-generated ripples) on the observed brightness temperature has to be corrected to better than one tenth of a degree Kelvin. With this algorithm, the accuracy of retrieved wind speed will be high, varying from a few tenths to 0.6 m/s. The expected direction accuracy is also excellent (less than 10 ) for mid to high winds, but degrades for lower speeds (less than 7 m/s).

Yueh, Simon H.↗

Stochastic tensor contraction for quantum chemistry

Many computational methods in ab initio quantum chemistry are formulated in terms of high-order tensor contractions, whose cost determines the size of system that can be studied. We introduce stochastic tensor contraction to perform such operations with greatly reduced cost, and present its application to the gold-standard quantum chemistry method, coupled cluster theory with up to perturbative triples. For total energy errors more stringent than chemical accuracy, we reduce the computational scaling to that of mean-field theory, while starting to approach the mean-field absolute cost, thereby challenging the existing cost-to-accuracy landscape. Benchmarks against state-of-the-art local correlation approximations further show that we achieve an order-of-magnitude improvement in both total computation time and error, with significantly reduced sensitivity to system dimensionality and electron delocalization. We conclude that stochastic tensor contraction is a powerful computational primitive to accelerate a wide range of quantum chemistry.

Chemical Physics (physics.chem-ph)↗

Region Spherical Harmonic Magnetic Modeling from Near-Surface and Satellite-Altitude Anomlaies

The compiled near-surface data and satellite crustal magnetic measured data are modeled with a regionally concentrated spherical harmonic presentation technique over Australia and Antarctica. Global crustal magnetic anomaly studies have used a spherical harmonic analysis to represent the Earth's magnetic crustal field. This global approach, however is best applied where the data are uniformly distributed over the entire Earth. Satellite observations generally meet this requirement, but unequally distributed data cannot be easily adapted in global modeling. Even for the satellite observations, due to the errors spread over the globe, data smoothing is inevitable in the global spherical harmonic presentations. In addition, global high-resolution modeling requires a great number of global spherical harmonic coefficients for the regional presentation of crustal magnetic anomalies, whereas a lesser number of localized spherical coefficients will satisfy. We compared methods in both global and regional approaches and for a case where the errors were propagated outside the region of interest. For observations from the upcoming Swarm constellation, the regional modeling will allow the production a lesser number of spherical coefficients that are relevant to the region of interest

Modeling↗

Global Aerodynamic Modeling Using Automated Local Model Networks in Real Time

A novel method is presented for automated real-time global aerodynamic modeling using local model networks, known as Smoothed Partitioning with Localized Trees in Real Time (SPLITR), as part of NASA’s Learn-to-Fly technology development initiative. The global nonlinear aerodynamics are partitioned into several local regions known as cells, with the dimension, location, and timing of each partition automatically selected based on a residual characterization procedure, under the constraints of real-time operation. Regression trees represent the successive partitioning of the global flight envelope and describe the evolution of the cell structure. Recursive equation-error least-squares parameter estimation in the time domain is used to estimate a model that represents the local aerodynamics in each region, so that it can be updated independently with non-contiguous data in the range of each cell over time. A weighted superposition of these piecewise local models across the flight envelope forms a global nonlinear model that also accurately captures the local aerodynamics. The SPLITR approach is demonstrated using both simulation and flight data, and the results are analyzed in terms of model predictive capabilities as well as interpretability. The results show that SPLITR can be used to automatically partition complex nonlinear aerodynamic behavior, produce an accurate model, and provide valuable physical insight into the local and global aerodynamics.

Weinstein, Rose↗

The Uranian aurora and its relationship to the magnetosphere

About 32 h of Voyager Ultraviolet Spectrometer (UVS) observations of Uranus H2 band airglow emission (875 less than or equal to lambda less than or equal to 1115 A) have been analyzed using the singular value decomposition (SVD) approach to inversion, producing an intensity map showing aurora at both magnetic poles. An H Lyman alpha aurora may also be present but is difficult to separate from scattered solar and local interstellar medium components. SVD analysis of variance shows that the intensity estimate is significantly larger than the error estimate over both Uranographic poles and part of the equatorial region, fortuitously including both magnetic polar regions. The Goddard Space Flight Center Q(sub 3) magnetic field model correctly predicts that the aurora should be larger in area and emit more power at the weaker N magnetic pole than at the stronger S magnetic pole. However, the auroral emissions are quite localized in magnetic longitude and so do not form complete auroral ovals. The brightest auroral emission at each magnetic pole is confined to a range of approximately 90 deg of magnetic longitude centered on the magnetotail direction, at moderate magnetic L parameter (5 less than or equal to L less than or equal to 10), but some emission at each pole is distributed over a range of more than 180 deg of longitude. The magnetic longitudes of the aurora are completely inconsistent with the 'windshield wiper' effect for either ions or electrons, indicating that some other effect, such as rapid depletion of the population of precipitating particles of highly localized strong pitch-angle diffusion, may be acting to localize emission. The low apparent L of the precipitating particles indicates that their energies may be less than or equal to 10 keV. Hence magnetospheric convection is likely to be important, and thus particles exciting the aurora may not remain on constant L shells. The precipitating particles may be a relatively low-energy population at high L that is heated to aurora-exciting energy by adiabatic compression during convection to low L. We estimate that the total auroral power output at H Lyman alpha and shorter wavelengths is about 3 x 10(exp 9) to 7 x 10(exp 9) W, requiring about 10 times that much power for excitation.

Herbert, Floyd↗

A Particle-in-Cell Method for Plasmas with a Generalized Momentum Formulation, Part II: Enforcing the Lorenz Gauge Condition

In a previous paper Christlieb et al. (A particle-in-cell method for plasmas with a generalized momentum formulation, part I: Model formulation, 2024), we developed a new particle-in-cell (PIC) method for the relativistic Vlasov–Maxwell system in which the electromagnetic fields and the equations of motion for the particles were cast in terms of scalar and vector potentials through a Hamiltonian formulation. This new method evolved the potentials under the Lorenz gauge using integral equation methods. New methods to construct spatial derivatives of the potentials that converge at the same rates as the fields were also presented. The new particle method was compared against standard explicit discretizations, including the well-known FDTD-PIC method, for a range of applications involving sheaths and particle beams. Here, this paper extends this new class of methods by focusing on the enforcement the Lorenz gauge condition in both exact and approximate forms using co-located meshes. A time-consistency property of the proposed field solver for the vector potential form of Maxwell’s equations is established, which is shown to preserve the equivalence between the semi-discrete Lorenz gauge condition and the analogous semi-discrete continuity equation. Using this property, we present three methods to enforce a semi-discrete gauge condition. The first method introduces an update for the continuity equation that is consistent with the discretization of the Lorenz gauge condition. Both the finite difference and spectral implementations satisfy this discrete gauge condition to machine precision. The second approach we propose enforces a semi-discrete continuity equation using the boundary integral solution to the field equations. The potential benefit of this approach is that it eliminates spatial derivatives that appear on the particle data, namely the current density, which is often calculated by linear combinations of low-order spline basis functions. This method is ideally suited to boundary integral equation methods that invert multi-dimensional operators without dimensional splitting techniques and will be the subject of future work. The third approach introduces a gauge correcting method that makes direct use of the gauge condition to modify the scalar potential and uses local maps for both the charge and current densities. This results in a gauge error, as the maps do not enforce the continuity equation. The vector potential coming from the current density is taken to be exact, and using the Lorenz gauge, we compute a correction to the scalar potential that makes the two potentials satisfy the gauge condition. This method also enforces the gauge condition to machine precision. We demonstrate two of the proposed methods in the context of periodic domains. Problems defined on bounded domains, including those with complex geometric features remain an ongoing effort. However, this work shows that it is possible to design computationally efficient methods that can effectively enforce the Lorenz gauge condition in a non-staggered PIC formulation.

97 MATHEMATICS AND COMPUTING↗

Interaction of the equatorial midnight pressure bulge and thermospheric zonal winds

The zonal component of the thermospheric neutral wind in the equatorial region is found to flow eastward all night, having a minimum near midnight. The data, obtained by Spencer et al. on the Dynamics Explorer-2 (DE-2) satellite, and reported more recently by Wharton et al., indicate that the minimum feature is significant. The minimum is illustrated by a 4th order Fourier series fit to the data points reported by Wharton et al. A table gives the amplitudes and phases of the four harmonic components used with the estimated error for each amplitude. It is asked whether this behavior is due to the local passage of the equatorial midnight pressure bulge associated with the midnight temperature anomaly. If so, it is to be expected that other data characterizing the midnight temperature anomaly should show consistency with this observation, especially in view of the large changes observed in the zonal velocity in the midnight sector. Consistency in momentum conservation may be checked using available data from previous independent experiments. Such data, taken under similar conditions of solar activity, was substituted into the momentum equation and found to be consistent using a priori estimates of the effect of viscosity.

Herrero, F. A.↗

Topology of modified helical gears and Tooth Contact Analysis (TCA) program

The contents of this report covers: (1) development of optimal geometries for crowned helical gears; (2) a method for their generation; (3) tooth contact analysis (TCA) computer programs for the analysis of meshing and bearing contact of the crowned helical gears; and (4) modelling and simulation of gear shaft deflection. The developed method for synthesis was used to determine the optimal geometry for a crowned helical pinion surface and was directed to localize the bearing contact and guarantee favorable shape and a low level of transmission errors. Two new methods for generation of the crowned helical pinion surface are proposed. One is based on the application of a tool with a surface of revolution that slightly deviates from a regular cone surface. The tool can be used as a grinding wheel or as a shaver. The other is based on a crowning pinion tooth surface with predesigned transmission errors. The pinion tooth surface can be generated by a computer-controlled automatic grinding machine. The TCA program simulates the meshing and bearing contact of the misaligned gears. The transmission errors are also determined. The gear shaft deformation was modelled and investigated. It was found that the deflection of gear shafts has the same effect as gear misalignment.

Litvin, Faydor L.↗