Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “dynamic load model”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 1,171 records · Page 65
All sourcesNASA NTRSDOE OSTI
NASA NTRS1999

Control Design Strategies to Enhance Long-Term Aircraft Structural Integrity

Over the operational lifetime of both military and civil aircraft, structural components are exposed to hundreds of thousands of low-stress repetitive load cycles and less frequent but higher-stress transient loads originating from maneuvering flight and atmospheric gusts. Micro-material imperfections in the structure, such as cracks and debonded laminates, expand and grow in this environment, reducing the structural integrity and shortening the life of the airframe. Extreme costs associated with refurbishment of critical load-bearing structural components in a large fleet, or altogether reinventoring the fleet with newer models, indicate alternative solutions for life extension of the airframe structure are highly desirable. Increased levels of operational safety and reliability are also important factors influencing the desirability of such solutions. One area having significant potential for impacting crack growth/fatigue damage reduction and structural life extension is flight control. To modify the airframe response dynamics arising from command inputs and gust disturbances, feedback loops are routinely applied to vehicles. A dexterous flight control system architecture senses key vehicle motions and generates critical forces/moments at multiple points distributed throughout the airframe to elicit the desired motion characteristics. In principle, these same control loops can be utilized to influence the level of exposure to harmful loads during flight on structural components. Project objectives are to investigate and/or assess the leverage control has on reducing fatigue damage and enhancing long-term structural integrity, without degrading attitude control and trajectory guidance performance levels. In particular, efforts have focused on the effects inner loop control parameters and architectures have on fatigue damage rate. To complete this research, an actively controlled flexible aircraft model and a new state space modeling procedure for crack growth have been utilized. Analysis of the analytical state space model for crack growth revealed the critical mathematical factors, and hence the physical mechanism they represent, that influenced high rates of airframe crack growth. The crack model was then exercised with simple load inputs to uncover and expose key crack growth behavior. To characterize crack growth behavior, both "short-term" laboratory specimen test type inputs and "long-term" operational flight type inputs were considered. Harmonic loading with a single overload revealed typical exponential crack growth behavior until the overload application, after which time the crack growth was retarded for a period of time depending on the overload strength. An optimum overload strength was identified which leads to maximum retardation of crack growth. Harmonic loading with a repeated overload of varying strength and frequency again revealed an optimum overload trait for maximizing growth retardation. The optimum overload strength ratio lies near the range of 2 to 3 with dependency on frequency. Experimental data was found to correlate well with the analytical predictions.

Newman, Brett A.↗
DOE OSTI2021

Focused wave interactions with floating structures: a blind comparative study

The paper presents results from the Collaborative Computational Project in Wave Structure Interaction (CCP-WSI) Blind Test Series 2. Without prior access to the physical data, participants, with numerical methods ranging from low-fidelity linear models to fully non-linear Navier-Stokes (NS) solvers, simulate the interaction between focused wave events and two separate, taut-moored, floating structures: a hemispherical-bottomed cylinder and a cylinder with a moonpool. The ‘blind’ numerical predictions for heave, surge, pitch and mooring load, are compared against physical measurements. Dynamic time warping is used to quantify the predictive capability of participating methods. In general, NS solvers and hybrid methods give more accurate predictions; however, heave amplitude is predicted reasonably well by all methods; and a WEC-Sim implementation, with CFD-informed viscous terms, demonstrates comparable predictive capability to even the stronger NS solvers. Large variations in the solutions are observed (even among similar methods), highlighting a need for standardisation in the numerical modelling of WSI problems.

42 ENGINEERING↗
DOE OSTI2026

The high explosives & affected targets (HEAT) dataset

Artificial Intelligence (AI) surrogate models offer a computationally efficient alternative to full-physics simulations, yet no existing datasets are publicly available for training, testing, and validation of machine learning models of the dynamics of high-explosive driven shocks through multiple materials. Shock propagation through materials is a computationally challenging problem because simulations must include material-specific equations of state (EOS) along with descriptions of other physical processes such as plastic deformation, phase change, damage processes, fluid instabilities, and multi-material interactions. Shocks are typically initiated by high-velocity impacts or explosive loading. The latter case necessitates the addition of models of reactive materials to represent high-explosive (HE) detonation. Here, to address the lack of an expansive dataset for multi-material shock propagation in the AI/ML community, we present the High-Explosives and Affected Targets (HEAT) Dataset. HEAT is a physics-rich collection of two-dimensional, cylindrically symmetric, simulations generated using an Eulerian, multi-material, shock-propagation code developed at Los Alamos National Laboratory. The dataset includes two partitions: (1) the expanding shock-cylinder (CYL) simulations, Figs. 1, and (2) the Perturbed Layered Interface (PLI) simulations, Fig. 2. Entries in both partitions consist of time series of arrays of thermodynamic fields (pressure, density, and temperature), kinematic fields (position and velocity), and additional fields that depend on thermodynamic and/or kinematic fields (e.g., material stress). Materials in the CYL partition include solids (aluminium, copper, depleted uranium, stainless steel, tantalum, and a generic polymer), a liquid (water), gases (air, nitrogen), and a generic detonating material (high explosive, HE). The PLI partition spans a highly varying geometry but consists of fixed materials across entries: Copper, aluminium, stainless steel, generic polymer, and generic HE. HEAT captures critical phenomena such as momentum transfer, shock propagation, plastic deformation, and thermal effects, making HEAT a valuable benchmark for development of AI/ML emulation of multi-material shock propagation.

36 MATERIALS SCIENCE↗
NASA NTRS2008

A Theoretical Solid Oxide Fuel Cell Model for Systems Controls and Stability Design

As the aviation industry moves toward higher efficiency electrical power generation, all electric aircraft, or zero emissions and more quiet aircraft, fuel cells are sought as the technology that can deliver on these high expectations. The hybrid solid oxide fuel cell system combines the fuel cell with a micro-turbine to obtain up to 70% cycle efficiency, and then distributes the electrical power to the loads via a power distribution system. The challenge is to understand the dynamics of this complex multidiscipline system and the design distributed controls that take the system through its operating conditions in a stable and safe manner while maintaining the system performance. This particular system is a power generation and a distribution system, and the fuel cell and micro-turbine model fidelity should be compatible with the dynamics of the power distribution system in order to allow proper stability and distributed controls design. The novelty in this paper is that, first, the case is made why a high fidelity fuel cell mode is needed for systems control and stability designs. Second, a novel modeling approach is proposed for the fuel cell that will allow the fuel cell and the power system to be integrated and designed for stability, distributed controls, and other interface specifications. This investigation shows that for the fuel cell, the voltage characteristic should be modeled but in addition, conservation equation dynamics, ion diffusion, charge transfer kinetics, and the electron flow inherent impedance should also be included.

Kopasakis, George↗
NASA NTRS1999

Development of a 5-Component Balance for Water Tunnel Applications

The principal objective of this research/development effort was to develop a multi-component strain gage balance to measure both static and dynamic forces and moments on models tested in flow visualization water tunnels. A balance was designed that allows measuring normal and side forces, and pitching, yawing and rolling moments (no axial force). The balance mounts internally in the model and is used in a manner typical of wind tunnel balances. The key differences between a water tunnel balance and a wind tunnel balance are the requirement for very high sensitivity since the loads are very low (typical normal force is 90 grams or 0.2 lbs), the need for water proofing the gage elements, and the small size required to fit into typical water tunnel models. The five-component balance was calibrated and demonstrated linearity in the responses of the primary components to applied loads, very low interactions between the sections and no hysteresis. Static experiments were conducted in the Eidetics water tunnel with delta wings and F/A-18 models. The data were compared to forces and moments from wind tunnel tests of the same or similar configurations. The comparison showed very good agreement, providing confidence that loads can be measured accurately in the water tunnel with a relatively simple multi-component internal balance. The success of the static experiments encouraged the use of the balance for dynamic experiments. Among the advantages of conducting dynamic tests in a water tunnel are less demanding motion and data acquisition rates than in a wind tunnel test (because of the low-speed flow) and the capability of performing flow visualization and force/moment (F/M) measurements simultaneously with relative simplicity. This capability of simultaneous flow visualization and for F/M measurements proved extremely useful to explain the results obtained during these dynamic tests. In general, the development of this balance should encourage the use of water tunnels for a wider range of quantitative and qualitative experiments, especially during the preliminary phase of aircraft design.

Suarez, Carlos J.↗
NASA NTRS1989

Flight motor set 360L001 (STS-26R). (Reconstructed dynamic loads analysis)

A transient analysis was performed to correlate the predicted versus measured behavior of the Redesigned Solid Rocket Booster (RSRB) during Flight 360L001 (STS-26R) liftoff. Approximately 9 accelerometers, 152 strain gages, and 104 girth gages were bonded to the motors during this event. Prior to Flight 360L001, a finite element model of the RSRB was analyzed to predict the accelerations, strains, and displacements measured by this developmental flight instrumentation (DFI) within an order of magnitude. Subsequently, an analysis has been performed which uses actual Flight 360L001 liftoff loading conditions, and makes more precise predictions for the RSRB structural behavior. Essential information describing the analytical model, analytical techniques used, correlation of the predicted versus measured RSRB behavior, and conclusions, are presented. A detailed model of the RSRB was developed and correlated for use in analyzing the motor behavior during liftoff loading conditions. This finite element model, referred to as the RSRB global model, uses super-element techniques to model all components of the RSRB. The objective of the RSRB global model is to accurately predict deflections and gap openings in the field joints to an accuracy of approximately 0.001 inch. The model of the field joint component was correlated to Referee and Joint Environment Simulation (JES) tests. The accuracy of the assembled RSRB global model was validated by correlation to static-fire tests such DM-8, DM-9, QM-7, and QM-8. This validated RSRB global model was used to predict RSRB structural behavior and joint gap opening during Flight 360L001 liftoff. The results of a transient analysis of the RSRB global model with imposed liftoff loading conditions are presented. Rockwell used many gage measurements to reconstruct the load parameters which were imposed on the RSRB during the Flight 360L001 liftoff. Each load parameter, and its application, is described. Also presented are conclusions and recommendations based on the analysis of this load case and the resulting correlation between predicted and measured RSRB structural behavior.

Call, V. B.↗
DOE OSTI2024

Modeling and Validation of a Residential Multi-Functional Variable Refrigerant Flow Heat Pump System with Heat Recovery

To bridge the existing gap in modeling the variable refrigerant flow heat pump systems with heat recovery (VRFHR), we developed a suite of dynamic VRFHR system models in Modelica. These models are specifically tailored for residential multi-functional VRFHR (MF-VRFHR) applications, including space conditioning and domestic hot water (DHW) heating, utilizing both the TIL library for HVAC equipment and the Buildings library for thermal load calculations. The development comprises essential component models, including the newly developed heat recovery unit (HRU), along with system models that integrate the heat pump system and building envelope. These system models accommodate various operational modes such as heating-only, cooling-only, and heating-recovery (including heating-dominant and cooling-dominant) modes. Furthermore, we propose an efficient optimization-based model calibration method that identifies critical model parameters while utilizing a small amount of data obtained from either real systems or manufacturer's specifications. We demonstrate the effectiveness of these models and the proposed calibration method for a MF-VRFHR system installed in Richland, WA. The developed models are calibrated and validated using data collected under different operational modes during both heating and cooling seasons. The results show that the models capture the system dynamics and achieve high accuracy, with the coefficient of the variation of the root-mean-square-error less than 15% for variables such as outdoor unit power consumption, compressor speed, space temperature and DHW temperature. The validated models serve as a reliable representation of the MF-VRFHR system, facilitating the development and validation of optimized controls needed to realize the full benefits of integrated heat pump systems. Future research will utilize these models to develop advanced controls and optimize system performance for improved energy efficiency and demand flexibility.

Modeling, Variable refrigerant flow (VRF) systems,↗
NASA NTRS2000

Propulsion Aeroelastic Analysis Developed for Flutter and Forced Response

The NASA Glenn Research Center at Lewis Field develops new technologies to increase the fuel efficiency of aircraft engines, improve the safety of engine operation, reduce emissions, and reduce engine noise. With the development of new designs for fans, compressors, and turbines to achieve these goals, the basic aeroelastic requirements are that there should be no flutter (self-excited vibrations) or high resonant blade stresses (due to forced response) in the operating regime. Therefore, an accurate prediction and analysis capability is required to verify the aeroelastic soundness of the designs. Such a three-dimensional viscous propulsion aeroelastic analysis capability has been developed at Glenn with support from the Advanced Subsonic Technology (AST) program. This newly developed aeroelastic analysis capability is based on TURBO, a threedimensional unsteady aerodynamic Reynolds-averaged Navier-Stokes turbomachinery code developed previously under a grant from Glenn. TURBO can model the viscous flow effects that play an important role in certain aeroelastic problems such as flutter with flow separation, flutter at high loading conditions near the stall line (stall flutter), flutter in the presence of shock and boundary-layer interaction, and forced response due to wakes and shock impingement. In aeroelastic analysis, the structural dynamics representation of the blades is based on normal modes. A finite-element analysis code is used to calculate these in-vacuum vibration modes and the associated natural frequencies. In an aeroelastic analysis using the TURBO code, flutter and forced response are modeled as being uncoupled. To calculate if a blade row will flutter, one prescribes the motion of the blade to be a harmonic vibration in a specified in-vacuum normal mode. An aeroelastic analysis preprocessor is used to generate the displacement field required for the analysis. The work done by aerodynamic forces on the vibrating blade during a cycle of vibration is calculated. If this work is positive, the blade is dynamically unstable, since it will extract energy from the flow, leading to an increase in the blade s oscillation amplitude. The forced-response excitations on a blade row are calculated by modeling the flow through two adjacent blade rows using the TURBO code. The blades are assumed to be rigid. As an option, a single blade row can be modeled with the upstream blade row influence represented by a time-varying disturbance (gust) at the inlet boundary. The unsteady forces on a blade row from such analyses are used in a structural analysis along with the blade structural dynamics characteristics and aerodynamic damping associated with blade vibration to calculate the resulting dynamic stresses on the blade.

Bakhle, Milind A.↗
NASA NTRS2017

Extraction-Separation Performance and Dynamic Modeling of Orion Test Vehicles with Adams Simulation: 3rd Edition

NASA's Orion Capsule Parachute Assembly System (CPAS) Project is now in the qualification phase of testing, and the Adams simulation has continued to evolve to model the complex dynamics experienced during the test article extraction and separation phases of flight. The ability to initiate tests near the upper altitude limit of the Orion parachute deployment envelope requires extractions from the aircraft at 35,000 ft-MSL. Engineering development phase testing of the Parachute Test Vehicle (PTV) carried by the Carriage Platform Separation System (CPSS) at altitude resulted in test support equipment hardware failures due to increased energy caused by higher true airspeeds. As a result, hardware modifications became a necessity requiring ground static testing of the textile components to be conducted and a new ground dynamic test of the extraction system to be devised. Force-displacement curves from static tests were incorporated into the Adams simulations, allowing prediction of loads, velocities and margins encountered during both flight and ground dynamic tests. The Adams simulation was then further refined by fine tuning the damping terms to match the peak loads recorded in the ground dynamic tests. The failure observed in flight testing was successfully replicated in ground testing and true safety margins of the textile components were revealed. A multi-loop energy modulator was then incorporated into the system level Adams simulation model and the effect on improving test margins be properly evaluated leading to high confidence ground verification testing of the final design solution.

Varela, Jose G.↗
DOE OSTI2022

Vehicle-to-manufacturing (V2M) system: A novel approach to improve energy demand flexibility for demand response towards sustainable manufacturing

The U.S. manufacturing sector accounts for 77% of the industrial energy consumption, but its participation in demand response (DR) programs is largely lagged behind. The limited flexibility in production scheduling under high capacity utilization and the lack of DR schemes that incorporate energy demand flexibility measures are deemed as major barriers. In this study, a framework of the interactive vehicles-to-manufacturing (V2M) energy sharing system is proposed to improve the energy demand flexibility of the aggregated system and enhance the DR effectiveness for manufacturers. The V2M-based DR scheme aims to reduce the energy cost by load shifting through joint production and energy sharing control, which can eventually promote manufacturing DR implementation even under high capacity utilization requirements and dynamic real-time electricity prices. The V2M system is modeled based on a discrete-Markov chain considering the complex interconnections among various manufacturing resources and multi-directional energy flows among manufacturing facilities, electric vehicles, and the power grid. Based on the system model, a mixed-integer nonlinear programming (MINLP) problem is formulated to identify the optimal DR scheme. The effectiveness of the proposed approach is validated through comparisons with traditional manufacturing DR schemes. Finally, the results show that a 2.1 to 6.5 times energy demand flexibility improvement and an additional 4.7% to 6.9% energy cost reduction can be achieved by the proposed approach.

32 ENERGY CONSERVATION, CONSUMPTION, AND UTILIZATI↗
DOE OSTI2024

Virtual texture analysis to understand microstructure effects on deformation twinning and detwinning behavior in BCC metals

Understanding and predicting deformation twinning contributions to plastic deformation in BCC metals has been a long-standing challenge due to the interplay with dislocation slip and non-Schmid effects that render an asymmetry under tension and compression. This paper uses molecular dynamics simulations to understand the effect of unloading and grain orientation on deformation twinning in a nanocrystalline Fe (nc-Fe) system as a model BCC metal. The nc-Fe system is loaded under uniaxial stress tension and compression to understand the effect of grain orientation (Schmid effects) on deformation twinning behavior and the tension/compression asymmetry (non-Schmid effects). A new virtual texture analysis “VirTex” tool is used to understand the role of grain orientation on the nucleation of twins and their contributions to the observed stress–strain response. For certain grain orientations, the twinnability is observed to be different in tension and compression. In addition, the flow stress accommodation from twins in certain grains is observed to be different in tension and compression and different from that for the grains. Subsequent unloading leads to detwinning in the deformed microstructures, where the extent of detwinning depends on the strain from which the system is unloaded and on the morphology of the twin. Lastly, the simulations are carried out to analyze the role of the Schmid factor on the twinnability and asymmetry in tension and compression.

Kannan, Aadhithyan [University of Connecticut, Sto↗
DOE OSTI2026

Inundation and Sediment Supply Predict Annual but Not Centennial Carbon Accumulation in Tidal Wetlands Across Land‐Use Types

Despite growing interest in the contributions of tidal wetlands to natural climate solutions, data remain scarce on how land use affects their carbon accumulation rates (CAR). Additionally, environmental factors driving the large observed variability in CAR among sites are poorly understood. To address these knowledge gaps, we measured short-term (∼1 year) and long-term (∼100 years) CAR using feldspar marker horizons, 210 Pb profiles, and surface elevation tables from 39 sites in seven estuaries in Oregon and Washington, USA. Sites varied in wetland type (tidal marsh, tidal swamp, nontidal pasture), land-use history (reference, restored, and disturbed), and salinity (fresh to polyhaline). Across both timescales, CAR was lowest in disturbed pastures and highest in restored marshes, reflecting patterns in sediment accretion rates. Short-term CAR exceeded long-term CAR by approximately 50% on average and was only weakly correlated with long-term CAR, with differences among hydrogeomorphic setting. Short-term CAR was well predicted by summer inundation frequency and relative sediment load, together explaining 60% of variance, whereas long-term CAR was poorly predicted by contemporary environmental conditions. These findings highlight the importance of aligning predictor and response timescales when modeling blue carbon dynamics, and reinforce the climate mitigation potential of tidal wetland restoration.

land use and land cover change↗
DOE OSTI2020

Numerical Model Development of a Variable-Geometry Attenuator Wave Energy Converter

Because the wave energy industry is still in its infancy, an optimal design for wave energy converters (WECs) has yet to be established; more work is needed to explore various cost-reduction pathways. The primary cost-reduction pathway considered for this work is the optimization of the geometric profile on an attenuator WEC to maximize power production while, at the same time, minimizing capital expenditures through the use of variable-geometry modules. In this investigation, the variable-geometry modules consist of inflatable bags placed on either side of a base central steel cylinder that would be inflated in low-moderate sea states to maximize power capture and then deflated in moderate-extreme sea states to minimize wave loading. The numerical model and simulation of the attenuator WEC were developed and completed using WEC-Sim, which is an open-source code that is appropriate for use in evaluating the dynamic response of the different WEC models in operational seas. The power production estimates were obtained from the Wave Energy Prize (WEP) sea states, which are representative of U. S. deployment sites, to calculate the average climate capture width that is used in the WEP ACE calculation. Preliminary capital expenditure costs were obtained assuming the base central steel cylinder mass was equal to the fluid displaced mass, minus the mass of the variable-geometry bags. The additional weight required to offset the additional buoyancy from the variable-geometry bags was assumed to come from the addition of seawater ballast. The variable-geometry attenuator model was found to have a similar power capture efficiency as a fixed-body model, but is expected to have a lower characteristic capital expenditure given its more streamlined profile, which demonstrates that variable-geometry modules may provide a realistic cost-reduction pathway to help design a more cost-competitive WEC.

50 EE - Wind and Water Power Program - Water (EE-4↗
DOE OSTI2023

Development of a hardware-in-the-loop testbed for laboratory performance verification of flexible building equipment in typical commercial buildings

The goals of reducing energy costs, shifting electricity peaks, increasing the use of renewable energy, and enhancing the stability of the electric grid can be met in part by fully exploiting the energy flexibility potential of buildings and building equipment. The development of strategies that exploit these flexibilities could be facilitated by publicly available high-resolution datasets illustrating how control of HVAC systems in commercial buildings can be used in different climate zones to shape the energy use profile of a building for grid needs. This article presents the development and integration of a Hardware-In-the-Loop Flexible load Testbed (HILFT) that integrates physical HVAC systems with a simulated building model and simulated occupants with the goal of generating datasets to verify load flexibility of typical commercial buildings. Compared to simulation-only experiments, the hardware-in-the-loop approach captures the dynamics of the physical systems while also allowing efficient testing of various boundary conditions. The HILFT integration in this article is achieved through the co-simulation among various software environments including LabVIEW, MATLAB, and EnergyPlus. Although theoretically viable, such integration has encountered many real-world challenges, such as: 1) how to design the overall data infrastructure to ensure effective, robust, and efficient integration; 2) how to avoid closed-loop hunting between simulated and emulated variables; 3) how to quantify system response times and minimize system delays; and 4) how to assess the overall integration quality. Lessons-learned using the examples of an AHU-VAV system, an air-source heat pump system, and a water-source heat pump system are presented.

Chen, zhelun↗
NASA NTRS1984

Structural analysis of the Space Shuttle Orbiter

The structural analysis of the Space Shuttle orbiter was planned with two concepts in mind: use derivatives or subsets of the same basic finite element model whenever feasible, and substantiate the model's predictive capability by performing ground tests. The analysis cycle (model modal loads stress (MMLS)) starts with the finite element model conception and ends with the detailed stress analysis and margins of safety. The structural analysis of the orbiter encompasses a variety of static and dynamic problems. The salient features of these problems and their solutions are examined.

Henry R. Grooms↗
NASA NTRS1985

A study of the compatibility of an existing CFD package with a broader class of material constitutions

The flexibility of the PHOENICS computational fluid dynamics package was assessed along two general avenues; parallel modeling and analog modeling. In parallel modeling the dependent and independent variables retain their identity within some scaling factors, even though the boundary conditions and especially the constitutive relations do not correspond to any realistic fluid dynamic situation. PHOENICS was used to generate a CFD model that should exhibit the physical anomalies of a granular medium and permit reasonable similarity with boundary conditions typical to membrane or porous piston loading. A considerable portion of the study was spent prying into the existing code with a prejudice toward rate type and disarming any inherent fluid behavior. The final stages of the study were directed at the more specific problem of multiaxis loading of cylindrical geometry with a concern for the appearance of bulging, cross slab shear failure modes.

French, K. W., Jr.↗
NASA NTRS1987

Computer modeling and simulation of a 20kHz ac distribution system for Space Station

A computer model of a 20 kHz, ac distribution testbed for Space Station is presented. The system consists of six resonant inverters, a one-hundred-meter transmission line, and three load receivers: a dc receiver, a bidirectional receiver, and an ac receiver. A model library is generated characterizing all system components. The system's local and global behaviors are investigated using the EASY5 dynamic analysis program.

Tsai, Fu-Sheng↗
NASA NTRS1991

Modeling an electric motor in 1-D

Quite often the dynamicist will be faced with having an electric drive motor as a link in the elastic path of a structure such that the motor's characteristics must be taken into account to properly represent the dynamics of the primary structure. He does not want to model it so accurately that he could get detailed stress and displacements in the motor proper, but just sufficiently to represent its inertia loading and elastic behavior from its mounting bolts to its drive coupling. Described here is how the rotor and stator of such a motor can be adequately modeled as a colinear pair of beams.

Butler, Thomas G.↗