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At least 541 records · Page 30

SAFEGUARD: An Assured Safety Net Technology for UAS

As demands increase to use unmanned aircraft systems (UAS) for a broad spectrum of commercial applications, regulatory authorities are examining how to safely integrate them without loss of safety or major disruption to existing airspace operations. This work addresses the development of the Safeguard system as an assured safety net technology for UAS. The Safeguard system monitors and enforces conformance to a set of rules defined prior to flight (e.g., geospatial stay-out or stay-in regions, speed limits, altitude limits). Safeguard operates independently of the UAS autopilot and is strategically designed in a way that can be realized by a small set of verifiable functions to simplify compliance with regulatory standards for commercial aircraft. A framework is described that decouples the system from any other devices on the UAS as well as introduces complementary positioning source(s) for applications that require integrity and availability beyond what the Global Positioning System (GPS) can provide. Additionally, the high level logic embedded within the software is presented, as well as the steps being taken toward verification and validation (V&V) of proper functionality. Next, an initial prototype implementation of the described system is disclosed. Lastly, future work including development, testing, and system V&V is summarized.

Dill, Evan T.↗

Operational Modal Analysis of the Space Launch System Mobile Launcher on the Crawler Transporter ISVV-010 Rollout

NASA is developing an expendable heavy lift launch vehicle capability, the Space Launch System, to support lunar and deep space exploration. To support this capability, an updated ground infrastructure is required including modifying an existing Mobile Launcher system. The Mobile Launcher is a very large heavy beam/truss steel structure designed to support the Space Launch System during its buildup and integration in the Vehicle Assembly Building, transportation between the Vehicle Assembly Building and launch pad 39B by the Crawler Transporter, and provides the launch platform at the launch pad. As part of the Verification and Validation of the Mobile Launcher and Crawler Transporter, two rollouts of the Mobile Launcher transported by the Crawler Transporter, Integrated System Verification and Validation (ISVV) 005 and 010, have been performed to demonstrate the Crawler Transporter’s ability to transport the Mobile Launcher. ISVV-005 occurred in September 2018 and ISVV-010 occurred in late June 2019. ISVV-005 and ISVV-010 also provided the opportunity to gather data that can be used identify the Mobile Launcher on the Crawler Transporter rollout modal characteristics and refine the estimates of the Artemis I integrated launch vehicle rollout forcing functions. While the rollout environment has historically produced relatively small launch vehicle structural loads for the Saturn/Apollo and Space Shuttle programs in comparison to launch and ascent loads, these relatively small structural loads are inputs to structural fatigue analyses. The same holds true for the Space Launch System. Because the rollout forces acting on the Mobile Launcher and the Crawler Transporter are not directly measurable, Operational Modal Analysis techniques, instead of traditional Experimental Modal Analysis techniques, provide an empirical means to identify the Mobile Launcher on the Crawler Transporter rollout modal characteristics. The ISVV-010 rollout modal characteristics provide important supplemental modal information, which along with the Mobile Launcher modal test that was performed in June 2019 immediately prior to ISVV-010 rollout, combine to reduce uncertainty in the test correlated Mobile Launcher on the Crawler Transporter finite element model. A well test correlated finite element model will play a key role in the Building Block approach the Space Launch System program has implemented as part of its certification process for the Artemis I flight and in refining the Artemis I rollout forcing functions. At the time of the ISVV-005 rollout in September 2018, the Mobile Launcher was still undergoing construction, and therefore its modal characteristics are not directly comparable to those of the Mobile Launcher during the June 2019 Mobile Launcher modal test and subsequent ISVV-010 rollout. Hence the ISVV-005 rollout modal characteristics will not be looked at in this paper. This paper will briefly describe the Mobile Launcher and Crawler Transporter physical characteristics, ISVV-010 rollout data collection, the challenges in implementing Operational Modal Analysis techniques due in part to the Crawler Transporter harmonics, and how these challenges were overcome to obtain the ISVV-010 Mobile Launcher on the Crawler Transporter rollout modal characteristics.

Apollo↗

Operational Modal Analysis of the Space Launch System Mobile Launcher on the Crawler Transporter ISVV-010 Rollout

NASA is developing an expendable heavy lift launch vehicle capability, the Space Launch System, to support lunar and deep space exploration. To support this capability, an updated ground infrastructure is required including modifying an existing Mobile Launcher system. The Mobile Launcher is a very large heavy beam/truss steel structure designed to support the Space Launch System during its buildup and integration in the Vehicle Assembly Building, transportation between the Vehicle Assembly Building and launch pad 39B by the Crawler Transporter, and provides the launch platform at the launch pad. As part of the Verification and Validation of the Mobile Launcher and Crawler Transporter, two rollouts of the Mobile Launcher transported by the Crawler Transporter, Integrated System Verification and Validation (ISVV) 005 and 010, have been performed to demonstrate the Crawler Transporter’s ability to transport the Mobile Launcher. ISVV-005 occurred in September 2018 and ISVV-010 occurred in late June 2019. ISVV-005 and ISVV-010 also provided the opportunity to gather data that can be used identify the Mobile Launcher on the Crawler Transporter rollout modal characteristics and refine the estimates of the Artemis I integrated launch vehicle rollout forcing functions. While the rollout environment has historically produced relatively small launch vehicle structural loads for the Saturn/Apollo and Space Shuttle programs in comparison to launch and ascent loads, these relatively small structural loads are inputs to structural fatigue analyses. The same holds true for the Space Launch System. Because the rollout forces acting on the Mobile Launcher and the Crawler Transporter are not directly measurable, Operational Modal Analysis techniques, instead of traditional Experimental Modal Analysis techniques, provide an empirical means to identify the Mobile Launcher on the Crawler Transporter rollout modal characteristics. The ISVV-010 rollout modal characteristics provide important supplemental modal information, which along with the Mobile Launcher modal test that was performed in June 2019 immediately prior to ISVV-010 rollout, combine to reduce uncertainty in the test correlated Mobile Launcher on the Crawler Transporter finite element model. A well test correlated finite element model will play a key role in the Building Block approach the Space Launch System program has implemented as part of its certification process for the Artemis I flight and in refining the Artemis I rollout forcing functions. At the time of the ISVV-005 rollout in September 2018, the Mobile Launcher was still undergoing construction, and therefore its modal characteristics are not directly comparable to those of the Mobile Launcher during the June 2019 Mobile Launcher modal test and subsequent ISVV-010 rollout. Hence the ISVV-005 rollout modal characteristics will not be looked at in this paper. This paper will briefly describe the Mobile Launcher and Crawler Transporter physical characteristics, ISVV-010 rollout data collection, the challenges in implementing Operational Modal Analysis techniques due in part to the Crawler Transporter harmonics, and how these challenges were overcome to obtain the ISVV-010 Mobile Launcher on the Crawler Transporter rollout modal characteristics.

Apollo↗

An on-line equivalent system identification scheme for adaptive control

A prime obstacle to the widespread use of adaptive control is the degradation of performance and possible instability resulting from the presence of unmodeled dynamics. The approach taken is to explicitly include the unstructured model uncertainty in the output error identification algorithm. The order of the compensator is successively increased by including identified modes. During this model building stage, heuristic rules are used to test for convergence prior to designing compensators. Additionally, the recursive identification algorithm as extended to multi-input, multi-output systems. Enhancements were also made to reduce the computational burden of an algorithm for obtaining minimal state space realizations from the inexact, multivariate transfer functions which result from the identification process. A number of potential adaptive control applications for this approach are illustrated using computer simulations. Results indicated that when speed of adaptation and plant stability are not critical, the proposed schemes converge to enhance system performance.

Sliwa, S. M.↗

Use of nonlinear identification in robust attitude and attitude rate estimation for SAMPEX

A method is described for obtaining optimal attitude estimation/identification algorithms for spacecraft lacking attitude rate measurement devices (rate gyros), and then demonstrated using actual flight data from the Solar, Anomalous, and Magnetospheric Particle Explorer (SAMPEX) spacecraft. SAMPEX does not have on-board rate sensing, and relies on sun sensors and a three-axis magnetometer for attitude determination. The absence of rate data normally reduces both the total amount of data available and the sampling density (in time) by a substantial fraction. In addition, attitude data is occasionally unavailable (for example, during sun occultation). As a result, the sensitivity of the estimates to model uncertainty and to measurement noise increases. In order to maintain accuracy in the attitude estimates, there is an increased need for accurate models of the rotational dynamics. The Minimum Model Error(MME)/Least Square Correlation(LSC) algorithm accurately identifies an improved model for SAMPEX to be used during periods of complete data loss or extreme noise. The model correction is determined by estimating only one orbit(the identification pass) just prior to the assumed data loss(the prediction pass). The MME estimator correctly predicted the states during the identification phase, but more importantly determines the necessary model correction trajectory, d(t). The LSC algorithm is then used to find this trajectory's functional form, H(x(t)). The results show significant improvement of the new corrected model's attitude estimates as compared to the original uncorrected model's estimates. The possible functional form of the correction term is limited at this point in the study to functions strictly of the estimated states. The results, however, strongly suggest that functions based on the relative position of the satellite may also be possible candidates for future consideration.

Mook, D. Joseph↗

Configurable Hardware And Software For Multiple Related Uses

Control Monitor Unit (CMU) is system of configurable hardware and software undergoing development for use in controlling and monitoring complex systems of equipment. Provides comprehensive array of capabilities for such functions as processing equipment-test data for calibration and diagnosis, controlling operation of equipment in real time, simulating operation of equipment, and processing large streams of scientific-measurement data. Automates many of ground operations involved in preparing and testing spacecraft prior to launch. Also useful in variety of similar applications; for example, testing aircraft, ships, power plants, and automated production lines.

Uhrlaub, David R.↗

Locomotion in Lymphocytes is Altered by Differential PKC Isoform Expression

Lymphocyte locomotion is critical for proper elicitation of the immune response. Locomotion of immune cells via the interstitium is essential for optimal immune function during wound healing, inflammation and infection. There are conditions which alter lymphocyte locomotion and one of them is spaceflight. Lymphocyte locomotion is severely inhibited in true spaceflight (true microgravity) and in rotating wall vessel culture (modeled microgravity). When lymphocytes are activated prior to culture in modeled microgravity, locomotion is not inhibited and the levels are comparable to those of static cultured lymphocytes. When a phorbol ester (PMA) is used in modeled microgravity, lymphocyte locomotion is restored by 87%. This occurs regardless if PMA is added after culture in the rotating wall vessel or during culture. Inhibition of DNA synthesis also does not alter restoration of lymphocyte locomotion by PMA. PMA is a direct activator of (protein kinase C) PKC . When a calcium ionophore, ionomycin is used it does not possess any restorative properties towards locomotion either alone or collectively with PMA. Since PMA brings about restoration without help from calcium ionophores (ionomycin), it is infer-red that calcium independent PKC isoforms are involved. Changes were perceived in the protein levels of PKC 6 where levels of the protein were downregulated at 24,72 and 96 hours in untreated rotated cultures (modeled microgravity) compared to untreated static (1g) cultures. At 48 hours there is an increase in the levels of PKC & in the same experimental set up. Studies on transcriptional and translational patterns of calcium independent isoforms of PKC such as 8 and E are presented in this study.

Sundaresan, A.↗

The TechSat 21 Autonomous Sciencecraft Experiment

Software has been developed to perform a number of functions essential to autonomous operation in the Autonomous Sciencecraft Experiment (ASE), which is scheduled to be demonstrated aboard a constellation of three spacecraft, denoted TechSat 21, to be launched by the Air Force into orbit around the Earth in January 2006. A prior version of this software was reported in Software for an Autonomous Constellation of Satellites (NPO-30355), NASA Tech Briefs, Vol. 26, No. 11 (November 2002), page 44. The software includes the following components: Algorithms to analyze image data, generate scientific data products, and detect conditions, features, and events of potential scientific interest; A program that uses component-based computational models of hardware to analyze anomalous situations and to generate novel command sequences, including (when possible) commands to repair components diagnosed as faulty; A robust-execution-management component that uses the Spacecraft Command Language (SCL) software to enable event-driven processing and low-level autonomy; and The Continuous Activity Scheduling, Planning, Execution, and Replanning (CASPER) program for replanning activities, including downlink sessions, on the basis of scientific observations performed during previous orbit cycles.

Sherwood, Robert↗

BioSentinel Deep Space CubeSat Mission

The BioSentinel mission was recently launched aboard the SLS launch vehicle (LV) as part of the Artemis-1 campaign. This 6U CubeSat carries yeast cells to analyze the effects of radiation at large distances from Earth, becoming the first biological payload in Deep Space. Prelaunch activities included mission design updates, orbit determination rehearsals and the development of a tracking schedule in coordination with the Artemis-1 payload office and the Deep Space Network (DSN). An important influence on the trajectories of Artemis I secondaries was the uncertainty associated with deployment from the Interim Cryogenic Propulsion System (ICPS), the upper stage of the SLS LV. The ICPS was rotating at a rate of 1 rpm; there was also uncertainty in the spin axis attitude, which translated into an unknown clock angle of deployment. The variability in this angle and magnitude of deployment implied the existence of a non-negligible risk of a lunar impact, which was evaluated for various potential launch dates. On November 16 th 2022 BioSentinel successfully deployed from ICPS and the navigation team started to receive tracking data from the DSN and ESA antennas. Soon after deployment, the spacecraft was tumbling and entered safe mode. The mission team recovered the spacecraft and after four tracking passes, we solved for a first ephemeris that was sent to the DSN for better tracking of the spacecraft. After propagating this first ephemeris solution, we determined that we avoided impact with a margin of a few hundred km from the lunar surface. More tracking data over the next few days allowed for a more refined orbit solution predicting a periselene altitude of 406 km and a lunar eclipse lasting 36.5 minutes. Therefore, BioSentinel operators avoided any correction maneuvers on the trajectory and successfully tracked and guide the spacecraft. The spacecraft performed a nominal lunar flyby which provided the pertinent energy to achieve a final Earth-trailing heliocentric orbit. Over the course of two weeks, the mission operators corroborated that the subsystems were functioning as expected after the lunar eclipse and the large ΔV incurred. Science operations started once the mission achieved the nominal orbit in Deep Space. This paper discusses in detail the BioSentinel flight performance, as well as the challenges and lessons learned prior to and during this CubeSat mission.

Andres Dono Perez↗

A Deeper Dive Into the Meaning and Implications of Interoperability for LunaNet Communications and Navigation Services

As part of planning efforts for cislunar exploration and science missions, space agencies have been collaborating with each other to enable communications, networking, Position, Navigation, and Timing (PNT) systems to exchange information and provide services to spacecraft and space systems in transit, in orbit, and on the surface, thus helping each other to achieve their common goals. To achieve commonality and lower cost for mutual benefit, the strategy of interoperability is being adopted to help all the pieces fit together and function smoothly. Interoperability gives cislunar users the ability to operate in a collaborative environment similar to the terrestrial Internet, allowing them to share information, navigate safely despite increasing radio frequency congestion, and follow common processes and procedures for effective joint operations. Unlike prior government-dominated efforts, this ecosystem is expected to include commercial for-profit businesses, non-profit organizations, and academic institutions. Ultimately, the goal is to enable a cislunar ecosystem of service providers and users to contribute and/or utilize infrastructure and capabilities to accomplish mission objectives spanning the full range of human endeavours while supporting a variety of business models. This paper reports on the results of an effort to assist in efforts to frame the development of the international LunaNet architecture by providing a canonical definition of interoperability broad enough to meet these needs, examine architectural and operational implications of the definition, and explore interoperability strategies and tactics for deploying and evolving these services. It describes key systems-of-systems (SoS) (Network-of-Networks) interoperability concepts in the context of sustainment of the ecosystem over time as systems evolve in technologies, standards and Standards Development Organizations, component and subsystem upgrades, and user applications

LunaNet↗

The interaction of radio frequency electromagnetic fields with atmospheric water droplets and applications to aircraft ice prevention

The feasibility of computerized simulation of the physics of advanced microwave anti-icing systems, which preheat impinging supercooled water droplets prior to impact, was investigated. Theoretical and experimental work performed to create a physically realistic simulation is described. The behavior of the absorption cross section for melting ice particles was measured by a resonant cavity technique and found to agree with theoretical predictions. Values of the dielectric parameters of supercooled water were measured by a similar technique at lambda = 2.82 cm down to -17 C. The hydrodynamic behavior of accelerated water droplets was studied photograhically in a wind tunnel. Droplets were found to initially deform as oblate spheroids and to eventually become unstable and break up in Bessel function modes for large values of acceleration or droplet size. This confirms the theory as to the maximum stable droplet size in the atmosphere. A computer code which predicts droplet trajectories in an arbitrary flow field was written and confirmed experimentally. The results were consolidated into a simulation to study the heating by electromagnetic fields of droplets impinging onto an object such as an airfoil. It was determined that there is sufficient time to heat droplets prior to impact for typical parameter values. Design curves for such a system are presented.

Hansman, R. J., Jr.↗

Investigation of acceleration characteristics of a single-spool turbojet engine

Operation of a single-spool turbojet engine with constant exhaust-nozzle area was investigated at one flight condition. Data were obtained by subjecting the engine to approximate-step changes in fuel flow, and the information necessary to show the relations of acceleration to the sensed engine variables was obtained. These data show that maximum acceleration occurred prior to stall and surge. In the low end of the engine-speed range the margin was appreciable; in the high-speed end the margin was smaller but had not been completely defined by these data. Data involving acceleration as a function of speed, fuel flow, turbine-discharge temperature, compressor-discharge pressure, and thrust have been presented and an effort has been made to show how a basic control system could be improved by addition of an override in which the acceleration characteristic is used not only to prevent the engine from entering the surge region but also to obtain acceleration along the maximum acceleration line during throttle bursts.

Oppenheimer, Frank L↗

International Space Station Internal Thermal Control System Lab Module Simulator Build-Up and Validation

As part of the Sustaining Engineering program for the International Space Station (ISS), a ground simulator of the Internal Thermal Control System (ITCS) in the Lab Module was designed and built at the Marshall Space Flight Center (MSFC). To support prediction and troubleshooting, this facility is operationally and functionally similar to the flight system and flight-like components were used when available. Flight software algorithms, implemented using the LabVIEW(Registered Trademark) programming language, were used for monitoring performance and controlling operation. Validation testing of the low temperature loop was completed prior to activation of the Lab module in 2001. Assembly of the moderate temperature loop was completed in 2002 and validated in 2003. The facility has been used to address flight issues with the ITCS, successfully demonstrating the ability to add silver biocide and to adjust the pH of the coolant. Upon validation of the entire facility, it will be capable not only of checking procedures, but also of evaluating payload timelining, operational modifications, physical modifications, and other aspects affecting the thermal control system.

Paul Wieland↗

Progress in Development of Improved Ion-Channel Biosensors

Further improvements have recently been made in the development of the devices described in Improved Ion-Channel Biosensors (NPO-30710), NASA Tech Briefs, Vol. 28, No. 10 (October 2004), page 30. As discussed in more detail in that article, these sensors offer advantages of greater stability, greater lifetime, and individual electrical addressability, relative to prior ion-channel biosensors. In order to give meaning to a brief description of the recent improvements, it is necessary to recapitulate a substantial portion of the text of the cited previous article. The figure depicts one sensor that incorporates the recent improvements, and can be helpful in understanding the recapitulated text, which follows: These sensors are microfabricated from silicon and other materials compatible with silicon. Typically, the sensors are fabricated in arrays in silicon wafers on glass plates. Each sensor in the array can be individually electrically addressed, without interference with its neighbors. Each sensor includes a well covered by a thin layer of silicon nitride, in which is made a pinhole for the formation of a lipid bilayer membrane. In one stage of fabrication, the lower half of the well is filled with agarose, which is allowed to harden. Then the upper half of the well is filled with a liquid electrolyte (which thereafter remains liquid) and a lipid bilayer is painted over the pinhole. The liquid contains a protein that forms an ion channel on top of the hardened agarose. The combination of enclosure in the well and support by the hardened agarose provides the stability needed to keep the membrane functional for times as long as days or even weeks. An electrode above the well, another electrode below the well, and all the materials between the electrodes together constitute a capacitor. What is measured is the capacitive transient current in response to an applied voltage pulse. One notable feature of this sensor, in comparison with prior such sensors, is a relatively thick dielectric layer between the top of the well and the top electrode. This layer greatly reduces the capacitance of an aperture across which the ion channels are formed, thereby increasing the signal-to-noise ratio. The use of a relatively large aperture with agarose support makes it possible to form many ion channels instead of only one, thereby further increasing the signal-to-noise ratio and effectively increasing the size of the available ionic reservoir. The relatively large reservoir makes it possible to measure AC rather than DC. This concludes the recapitulation from the cited previous article.

Nadeau, Jay L.↗

Validation of the Pulmonary Function System for Use on the International Space Station

Aerobic deconditioning occurs during long duration space flight despite the use of exercise countermeasures (Convertino, 1996). As a part of International Space Station (ISS) medical operations, periodic tests designed to estimate aerobic capacity are performed to track changes in aerobic fitness and to determine the effectiveness of exercise countermeasures. These tests are performed prior to, during, and after missions of greater than 30 days in duration. Crewmembers selected for missions aboard the ISS perform a graded exercise test on a cycle ergometer approximately 270 days prior to their scheduled launch date in order to measure peak oxygen consumption (VO2PK) and peak heart rate (HRpk). Approximately 30 to 45 days prior to launch, crewmembers perform a submaximal cycle ergometer test at work rates set to elicit 25, 50 and 75% of their pre-flight VO2PK. This test, known as the Periodic Fitness Evaluation (PFE), serves as a baseline measure to which subsequent in-and post-flight exercise tests are compared. While onboard the ISS, crewmembers are normally scheduled to perform the PFE beginning with flight day (FD) 14 and every 30 days thereafter. The PFE is also conducted 5 and 30 days following flight. Using PFE data, aerobic fitness is estimated by quantifying the VO2 vs. HR relationship using linear regression and calculating the VO2 that would occur at the crewmember s previously measured HRpk. Currently, for data collected during flight, this technique assumes that the pre- vs. in-flight oxygen consumption per given cycle workload is similar. However, the validity of this assumption is based upon a sparse amount of data collected during the Skylab era (Michel, et al. 1977). The method of using heart rate and cycle ergometer work rates has been used to estimate aerobic fitness in normal gravity (Astrand and Ryhming, 1954; Lee, 1993). Due to spaceflight induced physiological alterations, such as shifts in extracellular fluid (e.g. plasma) volume, this method may not be valid during space flight. In addition, the ergometer onboard ISS is vibration-isolated and moves with the astronaut s application of force into the pedals. The effect of this movement on the VO2 of cycle exercise on ISS has not been quantified.

McCleary, Frank A.↗

Adjustment of saccade characteristics during head movements.

Saccade characteristics have been studied during coordinated eye-head movements in monkeys. Amplitude, duration, and peak velocity of saccades with head turning were compared with saccades executed while the head was artificially restrained. The results indicate that the saccade characteristics are modulated as a function of head movement, hence the gaze movement (eye+head) exactly matches saccades with head fixed. Saccade modulation is achieved by way of negative vestibulo-ocular feedback. The neck proprioceptors, because of their longer latency, are effective only if the head starts moving prior to the onset of saccade. It is concluded that saccades make with head turning are not 'ballistic' movements because their trajectory is not entirely predetermined by a central command.

Morasso, P.↗

The use of dispersion to eliminate multipath in planetary radio occultations

Occultation geometry is analyzed with a view toward eliminating all multipath resulting from layer effects. It is assumed that the spacecraft moves behind the planet with a constant velocity perpendicular to the direction from the spacecraft to the earth, and that for radio frequencies of 1 to 2 GHz the angle of refraction of a radio ray passing through the ionosphere is so small as to be imperceptible if drawn to scale. It is shown that if a particular amount of dispersion is inserted into the process at any stage prior to signal detection, the frequency vs time curve will be converted to a frequency vs ray asymptote height curve. Because there can be only one mode from any one ray asymptote height in a layered medium, frequency must be a single-valued function of this parameter. The introduction of an appropriate dispersion will therefore eliminate all multipath insofar as it is caused by spherically symmetric planetary layers. The degree of accuracy to be provided in flight and during postflight analysis is discussed.

Croft, T. A.↗

Shear buckling of specially orthotropic plates with centrally located cutouts

The shear loading of a composite rectangular plate with a centrally located circular cutout was analyzed in order to predict the buckling load of the plate. The first step in this analysis is to calculare the in-plane stress distribution of the plate prior to buckling. This problem can be solved using finite element methods, but a quicker and more cost efficient method is desired. The method chosen to determine the pre-buckling stresses is that of boundary collocation using complex variables. The real valued force functions are written in terms of two complex valued functions, each of which is a function of a different complex variable. The solution of the generalized biharmonic function is a superposition of functions of these two complex variables. For this analysis, the force functions are represented with a Laurent series. The constants in these series are found using boundary collocation. Two force equations for complex variables are satisfied over the plate boundaries to obtain these constants. The stresses in the plate are then known for any particular locations on the plate given the applied shear on the edges of the plate.

Klang, Eric C.↗