Search NASA⌕ Search

SEARCH · Search NASA

Results for “mixed methods”

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 163 records · Page 9
All sourcesNASA NTRSDOE OSTI
NASA NTRS1988

Data compression using adaptive transform coding. Appendix 1: Item 1

Adaptive low-rate source coders are described in this dissertation. These coders adapt by adjusting the complexity of the coder to match the local coding difficulty of the image. This is accomplished by using a threshold driven maximum distortion criterion to select the specific coder used. The different coders are built using variable blocksized transform techniques, and the threshold criterion selects small transform blocks to code the more difficult regions and larger blocks to code the less complex regions. A theoretical framework is constructed from which the study of these coders can be explored. An algorithm for selecting the optimal bit allocation for the quantization of transform coefficients is developed. The bit allocation algorithm is more fully developed, and can be used to achieve more accurate bit assignments than the algorithms currently used in the literature. Some upper and lower bounds for the bit-allocation distortion-rate function are developed. An obtainable distortion-rate function is developed for a particular scalar quantizer mixing method that can be used to code transform coefficients at any rate.

Rost, Martin Christopher↗
NASA NTRS1999

Interplanetary Fast Shocks and Associated Drivers Observed through the Twenty-Third Solar Minimum by WIND Over its First 2.5 Years

A list of the interplanetary (IP) shocks observed by WIND from its launch (in November 1994) to May 1997 is presented. Forty two shocks were identified. The magnetohydrodynamic nature of the shocks is investigated, and the associated shock parameters and their uncertainties are accurately computed using a practical scheme which combines two techniques. These techniques are a combination of the "pre-averaged" magnetic-coplanarity, velocity-coplanarity, and the Abraham-Schrauner-mixed methods, on the one hand, and the Vinas and Scudder [1986] technique for solving the non-linear least-squares Rankine-Hugoniot shock equations, on the other. Within acceptable limits these two techniques generally gave the same results, with some exceptions. The reasons for the exceptions are discussed. It is found that the mean strength and rate of occurrence of the shocks appears to correlated with the solar cycle. Both showed a decrease in 1996 coincident with the time of the lowest ultraviolet solar radiance, indicative of solar minimum and start of solar cycle 23, which began around June 1996. Eighteen shocks appeared to be associated with corotating interaction regions (CIRs). The distribution of their shock normals showed a mean direction peaking in the ecliptic plane and with a longitude (phi(sub n)) in that plane between perpendicular to the Parker spiral and radial from the Sun. When grouped according to the sense of the direction of propagation of the shocks the mean azimuthal (longitude) angle in GSE coordinates was approximately 194 deg for the fast-forward and approximately 20 deg for the fast-reverse shocks. Another 16 shocks were determined to be driven by solar transients, including magnetic clouds. These shocks had a broader distribution of normal directions than those of the CIR cases with a mean direction close to the Sun-Earth line. Eight shocks of unknown origin had normal orientation well off the ecliptic plane. No shock propagated with longitude phi(sub n) >= 220 +/- 10 deg, this would suggest strong hindrance to the propagation of shocks contra a rather tightly winding Parker spiral. Examination of the obliquity angle theta(sub Bn) (that between the shock normal and the upstream interplanetary magnetic field) for the full set of shocks revealed that about 58% was quasi-perpendicular, and some were very nearly perpendicular. About 32% of the shocks were oblique, and the rest (only 10%) were quasi-parallel, with one on Dec. 9, 1996 that showed field pulsations. Small uncertainty in the estimated angle theta(sub Bn) was obtained for about 10 shocks with magnetosonic Mach numbers between 1 and 2, hopefully significantly contributing to studies researching particle acceleration mechanisms at IP shocks, and to investigations where accurate values of theta(sub Bn) are crucial.

Mariani, F.↗
NASA NTRS1956

On the Calculation of Shallow Shells

This paper considers a sufficiently thin shallow shell of nonzero Gaussian curvature. It also presents a system of symmetrically constructed differential equations, constructed by the mixed method through the stress function and the displpacement function.

Ambartsumyan, S. A.↗
NASA NTRS2004

A Practical Approach to Obstacle Field Route Planning for Unmanned Rotorcraft

Autonomous helicopter operations in the near-earth environment will require robust and efficient obstacle field route planning. A method for obstacle field route planning is presented, which is composed of a mesh generation phase, a graph search phase, and a route refinement phase. The method mixes optimization and heuristics to obtain a satisfactory solution quickly. Simulations based on an unmanned helicopter model are presented.

Howlet, Jason K.↗
NASA NTRS2012

Analog Design for Digital Deployment of a Serious Leadership Game

This paper presents the design, development, and user testing of a leadership development simulation. The authors share lessons learned from using a design process for a board game to allow for quick and inexpensive revision cycles during the development of a serious leadership development game. The goal of this leadership simulation is to accelerate the development of leadership capacity in high-potential mid-level managers (GS-15 level) in a federal government agency. Simulation design included a mixed-method needs analysis, using both quantitative and qualitative approaches to determine organizational leadership needs. Eight design iterations were conducted, including three user testing phases. Three re-design iterations followed initial development, enabling game testing as part of comprehensive instructional events. Subsequent design, development and testing processes targeted digital application to a computer- and tablet-based environment. Recommendations include pros and cons of development and learner testing of an initial analog simulation prior to full digital simulation development.

Maxwell, Nicholas↗
NASA NTRS2021

Improved Potting of Litz Wires for High Power Density Electric Motor

Specific details including selections of process materials, tooling/fixture setups, and a step-by-step procedure for scaling-up demonstrations of the newly developed potting process, namely vacuum-assisted axial injection potting (VaAIP) have been fully developed for the stator winding of high power density electric motors for the future electrified aircrafts. Various essential pre-trial subtasks identified for the full-scale demonstrations were carried out. To date, optimization of cure condition and thermal stability of the coating epoxy and determination of optimum application conditions and thermal stability of the high temperature encapsulant wax were completed successfully and other subtasks were under way. The efforts to enhance thermal conductivity of the selected potting compound by adding conductive nano fillers such as boron nitride nanosheets (BNNS) have also been advanced. Initially, selection of the best mixing method and procedures was attempted via extensive and systematic experimental design.

Litz wire↗
NASA NTRS2021

Secondary Science Teachers’ Implementation of a Curricular Intervention When Teaching With Global Climate Models

In the past decade, emphasis on promoting “climate literacy” in K-16 science classrooms has increased. Teachers play a critical role in cultivating these opportunities, especially in secondary science classrooms. However, most prior climate education research has focused on students and student learning; little is known about how teachers implement climate-focused curricular interventions. Here, we report findings from a concurrent mixed methods, multiple-case study of four secondary science teachers’ implementation of a new, NGSS-aligned, model-centric climate curriculum module grounded in the use of a data-driven, computer-based climate modeling tool—Easy Global Climate Model (EzGCM). We employ multiple data sources, including video-recorded classroom observations, interviews, and instructional artifacts, and both qualitative and quantitative analyses, to investigate how teachers implemented the curriculum. Findings show that, overall, teachers implemented the curriculum in ways that were less model-centric than designed, placing greater emphasis on EzGCM itself rather than using the model to investigate Earth’s changing climate. Additionally, we present detailed single-case studies of each participant teacher that highlight differences in teachers’ implementation of the curriculum module and their reasoning for making observed instructional decisions. This research sheds light on the design of secondary science learning environments by illustrating the varied ways teachers implement a climate-focused curriculum to support students’ developing climate literacy. This has important implications for the design of climate-focused curriculum and supports for teachers.

Secondary science teaching↗
NASA NTRSDate not supplied

An Interim Set of TNT Curves for LOX/LNG Explosions

Space launch companies are actively developing, or in some cases have already developed, new vehicles that use large quantities of liquid oxygen (LOX) and liquid natural gas (LNG) propellants. This propellant mixture currently lacks sound/verified explosive safety standards for use in explosive siting and flight safety analysis. In the U.S., multiple government and commercial organizations have conducted limited testing, or are preparing to conduct tests, intended to provide a sound technical basis for LOX/LNG explosive standards applicable to space launch vehicle ground and flight operations. The limited testing and analysis conducted so far indicates the potential to produce extremely energetic explosions due to the miscibility of LOX/LNG (methane), a unique feature relative to previously used propellant mixtures such as hydrogen/oxygen or kerosene/oxygen. MOX is a homogenous mixture of methane and oxygen that is possible because of methane’s 100% miscibility in LOX. Preliminary testing by N2L, Inc. described in this paper shows that it is possible to form MOX with a variety of mixing methods, and that MOX is a sensitive high explosive capable of producing overpressures greater than comparable masses of C-4. The limited amount and quality of large-scale LOX/LNG explosive test data, and the potential formation of high-explosive MOX, create significant unknowns in the determination of hazard areas (cleared of the public) for ground operations (such as a static fire test) and flights of launch vehicles with LOX/LNG propellant. This paper will review the methods used by NASA to develop and validate the LOX/LH2 blast model using large-scale explosive test programs such as Project PYRO, the Hydrogen-Oxygen Vertical Impact (HOVI), and the Large-Scale Hydrogen-Oxygen Explosion (LSHOE) tests. This paper will describe how the same process was used to prepare an interim LOX/LNG blast model that applies to various LV failure scenarios and conservatively accounts for potential MOX formation. This paper also summarizes past and future testing and modeling efforts funded by a consortium of NASA, the US Federal Aviation Administration (FAA), and the US Space Force (USSF). These test programs should be concluded within the next 3 years and are intended to provide empirical data for model verification and validation.

Liquid Oxygen↗
NASA NTRSDate not supplied

An Interim Set of TNT Curves for LOX/LNG Explosions

Space launch companies are actively developing, or in some cases have already developed, new vehicles that use large quantities of liquid oxygen (LOX) and liquid natural gas (LNG) propellants. This propellant mixture currently lacks sound/verified LOX/LNG explosive safety standards for use in explosive siting and flight safety analysis. In the U.S., multiple government and commercial organizations have conducted limited testing, or are preparing to conduct tests, intended to provide a sound technical basis for explosive standards applicable to space launch vehicle ground and flight operations. The limited testing and analysis conducted so far indicates the potential to produce extremely energetic explosions due to the miscibility of LOX/LNG (methane), a unique feature relative to previously used propellant mixtures such as hydrogen/oxygen or kerosene/oxygen. MOX is a homogenous mixture of methane and oxygen that is possible because of methane’s 100% miscibility in LOX. Preliminary testing by N2L, Inc. described in this paper shows that it is possible to form MOX with a variety of mixing methods, and that MOX is a sensitive high explosive capable of producing overpressures greater than comparable masses of C-4. The limited amount and quality of large-scale LOX/LNG explosive test data, and the potential formation of high-explosive MOX, create significant unknowns in the determination of hazard areas (cleared of the public) for ground operations (such as a static fire test) and flights of launch vehicles with LOX/LNG propellant. This paper will review the methods used by NASA to develop and validate the LOX/LH2 blast model using large-scale explosive test programs such as Project PYRO, the Hydrogen-Oxygen Vertical Impact (HOVI), and the Large-Scale Hydrogen-Oxygen Explosion (LSHOE) tests. This paper will describe how the same process was used to prepare an interim LOX/LNG blast model that applies to various LV failure scenarios and conservatively accounts for potential MOX formation. This paper also summarizes past and future testing and modeling efforts funded by a consortium of NASA, the US Federal Aviation Administration (FAA), and the US Space Force (USSF). These test programs should be concluded within the next 3 years and are intended to provide empirical data for model verification and validation.

Liquid Oxygen↗
DOE OSTI2017

Alkali-ion battery with enhanced transition metal cyanometallate electrode structure

An alkali-ion battery is provided with a transition metal cyanometallate (TMCM) sheet cathode and a non-alkaline metal anode. The fabrication method mixes TMCM powders, conductive additives, and a polytetrafluoroethylene binder with a solution containing water, forming a wet paste. The wet paste is formed into a free-standing sheet of cathode active material, which is laminated to a cathode current collector, forming a cathode electrode. The free-standing sheet of cathode active material has a thickness typically in the range of 100 microns to 2 millimeters. The cathode electrode is assembled with a non-alkaline metal anode electrode and an ion-permeable membrane interposed between the cathode electrode and anode electrode, forming an assembly. The assembly is dried at a temperature of greater than 100 degrees C. The dried assembly is then inserted into a container (case) and electrolyte is added. Thick anodes made from free-standing sheets of active material can be similarly formed.

25 ENERGY STORAGE↗
DOE OSTI2020

The Gulf Nuclear Energy Infrastructure Institute: A Multidisciplinary Educational Approach for Integrated Nuclear Energy Safety Security and Safeguards in the Middle East

The Gulf Nuclear Energy Infrastructure Institute (GNEII) at Khalifa University of Science and Technology was created as a regional institute offering education, research and technical services to support nuclear energy safety, security and safeguards (3S) objectives. A mixed methods approach—using the (1) Course Evaluation, (2) GNEH Alumni Survey, (3) Capstone Project and, (4) GNEII-Related Literature data sets—was used to evaluate the effect of implementing this multidisciplinary `3S' educational program and the broader impact of the associated `3S' multidisciplinary institute on nuclear energy human resource development. Data sets (1), (2) and (3) illustrate how well GNEII implemented this novel 3S curriculum and resulted in successful knowledge transfer. Data sets (2), (3) and (4) illustrate how well GNEII's impact has positively influenced professional workplace behaviors and the institute's broader reputation to support responsible nuclear energy program education. Furthermore, GNEII demonstrates one option for successfully providing a multidisciplinary, 3S curriculum to support broader nuclear infrastructure and human resource development aims.

11 NUCLEAR FUEL CYCLE AND FUEL MATERIALS↗
DOE OSTI2019

I Asked 30,000 People in the Nuclear Industry About Safety Culture; This is What I Learned - 19204

Organizations in the nuclear industry place a high value on maintaining the safety of the workforce and community. Some of these organizations do a better job than others when it comes to developing and sustaining a healthy safety culture. We spent six years studying nuclear organizations to better understand the factors that influence safety culture. We used mixed methods to evaluate perceptions of safety culture in these organizations including administering a valid and reliable safety culture survey, conducting one-on-one interviews with managers, and facilitating focus groups with members of the workforce. We analyzed the quantitative and qualitative data collected from more than 30,000 workers in U.S. based organizations. All of the organizations had a nuclear component to their operations. They operated nuclear reactors, processed nuclear materials, decontaminated and decommissioned buildings with radiological contamination, cleaned-up hazardous/mixed waste, and conducted basic science research and development. We talked to craft workers, technicians, engineers, scientific professionals, support personnel, production workers, supervisors and managers. We conducted extensive statistical analysis and learned that, with rare exception, there is a gap between the perceptions of managers and the workforce that they manage. Managers almost always have a more positive perception of safety culture than the people that are actually performing the work. It is not a question of whether there will be a gap; it is simply a matter of the magnitude of the gap. We also learned through exploratory factor analysis that there is one dominant factor that accounted for almost all (75%) of the variance in the perceptions of safety culture. It was the behavior of the leaders, the decisions they made, and the systems they implemented that most influenced the workforce's perception of safety culture. Personal accountability for safety accounted for 14% of the variance, communication and reporting accounted for another 6%. Everything else accounted for the remaining 5%. Management systems, decision making and behavior is the dominant factor and managers almost always have a more positive perception than the people that work for them. People that work in organizations where management exhibits positive decision-making and safety-related behaviors tend to have more positive perceptions of safety culture. Based upon these results, we developed a theory that there were a few important factors on which management could focus on if they sought to influence the safety culture in their organization. Our experience and research suggests that management often fails to recognize five root causes of employee discontent. Employees seek: - To be recognized as an individual; - To be respected; - A sense of stability; - Fairness; - To make a difference. Understanding the perceptions of the workforce is critical because perceptions influence attitudes, which contribute to behaviors that determine individual performance. An organisation's performance is simply the sum of the performance of all of the members of that organization. To change an organisation's safety culture, interventions should target management behavior and decision making for the greatest return on investment. (authors)

12 MANAGEMENT OF RADIOACTIVE AND NON-RADIOACTIVE W↗
DOE OSTI2024

Angular Distribution of Dimuons from Drell-Yan Production in p+Fe Interactions at 120 GeV Beam Energy

In the E906/SeaQuest Fermilab experiment, we report a measurement of the angular distributions by measuring the angular parameters $\lambda$, $\mu$, and $\nu$ of Drell-Yan dimuons produced using a 120 GeV proton beam incident on an iron target. The angular distribution in the naive Drell-Yan model does not show any $\cos2\phi$ dependency, where $\phi$ denotes the azimuthal angle of dimuons in the Collins-Soper frame. However, pion-induced Drell-Yan experiments, such as NA10 and E615, have observed a significant dependence on $\cos2\phi$. The Boer–Mulders function, a transverse momentum-dependent distribution function, represents the correlation between the transverse spin and the transverse momentum of the quark. A non-zero Boer-Mulders function or an improved higher-order Drell-Yan model considering QCD effects can produce a $\cos 2\phi$ modulation in the Drell-Yan angular distribution. To measure the angular distributions, we have used an event mixing method to construct the combin atorial background, which was then subtracted from the data to isolate the Drell-Yan signal. Following this, we corrected the detector, trigger, and reconstruction efficiencies using a doubly-iterative Bayesian Unfolding method. This iterative unfolding technique improves the response matrix based on the results of the previous unfolding step, ensuring robust convergence without exaggeration of uncertainties. The angular distributions of the dimuons were measured over the invariant mass range $5.0 < M_{\mu^+ \mu^-} < 8.0$ $GeV/c^2$, with dimuon transverse momentum $P_T < 2$ GeV/c and Feynman-x $-0.18 < x_F < 0.9$. The measured angular distributions are then compared with the QCD calculations for $p + \text{Fe}$ interactions, and proton-induced angular distribution measurements from other experiments. We have observed weak $\cos 2\phi$ modulations as a function of $P_T$. For $P_T > 1.0 \, \text{GeV}/c$, the predicted NNLO perturbative QCD value of $\nu$ is larger than what we have me asured at E906/SeaQuest. Moreover, we have not observed a strong dependence of $\nu$ on the kinematic variables, such as dimuon mass $M_{\mu^+ \mu^-}$ and Bjorken-$x$. The spin alignment of the virtual photon, $\lambda$, measured from the SeaQuest Drell-Yan $p+\text{Fe}$ data, is found to be strongly dependent on $P_T$, decreasing as $P_T$ increases. $\lambda$ also holds to the upper bound condition $\lambda < 1.0$ within the statistical uncertainty, showing a trend similar to that predicted by NNLO perturbative QCD. However, for $1.0 < P_T < 2.0 \, \text{GeV}/c$, the extracted $\lambda$ value from SeaQuest is smaller than that predicted by perturbative QCD at NNLO.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
NASA NTRS2001

Development of a Hybrid RANS/LES Method for Turbulent Mixing Layers

Significant research has been underway for several years in NASA Glenn Research Center's nozzle branch to develop advanced computational methods for simulating turbulent flows in exhaust nozzles. The primary efforts of this research have concentrated on improving our ability to calculate the turbulent mixing layers that dominate flows both in the exhaust systems of modern-day aircraft and in those of hypersonic vehicles under development. As part of these efforts, a hybrid numerical method was recently developed to simulate such turbulent mixing layers. The method developed here is intended for configurations in which a dominant structural feature provides an unsteady mechanism to drive the turbulent development in the mixing layer. Interest in Large Eddy Simulation (LES) methods have increased in recent years, but applying an LES method to calculate the wide range of turbulent scales from small eddies in the wall-bounded regions to large eddies in the mixing region is not yet possible with current computers. As a result, the hybrid method developed here uses a Reynolds-averaged Navier-Stokes (RANS) procedure to calculate wall-bounded regions entering a mixing section and uses a LES procedure to calculate the mixing-dominated regions. A numerical technique was developed to enable the use of the hybrid RANS-LES method on stretched, non-Cartesian grids. With this technique, closure for the RANS equations is obtained by using the Cebeci-Smith algebraic turbulence model in conjunction with the wall-function approach of Ota and Goldberg. The LES equations are closed using the Smagorinsky subgrid scale model. Although the function of the Cebeci-Smith model to replace all of the turbulent stresses is quite different from that of the Smagorinsky subgrid model, which only replaces the small subgrid turbulent stresses, both are eddy viscosity models and both are derived at least in part from mixing-length theory. The similar formulation of these two models enables the RANS and LES equations to be solved with a single solution scheme and computational grid. The hybrid RANS-LES method has been applied to a benchmark compressible mixing layer experiment in which two isolated supersonic streams, separated by a splitter plate, provide the flows to a constant-area mixing section. Although the configuration is largely two dimensional in nature, three-dimensional calculations were found to be necessary to enable disturbances to develop in three spatial directions and to transition to turbulence. The flow in the initial part of the mixing section consists of a periodic vortex shedding downstream of the splitter plate trailing edge. This organized vortex shedding then rapidly transitions to a turbulent structure, which is very similar to the flow development observed in the experiments. Although the qualitative nature of the large-scale turbulent development in the entire mixing section is captured well by the LES part of the current hybrid method, further efforts are planned to directly calculate a greater portion of the turbulence spectrum and to limit the subgrid scale modeling to only the very small scales. This will be accomplished by the use of higher accuracy solution schemes and more powerful computers, measured both in speed and memory capabilities.

Georgiadis, Nicholas J.↗
NASA NTRS1993

Elastic interactions of a fatigue crack with a micro-defect by the mixed boundary integral equation method

In this paper, the mixed boundary integral equation method is developed to study the elastic interactions of a fatigue crack and a micro-defect such as a void, a rigid inclusion or a transformation inclusion. The method of pseudo-tractions is employed to study the effect of a transformation inclusion. An enriched element which incorporates the mixed-mode stress intensity factors is applied to characterize the singularity at a moving crack tip. In order to evaluate the accuracy of the numerical procedure, the analysis of a crack emanating from a circular hole in a finite plate is performed and the results are compared with the available numerical solution. The effects of various micro-defects on the crack path and fatigue life are investigated. The results agree with the experimental observations.

Lua, Yuan J.↗
DOE OSTI2026

Particle-hole symmetric slave-boson method for the mixed valence problem

We introduce an analytic slave-boson method for treating the finite-𝑈 Anderson impurity model. Our approach introduces two bosons to track both 𝑄 ⇌ 𝑄 ± 1 valence fluctuations and reduces to a single symmetric 𝑠 boson in the effective action, encoding all the high-energy atomic physics information in the boson's kinematics, while the low-energy part of the action remains unchanged across finite-𝑈, infinite-𝑈, and Kondo limits. We recover the infinite-𝑈- and Kondo-limit actions from our approach and show that the Kondo resonance already develops in the normal state when the slave boson has yet to condense. We show that the slave rotor and 𝑠 boson have the same algebraic structure, and we establish a unified functional integral framework connecting the 𝑠-boson and slave-rotor representations for the single-impurity Anderson model.

Anderson impurity model↗
NASA NTRS1991

A PC-based inverse design method for radial and mixed flow turbomachinery

An Inverse Design Method suitable for radial and mixed flow turbomachinery is presented. The codes are based on the streamline curvature concept; therefore, it is applicable for current personal computers from the 286/287 range. In addition to the imposed aerodynamic constraints, mechanical constraints are imposed during the design process to ensure that the resulting geometry satisfies production consideration and that structural considerations are taken into account. By the use of Bezier Curves in the geometric modeling, the same subroutine is used to prepare input for both aero and structural files since it is important to ensure that the geometric data is identical to both structural analysis and production. To illustrate the method, a mixed flow turbine design is shown.

Skoe, Ivar Helge↗
NASA NTRS2001

A Unified Development of Basis Reduction Methods for Rotor Blade Analysis

The axial foreshortening effect plays a key role in rotor blade dynamics, but approximating it accurately in reduced basis models has long posed a difficult problem for analysts. Recently, though, several methods have been shown to be effective in obtaining accurate,reduced basis models for rotor blades. These methods are the axial elongation method,the mixed finite element method, and the nonlinear normal mode method. The main objective of this paper is to demonstrate the close relationships among these methods, which are seemingly disparate at first glance. First, the difficulties inherent in obtaining reduced basis models of rotor blades are illustrated by examining the modal reduction accuracy of several blade analysis formulations. It is shown that classical, displacement-based finite elements are ill-suited for rotor blade analysis because they can't accurately represent the axial strain in modal space, and that this problem may be solved by employing the axial force as a variable in the analysis. It is shown that the mixed finite element method is a convenient means for accomplishing this, and the derivation of a mixed finite element for rotor blade analysis is outlined. A shortcoming of the mixed finite element method is that is that it increases the number of variables in the analysis. It is demonstrated that this problem may be rectified by solving for the axial displacements in terms of the axial forces and the bending displacements. Effectively, this procedure constitutes a generalization of the widely used axial elongation method to blades of arbitrary topology. The procedure is developed first for a single element, and then extended to an arbitrary assemblage of elements of arbitrary type. Finally, it is shown that the generalized axial elongation method is essentially an approximate solution for an invariant manifold that can be used as the basis for a nonlinear normal mode.

Ruzicka, Gene C.↗