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NASA NTRS1999

A Kinematic Link Between Boxy Bulges, Stellar Bars, and Nuclear Activity in NGC 3079 and NGC 4388

We present direct kinematic evidence for bar streaming in two active galaxies with boxy stellar bulges. The Hawaii Imaging Fabry-Perot Interferometer was used on the Canada-France-Hawaii 3.6-m telescope and the University of Hawaii 2.2-m telescope to derive the two-dimensional velocity field of the line-emitting gas in the disks of the Sc galaxy NGC 3079 and the Sb galaxy NGC 4388. In contrast to previous work based on long-slit data, the detection of the bar potential from the Fabry-Perot data does not rely on the existence of inner Lindblad resonances or strong bar-induced shocks. Simple kinematic models which approximate the intrinsic gas orbits as nonintersecting, inclined elliptical annuli that conserve angular momentum characterize the observed velocity fields. In NGC 3079, bar streaming motions with moderately eccentric orbits (e = b/a approx. 0.7) aligned along PA = 130 deg. intrinsic to the disk (PA = 97 deg. on the sky) are detected out to R(sub b) = 3.6 kpc. The orbits become increasingly circular beyond that radius (e = 1 at R(sub d) approx. = 6 kpc). The best model for NGC 4388 includes highly eccentric orbits (e approx. 0.3) for R(sub) less than or equal to 1.5 kpc which are aligned along PA = 135 deg. intrinsic to the disk (PA = 100 deg. on the sky). The observed "spiral arms" are produced by having the orbits become increasingly circular from the ends of the bar to the edge of the disk (R(sub d) approx. = 5 kpc), and the intrinsic bar PA shifting from 135 deg. to 90 deg.. Box-shaped bulges in both NGC 3079 and NGC 4388 are confirmed using new near-infrared images to reduce dust obscuration. Morphological analysis of starlight in these galaxies is combined with the gas kinematics derived from the Fabry-Perot spectra to test evolutionary models of stellar bars that involve transitory boxy bulges, and to quantify the importance of such bars in fueling active nuclei. Our data support the evolutionary bar models, but fail to prove convincingly that the stellar bars in NGC 3079 and NGC 4388 directly trigger or sustain the nuclear activity.

Veilleux, S.↗
NASA NTRS2001

Toward an Efficient Icing CFD Process Using an Interactive Software Toolkit: Smagglce 2D

Two-dimensional CID analysis for iced airfoils can be a labor-intensive task. The software toolkit SmaggIce 2D is being developed to help streamline the CID process and provide the unique features needed for icing. When complete, it will include a combination of partially automated and fully interactive tools for all aspects of the tasks leading up to the flow analysis: geometry preparation, domain decomposition. block boundary demoralization. gridding, and linking with a flow solver. It also includes tools to perform ice shape characterization, an important aid in determining the relationship between ice characteristics and their effects on aerodynamic performance. Completed tools, work-in-progress, and planned features of the software toolkit are presented here.

Vickerman, Mary B.↗
NASA NTRS2002

Toward an Efficient Icing CFD Process Using an Interactive Software Toolkit--SmaggIce 2D

Two-dimensional CFD analysis for iced airfoils can be a labor-intensive task. The software toolkit SmaggIce 2D is being developed to help streamline the CFD process and provide the unique features needed for icing. When complete, it will include a combination of partially automated and fully interactive tools for all aspects of the tasks leading up to the flow analysis: geometry preparation, domain decomposition, block boundary discretization. gridding, and linking with a flow solver. It also includes tools to perform ice shape characterization, an important aid in determining the relationship between ice characteristics and their effects on aerodynamic performance. Completed tools, work-in-progress, and planned features of the software toolkit are presented here.

Vickerman, Mary B.↗
NASA NTRS2002

Enlargement of Step-Free SiC Surfaces by Homoepitaxial Web-Growth of Thin SiC Cantilevers

Lateral homoepitaxial growth of thin cantilevers emanating from mesa patterns that were reactive ion etched into on-axis commercial SiC substrates prior to growth is reported. The thin cantilevers form after pure stepflow growth removes almost all atomic steps from the top surface of a mesa, after which additional adatoms collected by the large step-free surface migrate to the mesa sidewall where they rapidly incorporate into the crystal near the top of the mesa sidewall. The lateral propagation of the step-free cantilevered surface is significantly affected by pregrowth mesa shape and orientation, with the highest lateral expansion rates observed at the inside concave comers of V-shaped pregrowth mesas with arms lengthwise oriented along the {1100} direction. Complete spanning of the interiors of V's and other mesa shapes with concave comers by webbed cantilevers was accomplished. Optical microscopy, synchrotron white beam x-ray topography and atomic force microscopy analysis of webbed regions formed over a micropipe and closed-core screw dislocations show that c-axis propagation of these defects is terminated by the webbing. Despite the nonoptimized process employed in this initial study, webbed surfaces as large as 1.4 x 10(exp -3) square centimeters, more than four times the pregrowth mesa area, were grown. However, the largest webbed surfaces were not completely free of bilayer steps, due to unintentional growth of 3C-SiC that occurred in the nonoptimized process. Further process optimization should enable larger step-free webs to be realized.

Neudeck, Philip G.↗
NASA NTRS2003

An Analysis of Model Scale Data Transformation to Full Scale Flight Using Chevron Nozzles

Ground-based model scale aeroacoustic data is frequently used to predict the results of flight tests while saving time and money. The value of a model scale test is therefore dependent on how well the data can be transformed to the full scale conditions. In the spring of 2000, a model scale test was conducted to prove the value of chevron nozzles as a noise reduction device for turbojet applications. The chevron nozzle reduced noise by 2 EPNdB at an engine pressure ratio of 2.3 compared to that of the standard conic nozzle. This result led to a full scale flyover test in the spring of 2001 to verify these results. The flyover test confirmed the 2 EPNdB reduction predicted by the model scale test one year earlier. However, further analysis of the data revealed that the spectra and directivity, both on an OASPL and PNL basis, do not agree in either shape or absolute level. This paper explores these differences in an effort to improve the data transformation from model scale to full scale.

Brown, Clifford↗
NASA NTRS2000

The origin of cellular life

This essay presents a scenario of the origin of life that is based on analysis of biological architecture and mechanical design at the microstructural level. My thesis is that the same architectural and energetic constraints that shape cells today also guided the evolution of the first cells and that the molecular scaffolds that support solid-phase biochemistry in modern cells represent living microfossils of past life forms. This concept emerged from the discovery that cells mechanically stabilize themselves using tensegrity architecture and that these same building rules guide hierarchical self-assembly at all size scales (Sci. Amer 278:48-57;1998). When combined with other fundamental design principles (e.g., energy minimization, topological constraints, structural hierarchies, autocatalytic sets, solid-state biochemistry), tensegrity provides a physical basis to explain how atomic and molecular elements progressively self-assembled to create hierarchical structures with increasingly complex functions, including living cells that can self-reproduce.

Review, Academic↗
NASA NTRS2003

2003 NASA Faculty Fellowship Program at Glenn Research Center

The Office of Education at NASA Headquarters provides overall policy and direction for the NASA Faculty Fellowship Program (NFFP). The American Society for Engineering Education (ASEE) and the Universities Space Research Association (USRA) have joined in partnership to recruit participants, accept applications from a broad range of participants, and provide overall evaluation of the NFFP. The NASA Centers, through their University Affairs Officers, develop and operate the experiential part of the program. In concert with co-directing universities and the Centers, Fellows are selected and provided the actual research experiences. This report summarizes the 2003 session conducted at the Glenn Research Center (GRC).Research topics covered a variety of areas including, but not limited to, biological sensors, modeling of biological fluid systems, electronic circuits, ceramics and coatings, unsteady probablistic analysis and aerodynamics, gas turbines, environmental monitoring systems for water quality, air quality, gaseous and particulate emissions, bearings for flywheel energy storage, shape memory alloys,photonic interrogation and nanoprocesses,carbon nanotubes, polymer synthesis for fuel cells, aviation communications, algorithm development and RESPlan Database.

Prahl, Joseph M.↗
NASA NTRS2000

Engine-Scale Combustor Rig Designed, Fabricated, and Tested for Combustion Instability Control Research

Low-emission combustor designs are prone to combustor instabilities. Because active control of these instabilities may allow future combustors to meet both stringent emissions and performance requirements, an experimental combustor rig was developed for investigating methods of actively suppressing combustion instabilities. The experimental rig has features similar to a real engine combustor and exhibits instabilities representative of those in aircraft gas turbine engines. Experimental testing in the spring of 1999 demonstrated that the rig can be tuned to closely represent an instability observed in engine tests. Future plans are to develop and demonstrate combustion instability control using this experimental combustor rig. The NASA Glenn Research Center at Lewis Field is leading the Combustion Instability Control program to investigate methods for actively suppressing combustion instabilities. Under this program, a single-nozzle, liquid-fueled research combustor rig was designed, fabricated, and tested. The rig has many of the complexities of a real engine combustor, including an actual fuel nozzle and swirler, dilution cooling, and an effusion-cooled liner. Prior to designing the experimental rig, a survey of aircraft engine combustion instability experience identified an instability observed in a prototype engine as a suitable candidate for replication. The frequency of the instability was 525 Hz, with an amplitude of approximately 1.5-psi peak-to-peak at a burner pressure of 200 psia. The single-nozzle experimental combustor rig was designed to preserve subcomponent lengths, cross sectional area distribution, flow distribution, pressure-drop distribution, temperature distribution, and other factors previously found to be determinants of burner acoustic frequencies, mode shapes, gain, and damping. Analytical models were used to predict the acoustic resonances of both the engine combustor and proposed experiment. The analysis confirmed that the test rig configuration and engine configuration had similar longitudinal acoustic characteristics, increasing the likelihood that the engine instability would be replicated in the rig. Parametric analytical studies were performed to understand the influence of geometry and condition variations and to establish a combustion test plan. Cold-flow experiments verified that the design values of area and flow distributions were obtained. Combustion test results established the existence of a longitudinal combustion instability in the 500-Hz range with a measured amplitude approximating that observed in the engine. Modifications to the rig configuration during testing also showed the potential for injector independence. The research combustor rig was developed in partnership with Pratt & Whitney of West Palm Beach, Florida, and United Technologies Research Center of East Hartford, Connecticut. Experimental testing of the combustor rig took place at United Technologies Research Center.

DeLaat, John C.↗
NASA NTRS1994

(abstract) A Low-Cost Mission to 2060 Chiron Based on the Pluto Fast Flyby

The Pluto Fast Flyby-based mission to Chiron described in this paper is a low cost, scientifically rewarding, focused mission in the outer solar system. The proposed mission will make a flyby of 2060 Chiron, an active 'comet' with over 10(sup 4) times the mass of Halley, and an eccentric, Saturn-crossing orbit which ranges from 8.5 to 19 AU. This mission concept achieves the flyby 4.2 years after launch on a direct trajectory from Earth, is independent of Jupiter launch windows, and fits within Discovery cost guidelines. This mission offers the scientific opportunity to examine a class of object left unsampled by the trail-blazing Mariners, Pioneers, Voyagers, and missions to Halley. Spacecraft reconnaissance of Chiron addresses unique objectives relating to cometary science, other small bodies, the structure of quasi-bound atmospheres on modest-sized bodies, and the origin of primitive bodies and the giant planets. Owing to Chiron's large size (180<D<370 km), unique nature, and unusual orbit, this mission is likely to draw significant public interest. As described by COMPLEX, the SSEC, and later the SSES, flybys are the appropriate scale missions for initial reconnaissance missions. Carrying three sophisticated instruments, the proposed flyby will return critical data about Chiron's size, shape, polar obliquity, atmosphere, surface morphology, surface composition, internal structure, surface activity (including the nature of Chiron's outbursts), and origin. Engineering analysis indicates that the spacecraft is capable of navigating to and encountering Chiron at close approach distances of less than 5 000 km, well inside the 50 000 to 150 000 km coma, and perhaps within the collisional chemistry zone of the coma. The low cost of the proposed Chiron mission is based on the opportunity to use the planned Pluto Flyby spare spacecraft and a Proton Expendable Launch Vehicle (ELV) (the pluto spacecraft is being designed to be compatible with a Proton launch). Backup launch opportunities on Delta II and Atlas ELVs are available. The Pluto Fast Flyby mission plans to develop a low cost ($150M), lightweight (<150 kg) outer planet spacecraft which is well suited to flyby reconnaissance in the outer solar system. This Chiron flyby mission is designed to leverage SSED's Pluto spacecraft investment into a scientifically valuable successor mission within the Discovery cost cap. Taking advantage of the spare spacecraft, we estimate the Discovery Chiron intercept mission can be launched for less than $100 M. We believe this makes a highly attractive proposal, which maintains US presence and leadership in the study of the outer solar system, and enables the first-time exploration of a wholly new class of planetary target, an outer solar system planetesimal.

2060 Chiron planetesimal flyby Pluto Fast Flyby sp↗
NASA NTRS2006

Coherent Detection of High-Rate Optical PPM Signals

A method of coherent detection of high-rate pulse-position modulation (PPM) on a received laser beam has been conceived as a means of reducing the deleterious effects of noise and atmospheric turbulence in free-space optical communication using focal-plane detector array technologies. In comparison with a receiver based on direct detection of the intensity modulation of a PPM signal, a receiver based on the present method of coherent detection performs well at much higher background levels. In principle, the coherent-detection receiver can exhibit quantum-limited performance despite atmospheric turbulence. The key components of such a receiver include standard receiver optics, a laser that serves as a local oscillator, a focal-plane array of photodetectors, and a signal-processing and data-acquisition assembly needed to sample the focal-plane fields and reconstruct the pulsed signal prior to detection. The received PPM-modulated laser beam and the local-oscillator beam are focused onto the photodetector array, where they are mixed in the detection process. The two lasers are of the same or nearly the same frequency. If the two lasers are of different frequencies, then the coherent detection process is characterized as heterodyne and, using traditional heterodyne-detection terminology, the difference between the two laser frequencies is denoted the intermediate frequency (IF). If the two laser beams are of the same frequency and remain aligned in phase, then the coherent detection process is characterized as homodyne (essentially, heterodyne detection at zero IF). As a result of the inherent squaring operation of each photodetector, the output current includes an IF component that contains the signal modulation. The amplitude of the IF component is proportional to the product of the local-oscillator signal amplitude and the PPM signal amplitude. Hence, by using a sufficiently strong local-oscillator signal, one can make the PPM-modulated IF signal strong enough to overcome thermal noise in the receiver circuits: this is what makes it possible to achieve near-quantum-limited detection in the presence of strong background. Following quantum-limited coherent detection, the outputs of the individual photodetectors are automatically aligned in phase by use of one or more adaptive array compensation algorithms [e.g., the least-mean-square (LMS) algorithm]. Then the outputs are combined and the resulting signal is processed to extract the high-rate information, as though the PPM signal were received by a single photodetector. In a continuing series of experiments to test this method (see Fig. 1), the local oscillator has a wavelength of 1,064 nm, and another laser is used as a signal transmitter at a slightly different wavelength to establish an IF of about 6 MHz. There are 16 photodetectors in a 4 4 focal-plane array; the detector outputs are digitized at a sampling rate of 25 MHz, and the signals in digital form are combined by use of the LMS algorithm. Convergence of the adaptive combining algorithm in the presence of simulated atmospheric turbulence for optical PPM signals has already been demonstrated in the laboratory; the combined output is shown in Fig. 2(a), and Fig. 2(b) shows the behavior of the phase of the combining weights as a function of time (or samples). We observe that the phase of the weights has a sawtooth shape due to the continuously changing phase in the down-converted output, which is not exactly at zero frequency. Detailed performance analysis of this coherent free-space optical communication system in the presence of simulated atmospheric turbulence is currently under way.

Vilnrotter, Victor↗
NASA NTRS2012

Transition Within Leeward Plane of Axisymmetric Bodies at Incidence in Supersonic Flow

Boundary layer transition along the leeward symmetry plane of axisymmetric bodies at nonzero angle of attack in supersonic flow was investigated experimentally and numerically as part of joint research between the Japan Aerospace Exploration Agency (JAXA) and National Aeronautics and Space Administration (NASA). Transition over four axisymmetric bodies (namely, Sears-Haack body, semi-Sears-Haack body, straight cone and flared cone) with different axial pressure gradients was measured in two different facilities with different unit Reynolds numbers. The semi-Sears-Haack body and flared cone were designed at JAXA to broaden the range of axial pressure distributions. For a body shape with an adverse pressure gradient (i.e., flared cone), the experimentally measured transition patterns show an earlier transition location along the leeward symmetry plane in comparison with the neighboring azimuthal locations. For nearly zero pressure gradient (i.e.,straight cone), this feature is only observed at the larger unit Reynolds number. Later transition along the leeward plane was observed for the remaining two body shapes with a favorable pressure gradient. The observed transition patterns are only partially consistent with the numerical predictions based on linear stability analysis. Additional measurements are used in conjunction with the stability computations to explore the phenomenon of leeward line transition and the underlying transition mechanism in further detail.

Tokugawa, Naoko↗
NASA NTRSDate not supplied

Sonic Boom Ground Noise Minimization Via the Adjoint Method

A multidisciplinary design optimization methodology to directly minimize the ground-level noise generated by the sonic boom of a high altitude supersonic body is presented. A Cartesian Euler flow solver is coupled with an atmospheric propagation tool to create a ground-noise analysis capability for supersonic bodies. Adjoint formulations for both the flow solver and propagation tool are also coupled to compute sensitivities of shape variations in the body in a highly efficient manner. A gradient-based optimizer is then introduced to forge a valuable design capability. Output-based mesh adaptation that is driven directly by ground-level noise is employed to increase accuracy and provide error estimation. The design method is demonstrated first on a simple axisymmetric body with few design variables to evaluate the efficacy of the optimization scheme. Guided by the results of this initial case, the problem is then repeated with somewhat different design variables to further demonstrate the capabilities of the design method. Finally, the method is applied to a real-world problem by optimizing control surface deflection settings of a low-boom aircraft to minimize ground noise while maintaining trimmed, level flight.

ARMD↗
NASA NTRS1995

Three-dimensional CTOA and constraint effects during stable tearing in a thin-sheet material

A small strain theory, three-dimensional elastic-plastic finite element analysis was used to simulate fracture in thin sheet 2024-T3 aluminum alloy in the T-L orientation. Both straight and tunneled cracks were modeled. The tunneled crack front shapes as a function of applied stress were obtained from the fracture surface of tested specimens. The stable crack growth behavior was measured at the specimen surface as a function of applied stress. The fracture simulation modeled the crack tunneling and extension as a function of applied stress. The results indicated that the global constraint factor, alpha(sub g), initially dropped during stable crack growth. After peak applied stress was achieved, alpha(sub g) began to increase slightly. The effect of crack front shape on alpha(sub g) was small, but the crack front shape did greatly influence the local constraint and through-thickness crack-tip opening angle (CTOA) behavior. The surface values of CTOA for the tunneled crack front model agreed well with experimental measurements, showing the same initial decrease from high values during the initial 3mm of crack growth at the specimen's surface. At the same time, the interior CTOA values increased from low angles. After the initial stable tearing region, the CTOA was constant through the thickness. The three-dimensional analysis appears to confirm the potential of CTOA as a two-dimensional fracture criterion.

Dawicke, D. S.↗
NASA NTRS2017

Computational and Experimental Fluid-Structure Interaction Analysis of a High-Lift Wing with a Slat-Cove Filler for Noise Reduction

Airframe noise is a significant component of overall noise produced by transport aircraft during landing and approach (low speed maneuvers). A significant source for this noise is the cove of the leading-edge slat. The slat-cove filler (SCF) has been shown to be effective at mitigating slat noise. The objective of this work is to understand the fluid-structure interaction (FSI) behavior of a superelastic shape memory alloy (SMA) SCF in flow using both computational and physical models of a high-lift wing. Initial understanding of flow around the SCF and wing is obtained using computational fluid dynamics (CFD) analysis at various angles of attack. A framework compatible with an SMA constitutive model (implemented as a user material subroutine) is used to perform FSI analysis for multiple flow and configuration cases. A scaled physical model of the high-lift wing is constructed and tested in the Texas A&M 3 ft-by-4-foot wind tunnel. Initial validation of both CFD and FSI analysis is conducted by comparing lift, drag and pressure distributions with experimental results.

Scholten, William D.↗
NASA NTRS2004

Unique Approaches to Analysis of Time-Variable Gravity from GRACE

We have developed an innovative analysis strategy for analysis of GRACE data. We have developed a capability to recover local/regional gravity changes using non-global functional representations (Le. surface anomalies vs. global spherical harmonics) h m the GRACE data. Our approach can take regularly or irregularly shaped regions, populate them with surface anomaly blocks of suitable area and solve for the resulting mass flux with respect to a mean field. The surface mass or gravity anomalies benefit from the application of spatial and temporal constraints to add stability to the solution. In this paper we discuss the analysis of four months of GRACE Level 1B data (accelerometry, intersatellite data, attitude information and precise orbits) from July to October 2003, recently released to the GRACE Science Team. We compare and contrast this local approach to gravity recovery, with the more conventional approach using global spherical harmonics. We review simulations of this technique which allow us to pinpoint optimum strategies for applications of this local gravity recovery approach.

Lemoine, Frank G.↗
NASA NTRS2016

Hybrid Reality Lab Capabilities - Video 2

Our Hybrid Reality and Advanced Operations Lab is developing incredibly realistic and immersive systems that could be used to provide training, support engineering analysis, and augment data collection for various human performance metrics at NASA. To get a better understanding of what Hybrid Reality is, let's go through the two most commonly known types of immersive realities: Virtual Reality, and Augmented Reality. Virtual Reality creates immersive scenes that are completely made up of digital information. This technology has been used to train astronauts at NASA, used during teleoperation of remote assets (arms, rovers, robots, etc.) and other activities. One challenge with Virtual Reality is that if you are using it for real time-applications (like landing an airplane) then the information used to create the virtual scenes can be old (i.e. visualized long after physical objects moved in the scene) and not accurate enough to land the airplane safely. This is where Augmented Reality comes in. Augmented Reality takes real-time environment information (from a camera, or see through window, and places digitally created information into the scene so that it matches with the video/glass information). Augmented Reality enhances real environment information collected with a live sensor or viewport (e.g. camera, window, etc.) with the information-rich visualization provided by Virtual Reality. Hybrid Reality takes Augmented Reality even further, by creating a higher level of immersion where interactivity can take place. Hybrid Reality takes Virtual Reality objects and a trackable, physical representation of those objects, places them in the same coordinate system, and allows people to interact with both objects' representations (virtual and physical) simultaneously. After a short period of adjustment, the individuals begin to interact with all the objects in the scene as if they were real-life objects. The ability to physically touch and interact with digitally created objects that have the same shape, size, location to their physical object counterpart in virtual reality environment can be a game changer when it comes to training, planning, engineering analysis, science, entertainment, etc. Our Project is developing such capabilities for various types of environments. The video outlined with this abstract is a representation of an ISS Hybrid Reality experience. In the video you can see various Hybrid Reality elements that provide immersion beyond just standard Virtual Reality or Augmented Reality.

Delgado, Francisco J.↗
NASA NTRS1994

Experimental Water Droplet Impingement Data on Airfoils, Simulated Ice Shapes, an Engine Inlet and a Finite Wing

An experimental method has been developed to determine the water droplet impingement characteristics on two- and three-dimensional aircraft surfaces. The experimental water droplet impingement data are used to validate particle trajectory analysis codes that are used in aircraft icing analyses and engine inlet particle separator analyses. The aircraft surface is covered with thin strips of blotter paper in areas of interest. The surface is then exposed to an airstream that contains a dyed-water spray cloud. The water droplet impingement data are extracted from the dyed blotter paper strips by measuring the optical reflectance of each strip with an automated reflectometer. Experimental impingement efficiency data represented for a NLF (1)-0414 airfoil, a swept MS (1)-0317 airfoil, a Boeing 737-300 engine inlet model, two simulated ice shapes and a swept NACA 0012 wingtip. Analytical impingement efficiency data are also presented for the NLF (1)-0414 airfoil and the Boeing 737-300 engine inlet model.

Papadakis, M.↗
NASA NTRS1990

Computational aspects of sensitivity calculations in linear transient structural analysis

A study was performed focusing on the calculation of sensitivities of displacements, velocities, accelerations, and stresses in linear, structural, transient response problems. One significant goal of the study was to develop and evaluate sensitivity calculation techniques suitable for large-order finite element analyses. Accordingly, approximation vectors such as vibration mode shapes are used to reduce the dimensionality of the finite element model. Much of the research focused on the accuracy of both response quantities and sensitivities as a function of number of vectors used. Two types of sensitivity calculation techniques were developed and evaluated. The first type of technique is an overall finite difference method where the analysis is repeated for perturbed designs. The second type of technique is termed semi-analytical because it involves direct, analytical differentiation of the equations of motion with finite difference approximation of the coefficient matrices. To be computationally practical in large-order problems, the overall finite difference methods must use the approximation vectors from the original design in the analyses of the perturbed models. In several cases this fixed mode approach resulted in very poor approximations of the stress sensitivities. Almost all of the original modes were required for an accurate sensitivity and for small numbers of modes, the accuracy was extremely poor. To overcome this poor accuracy, two semi-analytical techniques were developed. The first technique accounts for the change in eigenvectors through approximate eigenvector derivatives. The second technique applies the mode acceleration method of transient analysis to the sensitivity calculations. Both result in accurate values of the stress sensitivities with a small number of modes and much lower computational costs than if the vibration modes were recalculated and then used in an overall finite difference method.

Greene, William H.↗