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At least 1,081 records · Page 60

Direct simulation of compressible turbulence in a shear flow

Compressibility effects on the turbulence in homogeneous shear flow are investigated. The growth of the turbulent kinetic energy was found to decrease with increasing Mach number: a phenomenon which is similar to the reduction of turbulent velocity intensities observed in experiments on supersonic free shear layers. An examination of the turbulent energy budget shows that both the compressible dissipation and the pressure-dilatation contribute to the decrease in the growth of kinetic energy. The pressure-dilatation is predominantly negative in homogeneous shear flow, in contrast to its predominantly positive behavior in isotropic turbulence. The different signs of the pressure-dilatation are explained by theoretical consideration of the equations for the pressure variance and density variance. Previously, the following results were obtained for isotropic turbulence: (1) the normalized compressible dissipation is of O(M(sub t)(exp 2)); and (2) there is approximate equipartition between the kinetic and potential energies associated with the fluctuating compressible mode. Both of these results were substantiated in the case of homogeneous shear. The dilatation field is significantly more skewed and intermittent than the vorticity field. Strong compressions seem to be more likely than strong expansions.

Sarkar, S.↗

Impingement tones of large aspect ratio supersonic rectangular jets

The frequency structure of large aspect ratio jet impingement tones is analyzed. Feedback loops consisting of downstream propagating instability waves of the jet flow and upstream propagating acoustic waves from the wall to the nozzle lip produce as many as 20 or more tones when a supersonic rectangular jet of large aspect ratio is directed normal on a wall. It is seen that there are only two basic tones. The basic tone of the lower frequency is associated with a symmetric instability feedback mode of the jet, while that of the higher frequency is associated with an antisymmetric instability feedback mode. The remaining tones are combination tones of the two basic frequencies and their harmonics. It is determined that the measured basic impingement tone frequencies fall in the theoretical permissible frequency bands over a wide range of jet Mach number.

Tam, Christopher K. W.↗

Candle flames in microgravity

The candle flame in both normal and microgravity is non-propagating. In microgravity, however, the candle flame is also non-convective where (excepting Stefan flow) pure diffusion is the only transport mode. It also shares many characteristics with another classical problem, that of isolated droplet combustion. Given their qualitatively similar flame shapes and the required heat feedback to condensed-phase fuels, the gas-phase flow and temperature fields should be relatively similar for a droplet and a candle in reduced gravity. Unless the droplet diameter is maintained somehow through non-intrusive replenishment of fuel, the quasi-steady burning characteristics of a droplet can be maintained for only a few seconds. In contrast, the candle flame in microgravity may achieve a nearly steady state over a much longer time and is therefore ideal for examining a number of combustion-related phenomena. In this paper, we examine candle flame behavior in both short-duration and long-duration, quiescent, microgravity environments. Interest in this type of flame, especially 'candle flames in weightlessness', is demonstrated by very frequent public inquiries. The question is usually posed as 'will a candle flame burn in zero gravity', or, 'will a candle burn indefinitely (or steadily) in zero gravity in a large volume of quiescent air'. Intuitive speculation suggests to some that, in the absence of buoyancy, the accumulation of products in the vicinity of the flame will cause flame extinction. The classical theory for droplet combustion with its spherically-shaped diffusion flame, however, shows that steady combustion is possible in the absence of buoyancy if the chemical kinetics are fast enough. Previous experimental studies of candle flames in reduced and microgravity environments showed the flame could survive for at least 5 seconds, but did not reach a steady state in the available test time.

Dietrich, D. L.↗

Determination of the Fracture Parameters in a Stiffened Composite Panel

A modified J-integral, namely the equivalent domain integral, is derived for a three-dimensional anisotropic cracked solid to evaluate the stress intensity factor along the crack front using the finite element method. Based on the equivalent domain integral method with auxiliary fields, an interaction integral is also derived to extract the second fracture parameter, the T-stress, from the finite element results. The auxiliary fields are the two-dimensional plane strain solutions of monoclinic materials with the plane of symmetry at x(sub 3) = 0 under point loads applied at the crack tip. These solutions are expressed in a compact form based on the Stroh formalism. Both integrals can be implemented into a single numerical procedure to determine the distributions of stress intensity factor and T-stress components, T11, T13, and thus T33, along a three-dimensional crack front. The effects of plate thickness and crack length on the variation of the stress intensity factor and T-stresses through the thickness are investigated in detail for through-thickness center-cracked plates (isotropic and orthotropic) and orthotropic stiffened panels under pure mode-I loading conditions. For all the cases studied, T11 remains negative. For plates with the same dimensions, a larger size of crack yields larger magnitude of the normalized stress intensity factor and normalized T-stresses. The results in orthotropic stiffened panels exhibit an opposite trend in general. As expected, for the thicker panels, the fracture parameters evaluated through the thickness, except the region near the free surfaces, approach two-dimensional plane strain solutions. In summary, the numerical methods presented in this research demonstrate their high computational effectiveness and good numerical accuracy in extracting these fracture parameters from the finite element results in three-dimensional cracked solids.

Lin, Chung-Yi↗

CIMEL Measurements of Zenith Radiances at the ARM Site

Starting from October 1, 2001, Cimel at the ARM Central Facility in Oklahoma has been switched to a new "cloud mode." This mode allows taking measurements of zenith radiance when the Sun in blocked by clouds. In this case, every 13 min. Cimel points straight up and takes 10 measurements with 9 sec. time interval. The new Cimel's mode has four filters at 440, 670, 870 and 1020 nm. For cloudy conditions, the spectral contrast in surface albedo dominates over Rayleigh and aerosol effects; this makes normalized zenith radiances at 440 and 670 as well as for 870 and 1020 almost indistinguishable. We compare Cimel measurements with other ARM cart site instruments: Multi-Filter Rotating Shadowband Radiometer (MFRSR), Narrow Field of View (NFOV) sensor, and MicroWave Radiometer(MWR). Based on Cimel and MFRSR 670 and 870 nm channels, we build a normalized difference cloud index (NDCI) for radiances and fluxes, respectively. Radiance NDCI from Cimel and flux NDCI from MFRSR are compared between themselves as well as with cloud Liquid Water Path (LWP) retrieved from MWR. Based on our theoretical calculations and preliminary data analysis,there is a good correlation between NDCIs and LWP for cloudy sky above green vegetation. Based on this correlation, an algorithm to retrieve cloud optical depth from NDCI is proposed.

Marshak, Alexander↗

Determination of Interlaminar Toughness of IM7/977-2 Composites at Temperature Extremes and Different Thicknesses

Composite materials are being used in the aerospace industry as a means of reducing vehicle weight. In particular, polymer matrix composites (PMC) are good candidates due to their high strength-to-weight and high stiffness-to-weight ratios. Future reusable space launch vehicles and space exploration structures will need advanced light weight composites in order to minimize vehicle weight while demonstrating robustness and durability, guaranteeing high factors of safety. In particular, the implementation of composite cryogenic propellant fuel tanks (cryotanks) for future reusable launch vehicles (RLVs) could greatly reduce the vehicle's weight versus identically sized cryotanks constructed of metallic materials. One candidate composite material for future cryotank designs is IM7/977-2, which is a graphite/epoxy system. A successful candidate must demonstrate reasonable structural properties over a wide range of temperatures. Since the matrix material is normally the weak link in the composite, tests that emphasize matrix-dominated behavior need to be conducted. Therefore, the objective of this work is to determine the mode I interlaminar fracture toughness of "unidirectional" 8-ply and 16-ply IM7/977-2 through experimental testing. Tests were performed at -196 degrees Celsius (-320 degrees Fahrenheit), 22 degrees Celsius (72 degrees Fahrenheit), 93 degrees Celsius (200 degrees Fahrenheit) and 160 degrees C (320 degrees Fahrenheit). Low temperature testing was completed while the specimen was submerged in a liquid nitrogen bath. High temperature testing was completed in a temperature-controlled oven.

Johnson, W. S.↗

Self-Consistent Model of Magnetospheric Ring Current and Propagating Electromagnetic Ion Cyclotron Waves: Waves in Multi Ion Magnetosphere - 1

The further development of a self-consistent theoretical model of interacting ring current ions and electromagnetic ion cyclotron waves [Khazanov et al., 2003] is presented. In order to adequately take into account the wave propagation and refraction in a multi-ion plasmasphere, we explicitly include the ray tracing equations in our previous self-consistent model and use the general form of the wave kinetic equation. This is a major new feature of the present model and, to the best of our knowledge, the ray tracing equations for the first time are explicitly employed on a global magnetospheric scale in order to self-consistently simulate spatial, temporal, and spectral evolutions of the ring current and electromagnetic ion cyclotron waves. To demonstrate the effects of EMIC wave propagation and refraction on the EMIC wave energy distributions and evolution we simulate the May 1998 storm. The main findings of our simulation can be summarized as follows. First, due to the density gradient at the plasmapause, the net wave refraction is suppressed, and He(+)-mode grows preferably at plasmapause. This result is in a total agreement with the previous ray tracing studies, and very clear observed in presented B-field spectrograms. Second, comparison the global wave distributions with the results from other ring current model [Kozyra et al., 1997] reveals that our model provides more intense and higher plasmapause organized distributions during the May, 1998 storm period. Finally, the found He(+)-mode energy distributions are not Gaussian distributions, and most important that wave energy can occupy not only the region of generation, i. e. the region of small wave normal angles, but the entire wave normal angle region and even only the region near 90 degrees. The latter is extremely crucial for energy transfer to thermal plasmaspheric electrons by resonant Landau damping, and subsequent downward heat transport and excitation of stable auroral red arcs.

Khazanov, G. V.↗

Episodic Aging and End States of Comets

It is known that comets are aging very rapidly on cosmic scales, because they rapidly shed mass. The processes involved are (i) normal activity - sublimation of ices and expulsion of dust from discrete emission sources on and/or below the surface of a comet's nucleus, and (ii) nuclear fragmentation. Both modes are episodic in nature, the latter includes major steps in the comet's life cycle. The role and history of dynamical techniques used are described and results on mass losses due to sublimation and dust expulsion are reviewed. Studies of split comets, Holmes-like exploding comets, and cataclysmically fragmenting comets show that masses of 10 to 100 million tons are involved in the fragmentation process. This and other information is used to investigate the nature of comets' episodic aging. Based on recent advances in understanding the surface morphology of cometary nuclei by close-up imaging, a possible mechanism for large-scale fragmentation events is proposed and shown to be consistent with evidence available from observations. Strongly flattened pancake-like shapes appear to be required for comet fragments by conceptual constraints. Possible end states are briefly examined.

fragmentation↗

22” ADP Fan Rig Liners Design Report

A liner design study was conducted as part of a cooperative effort between five (5) government/industry teams that together seek to demonstrate the technology for designing and manufacturing acoustic liners that are twenty five percent (25%) more efficient than 1992 technology liners. The study emphasized teaming collaboration. The improved liners were designed by Pratt & Whitney and Boeing Airplane Company and built by Rohr Inc, and will be tested in the NASA/P& W 22-inch ADP fan rig at NASA Lewis Research Center's 9' x 15' wind tunnel. Design guidelines and decisions were made collectively during monthly design review telecons and at the formal final design review. The tools that were used to design the new improved liners were not new, but the process that was developed as part of this study that led into the evaluation and selection of the final and best liner designs is new. Until now, the liner design process as practiced by different industry teams varied greatly. Some procedures were based on empirical liner attenuation databases which could not adequately account for engine-to-engine hardwall fan noise spectral differences, while others were entirely theoretical. And regardless of which method one chose, the major difficulty was still the lack of understanding and knowledge of the actual hardwall fan source noise modal structure and farfield SPL spectra. The NASA-led effort to attempt actual measurements of the fan source noise modal structure by means of the rotating microphone array is expected to contribute significantly to the understanding of the nature of the fan tone noise modes, but the application to broadband noise is still a long way off. The new design process included consideration of the measured fan tone modes, but for the majority of the spectra a separate, systematic process was used. It is a common practice in the engine/nacelle industry that acoustic liners for new products are designed before actual measured hardwall engine far-field noise spectra are available. Target noise spectra are normally derived from existing engine noise databases with some adjustments to absolute SPLs and frequencies. This practice has been acceptable as long as the new engine was a derivative of the base engine. However, in the case of the ADP, the transition from a current engine base is too - great, and the adjustments could not adequately account for the quantum changes in SPL spectral differences. Examples were the 1992 single degree of freedom (SDOF) inlet and aft liners (designed by P&W) that would be used as the baseline liners against which the new improved liners' efficien¬cies would -be measured. As will be shown, these baseline liners have been found to be deeper than the desire optimum depths. The new liner design process begins with the selection of the target hard wall fan noise spectra. These spectra were obtained by scaling up (5.91 scale factor) the measured hardwall fan spectral data from the 22” ADP fan rig. Next, the modal energy contents for each 1/3-octave band center frequency of these hardwall fan noise spectra are estimated. (Within each 1/3-octave band center frequency, the model energy distribution approximation is for both tone and broadband noise). For the inlet noise, P&W uses a derivative of the NASA Lewis (Ed Rice's) method of classifying; and grouping propagating modes by their cutoff ratios. The next step is to assign energy level to each group of modes having the same cutoff ratio. In Ed Rice's model, the modes were grouped according to ten (10) cutoff ratio intervals with center values located at 1.026, 1.085, 1.155, 1.24, 1.35, 1.49, 1.69, 2.0, and 4.47 (with equal number of modes in each interval). The modes were assumed to have equal energy. P&W’s model expands the cutoff ratio-mode grouping into two hundred (200) smaller cutoff ratio intervals with center values located f1".'m 1.003 to 11.5 in 199 increasing incremental intervals. Next, P&W's model uses a "2-parameter" normal distribution as a template to assign energy levels to these 200 pre-determined cutoff ratio values (for each 1/3-octave band center frequency). In the past, P&W had conducted an extensive study to determine what "2-parameter" values are appropriate for fullscale inlet liners, and had developed a set of twenty-four (24) "2-pararneter" values (i.e. one for each 1/3-octave frequency band) that when used in Ed Rice's Inlet Attenuation Prediction Method produced predicted liner attenuations that closely matched measured liner attenuations from several P&W's engines. In the absence of actual measured tone and broadband modal data from the 22" ADP rig, the process will use P&W's proprietary set of "2-parameter" values for this liner design study. For the aft noise, Boeing uses a modal energy approximation that the "transport energy" of each propagating mode is equal. This approximation is almost the same as the equal energy per mode approximation, except for modes that are near cutoff. Boeing's model forces these modes to have lower energy levels. Both P&W and Boeing agreed that the "transport energy" approximation should work well for the ADP aft fan noise which appears to be dominated by broadband noise. The design process then proceeds to calculate the optimum liner impedances for each frequency in both the inlet and the aft. These optimum impedances represent the target impedances that the designed liners should have. Liners with impedances matching the optimum impedances at all frequencies are "ideal" liners. These ideal liners are theoretically the best liners. Unfortunately, it has been showed that it is impossible to design and build such ideal liners. The next best liners are ones that have impedances matching the optimum impedances over some frequency range (not all frequencies as for the ideal). This is accomplished by the use of "frequency weightings". Several of P&W's and Boeing's existing computer decks were used for optimizing and matching the designed liner impedances to the target optimum values (with the various frequency weightings specified). The optimization produces liner candidates with predicted liner impedances and descriptions of their physical liner characteristics. These candidate liner impedances are then used to predict their spectral attenuation characteristics which are then used together with the target hardwall fan noise spectra to determine the resulting treated noise spectra and PNLT values. Further optimization around the selected candidate designs yield final designs that are best in PNLT attenuations. Use of the optimization decks allowed a large number of liner candidates to be screened in a relatively short period of time. This design process is systematic and is efficient. The new inlet SDOF liner design obtained from the improved process was predicted to be 34% more effective (per unit area) than the 1992 baseline inlet liner. This inlet liner design is a 112 rayl wovenwiremesh facesheet over a 0.312-inch deep honeycomb core. The new aft SDOF liner design was predicted to be 52% more efficient than the aft baseline liner. The new aft SDOF liner is "segmented" with a shallower liner on the core cowl (inner duct wall) and a deeper liner on the fan cowl (outer duct wall). The shallower liner is a 70.6 rayl woven-wiremesh facesheet over a 0.141-inch deep honeycomb core. The deeper liner is a 68 rayl woven-wiremesh facesheet over a 0.309-inch deep honeycomb core. All liner dimensions are for the 22-inch model-scale ADP fan rig liners. The selected advanced liners are "segmented" double-layer (DDOF) liners for the inlet and aft locations. Also, for the inlet, a bulk liner with ceramic foam for wider broadband noise absorption was also selected. Triple-layer liner designs were not considered since the model scaled liners ( 1/5. 91) were dimensionally too small to be built correctly, and irrin earlier concept study, Boeing found a triple-layer to have only very small benefits over a double-layer. The inlet DDOF liner was predicted to be 83% more effective than the baseline. The inlet DDOF design is a 78 rayl facesheet over a 0.080 top cavity depth, a 68 rayl septum and a 0.227-inch bottom cavity depth. The inlet bulk liner is a 60 rayl facesheet over a 0.33-inch deep honeycomb filled with high temperature (HTP) ceramic foam with a density of 4.8 lb/cu.ft and a flow resistivity of 167 rayl/cm. The inlet bulk liner was predicted to be 83% more effective than the baseline liner. The aft DDOF liners are segmented with a shallower liner on the core cowl and a deeper liner on the fan cowl. The shallow DDOF liner is a 49.8 rayl facesheet over a 0.093-inch top cavity depth, a septum of 88.2 rayls over a 0.181-inch bottom cavity depth. The deep DDOF liner is a 12.9 rayl facesheet over a 0.140-inch top cavity depth, a septum of 53.1 ray ls over a 0.258-inch botom cavity depth. The segmented aft DDOF liners were predicted to be 86% more efficient than the baseline liner.

turbofan acoustic treatment↗

A real-time guidance algorithm for aerospace plane optimal ascent to low earth orbit

Problems of onboard trajectory optimization and synthesis of suitable guidance laws for ascent to low Earth orbit of an air-breathing, single-stage-to-orbit vehicle are addressed. A multimode propulsion system is assumed which incorporates turbojet, ramjet, Scramjet, and rocket engines. An algorithm for generating fuel-optimal climb profiles is presented. This algorithm results from the application of the minimum principle to a low-order dynamic model that includes angle-of-attack effects and the normal component of thrust. Maximum dynamic pressure and maximum aerodynamic heating rate constraints are considered. Switching conditions are derived which, under appropriate assumptions, govern optimal transition from one propulsion mode to another. A nonlinear transformation technique is employed to derived a feedback controller for tracking the computed trajectory. Numerical results illustrate the nature of the resulting fuel-optimal climb paths.

Calise, A. J.↗

Multiple switch-off slow shock solutions

A new electromagnetic quasi-neutral time-dependent implicit hybrid model for simulating the slow-mode switch-off shock is developed, in which the electrons are treated as a finite-mass adiabatic fluid (gamma sub e = 5/3), whose motion along the shock normal provides local charge neutrality. The model is used to examine the effects of various boundary conditions applied during shock formation. It is shown that, for each upstream condition, two distinctly different slow-shock structures can be obtained: a low-shock structure with a trailing magnetic wave (TMW) and a slow-shock structure without a TMW, depending strongly on the boundary conditions for the electric and magnetic fields at the physical piston which generates the shock. The results indicate that the slow-shock structure with a TMW is nonlinearly unstable with respect to changes in the upstream magnetic field, suggesting that the downstream structure of slow shocks need not be associated with a TMW.

Vu, H. X.↗

New Propagating Mode near the Superfluid Transition in He-4

We have observed a new temperature-entropy wave that propagates opposite to the direction of a steady heat flux Q when the helium column is heated from above. This propagating mode is due to non-linear thermo-conductance of the helium sample in the self-organized state. Such a mode had been predicted to exist on the self-organized heat transport state for Q less than about 100 nW/sq cm. We confirm that this mode exists in this regime. However, we also observe that it propagates even when the helium is pushed away from the self-organized heat transport state into the normal state.

Sergatskov, Dmitri A.↗

Spacecraft Design Thermal Control Subsystem

The Thermal Control Subsystem engineers task is to maintain the temperature of all spacecraft components, subsystems, and the total flight system within specified limits for all flight modes from launch to end-of-mission. In some cases, specific stability and gradient temperature limits will be imposed on flight system elements. The Thermal Control Subsystem of "normal" flight systems, the mass, power, control, and sensing systems mass and power requirements are below 10% of the total flight system resources. In general the thermal control subsystem engineer is involved in all other flight subsystem designs.

Jupiter Icy Moon Orbiter (JIMO)↗

Microstructure of the interplanetary magnetic field near 4 and 5 AU

Interplanetary magnetic field fluctuations near 4 and 5 AU are investigated on the basis of Pioneer 10 and 11 vector magnetic field measurements in order to identify extant wave modes and propagation directions as functions of frequency in the range 0.00004 to 0.009 Hz. Analysis of the approximately 500 spectral matrices as functions of frequency obtained from each spacecraft indicates the distributions of maximum variance directions to be peaked along the normal to the plane of the minimum variance direction and the mean magnetic field, as would be expected for a planar, small-amplitude Alfven wave with a minimum variance direction close to the wave vector. It is thus inferred that single small-amplitude Alfven waves are occasionally present in the temporal and frequency bands analyzed, however waves with wave vectors not well approximated by the minimum variance direction, Alfven wave ensembles, finite-amplitude waves and nonplanar fluctuations may also be present. Observations are also noted to suggest the predominance of the Alfven mode over the fast and slow modes, however do not support theoretical models of outward propagating plane Alfven waves of solar origin.

Parker, G. D.↗

In Situ Surface Characterization

Operation of in situ space assets, such as rovers and landers, requires operators to acquire a thorough understanding of the environment surrounding the spacecraft. The following programs help with that understanding by providing higher-level information characterizing the surface, which is not immediately obvious by just looking at the XYZ terrain data. This software suite covers three primary programs: marsuvw, marsrough, and marsslope, and two secondary programs, which together use XYZ data derived from in situ stereo imagery to characterize the surface by determining surface normal, surface roughness, and various aspects of local slope, respectively. These programs all use the Planetary Image Geometry (PIG) library to read mission-specific data files. The programs themselves are completely multimission; all mission dependencies are handled by PIG. The input data consists of images containing XYZ locations as derived by, e.g., marsxyz. The marsuvw program determines surface normals from XYZ data by gathering XYZ points from an area around each pixel and fitting a plane to those points. Outliers are rejected, and various consistency checks are applied. The result shows the orientation of the local surface at each point as a unit vector. The program can be run in two modes: standard, which is typically used for in situ arm work, and slope, which is typically used for rover mobility. The difference is primarily due to optimizations necessary for the larger patch sizes in the slope case. The marsrough program determines surface roughness in a small area around each pixel, which is defined as the maximum peak-to-peak deviation from the plane perpendicular to the surface normal at that pixel. The marsslope program takes a surface normal file as input and derives one of several slope-like outputs from it. The outputs include slope, slope rover direction (a measure of slope radially away from the rover), slope heading, slope magnitude, northerly tilt, and solar energy (compares the slope with the Sun s location at local noon). The marsuvwproj program projects a surface normal onto an arbitrary plane in space, resulting in a normalized 3D vector, which is constrained to lie in the plane. The marsuvwrot program rotates the vectors in a surface normal file, generating a new surface normal file. It also can change coordinate systems for an existing surface normal file. While the algorithms behind this suite are not particularly unique, what makes the programs useful is their integration into the larger in situ image processing system via the PIG library. They work directly with space in situ data, understanding the appropriate image metadata fields and updating them properly. The secondary programs (marsuvwproj, marsuvwrot) were originally developed to deal with anomalous situations on Opportunity and Spirit, respectively, but may have more general applicability.

Deen, Robert G.↗

Orifice Blocks Heat Pipe in Reverse Mode

High forward-mode conductance is combined with rapid reverse-mode shutoff in a heat pipe originally developed to cool spacecraft payloads. A narrow orifice within the pipe "chokes off" the evaporator if heat sink becomes warmer than source. During normal operation, with source warmer than sink, orifice has little effect. Design is simpler and more compact than other thermal-diode heat pipes and requires no special materials, forgings, or unusual construction techniques.

Alario, J. P.↗

Ray tracing of broadband bursty radio emissions from Uranus

To determine the source position of the broadband bursty emission, rays of X-mode emissin were traced from source positions along magnetic field lines with footprints that form a large grid centered approximately on the south magnetic pole of Uranus. For large wave normal angles, source regions different from those producing b-smooth emission were found. The emission observed prior to closest approach has a source along field lines that are distinct from those which generate emissions observed after closest approach.

Curran, D. B.↗

The direct effects of gravity on the control and output matrices of controlled structure models

The effects of gravity on the dynamic performance of structural control actuators and sensors are dual forms of an additive perturbation that can attenuate or amplify the device response (input or output). The modal modeling of these perturbations is derived for the general case of arbitrarily oriented devices and arbitrarily oriented planes of deformation. A nondimensional sensitivity analysis to identify the circumstances under which the effects of gravity are important is presented. Results show that gravity effects become important when the product of the ratio of the normalized modal slope and the modal displacement is comparable to the ratio of the gravitational acceleration and the product of the beam length and the squared eigenfrequency for a given mode.

Rey, Daniel A.↗