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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.

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At least 217 records · Page 12
All sourcesNASA NTRSDOE OSTI
DOE OSTI2024

Impact of beam coupling impedance on crab cavity noise induced emittance growth

Crab cavities will be deployed as a part of the High Luminosity Large Hadron Collider (HL-LHC) upgrade to mitigate the luminosity reduction induced by the crossing angle at the main experiments (ATLAS and CMS). Two prototype crab cavities have been installed in the CERN Super Proton Synchrotron (SPS) in 2018 for studies with proton beams. An issue of concern is the transverse emittance growth induced by noise in the crab cavity radio frequency (rf) system, which is anticipated to limit the performance of the HL-LHC. In measurements conducted in the SPS in 2018, the crab cavity noise-induced emittance growth was measured to be a factor of 4 lower than predicted from the existing analytical models. In this paper, it is shown that the observed discrepancy is explained by damping effects from the beam coupling impedance, which were not included in the models up to now. Using the van Kampen mode approach, a new theory is developed, suggesting that the impedance can separate the coherent tune from the incoherent spectrum leading to an effective reduction of the crab cavity rf noise-induced emittance growth. This mechanism is validated in tracking simulations using the SPS impedance model as well as in dedicated experimental measurements conducted in the SPS in 2022. The implications for the HL-LHC project are discussed. Published by the American Physical Society 2024

Physics↗
NASA NTRS1979

Control of large space structures

The control of large space structures was studied to determine what, if any, limitations are imposed on the size of spacecraft which may be controlled using current control system design technology. Using a typical structure in the 35 to 70 meter size category, a control system design that used actuators that are currently available was designed. The amount of control power required to maintain the vehicle in a stabilized gravity gradient pointing orientation that also damped various structural motions was determined. The moment of inertia and mass properties of this structure were varied to verify that stability and performance were maintained. The study concludes that the structure's size is required to change by at least a factor of two before any stability problems arise. The stability margin that is lost is due to the scaling of the gravity gradient torques (the rigid body control) and as such can easily be corrected by changing the control gains associated with the rigid body control. A secondary conclusion from the study is that the control design that accommodates the structural motions (to damp them) is a little more sensitive than the design that works on attitude control of the rigid body only.

Gran, R.↗
NASA NTRS1987

Passive isolation/damping system for the Hubble space telescope reaction wheels

NASA's Hubble Space Telescope contain large, diffraction limited optics with extraordinary resolution and performance for surpassing existing observatories. The need to reduce structural borne vibration and resultant optical jitter from critical Pointing Control System components, Reaction Wheels, prompted the feasibility investigation and eventual development of a passive isolation system. Alternative design concepts considered were required to meet a host of stringent specifications and pass rigid tests to be successfully verified and integrated into the already built flight vehicle. The final design employs multiple arrays of fluid damped springs that attenuate over a wide spectrum, while confining newly introduced resonances to benign regions of vehicle dynamic response. Overall jitter improvement of roughly a factor of 2 to 3 is attained with this system. The basis, evolution, and performance of the isolation system, specifically discussing design concepts considered, optimization studies, development lessons learned, innovative features, and analytical and ground test verified results are presented.

Hasha, Martin D.↗
NASA NTRS1959

Stability and Control Characteristics at Subsonic Speeds of a Flat-Top Arrowhead Wing-Body Combination

A wind-tunnel investigation was made to determine the longitudinal- and lateral-stability derivatives of a flat-top wing-body configuration at Mach numbers from 0.22 to 0.90 and Reynolds numbers of 3.5 and 17 million. The wing had a leading-edge sweepback of 78.9 deg and a cathedral of 45 deg on the outer panels. The tests included the determination of the effectiveness of elevon and rudder controls and also an investigation of ground effects. The model was tested at angles of attack up to 28 deg and angles of sideslip up to 18 deg. The dynamic response of this configuration has been determined from the wind-tunnel data for a simulated airplane having a wing loading of 17.7 pounds per square foot. The longitudinal data show a forward shift in aerodynamic center of 10 percent of the mean aerodynamic chord as the lift coefficient is increased above 0.1. Although flown in the lift range of decreasing stability, the simulated airplane did not encounter pitch-up in maneuvers initiated from steady level flight with zero static margin unless a load factor of 2.2 was exceeded. This maneuver margin was provided by a large value of pitching moment due to pitching velocity. The number of cycles to damp the Dutch roll mode to half amplitude, the time constants of the roll subsidence and spiral divergence modes, and control effectiveness in roll are computed. The lateral stability is shown to be positive but is marginal in meeting the military specifications for today's aircraft. An analog computer study has been made in five degrees of freedom (constant velocity) which illustrates that the handling characteristics are satisfactory. Several programed rolling maneuvers and coordinated turns also illustrate the handling qualities of the airplane.

Buell, Donald A.↗
NASA NTRS1990

Calculation of flight vibration levels of the AH-1G helicopter and correlation with existing flight vibration measurements

NASA-Langley is sponsoring a rotorcraft structural dynamics program with the objective to establish in the U.S. a superior capability to utilize finite element analysis models for calculations to support industrial design of helicopter airframe structures. In the initial phase of the program, teams from the major U.S. manufacturers of helicopter airframes will apply extant finite element analysis methods to calculate loads and vibrations of helicopter airframes, and perform correlations between analysis and measurements. The aforementioned rotorcraft structural dynamics program was given the acronym DAMVIBS (Design Analysis Method for Vibrations). Sikorsky's RDYNE Rotorcraft Dynamics Analysis used for the correlation study, the specifics of the application of RDYNE to the AH-1G, and comparisons of the predictions of the method with flight data for loads and vibrations on the AH-1G are described. RDYNE was able to predict trends of variations of loads and vibrations with airspeed, but in some instances magnitudes differed from measured results by factors of two or three to one. Sensitivities were studied of predictions to rotor inflow modeling, effects of torsional modes, number of blade bending modes, fuselage structural damping, and hub modal content.

Sopher, R.↗
NASA NTRS1993

Numerical solution of flame sheet problems with and without multigrid methods

Flame sheet problems are on the natural route to the numerical solution of multidimensional flames, which, in turn, are important in many engineering applications. In order to model the structure of flames more accurately, we use the vorticity-velocity formulation of the fluid flow equations, as opposed to the streamfunction-vorticity approach. The numerical solution of the resulting nonlinear coupled elliptic partial differential equations involves a pseudo transient process and a steady state Newton iteration. Rather than working with dimensionless variables, we introduce scale factors that can yield significant savings in the execution time. In this context, we also investigate the applicability and performance of several multigrid methods, focusing on nonlinear damped Newton multigrid, using either one way or correction schemes.

Douglas, Craig C.↗
NASA NTRS1998

Robust Multigrid Smoothers for Three Dimensional Elliptic Equations with Strong Anisotropies

We discuss the behavior of several plane relaxation methods as multigrid smoothers for the solution of a discrete anisotropic elliptic model problem on cell-centered grids. The methods compared are plane Jacobi with damping, plane Jacobi with partial damping, plane Gauss-Seidel, plane zebra Gauss-Seidel, and line Gauss-Seidel. Based on numerical experiments and local mode analysis, we compare the smoothing factor of the different methods in the presence of strong anisotropies. A four-color Gauss-Seidel method is found to have the best numerical and architectural properties of the methods considered in the present work. Although alternating direction plane relaxation schemes are simpler and more robust than other approaches, they are not currently used in industrial and production codes because they require the solution of a two-dimensional problem for each plane in each direction. We verify the theoretical predictions of Thole and Trottenberg that an exact solution of each plane is not necessary and that a single two-dimensional multigrid cycle gives the same result as an exact solution, in much less execution time. Parallelization of the two-dimensional multigrid cycles, the kernel of the three-dimensional implicit solver, is also discussed. Alternating-plane smoothers are found to be highly efficient multigrid smoothers for anisotropic elliptic problems.

Llorente, Ignacio M.↗
NASA NTRS1997

Variable Dynamic Testbed Vehicle: Dynamics Analysis

The Variable Dynamic Testbed Vehicle (VDTV) concept has been proposed as a tool to evaluate collision avoidance systems and to perform driving-related human factors research. The goal of this study is to analytically investigate to what extent a VDTV with adjustable front and rear anti-roll bar stiffnesses, programmable damping rates, and four-wheel-steering can emulate the lateral dynamics of a broad range of passenger vehicles.

Variable Dynamic Testbed Vehicle VDTV collision av↗
NASA NTRS2023

Fully Coupled Aeroelastic Stability Analysis of Adaptive Shape Memory Alloy Structural Technologies for Airframe Noise Reduction

This final report documents work performed by ATA Engineering, Inc., (ATA) to develop computational models and analyze the coupled fluid-structure response of two types of noise treatments applied to the leading-edge-slat component of a high-lift system typical of modern transport aircraft. The first treatment is a slat-gap filler (SGF), which closes the gap between the suction surfaces of a deployed slat and an aircraft main wing, and the second treatment is a slat-cove filler (SCF), which replaces the recirculating flow on the slat cove with a surface that promotes flow attachment. The representative airframe chosen for this work was NASA’s High-Lift Common Research Model (CRM-HL) in a baseline high-lift configuration. Superelastic shape memory alloys (SMAs) have been identified as enabling materials for these structural treatments. Since the technology elements rely upon having a highly reconfigurable structure, designs were assessed for their static aeroelastic deflection as well as their dynamic aeroelastic stability using coupled computational fluid dynamics (CFD) and nonlinear computational structural dynamics (NL-CSD) tools. Specifically, fluid-structure interaction (FSI) problems were solved computationally using the CFD solver Loci/CHEM and the NL-CSD solver Abaqus. As a part of the overall project, a similar capability was implemented using the CFD solver FUN3D coupled to Abaqus, although that work is documented in a separate report and that FSI framework was not used to analyze any of NASA’s SGF and SCF configurations. The technical approach consisted of solving for the flow field around the entire vehicle using a global CFD model, followed by extraction of relevant local subdomain data for CFD and NL-CSD co-simulations. The SGF design was analyzed using both 2D and 3D co-simulations to predict quasi-static aeroelastic deformations and to assess dynamic aeroelastic stability, whereas the SCF was analyzed in 2D only. SGF static aeroelastic response predictions focused on characterizing the deformed shape, with maximum displacements predicted to be on the order of magnitude of the technology element panel thickness. SGF dynamic aeroelastic response predictions used Partial Floquet analysis of the temporal evolution of selected nodal displacements to quantify the sign and magnitude of aeroelastic damping. Results suggest that the CRM-HL operating conditions would result in a dynamically stable response. The simulated dynamic pressure was also increased up to a factor of about four, and resulting responses suggest that predicted dynamic stability would be achieved with some margin.

Fluid Structure Interaction↗
DOE OSTI2024

Cross-scale interaction between microturbulence and meso-scale reversed shear Alfvén eigenmodes in DIII-D plasmas

Abstract This paper reports global nonlinear gyrokinetic simulations that couple meso-scale reversed shear Alfvén eigenmodes (RSAEs) driven by energetic particles (EPs) and ion temperature gradient (ITG) microturbulence driven by thermal plasma, using equilibrium and profiles from DIII-D discharge #159243. In simulations focusing only on the ITG, electrostatic ITG drives a huge thermal ion heat transport, which is reduced by a factor of 10 to a level close to the experimental value in electromagnetic simulation due to finite β effect. In the simulations coupling the RSAE and ITG, ITG can scatter the resonant EP nonlinearly trapped by the RSAE and damp the zonal flows generated by the RSAE. The regulation of the RSAE by the ITG greatly reduces the initial saturation amplitude of the RSAE but increases the RSAE amplitude and associated EP transport to experimental levels in the quasi-steady state. The RSAE effects on the ITG, specifically the stronger zonal flows generated by the RSAE and the RSAE frequency modulation of the ITG-induced thermal ion heat transport, in turn, leads to a reduction of the thermal ion heat transport by more than a factor of 2 . For a stronger background ITG, the regulation of the RSAE by the ITG is stronger, while the RSAE effects on the ITG are weaker. This work highlights the importance of cross-scale coupling in the dynamics of the AE turbulence and EP transport.

Physics↗
NASA NTRS1947

Measurements in Flight of the Flying Qualities of a Chance Vought F4U-4 Airplane: TED No. NACA 2388

The results of flight tests to determine flying qualities of a Chance Vought F4U-4 airplane are presented and discussed herein. In addition to comprehensive measurements at low altitude (about 8000 ft), tests of limited scope were made at high altitude (about 25,000 ft). The more important characteristics, based on a comparison of the test results and opinions of the pilots with the Navy requirements, can be summarized as follows: 1. The short-period control-free oscillations of the elevator angle and the normal acceleration were satisfactorily damped. 2. The most rearward center-of-gravity locations for satisfactory static longitudinal stability with power on, as determined by the control-force variations, were approximately 30 and 27 percent M.A.C. with flaps and gear up and down, respectively. 3. In maneuvering flight the conditions for which control-force gradients of satisfactory magnitude were obtained were seriously limited by sizable changes in the gradient with center-of-gravity location, airspeed, altitude, acceleration factor, and direction of turn. 4. The elevator and rudder controls were satisfactory for landings and take-offs. 5. The trim tabs were sufficiently effective for all controls. 6. The directional and lateral dynamic stability was positive, but the rudder oscillation did not damp within one cycle. The airplane oscillation damped sufficiently at low altitude but not at high altitude. 7. Both rudder-fixed and rudder-free static directional stability were positive over a sideslip range of +/-15 deg. However, the rudder force tended to reverse at high angles of right sideslip with flaps and gear up, power on, at low speeds. 8. The stick-fixed static lateral stability (dihedral effect) was positive in all conditions, but the stick-free dihedral effect was neutral at low speeds with flap and gear down, power on. 9. The yaw due to abrupt full aileron deflection at low speed was mot excessive, and the rudder control was adequate to hold trim sideslip. 10. In abrupt rudder-fixed aileron rolls in the clean configuration the maximum pb/2V for full aileron deflection at low and normal speeds was only 0.064. 11. The stalling characteristics were considered unsatisfactory in all configurations in both straight and turning flight due to inadequate stall warning. The motions in the stalls were not unduly severe, and recovery could be effected promptly by normal use of the controls.

Liddell, Charles J., Jr.↗
NASA NTRSDate not supplied

Gimbal Bearing Friction in the SLS Core Stage Thrust Vector Control System

The Space Launch System (SLS) Core Stage Thrust Vector Control (TVC) system is comprised of eight mechanical feedback Shuttle heritage Type III TVC actuators that vector the four Shuttle heritage RS-25 engines about a Shuttle heritage gimbal block/bearing. The MSFC Controls community has long regarded gimbal friction to be a negligible effect on the overall control of gimbaled RS-25 engines. This is corroborated by Space Shuttle test and flight data that does not appear to show degraded effects, nor limit cycling at the end of the shuttle flight. For this reason, friction was not expected to be a driving factor of performance and control of the reused RS-25 engines aboard the SLS. However, after test data showed a large shift in frequency behavior and a highly damped step-response in the time series, there was further investigation into what could have caused this behavior. Heritage friction models used in previous gimbal and ball bearings were evaluated such as Coulomb, Dahl and LuGre, but the single degree of freedom friction models alone were not enough to explain the behavior and shifts seen in the test data. This paper presents the additional findings and modeling efforts regarding friction on the RS-25 engines. Using the Two Actuator Operational Simulation (TAOS), the difference from modeling separate friction degrees of freedom to coupled degrees of freedom was investigated to deduce the effects of one axis’s movement on the other. Next, due to the vibration environment, a modified LuGre model has been proposed that adds an additional term to decrease the friction coefficient at low engine velocity amplitudes. Lastly, the addition of the stiffness in each half of the gimbal bearing has increased modeling fidelity by also adding the effect on the gimbal bearing bending in compliance to both the friction torque on the surface of the gimbal bearing and the actuator force that is forcing the engine in a specified direction. Through these effects, the time and frequency domain behavior seen in test can be characterized accurately.

Friction↗
NASA NTRSDate not supplied

Gimbal Bearing Friction in the SLS Core Stage Thrust Vector Control System

The Space Launch System (SLS) Core Stage Thrust Vector Control (TVC) system is comprised of eight mechanical feedback Shuttle heritage Type III TVC actuators that vector the four Shuttle heritage RS-25 engines about a Shuttle heritage gimbal block/bearing. The MSFC Controls community has long regarded gimbal friction to be a negligible effect on the overall control of gimbaled RS-25 engines. This is corroborated by Space Shuttle test and flight data that does not appear to show degraded effects, nor limit cycling at the end of the shuttle flight. For this reason, friction was not expected to be a driving factor of performance and control of the reused RS-25 engines aboard the SLS. However, after test data showed a large shift in frequency behavior and a highly damped step-response in the time series, there was further investigation into what could have caused this behavior. Heritage friction models used in previous gimbal and ball bearings were evaluated such as Coulomb, Dahl and LuGre, but the single degree of freedom friction models alone were not enough to explain the behavior and shifts seen in the test data. This paper presents the additional findings and modeling efforts regarding friction on the RS-25 engines. Using the Two Actuator Operational Simulation (TAOS), the difference from modeling separate friction degrees of freedom to coupled degrees of freedom was investigated to deduce the effects of one axis’s movement on the other. Next, due to the vibration environment, a modified LuGre model has been proposed that adds an additional term to decrease the friction coefficient at low engine velocity amplitudes. Lastly, the addition of the stiffness in each half of the gimbal bearing has increased modeling fidelity by also adding the effect on the gimbal bearing bending in compliance to both the friction torque on the surface of the gimbal bearing and the actuator force that is forcing the engine in a specified direction. Through these effects, the time and frequency domain behavior seen in test can be characterized accurately.

Friction↗
NASA NTRS1991

Damping of structural vibrations with piezoelectric materials and passive electrical networks

The possibility of dissipating mechanical energy with piezoelectric material shunted with passive electrical circuits is investigated. The effective mechanical impedance for the piezoelectric element shunted by an arbitrary circuit is derived. The shunted piezoelectric is shown to possess frequency dependent stiffness and loss factor which are also dependent on the shunting circuit. The generally shunted model is specialized for two shunting circuits: the case of a resistor alone and that of a resistor and inductor. For resistive shunting, the material properties exhibit frequency dependence similar to viscoelastic materials, but are much stiffer and more independent of temperature. Shunting with a resistor and inductor introduces an electrical resonance, which can be optimally tuned to structural resonances in a manner analogous to a mechanical vibration arsorber. Techniques for analyzing systems which incorporate these shunting cases are presented and applied to a cantilevered beam experiment. The experimental results for both the resistive and resonant shunting circuits validate the shunted piezoelectric damping models.

Hagood, N. W.↗
NASA NTRS2001

An Implicit Characteristic Based Method for Electromagnetics

An implicit characteristic-based approach for numerical solution of Maxwell's time-dependent curl equations in flux conservative form is introduced. This method combines a characteristic based finite difference spatial approximation with an implicit lower-upper approximate factorization (LU/AF) time integration scheme. This approach is advantageous for three-dimensional applications because the characteristic differencing enables a two-factor approximate factorization that retains its unconditional stability in three space dimensions, and it does not require solution of tridiagonal systems. Results are given both for a Fourier analysis of stability, damping and dispersion properties, and for one-dimensional model problems involving propagation and scattering for free space and dielectric materials using both uniform and nonuniform grids. The explicit Finite Difference Time Domain Method (FDTD) algorithm is used as a convenient reference algorithm for comparison. The one-dimensional results indicate that for low frequency problems on a highly resolved uniform or nonuniform grid, this LU/AF algorithm can produce accurate solutions at Courant numbers significantly greater than one, with a corresponding improvement in efficiency for simulating a given period of time. This approach appears promising for development of dispersion optimized LU/AF schemes for three dimensional applications.

Beggs, John H.↗
NASA NTRS1993

On the delivery of planetesimals to a protoplanet in the solar nebula

A planetesimal moving in the solar nebula experiences an aerodynamic drag which causes its orbit to circularize and shrink. However resonant perturbations from a protoplanet interior to the planetesimal's orbit can counteract both the orbital decay and the damping of the eccentricity: the planetesimal can be captured into an orbital resonance and its eccentricity pumped up to an equilibrium value. The resonance trapping phenomenon is sensitive to several factors and we elaborate on these below. Orbital resonances form (partial) barriers to the transport of planetesimals into the feeding zone of the protoplanet. Here we report on a systematic study of the idealized system of a single protoplanet of mass much larger than the masses of the planetesimals, comparing relevant analytic estimates with numerical simulations. Our principal conclusion is that in a gas-rich environment, the bulk of the solid material delivered to the feeding zone of the protoplanet is in the form of bodies of small radius.

Malhotra, Renu↗
NASA NTRS2002

Development of a New Membrane Casting Apparatus for Studying Macrovoid Defects in Low-G

A new membrane-casting apparatus is developed for studying macrovoid defects in polymeric membranes made by the wet- and dry-casting process in low-gravity. Macrovoids are large (10-50 micron), open cavities interspersed among the smaller pores in the substructure under the gelled skin surface layer of the cast membrane. Although their occurrence is considered endemic to the wet- and dry-casting process since they can lead to compaction or skin rupture in the membrane process, recent studies suggest several useful applications such as transdermal and osmotic drug delivery systems, miniature bioreactors, etc. However, lack of knowledge about the macrovoid formation mechanism is an obstacle to further development of applications using them. An on-going debate is the role of the surface-tension-driven solutocapillary convection during macrovoid formation. The rapid growth of macrovoids within 1-5 seconds and the high polymer concentration in and near macrovoids make it difficult to explain the mechanism of macrovoid growth by diffusion alone, which is the widely accepted hypothesis proposed by Reuvers et al. The hypothesis advanced by our research group can explain this rapid growth via a mechanism that involves diffusion from the casting solution in the meta-stable region to the macrovoid enhanced by solutocapillary convection induced by the steep nonsolvent concentration gradient in the vicinity of the macrovoid. Since macrovoid growth is hypothesized to be the interplay of a solutocapillary-induced driving force counteracted by viscous drag and buoyancy, eliminate the latter provides a means for testing this hypothesis. Moreover, free convection mass transfer in the nonsolvent immersion bath used to cause phase-separation in membrane casting complicates developing a model for both the wet-casting process and macrovoid growth. The low-g environment minimizes gravitationally induced free convection thereby permitting a tractable solution to the ternary diffusion equations that characterize membrane formation. NASA's Parabolic Flight Research Aircraft provides a small window of low-g (approximately 25 s) that can be used to study macrovoid development in both wet- and dry-cast membranes if an appropriate casting apparatus is used. This casting apparatus should be able to cast the membrane in both low- and high-g in a manner so that essential one-dimensional mass transfer conditions are achieved to insure lateral uniformity in the membrane. The apparatus used in previous research on membrane casting in low-gravity was operated with the plunger driven mechanism. The spring-loaded plunger pushes the bottom block containing the polymer casting solution well directly under the absorbent chamber located in the upper stationary block. However, membranes made via this casting apparatus often displayed lateral nonuniformities that precluded obtaining quantitative information on the macrovoid growth process. Thus, it was necessary to determine the reason for these structural irregularities observed in the low-g casting apparatus. Both experimental as well as computer simulation studies of the low-g casting apparatus established that the impulsive action of the plunger caused the undesired structural nonuniformities. The simulation results showed that the width-to-depth aspect ratio of the shallow well that contains the casting solution in this apparatus was not an important factor in minimizing this problem. Even for a 40:1 (width : depth) aspect ratio, any convection induced by the horizontal motion of the interface of the casting solution will be damped out within 6.25x10(exp 4) seconds. However, the experimental studies revealed that the impulsive motion of the plunger caused a 'sloshing' of the casting solution that had to be eliminated. Therefore, the plungerdriven mechanism was changed to a cam-driven mechanism that did not cause any impulsive motion of the casting solution. Other refinements to this new membrane-casting apparatus include provision for removing the membranes from the casting wells in a less destructive manner. This was accomplished by using a slit geometry for the casting well that permitted disassembly for removal of the cast membrane. The materials used in the construction of this casting apparatus were chosen to insure wetting at the side walls and to maintain precise control of the thickness of the polymer solution in the casting well. An additional provision in this new casting apparatus is the ability to carry out both wet- as well as dry-casting. As such, this apparatus permitted the first studies of the wet-casting of polymeric membranes in low-g. Both wet- and dry-casting experiments on NASA's KC-135 research aircraft employing this new membrane-casting apparatus are scheduled in July 2002. The morphology of the resulting membranes will be characterized using an environmental scanning electron microscope (ESEM). The results of these low-g studies will be reported later.

Lee, Hanyong↗
NASA NTRS1987

A simplified computer solution for the flexibility matrix of contacting teeth for spiral bevel gears

A computer code, FLEXM, was developed to calculate the flexibility matrices of contacting teeth for spiral bevel gears using a simplified analysis based on the elementary beam theory for the deformation of gear and shaft. The simplified theory requires a computer time at least one order of magnitude less than that needed for the complete finite element method analysis reported earlier by H. Chao, and it is much easier to apply for different gear and shaft geometries. Results were obtained for a set of spiral bevel gears. The teeth deflections due to torsion, bending moment, shearing strain and axial force were found to be in the order 10(-5), 10(-6), 10(-7), and 10(-8) respectively. Thus, the torsional deformation was the most predominant factor. In the analysis of dynamic load, response frequencies were found to be larger when the mass or moment of inertia was smaller or the stiffness was larger. The change in damping coefficient had little influence on the resonance frequency, but has a marked influence on the dynamic load at the resonant frequencies.

Hsu, C. Y.↗