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NASA NTRS2012

Aeroelastic Modeling of Elastically Shaped Aircraft Concept via Wing Shaping Control for Drag Reduction

Lightweight aircraft design has received a considerable attention in recent years as a means for improving cruise efficiency. Reducing aircraft weight results in lower lift requirements which directly translate into lower drag, hence reduced engine thrust requirements during cruise. The use of lightweight materials such as advanced composite materials has been adopted by airframe manufacturers in current and future aircraft. Modern lightweight materials can provide less structural rigidity while maintaining load-carrying capacity. As structural flexibility increases, aeroelastic interactions with aerodynamic forces and moments become an increasingly important consideration in aircraft design and aerodynamic performance. Furthermore, aeroelastic interactions with flight dynamics can result in issues with vehicle stability and control. Abstract This paper describes a recent aeroelastic modeling effort for an elastically shaped aircraft concept (ESAC). The aircraft model is based on the rigid-body generic transport model (GTM) originally developed at NASA Langley Research Center. The ESAC distinguishes itself from the GTM in that it is equipped with highly flexible wing structures as a weight reduction design feature. More significantly, the wings are outfitted with a novel control effector concept called variable camber continuous trailing edge (VCCTE) flap system for active control of wing aeroelastic deflections to optimize the local angle of attack of wing sections for improved aerodynamic efficiency through cruise drag reduction and lift enhancement during take-off and landing. The VCCTE flap is a multi-functional and aerodynamically efficient device capable of achieving high lift-to-drag ratios. The flap system is comprised of three chordwise segments that form the variable camber feature of the flap and multiple spanwise segments that form a piecewise continuous trailing edge. By configuring the flap camber and trailing edge shape, drag reduction could be achieved. Moreover, some parts of the flap system can be made to have a high frequency response for roll control, gust load alleviation, and aeroservoelastic (ASE) modal suppression control. Abstract The aeroelastic model of the ESAC is based on one-dimensional structural dynamic theory that captures the aeroelastic deformation of a wing structure in a combined motion that involves flapwise bending, chordwise bending, and torsion. The model includes the effect of aircraft propulsion due to wing flexibility which causes the propulsive forces and moments to couple with the wing elastic motion. Engine mass is also accounted in the model. A fuel management model is developed to describe the wing mass change due to fuel usage in the main tank and wing tanks during cruise. Abstract The model computes both static and dynamic responses of the wing structures. The static aeroelastic deflections are used to estimate the effect of wing flexibility on induced drag and the potential drag reduction by the VCCTE flap system. A flutter analysis is conducted to estimate the flutter speed boundary. Gust load alleviation via adaptive control has been recently investigated to address flexibility of aircraft structures. A multi-objective flight control approach is presented for drag reduction control. The approach is based on an optimal control framework using a multi-objective cost function. Future studies will demonstrate the potential benefits of the approach.

Aeroelasticity↗
NASA NTRS1988

Equilibrium shape and location of a liquid drop acoustically positioned in a resonant rectangular chamber

The effect of a standing wave field in a rectangular chamber on the shape and location of an acoustically positioned drop or bubble is calculated. The sample deformation and equilibrium position are obtained from an analysis of the spherical harmonic projections of the total surface stress tensor. The method of calculation relies on the assumed condition that the sample is only slightly distorted from a spherical form. The equilibrium location of a levitated drop is combined with a formula introduced by Hasegawa (1979) to calcualte the ka dependence of the radiation force function. The present theory is valid for large as well as small ka values. Calculations in the small ka limit agree with previous theories and experimental results. Examples are presented for nonplane-wave modes as well as plane-wave rectangular modes.

Jackson, H. W.↗
NASA NTRS1999

Aeolian Sand Transport in the Planetary Context: Respective Roles of Aerodynamic and Bed-Dilatancy Thresholds

The traditional view of aeolian sand transport generally estimates flux from the perspective of aerodynamic forces creating the airborne grain population, although it has been recognized that "reptation" causes a significant part of the total airborne flux; reptation involves both ballistic injection of grains into the air stream by the impact of saltating grains as well as the "nudging" of surface grains into a creeping motion. Whilst aerodynamic forces may initiate sand motion, it is proposed here that within a fully-matured grain cloud, flux is actually governed by two thresholds: an aerodynamic threshold, and a bed-dilatancy threshold. It is the latter which controls the reptation population, and its significance increases proportionally with transport energy. Because we only have experience with terrestrial sand transport, extrapolations of aeolian theory to Mars and Venus have adjusted only the aerodynamic factor, taking gravitational forces and atmospheric density as the prime variables in the aerodynamic equations, but neglecting reptation. The basis for our perspective on the importance of reptation and bed dilatancy is a set of experiments that were designed to simulate sand transport across the surface of a martian dune. Using a modified sporting crossbow in which a sand-impelling sabot replaced the bolt-firing mechanism, individual grains of sand were fired at loose sand targets with glancing angles typical of saltation impact; grains were projected at about 80 m/s to simulate velocities commensurate with those predicted for extreme martian aeolian conditions. The sabot impelling method permitted study of individual impacts without the masking effect of bed mobilization encountered in wind-tunnel studies. At these martian impact velocities, grains produced small craters formed by the ejection of several hundred grains from the bed. Unexpectedly, the craters were not elongated, despite glancing impact; the craters were very close to circular in planform. High-speed photography showed them to grow in both diameter and depth after the impactor had ricochetted from the crater site. The delayed response of the bed was "explosive" in nature, and created a miniature ejecta curtain spreading upward and outward for many centimeters for impact of 100-300 micron-diameter grains into similar material. Elastic energy deposited in the bed by the impacting grain creates a subsurface stress regime or "quasi-Boussinesq" compression field. Elastic recovery of the bed occurs by dilatancy; shear stresses suddenly convert the grains from closed to open packing, and grains are consequently able to eject themselves forcefully from the impact site. Random jostling of the grains causes radial homogenization of stress vectors and a resulting circular crater. There is a great temptation to draw parallels with cratering produced by meteorite impacts, but a rigorous search for common modelling ground between the two phenomena has not been conducted at this time. For every impact of an aerodynamically energized grain, there are several hundred grains ejected into the wind for the high-energy transport that might occur on Mars. Many of these grains will themselves become subject to the boundary layer's aerodynamic lift forces (their motion will not immediately die and add to the creep population), and these grains will become indistinguishable from those lifted entirely by aerodynamic forces. As each grain impacts the bed, it will eject even more grains into the flow. A cascading effect will take place, but because it must be finite in its growth, damping will occur as the number of grains set in motion causes mid-air collisions that prevent much of the impact energy from reaching the surface of the bed -thus creating a dynamic equilibrium in a high-density saltation cloud. It is apparent that for a given impact energy, the stress field permits a smaller volume of grains to convert to open packing as the size of the bed grains increases, or as the energy of the "percussive" grain decreases (by decrease in velocity or mass). Thus, the mass of the "repercussive" grain population that is ejected from the impact site becomes a function of the scale of the stress field in relation to the scale of the bed material (self-similarity being applicable if both bed size and energy are simultaneously adjusted). In other words, in a very high energy aeolian system where an aerodynamically raised grain can ballistically raise many more grains, the amount of material lifted into the wind becomes largely a function of a dilatancy threshold. If this threshold is exceeded, grains are repercussively injected into the saltation cloud. The "dilatancy threshold" may be defined in terms of the saltation percussive force required to convert the bed, through elastic response, from a closed to an open packing system. If open packing cannot be created, the grains cannot escape from the impact site, even though the elastic deformation and percussive force may be able to reorganize the grains with respect to one another. As the crossbow experiments showed, for an ever-increasing bed grain size, a point is reached when no material can be moved because the energy of the percussive grain is insufficient to dilate the relatively coarse bed. Although this seems to be stating the obvious -- that too little energy will not cause the bed to splash -- the consequences of exceeding the "splash threshold" by dilatancy are not so obvious for high-energy aeolian transport. It is noted that the force required to elastically dilate the bed has to overcome Coulombic grain attractions such as dipole-dipole coupling, dielectric, monopole, contact-induced dipole attractions, van der Waals forces, molecular monolayer capillary forces, as well as the mechanical interlocking frictional resistance of the grains. On Mars, it is predicted that the dilatancy threshold may be the prime control of grain flux. Additional information is contained in the original.

Marshall, J. R.↗
NASA NTRS2012

The Exact Solution for Linear Thermoelastic Axisymmetric Deformations of Generally Laminated Circular Cylindrical Shells

A detailed exact solution is presented for laminated-composite circular cylinders with general wall construction and that undergo axisymmetric deformations. The overall solution is formulated in a general, systematic way and is based on the solution of a single fourth-order, nonhomogeneous ordinary differential equation with constant coefficients in which the radial displacement is the dependent variable. Moreover, the effects of general anisotropy are included and positive-definiteness of the strain energy is used to define uniquely the form of the basis functions spanning the solution space of the ordinary differential equation. Loading conditions are considered that include axisymmetric edge loads, surface tractions, and temperature fields. Likewise, all possible axisymmetric boundary conditions are considered. Results are presented for five examples that demonstrate a wide range of behavior for specially orthotropic and fully anisotropic cylinders.

Nemeth, Michael P.↗
NASA NTRS1992

A dynamic study of fragmentation and energy loss during high velocity impact

Research conducted under this contract can be divided into two main areas: hypervelocity (in the range up to 7 km/s) and high velocity (less than 1 km/s). Work in the former was performed at NASA-MSFC using the Light Gas Gun Facility. The lower velocity studies were conducted at Auburn University using the ballistic gun. The emphasis of the project was on the hypervelocity phenomenon especially in the characterization of the debris cloud formed by the primary impact events. Special devices were made to determine the angular distributions of momentum and energy of the debris cloud as a function of impact conditions. After several iteration processes, it was decided to concentrate on the momentum effort. Prototype devices were designed, fabricated, and tested. These devices were based on the conservation of momentum. Distributions of the debris cloud formed were measured by determining the amount of momentum transferred from the debris cloud to strategically placed pendulum measurement devices. The motion of the pendula was monitored using itegrated opto-interrupters. The distribution of momentum in the debris cloud was found to be a strong function of the impact condition. Small projectiles at high velocities were observed to produce finely dispersed debris whereas large projectiles generated discrete particles in the debris. Results also show that the momentum in the forward direction was enhanced due to the impact. This phenomenon of momentum multiplication was also observed in other studies and in computer simulations. It was initially planned to determine the energy distribution using deformation energy in a rod with strain gauges. Results from preliminary studies show that this technique is acceptable but too tedious. A new technique was explored based on measuring the heating effect of the debris cloud using an IR camera. The feasibility and sensitivity was established at Auburn University. This type of energy distribution measurement method can easily be adapted to the gas gun facility at MSFC. The objective of the lower velocity studies at Auburn was to simulate the damage produced in advanced materials by the lower energy debris cloud.

Zee, Ralph H.↗
NASA NTRS1995

Sensitivity Analysis of Wing Aeroelastic Responses

Design for prevention of aeroelastic instability (that is, the critical speeds leading to aeroelastic instability lie outside the operating range) is an integral part of the wing design process. Availability of the sensitivity derivatives of the various critical speeds with respect to shape parameters of the wing could be very useful to a designer in the initial design phase, when several design changes are made and the shape of the final configuration is not yet frozen. These derivatives are also indispensable for a gradient-based optimization with aeroelastic constraints. In this study, flutter characteristic of a typical section in subsonic compressible flow is examined using a state-space unsteady aerodynamic representation. The sensitivity of the flutter speed of the typical section with respect to its mass and stiffness parameters, namely, mass ratio, static unbalance, radius of gyration, bending frequency, and torsional frequency is calculated analytically. A strip theory formulation is newly developed to represent the unsteady aerodynamic forces on a wing. This is coupled with an equivalent plate structural model and solved as an eigenvalue problem to determine the critical speed of the wing. Flutter analysis of the wing is also carried out using a lifting-surface subsonic kernel function aerodynamic theory (FAST) and an equivalent plate structural model. Finite element modeling of the wing is done using NASTRAN so that wing structures made of spars and ribs and top and bottom wing skins could be analyzed. The free vibration modes of the wing obtained from NASTRAN are input into FAST to compute the flutter speed. An equivalent plate model which incorporates first-order shear deformation theory is then examined so it can be used to model thick wings, where shear deformations are important. The sensitivity of natural frequencies to changes in shape parameters is obtained using ADIFOR. A simple optimization effort is made towards obtaining a minimum weight design of the wing, subject to flutter constraints, lift requirement constraints for level flight and side constraints on the planform parameters of the wing using the IMSL subroutine NCONG, which uses successive quadratic programming.

Issac, Jason Cherian↗
NASA NTRS2005

Improved Gear Shapes for Face Worm Gear Drives

Shapes different from the traditional ones have been proposed for face worm gears and for conical and cylindrical worms that mesh with them. The proposed shapes are based on the concept of generating a face worm gear surface by use of a tilted head cutter instead of by the traditional use of a hob. (As used here, head cutter is also meant to signify, alternatively, a head grinding tool.) The gear-surface-generation equipment would be similar to that used for generation of spiral bevel and hypoid gears. In comparison with the corresponding traditional hob, a tilted head cutter according to the proposal would be larger, could be fabricated with greater precision, and would enable the generation of gear surfaces with greater precision and greater productivity. A face worm gear would be generated (see figure) by use of a tilted head cutter, the blades or grinding surfaces of which would have straight-line profiles. The tilt of the head cutter would prevent interference with teeth adjacent to the groove being cut or ground. A worm to mesh with the face worm gear would be generated by use of a tilted head cutter mounted on the cradle of a generating machine. The blades or grinding surfaces of the head cutter would have a parabolic profile and would deviate from the straight-line profiles of the head cutter for the face worm gear. The shortest distance between the worm and the cradle would follow a parabolic function during the cycle of meshing in the generating process to provide a parabolic function of transmission errors to the gear drive. The small mismatch between the profiles of the face-worm-gear and worm head cutters would make it possible to localize the bearing contact in the worm gear drive. The parabolic function of transmission errors could absorb discontinuous linear functions of transmission errors caused by errors of alignment; this could afford a significant benefit, in that such errors are main sources of noise and vibration in gear drives. The main advantage of using tilted head cutters is that cutting speeds are independent of the shape-generation processes, making it possible to choose cutting speeds that are optimum with respect to requirements to minimize temperatures and deformations during fabrication and improve the quality of finished parts.

Litvin, Faydor L.↗
NASA NTRS2012

General MACOS Interface for Modeling and Analysis for Controlled Optical Systems

The General MACOS Interface (GMI) for Modeling and Analysis for Controlled Optical Systems (MACOS) enables the use of MATLAB as a front-end for JPL s critical optical modeling package, MACOS. MACOS is JPL s in-house optical modeling software, which has proven to be a superb tool for advanced systems engineering of optical systems. GMI, coupled with MACOS, allows for seamless interfacing with modeling tools from other disciplines to make possible integration of dynamics, structures, and thermal models with the addition of control systems for deformable optics and other actuated optics. This software package is designed as a tool for analysts to quickly and easily use MACOS without needing to be an expert at programming MACOS. The strength of MACOS is its ability to interface with various modeling/development platforms, allowing evaluation of system performance with thermal, mechanical, and optical modeling parameter variations. GMI provides an improved means for accessing selected key MACOS functionalities. The main objective of GMI is to marry the vast mathematical and graphical capabilities of MATLAB with the powerful optical analysis engine of MACOS, thereby providing a useful tool to anyone who can program in MATLAB. GMI also improves modeling efficiency by eliminating the need to write an interface function for each task/project, reducing error sources, speeding up user/modeling tasks, and making MACOS well suited for fast prototyping.

Sigrist, Norbert↗
NASA NTRS1991

Compression studies on particulate composites of ternary Al-Ti-Fe, and quaternary Al-Ti-Fe-Nb and Al-Ti-Fe-Mn L1(2) compounds

Compression studies were conducted on monolithic and TiB2 particulate-reinforced composites of Al22Fe3Ti8, both with and without minor quaternary alloying additions (2 at. pct Nb and 2 at. pct Mn) as a function of temperature and as a function of strain rate at high temperature. The volume fraction of reinforcement was varied between 0 and 20 percent. The particulate reinforcements were found to be effective in increasing ambient- and warm-temperature strength; at high temperatures, the monolithic material is stronger than the composites, although the composites are superior at slow strain rates. The microstructures of the monolithic and composite specimens were examined before and after deformation to explain these observations.

Kumar, K. S.↗
NASA NTRS1989

Optical joint correlation using the deformable mirror device

An experimental investigation of the Deformable Mirror Device (DMD) developed by Texas Instruments at Dallas for use in object identification was completed. The DMD was tested as a joint correlator. The DMD was used as a spatial light modulator on which the squared modulus of the Fourier transform of test object pairs was written. The squared modulus was phase encoded on the DMD after it had been thresholded and rewritten as a binary phase function. The thresholding was found to produce a sharp peak in the autocorrelation when the test objects were matched and no significant peak in the case of distinct objects. It was concluded that the use of the DMD as a joint correlator looks promising and further studies should be carried out.

Knopp, Jerome↗
NASA NTRS2001

Finite Element Methods for real-time Haptic Feedback of Soft-Tissue Models in Virtual Reality Simulators

We have applied the linear elastic finite element method to compute haptic force feedback and domain deformations of soft tissue models for use in virtual reality simulators. Our results show that, for virtual object models of high-resolution 3D data (>10,000 nodes), haptic real time computations (>500 Hz) are not currently possible using traditional methods. Current research efforts are focused in the following areas: 1) efficient implementation of fully adaptive multi-resolution methods and 2) multi-resolution methods with specialized basis functions to capture the singularity at the haptic interface (point loading). To achieve real time computations, we propose parallel processing of a Jacobi preconditioned conjugate gradient method applied to a reduced system of equations resulting from surface domain decomposition. This can effectively be achieved using reconfigurable computing systems such as field programmable gate arrays (FPGA), thereby providing a flexible solution that allows for new FPGA implementations as improved algorithms become available. The resulting soft tissue simulation system would meet NASA Virtual Glovebox requirements and, at the same time, provide a generalized simulation engine for any immersive environment application, such as biomedical/surgical procedures or interactive scientific applications.

Frank, Andreas O.↗
NASA NTRS1972

Detection of fiber cracking by acoustic emission.

A theoretical model is presented that relates acoustic emission to fiber cracking which occurs during a rising load tension test on a fiber reinforced composite. The percentage of broken fibers in an Al3Ni fiber reinforced aluminum was measured as a function of tensile strain by optical inspection of the polished surface of strained specimens. This information was used in conjunction with the proposed model to predict the acoustic emission response of the composite material. These predictions were compared with experimental observations, and a good agreement was obtained between the two sets of results. These results indicate that it is possible to relate acoustic emission quantitatively to the micromechanics of the deformation processes occurring within fiber reinforced composites, thereby demonstrating the applicability of acoustic emission to materials studies and also to nondestructive evaluation of the integrity of composite materials.

Harris, D. O.↗
NASA NTRS1993

The production of premixed flame surface area in turbulent shear flow

In the present work, we use three-dimensional Direct Numerical Simulation (DNS) of premixed flames in turbulent shear flow to characterize the effect of a mean shear motion on flame surface production. The shear is uniform in the unburnt gas, and simulations are performed for different values of the mean shear rate, S. The data base is then used to estimate and compare the different terms appearing in the Sigma-equation as a function of S. The analysis gives in particular the relative weights f the turbulent flow and mean flow components, a(sub T) and A(sub T), of the flame surface production term. This comparison indicates whether the dominant effects of a mean flow velocity gradient on flame surface area are implicit and scale with the modified turbulent flow parameters, kappa and epsilon, or explicit and scale directly with the rate of deformation.

Trouve, A.↗
NASA NTRS2006

Airframe Noise Prediction by Acoustic Analogy: Revisited

The present work follows a recent survey of airframe noise prediction methodologies. In that survey, Lighthill s acoustic analogy was identified as the most prominent analytical basis for current approaches to airframe noise research. Within this approach, a problem is typically modeled with the Ffowcs Williams and Hawkings (FW-H) equation, for which a geometry-independent solution is obtained by means of the use of the free-space Green function (FSGF). Nonetheless, the aeroacoustic literature would suggest some interest in the use of tailored or exact Green s function (EGF) for aerodynamic noise problems involving solid boundaries, in particular, for trailing edge (TE) noise. A study of possible applications of EGF for prediction of broadband noise from turbulent flow over an airfoil surface and the TE is, therefore, the primary topic of the present work. Typically, the applications of EGF in the literature have been limited to TE noise prediction at low Mach numbers assuming that the normal derivative of the pressure vanishes on the airfoil surface. To extend the application of EGF to higher Mach numbers, the uniqueness of the solution of the wave equation when either the Dirichlet or the Neumann boundary condition (BC) is specified on a deformable surface in motion. The solution of Lighthill s equation with either the Dirichlet or the Neumann BC is given for such a surface using EGFs. These solutions involve both surface and volume integrals just like the solution of FW-H equation using FSGF. Insight drawn from this analysis is evoked to discuss the potential application of EGF to broadband noise prediction. It appears that the use of a EGF offers distinct advantages for predicting TE noise of an airfoil when the normal pressure gradient vanishes on the airfoil surface. It is argued that such an approach may also apply to an airfoil in motion. However, for the prediction of broadband noise not directly associated with a trailing edge, the use of EGF does not appear to offer any advantages over the use of FSGF at the present stage of development. It is suggested here that the applications of EGF for airframe noise analysis be continued. As an example pertinent to airframe noise prediction, the Fast Scattering Code of NASA Langley is utilized to obtain the EGF numerically on the surface of a three dimensional wing with a flap and leading edge slat in uniform rectilinear motion. The interpretation and use of these numerical Green functions are then discussed.

Farassat, F.↗
NASA NTRS1989

Joint-transform correlator systems using deformable-mirror spatial light modulators

Two joint-transform correlator systems based on deformable-mirror-device (DMD) spatial light modulators and CCD video cameras are described. The system designs provide for real-time correlation between video-formatted input scenes and stored reference images. The DMD light modulators are used exclusively for inputting data in the optical stages of these systems, and the CCD cameras are used to provide the critical square-law detection required in joint-transform achitectures. Experimental verification of the functionality of these DMD/CCD joint-transform systems is also presented.

Florence, James M.↗
NASA NTRS1960

Criteria for Yielding of Dispersion-Strengthened Alloys

A dislocation model is presented in order to account for the yield behavior of alloys with a finely dispersed second-phase. The criteria for yielding used in the model, is that appreciable yielding occurs in these alloys when the shear stress due to piled-up groups of dislocations is sufficient to fracture or plastically deform the dispersed second-phase particles, relieving the back stress on the dislocation sources. Equations derived on the basis of this model, predict that the yield stress of the alloys varies as the reciprocal square root of the mean free path between dispersed particles. Experimental data is presented for several SAP-Type alloys, precipitation-hardened alloys and steels which are in good agreement with the yield strength variation as a function of dispersion spacing predicted by this theoretical treatment.

Ansell, G. S.↗
NASA NTRS1996

Computational Methods for Frictional Contact With Applications to the Space Shuttle Orbiter Nose-Gear Tire

A computational procedure is presented for the solution of frictional contact problems for aircraft tires. A Space Shuttle nose-gear tire is modeled using a two-dimensional laminated anisotropic shell theory which includes the effects of variations in material and geometric parameters, transverse-shear deformation, and geometric nonlinearities. Contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of the stress resultants, the generalized displacements, and the Lagrange multipliers associated with both contact and friction conditions. The contact-friction algorithm is based on a modified Coulomb friction law. A modified two-field, mixed-variational principle is used to obtain elemental arrays. This modification consists of augmenting the functional of that principle by two terms: the Lagrange multiplier vector associated with normal and tangential node contact-load intensities and a regularization term that is quadratic in the Lagrange multiplier vector. These capabilities and computational features are incorporated into an in-house computer code. Experimental measurements were taken to define the response of the Space Shuttle nose-gear tire to inflation-pressure loads and to inflation-pressure loads combined with normal static loads against a rigid flat plate. These experimental results describe the meridional growth of the tire cross section caused by inflation loading, the static load-deflection characteristics of the tire, the geometry of the tire footprint under static loading conditions, and the normal and tangential load-intensity distributions in the tire footprint for the various static vertical loading conditions. Numerical results were obtained for the Space Shuttle nose-gear tire subjected to inflation pressure loads and combined inflation pressure and contact loads against a rigid flat plate. The experimental measurements and the numerical results are compared.

Tanner, John A.↗
NASA NTRS1996

Computational methods for frictional contact with applications to the Space Shuttle orbiter nose-gear tire: Comparisons of experimental measurements and analytical predictions

A computational procedure is presented for the solution of frictional contact problems for aircraft tires. A Space Shuttle nose-gear tire is modeled using a two-dimensional laminated anisotropic shell theory which includes the effects of variations in material and geometric parameters, transverse-shear deformation, and geometric nonlinearities. Contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with the fundamental unknowns consisting of the stress resultants, the generalized displacements, and the Lagrange multipliers associated with both contact and friction conditions. The contact-friction algorithm is based on a modified Coulomb friction law. A modified two-field, mixed-variational principle is used to obtain elemental arrays. This modification consists of augmenting the functional of that principle by two terms: the Lagrange multiplier vector associated with normal and tangential node contact-load intensities and a regularization term that is quadratic in the Lagrange multiplier vector. These capabilities and computational features are incorporated into an in-house computer code. Experimental measurements were taken to define the response of the Space Shuttle nose-gear tire to inflation-pressure loads and to inflation-pressure loads combined with normal static loads against a rigid flat plate. These experimental results describe the meridional growth of the tire cross section caused by inflation loading, the static load-deflection characteristics of the tire, the geometry of the tire footprint under static loading conditions, and the normal and tangential load-intensity distributions in the tire footprint for the various static vertical-loading conditions. Numerical results were obtained for the Space Shuttle nose-gear tire subjected to inflation pressure loads and combined inflation pressure and contact loads against a rigid flat plate. The experimental measurements and the numerical results are compared.

Tanner, John A.↗