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

AbstractA computational procedure is presented for the solution of frictional contactproblems for aircraft tires. A Space Shuttle nose-gear tire is modeled using a two-dimensional laminated anisotropic shell theory which includes the effects of varia-tions in material and geometric parameters, transverse-shear deformation, andgeometric nonlinearities. Contact conditions are incorporated into the formulation byusing a perturbed Lagrangian approach with the fundamental unknowns consisting ofthe stress resultants, the generalized displacements, and the Lagrange multipliersassociated with both contact and friction conditions. The contact-friction algorithm isbased on a modified Coulomb friction law. A modified two-field, mixed-variationalprinciple is used to obtain elemental arrays. This modification consists of augmentingthe functional of that principle by two terms: the Lagrange multiplier vector associ-ated with normal and tangential node contact-load intensities and a regularizationterm that is quadratic in the Lagrange multiplier vector. These capabilities and com-putational features are incorporated into an in-house computer code. Experimentalmeasurements were taken to define the response of the Space Shuttle nose-gear tire toinflation-pressure loads and to combined inflation-pressure loads and static normalloads against a rigid flat plate. These experimental results describe the meridionalgrowth of the tire cross section caused by inflation loading, the static load-deflectioncharacteristics of the tire, the geometry of the tire footprint under static loading con-ditions, and the normal and tangential load-intensity distributions in the tire footprintfor the various static vertical-loading conditions. Numerical results were obtained forthe Space Shuttle nose-gear tire subjected to inflation-pressure loads and combinedinflation-pressure and contact loads against a rigid flat plate. The experimental mea-surements and the numerical results are compared.IntroductionContact-friction problems are inherently nonlinearand path dependent. Nonlinearity occurs partly becauseboth the contact area and the contact-load intensities arenot known beforehand and vary during the loadinghistory. Path dependency is a result of the nonconserva-tive (irreversible dissipative) character of the frictionalforces.A review of static contact problems presented in ref-erence 1, which includes a bibliography of approxi-mately 700 papers, points out that contact problems areimportant to thermomechanical stress analyses, fracturemechanics, mechanical problems involving elastic foun-dations, the mechanics of joints, geomechanics, and tires.Contact problems occupy a position of specialimportance in aircraft tire mechanics because the contactzone is where the forces are generated to support, guide,and maneuver the airplane. Distributions of contact loadsand frictional forces define the moments and shears thatare applied to the landing gear system (ref. 2). Underrolling conditions, the distribution of sliding velocitieswithin the tire footprint combined with the frictionalforces developed by the tire defines the rate of energydissipation associated with the loading conditions andprovides a measure of tire wear (refs. 3 and 4). In thecase of the Space Shuttle orbiter, this wear mechanism isstrong enough to cause tire failures during individuallanding operations (refs. 5 and 6). Therefore, an under-standing of these tire friction forces and the resulting slipvelocities is critical to the design of aircraft tires for thenext generation of high-performance aircraft, such as theNational Aero-Space Plane and the High-Speed CivilTransport.Modeling contact phenomena in the tire footprint is aformidable task partly because of difficulty of modelingtire response. Distribution of tractions and the footprintgeometry are both functions of the normal, frictional, andinflation tire loads. Moreover, the complex mechanismsof dynamic friction, which allow the tire to develop thenecessary steering and braking forces for aircraft controlduring ground operations, are not fully understood(ref. 7). The tire analyst thus is forced to choose amongseveral friction theories. When the tire contact problemincludes frictional effects, the solution becomes pathdependent and a unique solution is not guaranteed.The aircraft tire is a composite structure of rubberand textile constituents that exhibit anisotropic and non-homogeneous material properties. Normal tire operatingconditions create loads that can produce large deforma-tions. Elevated operating temperatures from the com-bined effects of material hysteresis and frictional heatingcan cause variations in the material characteristics of the