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184 APPENDIX F Tire Models In order to predict landing gear loads effectively, computer steady-state depth in the material, producing a rut of the cor- models were required for both the arrestor materials and the rect depth and width for that loading condition. tires of the three subject aircraft. This appendix discusses the For a tire model to create such a rut, it would have to repli- tire model development, while Chapters 9 through 12 discuss cate two critical behaviors: the modeling of the candidate arrestor systems. 1. Correct deformation shape under loading, and 2. Correct interface loading (ground pressure) against the F.1. Subject Tires Modeled arrestor material. Manufacturer references give the tire sizes, pressures, and load limits for the three aircraft of interest (60). Table F-1 lists After a review of available tire modeling methods and the main- and nose-gear tires for each aircraft, and Table F-2 mechanical phenomena, the tire model priorities were set per provides specifications for the tires. The tire designations give Table F-3. three numbers, referring to the diameter, width, and ply rat- ing, respectively. The leading "H" appearing in some tires F.2.1. Tire Dimensions indicates a bias-ply tire designed for high deformations. Goodyear supplied load-deflection curves for the tires listed, The tire models needed to accurately represent the gross which were used for later tire calibration and validation. dimensions of the actual aircraft tires. Ensuring such a match was actually more complicated than it appeared. During infla- tion, the tire model stretched, which altered its original gross F.2. Tire Modeling Approach dimensions. Therefore, the dimensional criterion applied to Tire modeling can be a highly detailed process, involving the match of the inflated model tire with the inflated actual tire. discrete representation of the treads, carcass material, reinforc- ing fiber layers, the tirerim interface, inflation pressures, and F.2.2. Ground Pressure so on. Modeling these facets is required for accurate prediction of stresses within the tire, oscillatory behavior, tread wear, and The ground pressure that the tire produced strongly so on. However, for arrestor bed modeling, such detail did not depended on its inflation pressure and governed the penetra- necessarily offer added value. A number of these facets would tion of the tire into the arrestor material. Provided that the gross constitute higher order (and lower importance) effects when dimensions were correct and the load-deflection performance applied to the simulation of crushable material interactions. matched, the model would produce a ground pressure that A tire model was needed that could produce a realistic rut would be sufficiently accurate for the soft-ground interaction. through an arrestor bed. A realistic rut would feature the cor- rect penetration depth and cross-sectional shape. Since the F.2.3. Load-Deflection Performance energy dissipation is largely based on the crushed volume of material, a rut of the correct dimensions would tend to pro- Goodyear provided load versus deflection curves, both in duce the correct energy dissipation and hence the correct quasi-static and dynamic loading regimes. The load-deflection drag load. Under a static vertical load with steady forward performance is analogous to a spring deflection curve. Typi- motion, this means that the tire would deform and settle to a cal efficiencies for a tire are slightly less than an ideal spring,

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185 Table F-1. Tires for subject aircraft. material (61). The smeared properties represented the com- bination of all materials in the actual tire: tread, bulk rubber, Aircraft Tire Tire Designation reinforcement plies, and so on. This approach was attractive CRJ-200 Main Tire H29x9.0-15 due to its simplicity, and it produced correct deformation Auxiliary Tire 18x4.4 within the loading range of interest. It was also practical since B737-800 Main Tire H44.5x16.5-21 little detail was known about the inner construction of the Auxiliary Tire 27x7.75R15 actual tires: proprietary manufacturer data would be required to effectively model the various components individually. B747-400 Main Tire H49x19.0-22 The tire model used two-dimensional shell elements for the Auxiliary Tire H49x19.0-22 carcass in order to obtain the best computational efficiency. For a full tire, the carcass contained 4,608 shell elements in 2.5-degree increments; usually a quarter- or half-symmetry at nominally 45 to 47%, a quasi-linear behavior that applies version was used, with the element count reduced accordingly to the bulk of the loading domain (14). This linear behavior (Figure F-2). ceases during very high "bottoming" loading of the tire, when A constant pressure was applied to the inner surface of the the deflection is high enough to allow the wheel rim to "bot- tire to simulate pneumatic pressure. When loaded vertically, tom out." Bottoming loads are most likely during hard land- the tire models deflected realistically, forming an increasingly ing impacts, but do not occur in an aircraft in a steady roll large flat contact area with the ground (Figure F-3). with properly inflated tires. As such, replication of the tire performance during bottoming was deemed non-essential to this research. The domain of concern was limited to 80% of F.3.2. Model Calibration bottoming loads for each tire (Figure F-1). In using a smeared approach, the material properties of the tire carcass required calibration, such that the load-deflection F.3. Model Construction behavior of the model matched that of the actual tire. Using LS-OPT, this calibration was accomplished in a systematic F.3.1. Modeling Methods fashion. Two optimization criteria were defined: Finite element tire models for each aircraft were con- structed in LS-DYNA. The tire models were developed using 1. Match the load-deflection curve for the actual tire as closely a simple tire carcass with a smeared-property orthotropic as possible, and Table F-2. Data for subject aircraft main-gear tires (60). Tire Designation 18x4.4 27x7.75R15 H29x9.0-15 H44.5x16.5-21 H49x19-22 Units Rated Speed 210 225 210 225 235 mph Rated Load 4,350 9,650 14,500 44,700 56,600 lbf Rated Inflation Pressure 225 200 196 214 205 psi Maximum Bottoming Load 13,000 28,950 39,200 121,000 152,800 lbf Table F-3. Aspects of tire dynamics for inclusion and exclusion. Aspects to Replicate Aspects to Neglect Tire dimensions Internal tire stresses Ground pressure Heat generation Load versus vertical deflection performance Tread features Correct mode of deformation Ground traction and slip Computational efficiency Lateral loading deformations High frequency effects, noise, vibration

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186 Vertical Bottoming Load Bottoming Load Linear Spring Performance Domain of Matching for Model: 0% to 80% of Bottoming Load Energy Absorbed in Compression ~47% Efficiency Vertical Deflection Figure F-1. Tire load curve and modeling domain of interest. 2. Match the inflated dimensions for the actual tire as closely Many simulations were conducted in batches over multiple as possible. iterations. Using a sequential response surface methodology (SRSM), the design parameters were gradually narrowed until The load-deflection objective was defined by the mean- LS-OPT had determined an optimum set of material proper- squared-error (MSE) between the load curves of the simulated ties (62). These best-case properties produced the closest over- and actual tires. The inflated dimensions were compared by all match between the model and the actual tire performance. measuring the tire just after the inflation process completed. Table F-4 summarizes the tire calibration results. As shown in the lower half of the table, nearly all the criteria were met to within 2% of the objective values. The RMSE values indicate how closely the load-deflection curves match the real tires (Fig- ure F-4). The inflated dimension error represents how closely the final tire diameter matched the actual tire. Practically speak- ing, these values indicate that the tires have the correct shape and respond with the correct loading as the tire is compressed. F.4. Summary of Tire Model Development The selected tire modeling approach struck a balance between accuracy and efficiency. The orthotropic smeared- property method neglected details required for nuanced tire design models, but it still replicated the overall tire deformation and loading pertinent to arrestor simulation. When compared with the prior state-of-the-art in arrestor modeling--the radial spring tire model--the FEM approach adopted herein offers an increase in fidelity. This three- dimensional model does not depend on ellipsoid contact sur- face assumptions and includes lateral tire bulging. The five tires developed in this task replicate the main- and nose-gear tires of the three subject aircraft. Each mimics the actual tire performance well, typically with less than 2% error for inflated dimensions and load-deflection behavior. The iterative optimization process using LS-OPT made this accu- Figure F-2. 44.5-in. tire model. rate calibration possible.

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187 Figure F-3. Quarter symmetry 44.5-in. tire model undergoing vertical deflection. Table F-4. Summary of tire calibration. Tire 18x4.4 27x7.75R15 H29x9- H44.5x16.5- H49x19- 15 21 22 Simulation RSM Type Quadratic Quadratic Quadratic Linear Linear Parameters Open Variables 3 3 2 3 3 Simulations Per Iteration 16 16 10 7 7 Number of Iterations (Pre Final) 10 20 4 5 7 Total Simulations 161 321 41 36 50 Quality of RMSE for Load-Deflection 1.6% 7.4% 2.0% 2.0% 1.7% Optimized Curve Match Design Error for Targeted Inflation 2.65% 1.1% 2.2% 1.9% 0.08% Dimensions History comparison for MSE (Experiment 8.1) A Target (TireLCF) B B Computed (Force_Dist) Normalized Load BA B A B A AB Normalized Vertical Deflection Figure F-4. Sample load-deflection curve comparison to 80% of bottoming load (units omitted).