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Constant Vertical Load Settled Steady-State:
Acceleration/Spin-Up Constant Forward Velocity Take Measurements
Figure 12-6. Sequencing method for depth-varying simulations.
permutations with various bed depths, strengths, and strength the 0.8 factor became the optimization criterion; an optimal
gradients. It sequentially executed the simulations and extracted design would achieve a main-gear drag loading factor of 0.8
the load data from them. Generally, the batches were conducted for both aircraft.
with one iteration of 50 simulations. All nose-wheel loading was neglected, although in practice
it typically limits the arrestor bed design. Since the results of
the analysis would not be integrated with the APC, it was fea-
12.4. Metamodel Analysis
sible to make this simplification.
12.4.1. Metamodel Method
The output from the batch simulations was extracted and 12.4.3. Metamodel Interpretation
assembled automatically by LS-OPT, where metamodels were
A visual review of the metamodel proved informative for
constructed for the drag and vertical load forces. Metamodel-
comparing the material performance. Because this optimiza-
ing is analogous to fitting a curve through experimental data
tion study had two simultaneous constraints, one for each
except that it is applied to multi-dimensional data sets. These
tire, the surface plots show two constraint curves and three
data sets were four-dimensional, including depth, strength,
shaded regions. As a key for understanding the other plots in
strength gradient, and load (either vertical or drag). The meta-
this section, Figure 12-7 gives an example of such a surface.
models were RBF networks, which can effectively capture
Two curves are drawn across the surface as boundaries to
non-linear behaviors including multiple concavity changes
across the data set. the shaded regions. One curve indicates the constraint
boundary for the 29.0-in. tire. The other curve indicates the
constraint boundary for the 44.5-in. tire. For each case, the
12.4.2. Optimization Criteria curve indicates the ideal case of a 0.8 strut load factor.
and Constraints The darkest region violates the constraints for both tires
The LS-OPT simulation sought to maximize the decelera- (strut overload). The two medium-shaded regions violate the
tion for both aircraft simultaneously, assuming that the main constraints for one tire or the other. The light-shaded region
gear were responsible for the deceleration. Therefore, the is acceptable for both constraints.
drag load on both tires was maximized. Where the region boundary curves intersect, a dual-case
Landing gear loading limits constrained the optimization. optimum occurs, providing an ideal load for both aircraft
From the tire drag load, multiplied by two tires per strut, the struts. The strut load factor would be 0.8 for both aircraft in
overall rearward strut loading was calculated. This loading this case. If the material was designed with the depth, strength,
was then limited to be no greater than acceptable horizontal and strength gradient from this intersection point, the struts
loadings as defined by the FAR. would be ideally loaded for maximum deceleration.
The loading requirements of the FAR define two types of
criteria: (1) limit, which the gear should withstand without 12.4.4. Linear and Quadratic
suffering damage, and (2) ultimate, which the gear should Profile Comparison
withstand without collapsing. For the evaluation, the limit
strength criterion was used. FAR Section 25.493 requires hor- For either the linear or quadratic depth-varying material,
izontal main-gear strength equivalent to a 0.8 braking factor, the performance for the 29.0-in. tire improved. In the homo-
applied to the maximum taxi weight on the strut. Therefore, geneous material, drastic overloading could occur if the bed

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Dark Region:
Medium Region:
Both Struts Overloaded
One Strut Overloaded
Strut Load Factor
Light Region:
Loading Acceptable
for Both Struts
Optimum
Design Point
Material Depth
Strength
Figure 12-7. Example figure for dual-constraint surface plot.
was too deep; both depth-varying methods eliminated this 12.4.5. Linear Gradient Material Optimum
drastic overloading regime.
Both methods also reduced the sensitivity to the material Figure 12-8 and Figure 12-9 compare 29-in. tire metamodel
strength. This provided a serendipitous advantage, since surfaces for homogeneous (left images) and linear (right
crushable foam materials can have a degree of scatter in their images) depth gradient materials. Figure 12-8 compares the
strength values based on manufacturing variation. The sensi- penetration depth of the tires, while Figure 12-9 compares the
tivity reduction implied that a robust design can be achieved strut loading ratios. The surfaces for the 44.5-in. tire (not
with less dependence on precision materials. shown) reflected similar trends.
The linear depth-varying approach seemed slightly better The right hand image of Figure 12-8 shows an interesting
than the quadratic. While the qualitative trends are similar, the behavior: for the linear gradient material, the response becomes
linear method appeared to offer simultaneous optimal condi- insensitive to the arrestor bed depth, becoming essentially
tions for both large and small tires. Hence, the linear profile flat in the depth-wise direction. The tires settled to their own
was chosen for advancement to the final optimization stage. natural depth in the material and were no longer prone to
Homogeneous Linear Gradient
-4 -4
Penetration Depth
Penetration Depth
-22 -22
100 18 100 18
Strength Depth Strength Depth
37.5 30 37.5 30
Figure 12-8. Penetration depth trends for 29.0-in. tire with homogeneous (left)
and linear (right) depth gradient.