The paper concerns the optimization of the skid landing gear of a helicopter using the response surface methodology.

1. 1 Copyright © 2016 by CSME
Proceedings of The Canadian Society for Mechanical Engineering International Congress 2016
CSME International Congress 2016
June 26-29, 2016, Kelowna, BC, Canada
CRASHWORTHINESS DESIGN OPTIMIZATION OF A CONVENTIONAL SKID
LANDING GEAR USING RESPONSE SURFACE METHOD
Muftah Saleh, Ramin Sedaghati and Rama Bhat
Mechanical and Industrial Engineering
Concordia University
Montreal, Canada
mu_saleh@encs.concordia.ca
Abstract—The present study concerns with increasing the
energy absorption capability of a helicopter’s skid landing gear
(SLG) system in order to reduce the level of acceleration
experienced by helicopter occupants in the event of hard
landing. A nonlinear finite element model of the baseline SLG
has been developed using ABAQUS software. The simulation
for drop test according to the airworthiness regulation has then
been carried out to evaluate its crashworthiness performance.
Considering diameter and thickness of the front and rear tubes
as design variables, the developed finite element model has
been utilized to develop response surface functions (meta-
models) using design of experiment and response surface
methods. A design optimization has then been formulated to
maximize the specific energy absorption (SEA) of the baseline
SLG under weight, deflection, flight load factor and size
constraints based on North American crashworthiness
regulations. The results show that the size of the forward and
rear cross members could be successfully optimized for lower
weight and larger SEA while meeting the crashworthiness
requirements.
Keywords: crashworthiness; skid landing gear; size optimization;
specific energy absorption; resposne surface method
I. INTRODUCTION
Due to the limited stroke of the helicopter’s sub-floor
structure, significant effort has been devoted toward increasing
the energy absorption capabilities of the remaining crashworthy
systems, i.e. the landing gear and the occupant seat. From
crashworthiness perspective, several standards have been
developed to guide the design, testing, and certification
procedure of aircraft crashworthiness systems. In North
America, for instance, any normal category civil helicopter
equipped with a skid landing gear (SLG) has to meet the
requirements of AWM 527 (chapter 527 of Airworthiness
Manual) of Transport Canada Civil Aviation (TCCA) in Canada
and Part 27 of Federal Aviation Regulation (FAR) of the Federal
Aviation Administration (FAA) in United States [1] for vertical
impact velocity of up to 7.9 m/s [2]. According to these
regulations, the SLG has to be successfully tested against the
following two level drop conditions: a) Limit drop at velocity of
2 m/s at the instant of touchdown, and b) Reserve energy at 2.44
m/s. However, Army standards impose more stringent
requirements. For instance, the reserve energy drop condition at
3.7 m/s is considered as the design condition of the landing gear
on the OH-58 helicopter is considered [3]. Simultaneously, the
airworthiness principle of lightweight aircraft structure has to be
considered. In these vertical impacts, the crosstubes which are
shown in Fig. 1 are the only parts allowed to deform plastically
to absorb impact energy. During the event of hard landing or
crash, excessive accelerative loads beyond 15 g’s should be
attenuated progressively [4] in order to prevent these loads from
being transmitted to the occupants. Moreover, in these drop tests
a load may be applied at the center of gravity of the helicopter at
the instant of touchdown to simulate the effect of the upward lift
force during the landing. The rotor lift factor (L) determines the
ratio of the lift force to the maximum gross weight of the
helicopter. The introduced load reduces the inertial effect of the
impacting mass to the so called “effective mass”. The above
civilian regulations state that rotor lift factor should be two-third
for limit drop test and unity in energy reserve drop test.
However, it is required under all circumstances to maintain a
positive clearance between the fuselage underneath and the
ground.
In the present article a design optimization study is conducted
on the SLG shown in Fig. 2. The geometry and the mass
properties of the helicopter are adopted from [5]. The objective
Fig. 1. Skid landing gear assembly.

2. 2 Copyright © 2016 by CSME
of this work is to optimize the given SLG system without
violating the requirements of AWM527 and FAR Part 27. This
could be achieved by identifying the optimal cross sectional
dimensions of the crosstubes that would maximize the specific
energy absorption (SEA) capability under hard level landing at
velocity of 2.44 m/s. SEA is defined as the ratio of the internal
energy over the weight of energy absorbing members, which
are the crosstubes in this study. This will result in a significant
reduction in the level of vertical acceleration experienced by the
occupant in the event of impact.
A. Beseline Configuration of the SLG
The SLG investigated here is manufactured from aluminum
alloy 7075-T6. This alloy is commonly used in conventional
skid gears due to its high strength and lightweight properties.
The geometrical dimensions [5] and the elastic properties [6] of
the SLG are tabulated in Table 1.
Due to the high cost of setting up and drop testing SLGs,
nonlinear finite element softwares such as ABAQUS, are
usually used in the initial design phase to evaluate the dynamic
response of the SLG and modify the design parameters
accordingly.
B. Finite Element dynamic Modelling of the SLG
In this study, ABAQUS explicit solver is used to accurately
simulate the response of both the baseline SLG design and the
optimum design configurations. Assessing energy absorption
capability of the SLG, requires frictionless contact between the
skids and the impact surface [7]. To reduce the computation
burden, it is common practice to model the skid gear in detail
and lump the mass properties of the remainder of the helicopter
structure at the center of gravity. Here, the SLG members are
modelled using two-node space beam element “B31” available
in ABAQUS material library [8]. The developed finite element
model of the SLG structure contains a total number of 452
nodes and 138 beam elements. A gravity load of “one g” is
applied at the C.G. to initiate the free fall of the helicopter under
gravitational load from distance (H). The fuselage’s forward
and rear attachments are modeled as rigid beam connectors. The
downward vertical velocity of the vehicle when the skids touch
the ground is computed as 𝑣 = √2𝑔𝐻 . The duration of
simulating the impact of SLG is set as 1.2 seconds, which is
adequate as the impact event takes around 50-200 ms. The mass
properties of the helicopter are tabulated in Table 2.
II. RESPONSE SURFACE METHOD
Conducting a design optimization using the full nonlinear
finite element (FE) model is computationally very expensive as
at each optimization iteration, the FE model may be run several
times. Moreover, since the dynamic responses obtained using
FE model are typically noisy, the gradient based optimization
algorithms may not be used since the accurate evaluation of the
gradient of responses is not possible. In these situations, the
design of experiment (DOE) and response surface method
(RSM) techniques can be utilized to develop accurate objective
and constraint response functions. In the current study, the
derived response functions will be used in design optimization
of the forward and aft crosstubes to improve their energy
absorption capabilities. Herein, the outer radii and thicknesses
of each crosstube are considered as design variables that are to
be optimized to increase the SEA of the crosstubes. The design
constraints are: crosstubes mass (m), maximum deformation of
the C.G. (δmax), maximum load factor (nj,max), and the upper and
lower bounds on design variables. The lower and upper bounds
of outer tube radius are set as 30 mm and 44 mm, respectively,
while the lower and upper bounds of thickness are 2 mm and 8
mm, respectively. Using DOE technique in Minitab software,
27 design points were identified in the design space in which x1
and x2 are the external radius and thickness of the aft cross
member, whereas x3 and x4 represent those for the forward tube,
respectively. Then, the developed finite element model, was
utilized to perform 27 simulation runs on the selected design
points. The desired responses of the helicopter’s FE model are
tabulated against the corresponding design points in terms of
SEA, nj,max, and δmax. Sample of these data is shown in Table 3.
Thereafter, RSM was employed to develop approximate
response functions for the above mentioned responses in
addition to the mass of the crosstubes. To estimate these
responses more accurately, full quadratic terms were used as
[9]:
𝑦̂ = 𝛽0 + ∑ 𝛽𝑗 𝑥𝑗 +
𝑘
𝑗=1
∑ 𝛽𝑗𝑗 𝑥𝑗
2
+ ∑ ∑ 𝛽𝑖𝑗 𝑥𝑖 𝑥𝑗
𝑘
𝑗=2𝑖<
𝑘
𝑗=1
(1)
Figure 2. Configuration of the skid landing gear [5].
TABLE 1. DIMENSIONS AND ELASTIC PROPERTIES
OF THE BASELINE SLG SYSTEM [5,6]
Front crosstube cross sectional dimensions Ø 652.5 mm
Aft crosstube cross sectional dimensions Ø 802.5 mm
Skid tube cross sectional dimensions Ø 752 mm
h 1.602 m
a 0.483 m
b 1.687 m
Modulus of elasticity (E) 72 GPa
Yield stress (σy) 531 MPa
Rupture stress (σr) 658 MPa
Poison’s ratio () 0.3
Mass of crosstubes 9.06 kg
TABLE 2. MASS PROPERTIES OF THE HELICOPTER [5]
Helicopter mass ( Except SLG) 2243.4 kg
Density of alloy 7075-T6 2810 kg/m3
Ixx about C.G. ( Except SLG) 3199 kg.m2
Iyy about C.G. ( Except SLG) 10197 kg.m2
Izz about C.G. ( Except SLG) 11638 kg.m2

3. 3 Copyright © 2016 by CSME
where ŷ is the approximated response determined by the RSM,
β0, βj, βjj, βij, are constant coefficients to be identified, and xi,
and xj are the design variables. Here k=4 and as explained
earlier variables x1 and x2 are defined as the external radius and
the thickness of the aft crosstube, respectively, and x3 and x4 are
the external radius and the wall thickness of the forward
crosstube, respectively. Vector β contains constant coefficients
that are estimated using the method of least squares [9]. The
derived objective functions using RSM can be described as:
𝑆𝐸𝐴 = −89355 𝑥1
2
+ 1774821 𝑥2
2
− 189405 𝑥3
2
+ 2380650 𝑥4
2
+
12187 𝑥1 − 42407 𝑥2 + 15152 𝑥3 − 59393 𝑥4 − 1011905 𝑥1 𝑥2 −
41500 𝑥1 𝑥3 + 249205 𝑥1 𝑥4 + 401820 𝑥2 𝑥3 + 5863889 𝑥2 𝑥4 −
788095 𝑥3 𝑥4 + 183 (2)
Constraint functions are also identified as:
i. Mass
𝑚 = 8.5 (𝑥1
2
+ 𝑥3
2) − 25926( 𝑥2
2
+ 𝑥4
2) + 0.09(𝑥1 + 𝑥3) +
3.7(𝑥2 + 𝑥4) + 51785.7(𝑥1 𝑥2 + 𝑥3 𝑥4) − 119(𝑥1 𝑥4 + 𝑥2 𝑥3) −
0.024 (3)
ii. Maximum Deflection
𝛿 𝑚𝑎𝑥 = 227.3 𝑥1
2
+ 2100 𝑥2
2
+ 181.1 𝑥3
2
+ 1862 𝑥4
2
− 23.55 𝑥1 −
27.12 𝑥2 − 20.76 𝑥3 − 28.49 𝑥4 − 483 𝑥1 𝑥2 + 128.1 𝑥1 𝑥3 +
311 𝑥1 𝑥4 + 364 𝑥2 𝑥3 + 1031 𝑥2 𝑥4 − 326 𝑥3 𝑥4 + 1.215 (4)
ii. Maximum flight load factor
𝑛𝑗,𝑚𝑎𝑥 = 1021 𝑥1
2
− 114435 𝑥2
2
+ 400 𝑥3
2
− 14769 𝑥4
2
− 31.1 𝑥1 +
19.1 𝑥2 + 14.5 𝑥3 + 68.7 𝑥4 + 11429 𝑥1 𝑥2 − 357 𝑥1 𝑥3 +
2143 𝑥1 𝑥4 − 2143 𝑥2 𝑥3 − 7500 𝑥2 𝑥4 + 10238 𝑥3 𝑥4 + 0.23 (5)
The coefficient of multiple determination R2
[10] is utilized
to verify the accuracy of the derived approximate functions in
predicting the response of the real system. R2
varies from zero
to one where values close to one mean the function is able to
predict the response more accurately. The value of R2
always
increases when adding more terms to the model’s approximate
function. However, the additional variables may not be
statistically significant [11]. Therefore, an additional statistical
measure called adjusted R2
(R2
-adj) may be considered since it
does not increase when unnecessary terms are added to the
function. Minitab provides these measures based on hypothesis
testing of the regression model derived using RSM. The values
of R2
and R2
-adj for equations (2), (3), (4), and (5) are presented
in Table 4. It can be realized that the derived response functions
can accurately estimate their respective responses without using
finite element model.
It should be noted that R2
and R2
-adj can be computed from the
following formulas, respectively [11]
𝑅2
= 1 −
𝑆𝑆 𝐸
𝑆𝑆 𝑇
(6)
𝑅2
− adj = 1 −
𝑛−1
𝑛−𝑝
(1 − 𝑅2
) (7)
where 𝑆𝑆 𝐸 = ∑ (𝑦𝑖 − 𝑦̂)2𝑛
𝑖=1 (8)
𝑆𝑆 𝑇 = ∑ (𝑦𝑖
− 𝑦̅)
2𝑛
𝑖=1 (9)
in which yi is the real response computed by ABAQUS, 𝑦̅ is the
average value of real responses, n is the number of design
points, and p is the number of regression coefficients.
III. DESIGN OPTIMIZATION OF THE SKID LANDING GEAR
A. Formulation of the Optimization Problem
The approximate objective and constraint functions of the
model, i.e. SEA, m, δmax, and nj,max can now be used with size
constraint functions to formulate the optimization problem of the
SLG. Here, the design objective is to maximize the SEA of the
crosstubes by increasing the amount of internal energy the tubes
can absorb. The applicable constraints to the optimization
problem are: 1) Crosstubes mass should not exceed the mass of
the baseline SLG design, i.e 9 kg; 2) Flight load factor should
not exceed 3.6 to preserve the integrity of the helicopter structure
during landing; 3) Deformation of the C.G. should not exceed
the height of the SLG system, i.e. 460 mm; 4) External radii (x1
and x3) of the crosstubes should be in 30-44 mm range, and; 5)
Tubes thicknesses (x2 and x4) are from 2 to 8 mm. The design
optimization problem can be formally formulated as:
Find design variable vector X*:
to maximize SEA (of the crosstubes)
Subject to: Mass– Mass (baseline)  0
nj,max – nj,max (baseline)  0
δmax– δmax (baseline)  0
where, xL  xi  xU
Now, considering above and (2)-(5), one can write:
Find design variable vector X*:
to maximize
𝑆𝐸𝐴 = −89355 𝑥1
2
+ 1774821 𝑥2
2
− 189405 𝑥3
2
+ 2380650 𝑥4
2
+
12187 𝑥1 − 42407 𝑥2 + 15152 𝑥3 − 59393 𝑥4 − 1011905 𝑥1 𝑥2 −
41500 𝑥1 𝑥3 + 249205 𝑥1 𝑥4 + 401820 𝑥2 𝑥3 + 5863889 𝑥2 𝑥4 −
788095 𝑥3 𝑥4 + 183 (2)
Subject to the following constraints:
TABLE 4. COEFFICIENTS OF DETERMINATION
Approximate RSM function R2
, % R2
-adj, %
SEA 93.96 86.92
m 100 100
δ 96.73 92.92
nj,max 99.82 99.61
TABLE 3. SAMPLE OF DOE DESIGN POINTS AND
FE MODEL RESPONSES
Design Points FE Model Responses
x1
mm
x2
mm
x3
mm
x4
mm
SEA
J/kg
m
kg
nj,max
g’s
δmax
mm
44 5 44 5 268.5 21.5 4.98 124.0
37 8 37 2 298.5 17.4 3.77 158.8
37 8 44 5 225.6 24.42 4.87 128.7
30 5 37 8 222.4 20.8 3.8 159.3
37 5 44 2 361.3 13.39 3.45 160
37 2 30 5 370.9 10.85 2.62 204.8
37 5 30 8 254.6 19.71 3.63 165.8
30 2 37 5 305.8 11.94 2.89 178.4

4. 4 Copyright © 2016 by CSME
𝑔1(𝑥) = 8.5 (𝑥1
2
+ 𝑥3
2) − 25926( 𝑥2
2
+ 𝑥4
2) + 0.09(𝑥1 + 𝑥3) +
3.7(𝑥2 + 𝑥4) + 51785.7(𝑥1 𝑥2 + 𝑥3 𝑥4) − 119(𝑥1 𝑥4 + 𝑥2 𝑥3) −
9.024 ≤ 0 (10)
𝑔2(𝑥) = 227.3 𝑥1
2
+ 2100 𝑥2
2
+ 181.1 𝑥3
2
+ 1862 𝑥4
2
− 23.55 𝑥1 −
27.12 𝑥2 − 20.76 𝑥3 − 28.49 𝑥4 − 483 𝑥1 𝑥2 + 128.1 𝑥1 𝑥3 +
311 𝑥1 𝑥4 + 364 𝑥2 𝑥3 + 1031 𝑥2 𝑥4 − 326 𝑥3 𝑥4 + 0.755 −
0.460 ≤ 0 (11)
𝑔3(𝑥) = 1021 𝑥1
2
− 114435 𝑥2
2
+ 400 𝑥3
2
− 14769 𝑥4
2
− 31.1 𝑥1 +
19.1 𝑥2 + 14.5 𝑥3 + 68.7 𝑥4 + 11429 𝑥1 𝑥2 − 357 𝑥1 𝑥3 +
2143 𝑥1 𝑥4 − 2143 𝑥2 𝑥3 − 7500 𝑥2 𝑥4 + 10238 𝑥3 𝑥4 −
3.37 ≤ 0 (12)
𝑔4(𝑥) = − 𝑥1 + 0.03 ≤ 0 (13)
𝑔5(𝑥) = 𝑥1 − 0.044 ≤ 0 (14)
𝑔6(𝑥) = 𝑥2 − 0.002 ≤ 0 (15)
𝑔7(𝑥) = 𝑥2 − 0.008 ≤ 0 (16)
𝑔8(𝑥) = −𝑥3 + 0.03 ≤ 0 (17)
𝑔9(𝑥) = 𝑥3 − 0.044 ≤ 0 (18)
𝑔10(𝑥) = 𝑥4 − 0.002 ≤ 0 (19)
𝑔11(𝑥) = 𝑥4 − 0.008 ≤ 0 (20)
B. Solution of the Optimization Problem
The Sequential Quadratic Programming (SQP) numerical
technique has been proven as a powerful and robust algorithm
in solving the nonlinear constrained optimization problems.
Thus SQP method has been utilized to solve the optimization
problem formulated in previous section using MATLAB
optimization toolbox. SQP is a local optimizer which may trap
in a local optimum. To ensure catching the global optimum,
different initial design points are selected randomly.
IV. RESULTS AND DISCUSSION
The optimum design parameters (x1, x2, x3, x4) and their
corresponding optimal SEA for different initial points are
provided in Table 5. As it can be realized, different initial points
yield the same optimum solution confirming that the global
solution has been caught. The iteration history is also shown in
Fig. 3.
Comparison between optimal and baseline design are provided
in Table (6). The results show an increase in SEA for the
optimized SLG of about 17.3% with reduction of 14.1% in
SLG mass.
The RSM response function for SEA is also verified against
finite element simulation by varying the variable x2 from 2 to 8
mm, while keeping other optimum design variables constant.
The results are shown in Fig. 4.
The behavior of the optimized SLG is compared against the
baseline SLG structure under impact velocity of 2.44 m/s in
terms of SEA, flight load factor, plastic dissipation, and Von-
Mises stress. These results are shown in Figs. (5) to (10).
The time history of SEA and flight load factor for the reserve
energy impact condition are depicted in Figs. (5) and (6) over a
duration of 1.2 seconds. The helicopter starts falling freely under
gravity from height (H) = 303.5 mm. In both SLG configurations
the gravitational effect dominates until the skids hit the landing
surface at instant 0.2487 seconds (A). During the impact strokes,
as shown in Fig. (5), an increase in peak SEA is noticed in the
optimized SLG model over the baseline one. Since a frictionless
contact is assumed throughout the simulation, the absorption of
impact energy is attributed to the bending of the cross members.
The energy absorption results in a reduction of the peak vertical
acceleration and consequently a slight enhancement in nj of
(3.4%) during the first impact from 2.63 (C) to 2.54 g’s (B), as
shown in Fig. (6). Curve segments (B-D) for the optimized
version and (C-D) for the baseline model represent gradual
springing back of the crosstubes. At instant 0.493 second, the
SLG bounces up and departs the rigid surface for about 0.325 s
before the SLG hits the ground again. During this very short
period of time (D-E), factor nj decreases to “one g”
approximately and the crosstubes dissipate part of the impact
energy plastically (≈1400 J for original and 1724 J for optimal).
The second impact of the SLG takes places at instant t = 0.818 s
TABLE 5. CONVERGENCE OF SQP TO THE MAXIMUM GLOABLE
Initial Point (X0), mm SEA, j/kg Optimum X, mm
(38.0,2.0,36.0,2.0) 523.6 (44.0,2.0,33.1,2.0)
(42.5,4.5,30.0,3.5) 523.6 (44.0,2.0,33.1,2.0)
(39.5,7.5,44.0,8.0) 523.6 (44.0,2.0,33.1,2.0)
(33.5,2.0,33.5,2.0) 523.6 (44.0,2.0,33.1,2.0)
(44.0,5.0,44.0,5.0) 523.6 (44.0,2.0,33.1,2.0)
TABLE 6. COMPARISON OF THE KEY RESULTS FOR THE BASELINE AND
OPTIMIZED SLG
SLG Version SEA, J/kg m, kg δmax, mm nj,max, g’s
Baseline 453.6 9.06 191.0 2.63
Optimum 532.2 7.78 195.1 2.54
Figure 3. Iteration history of the optimized meta-model.
150
250
350
450
550
0 1 2 3 4 5 6
SEA,J/kg
No. of Iterations
Figure 4. Iteration history of the optimized meta-model.
0
100
200
300
400
500
600
0 2 4 6 8 10
SEA,J/kg
Aft crosstube thickness (x2), mm
RSM
Abaqus

5. 5 Copyright © 2016 by CSME
(E) and lasts for about 0.31 s. By the end of 2nd
impact, the
original design dissipates total of 1900 J in the form of plastic
energy and the optimized design dissipates 2186 J plastically, as
shown in Fig. 7. This means that 32.5% of impact energy is
dissipated plastically in the case of optimal SLG impact, whereas
the baseline design dissipated 28.4% only. The remaining
amount of impact energy is recovered as elastic energy. To avoid
aliasing results, the acceleration output is filtered using
ABAQUS internal low pass Butterworth filter of 2nd degree.
The cutoff frequency is set at 75 Hz. Fig. 8 depicts the time
history of filtered and raw vertical acceleration for the optimal
SLG.
Fig. 9 illustrates the distribution of von Mises stress in the
crosstubes. Elements adjacent to the fuselage attachments
experience the highest stress. In the baseline SLG, the plastic
deformation of the tubes generates a residual (plastic) stress of
about 274 MPa in these elements after removing the external
load. The back crossbeam is stressed more than the front
crosstubes because of its relative closeness to the helicopter C.G.
In the optimized configuration, this stress relaxes slightly to
255.3 MPa as the tubes are allowed to deform a bit more than
the original design, as shown in Fig. 10. Under this impact
condition, standards AWM 27 and FAR Part 27 prohibit the
fuselage underneath structure from contacting the ground.
Fortunately, this requirement was successfully met at the cost of
replacing the plastically deformed crosstubes.
IV. CONCLUSION
The objective of the conducted design optimization study
was to increase the specific energy absorption of the crosstubes
of the skid landing gear under mass and crashworthiness
requirements. The objectives were achieved successfully as per
reserve energy drop test requirements of regulations AWM 27
and FAR Part 27. Instead of carrying out a computationally
expensive design simulations in ABAQUS, DOE and RSM were
employed to develop the approximate functions for the response
(SEA) and the constraints. These functions were used to
formulate design optimization problem which is then solved
using SQP on MATLAB platform. The sizes of the aft and
forward crosstubes are ø80×2.5 mm and ø65×2.5 mm for
baseline design, respectively. These tubes were optimized to
ø88×2 mm and ø66.2×2 mm, respectively. As a result, the
SEA increased remarkably from 453.6 for the original SLG
design to 532.2 J/kg for the optimum SLG configuration.
Meanwhile, the mass of these parts was reduced considerably by
15% from 9.06 to 7.78 kg. The SLG structure is undergone a
plastic deformation which requires a replacement after this hard
landing.
Figure 5. Time change of SEA.
Figure 6. Time history of flight load factor.
Figure 7. Time history of plastic dissipation.
Figure 8. Time history of vertical acceleration for the optimized SLG.

6. 6 Copyright © 2016 by CSME
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Figure 9. Von-Mises stress distribution in the original SLG.
Figure 10. Von-Mises stress distribution in the optimized SLG.