Research ArticleAnalysis of Global Sensitivity of Landing Variables on LandingLoads and Extreme Values of the Loads in Carrier-Based Aircrafts

Jin Zhou , Jianjiang Zeng , Jichang Chen , and Mingbo Tong

College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing, China

Correspondence should be addressed to Mingbo Tong; tongw@nuaa.edu.cn

Received 12 June 2017; Revised 26 October 2017; Accepted 9 November 2017; Published 14 January 2018

Academic Editor: Kenneth M. Sobel

Copyright © 2018 Jin Zhou et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

When a carrier-based aircraft is in arrested landing on deck, the impact loads on landing gears and airframe are closely related tolanding states. The distribution and extreme values of the landing loads obtained during life-cycle analysis provide an importantbasis for buffering parameter design and fatigue design. In this paper, the effect of the multivariate distribution was studiedbased on military standards and guides. By establishment of a virtual prototype, the extended Fourier amplitude sensitivity test(EFAST) method is applied on sensitivity analysis of landing variables. The results show that sinking speed and rolling angle arethe main influencing factors on the landing gear’s course load and vertical load; sinking speed, rolling angle, and yawing angleare the main influencing factors on the landing gear’s lateral load; and sinking speed is the main influencing factor on thebarycenter overload. The extreme values of loads show that the typical condition design in the structural strength analysis issafe. The maximum difference value of the vertical load of the main landing gear is 12.0%. This research may provide somereference for structure design of landing gears and compilation of load spectrum for carrier-based aircrafts.

1. Introduction

Carrier-based aircraft is the most important method foraircraft carrier strike group to control sea supremacy, and itis also an indispensable power for modern navies. Due tolimited area in deck landing zone and the demand for boltingand go-around, carrier-based aircraft usually lands on deckvia impact method under high sinking speed and high engag-ing speed along a fixed glide path angle [1]. The impact load,braking load of arresting cable [2, 3], and other loads at themoment when the aircraft touches deck put forward higherrequirements for design and analysis of landing gears andairframe structure, especially for the structures closely relatedto landing [4].

At the primary design stage of an aircraft landing gear,the landing loads and the most severe landing conditions ofthe landing gear under different landing conditions shouldbe determined according to design outline, which will beadopted in the parameter design of landing gear buffers andstructural strength analysis. At the detailed design stage,

optimization design will be carried out to balance perfor-mance and structure of the landing gear according to therelation between landing conditions and landing loads.Finally, the maneuvering envelope will be determined andthe design of service load spectrum will be compiled for theassessment of fatigue life [5, 6].

At present, the research literatures and reports on defini-tion of arrested deck landing conditions are mainly militarystandards and guides [7, 8]. MIL-A-8863C (AS) providesthe landing variables, their distribution form, and empiricalformula of mean value and standard deviation that shouldbe considered in deck landing. Micklos [9] provided ameasurement report on the landing variables of all kinds ofcarrier-based aircrafts landed on the Enterprise AircraftCarrier in both day and night. The measurement of landingvariables and loads is a costly and tedious task, thus a largeamount of researches have been carried out on the simulationand analysis of landing dynamics to analyze different landingconditions and loads in recent years. Zhang [10] introduceddeck motion into a dynamics model through the wheel-

HindawiInternational Journal of Aerospace EngineeringVolume 2018, Article ID 2105682, 14 pageshttps://doi.org/10.1155/2018/2105682

deck coordinate system and simulated three landing situa-tions to verify the model. Mikhaluk [11] and Lihua et al.[12] used finite element model to analyze forces on cablesunder different initial velocities and masses. Sati et al. [13]built a detailed aircraft arresting system using bond graphapproach and studied the landing performance at differentengagement speeds and masses.

The studies mentioned above give a simple trend of therelationship between landing variables and loads under afew landing conditions. However, the conditions empiricallyselected are unrepresentative. During the compilation ofdesign service load spectrum, representative values of typicallanding conditions can be obtained by discretizing the inter-val of landing parameters through sensitivity analysis. Incor-rect structural load analysis may lead to excessive design,which would induce large weight and high cost at theprimary stage of structure design. A further determinationof the relationship between landing variables as well as theextreme value of the loads is required. Sebastian [14] ana-lyzed the condition of free flight engagement (FFE), and theresults show that the most severe loading condition of thelanding gear takes place under the condition of free flightengagement at small sinking speed. Chester [15] researchedthe response of the main and nose gears via simulation oflanding impact considering pitching and heaving degrees offreedom of the aircraft motion. The simulation indicates thatthe maximum vertical loads of main gears are almost linearlydependent on the sinking speed, while the response of nosegear is very sensitive to the initial values of pitch angle andpitching inertia. Yunwen et al. [16] explored the effect ofdifferent landing variables on sinking velocity based on thelanding data measured from an E-2C. The results show thataircraft path angle and deck pitch angle are highly correlatedwith sinking velocity. Although they have studied part of therelevance between landing variables and loads, there is nodirect conclusion on method accurate enough for determi-nation of the representative values that can reflect typicallanding conditions. Therefore, sensitivity and extremevalue analysis need to be studied and it provides animportant basis for the parameter design of landing gearbuffer, structural optimization design [17], and fatigueanalysis. However, theoretical basis and analysis methodstill need to be further investigated.

The extended Fourier amplitude sensitive test (EFAST)analysis method [18] is adopted in this paper to study thecoupling and sensitivity between landing variables andlanding loads. The method has been well applied in thefield of hydrology [19], physical model [20], and so on.Based on military standards and guides, the effects of themultivariate distribution on the probability density func-tion of single landing variable were analyzed and thedistribution of single landing variable was fitted. By usingthe simplified landing virtual prototype and analysis of alarge amount of landing conditions with the multicondi-tion automatic simulation technology, the first-order andglobal sensitivity coefficients of landing variables on land-ing loads were calculated quantitatively, and the extremevalue conditions and the frequency curves of the landingload were obtained.

2. Studies on Landing Variables inMilitary Standards

Many factors would affect the carrier-based aircrafts duringthe arrested deck landing. The landing variables and theirrelationships are mainly stipulated by related militarystandards and specifications.

2.1. Landing Variables and Multivariate Distribution. Themilitary standard MIL-A-8863C (AS) stipulates the landingvariables and their distributions. The condition of the land-ing variables needs to satisfy the demands of the multivariatedistributions. The joint probability of the eight landingvariables PT is calculated by

PT = P VTD /<VTDi P VE /<VEi P VV /<VVi

P θP /<θPi P θR /<θRi P θR /<θRi

P θY /<θYi P d /<di ,

1

where VTD is the approaching speed, VE is the engagingspeed, VV is the sinking speed, θP is the aircraft pitchingangle, θR is the aircraft rolling angle, θR is the aircraft rollingrate, θY is the aircraft yawing angle, and d is the off-centerarresting distance. The subscript i denotes the initial valueof each landing variable. The symbol /< is chosen to beeither > or < according to the initial value of the variable.When the initial value of any landing variable is greater thanthe average one, the symbol > is chosen and it represents thatthe cumulative probability of that landing variable is greaterthan the initial value. Conversely, when the symbol < ischosen, it represents that the cumulative probability of thatlanding variable is less than the initial value.

The evaluation of the landing variables is determined bythe extreme value of the cumulative probability P0, namely,

P0 = P x > xmax i = P x < xmin i , 2

where xmax i and xmin i are the maximum and minimumvalue of the variables. In this paper, only the touch-and-goand arrested landing conditions are considered and the land-ing variables follow a normal distribution. The cumulativeprobability P0 = 1 × 10−3, and the multivariate joint probabil-ity PT = 7 8125 × 10−6

2.2. The Effects of the Multivariate Distribution on theLandingVariables.The relationships among landing variablesare according to the multivariate distribution. The product ofthe joint probability of the landing variables does not reflectthe effects on the distribution of the single landing variabledirectly. Therefore, the distribution of landing variables isinvestigated according to the multivariate distribution.

Take the approaching speed VTD as an example. Thereexists a relationship:

P VTD /<VTDi ≤ P μ VTD = 0 5, 3

where μ VTD is the average of the carrier engaging speeds.Take VTD and VE as an example. The extreme value of the

2 International Journal of Aerospace Engineering

joint probability of any two landing variables is given byinserting (3) into (1).

PT′ = P VTD /<VTDi P VE /<VEi

≥PT

0 5 ⋅ 0 5 ⋅ 0 5 ⋅ 0 5 ⋅ 0 5 ⋅ 0 5

=PT

0 015625= 0 0005

4

The distribution of the landing variable is mapped to astandard normal distribution in order to unify the range ofthe variables. The mutual effects among the value ranges ofthe landing variables are determined by the multivariate dis-tribution [21]. In Figure 1, there are six equiprobable designenvelopes of any two landing variables. The probability ofeach equiprobable envelope is shown in the figure.

The equiprobable envelope 6 represents the conditionthat the joint probability of two landing variables reachesthe minimum, the value of which is PT′ = 0 0005 Theequiprobable envelope 1 degenerates to a point which repre-sents the condition that the joint probability of two landingvariables reaches the maximum.

The evaluation of the eight landing variables of fourgroups on the equiprobable envelopes must ensure that thesum of their sequence number is nine, satisfying the demandof the joint probability. For example, when one group ofvariables is the equiprobable envelope 5, then other threegroups must be the equiprobable envelope 1, 1 and 2.

It is difficult to make analytical calculations on the effectsof the multivariate distribution on the single landing variable.Monte Carlo method is adopted to sample simulation for thecomplex probability constraints. Before sampling, the distri-bution of each variable should be mapped to the standardnormal distribution. 105 simulate samples are extractedaccording to the distribution of each landing variable. Thesimulating sample screening is according to the multivariatedistribution determined by (1). Figures 2 and 3 show theMonte Carlo sampling condition and the condition of apply-ing multivariate distribution.

From the screening results, it can be seen that only 29,194out of 105 samples remain after adopting multivariate distri-bution and the 71% of the samples cannot satisfy the require-ments of the joint probability. By comparing Figure 2 withFigure 3, it can be known that the multivariate distributiondeletes lots of samples that landing variables approachingthe extreme value and keeps the samples that landing vari-ables closing to the average. In Figure 3, the equiprobableenvelope 6 is the envelope curve of the minimum value ofthe multivariate distribution shown in Figure 1.

The normal test is carried out by Kolmogorov-Smirnovmethod. D = 0 005, P = 0 372 > 0 05, and the landing vari-ables after screening obey the normal distribution. Thecomparisons between distributions of landing variablesbefore and after screening as well as the fitting conditionsare shown in Figure 4. It is seen that after adopting themultivariate distribution, the probability density function ofthe single landing variable changes a lot.

From the analysis above, it is seen that the requirement ofthe multivariate distribution restricts the optional values

3

2

1

0

0Approach speed VTD

1 2 3

−1

−1

−2

−2−3

−3

P = 5.0 × 10−4

P = 1.732862108 × 10−3

P = 6.00562217 × 10−3

P = 2.081383019 × 10−2

P = 7.21349953 × 10−2

P = 0.251

23

45

6

Enga

ging

spee

d V E

Figure 1: Equiprobable design envelope graph.

3

2

1

0

−1

−2

−3

Enga

ging

spee

d V E

0Approach speed VTD

1 2 3−1−2−3

Figure 2: Multivariate Monte Carlo scatter diagram.

3

2

1

0

−1

−2

−3

Enga

ging

spee

d V E

0Approach speed VTD

1 2 3−1−2−3

P = 5.0 × 10−4

1

Figure 3: Multivariate distribution screening scatter diagram.

3International Journal of Aerospace Engineering

among the variables and has a large effect on the standarddeviation of the single landing variable. The results of thesensitivity analysis are closely related to the probability den-sity function of the input variables. The adoption of the dis-tribution of the landing variables before their multivariatedistribution would overestimate the probability that landingvariables reach the maximum or minimum values, whichmay cause the errors of the global sensitivity analysis of thelanding load. In this section, based on the requirements ofthe multivariate distribution, the equiprobable design enve-lopes of the landing variables are obtained. The distributionof the single landing variable for global sensitivity analysisis studied by Monte Carlo sampling method.

3. The Landing Virtual Prototype of theCarrier-Based Aircraft and theMulticondition Automatic Simulation

The attitudes of the aircrafts differ a lot at the moment ofdeck landing. Under the effects of the deck reacting force,the arresting force of the arresting hook, the aerodynamicforce, and other forces, the load-time history of the landinggear is rather complicated during the landing process. Theestablishment of the accurate landing simulation model ofthe aircraft is the basis of the sensitivity analysis of the land-ing load. The accuracy of the sensitivity analysis is relatedwith the sample quantity. In order to improve the reliabilityof the sensitivity analysis, it is necessary to consider the com-putation efficiency during simulating different landing con-ditions. The constraints and loads of the aircrafts duringlanding are analyzed, and the landing virtual prototype ofthe carrier-based aircraft based on LMS Virtual.Lab is estab-lished. For the computation efficiency problem of the massivelanding conditions, the secondary development technology isapplied to realize the multicondition automatic simulationand data processing of the landing conditions.

3.1. Hypothesis of the Aircraft Virtual Prototype and ModelEstablishment. The aircraft virtual prototype consists of noselanding gear subsystem, main landing gear subsystem, arrest-ing hook system, airframe model, and deck model. The exter-nal forces on the aircraft during landing are body load whichincludes aircraft gravity and engine thrust, aerodynamic force,acting force between tires, and deck and arresting force.

The virtual prototype has the following reasonableassumptions:

(1) Ignore the motion of the aircraft carrier and off-center landing.

(2) Assume that all components of the subsystems arerigid body, and themass concentrates on the centroid.

(3) Ignore the effects of yawing landing on the arrestingforce curve.

According to the real motion of each component of thesubsystems during the aircraft landing, the relating motionpairs such as swirl joint, cylinder joint, and spherical jointin subsystems are established. The constraints and load con-ditions of the subsystems are shown in Figure 5.

The initial condition of the virtual prototype is themoment that the aircraft lands on the deck, and the landinggears just touch the deck. During landing, the engine thrustmaintains the maximum (twin engine). Arresting force-distance curve refers to military standards [8]. MK7-3 arrest-ing machine is selected. The curve is obtained by weightinterpolation and velocity compensation, shown in Figure 6.

The virtual prototype is established according to E-2 air-borne early warning aircraft, shown in Figure 7.

3.2. Landing Multicondition Automatic Simulation. In orderto improve the computation efficiency of the massive landingconditions, the automation technology is applied to realizethe multicondition automatic simulation and data processingof the landing conditions. The auto-set, renewal, and analysisof the virtual prototype parameters are carried out by LMS

0

Histogram after filtrateNormal distribution after filtrateNormal distribution before filtrate

Engaging speed VE

1 2 3−1−2−30

500

1000

1500

2000

2500

3000

3500

Cou

nt

𝜇 = 0, 𝜎 = 0.69

𝜇 = 0, 𝜎 = 1

Figure 4: Comparisons of sample distributions before and afterscreening.

Engine thrust Airframe model Aerodynamic force

Nose landinggear subsystem

Main landinggear subsystem

Outerbarrel

Outerbarrel

Up torquearm

Down torque arm

Down torque arm

Up torquearm

Pistonrod

Pistonrod

Wheelaxel

WheelaxelWheel Wheel

Arresting hooksubsystem

Arresting hookheaf

Arresting hookbody

Deck model

FARREST

R

FTIREFTIRE

S

FFF

R

R CFAIR FOILC

R

FR

R

FR

FAIR FOIL R

R

R

Figure 5: Landing virtual prototype constraints and loads. In thefigure: R: swirl joint; C: cylindrical joint; S: spherical joint; F: fixedjoint; FAIR: air spring force; FOIL: oil damping force; FTIRE: tireforce; FARREST: arresting force; omit the gravity.

4 International Journal of Aerospace Engineering

secondary development to realize the multicondition auto-matic simulation of the virtual prototype.

The virtual prototype of LMS Virtual.Lab motion can bedivided into the product model and analysis model by func-

tion. The secondary development of the virtual prototypecomputes the initial landing attitude and location of theaircraft. The flow chart of the multicondition automaticsimulation of the virtual prototype is shown in Figure 8.

The initial landing attitude of the carrier-based aircraftis determined by the multivariable pitching angle θP, roll-ing angle θR, and yawing angle θY The attitude transitionmatrix from airframe coordinate system to ground coordi-nate system is calculated by Tait-Bryan angles [22]. Thetransition matrix is used to update the aircraft attitude ofthe product model as follows:

where the pitching angle θ = θP, rolling angle φ = θR, andyawing angle ψ = θY

When the aircraft attitude is determined, the aircraftlocation can be determined just with the constraints betweenthe aircraft and the deck. When the rolling angle θR is posi-tive, the aircraft tilts to the right and the right main landinggear just touches the deck, otherwise the opposite. The initialcondition of the carrier-based aircraft includes the carrierengaging speed VE, the sinking speed VV, and the aircraftrolling rate θR These variables can be set by analyzing themodel. In the eight landing variables stipulated by militarystandards and guides, VTD is not the variable at the momentof landing, so it cannot be considered in the virtual proto-type. Besides, off-center condition is not taken into consider-ation. Therefore, both the ground critical velocity VTD andoff-center arresting distance d are simplified by using theiraverage ones.

The sinking speed of the carrier-based aircraft is largeduring landing, and the landing gear bears rather largeimpact load. Referring to the analysis of the land-based land-ing load of the aircraft [23], the three directional loads of thelanding gear and barycenter overload are selected as landingload to carry out the study. The course load and vertical loadof the landing gear act on the center of the tire axle, while thelateral load acts on the contact point between the tires and

the ground. The bounce landing is not permitted, so the max-imum values of the three directional load and barycenteroverload are selected as landing loads for analysis. After theBDF solver finishing solving the landing process, the extremevalues of the three directional load and barycenter overloadunder this condition are output by calling the analysis model.

4. EFAST Global Sensitivity Analysis Method

During the aircraft landing, there are many landing variablesand landing loads. The local sensitivity analysis method islimited by the variable selection and model, so the analyticalresults are usually not comprehensive or accurate enough.The variable range of the global sensitivity analysis can beextended to the entire variable domain, which can providerather integrated quantitative analysis results. EFASTmethod is used to study the global sensitivity of the landingvariables to the landing loads.

EFAST is the quantitative global sensitivity analysismethod which is developed by Saltelli et al. [18]. Based onclassical Fourier amplitude sensitivity test (FAST), Saltelliet al. combined Sobol variance decomposition thought todevelop themethod. Thismethodmaps themultidimensionalvariable to independent variable by appropriate searchingcurves. Then the search within the multidimensional variable

Lbg =

cos θ cos ψ cos θ sin ψ −sin θ

sin ϕ sin θ cos ψ − cos ϕ sin ψ sin ϕ sin θ sin ψ + cos ϕ cos ψ sin ϕ cos θ

cos ϕ sin θ cos ψ + sin ϕ sin ψ cos ϕ sin θ sin ψ − sin ϕ cos ψ cos ϕ cos θ

, 5

Arresting distance (m)

Arr

estin

g fo

rce (

kN)

00

100

200

300

500

600

400

20 40 60 80 100 120

Figure 6: The arresting force-distance curve.

Figure 7: Carrier-based aircraft landing virtual prototype.

5International Journal of Aerospace Engineering

space of the model is changed into the periodic function con-trolled by independent variable. Fourier transform is used tocalculate the amplitude output by the model. The greater thechange is, the more sensitive the variables are.

Assume that the landing analytical model is described asy = f X , where X = X x1, x2,… , xn , n = 6 is the combina-tion of the landing variables. Before performing the globalsensitivity analysis, all variables have to be mapped to theinterval [0, 1] to constitute the n-dimensional unit supercube space.

Kn = X∣0 ≤ xi ≤ 1 i = 1, 2,… , n 6

Define an independent variable s and introduce the samemapping s→ X, so

xi s =Gi sin ωis , ∀i = 1, 2,… , n, 7

where s −∞ < s < +∞ is the scalar variable, ωi ∈ Ζ =−∞,… , − 1, 0, 1,… , +∞ , and its value changes to suit-able value according to xi With the change of s, all xichange in the value space Kn The oscillation frequency isωi Gi is the searching curve and recommended by Satiet al. [13] as

xi =12+

1πarcsin sin ωis + φ , ∀i = 1, 2,… , n, 8

where φ is the random value in 0, 2π

Expand f X by Fourier series.

y = f X = f x1, x2,… , xn= f x1 s ,… , xn s

= 〠j=+∞

j=−∞Aj cos js + Bj sin js ,

9

where

Aj =1π

π

−πf s cos jsds,

Bj =1π

π

−πf s sin jsds

10

j ∈ z = −∞,… , − 1, 0, 1,… , +∞ , and the amplitudeof the Fourier series is Λj = A2

j + B2j

From the properties of Aj and Bj, it is known thatA−j = Aj, B−j = −Bj, and Λ−j = Λj By calculating all fre-quencies, the total variance caused by each variable xi canbe obtained, namely,

V E y∣xi = 〠p∈Z0

Λpωi= 2〠

+∞

p=1Λpωi

, 11

where Z0 = Z − 0

LMS secondarydevelopment flow

Obtain product model Obtain analysis model

Compute the airframeattitude matrix

Inputlandingvariableparameters

Update the airframeattitude matrix

Update the aircraftsubsystem constraint

Set the initial motioncondition

Update the solvingparameters

Motion simulation solversolves

Validity verification

Extract the motionsimulation analyticalresults

PassOutput thedynamicresponse dataof the impactload

Quick data postprocessing

Complete allconditions

No

End

Acquire LMSprocess

Yes

NoNo

Figure 8: Flow chart of the virtual prototype multicondition automatic simulation.

6 International Journal of Aerospace Engineering

The total variance of the model is

V E y = 〠j∈Z0

Λj = 2〠+∞

j=1Λj 12

According to Sobol variance decomposition thought, themodel can be decomposed into functions of single variableand combined variable, namely,

V y = 〠n

i=1Vi +〠

i≠jVi,j + 〠

i≠j≠kVi,j,k +⋯ +V1,2,3,…,n, 13

where Vi =V E y∣xi which represents the variance of xi,Vi,j =V E y∣xi, xj −Vi − V j which represents the varianceof xi contributed by xj, and Vi,j,k~V1,2,…,n are by analogy.The first-order sensitivity coefficient Sxi of the variable xi isdefined as below.

Sxi =Vi

V y14

The total effects of the variable xi include the single effectand interacting effects on the output of the model. The globalsensitivity coefficient of the variable xi is defined as

STxi = Sxi +〠i≠jSxi ,xj + 〠

i≠j≠kSxi ,xj ,xk + Sx1,x2,…,xn 15

The difference between global sensitivity coefficient STxiand first-order sensitivity coefficient Sxi reflects the degreeof interaction between xi and other variables.

5. Landing Variable Sensitivity and ExtremeValue Analysis

When using EFAST method to analyze the global sensitivity,the input of probability distribution of the variable and theoutput of the model corresponding to the inputting samplesare required [18]. The landing variables given by standards[7] should be modified according to practical situation. Thetype of the landing variable distribution maintains constant,then the average and standard deviations of the landing var-iables need to be modified to satisfy the design requirements.Table 1 shows the distribution of the six landing variablesunder the given design requirements. Due to the effects ofthe deck layout, the value range of the rolling angle and yaw-ing angle are not bilateral symmetric.

The input parameters of EFAST analysis method use thedistribution of the landing variable and its value rangeobtained after adopting multivariate distribution presentedin Section 2.2. The EFAST sampling is carried out by using(8), then 6438 samples are obtained. The samples in the gen-erated interval [0, 1] are mapped back to the distribution ofthe landing variable, used to input the variables of the virtualprototype. For all landing conditions, the virtual prototypeand its secondary development technology are used to ana-lyze the landing loads automatically. According to the analyt-ical results of the landing load, the first-order and global

sensitivity coefficients of landing variables to landing loadsare calculated.

5.1. The Sensitivity Analysis of Main Landing Gear Load.When the carrier-based aircraft lands on the deck, mainlanding gear touches the deck first in most cases. The verticalimpact load is the main load that acts on the main landinggear. The sensitivity analysis results of the vertical load ofthe main landing gear are shown in Figure 9.

The analysis results show that sinking speed and rollingangle are the main affecting variables of the vertical load ofthe main landing gear. The total contribution of the two var-iables is more than 82%, while the remaining landing vari-ables only have little coupling effects on the vertical load.The sinking speed has larger first-order sensitivity coefficientin the analysis of the left main landing gear while the rollingangle has larger first-order sensitivity coefficient in the anal-ysis of the right main landing gear. When the right mainlanding gear touches the deck first, the rolling angle is largeand the landing load is relatively severe. When the left mainlanding gear touches the deck first, the rolling angle is smalland the sinking speed is the main effect of the vertical load.

Figure 9 also shows the distribution of the cumulativefrequency of the main landing gear’s vertical load. The larg-est vertical load (853 kN) condition of the left main gear isVV = 4 76m/s and θR = −1 99° The largest vertical load(1278 kN) condition of the right main gear is VV = 4 77m/s and θR = 7 09° The extreme value of the vertical load ofthe right landing gear is 1.5 times of the one of the left land-ing gear.

The sensitivity analysis results of the main landing gear’slateral load are shown in Figure 10. It is seen that the sinkingspeed, rolling angle, and yawing angle are the main affectingfactors of the main landing gear’s lateral load. The total con-tribution of the three variables is over 91% while remainingvariables barely have direct effects on the lateral load.

From the cumulative frequency curve of the lateral loadshown in Figure 10, it is seen that the extreme value of the lat-eral load of the left main landing gear is nearly the same asthat of the right main landing gear. However, the averageload of the right main landing gear is larger than that of theleft main landing gear. Under the condition that the largestlateral load of the left main landing gear is 120 kN, the sink-ing speed VV = 4 62m/s, rolling angle θR = 7 36°, and yawingangle θY = 0 84° Under the condition that the largest lateralload of the right main landing gear is 122 kN, the sinking

Table 1: Aircraft landing variables distribution.

Landing variables Mean Deviation

VE 166.164 9.26

VV 3.5052 0.9184

θP 4.5 0.85

θR 2.2 2.5

θR 0 2.8

θY 0.8 0.7

7International Journal of Aerospace Engineering

speed VV = 4 61m/s, rolling angle θR = 7 36°, and yawingangle θY = 0 84°

The sensitivity analysis results of the main landing gear’scourse load are shown in Figure 11. The analysis results showthat the sinking speed and rolling angle are the main affectingvariables of the main landing gear’s course load. The two var-iables’ total contribution is 76% while the remaining landingvariables have little coupling effects on the course load.Known from the cumulative frequency curve, the load distri-bution and extreme load of the left main landing gear isalmost the same as that of the right main landing gear. Underthe condition that the largest course load of the left mainlanding gear is 609 kN, the sinking speed VV = 4 59m/s androlling angle θR = −2 04° Under the condition that the larg-est course load of the right main landing gear is 721 kN, the

sinking speed VV = 4 76m/s and rolling angle θR = 7 09°The course loads of the left and right main landing gears havelittle difference. The extreme value of the course load alsooccurs under the condition of large sinking speed and largerolling angle.

Figures 9 and 11 show that the sensitivity analysis resultsof the course load are similar to that of the vertical load. Thesensitivity coefficients of sinking speed and rolling angle arenearly the same. Besides, the course loads of the right mainlanding gear are all greater than that of the left main landinggear. The course load has rather high correlation with thevertical load.

5.2. Nose Landing Gear Load and Barycenter OverloadAnalysis. The sensitivity analysis results of the nose landing

VE VV 𝜃P 𝜃R 𝜃R 𝜃Y0.0

0.1

0.2

0.3

0.4

Sens

itivi

ty fa

ctor 0.5

0.6

0.7

Landing parameter

First-order sensitivityGlobal sensitivity

(a) Sensitivity factor of vertical load of the left main gear

Frequency1 10 100 1000

Load

(N)

9×105

8

7

6

5

4

3

2

1

(b) Frequency curve of vertical load of the left main gear

VE VV 𝜃P 𝜃R 𝜃YLanding parameter

0.0

0.1

0.2

0.3

0.4

Sens

itivi

ty fa

ctor

0.5

0.6

0.7

First-order sensitivityGlobal sensitivity

𝜃R

(c) Sensitivity factor of vertical load of the right main gear

Frequency1 10 100 1000

2.0 × 105

4.0 × 105

6.0 × 105

8.0 × 105

1.0 × 106

1.2 × 106

Load

(N)

(d) Frequency curve of vertical load of the right main gear

Figure 9: Sensitivity and cumulative frequency curve of vertical load of the main landing gear.

8 International Journal of Aerospace Engineering

gear’s three directional loads are shown in Figure 12. It canbe known that the sinking speed and rolling angle are themain affecting variables of the nose landing gear’s courseload and their first-order sensitivity coefficients are 0.59and 0.14, respectively. The pitching angle is the secondaryaffecting variable, and its first-order sensitivity coefficient is0.004. Other variables have a few coupling effects on thecourse load.

For the lateral load of the nose landing gear, the rollingangle is the main affecting variable and its first-order sensi-tivity coefficient is 0.6. The sinking speed, pitching angle,and yawing angle have a few effects on the lateral load. Forthe vertical load of the nose landing gear, the pitching angle,rolling angle, and sinking speed are the main affecting

variables and their first-order sensitivity coefficients are0.53, 0.14, and 0.07, respectively. Other variables do not haveappreciated effects on the vertical load.

From the cumulative frequency curve of the nose landinggear’s three directional loads, the sinking speed VV = 5 41m/s, the pitching angle θP = 0 62°, and the rolling angleθR = 3 93° under the condition that the largest course loadof the nose landing gear is 133 kN. Under the condition thatthe largest lateral load of the nose landing gear is 93 kN, thesinking speed VV = 4 77m/s, the pitching angle θP = 3 09°,and the rolling angle θR = 7 09°, which is the large rollingangle condition. Under the condition that the largest verticalload of the nose landing gear is 196 kN, the sinking speedVV = 3 44m/s, the pitching angle θP = 6 23°, and the rolling

VE VV 𝜃P 𝜃R 𝜃Y

0.0

0.1

0.2

0.3

0.4

Sens

itivi

ty fa

ctor 0.5

0.6

0.7

Landing parameter

First-order sensitivityGlobal sensitivity

𝜃R

(a) Sensitivity factor of lateral load of the left main gear

Frequency1 10 100 1000

Load

(N)

1.2 × 105

1.0 × 105

8.0 × 104

6.0 × 104

4.0 × 104

2.0 × 104

0.0

(b) Frequency curve of lateral load of the left main gear

0.0

0.1

0.2

0.3

0.4

Sens

itivi

ty fa

ctor 0.5

0.6

0.7

VE VV 𝜃P 𝜃R 𝜃Y

Landing parameter

First-order sensitivityGlobal sensitivity

𝜃R

(c) Sensitivity factor of lateral load of the right main gear

Load

(N)

1.2 × 105

1.0 × 105

8.0 × 104

6.0 × 104

4.0 × 104

2.0 × 104

0.0

Frequency1 10 100 1000

(d) Frequency curve of lateral load of the right main gear

Figure 10: Sensitivity and cumulative frequency curve of lateral load of the main landing gear.

9International Journal of Aerospace Engineering

angle θR = 6 07°, which is the condition of medium sinkingspeed and large pitching angle.

The sensitivity analysis results of the barycenter overloadof the airframe are shown in Figure 13. It is seen that sinkingspeed and rolling angle are the main affecting variables andtheir first-order sensitivity coefficients are 0.74 and 0.18,respectively. Under the condition that the largest barycenteroverload is 5.11, the sinking speed VV = 4 77m/s and therolling angle θR = 7 09°, which is the condition of large roll-ing angle and relatively large sinking speed. The barycenteroverload of the airframe mainly concentrates between 2.0and 4.0.

5.3. Sensitivity Analysis of the Landing Load of the Carrier-Based Aircraft. According to the sensitivity analysis resultsof the main landing gear, nose landing gear, and barycenteroverload, the influence degree of the landing variables onlanding loads can be classified as shown in Table 2. The sink-ing speed and pitching angle are the main affecting factors ofthe landing loads. The pitching angle and yawing angle havesignificant effects on some loads. The ground critical velocityand rolling rate have little effect on the landing loads, andthey are influenced a lot by the change of the arresting cableload and the motion of the deck due to the restrictions of thesimplified condition of the virtual prototype.

VE VV 𝜃P 𝜃R 𝜃Y

Landing parameter

First-order sensitivityGlobal sensitivity

0.0

0.1

0.2

0.3

0.4

Sens

itivi

ty fa

ctor

0.5

0.6

0.7

𝜃R

(a) Sensitivity factor of course load of the left main gear

×105

6

5

4

3

2

Load

(N)

Frequency1 10 100 1000

(b) Frequency curve of course load of the left main gear

0.0

0.1

0.2

0.3

0.4

Sens

itivi

ty fa

ctor

0.5

0.6

0.7

0.8

VE VV 𝜃P 𝜃R 𝜃Y

Landing parameter

First-order sensitivityGlobal sensitivity

𝜃R

(c) Sensitivity factor of course load of the right main gear

Frequency1 10 100 1000

6

7

8

5

4

3

2

Load

(N)

×105

(d) Frequency curve of course load of the right main gear

Figure 11: Sensitivity and cumulative frequency curve of course load of the main landing gear.

10 International Journal of Aerospace Engineering

VE VV 𝜃P 𝜃R 𝜃Y0.0

0.1

0.2

0.3

0.4

Sens

itivi

ty fa

ctor

0.5

0.6

0.7

0.8

Landing parameterFirst-order sensitivityGlobal sensitivity

𝜃R

(a) Sensitivity factor of the course load of nose gear

Frequency1 10 100 1000

Load

(N)

1.2 × 105

1.4 × 105

1.0 × 105

8.0 × 104

6.0 × 104

(b) Frequency curve of the course load of nose gear

VE VV 𝜃P 𝜃R 𝜃Y0.0

0.1

0.2

0.3

0.4

Sens

itivi

ty fa

ctor

0.5

0.6

0.7

0.8

Landing parameter

First-order sensitivityGlobal sensitivity

𝜃R

(c) Sensitivity factor of the lateral load of nose gear

Frequency1 10 100 1000

1.0 × 105

8.0 × 104

6.0 × 104

4.0 × 104

2.0 × 104

Load

(N)

(d) Frequency curve of the lateral load of nose gear

VE VV 𝜃P 𝜃R 𝜃R 𝜃Y0.0

0.1

0.2

0.3

0.4

Sens

itivi

ty fa

ctor

0.5

0.6

0.7

0.8

Landing parameter

First-order sensitivityGlobal sensitivity

(e) Sensitivity factor of the vertical load of nose gear

Frequency1 10 100 1000

1.5 × 105

1.8 × 105

1.2 × 105

9.0 × 104

6.0 × 104

Load

(N)

(f) Frequency curve of the vertical load of nose gear

Figure 12: Sensitivity and cumulative frequency curve of three directional loads of the nose landing gear.

11International Journal of Aerospace Engineering

When selecting the representative landing conditions ofthe carrier-based aircraft, the sinking speed, rolling angle,and pitching angle which have remarkable sensitivity shouldbe discretized into several intervals. The intervals are selectedto represent the values of variables. Different representativelanding conditions are formed by the combination of the var-iables to reflect the landing load condition during the entirelanding process.

5.4. Extreme Value Analysis of the Landing Loads of theCarrier-Based Aircraft. In the structure design stage, it is dif-ficult to obtain the most severe load condition of the air-frame during its service. When determining the mostsevere parameter combination, some conservative condi-tions are usually selected. For example, a certain or severalextreme values of landing variables and the average ofremaining landing variables are selected as extreme valueconditions of the load for the researches. The conditionsrepresent different landing conditions such as tail sinking,yawing landing, and landing with single landing gear. Someconditions will not happen during the service period, whichleads to structural overdesign.

There are many landing samples selected in the sensitiv-ity analysis of the landing load of the carrier-based aircraft.Approximately, the extreme load conditions of these samplescan be used as extreme load conditions of all landing

conditions to carry out the comparative analysis. From thecomputed results of the 6438 samples in EFAST sensitivityanalysis method, the extreme value conditions of the threedirectional loads of the main and nose landing gear and bar-ycenter overload are usually the condition that the sinkingspeed, rolling angle, or pitching angle are large and the land-ing variables do not pass the maximum of the value range.For further study in the rationality and security of the conser-vative condition design, typical condition design is selectedfor the comparative calculation.

Table 3 shows eight kinds of typical design conditionsselected to compare with samples of EFAST sensitivity anal-ysis in extreme loads of each direction. From the comparison,it is seen that the maximum gravity overload is slightly largerthan that of the EFAST samples and the difference value is23.5% approximately.

The extreme vertical and course loads of the landing gearof the typical design condition are nearly the same as theextreme values of EFAST samples. The maximum courseload of the left and right main landing gear of the typicaldesign condition is almost the same as the maximum loadof the EFAST samples, and the percentage differences areabout 6.2% and 16.0%, respectively. The maximum verticalload of the left and right main landing gear of the typicaldesign condition is almost the same as the maximum loadof the EFAST samples, and the percentage differences areabout 2.5% and 12.0%, respectively. The maximum courseand vertical loads of the nose landing gear of the typicaldesign condition are quite different from the maximum loadsof the EFAST samples, and the percentage differences areabout 26.3% and 28.6%, respectively.

For the lateral load which is the secondary load, the max-imum load of the typical design condition is quite differentfrom that of the EFAST samples and the percentage differ-ence is about 35.2%. From the overall analysis, the gravityoverload and extreme load in each direction of the eight

Table 2: Sensibility of landing variables to landing loads.

Type High Moderate Low

Main gear VVθR θY VEθPθR

Nose gear VVθRθP θY VEθR

Gravity overload VVθR — VEθPθRθY

VE VV 𝜃P 𝜃R 𝜃Y0.0

0.1

0.2

0.3

0.4

Sens

itivi

ty fa

ctor

0.5

0.6

0.7

0.8

Landing parameter

First-order sensitivityGlobal sensitivity

𝜃R

(a) Sensitivity factor of barycenter overload

Frequency1 10 100 1000

1.0

2.0

3.0

4.0

5.0

Load

(N)

(b) Frequency curve of barycenter overload

Figure 13: Sensitivity and cumulative frequency curve of the barycenter overload.

12 International Journal of Aerospace Engineering

kinds of typical design conditions are nearly the same asthose of the EFAST samples, and the key loads of the typicaldesign conditions are slightly larger. The extreme loadsobtained by the conservative condition design are safe to beused in structure strength analysis and would not causeoverdesign.

6. Conclusions

Based on the results reported, several conclusions canbe drawn.

Firstly, the multivariate distribution has a large influenceon the distribution of the landing variables. After multivari-ate distribution processing, the distribution of the landingvariables is still close to the normal distribution, and themean values of the variables maintain constant while thestandard deviation decreases.

Secondly, the results of global sensitivity analysis showthat all the six landing variables have effects on landing loads.The sinking speed and rolling angle almost influence allkinds of landing loads significantly and are the main affectingfactors. The pitching angle and yawing angle may only influ-ence some landing variables a lot, and they are the secondaryaffecting factors. The results of extreme value analysis showthat it is safe to adopt typical condition design in the struc-tural strength analysis.

Conflicts of Interest

The authors declare that there is no conflict of interestregarding the publication of this paper.

References

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Table 3: Extremum loads of typical landing conditions.

Condition of maximum variable(remaining variables are mean)

Gravity overloadPort main gear (kN) STBD main gear (kN) Nose gear (kN)

Course Lateral Vertical Course Lateral Vertical Course Lateral Vertical

VV 5.5 564 81 787 692 40 1178 141 43 192

θR 3.9 340 63 398 586 52 1000 115 59 173

θP 2.8 328 53 371 459 26 645 113 33 195

VVθR 6.3 571 115 829 752 75 1431 166 74 219

VVθP 5.6 556 90 784 556 46 1166 140 46 215

VVθRθP 6.2 564 118 827 649 75 1398 148 68 246

VVθRθY 6.2 564 158 832 836 69 1368 168 69 231

VVθRθPθY 6.1 558 160 830 649 79 1333 154 84 252

EFAST samples 5.1 609 120 853 721 122 1278 133 93 196

Difference of maximum variable 23.5% −6.2% 33.3% −2.5% 16.0% −35.2% 12.0% 26.3% −9.7% 28.6%

13International Journal of Aerospace Engineering

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14 International Journal of Aerospace Engineering

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