Research ArticleKinematic Singularity and Bifurcation Analysis of SidestayLanding Gear Locking Mechanisms

Kui Xu , Yin Yin, Yixin Yang, Hong Nie , and Xiaohui Wei

State Key Laboratory of Mechanics and Control of Mechanical Structures, Key Laboratory of Fundamental Science for NationalDefense-Advanced Design Technology of Flight Vehicle, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Correspondence should be addressed to Hong Nie; hnie@nuaa.edu.cn

Received 19 November 2020; Revised 16 March 2021; Accepted 17 June 2021; Published 4 August 2021

Academic Editor: Jun-Wei Li

Copyright © 2021 Kui Xu 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.

A dual-sidestay landing gear is prone to locking failure in the deployed state due to the restriction of movement between twosidestays. However, the principle of its locking movement still remains unclear. The present study is aimed at investigating thesynchronous locking performance of the dual-sidestay landing gear based on the singularity and bifurcation theory. From theperspective of the kinematic mechanism, the reason for high sensitivity to structural dimensions in the locking process isexplained, and the locked position is investigated by employing the numerical continuation method in the case of a single-sidestay landing gear. Afterwards, the reason for the locking failure of the dual-sidestay landing gear is analyzed, and akinematic optimization method for the synchronous locking is proposed. The results reveal that the lock links must reach thelower overcenter singular point to fully lock the landing gear, and the singular point is sensitively affected by structuralparameters. Owing to the different positions of singular points, the movements of fore and aft sidestays seriously restrict eachother, causing locking failure of the dual-sidestay landing gear. The singular points of two sidestays can be optimized to beapproximately identical, making their movements more coordinated.

1. Introduction

A supercritical wing is extensively used in large aircraft forattaining better aerodynamic performance [1, 2]. For thesupercritical wing, however, the stowage for the landing gearis considerably smaller than that for the conventional wing.Narrow space signifies that a conventional planar (2D) land-ing gear can hardly comply with the design requirements,since the sidestay links are located above the main strut afterfolding, which occupy a large vertical space, as shown inFigure 1(a). In contrast, for a three-dimensional (3D) landinggear, the sidestay plane can be rotated about a specific axis, sothat it is overall folded over the side of the main strut. As aresult, the 3D landing gear excellently reduces the size ofthe vertical space, as shown in Figure 1(b).

The extensive use of composite material has contributedhugely to the reduction in aircraft weight. A composite wing,however, does not carry as much ground load as a metallicone. From the load transfer perspective, the ground impactload should not be borne entirely by the wing. Thus, a

dual-sidestay (DS) landing gear is used to transfer groundload to the wing and fuselage [3, 4]. Compared to a single-sidestay (SS) landing gear, the DS landing gear has anadditional sidestay, as shown in Figure 2, and the motioncoordination between fore and aft sidestays appears partic-ularly important. In the case of an unreasonable design ofstructural parameters, the two sidestays can hinder andconstrain each other, disallowing its complete locking.Hence, it is imperative to investigate the motion coordina-tion between two sidestays.

The multidisciplinary time-domain cosimulation tech-nology [5–7] has often been adopted to research landing gearmechanisms. However, this approach requires substantialiterations in handling highly nonlinear problems, which istime- and labour-consuming. Numerical continuation [8, 9]has been applied to aircraft design for several years [10],which can enable fast and efficient exploration of highly non-linear problems of the landing gear. Following the initial pro-posal of this method for analyzing landing gear mechanismsby Knowles et al. [11], the research approach integrating

HindawiInternational Journal of Aerospace EngineeringVolume 2021, Article ID 6685635, 15 pageshttps://doi.org/10.1155/2021/6685635

numerical continuation and bifurcation analysis has becomeincreasingly mature [12]. Exploiting this method, Knowleset al. [13–16] and Yin et al. [17] analyzed the bifurcationcharacteristics of different landing gear mechanisms sequen-tially and concluded that the landing gear would inevitablyundergo a jump phenomenon at bifurcation points in thelocking process. However, when solving the dynamic equa-tions of the DS landing gear using the numerical continua-tion method, it has been found that the appearance ofbifurcation points does not necessarily imply that twosidestays can be locked synchronously, which indicates thelimitation of using the dynamic equations in solving DSmechanisms with extremely high motion sensitivity.

The DS landing gear has several singular points in itsretraction cycle, which cause extremely high sensitivity ofmechanism motion. Hence, the DS mechanism is analyzedfrom a kinematic singularity perspective in this paper. Exist-ing methods concerning singularity include the Jacobianmatrix method [18], the Grassmann line geometry [19],and the differential geometric theory [20], while attemptingto avoid the singular position in the mechanism as far aspossible [21, 22]. For instance, Xie et al. [23] and Li andHerve [24] used the above methods to explore the singular-ity characteristics of different mechanisms. Nevertheless, lit-tle literature focuses on the singularity of landing gearmechanisms, particularly the complex spatial DS mecha-nisms. Therefore, continuation analysis on the kinematicsequations of the DS landing gear is performed, and the

locked position as well as the underlying jump phenomenonof the mechanism is analyzed from the kinematic singularityand bifurcation perspective in this paper. Furthermore, thefundamental causes of mutual hindrance and restrictionbetween fore and aft sidestays are explained, which can pro-vide crucial guidance for the design of DS landing gearmechanisms.

Initially, the locking performance of a 3D SS mechanismis analyzed. On this basis, the reasons for incomplete lockingcaused by mutual restriction between two sidestays are inves-tigated. Finally, a novel method for analyzing the synchro-nous locking of the DS landing gear is developed, which isbased on the singularity and bifurcation theory. The struc-ture of this paper is organized as follows: In Section 2, thekinematic model of a 3D SS landing gear mechanism is built,after analyzing its bifurcation and singularity characteristics,and the jump phenomenon and the locked position are ana-lyzed in Section 3. Meanwhile, the effects of structuralparameters on the locked position are explored. In Section4, the principle of mutual hindrance between fore and aftsidestays for the 3D DS landing gear mechanism is investi-gated based on the foregoing analysis, and the fundamentalcauses of asynchronous locking are identified. Furthermore,a set of optimal solutions satisfying constraints are given withthe aid of multiobjective particle swarm optimization (PSO).The application of the proposed method can effectivelyachieve the synchronous locking at the initial design phaseof DS landing gear mechanisms.

Main strut

Sidestay plane

(a) (b)

Verticalspace

Verticalspace

Figure 1: Vertical storage space of 2D versus 3D landing gear.

Direction of travel

Fore sidestay plane

Aft sidestay plane

Sidestay plane

Main strut

Direction of travel

Figure 2: Changes in the DS versus SS landing gears.

2 International Journal of Aerospace Engineering

2. Kinematic Model of the LandingGear Mechanism

In Figure 3, a certain 3D landing gear mechanism is illus-trated, which consists of five links (two sidestay links, twolock links, and a main strut), as well as three nodes (two side-stay nodes and a lock node). These various components areconnected via planar revolute joints. To achieve the 3Dretraction function, the axes of sidestay nodes (axes 2 and 4in Figure 3), the axis of the lock node (axis 1), and the rota-tion axis of the main strut (axis 3) must intersect at one point(point o) [25]. In this way, the motion of sidestay can bealways within the same plane. During the retraction cycle ofthe landing gear, the main strut rotates along the axis 3, whilethe sidestay not only performs the folding motion but alsorotates about the axis 2, so that the rotation plane of the mainstrut is not in the same plane as the sidestay. The entireretraction and extension processes can be summarized as fol-lows: When retracting the gear, the unlock actuator is neededto unlock the lock links. Then, the retraction actuator can beengaged to allow the main strut to rotate about axis 3, and thesidestay moves accordingly. Finally, the landing gear reachesthe deployed position and gets locked. When extending thegear, after upper unlocking, the retraction actuator suppliesa pushing force to enable slow extension of the main strutaround axis 3. When approaching the lower locked position,the lock links jump from the upper to the lower overcenterstate by the action of locking springs, thereby reaching adownlock state. The lock links are equipped with a stop blockto prevent excessive downward folding. Eventually, thelanding gear stops at the deployed position to complete thelocking procedure.

2.1. Coordinate Transformation. Since the rotation plane ofthe main strut is not in the same plane as the sidestay, itcan hardly analyze the mechanism in the same coordinatesystem. Owing to the fact that the motion of the sidestay isalways within the same plane [6, 26], the mechanism modelcan be divided into different coordinate systems to describe

the motion of the main strut and sidestay. The transforma-tion matrix is employed to accomplish the data exchangebetween them.

Figure 3 presents the establishment of the coordinate sys-tems. In the global coordinate system O − XYZ, the origin Ois the intersection between the main strut and the axis 3; theX-axis is parallel to axis 3 (positive forward); the Z-axis isaligned with the gravity vector (positive down); and thedirection of the Y-axis is determined by the right-hand rule.In the local coordinate system o − xyz, the origin o is theintersection point of four axes (axes 1-4); the y-axis is thedirection of axis 2; the z-axis is defined to be perpendicularto the y-axis within the sidestay plane; and the direction ofthe x-axis is determined by the right-hand rule.

With the operation of the landing gear, the main strutrotates about the X-axis, and the sidestay plane rotates aboutthe y-axis. Accordingly, the total transformation matrix T forthe global-to-local coordinate system is

T = T0 ⋅ T1, ð1Þ

where T0 is the transformation matrix of the global to thelocal coordinate system in the initial state (extended, lockedposition) and T1 is the transformation matrix of the localcoordinate system rotating about the y-axis.

In the initial state, given the presence of both transla-tional and rotational transformations during the global-to-local system conversion, the homogeneous coordinatetransformation is adopted herein. Let e1 = ½e11 e12 e13�T ande2 = e21 e22 e23½ �T be the base vectors in the global andlocal systems, respectively, and oO be the vector of origin Oin the local system; then,

T0 =e1 ⋅ e2T oOT

0 1

" #: ð2Þ

Z

Y X

Z

X

Y

y z

x

y

x

z

y

xz

Sidestay links

Lock links

LockingSpring

Unlockactuator

O (o)O o

Oo

Main strut

Wheel

Sidestay nodes

Lock node

Axis 3

Axis 1 Axis 2

Axis4

Figure 3: A certain 3D landing gear mechanism.

3International Journal of Aerospace Engineering

Assuming α is the rotation angle of local system about they-axis, then

T1 =

cos α 0 sin α 00 1 0 0

−sin α 0 cos α 00 0 0 1

2666664

3777775: ð3Þ

2.2.Motion Constraint Equations. In Figure 4, the schematic ofthe landing gear mechanism is presented. The constraint rela-tions between various links can be derived from eight variablesðXi, Yi, Zi,Θi, xi, yi, zi, and θiÞ , where ðXi, Yi, ZiÞ and ðxi, yi,ziÞare the global and local coordinates of the gravity centerof the i-th link, respectively; Θi denotes the angle betweenthe i-th link and the Z-axis (specifically, Θ1 is the retractionangle of the main strut); and θi is the angle between the i-thlink and the y-axis.

Since the global and local coordinates can be intercon-verted through the coordinate transformation, the positionof various links can be identified independently with ðXi, Yi,Zi,ΘiÞ or ðxi, yi, zi, θiÞ . There are twenty independent geo-metric variables for the SS mechanism. As the analysisreveals, the mechanism has nineteen independent constraintequations (see Equation (4) and Table 3 in the Appendix), sothe degree of freedom (DOF) is one.

X1

Y1‐0:5l1 ⋅ cos Θ1

Z1‐0:5l1 ⋅ sin Θ1

x2

y2 − 0:5l2 ⋅ cos θ2 − yA

z2 − 0:5l2 ⋅ sin θ2 − zA

x3

y3 − 0:5l3 ⋅ cos θ3‐y2 − 0:5l2 ⋅ cos θ2z3 − 0:5l3 ⋅ sin θ3‐z2 − 0:5l2 ⋅ sin θ2

x4

y4 + 0:5l4 ⋅ cos θ4‐y2 − 0:5l2 ⋅ cos θ2z4 − 0:5l4 ⋅ sin θ4‐z2 − 0:5l2 ⋅ sin θ2

x5

y5 + 0:5l5 ⋅ cos θ5‐y4 + 0:5l4 ⋅ cos θ4z5 − 0:5l5 ⋅ sin θ5‐z4 + 0:5l4 ⋅ sin θ4

y5‐0:5l5 ⋅ cos θ5‐yEz5 − 0:5l5 ⋅ sin θ5‐zEy3 + 0:5l3 ⋅ cos θ3‐yCz3 + 0:5l3 ⋅ sin θ3‐zC

2666666666666666666666666666666666666666666666666664

3777777777777777777777777777777777777777777777777775

= 0, ð4Þ

where li denotes the length of the i-th link; y∗, z∗ denote thelocal coordinates of the points ∗ (points A/C/E in Figure 4),which can be derived from the node parameters as follows:

yA

zAyC

zC

yE

zE

2666666666664

3777777777775=

yF + lAF ⋅ cos φAF

lAF ⋅ sin φAF

yG‐lCG ⋅ cos φCG

zG‐lCG ⋅ sin φCG

yH‐lEH ⋅ cos φEH

zH‐lEH ⋅ sin φEH

2666666666664

3777777777775, ð5Þ

where lAF, lCG, lEH denote the node lengths and φAF, φCG, φEHdenote the angles between the node axes and the y-axis.Meanwhile, the local coordinates of points G and H areobtained through coordinate transformation as follows:

xG

yG

zG

1

2666664

3777775 = T ⋅

XG

YG

ZG

1

2666664

3777775, ð6Þ

xH

yH

zH

1

2666664

3777775 = T ⋅

XH

YH

ZH

1

2666664

3777775: ð7Þ

2.3. Bifurcation Equation of Mechanism Motion. Given theone DOF of the 3D mechanism, Θ1 is selected as the inde-pendent control parameter, while the other variables can beregarded as the state variables. Accordingly, Equation (4)can be rewritten as

F x, λð Þ = 0, ð8Þ

l1

l2

l3

l4l5

yz

o

Z

O (o)y

Y

z

xx

(x3, y3, z3)

(x5, y5, z5)

lEH

lCG

lAF

A

B

C

D

E

F

G

HZ

XO

𝜑AF

𝜃3

𝜃2𝜃4

𝜃5

𝛩1

Figure 4: Schematic diagram of the 3D landing gear mechanism.

4 International Journal of Aerospace Engineering

where x = ðX1, Y1, Z1, x2, y2, z2, θ2,⋯,x5, y5, z5, θ5Þ and λ =Θ1.

Equation (8) expresses the variation of state parameter xwith the bifurcation control parameter λ, which is called thebifurcation equation of the 3D mechanism. Utilizing thisequation, the bifurcation characteristics of the mechanismcan be analyzed.

3. Analysis of the SS Landing Gear Mechanism

3.1. Bifurcation Analysis. The retraction and extension pro-cesses of the landing gear can be simulated through numeri-cally continuing Equation (8), and the variation of stateparameters (e.g., the overcenter angle θoc = θ4‐θ5 of locklinks) with the control parameter Θ1 can be derived, asshown in Figure 5. For convenience of observation, the kine-matic sketches of the mechanism are given in Figure 6, whichcorrespond to various points in Figure 5.

In Figure 5, the motion bifurcation curve is symmetricalabout θoc = 0. The upper half of the curve represents theupward folding of lock links (U1, U2, and S1), while thelower half represents their downward folding (D1, D2, andS3). Three bifurcation points (S1, S2, and S3) appear in thefigure, at which two types of motion trends of links are pres-ent. As the landing gear extends, it moves along the curvefrom U1 to U2. At the beginning, Θ1 changes drastically,while θoc changes a little. On the contrary, when the landinggear gets close to the deployed state,Θ1 is almost unchanged,while θoc changes drastically. Bifurcation occurs when thelanding gear moves to the point S1, and two motion branchesappear, which correspond to the two motion trends of side-stay links (the dashed lines in Figure 6(c)). The point S1 cor-responds to the collinear position of upper and lower sidestaylinks. At this time, Θ1 is the smallest, implying that the mainstrut is extended to the farthest position. By the favorableaction of locking springs, the landing gear will move to pointS2, at which the upper and lower lock links are collinear. Sim-ilarly, two motion branches appear when the mechanismmoves to S2, which correspond to the two motion trends oflock links (the dashed lines in Figure 6(d)). When moving

to S3 by the action of springs, the upper and lower sidestaylinks are collinear again, and the lock links are folded down.The appearance of two motion branches of sidestay links isnoted (the dashed lines in Figure 6(g)). Generally, the landinggear is provided with a stop block here to prevent furtherdownward motion of lock links, so it is impossible to reachthe states D1 and D2 in reality.

3.2. Singularity Analysis. Derivation of Equation (8) withrespect to time yields is as follows:

Ax: + B λ:

= 0, ð9Þ

where A = ∂F/∂x and B = ∂F/∂λ. Let J = ‐A‐1 ⋅ B; then,

x: = J ⋅ λ:

: ð10Þ

When the mechanism is in a singular position, detðAÞ = 0 [27], i.e., det ðJÞ⟶∞. Since the bifurcation pointsof a mechanism must be in a singular position [28], thepoints S1, S2, and S3 are the singular positions of the landinggear mechanism.When considering the locking status of locklinks, Equation (10) can be rewritten as

θ:

oc = J ⋅Θ:

1: ð11Þ

It can be seen from Equation (11) that in the vicinity ofsingular positions, due to J ⟶∞, the lock link speed θ

:

ocis considerably high even if Θ

:

1 is very low. Accordingly, thelock links undergo a quick jump when the landing gear isclose to locking. The slope of the curve in Figure 5 can bederived as

k = ∂θoc∂Θ1

= ∂θoc/∂t∂Θ1/∂t

= θ:

oc

Θ:

1: ð12Þ

The equation of k = J can be derived from Equations (11)and (12). That is, the value of J corresponds to the slope of

0 20 40 60 80 100

Θ1 (deg)

–160

–80

0

80

160

𝜃oc

(deg

)

U1U2

D1D2

(a)

Θ1 (deg)

𝜃oc

(deg

)

–1 0 1 2 3×10–5

–6

–3

0

3

6

S2

S3

S1

(b)

Figure 5: Bifurcation curve of the 3D SS landing gear ((b) presents an enlarged view of the rectangular frame in (a)).

5International Journal of Aerospace Engineering

motion trajectory in Figure 5. Meanwhile, using the principleof virtual work and combining with Equation (11), Equation(13) can be obtained.

M1 = J ⋅Moc, ð13Þ

whereM1 denotes the external moment of the main strut andMoc denotes the internal moment of lock links (centered onthe intersection of the lock links).

In the unlocking process of the landing gear, the sidestaylinks and lock links must be folded. Since the stop blockrestricts the downward folding of lock links, the lock linkscan only be folded upwards. Unlocking is impossible toachieve without unlocking the actuator force at the singularpoints (Figure 7). If the unlocking force is not provided, over-coming the drag load from the upward folding of lock linkswill be required. This drag load is the equivalent momentMoc of locking springs. From Equation (13), it is clear thatJ ⋅Moc ⟶∞ when J ⟶∞. In theory, the landing gearcan be unlocked only when the external moment M1 is infi-nite. In other word, the landing gear will never be unlockedat the singular point. Therefore, the singular feature can beexploited to lock the landing gear near the singular point.

3.3. Analysis of Locked Position. As is clear from the aboveanalysis, the landing gear should be locked near a singularposition. A total of three singular positions are present. Thissection analyzes at which singular position the landing gearshould be locked.

The overcenter angle of sidestay links is defined as θd =θ2‐θ3. Figure 8 shows the curves obtained by numericallycontinuing Equation (8), where θd is used as the state param-eter andΘ1 is used as the control variable. The black line rep-resents the upward folding of lock links (U1, U2, and S1),whereas the grey line represents the downward folding oflock links (D1, D2, and S3). The black and grey lines overlap

each other. Corresponding to Figures 5 and 6, there are threesingular points on the motion trajectories. The point S2 is atthe intersection of black and grey lines.

The landing gear is highly likely to vibrate during the air-craft take-off, landing, and taxiing [29, 30], so the downlockmechanism must possess a certain function of perturbationresistance. The stop block, as the important component ofthe downlock mechanism, is mainly used to prevent theexcessive motion of lock links. Eventually, the landing gearis locked by the stop block. The locked position can be con-trolled by setting up the stop block, which is generallyarranged at the lock links. To achieve complete locking, themotion of lock links must be restricted in both upward anddownward directions. The stop block restricts the downwardmotion of lock links, while the upward motion is restricted bythe structural design and locking springs. In this process, theunlocking of the downlock mechanism can be avoided.

The locked position of landing gear can be analyzedbased on Figure 8 combined with Figure 6.

U1

(a)

U2

(b)

S1

(c)

S2

(d)

D1

(e)

D2

(f)

S3

(g)

Figure 6: Sketches of the retraction and extension motion of a landing gear: (a) U1; (b) U2; (c) S1; (d) S2; (e) D1; (f) D2; (g) S3.

M1

Fu

Fs Lockingspring

Unlockingactuator

Figure 7: Load analysis at the singular position of landing gear.

6 International Journal of Aerospace Engineering

(1) In case the landing gear is locked at the point S1, thestop block will restrict the further downward foldingof the lock links and sidestay links. At this time, theupper and lower sidestay links are collinear, whilethe lock links are in the upper overcenter state. Whenthe sidestay links are subjected to a perturbation toleave the collinear state, they will drive the lock linksto fold upward and get unlocked. As is clear fromFigure 8, when the sidestay links are subjected to aperturbation of Δθd > 0, the landing gear will leavethe singular point S1, thereby resulting in unlocking

(2) In case the landing gear is locked at the point S2, thestop block will restrict the downward folding of thelock links. At this time, the upper and lower lock linksare collinear. It is not difficult to find that when thelock links are subjected to a perturbation to leavethe collinear state, they will drive the sidestay linksto make Δθd > 0. Thus, the landing gear will leavethe singular point S2, resulting in unlocking. Further-more, according to the force transmission relation-ship, the lock links need to bear ground load, whichwill compromise the reliability of the downlockmechanism [31]

(3) In case the landing gear is locked at the point S3, thestop block will restrict the downward folding of locklinks and the upward folding of sidestay links. At thistime, the upper and lower sidestay links are collinearagain, while the lock links are in the lower overcenterstate. When the sidestay links are subjected to adownward perturbation of Δθd < 0, the lock links willleave point S3 and move to point S2. Although thelanding gear will not be unlocked since the lock linkswill not go beyond point S2, the main strut may rotatedue to the perturbation. To diminish the rotationangle, the value of ðθocÞS3 should not be excessivelylarge in the deployed state. According to Figure 5,the landing gear is designed to have ðθocÞS3 = 2°.

To sum up, the locked position of the landing gear shouldbe at the point S3. As described in the foregoing analysis,

when the landing gear is at the singular position, the mainstrut is unable to move regardless of how large the load ofthe retraction actuator is. If the landing gear is locked at thepoint S3, it needs to pass the singular points S1 and S2. At thistime, the action of locking springs assists in passing the sin-gular points, so that the landing gear does not return alongthe original trajectory.

3.4. Effects of Structural Parameters on the Locked Position.Once the installation location of the landing gear is decided,the link length parameters will directly affect its locked posi-tion (point S3). With changes in the link lengths, the numberof singular points also changes accordingly. Thus, selectingappropriate link lengths to ensure the smooth and completelocking of the landing gear is necessary. When consideringthe effects of link lengths on the point S3, Θ1 can be treatedas a fixed parameter (Θ1 = 0), where θoc is used as the stateparameter, and the link lengths l2 and l4 are used as the controlparameters. Figure 9 presents the variation curves of the sin-gular point S3 through numerically continuing Equation (8).

The singular points exhibit a consistent variation trendwith the link lengths in the two subfigures. As l2/l4 decreases,the points S1 and S3 come together and disappear at thepoint SN. A new bifurcation point SN appears. As describedin the previous section, the landing gear should be locked atthe point S3. If the link lengths are decreased past the pointSN, the point S3 will disappear, which means failure of lock-ing. Therefore, the feasible range of l2 > 549:9mm or l4 >249:9mm should be selected to accomplish the locking ofthe landing gear. In addition, it is clear that l4 is more sensi-tive to the influence of θoc.

In actuality, the point SN corresponds to the state ofθoc = θd = 0, as shown in Figure 10(b). The state wherein linklengths are not in the feasible range is shown in Figure 10(a).In this scenario, the landing gear cannot be locked in thedeployed position. The point SN is determined jointly bythe link lengths. When one of the parameters is changed,the remaining several parameters may also undergo corre-sponding changes. Their mutual coupling together influencesthe landing gear locked position. For a more detailed analysisof the effects of link lengths, it can be computed directly by

0 20 40 60 80 100–40

0

40

80

120

160

U2 (D2)U1 (D1)

Θ1 (deg)

𝜃d (

deg)

(a)

–1 0 1 2 3–0.05

0

0.05

0.1

S2

S1 (S3)

Θ1 (deg)

𝜃d (

deg)

×10–5

(b)

Figure 8: Bifurcation curves for variation of θd with Θ1.

7International Journal of Aerospace Engineering

using two-parameter continuation, thereby obtaining a two-parameter continuation of the point SN locus, as shown bythe black line in Figure 11. Figure 11(a) is a 3D view, andFigure 11(b) is the projections of the black curve in the l2‐l4plane. With changes in the parameters, the bifurcation pointSN changes accordingly, which in turn affects the point S3 aswell as the locked position.

From Figure 11(b), it is clear that when the values of l2and l4 locate at the left side of the black curve, points S1and S3 are absent, implying that the landing gear cannot belocked at the S3 position. In contrast, when the values of l2and l4 locate at the right side of black curve, points S1 andS3 are present, and the landing gear can be fully locked. Con-sequently, the right side of the curve is the feasible region oflink lengths.

The research on the SS landing gear suggests that thelock links must be folded down, in order to achieve full lock-ing. For the SS landing gear, the downward folding motionof lock links will not be hindered due to the promotion bylocking springs. In the case of the DS landing gear, however,the sidestay links on both sides may affect each other. Evenwith the beneficial effect of locking springs, it is highly likelythat lock links on the one side undergo locking failure.Hence, the next section carries out research focusing onthe DS landing gear.

4. Analysis of DS Landing Gear Mechanism

Compared to the SS landing gear, the DS landing gear willencounter difficulty in synchronous locking of fore and aftsidestays if the additional sidestay is unreasonably designed.An obvious conclusion is that if the fore and aft sidestays arefully symmetrical, their motion trajectories will completelyoverlap. However, since the ground load needs to be trans-ferred separately to the fuselage and wing via the two side-stays, full symmetry of sidestays can hardly be achieveddue to the difference in attachment points. In case the unrea-sonable design of structural parameters causes mutualrestriction between the fore and aft sidestays, incompletelocking will be unavoidable. Thus, the motion coordinationbetween the two sidestays appears particularly important.

4.1. Constraint Equations of DS Landing Gear. Figure 12 plotsthe schematic diagrams of a DS landing gear mechanism.Clearly, the mechanism comprises nine links, which hasthirty-six independent geometric variables, and thirty-fiveindependent constraint equations, as shown in the appendix.Compared to the SS mechanism, the number of constraintequations increases by sixteen. The superscripts f and a rep-resent the relevant parameters of fore and aft sidestays,respectively, and the definitions of other parameters are con-sistent with those of the SS landing gear.

4.2. Singularity Analysis of DS Landing Gear. The retractionand extension processes of the DS landing gear can be simu-lated through numerically continuing the constraint equa-tions, and the singular points and bifurcation curves of twosidestays can be derived, as shown in Figure 13. The ordinateθoc denotes the difference between angles of lock links, whichis θoc = θ4 − θ5 for the fore lock links and θoc = θ8 − θ9 for theaft lock links.

Figure 13 shows the relationship between θoc and theretraction angle, where the grey and black lines representthe motion trajectories of the fore and aft lock links, respec-tively. The fore sidestay has three singular points (S1f, S2f,and S3f). In contrast, the aft sidestay has only one singularpoint (S1a). It can be seen intuitively that the fore lock links

540 560 580 600

l2 (mm)

–60

–40

–20

0

20

40

60

SN

S1

S3𝜃

oc (d

eg)

(a)

240 260 280 300

l4 (mm)

–60

–40

–20

0

20

40

60

SN

S1

S3

𝜃oc

(deg

)

(b)

Figure 9: Variations of points S1 and S3 with the link length parameters.

Parallel

(a) (b)

Figure 10: Landing gear mechanism corresponding to the vicinityof point SN.

8 International Journal of Aerospace Engineering

can move to the area of θoc < 0, while the aft lock links onlymove within the area of θoc > 0. This indicates that the forelock links can be locked, while the aft lock links cannot belocked. To better explain the motion of the studied landinggear, Figure 14 provides the kinematic sketches of mecha-nism that correspond to various points in Figure 13.

In Figure 14, the grey lines represent the fore sidestay,whereas the black lines represent the aft sidestay. It is notdifficult to find that the sketches of fore and aft sidestaysare similar in Figures 14(a)–14(c). When the fore sidestayreaches the singular point S2f, the aft sidestay moves topoint A2a, as shown in Figure 14(d). At this time, the fore

280–40

–20

0

l4 (mm)520 250

20

40

l2 (mm)560 220600

𝜃oc

(deg

)

(a)

520 560 600

l2 (mm)

220

240

260

280

l 4 (m

m)

Feasibleregion

(b)

Figure 11: Bifurcation point SN variation under two-parameter continuation.

l2

l3

l4l5

zf zf

of xf

Z

XOy

xa

a

l6

l7

l8l9

A

B

C

D

E

F

G

H

oa

IQ

K

M

R

N S

P

yf

Z

yfzf

xf Y

A

B

C

D

F

Z

yaY

za

xa

M

GI

(xf4, yf4, zf4)

(xf3, yf3, zf3) (xa7, ya7, za7)

(xa6, ya6, za6)

(xa8, ya8, za8)(xa9, ya9, za9)

(xf2, yf2, zf2)

𝜃7

𝜃6

𝜃7

𝜃f2l1

lAF

𝜃f3

𝜃f3

Figure 12: Schematic diagrams of the DS mechanism.

0 20 40 60 80 100–160

–80

0

80

160

U2a U1a

U1f

D2f D1f

U2f

𝜃oc

(deg

)

Θ1 (deg)

–5 0 5 10 15 20–4

–2

0

2

4

(S1a)S1f

S3f

S2f

A2a

𝜃oc

(deg

)

Θ1 (deg)×10–7

Figure 13: Bifurcation curves of DS landing gear.

9International Journal of Aerospace Engineering

lock links are collinear, while the aft lock links are still inthe upper overcenter state. By the beneficial action of lock-ing springs, the fore sidestay reaches the singular point S3f,while the aft sidestay can only return to the singular pointS1a, as shown in Figure 14(g). As a result, only the foresidestay is locked. Due to stop block, the landing gear doesnot move to the states shown in Figures 14(e) and 14(f) inreality.

To explain the reason that the aft sidestay cannot be fullylocked, numerical continuation independently on the foreand aft sidestays is performed, as shown in Figure 15.

According to Figure 15, the motion trajectory of the aftsidestay also has three singular points when the two sidestaysare computed separately, with S2a appearing later than S2f.When the landing gear reaches the point S1f(S2a), the mainstrut rotates to the farthest end, then begins to move back-wards. At the time the fore sidestay reaches the point S2f,

the main strut has moved backwards to the farthest position(Figure 14(d)). To allow the aft sidestay to reach the S2a

point, the main strut needs to move continuously backwardsto a position farther than S2f. However, such motion isrestricted by the fore sidestay, thereby creating a contradic-tion in motion coordination. Precisely, this contradiction ofmutual restriction between the fore and aft sidestays is thefundamental reason why the DS landing gear is unable tobe fully locked. To achieve complete locking, the retractionangle corresponding to the singular points S2f and S2a mustbe identical (i.e., ðΘ1ÞS2 f = ðΘ1ÞS2a).

4.3. Effects of Link Length Parameters on the SynchronousLocking. In this section, the effects of link lengths on the sin-gular points S2f and S2a are analyzed. It can be seen fromFigure 15 that θfoc and θaoc are equal to zero at the points S2f

and S2a. Hence, when considering the effects of points S2f

and S2a, the parameters θfoc and θaoc can be used as the fixedparameters (θfoc = θaoc = 0), where Θ1 is used as the stateparameter, and the link lengths are used as the controlparameters. Figure 16 plots the analytical results.

In Figure 16, the grey solid lines represent the influenceof l2 and l4 on the singular point S2f, while the black dashedlines represent the influence of l6 and l8 on the singularpoint S2a. It is clear that the singular points exhibit gener-ally consistent trends of variation with the link lengths. Asdescribed in the previous section, the synchronous lockingcan only be achieved when ðΘ1ÞS2 f = ðΘ1ÞS2a . So once thevalue of Θ1 is determined (e.g., Θ1 = 0:1°), the link lengthscan be obtained intuitively, as shown by the dotted lines inFigure 16. The effects of variations in other link lengths canbe analyzed in a similar way, which are thus not detailedherein.

Although changing the link lengths can yield ðΘ1ÞS2 f =ðΘ1ÞS2a , it also changes the θfoc and θaoc in a deployed state.

U1f (U1a)

(a)

U2f (U2a)

(b)

S1f (S1a)

(c)

S2f (A2a)

(d)

D1f (U1a)

(e)

D2f (U2a)

(f)

S3f (S1a)

(g)

Figure 14: Sketches of DS mechanism motions: (a) U1f(U1a); (b) U2f(U2a); (c) S1f(S1a); (d) S1f(A2a); (e) D1f(U1a); (f) D2f(U2a); (g) S3f(S1a).

–5 0 5 10 15 20–4

–2

0

2

4

S1f(S1a)

S3f(S3a)

S2aS2f

Θ1 (deg)

𝜃oc

(deg

)

×10–7

Figure 15: Independent computational results of the fore and aftsidestays.

10 International Journal of Aerospace Engineering

Hence, a comprehensive consideration of the link lengths isneeded based on multiple parameters.

4.4. Optimization of Synchronous Locking. In this section, anoptimization is carried out to accomplish fully locking of theDS landing gear via the PSO, which is achieved by integratingISIGHT with MATLAB for cosimulation analysis. Throughoptimization of link lengths, the fore and aft sidestays canbe locked synchronously. PSO is a population-based optimi-zation algorithm proposed by Kennedy and Eberhart [32] in1995, who got inspiration from birds hunting for food. Itinitializes a system with a substantial number of randomsolutions and performs search for the optimal solution byupdating generations. The PSO algorithm features veryhigh probability of convergence to a global optimal solu-tion, very fast computation, and strong global search ability.Existing research and applications indicate that PSO is apromising and effective optimization method [33–35].Hence, the PSO algorithm is used for the optimization pur-pose in this paper.

As suggested in the foregoing analysis, θfoc and θaoc should

not be excessively large when the landing gear is locked.Thus, the invariant overcenter angles of the lock links is usedas one of the constraints, i.e., ðθfocÞS3 f = ðθaocÞS3 f = 2°. LetΔΘ =jðΘ1ÞS2 f ‐ðΘ1ÞS2a j; then, the synchronous locking of fore andaft sidestays necessitates ΔΘ = 0. The smaller the value of ΔΘ, the lesser the mutual restraint effect between the fore andaft sidestays. Accordingly, the optimization objective is set asmin ΔΘ. The optimization variables are the lengths of variouslinks. Due to the constraints, there are only four independentoptimization variables, i.e., liði = 2, 4, 6, 8Þ.

In Figure 17, the cosimulation-based optimization pro-cess is displayed, where the operators include the ISIGHTalgorithmmodule and the MATLAB numerical continuationmodule. As a first step, the ISIGHT generates a set of initialparameters li based on the PSO algorithm and then passesthe parameters to the MATLAB module. Afterwards, theMATLAB calculates the bifurcation points of the fore and

aft sidestays based on the numerical continuation, as well asΔΘ, ðθfocÞS3 f , and ðθaocÞS3 f , which are then transferred backto the ISIGHT. Finally, the ISIGHT identifies the optimalresults that conform to the constraints.

Since the change in link lengths leads to changed solu-tions of constraint equations, the starting point that sat-isfies the constraint equations is solved initially based onthe PSO algorithm. To reduce the computational burden,the solution at Θ1 = 5° is chosen as the starting point.Afterwards, the singular points S2f and S2a are obtainedthrough the parameter continuation analysis for reducingΘ1, and the corresponding Θ1 is identified. Finally, ΔΘis solved.

Based on the consideration of the mechanism motioninterference, the optimization variables are set to l2 ∈ ½450mm, 650mm� and l6 ∈ ½450mm, 650mm�, and the lock linklengths are set to l4 ∈ ½200mm, 300mm� and l8 ∈ ½200mm,300mm�. The optimization results are shown in Tables 1and 2 and Figure 18.

According to the optimization results in Figure 18, thepoints S2f and S2a are almost completely coincident, whichallow simultaneous locking of the fore and aft sidestays,respectively, at S2f and S2a. When the two sidestays are in alocked state, the angle between lock links remains constantat 2°.

500 520 540 560 580 600

l2 or l6 (mm)

–0.05

0

0.05

0.1

0.15

0.2

0.25

l6 l2

Θ1 (

deg)

(a)

200 220 240 260 280 300

l4 or l8 (mm)

–0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

l4l8

Θ1 (

deg)

(b)

Figure 16: Curves describing the effects of link lengths on the second singular point.

ISIGHT: PSO

Fore planebifurcation

point

Starting point

Variables: li, i = 2, 4, 6, 8

Objective:

MATLAB: numerical continuation

Constraint:

minΔΘ1

Aft planebifurcation

point

The difference of the secondbifurcation point

(𝜃foc)S3f

(𝜃aoc)S3f

Figure 17: Computational flow of cosimulation-based optimization.

11International Journal of Aerospace Engineering

5. Conclusion

A bifurcation theory-based method for analyzing the kine-matics of sidestay landing gear locking mechanisms is pro-posed in this paper. Initially, the simple single-sidestay (SS)mechanism is analyzed, and then, the complex dual-sidestay (DS) mechanism is investigated based on SS. Themain conclusions are as follows:

(1) Based on the bifurcation theory and singularity, thelocked position of the SS landing gear is analyzed,clarifying that the lock links must be at the singularpoint of a lower overcenter angle, in order to achievefully locking. Besides, the effect of link lengths on thelocked position is analyzed, concluding that the linklengths should be within a feasible range

(2) For DS, the presence of an additional sidestay poses anadverse impact on the downward folding motion oflock links. The kinematic cause underlying the adverseimpact is analyzed. Owing to the different positions ofsingular points, the movements of fore and aft side-stays seriously restrict each other, causing lockingfailure of the DS landing gear

(3) A kinematic optimization method for the synchro-nous locking of the DS landing gear is proposed. Thisstudy indicates that the full locking of the DS gear isachievable on condition that the overcenter angle oflock links is invariant.

Table 1: Synchronous locking optimization results.

l2 l4 l6 l8 ΔΘ

Before 550mm 250mm 550mm 250mm 7:72e − 08°

Optimization 532.4655mm 258.8501mm 538.6727mm 288.9462mm 2:24e − 10°

Table 2: Synchronous locking optimization results (other parameters).

l3 l5 l7 l9Before 550mm 260mm 550mm 260mm

Optimization 567.5345mm 261.2052mm 561.3273mm 227.5027mm

0 20 40 60 80 100–150

–100

–50

0

50

100

150

Θ1 (deg)

𝜃oc

(deg

) S2f (S2a)

–5 0 5 10 15–4

–2

0

2

4

S1f (S1a)

S3f (S3a)

Θ1 (deg)

𝜃oc

(deg

)

×10–7

Figure 18: Singular points of dual-sidestay links after optimization.

Table 3: Parameters of the SS landing gear.

Parameters Values

l1 (mm) 1800

l2 (mm) 550

l3 (mm) 550

l4 (mm) 250

l5 (mm) 260

αð ÞΘ1=90 (deg) 115.3917

lAF (mm) 98.6181

lCG (mm) 60

lEH (mm) 40

φAF (deg) 130.0157

φCGð ÞΘ1=0 (deg) 125

φEHð ÞΘ1=0 (deg) 105

oO (mm) (-69.2820, 39.1058, 8.4104)

e21ð ÞΘ1=0 (mm) (0.8660, -0.5000, 0)

e22ð ÞΘ1=0 (mm) (0.4888, 0.8467, 0.2103)

e23ð ÞΘ1=0 (mm) (-0.1051, -0.1821, 0.9776)

Gð ÞΘ1=0 (mm) (105, 43.3013, 1000)

Hð ÞΘ1=0 (mm) (92.5, 21.6506, 500)

12 International Journal of Aerospace Engineering

AppendixThe motion constraint equations of the DS landing gear areas follows (Tables 3 and 4):

X1

Y1‐0:5l1 ⋅ cos Θ1

Z1‐0:5l1 ⋅ sin Θ1

xf2

yf2 − 0:5l2 ⋅ cos θf2 − yfA

zf2 − 0:5l2 ⋅ sin θf2 − zfA

xf3

yf3 − 0:5l3 ⋅ cos θf3‐y

f2 − 0:5l2 ⋅ cos θ

f2

zf3 − 0:5l3 ⋅ sin θf3‐zf2 − 0:5l2 ⋅ sin θf2

xf4

yf4 + 0:5l4 ⋅ cos θf4‐y

f2 − 0:5l2 ⋅ cos θ

f2

zf4 − 0:5l4 ⋅ sin θf4‐zf2 − 0:5l2 ⋅ sin θf2

xf5

yf5 + 0:5l5 ⋅ cos θf5‐y

f4 + 0:5l4 ⋅ cos θ

f4

zf5 − 0:5l5 ⋅ sin θf5‐zf4 + 0:5l4 ⋅ sin θf4

yf5‐0:5l5 ⋅ cos θf5‐y

fE

zf5 − 0:5l5 ⋅ sin θf5‐zfE

yf3 + 0:5l3 ⋅ cos θf3‐y

fC

zf3 + 0:5l3 ⋅ sin θf3‐zfC

xa6

ya6 − 0:5l6 ⋅ cos θa6 − yaI

za6 − 0:5l6 ⋅ sin θa6 − zaI

xa7

ya7 − 0:5l7 ⋅ cos θa7‐ya6 − 0:5l6 ⋅ cos θa6za7 − 0:5l7 ⋅ sin θa7‐za6 − 0:5l6 ⋅ sin θa6

xa8

ya8 + 0:5l8 ⋅ cos θa8‐ya6 − 0:5l6 ⋅ cos θa6za8 − 0:5l8 ⋅ sin θa8‐za6 − 0:5l6 ⋅ sin θa6

xa9

ya9 + 0:5l9 ⋅ cos θa9‐ya8 + 0:5l8 ⋅ cos θa8za9 − 0:5l9 ⋅ sin θa9‐za8 + 0:5l8 ⋅ sin θa8

ya9‐0:5l9 ⋅ cos θa9‐yfP

za9 − 0:5l9 ⋅ sin θa9‐zfP

26666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666666664

ya7 + 0:5l7 ⋅ cos θa7‐yfM

za7 + 0:5l7 ⋅ sin θa7‐zfM

� = 0, ðA:1Þ

Table 4: Parameters of the DS landing gear.

Parameters Values

l1 (mm) 1800

l2 (mm) 550

l3 (mm) 550

l4 (mm) 250

l5 (mm) 260

αf� �

Θ1=90(deg) 115.3917

l6 (mm) 550

l7 (mm) 550

l8 (mm) 250

l9 (mm) 260

αað ÞΘ1=90 (deg) 111.4145

lAF (mm) 98.6181

lCG (mm) 60

lEH (mm) 40

φAF (deg) 130.0157

φCGð ÞΘ1=0 (deg) 125

φEHð ÞΘ1=0 (deg) 105

lIQ (mm) 98.6181

lMR (mm) 60

lPS (mm) 40

φIQ (deg) 130.0157

φMRð ÞΘ1=0 (deg) 125

φPSð ÞΘ1=0 (deg) 105

ofO (mm) (-69.2820, 39.1058, 8.4104)

ef21� �

Θ1=0(mm) (0.8660, -0.5000, 0)

ef22� �

Θ1=0(mm) (0.4888, 0.8467, 0.2103)

ef23� �

Θ1=0(mm) (-0.1051, -0.1821, 0.9776)

Gð ÞΘ1=0 (mm) (105, 43.3013, 1000)

Hð ÞΘ1=0 (mm) (92.5, 21.6506, 500)

oaO (mm) (65.5322, -44.8604, 9.6280)

ea21ð ÞΘ1=0 (mm) (0.8192, -0.5736, 0)

ea22ð ÞΘ1=0 (mm) (-0.5608, 0.8008, 0.2103)

ea23ð ÞΘ1=0 (mm) (0.1206, -0.1722, 0.9776)

Rð ÞΘ1=0 (mm) (-108.6788, 40.9576, 1000)

Sð ÞΘ1=0 (mm) (-94.3394, 20.4788, 500)

13International Journal of Aerospace Engineering

where

yfA

zfAyfC

zfC

yfE

zfE

yaI

zaIyaM

zaM

yaP

zaP

266666666666666666666666666666664

377777777777777777777777777777775

=

yfF + lAF ⋅ cos φAF

lAF ⋅ sin φAF

yfG‐lCG ⋅ cos φCG

zfG‐lCG ⋅ sin φCG

yfH‐lEH ⋅ cos φEH

zfH‐lEH ⋅ sin φEH

yaQ + lIQ ⋅ cos φIQ

lIQ ⋅ sin φIQ

yaM‐lMR ⋅ cos φMR

zaM‐lMR ⋅ sin φMR

yaP‐lPS ⋅ cos φPS

zaP‐lPS ⋅ sin φPS

266666666666666666666666666666664

377777777777777777777777777777775

,

xfG

yfG

zfG

1

26666664

37777775= T f ⋅

XG

YG

ZG

1

2666664

3777775,

xfH

yfH

zfH

1

26666664

37777775= T f ⋅

XH

YH

ZH

1

2666664

3777775,

xaM

yaM

zaM

1

2666664

3777775 = Ta ⋅

XM

YM

ZM

1

2666664

3777775,

xaP

yaP

zaP

1

2666664

3777775 = Ta ⋅

XP

YP

ZP

1

2666664

3777775: ðA:2Þ

Notations

T0, T1, T: Transformation matrix of the global coordi-nate system to the local system

lAF, lCG, lEH: The node lengthsli: The length of link ie1, e2: The base vectors in the global and local

coordinate systems

xi, yi, zi: The local coordinate values at the center ofgravity of the i-th link

Xi, Yi, Zi: The global coordinate values at the center ofgravity position of the i-th link

lAF, lCG, lEH: The node lengthsφAF, φCG, φEH: The angles of node axes with respect to the y

-axis of local coordinatesΘi: The angle of the i-th link with respect to the

Z-axis of global coordinatesθi: The angle of the i-th link with respect to the y

-axis of local coordinatesα: The rotation angle of local coordinate system

about the y-axisθd: Overcenter angle of sidestay linksθoc: Overcenter angle of lock linksy∗, z∗: The local coordinate values at the positions

of node ∗Superscript f: The relevant parameters of fore sidestaySuperscript a: The relevant parameters of aft sidestay.

Data Availability

The geometry data used to support the findings of this studyare included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

Kui Xu and Yin Yin contributed equally to this work.

Acknowledgments

This study was financially supported by the National NaturalScience Foundation of China (51805249), the Natural Sci-ence Foundation of Jiangsu Province (BK20180436), theFundamental Research Funds for the Central Universities(NF2018001), and the Priority Academic Program Develop-ment of Jiangsu Higher Education Institutes.

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