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Research Papers

# Design and Folding/Unfolding Dynamics of an Over-Constrained Airplane's Landing Gear With Four Side StaysPUBLIC ACCESS

[+] Author and Article Information
Camille Parat

Department of Mechanical Engineering,
Tsinghua University,
Beijing 1000084, China
e-mail: pkm16@mails.tsinghua.edu.cn

Zu-Yun Li

Department of Mechanical Engineering,
Tsinghua University,
Beijing 1000084, China
e-mail: lizuyun16@mails.tsinghua.edu.cn

Jing-Shan Zhao

Department of Mechanical Engineering,
Tsinghua University,
Beijing 1000084, China
e-mail: jingshanzhao@mail.tsinghua.edu.cn

1Corresponding author.

Contributed by the Mechanisms and Robotics Committee of ASME for publication in the JOURNAL OF MECHANISMS AND ROBOTICS. Manuscript received March 20, 2018; final manuscript received September 9, 2018; published online November 12, 2018. Assoc. Editor: Guimin Chen.

J. Mechanisms Robotics 11(1), 011001 (Nov 12, 2018) (10 pages) Paper No: JMR-18-1074; doi: 10.1115/1.4041485 History: Received March 20, 2018; Revised September 09, 2018

## Abstract

This paper introduces the design of a specific landing gear retraction system presenting a mechanism with four redundant side stays and examines its dynamic behavior during the folding and unfolding processes. First, a concept design of a four-side-stay landing gear retraction system is presented. To get the particular motion during folding and unfolding, the main kinematics parameters are given. Then, the influence of the side stay's kinematic redundancy on the mechanism parameters is examined. Because the mechanism is overconstrained, the allowable parameters belong to a specific region of the space called feasible region. Finally, a dynamic analysis of the over-constrained system is executed by using the Newton–Euler approach and compliant equations. Numerical simulations indicate that this kind of landing gear retraction system equitably share the loads between different side stays, and therefore, the total load at one side stay is greatly reduced.

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## Introduction

Landing gear, also referred to as undercarriage, is one of the most important functional parts of an airplane to be considered during its design. It not only ensures the smoothness of the landing and take-off, the taxiing on the landing strip, but also is the part of the airplane where the maximum load, static or dynamic, is reached. Despite being a crucial part of the airplane, landing gear had various designs and mechanisms over the years that change from one manufacturer to another, from one model of airplane to another [110]. The literature for the design of an aircraft and its systems, notably of landing gear's mechanisms, is relatively old [1115] and mainly focuses on its kinematic analysis. Moreover, it mainly explores the conventional one-side-stay landing gear retraction system and few references examine some complex designs. Recently, Sadraey developed a system engineering approach [16] to design an aircraft and its systems. Rajesh and Abhay analyzed the stress distribution and deformation on the aircraft's nose landing gear [17] when being subject to the external loads. It appears with finite element analyses that the deformations at the attachment point of the side stay are relatively important compared to the rest of the structure. One solution to reduce the loads at the attachment points between the landing gear and the aircraft frame is to add a second side stay to the retraction mechanism as done by Knowles et al. who modeled a dual-side stay landing gear mechanism and analyzed its behavior with respect to the side stay angle [18]. Luo and Zhao designed a double-lock landing gear mechanism [19] to achieve higher stiffness, strength, and reliability of the system.

The Sarrus mechanism [20], represented in Fig. 1 and known as one of spatial overconstrained mechanisms, only composed of revolute joints, is capable of transforming rotational movement into a translational one. This special mechanism was proposed by Sarrus in 1853. The square end effector C1C2, connected through two kinematic chains to a fixed square base A1A2, follows a rectilinear and straight line along z-axis. This mechanism allows the end effector to have a limited workspace and has better mechanical characteristics compared to its similar counterparts such as the Peaucellier–Lipkin mechanism. Although the stiffnesses along x-direction and y-direction are very high, there is virtually no stiffness along the movement of the end effector C1C2 with respect to A1A2.

The present contribution is a novel overconstrained landing gear retraction system that presents four side stays symmetrically arranged around the strut to ensure the folding and unfolding processes of the landing gear. The linkage is similar to an overconstrained Sarrus mechanism presented in Fig. 1, but here A1C1 is not parallel to the z-axis, i.e., the difference of abscissa between A1 and C1 is not zero. Doing that, the loads applied at the attachment points are reduced—because there are more attachment points—and the mechanism has different properties from the conventional one. Examining the dynamics of such a redundant mechanism is quite challenging and complex. Recently, García and Gutierrez-Lopez [21] consider positive-semidefinite inertia matrices and redundant constraint equations to give a rule about the existence, uniqueness and solving of the dynamic equations when the constraint forces are not entirely determined. Wojtyra et al. [22,23] review and compare different methods of handling redundant constraints when calculating the dynamics of multibody systems (elimination of redundant constraints, pseudoinverse-based calculations, and augmented Lagrangian formulation). The pseudoinverse-based approach includes the Moore–Penrose pseudoinverse matrix used by Udwadia and Kalaba [24] to determine a solution to the system of equations with initial infinite solutions. Callejo et al. [25] propose a unique minimum norm determination of the reaction forces in multibody systems where it exists in theory infinite solutions to the system of dynamic equations. The conditions where the problem has nonunique solution are given and a normalization minimum solution is examined in order to obtain one unique solution. González and Kovecses [26] give a method using penalty formulations to calculate the dynamics of multibody systems that have redundant constraints. The structural properties of the structure are then used to model compliant equations that can be used to determine a unique and simple solution that accurately approach the constraint forces of the multibody system.

Although these models are good to predict the dynamic behavior of complex systems, they are general and do not consider specific situations, symmetrical arrangements as the one examined here. In this paper, after describing and giving the characteristics of the proposed landing gear retraction system, its kinematics parameters are investigated. Because the system is over-constrained, the parameters of the mechanism cannot be randomly chosen and the allowable values for these parameters will be calculated. Finally, the dynamics of the retraction system is examined and numerically calculated through the Newton–Euler approach in order to obtain the forces and torques at the different joints of the mechanism.

## Characteristics of the Landing Gear Retraction Mechanism

The new design of a landing gear retraction system presented in this paper does not have one unique side stay to fold the strut like the conventional landing gear retraction systems have, but has four side stays symmetrically arranged around the landing gear strut as shown in Figs. 2 and 3. Therefore, the landing gear strut is connected to the aircraft frame through four kinematics chains AiBiCi at point Ci (i = 1, 2, 3, 4), each of the chains is composed of two links AiBi and BiCi and three revolute joints at points Ai, Bi, and Ci. The coordinate system used in the paper is such that the origin is the intersection point between lines A1A3 and A2A4, the z-axis is along the landing gear main axis, the x-axis is along the line A1A3 and the y-axis is chosen to obtain a right-handed coordinate system as shown in Fig. 3. The arrangement of the side stays is such that the side stay A3B3C3 is the symmetry of the side stay A1B1C1 with respect to the plane $π2$ composed by the side stays A2B2C2 and A4B4C4 and the strut's main axes. Similarly, as displayed in Fig. 3, it appears that the side stay A4B4C4 is the symmetry of the branch A2B2C2 with respect to the plane $π1$ composed by the side stays A1B1C1 and A3B3C3 and the strut's main axes. The angle between the planes $π1$ and $π2$ is denoted as $2β$ in Fig. 3. The whole retraction system is then connected to the aircraft frame and actuated at points Ai (i = 1, 2, 3, 4) via journal bearings. A journal bearing is a journal rotating in a bearing with a thin lubricant film separating the two elements. The lubricant layer limits the metal-to-metal contact, removes instabilities, and subsynchronous vibration by providing damping and is able to absorb impact loads. In addition, it gives flexibility for the positioning of the side stays during the assembling of overconstrained mechanism.

The motion of the landing gear strut can be geometrically determined. The side stays A1B1C1 and A3B3C3, only composed of revolute joints, constrain the mechanism and only authorize a motion of the strut within the plane $π1$. On the other hand, the motion of the end effector is similarly constrained by the side stays A2B2C2 and A4B4C4 so that the strut can only move within the plane $π2$. Thus, the motion of the end effector is limited to the intersection of the two planes $π1$ and $π2$, which is a straight vertical line passing through the main axis of the strut. Thus, the landing gear using the proposed four-side-stay mechanism can only go upward and downward along the z-axis while folding and unfolding, respectively. There is no rotation and translation of the strut allowed by the retraction system except the translation along the main axis of the landing gear strut. Because there are at least one left and one right main landing gears in an aircraft, using this four-side-stay mechanism, which makes the landing gear strut move along a vertical direction, leads to a constant horizontal distance between landing gears. Doing that, the wear of the wheels is reduced and the handling and ride of the aircraft on the landing strip is improved. In order to obtain a retraction mechanism with the same stiffness and strength that the current systems have, the design of the mechanism can be made more compact so that the weight of the whole system is reduced, which is an important criterion in aerospace and aeronautic applications.

## Kinematics and Influence of Over-Constraints on the Mechanism

To numerically investigate the folding/unfolding dynamics of the landing gear retraction system, the kinematic characteristics of the mechanism should be first examined. In this section, the kinematic parameters and their corresponding derivatives are obtained from the constraint equations of the mechanism. These constraint equations also limit the range of allowable parameters that can be used to realize the mechanism.

###### Kinematic Analysis of the Mechanism.

From the concept design above, it has been shown that the landing gear strut only moves in a vertical and straight direction along the z-axis: the closed chains in the mechanism impose four kinematic constraints that reduce the degree-of-freedom of the retraction system and limit its motion. These four kinematic constraints reflect the fact that the distances between the fixed points Ai (i = 1, 2, 3, 4) are fixed on the fuselage of the aircraft. Because the links AiBi and BiCi (i = 1, 2, 3, 4) are exactly the same and they are symmetrically arranged around the strut, no distinction between the four elements will be made so that the simplified notation AB and BC will be used in this paper. Moreover, the rotation invariance of the mechanism about the z-axis makes the four kinematic constraints identical and they are expressed as Display Formula

(1)$xA−xC=−l1cosa−l2cosφ$

where $xA$ and $xC$ are the abscissae of points A and C, $l1$ and $l2$ are the lengths of links AB and BC, $θ$ and $φ$ are the angles between the horizontal axis and the main axes of the links AB and BC, respectively.

Since the motion of the landing gear strut depends on both angles $θ$ and $φ$, and is constrained with Eq. (1), there is only one degree-of-freedom for the proposed mechanism. The degree-of-freedom is here chosen to be $θ$ because its value can directly be controlled by the actuator at point A. Therefore, the expression of $θ(t)$ is independent of the mechanism and only depends on the input of the actuator. For the purpose of numerical simulations, $θ(t)$ will be a sinusoid whose minimum and maximum points are the initial and final values of $θ$ during the folding and unfolding of the landing gear strut. The minimum value of $θ(t)$ is when the strut is fully unfolded and $θ$ equals $φ$ as shown in Fig. 4(a). The minimum value is then calculated with classic trigonometrical formula by considering that the initial values $θmin$ and $φmin$ are equal Display Formula

(2)$θmin=π−acosxA−xCl1+l2$

On the other hand, the maximum value corresponds to the situation where the strut is at its lowest altitude zmin as shown in Fig. 4(b). First, the expression of the ordinate of the landing gear strut is obtained from the system of two equations connecting the position of the landing gear strut to the parameters of the retraction mechanism Display Formula

(3)$zmin=l1sinθmax+l2sinφ(θ=θmax)xA−xC=−l1 cos θmax−l2 cos φ(θ=θmax)$

where $φ(θ=θmax)$ is the value of angle $φ$ when $θ=θmax$.

By squaring the system of Eq. (3), one can remove $φ(θ=θmax)$ and obtain

$l22=l12+zmin2+xA−xC2+2xA−xCl1 cos θmax−2zminl1 sin θmax$

Using the two half-angle identities $cos θmax=1−t2/1+t2$ and $sin θmax=2t/1+t2$ where $t=tanθmax/2$ leads to Display Formula

(4)$t2l12−l22+zmin2+xA−xC2−2xA−xCl1−4zminl1t+l12−l22+zmin2+xA−xC2+2xA−xCl1=0$

Finally, one can obtain the expression of $θmax$ by solving the second-order equation (4) with respect to the variable $t=tanθmax/2$Display Formula

(5)$θmax=2π+atan4zminl1+Δ2l12−l22+zmin2+xA−xC2−2xA−xCl1$
where
$Δ=4zminl12−4l12−l22+zmin2+xA−xC22−4xA−xC2l12$

Therefore, a sinusoidal expression of θ(t) is gained Display Formula

(6)$θt=θmin+θmax2+θmax−θmin2 cos 2πTt$
where T is the period of $θ(t)$ and $θmin$ and $θmax$ are the expressions derived in Eqs. (2) and (5).

The first and second derivatives of $θ(t)$ are then obtained Display Formula

(7)$θ˙(t)=−2πTθmax−θmin2 sin 2πTtθ¨(t)=−2πT2θmax−θmin2 cos 2πTt$

Because the system only has one degree-of-freedom, the expression of $φ(t)$ only depends on the expression of $θ$(t). To get this expression, $φ(t)$ has to be isolated from Eq. (1)Display Formula

(8)$φ=acosxC−xA−l1 cos θl2$

Then, the expressions of $φ˙(t)$ and $φ¨(t)$ are then obtained by calculating the first and second orders of derivatives of constraint Eq. (8) with respect to time Display Formula

(9)$φ˙=−θ˙l1 sin θl2 sin φφ¨=−l1θ¨ sin θ−l1θ˙2 cos θ−l2φ˙2 cos φl2 sin φ$

The position of the masse centers r1 of link AB, r2 of link BC and r3 of the landing gear strut are given by Display Formula

(10)$r1=xA+l12 cos θ0l12 sin θr2=xA+l1 cos θ+l22 cos φ0l1 sin θ+l22 sin φr3=xA+l1 cos θ+l2 cos φ0l1 sin θ+l2 sin φ$

The acceleration of links AB and BC are named a1, a2 and a3, respectively, and can be obtained by deriving their mass center's position vectors whose expressions are given in Eq. (10)Display Formula

(11)$a1=−l12θ¨ sin θ+θ˙2 cos θ0l12θ¨ cos θ−θ˙2 sin θ$
Display Formula
(12)$a2=−l1θ¨ sin θ+θ˙2 cos θ−l22φ¨ sin φ+φ˙2 cos φ0l1θ¨ cos θ−θ˙2 sin θ−l22φ¨ cos φ−φ˙2 sin φ$
Display Formula
(13)$a3=−l1θ¨ sin θ+θ˙2 cos θ−l2φ¨ sin φ+φ˙2 cos φ0l1θ¨ cos θ−θ˙2 sin θ−l2φ¨ cos φ−φ˙2 sin φ$

Since the motion of the landing gear strut is constrained by redundant links, the speed and the acceleration of the landing gear strut along x-direction equal zero so that expression (13) turns into

$a3=00l1θ¨ cos θ−θ˙2 sin θ−l2φ¨ cos φ−φ˙2 sin φ$

###### Motion Constraint Equation and Feasible Region of the Mechanism.

In addition to limit the motion of the landing gear strut, the overconstraints also limit the allowable parameters of the mechanism. Indeed, by isolating the term depending on the angle $φ$ of link BC given in Eq. (1), one can obtain Eq. (14). Then, because the cosine function is bounded between −1 and 1, one inequality which embodies a limited region of space is obtained from the constraint equation shown in Eq. (15)Display Formula

(14)$l2 cos φ=xC−xA−l1 cos θ$
Display Formula
(15)$xC−xA−l1 cos θ≤l2$

Besides the aforementioned inequality, other inequalities have to be considered in order to get a feasible mechanism and display its allowable parameters: all of the distances ($l1$, $l2$ and $xA−xC$) have to be positive to get a realistic model. Finally, the allowable parameters of the mechanism considering its kinematic constraints comply with the following inequalities: Display Formula

(16)$xC−xA−l1 cos θ≤l20

The set of inequalities (16) represent a region of the space where the parameters $l1$, $l2$ and $xA−xC$ have to be chosen to obtain a feasible mechanism. In order to get a more accurate feasible region for the parameters of the mechanism and to display it, the minimum and maximum values of $cos θ$ during the motion have to be found. The minimum reachable value of $cos θ$ is −1 which is reached when $θ=π$. The maximum value of $cos θ$ depends on the expression of the final angle $θmax$ of the link AB given in Eq. (5). Figure 5 displays the maximum value of cos$θ$ with respect to $θmax$ and $θmin$ on the trigonometrical circle. If $θmax>θmin$ and $θmax≤2π−θmin$, then the maximum value of cos$θ$ is when $θ=θmin$, as displayed in dark gray. On the other hand, if $θmax$ does not belong to this interval, the maximum value of cos$θ$ is when $θ=θmax$ as displayed in light gray.

Using this result, the system of Eq. (16) can be restricted to Display Formula

(17)$xC−xA−l1 cos θmin≤l2ifθmax>θminandθmax≤2π−θminxC−xA−l1 cos θmax≤l2else0

The feasible region of the mechanism obtained from the system of Eq. (17) is a complex three-dimensional polygon as it has been displayed in Fig. 6 using MATLAB R2016b. If the parameters do not belong to this region of the space, the mechanism is not possible and the kinematic and dynamic expressions are complex functions. Therefore, the parameters $l1$, $l2$ and $xA−xC$ have to belong to this region of the space to obtain a feasible mechanism. These results will be used in further numerical applications in order to model a realistic mechanism for the retraction system's dynamic analysis.

## Folding/Unfolding Dynamics and Numerical Simulations of the Retraction Mechanism

The retraction mechanism with four side stays proposed in Fig. 1 constrains the end effector so that it can only be raised and lowered along a straight and vertical line as shown in the concept design of Sec. 2. In this section, an investigation on the dynamics of the retraction system will be discussed through the Newton–Euler approach. Contrary to the Lagrangian approach which is an energy-based method that does not need to consider the internal forces of the mechanism, the Newton–Euler approach has to consider the mechanism as a multirigid body and leads to, among other things, the expression of the forces acting on the different joints of the mechanism. In three dimensions, there are three Newton equations describing the translational dynamics and three Euler equations describing the rotational dynamics for each rigid body. Zhao et al. proposed a method to investigate the dynamics of an overconstrained suspension mechanism [27] that can be used to determine the forces and actuation moment describing the dynamics of the retraction system.

###### Theoretical Dynamics of the Landing Gear Retraction System.

Dynamics is the study of a moving mechanical system under the influence of applied torques and forces. The present dynamic analysis concerns the unfolding and folding processes of the landing gear, i.e., during flight so that there are no external forces acting on the landing gear structure. Assuming that the structures and the motions of the four-side-stay mechanism are strictly identical, one can first consider the dynamics of one side stay and then, extrapolate the result to the other side stays to get the analysis for the whole system. When using revolute joints, there are three internal forces and two internal moments at the joints. However, in this situation, the mechanism of one side stay is a planar mechanism so that there are only three internal forces and one moment perpendicular to the mechanism's plane at each joint. Given that all the joints are revolute ones, there is no moment around the joint direction, i.e., there is no internal moments to be considered here. The links AB and BC, and the landing gear strut are assumed to be rigid and uniform so that the mass center of each coincides with its geometric center of gravity.

The rotations of links AB and BC, that can be regarded as rods, are only around the y-direction. In this situation, the inertial accelerations of the links are given by

$IxxIxyIxzIyxIyyIyzIzxIzyIzz0ω˙0=IxyIyyIzyω˙$

where $Iij$ represents the moment of inertia about the ith direction when the rod is rotating around the jth direction and $ω˙$ is the rotational acceleration of the rod around the y-direction. Because the links remain within the xOz-plane, $Iyy≫Ixy$ and $Iyy≫Izy$ so that only the inertial acceleration along the y-direction will be considered in the following analysis and noted as Ji (i = 1 for link AB and i = 2 for link BC).

The Newton–Euler equations of the link AB can be obtained from its free-body diagram illustrated in Fig. 7(a). There are three Newton equations (18) and three Euler equations (19) to represent the translational and the rotational dynamics of the link, respectively, Display Formula

(18)$FAB+m1g−m1a1=0$
Display Formula
(19)$l1eAB×FB−m1l12eAB×a1+m1l12eAB×g−T1−c1ω1−c2ω1−ω2−M1=0$

where $FAB$ is the resultant of the internal forces acting on link AB; $m1$, $l1$ and $a1$ are the mass, length and acceleration of link AB, respectively; $T1$ and $M1$ are the inertial torques of the link and the actuation moments at point A, respectively; $eAB$ is the unit vector passing through points A and B, and g the acceleration of gravity; $c1$ and $c2$ are the viscous friction coefficients of moment of revolute joints A and B, respectively; $ω1=θ˙$ and $ω2=φ˙$ are the absolute angular velocity of links AB and BC, individually.

Using Eq. (11), the Newton equations (18) can be developed into Display Formula

(20)$FAx+FBx+m1l12θ¨ sin θ+θ˙2 cos θ=0FAy+FBy=0−FAz−FBz−m1l12θ¨ cos θ−θ˙2 sin θ+m1g=0$

and the Euler equations (19) can be transformed into Display Formula

(21)$−FByl1 sin θ=0FBxl1 sin θ+FBzl1 cos θ−m1gl12 cos θ+m1l122θ¨−J1θ¨−c1θ˙−c2θ˙−φ˙−M1=0FByl1 cos θ=0$

where $FAx,FAy,FAZ$ and $FBx,FBy,FBZ$ represent the forces along the three directions x, y, and z, as displayed in Fig. 3, acting at points A and B, individually; $J1$ is the moment of inertia of the link about the y-direction Display Formula

(22)$FBC+m2g−m2a2=0$
Display Formula
(23)$l2eBC×FC−m2l22eBC×a2+m2l22eBC×g−T2−M2−c2ω2−ω1−c3ω2=0$

where $FBC$ is the resultant of the forces acting on link BC; $m2$, $l2$, and $a2$ are the mass, length and acceleration of link BC, respectively; $T2$ and $M2$ are the inertial torques of the link and the actuation moments at point C, respectively; and $eBC$ is the unit vector passing through points B, $c3$ is the viscous friction coefficient of moment of revolute joint C. There is no actuation torque acting on link BC so that M2 = 0. Then, using Eq. (12), the Newton equations (22) can be simplified into Display Formula

(24)$−FBx−FCx+m2l1θ¨ sin θ+θ˙2 cos θ+l22φ¨ sin φ+φ˙2 cos φ=0−FBy−FCy=0FBz+FCz−m2l1θ¨ cos θ−θ˙2 sin θ+l22φ¨ cos φ−φ˙2 sin φ+m2g=0$

and the Euler equations (23) are then developed into Display Formula

(25)$FCyl2 sin φ=0−FCxl2 sin φ−FCzl2 cos φ−m2gl22 cos φ−J2θ¨+φ¨+m2l222φ¨+l1l22θ¨ cos φ−θ+θ˙2 sin φ−θ−c2φ˙−θ˙−c3φ˙=0−FCyl2 cos φ=0$

where $J2$ is the moment of inertia of the link BC about y-axis and $FCx,FCy,FCz$ represents the forces acting at point C along the three directions x, y and z, as displayed in Fig. 7(b).

The Newton–Euler equations for the landing gear strut are then given in the matrix form Display Formula

(26)$∑i=14FCi+m3g−m3a3=0$
Display Formula
(27)$l3∑i=14eCi×FCi−T3−M3+∑i=14c3ω2,i=0$

where Fstrut is the resultant of the internal forces of the structure acting on the strut, FCi are the resultants of forces acting at point Ci, eCi are the unit vectors passing through the mass centers and the points Ci (i = 1, 2, 3, 4), l3 is the length, m3 is the mass, a3 is the acceleration, T3 is the inertial acceleration of the landing gear strut, M3 the actuation torque on the strut, and $ω2,i$ is the angular velocity of joint Ci. There is no rotation and no actuation on the strut so that T3 = 0 and M3 = 0.

Considering there are only three forces and two moments acting on each of the 12 revolute joints in a planar configuration, the landing gear retraction proposed in this paper has $12×5=60$ reaction components. The mechanism comprises $2×4+1=9$ links so that there are $9×6=54$ equilibrium equations and one actuation torque, which have to be computed to describe the system. Therefore, the system has a degree of indeterminacy equalling 7. A sensible solution of the problem cannot be computed without considering the link and joint compliances. It can be achieved by examining the symmetry properties of the structure. During the folding/unfolding processes of the landing gear, the only load on the structure is its self-weight that is along the rotational axis of the structure. Thus, there is no reason to consider asymmetric forces, i.e., each force equals the corresponding one in another side stay and

$FC1x=FC2x=FC3x=FC4x=FCxFC1y=FC2y=FC3y=FC4y=FCyFC1z=FC2z=FC3z=FC4z=FCz$
$MC1x=MC2x=MC3x=MC4x=MCxMC1y=MC2y=MC3y=MC4y=MCyMC1z=MC2z=MC3z=MC4z=MCz$

where $FCix,FCiy,FCiz$ and $MCix,MCiy,MCiz$ represent the components of the force and moment acting at point Ci (i = 1,2,3,4) as shown in Fig. 8 and can be summarized as $FCx,FCy,FCz$. Then, by using the compliance of the mechanism, one can obtain the simplified Newton–Euler equations for the landing gear strut Display Formula

(28)$−4FCz+m3g−m3a3=04FCyl3=0$

Because of the symmetry of the four side stays around the landing gear strut in the design proposed and the absence of external forces during unfolding and folding processes, there is no torsion or bending moment of the strut so that no torques have been considered at joints B and C. This system of equations does not explicitly depend on the side stay angle 2$β$. Therefore, the angle between sidestays has no significant influence on the internal loads applied on the mechanism like what has been demonstrated with the dual side stays landing gear mechanism by Knowles et al. [18]. Finally, there are ten nonzero equations among the systems of Eqs. (20), (24), and (28) and ten unknowns (nine forces and one actuation torque). Having considered the symmetry of the structure while dealing with an overconstrained mechanism has allowed to remove the uncertainties of the problem. To solve these equations, one can put the systems of equations into the matrix form $Ax=b$, where A is a $10×10$ square matrix (29) and x is the $1×10$ column matrix (30) containing ten unknown and nonzero forces and torques ($FAx,FAy,FAz,FBx,FBy,FBz,FCx,FCy,FCz$ and M), and B is a $1×10$ column matrix (31)Display Formula

(29)
$A=[1001000000010010000000−100−10000000l1sin θ0l1sin θ000−1000−100−10000000−100−1000000010010000000−l2sin φ0−l2sin φ000000000−4000000004l300]$
Display Formula
(30)$x=FAxFAyFAzFBxFByFBzFCxFCyFCzM$
Display Formula
(31)$b=−m1l12θ¨ sin θ+θ˙2 cos θ0m1l12θ¨ cos θ−θ˙2 sin θ−G1G1l12 cos θ−m1l122θ¨+T1+c1θ˙+c2θ˙−φ˙−m2l1θ¨ sin θ+θ˙2 cos θ+l22φ¨ sin φ+φ˙2 cos φ0m2l1θ¨ cos θ−θ˙2 sin θ+l22φ¨ cos φ−φ˙2 sin φ−G2G2l22 cos φ+T2−m2l222φ¨+l1l22θ¨ cos φ−θ+θ˙2 sin φ−θ+c2φ˙−θ˙+c3φ˙−G3+m3a30$

It is very convenient to get the solutions of a system in the form of matrices (29), (30) and (31). The system of equations will be numerically solved in order to obtain the expressions of the forces and torques at different joints of the mechanism.

###### Numerical Simulations and Results.

The following simulations have been executed using MATLAB R2016b. For the purpose of numerical simulations, the parameters $l1$, $l2,$ and $xA−xC$ of the retraction system previously described have been chosen so that the constraint Eq. (1) is respected and that these parameters belong to the feasible region displayed in Fig. 6. Therefore, we have $l1$ = 0.6 m, $l2$= 0.4 m, and $xA−xC$ = 0.4 m. Then, the material of the structure chosen is the Ti-6Al-6V-2Sn alloy, an alloy of titanium with 5–6 mass percent of aluminum, 1–2 mass percent of tin and 5–6 mass percent of vanadium. This material is frequently used for aircraft applications because it has a high yield stress of 1210 MPa but also has a low density of 4.54 g cm−3. Moreover, that specific alloy can also be subject to heat treatments in order to strengthen it. The mass of link AB is $m1=6.4kg$, the mass of link BC is $m2=4.5kg$ and the mass of the strut and wheels is $m3=502kg$ with moments of inertias of J1 = 0.222 kg m2 and J2 = 0.065 kg m2, individually. Finally, the viscous friction coefficients of moment of the revolute joints are identically equal to $c=c1=c2=c3=1Nms/rad$. From these data, the variations of the kinematic parameters are obtained in Fig. 9, and then, the variations of the reaction forces and actuation torques are deduced and displayed in Fig. 10 during two cycles of folding inside and unfolding outside the wheel wells. As shown with an increase of $θ$ in Fig. 9, the landing gear strut is first folded. During the folding motion, i.e., while $θ$ is increasing, $φ$ decreases first, but then increases irrespective of the chosen shape for $θ$ (here sinusoidal).

From Fig. 10, it appears that the reaction forces of joints A, B and C have similar shapes and intensities. The components along the y-axis of the forces at different joints have been shown to equal zero. One can notice from Fig. 10 that the components of the forces along the z-direction at points A, B and C are almost always constant. Moreover, when the landing gear is fully folded, the components along the z-axis reach local maximum. On the other hand, when the landing gear is fully deployed, the components along the x-axis reach global minimum. In the four-side-stay mechanism, the forces, and thus the loads, are evenly distributed between the links AB and BC. These results can be used in further analyses to optimize the dynamic stiffness or strength of the mechanism.

## Prototype Fabrication

When manufacturing and assembling overconstrained mechanisms, some axial misalignments, link dimension errors or other unpredictable issues can occur and lead to a decrease of the number of degrees-of-freedom of the mechanism or an impossibility of assembling for larger deployable structure. Joint clearance [28] is one of the solution used to tackle these issues and facilitate the assembly of the mechanism without weakening the structure. A prototype of the landing gear retraction system with four side stays presented in this paper has been manufactured by using a 3D printer in order to address any potential manufacturing issues. The prototype shown in Fig. 11 has been realized by respecting the mechanism's feasible region calculated in Sec. 3. In order to obtain identical structural and dynamic properties for each of the side stay, the angle between the two planes $π1$ and $π2$ is selected to be $2β=π/2$. In this specific configuration, the structure is invariant through a rotation of $π/2$ around the landing gear main axis. Moreover, the support of the system can here be chosen as a square to reduce the storage space of the mechanism shown in Figs. 11(c) and 11(d).

As predicted in Sec. 2, an actuation of the prototype at joint A results in a vertical translation of the landing gear strut. In addition, several experiments have shown that the actuation of one unique joint A is sufficient to activate the whole mechanism and complete the desired motion and that the system can be deployed under the action of its own weight. Then, redundant and independent actuators can be used to realize the motion, which augments the reliability of the folding and unfolding processes. This characteristic is specific to the overconstrained/redundant mechanisms. The distance $xA$ from the center of the strut to point A, the lengths $l1$ and $l2$ of the links AB and BC, the radius of the strut $xC$ and the shape of the support are parameters that can be adjusted from one design of aircraft to another one in order to comply with its requirements such as its ground clearance, the sort of landing gear system used, and its number of wheels, etc., on conditions that these parameters still belong to the feasible region displayed in Fig. 6.

## Conclusions

A novel design of landing gear retraction system has been presented in this paper. Instead of having only one side stay to fold the landing gear into the wheel wells, this design has four redundant side stays symmetrically situated around the strut to ensure the motions. Then, the way this mechanism acts on the landing gear differs from the existing ones used in the airplanes. Indeed, instead of rotating to be unfolded outside or folded inside the fuselage, the landing gear using such a new retraction mechanism only goes up and down while being folded and unfolded because of the redundant side stays. The impact of those redundant constraints on the mechanism and its properties has been studied. It appears that the parameters of a mechanism with redundant constraints cannot be arbitrarily chosen and that these parameters have to be located inside a specific domain called feasible region.

An investigation was then conducted to determine the influence of the over-constrained mechanism on the dynamics of the system when the landing gear is unfolded and then folded. For this purpose, the Newton–Euler approach considering both translational and rotational dynamics has been used. The advantages of this approach over the Lagrangian approach is that it is not considering the mechanism as a whole system, but leads to the expressions of torques and forces acting on each joint of the mechanism. Finally, numerical simulations have been conducted to display the dynamics of the retraction system. Numerical simulations show that the loads are equitably distributed throughout the four side stays and that the forces and torques at the different joints are in the same order of magnitude, reducing the total wear on the landing gear and enhancing the global strength of the retraction mechanism. This landing gear retraction mechanism can then improve the ride and handling properties of advanced airplanes.

## Funding Data

• National Natural Science Foundation of China (Grant No. 51575291).

## References

Minshall, R. , 1932, “ Retractable Landing Gear,” Boeing Co, Chicago, IL, U.S. Patent No. US1874570A.
Armstrong, W. , 1932, “ Aircraft Landing Gear,” Armstrong Whitworth Co., Newcastle upon Tyne, UK, U.S. Patent No. GB365364.
Dowty, H. , 1936, “ Improvements Relating to Retractable Undercarriages for Aircraft,” U.S. Patent No. GB45187.
Robert, R. , 1945, “ Retractable Landing Gear for Aircraft,” U.S. Patent No. US2589434.
Lucien, R. , 1955, “ Retractable Aircraft Landing Gear,” Society for Industrial and Applied Mathematics, Philadelphia, PA, U.S. Patent No. US2859006.
Bendix Corporation, 1964, “ Aircraft Landing Gear,” U.S. Patent No. GB1067088.
United Aircraft Corporation, 1970, “ Aircraft Landing Gear,” U.S. Patent No. GB1215552.
Aerostar, S. A. B. , 2012, “ Jamba principal pentru trenul de aterizare al aeronavei,” U.S. Patent No. RO201100035.
Bell Helicopter Textron Inc., 2014, “ Retractable Aircraft Landing Gear,” Bell Helicopter Textron Inc., Fort Worth, TX, U.S. Patent No. EP2708464A1.
Sagem Defense Securite, 2015, “ Aircraft Comprising a Retractable Arm Equipped With an Obstacle Detector,” Messier-Bugatti-Dowty, Vélizy-Villacoublay, France, U.S. Patent No. WO2015158892A1.
Currey, N. , 1988, Aircraft Landing Gear Design: Principle and Practice, AIAA, Washington, DC.
Conway, H. , 1958, Landing Gear Design, Chapman and Hall, London, pp. 175–195.
Veaux, J. , 1986, “ New Design Procedures Applied to Landing Gear Development,” J. Aircr., 25(10), pp. 904–910.
Daniels, J. , 1996, “ A Method for Landing Gear Modeling and Simulation With Experimental Validation,” National Aeronautics and Space Administration, Hampton, VA, Technical Report No. NASA/TM-1996-201601 NCC1-208.
Hac, M. , and From, K. , 2008, “ Design of Retraction Mechanism of Aircraft Landing Gear,” J. Mech. Mech. Eng., 12(4), pp. 357–373.
Sadraey, M. , 2012, “ Landing Gear Design,” Aircraft Design: A System Engineering Approach, Wiley, New York, pp. 479–544.
Rajesh, A. , and Abhay, B. , 2015, “ Design and Analysis Aircraft Nose and Nose Landing Gear,” J. Aeronaut. Aerosp. Eng., 4(2), pp. 74–78.
Knowles, J. , Krauskopf, B. , and Lowenberg, M. , 2014, “ Numerical Continuation Analysis of a Dual-Sidestay Main Landing Gear Mechanism,” J. Aircr., 51(1), pp. 129–143.
Luo, H. , and Zhao, J. , 2018, “ Synthesis and Kinematics of a Double-Lock Overconstrained Landing Gear Mechanism,” J. Mech. Mach. Theory, 121, pp. 245–258.
Nolle, H. , 1974, “ Linkage Coupler Curve Synthesis: A Historical Review—I: Developments Up to 1875,” Mech. Mach. Theory, 9(2), pp. 147–168.
Garcia, J. , and Gutierrez-Lopez, M. , 2013, “ Multibody Dynamics With Redundant Constraints and Singular Mass Matrix: Existence, Uniqueness, and Determination of Solutions for Accelerations and Constraint Forces,” Multibody Syst. Dyn., 30(3), pp. 311–341.
Wojtyra, M. , and Fraczek, J. , 2013, “ Comparison of Selected Methods of Handling Redundant Constraints in Multibody Systems Simulations,” ASME J. Comput. Nonlinear Dyn., 8(2), p. 021007.
Wojtyra, M. , 2009, “ Joint Reactions in Rigid Body Mechanisms With Dependent Constraints,” Mech. Mach. Theory, 44(12), pp. 2265–2278.
Udwadia, F. , and Kalaba, R. , 1996, Analytical Dynamics: A New Approach, Cambridge University Press, Cambridge, UK.
Callejo, A. , Gholami, F. , Enzenhofer, A. , and Kovecses, J. , 2017, “ Unique Minimum Norm Solution to Redundant Reaction Forces in Multibody Systems,” Mech. Mach. Theory, 116, pp. 310–325.
Gonzalez, F. , and Kovecses, J. , 2013, “ Use of Penalty Formulations in Dynamic Simulation and Analysis of Redundantly Constrained Multibody Systems,” Multibody Syst. Dyn., 29(1), pp. 57–76.
Zhao, J. , Li, L. , Chen, L. , and Zhang, Y. , 2010, “ The Concept Design and Dynamics Analysis of a Novel Vehicle Suspension Mechanism With Invariable Orientation Parameters,” J. Veh. Syst. Dyn., 48(12), pp. 1495–1510.
Meng, J. , Zhang, D. , and Li, Z. , 2009, “ Accuracy Analysis of Parallel Manipulators With Joint Clearance,” ASME J. Mech. Des., 131(1), p. 001013.
View article in PDF format.

## References

Minshall, R. , 1932, “ Retractable Landing Gear,” Boeing Co, Chicago, IL, U.S. Patent No. US1874570A.
Armstrong, W. , 1932, “ Aircraft Landing Gear,” Armstrong Whitworth Co., Newcastle upon Tyne, UK, U.S. Patent No. GB365364.
Dowty, H. , 1936, “ Improvements Relating to Retractable Undercarriages for Aircraft,” U.S. Patent No. GB45187.
Robert, R. , 1945, “ Retractable Landing Gear for Aircraft,” U.S. Patent No. US2589434.
Lucien, R. , 1955, “ Retractable Aircraft Landing Gear,” Society for Industrial and Applied Mathematics, Philadelphia, PA, U.S. Patent No. US2859006.
Bendix Corporation, 1964, “ Aircraft Landing Gear,” U.S. Patent No. GB1067088.
United Aircraft Corporation, 1970, “ Aircraft Landing Gear,” U.S. Patent No. GB1215552.
Aerostar, S. A. B. , 2012, “ Jamba principal pentru trenul de aterizare al aeronavei,” U.S. Patent No. RO201100035.
Bell Helicopter Textron Inc., 2014, “ Retractable Aircraft Landing Gear,” Bell Helicopter Textron Inc., Fort Worth, TX, U.S. Patent No. EP2708464A1.
Sagem Defense Securite, 2015, “ Aircraft Comprising a Retractable Arm Equipped With an Obstacle Detector,” Messier-Bugatti-Dowty, Vélizy-Villacoublay, France, U.S. Patent No. WO2015158892A1.
Currey, N. , 1988, Aircraft Landing Gear Design: Principle and Practice, AIAA, Washington, DC.
Conway, H. , 1958, Landing Gear Design, Chapman and Hall, London, pp. 175–195.
Veaux, J. , 1986, “ New Design Procedures Applied to Landing Gear Development,” J. Aircr., 25(10), pp. 904–910.
Daniels, J. , 1996, “ A Method for Landing Gear Modeling and Simulation With Experimental Validation,” National Aeronautics and Space Administration, Hampton, VA, Technical Report No. NASA/TM-1996-201601 NCC1-208.
Hac, M. , and From, K. , 2008, “ Design of Retraction Mechanism of Aircraft Landing Gear,” J. Mech. Mech. Eng., 12(4), pp. 357–373.
Sadraey, M. , 2012, “ Landing Gear Design,” Aircraft Design: A System Engineering Approach, Wiley, New York, pp. 479–544.
Rajesh, A. , and Abhay, B. , 2015, “ Design and Analysis Aircraft Nose and Nose Landing Gear,” J. Aeronaut. Aerosp. Eng., 4(2), pp. 74–78.
Knowles, J. , Krauskopf, B. , and Lowenberg, M. , 2014, “ Numerical Continuation Analysis of a Dual-Sidestay Main Landing Gear Mechanism,” J. Aircr., 51(1), pp. 129–143.
Luo, H. , and Zhao, J. , 2018, “ Synthesis and Kinematics of a Double-Lock Overconstrained Landing Gear Mechanism,” J. Mech. Mach. Theory, 121, pp. 245–258.
Nolle, H. , 1974, “ Linkage Coupler Curve Synthesis: A Historical Review—I: Developments Up to 1875,” Mech. Mach. Theory, 9(2), pp. 147–168.
Garcia, J. , and Gutierrez-Lopez, M. , 2013, “ Multibody Dynamics With Redundant Constraints and Singular Mass Matrix: Existence, Uniqueness, and Determination of Solutions for Accelerations and Constraint Forces,” Multibody Syst. Dyn., 30(3), pp. 311–341.
Wojtyra, M. , and Fraczek, J. , 2013, “ Comparison of Selected Methods of Handling Redundant Constraints in Multibody Systems Simulations,” ASME J. Comput. Nonlinear Dyn., 8(2), p. 021007.
Wojtyra, M. , 2009, “ Joint Reactions in Rigid Body Mechanisms With Dependent Constraints,” Mech. Mach. Theory, 44(12), pp. 2265–2278.
Udwadia, F. , and Kalaba, R. , 1996, Analytical Dynamics: A New Approach, Cambridge University Press, Cambridge, UK.
Callejo, A. , Gholami, F. , Enzenhofer, A. , and Kovecses, J. , 2017, “ Unique Minimum Norm Solution to Redundant Reaction Forces in Multibody Systems,” Mech. Mach. Theory, 116, pp. 310–325.
Gonzalez, F. , and Kovecses, J. , 2013, “ Use of Penalty Formulations in Dynamic Simulation and Analysis of Redundantly Constrained Multibody Systems,” Multibody Syst. Dyn., 29(1), pp. 57–76.
Zhao, J. , Li, L. , Chen, L. , and Zhang, Y. , 2010, “ The Concept Design and Dynamics Analysis of a Novel Vehicle Suspension Mechanism With Invariable Orientation Parameters,” J. Veh. Syst. Dyn., 48(12), pp. 1495–1510.
Meng, J. , Zhang, D. , and Li, Z. , 2009, “ Accuracy Analysis of Parallel Manipulators With Joint Clearance,” ASME J. Mech. Des., 131(1), p. 001013.

## Figures

Fig. 4

Initial (a) and final (b) positions of one arm of the retraction system

Fig. 5

Maximum value of cosθ

Fig. 6

Feasible region of the mechanism depending on the parameters a, b, and xA–xC

Fig. 7

Dynamic schemes of AB (a) and BC (b) links

Fig. 3

Front (a) and top (b) view of the retraction system

Fig. 2

Landing gear using a four-side-stay retraction mechanism: (a) unfolded position and (b) folded position

Fig. 9

Kinematic curves when the strut goes up and down during two periods

Fig. 10

Forces and torques when the strut goes up and down during two periods (a) unfolded position, (b) semi-folded position, (c) folded position from isometric view and (d) folded position from top view

Fig. 11

Views of the folding process: (a) unfolded position, (b) semifolded position, (c) folded position from isometric view, and (d) folded position from top view

Fig. 1

Structure of Sarrus mechanism (a) unfolded position (b) folded position

Fig. 8

Free-body diagram of landing gear strut

## Errata

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