Optimization of a main landing gear locking mechanism using bifurcation analysis

Transcription

1 Loghborogh Universit Instittional Repositor Optimiation of a main landing gear locking mechanism sing bifrcation analsis This item was sbmitted to Loghborogh Universit's Instittional Repositor b the/an athor. Citation: YIN, Y.... et al, 017. Optimiation of a main landing gear locking mechanism sing bifrcation analsis. Jornal of Aircraft, 54(6), pp Additional Information: This paper was accepted for pblication in the jornal Jornal of Aircraft and the definitive pblished version is available at Metadata Record: Version: Accepted for pblication Pblisher: c American Institte of Aeronatics and Astronatics Rights: This work is made available according to the conditions of the Creative Commons Attribtion-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Fll details of this licence are available at: Please cite the pblished version.

2 Optimiation of a Main Landing Gear Locking Mechanism Using Bifrcation Analsis Yin Yin 1 Nanjing Universit of Aeronatics and Astronatics, Nanjing, Jiangs, 10016, China Universit of Bristol, Bristol, BS8 1TR, United Kingdom Simon A. Neild and Jason Zheng Jiang 3 Universit of Bristol, Bristol, BS8 1TR, United Kingdom James A. C. Knowles 4 Loghborogh Universit, Leicestershire, England LE11 3TU, United Kingdom and Hong Nie 5 Nanjing Universit of Aeronatics and Astronatics, Nanjing, Jiangs, 10016, China A ke part of the main landing gear (MLG) of a civil aircraft is its locking mechanism that holds the gear in the deploed or down-locked state. The locking is driven b a spring mechanism and its release b the nlock actator. This paper considers this mechanism in terms of its stabilit and the locking and nlocking forces reqired for down-locking. To std this an analtical model was developed. The eqations, consisting of geometric constraints and force/moment eqilibrims, were derived sing the coordinate transformation method. Using nmerical contination to solve these eqations, the effect of the nlock force on the MLG retraction ccle was analed. The variation of a fold bifrcation point, which indicates the transition between the locked state and the nlocked state, gives frther insight into the reqired nlock force that governs the siing of the nlock actator. Moreover, some important information, sch as the critical position for the lock-links stops, the nlock position and the nlock force, are discssed sing the bifrcation diagrams for the MLG retraction/extension ccle. Then, the effect of three ke geometr parameters of the locking spring (the spring stiffness, nstrained spring length and spring attachment point) on the critical over-center angle and the nlock force are investigated. Finall, an optimiation of the critical nlock force is carried ot with a constraint on the initial over-center angle. The reslts show that the spring parameters have significant effects on the MLG s retraction performance. A 37% redction of the reqired nlock force is obtained throgh optimiing for the gear considered here. 1 Ph.D. Stdent, College of Aerospace Engineering/Department of Mechanical Engineering, ininjordan@163.com. Professor of Dnamic and Control, Department of Mechanical Engineering, simon.neild@bristol.ac.k. 3 Lectrer of Dnamic and Control, Department of Mechanical Engineering,.jiang@bristol.ac.k. 4 Lectrer of Dnamic and Control, Department of Aeronatical and Atomotive Engineering. 5 Professor of of Dnamic and Control, College of Aerospace Engineering, hnie@naa.ed.cn.

3 Nomenclatre ADAMS = Atomatic Dnamic Analsis of Mechanical Sstem CAE FP LP MIGA MLG NLG PIDO = Compter Aided Engineering = Fold Point Bifrcation = Limit Point = Mlti-Island Genetic Algorithm = Main Landing Gear = Nose Landing Gear = Process Integration and Design Optimiation ˆn * = the normal vector of side-sta plane OAB Tg l = transformation matrix between the global coordinate sstem and the local coordinate sstem l i = the length of i th link; m i = the mass of i th link; k = stiffness of spring l = nstrained length of spring s l = distance between the lower attachment point of spring and the joint point of lower side-sta s = distance between the pper attachment point of spring and the gravit center of pper side-sta l = distance between the lower attachment point of nlock actator and the gravit center of lower lock-link = distance between the pper attachment point of nlock actator and the gravit center of pper side-sta A I. Introdction N aircraft main landing gear (MLG) is a complex mechanism that mst be capable of being locked in both the deploed and retracted states. Tpicall, in the down-lock state the main strt is spported with a two-element sidesta that is locked in place sing lock-links [1]. The lock-links snap into the locked state b passing the overcenter point and hitting stops which prevent frther movement []. A spring is sed to trigger the snapping and the mechanism can onl be released from its down-locked state, allowing retraction of the landing gear, sing an nlock actator. Once nlocked, the landing gear can be retracted and fixed in the retracted state sing an p-lock mechanism [1, 3]. Dnamic simlation can be an effective method for the performance analsis of landing gear retraction mechanisms in the aviation indstr [4, 5]. Commercial dnamic simlation software (sch as ADAMS, Siemens Virtal Lab and SIMPACK) [3, 6, 7] is tpicall sed to establish a time domain dnamic simlation model of the landing gear s locking mechanism, which allows investigation of the changes of stead-state soltions, namel bifrcation points, via appling slowl-varing forces on the lock links. However, the time-domain dnamic

4 simlation approach is inefficient for finding the stead-state soltions becase a new simlation mst be carried ot for each parameter vale being changed. For these investigations, dnamic simlations reqire a large amont of comptation time. An alternative approach is to make se of dnamic sstem theor [8] to std the bifrcations and topological changes in the soltion space as parameters var, b sing a nmerical contination solver [9-11]. The ke to the efficienc of nmerical contination is that it can be sed to follow ke featres, sch as bifrcation points, directl in the parameter space of interest with standard nmerical contination software sch as COCO [1]. Bifrcation analsis and nmerical contination method have been widel applied to aircraft dnamics analsis for man ears, as discssed in the review b Sharma et al. [13]. For example, landing gear shimm oscillations have been stdied to identif regions in which the onl stable stead-state soltion is that where these is no shimm motion and their bondaries with regions where shimm exists considering two ke parameters: the forward velocit of aircraft and the vertical force acting on the landing gear [14-16]. Similarl, the characteriation of aircraft grond handling behaviors has been stdied sing bifrcation analsis method. Bifrcation diagrams in terms of the aircraft velocit and the steering angle were obtained b condcting two-parameter continations of saddle-node and Hopf bifrcations [17-19]. Regarding the contination analsis of landing gear mechanism, it is necessar to assme that the retraction/deploment motion is sfficientl slow to be considered as qasi-static. This assmption was shown to be acceptable when compared to fll dnamic models solved sing time-stepping method - both the dnamic simlation and static eqations exhibit limit points (LPs) in the locking mechanism: the prior method provides a dnamic reslt of jmp at the limit point, while the later one gives a Fold Point (FP) bifrcation point []. Using this approach, an analsis of the snapping of the MLG lock-links into the deploed state has been condcted sing bifrcation theor and the effects of geometr parameters on the locking performance stdied b tracing ke fold points in parameter space [, 0]. This method will be sed for present std. The snapping of the lock-links into the locked position is observed as a jmp, triggered b a fold in the soltion srface, from the stable soltion branch of the above-over-center state to the stable branch of the below-over-center state. Unlocking the mechanism from the locked below-over-center state reqires se of the nlock actator, which drives the lock-links back into the nlocked above-over-center state. The retraction actator is then able to move the landing gear between the deploed and retracted states. The nlock force has a significant effect on the retraction

5 trajector of the landing gear, and is affected b mltiple parameters sch as the nstrained length of lock spring, the spring stiffness and the spring attachment point. In this work, the nlock force is stdied b (1) analing the effects of different nlock forces on the retraction performance of the landing gear and () analing the effects of different geometric parameters on the nlock force. To this end, the strctre of this paper is organied as follows. A set of copled kinematic and force eqilibrim eqations for a tpical three-dimensional (3-D) retraction MLG is derived in Section. The retraction and deploment ccle, sing the nominal MLG parameter vales, is described in Section 3. In Section 4, the effects of appling different nlock forces on the MLG retraction trajector are analed in detail and two important featres, the critical nlock position and the critical nlock force, are defined. Following this, in Section 5, the effects of selecting different geometric parameters on the reqired nlock force are analed, allowing parameters to be optimied for a lower nlock force and hence a redced specification for the nlock actator. Finall, Section 6 provides some conclsions. II. Mechanism and Eqations A spatial-retraction MLG shown in Fig. 1 consists of five links. The main strt is a bffer mechanism for absorbing vertical energ dring the landing process and spporting the aircraft dring grond manevers. The pper and lower side-stas are the main parts of spporting strctres. The MLG has two separate locking mechanisms to fix the main strt in the retracted and deploed states: one is the down-lock mechanism, consisting of pper and lower lock-links; the other is a hook lock box which is applied as the p-lock mechanism (not shown). Three actators are needed to realie the movements of retraction and deploment: two nlock actators are reqired to nlock the two locking mechanisms and the retraction actator is sed to move the MLG between the deploed and retracted states. This work focses on the down-locking mechanism, namel the lock-links, rather than the hook lock box device, as the latter device has little effect on the MLG retraction performance. A. Definition of Coordinate Sstem In previos stdies a two-dimensional (D) retraction landing gear, which can be analed as a planar linkage mechanism problem withot transformation between different coordinate sstems, has been considered []. Compared to this the 3D retraction MLG is mch more complicated becase the side-stas and lock-links follow a compond motion consisting of fold and rotation. In addition, the two moving planes of side-stas and main strt are

6 non-coplanar, which leads to difficlties in the kinematic and dnamic analsis of retraction mechanisms. However, an interesting motion law, which states side-stas and lock-links are alwas in the same plane dring the whole retraction or deploment process, makes it possible to redce the complexit [4, 5]. Considering the side-stas and main strt in their own local coordinate sstems, the spatial problem is changed to two planar problems with the variables being expressed in the different coordinates and linked via a transformation matrix. Based on the above analsis, the global coordinate sstem O XYZ is set p as shown in Fig.1. The origin O is at the cross point of the three rotation axes. The OX axis is defined as the rotation shaft of the main strt and points in the direction of aircraft forward direction, the OZ axis points verticall downwards and the OY axis is decided b the right-hand rle. The local coordinate sstem for the side-stas, O x, is defined as follows: the O axis is collinear with the rotation axis of the pper node; the O axis is perpendiclar to the O axis in the side-sta plane and points downward; and the Ox axis completes the right-handed coordinate sstem. B. Coordinate Transformation Fig. 1 Definition of coordinate sstem The transformation matrix Tg lbetween the global coordinate sstem and the local coordinate sstem for the down-lock position (as shown in Fig. ) can be obtained sing the Carl Dan coordinate transformation method [8]. The order of the Carl Dan rotations is as follows. The global coordinate[ XY,,Z] sstem begins with a rotation of g labot the OX-axis, then a rotation g labot the new OY'-axis, and finall a rotation of g l abot the

7 reslting OZ"-axis into the local coordinate sstem [ x,, ]. The reslting transform is given b cgl c gl cgl s gl sgl T ' s s c c s s s s c c s c c s c s s c s s s c c c where the shorthand s sin and c cos has been sed. gl gl gl gl gl gl gl gl gl gl gl gl gl g l g l g l g l g l g l g l g l g l g l g l g l, (1) The local coordinate sstem will be rotating abot O axis as the MLG retracts, that is to sa, the down-lock position transformation matrix, Tg l, is followed b transformation matrix T 1, which is given b gl g l cos 0 sin T 0 1 0, () 1 sin g l 0 cos gl Where indicates the rotation angle of the local coordinate sstem abot the O axis. g l The rotation angle g l, can be obtained via calclating the angle change of the normal vector of plane OAB sing gl =cos where ˆn 0 indicates the initial normal vector of plane OAB; 1 nˆ 0 nˆ 1, (3) nˆ 0 nˆ 1 ˆn denotes the normal vector of plane OAB with a rotation angle 1 of the main strt. The final transformation matrix T g l can now be written as: 1 1 T T T '. (4) gl gl C. Model Formlation Before the geometric constraint and eqilibrim eqations are discssed, it is worth considering how the model will be constrcted in terms of the nknowns and the eqations relating them. The model will be derived withot the inclsion of the lock-link stops, the effect of which can be imposed on the mechanism later as a constraint. There are five elements in the mechanism, the locations of which can be described b x,,, in the local i i i i coordinate axis, where i is the angle of the element in the -axis. Alongside these 0 degrees-of-freedom there are 19 geometric constraint eqations. This indicates that the mechanism laot can be described completel b one angle or location. In addition, there are 39 forces and force balance eqations. These forces are made p of 30 interlink forces (x, and directions for either side of the 5 inter-link joints), 6 grond-link forces from the external spports, a spring force and an nlock actator force, as well as a moment applied b the retraction actator. Alongside these there are 39 force and moment balance eqations.

8 The reslt is that, for a given retraction actator moment and nlock actator force, all the other forces (or a sbset of them) and the mechanism position can be fond. D. Geometric Constraints According to the principle of 3D retraction mechanism, joints O, C, D, and E are revolte joints, whilst joints A, B and H are niversal joints. This indicates that the degree of freedom of the retraction mechanism eqals 1. Both the side-stas and the main strt can be considered as planar problems based on the definition of their coordinate sstems, ths each link can be described in terms of for degrees-of-freedom, L x,,, i i i i i L X, Y, Z,, as shown in Fig.. or i i i i i x,, and,, i i i i i i Fig. Sketch of spatial retraction mechanism X Y Z are the local coordinates and the global coordinates respectivel, both of which describe the position of L i s center of gravit. The local and global coordinate sstems are related via the transformation matrix Tg l. i indicates the relative angle between the i th link and axis. Given the conditions of 5 links, 0 states and 1degree of freedom, 19 geometric constraint eqations are needed to describe the phsical constrains in the landing gear mechanism. The geometric constraints can be expressed as follows:

9 Here B and X1 C Y1 l1 sin 1 Z1 l1 cos1 x l cos A l sin A x3 l l3 cos 3 cos3 l l3 sin 3 sin 3 l3 3 cos3 B l3 3 sin 3 B 0. (5) x4 l4 l4 4 cos4 cos4 l4 l 4 4 sin 4 sin 4 x5 l5 l4 5 cos5 4 cos4 l5 l4 5 sin 5 4 sin 4 l 5 5 cos5 H l 5 3 sin 5 H B are the vale and vale (respectivel) of point B nder the local coordinate sstem. The can be calclated b the expression: B x BX B T gl B Y. Similarl, H, H and B B Z A, A are the vale and vale of points H and A nder the local coordinate sstem respectivel.. C indicates the X vale of the center of the main strt s gravit in the global coordinate sstem. E. Force/ Moment Eqilibrim Eqations The force elements of the MLG consists of:

10 x ;, ;, i j i j i ; j F F F indicate the internal force between adjacent two links, with i and j denoting the link that is experiencing the force and the link that is appling the force respectivel. 30 parameters are needed to describe the internal forces; 6 grond-link forces: these are the internal forces between the strt and the aircraft bod, and the internal forces between the pper side-sta and the aircraft bod; Fs indicates the lock spring force; F denotes the nlock actator force; M act is the retraction actator moment. Considering all 39 forces of the retraction mechanism increases the comptational complexit, so some simplifications are pt forward here. Firstl, half the internal joint reaction forces can be removed b describing x x them as eqal and opposite (sch as F3;1 F1;3 ). As the side-stas, lock-links and main strt are planar (as shown in Fig. 3(a) and 3(b)), in general the x components of the internal forces are not needed. There are two exceptions to Y Z this de to the fact that the force elements [ F1;5, F1;5 ] and [ F1;3, F 1;3 ] are needed to calclate the moment eqilibrim Y Z x x of the main strt (as shown in Fig. 3(b)). This reqires the fll local coordinate force elements, hence F3;1 and F 5;1 are needed. Knowles et al. [] points ot that most of the grond-link forces can be removed as the are of no interest. The exception here is the x component of x F ;RA, ; RA F, which is needed to establish the mechanism x x eqilibrim eqations in the x direction combined with the force elements F3;1 and F 5;1 (as shown in Fig. 3(c)). In addition, the lock spring force, which can be determined b the geometric constraints and Hooke s Law, can be easil eliminated. After eliminating these, 14 force state variables are left along with the retraction actator moment. Treating the retraction actator moment and nlock actator force as inpts, 14 force and moment balance eqations can be derived (detailed below) allowing the forces and the mechanism position at eqilibrim to be fond.

11 Fig. 3 Force diagram of spatial retraction mechanism. (1) Force and moment eqilibrim of the side-stas and lower lock link 10 eqations Each of the links shold be in force and moment eqilibrim. The eqilibrim eqations can be written as: Fi; j=0 j Fi; j=0, (6) j * * Mi; j li; j Fi ; j 0 j where length l * i; jis the moment arm of force F i; jfrom position *. Lock spring force and nlock force can be expressed as and p low p low Fs s s s s l k low p s s Fs ;,4 Fs p low p low s s s s Spring force: low p, (7) s s Fs;,4 Fs p low p low s s s s Fs;,4 Fs;4, Fs;,4 Fs;4,

12 low p F ;,4 F p low p low low p Unlock force: F;,4 F. (8) p low p low F;,4 F;4, F;,4 F;4, Here p, p, low, low s s s s and p, p, low, low are the coordinates of the lock spring s pper and lower attachment points and the nlock actator s pper and lower attachment points, respectivel. () Moment eqilibrim of the main strt 1 eqation As shown in Fig. 3(b), the moment eqilibrim eqation of the main strt can be written as Y Z Here 1;3, 1;3 Y Z Y Z 1;3 Z 1;3 Y 1;5 Z 1;5 Y 1 1 act =0 F B F B F H F H m g Y M. (9) F F represent the forces on the main strt applied b the lower side-sta, F1;5, F 1;5 represent those on the Y Z main strt applied b the pper lock link and M is the retraction actator moment. The inverse act 1 T g l of the transformation matrix Tg lis sed to express the side-sta and lock-link local force 1;3, 1;3 1;5, F F and F F1;5 in the global coordinate sstem. The transformation expressions are given b Here 1 g l X x x F 1;3 F 1;3 F 3;1 Y 1 F1;3 Tgl F1;3 t F3;1, Z F1;3 F1;3 F3;1 (10) X x x F 1;5 F 1;5 F 5;1 Y 1 F1;5 Tgl F1;5 t F5;1. Z F1;5 F1;5 F5;1 (11) T t denotes the inverse of the transformation matrix g l. In order to get the specific expression of T Y Z 1;3, 1;3 Y Z F F and F1;5, F 1;5, eqation (10) and (11) can be expanded as follows F t F t F t F F t F t F t F F t F t F t F F t F t F t F (3)Force and moment eqilibrim in the x direction 3 eqations Y x 1;3,1 3;1, 3;1,3 3;1 Z x 1;3 3,1 3;1 3, 3;1 3,3 3;1 Y x 1;5,1 5;1, 5;1,3 5;1 Z x 1;5 3,1 5;1 3, 5;1 3,3 5;1. (1)

13 x x The force state variables contain onl three nknown components in the x direction, namel, F 1;3, F1;5 and x F ;RA. As shown in Fig. 3(c), the three nknown state variables can be obtained b the eqations as follows x x x x x F; RA lob; A m g lob; m3 g lob;3 m4 g lob;4 m5 g lob;5 0 x x x x x x x F; RA m g m3 g m4g m5g F3;1 F5;1 0. (13) x l5 x l3 x x x x F5;1 5 sin5 F3;1 3 sin3 m g m3 g 3 m4g 4 m5g 5 0 Here l OB;* indicates the distance from point * to axis OB, namel the moment arm abot axis OB and i denotes the shortest length from L i s center of gravit to axis OA. These force and moment eqilibrim eqations can now be assembled in the matrix form AF B 0 (14) Where F is a vector of the inter link forces, A is a matrix of force coefficients, and B is a vector of the remaining terms (the spring, actator and gravitational forces). These vectors and matrices are given in the Appendix. III. Retraction and Deploment Ccle In order to retract or deplo the MLG in an ordered manner, the retraction actator mst work with the nlock actator. For the retraction process, an nlock actator is needed to drive the lock-links to fold pwards and release first, then a retraction actator is engaged to retract the MLG to the p-lock position. Usall, the nlock actator can be switched off part-wa throgh the retraction ccle. However, it is worth pointing ot that the time at which this occrs has little effect on the retraction as the retraction actator generates sfficient force to conteract the external loads [0]. Hence, in this work, the nlock actator is switched off after the MLG has been locked in the retracted position, allowing the effects of the actators forces on the retraction performance to be stdied separatel. For the deploment process, the third actator is engaged to nlock the p-lock mechanism (As this mechanism does not inclde the lock-links, it is not considered here). Once nlocked, the retraction actator provides resistive moment to enable the main strt to deplo slowl. Finall, the MLG is locked at the down-lock position with the help of a spring force. Hence, a retraction schedle consists of an nlock stage, retraction stage, nlock force release stage and extension stage this is shown in Fig. 4. Note that the timescales in Fig. 4 are arbitrar, the onl real reqirement is that the ramps are sfficientl slow for the inertial loading to be negligible.

14 Given the ke roles the nlock actator force Fig.4 Schedle of landing gear retraction and extension F and the retraction moment M act pla in the retraction and extension ccle, this ccle is now considered. The static eqilibrim eqations can be solved sing the nmerical contination method, revealing how the over-center angle ov varies as a fnction of the nlock actator force F and the retraction moment M act. This over-center angle, ov 4 5, represents the angle between the two locklinks. Given the one degree-of-freedom natre of the mechanism, this angle niqel describes the state of the gear, sch that the retraction angleq 1, for example, can be calclated directl from it. De to the geometr of the locklinks, which can rotate clockwise or anticlockwise relative to each other, the reverse cannot be calclated. The parameter vales for the landing gear mechanism sed in the simlations presented here are for a mid-sie passenger aircraft and are given in Table 1. Table 1 Main parameters of retraction mechanism Parameters Vales Parameters Vales Parameters Vales l m 1.94 m1, kg s, m 0. 1, 3, l, m m, kg l, m 0.03 l m m 3, kg 69.96, m 0.1 l m , 4, l 5, m , 4,deg 18.37, / l m , 4, m kg AX, AY, AZ, m ,1.078,0.933 m kg BX, BY, BZ, m , ,1.657 k N m 6000 FX, FY, FZ, m , 0.0, gl, gl, g l, deg 10.35, 5.01, 7.75 l m 0.34 Cm, s, m l

15 Fig. 5 shows the nmerical contination reslts of a tpical retraction and extension ccle. The black crves represent the eqilibrim soltions where the lock-links are above-over-center. The dark gre crves show the eqilibrim soltions where the lock-links are below-over-center. The light gre crves denotes the MLG s response nder the sole actation of the nlock force. Frthermore, solid crves denote stable eqilibrim soltions, while dashed crves indicate nstable eqilibrim soltions. A fold bifrcation (labeled FP) occrs on the interface between the solid and dashed crves, and indicates the eqilibrim soltion changes stabilit via the FP. Fig.5 Nmerical contination reslts of a tpical retraction and extension ccle Combined with the schedles of landing gear retraction and extension, the retraction process can be inferred from the eqilibrim soltion crves. With reference to Fig 4, the single arrows on the crves denote the direction of the MLG s motion. In the nlock stage, the MLG moves from the starting position D 1 ( F 0kN ) to the nlocked position U 1 ( F 6kN ) driven b the nlock actator. After that, the main strt retracts smoothl from position U 1 to position U 4 as the retraction actator moment increases. Once the MLG has been locked at the stowed position U 4, the nlock force cold be released shown as the light gre crve, and the main strt sta at the same position U 4 dring this process. It can be fond that, removing the nlock force has little effect on the MLG s position, with the landing gear staing at point U 4. In addition, an inference can be made that the nlock actator can be switched off with no inflence once the retraction actator generates sfficient force to conteract the external loads.

16 For the extension stage, the MLG follows the black stable crve ntil the bifrcation point with the help of the resistive moment spplied b the retraction actator. Rather than following the nstable eqilibrim, the trajector jmps from the stable branch of the above-over-center crve to the stable branch of the below-over-center crve and reaches position D. After that decreasing the retraction actator force cases the MLG to go back to the initial position D 1. In realit this jmp to D is not completed de to stops, which are sed to prevent the lock-links from folding downwards beond a certain limit locking them together. Hence the eqilibrim soltion of the retraction motion on the dark gre crve wold not be phsicall realiable in an actal MLG mechanism; instead, the mechanism hits the lock-link stops, and locks at the point labeled stop. Contrar to the MLG s response with nlock force F 6kN, onl one stable eqilibrim soltion D 1 at M act = 0kNm can be seen on the dark gre crve in the case of F = 0kN. Increasing the retraction actator moment cases the MLG to follow the stable below-over-center crve with the lock-links folding downwards and reach the retracted state D 4 finall. The trajector described b the dark gre crves wold also not be phsicall realiable in realit de to the stops. For clarit, all the MLG positions corresponding to the tpical eqilibrim points (U 1 -U 4 above-over-center) and (D 1 -D 4 below-over-center) are shown in Fig. 6, panels (U 1 -U 4 ) and (D 1 -D 4 ). Fig. 6 Diagrams of the MLG retracts as the lock-links folds pwards (U 1 - U 4 ) and downwards (D 1 - D 4 )

17 IV. Effect of the nlock force on the MLG retraction ccle Discssions in the previos sections have shown that variation of nlock force cold change the folding direction and the FP bifrcation s position of the lock-links, which represents the changing of some ke properties of the MLG mechanism. As the focs of this section is to investigate the critical nlock force, nder which the landing gear cold retract normall with the lock-links folding pward, the effects of different nlock forces on the behavior of the MLG will be discssed in detail. A. Changes of the MLG Retraction Ccle with Different Unlock Force The eqilibrim soltions of the retraction mechanism nder 6 different nlock force cases are shown in Fig. 7, with the figres (a) and (f) corresponding to the behaviors shown in Fig. 6. The solid crves indicate stable soltions; dashed crves indicate nstable soltions. The black crve shows soltions where the lock-links are above-overcenter, whilst the gre crve denotes soltions where the lock-links are below-over-center. Single arrows indicate the direction of motion for an increasing retraction actator moment starting from M 0kN m. Fold point bifrcations are indicated b points force. FP and FP d. For clarit, the reslts are discssed in order of decreasing nlock As shown in Fig. 7(a), the MLG can be nlocked with the nlock force F 6kN and retracted along the black stable above-over-center crve as the retraction moment increase. Compared with the reslts in Fig. 7(a), the retraction trajector with the nlock force F 5kN shown in Fig. 7(b) changes little except the beginning phase of the retraction trajector, where the gradient of the crve has increased. Decreasing the nlock force to F 3.5kN, as shown in Fig. 7(c), a single stable eqilibrim point at M 0kN m can still be fond on the stable above-overcenter crve, which means the MLG cold be retracted b the retraction actator. However, nlike the stable crves act act in Fig. 7(a) and Fig. 7(b), two fold bifrcation points 1 FP and FP appear and divide the stable above-over-center crve into two stable parts (pper stable branch and lower stable branch), which can form a hsteresis loop shown in 1 Fig. 7(c). Before reaching the bifrcation FP, the MLG follows the lower black stable branch as the retraction moment increases, bt small changes of over-center angle indicate the MLG hardl moves while on the lower 1 branch. Increasing the retraction moment past FP, the retraction trajector jmps from the lower stable branch to

18 the pper stable branch, which means the lock-links fold sharpl, and the MLG gets into the reglar retraction crve. The reason for the two bifrcation points and the reslting hsteresis loop is that the MLG stas at the critical state between nlocked and locked states. In other words, the MLG is incompletel nlocked with the nlock force of F 3.5kN ntil the jmp where the links snap nlocked. Fig. 8(d) shows the eqilibrim soltion with the nlock force of F.5kN. In this case, the MLG can onl follow the light gre stable crve between the aboveover-center and below-over-center crves. As shown in the Fig. 8(d), increasing retraction moment hardl moves the mechanism, which means the MLG cold not be nlocked when the nlock force F.5kN. Regarding the reslts shown in Fig. 7(e) and Fig. 7(f), the conclsions are similar with the ones shown in Fig. 7(c) and Fig. 7(b) respectivel, with the onl difference of the lock-links folding direction. To be specific, two bifrcation points occr on the below-over-center crve and case the forming of hsteresis loop in the case of F 1.5kN (as in Fig. 7(e)). When the nlock force redces to ero, the MLG can be retracted with the lock-links folding downwards.

19 Fig 7 Nmerical contination reslts nder different nlock actator force B. The Definition of Different Unlock Region and Critical Unlock Positions Based on the above analsis, the effect of nlock force on the MLG s retraction behavior can be divided to five regions, which are shown in Fig. 8: (1) The above-over-center, completel nlocked region: in this region, the MLG has been nlocked and can be retracted smoothl b the retraction actator. () The above-over-center, incompletel nlocked region: in this region, the MLG will go throgh a gentle-slope trajector, and will onl retract smoothl once it jmps past the bifrcation point. (3) The locked region: in this region, the MLG can never be retracted b the retraction actator. (4) The below-over-center, incompletel nlocked region: in this region, the MLG will go throgh a gentle-slope trajector, and will onl retract smoothl once it jmps past the bifrcation point. However, the motion that the locklinks fold downwards wold not happen in realit owing to the stops. (5) The below-over-center, completel nlocked region: in this region, the MLG cold be retraced b the retraction actator with the lock-links folding downward. Again, this is not phsicall realiable in an actal retraction mechanism de to the lock-links stops.

20 Fig. 8 Bifrcation diagrams and projection of the MLG retraction ccle with different nlock forces As shown in Fig. 8(b), two important definition of critical positions are provided in this section: one is the critical position for the design of the lock-links stops; the other is the critical nlocked position which means the MLG has been completel nlocked. Stops can be defined as the locking point where the lock-link stops contact with each other to prevent the locklinks folding downwards. The locking point is a position where the MLG is totall locked, ths the locking point shold locate in the completel locked region or below this region (with the stops locking the mechanism). As a reslt, the critical position for the design of the lock-links stops is the pper bondar of the completel locked region shown in Fig. 8(b). The nlocked state indicates where the MLG can be retracted smoothl with the retraction actator, and the critical nlocked position indicates the interface between the above-over-center, completel nlocked and the aboveover-center, incompletel nlocked regions. From the locs of the FPs as a fnction of nlock force F and

21 retraction moment M act shown in Fig. 8(b), it can be seen that the two fold bifrcation points come together and disappear at a csp point (indicated b *) as the nlock force F increases. The disappearance of the FP means the retraction mechanism enters the completel nlocked region. Therefore, the position corresponding to the csp point is the critical nlocked position and the nlock force corresponding to the csp point is the critical nlock force representing the minimm force that shold be sed to nlock the mechanism. C. The Transition Between Different Unlock Regions Firstl, consider the transition between the above-over-center, completel nlocked and the above-over-center, incompletel nlocked regions which captres the critical force reqired for nlocking: In the above-over-center, incompletel nlocked region, two fold bifrcation points appear on the above-over-center retraction trajector. From the detailed graphs of the locs of the FPs shown in Fig. 9, it can be seen that the two fold bifrcation points come together and disappear at a csp point as the nlock force incompletel nlocked region transfers to the completel nlocked region. F increases. The disappearance of FP means the Fig. 9 Fold bifrcation point trace varied with nlock forces on the above-over-center branch Secondl, consider the transformation between the above-over-center, incompletel nlocked and the completel locked regions which captres the critical position for the stops: In the above-over-center, incompletel nlocked region, the MLG can be retracted on the condition that the retraction moment passes the bifrcation point namel, the critical vale. As shown in Fig. 8(b), the locs of bifrcation points in the above-over-center, incompletel nlocked region shows the 1 FP, 1 FP arbitraril close to the pper bondar of completel locked region and the corresponding moment tending to infinit. That is to sa, if the nlock force is located in the completel locked

22 region, the critical moment corresponding to hdralic sstem, ths the landing gear cold not retract in this region. 1 FP is ver large and reqiring an nrealistic level of power from the V. Optimiation of Critical Unlock Force Considering Mltiple Geometric Parameters The conclsions in section 4 show that the nlock actator needs to provide sfficient force to drive the locklinks past the critical angle before the MLG can be retracted smoothl, which indicates that the nlock force has a great effect on the landing gear s retraction performance. Additionall, previos work [] has shown that the nlock force is affected b geometr parameters, sch as the spring stiffness k, the nstrained spring length l and the attachment point of spring s l. Therefore, in this section, the dependence of the critical nlock angle and the nlock force on different parameters of interest is analed in detail. Following this, an optimiation std is carried ot to redce the force needed to nlock the mechanism. A. Effect of Mltiple Parameters on the Critical Unlock Angle Before analing the effect of mltiple parameters on the nlock force, the variation of critical nlock angle de to changes in gear parameters is considered first. It shold be noted that three ke parameters are of interest: inclding spring stiffness k ; nstrained spring length l ; attachment point s l. The parameter vales of three different test cases are shown in Table. Table Geometr parameters vales of different cases Case nmber Spring stiffness k Unstrained length l 1 / 4 / 3 6 / Attachment point s l kn m 0.34m 0.05m kn m 0.365m 0.1m kn m 0.39m 0.15m Bifrcation diagrams in the above-over-center, incompletel nlocked region are shown in Fig. 10. For the different cases, thin solid crves indicate the nlock process; dashed crves indicate the locs of FPs in the aboveover-center, incompletel nlocked region; heav solid crves indicate the over-center angle varied with retraction moment. The nlock crve, represented b the thin solid crves, moves down as the parameters are changed, which means the reqired nlock force corresponding to the same nlock angle increases. As sggested b the variation in the dashed crves, the nlock force corresponding to the nlock csp point, namel critical nlock force, increases with

23 the three ke parameters increasing. However, the critical nlock angles corresponding to the csp points, represented b black filled circles, do not change with parameters of interest. The reslts demonstrate that the three ke parameters ( k, l, s ) ma change the critical nlock force, bt the have little effect on the critical angle which l is related onl to the strctral parameters of the landing gear links. Fig. 10 Nmerical contination reslts on the space of k, l and s l. B. Effect of Mltiple Parameters on the Unlock Force This section demonstrates how the nlock force varies with the ke parameters of interest. Taking the horiontal state of the lock-links as an example to investigate how all the parameters ( k, l, s ) affect the nlock force F [], the following reslts can be obtained b sing ov as a fixed parameter ( ov 0deg ) and setting nlock actator force F as the state variable. l

24 Fig. 11 varies with three ke geometric parameters As shown in Fig. 11(a) and Fig. 11(b), the spring stiffness k and nstrained length l both have linear effects on the nlock force F. The increase of spring stiffness k and the decrease of nstrained length l increase the moment created b the spring force abot the lock-link-side-sta joint, reslting in an increase of the reqired nlock force. The relationship between the nlock force and the position of the spring attachment point, shown in Fig. 11(c), is nonlinear and two LPs appear at the peak and thogh of the crve. To be specific, the trend of nlock force as a fnction of nstrained spring length can be divided into three parts: first it increases, then it decreases, and finall it increases again. The reason for this trend can be explained with Fig. 1 as follows. The reqired nlock force to maintain eqilibrim changes with the moment prodced b the spring force, which is related to the attachment point s l. The increase of s l cases the distance between the spring ends and spring force to firstl decrease and then increase. In contrast to this, the moment arm of spring firstl increases and then decreases with the increase of s l. Ths the spring length and the moment arm of spring make p a pair of conflicting factors, and the weight change of the effect on nlock force between the two conflicting factors leads to the nonlinear relationship shown in Fig. 11(c). Fig 1 Force diagram of lock spring

25 C. Optimiation of the Unlock Force In this section, an optimiation is carried ot to minimie the critical nlock force throgh selection of three design variables: spring stiffness k ; nstrained spring length l and spring attachment point s l. The Mlti-Island Genetic Algorithm (MIGA) method [1] is adopted to optimie the nlock force based on a Compter Aided Engineering (CAE) software Isight, combined with MATLAB. Isight is a Process Integration and Design Optimiation (PIDO) software framework [], which enables varios applications to be easil integrated inclding MATLAB. The process of optimiation is illstrated b the flow chart shown in Fig. 13. The framework consists of Isight and COCO, which are sed as a data processing toolbox and a contination algorithm toolbox respectivel. At the beginning of each iteration, a selection of k, l, s based on the rles of MIGA is condcted b Isight and transferred l to COCO. B sing nmerical contination algorithm, the critical nlock force (object variable) over-center angle c F and the initial 0 ov can be obtained and transferred back to Isight for checking the convergence propert of object variable. After that, the iterative comptation is performed again ntil the convergence propert meets the reqirements. The target of contination algorithm is to obtain the csp point in the above-over-center, incompletel nlocked region, which shold be fond exactl nder a specific set of parameters k, l, s. Based on the above analsis, it can be seen that k, l, s l affect the nlock force F in the above-over-center, incompletel nlocked region, bt have little effect on the over-center angle ov. Ths the location of above-over-center, incompletel nlocked region can be fond based on the over-center angle ov. According to the details shown in Fig. 10, the landing gear will operate in the above-over-center, incompletel nlocked region within the scope of over-center angle 0 o o ov ~ 33.6, and two fold bifrcation points appear on the retraction trajector. Ths the flow of each iteration can be indicated as follows: firstl, calclate the nlock crve b sing contination algorithm and choose 5 o as the start point where the landing gear is in the incompletel ov nlocked state. Secondl, two 1-parameter contination slices ( FPs) can be obtained b increasing the retraction moment. Finall, -parameters contination is performed to get the locs of FPs and the csp point. The nlock force corresponding to the csp point is the critical nlock force c F. l

26 Fig. 13 Flow chart of optimiation combined with contination algorithm The bonds of variables X k, l, s are set as X /,0.,0.05 and 8 /,0.5,0.4 l lb KN m m m 0 ov X b KN m m m. Additionall, the constraint of initial over-center angle ( 45.0deg ) needs to be inclded in the optimiation in order to garantee the locking mechanism s abilit to lock. The objective variables shold not onl realie the minimm critical nlock force, bt also make the lock-links reach the initial over-center angle, which needs to be the same as the initial over-center angle realied b the defalt parameters with no stops. Table 3 Optimiation reslts for minimiing critical nlock force Spring stiffness k Unstrained length l Attachment point s l Lock abilit Initial angle 0 ov Critical nlock c force F Defalt 6 kn / m 0.34m 0.5m 45.0deg 4.7kN Optimiation 4.8 kn / m 0.5m 0.10m 45.deg.664kN Fig. 14 Optimiation reslts of critical nlock force

27 The optimiation reslts are shown in Fig. 14, with nmerical reslts given in Table 3. The reslts show that, the optimied parameters make the critical nlock force decrease greatl b 37% while ensring the same initial overcenter angle of the lock-links with no stops. VI. Conclsion With nmerical contination, the effects of nlock force and parameters of interest on a MLG s retraction performance can be analed efficientl. The nlock force has a great effect on the locking mechanism: the main strt can be retracted smoothl onl when the lock-links pass a critical angle. For different nlock forces, nlock states of the MLG can be divided into five separate regions. Two important critical positions are identified throgh bifrcation analsis method and can be described as follows: the critical position for the lock-links stops is the pper bondar of the completel locked region; the critical nlocked position locates at the interface between the above-over-center, completel nlocked and the above-over-center, incompletel nlocked regions. Analing the effects of ke parameters on locking mechanism shows that spring parameters do not affect the critical nlock overcenter angle, bt do have effect on the critical nlock force. The spring stiffness k and nstrained length l both have linear effects, while the attachment point s l has nonlinear effect, on the nlock force F. Based on optimiation of nlock force, an appropriate set of spring parameters can be obtained to make a 37% redction on nlock force F, whilst maintaining the same level of locking abilit, with an initial over-center angle of 0 ov 45.deg. With regards to ftre work, this techniqe for optimiation of critical nlock force cold be applied to nose landing gear (NLG) which se the same locking mechanism to down-lock and p-lock the strt [, 0]. The optimiation design of the NLG s locking spring shold be more complex since it is a mlti-objective and mlti-constrained optimiation problem. Three optimiation objects wold be considered: the minimied critical nlock force for both the p-lock and down-lock mechanism; the minimied spring stiffness to provide sfficient force to p-lock the NLG in the stowed position. Acknowledgments This work was financiall spported b the National Natral Science Fondation of China (113719), the Priorit Academic Program Development of Jiangs Higher Edcation Instittes, and the China Scholarship Concil Stdentship. In additions S. A. N. was spported b an EPSRC fellowship (EP/K005375/1).

28 Appendix Based on the description given in section II.E, the internal forces ma be derived from the actator forces and landing gear position sing the matrix expression AF B 0 (Eq. 14) where A 0 C1 C ls lc C l3s3 l3c l4s4 l4c C4, l5s5 l5c ; C C C7 C8 C9 0 0 C10 C11 C1 0 l OB A F F F F F F F F F F F F F F F x ; RA ;4 ;4 ;3 ;3 x 3;1 3;1 3;1 4;5 4;5 x 5;1 5;1 5;1 l l l l m g s m g c F s s F s c mg 3 mg 3 l3 l3 m3 g s3 m3g c3 Fs ;4, m4 g ;4, 4 Fs m g l4 l4, B Fs ;4,sl s 4 Fs ;4,slc 4 m4 g s 4 m4 g c 4 mg 5 mg l l m5 g s5 m5 g c5 x x x x m g m3g m4g m5g x x x x m g lob; m3 g lob;3 m4 g lob;4 m5 g lob;5 x x x x m g m3g 3 m4g 4 m5 g 5 m1gy1 M act s;,4 s;,4 and the shorthand s i = sinq, c i i = cosq i for i=1,,3,4,5 has been sed along with l l C1 l4 sin 4 sin C l4 cos 4 cos,

29 l l l4 l4 C3 n sin n cos C4 m sin4 m cos4, l3 l5 C5 3 sin3 C6 5 sin5 C7 t,1 BZ t3,1 BY C8 t, BZ t3, BY,,, C9 t,3 BZ t3,3 BY C10 t,1 FZ t3,1 FY C11 t, FZ t3, FY C1 t,3 FZ t3,3 FY,,, References 1. Crr, N. S. Aircraft Landing gear Design: Principle and Practice. Washington, D.C.: AIAA, Knowles, J. A. C., Kraskopf, B., and Lowenberg, M. H. "Nmerical Contination Applied to Landing Gear Mechanism Analsis," Jornal of Aircraft Vol. 48, No. 4, 011, pp doi: /1.C Wei, X. H., Yin, Y., Chen, H., and Nie, H. "Modeling and simlation of Aircraft Nose Landing Gear Emergenc Lowering Using Co-simlation Method," Advances in Design Technolog, Vols 1 and Vol , 01, pp doi: / 4. Hać, M., and Form, K. "Design of Retraction Mechanism of Aircraft Landing Gear," Mechanics and Mechanical Engineering Vol. 1, No. 4, 008, pp Yin, Y., Nie, H., Ni, H. J., and Zhang, M. "Reliabilit Analsis of Landing Gear Retraction Sstem Inflenced b Mltifactors," Jornal of Aircraft Vol. 53, No. 3, 016, pp doi: /1.C Zhang, H., Ning, J., and Schmeler, O. "Integrated Landing Gear Sstem Retraction/Extension Analsis Using ADAMS," International ADAMS User Conference. Orlando, FL, Yin, Y., Nie, H., Wei, X. H., and Ni, H. J. "Retraction sstem performance analsis of landing gear with the inflence of mltiple factors," Jornal of Beijing Universit of Aeronatics and Astronatics Vol. 41, No. 5, 015, pp Amiroche, F. M. L. Fndamentals of Mltibod Dnamics: Theor and Applications. Chicago: Birkhäser Basel, Kraskopf, B., Osinga, H. M., and Galán-Vioqe, J. Nmerical Contination Methods for Dnamical Sstems. Netherlands: Springer, 007.

30 10. Knowles, J. A. C., Kraskopf, B., and Lowenberg, M. "Nmerical contination analsis of a threedimensional aircraft main landing gear mechanism," Nonlinear Dnamics Vol. 71, No. 1-, 013, pp doi: /s Knowles, J. A. C., Kraskopf, B., Lowenberg, M. H., Neil, S. A., and Thota, P. "Nmerical Contination Analsis of a Dal-Sidesta Main Landing Gear Mechanism," Jornal of Aircraft Vol. 51, No. 1, 014, pp doi: /1.C Dankowic, H., and Schilder, F. Recipes for Contination. Philadelphia: Societ for Indstrial and Applied Mathematics, Sharma, S., Coetee, E. B., Lowenberg, M. H., Neild, S. A., and Kraskopf, B. "Nmerical contination and bifrcation analsis in aircraft design: an indstrial perspective," Philosophical Transactions of the Roal Societ a-mathematical Phsical and Engineering Sciences Vol. 373, No. 051, 015. doi: ARTN /rsta Thota, P., Kraskopf, B., and Lowenberg, M. "Mlti-parameter bifrcation std of shimm oscillations in a dal-wheel aircraft nose landing gear," Nonlinear Dnamics Vol. 70, No., 01, pp doi: /s Howcroft, C., Kraskopf, B., Lowenberg, M. H., and Neild, S. A. "Inflence of Variable Side-Sta Geometr on the Shimm Dnamics of an Aircraft Dal-Wheel Main Landing Gear," Siam Jornal on Applied Dnamical Sstems Vol. 1, No. 3, 013, pp doi: / Feng, F., Nie, H., Zhang, M., and Peng, Y. M. "Effect of Torsional Damping on Aircraft Nose Landing- Gear Shimm," Jornal of Aircraft Vol. 5, No., 015, pp doi: /1.C Rankin, J., Desroches, M., Kraskopf, B., and Lowenberg, M. "Canard ccles in aircraft grond dnamics," Nonlinear Dnamics Vol. 66, No. 4, 011, pp doi: /s