Research Article A Simplified Flexible Multibody Dynamics ...Research Article A Simplified Flexible...

Research ArticleA Simplified Flexible Multibody Dynamics for a Main LandingGear with Flexible Leaf Spring

Zhi-Peng Xue,1,2 Ming Li,1 Yan-Hui Li,1 and Hong-Guang Jia1

1 Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, 3888, Dong Nanhu Road,Changchun 430033, China

2University of Chinese Academy of Sciences, 19, A Yuquan Road, Beijing 100049, China

Correspondence should be addressed to Hong-Guang Jia; [email protected]

Received 5 March 2014; Revised 1 July 2014; Accepted 6 July 2014; Published 23 July 2014

Academic Editor: Hassan Haddadpour

Copyright © 2014 Zhi-Peng Xue et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The dynamics of multibody systems with deformable components has been a subject of interest in many different fields such asmachine design and aerospace. Traditional rigid-flexible systems often take a lot of computer resources to get accurate results.Accuracy and efficiency of computation have been the focus of this research in satisfying the coupling of rigid body and flex body.Themethod is based on modal analysis and linear theory of elastodynamics: reduced modal datum was used to describe the elasticdeformation which was a linear approximate of the flexible part. Then rigid-flexible multibody system was built and the highlynonlinearity of the mass matrix caused by the limited rotation of the deformation part was approximated using the linear theoryof elastodynamics. The above methods were used to establish the drop system of the leaf spring type landing gear of a small UAV.Comparisons of the drop test and simulation were applied. Results show that the errors caused by the linear approximation areacceptable, and the simulation process is fast and stable.

1. Introduction

Automatic takeoff, landing, and taxiing are very importantparts of completely autonomous flight of UAV, and taxiingtests on UAVs are obviously expensive and risky as it iseasy to bring about accidents under high-speed taxiing.Simulation on the other hand is particularly important beforethe prototype taxiing experiment. During the development ofcontrol systems, real-time simulations such as man-in-the-loop (MIL) and hardware-in-the-loop (HIL) are also usedto take place between design level simulations and costlyexperiments with the real plant [1–3]. Obviously accurate andefficient computer simulations of UAV on-ground modelsplay an important role in the control system design, per-formance evaluations, and dynamics analysis of such vehiclesystems. Meanwhile research on ground dynamics in the pastdecades has achieved significant results, and the design ofthese dynamics traditionally relies on mathematical model.

Ro [4] has studied aircraft-runway dynamics in detail. Ingeneral mathematical equations of ground dynamics arecomplex, highly coupled, and nonlinear; their derivation andnumerical implementation demand considerable time andcomputer cost. In recent years, the accuracy and compu-tational efficiency of flexible multibody has been the focusof research [5, 6]. The simplified flexible multibody methodenables the dynamic response to simulate effectively, even beused in many real-time simulations [7–9].

Nonretractable type landing gear has been widely usedin small UAVs, a leaf spring is used to provide stiffnessand damping which absorbs shock and energy by its spacedeformations [10, 11]. A common way to model body flex-ibility is the finite element (FE) method which is basedon body geometry and can solve deformations stress andstrain precisely [12, 13]. But the finite element method oftenproduces a large number of degrees of freedom (DOF) whichleads to the computer computation very costly [14]. To reduce

Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 595964, 10 pageshttp://dx.doi.org/10.1155/2014/595964

2 Shock and Vibration

the amount of DOF a simplified method was applied whichis called modal method that a body’s deformation can beapproximated by a linear combination of preselected modeshape functions [15]. Hauser et al. [7] have studied thecomputational efficiency and error of modal method, andit has been applied in video games, surgery training, andother simulation systems. In this paper modal superpositionmethod was used to establish the flexible model of the leafspring landing gear.

Flexible body, experiencing large space motion whileundergoing small elastic deformations, theoretically willcause the inertial coupling between elastic deformation andrigid body motion; the most common approach is to usea time-varying mass matrix which is highly nonlinear andneeds to be recomputed in every time step. Many methods[16] are used to simplify the mass matrix. A more simple wayis called the linear theory of elastodynamics that the impact ofelastic deformation at themultibody dynamics is ignored andmay not be suitable for some cases like high-speed rotation.

Tire is another elastic part of the system.The most repre-sentative models are based on empirical functions, designedand tuned to fit measure data. Some mathematical formula-tions can be already found in recent research. Pacejka’s MagicFormula (trigonometric formulation) [17] is a popular modelin the automotive industry. The NASA, through an AGARDproject, proposed a mathematical formulation applied tocommercial airplanes and based on polynomial equations[3, 18]. Linear parameter varying (LPV) and linear fractionaltransformation (LFT) modeling are also used together witha novel force identification approach based on nonlineardynamic inversion theory to obtain a simplified tire model[19]. To decrease the complexity of themodel a typical, widelyused tire model is a point contact spring model with eitherlinear or nonlinear stiffness and damping in parallel [4].

The idea which motivates this work is to propose ageneral modeling method for elastic leaf spring. This paperalso proposes a simplified flexible multibody to study thedynamics of a leaf spring landing gear, aiming at efficientformulation and solution of equations where a compromisebetween accuracy and computational efficiency has to bereached. Another contribution of this paper is to discussthe computer implementation of the proposed formulationand demonstrate its use by developing a drop system model.Drop test and simulation are conducted to analyze the modelaccuracy and efficiency.

2. Flexible Model

Modal method is a linear approximation of elastic deforma-tion; complex nonlinear differential equations are decoupledinto a set of single DOF linear equations bymodal coordinatetransformation. These equations can be further transformedinto a more briefly state space form as shown in Figure 1.Since the decoupled equations can be solved by computerdirectly, the main advantage of modal method is that itscomputational efficiency is very high.

2.1. Modal Superposition. Continuum body can be describedby its mass matrices 𝑀

𝑢and stiffness matrices 𝐾

𝑢, with a

Generate mesh

Discarding modes

Construct state-space

Flexible part

Stiffness and mass matrices

Modal results

Deformation

Oscillatorydifferential equation

Modaldecomposition

Coordinatestransformation

Modal analysis

Modal dataextraction

Modal coordinates differential equation

FE analysis

x = Ax + Buy = Cx + Du

Figure 1: Block diagram of modal analysis method.

damping matrix 𝐶𝑢; then the general vibration differential

equation is as

𝑀𝑢��𝑓+ 𝐶𝑢��𝑓+ 𝐾𝑢𝑢𝑓= 𝑓, (1)

where 𝑢𝑓represents the displacements vector of the node

degrees of freedom (DOF), 𝑓 represents the force vector,and generally 𝑢

𝑓has a large amount of DOF 𝑛. The modal

coordinate transformation as shown in (2) is used to decouplevibration differential equations into linear equations:

𝑢𝑓= Φ𝑞𝑓, (2)

where 𝑞𝑓represents the modal coordinates andB represents

the reduced modal matrix. By substituting (2) into (1) andpremultiplying the transpose of matrix B, one gets thefollowing reduced system with the dimension 𝑛

𝑚(𝑛𝑚< 𝑛):

𝑀𝑞𝑞𝑓+ 𝐶𝑞𝑞𝑓+ 𝐾𝑞𝑞𝑓= 𝑄 (3)

in which

𝑀𝑞= Φ𝑇

𝑀Φ = 𝐼𝑛𝑚×𝑛𝑚

,

𝐾𝑞= Φ𝑇

𝐾Φ = diag (𝜔2𝑖) ,

𝐶𝑞= Φ𝑇

𝐶Φ,

𝑄 = Φ𝑇

𝑓,

(4)

where 𝜔𝑖is the natural frequency of the 𝑖th mode (1 ≤

𝑖th ≤ 𝑛𝑚), 𝑀𝑞is identity matrix, and 𝐾

𝑞and 𝐶

𝑞are

diagonal matrices. Thus, the vibration differential equationsare approximated by (3). Each modal coordinate can becalculated independently; then modal coordinates (Figure 2)are used to solve for node displacements using (2).

Shock and Vibration 3

X

Y

Z

O

B

riB0

uBi uBf

X1

Y1

Z1

rBi

Figure 2: Coordinates of flexible body.

2.2. State-Space Form. The mathematical calculationsthrough matrix-vector form by computer will be veryefficient by transforming (3) into the state space form. Thelinear state space is as follows:

�� = 𝐴𝑥 + 𝐵𝑢,

𝑦 = 𝐶𝑥 + 𝐷𝑢,

(5)

where 𝑥 is the state vector as (6) and �� is the derivative ofthe state vector. 𝐴, 𝐵, 𝐶, and 𝐷, respectively, are the systemmatrix, input matrix, output matrix, and transfer matrix; 𝑢and 𝑦 are the I/O of system:

𝑥 = [𝑞𝑓

𝑞𝑓

] . (6)

Then (3) is transformed as follows:

𝑞𝑓+ 2Γ 𝑞

𝑓+ Ω𝑞𝑓= Σ𝑢, (7)

Σ = Φ𝑇

𝐹, (8)

where Σ is defined in (8); 𝐹 is used to define the degrees offreedom of the system input.

According to (5), the displacements, velocities, and accel-erations are described by the modal coordinates:

𝑦 = [

[

Θ𝑞𝑓

Θ 𝑞𝑓

Θ 𝑞𝑓

]

]

, Θ = 𝑇Φ, (9)

wherematrix𝑇 is used to define the system output.Then eachmatrix of state space can be determined according (7):

𝐴 = [0 𝐼

−Ω −2Γ] , 𝐵 = [

0

Σ] ,

𝐶 = [

[

Θ 0

0 Θ

−ΘΩ −2ΘΓ

]

]

, 𝐷 = [

[

0

0

ΘΣ

]

]

.

(10)

Thematrices of state space are constructed by the reducedmodal data, its output is the deformation, and input is thestress of the flexible body.

2.3. Modal Reduction. One gets the same number of modeswith the number of DOF bymodal analysis, and the accuracyofmodalmethod depends on a user-defined frequency range,though themoremodalmodes are contained, the greater costof calculation is required. Modal reduction is based on thefrequency response analysis which is used to determine thecontribution of each mode to the displacement response.

Taking Laplace transforms with zero initial conditions of(5),

𝑥 (𝑠) = (𝑠𝐼 − 𝐴)−1

𝐵𝑢 (𝑠) ,

𝑦 (𝑠) = 𝐶(𝑠𝐼 − 𝐴)−1

𝐵𝑢 (𝑠) + 𝐷𝑢 (𝑠) .

(11)

Solving the transfer functions

𝑦 (𝑠)

𝑢 (𝑠)= 𝐶(𝑠𝐼 − 𝐴)

−1

𝐵 + 𝐷 (12)

for combinations of degrees of freedom where forces areapplied and where displacements are taken can get theamplitude of the response of each mode.

3. Rigid-Flexible Multibody Dynamics

3.1. Multibody Dynamics. The most frequent approach usedin multibody system to consider flexibility is the floatingframe of reference formulation (FFRF), in which two setsof coordinate are used to describe a deformable body asdiscussed by Shabana [20]. One set is reference coordinatewhich describes the location and orientation of a selectedbody coordinate system, while the second one is elasticcoordinate which describes the deformation of the body withrespect to its coordinates. The global position of an arbitrarypoint 𝐵 on the flexible body is as follows:

𝑟𝐵

𝑖= 𝑟𝑖+ 𝐴𝑖(𝑢𝐵

𝑖+ 𝑢𝐵

𝑓) , (13)

where 𝑟𝑖is the local position vector, 𝐴

𝑖is the transformation

matrix, 𝑢𝐵𝑖is the local position of point 𝐵 in the undeformed

state, and 𝑢𝐵𝑓is the deformation vector of point 𝐵 which can

be approximated in the modal coordinate using (2).Thus, we can define the coordinates of body 𝑖 as

𝑞𝑖

=

[[[

[

𝑅𝑖

𝜃𝑖

𝑞𝑖

𝑓

]]]

]

, (14)

where 𝑅𝑖 and 𝜃𝑖 are the reference coordinates respect dis-placement and rotation, respectively; 𝑞𝑖

𝑓is the vector of elastic

coordinates.


The equations of motion of Lagrangian formalism inmultibody systems have been proposed in [20]. The aug-mentedmatrix equation expressed in generalized coordinatesis

[[

[

𝑚𝑖

𝑅𝑅𝑚𝑖

𝑅𝜃𝑚𝑖

𝑅𝑓

𝑚𝑖

𝜃𝜃𝑚𝑖

𝜃𝑓

symmetric 𝑚𝑖

𝑓𝑓

]]

]

[[

[

��𝑖

𝜃𝑖

𝑞𝑖

𝑓

]]

]

+ [

[

0 0 0

0 0 0

0 0 𝐾𝑖

𝑓𝑓

]

]

[[

[

𝑅𝑖

𝜃𝑖

𝑞𝑖

𝑓

]]

]

+

[[[[

[

𝐶𝑇

𝑅𝑖

𝐶𝑇

𝜃𝑖

𝐶𝑇

𝑞𝑖

𝑓

]]]]

]

𝜆 =

[[[[

[

(𝑄𝑖

𝑒)𝑅

(𝑄𝑖

𝑒)𝜃

(𝑄𝑖

𝑒)𝑓

]]]]

]

+

[[[[

[

(𝑄𝑖

V)𝑅

(𝑄𝑖

V)𝜃

(𝑄𝑖

V)𝑓

]]]]

]

(15)

under the constraints

𝐶 (𝑅, 𝜃, 𝑞𝑓) = 0, (16)

where 𝑚𝑖𝑖represents mass submatrices, 𝐾

𝑓𝑓is the stiffness

matrix, 𝜆 is the vector of Lagrange multiplier, matrix 𝐶𝑇𝑞rep-

resents the transposition of the constraint Jacobianmatrix,𝑄𝑒

is the vector of external force, and𝑄V is the vector of quadraticvelocity.

3.2. Coupling between Reference and Elastic Displacements

3.2.1. Inertia of Deformable Bodies. The following definitionof the kinetic energy is used:

𝑇𝑖

=1

2∫𝑉𝑖

𝜌𝑖

𝑟𝐵𝑇

𝑖𝑟𝐵

𝑖𝑑𝑉𝑖

, (17)

where 𝑇𝑖 is the kinetic energy of body 𝑖; 𝜌𝑖 and 𝑉𝑖 are,respectively, the mass density and volume of body 𝑖; and 𝑟

𝐵

𝑖

is the global velocity vector of point 𝐵 on the body. Theexpression of 𝑟𝐵

𝑖is

𝑟𝐵

𝑖= 𝑟𝑖+ ��𝑖(𝑢𝐵

𝑖+ 𝑢𝐵

𝑓) + 𝐴

𝑖(��𝐵

𝑖+ ��𝐵

𝑓) , (18)

where ��𝐵𝑖= 0 is used, the central term on the right-hand side

of (18) can in general be written as

��𝑖(𝑢𝐵

𝑖+ 𝑢𝐵

𝑓) = 𝐵𝑖

𝜃𝑖, (19)

where 𝜃𝑖is the vector of the time derivatives of the rotational

coordinates of the body reference and 𝐵𝑖 is defined as

𝐵𝑖= [

𝜕

𝜕𝜃𝑖

1

𝐴𝑖(𝑢𝐵

𝑖+ 𝑢𝐵

𝑓) ⋅ ⋅ ⋅

𝜕

𝜕𝜃𝑖𝑛𝑟

𝐴𝑖(𝑢𝐵

𝑖+ 𝑢𝐵

𝑓)] , (20)

where 𝑛𝑟is the total number of rotation coordinates of the

reference of body 𝑖. So (18) can be written as

𝑟𝐵

𝑖= [𝐼 𝐵

𝑖𝐴𝑖Φ𝑖] [

[

𝑟𝑖

𝜃𝑖

𝑞𝑖

𝑓

]

]

. (21)

Equation (17) can be written in a more compact form as

𝑇𝑖

=1

2𝑞𝑇

𝑖𝑀𝑖

𝑞𝑖

. (22)

The expression of mass matrix of body 𝑖 is

𝑀𝑖= ∫𝑉𝑖

𝜌𝑖 [

[

𝐼

𝐵𝑇

𝑖

(𝐴𝑖Φ𝑖)𝑇

]

]

[𝐼 𝐵𝑖𝐴𝑖Φ𝑖] 𝑑𝑉𝑖

= ∫𝑉𝑖

𝜌𝑖 [

[

𝐼 𝐵𝑖

𝐴𝑖Φ𝑖

𝐵𝑇

𝑖𝐵𝑖𝐵𝑇

𝑖𝐴𝑖Φ𝑖

symmetric Φ𝑇

𝑖Φ𝑖

]

]

𝑑𝑉𝑖

.

(23)

The mass matrix can also be written as

𝑀𝑖=[[

[

𝑚𝑖

𝑅𝑅𝑚𝑖

𝑅𝜃𝑚𝑖

𝑅𝑓

𝑚𝑖

𝜃𝜃𝑚𝑖

𝜃𝑓

symmetric 𝑚𝑖

𝑓𝑓

]]

]

. (24)

As given by (23) and (24), the submatrix 𝑚𝑖𝑅𝑅

associatedwith the translations of the origin of the body reference is aconstant matrix; the submatrices 𝑚𝑖

𝑅𝑓and 𝑚𝑖

𝜃𝑓represent the

coupling between reference motion and elastic deformation;and submatrices𝑚𝑖

𝜃𝜃and𝑚𝑖

𝑅𝜃depend on both the rotational

reference coordinates and the elastic coordinates of the body𝑖. As (20) shows

𝐵𝑖= 𝐵𝑖( 𝜃𝑖, 𝑞𝑖

𝑓) . (25)

If the body is rigid and the origin of the body reference isattached to the center of mass, the translation and rotationof the rigid body are dynamically decoupled. In this casethe matrix 𝑚𝑖

𝑅𝜃is the null matrix which means that the

matrix 𝐵𝑖is also the null matrix. This, however, is not the

case when deformable bodies are considered. As (25) shows,there is no guarantee that 𝐵

𝑖is the null matrix because of

the time-dependent coordinates 𝑞𝑖

𝑓. Then vector 𝐵

𝑖must be

iteratively updated. Therefore, submatrices 𝑚𝑖𝑅𝜃, 𝑚𝑖𝜃𝑓, 𝑚𝑖𝑅𝑓,

and 𝑚𝑖𝜃𝜃

must be iteratively updated, respectively. Finally,the submatrix 𝑚𝑖

𝑓𝑓in (24) is independent of the generalized

coordinates of the body, therefore, is constant.

3.2.2. Linear Theory of Elastodynamics. As above sectionshows, the mixed set of reference and elastic coordinates ofdeformable body leads to a highly nonlinear mass matrix asa result of the inertia coupling between the reference andthe elastic displacements.Themassmatrixmust be iterativelyupdated.

Rigid bodymotion and flexible deformation interact witheach other though inertial effects. The deformation body isrotating rapidly; neither effect will be significant. Thereforea solution strategy that has been used to omit these effects.The multibody system was first as rigid body systems as (26).General multi-rigid-body computer programs then be usedto solve for the inertia and reaction forces. These forces arethen introduced to the state space form of elastic deformable


State spacesimulation

Rigid bodysimulation

Reaction force

PreprocessingMotion superimposing

Deformation Rigid motion

Flexible body motion

State space offlexible body

Deflect geometry

Multi-rigid-bodysystem

Multibody system

Figure 3: Block diagram of rigid-flexible multibody system.

54

3

21

(1) Carbon cloth(2) Glass fibre(3) Glass prepreg(4) Glass fibre(5) Carbon cloth

YC

XC

Figure 4: Five-layer leaf spring landing gear.

model to solve the deformation displacements. The totalmotion of a body is then obtained by superimposing the smallelastic deformation on the gross rigid body motion as shownin Figure 3:

[𝑚𝑖

𝑅𝑅

𝑚𝑖

𝜃𝜃

] [��𝑖

𝜃𝑖] +

[[

[

𝐶𝑇

𝑅𝑖

𝐶𝑇

𝜃𝑖

]]

]

𝜆 =[[

[

(𝑄𝑖

𝑒)𝑅

(𝑄𝑖

𝑒)𝜃

]]

]

+[[

[

(𝑄𝑖

V)𝑅

(𝑄𝑖

V)𝜃

]]

]

.

(26)

4. Simulation and Test

Sections above have presented a very effective modelingmethod for rigid-flexible multibody system which will beused in this section to build the main landing gear of a smallUAV inMatlab/Simulink. Tests are implemented to verify theaccuracy of this method.

4.1. Modeling of the Leaf Spring. Cantilever type leaf springwas manufactured by composite material layers as shown inFigure 4 and the parameters of each material are shown in

Frequency (rad/s)

Mag

nitu

de (d

B)

Z-YZ-Z

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

102 103 104 105

X: 1927Y: 2.17e − 5

X: 107Y: 3.2e − 2

X: 3849Y: 1.15e − 5

Figure 5: Frequency response.

Table 1: Layer parameters.

Parameters MaterialsCarbon cloth Glass fibre Glass prepreg

Thickness (mm) 0.2 0.2 0.2Young’s modulus 𝐸

𝑥(Gpa) 57–66 17–20 42–52

Young’s modulus 𝐸𝑦(Gpa) 57–66 17–20 8–10

Young’s modulus 𝐸𝑧(Gpa) 8–10 8–10 8–10

Shear modulus 𝐺 (Gpa) 4.8 4 3.5Poisson’s parameters 0.3 0.3 0.3Destiny (g/cm3) 1.6 2 2

Table 1. The FE model of leaf spring has 2868 nodes whichmeans that it has 2868 × 3 modes. The first 6 modes are rigidbodymodes resulting from a free-freemodal analysis and the7th to 17th modes are shown in Table 2.

Frequency response analysis for first 50 modes exceptthe rigid body modes is performed at the joint connectedthe landing gear and tire. The results are shown in Figure 5,where𝑍-𝑍means the response of vertical when vertical input


7th mode 10th mode

11th mode 17th mode

Figure 6: Parts of mode shapes.

and 𝑍-𝑌 means the response of lateral when vertical input.The magnitude of lateral response (𝑌 direction) is greaterthan that of vertical response (𝑍 direction) at the formermodes, which means that the lateral stiffness is lower thanthe vertical stiffness. For the reducedmodes must include themajor modes of the two directions, mode reduce is based on𝑍-𝑍.

The 7th mode has the largest frequency magnitude, and,as shown in Figure 6, 7th mode shape is the first bendingmode; this bending shape was appearing repeatedly duringthe compressing of the leaf spring of drop tests, whichindicated that the 7th mode is the most important mode.Themagnitude response decreases as the frequency increases,and the first area peak appears at the 10th mode, and itsmode shape is second bending mode. The mode which hasa resonant area peak is the important mode, such as the 11thmode; it is the third bendingmode.Themagnitude of the 17thmode is about 103 times smaller than that of the 7th mode,and the magnitude of the resonant peaks after 17th modesis far smaller than that of the 17th mode. So the 7th to 17thmodes are the major modes.

Then 11 modes are used to construct the elastic model,and each mode shape of the reduced modes still containsthe modal datum of every node. For we only concern thenodes where forces are applied and where displacementsare taken, the nodes except these located on the joints andwhere we want to measure the deformations are excluded. Sothe flexible model is constructed by the reduced modes andreduced nodes.

4.2. Rigid-Flexible Multibody Model. A penetration formula-tion is used to calculate the normal force between the tire and

Table 2: Modal analysis results.

Modes Frequency rad/s7 1078 3229 49410 64111 126112 129413 192714 294515 339316 345317 3849

ground as shown in Figure 7(a). Similar to a spring/damperformula, the force applied to the tire is proportional to thepenetration depth and the penetration rate.The normal forceis described as

𝐹nom = 𝐾nom𝜏 + 𝐶nom 𝜏, (27)

where 𝐾nom and 𝐶nom are, respectively, the vertical stiffnessand damping of tire. 𝜏 is the penetration depth.

The lateral force exerted by the tires is described as afunction of slip angle. As shown in Figure 7(b), slip angle 𝛽is the orientations of the velocity of the tire’s points of contactthe surface of motion with respect to the tires’ longitudinalaxes.The lateral force is applied to account for the side loadingon the tire. It is applied in a direction alignedwith the rotationaxis of the wheel. This force acts in the opposite direction tothe lateral motion of the wheel and can be written as

𝐹lat = 𝐶1𝛽, (28)

where 𝐶1is the lateral stiffness factors and 𝛽 is the slip angle.

The measured data of tire parameters are shown in Table 3.The drop multibody system includes leaf spring, load

masses, and tires as in Figure 8. The leaf spring and loadmasses are connected by 2 pin joints a and b, the leaf springand tires are connected by 2 revolute joints c and d, and thelanding gear system is connected with ground by a prismaticjoint.

As introduced in Section 3.2, both rigid model andflexible model are used to build the leaf spring, and theinformation exchanged between them is reaction forces anddeformations; the external forces of the flexible model arethe reaction forces of the joints. Massless bodies are used totransfer information between rigid body and flexible body.The entire model experiences free falling if the tires have notcontacted the ground. If contact happens, reaction forces aresolved by rigid system simulation; then these forces are as theinputs of the state space to calculate the displacements of thejoint nodes which are used to correct the global positions ofthese joint nodes in the next simulation step; rigid-flexiblemultibody simulation is accomplished as this iterative loop.


Ground

C

Zi

Xi Vi

Fnom

Viroll

𝜏

(a) Vertical force

Vi

𝛽

Flat

(b) Lateral force

Figure 7: Tire model.

Load mass

Tire

YZa b

c d

e

1565mm

C. g

520

mm

289

mm

Figure 8: Drop multibody system.

Measuringpoint

Load mass

Figure 9: Statics test.

4.3. Statics Test. Statics test was conducted before the droptest as shown in Figure 9. As the results show in Table 4, theerrors of vertical displacement are below 3% for the staticsconditions. So the present flexible model of leaf spring leadsto excellent accuracy under the statics situation.

4.4. Drop Test. As shown in Figure 10(a), the drop test systemincludes supporting subsystem, slide track subsystem, liftsubsystem, and data acquisition subsystem. The drop mass(100 kg) is attached to the top of the landing gear to simulatethe landing mass. For the test results are used to comparewith the simulation results, so only the vertical velocity isconsidered.

Table 3: Tire parameters.

Parameter ValueRadius 0.1mWidth 0.075mVertical stiffness 75000N/mLateral stiffness 166.55N/∘

Mass 1.5 kg𝐼𝑥𝑥/𝐼𝑦𝑦

3.45 ∗ 10−3 Kgm2

𝐼𝑧𝑧

5.62 ∗ 10−3 Kgm2

Damping 175N/(m/s)

Table 4: Statics results.

Load mass Tests (mm) Simulations (mm) Errors10 kg 5.125 5.079 −0.9%20 kg 10.250 10.11 −0.6%30 kg 15.750 15.754 040 kg 21.000 21.4145 2%50 kg 26.625 27.3015 2.5%

Table 5: Parameters of drop tests.

Number Drop velocity Load mass Drop height Drop attitude1 1.8m/s 180 kg 0.165m Level2 3m/s 180 kg 0.46m Level

According to the use-velocity and limit-velocity of land-ing, set the drop heights as Table 5; the tires are still at thebeginning of the drop test.

Video footage is used to capture the vertical and lateraldisplacements during drop tests by the vertical and lateralfixed rulers.

4.5. Results. The flexible body motion of landing gear isshown in Figure 11. Results including the displacements and


(a) Drop test system (b) Measuring system

Figure 10: Devices of drop test system.

0 0.5 1 1.5 2 2.5 3100

200

300

400

500

600

700

800

Time (s)

Disp

lace

men

t (m

m)

Test 1Test 2

Figure 11: Motions of joint b of landing gear.

Table 6: Displacements of joint b in test 2 (mm).

Drop test Simulation Error−155.4 −141.9 8.7%122.5 132.7 8.3%−61.3 −64.7 5.5%26.6 28.7 7.9%−10.8 −11.4 5.6%

reaction forces of simulation 2 are shown in Figure 12. Thesimulation step is 1ms. It is because of the loss of energyduring impact which is caused by the frictions of drop testthat the shock time of drop test is shorter than that of thedrop simulation.

In test 2 the vertical displacements of joint b from themeasuring results of video footage and simulation resultswhen the landing gear undergoing fully compressed aregiven in Table 6. In which a negative displacement meansthe landing gear is under fully compress and a positivedisplacement means the landing gear is on the rebound.

As shown in Table 6 the landing gear absorbed majorityenergy after 3 compressions, and the errors are less than 10%.

Table 7: Error of elastic deformations of joint d in test 2 (mm).

Compression number Vertical Lateral1 8.0% 7.6%2 6.4% 6.1%3 5.3% 4.9%

Elastic deformations of joint d during the three compressionsare shown in Table 7.

5. Conclusions

A simplified rigid-flexible multibody method is presentedand used in modeling the main landing gear system of asmall UAV. The computational efficiency of the vibrationdifferential equations which has been decoupled by modalcoordinate transformation is very high. Modal superpositionhas greatly reduced the model that only 11 modes are used tobuild the flexible model of the leaf spring, while the FEmodelhas 17208 DOFs. The highly nonlinear mass matrix becauseof the finite rotation of the flexible part is eliminated by linearelastic dynamics. So the whole model is characterized by avery small number of equations, and there are no significantnonlinear factors that a very high computational efficiencycan be reached and it is very suitable for real-time simulationsystems.

The comparisons between test and simulations areapplied to verify the accuracy of this approach. Static errorsare less than 5%, and dynamics errors are less than 10%;meanwhile the simulation step has reached to 1ms. So thecomputation of this approach is accuracy and efficiency. Themodal datum used by this approach can be get from not onlythe FE modal analysis, but also the modal experiments tomodify the first fewmodes, to get amore accurate and reliableresult. Results have shown that the proposed method has agood potential to model UAV with elastic leaf spring landinggear on-ground multibody system which can be used in real-time simulations such as MIL and HIL simulations in thefuture.

Conflict of Interests

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


0 0.5 1 1.5 2 2.5 3Time (s)

Disp

lace

men

t (m

m)

−80

−60

−40

−20

0

20

y-axisz-axis

(a) Displacement of joint b

0 0.5 1 1.5 2 2.5 3

0

Time (s)

Forc

e (N

)

−6000

−4000

−2000

2000

y-axisz-axis

(b) Reaction force of joint b

Time (s)0 0.5 1 1.5 2 2.5 3

Disp

lace

men

t (m

m)

−50

0

50

100

y-axisz-axis

(c) Displacement of joint d

Time (s)0 0.5 1 1.5 2 2.5 3

Forc

e (N

)

−2000

0

2000

4000

6000

y-axisz-axis

(d) Reaction force of joint d

Figure 12: Simulation results of leaf spring of simulation 2.

Acknowledgments

The project was supported by the Knowledge InnovationProgram of the Chinese Academy of Sciences (Grant no.YYYJ-1122) and the Innovation Program of UAV fundedby the Changchun Institute of Optics, Fine Mechanics andPhysics, ChineseAcademyof Sciences,(CIOMP).The authorsalso gratefully acknowledge the support from the ScienceFund for Young Scholars of the National Natural ScienceFoundation of China (Grant no. 51305421).

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