a numerical-experimental method for drop impact analysis ... · identification of composite landing...

Research ArticleA Numerical-Experimental Method for Drop Impact Analysis ofComposite Landing Gear

Yongliang Guan12 Zhipeng Xue13 Ming Li13 and Hongguang Jia13

1Changchun Institute of Optics Fine Mechanics and Physics Chinese Academy of Sciences 3888 Dong Nanhu RoadChangchun 130033 China2University of Chinese Academy of Sciences 19 A Yuquan Road Beijing 100049 China3Chang Guang Satellite Technology Co Ltd Bei Yuanda Street and Longhu Road Interchange Changchun 130052 China

Correspondence should be addressed to Hongguang Jia 395227649qqcom

Received 4 January 2017 Revised 17 February 2017 Accepted 15 May 2017 Published 13 June 2017

Academic Editor M I Herreros

Copyright copy 2017 Yongliang Guan et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The special material performance manufacturing process machining behavior and operating condition of composite materialscause uncertainties to inevitably appear in the mechanical properties of composite structures Therefore variability in mechanicalproperties must be considered in a mechanical response analysis of composite structures A method is proposed in this paperto predict the dynamic performance of composite landing gear with uncertainties using experimental modal analysis data andnonlinear static test data In this method the nonlinear dynamic model of the composite landing gear is divided into two partsthe linear and the nonlinear parts An experimental modal analysis is employed to predict the linear parameters with a frequencyresponse function and the nonlinear parameters caused by large deflection are identified by a nonlinear static test with the nonlinearleast squares method To check the accuracy and practicability of the method it is applied to drop impact simulations and tests ofcomposite landing gear The results of the simulations are in good agreement with the test results which shows that the proposedmethod is perfectly suited for the dynamic analysis of composite landing gear

1 Introduction

Landing gear is the undercarriage of an aircraft or spacecraftandmay be used for either takeoff or landing For aircraft thelanding gear supports the craft when it is not flying allowingit to take off land and taxi without damage The impactloads are extremely high when the aircraft is landing [1ndash4]and the aircraft equipment would be damaged if the high-impact loads were transmitted to the airframeTherefore thelanding gear is designed to absorb and dissipate the kineticenergy of the landing impact reducing the impact loadstransmitted to the airframe The need to design landing geartakes into account various requirements of strength stabilityand stiffness so accurate analyses are needed in the design oflanding gear

Composite materials are widely used in aerospace air-craft marine civil automotive sport [5] and even opticalengineering [6ndash10] This is due to their outstanding physicalmechanical and thermal properties particularly their high

stiffness and strength to weight ratios good fatigue strengthexcellent corrosion resistance and dimensional stabilityComposite materials frequently appear in impact conditionIncreasing numbers of landing gears are being made ofcomposite materials in the aviation field

The dynamic characteristic analysis of the compositestructures has recently attracted more attention The finiteelement method is generally used for the analysis of complexstructural behaviors of the composite structures Many excel-lent theories are available on the finite element method forcomposite structural analysis The classical analysis theory isbased on the Kirchhoff plate theory which is the simplest the-ory among themThe first-order shear deformation theory issuitable for the global structural behavior of thin and moder-ately thick laminated composite Various higher-order sheardeformation theories have overcome limitations in both theclassical and first-order shear deformation theories Layer-wise lamination theory which can predict the interlaminarstresses accurately assumes a displacement representation

HindawiShock and VibrationVolume 2017 Article ID 3073524 11 pageshttpsdoiorg10115520173073524

2 Shock and Vibration

formula in each layer and the theory based on the 3Dcontinuum can predict a composite laminatersquos interlaminarstress [11]

The theories have made significant progress but are basedon accurate composite material parameters The materialproperties determined from standard specimens tested in thelaboratorymay deviate significantly from those of actual lam-inated composite componentsmanufactured in a factory [12]Composite structures exhibit variability and uncertainties intheir material properties with relation to their compositionsand manufacturing processes [13] Operating conditionsquality control procedures and environmental effects areoften difficult to control which affect the performance ofcomposite structures [14] Delamination is a major problemin composite manufacturing which brings uncertainties inthe material properties of composites [15] The nonhomoge-neous and anisotropic nature of composite materials maketheir machining behavior differ from metal machining [16]which can affect the mechanical properties of the compositestructures [17] Adhesive bonding is common practice in theinteraction of composite parts and its uncontrollable fea-ture of consumption makes the partsrsquo preformation discrete[18] To ensure high reliability of the structures the actualbehaviors of the composite parts in servicemust be accuratelypredicted and carefully monitoredTherefore it is essential tocapture and model the variability and uncertainties inherentin these material properties

The properties of composites can be obtained accu-rately via experimental tests [19] so the mixed numerical-experimental technique is applied to extract the physicalinformation of composites This usually involves minimizingan error function between the experimental and numericaloutputs [13] Constructing of predictive computational mod-els for analysis and design of many complex engineeringsystems requires not only a fine representation of the rel-evant physics and their interactions but also a quantitativeassessment of underlying uncertainties and their impacton design performance objectives [20] Hence a thoroughcharacterization of the mechanical properties of these struc-tures is needed to establish reliable designs Soares et alidentified elastic properties of laminated composites by usingexperimental eigenfrequenciesThe stiffness parameters wereidentified from the measured natural frequencies of thelaminated composites plate by direct minimization of theidentification function [21] A similar method was used byFrederiksen and Araujo Frederiksen improved the modelused for identification [22] Araujo used a mixed numerical-experimental technique to identify the damping propertiesof plyometric composites [23] Cunha et al identified severalproperties of a composite plate froma single test [24]Diveyevet al predicted elastic and damping properties of compositelaminated plates on the basis of static three-point bendingtests measured eigenfrequencies and refined calculationschemes [25] Rikards et al identified the elastic propertiesof cross-ply laminates from the measure eigenfrequencies ofcomposite plates [26]

The mixed numerical-experimental technique makes thefinite element method suitable for dynamic analysis of thelaminated composites However the tests for measuring

eigenfrequencies of composite plates are almost all linearas in the works mentioned above and do not consider thestructural nonlinearities of the laminated composites Thereare obvious geometrical nonlinearities in the work processof landing gears so the nonlinear system identification isneeded in the nonlinear structural models of landing gearsThe nonlinear system identification in structural dynamicshas been studied since the 1970s many excellent methodsare available for identification of nonlinear structuralmodelsTime-domain methods such as the restoring force surfacemethod the approach based on nonlinear autoregressivemoving average with exogenous input model and Hilberttransform-based data decomposition [27] have the advan-tage that the signals are directly provided by currentmeasure-ment devices less time and effort is spent on data acquisitionand processing [28] The frequency-domain methods con-sider the data from Fourier spectra frequency response andtransmissibility functions or power spectral densities andextend modal analysis to nonlinear structures [28] Time-frequency analysis offers useful insight into the dynamics ofnonlinear systems [27] Modal methods develop nonlinearsystem identification techniques based on nonlinear modes[28] Black-box modeling concentrates on function whichmaps the input to the output [28] Model updating methodsimprove the nonlinear dynamic modeling with test results[27]

Our study is aimed at expanding a numerical-experi-mental method based on linear and nonlinear tests for non-linear modeling and drop impact analysis of composite land-ing gear

2 Numerical-Experimental Method

The equation of motion of an 119899-degree-of-freedom nonlinearsystem can be written in the following form

Mx (119905) + Cx (119905) + (K + Ku) x (119905) = F (119905) (1)

where M and K are the real symmetric mass and stiffnessmatrices respectively C is the damping of the system x(119905)and F(119905) are the displacements and exciting load vectorsrespectively and Ku is the geometrically nonlinear stiffnesscomponent dependent on displacements For the relevantproblems the nonlinear stiffness force vector Kux(119905) repre-sents a deviation from the linear stiffness force vector Kx(119905)and is more than adequately represented by second- andthird-order terms in x(119905) When displacements are small thesecond- and third-order terms become negligible and thetotal stiffness-related force vector is reduced to the regularlinear term Kx(119905)

Any solution to (1) requires knowledge of the systemmatrices In the linear identification works mentioned aboveM K and C are generally available The geometricallynonlinear stiffness is related to Ku which is typically notavailable within a linear identification work Therefore amean of numerically evaluating M K C and Ku was devel-oped

Shock and Vibration 3

A set of coupled modal equations with reduced degreesof freedom is first obtained by applying the modal coordinatetransformation

x (119905) = Φq (119905) (2)

where Φ is eigenvector matrices of the model defined in (1)without Ku and q(119905) is the vector of the displacement modalcoordinates for each time instance 119905

Generally a subset of 119897 eigenvectors are included in thesolution such that 119897 le 119899 and 119899 is the number of physicaldegrees of freedom

Then (1) can be expressed as

Mq (119905) + Cq (119905) + Κq (119905) + 120574 (1199021 1199022 119902119897) = F (119905) (3)

where

M = Φ119879MΦ = ΙC = Φ119879CΦ = 2120585120596Κ = Φ119879KΦ = 1205962120574 = Φ119879KuΦ

F (119905) = Φ119879F (119905)

(4)

1199021 1199022 119902119897 are the components of q(119905) 120596 is the eigen-value matrices of the model defined in (1) and 120585 is thedamping ratio of the model Then (3) can be written as

q (119905) + 2120585120596q (119905) + 1205962q (119905) + 120574 (1199021 1199022 119902119897)= ΦΤF (119905) (5)

Equation (5) can be divided into two parts the linear part

q (119905) + 2120585120596q (119905) + 1205962q (119905) = ΦΤF119898 (119905) (6)

and the nonlinear part

120574 (1199021 1199022 119902119897) = ΦΤF119904 (119905) minus 1205962q (119905) (7)

where F119898(119905) is linear identification load and F119904(119905) is nonlinearidentification load

In the numerical-experimental method of this paperthe linear part was identified by experimental eigenvaluematrices eigenvectormatrices and others from experimentalmodal analysis and the nonlinear part was identified bya nonlinear static test The flowchart of the numerical-experimental method is shown in Figure 1

21 Linear Parameters Identification The linear parameterswere identified by experimental modal analysis The exper-imental modal analysis is a method to describe a structurein terms of its natural characteristics which are the frequen-cies damping and mode shapes By using signal-analysistechniques one can easily measure vibration on operatingstructures and make a frequency analysis

The frequency spectrum is the description of how vibra-tion levels vary with frequencies which can then be checked

against a specification The frequency response functionmeasurement removes forces spectrum from the data anddescribes the inherent structural response between definedpoints on the structure

In the time domain structural properties are given interms of mass stiffness and damping The dynamic equilib-rium of an 119899-degree-of-freedom system is generally given by

Mx (119905) + Cx (119905) + Kx (119905) = F (119905) (8)

The corresponding dynamic equilibrium in frequencydomain can be described as

(K + 119894120596C minus 1205962M)X (120596) = F (120596) (9)

where 120596 is the circular frequency of exciting loads and X(120596)and F(120596) are the Fourier transforms of output and input

X (120596) = int+infinminusinfin

x (119905) 119890minus119894120596119905119889119905F (120596) = int+infin

minusinfinF (119905) 119890minus119894120596119905119889119905

(10)

The frequency response vector X(120596) could be expressedas

X (120596) = (K + 119894120596C minus 1205962M)minus1 F (120596) (11)

The matrixH(120596) is usually called the frequency responsematrix

H (120596) = (K + 119894120596C minus 1205962M)minus1 (12)

and the frequency response function model is the ratioof outputinput spectra Then 120596 and Φ which are theeigenvalue and eigenvector matrices of the model could beobtained fromH(120596)22 Nonlinear Parameters Identification The nonlinearparameters were identified by a nonlinear static test Thistest measures the nonlinear displacements of the compositestructure x(119905) under different loads Fs(119905)

The equation of an n-degree-of-freedom under staticloads can be seen as in (7) shown above

120574 (1199021 1199022 119902119897) = ΦΤF119904 (119905) minus 1205962q (119905) (13)

120596 and Φ being the eigenvalue and eigenvector matrices ofthe dynamic system are obtained from linear parametersidentification Fs(119905) and x(119905) are obtained from the nonlinearstatic test and q(119905) is converted from

x (119905) = Φq (119905) (14)

Muravyov [29] and Kuether et al [30] wrote the dynamicequations in modal coordinates and determined nonlinearstiffness coefficients by combining the linear modes For thedynamic analysis the process of obtaining the nonlinearstiffness coefficients replaces the procedure of solving thechange ofmodal shapes and frequencies caused by deflection


Experimental modal analysis Experimental static analysis

Nonlinear dynamic model

Nonlinear dynamic equation

Linear part Nonlinear part

Linear parametersidentification

Nonlinear parametersidentification

q(t) + 2흃흎q(t) + 흎ퟐq(t) + 훾 (q1 q2 ql) = 횽TF(t)

q(t) + 2흃흎q(t) + 흎ퟐq(t) = 횽TFm(t) 훾 (q1 q2 ql) = 횽TFs(t) minus 흎ퟐq(t)

Figure 1 The flowchart of the numerical-experimental method

and the verification results in their works show that themethod has enough accuracy for the dynamic analysis

The nonlinear force vector may be expressed in thefollowing form

120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= ΦΤF119904 (119905) minus 1205962q (119905)

(15)

where 119886119903 and 119887119903 are the 119903th line identification parametersof 120574 and 119903 = 1 2 119897 119897 is the number of modes used inparameters identification

The nonlinear parameters 119886 and 119887 in (15) were identifiedby the nonlinear static test based on the nonlinear leastsquares method The nonlinear least squares method is oneof the most common methods used in practical data analysisand involves the fitting of a theoretical model to experimentaldata Frequently the model takes the form of a dependentvariable expressed as a function of several independentvariables Often themodel will contain one ormore estimatedparameters This estimation occurs on the basis of fitting themodel to observations using the least squares concept

There is a model of the following form

Q = f (119870 119871 120572 120573) (16)

whereQ is viewed as a function of two independent variables119870 and 119871 and 120572 and 120573 are two constants The objective isto estimate the values of the parameters 120572 and 120573 which weshall do by minimizing the sum of squared errors 119878 over theallowable values of 120572 and 120573

119878 = 119878 (120572 120573) = sum[119891 (119870 119871 120572 120573) minus 119876]2 (17)

1400 mm

7mm

400

mm

Junction to fuselageJunction to wheel

Figure 2 The composite landing gear

The first-order necessary condition for a minimum is

120597119878120597120572 = 0120597119878120597120573 = 0

(18)

Then the values of119870 and 119871 are solved outThe nonlinear dynamic model is built after the linear

parameters 120596 120585 andΦ and nonlinear parameters a and b areidentified

3 Identification of Composite Landing Gear

31 Composite Landing Gear The composite landing gear asshown in Figure 2 is considered as follows The length ofthe landing gear is 1400mm the height is 400mm and thethickness is 17mmThe top of the landing gear is fixed to thefuselage and the bottom is the junction to wheels

Three composite materials the carbon cloth the glasscloth and the glass fiber with polymeric matrix are used inthe manufacture of the landing gear as shown in Figure 3

32 Linear Parameters Identification of the Landing GearThe linear parameters of the composite landing gear wereidentified by experimental modal analysis In general thedatabase is obtained from direct measurements The fre-quency response functions are the measured output that willbe used to construct the experimental realizations of thelinear parameters identification

The landing gear was tested in fixed constraint at thejunction to the fuselage as shown in Figure 4


(5)(4)(3)(2)(1)

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

XC

YC

ZC

Figure 3 Composition of the landing gear

Figure 4 Setup of experimental modal analysis

Sensor

Figure 5 Locations of sensors

The sensors were located in the center of both sides of thelanding gear and separated by 20mm as shown in Figure 5

The FRFs (frequency response functions) were computedby a data acquisition and analysis system LMS SCADASIII and the experimental natural frequencies and modalshapes were obtained by modal analyses using LMS TestLabStructure analysis

The first six modal shapes are shown in Figure 6Table 1 gives the first twenty frequencies of each mode

33 Nonlinear Parameters Identification of the Landing GearThe nonlinear parameters were identified by nonlinear statictestThe test was performed in fixed constraint at the junctionto the fuselage and vertical downward loads were applied onthe junction to the wheel as shown in Figure 7

The displacement of the landing gear changed as theload changed The loads varied from 100N to 1000N with

Table 1 Frequency versus mode number

Mode FrequencyHz1 34092 78523 84004 137815 142066 214937 288728 356389 4334910 4356411 5906612 6587813 6665214 7692715 8243616 9616317 10687118 11652019 11729320 130338

Table 2 Results of the nonlinear static test

LoadN Displacementmm100 268200 536300 798400 1076500 1412600 1728700 1935800 2355900 25491000 2861

increment of 100N The displacements were measured byheight caliper and the displacements under different loadsare shown in Table 2

The nonlinear parameters of the composite landing gearwere identified by (15) with the nonlinear least squaresmethod based on the results of the nonlinear static test Fourmodes were used to identify the nonlinear parameters of thecomposite landing gear

120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= ΦΤF119904 (119905) minus 1205962q (119905)

(19)


Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

0505

05050 0

00minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050minus05

050minus05

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

Figure 6 Modal shapes of the composite landing gear

Load point

Fixed constraint

Figure 7 Setup of nonlinear static test

The displacements of the landing gear under differentloads were recalculated after the nonlinear parameters wereidentified The results of the identification were comparedwith the displacement-loading curve of the nonlinear statictest as shown in Figure 8

Figure 8 shows that the results of the identification arenot obviously different from the results of the nonlinear staticstructural test

100 200 300 400 500 600 700 800 900 10000Load (N)

0

5

10

15

20

25

30

35

Disp

lace

men

t (m

m)

IdentificationTest

Curve of test

Figure 8 Results comparison of identification and test

34 Tire Modeling The tire modeling of the land gear is clas-sical tire theory which is based on the work of Mitschke andWallentowitz [31] In their work a penetration formulation is


Tire

Ground

Flat

Vroll

Fnom

휃V

x

z

휏

Figure 9 Tire modeling

Table 3 Tire parameters

Parameter ValueRadius 01mWidth 0075mVertical stiffness 75000NmLateral stiffness 16655N∘

Mass 15 kg119868119909119909119868119910119910 345 times 10minus3 kgsdotm2119868119911119911 562 times 10minus3 kgsdotm2Damping 175N(ms)

used to calculate the normal force of the tire applied by theground as shown in Figure 9

The normal force applied to the tire is proportional to thepenetration depth and the penetration rate described as

119865nom = 119870nom120591 + 119862nom 120591 (20)

where 119870nom and 119862nom are the vertical stiffness and dampingof tire respectively and 120591 is the penetration depth

The lateral force of the tire is described as a function ofslip angle The slip angle 120579 is the included angle of the tirersquosvelocity and the tiresrsquo longitudinal axes as shown in Figure 9The lateral force is applied to account for the side loading onthe tire and can be written as

119865lat = 119862lat120579 (21)

where 119862lat is the lateral stiffness factor of the tireThe parameters of the tire used in our work were mea-

sured by experimental method and are shown in Table 3

4 Drop Impact Analysis of the CompositeLanding Gear

The landing gear is designed to absorb and dissipate thekinetic energy of the landing impact reducing the impactloads transmitted to the airframe Drop impact analysis isthe method most widely used to check performance of thelanding gear Therefore three drop impact simulations of thecomposite landing gear were carried out with the nonlineardynamicmodel identified in Section 3 To check the accuracyof the method three drop impact tests were also conducted

41 Method for Drop Simulations The drop impact simula-tions were carried out with the identified nonlinear dynamicmodel In the simulations the composite landing gear wasdropped froma certain height and thewhole process dynamicresponse of the landing gear was calculated

To solve the nonlinear dynamic model (5) was projectedin the modal space The modal space could reduce thedifferential equation to first order

The nonlinear part of (7) was written as

120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= 120594 (1199021 1199022 119902119897) q (119905)

(22)

Then the modal space of (5) was

z (119905) = Dz (119905) +D119909z (119905) + VF (119905) (23)

where

z (119905) = q (119905)q (119905)

V = 0ΦΤ

D = [ 0 119868minus1205962 minus2120585120596]

D119909 = [ 0 0minus120594 (1199021 1199022 119902119897) 0]

(24)

Cacciola and Muscolino put forward a new approach toconvert the modal space to iterative equation in discrete time[32] Divide the time axis in small intervals of equal lengthΔ119905 and let 1199050 1199051 119905119896 119905119896+1 be the division timeThen thenumerical solution of (23) can be written as follows

z119896+1 = Γ1 (Δ119905)Dx (z119896+1) z119896+1+ (Θ0 (Δ119905) + Γ0 (Δ119905)Dx (z119896)) z119896+ Γ0 (Δ119905)VF119896 + Γ1 (Δ119905)VF119896+1

(25)

where

z119896 = z (119905119896)F119896 = F (119905119896)

L (Δ119905) = [Θ0 (Δ119905) minus I]Dminus1Γ0 (Δ119905) = [Θ0 (Δ119905) minus 1Δ119905L (Δ119905)]Dminus1Γ1 (Δ119905) = [ 1Δ119905L (Δ119905) minus I]Dminus1

(26)


Θ0(119905) is the so-called transition or fundamental matrixwhich is given as

Θ0 (119905) = [minusg (119905)1205962 h (119905)

minush (119905)1205962 h (119905)] (27)

where g(119905) h(119905) g(119905) and h(119905) are diagonal matrices whose119895th elements are given respectively as

119892119895 (119905) = minus 11205962119895 119890minus120585119895120596119895119905 [cos (120596119863119895119905) + 120585119895120596119895120596119863119895 sin (120596119863119895119905)]

ℎ119895 (119905) = 119892119895 (119905) = 1120596119863119895 119890minus120585119895120596119895119905 sin (120596119863119895119905)

ℎ119895 (119905) = 119890minus120585119895120596119895119905 [cos (120596119863119895119905) minus 120585119895120596119895120596119863119895 sin (120596119863119895119905)] (28)

and

120596119863119895 = 120596119895radic1 minus 1205852119895 (29)

is the 119895th damped natural frequencyThen (25) can be written as follows

z119896+1 = (119868 minus Γ1 (Δ119905)Dx (z119896+1))minus1sdot (Θ0 (Δ119905) + Γ0 (Δ119905)Dx (z119896)) z119896+ (119868 minus Γ1 (Δ119905)Dx (z119896+1))minus1sdot (Γ0 (Δ119905)VF119896 + Γ1 (Δ119905)VF119896+1)

(30)

The solution process of (30) was nonlinear so an iterativemethod was needed The iterative method used here was theNewton Raphson method

In general we will be searching for one or more solutionsto the equation

F (x) = 0dArr

1198911 (1199091 1199092 119909119899) = 01198912 (1199091 1199092 119909119899) = 0

119891119899 (1199091 1199092 119909119899) = 0

(31)

Consider the Taylor-series expansion of the functionF(119909)about a value x = x119895

F (x) = F (x119895) + (120597F (x)120597x )x119895(x minus x119895)

+ 12 (1205972F (x)120597x2 )

x119895(x minus x119895)2 + 119874 (x3) x119895 = 0

(32)

Figure 10 System of drop tests

Using only the first two terms of the expansion a firstapproximation to the root of (31) can be obtained from

F (x) = F (x119895) + (120597F (x)120597x )x119895(x minus x119895) asymp 0 (33)

Such approximation is given by

x asymp x119895 minus (120597F (x)120597x )minus1x119895F (x119895) (34)

The Jacobian matrix of F(x) is J(x)

J (x) =[[[[[[[[[[[[

1205971198911120597119909112059711989111205971199092 sdot sdot sdot

12059711989111205971199091198991205971198912120597119909112059711989121205971199092 sdot sdot sdot

1205971198912120597119909119899 1205971198911198991205971199091

1205971198911198991205971199092 sdot sdot sdot120597119891119899120597119909119899

]]]]]]]]]]]]

(35)

Then the solution of (31) is

x = x minus 119869 (x)minus1 F (x) (36)

The convergence criterion of the iterative process wasbased on the minimum 2-norm In the solution process of(30) while

1003817100381710038171003817z119896+1 minus z11989610038171003817100381710038172 le CRIT (37)

the results are convergent

42 Setup of Drop Tests The drop tests were carried out withthe system shown in Figure 10 and the degrees of freedomexcept the vertical direction of the composite landing gearwere limited by the drop-test platform

The wheels were fixed to the bottom of the landing gearand the equivalent mass of the aircraft was fixed to thejunction to fuselage as shown in Figure 11


Figure 11 Setup of drop tests

Table 4 Parameters of the drop impact analysis

Number Equivalent masskg Heightm AOA∘

A 120 0136 0B 131 046 0C 136 046 12

In drop tests the composite landing gear was droppedfrom the same height as the simulations and impact loadsof the ground were applied to the wheels The whole processdynamic response was collected by sensors and the datacollection system was LMS SCADAS III

43 Drop Impact Analysis To check the accuracy of theidentified nonlinear dynamic model three groups of dropimpact analysis were carried out Every group contained adrop simulation and a drop test The drop simulations werebased on the identified nonlinear dynamic model and theexperimental method mentioned above was used in the droptests

The analysis parameters of drop impact analysis areshown in Table 4

Since the fuselage of the aircraft was replaced by theequivalent mass the dynamic responses of the equivalentmass were measured in the drop impact analysis In dropanalysis A the dynamic responses of the equivalentmass afterboth the drop simulation and drop test were compared andthe results are shown in Figure 12

Figure 12 shows that the result of the simulation is in goodagreement with the result of the test and the maximum errorof the simulation compared to the test is 433

The analysis parameters of the second drop impactanalysis are different from those of the first analysis Thedynamic responses of the equivalent mass after both dropsimulation and the drop test were compared the results areshown in Figure 13

Figure 13 shows that the result of the simulation isnot obviously different from the result of the test and the

Drop simulation A Drop test A

05 15 210minus03

minus025

minus02

minus015

minus01

minus005

0

Disp

lace

men

t (m

)

t (s)

Figure 12 Comparison of results for drop analysis A

maximum error of the simulation compared to the test is536

In the third drop impact analysis the dynamic responsesof the equivalent mass after both the drop simulation anddrop test were compared and the results are shown inFigure 14


5 Conclusion

A method is proposed in this paper to predict the dynamicperformance of composite landing gear with uncertaintiesusing experimental modal analysis data and nonlinear statictest data In the method the nonlinear dynamic model of thecomposite landing gear is divided into two parts the linearand the nonlinear parts Experimental modal analysis is


Drop simulation B Drop test B

05 1 15 20

t (s)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

Figure 13 Comparison of results for drop analysis B

Drop simulation CDrop test C

05 15 210minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

t (s)

Figure 14 Comparison of results for drop analysis C

employed to predict the linear parameters with a frequencyresponse function and the geometrically nonlinear parame-ters are identified by nonlinear static test with the nonlinearleast squares method

To check its accuracy and practicability the method isapplied to drop impact analysis of composite landing gearBoth simulations and tests are conducted for the drop impactanalysis and the errors of the analyses are extremely smallThemaximum error of the simulation compared to the test is433 in drop analysis A themaximum error of drop analysisB is 536 and the maximum error of drop analysis C is455 The results of the simulations are in good agreementwith the test results which shows that the proposed methodcan accurately model the dynamic performance of compositelanding gear and themethod is perfectly suitable for dynamicanalysis of composite landing gear

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The project was supported by the Knowledge InnovationProgram of the Chinese Academy of Sciences (Grant noYYYJ-1122) and the Innovation Program of UAV funded bythe Chang Guang Satellite Technology Co Ltd The authorsalso gratefully acknowledge the support from the ScienceFund for Young Scholars of the National Natural ScienceFoundation of China (Grant no 51305421)

References

[1] Z-P Xue M Li Y-H Li and H-G Jia ldquoA simplified flexiblemultibody dynamics for a main landing gear with flexible leafspringrdquo Shock and Vibration vol 2014 Article ID 595964 10pages 2014

[2] X M Dong and G W Xiong ldquoVibration attenuation of mag-netorheological landing gear system with human simulatedintelligent controlrdquoMathematical Problems in Engineering vol2013 Article ID 242476 13 pages 2013

[3] G Mikułowski and Ł Jankowski ldquoAdaptive landing gear opti-mum control strategy and potential for improvementrdquo Shockand Vibration vol 16 no 2 pp 175ndash194 2009

[4] F A C Viana V Steffen Jr M A X Zanini S A Magalhaesand L C S Goes ldquoIdentification of a non-linear landing gearmodel using nature-inspired optimizationrdquo Shock and Vibra-tion vol 15 no 3-4 pp 257ndash272 2008

[5] M Ansar W Xinwei and Z Chouwei ldquoModeling strategies of3D woven composites a reviewrdquo Composite Structures vol 93no 8 pp 1947ndash1963 2011

[6] G Z Dai H X Xiao W Wu et al ldquoPlasmon-driven reactioncontrolled by the number of graphene layers and localizedsurface plasmon distribution during optical excitationrdquo Light-Science amp Applications vol 4 2015

[7] P Lutsyk R Arif J Hruby et al ldquoA sensing mechanism for thedetection of carbon nanotubes using selective photolumines-cent probes based on ionic complexes with organic dyesrdquo Light-Science amp Applications vol 5 2016

[8] C X Ma Y Dai L Yu et al ldquoEnergy transfer in plasmonicphotocatalytic compositesrdquo Light-Science amp Applications vol 52016

[9] S Malobabic M Jupe and D Ristau ldquoSpatial separation effectsin a guiding procedure in a modified ion-beam-sputteringprocessrdquo Light-Science and Applications vol 5 article e160442016

[10] F J Rodrıguez-Fortuno and A V Zayats ldquoRepulsion of polar-ised particles from anisotropic materials with a near-zero per-mittivity componentrdquo Light-Science and Applications vol 5article e16022 2016

[11] Y X Zhang and C H Yang ldquoRecent developments in finiteelement analysis for laminated composite platesrdquo CompositeStructures vol 88 no 1 pp 147ndash157 2009

[12] W T Wang and T Y Kam ldquoMaterial characterization of lami-nated composite plates via static testingrdquo Composite Structuresvol 50 no 4 pp 347ndash352 2000


[13] L Mehrez D Moens and D Vandepitte ldquoStochastic identifi-cation of composite material properties from limited experi-mental databases part I experimental database constructionrdquoMechanical Systems and Signal Processing vol 27 no 1 pp 471ndash483 2012

[14] S Sriramula and M K Chryssanthopoulos ldquoQuantification ofuncertainty modelling in stochastic analysis of FRP compos-itesrdquoComposites Part A Applied Science andManufacturing vol40 no 11 pp 1673ndash1684 2009

[15] P K Rakesh V Sharma I Singh and D Kumar ldquoDelaminationin fiber reinforced plastics a finite element approachrdquo Engineer-ing vol 3 no 05 pp 549ndash554 2011

[16] C C Tsao and H Hocheng ldquoTaguchi analysis of delaminationassociated with various drill bits in drilling of composite mate-rialrdquo International Journal of Machine Tools and Manufacturevol 44 no 10 pp 1085ndash1090 2004

[17] A TMarques LMDurao AGMagalhaes J F Silva and JMR S Tavares ldquoDelamination analysis of carbon fibre reinforcedlaminates Evaluation of a special step drillrdquo Composites Scienceand Technology vol 69 no 14 pp 2376ndash2382 2009

[18] S Pantelakis and K I Tserpes ldquoAdhesive bonding of compositeaircraft structures Challenges and recent developmentsrdquo Sci-ence China Physics Mechanics and Astronomy vol 57 no 1 pp2ndash11 2014

[19] D Jiang Y Li Q Fei and SWu ldquoPrediction of uncertain elasticparameters of a braided compositerdquo Composite Structures vol126 pp 123ndash131 2015

[20] L Mehrez A Doostan D Moens and D Vandepitte ldquoStochas-tic identification of composite material properties from lim-ited experimental databases part ii uncertainty modellingrdquoMechanical Systems and Signal Processing vol 27 pp 484ndash4982012

[21] C M M Soares M M de Freitas A L Araujo and PPedersen ldquoIdentification of material properties of compositeplate specimensrdquoComposite Structures vol 25 no 1ndash4 pp 277ndash285 1993

[22] P S Frederiksen ldquoApplication of an improved model for theidentification of material parametersrdquo Mechanics of CompositeMaterials and Structures vol 4 no 4 pp 297ndash316 1997

[23] A L Araujo C M Mota Soares and M J Moreira De FreitasldquoCharacterization of material parameters of composite platespecimens using optimization and experimental vibrationaldatardquo Composites Part B Engineering vol 27 no 2 pp 185ndash1911996

[24] J Cunha S Cogan and C Berthod ldquoApplication of geneticalgorithms for the identification of elastic constants of com-posite materials from dynamic testsrdquo International Journal forNumerical Methods in Engineering vol 45 no 7 pp 891ndash9001999

[25] B Diveyev I Butiter andN Shcherbina ldquoIdentifying the elasticmoduli of composite plates by using high-order theoriesrdquoMech-anics of Composite Materials vol 44 no 1 pp 25ndash36 2008

[26] R Rikards A Chate andGGailis ldquoIdentification of elastic pro-perties of laminates based on experiment designrdquo InternationalJournal of Solids and Structures vol 38 no 30 pp 5097ndash51152001

[27] J P L No and G Kerschen ldquoNonlinear system identificationin structural dynamics 10 more years of progressrdquo MechanicalSystems amp Signal Processing vol 83 pp 2ndash35 2016

[28] G Kerschen K Worden A F Vakakis and J GolinvalldquoPast present and future of nonlinear system identification in

structural dynamicsrdquoMechanical Systems and Signal Processingvol 20 no 3 pp 505ndash592 2006

[29] A A Muravyov ldquoDetermination of nonlinear stiffness coeffi-cients for finite element models with application to the randomvibration problemrdquo MSC Worldwide Aerospace Conference pp1ndash14 1999

[30] R J Kuether B J Deaner J J Hollkamp and M S AllenldquoEvaluation of geometrically nonlinear reduced-order modelswith nonlinear normal modesrdquoAIAA Journal vol 52 no 11 pp3273ndash3285 2015

[31] MMitschke andHWallentowitzDynamik der KraftfahrzeugeSpringer Berlin Germany 1972

[32] P Cacciola and G Muscolino ldquoDynamic response of a rectan-gular beam with a known non-propagating crack of certain oruncertain depthrdquo Computers and Structures vol 80 no 27 pp2387ndash2396 2002

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



formula in each layer and the theory based on the 3Dcontinuum can predict a composite laminatersquos interlaminarstress [11]

The theories have made significant progress but are basedon accurate composite material parameters The materialproperties determined from standard specimens tested in thelaboratorymay deviate significantly from those of actual lam-inated composite componentsmanufactured in a factory [12]Composite structures exhibit variability and uncertainties intheir material properties with relation to their compositionsand manufacturing processes [13] Operating conditionsquality control procedures and environmental effects areoften difficult to control which affect the performance ofcomposite structures [14] Delamination is a major problemin composite manufacturing which brings uncertainties inthe material properties of composites [15] The nonhomoge-neous and anisotropic nature of composite materials maketheir machining behavior differ from metal machining [16]which can affect the mechanical properties of the compositestructures [17] Adhesive bonding is common practice in theinteraction of composite parts and its uncontrollable fea-ture of consumption makes the partsrsquo preformation discrete[18] To ensure high reliability of the structures the actualbehaviors of the composite parts in servicemust be accuratelypredicted and carefully monitoredTherefore it is essential tocapture and model the variability and uncertainties inherentin these material properties

The properties of composites can be obtained accu-rately via experimental tests [19] so the mixed numerical-experimental technique is applied to extract the physicalinformation of composites This usually involves minimizingan error function between the experimental and numericaloutputs [13] Constructing of predictive computational mod-els for analysis and design of many complex engineeringsystems requires not only a fine representation of the rel-evant physics and their interactions but also a quantitativeassessment of underlying uncertainties and their impacton design performance objectives [20] Hence a thoroughcharacterization of the mechanical properties of these struc-tures is needed to establish reliable designs Soares et alidentified elastic properties of laminated composites by usingexperimental eigenfrequenciesThe stiffness parameters wereidentified from the measured natural frequencies of thelaminated composites plate by direct minimization of theidentification function [21] A similar method was used byFrederiksen and Araujo Frederiksen improved the modelused for identification [22] Araujo used a mixed numerical-experimental technique to identify the damping propertiesof plyometric composites [23] Cunha et al identified severalproperties of a composite plate froma single test [24]Diveyevet al predicted elastic and damping properties of compositelaminated plates on the basis of static three-point bendingtests measured eigenfrequencies and refined calculationschemes [25] Rikards et al identified the elastic propertiesof cross-ply laminates from the measure eigenfrequencies ofcomposite plates [26]

The mixed numerical-experimental technique makes thefinite element method suitable for dynamic analysis of thelaminated composites However the tests for measuring

eigenfrequencies of composite plates are almost all linearas in the works mentioned above and do not consider thestructural nonlinearities of the laminated composites Thereare obvious geometrical nonlinearities in the work processof landing gears so the nonlinear system identification isneeded in the nonlinear structural models of landing gearsThe nonlinear system identification in structural dynamicshas been studied since the 1970s many excellent methodsare available for identification of nonlinear structuralmodelsTime-domain methods such as the restoring force surfacemethod the approach based on nonlinear autoregressivemoving average with exogenous input model and Hilberttransform-based data decomposition [27] have the advan-tage that the signals are directly provided by currentmeasure-ment devices less time and effort is spent on data acquisitionand processing [28] The frequency-domain methods con-sider the data from Fourier spectra frequency response andtransmissibility functions or power spectral densities andextend modal analysis to nonlinear structures [28] Time-frequency analysis offers useful insight into the dynamics ofnonlinear systems [27] Modal methods develop nonlinearsystem identification techniques based on nonlinear modes[28] Black-box modeling concentrates on function whichmaps the input to the output [28] Model updating methodsimprove the nonlinear dynamic modeling with test results[27]

Our study is aimed at expanding a numerical-experi-mental method based on linear and nonlinear tests for non-linear modeling and drop impact analysis of composite land-ing gear

2 Numerical-Experimental Method

The equation of motion of an 119899-degree-of-freedom nonlinearsystem can be written in the following form

Mx (119905) + Cx (119905) + (K + Ku) x (119905) = F (119905) (1)

where M and K are the real symmetric mass and stiffnessmatrices respectively C is the damping of the system x(119905)and F(119905) are the displacements and exciting load vectorsrespectively and Ku is the geometrically nonlinear stiffnesscomponent dependent on displacements For the relevantproblems the nonlinear stiffness force vector Kux(119905) repre-sents a deviation from the linear stiffness force vector Kx(119905)and is more than adequately represented by second- andthird-order terms in x(119905) When displacements are small thesecond- and third-order terms become negligible and thetotal stiffness-related force vector is reduced to the regularlinear term Kx(119905)

Any solution to (1) requires knowledge of the systemmatrices In the linear identification works mentioned aboveM K and C are generally available The geometricallynonlinear stiffness is related to Ku which is typically notavailable within a linear identification work Therefore amean of numerically evaluating M K C and Ku was devel-oped



x (119905) = Φq (119905) (2)




Mq (119905) + Cq (119905) + Κq (119905) + 120574 (1199021 1199022 119902119897) = F (119905) (3)

where


F (119905) = Φ119879F (119905)

(4)


q (119905) + 2120585120596q (119905) + 1205962q (119905) + 120574 (1199021 1199022 119902119897)= ΦΤF (119905) (5)


q (119905) + 2120585120596q (119905) + 1205962q (119905) = ΦΤF119898 (119905) (6)


120574 (1199021 1199022 119902119897) = ΦΤF119904 (119905) minus 1205962q (119905) (7)







Mx (119905) + Cx (119905) + Kx (119905) = F (119905) (8)


(K + 119894120596C minus 1205962M)X (120596) = F (120596) (9)





(10)




H (120596) = (K + 119894120596C minus 1205962M)minus1 (12)



120574 (1199021 1199022 119902119897) = ΦΤF119904 (119905) minus 1205962q (119905) (13)


x (119905) = Φq (119905) (14)














120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= ΦΤF119904 (119905) minus 1205962q (119905)

(15)




Q = f (119870 119871 120572 120573) (16)


119878 = 119878 (120572 120573) = sum[119891 (119870 119871 120572 120573) minus 119876]2 (17)

1400 mm

7mm

400

mm




120597119878120597120572 = 0120597119878120597120573 = 0

(18)









(5)(4)(3)(2)(1)


XC

YC

ZC



Sensor













120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= ΦΤF119904 (119905) minus 1205962q (119905)

(19)


Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

0505

05050 0

00minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050minus05

050minus05

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0


Load point

Fixed constraint




100 200 300 400 500 600 700 800 900 10000Load (N)

0

5

10

15

20

25

30

35

Disp

lace

men

t (m

m)

IdentificationTest

Curve of test




Tire

Ground

Flat

Vroll

Fnom

휃V

x

z

휏







119865nom = 119870nom120591 + 119862nom 120591 (20)



119865lat = 119862lat120579 (21)








120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= 120594 (1199021 1199022 119902119897) q (119905)

(22)


z (119905) = Dz (119905) +D119909z (119905) + VF (119905) (23)

where

z (119905) = q (119905)q (119905)

V = 0ΦΤ

D = [ 0 119868minus1205962 minus2120585120596]

D119909 = [ 0 0minus120594 (1199021 1199022 119902119897) 0]

(24)



(25)

where

z119896 = z (119905119896)F119896 = F (119905119896)


(26)



Θ0 (119905) = [minusg (119905)1205962 h (119905)

minush (119905)1205962 h (119905)] (27)



ℎ119895 (119905) = 119892119895 (119905) = 1120596119863119895 119890minus120585119895120596119895119905 sin (120596119863119895119905)


and

120596119863119895 = 120596119895radic1 minus 1205852119895 (29)



(30)



F (x) = 0dArr

1198911 (1199091 1199092 119909119899) = 01198912 (1199091 1199092 119909119899) = 0

119891119899 (1199091 1199092 119909119899) = 0

(31)



+ 12 (1205972F (x)120597x2 )

x119895(x minus x119895)2 + 119874 (x3) x119895 = 0

(32)







J (x) =[[[[[[[[[[[[

1205971198911120597119909112059711989111205971199092 sdot sdot sdot

12059711989111205971199091198991205971198912120597119909112059711989121205971199092 sdot sdot sdot

1205971198912120597119909119899 1205971198911198991205971199091

1205971198911198991205971199092 sdot sdot sdot120597119891119899120597119909119899

]]]]]]]]]]]]

(35)




1003817100381710038171003817z119896+1 minus z11989610038171003817100381710038172 le CRIT (37)








A 120 0136 0B 131 046 0C 136 046 12









05 15 210minus03

minus025

minus02

minus015

minus01

minus005

0

Disp

lace

men

t (m

)

t (s)





5 Conclusion




05 1 15 20

t (s)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)



05 15 210minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

t (s)






Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











x (119905) = Φq (119905) (2)




Mq (119905) + Cq (119905) + Κq (119905) + 120574 (1199021 1199022 119902119897) = F (119905) (3)

where


F (119905) = Φ119879F (119905)

(4)


q (119905) + 2120585120596q (119905) + 1205962q (119905) + 120574 (1199021 1199022 119902119897)= ΦΤF (119905) (5)


q (119905) + 2120585120596q (119905) + 1205962q (119905) = ΦΤF119898 (119905) (6)


120574 (1199021 1199022 119902119897) = ΦΤF119904 (119905) minus 1205962q (119905) (7)







Mx (119905) + Cx (119905) + Kx (119905) = F (119905) (8)


(K + 119894120596C minus 1205962M)X (120596) = F (120596) (9)





(10)




H (120596) = (K + 119894120596C minus 1205962M)minus1 (12)



120574 (1199021 1199022 119902119897) = ΦΤF119904 (119905) minus 1205962q (119905) (13)


x (119905) = Φq (119905) (14)














120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= ΦΤF119904 (119905) minus 1205962q (119905)

(15)




Q = f (119870 119871 120572 120573) (16)


119878 = 119878 (120572 120573) = sum[119891 (119870 119871 120572 120573) minus 119876]2 (17)

1400 mm

7mm

400

mm




120597119878120597120572 = 0120597119878120597120573 = 0

(18)









(5)(4)(3)(2)(1)


XC

YC

ZC



Sensor













120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= ΦΤF119904 (119905) minus 1205962q (119905)

(19)


Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

0505

05050 0

00minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050minus05

050minus05

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0


Load point

Fixed constraint




100 200 300 400 500 600 700 800 900 10000Load (N)

0

5

10

15

20

25

30

35

Disp

lace

men

t (m

m)

IdentificationTest

Curve of test




Tire

Ground

Flat

Vroll

Fnom

휃V

x

z

휏







119865nom = 119870nom120591 + 119862nom 120591 (20)



119865lat = 119862lat120579 (21)








120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= 120594 (1199021 1199022 119902119897) q (119905)

(22)


z (119905) = Dz (119905) +D119909z (119905) + VF (119905) (23)

where

z (119905) = q (119905)q (119905)

V = 0ΦΤ

D = [ 0 119868minus1205962 minus2120585120596]

D119909 = [ 0 0minus120594 (1199021 1199022 119902119897) 0]

(24)



(25)

where

z119896 = z (119905119896)F119896 = F (119905119896)


(26)



Θ0 (119905) = [minusg (119905)1205962 h (119905)

minush (119905)1205962 h (119905)] (27)



ℎ119895 (119905) = 119892119895 (119905) = 1120596119863119895 119890minus120585119895120596119895119905 sin (120596119863119895119905)


and

120596119863119895 = 120596119895radic1 minus 1205852119895 (29)



(30)



F (x) = 0dArr

1198911 (1199091 1199092 119909119899) = 01198912 (1199091 1199092 119909119899) = 0

119891119899 (1199091 1199092 119909119899) = 0

(31)



+ 12 (1205972F (x)120597x2 )

x119895(x minus x119895)2 + 119874 (x3) x119895 = 0

(32)







J (x) =[[[[[[[[[[[[

1205971198911120597119909112059711989111205971199092 sdot sdot sdot

12059711989111205971199091198991205971198912120597119909112059711989121205971199092 sdot sdot sdot

1205971198912120597119909119899 1205971198911198991205971199091

1205971198911198991205971199092 sdot sdot sdot120597119891119899120597119909119899

]]]]]]]]]]]]

(35)




1003817100381710038171003817z119896+1 minus z11989610038171003817100381710038172 le CRIT (37)








A 120 0136 0B 131 046 0C 136 046 12









05 15 210minus03

minus025

minus02

minus015

minus01

minus005

0

Disp

lace

men

t (m

)

t (s)





5 Conclusion




05 1 15 20

t (s)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)



05 15 210minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

t (s)






Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation





















120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= ΦΤF119904 (119905) minus 1205962q (119905)

(15)




Q = f (119870 119871 120572 120573) (16)


119878 = 119878 (120572 120573) = sum[119891 (119870 119871 120572 120573) minus 119876]2 (17)

1400 mm

7mm

400

mm




120597119878120597120572 = 0120597119878120597120573 = 0

(18)









(5)(4)(3)(2)(1)


XC

YC

ZC



Sensor













120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= ΦΤF119904 (119905) minus 1205962q (119905)

(19)


Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

0505

05050 0

00minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050minus05

050minus05

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0


Load point

Fixed constraint




100 200 300 400 500 600 700 800 900 10000Load (N)

0

5

10

15

20

25

30

35

Disp

lace

men

t (m

m)

IdentificationTest

Curve of test




Tire

Ground

Flat

Vroll

Fnom

휃V

x

z

휏







119865nom = 119870nom120591 + 119862nom 120591 (20)



119865lat = 119862lat120579 (21)








120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= 120594 (1199021 1199022 119902119897) q (119905)

(22)


z (119905) = Dz (119905) +D119909z (119905) + VF (119905) (23)

where

z (119905) = q (119905)q (119905)

V = 0ΦΤ

D = [ 0 119868minus1205962 minus2120585120596]

D119909 = [ 0 0minus120594 (1199021 1199022 119902119897) 0]

(24)



(25)

where

z119896 = z (119905119896)F119896 = F (119905119896)


(26)



Θ0 (119905) = [minusg (119905)1205962 h (119905)

minush (119905)1205962 h (119905)] (27)



ℎ119895 (119905) = 119892119895 (119905) = 1120596119863119895 119890minus120585119895120596119895119905 sin (120596119863119895119905)


and

120596119863119895 = 120596119895radic1 minus 1205852119895 (29)



(30)



F (x) = 0dArr

1198911 (1199091 1199092 119909119899) = 01198912 (1199091 1199092 119909119899) = 0

119891119899 (1199091 1199092 119909119899) = 0

(31)



+ 12 (1205972F (x)120597x2 )

x119895(x minus x119895)2 + 119874 (x3) x119895 = 0

(32)







J (x) =[[[[[[[[[[[[

1205971198911120597119909112059711989111205971199092 sdot sdot sdot

12059711989111205971199091198991205971198912120597119909112059711989121205971199092 sdot sdot sdot

1205971198912120597119909119899 1205971198911198991205971199091

1205971198911198991205971199092 sdot sdot sdot120597119891119899120597119909119899

]]]]]]]]]]]]

(35)




1003817100381710038171003817z119896+1 minus z11989610038171003817100381710038172 le CRIT (37)








A 120 0136 0B 131 046 0C 136 046 12









05 15 210minus03

minus025

minus02

minus015

minus01

minus005

0

Disp

lace

men

t (m

)

t (s)





5 Conclusion




05 1 15 20

t (s)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)



05 15 210minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

t (s)






Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










(5)(4)(3)(2)(1)


XC

YC

ZC



Sensor













120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= ΦΤF119904 (119905) minus 1205962q (119905)

(19)


Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

0505

05050 0

00minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050minus05

050minus05

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0


Load point

Fixed constraint




100 200 300 400 500 600 700 800 900 10000Load (N)

0

5

10

15

20

25

30

35

Disp

lace

men

t (m

m)

IdentificationTest

Curve of test




Tire

Ground

Flat

Vroll

Fnom

휃V

x

z

휏







119865nom = 119870nom120591 + 119862nom 120591 (20)



119865lat = 119862lat120579 (21)








120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= 120594 (1199021 1199022 119902119897) q (119905)

(22)


z (119905) = Dz (119905) +D119909z (119905) + VF (119905) (23)

where

z (119905) = q (119905)q (119905)

V = 0ΦΤ

D = [ 0 119868minus1205962 minus2120585120596]

D119909 = [ 0 0minus120594 (1199021 1199022 119902119897) 0]

(24)



(25)

where

z119896 = z (119905119896)F119896 = F (119905119896)


(26)



Θ0 (119905) = [minusg (119905)1205962 h (119905)

minush (119905)1205962 h (119905)] (27)



ℎ119895 (119905) = 119892119895 (119905) = 1120596119863119895 119890minus120585119895120596119895119905 sin (120596119863119895119905)


and

120596119863119895 = 120596119895radic1 minus 1205852119895 (29)



(30)



F (x) = 0dArr

1198911 (1199091 1199092 119909119899) = 01198912 (1199091 1199092 119909119899) = 0

119891119899 (1199091 1199092 119909119899) = 0

(31)



+ 12 (1205972F (x)120597x2 )

x119895(x minus x119895)2 + 119874 (x3) x119895 = 0

(32)







J (x) =[[[[[[[[[[[[

1205971198911120597119909112059711989111205971199092 sdot sdot sdot

12059711989111205971199091198991205971198912120597119909112059711989121205971199092 sdot sdot sdot

1205971198912120597119909119899 1205971198911198991205971199091

1205971198911198991205971199092 sdot sdot sdot120597119891119899120597119909119899

]]]]]]]]]]]]

(35)




1003817100381710038171003817z119896+1 minus z11989610038171003817100381710038172 le CRIT (37)








A 120 0136 0B 131 046 0C 136 046 12









05 15 210minus03

minus025

minus02

minus015

minus01

minus005

0

Disp

lace

men

t (m

)

t (s)





5 Conclusion




05 1 15 20

t (s)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)



05 15 210minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

t (s)






Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Mode 1 Mode 2

Mode 3 Mode 4

Mode 5 Mode 6

0505

05050 0

00minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050

minus05

050

minus05

minus05

050minus05

050minus05

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0

minus06

minus04

minus02

0


Load point

Fixed constraint




100 200 300 400 500 600 700 800 900 10000Load (N)

0

5

10

15

20

25

30

35

Disp

lace

men

t (m

m)

IdentificationTest

Curve of test




Tire

Ground

Flat

Vroll

Fnom

휃V

x

z

휏







119865nom = 119870nom120591 + 119862nom 120591 (20)



119865lat = 119862lat120579 (21)








120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= 120594 (1199021 1199022 119902119897) q (119905)

(22)


z (119905) = Dz (119905) +D119909z (119905) + VF (119905) (23)

where

z (119905) = q (119905)q (119905)

V = 0ΦΤ

D = [ 0 119868minus1205962 minus2120585120596]

D119909 = [ 0 0minus120594 (1199021 1199022 119902119897) 0]

(24)



(25)

where

z119896 = z (119905119896)F119896 = F (119905119896)


(26)



Θ0 (119905) = [minusg (119905)1205962 h (119905)

minush (119905)1205962 h (119905)] (27)



ℎ119895 (119905) = 119892119895 (119905) = 1120596119863119895 119890minus120585119895120596119895119905 sin (120596119863119895119905)


and

120596119863119895 = 120596119895radic1 minus 1205852119895 (29)



(30)



F (x) = 0dArr

1198911 (1199091 1199092 119909119899) = 01198912 (1199091 1199092 119909119899) = 0

119891119899 (1199091 1199092 119909119899) = 0

(31)



+ 12 (1205972F (x)120597x2 )

x119895(x minus x119895)2 + 119874 (x3) x119895 = 0

(32)







J (x) =[[[[[[[[[[[[

1205971198911120597119909112059711989111205971199092 sdot sdot sdot

12059711989111205971199091198991205971198912120597119909112059711989121205971199092 sdot sdot sdot

1205971198912120597119909119899 1205971198911198991205971199091

1205971198911198991205971199092 sdot sdot sdot120597119891119899120597119909119899

]]]]]]]]]]]]

(35)




1003817100381710038171003817z119896+1 minus z11989610038171003817100381710038172 le CRIT (37)








A 120 0136 0B 131 046 0C 136 046 12









05 15 210minus03

minus025

minus02

minus015

minus01

minus005

0

Disp

lace

men

t (m

)

t (s)





5 Conclusion




05 1 15 20

t (s)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)



05 15 210minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

t (s)






Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Tire

Ground

Flat

Vroll

Fnom

휃V

x

z

휏







119865nom = 119870nom120591 + 119862nom 120591 (20)



119865lat = 119862lat120579 (21)








120574 (1199021 1199022 119902119897) = 119897sum119895=1

119897sum119896=119895

119886119903119895119896119902119895119902119896

+ 119897sum119895=1

119897sum119896=119895

119897sum119898=119896

119887119903119895119896119898119902119895119902119896119902119898= 120594 (1199021 1199022 119902119897) q (119905)

(22)


z (119905) = Dz (119905) +D119909z (119905) + VF (119905) (23)

where

z (119905) = q (119905)q (119905)

V = 0ΦΤ

D = [ 0 119868minus1205962 minus2120585120596]

D119909 = [ 0 0minus120594 (1199021 1199022 119902119897) 0]

(24)



(25)

where

z119896 = z (119905119896)F119896 = F (119905119896)


(26)



Θ0 (119905) = [minusg (119905)1205962 h (119905)

minush (119905)1205962 h (119905)] (27)



ℎ119895 (119905) = 119892119895 (119905) = 1120596119863119895 119890minus120585119895120596119895119905 sin (120596119863119895119905)


and

120596119863119895 = 120596119895radic1 minus 1205852119895 (29)



(30)



F (x) = 0dArr

1198911 (1199091 1199092 119909119899) = 01198912 (1199091 1199092 119909119899) = 0

119891119899 (1199091 1199092 119909119899) = 0

(31)



+ 12 (1205972F (x)120597x2 )

x119895(x minus x119895)2 + 119874 (x3) x119895 = 0

(32)







J (x) =[[[[[[[[[[[[

1205971198911120597119909112059711989111205971199092 sdot sdot sdot

12059711989111205971199091198991205971198912120597119909112059711989121205971199092 sdot sdot sdot

1205971198912120597119909119899 1205971198911198991205971199091

1205971198911198991205971199092 sdot sdot sdot120597119891119899120597119909119899

]]]]]]]]]]]]

(35)




1003817100381710038171003817z119896+1 minus z11989610038171003817100381710038172 le CRIT (37)








A 120 0136 0B 131 046 0C 136 046 12









05 15 210minus03

minus025

minus02

minus015

minus01

minus005

0

Disp

lace

men

t (m

)

t (s)





5 Conclusion




05 1 15 20

t (s)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)



05 15 210minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

t (s)






Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Θ0 (119905) = [minusg (119905)1205962 h (119905)

minush (119905)1205962 h (119905)] (27)



ℎ119895 (119905) = 119892119895 (119905) = 1120596119863119895 119890minus120585119895120596119895119905 sin (120596119863119895119905)


and

120596119863119895 = 120596119895radic1 minus 1205852119895 (29)



(30)



F (x) = 0dArr

1198911 (1199091 1199092 119909119899) = 01198912 (1199091 1199092 119909119899) = 0

119891119899 (1199091 1199092 119909119899) = 0

(31)



+ 12 (1205972F (x)120597x2 )

x119895(x minus x119895)2 + 119874 (x3) x119895 = 0

(32)







J (x) =[[[[[[[[[[[[

1205971198911120597119909112059711989111205971199092 sdot sdot sdot

12059711989111205971199091198991205971198912120597119909112059711989121205971199092 sdot sdot sdot

1205971198912120597119909119899 1205971198911198991205971199091

1205971198911198991205971199092 sdot sdot sdot120597119891119899120597119909119899

]]]]]]]]]]]]

(35)




1003817100381710038171003817z119896+1 minus z11989610038171003817100381710038172 le CRIT (37)








A 120 0136 0B 131 046 0C 136 046 12









05 15 210minus03

minus025

minus02

minus015

minus01

minus005

0

Disp

lace

men

t (m

)

t (s)





5 Conclusion




05 1 15 20

t (s)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)



05 15 210minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

t (s)






Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













A 120 0136 0B 131 046 0C 136 046 12









05 15 210minus03

minus025

minus02

minus015

minus01

minus005

0

Disp

lace

men

t (m

)

t (s)





5 Conclusion




05 1 15 20

t (s)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)



05 15 210minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

t (s)






Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











05 1 15 20

t (s)

minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)



05 15 210minus07

minus06

minus05

minus04

minus03

minus02

minus01

Disp

lace

men

t (m

)

t (s)






Acknowledgments


References



































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









a numerical-experimental method for drop impact analysis ... · identification of composite landing...

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