Variation modeling of aeronautical thin-walled structures with multi-state riveting

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Journal of Manufacturing Systems 30 (2011) 101– 115

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Journal of Manufacturing Systems

jo u r n al hom epa ge: www.elsev ier .com/ locate / jmansys

echnical paper

ariation modeling of aeronautical thin-walled structures with multi-stateiveting

ui Chenga,∗, Yuan Lia, Kai-fu Zhanga, Wei-qiang Mua, Bo-feng Liub

The Ministry of Education Key Lab of Contemporary Design and Integrated Manufacturing Technology, Northwestern Polytechnical University, Xi’an, ChinaXi’an Aircraft Industrial (Group) Co. Ltd., Xi’an, China

r t i c l e i n f o

rticle history:eceived 19 November 2009eceived in revised form 23 May 2011ccepted 23 May 2011vailable online 22 June 2011

eywords:erospace industryhin-walled structuresivetingariation

a b s t r a c t

Assembly variation is inevitable in aeronautical thin-walled structure (ATWS) assembly, especially instructures joined with “C-type” automated riveting system (CARS). The variation propagates along theassembly process flow and will influence final product performance such as dimensional quality andfatigue durability. This paper represents a new variation model of ATWS with multi-state riveting. Firstly,a novel multi-state process of ATWS riveting with CARS called PDJR to PDRR (P to P) is developed and itcontains two stages and eight states. Secondly, based on the P to P process, the variation model is dividedinto three sub-models (feature, displacement and propagation) to represent the assembly variation ofATWS multi-state riveting. For the feature sub-model, three important features are discussed, and thegeometric and topological information is represented by a hierarchical method. For the displacementsub-model, the variation of eight-state in P to P is divided into four types, and the displacement of eachtype is analyzed separately according to the coordinate transformation and finite element method (FEM).
For the propagation sub-model, a translation matrix considering the disturbance factor of every state isdeveloped to obtain the final variation. Lastly, a multi-state riveting process of a wing panel which is madeup of a skin and four stringers is modeled as a case study. The FEM is integrated into the Monte Carlosimulation (MCS) to analyze the variation, and the result proves that the proposed variation modelingof ATWS multi-state riveting can solve the problem of variation analysis in ATWS multi-state rivetingefficiently.
iety o
© 2011 The Soc
. Introduction

Thin-walled structures are widely used in the aircraft indus-ry in assembling important subsystems such as fuselage, wings,orizontal stabilizer, and vertical stabilizer [1]. We call such struc-ures aeronautical thin-walled structures (ATWSs). Riveting is onef the most popular joining methods for ATWS by virtue of itsow production cost, high strength and easy inspection. Under thenteraction of positioning errors, drilling force, clamping force, fric-ion force, as well as the force supplied by the plastic deformationf the rivets, ATWS will deform during the assembly process andemain deformed after assembly. These deformations cause vari-tions, i.e., dimensional error [2,3]. The variation may adverselyffect the fatigue durability and damage tolerance characteristicsf the aircraft. An effective variation model and intelligent usage of
nowledge attained from it will increase the quality of the prod-cts. Therefore it is important to model and analyze the variation
n the riveting process of ATWS.

∗ Corresponding author. Tel.: +86 029 88460580; fax: +86 029 88460580.E-mail address: [email protected] (H. Cheng).

278-6125/$ – see front matter © 2011 The Society of Manufacturing Engineers. Publisheoi:10.1016/j.jmsy.2011.05.004

f Manufacturing Engineers. Published by Elsevier Ltd. All rights reserved.

The requirements for assembly efficiency and accuracy madeautomated solutions method more and more popular in modernaerospace industry. The current automation solutions for the riv-eting of ATWS are based on large dedicated machines, such asautomatic riveting system (ARS). A typical “C-type” automated riv-eting system (CARS) which is often used in wing panel riveting isshown in Fig. 1(A) [4]. As one of the most commonly used ATWS, thepanel is always made up of skin and stringers. As is shown in Fig. 1,splints are used to support and clamp the panel. Normally, once thesplints are fixed on the frame, the positions of stringers are deter-mined, so is the position of the panel. The blocks on a splint, whichare shown in Fig. 1(B), are used to limit the position of stringers inriveting.

According to Fig. 1, the skin and stringers cannot be loaded ontothe splints separately, as the splints can only locate the position ofthe stringers. Hence the panel should be pre-joined by some screwson a special fixture (panel assembly fixture) before it is loaded onthe CARS. The process is shown in Fig. 2.

The riveting process of ATWS with CARS usually contains twoassembly stations with each assembly station needing more thanone fixture. Different fixtures have different assembly datum,which in turn cause riveting variation. The interactions of drilling

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102 H. Cheng et al. / Journal of Manufacturing Systems 30 (2011) 101– 115

” auto

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Fig. 1. A typical “C-type

orce, clamping force, friction and the forces supplied by rivetsnto account make variation modeling in ATWS riveting very com-licated. In order to ensure the assembly accuracy of the ATWS,iveting variation must be analyzed and controlled well. Rivetingariation is inevitable, but using a proper variation analysis methodan help understand the root causes of variation and reduce it.

Various methods for variation analysis have been proposed, butvery variation analysis method is based on an effective variationodel, hence the demand for variation modeling of ATWS rivetingith CARS is extremely urgent. In order to ensure the high accu-

acy of aircraft, a lot of efforts have done into the variation analysisf ATWS riveting. Researchers in University of Birmingham usednite element method (FEM) and experiment to predict the varia-ion in wing box assembly [5] and the geometric deformation on theext joining location with respect to the present joining location2]. They also developed a prediction method with FEM capabili-ies to estimate the positional error encountered in the part-to-partssembly of the rib components within a large Airbus wing struc-ure [6]. Blanchot and Daidie [7] represented the simulation ofiveting process and its influence on the riveted link behavior. Theyroposed that riveting affects the behavior of the overall assemblyhen subjected to an exterior load since the rivet and the plates are

n state of pretension subsequent to the riveting operation whichill affect the variation. By considering the effect of variation inrilled holes size, diameter of rivet, length and squeeze force on theiveting process, Cheraghi et al. developed a finite element simu-ation method to study the effect of the riveting parameters on theariation of riveting [3]. Fernlund et al. developed a methodologyor modeling a large complex part such as the aft strut fairing, they

easured deformations for a number of parts and compared toredict deformations based on models developed using a FE basedomposites processing software [8]. Zhao et al. developed a method
o move assembly variation analysis into the early stages of aircraftevelopment where critical partitioning, sourcing, and productionecisions are often made for component parts that have not yeteen designed, they used regression analysis and artificial neural
Fig. 2. A typical 3D model of

mated riveting system.

networks to build variation transfer functions between design andmanufacturing [9]. Most of these methods and models are aimed atparts or products themselves and solved their respective questionquiet well. At the same time, the effects of assembly process (espe-cially the riveting process) and fixtures have not been considered,which makes the problem of predicting and controlling variationsin ATWS riveting with CARS an important one.

Besides aircraft, the thin-walled structures are widely used inautomotive and other mechanical industries. The variation modelsof thin-wall structure assembly in these industries are valuable formodeling ATWS riveting with CARS. Liu and Hu [10] presented anoffset beam element model for predicting the assembly variationof deformable thin-walled sheet metal parts joined by resistancespot welding. On the basis of the offset beam element model, they[11] developed a model based on the method of influence coef-ficients (MIC) to analyze the effect of component deviations andassembly springback on spot welding variation by applying linearmechanics and statistics. Using FEM, they constructed a sensitivitymatrix for a thin-walled structure of complex shapes. The sensitiv-ity matrix established a linear relationship between the incomingpart deviation and the output assembly deviation. Chang and Gos-sard [12] summarized the welding process of automotive bodiesinto a PCFR/PCMR (place clamp fasten release/place clamp mea-sure release) cycles by contact chains. The geometric compatibility,force continuity, and constitutive relations at the nodes of con-tact chains are represented by vector equations. They also used themodel to simulate the propagation of variations during assemblyprocesses and predict variations in the resulting assembly. Hu et al.[13] developed a numerical simulation method for the assemblyprocess incorporating compliant non-ideal parts. They consideredthe interaction and interference between compliant parts due topart variation, assembly tooling variation, welding distortion, and
springback effects. By analyzing the interaction between assemblyvariations and part geometric variations, as well as fixture vari-ations and welding gun variations influence, Camelio et al. [14]developed a variation propagation model for multi-station assem-
plane assembly fixture.

facturing Systems 30 (2011) 101– 115 103

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H. Cheng et al. / Journal of Manu

ly systems with thin-walled structure parts in automotive body.ccording to the propagation model, a new method for varia-

ion propagation analysis in compliant assembly was developedy using the covariance matrix of the components [15,16]. Theethod replaces the MIC by using variation vectors defined for

ach deformation pattern identified from the covariance of theomponents and can be used to determine the part-to-part andooling interaction in the assembly system. At the same time, inrder to determine the optimal fixture position so that assem-ly variation is minimized, they also analyzed the effect of partariation, tooling variation and assembly springback [17]. Takinghe advantage of MIC and contact modeling, Dahlström and Lind-vist [18] presented a simple method for contact modeling, whichonsists of algorithms for contact detection and contact equilib-ium search. Such algorithms can be implemented in MIC to avoidenetrations. In order to provide an interface between rigid andompliant assembly models, Huang and Kong [19] simulated parteometric errors (PGE) and rigid body kinematics stackup error (RE)n rigid assembly processes. A covariance matrix of PGE and RE

as constructed providing input to subsequent compliant assem-ly models. Yu et al. [20] developed a new variation model ofhin-walled structure assembly with sources of material variations,art geometric variations and fixture variations on the base ofIC. Cai et al. [21] presented a hybrid method for incorporating

iveting deformation in assembly simulation using experimentalata and finite element analysis to prediction assembly dimen-ions of self-piercing riveted aluminum panels. According to theool variation, Cai [22] formulated a fixture optimization modelo minimize the assembly dimensional variations under weld-ng gun variations. He developed linear and non-linear assemblyimulation models and analysis methods through a two-step lin-arization with applications in automotive body assembly [23].iao and Wang [24] studied the influence of the surface micro-eometry of assembly components on the final assembly variation,mployed fractals and FEM for detailed variation analysis of thin-all structure. They also take the part variation as a signal and

pplies wavelets transform to decompose it into components atifferent scales. The deformation of thin-wall structure assemblieshat corresponds to these components at different scales is cal-ulated using FEM [25]. Furthermore, they studied of the contactroblem of the thin-walled metal structure assemblies, developed aystematic procedure of the non-linear dimensional variation anal-sis for the thin-walled metal structure assemblies by using theontact finite element method [26]. Wang and Ceglarek [27] rep-esented a beam-based model based on the assumption that onlyelected critical points/features in the assembly are important toariation. They also deduced a state-of-the-art variation propaga-ion model by integrating the influence of join error on variationropagation and incorporating them into a vector-based model.

As described above, most of the assembly variation analy-is methods are aimed at the assembly variation of thin-walltructures joined by spot welding or self-piercing riveting in auto-obile industry. The thin-wall structures in automotive and otherechanical industry are not as complex as ATWS (most of panels

re double-curvature surface). In addition, the process of auto-ated riveting is quite different from the process of spot welding

r self-piercing riveting. Since different products have differentttributes, the existing methods and experiences in automobile orther mechanical products are not very suitable for riveting varia-ion analysis of ATWS with CARS. As CARS are used more and moreidely in the modern aircraft industry, new variation models ofTWS riveting with CARS are essential.

This paper develops a novel multi-state process to form a gen-ral process of ATWS with CARS by taking the riveting process ofkin and stringers in ATWS into account. Taking the positioningrror and deformation under assembly force into account, the vari-

Fig. 3. A wing panel made up of skin and stringers.

ation modeling of ATWS multi-state riveting is built on the baseof the PDJR to PDRR (P to P) process. The model is consisted of afeature sub-model, a displacement sub-model and a variation prop-agation sub-model. The feature sub-model represents the geometryand topology information of parts which take part in the riveting.The displacement sub-model contains the method to calculate thevariation in different state of P to P. The propagation sub-modelpresents the relationship among the variation in different state ofP to P, and integrates the FEM and Monte Carlo simulation (MCS)to get the final variation. The model developed in this paper canbe used to predict variations of the ATWS riveting assembly withCARS, anticipate potential assembly problems before expensivethin-walled parts and fixtures are made, identify the dimensions inparts, fixture, CARS, and processes which are critical for the qual-ity of the final aeronautical thin-walled structures, and improvethe design of the aeronautical thin-walled structures, fixtures andCARS so that the final product is robust to variation.

2. Process analysis for ATWS with multi-state riveting

Panel assemblies are the most common components in ATWSsince wings and fuselages are all made up of skins and stringers. Atypical wing assembly made up of a skin and four stringers is shownas Fig. 3.

In order to ensure the aerodynamic shape of the aircraft, theskin and stringers are pre-joined by some screw bolts before rivet-ing. The pre-joining process always takes place on a special fixturewhich is shown in Fig. 2. The whole process can be divided into twostages by considering the effect of the fixtures as shown in Fig. 4.Each stage can be further divided into four operational states, sothe total process is multi-state. States take place on the special fix-ture belong to the pre-joining stage (PJS), and states taking placeon the CARS belong to joining stage (JS). The eight states of ATWSmulti-state riveting are:

(A) Positioning state (P): positioning the skin and stringers in PJS.(B) Drilling state (D): drilling the Positioning Holes in PJS.(C) Joining state (J): joining the skin and stringers by screw bolts

in PJS.(D) Releasing state (R): releasing the panel from the special fixture.(E) Re-Positioning State (P): re-positioning the panel on CARS in

JS.(F) Drilling state (D): drilling the Riveting Holes in JS.(G) Riveting state (R): riveting the panel in JS.(H) Re-Releasing state (R):re-releasing the panel from CARS.

The whole multi-state riveting process of can be called as PDJRto PDRR (P to P). The states PDJR belong to the PJS and the statesPDRR belong to the JS. According to the P to P process in Fig. 4, somedefinition can be made as follows:


of AT

DrTf

Drat

Di

DtF{

Dpp

boaaiJBiowss

p

(

Fig. 4. The main process

efinition 1. The variation of ATWS multi-state riveting are rep-esented by the variation at the key characteristic points (KCPs).he variation is caused by positioning error, clamping force, joiningorce, etc. in PJS and JS.

efinition 2. The key characteristic points (KCPs) are closelyelated to the shape of the thin-walled structure. In a sense, theccuracy of ATWS is the accuracy of related KCPs. KCPs are alwayshe joining points and anchoring points in assembly of ATWS.

efinition 3. The positioning holes (PHs) are joined by screw boltsn PJS.

efinition 4. The riveting holes (RHs) are homogeneous dis-ributed between PHs, the principle of RHs planning is shown inig. 5 and Eq. (1).

h > 2db > 3d

(1)

efinition 5. The anchoring points (APs) are used to clamp theanel. They are build on the“N-2-1′′ locating principle for sheet-anel locating [28].

In PJS, the main task is to join the stringers and skins by screwolts to form an initial panel. Pallets are used to ensure the skin’suter shape precision when the skins and stringers are locating on

dedicated panel assembly fixture, as is shown in Fig. 2. So thessembly datum of PJS is the outer shape of skin, and the variations propagated from skin to stringer, which can be shown in Fig. 6. InS, the main task is to further join the stringers and skin by rivets.ecause of the special configuration of CARS shown in Fig. 1, the

nitial panel formed in PJS can only be put on the frame, and splintsn frame are used to ensure the accuracy of stringers inner shape,hich is shown in Fig. 6 too. So the assembly datum of JS is the inner

hape of stringer, and the variation is propagated from stringer tokin.

Five observations can be summarized according to the P to P
rocess presented in Figs. 4 and 6:
1) The process of ATWS riveting is a multi-state process and vari-ation can propagate from state to state.

Fig. 5. The principle of RHs deciding.

WS multi-state riveting.

(2) Positioning errors of both PJS and JS will cause assembly varia-tion.

(3) Due to the difference of assembly datum, variation on specialfixture will propagate to CARS, sometimes even be magnified.

(4) At a certain extent, splints and pallets have effects on control-ling the shape of ATWS in assembly process, which means wecan only consider the dimension variation of ATWS.

(5) The final assembly variation is determined by the integratedvariation of PJS and JS.

3. Variation analysis model for ATWS riveting

Whether a feature is a PH, RH or AP, each element in the P to Pprocess is a geometry feature. The feature which includes geomet-ric and topological information should be modeled first in orderto analyze the riveting variation. In a certain sense, variation canbe regarded as the displacements of the KCPs, so the displace-ment of every state should be modeled using the feature model.The multi-state riveting process has two stages and eight states,and variation propagates from state to state, so the propagation ofvariation should be taken into account.

In summary, in order to analyze the variation of ATWS withmulti-state riveting three important sections should be includedin the variation analysis model in addition to besides the P to Pprocess:

(1) Geometric and topological information of the ATWS.(2) Displacement of KCPs in every state of P to P.(3) The propagation process of variation generated in each state.

These three important sections are specified as three sub-models of the variation analysis model for ATWS riveting in thispaper, and the framework of the variation model is given in Fig. 7,where � is the total variation of the P to P process, �i (i = 1, 2, . . .,8) is the variation of each state, Ri (i = 1, 2, . . ., 8) is the transformmatrix of �i.

3.1. Feature sub-model for geometric and topological information

The feature sub-model provides the basis for developing thevariation model. All analysis of variation is based on the informa-tion supplied by the feature sub-model. Taking the panel shown in
Fig. 3 as an example, stringer and skin are made up of three types offeatures which are Coordinate, Surface and Point. To represent thegeometric and topological information, each feature should haveits own parameters, which is shown in Table 1. The architecture

H. Cheng et al. / Journal of Manufacturing Systems 30 (2011) 101– 115 105

riation

og

(

(

Fig. 6. The location mode and va

f feature sub-model for geometric and topological information isiven by Fig. 8.

1) Coordinate: The riveting datum is different in PJS and JS, sothe value of Coordinate in each stage is different. RegardingCoordinate of each feature as a typical Cartesian coordinate sys-tem (CCS), once the X direction, Z direction and the origin areconfirmed, the Coordinate is confirmed. So parameters of Coor-
dinate are “X”, “Z” and “O”, which are shown in Table 1.
2) Surface: The shape of ATWS is different from other mechanicalpart. Most of ATWS are double-curvature. Taking the Surface ofskin or stringer as the NURBS (non-uniform rational B-splines)

Fig. 7. The frame of the

propagation of both PJS and JS.

surface, the Surface can be represented as Eq. (2) [29], where�Pij(i = 0, 1, . . . , n; j = 0, 1 . . . , n′) is the control vertex, n andn′ is the number of control vertex on u and v direction, wij is theweighting factor of �Pij , Ni,k(u) and Nj,k(v) are the basis functionon u and v. So the parameters a Surface has are “U”, “V”, “Vertices”and “Weights”, which are also shown in Table 1.

C(u, v) =∑n

i=0

∑n′j=0wijNi,k(u)Nj,k(v)�Pij∑ ∑ (2)

ni=0

n′j=0wijNi,k(u)Nj,k(v)

(3) Point: Point is the main feature of KCPs. There are three typesof points according to Definitions 2–5: RH, PH and AP. PHs and

variation model.


Table 1The parameters instruction.

Features Parameters Instructions

CoordinateO The origin of the coordinate [x, y, z]T

X The X direction of the coordinateZ The Z direction of the coordinate

Surface

U The knot vector for the polynomial basis definition on the surface first directionV The knot vector for the polynomial basis definition on the surface second directionVertices The set of control pointsWeights The weights array

Point

ID The ID of the pointType The type of point (RH, PH, AP)Surface The surface of point, which has the same parameters as the feature “Surface”Position The academic position of point on the surface [u, v]T

�Position Variation between academic position and actual position of Point [�u, �v]T

geom
Fig. 8. The feature sub-model for
RHs are on both the skin and stringer, and are constant in bothPJS and JS. APs are different in two stages, in PJS they are on theouter surface of a skin, in JS they are on the inner surface of askin and stringers. The actual positions of a Point on a Surface arealways changing because of positioning error, assembly defor-mation, etc. So a Point can only present the KCPs on a Surfacewhich is shown in Fig. 9, where “U” and “V” are knot vectors forthe polynomial basis definition on the surface’s first and sec-
ond direction. Parameters of Point are “ID”, “Type”, “Surface”,“Position” and “�Position”, which are shown in Table 1 too.
Fig. 9. Point on Surface.

etric and topological information.

As described above, the riveting process of ATWS is multi-state,so the values of parameters for every feature are varying as the Pto P process flows down. Each stage has an assembly datum, so the“O”, “X” and “Z” of Coordinate are consistent in each stage. No mat-ter RH, PH, or AP, “ID”, “Type” “Surface” and “Position” are constantparameters of a Point, so they are constant during the process of Pto P. “�Position” is the parameter to record the deviation betweenpredicted and actual position, so in every state of P to P, whenvariation comes out, the value of “�Position” changes. Point hasdifference so is the Surface. The relationship between Surface andPoint is assumed to be invariable in each state, in order to sim-plify the calculation. When a state is finished, the Surface shouldbe reconstructed according to the variation of Points caused in thestate. According to the definition of NURBS surface, “U”, “V” and“Weights” of Surface are the same in states of the same stage. “Ver-tices” will change slightly in each states of P to P. Consequently, thevariation of ATWS can be regarded as the variation of KCPs, further,the “�Position” of Point.

3.2. Displacement sub-model

As described above, variation can be represented as the“�Position” of a Point. Hence the displacement sub-model is to rep-
resent the “�Position” of KCPs in every state of P to P. According toFigs. 4 and 6, in the Positioning state of PJS and the Re-Positioningstate of JS, variations are caused by the positioning error; in theDrilling state of both PJS and JS, variations are caused by the drilling

H. Cheng et al. / Journal of Manufactur

fRisdt

3

bAitaApficttPeaeF

r“iTas

[

wcr“std

P�

�

�

Fig. 10. Variations in Positioning and Re-Positioning states.

orce as well as the clamping force; in the Joining state of PJS and theiveting state of JS, variations are caused by jointing force and the

nteraction of rivets; in the Releasing state of PJS and Re-Releasingtate of JS, variations are caused by releasing force. Therefore theisplacement sub-model can be divided into four parts accordingo the four types of state.

.2.1. The variation of Positioning and Re-Positioning statesIn PJS and JS of P to P process, positioning error is inevitable

ecause of location accuracies of the special fixture and CARS.TWS, special fixture as well as the CARS all have their own datum

n P to P process, and the datum will be transformed as the Posi-ioning and Re-Positioning states are carried out. When a skin or

stringer is located on the special fixture, the initial datum is onTWS, and final datum is on the fixture. In the same way, when theanel is located on CARS, the initial datum is the datum of specialxture; the final datum is the datum of CARS. Different datum mayause mismatch error, and then variation accumulates. At the sameime, incorrect manipulation, equipment failure and other uncer-ain factor will make significant contributions to the variation ofositioning and Re-Positioning states. As small probability events,ffects of these situations are ignored in our model, and the vari-tion of Position and Re-Position states is regarded as mismatchrror, which is the result of datum changing and can be shown inig. 10.

As shown in Fig. 10, the global coordinate system “OXYZ” rep-esents the final datum of a state, the local coordinate systemOtXtYtZt” represents the initial datum of a state. Pn (n = 1, 2, . . .,) represents the KCP n of the panel, where i is the number of KCPs.herefore, for each Pn (n = 1, 2, . . ., i), it should satisfy Eq. (3), byssuming that there is no variation in the Position or Re-Positiontate.

nti ]

T AT (r0i − ri) = [nt

i ]T · rt

i (3)

here ri = [xi, yi, zi]T represents the position of KCPs in globaloordinate “OXYZ”, which is the final value of a particular KCP,ti= [xt

i, yt

i, zt

i]T represents the position of KCPs in local coordinate

OtXtYtZt”, which is the initial value of a particular KCP, nti

repre-ents the normal direction of KCPs in global coordinate, A is theransformation matrix from local coordinate system to globe coor-inate system.

Therefore the variation of KCPs in Positioning and Re-ositioning states can be represented as Eqs. (4) and (5), where

(x , y , z ) is the interpolation function of KCPs.
ki ki ki
i = [nti ]

T AT (r0i − ri) − [nt

i ]T · rt

i (4)

i = � (xki, yki, zki) (5)

ing Systems 30 (2011) 101– 115 107

3.2.2. Variation of Drilling stateWhether the joining process is bolting or riveting, the ATWS

need to be drilled ahead, and the model for variation caused inDrilling state in both PJS and JS are the same. When ATWS aredrilled, it not only needs drilling force but also clamping force, soin the Drilling state of both PJS and JS, the variation is caused by theintegrative action of clamping force and drilling force.

Taking the clamping force into account, for each KCP of ATWS,it sustains equivalent nodal clamping force by using FEM. In gen-eral, the clamping force Fc is a vector when there is more than onevariation source. So the variation of KCPs in Drilling state of bothPJS and JS caused by clamping force can be given by Eq. (6).

Vc =

⎧⎪⎪⎨⎪⎪⎩

Vc1Vc2

...Vcn

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

B11 B12 · · · B1n

B21 B22 · · · B2n

......

. . ....

Bn1 Bn2 · · · Bnn

⎫⎪⎪⎬⎪⎪⎭

·

⎧⎪⎪⎨⎪⎪⎩

Fc1Fc2

...Fcn

⎫⎪⎪⎬⎪⎪⎭

= [B] · {Fc} (6)

where Vc = {Vc1, Vc2, . . ., Vcn} T stands for the displacement of eachKCP nodes caused by clamping force, n represents the number ofKCPs, Fc = {Fc1, Fc2, . . ., Fcn} T is the equivalent nodal clamping forceafford to KCPs, and [B] is the effect of equivalent nodal clampingforce on KCP. Considering the effect of a part stiffness matrix Kc therelationship between Fc and Vc can be given by Eq. (7). So the formof [B] can be given by Eq. (8).

{Fc} = [Kc] · {Vc} (7)

{Fc} = [Kc] · {Vc} ⇒ [B] = [Kc]−1 (8)

The variation caused by the clamping force can be given by aclassical elasticity equation, so is the variation caused by drillingforce. Assuming the equivalent nodal drilling force is Fd = {Fd1, Fd2,. . ., Fdn}T, where n is the number of KCPs. Displacement of each KCPcaused by the drilling force can be represented as Vd = {Vd1, Vd2, . . .,Vdn} T, and can be given by Eq. (9), where D is the relational matrix.

Vd=

⎧⎪⎪⎨⎪⎪⎩

Vd1Vd2

...Vdn

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

D11 D12 · · · D1n

D21 D22 · · · D2n

......

. . ....

Dn1 Dn2 · · · Dnn

⎫⎪⎪⎬⎪⎪⎭

·

⎧⎪⎪⎨⎪⎪⎩

Fd1Fd2

...Fdn

⎫⎪⎪⎬⎪⎪⎭

= [D] ·{

Fd

}(9)

The clamping force and drilling force are loaded by differentindependent source. Hence the total variation of KCPs in Drillingstate of both PJS and JS (�) can be given by Eq. (10). For KCP i, �i canbe given by Eq. (11), where n is the number of KCPs.

� =

⎧⎪⎪⎨⎪⎪⎩

�k1�k2

...�kn

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

Vc1Vc2

...Vcn

⎫⎪⎪⎬⎪⎪⎭

+

⎧⎪⎪⎨⎪⎪⎩

Vd1Vd2

...Vdn

⎫⎪⎪⎬⎪⎪⎭

= Vc + Vd (10)

�i = {Bi1, Bi2, . . . , Bin} · Fc + {Di1, Di2, . . . , Din} · Fd (11)

3.2.3. Variation of Joining and Riveting statesBoth the Joining and Riveting states happen after a particular

Drilling state, the main target of the Joining and Riveting states isto close the gaps which are caused by states in advance. In a Joiningstate, screw bolts are utilized to close the gap which is caused inPositioning and Drilling sate. However, in a Riveting state, besides,plastic deformation of rivet will also cause additional variation inaddition to gaps caused in forward states. So the analysis modelsof variation for these two states are different. Variation in Joiningand Riveting states is shown in Fig. 11.

The purpose of a Joining state in PJS is to join the skins andstringers by screw bolts to form an initial panel which can be loadedon the CARS. As is shown in Fig. 11(A), joining the skin and stringersby screws in PJS is considered as a process of gap-closing. So we can


f Joint

oJ

atcob“sts2sa˛

�

(

iJervbst(J

ε

(

3

stw

Fig. 11. The variation o

nly consider the effect of jointing force, to obtain the variation inoining state.

In the coordinate system “OXY” of Fig. 11(A), point “A” represents KCP of ATWS. “A0” is the theoretical position of “A”, and “A′” ishe actual position of “A”, when the initial panel is formed. So inoordinate system “OXY”, vector

−−→A0A′ can represent the variation

f KCP “A” in Joining state. A new coordinate system “O′X′Y′” cane formed by contra-rotating the coordinate system “OXY” by ̨ inO′X′Y′” the projection of

−−→A0A′ on the X′-axis of the new coordinate

ystem is a point, which means “A0” and “A′” are both on Y′-axis ofhe new coordinate system. So the variation of KCP “A” in Joiningtate can be given by Eq. (12). At the same time, expanding theD coordinate system “OXY” to the coordinate described in featureub-model,

−−→A0A′ can be replaced by an input variation “ri (i = x, y, z)”,

equilibrium equation of KCP “A” can be gained as Eq. (13), wherei (i = 1, 2, 3) is the rotation angle of X-axis, Y-axis and Z-axis.

ki = −−→A0A′ = −−→

OA0 − −→OA′ = −−→

OA0 − (−→OA + −→

AA′) = −−→A0A − −→

AA′ = −−→A0A − uA

(12)

rx − uAx) cos ˛1 + (ry − uAy) cos ˛2 + (rz − uAz) cos ˛3 = 0 (13)

As is shown in Fig. 11(B), riveting the skin and stringers in JSs considered as a process of gap-closing. But different from theoining state, the composition of input variation is changed. Riv-ting is the joining method based on the plastic deformation ofivets, besides the variation caused by clamping and drilling, theariation caused by the plastic deformation of rivet “

−→AAt” should

e considered too. So there are two source of variation in Rivetingtate, which are “uA” and uAt and is shown in Fig. 11(B). Applyinghe same method, the variation of Riveting state can be given by Eq.14). At the same time the equilibrium equation of Riveting state inS can be given by Eq. (15).

ki = −−→A0A − uA + uAt = −−→

A0A′ (14)

rx + r′x − uAx) cos ˛1 + (ry + r′

y − uAy) cos ˛2 + (rz + r′zuAz) cos ˛3 = 0

(15)

.2.4. Variation of Releasing and Re-Releasing states
In Releasing and Re-Releasing states, ATWS are unloaded from a
pecial fixture or CARS after the assembly. According to the charac-eristic of ATWS, when the clamping force is released, the structureill spring back consequently. Therefore, the releasing force Fr is

ing and Riveting states.

closely related to the clamping force in drilling state, and can begiven by Eq. (16), where T is the transform matrix.

Fr =

⎧⎪⎪⎨⎪⎪⎩

Fr1Fr2

...Frn

⎫⎪⎪⎬⎪⎪⎭

=

⎡⎢⎢⎣

T11 T12 · · · T1n

T21 T22 · · · T2n

......

. . ....

Tn1 Tn2 · · · Tnn

⎤⎥⎥⎦ ·

⎧⎪⎪⎨⎪⎪⎩

Fc1Fc2

...Fcn

⎫⎪⎪⎬⎪⎪⎭

= [T] · Fc (16)

For the releasing force of every KCP in ATWS, it sustains theequivalent nodal releasing force by using FEM, which is the sameas clamping force. So the variation of KCPs in Releasing state of PJSand Re-Releasing state of JS caused by releasing force can be givenby Eq. (17).

Vr =

⎧⎪⎪⎨⎪⎪⎩

Vr1Vr2

...Vrn

⎫⎪⎪⎬⎪⎪⎭

=

⎧⎪⎪⎨⎪⎪⎩

M11 M12 · · · M1n

M21 M22 · · · M2n

......

. . ....

Mn1 Mn2 · · · Mnn

⎫⎪⎪⎬⎪⎪⎭

·

⎧⎪⎪⎨⎪⎪⎩

Fr1Fr2

...Frn

⎫⎪⎪⎬⎪⎪⎭

= [M] · {Fr}

(17)

In Eq. (17) Vr = {Vr1, Vr2, . . ., Vrn} T stands for the displacement ofeach KCP caused by releasing force, where n is the number of KCPs,Fr = {Fr1, Fr2, . . ., Frn} T is the equivalent nodal releasing force affordto KCPs. M is inverse matrix of stiffness matrix Kr.

At the same time, the releasing force is closing related to clamp-ing force, as is shown in Eq. (16). Taking Eq. (7) into account, therelationship between variation caused by clamping force and thevariation caused by releasing force can be built which is presentedas Eq. (18).

Vr = [M] · [T] · {Fc} = [M] · [T] · [Kc] · {Vc} (18)

3.3. Variation propagation sub-model

The multi-state riveting process of ATWS is a hierarchical pro-cess with two-stage and eight-state, and variation exists in eachstate. So far, the geometric and topological information are inte-grated according to the feature sub-model, and variation of everystate is modeled according to the displacement sub-model. Duringthe assembly process, variation of ATWS will propagate as the P toP process flows down. Each variation is the input to the subsequentstate. The state-to-state interactions cause an increase or some-
times decrease of variation. The process of variation propagationcan be shown as Fig. 12.
As is shown in Fig. 12, the variation is made up of three elementsin each state of P to P process.


n prop

(((

vt1gwt�{

sst(

(

(

cb

R

fpc

�

MCS to get the final variation, and the whole analysis procedureis shown as Fig. 15, where p is the number of state. In Positioningand Re-Positioning states, the variations are randomly generated

Table 2The parameters of KCP 1 and KCP 2.

Fig. 12. The variatio

1) The variation caused in the state.2) The variation propagated from the priorstate.3) The disturbance factor of the state.

In Fig. 12, �pki

(p = 1, 2, . . . , 8), (i = 1, 2, . . . , n) stands for theariation of KCP “i” in state “p” of P to, tpi(p = 1, 2, . . ., 8) stands forhe disturbance factor of variation in the particular state, ��p

ki(p =

, 2, . . . , 8), (i = 1, 2, . . . , n) is variation which state “p-1′′ propa-ates to the state “p”, and the form of ��p

kican be given as Eq. (19),

here Rp (p = 1, 2, . . ., 8) is the translation matrix of state “p”. Athe same time, the Surface should be reconstructed in state “p” after

�pki

is imported, according to the feature sub-model of this paper.

��pki

= Rp · (�pki

+ tpi) (when p = 1)��p

ki= Rp · (�p

ki+ tpi + ��p−1

ki) (when p = 2, 3, . . . , 8)

(19)

As is described above, the assembly datum of PJS is the outerhape of skin, and the assembly datum of JS is the inner shape oftringer. So when the ATWS are moved to the CARS, the manipula-ion of re-location is needed. And the translation matrix Rp in Eq.19) is made up of two parts:

1) The re-location translation matrix Rrp.

2) The location translation matrix Rlp.

Rrp is the component for re-location, and the inequality “Rr

p /= 0”an come into existence only when p = 4. So the Rp in Eq. (19) cane given by Eq. (20).

p = [Rrp] + [Rl

p] ={

Rlp (when p /= 4)

Rrp + Rl

p (when p = 4)(20)

So there is a different Rp(p = 1, 2, . . ., 8) and tp(p = 1, 2, . . ., 8)or every KCP in each state. According to Fig. 12, when the rivetingrocess is finished, the total variation of a KCP is equal to ��8

kand

an be given by Eq. (21).

�8k =

⎧⎪⎪⎨⎪⎪⎩

��8k1

��8k2

...

⎫⎪⎪⎬⎪⎪⎭

= [R7] ·

⎛⎜⎜⎝

⎧⎪⎪⎨⎪⎪⎩

�8k1

�8k2...

⎫⎪⎪⎬⎪⎪⎭

+

⎧⎪⎪⎨⎪⎪⎩

t8k1

t8k2...

⎫⎪⎪⎬⎪⎪⎭

+

⎧⎪⎪⎨⎪⎪⎩

��7k1

��7k2

...

⎫⎪⎪⎬⎪⎪⎭

⎞⎟⎟⎠

��8kn

�8kn

t8kn

��7kn

= [R7] · (�8k + t8

k + ��7k) (21)

agation sub-model.

4. Case study

In order to demonstrate the effectiveness and robustness of thevariation model of ATWS with multi-state riveting, we choose awing panel, which is made up of a skin and four stringers as theexample. The CAD model of the panel is developed in CATIA V5R17, which is shown in Fig. 13. “TEST PANEL” is the ID of the panel;“TEST 017′′ is the ID of the skin and “TEST 013′′–“TEST 016′′ areIDs of stringers. Coordinate, Surface as well as Point are developed,according to Fig. 13. Three relational Coordinate are included in thecase panel, which are “Riveting Machine Coordinate System”, “Fix-ture Coordinate System” and “Wing Coordinate Systems” in Fig. 13,and each coordinate is formed according to the data measured bylaser tracker. The skin is formed as an extruded part of a NURBS sur-face which is developed in Eq. (2). The panel is parametric modeled,so that it can be reconstructed exactly to support the FEM, accord-ing to the variation calculated by our model. Points of the exampleare organized by the ID of stringers, and are projected onto theskin to form the RHs and PHs of the feature sub-model. We selectthe riveting between stringer (TEST 016) and skin to analysis thevariation during the P to P process.

The location condition of the panel in PJS and JS is shown inFig. 14. “KCP 1′′ and “KCP 2′′ are chosen as the points to analyzethe variation during the multi-state riveting process. The splintsare used to keep the outer shape of the panel in PJS, and palletsare used to keep the inner shape of the panel in JS. So the panel islocated by the typical “N-2-1′′ principle [30] for deformable sheetpanel, and mark “AP n” in Fig. 14 represents APs of the panel. Theparameters of KCP 1 and KCP 2 can be shown in Table 2.

At this point, the feature sub-model is developed in order toget the final variation of ATWS with multi-state riveting the dis-placement and the propagation of variation should be analyzed.Considering the randomness of the positioning error, we use the

ID Type Surface Position

KCP 1 RH TEST 017 [345, 406]KCP 2 RH TEST 017 [432, 397]


Fig. 13. The feature model in CATIA V5.

dition

asgoowp

sysoaafips

saT

The variation caused in Positioning and Re-Positioning stateswill change the initial appearance of ATWS in Drilling state. Thevariation ahead would not lead to the shape of ATWS changing

Table 3The positioning accuracy of related fixture.

Fixture Direction Positioning accuracy

The panel fixtureu N(0.4, 0.032)v N(0.2, 0.022)

CARSu N(0.4, 0.042)v N(0.3, 0.022)

Table 4Force supplied to the panel.

Parameters Magnitude

Fig. 14. The location con

ccording to the statistics of the production. The variation of resttates are calculated by FEM, in process of which random variableso into the FE simulation of each state and random variables comeut. In process of our analysis, the feature and boundary conditionf FE are varied, according to Fig. 15. With the feature sub-model,e can easily adjust the value of the Surface and Point to form theroper feature and boundary condition in FE.

In order to obtain the variation in Positioning and Re-Positioningtates, the positioning accuracy of APs should be input into the anal-sis procedure firstly. At the same time, APs are pressed to fixtures,o the accuracy of APs is assumed equal to which of fixture. Becausef the large wingspan of the panel, the positioning accuracy on und v direction of a fixture are different, according to the statistics ofctual production, the positioning accuracy of the panel assemblyxture and CARS on u and v direction are all obeying the normalrobability distribution basically, and the detail situation can behown in Table 3.

As is shown in Fig. 15, except the Positioning and Re-Positioningtates, variations in rest states can be calculated by FEM. Materialnd joining parameters which are needed for FEM are shown inables 4 and 6.

in PJS and JS of the case.

Drilling force 3 KNRiveting force 35 KNClamping force 300 NReleasing force 300 N


chart

apawvcs

TT

Fig. 15. The flow

ccording to our assumption. In Drilling state we just change thearameters of Point by adding ��1

kion both u and v direction, which

re gained by MCS. KCP 1 and KCP 2 are both RHs, so in our case
e just calculate the variation in Drilling state of JS, and ignore the
ariation in Drilling state of PJS. In Drilling state, ATWS are affordedlamping force and drilling force, according to the displacementub-model of this paper. The splints of CARS can keep the relative

able 5he disturbance factor in riveting.

Disturbance factor Value

t1i 0.03t2i 0.05t3i 0t4i 0t5i 0.04t6i 0.02t7i 0t8i 0

of MCS process.

position of stringers and skin well, so to simply the calculation wejust choose the skin to analyze the variation in FEM. The boundarycondition of the skin is following the “N-2-1′′ positioning principle,which is shown in Fig. 16.

Joining and Riveting states are both following a special Drillingstate, so the shape of skin should be reconstructed according tothe variation caused in Drilling state (��2

kiand ��6

ki). KCP 1 and

KCP 2 are both RHs, so we only consider the variation in Rivetingstate, and the boundary conditions and force can be given by Fig. 17according to our model.

In Releasing and Re-Releasing states, the skin and stringers areappeared as a panel, so they should be considered as a componentin our simulation, and the shape of the panel should be recon-structed according to the variation propagated from the Joining
and Riveting states (��3
kiand ��7

ki). We only consider the vari-

ation in Re-Releasing state, and the boundary conditions and forcesupplied to the panel can be given by Fig. 18 according to ourmodel.


Fig. 16. The boundary condition of the skin in Drilling state.

Fig. 17. The boundary conditions of Riveting and Joining states.

Fig. 18. The boundary conditions of Releasing and Re-Releasing states.

Table 6The material and parameters of the case.

Part Material Diameter (mm) Thickness (mm)

Screws STEEL 6.2 –Rivets 2117-T4 5.8 –Skin 7055-T7751 – 5

Fig. 19. The variation of KC

Stringers 7055-T7651 – 5

As is shown in Fig. 12, variation of each state in P to P processwill affect the variation in following states in forms of ��p

ki. Accord-

ing to the assembly experience, in states A, B, E and F of Fig. 4, thedisturbance factor is more than states C, D, G and H of Fig. 4. Consid-ering the Riveting and Joining states are taken place by automatedtool, and the releasing of panel is the inverse process of clamping,the value of disturbance factor in our calculation can be given byTable 5.

As is shown in Fig. 15, once an analysis procedure is finished afinal variation would be gained. In our case, 10 simulation processesare studied to prove the effective of our variation model.

According to Eqs. (3)–(5), the variation of KCP 1 and KCP 2caused in Positioning state can be solved by MCS. 200 sam-ples are used in every MCS to get the means and variances ofthe two KCPs, and the results of 10 times MCS are shown inTables 7 and 8.

The variation of KCP 1 and KCP 2 in PJS cannot be calculateddirectly, but the variation of related PHs and be gained by the samemethod mentioned above, so ��4

kiof KCP 1 and KCP 2 can be inter-

polated by the variation of related PHs, and imported into the MCSin JS.

The variation in Re-Positioning state can be gained in our simu-lation, according to Eqs. (3)–(5) and the same method of Positioningstate. Means and variances of the two KCPs’ variation simulated byus can be shown as Tables 9 and 10.

We developed our displacement sub-model into the MCS pro-cess shown in Fig. 15 by the User Subroutines of ABAQUS, andintegrated the propagation sub-model by ISight. The final variationsimulated by us can be shown as Figs. 19 and 20.

The panel shown in Fig. 13 is riveted for 10 times in our exper-iments, in order to demonstrate the simulation result. Fixturesshown in Figs. 1 and 2 are the CARS and panel assembly fix-
ture of our case. Material of ATWS and the forces supplied to thepanel in our experiment are in accordance with the parameters inTables 4 and 6, so that the results of experiments are comparabil-
P 1 according to MCS.


Table 7The variations of KCP 1 in Positioning state.

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th

Mean (u) 0.3910 0.4006 0.4009 0.3912 0.4005 0.4006 0.4004 0.4010 0.3909 0.4005Variance (u) 9.13e−5 9.02e−5 9.04e−5 8.92e−5 9.10e−5 9.05e−5 8.97e−5 9.03e−5 9.10e−5 9.11e−5Mean (v) 0.2025 0.1969 0.2005 0.2014 0.1973 0.2015 0.2008 0.2010 0.1992 0.1984Variance (v) 4.15e−5 4.08e−5 4.14e−5 4.02e−5 4.12e−5 4.11e−5 4.03e−5 3.96e−5 4.26e−5 3.88e−5

Table 8The variations of KCP 2 in Positioning state.



Table 9The variations of KCP 1 in Re-Positioning state.



Table 10The variations of KCP 2 in Re-Positioning state.



Table 11Variation of KCP 1 and KCP 2 by experiments (mm).

1st 2nd 3rd 4th 5th 6th 7th 8th 9th 10th Mean Variance

�U1 0.935 0.931 0.937 0.936 0.929 0.932 0.945 0.941 0.931 0.942 0.9359 2.877e−511

55

53

iidt

�V1 0.614 0.619 0.611 0.604 0.609 0.6�U2 1.049 1.058 1.052 1.055 1.046 1.0�V2 0.652 0.657 0.659 0.664 0.661 0.6

ty. At the same time, because the curvature radius of the panels very large (larger than 25 m), the values of variation on normalirection of the panel are not measured in our experiments. Varia-ions of KCP 1 and KCP 2 are measured by laser track, and the result

Fig. 20. The variation of KCP

0.606 0.612 0.608 0.613 0.6107 1.823e−51.044 1.046 1.051 1.054 1.0510 2.156e−50.657 0.649 0.662 0.655 0.6569 2.254e−5

can be shown as Table 11, where �U1 is the variation of KCP 1 onu direction, �V1 is the variation of KCP 1 on v direction, �U2 is thevariation of KCP 2 on u direction, �V2 is the variation of KCP 2 onv direction.

2 according to MCS.


Fig. 21. The comparison of variation on U-direction of KCP 1.

iteoi

ivvo

TR

Fig. 23. The comparison of variation on U-direction of KCP 2.

Fig. 22. The comparison of variation on V-direction of KCP 1.

Variations of ATWS with multi-state riveting are basically obey-ng the normal probability distribution according to statistics. Sohe comparison between variations simulated by our method andxperiment date can be shown as Figs. 21–24, and the percent errorf the mean between the simulation and experimental data is givenn Table 12.

The results according to the model are matched with exper-ments results, according to Figs. 21–24. The percent error ofariation is very small (less than 0.2%) according to Table 12. So theariation model is efficient and can be used to analyze the variationf ATWS with multi-state riveting practically.

able 12epresents error of the mean between the simulation and experimental data.

Percent error

�U1 0.075%�V1 0.164%�U2 0.085%�V2 0.076%

Fig. 24. The comparison of variation on V-direction of KCP 2.

5. Conclusion

This paper presents a variation model for AWTS with multi-stateriveting on the base of a novel P to P riveting process. The variationmodel consists of three sub-models which are feature sub-model,displacement sub-model and variation propagation sub-model. Inthe feature sub-model, geometric and topological information isrepresented by a hierarchical method base on three important geo-metric features. In the displacement sub-model, the variation ofeight-state in P to P is divided into four types, and the displacementof each type is analyzed separately according to the coordinatetransformation and FEM. In the propagation sub-model, a trans-lation matrix considering the disturbance factor of every state isdeveloped to integrate variation. By integrated the FEM with MCS,variation in a multi-state riveting process of a special wing panel iscalculated and the results prove the correctness and applicabilityof the variation model. As this variation model can calculate thevariation early, it can be used for:

• Predicting variations of multi-state riveting assembly for theATWS with CARS precisely.

factur

•

•

•

A

SRa(etc

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

H. Cheng et al. / Journal of Manu

Anticipating potential riveting assembly problems caused byvariation before expensive thin-walled parts and fixtures aremade.Identifying the processes which are critical for the quality of thefinal ATWS.Providing the basic data for optimal fixture configuration design,including the optimal design of CARS.

cknowledgements

We gratefully acknowledge the support of National Naturalcience Foundation of China (50805119), the Nation High-Tech.&D Program of China (Grant No. 2007AA041903), the Doctor-te Foundation of Northwestern Polytechnical University, ChinaCX200809) and the Foundation for Basic Research of Northwest-rn Polytechnical University, China (JC201032). We would also likeo thank the editors and the anonymous referees for their insightfulomments.

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Variation modeling of aeronautical thin-walled structures with multi-state riveting

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