Research article
Variation propagation modeling and analysis
at preliminary design phase of multi-station
assembly systems
Haixia Wang
Departmentof AdvancedStructures,ManufacturingR&D,CaterpillarInc.,Peoria,Illinois,USA,and
Dariusz Ceglarek
TheDigitalLaboratory,WMG,Universityof Warwick,Coventry,UK
Abstract
Purpose- Dimensionalvariationmanagementis a majorchallengein multi-stationsheetmetal assemblyprocessesinvolvingcomplexproducts
such as automotivebody and aircraft fuselageassemblies.Very few studies have exploredit at a preliminarydesignphase taking into
consideration
effectsof part deformationon variationpropagation,sinceearlydesignphaseinvolvesthe developmentof imprecisedesignmodels
with scantor incompleteproductand processknowledge.Theobjectiveof this paperis to presenta variationmodelwhichcanbe built into the
preliminarydesignphasetaking into considerationall of the existinginteractionsbetweenflexiblepartsand tools in multi-stationsheetmetal
assemblyprocess.
Design/methodology/approach- Thepaperaddresses
thisproblembyfirst,presentinga beam-based
productandprocessmodelwhichsharesthe
samedatastructure
of theB-RepCAD models,andthereforecanbeembedded
in CADsystems
for automatic
productskeletaldesign;second,
determiningthe influenceof part deformation,for various,differingjoiningandreleasingschemes,on variationpropagation;andthird, utilizingthis
informationto generatea vector-based
variationpropagationmodelfor multistationsheetmetalassemblies.
Findings - Thispaperpresentsa beam-based
productand processmodelwhichsharesthe samedatastructureof the B-RepCADmodels,and
thereforecanbeembeddedin CADsystemsfor automaticproductskeletaldesign;determinesthe influenceof partdeformation,for various,differing
joiningandreleasingschemes,on variationpropagation;and utilizesthis informationto generatea vector-based
variationpropagationmodelfor
multistationsheetmetalassemblies.
Originality/value - A truckcabassemblyispresented
to demonstrate
the advantages
of the proposedmodeloverthe state-of-the-art
approachused
in industryfor sheetmetalassemblies.
Keywords Designfor assembly,Productdesign,Modelling,Dimensionalmeasurement
Paper type Research
paper
1. Introdudion
In the literature, research on developing a variation model can
be grouped into four categories, based on whether a particular
model is built for detailed or preliminary product design, or if
the parts considered in the model are rigid or compliant.
Designing a variation model involves tWo major tasks:
1 modeling of product and process elements; and
2 formulating a variation propagation model based on
models generated by task 1.
It has been reported that about tWo thirds of engineering
changes in automotive body and aircraft fuselage assemblies
are related to product dimensional variation (Shalon et al.,
1992; Ceglarek and Shi, 1995). Proper management of
dimensional variation requires a variation model that uses
quantitative methods to analyze and predict how parts and
tooling variations propagate through multi-station assembly
processes to the final product.
For task 1, both boundary representation (B-Rep) and
constructive solid geometry (CSG) representations of solid
parts have been widely used in the detailed design phase for
variation analysis of rigid part assemblies. Finite element
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/Ol44-5154.htm
I
. ..
The authors gratefully acknowledge the financial support of the Advanced
Technology Program (ATP) award provided by US National Institute of
Standards
and Technology
(ATP
Cooperative
Agreement
No. 70NANB3H3054), UK EPSRC Star Award EPIE044506/1 and US
NSF-CAREER Award DMll-0239244.
Assembly Automation
29/2 (2009) 154-166
Cl Emerald Group Publishing Limited [ISSN 0144-5154]
[DOl 10.1108101445150910945606]
154
Variation propagation modeling and analysis
Haixia W&ng and Dan'usz
Assembly
J,blume 29
Ceglarek
modeling and analyses have been used for variation analysis of
compliant part assemblies. Very few studies have explored
variation modeling at a preliminary design phase since it
involves the development of imprecise design models with
scant or incomplete product and process knowledge. Ullman
(1992) and Wood et al. (1992), among others, notes that
computers are not widely used in the early phases of
engineering design because:
existing computer tools need a very refined representation
of an object on which to operate; and
computers are primarily evaluation tools and of limited
value in generating concepts.
. Number
Automation
2'
2009
. 154-166
tolerancing design to a preliminary design stage. In aerospace
assembly system, Odi et al. (2000) emphasize the importance
of the design phase as the earliest opportunity to address
product dimensional quality wherein the total transfer
variation between parts accounts for the biggest portion of
total product dimensional
variation, and other detail
geometrical variations, such as those from mating surfaces
themselves, are rather small. At present, important anempts
are underway to model product and process at a preliminary
design phase for dimension control. The concept of key
characteristics (KC) [2] delivery and KC representation of
product and process has been utilized to study variation
propagation in multi-station assembly systems (Ceglarek et aI.,
1994; Lee and Thornton, 1996; Cunningham et al., 1996;
Thornton, 1999).
In an effort to advance variation analysis for compliant
sheet metal assemblies into a preliminary design phase, Shiu
et al. (1997), Ceglarek and Shi (1997), Rong et al. (2000) and
Shiu et al. (2003) develop a beam-based model by deriving
principles of decoupling automotive parts into beam
members, and modeling of part-to-part joint geometry and
part locating points using beam elements and nodes. The
method of decomposing parts into beams is also applied in
this paper. Furthermore, the data structure of the beam
model presented in this paper, shares the same data structure
as the B-Rep CAD systems, thus allowing automatic
connection of the CAD system with a variation model at a
preliminary design phase. Table I depicts how this paper fits
with, and expands upon, previous literature on modeling of
product and process elements.
Preliminary
design descriptions
are characteristically
imprecise: the designer has yet to make most of the
decisions that will allow for dimensional
variation
management. Nevertheless, a model built during an early
design phase will bring more benefit because it provides useful
information before all design characteristics and engineering
and cost requirements have been locked in. It is widely
believed that 80 per cent or more of the life-cycle cost of a
product is determined during the early design process
(EI-Haik, 2005). Early identification and prevention of
manufacturing
and assembly problems
can have an
enormous impact towards reducing product development
and launch time.
For task 2, a variety of analytical models have been
developed beyond the traditional Monte Carlo simulationbased models. Most of the models assume that parts are rigid.
For compliant part assemblies, the method of influence
coefficient (Liu et aI., 1995) has been presented with the
assumption of a linear relationship between part deviations
and assembly spring-back deviations. The method has greatly
reduced the prohibitive computational efforts associated with
numerous iterations of finite element analyses (FEAs). Later
on, the method has been extended by including interactions
between joints (Shiu et aI., 1997) and interactions between
flexible parts and fixture locators (Long and Hu, 1998).
However, interactions between flexible parts and tools in
multi-station assembly processes have yet to be thoroughly
studied and modeled.
Many mechanical products such as automotive bodies,
airplanes fuselages and household appliances are assembled at
multiple stations using numerous compliant sheet metal
parts[I]. Part deformations caused by assembly forces lead to
subassembly errors which accumulate on the final product.
Therefore, the objective of this paper is to present a variation
model which can be built into the preliminary design phase
taking into consideration all of the existing interactions
between flexible parts and tools in multi-station sheet metal
assembly process.
1.2 Formulating
a variation propagation
model
For rigid part assemblies, simulation software packages, such
as 3DCS, CE-TOL and VSA, have been widely used in
industries, while analytical models, such as vector-loop-based
models (Chase et al., 1997; Gao et aI., 1998), autoregressive
model (Lawless et aI., 1999), state transition model
(Mantripragada
and Whitney, 1999), and "stream-ofvariation model based on state space approach" ain and
Shi, 1999; Ding et aI., 2000; Huang et al., 2007) have been
actively explored in recent studies. A recent monograph (Shi,
2006) and a review paper (Ceglarek et aI., 2004) provide
detail descriptions of the existing research work on multistage
manufacturing processes.
For compliant part assemblies, some CAD software such
as CATIA VS Tolerance Analysis of Deformable Assembly
has anempted to integrate part deformation into tolerance
analysis for multistage assemblies. Interactions between
flexible parts and tools in multi-station assembly processes
often entail an enormous amount of computational work that is
frequently prohibitive for variation analysis. To mitigate the
computational effort, Liu et al. (1995, 1996) developed
variation analysis for compliant part assembly by presenting a
linear model, called "mechanic variation simulation" model,
for part-to-part joining process in a single assembly station.
This simulation reduces computation efforts by simplifying the
Monte Carlo portion of the analyses, and is often referred to as
the method of influence matrix. Subsequent research has since
been conducted with the inclusion of different level of
complexity. Long and Hu (1998) extend the analysis model
by incorporating position variations of fixture locators into the
model. Whitney (2008) presents proper constraints as a way to
support the goal of placing key parts in particular geometric
1.1 Modeling of product and process elements for
tolerance analysis
Most research is conducted in a detailed design phase since
the knowledge of detailed geometry of the assemblies is
readily available and existing CAD software can tackle
product and process modeling.
Recognizing the importance of conducting variation
analysis at a preliminary design phase, researchers have
presented on what an assembly design process should be at a
preliminary design stage taking into account dimensional
quality control. Narahari et al. (1999) advocate advancing
155
Variation
propagation
mode ling and analysis
Assembly
HJlume 29
Haixia Wfzng and Dariusz Ceglarek
Automation
. Number
2 .2009
. 154-166
Table I Briefliteraturereviewfor task1: modelingof productandprocesselementsfor toleranceanalysis
Assemblies
Designphases
Rigid parts
Compliant parts
Detaileddesign
Basedon B-Repor CSGrepresentationof parts
VSAManual(1998),3DCSManual (2002),Chaseet al.(1997),
Gao et al. (1998),Ceglareket al. (2004),Huanget al.(2007),ete.
Preliminary design
Basedon KCrepresentationof parts
Ceglareket al.(1994),Thomton (1999),Mantripragadaand
Whitney (1999),JinandShi(1999), Ding et al. (2000),ete.
Basedon generalfinite element(FE)representation
of parts
liu and Hu (1997),Hu(1997),Long and Hu(1998),
Merkley (1998),Changand Gossard(1997),
HsiehandOh(1997),Camelioet al. (2003)and
CATIA(Cozzens,2007)
Basedon beam-basedrepresentationof productand
processinformation
Shiuet at. (1997,2003), Ceglarekand Shi (1997) and
Wangand Ceglarek(2009)
that are addressed in more detail in Sections 2 and 3,
respectively:
1 A beam-based model is proposed to represent product
and process information at a preliminary design phase.
A product is decomposed into beam elements, with beam
nodes representing KC including part-to-part joints, partto-fixture mating surfaces, inspection features, and part
cross-section changes. Beams are represented as the same
data structure of the B-Rep CAD models.
2 A simulation
model,
the vector-based
variation
propagation model, is proposed by tracing coordinate
changes of beam nodes through typical sheet metal
assembly operation,
taking into consideration
the
influence on variation propagation from part and fixture
imperfections, part elastic properties, fixture locator
layouts, p art-to-p art joint orientation
and joining
kinematics.
relationship with respect to one another so that a DFC can
deliver KCs unambiguously. Merkley (1998) recognizes the
need to include covariance in statistical FEAs when random
variables are used for input. Hu (1997) describes the concept
of "stream-of-variation"
for a multi-station
compliant
structure assembly. Chang and Gossard (1997) present a
framework of a variation model for multi-station compliant
sheet metal assemblies using the method of vector addition.
Hsieh and Oh (1997) develop special-purpose finite element
simulation software, elastic assembly variation simulation,
which is used for digital panel assembly design within the
General Motors Corporation. Camelio et aL (2003) present a
"stream-of-variation" model based on state space approach for
sheet metal assemblies including flexible part interaction with
fixtures and welding guns. Soderberg et al. (2008) propose a
combination of the method of influence matrix and a technique
of contact modeling to accommodate some types of geometry
that leads to nonlinear behaviors. Table II depicts how this
paper fits with, and expands upon, previous literature on
formulating variation propagation model.
A limitation of the existing models for variation analysis of
compliant part assemblies is that they do not take into
account influences of part deformation for various differing
joining and releasing schemes. This paper addresses this
limitation by determining these influences and incorporating
them into a vector-based variation propagation model.
The proposed method is shown in Figure 1 where
subassembly variation information at station n is represented
as
X".
2. Beam-based produd model
This paper proposes a beam-based representation of a
product is proposed since, since it is able to estimate overall
deformation of vehicle body structure with 95 per cent
accuracy (Ch on, 1986), as well as the capability to model
product and process with incomplete product- and processrelated information.
A beam-based partlsubassembly is defined as shown in
Figure 2(a). It has the same data structure as B-Rep CAD
systems (Figure 2(b)), which allows to automatically generate
finite element models from CAD design models, running the
1.3 Proposed lIlethod
This paper presents a variation analysis model for multistation sheet metal assemblies, which takes into consideration
the influence of part deformation, and can be embedded into
existing CAD systems at a preliminary design phase for
product skeletal design. It involves the following two steps
Table 11 Brief literaturereviewfor task2: formulatinga variationpropagationmodel
Assemblies
Compliantparts
Assembly process
Rigidparts
Single station level
LeeandWoo(1990)andChaseandParkinson
(1991)
Multi-station level
Ceglarek
et at. (1994),
Chase
et al. (1997),
Gao et al. (1998), Thomton (1999), Mantripragadaand
Whitney (1999), Jin and Shi (1999), Ding et al. (2000),
Ceglarek et al. (2004) and Huang et al. (2007)
156
liu and Hu (1997),Ceglarekand Shi,1997),
Shiu et al. (1997),Long and Hu (1998)and Merkley (1998)
Shiu(1996), Hu (1997), Changand Gossard(1997),
Hsiehand Oh (1997),Camelio et al.(2003),Whitney (2008),
Soderberget al.(2008)and Wangand Ceglarek(2009)
Variation
propagation
modeling
and analysis
Assembly
Haixia Wang and Dariusz Ceglarek
HJlume 29
Automation
. Number
2 . 2009 . 154-166
Figure1 Outlineof the proposedbeam-based
multi-stationvariationpropagationmodel
~~i]'-'-'-'-'-'-'-'-'-'-'-'-'-~~-'-'-'-'-'-'-'-'-'-'-.-
,
;<
'
Figure2 Beam-based
part and subassembly
representationdefined
asa graph
Beam-based
= {PE}
I
11
11
Beam:
PE
= «Bv;, Bv), b)
Beam node: Bv = «X, Y, Z), c)
Before applying deformations
Topological CAD definition
part definition
Part: Ps
11
Figure 4 Beam deformationand node detection of dimensional
variation
I
I
Face table: S = {E}
Edge table: E - (Vi, Vi)
1
Vertex table: V = (X, Y, Z)
1
(a) Part!subassembly definition
proposed in this paper
*
I
. fixturelocator
BV2'
Beam nodes are extracted from four types of KC, Le.:
1 part-to-fixture mating features;
2 part-to-part mating features;
3 inspection features; and
4 turning areas of part cross-section changes.
variation analysis model within CAD software, and utilizing
available techniques developed in CAD software for product
skeletal design at the preliminary design phase. In addition, its
graph-based data structure allows utilizing algorithms in
graph theory to facilitate automatic product design and
assembly process planning.
A given part Ps is modeled as a set of beam elements PE'
= «Bv;,
A beam node is defined as Bv = «X, Y, Z), c), where (X, Y,
Z) represents the node's global coordinates. For a part feature
that is designed for locating the part by fixture or by another
part, attribute c of the defined beam node represents
translational degrees of freedom (DOFs) along x-, y-, and
z-axes and rotational DOFs around x-, y-, and z-axes,
respectively. It is a 6-tUple, as presented in equation (1), with
an entry value of "1" indicating the constraint applied, and
"0" otherwise:
Bvj), b), as shown in
Figure 3, where Bv; and BViare beam nodes and b represents
beam stiffness information. The stiffness information of a
beam is represented
as b = f(E, Iyy>Izz, A) and is a function
of
Young's modulus
(E) and cross-section
geometry
characteristics (inertia moments Iyy and Izz and cross section
area A).
Part deformation detected by coordinate changes of a node
position is shown in Figure 4 (in this case, node BV2becomes
c = (Tran-x, Tran-y, Tran-z,Rot-x, Rot-y,Rot-z)
.
BVi
b
(1)
For a part feature that is designed for mating the part with
other parts or subassemblies, attribute c is defined in equation
(2). It represents the normal direction of a mating surface in
the global coordinate system (as shown in Figure 5), and k1,2
represents a designed kinematic characteristic for joining
(details arc shown in Figure 11 and Section 3.3). The
attribute c of a beam node has a null value if it models a
feature stated in KC 3 or 4:
B~2)'
Figure 3 Illustration of beam element PE= ((Bv;.Bvj).b)
!8
BVI
I
(b) B-Rep definition in CAD
systems
A beam element is defined as PE
. ~
~
BV2
BVI
1
After applying deformations
.
BVj
c
157
= ((x,y,Z),k!,2)
(2)
Variation
propagation
Assembly
mode ling and analysis
Ullume
Haixia wang and Dariusz Ceglarek
X
Similar to a part CAD consisting of vertex table, edge table,
and surface table, a beam-based product consists of node
table, beam table and a partlsubassembly table (Figure 6).
Each entry in the node table is a list of coordinates defining
the related node and its related constraint parameter; each
entry in the beam table consists of a pointer to each beam
node and of its related stiffness parameter; each entry in the
partlsubassembly table represents a partlsubassembly with its
beam elements.
The beam-based model of the product is realized with
several beam-based subassemblies and parts related together;
Figure 7 shows beam representation of a truck cabinet
assembly.
3.1 Variation
vector after "place"
BVI: xI. YI. Zl. CJ
BV2: X2. Y2. Z2. C2
BV3: X3' Y3. Z3. C3
...................
. 154-166
operation
The part variation vector ~-l) caused by part re-orientation
in the place operation can be obtained through homogenous
matrix transformations as shown in equation (3):
Figure 6 Exampleof nodetable,beamtable andsubassembly
table
Node table
2 .2009
A vector-based variation propagation model is proposed by
tracing coordinate changes of beam nodes through typical
place, clamp, fasten/join and release (pCFR) operations in a
single-station sheet metal assembly process (Figure 8).
Placing a part to its fixtUre locators, usually following the
3-2-1 fixtUring locating principle (Menassa and DeVries,
1989), causes part rigid body movement. Clamping a sheet
part to its extra fixtUre locators (Cai et al., 1996) causes part
deformation.
Fastening/joining
parts might also cause
part deformation. Releasing a sub assembly from fixtUres
causes spring-back of partial or complete deformation.
Variation propagation due to rigid body movement of parts
can be solved by homogeneous matrix transformation, while
variation propagation due to part deformation can be
calculated by FEAs, as briefly summarized in Table m.
Vector-based variation tracing is shown in Figure 9.
The contribution of this vector-based model is that it
presents the joint error after the fastening/joining operations,
differentiates tWo sitUations of variation propagation for
releasing operations, and formulates the variation propagation
for a releasing operation with complete spring-back.
y
)---
Automation
. Number
3. Vedor-based simulation model for variation
analysis
Figure 5 Schematic
representation of part-ta-partjoint
Z
29
Partlsubassembly
Beam table
PEI: BVI> BV2. bl
PE2: BV2. BV3. b2
PE3: BV3. By4. b3
table
SE): PEI' PE2. PE3
SE2: PE4. PES. PE6. PE? PES
SE3: PE9. PElO> PEII. PEI2
..................
..................
By;: xi. Yi. Z;. Cj
Figure7 Beamrepresentationof a truckcabassemblyfor dimensionalvariationanalysis
. fixture locator
..l, ... part-to-part joint
. Measurement
158
Variation propagation
modeling
and analysis
Assembly
Automation
. Number
Hllume 29
Haixia Wbng and Dariusz Ceglarek
2 . 2009 . 154-166
Figure8 ThePCFR
operationsin a multi-stationsheetmetalassembly
_
-(I)
Multi-station
I Station
I
... --I
Station
m
_ (f)
_
-(f»
0
811 =
Final
t-- ...
0
( X(II-I)2+ I~I Vll2)- (X(II-l)l + I~I V,d ) (5)
- where X(II-I)I, V"I , X(II-I)2 and VII2(j= -1,0) represent
process
1-- product
(f)
(f)
the position deviation vectors from the previous station and
from the current "place" and "clamp" operations for node
BVI and Bv2, respectively.
Processes at Station n
Variation
trom parts
3.3 Error
vector
p(_I)§(-I)
_. "
n
--
-
V"(-I)
(3)
where p(-I)
is the homogeneous transformation matrix that is
-"
related to part rotations and translations due to alignment of
-(-I)
part to (3-2-1) fixture; and 8"
is the vector of gaps between
part and the (3-2-1) fixture locators after the part is released
from a previous station.
3.2 Error vector after "clamp" operation
. .
- (0)
..
Th e part vanatlOn vector V" cause d b y part d e fiormatlon m
the clamp operation can be obtained through mechanic
analysis, as represented by equation (4):
k1,2
where $(0) is the influence coefficient matrix presented by
"
-(0)
Long and Hu (1998); and 8" is the vector of gaps between
part and the extra fixture clamps.
The part variation vector after the place and clamp
operations is:
o
'"
~
1=-1
= 0.1
(media-filled).
Both
joining
machine
tips stop
when they touch the mating surfaces and fill the gap with
media (e.g. weld nugget). No part deformation occurs in
the joining process. That is, joint error comes from
variation accumulation at previous operations, as shown in
Figure 12(a), where the vector from B~I to B~2 represents
joint error.
k1,2= 0.2, 0.3, 0.4, 0.5, 0.6 (single-side-controlled). Only
one side of the joining machine tip moves for spotWeld.
The joint error comes from variation accumulation at both
of the parts, part deformation, and the location variation
of the stationary tip. kl,2 = 0.5 is shown in Figure 12(b).
The dotted lines and solid lines represent joints before and
after a joining process, respectively. Joint error, i.e.
position difference from B~l to Byl, can be calculated
(4)
X,,_I +
after "fast~nJjoin" operation
Referring to Figure 5, let BYI and BY2 denote the new
positions ofBv1 and BY2after assembly at a station. Here, the
misalignment between the mating surfaces after the assembly,
i.e. the position difference from B~I to B~2' is called a joint
error. A joint error is generated due to the interactions of joint
geometry, joining kinematics,
part stiffness, position
difference between BVI and Bv2, and dimensional variation
source from joining devices. These interactions are taken into
consideration for different cases of joining kinematics, as
shown in Figure 11.
As shown in Figure 11, kl,2 = 1 indicates that BVI partially
locates BV2,kl,2 = - 1 indicates that BV2partially locates BVI'
In both cases, usually there is no part deformation during the
joining process. The location deviation of the located part can
be achieved through a homogeneous transformation matrix
stated in equation (3).
kl,2 = 0 indicates the scenarios where both parts are located
separately by their own fixtures, which are briefed as follows:
(f)
VII .
The gap §~I) generated between two joint nodes, BVI and
Bv2, is shown in equation (5) and Figure 10:
TableIII Notationsandalgorithmsfor partdimensionalvariationvectorsduringPCFRcycles
Assembly at station n
Physics phenomenon
Algorithms
Place
Partsare reoriented
Clamp
Partsare deformed
Part input from station n
-
1
Fasten/join
Partsare deformedwhich
Homogeneous
matrix
transformation
Mechanic
analysis
Mechanic
analysis
Release
causejoint mating surfacesslip
Partdeformationis recovered
Mechanicanalysis
Part dimensional
variation vector"
X"-1
-(-1)
V"
partially or totally due to the
releaseof fixture holding forces
Subassembly
Xn= Xn-1 + Lf-:,+~1 V;,~
output to station n + 1
Notes: "Forsimplicity,the" dimensionalvariation" is written as "variation" in the restof the text. bw = 1, 2, . .., W",where WIIis the total numberof joints to
be finishedat station n
159
....
"
Assembly
Variation propagation modeling and analysis
H>lume29
Haixia ~ng and Dariusz Ceglarek
Automation
. Number
2
. 2009 . J 54-
J 66
Figure 9 Illustrationof variationpropagationin a multi-stationassemblyprocess
y
Part error input
from previous
stations
IStatr I I
Joining
machine
variation
y
Part variation output to
station n+I
,
x
3.4 Errorvector after "release" operation
Releasing the deformed subassembly from fixtures causes it to
spring-back. Here, two types of spring-back are identified:
1 partial spring-back with some lock-in stress left in the
subassembly, as shown in Figure 13; and
2 complete spring-back without any residual deformation
left in the subassembly, as shown in Figure 14.
by using FEA with the input of joint geometry, part
stiffness, and position difference between BV1 and Bv2.
kJ,2 = 0.7 (self-equalized force control). As shown in
Figure 12(c), equal-value forces with opposite directions
are applied to enable the two mating surfaces to come into
contact with each other. Part deformation
due to
fastening/joining can be obtained using the mechanic
equilibrium method (Merkley, 1998) from two mechanic
analyses. Joint error comes from variation accumulation at
previous operations and part deformation during the
joining.
..
.
l:
. V (w). th
th f:aste n/ Jom operation
Th e part d e.ormation
" mew
can be obtained through mechanic analysis, as represented by
equation (6):
Variation analysis that considers the possible occurrence of
complete spring-back has not been addressed in prior
literature, since the symptoms of joint error remains to be
identified.
In the case of partial spring-back, beam node position
=-1"'1-1)
deviation vector, V~ , is obtained through mechanic
analysis, as represented by equation (7):
~
(6)
(7)
where p(w) is the influence coefficient matrix related to
where
G(w+l)
is the inverse of stiffness matrix of the
-"
subassembly constrained by m-2-1 fixture locators;. and
l:
.
;;(w+l) .
U"
IS the vector 0f reIease .orces whose Items have
opposite directions to the reaction forces on fTh.'tUreIocators.
The part variation vector after the release operation is:
subasse~bly stiffness matrix (Liu and Hu, 1997); and §~w)is
the vector of gaps generated at the wth joint after the (w - l)th
joint is closed. The part variation vector after the place,
clamp, and fasten/join operations is:
w+l
w
XII-l
+
I: vi/).
1=-1
X" = XIl-1
+ '"
L-,II VU)
1=-1
160
(8)
Variation
propagation
H aixia
modeling
Wilng and Dariusz
and analysis
Assembly
Ceglarek
H>lume 29
Figure10 Illustrationof part-to-partgapgenerationat thefirstjoint on
stationn
X
Positiondeviationof joint
nodeBV! on PSI
o
X
z
Positiondeviationof joint
nodeBV2on PS2
.
..
-
(I)
-
(I)
In the case 0f complete sprmg- b ack, Jomt error Vn2 - V nl
plays a dominant role in variation propagation. The part
variation vector after the release operation is as follows:
Xn = X("-I)
{
- = X(n-I)
- + V,'2
- - v,,!
(I)
X"
2 . 2009
. 154-166
The aforementioned comparisons demonstrate that:
the beam-based model can analyze the multi-station
assembly process with much more accuracy than the stateof-the-art "Rigid Bend" method; and that
the beam-based model can be meaningfully used to
highlight relations between parts that are not adjacent
but influence each other in dimensional
variation
propagation.
for part 1
(9)
(I)
Automation
A five-station assembly is considered in this case study, as
shown in Figure 16.
Input source variation for fixtures is set to 0.625 mm
(standard deviations), according to Theodolites measuring
resolutions. Part or subassembly variations are obtained from
upstream stations with the fabricated part deviation as
0.5 mm. Comparative
analysis results are shown in
Figure 17, with the abscissa labeled with selected beam
nodes in x, y, z directions and ordinate representing the
position deviations.
The small differences occur at the points and directions
which are much more rigid. Figure 18 shows the ten beam
nodes with the smallest differences, as listed in Table IV.
Among the ten nodes, seven are on the underbody (the most
rigid subassembly of the vehicle structure); one is on the
A-Pillar, near the right door upper hinge and therefore, close
to underbody; and the other two are on both the C-Pillars,
which fit well with current engineering experience.
As can be seen in Figure 19 the large differences occur at
places and directions which are more compliant. Figure 19
shows the ten beam nodes with the largest differences, as
listed in Table V.
The figures and tables illustrate that only by using the
beam-based model can true significance of dimensional
variation be detected:
nodes 46x and 60x, on right and left A-Pillar, identify the
root cause termed "door upper hinge mating surface
tolerance", a critical problem for automotive assembly;
nodes 71z and 61z, on roof bows, are related to process
operations, where torsional moments are applied and
modeled in the compliant body structure; and
a positive X-shifting on right C-pillar and right front
fender hydro-form tube is identified by points 63x, 15x,
16x and 40x, which again are related to process
operations, where torsional moments are applied and
modeled in the compliant body structure.
y
o
. Number
for part 2
4. Case study
To illustrate the validity of the proposed beam-based model,
comparative analysis between 3D CS simulations and the
presented beam-based model for a truck cab assembly is
conducted.
Variation propagation
is analyzed using
commercial software 3DCS, as its results are considered to
be an appropriate baseline to the variation analysis results
from the beam-based model. "Rigid Bend" is used as a
strategyto approximatepart compliance in 3D CS, during the
detailed design phase, by avoiding the computation based on
product mechanics. The presented beam-based model
approximates part compliance using beam elements, which
allows for both the inclusion of the information of product
deformation as well as making the product mechanics
computationally efficient.
The beam representation of a truck cabinet (as shown in
Figure 15(a)) is shown in Figure 15(b), depicting 87 beam
elements with 65 nodes.
5. Summary
This paper presented a dimensional variation analysis model
for multi-station compliant part assemblies at a preliminary
design phase. The developed model has the following
characteristics:
A beam-based
model is presented
based on the
assumption that only selected critical points/features in
the assembly are important for variation study. The model
has the capability to model product and process
characteristics for an incomplete product, as well as
process-related
information
that allows conducting
variation propagation analysis at a preliminary design
phase. The model shares the same data structure of the BRep CAD models and therefore can be embedded in CAD
systems for automatic product skeletal design.
161
--
Variation propagation modeling and analysis
Assembly
Wllume 29
Haixia W&ngand Dariusz Ceglarek
Automation
. Number
2
. 2009
. 154-166
Figure11 Schematic
illustrationof joiningkinematics
Graphical Illustration
kl.2
Joiningoperation
BV) BV2 Fixture
-I
PSI
kl,2=-1
PS2
Id
Part 2 is located only
by fixture
Part I is partially located
by the joint
Bv, BV2
Fixture
g operation
PS2
PS)
kl,2= 1
Part 2 is partially located
by the joint
Part I is located only by
fixture
1
J
-t
PSI
BV) BV2 Joining operation
f-
PS2
Parts I and 2 are fullylocatedby fixture
Position-
k),2=
controlled0.1
weldgun
k),2=
0.2
k),2=
0.3
k),2= 0
. .
BV) BV2Keep BI and B2 stationary and the gap is filled
(as illustrated in Figure 6(a»
-.
BV) BV2 Move BI at a controlled position and keep B2
stationary (as illustrated in Figure 6(b»
. ...
BVI BV2Keep BI stationary and move B2at a controlled
position
k),2=
0.4
Force-
-...
-.
BVI BV2Move B, and B2 at controlled positions
(as illustrated in Figure 6(c»
controlled
0.5=
k),2
BVI BV2Move BI with a controlled force and keep B2
stationary (as illustrated in Figure 6(b»
weldgun
k),2 =
0.6
kl,2=
0.7
. ...
BV) BV2Keep B B) stationary and move B! with a
controlled force
-...
BV! BV2 Move both joints with equalized forces
(as illustrated in Figure I I (c»
Figure12 Illustrationof part-ta-partjoint kinematics
Tip ofweldgun
Force~\
Bv!.-\
.'
....
......
Force~mJ,
BVI\
......'..'
.'..'
~~.........
~....
(a) media-filled joint
(b) single-sidecontrolled joint
(c) self-equalized force
controlled joint
The case study demonstrates that the developed beam-based
model, to be applied at the preliminary design phase, can detect
the influence of part deformation on product dimensional quality
with much more accuracy than the state-of-the-art "Rigid Bend"
method used in the detailed design phase. This demonstrates the
advantage of the proposed model over the state-of-the-art
approach used in industry for tolerance analysis of multi-station
sheet metal assembly.
The state-of-the-art
variation propagation model is
expanded by integrating the influences of joint error on
variation propagation and incorporating them into a vectorbased model. With the inclusion of joint error, part
deformation influence on variation propagation is extended
to include open structure assemblies wherein the assembled
structUre experiences a complete spring-back, i.e. no residual
stress is left within the structure after fixture is released.
162
Variation
propagation
Haixia
modeling
Wang and Dariusz
Assembly
and analysis
JiJlume 29 . Number
Geglarek
Automation
2 . 2009
Figure13 Illustrations
of partdeformationfor the partialspring-backcase
(b) After a PCFR
operation cycle
(a) Before a PCFR
operation cycle
Figure14 Illustrationsof partdeformationfor the completespring-backcase
(b) After a PCFR
operation cycle
(a) Before a PCFR
operation cycle
Figure15 A truckcabandits beamrepresentation
(b) its beam representation
(a) A truck cabinet
Figure16 A five-stationassemblyprocessandits beam-based
subassemblies
Step I
Step 2
Step 3
Step 4
Step 5
Right Door
Frame
Clamping
Left Door
Frame
Clamping
Bow
From
Roof
Clamping
Central
Roof Bow
Clamping
Rear Roof
Bow
Clamping
.~....
163
I
. 154-166
Variation propagation
modeling
Assembly
and analysis
U,lume 29
Haixia W&ngand Dari.lSz Ceglarek
. Number
Automation
2 .2009
. 154-166
Figure 17 Comparativeanalysisbetween 3DCSsimulations and beam-basedmodel results
2.2
2.0
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
37y54y
8z 51z37z42x48x
14y44y73y27y21y
1-
lly24y
3DCS
-
Figure 18 KPCswith the smallestdifferenceof standarddeviations
between3DCSsimulationsandbeam-based
model
9z 2y 38z 16y 12z71y61y20x
Beam-based
8x 14x 19y40x 61z
I
Figure 19 KPCswith the biggestdifferenceof standarddeviations
between3DCSsimulationsandbeam-based
model
.48
Table IV Thesmallest difference of standard deviations between the
two models
Beam
node
37y
32y
5y
58z
64z
54y
50l
44z
60y
52y
From 3DCS
Software
From
beam-based
Table V Thebiggest difference of standard deviations between the two
models
Difference
(mm)
model (mm)
(Beam-3DCS)
(mm)
0.2405
0.2473
0.2400
0.2109
0.1949
0.1182
0.1543
0.0843
0.0818
0.1182
0.2407
0.2477
0.2406
0.2120
0.1961
0.1222
0.1590
0.0892
0.0868
0.1310
0.0002
0.0003
0.0006
0.0011
0.0012
0.0040
0.0047
0.0049
0.0049
0.0129
Beam
node
46x
71z
61z
60x
64x
63x
15x
16x
40x
164
From 3DCS
Software
From
beam-based
(mm)
model (mm)
(Beam-3DCS)
(mm)
Difference
0.1246
0.1945
0.3318
0.1243
0.1371
0.1519
0.3392
0.2209
0.3464
2.1084
1.8399
1.7841
1.2068
1.2043
1.1915
1.3657
1.2442
1.3697
1.9838
1.6454
1.4523
1.0825
1.0672
1.0396
1.0265
1.0233
1.0233
Variation propagation modeling and analysis
Assembly
Automation
Tfllume 29 . Number 2 . 2009 . 154-166
Haixia Wizngand Dariusz Ceglarek
Notes
Symposium on Flexible Automation, Ann Arbor, MI, USA,
July 23-26, 2000JUSFA-13146.
EI-Haik, B.S. (2005), Axiomatic Quality: Integrating Axiomatic
Design with Six-Sigrna, Reliability, and Quality Engineering,
Wiley, New York, NY.
Gao, J., Chase, KW and Magleby, S.P. (1998), "General 3-D
tolerance analysis of mechanical assemblies with small
kinematic
adjustments",
lIE Transactions, Vol. 30,
pp. 367-77.
Hsieh, C. and Oh, KP. (1997), "A framework for modeling
variation in vehicle assembly processes", International
Journal of vehicle Design, Vol. 18 No. 5, pp. 466-73.
Hu, S.J. (1997), "Stream-of-variation theory for automotive
body assembly", Annals of the CIRP, Vol. 46 No. 1, pp. 1-6.
Huang, W., Lin, J., Kong, Z. and Ceglarek, D. (2007),
"Stream-of-variation (SOVA) modeling 11: a generic 3D
variation model for rigid body assembly in multi station
assembly processes", ASME 7ransactions on Journal of
Manufacturing Science and Engineering, Vol. 129 No. 4,
pp. 832-42.
Jin, J. and Shi, J. (1999), "State transition modeling of sheet
metal assembly
for dimensional
control",
ASME
Manufacturing Science and Engineering, Vol. 121, pp. 756-62.
Lawless, J.F., Mackay, RJ. and Robinson, J.A. (1999),
"Analysis of variation transformation in manufacturing
processes - Part 1", Journal of Quality Technology,Vol. 31
No. 2, pp. 131-42.
Lee, nJ. and Thornton, A.C. (1996), "Enhanced key
characteristics
identification
methodology
for agile
design", Proceedings of the Agile ManufactUling Forum,
Boston, J\.1A, USA.
Lee, W.J. and Woo, 1:C. (1990), "Tolerances: their analysis
and synthesis", ASME Journal of Engineering for Industry,
Vol. 112, pp. 113-21.
Liu, S.C. and Hu, S.J. (1997), "Variation simulation for
deformable sheet metal assemblies using finite element
methods", ASME Journal of Manufacturing Science and
Engineering, Vol. 119, pp. 368-74.
Liu, S.C., Hu, S.]. and Woo, T.C. (1996), "Tolerance analysis
for sheet metal assemblies", ASME Journal of Mechanical
Design, Vol. 118, pp. 62-7.
Liu, S.C., Lee, H.W and Hu, S.J. (1995), "Variation
simulation for deformable sheet metal assemblies using
mechanistic models", 7ransactions of NAMRI/SME, Vol. 23
No. 5, pp. 235-41.
Long, Y. and Hu, S.]. (1998), "A unified model for variation
simulation of sheet metal assemblies", in EIMaraghy, H.A.
(Ed.), Geometric Design Tolerancing: Theoties, Standards and
Applications, Chapman & Hall, Boca Raton, PL.
Mantripragada, R and Whitney, D.E. (1999), "Modeling and
controlling variation propagation in mechanical assemblies
using state transition models", IEEE Transactions011Robotics
and Automations, Vol. 15, pp. 124-40.
Menassa, R and DeVries, W. (1989), "Locating point
synthesis in fixture design", CIRP Annal, Vol. 38, pp. 165-9.
Merkley, KG. (1998), "Tolerance analysis of compliant
assemblies", PhD thesis, Brigham Young University, Provo,
UT.
Narahari, Y., Sudarsan, R, Lyons, KW, Duffey, M.R. and
Sriram, RD. (1999), "Design for tolerance of electromechanical assemblies: an integrated approach", IEEETransactions on Robotics and Automation, Vol. 15 No. 6,
pp. 1062-79.
According to Shiu (1996), 37 per cent of all assembly
stations in automotive body structure manufacturing
assemble non-rigid (compliant) parts.
2 KCs are the product, subassembly, part and process
features that significantly
impact the final cost,
performance, or safety of a product when the KCs vary
from nominal (Lee and Thornton, 1996, p. 146).
References
3D CS Manual (2002), 3DCS Software, Minitab and Matlab,
available at: www.3dcs.com
Cai, W., Hu, S.J. and Yuan, J.X. (1996), "Deformable sheet
metal fixturing: principles, algorithms, and simulations",
ASME Journal of Manufacturing Science and Engineering,
Vol. 118, pp. 318-24.
Camelio, J.A., Hu, S.]. and Ceglarek, D. (2003), "Modeling
variation propagation of multi-station assembly systems
with compliant parts", Transactions of ASME, Journal of
Mechanical Design, Vol. 125 No. 4, pp. 673-81.
Ceglarek, D. and Shi, J. (1995), "Dimensional variation
reduction for automotive body assembly manufacturing",
Manufacturing Review, Vol. 8 No. 2, pp. 139-54.
Ceglarek, D. and Shi,]. (1997), "Tolerance analysis for sheet
metal assembly using a beam-based model", 1997 ASME
International Mechanical Engineering Congress and Exposition,
Vol. DE-94, pp. 153-9.
Ceglarek, D., Shi, ]. and Wu, S.M. (1994), "A knowledgebased diagnosis approach for the launch of the auto-body
assembly process", Transactions of ASME, Journal of
Engineering for Industry, Vol. 116 No. 4, pp. 491-9.
Ceglarek, D., Huang, W, Zhou, S., Ding, Y., Kumar, Rand
Zhou,
Y. (2004),
"Time-based
competition
in
manufacturing:
stream-of-variation
analysis (SOVA)
methodology - review", International Journal of Flexible
Manufacturing Systems, Vol. 16 No. 1, pp. 11-44.
Chang, M. and Gossard, D.C. (1997), "Modeling the
assembly of compliant, non-ideal parts", Computer Aided
Design, Vol. 29, pp. 701-8.
Chase, KW. and Parkinson, A.R (1991), "A survey of
research in the application of tolerance analysis to the
design of mechanical assemblies", Research in Engineers
Design, Vol. 3, pp. 23-37.
Chase, KW., Gao, J. and Magleby, S.P. (1997), "Tolerance
analysis of 2-D and 3-D mechanical assemblies with small
kinematic adjustments", Advanced Tolerancing Techniques,
Wiley, New York, NY.
Chon, c.T. (1986), "Generic 'stick' model of a vehicular
structure", SAE Paper No. 860825.
Cozzens, R (2007), Advanced CATIA VS WOrkbook,
Knowledgeware and WOrkBenches (Release 16), Southern
Utah University, Cedar City, UT.
Cunningham, T.W., Mantripragada, R, Lee, D., Thornton,
A. and Whitney, D.E. (1996), "Definition, analysis, and
planning of a flexible assembly process", Proceedings of the
Japan/USA Symposium on Flexible Automation, Boston, J\.1A,
USA, pp. 767-78.
Ding, Y., Ceglarek, D. and Shi, J. (2000), "Modeling and
diagnosis of multistage manufacturing processes: part I state transition model", Proceedingsof the 2000 Japan/USA
165
Variation
propagation
mode ling and analysis
Assembly
Automation
Haixia wang and DaritlSZ Ceglarek
H>lume 29 . Number 2 . 2009 . 154-166
Odi, R, Burley, G.J., Naing, S., Williamson, A. and Corbett, J.
(2000), "The role of error budgeting in the use of structurally
integrated features for aerospace assembly", SAE Paper
No. 2000-01-3025.
Rong, Q., Ceglarek, D. and Shi, J. (2000), "Dimensional fault
diagnosis for compliant beam structure "assemblies",
Transactions of ASME, Journal of Manufacturing Science
and Engineering, Vol. 122 No. 4, pp. 773-80.
Shalon, D., Gossard, D., Ulrich, K and Firzpatrick, D.
(1992), "Representing geometric variations in complex
structural assemblies on CAD systems", Proceedings of the
19th Annual ASME Advances in Design Automation
Conference, Vol. DE-44 No. 2, pp. 121-32.
Shi, J. (2006), Stream of Variation Modeling and Analysis for
Multistage Manufacturing Processes,CRC Press, Boca Raton,
FL, p. 469.
Shiu, B.W. (1996), "Modeling of an automotive body
assembly system for dimensional control", PhD thesis,
The University of Michigan, Ann Arbor, MI.
Shiu, B.W:, Ceglarek, D. and Shi, J. (1997), "Flexible beambased modeling of sheet metal assembly for dimensional
control", Transactions of NAMRI, Vol. XXV, pp. 49-54.
Shiu, B.W., Apley, D., Ceglarek, D. and Shi, J. (2003),
"Tolerance allocation for sheet metal assembly using beambased
model",
Transactions of lIE,
Design and
Manufacturing, Vol. 35 No. 4, pp. 329-42.
Soderberg, R, Wickman, C. and Lindkvist, L. (2008),
"Improving decision making by simulating and visualizing
geometrical variation in non-rigid assemblies", CIRP Annals
- Manufacturing Technology,Vol. 57 No. 1, pp. 175-8.
Thornton, A.C. (1999), "A mathematical framework for the
key characteristic process", Research in Engineers Design,
Vol. 11, pp. 145-6.
Ullman, D.G. (1992), The Mechanical Design Process,
McGraw-Hill, New York, NY.
VSA Manual (1998), VSA-3D Release 12.5 User Manual,
Variation System Analysis, Inc., St. Clair Shores, MI.
Wang, H. and Ceglarek, D. (2009), "Variation propagation
modeling and analysis at preliminary design phase of multistation assembly systems", Assembly Automation, Vol. 29
No. 2 (in press).
Whitney, D. (2008), "Proper constraint as a design principle
for assemblies", Assembly Automation, Vol. 28 No. 2,
pp. 120-5.
Wood, KL., Otto, KN. and Antonsson, E.K (1992),
"Engineering design calculations with fuzzy parameters",
Fuzzy Sets and Systems, Vol. 52 No. I, pp. 1-20.
Further reading
Barkmeyer, E.J. (Bd.) (1996), "SIMA reference architecture
part 1: activity models", The National Institute of
Standards and Technology Internal Report No. 5939,
National Bureau of Standards, Gaithersburg, MD, available
at: www.nist.gov/msidlibrary/doc/simarchl/manact.ps
Corresponding author
Dariusz Ceglarek
warwick.ac.uk
\
r
To purchase reprints of this article please e-mail: reprints@emeraldinsight.com
Or visit our web site for further details: www.emeraldinsight.comJreprints
166
can be contacted at: d.j.ceglarek@