Jaime Camelio
buz
S.Jack Hu Department of Mechanical Engineering,
The University of Michigan,
Ann Arbor,MI48109
e-mail:jackhu@umich.edu
Dariusz Ceglarek Department of Industrial Engineering,
The University of Wisconsin,
Madison,WI53706
e-mail:darek@engr.wisc.edu Modeling Variation Propagation of Multi-Station Asmbly Systems With Compliant Parts Products made of compliant sheet metals are widely ud in automotive,aerospace,ap-pliance and electronics industries.One of the most important challenges for the asmbly process with compliant parts is dimensional quality,which affects product functionali
ty and performance.This paper develops a methodology to evaluate the dimensional varia-tion propagation in a multi-station compliant asmbly system bad on linear mechanics and a state space reprentation.Three sources of variation:part variation,fixture varia-tion and welding gun variation are analyzed.The propod method is illustrated through a ca study on an automotive body asmbly process.͓DOI:10.1115/1.1631574
͔
1Introduction
Compliant sheet metal asmbly is a manufacturing process where two or more sheet metal parts are joined together using various joining techniques.The result of this process is a subas-mbly or afinal product.One of the most important challenges for sheet metal asmbly is the understanding of how dimensional variation propagates͓1,2͔.Due to the variability of the parts,fix-tures and joining methods in each station and their interactions,a sheet metal asmbly process can be considered as a variation ‘‘stack up’’process.
Dimensional variation can stem from both the design and manufacture of a product.Since some manufacturing induced variation is inevitable,it is important to minimize the level of inherent dimensional variation caud by product and process de-sign.Many of the problems associated with dimensional accuracy occur becau the capability of the manufacturing process is not considered when designing the product and process.The prob-lems may affect thefinal product functionality and process perfor-mance.For example,large product dimensional variation in an automotive body asmbly process may cau product problems such as water leakage and wind noi,as well as process difficul-ties such asfitting problems in subquent operations.Clearly, reducing dimensional variation in an asmbly process is of criti-cal importance to improve thefinal product quality.In addition,a better understanding of a process behavior can also bring a reduc-tion in the time needed
to launch a new manufacturing system. Early and accurate evaluations of inherent process variation are crucial factors in determining thefinal dimensional variation of an asmbled product.
In recent years,the importance of dimensional variation has been obrved by an increasing amount of rearch conducted in the area of sheet metal asmbly process.Since Takezawa͓3͔obrvation that for compliant sheet metal asmblies the tradi-tional additive theorem of variance is no longer valid,veral models have been propod to reprent the variation propagation on asmbly process.The models developed can be grouped into four different categories,depending on whether the model is for a single station or a multi-station process,or if the model considers rigid or compliant parts.Station level models treat the asmbly process as if it is conducted in one step.In contrast, multi-station models analyze the process recursively as the asm-bly is moved from one station to the next.Rigid part models do not consider part deformation during asmbly so that the part and tooling variation can be solely reprented by kinematic relation-ships.Compliant part models consider the possible deformation of the parts during the asmbly process.The models include a force analysis that take into consideration the stiffness of each part and the forces applied by each tool.
Dimensional variation modeling and analysis for multi-station manufacturing process has been de
veloped mainly for rigid parts.Multi-station asmbly process with rigid parts cover a large number of currently ud process such as power-train as-mbly and general asmbly in automotive industry.However,a large group of multi-station asmbly process consider non-rigid parts.For example,37%of all asmbly stations in automotive body structure manufacturing asmble nonrigid parts͓4͔.Varia-tion propagation analysis for a multi station asmbly process in-troduces new modeling challenges.In comparison to the station level approach,it is necessary to define an appropriate variation reprentation in order to track the variation propagation from station to station.The variation simulation process is quential, i.e.,to estimate the variation at station i,it is necessary to know the variation at station iϪ1.Moreover,there is a station-to-station interaction introduced by the relea of holdingfixtures and the u of newfixtures in subquent stations.Finally,compliant as-mbly variation analysis requires applyingfinite element meth-ods to calculate the deformation after asmbly.Therefore,the number of calculations increas with the number of stations. Recent publications in each of the areas are summarized in Table1.As can be en,most of the dimensional variation analy-sis has been conducted for single station or multi-station rigid part asmbly and some work exists at station level for compliant as-mbly.
At station level,Liu et al.͓5͔and Liu and Hu͓6͔propod a model to analyze the effect of deformation and
springback on asmbly variation by applying linear mechanics and statistics. Usingfinite element methods͑FEM͒,they constructed a nsitiv-ity matrix for compliant parts of complex shapes.The nsitivity matrix establishes the linear relationship between the incoming part deviation and the output asmbly deviation.Long and Hu͓7͔extended this model to a unified model for variation simulation by considering part variation and locatingfixture variation.Shiu et al.͓4͔prented a simplifiedflexible beam reprentation of body structures.Huang and Ceglarek͓8͔prented a discrete-cosine-transformation͑DCT͒bad decomposition method for modeling and control of compliant asmblies form error.The method decompos the dimensional errorfield into a ries of independent error modes.
At multi-station level,Lawless et al.͓9͔propod a method called Variation Driver Analysis using time ries analysis.The method is bad in tracking the characteristics of individual parts as they pass through multiple stations using autoregressive mod-
Contributed by the Design Theory and Methodology Committee for publication in the J OURNAL OF M ECHANICAL D ESIGN.Manuscript received July2001;revid April2003.Associate Editor:D.C.Thurston.
friends第一季
els.Mantripragada and Whitney͓10͔propod a variation propa-gation model using state transition models.The propod model considers rigid parts and the state space vector can be fully de-scribed by a translation and re-orientation.The state transition model allows the application of control systems theory to the analysis of multi-station asmbly system.Jin and Shi͓11͔pro-pod a state space modeling approach for dimensional control for in plane motion of rigid parts in a sheet metal asmbly process, where the state equation considers two types of dimensional varia-tion,the part error itlf and thefixture error.Ding et al.͓12,13͔developed a complete state space model for variation in the plane of rigidity for rigid components.
Comparatively,little rearch has been done in multi-station systems considering compliant,non-rigid parts.Liu and Hu͓14͔developed a model to evaluate the spot weld quence in sheet metal asmbly.This model considered a process where welding was carried out in multiple stages.Chang and Gossard͓15͔pre-nted a graphic approach for multi-station asmbly of compliant parts.However,there are no analytical models for variation analy-sis in multi-station compliant asmbly systems.
pascal
It is critical to develop realistic models for sheet metal asm-bly process that consider compliant parts and also include the station-to-station interaction in multi-station asmbly systems. Such mod
els can be quite uful during both the design and launch stage of the manufacturing system.During design,such models can be ud to predict product variation so that changes in parts or process can be made early.During the launch of the manufacturing system,such models can aid in the diagnosis of root caus of variation͓16,17͔.The purpo of this paper is to prent a methodology for modeling the impact of part and tooling variation on the dimensional quality on a multi-station asmbly system with compliant sheet metal parts and study how variation propagates from different subasmblies to thefinal product. The remainder of this paper is organized as follows.Section2 prents the model for multi-station variation analysis using a state space reprentation.The model is developed for three dif-ferent sources of variation:part variation,fixture variation and welding gun variation.In Sec.3,a simplified example for auto-motive sheet metal asmbly variation analysis is prented.Fi-nally,Sec.4draws conclusions.
2Multi-Station Asmbly Model
The automotive body asmbly process will be ud to develop the methodology for modeling and analyzing dimensional varia-tion propagation in multi-station systems.However,the developed model can be generalized into other multi-station asmbly process with compliant parts such as appliances or furniture manufacturing.
An automotive body asmbly process is a multi-leveled hier-archical process,in which sheet metal parts are joined together to form a subasmbly͓2,18͔.During the asmbly process,each part or subasmbly becomes an input for the subquent stations. While parts move from one station to another,dimensional varia-tion of the parts and subasmblies propagates through the system. The station-to-station interactions cau an increa or sometimes decrea of the dimensional variation.
An asmbly process can be considered a discrete-time dynami-cal system,where the independent variable time can reprent the station location.Then,a state space reprentation can be devel-oped to illustrate the part deviation͓10–12͔.In station k,part deviation after asmbly operations is function of the input parts deviation and tooling deviation,as shown in Fig.1.Then,part deviation can be calculated by the state equation,Eq.͑1͒,if func-tion f is known.In addition,extra measurement point deviation can be calculated using the obrvation equation,Eq.͑2͒,for a given function g.The objective of the model is to define the appropriate functions f and g.
X͑k͒ϭf k͑X͑kϪ1͒,U͑k͒͒(1)
Y͑k͒ϭg k͑X͑k͒͒(2) where X(k)reprents the part deviation for every part in the asmbly at station k,U(k)the tooling deviation at station k and Y(k)the key characteristic points deviation at station k.
The developed methodology will be prented in two steps. First,a short description of a single station model will be re-viewed and adapted to our methodology.Second,the multi-station model will be developed bad on the single station model and station-to-station variation propagation model.
2.1Single Station Asmbly Modeling.Traditionally, modeling of dimensional variation propagation for a single station assumes that all the process operations occur , quence of operations and interactions between operations are not taken into considerations.In this paper,the variation modeling approach at the station level is bad on the mechanistic simula-tion method developed by Liu and Hu͓6͔.This procedure as-sumes that:sheet deformation is in the linear elastic range;the material is isotropic;fixture and welding gun are rigid;there is no thermal deformation and stiffness matrix remains constant for non-nominal part shapes.Reprentation of the asmbly process with compliant parts is illustrated in Fig.2and can be described in the following steps:
1.Part loading and locating operation͑Fig.2a͒
2.Part holding operation͑Fig.2b͒
3.Part joining operation,such as spot welding͑Fig.2c͒
Table1Recent publications in dimensional variation
Rigid Parts Compliant Parts
Single Station
Level
Lee and Woo͓21͔
Cha and Parkinson͓22͔
Etc.
Liu et al.͓5͔
Cai et al.͓20͔
Liu and Hu͓6͔
Shiu et al.͓4͔
Long and Hu͓7͔
Huang and
Ceglarek͓8͔
Multi-Station
eleLevel
Shiu et al.͓23͔
Mantripragada
and Whitney͓10͔
Jin and Shi͓11͔
Ding et al.͓12͔
To be developed
in this
Paper
Fig.1Deviation propagation in a multi-station
system
Fig.2Sheet metal asmbly process
4.Part unloading,the clamp is relead and the subasmbly
springback͑Fig.2d͒.
The modeling of the four steps is prented as follows:
Step1The parts are loaded and located in the station using a
locating scheme͑Fig.2a͒.After locating part1,it has a deviation
V u from the nominal part shape.If more than one source of de-
viation is prent,a vector͕V u͖will reprent the deviation.Index u refers to unwelded parts.
Step2Part1deviation(V u)is clod by a welding gun or a
fixture applying a force F u͑Fig.2b͒.If there is more than one
洛莉塔
source of variation,͕F u͖will be a vector.Then considering a part stiffness matrix K u the force required to clo the gap V u will be given by Eq.͑3͒.
͕F u͖ϭ͓K u͔•͕V u͖(3) Step3The parts are joined together while the force F u is still being applied͑Fig.2c͒.
Step4The welding gun/fixture is removed͑Fig.2d͒.After removing the forces applied by the clamping system,the new asmbled structure will springback.The springback position is determined assuming that a force(F w)of the same magnitude of the clamping forces(F u)but in opposite direction is applied over the nominal welded structure.Knowing the asmbly stiffness ma-trix(K w),the value of the springback variation(V w)can be de-termined using Eq.͑4͒–͑7͒,
͕F w͖ϭ͓K w͔•͕V w͖(4)
͕F u͖ϭ͕F w͖(5)
͕V w͖ϭ͓K w͔Ϫ1•͓K u͔•͕V u͖(6)
͕V w͖ϭ͓S uw͔•͕V u͖(7) where,͓S uw͔is the nsitivity matrix,which reprents how n-sitive is the output asmbly deviation to an input part deviation, where index u reprents the input source of variation and w the output measurement points.As a result,the springback of the asmbly can be reprente
d by the mechanistic variation model as,
V wϭS"V u(8) The nsitivity matrix S can be determined using the method of influence coefficients as prented in Liu and Hu͓6͔.Then,con-sidering a linear relationship between the incoming parts deviation and thefinal asmbly deviation for compliant parts at the station level and usingfinite element analysis,it is possible to construct the nsitivity matrix for a specific station configuration.Finite element methods are ud to obtain the stiffness matrices for parts of complex shape.
2.2Multi-Station Asmbly Modeling.Considering the variation propagation process as a linear time varying discrete time system,where the variable time reprents station location,a state space reprentation can be ud to model the multi-station asmbly process.Therefore,the dimensional deviation of the as-mbly parts can be reprented by the following equations: X͑k͒ϭA͑k͒•X͑kϪ1͒ϩB͑k͒•U͑k͒ϩW͑k͒(9)
Y͑k͒ϭC͑k͒•X͑k͒ϩW͑k͒(10) where,X(k)is the state vector,A(k)is the state matrix,B(k)is the input matrix,U(k)is the input vector,C(k)is the obrvation matrix and W(k)is noi.The following ctions will develop the expressions for X(k),A(k),B(k)and U(k).
2.2.1State Vector,X(k).The discrete system state vector is a t of variables that allow reprentation of the system behavior. The state vector will include the reprentation of all the parts/ subasmblies at each station in the system.Compliant sheet metal parts require more than two points to reprent a state com-pared with the just two points required for rigid body reprenta-tion.In fact,to completely describe the real part shape it will be necessary to know an infinite number of points,similar to a mesh in FE analysis.
There is a balance between the accuracy of the model,the time necessary to conduct the simulation and the size of the matrices. The number of points lected to reprent a part will depend of the complexity of the parts and the accuracy necessary.Therefore, a limited number of points are ud to analyze how variation propagates through the asmbly line.The relevant points required to reprent a compliant part state are:the part deviation on the welding positions or welding locating points͑WLP͒,thefixture points or principal locating points͑PLP͒and any additional mea-surement point or measurement locating points͑MLP͒.
Due to the u offinite element methods͑FEM͒,a mesh is generated defining the particular points as nodes among the complete mesh.Then,for compliant part j,at station k,we will have a state vector reprented as:
X j͑k͒ϭ͓X WLP
1¯
X WLP i X PLP1¯
X PLP
j
X MLP
1¯
X MLP l͔T(11) Thus,for an asmbly of n parts,the state vector at station k,will be,
X͑k͒ϭͫX1͑k͒X2͑k͒ӇX n͑k͒
ͬ(12)
2.2.2State Transition Matrix,A(k),With No Tooling Variation. Using a modified mechanistic variation si
mulation method,it is possible to define the relation among the input parts deviation and the output subasmbly deviation.This relation is the result of three concutive operations͑Fig.3͒.First,incoming parts are re-located in the station using a3-2-1fixture layout.Second,the part is deformed when the welding guns and additional clamps on the primary plane are clod and the parts are welded to produce a subasmbly.Finally,the welding guns andfixture are relead causing sprinback.
Each of the operations can be reprented in a matrix form. The re-location/re-orientation effect is defined by the matrix M. The part deformation before welding is reprented by matrix P. The springback for unit deviations can be obtained from matrix S. The following ction will define matrices M,P,S and their re-lation with the state transition matrix A͑Eq.͑13͒͒
A͑k͒ϭf͑M͑K͒,P͑k͒,S͑k͒͒(13) Re-location matrix:As prented previously,the state space vector for station k reprents the components/subasmblies de-viation after station k in a global coordinate system.At each sta-tion the components/subasmblies are located considering a 3-2-1locating scheme.Considering the example of a2D-beam in Fig.4,the3-2-1locating scheme will be equivalent to add2 locators in z-direction and1locator in the x-direction.Then,we can study the relocation effect in the xz plane.In Fig.4,part j is located with locators P1and P2at station kϪ1.After the asm-bly process at station kϪ1isfinished,thefixture P1and P2are relead and the part is moved to the next st
ation͑station k).At station k a new locating scheme is utilized,the
沈阳新东方
components/ Fig.3Re-location and asmbly process
sculpturesubasmblies will be re-located using a new t of locators,Q1 and Q2.The state vector will be modified accordingly. Considering the part deviation in the z-direction at station k Ϫ1in points Q1and Q2,the position of point A͑part j)at station
k will change due to the re-location process.The coordinates of point A can be obtained using homogeneous transformation.As-suming small deviations,the displacements in the x-direction can be neglected.Therefore,the linear relationship for the part devia-tion in the z-direction due to re-orientation will
be,
(14) where M A j(k)is the re-location matrix for point A in part j at station(k).
Extending the same concept for each point of the state vector and a3-2-1locating scheme in3D,the state space vector after relocating in station k will be,
XЈ͑kϪ1͒ϭX͑kϪ1͒ϩM͑k͒•X͑kϪ1͒(15) where,the relocating matrix M has the
gabrielle aplin
form,
(16) where,n is the number of elements in state vector and m i is the
number of key points in part/subasmbly i.The zeros in the
matrix M(k)correspond to the null impact of others subasmbly知足常乐 英文
locators over one determined part or subasmbly.
The effect of the additional clamps in the primary plane͑N-2-1
locating scheme͒is not considered in the locating process.Once
the3-2-1locating scheme is ud to completely locate the part in
a station,any additional clamps͑N-3͒will deform the parts.
Therefore,the analysis of the additional clamps will be included
in the matrices P and S.
Deformation and nsitivity matrix:Using method of influence
coefficients͓6͔,it is possible to obtain the nsitivity matrix for a
specific station k.It should be noticed that sources of variation at
station k have a null effect over parts that are not being asmbled
at that station.Therefore,the elements of matrix S for the parts
that do not participate in the asmbly in that station are equal to
zero.Then,the matrix S has the form,
S͑k͒ϭͫS Subasmbly1000
0S Subasmbly
2
00
00 0
四级英语作文范文000S Subasmbly
nͬ(17)
The method of influence coefficients͓6͔assumed a specific
shape of the subasmbly components for a given source of varia-
tion vector.Deformation matrix P reprents this assumed com-
ponents deformation.Considering a different incoming part shape
only the net effect of the asmbly process must be added to the
incoming part variation.To calculate the net effect,Matrix P must
be subtracted from the nsitivity matrix S.Therefore,the relation
between the output asmbly deviation and the incoming part de-
viation can be expresd
as:
Fig.4Locating process from station…kÀ1…to station…k…
V w ϭ͑S ϪP ͒•V u ϩV u
Using method of influence coefficients;it is possible to obtain the deformation matrix P for the asmbly components as,
P ͑k ͒ϭ
ͫ
P Part 1
0000P Part 2
0000 00
P Part n
ͬ
(18)
Finally,using the state space reprentation and the matrices M ,P and S ,the model can be expresd by,
X Ј͑k Ϫ1͒ϭX ͑k Ϫ1͒ϩM ͑k ͒•X ͑k Ϫ1͒(19)X ͑k ͒ϭ͑S ͑k ͒ϪP ͑k ͒͒•X Ј͑k Ϫ1͒ϩX Ј͑k Ϫ1͒ϩW ͑k ͒(20)
and the state transition matrix will be,
A ͑k ͒ϭ͑S ͑k ͒ϪP ͑k ͒ϩI ͒•͑I ϩM ͑k ͒͒
(21)
2.2.3State Transition Matrix,A(k),With Tooling Variation Part deviation is only one of the variation contributors in sheet metal asmbly process.Ceglarek and Shi ͓1͔established that a high percent of all root caus failures for autobody asmbly process are due to fixture related problems.Conquently,it is necessary to consider the effect of tooling deviation over the as-mbly variation.Tooling variation impact can be decompod into two independent sources of variation:welding gun variation and fixture variation,including locators and clamps.
Welding gun variation:Variation of welding guns has been shown to have a large impact on the final asmbly variation ͓14͔.The influence on asmbly variation will depend on the welding gun type.In general,three types of welding guns are ud in sheet metal asmbly,position controlled welding gun,equalized weld-ing gun and force controlled welding gun.In this paper,a position controlled weldi
ng gun variation model is prented.Without loss of generality the methodology can be applied to the other two welding gun types.
A position controlled welding gun is ud to weld two parts at the gun/electrode position.Position controlled welding gun model assumes that the welding gun can apply a sufficient force over the part to clo the gap between the part deviation and its electrode position.As shown in Fig.5,part 1has a deviation of v 1and part 2has a deviation of v 2.In addition,the welding gun has a devia-tion from the nominal v g .The force required to clo the gap in part 1and 2will be:
F 1ϭK 1͑v 1Ϫv g ͒
(22)F 2ϭK 2͑v 2Ϫv g ͒
(23)
where K 1and K 2is the stiffness of part 1and part 2respectively.The resulting force that will produce springback over the as-mbly of welded parts 1and 2with stiffness K a will be F ϭF 1ϩF 2.Therefore,the springback will be,
v a ϭ
F K a ϭK 1K a •v 1ϩK 2K a •v 2Ϫ͑K 1ϩK 2͒K a
•v g (24)
Similarly,using the nsitivity matrix definition,
v a ϭS •
ͫ
v 1Ϫv g v 2Ϫv g ͬϭS •ͫv 1v 2ͬϪS •
ͫv g
v g
ͬ
Finally,considering the state space reprentation,and defining
an input vector U g for the welding gun deviation,Eq.͑20͒can be rewritten as:
X ͑k ͒ϭ͑S ͑k ͒ϪP ͑k ͒ϩI ͒•X Ј͑k Ϫ1͒Ϫ͑S ͑k ͒ϪP ͑k ͒͒•U g ϩW ͑k ͒
(25)where,U g ,the welding gun deviation vector,has the
form,
Fixture variation:The ͑3-2-1͒fixturing principle ͓19͔is a lo-cating principle for a rigid body.However,locating and holding compliant sheet metal workpiece requires a (N -2-1)fixturing principle ͓20͔.The variation model prented considers a decom-position of the fixture variation vector into two ts of fixtures,the ͑3-2-1͒locating fixtures,U t 3Ϫ2Ϫ1and the (N -3)additional holding fixtures,U t (N Ϫ3).The locating fixture variation effect is considered as a rigid body translation and rotation,and can be obtained by
kinematics analysis.The locating fixture variation directly impacts the state space equation at the re-location process.Therefore,Eq.͑19͒will be:
X Ј͑k Ϫ1͒ϭX ͑k Ϫ1͒ϩM ͑k ͒• X ͑k Ϫ1͒ϪU t 3Ϫ2Ϫ1͑k ͒ (26)where U t 3Ϫ2Ϫ1corresponds to the input vector for the location fixture deviation and has the
form,
On the other hand,the additional (N -3)clamps can be ana-lyzed equivalently as extra position-controlled welding guns.However,the FEA model does not consider any links at tho nodes.Therefore,the clamps apply a force over the part.The force is relead after asmbly,and then,it produces a springback.The method of influence coefficients propod by Liu and Hu ͓6͔
can
Fig.5Position welding gun variation impact
