simulation and analysis of assembly processes considering compliant, non-ideal parts and tooling...

International Journal of Machine Tools & Manufacture 41 (2001) 2233–2243

Simulation and analysis of assembly processes consideringcompliant, non-ideal parts and tooling variations

Min Hu a,*, Zhongqin Lina, Xinmin Lai a, Jun Ni b

a Institute of Mechanics and Engineering, Shanghai Jiaotong University, Shanghai, PR Chinab Department of Mechanical Engineering, The University of Michigan, Michigan, USA

Received 2 December 2000; accepted 2 April 2001

Abstract

Variations in parts and tooling are a major problem in automobile body assembly processes. Thosediscrepancies adversely affect body assembly quality, functionality, cost and time-to-market. Variationsimulation analysis has been used in the design stage to predict such uncertainties, but variation analysisbased on rigid body assumptions usually yields over-estimates of the assembly variations. This paperpresents a new numerical simulation method for the assembly process incorporating compliant non-idealparts. This method considers the interaction and interference between compliant parts due to part variation,assembly tooling variation, welding distortion, and spring back effects. A butt-to-butt joint assemblyexample is used to illustrate the effects of various variation causes. A three-parts assembly example exam-ines the assembly and welding sequences. Results from these two examples demonstrate that the proposedmethod is theoretically sound and practically useful. 2001 Published by Elsevier Science Ltd.

Keywords: Variation; Spot welding; Sheet metal; Assembly

1. Introduction

With increased competition in the automobile market, more attention has been given to manag-ing variations in automobile body assembly processes. Dimensional variation affects fit qualityand functionality. For example, variations in a body-in-white (BIW) can ultimately cause poorsealing, undue effort required for door closing, water leaks, excessive wind noise, prolongedtime-to-market and added manufacturing costs. Typically, the automobile body assembly processcomprises numerous steps, utilizing 300–500 compliant sheet metal parts, 50–120 assembly sta-

* Corresponding author.E-mail address: [email protected] (M. Hu).

0890-6955/01/$ - see front matter 2001 Published by Elsevier Science Ltd.PII: S0890-6955(01)00044-X

2234 M. Hu et al. / International Journal of Machine Tools & Manufacture 41 (2001) 2233–2243

tions and 3000–6000 spot welds. Each step in the process is capable of contributing a degree ofvariation. Those variations in turn act on one another to compound distortion in the final BIW.The complexity of this interaction places severe demands on the existing methods of simulation,which currently fall far short of satisfaction.

Traditional variation analysis focuses on either the worst case, or the statistical analysis, or theMonte Carlo simulation. The worst case analysis is primarily used for one-dimensional assemblythat assumes all parts take on their extreme values, but in fact the chances that all componentswill take their extreme values simultaneously are remote. As a result, tight tolerances to meet as-designed specifications are the norm. That strict standard inevitably inflates manufacturing costsneedlessly [1].

Statistical analysis specifies variations in parts as statistical distributions. Calculating designfunction distribution is the critical task. If that is accurate, the method is superior to the worst-case approach in modeling the interchangeability of mass production and more realistic in estimat-ing assembly tolerances [2].

Monte Carlo simulation samples points from the distribution being evaluated. The method sel-ects points randomly and analyzes them to determine the design function values. Accuracy there-fore depends on the number of samples and can require an extensive series of calculations. VSAand VSM are two commonly used packages for predicting assembly variation [3].

All three of those methods are inadequate for automobile body assembly. They proceed fromthe assumption that parts are rigid and never deformed during assembly. Yet, almost all BIWparts are made from sheet metal, which is compliant and subject to deformation during placing,clamping, locating and welding. When current simulations ignore this fact, they invite large-scaleinaccuracies. From his 1980 statistical study of the automotive body panel, Takezawa [4] con-cluded that “ the conventional addition theorem of variance is no longer valid for determining thepermissible limits for the auto-body assembly” . The success of the “2 mm Program” in the Amer-ican automobile industry [5] gave further support to the principle that deformation of compliantparts plays an important role in automobile body final assembly variation.

In Liu et al. [6], a cantilever beam model and mechanical variation simulation were used topredict deformable-part assemblies by combining engineering structural models with statisticalanalysis. Their research revealed important concepts and fundamental properties central to theassembly of deformable parts. In the same year, Liu and Hu [7] proposed a concept — the “offsetbeam element” . This was a revision of the stiffness matrix that took into consideration forcebehaviors at spot welding points. Subsequently, Liu et al. applied these two methods to the studyof variation propagating factors with such parameters as joint styles, part thickness, part and toolvariations. Their models confine spot welding to joined parts at beam-ends.

Later, Chang and Gossard [8] constructed a CAD model of compliant parts to predict variationsin the final product, but excluded from their models were the consequences of deformation-induced interaction and the effects of force. Moreover, most of their models were created usinga self-developed code.

Proposed in this paper is a general numerical method of simulating assembly processes involv-ing compliant parts and dimensional variations. This method combines elastic and contact analysesby using a popular FEM code ANSYS [9].

This paper is organized as follows. Section 2 examines an assembly process in details andpresents a simulation method based on contact and search analysis. Section 3 applies the method

2235M. Hu et al. / International Journal of Machine Tools & Manufacture 41 (2001) 2233–2243

to a butt welding joining assembly process, obtains some results to better understand this methodand prove its suitability. Section 4 applies the proposed method to analyze a practical assemblyof compliant sheet metals and draws some conclusions.

2. Simulation method

2.1. Causes of variation

It is necessary when building a model to make clear root causes of variation in the auto bodyassembly process. Any factor which can affect the manufacturing quality of part and tooling,should be included. Figure 1 illustrates the causes in details.

2.2. Assembly and simulation cycles

To incorporate part compliance in the simulation of automobile body assembly processes, thestiffness matrix of each part and each step is required. In a typical automobile body assemblystation, parts or subassemblies are assembled in the following four steps:

1. Positioning — parts are placed on work-holding fixtures. Part variation is a natural phenomenonin the sheet metal manufacturing and assembly process. Part variation {d} occurs in stamping,transportation or prior subassemblies. Therefore, the initial matching gap between the parts isan inevitable problem;

2. Clamping — parts are clamped to the fixtures, which can usually be assumed to be rigid relativeto the compliance of the part. The initial gaps at the clamping positions will be overcome byclamping forces. Variations in clamps and locators can also cause parts to deform and interact.It can be described as {Fu}=[Ku]{du}, in which {du} is the closed gap vector at the clampingpositions, {Fu} is the clamping force vector and [Ku] is the stiffness matrix;

3. Welding — two parts are joined with point-to-point connection at each welding spot. Defor-mation occurring as the gap between the parts is closed. Welding guns often add further defor-

Fig. 1. Root causes of assembly variation.


mation to the parts by pulling or pushing them. Frequently, the actual welding spot strays fromthe designed spot;

4. Releasing — clamps are released and the assembled part springs back to minimize the strainenergy stored during the preceding operations. Subassembly contact status varies. Spring-backdeformation is calculated by removing displacement boundaries at clamping points to simulateclamp release. It can be calculated by the equation: {Fw}=[Kw]{dw}, where {dw} is the spring-back deformation vector, {Fw} is equal to the negative reaction forces at the clamping positionsand [Kw] is the stiffness matrix of the welding assembly structure.

Moreover, the compliance of sheet metal makes it sensitive to exterior loading (from clampingand welding) and interior loading (such as from residual stresses), both with the potential toproduce deformation.

Fig. 2. Flowchart of the NVAM simulation based on contact and search analysis.


2.3. Simulation method

The four steps described above constitute an elementary assembly cycle called PCWR. A newsimulation procedure for automobile body assembly variation analysis is shown in Fig. 2, whichis named NVAM (new variation analysis method). It combines contact and interaction analysisbetween parts — an approach more in line with the actual PCWR process and more readilyunderstood. Thus, the entire automobile body assembly process can be formulated as mPCnWR,in which m equals the total number of assembly cycles and n is the number of welding points inone PCWR. For simplicity, the process is assumed to be friction free and linear.

In addition to the part variations, clamping processes often introduce variation due to variationsin clamps and locators. Thus, displacement boundaries can be applied to simulate the variationsin clamping and locating points. The deformation due to clamping and weld squeezing can becalculated in the displacement-based finite element formulation: {F}=[K]{d}. The minimum forceis dependent on the material properties and the dimensions of the parts.

For welding assembly with an initial gap and variations, it is very important to understand thecontact and interaction between parts under clamping and squeezing forces. Under the effect offorces, gaps at the positions with forces working on are closed, geometries of parts change andcontact will occur between parts. The contact will have a large effect on deformation in the furtherassembly. There will be an interactive force between parts.

In order to evaluate assembly design and quality, a set of standards must first be established.Usually, there are some key characteristics on parts which could significantly affect the targetvalue of controlled variation, the performance of part function and customer satisfaction. There-fore, the CPs (critical points or surfaces) related to those key characteristics are defined as con-trolled parameters to ensure manufacturing and assembly. In fact, the determination of CP shouldtake the geometry of parts, the practical assembly process, the performance of the part or subas-sembly function and the variation requirement into consideration.

According to the flowchart described in Fig. 2, a FEM model can be generated with ANSYS5.5.The model was then “map-meshed” with structural solid elements and surface contact elements,as in the following examples. Note the fine mesh at the matching interfaces, where parts will bejointed into a whole.

3. Verification example — butt joint welding

In order to verify the proposed simulation method, a simple example is selected for which ananalytical solution is available. We will compare the results obtained from the proposed numericalsimulation with the analytical solution.

Fig. 3. An illustration of butt joint assembly example.


Butt joints are widely used to weld two parts in automobile body assembly. In a butt-to-buttjoint, the flanges of two or more parts are welded together to form an assembly. An assemblywith a butt joint is shown in Fig. 3, and is used as an example to analyze the final assemblyvariation. In order to simplify, we assume only part variation exists. Then, the butt joint assemblyprocess is as shown in Fig. 4.

Fig. 4. The process of two parts butt joint assembly with part variation du. (a) Part variation du from nominal position;(b) clamping and welding gun closing; (c) butt joint welding; (d) releasing clamps; (e) dimensions of assembly structure.


Fig. 5. Meshes and constraints in FEM analysis of butt joint example.

Both parts shown in Fig. 4 are made of mild sheet metal with Young’s modulusE=207,000 N/mm2 and Poisson’s ratio n=0.3. The thickness of both parts is 1.2 mm. Assumedu=0.5 mm, L=100 mm, c=10 mm and b=8.533 mm.

3.1. Simulation in FEM

According to ANSYS code and NVAM proposed in Section 2.3, the finite element mesh,including contact analysis for the assembly model, is shown in Fig. 5 with 94 elements and 120nodes, including 18 pairs of contact elements at the matching surface with the butt joint. Eachwelding spot is simulated in the style of coupled nodes. The small triangles in Fig. 5 representthe constraints provided by fixtures.

3.2. Analytical solution

According to Hooke’s law, the theoretic variation of the above assembly structure can be calcu-lated using Eq. (1):

uy�3a2

(4a+b)bdu (1)

3.3. Comparison

The same parameters are used in simulative and analytical solutions. A comparison of resultsis summarized in Table 1. The simulation results are in close agreement with the results calculatedby the analytical solution using Eq. (1).

It is clear from Table 1 that the NVAM that integrates contact, search and couple analysis issufficiently accurate. In the following section, this method will be applied to examine a practi-cal assembly.

Table 1Comparison of results calculated by analytical solution and simulation method

Variation Theoretical result Simulation result Error

ux �0.5 �0.497027 0.59%uy 0.83168 0.834663 0.36%


Fig. 6. Show of parts and constraints.

4. Application example: a dash panel cover assembly problem

In this section, the NVAM proposed simulation method will be utilized to investigate a practicalassembly, which is the assembly of dash panel cover and dash panel as shown in Fig. 6. Figure6 also shows the finite element model for the subassembly with constraints from locators, clampsand welds connecting the instrument panel with the front-end body. As the original status, thereis a gap between two matching surfaces shown in Fig. 7, which will be assembled by butt-joint

Fig. 7. The original status.


Fig. 8. Welding points.

welds. These two panels are welded together at 26 points (see Fig. 8) after the panels are locatedand clamped. Excessive deformation in the subassembly of these two panels affects the assemblyquality of the front window glass and instrument panel. The deformation is calculated by simul-ation and measured by experiment at the critical points as shown in Fig. 9. Figure 10 shows theconstructed fixture and measuring method in the experiment.

According to the symmetric characteristic of parts, two welding gun is considered. Thus, thereare three commonly used welding sequences. They are, respectively: the current adopted sequence,from the middle to outwards, and from the outwards to middle, which can be referred to assequences 1, 2 and 3 as the following.

1. Sequence 1: 1–25–3–5–7–9–11–13–15–17–19–21–23 and 2–26–4–6–8–10–12–14–16–18–20–22–24;

2. Sequence 2: 26–24–22–20–18–16–14–12–10–8–6–4–2 and 25–23–21–19–17–15–13–11–9–7–5–3–1; and

3. Sequence 3: 1–3–5–7–9–11–13–15–17–19–21–23–25 and 2–4–6–8–10–12–14–16–18–20–22–24–26.

Then, two parts are assembled in terms of different assembly sequences. The results CP1, CP2from simulation and MP1, MP2 from experiment are shown in Fig. 11. In fact, it is very difficult


Fig. 9. Critical points.

Fig. 10. Experiment for assembly analysis.

to make the measurement point and critical point fully coincident. According to Fig. 11(a)–(d),the conclusion can be drawn that the results from the NVAM simulation and experiment are fairlyclose and the NVAM proposed above is very useful.

5. Conclusions

Outlined in this paper is the NVAM method and procedure for simulating and analyzing auto-mobile body assembly variations, considering both various tooling variations and compliant, non-ideal parts. Founded on contact, search and coupled node analysis, the method is a powerful toolfor comprehending and calculating the final assembly variation. Through a simple butt jointexample, the simulation is tested and verified against the theoretical solutions. From there, it hasbeen extended to variation simulation of more complex practical compliant sheet metal assemblies.


Fig. 11. Comparisons of assembly deformation in different sequences. (a) Assembly in three different sequences; (b)assembly in sequence 1; (c) assembly in sequence b; (d) assembly in sequence 3.

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simulation and analysis of assembly processes considering compliant, non-ideal parts and tooling...

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