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Optimal Fixture Design for Drilling Through Deformable Plate Workpieces Part I: Model Formulation

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Optimal Fixture Design for Drilling Through Deformable Plate Workpieces Part I: Model Formulation Journal of Manufacturing Systems Vol. 2a/No. 1 w 2001 0 ~ Optimal Fixture Design for Drilling Through Deformable Plate Workpieces Part I: Model Formulation K.R. Wardak, U. Tasch, and P.G. Charalambides, Dept. of Mechanical Engineering, University of Maryland Baltimore County, Baltimore, Maryland, USA Abstract in which the model formulation is discussed in Part I This is the first part of two manuscripts that address the and the results in Part II. (Part II follows in this development of scientifically based methodologies that use issue.) Part I further develops the scientific tools that finite element methods and optimization algorithms to design optimal fixturing layouts for the drilling process. Part I focuses capture the shape and dimensional characteristics of on the problem formulation and Part II discusses the results. the machined surfaces generated through drilling The fixturing problem formulation is posed as a constrained operations. These tools are based on FE analysis, optimization problem, in which the physical fixture constraints optimization, and schemes that handle material define the domain, and the desired fixturing characteristics are optimized in accordance with a selected objective func- removal strategies. In formulating the fixturing opti- tion. The optimal fixturing model developed includes a mater- mization problem, Part I presents five different ial removal strategy that enables one to calculate the shape objective fimctions that result in different fixture lay- and dimensions of the machined hole surface. The boundary outs. These layouts are tested and evaluated in Part value problem associated with the optimal fixturing simula- tions is first formulated. Five different objective functions II. A literature review that highlights the contribu- capable of describing various geometrical aspects of the tions of previous researchers follows. machined hole have been developed and tested. These func- Shirinzadeh2 introduced reconfigurable fixture tions evaluate the square differences between the simulated modules that are assembled by robotic manipulators and nominal radii and diameter values, minimize the maxi- and based on a CAD model of the workpiece. mum values of the latter quantities, and minimize the devia- tions of the drilled surface from a perfect cylinder. Automatically reconfigurable fixture designs are also addressed by earlier reports of Asada and By3 Keywords: Optimal, Fixture, Deformable, Workpiece, and Chou et a1.4; however, these concepts have been Drilling, Design, Objective Function, Finite Elements limited to laboratory applications. Automatic fixtur- ing design5 retains resting equilibrium and static sta- bility of the workpiece. Resting equilibrium condi- 1. Introduction tions are essential for stabilizing the workpiece Approximately 40% of rejected parts are due to when it is initially placed on the fixture locators, and dimensioning errors that are attributed to poor fix- static stability must be maintained when the clamp- turing design.’ Fixtures are used to accurately posi- ing and machining forces are applied. A resting tion and adequately constrain a workpiece in the equilibrium is obtained when the wrench formed by machine-tool coordinate system. Adequate con- the reacting locator-forces falls within the triangle straints of a deformable workpiece ensure that the formed by the three supporting locators; static sta- dimensions and shape of the machined surfaces are bility is obtained if the matrix formed by the static within the required tolerances. Research efforts that equilibrium equations is fully ranked. address the development of scientifically based fix- Brost and Goldberg6 formulated optimal fixturing ture design methodologies are reported in Refs. 2 designs that minimize the reaction forces at the loca- through 10. Mathematical tools that are based on tors. In this formulation, the workpiece is modeled computer-aided design (CAD) and finite element as a rigid body and the deformations induced by the (FE) analysis have been instrumental in advancing clamping and machining forces are not considered. state-of-the-art fixture design methodologies. Zhang et al.’ compared the optimal fixturing designs This paper is structured as a two-part manuscript obtained through rigid body analysis to fixtures cal- 23 Journal of Manufacturing Systems Vol. 2a/No. 1 2001 culated by elastically deformable workpiece models. the MIP model size strictly proportional to the num- Their findings demonstrate that the two formula- ber of machining response points and alleviates this tions result in substantially different optimal fixtur- severe limitation is reported in Sayeed and De ing layouts. Meter.16 Another fixture optimization algorithm was Lee and Haynes8 and Menassa and DeVries’ suggested by De Meter,” who presented Fast employed FE models to, respectively, obtain fixtures Support Layout Optimization (FSLO) models. that minimize the work done by the clamping and FSLO determines support locations that minimize machining forces and minimize the displacements the maximum displacement-to-tolerance ratio of a of selected nodes of the FE mesh. Pang” introduced set of workpiece features subject to machining frictional contact elements to the FE models and loads. This work is highly relevant to the work pre- incorporated formulations that minimize the dis- sented in this manuscript. placements of selected FE nodes as well. The focus of the current research project is to De Meter” applied optimization techniques to develop scientific methods that capture the shape fixturing design based on contact region loading, and geometry of machined surfaces generated where the workpiece is treated as a rigid body. Here, through drilling operations and to design fixtures the selected objective function is the summation of that maintain dimensional errors within allowable the magnitudes of the loads in the contact regions. tolerances. Part I is organized as follows: the finite Melkote,12 on the other hand, used genetic optimiza- element formulation of the surface geometry gener- tion algorithms to determine the layout of fixtures in ated through the drilling process is presented, and which the deformation of the machined surface due automatic mesh generation procedures are dis- to clamping and machining forces is minimized over cussed. This is followed by a derivation of the opti- the entire tool path. mal fixturing problem for which five different objec- Liao, Hu, and Stephenson13 optimized fixture lay- tive functions are suggested. Then, conclusions are outs by minimizing workpiece deflections due to reached. clamping forces; nevertheless, the effects of machin- ing forces are not accounted for. In this model, the deformations of the workpiece and locators are 2. Finite Element Modeling accounted for, and the interface between the locator In this study, the method of finite elements is and the workpiece is modeled as a surface contact employed for the purpose of calculating the work- pair. 3-2-l fixturing scenarios are considered, and piece deformations induced by the drilling loads. It FEA is used to solve the deformation field. The is understood that the actual drilling process may be deformations of selected nodes constitute the objec- governed by nonlinear interactions between the pen- tive function. In addressing fixturing problems of etrating drill and the deforming workpiece. For deformable sheet metal that is neither prismatic nor example, the removal of material during drilling solid, an N-2-l fixture design has been proposed by alters the geometry and thus the structural stiffness Cai, Hu, and Yuan.14 They report algorithms for find- of the workpiece, which in turn leads to higher ing the best N locating points such that total defor- deformations, both in the near-drill proximity as mation of a sheet metal is minimized. These algo- well as in remote regions of the workpiece. In addi- rithms are based on the N-2-l fixturing scenarios. tion to the local and global effects of material Here, FEA models and nonlinear programming are removal, nonlinear damage evolution in the near- utilized to obtain the optimal fixture layout. drill proximity is also expected to augment the local Sayeed and De Meter” introduced a Mixed deformation field, which determines the quality of Integer Programming (MIP) model for determining the drilling process. This may include several types the optimal locations of locator buttons and their of damage such as microcracking in ceramics, opposing clamps for minimizing the effect of static delamination in composite laminates, plasticity in workpiece deformations on machined feature geo- metals, fiber debonding and pull-out in fiber rein- metric error. This model utilizes finite element forced composites, and other potential forms of analysis of the workpiece, yet its solution is size sen- damage. To account for all of the above, a robust sitive to the square of the number of machining nonlinear finite element model needs to be devel- response points considered. A technique that makes oped. Such a nonlinear and computationally inten- 24 Journal of Manufacturing Systems Vol. 20/No. 1 2001 Iocator drilled hole 1 t / ,-/ o,/= i 4 ................ / / a,,mp i i Cj'"! : Ls ~," ..... i '. i /~ x~ ~ i i x Figure 2 1/2 in. and 3/4 in. Holes are Drilled Through a 3x4xi/4 in. Plate. Centers are located at (2.5, 0.5) and 0.0, 3.0) (in.), respectively. ed by C1(Xcl, Yc,, h), C2(Lx, Yq, ~), and C3(Xc3, Ly, ~) are introduced to ensure total restraint. This applica- tion of the fixturing
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