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科技譯文
Improved Workpiece Location Accuracy Through Fixture Layout Optimization
Abstract
Inaccuracies in workpiece location lead to errors in position and orientation of a machined feature on the workpiece. The ability to accurately locate a workpiece in a machining fixture is strongly influenced by rigid body displacements of the workpiece caused by elastic deformation of loaded fixture-workpiece contacts. This paper presents a model for improving workpiece location accuracy in fixturing. A discrete elastic contact model is used to represent each fixture-workpiece contact. Reduction in workpiece locating error due to rigid body displacements is achieved through optimal placement of locators and clamps around the workpiece. The layout optimization model is also shown to improve the overall workpiece deflection and reaction force characteristics.
1.Introduction
The accuracy of location of a machined feature depends on the machining fixture’s ability to precisely locate the workpiece relative to the machine tool axes. Workpiece location in a fixture is significantly influenced by localized elastic deformation of the workpiece at the fixturing points. These deformations are caused by the clamping force(s) applied to the workpiece. For a relatively rigid workpiece the localized elastic deformations cause it to undergo rigid body translations and rotations which alter its location with respect to the cutting tool. It is therefore important to minimize such effects through optimal design of the fixture layout.
Previous work in fixture layout optimization has focused on the use of finite element and rigid body models. Menassa and DeVries [1], Rearick et al. [2], Trappey et al. [3], and Cai et al [4] used finite element models of the fixture-workpiece system as input to the layout optimization. In these works the fixture layout design is formulated as a constrained nonlinear optimization problem. The goal is to determine the positions of locator-clamp pairs that will minimize a nonlinear function of the elastic deformation at selected points on the workpiece. Such a formulation requires solution of the complete finite element model during each iteration of the optimization process. Hence, the technique is computationally intensive.
DeMeter [5] presented a min-max algorithm to determine the optimal fixture layout and clamping force intensity that minimizes the maximum contact force. In this study the workpiece and fixture were assumed to be perfectly rigid. Such a formulation does not allow the effect of workpiece displacement on locating errors to be minimized directly. Recently, Gui et al [6] reported a model for improving workpiece location accuracy by optimizing the clamping force. They model the elasticity of fixtureworkpiece contacts using linear springs of known stiffness. However, methods for determining the contact stiffness are not addressed. In addition, the fixture layout is assumed fixed for a given workpiece and cutting force system.
This paper presents a method that directly minimizes workpiece location errors due to localized elastic deformation of the workpiece at the fixturing points by optimally placing the locators and clamps around the workpiece. The method considers the fixtureworkpiece contact to be linearly elastic and uses closed-form contact stiffness models derived from well-known contact mechanics problems. Also, the method outlined here is computationally less intensive than the finite element approach.
The following sections give details of the underlying models and constraints used to formulate the fixture layout optimization procedure. Model simulations are presented to demonstrate the ability of the method to minimize workpiece location errors through optimal arrangement of locators and clamps.
2.Fixture Layout Optimization
Fixture layout optimization requires formulation of an objective function and constraints. In this paper our objective is to minimize the effect of localized elastic deformation of the workpiece at the fixturing points on workpiece location. As stated earlier, the elastic deformations cause the workpiece to undergo a rigid body motion, which in turn shifts the workpiece location. The objective function for optimization is constructed as follows. Objective Function Formulation. Consider a solid rectangular workpiece held in a fixture consisting of several locators and clamps (see Figure 1). The fixture is typically very rigid compared to the workpiece. It can hence be assumed that the locators do not undergo any rigid body displacement. In contrast, forces acting on the workpiece at the locating and clamping points cause the workpiece to translate and rotate in the global coordinate system. Assume that the rigid body motion of the workpiece due to normal and tangential elastic deformations at the ith fixturing point is given by vector δ i=[δ δ δ ] xiyizi T . Note that the components of δ i are expressed in the local coordinate system fixed to the ith point. Geometric transformations are applied so that the rigid bodymotion due to deformation at the ith fixturing point is expressed in the global coordinatesystem as:
where Tgi is a general rotation matrix that transforms quantities expressed in the ith local coordinate system into the global coordinate system. Thus, the total rigid body motion ofthe workpiece due to elastic deformations at all the fixturing points is:
where N is the total number of locators and clamps.
In order to minimize the effect of rigid body motion on workpiece location, a quadratic objective function for fixture layout optimization can now be formulated as follows:
Note that the above expression is not an explicit function of the fixture element positions.But the rigid body motion δ i , and therefore Δ , is dependent on the fixturing forces which are in turn uniquely determined by the layout of fixturing points and elastic contact properties. Hence, changing the fixture layout changes the value of the objective function indirectly.
Fixture-Workpiece Contact Constraints. The fixture-workpiece system is subject toseveral contact constraints that the optimum fixture layout must also satisfy. In particular, constraints specifying the geometric compatibility of elastic deformation andfrictional resistance are needed. These constraints are developed using a discrete elasticcontact modeling approach similar to that of Conry and Seireg [7], and Sinha and Abel[8].
The workpiece is assumed to be elastic in the contact region and rigid elsewhere.The fixture is assumed to be completely rigid. At each fixturing point a square contact surface tangent to the fixture and workpiece surfaces is assumed. The contact surface isdiscretized into a grid containing M square elements as shown in Figure 2. A distributed normal force of intensity p ji and a distributed friction force of intensity (q) (q) xjiyj2 i 2 +are assumed to act across an arbitrary element j of the ith contact surface. The total normal ( Pi ) and friction (Qi ) forces acting at the ith fixturing point are then given by:
The localized deformation at a fixturing point causes distant points in the workpiece to undergo a rigid body motion in the normal direction given by δ zi . If s ji isthe initial separation of the fixture and workpiece surfaces for the jth element at the ith fixturing point, the normal deformation, wji , must satisfy the following contact condition[9]:
The equality sign applies to points that lie inside the equilibrium contact area and the
inequality sign for outside points.
Orthogonal components of tangential deformation, uji and v ji , produced by the frictional forces acting at a fixturing point lead to tangential rigid body motions δ δ xiy, i ,respectively. The deformation and rigid body motion should satisfy the following geometric compatibility conditions [9]:
where the equality and inequality signs apply for slip and no slip cases, respectively.
Contact Deformation Model.The workpiece is assumed to be a linear elastic solid in the vicinity of the fixturing points. Hence, by linear superposition, the normal deformation in the jth element of the ith contact region can be written as:
where e e e jknjkxjk, , y are the flexibility influence coefficients for deformation in the normal direction due to fixturing forces in the normal (n) and tangential directions (x, y).Similarly, the x and y components of tangential deformation are given by:
where are the normal and tangential flexibility influence coefficients for workpiece deformation in the local x and y directions at the ith fixturing point, respectively.
In this paper the influence coefficients are derived from closed-form solutions for
the contact compliance of an elastic half-space subjected to distributed normal and tangential loads. Details of the influence coefficient models may be found in [8, 9].
Contact Friction Constraint. Coulomb friction is assumed to apply at each fixturing point. This implies a nonlinear relation between the normal and frictional forces acting at a fixturing point, i.e., (q) (q) p xj where μ s is the coefficient of static friction for the fixture-workpiece material pair. For simplicity, a linearized version of this constraint is used:
Static Equilibrium Constraint. The workpiece must be in static equilibrium after application of fixturing forces at the selected points. This constraint is given by the following force and moment equilibrium equations:
ΣF = 0 (12)
ΣM = 0 (13)
where the forces and moments are expressed in terms of the elemental normal ( p ji ) and tangential forces ( qxji , qyji ) acting at the contact surface for each fixturing point.
Clamping Force Constraint. When the clamping force applied by the clamps isspecified, it is necessary that the sum of the elemental normal forces at the clampingpoint equal the specified force. This constraint is expressed as follows:
where C is the number of clamps in the fixture. In this paper the clamping force isassumed to be known. In general however the clamping force could be treated as adesign variable in the layout optimization process [5].
Fixture Element Position Constraints. The fixture layout optimization procedure seeksto find the optimal locations of the fixturing points. In general, fixture element positionson a workpiece datum surface cannot be chosen randomly and are often constrained bythe geometric complexity of the workpiece surfaces, size and location of the features tobe machined, and other process related issues. Hence, the position of a fixture element isrestricted to a bounded region on the datum surface. In this paper each fixture elementposition is constrained to lie inside a convex polygonal region. A sequence of orderedstraight edges represents each convex polygon. Mathematically, the system of linearinequalities constructed from the line equations for all ordered edges (for N fixturingpoints) is used to specify the bounded region:
A X C p p p ≤ (15)
Where
And
The elements of Ai and ci are coefficients of the line equations of the polygon edges usedto specify the polygon boundary for the ith point, xi is the position vector (global) to the ithpoint on the workpiece surface, and li is the number of ordered edges making up thebounding polygon for the ith point.The above inequalities can now be used to easily establish the location of afixturing point relative to its polygon boundary. Points inside or on the boundarycompletely satisfy the above inequalities whereas points outside the bounded region do not [10].
Layout Optimization Model. The complete fixture layout optimization problem cannow be formally stated as follows:
Minimize ：
Subject to:
Bounds: pji ≥ 0
i = 1, …, N; j = 1, …, M; k = 1, …, C
Note that the normal compatibility constraint has been multiplied by -1 to convert it intoa ≤ type inequality. Also, by definition, the friction force components qxji and qxji lie inthe contact surface plane, and p ji is assumed to be positive when directed into the9workpiece surface.
3.Solution Method
A nonlinear programming method is used to solve the above layout optimizationproblem. Specifically, Zoutendijk’s method of feasible direction [11] is used. Thismethod is similar to that used by DeMeter [5] and involves the solution of the followinggeneral nonlinear program:
Minimize f(x)
Subject to Gx ≤ b (linear inequality constraint)
H(x) = 0 (nonlinear equality constraint)
Ex = e (linear equality constraint)
where x is the feasible solution. For the nonlinear program given in equation (16) the
solution x =[F δ X ] pT
where:
Note that in addition to position of the fixturing points, Xp, the solution procedure treats
the fixturing forces F and rigid body motion δ also as design variables during theoptimization process. This is because the fixturing forces and rigid body motion dependon the fixture layout and are determined uniquely for each layout by the physics of theproblem.
The first linear inequality constraint is constructed by combining all the inequality constraints given in equation (16). The second nonlinear constraint arises from themoment equilibrium equation in (16). Finally, the linear equality constraint equation isconstructed by combining all the equality constraints listed in (16). For the problem athand, G, H, and E result in matrices with the following sizes: [(4MN+N liiN= Σ1) x(3MN+6N)], [3 x (3MN+6N)], and [(3+C) x (3MN+6N)]. Note that x is a [(3MN+6N) x1] column vector.
The method of feasible directions solves the nonlinear program by moving from a initial feasible solution to an improved feasible solution. This is accomplished in four steps: a) find initial feasible solution, b) determine line search direction, c) determine step size, d) solve quadratic program. By iterating between steps (b) and (d) furtherimprovements in the feasible solution can be obtained. Mathematical details of step (b)through (d) can be found in reference [11].
The initial feasible solution x is obtained by solving the elastic fixture-workpiececontact model for the initial layout. This is done by minimizing the total complementaryenergy for the fixture-workpiece system. Details of the solution procedure andexperimental validation can be found in [7, 8, 12]. Note that the contact model needs tobe solved only once at the beginning to obtain the initial feasible solution. Thereafter,the layout optimization model relies on the contact constraints and the contactdeformation model to compute valid rigid body displacements and fixturing forces.
4.Results and Discussion
The fixture layout optimization model and solution algorithm has been implemented in MATLAB (version 5.0). The capability of the model is illustrated through an example. Consider the initial fixture layout shown in Figure 3. This layout uses a "4-2-1" location scheme with two simultaneously actuated hydraulic clamps to hold the workpiece against the locators. Table 1 lists the positions and orientations of thefixture elements in the initial layout. Locators L1-L4 and clamps C1-C2 have sphericaltips while locators L5-L7 have small area planar tips (area = 63 mm2). A clamping forceof 703 N is assumed to act at each clamping point. The workpiece is a 127mm x 127mmx 382 mm block of Aluminum 7075-T6. The Young's modulus (E) and Poisson’s ratio(ν) for the workpiece are 70.3 GPa and 0.354 respectively, and 201 GPa and 0.296respectively for the fixture elements.
The initial feasible solution vector x is computed by solving the fixture-workpiececontact model for the initial fixture layout using the minimum complementary energymethod. The layout optimization problem is then solved using the four step iterationprocedure outlined in the previous section. The fixture element position constraints usedfor this problem are given in Table 2. The improved fixture layout that minimizes the11effects of rigid body motion is given in Table 1. The objective function value is reducedfrom 528 μm2 to 426 μm2.
The impact of the optimization process on the fixture layout is shown in Figure 4.The initial fixture layout was intentionally designed to violate well-known empirical“l(fā)ocating rules” [13]. For instance, it is standard practice to position the locators on adatum surface as far apart as possible. This is done to ensure the best possible locationalstability of the workpiece. In the initial layout, locators L1-L2 and L4-L7 clearly do notsatisfy this rule. Also, the initial position of clamps C1 and C2 do not provide adequateclamping stability. It is clear from Figure 4 and Table 1 that the layout optimizationmodel gives a solution that supports the empirical rules. Specifically, L1 and L2 arepushed as far apart as possible. Also, locators L4-L7 are spread out on the primarydatum plane so as to include the projected center of gravity of the workpiece inside thebounding polygon formed by joining L4-L7. This improves workpiece stability in thefixture. The new position of clamp C1 is approximately half-way between locators L1and L2. Similarly, clamp C2 and locator L3 directly oppose each other in the improved lay out.
If, for simplicity, only the normal component of rigid body motion ( δ z ) isconsidered, it can be shown through suitable geometric transformations that the locationerror, Ep, of a point P on the workpiece is reduced by the optimization process surface(see Figure 5). For instance, the location error of the point (30, 100, 19.1) decreases from15.3 μm for the initial fixture layout to 11.7 μm for the improved layout. Thus, thefixture layout optimization model and solution procedure described above improve workpiece location accuracy by minimizing the effect of workpiece rigid bodydisplacement.
Finite Element Analysis. In order to further analyze the effect of the fixturelayout optimization process on overall workpiece deformation a finite element model wasconstructed using ANSYS? (version 5.3). The locators were modeled as displacementconstraints that prevent workpiece translation in the normal direction. The clampingforce was modeled as a uniformly distributed force acting over the workpiece-clampcontact area.
The deflection of the top surface of the workpiece (i.e., the surface to bemachined) is shown for the initial and improved fixture layouts in Figures 6 and 7,respectively. The initial fixture layout shows a significant deflection gradient across thetop surface of the workpiece. Deflection magnitudes range from 0.25 x 10-4 mm to 0.76 x10-2 mm. In general a large variation in deflection magnitudes is not desirable. On theother hand, the improved fixture layout produces a relatively uniform distribution ofdeflections that range from 0.10 x 10-2 mm to 0.19 x 10-2 mm. The maximum deflectionof the top surface is much less for the improved layout (0.19 x 10-2 mm compared to 0.76x 10-2 mm). Also, the reaction forces at L1 and L2 are 638.05 N and 65.31 Nrespectively for the initial layout, and 327.20 N and 376.16 N respectively for theimproved layout. Thus reaction forces in the improved layout are more uniformlydistributed than the initial layout.
Therefore the optimization process produces a fixture layout that improves theoverall workpiece deflection and reaction force characteristics in addition to improvingworkpiece location accuracy.
5.Conclusions
The paper presented a fixture layout optimization model for improving thelocation accuracy of the workpiece when clamped in a machining fixture. Theinaccuracy in workpiece location was due to rigid body motion of the workpieceproduced by the localized elastic deformation at the fixturing points. A discretizedelastic contact model of the fixture-workpiece interaction was used to develop t