A Second-Order Method for Assembly Tolerance …

1 1 Copyright 1999 by ASMEP roceedings of the1999 ASME Design Engineering Technical ConferencesSeptember 12-15, 1999, Las Vegas, NevadaDETC99/DAC-8707A Second-Order Method FOR Assembly Tolerance ANALYSISC harles G. GlancyConcurrent Engineering Products GroupRaytheon Systems CompanyDallas, W. ChaseMechanical Engineering DepartmentBrigham Young UniversityProvo, analysis and Monte Carlo simulation are two well-established methods for statistical Tolerance analysis ofmechanical assemblies. Both methods have advantages anddisadvantages. The Linearized Method , a form of linearanalysis, provides fast analysis, Tolerance allocation, and thecapability to solve closed loop constraints. However, theLinearized Method does not accurately approximate nonlineargeometric effects or allow for non-normally distributed input oroutput distributions. Monte Carlo simulation, on the other hand,does accurately model nonlinear effects and allow for non-normally distributed input and output distributions.

2 Of course,Monte Carlo simulation can be computationally expensive andmust be re-run when any input variable is Second-Order Tolerance analysis (SOTA) methodattempts to combine the advantages of the Linearized Methodwith the advantages of Monte Carlo simulation. The SOTA Method applies the Method of System Moments to implicitvariables of a system of nonlinear equations. The SOTA methodachieves the benefits of speed, Tolerance allocation, closed-loopconstraints, non-linear geometric effects and non-normal inputand output distributions. The SOTA Method offers significantbenefits as a nonlinear analysis tool suitable for use in comparison was performed between the LinearizedMethod, Monte Carlo simulation, and the SOTA Method . TheSOTA Method provided a comparable nonlinear analysis toMonte Carlo simulation with 106 samples. The analysis time ofthe SOTA Method was comparable to the Linearized INTRODUCTIONT olerance analysis is increasingly becoming an importanttool for mechanical design.

3 This seemingly arbitrary task ofassigning tolerances can have a large effect on the cost andperformance of manufactured products. With the increase incompetition in today s marketplace, small savings in cost orsmall increases in performance may determine the success of paper proposes a new Second-Order Tolerance analysis(SOTA) Method . The development of the SOTA Method wasmotivated by the differences in capabilities between two well-established Tolerance analysis methods : the Linearized Methodand Monte Carlo simulation. The SOTA Method specificallyaddresses Tolerance analysis of vector-loop Tolerance following three sections introduce vector-loop tolerancemodels, the Linearized Method and Monte Carlo Vector-loop Tolerance ModelsVector loops can be used to model manufacturedassemblies. Figure 1 shows an example of a two-dimensionalassembly described by three vector loops.

4 A vector-looptolerance model mathematically establishes how themanufactured lengths and angles of each component combine inorder to properly assemble together. The vector loops are ableto model dimensional, form and kinematic 2 Loop 1 Loop 3 Figure 1: Vector-loop Assembly ModelVector-loop closure is an important condition for assemblytolerance analysis. Closure simply refers to the condition whenthe beginning of the vector loop is the same position andorientation as the end of the loop. Loop closure is themathematical equivalent of an Assembly fitting together with no2 Copyright 1999 by ASME clearance between parts. The loop closure condition can bewritten as the system of nonlinear equations:h(x,u) = 0(1)where h is the system of loop equations, x is the set of vectorsrepresenting manufactured component dimensions, and u is theset of vectors representing unknown Assembly lengths andangles. The unknown Assembly lengths and angles are thekinematic Assembly dimensions that change as a function of thecomponent Linearized MethodThe Linearized Method is a vector-loop-based Method ofassembly Tolerance analysis.

5 The Method s name comes fromthe fact that the nonlinear equations of the vector-loop modelare linearized for the analysis. The linearized equationsdetermine how small changes of the component dimensions,form and contact affect an Assembly . For this Method only oneassembly needs to be analyzed statistically. Linear analysis isextremely fast and allows for Tolerance allocation and designiteration. It is, however, limited to normal componentdistributions and cannot be applied to non-normal tolerances are small compared to the nominaldimension, on the order of 1/100 to 1/1000, the LinearizedMethod gives excellent results. A comparison [Gao 1995]between the Linearized Method and Monte Carlo simulationfound that the accuracy of the Linearized Method correspondedto Monte Carlo simulation with a sample size of 30,000, forquality levels near three sigma.

6 However, for highly nonlinearassemblies or highly skewed distributions, the LinearizedMethod loses Linearized Method expands the loop closure equation,Equation 1, for small variations about the nominal by Taylor'sseries expansion, retaining first order derivatives. Thisexpansion yields:011= + = ==mjjjinjjjiiduuhdxxhdh(2)where dxj are the specified tolerances of the componentdimensions and duj are the resultant variations in the dependentassembly dimensions. This expression can be put in vectorform by forming the matrix A of partial derivatives jixh and thematrix B of the partial derivatives jiuh .[ A ] {dx} + [ B ] {du} = { 0 } (3)Solving for du:{du} = -[ B-1] [ A ] {dx} (4)Therefore, the product of the matrices -B-1A gives thesensitivities of the dependent Assembly dimension with respectto the component dimensions. Having established thisrelationship, the Standard Deviation of the dependent assemblydimension variations may be estimated by the root sum squaresexpression: = =njjjiidxxudu12(5)where jixu are the elements of the -B-1A formulation of the Linearized Method allows theimplicit Assembly dimensions in the loop equations to beexpressed as an explicit, statistical function of the StandardDeviationFit Normal DistributionCalculate SensitivitiesFigure 2: Steps of the Linearized MethodFigure 2 shows the steps of the Linearized Method .

7 Step 1is the calculation of the sensitivities, the elements of the -B-1 Amatrix. Step 2 uses Equation 5 to calculate the StandardDeviation. Steps 3 applies a Normal distribution assumption tothe Standard Deviation calculated in Step 2. Finally, Step 4calculates the rejects given the Normal distribution andspecification Monte Carlo SimulationMonte Carlo simulation is a random number based methodfor performing Assembly Tolerance analysis. The manufacture ofan Assembly is simulated, for example, by creating a set ofcomponent dimensions with small random changes to simulatenatural process variations. Next, the resulting assemblydimensions are calculated from the simulated set of componentdimensions. The number of rejects that fall outside thespecification limits are then counted. These three steps areillustrated in Figure sizes generally range between 5,000 to 100,000based on the required accuracy of the simulation.

8 The accuracy3 Copyright 1999 by ASMEof Monte Carlo simulation increases with larger sample , the computational effort of large sample sizes can besignificant, but Monte Carlo simulation offers many advantagesbecause of its flexibility. Monte Carlo simulation allows anycomponent distribution to be specified and will calculate theresulting Assembly each sampleRandomly Change AllModel VariablesEvaluate Model FunctionCount RejectsFigure 3: Steps of Monte Carlo SimulationMonte Carlo simulation and the Linearized Method providedifferent capabilities. The Linearized Method can perform ananalysis and a Tolerance allocation quickly, so it is suitable fordesign iteration. The Linearized Method is limited in that itcannot output non-normal distributions or handle non-normalcomponent distributions. Also, the Linearized Method will notbe accurate for highly nonlinear assemblies. Monte Carlosimulation allows non-normal input distributions and anonlinear analysis.

9 However, Monte Carlo simulation iscomputationally expensive and does not accommodate rapiddesign iteration. For example, if a single input parameter ismodified, the entire Monte Carlo simulation must be 1: Comparison of Method FeaturesFeaturesLinearizedMethodMonte CarloSimulationSOTAM ethodSpeed Tolerance allocation Closed-loop constraints Nonlinear approximation Non-normal inputdistributions Non-normal outputdistributions Table 1 summarizes the features of the Linearized Method ,Monte Carlo simulation and the Second-Order ToleranceAnalysis (SOTA) Method proposed in this paper. The SOTA Method attempts to combine the features of the LinearizedMethod and Monte Carlo next section of this paper, Section 2, discusses researchrelated to the SOTA Method . Section 3 presents the SOTA Method . Section 4 compares the results of the SOTA methodwith the Linearized Method and Monte Carlo simulation for asample RESEARCH Linearized MethodThe Linearized Method , explained in Section , providesa quick way to perform nonlinear Tolerance analysis for bothexplicit and implicit Assembly dimensions of a vector-looptolerance model.

10 Because of its speed, the Linearized Methodis ideal for design iteration and Tolerance allocation. Multipleresearch studies have continued to refine the LinearizedMethod, making it more general and Direct Linearization Method (DLM) [Marler 1988]prescribed a systematic approach to vector-loop modeltolerance analysis. DLM has enabled the Linearized Method tobe applied to a broad range of Tolerance problems. Mostimportantly, DLM has allowed a general Tolerance analysismethodology to be incorporated into a computer programsuitable for integration with a CAD recently, the Global Coordinate Method [Gao 1993]for determining the partial derivatives of the loop equations wasdeveloped. This Method simplified the calculations of thesederivatives. In the same paper, Gao benchmarked theLinearized Method against a comparable Monte Carlosimulation system.