Carlson

A Closed Form Solution for Nonlinear Tolerance AnalysisGeoff CarlsonBrigham Young University

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OutlineStatistical Tolerance Analysis OverviewAssembly FunctionsTolerance Analysis MethodsAcceleration Analysis MethodSkewness ApproximationSummary

Statistical Tolerance AnalysisComponent Variation:xi dxi QualityFractionGivenFindAssemblyToleranceAssemblyFunctionLLULAssembly Variation:ui dui

Statistical Tolerance Analysis:Two and Three-dimensional assembliesu, represents the dependant assembly dimension xi, represents the component dimensions in the assembly , represents the contribution of each component dimension

Statistical Tolerance Analysis: Two-dimensional Example

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Statistical Tolerance Analysis: Example: Nonlinear Assembly FunctionLimits of the input variable, , are symmetrically distributedDistribution of the output variable, u, is skewed

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Four Moments of a DistributionFirst Moment:mean - measure of location

Second Moment:standard deviation - measure of spread

Third Moment:skewness - measure of symmetry

Fourth Moment:kurtosis - measure of peakedness+s-sx

Nonlinear Assemblies:Assembly Function RepresentationExplicit: xi = set of input variablesuj = set of output variablesImplicit:xi = set of input variablesuj = set of output variables

Nonlinear Assemblies:Assembly Function RepresentationLinearized:

dh = change in the assembly functiondxi = small changes in the assembly dimensions, xiduj = the corresponding kinematic changes, uj

Nonlinear Assemblies:Assembly Function RepresentationLinearized:

A = assembly dimension sensitivitiesB = kinematic sensitivitiesdX = column vector of assembly dimension variations dU = column vector of kinematic variations B-1A = tolerance sensitivity matrix

Example: Nonlinear AssembliesOne-way Clutch Assembly Function Explicit:

Implicit:

Linearized:

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Solutions to Nonlinear AssembliesMonte Carlo Simulation (MCS)Method of System Moments (MSM)Second Order Tolerance Analysis (SOTA)Tolerance Analysis Using Kinematic Sensitivities (TAKS)Acceleration Analysis Method

Solutions to Nonlinear Systems:MCS 10,000 Sets of PartsAssembly HistogramCount the Rejects Random No. GeneratorAssembly Function

Solutions to Nonlinear Systems:MSMComponent Input MomentsAssembly Output MomentTaylor Series Expansion--Second Order

Solutions to Nonlinear Systems:MSMThe first four raw moments of the output distribution can be found by applying the expected value operator to y:

Solutions to Nonlinear Systems:MSMThe first four raw moments can be centralized using the following equations:

where, i is the ith central moment of R

Solutions to Nonlinear Systems:SOTAFirst and second order sensitivities are found using finite difference formulas:

Solutions to Nonlinear Systems:TAKSVariation ModelKinematic ModelVelocity equation:Small displacement equation:

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Solutions to Nonlinear Systems:Method ComparisonRelative Effort

One-way Clutch

MCS 100k

340,000

MCS 30k

102,300

SOTA

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Linear

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Acceleration Analysis Method:Can we extend the kinematic velocity analogy?Can second order sensitivities be obtained from an acceleration analysis of a kinematic model?Can skewness be approximated from the acceleration analysis?

Acceleration Analysis MethodExample: One-way Clutch

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Acceleration Analysis MethodExample: One-way ClutchAcceleration Equation:

Acceleration Analysis MethodExample: One-way ClutchResolving the acceleration equation into real and imaginary parts and organizing into matrix form yields:

where,

Acceleration Analysis MethodExample: One-way Clutch

Closed Form Sensitivity Kinematic Acceleration Coefficients

Acceleration Analysis Method:SkewnessVariance from TAKS method

Skewness from Acceleration Method?

Skewness Approximation:Acceleration AnalysisSolving the acceleration equation for :

Simplifying:

Skewness Approximation:Acceleration AnalysisRaw moments with terms directly from the acceleration equation

Acceleration Equation GroupsOperationTransformed Acceleration Equation GroupsRelated Raw Moment

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Skewness Approximation:Acceleration Analysis

Standardized Skewness InputsStandardized Skewness of a3c3f3MSMKinematic Approx.Error (%)0.00.00.0-0.093555-0.457E-699.9990.10.10.1-0.170698-0.07773754.4590.50.50.5-0.473756-0.38868517.9571.41.41.4-1.124679-1.0883163.233Number of Terms809-

Skewness Approximation:MSM Raw MomentsFirst three raw moments:

Skewness Approximation:Truncated MSM Raw MomentsSecond raw moment blocksTruncated second raw moment:

E(y2) Blocks Neglected Termsblock 1i4block 2entire block

Skewness Approximation:Truncated MSM Raw MomentsThird raw moment blocks

Skewness Approximation:Truncated MSM Raw MomentsTruncated third raw moment:

E(y3) Blocks Neglected Termsblock 1i5, i6block 2i3j3block 3i2j4block 4entire block

Skewness Approximation:Truncated MSM Raw Moments

Standardized Skewnessi3 MSMTruncation Approx.Error (%)Kinematic Approx.Error (%)0.0-0.93555-0.0936000.049-0.457E-699.9990.1-0.170698-0.1708000.067-0.07773754.4590.5-0.473756-0.4740750.068-0.38868517.9571.4-1.124679-1.1254480.068-1.0883163.233Number of terms8042-9

Acceleration Analysis MethodSummarySecond order sensitivities can be obtained directly from acceleration analysisSensitivities can be used with MSMIncreased efficiencyNo iteration requiredTruncated MSM equation provide a good estimate of output skewness

Need to characterize variation (mean & std. dev.) of each contributing dimensionSum them statistically to get assembly variation.Assembly function: system of equations that describe the relationships between the component parts of an assemblyIndividual part tolerances must be multiplied by their corresponding sensitivitiesSensitivities reflect the degree of change in the dependant variable, u, caused by a change in the component dimension, x.U is criticalU is dependant on phi and ydphi and dy represent the dimensional variations of theta and yFor small changes, the sensitivities can be evaluated at the nominal values

This is an example of a nonlinear assembly functionLinear assembly functions produce symmetric distributionsNonlinear assembly functions produce skewed distributionsFor a nonlinear assembly function, input variables with symmetric distributions will produce a skewed output distribution.

Intro for this slide:For nonlinear assemblies, need analysis methods that can compute all four moments of the output distributionGenerally, cannot get explicit assembly functionsIf possible, derivatives can be cumbersomeDerivatives are also difficult in implicite systems iterative solution and finite differenceFirst order Taylor series expansionMatrix formsolve by linear algebraConverts implicit assembly functions to explicit assembly functionsBinvA is a matrix of tolerance sensitivities (first order)

AdvantagesEfficientClosed formFinds first order sens in a single matrix operationCan be automatedDisadvantagesCant predict skewness due to:Nonlinear assembly functionsNonlinear inputsRandom number generator generates component dimensions with small variationsResultant assembly dimensions are recalculated for each change in a component dimensionAccuracy of analysis depends on number of simulations performedNumber of simulations ranges from 5,000 to 100,000Efficiency is poor due to number of calculations that must be performedAdditional simulations needed to determine sensitvities

AdvantagesInput variables can be of any distributionCan directly calculate number of rejectsDisadvantagesLarge numbers of simulationsPoor efficiencyIterative solution required for implicit assembly functions

Finding sensitivities can be challenging if the assembly function is nonlinear and implicitLinear analysis first order sens. onlyNonlinear analysis first and second order sens. NeededDifficulty lies in obtaining the sensitivities

MSM AdvantagesSingle assembly at nominalWorks well for explicit assembly functionsMSM DisadvantagesImplicit assembly functions: sensitivities require iterative solutionNumber of terms increases rapidly with number of parts.

For implicit and/or nonlinear assembly functionsfinite differenceChange one variable at a timesolve nonlinear equations to determine the change in the dependant variablesApply the finite difference equations to MSM

Same vector loopVariation model: component dimensions varied calculate variation in dependant variable, phiKinematic model: Rigid bodies; must add variational elements to the model; perform velocity analysis as shown by FaerberVariation analysis is analogous to velocity analysis

SOTA method also includes calculations to obtain the momentsthe 41 only takes into account finding the sensitivitiesRadial acceleration: d2rCoriolis acceleration: 2w*Vrel Normal acceleration: rw2 *Emphasize that desired approximation comes DIRECTLY from the kinematic equation