Geometric Covariance in Compliant Assembly Tolerance Analysis Jeffrey B. Stout Brigham Young...

Geometric Covariancein

Compliant AssemblyTolerance Analysis

Jeffrey B. StoutBrigham Young University

ADCATSJune 16, 2000

Presentation Outline

PIntroductionPBackgroundPCompliant Assembly Tolerance AnalysisPGeometric CovariancePSimulations, Results, and VerificationPConclusions and Recommendations

IntroductionTraditional Statistical Assembly Tolerance Analysis Methods

PTraditional tolerance analysis methods predict the accumulation of manufacturing variations in assemblies.PBased on the assumptions that the variations are random and all of the parts are rigid.

Loop 1

Loop 2

Two closed loops of a stacked block problem

PVector Loop Method (CE/TOL)PMonte Carlo Analysis (VSA)

IntroductionA new method must be used to analyze assemblies with compliant parts.

FEA and STA can be combined to analyze assembly tolerances with compliant parts.

BackgroundCompliant Assembly Tolerance Analysis - Related Research

PVariation Modeling in Vehicle Assembly by Hsieh and OhPTolerance Analysis of Flexible Sheet Metal Assemblies by Liu, Hu, and WooPFEA/STA Method by Merkley and Chase, contributions by Soman, Bihlmaier, and Stout

Compliant Assembly Tolerance AnalysisMethod Outline - Combining STA with FEA

PFind Misalignment of Rigid PartsPModel the Flexible Parts with FEAPObtain the Condensed Stiffness MatricesPApply Boundary ConditionsPSolve for Closure ForcesMean forceVariation of force about the mean

Find Misalignment of Rigid Parts

PCreate closed vector loop. Determine mean gap/interference.PEstimate variation due to stack-up.PInclude dimensional variation of both rigid and flexible parts.

Compliant Assembly Tolerance Analysis

B D

A

EC

A

B C D E

t = TA2 + TB

2 +TC2 +TD

2 +TE2

t

Model the Flexible Parts with FEA

PCreate geometryPMesh each compliant part, place nodes at fastener locationsPApply displacement boundary conditionsPOutput the global stiffness matrix


Obtain Condensed Stiffness MatricesThe Global Stiffness Matrix

PSum of element stiffnessesPIncludes all model degrees of freedomPSymmetricPVery large and sparse

Ka

12

34

56

78

910

Ka =


Obtain Condensed Stiffness MatricesMatrix Partitioning

Ka = =

Boundary DOF KbbCoupled DOF Kbi

Coupled DOF Kib Interior DOF Kii

Kbb

Kib

Kbi

Kii

PSort the stiffness matrix for each part to group the boundary node DOF and interior node DOFPPartition the matrix


Obtain Condensed Stiffness MatricesCreate Super-Element. Condense the stiffness matrix for each part to include only the boundary nodes.

Kbb

Kib

Kbi

Kii

Fb = Kbb*b + Kbi*i

Fb

Fi

=

b

i0 = Kib*b + Kii*i

No forces on interior nodes, Fi =0

Solve second equation for i : i = -Kii

-1*Kib*b

Substitute this into the first equation

Fb = Kbb*b - Kbi*Kii-1*Kib*b = [Kbb - Kbi*Kii

-1*Kib]*b

Kse = Kbb - Kbi*Kii-1*Kib

Super-element stiffness matrix:

Fb and i are unknown


Obtain Condensed Stiffness MatricesAdvantages of Super-Elements

PSuper-elements are well suited for assembly analysisEach part can be represented by a super-elementPReduces part stiffness matrix DOF to boundary node DOFPSimplify matrix algebra, and reduce computation timePProvide sensitivities for statistical analysis


Apply Boundary Conditions

PCompliant members A and B are to be joined. The dimensional variations va and vb of each part are given.PAt the point of equilibrium, the assembly forces are equal and opposite.PSolve for a and b. These are the deflections required to assemble the parts at equilibrium.PSolve for the assembly closure forces from the part stiffness matrices.

a bKa

Vb

Kb

Va

equilibrium position

nominal position


Apply Boundary Conditions2-D Assembly Closure Force -Derivation

Ka Vb

Kb

Va

nominal position

Misaligned Parts

Define Gap Vector: o = vb -va

o

a

b

KaKb

equilibrium position

Force-Closed Assembly

Define Closure Displacements: a - b = o

Closure Forces: fa = -fb

o


NOTE: Displacements and stiffness for all degrees of freedom must be linear and elastic.

Solve for Closure Forces2-D Assembly Closure Force -Derivation

a

b

Ka Vb

Kb

Va

Equilibrium

Nominal

fa = -fb

Kaa = -Kbb

since b = a - o, then

Kaa = Kbo - Kba

Kaa + Kba = Kbo

(Ka + K

o

Closure Deflections:a = (Ka + Kb)-1 Kbo

b = o - a

Closure Forces:fa = Kaa

fa = Ka(Ka + Kb)-1Kbo


Solve for Closure Forces

PFour types of assembly solutions:

Single Solution - Individual assembly, measured dimensional variation

Average - mean of a production lot of assemblies

Worst Case - stack-up analysis assuming all dimensions are at size limits

RSS - statistical analysis using root-sum-squares stack-up, statistical variation.


Solve for Closure ForcesSummary of Key Equations

fa = Krao

Use the following equations for single and mean cases

Use the covariance equation for statistical cases

Define the stiffness matrix ratio

Kra = Ka(Ka + Kb)-1Kb

fa = KraoKraT

fa = Krao

fa = Ka(Ka + Kb)-1Kbo


Solve for Closure ForcesDerivation of Statistical FEA/STA Equations

E[fa] = E[Kra

fara

E[(fa-fa)(fa-fa)T] = E[(KraKrao)(KraKrao)T]

fa = E[Kra(o)[Kra(o)]T]

fa = E[Kra(o)(o)TKraT]

fa = KraoKraT

First statistical moment: mean

Second statistical moment: variance

E = statistical expectation operator

fa = Krao


Create Super-Element Stiffness Matrices

Two Plate Example

PPlate are modeled using plane stress theoryP10 x 10 element grid giving 11 nodes along the coincident edgesPThe global stiffness matrix of each part contains 121 nodes with two DOF per node, giving 242 total DOFPThe super-element of each part only contained 22 DOF

Geometric Covariance

PThe Need for a Covariant SolutionPMaterial and Geometric Covariance DefinedPCombining Geometric Covariance and FEA/STAPThe Curve Fit Polynomial MethodPForms of Geometric Covariance

Material and Geometric Covariance DefinedGeneral Covariance

A coupling of two variables


y

x

y

x

y

x

Uncorrelated Partially Correlated

Fully Correlated

Random variables x and y

Ellipses indicate constant probability

Material and Geometric Covariance Defined

Material Covariance


x = /E

y = y = x

Strain is fully correlated

= KfThe stiffness matrix defines the correlation of neighboring point displacements when a single point is subjected to a force.

Force

Orig

inal

Sur

face

Def

orm

ed S

urfa

ce

Material and Geometric Covariance Defined



Geometric variation of each point is unlikely to be completely random

Variation of points will be in the form of a continuous surface

Combining Geometric Covariance and FEA/STACombination Called CoFEA/STA Method


fa = KraGoKraT

Ga= SaaSaT

Define the part geometric covariance matrix to be a function of the displacement sensitivities and the part variation.

Go= SooSoT

The same rule applies to the whole gap as well.

The geometric covariance term replaces the variance term to form the CoFEA/STA equation:

The sensitivity matrix Sa defines the sensitivity of each DOF's position with respect to all other DOF. Sa is symmetric by nature.

The Curve Fit Polynomial MethodGeometric Covariance

Variation of points is constrained to be in the form of a polynomial curve

i i+1i-1i-2

x

y

yi = co + c1xi + c2xi2 + c3xi

3

Find sensitivity of other y values with respect to yi

yi = ... + si-2yi-2 + si-1yi-1 + siyi + si+1yi+1 + ...

PThe set of y's to be considered can be either a local band or all the points on the mating edge (banded or truncated).PFor curve fit polynomials, the s terms turn out to be purely a function of the x-spacing between the nodes.PThe s terms fill the sensitivity matrix S; each yi has a set of s terms that fill a row or column of S.

Forms of Geometric CovariancePZero CovarianceIndependent variation, see previous examplePTotal CovarianceMating surfaces constrained to displace as a unitPCurve Fit Polynomial CovarianceMating surfaces constrained to be in form of a polynomialCan be any order of polynomial including a 1st order line fitPSinusoidal CovarianceApplied constraints of varying amplitudes and wavelengthsCapable of higher order wavesFourier series = ability to simulate almost any surface


Simulations, Results, and Verification

PUse of Monte Carlo SimulationsPZero Covariance CasePFull Covariance CasePCurve Fit Polynomial Covariance CasePComparison of Results

Two Plate Example

A

y

z

x

o

B

Po is a vector of gaps between the nodes along the mating surfaces.Pa and b are vectors of the displacements at each node required to assemble the parts in equilibrium -to close the gap.PTolerance on each mating edge = 0.05". Uniform distribution gives o= {0.00167 0 0.00167 0 ...} in2

part A part B

10.00" 10.00"

10.0

0"

0.10" thick alum. typ.

fixed edge, typ.

mat

ing

edge

x

y


Use of Monte Carlo SimulationsMethod to Verify CoFEA/STA

PA Monte Carlo simulation can be used as an alternate method to analyze the statistical closure forces of a compliant assembly.PLarge populations of individual assemblies with random variations are solved.P Results are obtained by examination of the mean and variance of the population of solutions.


Zero Covariance CaseTwo Plate Problem with 40 Elements Along Part Edges

PGap variation: each node on the mating edge is allowed to randomly vary within the tolerance zone.PGeometric covariance matrix has the constant variance magnitude down the diagonal, 0.00167in2

PSolved using both CoFEA/STA and Monte Carlo simulation


Closure Force Covariance MatrixSolution to Monte Carlo simulation

Zero Covariance

Case


Force Covariance SolutionSolution using CoFEA/STA

Two Plate Example

Zero Covariance

Case

Zero Covariance CaseCoFEA/STA and MCS Solution - Closure Force Standard Deviation

5 10 15 20 25 30 35 400

2000

4000

6000

8000

10000

12000

Standard Deviation of Closure Forces - Using FEA/STA

node number

forc

e (

lb)

max force = 12170.2 lb


Full Covariance CaseTwo Plate Problem with 40 Elements Along Part Edges

PGap variation: all nodes along the mating edge displace the same magnitudePGeometric covariance matrix is populated entirely by the gap variance magnitude, 0.00167in2

PSolved using both CoFEA/STA and Monte Carlo simulation


Full Covariance CaseSolution - Closure Force Covariance Matrix


Full Covariance CaseSolution - Closure Force Standard Deviation

5 10 15 20 25 30 35 400

200

400

600

800

1000

1200

1400

1600

1800

2000

Standard Deviation of Closure Forces - Using CoFEA/STA

node number

forc

e (

lb)



Curve Fit Polynomial Covariance Case

Two Plate Problem with 40 Elements Along Part Edges

PMating edge constrained to conform to 3rd order polynomialPSet of points which affect geometric covariance include all points along the mating edge (truncated ends algorithm)PSolved using both CoFEA/STA and Monte Carlo simulation


Curve Fit Polynomial Covariance CaseGap Covariance Matrix -CoFEA/STA Method


Curve Fit Polynomial Covariance CaseClosure Force Covariance Matrix - CoFEA/STA Method


Curve Fit Polynomial Covariance CaseClosure Force Standard Deviation - CoFEA/STA Method

5 10 15 20 25 30 35 400

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

Standard Deviation of Closure Forces - using CoFEA/STA

node number

forc

e (lb

)


Curve Fit Polynomial Covariance CaseGap Covariance Matrix - Monte Carlo Simulation


Curve Fit Polynomial Covariance CaseGap Histogram - Monte Carlo Simulation


Curve Fit Polynomial Covariance CaseClosure Force Covariance Matrix - Monte Carlo Simulation


Curve Fit Polynomial Covariance CaseClosure Force Histogram - Monte Carlo Simulation


Curve Fit Polynomial Covariance CaseClosure Force Standard Deviation - Monte Carlo Simulation

5 10 15 20 25 30 35 400

200

400

600

800

1000

1200

1400

1600

1800

2000

node number

forc

e (

lb)



Comparisonof Results

5 10 15 20 25 30 35 400

200

400

600

800

1000

1200

1400

1600

node number

1.12

1.14

1.16

1.18

1.2

1.22

1.24

x 104 Standard Deviation of Closure Forces - using CoFEA/STA

forc

e (lb

)

NO COVARIANCE

5TH ORDER POLYNOMIAL COVARIANCE

3RD ORDER POLYNOMIAL COVARIANCE

TOTAL COVARIANCE

forc

e (lb

)

PCoFEA/STA results compare almost perfectly with Monte Carlo simulations when boundary conditions are set up properlyPInclusion of geometric covariance dramatically reduces the variance of the closure forces


Conclusions

PThe FEA/STA method was introduced and found to be incomplete without consideration of surface continuityPGeometric covariance was introduced, derived, and incorporated into FEA/STA (CoFEA/STA)PThe effects of three forms of geometric covariance were investigated using the CoFEA/STA methodZero covariance, total covariance, and curve fit polynomial covariancePThe CoFEA/STA method was demonstrated and verified using Monte Carlo simulations

ConclusionsA variety of gap cases can be analyzed using the same FEA model, no additional iterations necessary.

Ka

Kb

o

y

z

x

Ka

Kb

y

z

x

Ka

Kb

o

y

z

x

Ka

Kby

z

x

Uniform X-Gap Twisted Offset

Uniform Y-Gap Rotated Gap/Interference

LimitationsDrawbacks of the CoFEA/STA method

PSmall deformation theory applies, elastic behaviorPAssemblies and parts can only be analyzed in their linear rangePNeed access to full stiffness matrices of compliant partsPNeed to be able to manipulate the stiffness matricesPNeed to include effects of covariance

Applications

PThe CoFEA/STA theory and method of tolerance analysis for assemblies containing compliant parts allows:Prediction of mean and variance closure forces in assemblies with compliant partsPrediction of statistical deformations and equilibrium position of compliant parts after assemblySensitivity analysis of assemblies containing compliant parts which allows improvement of design and increased robustnessModeling and improvement of manufacturing process variation

Future Work

PRefine CoFEA/STA for plates and shellsPExpand capabilities to models with more DOF per node - define geometric covariance with rotational DOFPPursue geometric covariance using sinusoidsPCreate database of manufacturing variation that can be used to characterize geometric covariance for common processesPCombine CoFEA/STA with optimization tools to find best order of assembly

Geometric Covariance in Compliant Assembly Tolerance Analysis Jeffrey B. Stout Brigham Young...

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