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

June 16, 2000
Presentation Outline

PIntroduction
PBackground
PCompliant Assembly Tolerance Analysis
PGeometric Covariance
PSimulations, Results, and Verification
PConclusions and Recommendations
Introduction
Traditional Statistical Assembly Tolerance Analysis Methods

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

analysis methods predict
the accumulation of                Loop
manufacturing variations in        1

assemblies.
PBased on the assumptions                             Loop 2

that the variations are
random and all of the parts    Two closed loops of a stacked block problem
are rigid.
Introduction
A new method must be used to analyze assemblies with
compliant parts.

FEA and STA can
be combined to
analyze assembly
tolerances with
compliant parts.
Background
Compliant Assembly Tolerance Analysis - Related Research

PVariation Modeling in Vehicle Assembly by Hsieh and
Oh
PTolerance Analysis of Flexible Sheet Metal
Assemblies by Liu, Hu, and Woo
PFEA/STA Method by Merkley and Chase,
contributions by Soman, Bihlmaier, and Stout
Compliant Assembly Tolerance Analysis
Method Outline - Combining STA with FEA

PFind Misalignment of Rigid Parts
PModel the Flexible Parts with FEA
PObtain the Condensed Stiffness Matrices
PApply Boundary Conditions
PSolve for Closure Forces
<Mean force
<Variation of force about the mean
Compliant Assembly Tolerance Analysis

Find Misalignment of Rigid Parts

B           C                   D      E

A
t
B           C                    D      E

A
t = TA2 + TB2 +TC2 +TD2
+TE2
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

Model the Flexible Parts with FEA

PCreate geometry
PMesh each compliant part, place nodes at fastener
locations
PApply displacement boundary conditions
POutput the global stiffness matrix
Compliant Assembly Tolerance Analysis

Obtain Condensed Stiffness Matrices
The Global Stiffness Matrix

Ka
10
9
8
7
6
5
4
3

PSum of element stiffnesses                    1
2

PIncludes all model degrees
of freedom
PSymmetric
PVery large and sparse
Ka =
Compliant Assembly Tolerance Analysis

Obtain Condensed Stiffness Matrices
Matrix Partitioning
PSort the stiffness matrix for each part to group the
boundary node DOF and interior node DOF
PPartition the matrix

Boundary DOF   Kbb                      Coupled DOF    Kbi

Kb    Kbi
b
Ka =                              =
Kib   Kii

Coupled DOF Kib                        Interior DOF Kii
Compliant Assembly Tolerance Analysis

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

Fb and di are unknown
Fb         Kb          Kbi       *
b                     b
Fb = Kbb**b +
=                                            Kbi**i
No forces on interior nodes, F   i   =0
Fi         Kib         Kii       *
i            0 = Kib**b +
Kii**i
Solve second                             Substitute this into
equation for *i   *i =   -Kii-1*Kib**b     the first equation
:
Fb = Kbb**b - Kbi*Kii-                = [Kbb - Kbi*Kii-
1*K **                            1*K ]**
ib  b                             ib     b
Super-element stiffness matrix:       Kse = Kbb - Kbi*Kii-1*Kib
Compliant Assembly Tolerance Analysis

Obtain Condensed Stiffness Matrices

PSuper-elements are well suited for assembly analysis
<Each part can be represented by a super-element
PReduces part stiffness matrix DOF to boundary node
DOF
PSimplify matrix algebra, and reduce computation time
PProvide sensitivities for statistical analysis
Compliant Assembly Tolerance Analysis

Apply Boundary Conditions
equilibrium position
**
Ka        a   b
Kb

Va Vb
nominal position
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.
Compliant Assembly Tolerance Analysis

Apply Boundary Conditions
2-D Assembly Closure Force -Derivation
*
o
Ka                                                 Kb
Vb
Va
nominal position

Misaligned Parts
NOTE: Displacements
and stiffness for all
Define Gap Vector: *o = vb -va
degrees of freedom must
be linear and elastic.     equilibrium position       *
*          b
a          *                  Kb
Ka                              o

Force-Closed Assembly
Define Closure Displacements: *a - *b = *o
Closure Forces: fa = -fb
Compliant Assembly Tolerance Analysis

Solve for Closure Forces
2-D Assembly Closure Force -Derivation

Equilibrium       *
*          b
a      *                         Kb
Ka                               Vb
Va     o

Nominal

fa = -fb                                       Closure Deflections:
Ka@*a = -Kb@*b                                 *a = (Ka + Kb)-1 @Kb@*o
*b = *o - *a
since *b = *a - *o,
then                                           Closure Forces:
fa = Ka@*a
Ka@*a = Kb@*o -
Kb@*a                                          fa = Ka@(Ka + Kb)-
1
Compliant Assembly Tolerance Analysis

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.
Compliant Assembly Tolerance Analysis

Solve for Closure Forces
Summary of Key Equations

fa = Ka@(Ka + Kb)-
1@K @*
b    o
Define the stiffness matrix ratio
Kra = Ka@(Ka + Kb)-
1@K
b
Use the following equations for single and mean cases
fa =                          mfa = Kra@mo
Kra@*o
Use the covariance equation for statistical cases
Sfa = Kra@So@KraT
Compliant Assembly Tolerance Analysis

Solve for Closure Forces
Derivation of Statistical FEA/STA Equations
fa =
Kra@*o
First statistical moment: mean
Second statistical
mfa = KraAmo
moment: variance

KraAmo)T]
Sfa = E[KraA(do-mo)A[Kra(do-mo)]T]
E=
statistical     Sfa = E[KraA(do-mo)A(do-mo)TAKraT]
expectation
operator                    Sfa = KraASoAKraT
Two Plate Example

Create Super-Element Stiffness
Matrices

PPlate are modeled using plane stress theory
P10 x 10 element grid giving 11 nodes along the coincident
edges
PThe global stiffness matrix of each part contains 121
nodes with two DOF per node, giving 242 total DOF
PThe super-element of each part only contained 22 DOF
Geometric Covariance

PThe Need for a Covariant Solution
PMaterial and Geometric Covariance Defined
PCombining Geometric Covariance and FEA/STA
PThe Curve Fit Polynomial Method
PForms of Geometric Covariance
Geometric Covariance

Material and Geometric Covariance
Defined
General Covariance

A coupling of two variables

Random variables           y                  y                y
x and y
x                x                x
Ellipses indicate
constant
Uncorrelated       Partially        Fully
probability                               Correlated       Correlated
Geometric Covariance

Material and Geometric Covariance
Defined
Material Covariance
ex = s/E
ey = -
nAs/E
ey = -
nAex
Strain is fully correlated

Force
d = KAf
The stiffness matrix
defines the correlation
of neighboring point
displacements when a
single point is subjected
to a force.
Geometric Covariance

Material and Geometric Covariance
Defined
Geometric Covariance

Geometric                  Variation of
variation of               points will be
each point is              in the form of
unlikely to be             a continuous
completely                 surface
random
Geometric Covariance

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

Define the part geometric covariance matrix to be a function of the
displacement sensitivities and the part variation.
The sensitivity matrix Sa defines the sensitivity
Ga=                    of each DOF's position with respect to all other
Sa@Sa@SaT              DOF. Sa is symmetric by nature.

The same rule applies to the whole gap as well.
Go=
So@So@SoT
The geometric covariance term replaces the variance term
to form the CoFEA/STA equation:            S =  fa
Kra@Go@KraT
Geometric Covariance

The Curve Fit Polynomial Method

i-1       i   i+1
i-2
Variation of points
is constrained to be                                         y
in the form of a                                                   x
polynomial curve                          yi = co + c1Axi + c2Axi2 + c3Axi3

Find sensitivity of other y values with respect to yi
yi = ... + si-2Ayi-2 + si-1Ayi-1 + siAyi + si+1Ayi+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.
Geometric Covariance

Forms of Geometric Covariance
PZero Covariance
<Independent variation, see previous example
PTotal Covariance
<Mating surfaces constrained to displace as a unit
PCurve Fit Polynomial Covariance
<Mating surfaces constrained to be in form of a polynomial
<Can be any order of polynomial including a 1st order line fit
PSinusoidal Covariance
<Applied constraints of varying amplitudes and wavelengths
<Capable of higher order waves
<Fourier series = ability to simulate almost any surface
Simulations, Results, and Verification

PUse of Monte Carlo Simulations
PZero Covariance Case
PFull Covariance Case
PCurve Fit Polynomial Covariance Case
PComparison of Results
Simulations, Results, and Verification

Two Plate Example
*
o
P*o is a vector of gaps                                            A
between the nodes along the
mating surfaces.                             y                                          B
P*a and *b are vectors of                                z
the displacements at each                            x
node required to assemble                             10.00"                         10.00"
the parts in equilibrium -to
close the gap.
PTolerance on each mating
part A                          part B
edge = 0.05". Uniform
distribution gives *o=
{0.00167 0 0.00167 0 ...}
y                    0.10" thick alum. typ.
in2                                              x
fixed edge, typ.
Simulations, Results, and Verification

Use of Monte Carlo Simulations
Method 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.
Simulations, Results, and Verification

Zero Covariance Case
Two 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
Simulations, Results, and Verification

Closure Force Covariance Matrix
Solution to Monte Carlo simulation

Zero
Covariance
Case
Two Plate Example

Force Covariance Solution
Solution using CoFEA/STA

Zero
Covariance
Case
Simulations, Results, and Verification

Zero Covariance Case
CoFEA/STA and MCS Solution - Closure Force Standard Deviation
Standard Deviation of Closure Forces - Using FEA/STA

12000

10000

8000

6000

4000

2000

0
5         10         15        20        25           30         35   40
max force = 12170.2 lb              node number
Simulations, Results, and Verification

Full Covariance Case
Two Plate Problem with 40 Elements Along Part Edges

PGap variation: all nodes along the mating edge displace
the same magnitude
PGeometric covariance matrix is populated entirely by the
gap variance magnitude, 0.00167in2
PSolved using both CoFEA/STA and Monte Carlo
simulation
Simulations, Results, and Verification

Full Covariance Case
Solution - Closure Force Covariance Matrix
Simulations, Results, and Verification

Full Covariance Case
Solution - Closure Force Standard Deviation
Standard Deviation of Closure Forces - Using CoFEA/STA

2000

1800

1600

1400

1200

1000

800

600

400

200

0
5          10        15        20        25          30           35   40
node number
max force = 521.8 lb
Simulations, Results, and Verification

Curve Fit Polynomial Covariance Case
Two Plate Problem with 40 Elements Along Part Edges

PMating edge constrained to conform to 3rd order
polynomial
PSet 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
Simulations, Results, and Verification

Curve Fit Polynomial Covariance Case
Gap Covariance Matrix -CoFEA/STA Method
Simulations, Results, and Verification

Curve Fit Polynomial Covariance Case
Closure Force Covariance Matrix - CoFEA/STA Method
Simulations, Results, and Verification

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

Standard Deviation of Closure Forces - using CoFEA/STA

2200

2000

1800

1600

1400

1200

1000

800

600

400

200

0
5       10        15         20        25          30         35   40
node number
Simulations, Results, and Verification

Curve Fit Polynomial Covariance Case
Gap Covariance Matrix - Monte Carlo Simulation
Simulations, Results, and Verification

Curve Fit Polynomial Covariance Case
Gap Histogram - Monte Carlo Simulation
Simulations, Results, and Verification

Curve Fit Polynomial Covariance Case
Closure Force Covariance Matrix - Monte Carlo Simulation
Simulations, Results, and Verification

Curve Fit Polynomial Covariance Case
Closure Force Histogram - Monte Carlo Simulation
Simulations, Results, and Verification

Curve Fit Polynomial Covariance Case
Closure Force Standard Deviation - Monte Carlo Simulation
2000

1800

1600

1400

1200

1000

800

600

400

200

0
5          10   15     20        25   30   35   40
node number
max force = 862.7 lb
Simulations, Results, and Verification
4
x 10                Standard Deviation of Closure Forces - using CoFEA/STA

1.24

Comparison                               1.22
NO COVARIANCE

of Results                                1.2

1.18

PCoFEA/STA results                       1.16

compare almost                           1.14

perfectly with Monte                     1.12

Carlo simulations when                   1600

boundary conditions are                  1400

1200

set up properly
PInclusion of geometric                  1000                   5TH ORDER POLYNOMIAL COVARIANCE

800

covariance dramatically                   600

reduces the variance of                   400
TOTAL COVARIANCE

the closure forces                        200
3RD ORDER POLYNOMIAL COVARIANCE

0
5        10         15          20          25         30        35   40
node number
Conclusions

PThe FEA/STA method was introduced and found to be incomplete
without consideration of surface continuity
PGeometric 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 method
<Zero covariance, total covariance, and curve fit polynomial covariance
PThe CoFEA/STA method was demonstrated and verified using
Monte Carlo simulations
A variety of gap cases can be
Conclusions                                analyzed using the same FEA model,

*
o
Ka                                             Ka                  2

y                           Kb                 y                            Kb
z                                              z

x                                              x
Uniform X-Gap                                  Twisted Offset

*                                           2
Ka                                             Ka
o

y                           Kb                 y                            Kb
z                                              z

x       Uniform Y-Gap                          x       Rotated Gap/Interference
Limitations
Drawbacks of the CoFEA/STA method

PSmall deformation theory applies, elastic behavior
PAssemblies and parts can only be analyzed in their linear
range
PNeed to be able to manipulate the stiffness matrices
PNeed 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 parts
<Prediction of statistical deformations and equilibrium position of
compliant parts after assembly
<Sensitivity analysis of assemblies containing compliant parts
which allows improvement of design and increased robustness
<Modeling and improvement of manufacturing process variation
Future Work

PRefine CoFEA/STA for plates and shells
PExpand capabilities to models with more DOF per node -
define geometric covariance with rotational DOF
PPursue geometric covariance using sinusoids
PCreate database of manufacturing variation that can be
used to characterize geometric covariance for common
processes
PCombine CoFEA/STA with optimization tools to find
best order of assembly

```
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