Scott Pierce M.I. Technologies, L.L.C. Duluth, GA David Rosen George W. Woodruff School of Mechanical Engineering Georgia Institute of Technology Geometric Tolerance Analysis Methods for Imperfect-Form Assemblies
Dec 20, 2015
Scott Pierce
M.I. Technologies, L.L.C.
Duluth, GA
David Rosen
George W. Woodruff School of Mechanical Engineering
Georgia Institute of Technology
Geometric Tolerance Analysis Methods for Imperfect-Form Assemblies
Outline
• Introduction and Background– Motivation
– Our Approach: The Generate and Test Method
• Development of the Tolerance Analysis Module– The Variational Modeling Environment
– Simulation of Mating Between Imperfect-Form Components
• Case Study
• Conclusions and Future Work
Motivation
• The principal objective of this research is the development of a new, computer-aided approach to tolerance analysis– The purpose of tolerance
analysis is to define the relationships between tolerance values, product functionality and manufacturing cost
– In particular, we are interested in analysis of geometric tolerances that control form and orientation
A Very Simple Example: Two Squares in a Slot
Motivation
• A compound slider mechanism composed of several components
• Multiple mating surfaces
• Alignment between the driver, bender and bonnet affects functionality of the mechanism
• Manufacturer has quality assurance data and process experience that define “typical manufacturing errors”
A More Complex Example: The High-Speed Stapling Mechanism
How can the experience-based process knowledge be incorporated into this complex tolerance analysis problem?
Our Approach: Generate-and-Test
Step 2: Test the Effects of Manufacturing Errors by Simulating Mating Between As-Manufactured Components and Measuring Attributes of Functionality
Our Approach: Generate-and-Test
• This Generate-and-Test Process is Repeated for a Series of Error Geometries and Magnitudes That are Representative of the Proposed Manufacturing Processes
• Information Gained from this Analysis is Used to Guide the Tolerance Selection Process
Development of the Tolerance Analysis Module
• The Variational Modeling Environment
• Simulation of Mating Between Imperfect-Form Components
The Variational Modeling Environment
• The ACIS geometry engine is used as the modeling core
• Constructed as a set of C++ classes and extensions to the ACIS api’s
• Basic capabilities of the modeling environment allow:
– Creation of prismatic parts
– Use of Boolean operations to generate more complex prismatic geometry
– Application of rigid-body transformations
• We have built a CAD environment that uses a NURBS surface representation to construct models of imperfect-form variants of prismatic components
The Variational Modeling Environment
• Variational modeling capabilities allow:
– Definition of pointset classes that define variant surfaces
– Fitting of NURBS surfaces to a pointset to within a specified fitting tolerance.
– Replacing nominally planar faces of prismatic components with variant NURBS surfaces.
The Variational Modeling Environment
Table 3.1: Measurement of Fitting Accuracy of Nurbs Surfaces To PointsetMeasurements For A Series Of Milling Errors.
Manufacturing Error SurfaceLength(mm.)
SurfaceWidth(mm.)
MaximumDeviation of theMachining Datafrom a PerfectPlane (mm.)
AverageFittingError(mm.)
MaximumFittingError(mm.)
Side-Milling CutterDeflection - 0.25 mm.Depth of Cut
20.3 15.2 0.0458 0.00002 0.00004
Side-Milling CutterDeflection - 1.2 mm.Depth of Cut
20.3 15.2 0.0902 0.00002 0.00004
Side-Milling CutterDeflection - 0.5 mm.Depth of Cut
50.0 25.4 0.0695 0.00002 0.00008
Side-Milling CutterDeflection - 1.2 mm.Depth of Cut
50.0 25.4 0.1144 0.00000 0.00006
Transverse CutterDeflection of a SlotBottom Surface- Slot CutBy Side Milling WithVery Light Cut
70.0 40.0 0.0038 0.00000 0.00005
Tooth-to-Tooth Runoutin Upmilling Using anArbor Cutter (Note: Thissurface has a high-frequency periodic error)
24.0 22.0 0.0860 0.00001 0.00004
Verification: The Variational Modeling Module supports modeling of as-manufactured component variants to a resolution that is significant to tolerance analysis.
Typical machining errors for end milling are on the order of 0.01 mm.
Simulation of Mating Between Imperfect-Form Components
Minimize Z = total distance from perfect fits.t. non-interference between components
• Simulation of mating between surfaces that can be represented analytically is a well understood problem.
• Simulation of mating between non-analytic, freeform surfaces is a much more difficult problem.
Following a formulation originally proposed by Turner, we have chosen to formulate the mating problem as a mathematical programming problem of the form:
Simulation of Mating Between Imperfect-Form Components
• How should “perfect fit” be defined?– For perfect-form, planar surfaces perfect fit means that faces are
coplanar and that outward-facing normal vectors point in opposite directions.
– Coplanarity implies that the distance between any point on one surface and the corresponding closest point on the other surface is zero.
• This leads to the idea that for imperfect-form surfaces we should try to minimize the distance between any point on one surface and the corresponding closest point on the other surface
Simulation of Mating Between Imperfect-Form Components
• We have chosen to use sampling grids to perform distance measurements and interference detection between surfaces
• We find that the use of sampling grids is much more computationally efficient than the use of Boolean intersections
• Grid density can be adjusted so that the resolution is fine enough to represent any significant surface feature.
Mating Pair
Mating Pair
Non-Mating But Potentially Interfering Pair
Simulation of Mating Between Imperfect-Form Components
Using this sampling grid approach, we can formulate the mating problem as a constrained optimization problem:
Mating Pair
Mating Pair
Non-Mating But Potentially Interfering Pair
M = number of mating face pairsN = total number of mating face pairs, both mating and potentially interferingmF = number of gridpoints in the u-parameter direction for the given face pairnF = number of gridpoints in the v-parameter direction for the given face pair = minimum signed distance from gridpoint ij to the mating surface.
Find: = (roll, pitch, yaw, x, y, z) = six degrees of freedom of the movable body.x
d xF ij ( )
M in im ize Zm n
d x
F FF
M F ijj
n
i
m
F
M FF
: ( )
1
1
2
111
1
2
s t d F ij. . : 0 F = 1…N, i = 1…mF, j = 1…nF
Simulation of Mating Between Imperfect-Form Components
• Selection of a Solution Method:
• Finding the minimum distance between a grid point and the mating surface requires the solution of a point-projection problem
• Point projections are the most computationally-intensive part of the mating simulation
Closest Point on Mating Surface
Grid Point
d F ij
Selection of a solution method (continued):
We used published measurements of end-milled surfaces to generate test surfaces
We explored the topography of the solution space generated by mating these surfaces
We found the solution space to be nonlinear
We found the boundaries of the feasible region to be nonlinear and in some cases non-convex
Simulation of Mating Between Imperfect-Form Components
Simulation of Mating Between Imperfect-Form Components
• We examined several potential solution methods including:
– Successive linearization:– We have shown that both the objective and the constraints are
highly nonlinear. Successive linearization will generally not converge well under these conditions
– Generalized reduced gradient methods– Capable of handling nonlinear problems
– Requires solution of a prohibitively large number of point projection problems
Simulation of Mating Between Imperfect-Form Components
• We have chosen to modify the formulation of the mating problem to use the penalty function approach:
where:WF = mating/non-mating face switch
WINT = interference weighting factor (=100)
The penalty function formulation converts the constrained formulation into an unconstrained problem, allowing the use of unconstrained optimization algorithms
M in im ize Zm n
W d x W d x
F FF
N F ij
n o nerferin g
IN T ij
erfer in g
j
n
i
m
F
N FF
: in t in t
1
1
22
111
1
2
Simulation of Mating Between Imperfect-Form Components
To test potential solution algorithms we used two test problems:
End-milling cutter deflection with perfect-fit
End-milling cutter deflection with four different surfaces
“Correct” objective = 0.1484
Simulation of Mating Between Imperfect-Form Components
• We tested three different solution algorithms for use with the penalty-function formulation:
• Method 1: Simulated Annealing With Downhill Simplex:– Analogous to an annealing process, there is a finite probability of
accepting an “uphill” move
– This probability is reduced as the “temperature” is reduced– In theory, allows a more thorough search of the solution space so
that local minima are avoided
TZZCXP /exp)( 122 Where Z2 > Z1
In tests where we purposely introduced local minima into the solution space, simulated annealing was not very successful in avoiding them.
Simulation of Mating Between Imperfect-Form Components
• Method 2: Randomized Hooke-Jeeves pattern search– Direct search method does not require the calculation of numerical
gradients
– Explores the region around a test point for the steepest descent direction, then moves in that direction until descent stops
– When a downhill move cannot be found the step size is reduced and the exploration is repeated
– Very robust in the presence of nonlinearities
Simulation of Mating Between Imperfect-Form Components
• Method 3: Quasi-Newton Method– Gradient-based method
– Uses a quadratic approximation to the objective function
– Use first-order information to approximate the Hessian
– Use line searches to generate the step size
– We use the Broyden-Fletcher-Goldfarb-Shanno form of the quasi-Newton method.
– We used a line search that starts with a quadratic approximation, then reverts to a golden section search when convergence slows.
Simulation of Mating Between Imperfect-Form Components
Convergence of all three solution methods for the four surface cutter deflection example
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 200 400 600 800 1000 1200
Objective Evaluations
Bes
t Obj
ectiv
e (m
m.)
Simulated Annealing
Hooke-Jeeves
BFGS
Correct Answer:
Objective = 0.1484
Best Answer from This Test:
BFGS Objective = 0.1484
Simulation of Mating Between Imperfect-Form Components
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0 500 1000 1500
Objective Iterations
Be
st
Ob
jec
tiv
e (
mm
.)
Simulated Annealing
Hooke-Jeeves
BFGS
Convergence of all three solution methods for the perfect-fit cutter deflection example (correct objectivevalue = 0)
Simulation of Mating Between Imperfect-Form Components
Use of the hybrid BFGS/Hooke-Jeeves algorithm for the perfect-fit problem (correct objective value = 0)
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
0.016
0.018
0.02
0 50 100 150 200 250 300 350 400
Objective Evaluations
Bes
t Obj
ectiv
e (m
m.)
Hooke-Jeeves Algorithm Alone
Hybrid Algorithm When BFGS is Active
Hybrid Algorithm After “Fallback” to Hooke-Jeeves
BFGS Algorithm Alone
Tolerance Analysis Module - Summary
• Allows construction and manipulation of “as-manufactured” variant models
• Simulates assembly of imperfect-form component variants
• We now have a testbed that can be used to demonstrate the generate-and-test approach to tolerance analysis
Case Study
• The case study is built upon a simplified version of the high-speed stapling mechanism
• The components that have the most influence on the quality of the stapling process are included in the study:– Driver
– Bender
– Bonnet
BONNET
BENDER DRIVER
WIRE
PAGESTO BESTAPLED
Case Study
Attribute of Functionality - Z-axis Rotation
Z-Axis Rotation Attribute: Maximum possible difference between bender Z-rotation and driver Z-rotation
Functional Limit: If Z-axis rotation exceeds 1.7 mrad the staple will buckle
+X
+Y
DIRECTION OF FORCEON STAPLE LEG
Case Study
In order to control the attributes of functionality we assign geometric tolerances of form and orientation:
Bender
What tolerance values should be assigned in order to ensure that the mechanism will function?
Case Study
• Step 1: Group the surfaces of the stapling mechanism into four groups
• All surfaces within a group would be manufactured in a single setup, therefore they share a common level of precision
Case Study
• Step 2: Generate “as-manufactured” models of higher precision (higher cost) and lower precision (lower cost) variants of each surface group
• All error data comes from published measurements or results of end-milling simulations.
Case Study
• Step 2 (continued):
• Each component variant was measured using a functional gauging routine
• The results of each measurement were the tolerance values that the particular variant would meet
GROOVE AT HIGHER
PRECISION, OUTER
SURFACES AT HIGHER
PRECISION
GROOVE AT LOW ER
PRECISION, OUTER
SURFACES AT LOW ER PRECISION
OUTER BOTTOM FACE FLATNESS
(mm.) 0.05 0.12OUTER NEGATIVE Y
FACE PERPENDICULARITY
(mm.) 0.11 0.19
Case Study
• Mating simulation was performed for every combination of higher/lower precision surface groups
• The mechanism components were mated at a series of positions through the stapling process
• Functional attributes were measured
• The results were used to construct a 24 full-factorial analysis for each functional attribute
Case Study
Significant effects: Bender Groove, Bonnet
Setting both of these surface groups to higher precision results in a maximum Z-axis rotation of 1.92 mrad.
This is still above the functional limit of 1.7mrad, so the tolerance on the bender groove needs to be tightened further if possible
DRIVERBENDER GROOVE
BENDER OUTER BONNET
SINGLE FACTOR EFFECT -0.49713 -39.21363 -1.01513 -8.803
DRIVER/ BENDER GROOVE
DRIVER/ BENDER OUTER
BENDER GROOVE/ BENDER OUTER
DRIVER/ BONNET
BENDER GROOVE/ BONNET
BENDER OUTER/ BONNET
TWO FACTOR INTERACTION EFFECT -0.49713 0 -0.72031 -0.4569 -2.40009 -0.56839
Analysis of Variance for the Z-axis rotation attribute:
Case Study
Combining the results of the Z-axis rotation study with results from a study on a second functional attribute, we selected geometric tolerance values that strike a balance between the need for precision and manufacturing cost.
Conclusions
• I have described the development of an environment for computer-aided tolerance analysis.
• Allows the inclusion of experience-based manufacturing information through the use of the generate-and-test method of tolerance analysis.
• Development of an effective algorithm for simulation of mating between imperfect-form, non-analytic surfaces was key to the generate-and-test method.
• Through the case study, I have shown that this method can be used as an aid in the selection of geometric tolerances of form and orientation.
Possibilities for Future Work
• Link mating simulation with a kinematic analysis in order to bring force balance information into the picture.
• Extend to non-prismatic geometry.
• Apply the non-analytic surface mating methods in areas other than tolerance analysis (e.g. design of components whose perfect-form geometry is non-analytic).