Scott%20Pierce

1

Geometric Tolerance Analysis Methods for
Imperfect-Form Assemblies

• Scott Pierce
• M.I. Technologies, L.L.C.
• Duluth, GA
• David Rosen
• George W. Woodruff School of Mechanical
  Engineering
• Georgia Institute of Technology

2
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

3
Motivation
A Very Simple Example Two Squares in a Slot

• 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

4
Motivation
A More Complex Example The High-Speed Stapling
Mechanism

• 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

How can the experience-based process knowledge be
incorporated into this complex tolerance analysis
problem?
5
Our Approach Generate-and-Test
Step 1 Generate As Manufactured Component
Models
6
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
7
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

8
Development of the Tolerance Analysis Module

• The Variational Modeling Environment
• Simulation of Mating Between Imperfect-Form
  Components

9
The Variational Modeling Environment

• We have built a CAD environment that uses a
  NURBS surface representation to construct models
  of imperfect-form variants of prismatic components

• The ACIS geometry engine is used as the modeling
  core
• Constructed as a set of C classes and
  extensions to the ACIS apis

• 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

10
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.

11
The Variational Modeling Environment
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.
12
Simulation of Mating Between Imperfect-Form
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
Minimize Z total distance from perfect fit s.t.
non-interference between components
13
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

14
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.

15
Simulation of Mating Between Imperfect-Form
Components
Using this sampling grid approach, we can
formulate the mating problem as a constrained
optimization problem
Find (roll, pitch, yaw, x, y, z) six
degrees of freedom of the movable body.
M number of mating face pairs N total
number of mating face pairs, both mating and
potentially interfering mF number of
gridpoints in the u-parameter direction for the
given face pair nF number of gridpoints in the
v-parameter direction for the given face pair
minimum signed distance from gridpoint ij to
the mating surface.
F 1N, i 1mF, j 1nF
16
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-int
  ensive part of the mating simulation

17
Simulation of Mating Between Imperfect-Form
Components
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
18
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

19
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
20
Simulation of Mating Between Imperfect-Form
Components
To test potential solution algorithms we used two
test problems
End-milling cutter deflection with four
different surfaces Correct objective 0.1484
End-milling cutter deflection with perfect-fit
21
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

Where Z2 gt Z1
In tests where we purposely introduced local
minima into the solution space, simulated
annealing was not very successful in avoiding
them.
22
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

23
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.

24
Simulation of Mating Between Imperfect-Form
Components
Correct Answer Objective 0.1484 Best Answer
from This Test BFGS Objective 0.1484
Convergence of all three solution methods for the
four surface cutter deflection example
25
Simulation of Mating Between Imperfect-Form
Components
Convergence of all three solution methods for the
perfect-fit cutter deflection example (correct
objective value 0)
26
Simulation of Mating Between Imperfect-Form
Components
Use of the hybrid BFGS/Hooke-Jeeves algorithm for
the perfect-fit problem (correct objective value
0)
27
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

28
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

29
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
30
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?
31
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

32
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.

33
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

34
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

35
Case Study
Analysis of Variance for the Z-axis rotation
attribute
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
36
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.
37
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.

38
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).