Publication #04-2:

**Title: **Variation Simulation of Fixtured Assembly Processes for Compliant Structures Using Piecewise-Linear Analysis, 2004

**Author: **Michael Leon Stewart

** Abstract: **
While variation analysis methods for compliant assemblies are not new, little has been done to
include the effects of multi-step, fixtured assembly processes. This thesis introduces a new
method to statistically analyze compliant part assembly processes using fixtures. This method,
consistent with the FASTA method developed at BYU, yields both a mean and a variant solution.
The method, called Piecewise-Linear Elastic Analysis, or PLEA, is developed for predicting the
residual stress, deformation and springback variation in compliant assemblies. A comprehensive,
step-by-step analysis map is provided. PLEA is validated on a simple, laboratory assembly and a more
complex, production assembly. Significant modeling findings are reported as well as the comparison
of the analytical to physical results.

Publication #02-2:

**Title: **A Methodology for Statistical Tolerance Analysis of Flexible Assemblies, 2002

**Author: **Alan J. Mortensen

** Abstract: **
Statistical tolerance analysis has become a vital link between product design and manufacture,
aiding in the constant drive to increase quality, while lowering production costs. Although
conventional tolerance analysis methods are limited by the assumption of part rigidity, recent steps
have been taken to broaden this field of study to include flexible parts. Such steps include the
combination of statistical tolerance analysis with finite element analysis to predict closure
forces, deformations and internal stresses resulting from assembly processes. The achievement of
such would greatly aid aerospace, automotive and other industries that routinely use compliant parts
in design. The objective of this thesis is to present an integrated methodology for the statistical
tolerance analysis of flexible assemblies. To accomplish this objective, four main tasks were
completed:

- Integration of rigid STA with FEA for variation analysis of assemblies with compliant parts.
- Use of commercial FEA software for modeling of compliant assemblies.
- Statistical FEA, incorporating both material (elastic coupled) and geometric (continuity coupled) covariance.
- Inclusion of both compliant and rigid covariance components.

Publication #01-3:

**Title: **Constraint Analysis of Assemblies Using Screw Theory and Tolerance Sensitivities, 2001

**Author: **Daniel K. Smith

** Abstract: **
Constraint problems in assemblies due to parts with underconstrained or overconstrained degrees of
freedom need to be identified so they can be controlled by designers. The Variation-based Constraint
Analysis of Assemblies (VCAA) combines tolerance analysis and constraint analysis, making it
possible to perform both analyses on an assembly simultaneously. The variation sensitivity matrices
used in the Direct Linearization Method (DLM) of tolerance analysis contain constraint information
that can be interpreted using screw theory methods.

Previous work on screw theory constraint analysis is connected to tolerance analysis through the VCAA. The variation sensitivity matrices contain screw matrices that can be analyzed for under- and overconstraints. The sensitivities, calculated through the Global Coordinate Method, can show if parts have underconstrained, mobile degrees of freedom, or if the parts have redundant, overconstrained degrees of freedom. Using the steps outlined in the VCAA method, it is possible to extract screw matrices directly from the sensitivity matrices in order to analyze assemblies.

The development of the VCAA method is outlined as well as the analysis of several case study assemblies. The VCAA method correctly identifies underconstrained degrees of freedom using the dependent variation sensitivity matrix and overconstraints using the geometric feature variation sensitivity matrix. The case studies show that after a DLM tolerance analysis is performed on an assembly, it can successfully be analyzed for constraint problems using the information gained from the tolerance analysis. The advantage of the VCAA method is this ability to perform a constraint analysis coincident with a variation analysis.

Publication #01-2:

**Title: **Incorporating Geometric Feature Variation with Kinematic Tolerance Analysis of 3d Assemblies, 2001

**Author: **Jeffrey G. Dabling

** Abstract: **
As technology increases and performance requirements continually tighten, the cost and required
precision of assemblies increase as well. There exists a strong need for increased attention to
tolerance analysis to enable high-precision assemblies to be manuÂfactured at lower cost. Methods
for tolerance analysis of 2D and 3D assemblies have been developed and are in use in both research
and commercial applications. These methods are usually very complex, and are usually best
implemented in some type of automated software. However, existing variation analysis software does
not yet have an industryÂwide user base. In an effort to greatly increase the proportion of
companies with variation analysis capability, Paul Faerber [Faerber 1999] developed the TAKS method
(Tolerance Analysis Using Kinematically-Derived Sensitivities), allowing the creation of a tolerance
model from a kinematic mechanism using equivalent variational mechanisms (EVMs). Such a model can
then utilize much more widely used kinematic analysis software to perÂform tolerance analysis on an
assembly, and even increment the position of the mechaÂnism for conducting analyses at every desired
position very quickly. However, the TAKS method was limited to 2D and did not incorporate geometric
feature variation. This thesis expands upon FaerberÕs work to develop a kinematic analogy to 3D
tolerance analysis that also allows for inclusion of geometric feature variation sources. Libraries
of 3D EVMs are introduced that incorporate both dimensional and geometric variation in a kinematic
model. Methods of incorporating these equivalent joints into a kinematic model are set forth and
analysis techniques are explained. The use of a commercial kinematic software package to automate
the analysis is also demonstrated.

Publication #00-3:

**Title: **A Closed Form Solution For Nonlinear Tolerance Analysis, 2000

**Author: **Geoffrey K. Carlson

** Abstract: **
Predicting production yields for critical assembly features is computationally intensive when skewed
distributions or nonlinear assemblies are present. Current methods require iterative solutions for
nonlinear implicit assembly functions. Monte Carlo Simulation (MCS) requires an iterative solution
for each simulated assembly. Method of System Moments (MSM) only analyzes a single assembly, but
still requires an iterative solution for each dimension in the assembly function.

The Direct Second Order Method (DSO) is a new technique that eliminates the requirement for iterative solutions by providing the second order sensitivities in closed form. By extending the analogy between variation and kinematics, the second order sensitivities can be obtained by performing a kinematic acceleration analysis of a vector loop assembly model. Combining various terms of the acceleration analysis will yield the second order sensitivities in closed form, without iteration.

The second order sensitivities can be used in conjunction with MSM to calculate the statistical moments of the critical assembly features. By neglecting the components that contribute very little, the number of terms in the MSM raw moment equations can be greatly reduced. These simplified expressions provide accurate estimates of the assembly mean, variance, and skewness.

Applications of the DSO method and the truncated MSM raw moment equations are presented to illustrate how these tools can be used to perform nonlinear variation analysis more efficiently.

Publication #00-2:

**Title: **A Second-Order Method For Assembly Tolerance Analysis, 2000

**Author: **Charles G. Glancy and Kenneth W. Chase

** Abstract: **
Linear analysis and Monte Carlo simulation are
two well-established methods for statistical tolerance analysis of mechanical
assemblies. Both methods have advantages and disadvantages. The Linearized Method,
a form of linear analysis, provides fast analysis, tolerance allocation, and
the capability to solve closed loop constraints. However, the Linearized Method
does not accurately approximate nonlinear geometric effects or allow for non-normally
distributed input or output distributions. Monte Carlo simulation, on the other
hand, does accurately model nonlinear effects and allow for non-normally distributed
input and output distributions. Of course, Monte Carlo simulation can be computationally
expensive and must be re-run when any input variable is modified.

The second-order tolerance analysis (SOTA) method attempts to combine the advantages of the Linearized Method with the advantages of Monte Carlo simulation. The SOTA method applies the Method of System Moments to implicit variables of a system of nonlinear equations. The SOTA method achieves the benefits of speed, tolerance allocation, closed-loop constraints, non-linear geometric effects and non-normal input and output distributions. The SOTA method offers significant benefits as a nonlinear analysis tool suitable for use in design iteration.

A comparison was performed between the Linearized Method, Monte Carlo simulation, and the SOTA method. The SOTA method provided a comparable nonlinear analysis to Monte Carlo simulation with 106 samples. The analysis time of the SOTA method was comparable to the Linearized Method.

Publication #99-6:

**Title: **Tolerance Allocation Methods for Designers, 1999

**Author:** Kenneth W. Chase

** Abstract:**
Tolerance allocation is a design tool. It provides a rational basis for assigning tolerances to
dimensions. Several algorithms are described in this paper for performing tolerance allocation,
which is defined as the re-distribution of the “tolerance budget” within an assembly to reduce
over-all cost of production, while meeting target levels for quality.

The task of placing +/- tolerances on each dimension of a CAD model or set of engineering drawings may seem menial and of little consequence. However, it can have enormous impact on cost and quality. On the engineering design side, it affects the fit and function of the final product, which can cause poor performance and dissatisfied customers. On the manufacturing side, tolerance requirements determine the selection of machines, tooling and fixtures; operator skill levels and setup costs; inspection precision and gaging; and scrap and rework. In short, it affects nearly every aspect of the product life cycle.

Using the new CAD-based tolerance analysis tools, designers can perform variation analysis on the CAD model, before parts are made or tooling purchased. They can determine where tolerance controls are needed and how tight the limits must be set, to assist in process planning and tool design. Using the same CAD model, production personnel can use tolerance tools to determine whether marginal or out-of-spec parts can still be used. Tolerance analysis tools provide a quantitative basis for design for manufacture decisions, resulting in shorter product development time and increased quality. There is probably no other design improvement effort which can yield greater benefits for less cost than the careful analysis and assignment of tolerances.

Publication #99-5:

**Title: **Minimum-Cost Tolerance Allocation, 1999**
Author:** Kenneth W. Chase

*Abstract:*

Tolerance allocation is a design tool for reducing over-all cost of production, while meeting target levels for quality. Using allocation tools, a designer may re-distribute the “tolerance budget” within an assembly, systematically tightening tolerances on less expensive processes and loosening tolerances on costly processes, for a net reduction in cost. Several algorithms are described in this paper for performing tolerance allocation automatically, based on optimization techniques. A cost vs. tolerance function is used to drive the optimization to the minimum overall cost. The methods provide a rational basis for assigning tolerances to dimensions.

Publication #99-4:

**Title: **Multi-Dimensional Tolerance Analysis, 1999**
Author:** Kenneth W. Chase

** Abstract: ** Tolerance analysis of assemblies promotes concurrent engineering by bringing engineering requirements and manufacturing capabilities together in a common model. It provides a quantitative design tool for predicting the effects of manufacturing variation on performance and cost in a computer-based design environment.

A new method, called the Direct Linearization Method (DLM), is presented for tolerance analysis of 2D and 3-D mechanical assemblies. The models are constructed of common engineering elements: vector chains, kinematic joints, assembly datums, dimensional tolerances, geometric feature tolerances and assembly tolerance limits. The method is well suited for integration with commercial CAD systems.

The DLM applies matrix algebra and root sum squares error analysis to vector loop based models to estimate tolerance stackup in assemblies. Three sources of variation may readily be included in the models: dimensional, geometric and kinematic. Dimensional variations account for small changes in size due to manufacturing processes. Geometric variations describe changes in shape, location and orientation of features. Kinematic variations describe the propagation of variation through an assembly by small adjustments between mating parts.

The DLM has significant advantages over traditional tolerance analysis methods. It is not computationally intensive, so it is ideally suited for iterative design. Tolerance allocation algorithms may also be applied without repeating the analysis. Manual entry of equations is not needed. A systematic procedure for deriving the assembly equations directly from the CAD model eliminates this task for designers. Tolerance sensitivities and tolerance stackup expressions may also be derived automatically. Furthermore, the DLM does not require the functions describing the assembly variations to be explicit. Matrix methods make the algebraic manipulation of implicit equations unnecessary.

A general formulation is derived, with numerical examples and case studies to illustrate the integration of the DLM method with CAD systems and the design process.

Thesis #99-3:**
Title:
**Tolerance Analysis of Assemblies Using Kinematically Derived Sensitivities

** Abstract: **To estimate tolerance accumulation in an assembly requires the calculation of the tolerance sensitivity of critical assembly features to each source of dimensional variation in the
assembly. A Root-Sum-Squares expression may then be formulated to predict the variance and percent rejects to expect in production. To analyze accumulation in a mechanism,
rather than a static assembly, requires that this procedure be repeated in multiple positions, since the sensitivities change with the position geometry.

An analogy between tolerance analysis and velocity analysis is described which enables the calculation of the sensitivities in closed form using a kinematic model and classical kinematic analysis. The Jacobian matrix of kinematic analysis will be identical to the tolerance sensitivity matrix if the kinematic model is modified to include dimensional variation and if the kinematic variables are defined correctly. Commercial kinematics software may then be used to update the geometry and calculate the tolerance sensitivities in each position.

A set of variational model elements is presented, which may be added to a kinematic model, resulting in an “equivalent variational mechanism” (EVM), which includes dimensional variations as kinematic inputs. Demonstration of this method using the commercially available software ADAMS is presented. As a side benefit, the kinematic software can even analyze static assemblies, which have no moving parts.

Acrobat 4.0 (780k)

Thesis #99-2:**
Title:** Functional Surface Characterization for Tolerance Analysis of Flexible

** Abstract: **Statistical tolerance analysis
of flexible assemblies can predict the probable range of assembly forces and distortions
required to assemble warped or misaligned parts. The resulting distortion and stress
throughout the assembly are affected by the amplitude and wavelength of the surface waviness
of the mating surfaces. Therefore, the surface variations must be characterized statistically.
A new method is presented, called Functional Surface Characterization (FSC) method, to
characterize a population of surfaces statistically.

The FSC method classifies the surface variations as random and non-random. The method separates the two variations by statistical analysis and uses frequency spectrum methods to characterize the random variations. The outputs of this method, the mean surface and average autospectrum, can be used with a statistical finite element analysis model to predict assembly forces and distortions. The autospectrum is used to classify the spatial frequencies in three domains namely, low, medium and high, based on distinctly different effects. The method is demonstrated using real surface data acquired from sample flexible parts to evaluate the spectrum model.

Acrobat 4.0 (4.6MB)

Thesis #99-1:**
Title:
**Tolerance Analysis of Flexible Assemblies Using Finite Element and Spectral Analysis

** Abstract: **This
paper proposes a new method for modeling flexible assemblies,
called the Flexible Assembly Spectral Tolerance Analysis (FASTA)
method, which uses the
autocorrelation function from frequency spectrum analysis to model
random surface variations. Finite
element models are used to predict assembly forces and stresses
from known surface variations.
Variations at surface nodes are not independent variables.
Therefore, covariance must be
included in the statistical analysis of flexible assemblies.
Covariance between nodes
arises from both material elastic coupling and surface continuity
constraints. This paper shows how covariance is included in
statistical predictions, and how
to calculate the covariance matrix from autocorrelation functions
of part surfaces. Deterministic variations, such as part warping,
are separated from random variations.

Acrobat 4.0 (388k)

Publication #98-3:**
Title:
**Global Coordinate Method for Detemining sensitivity
in Assembly Tolerance Analysis

* Abstract: *Tolerance sensitivity indicates the
influence of individual component tolerances in an assembly on the variation of
a critical assembly feature or dimension. Important applications include assembly tolerance
analysis and tolerance allocation. This paper presents a new method for determining
tolerance sensitivity using vector loop assembly tolerance models. With a vector
loop model, the assembly kinematic constraints or assembly functions can be
established automatically as implicit functions.This method evaluates the derivatives
of the kinematic constraint equations with respect to both manufactured dimensions
and assembly variables. The derivative matrices are then used to calculate the
tolerance sensitivity matrix. This is a closed form method which relates the
derivatives of the assembly functions to the coordinates of the joints or nodes
and the orientations of the vectors and the local joint axes in an assembly.
It is accurate, simple and very suitable for design iterations.

Publication #98-2: **
Title:
**New Metrics for Evaluating Monte Carlo Tolerance Analysis of
Assemblies

** Abstract:** Monte Carlo simulation
may be used by engineering designers to predict the effects of manufacturing
variations in mechanical assemblies. Validity of the predictions depends on
the accuracy of the input variations, simulation sample size and the fit of
a statistical distribution to the resultant assembly data. New metrics are presented
for assessing the accuracy of variation analysis methods. Errors due to sample
size are estimated for the mean, variance, skewness and kurtosis of the resultant
distribution, and the predicted rejects.

Simple algebraic expressions are derived, which can be used to predict the error in the assembly variation parameters without having to repeat the simulation.

PhD Dissertation #98-1: **
Title:
**Tolerance Analysis Of Compliant Assemblies

*
Abstract: * This dissertation presents methods for combining tolerance analysis of assemblies and finite element analysis to predict assembly force, stress, and deformation in assemblies of compliant parts, such as airframes and automotive bodies. Assemblies of flexible parts are frequently subject to misalignment due to dimensional variation, gravity loads, and residual stress induced distortion. A steady-state contact solution is derived for closure of misaligned surfaces at known contact points, such as bolts or weld locations. This contact solution is expanded to include statistical variation in the contact point locations. An important finding is the covariance that arises from this statistical solution due to both material and geometric effects.

Material covariance is caused by elastic coupling between contact points and can be calculated from the stiffness matrix for the component parts. Geometric covariance arises form constraints on random variation of the mating surfaces due to surface continuity conditions. As part of the geometric covariance solution, the theory of random B´ ezier curves is developed. Random B´ ezier curves provide a method of mapping a profile tolerance on a curve to tolerance bands about the Bezier control points. Random B´ ezier curves may be subtracted to define the mean gap and covariance between two mating surfaces. This random quantity is used as input into the steady-state contact solution. Several example problems are given demonstrating the method and the results are validated by comparison with Monte Carlo simulations.

Publication #97-6:

** Abstract:** A technique to predict the effects of tolerance stack-up on assembly forces and stresses

combines a rigid body assembly model with a finite-element analysis. The technique includes several

assumptions. First small geometric variation in a part create insignificant changes in the part stiffness. In

addition, friction between the parts is negligible and the material in the assembly behaves linearly. A

demonstration of the technique is presented.

Publication #97-5:

** Abstract:** Assembly tolerance analysis
is a key element in industry for improving product quality and reducing overall
cost. It provides a quantitative design tool for predicting the effects of manufacturing
variation on performance and cost. It promotes concurrent engineering by bringing
engineering requirements and manufacturing requirements together in a common
model.

A new method, called the Direct Linearization Method (DLM), is presented for tolerance analysis of 2-D and 3-D mechanical assemblies, which generalizes vector loop-based models to include small kinematic adjustments. It has a significant advantage over traditional tolerance analysis methods in that it does not require an explicit function to describe the relationship between the resultant assembly dimension(s) and manufactured component dimensions. Such an explicit assembly function may be difficult or impossible to obtain for complex assemblies.

The DLM method is a convenient design tool. The models are constructed of common engineering elements: vector chains, kinematic joints, assembly datums, dimensional tolerances, geometric feature tolerances and assembly tolerance limits. It is well suited for integration with a commercial CAD system as a graphical front end. It is not computationally intensive, so it is ideally suited for iterative design.

A general formulation is derived, detailed modeling and analysis procedures are outlined, and the method is applied to two example problems.

Publication #97-4:

**Title: **A Comprehensive System for Computer-Aided Tolerance Analysis of
2-D and 3-D Mechanical Assemblies, April 27-29, 1997**
Authors:** Kenneth W. Chase, Spencer P. Magleby, & Charles G. Glancy

** Abstract: **Tolerance analysis of assemblies
promotes concurrent engineering by bringing engineering requirements and manufacturing
capabilities together in a common model. By further integrating the engineering
modeling and analysis with a CAD system, a practical tool for product and process
development is created. It provides a quantitative design tool for predicting
the effects of manufacturing variation on performance and cost in a computer-based
design environment.

A comprehensive method based on vector assembly models has been developed for modeling and analyzing variations in 2-D and 3-D mechanical assemblies. The models are constructed of common engineering elements: vector chains, kinematic joints, assembly datums, dimensional and geometric feature tolerances, and assembly tolerance limits. The method is consistent with engineering design practice and is well suited for integration with commercial CAD systems.

Three sources of variation may readily be included in vector models: dimensional, geometric and kinematic.

Dimensional variations account for small changes in size due to manufacturing processes. Geometric variations describe changes in shape, location and orientation of features. Kinematic variations describe the propagation of variation through an assembly by small adjustments between mating parts.

Design intent is expressed by assembly tolerance specifications, which may be added to the model and used in computing predicted quality levels. A basic set of assembly tolerances is described, patterned after ANSI Y14.5 geometric feature controls, for specifying design requirements for a wide variety of applications.

A complete computer-aided tolerancing system is described, which is tightly integrated with a commercial CAD system. Systematic modeling procedures and rules for creating vector assembly models are outlined. Vector assembly models are created graphically and analyzed for variation statistically in an interactive design environment. Examples of a 2-D and a 3-D assembly are presented with corresponding vector assembly models.

Thesis #97-3:

**Title: **Characterization of Assembly Variation Analysis Methods, December
1997**
Author:** Robert Cvetko

** Abstract: **This thesis is a study and comparison of current
assembly variation analysis methods. New metrics are presented for gauging the
accuracy of, not only the analysis methods, but also the problem information
without knowing the exact answer. A hybrid method using Monte Carlo with the
method of system moments is also presented and evaluated. Matching the accuracy
of the problem and the analysis method allows for simple models and an estimate
of confidence for the results. A framework for choosing the best method for
different analysis situations summarizes the findings.

Publication #96-1:

**Title: **Multivariate Statistical Analysis of Assembly Tolerance Specifications,
1996**
Author:** Marvin J. Law

** Abstract: ** Traditional variation analysis of mechanical
assemblies commonly represents functional requirements using one-dimensional
design specifications such as gaps and angles. A specification of this type
controls a single degree of freedom of an assembly feature. If there are multiple
specifications on an assembly, traditional practices call for analyzing each
specification in isolation from the others. This approach ignores the probable
covariance between the assembly variables. Ignoring the covariance may lead
to double counting out-of-specification assemblies. For example, an assembly
that fails two specifications may erroneously be counted as two rejects instead
of one. A more general approach is to define specifications that can directly
represent functional requirements that constrain multiple degrees of freedom
of an assembly feature, such as displacement variation in two orthogonal directions.
A further generalization is to account for the correlation between variables
of separate assembly specifications in order to more accurately predict the
true quality level of an assembly.

A statistical multivariate/multiple assembly specification analysis method is derived based on the linearized method of variation analysis. Covariance between critical assembly variables is correctly included in the analysis. A set of assembly tolerance specifications, including several multivariate examples, is also defined. Two practical test problems are modeled and analyzed using multiple assembly tolerance specifications to represent the functional requirements. The results are verified using a non-linear Monte Carlo simulation.

Thesis #95-3:**
Title:** Characterizing Kinematic Variation in Assemblies from Geometric Constraints

** Abstract: **Modeling the effects of process variation in
mechanical assemblies requires a kinematic model to describe the relative motion or degrees of
freedom between mating surfaces. Currently, the process if assembly joint characterization is a
manual process. The CAD system, however, already has the information necessary to infer the location
and orientation of joints from the model geometry and assembly constraints. A new system has been
developed to automate the identification of each joint or kinematic pair used to characterize mating
contact within an assembly as well as its type, location and orientation.

Kinematic assembly pairs are formed by mating two surfaces. In this system, three basic surfaces types are considered: plane, cylinder and toroid. Sphere, edge and point may also be obtained by choice of radii. Complex surfaces may be approximated locally by a general surface of two orthogonal radii of curvature at each point of contact. Three distinct surface parameterization were defined and combined to describe 16 different joint algorithm based on this approach was developed and implemented on a commercial CAD system. This program successfully demonstrated the joint identification algorithm on a set of problems that covered the range of joint types.

Acrobat 4.0 (5.1MB)

Publication
#95-2:**
Title: **Comparison of Assembly Tolerance Analysis by the Direct Linearization
and Modified Monte Carlo Simulation Methods

** Abstract: **Two methods
for performing statistical tolerance analysis of mechanical assemblies are compared:
the Direct Linearization Method (DLM), and Monte Carlo simulation. A selection
of 2-D and 3-D vector models of assemblies were analyzed, including problems
with closed loop assembly constraints. Closed vector loops describe the small
kinematic adjustments that occur at assembly time. Open loops describe critical
clearances or other assembly features.

The DLM uses linearized assembly constraints and matrix algebra to estimate the variations of the assembly or kinematic variables, and to predict assembly rejects. A modified Monte Carlo simulation, employing an iterative technique for closed loop assemblies, was applied to the same problem set. The results of the comparison show that the DLM is accurate if the tolerances are relatively small compared to the nominal dimensions of the components, and the assembly functions are not highly nonlinear. Sample size is shown to have great influence on the accuracy of Monte Carlo simulation.

Thesis #95-1:**
Title:** Statistical Models for Position and Profile Variation in Mechanical Assemblies

** Abstract: **Although ANSI Y 14.5 has previously defines
standards for position and profile tolerances, these are only defined for components. They have not
been designed to be combined into an assembly. One objective of this thesis was to define a method
to model profile and position form tolerances in a vector loop. Also, except for a Monte Carlo
analysis, no method to predict accurately the percent reject for a position assembly tolerance has
been found. The other objective of this thesis was to develop a statistical method to make this
prediction that will be much less computationally intensive than a Monte Carlo analysis.

The profile component tolerance was modeled by inserting a zero length vector between two contacting parts. Although the nominal length of the vector is zero, it is allowed to have a normally distributed variation in a direction normal to the surface. The position component tolerance was modeled by inserting two orthogonal vectors. When these two vectors are combined, they give the same distributions as a position component tolerance (Rayleigh for magnitude and uniform for direction.)

To predict the percent rejects for an assembly position tolerance, covariance analysis was used. This allowed the distribution of the position variation to be characterized. Once the variation is found, numerical methods were used to find the volume under the distribution, outside the tolerance limit.

Acrobat 4.0 (4.9MB)

Thesis #94-5:**
Title: **A Second-Order Method for Assembly Tolerance Analysis

** Abstract: **
There currently exists a wide gap between two established methods of assembly tolerance analysis.
The first, the Linearized Method, provides fast analysis, closed loop constraints, tolerances
allocation, and design iteration. The second, Monte Carlo simulation, accurately models nonlinear
effects of assembly geometry and non-normal component distributions, but requires intensive computation.
A new statistical, second-order method has been developed with the benefits of speed, tolerance
allocation, closed-loop constraints, non-linear geometric effects and non-normal component and
assembly resultant distributions. The Second-Order Tolerance Analysis (SOTA) method applies the
Method of System Moments to determine assembly variations described by a system of implicit nonlinear equations.

The SOTA method was compared to the Linearized Method and Monte Carlo simulation using five case studies. The SOTA method provided nonlinear analysis results comparable to Monte Carlo simulation with 100,000 samples. The analysis time of the SOTA method was comparable to the Linearized Method. The SOTA method offers significant advantages as a nonlinear analysis tool suitable for use in design iteration.

Acrobat 4.0 (4.9MB)

Publication #94-3:**
Title: **Including Geometric Feature Variations
in Tolerance Analysis of Mechanical Assemblies

* Abstract: *Geometric feature
variations are the result of variations in the shape, orientation or location
of part features as defined in ANSI Y14.5M-1982 tolerance standard [ANSI 1982].
When such feature variations occur on the mating surfaces between components
of an assembly, they affect the variation of the completed assembly. The geometric
feature variations accumulate statistically and propagate kinematically in a
similar manner to the dimensional variations of the components in the assembly.

The Direct Linearization Method (DLM) for assembly tolerance analysis provides a method for estimating variations and assembly rejects, caused by the dimensional variations of the components in an assembly.So far, no generalized approach has been developed to include all geometric feature variations in a computer-aided tolerance analysis system.

This paper introduces a new, generalized approach for including all the geometric feature variations in the tolerance analysis of mechanical assemblies. It focuses on how to characterize geometric feature variations in vector-loop-based assembly tolerance models. The characterization will be used to help combine the effects of all variations within an assembly in order to predict assembly rejects using the DLM.

Publication #94-2:

** Title: **General 3-D Tolerance Analysis of Mechanical Assemblies
with Small Kinematic Adjustments

**Authors:** Kenneth W. Chase, Spencer P. Magleby, Jinsong Gao

** Abstract: **The Direct Linearization
Method (DLM) for tolerance analysis of 3-D mechanical assemblies is presented.
Vector assembly models are used, based on 3-D vector loops which represent the
dimensional chains that produce tolerance stackup in an assembly. Tolerance
analysis procedures are formulated for both open and closed loop assembly models.
The method generalizes assembly variation models to include small kinematic
adjustments between mating parts.

Open vector loops describe critical assembly features. Closed vector loops describe kinematic constraints for an assembly. They result in a set of algebraic equations which are implicit functions of the resultant assembly dimensions. A general linearization procedure is outlined, by which the variation of assembly parameters may be estimated explicitly by matrix algebra.

Solutions to an over-determined system or a system having more equations than unknowns are included. A detailed example is presented to demonstrate the procedures of applying the DLM to a 3-D mechanical assembly.

Publication #94-1:**
Title:
**General 2-D Tolerance Analysis of Mechanical Assemblies with
small Kinematic Adjustments

** Abstract:** Assembly tolerance analysis is a key
element in industry for improving product quality and reducing overall cost.
It provides a quantitative design tool for predicting the effects of manufacturing
variation on performance and cost. It promotes concurrent engineering by bringing
engineering requirements and manufacturing requirements together in a common
model.

A new method, called the Direct Linearization Method (DLM), is presented for tolerance analysis of 2-D mechanical assemblies which generalizes vector loop-based models to include small kinematic adjustments. It has a significant advantage over traditional tolerance analysis methods in that it does not require an explicit function to describe the relationship between the resultant assembly dimension(s) and manufactured component dimensions. Such an explicit assembly function may be difficult or impossible to obtain for complex 2-D assemblies.

The DLM method is a convenient design tool. The models are constructed of common engineering elements: vector chains, kinematic joints, assembly datums, dimensional tolerances, geometric feature tolerances and assembly tolerance limits. It is well suited for integration with a commercial CAD system as a graphical front end. It is not computationally intensive, so it is ideally suited for iterative design.

A general formulation is derived, detailed modeling and analysis procedures are outlined and the method is applied to two example problems.

Publication #4:**
Title:
**A
New Monte Carlo Simulation Method for
Tolerance Analysis of Kinematically Constrained Assemblies

* Abstract: *A
generalized Monte Carlo simulation method is presented for tolerance analysis
of mechanical assemblies with small kinematic adjustments. This is a new tool
for assembly tolerance analysis based on vector-loop-based assembly models with
kinematic assembly constraints. Methods have been developed for simulating both
closed loop and open loop assembly constraints as well as overdetermined assembly
systems. Both dimensional and geometric feature variations of the components
may be included in the assembly simulation.

Compared to the traditional Monte Carlo simulation of mechanical assemblies, the new method does not require explicit functions to describe the assembly parameters. Explicite functions are often quite difficult or impossible for designers to obtain for 2-D and 3-D assemblies. An iterative nonlinear solution is required for each simulated assembly to assure that closed loop assembly constraints are satisfied. A general four parameter statistical distribution is fit to the resulting assembly data for estimating percent rejects.

Thesis #93-2:**
Title:
**A Comprehensive System for Modeling Variation in Mechanical Assemblies

** Abstract:** Tolerance analysis allows the
designer to quantitatively estimate the affects of variation on
design requirements in the early
design phases. Tolerances play a significant role in the development
and cost of manufactured products. By creating assemblies which
perform properly, are cost
efficient and readily manufacturable, engineers can assist in
producing high quality,
marketable products.

Research presented in this thesis focuses on developing a comprehensive 2-D system for modeling manufacturing variations. Previous studies included creating an engineering model based on part, datums, kinematic joints, vector loops, and form variations. Contributions from this thesis include: 1) A comprehensive system for defining assembly tolerance specifications. Just as feature controls are applied to individual parts, a similar system of assembly tolerance specifications may be applied to assemblies of parts. 2) A generalized model for manual process variations due to fastener clearances or other non-deterministic variations. 3) An algorithm for automatic open loop generation. Open vector loops are often necessary to represent design constraints on an assembly. 4) A method for detecting degree of freedom redundancy.

**Acrobat
4.0 **(226k)

Thesis #93-1:**
Title:
**A Comprehensive Method for Specifying Tolerance Requirements for Assemblies

** Abstract:** When
mechanical parts are assembled, their dimensional and form
variations accumulate statistically and propagate kinematically,
causing critical assembly features to vary.
The main objective of this thesis was to relate engineering
tolerance requirements to manufacturing constraints by means of a
common engineering model for variation in assemblies. A comprehensive system for defining assembly specifications
(engineering requirements) and for analytically predicting
assembly variations was developed, which will allow closer control
of product quality and performance.

Vector loops were used to model mechanical assemblies. The loops join relevant dimensions into chains. Closed loops define the kinematic constraints on the assembly. Open loops define engineering constraints. Linearized equations derived from the vector loops were used to solve for the assembly variations. The linearized equations are expressed in terms of scalar sensitivities and known component tolerances. Simplified methods were developed for determining sensitivities using nodal coordinates. (more in paper...)

**Acrobat
4.0 **(256k)

Thesis #92-1:**
Title: **Integrating Geometric Form Variations into Tolerance Analysis of 3-D Assemblies

** Abstract: **
Tolerance analysis is a vital part of any design project since tolerances directly affect cost,
quality and performance. Being able to determine the probability of successfully assembling parts
and meeting engineering requirements before any parts are manufactured is crucial to the success of
a product in todayÕs highly competitive market place. Having the ability to perform tolerance
analysis on the computer using a CAD model data base can assure an efficient and accurate tolerancing effort.

There are three main sources of variation in mechanical assemblies: 1) Dimensional, 2) form or feature, and 3) kinematic variations. Form variations arise form variations in shape, orientation or location as described by the geometric dimensioning and tolerancing standards, ANSI Y 14.5M. Form variations can have a significant effect on an assembly, since they can accumulate statistically and propagate kinematically the same dimensional variations.

The emphasis of this project is to expand the current computer-aided modeling system to accurately utilize form tolerances as define by the ANSI Y 14.5 standard and exchange this information with a 3-D tolerance analysis package. With all three variation sources accounted for, 3-D tolerance analysis can now be performed entirely using the computer.

Acrobat 4.0 (6.4MB)

Thesis #91-4:**
Title: **Representation and Modeling of Geometric Form Variations for 3-D Tolerance Analysis

** Abstract: **
The ANSI Y14.5M-1982 standard for dimensioning and tolerancing defines a set of geometric tolerances
which may be included on an engineering drawing to provide additional constraints on form variations
of produced parts. A geometric tolerance may be used to limit the variation of the form of a feature,
its relationship to other features on a part, and its orientation with respect to established datums.
In this thesis, geometric tolerances are collectively referred to as form tolerances.

There is no established standard for analyzing the cumulative effects of form variations on an entire assembly, nor is there a statistical model for form variation. In this thesis, each ANSI form tolerance has been modeled and categorized with respect to how the tolerance adds variation to the assembly model and how Rule #1 in the ANSI standard applies. Rule #1 states that form variations of any surface may not cause a part to exceed the limits specified by the size tolerances. This model of form variation is incorporated into an established vector loop method of analyzing the cumulative effect of tolerances.

A normal statistical distribution model is proposed for form variations, with a symmetric tolerance zone equal to the specified unilateral tolerance zone. For statistical analysis of form tolerances, it is proposed that Rule #1 not be applied since this rule is based on worst-case analysis and inspection. A c

A generalized assembly tolerance modeler has been created as an interface to the CATIA solid modeler. This interface incorporates the vector loop assembly model and allows the addition of geometric form tolerances to a solid model.

Acrobat 4.0 (6.9MB)

Thesis #91-3:**
Title: **A Generalized Approach to Kinematic Modeling for Tolerance Analysis of Mechanical Assemblies

** Abstract: **
Tolerance selection has a significant influence on the manufacturing cost of a mechanical assembly.
Tolerance analysis tools based on a CAD assembly model can help the design engineer evaluate the
consequences of tolerance specification early in the design process, when design changes are not
prohibitively critical or expensive. Research presented in this thesis focuses on developing a
generalized method of obtaining the additional kinematic information required for tolerance analysis.
The method is used to create an assembly tolerance model based on elements the designer should be
familiar with, including parts, datums, joints, and vectors which represent critical part and assembly
dimensions. Dimension vectors are connected in series to form vector loops, which follow only
controlled and kinematic dimensions in the model. The governing relationships for tolerance analysis
are determined from scalar equations of the vector loops.

Thesis contributions include:

- Generalized modeling approach for graphically creating a kinematic tolerance model for tolerance analysis of a mechanical assembly
- Introduction of the concept of datum paths to locate kinematic joints relative to the datum reference frame of each part in the model
- Complete automatic generation of the vector loops used to predict the effects of manufacturing variations on assembly performance
- Automatic identification of dependent variables in order to minimize user interaction and ensure proper setup for analysis
- Degree-of-freedom analysis and consistency checks to ensure solution is valid
- Implementation of automatic loop generation using C and of the generalized tolerance modeling method on the AutoCAD system using the AutoLISP application language

Acrobat 4.0 (7.7MB)

Publication #91-1:

**Title: **A Survey of Research in the Application of Tolerance
Analysis to the Design of Mechanical Assemblies

**Authors:** Kenneth W. Chase, Alan R. Parkinson

** Abstract:** Tolerance analysis is
receiving renewed emphasis as industry recognizes that tolerance management
is a key element in their programs for improving quality, reducing overall costs
and retaining market share. The specification of tolerances is being elevated
from a menial task to a legitimate engineering design function. New engineering
models and sophisticated analysis tools are being developed to assist design
engineers in specifying tolerances on the basis of performance requirements
and manufacturing considerations.

This paper presents an overview of tolerance analysis applications to design with emphasis on recent research that is advancing the state of the art. Major topics covered are:

- New models for tolerance accumulation in mechanical assemblies, including the Motorola Six Sigma model.
- Algorithms for allocating the specified assembly tolerance among the components of an assembly.
- The development of 2-D and 3-D tolerance analysis models.
- Methods which account for non-Normal statistical distributions and nonlinear effects.
- Several strategies for improving designs through the application of modern analytical tools.

Thesis #90-3:**
Title: **A Closed-Form Model for Tolerance Analysis of Two-Dimensional Mating Hole Patterns

** Abstract: **
In the design of mating hole patterns, the tolerances which are specified on the hole sizes and the
hole positions have a major influence on the cost and the quality of the assembly. The current
method for specifying the tolerances on mating hole patterns is the true position method of ANSI Y14.5
which is a worst case method for tolerance analysis. Because of the low probability of having a
worst case part, the tolerances can often be loosened substantially without significantly increasing
the number of rejected parts. Through statistical analysis of the tolerances on mating hole patterns,
designers can specify less conservative tolerances on hole sizes and hole positions and therefore
lower manufacturing costs.

Two new statistical methods for analyzing mating hole patterns have been developed. 1) A new Monte Carlo simulation of mating hole patterns which uses a multi-objective optimization program for assembling the randomly created hole patterns. 2) A closed form analytical solution which uses a kinematic model to represent the relative shift of the two mating patterns, followed by a multiple integration procedure to account for hole size and position variations. Both methods have proven to be consistent, fast, and accurate for analyzing mating hole patters, but the integration method is more efficient for moderate size problems.

Acrobat 4.0 (4.5MB)

Thesis #90-2:**
Title: **General System for Least Cost Tolerance Allocation in Mechanical Assemblies

** Abstract: **
This research report describes the development of an efficient method for determining the least-cost
component tolerances for mechanical assemblies. It combines a new optimization technique with statistical
tolerance analysis to insure lowest production cost of an assembly. Called the Equalized Gradient
method, it will give designers the ability to solve complex assembly problems with any of the
following features:

- Two-dimensional assemblies described by multiple vector loops and with dependent variables eliminated.
- Cost-vs-tolerance functions described by a power law with different exponent values for each independent dimension in an assembly.
- Mean shift values representing biased processes, which may be different for each component in the assembly.

The Equalized Gradient method is general enough that it may provide efficient solutions to other unrelated nonlinear problems.

Acrobat 4.0 (2.9MB)

Thesis #90-1:**
Title: **Automated Vector Loop Generation for Kinematic Models of Mechanical Assemblies

** Abstract: **
In the detailed design stage of a product, appropriate tolerance selection is crucial to the quality
and manufacturability of the final product. New modeling techniques are being developed for accurately
predicting the influence of manufacturing variations in mechanical assemblies. Assembly models are
based on a vector loop representation with kinematic constrains, to which statistical or worst case
variational analysis is applied.

Presented in this thesis is a new algorithm that greatly reduces the task of developing kinematic models. The algorithm automatically determines the number of required vector loops and their corresponding paths in an assembly model based on the application of network graph theory. The algorithm searches an incidence matrix which describes the parts of an assembly and the contact joints between them. It minimizes the number of joints and the length of each path in determining the optimal set of tolerance loops.

In conjunction with this algorithm, a graphical interface program has also been developed between CATS.BYU and AutoCAD. The interface gives a designer the tools required to create assembly tolerance models and store the information in the AutoCAD database.

Acrobat 4.0 (3.8MB)

Publication #89-4:

**Title: **Least Cost Tolerance Allocation for Mechanical Assemblies
with Automated Process Selection

**Authors:** Kenneth W. Chase, William H. Greenwood, Bruce G.
Loosli, Loren F. Hauglund

** Abstract: **The allocation of tolerances
among the components of a mechanical assembly can significantly affect the resulting
manufacturing costs. If cost versus tolerance data are available for each dimension,
the least cost tolerance allocation may be determined by optimization techniques.
However, when alternate manufacturing processes are available for some of the
components, a discrete optimization problem results. An exhaustive search of
all possible combinations of processes will determine the global optimum, but
the number of combinations increases geometrically, becoming very large for
assemblies of only moderate complexity.

Several methods were tested for systematically searching for the minimum cost process set for an assembly. Both discrete and continuous optimization schemes were compared to an exhaustive search, based on CPU times and the number of combinations required to find the global optimum. A new method is presented which is several orders of magnitude better than the exhaustive search.

Thesis #89-3:**
Title: **A Practical Method for Three-Dimensional Tolerance Analysis Using a Solid Modeler

** Abstract: **
The development of a 3-D non-linear tolerance analysis method practical for implementation in a
commercial solid modeler is described. Important issues include a method for modeling 3-D assembly
functions, automatically computing partial derivatives of assembly functions, accommodating geometric
tolerances, and solving for dependent assembly tolerances.

A vector-based method is presented for modeling 3-D mechanical assemblies which utilizes vectors to represent critical part features, and includes:

- modeling guidelines for identifying a complete and valid set of vector loops
- a set of kinematic joint types necessary for kinematic analysis of dependent variables and assembly rigidity
- incorporation of all ANSI Y14.5 feature tolerances
- a method for solving a matrix system of tolerance accumulation equations.

Acrobat 4.0 (9.6MB)

Thesis #88-7:**
Title: **Development of Two-Dimensional Tolerance Modeling Methods for CAD Systems

** Abstract: **
Research in the development of a two-dimensional geometric tolerance analysis modeler for CAD systems
is described. The objective was to determine the procedures and data requirements for an interactive
graphical assembly modeling tool which would permit the designer to graphically define 2-D vector
assembly loops and create an assembly model database for quantitative tolerance analysis.

Various CAD system limitations were encountered which required compromises in the preferred geometric modeling capability and user interface design.

Principle contribution of this research include:

- The development of a generalized system for modeling 2-D mechanical assemblies for tolerance analysis, based on a kinematic representation of assemblies.
- Successful incorporation of ANSI-Y14.5 feature control tolerances in the 2-D assembly model.
- The coding of a prototype program as a conceptual design tool on a commercial CAD system.
- The augmentation of the CAD data structure to include tolerance analysis data.

Acrobat 4.0 (7.0MB)

Thesis #88-6:**
Title: **Nonlinear Tolerance Analysis Using the Direct Linearization Method

** Abstract: **
An effective procedure for performing nonlinear tolerance analysis on complex mechanical assemblies
called the Direct Linearization Method (DLM) is presented. It is capable of modeling many of the
constraints that commonly occur in nonlinear tolerancing problems. It permits the use of multiple
vector loops with dependent variables. The DLM requires vectors and vector angles for input rather
than user-defined assembly equations.

Several example 2-D tolerancing examples are presented. The DLM is compared with current nonlinear tolerancing techniques. It is found to be much easier to use because it automatically computes the tolerancing sensitivities and eliminates dependent variables from the equations. It is a systematic procedure that is applicable to a wide range of assembly problems. Recommendations for future research in DLM are discussed.

Acrobat 4.0 (3.9MB)

Publication #87-5:

** Abstract: **The allocation of tolerances
among the components of a mechanical assembly can significantly affect the resulting
manufacturing costs. If cost versus tolerance data are available for each dimension,
the least cost tolerance allocation may be determined by optimization techniques.
However, when alternate manufacturing processes are available for some of the
components, a discrete optimization problem results. An exhaustive search of
all possible combinations of processes will determine the global optimum, but
the number of combinations increases geometrically, becoming very large for
assemblies of only moderate complexity.

Several methods were tested for systematically searching for the minimum cost process set for an assembly. Both discrete and continuous optimization schemes were compared to an exhaustive search, based on CPU times and the number of combinations required to find the global optimum. A new method is presented which is several orders of magnitude better than the exhaustive search.

Publication #87-4:

** Abstract: **The allocation of tolerances among the components of a mechanical assembly can significantly affect the resulting manufacturing costs. Using a cost-vs.-tolerance function for each dimension, the least cost tolerance allocation may be determined analytically by the method of Lagrange multipliers. When alternative manufacturing processes are available for some of the components, the analysis may be repeated for each of the possible combinations of processes to determine the least cost production method as well as tolerances. However, the number of combinations increase geometrically, becoming very large for assemblies of moderate complexity.

Several methods are described for systematically searching for the minimum cost process set without having to exhaustively try every possible combination. Two promising methods have been quantitatively tested and compared to an exhaustive search. The number of combinations and CPU time are greatly reduced.

When a preferred tolerance range is specified for each process, the process search methods have difficulty due to the presence of local minima. A procedure for applying process tolerance constraints was developed which greatly increases the likelihood of finding the absolute minimum cost.