SELECTION AND OPTIMIZATION OF SNAP-FIT FEATURES
VIA WEB-BASED SOFTWARE
DISSERTATION
Presented in Partial Fulfillment of the Requirement for
The Degree Doctor of Philosophy in the Graduate
School of The Ohio State University
By
Tieming Ruan, M.S.
* * * *
The Ohio State University
2005
Dissertation Committee:
Professor Anthony F. Luscher, Adviser
Professor Gary L. Kinzel
Professor Donald R. Houser Mechanical Engineering
Graduate Program

ABSTRACT
When used in plastics, snap-fits can be a simple, quick and cost-effective method of assembling two parts. When designed properly, parts with snap-fits can be assembled and disassembled numerous times without any adverse effect on the assembly. Snap-fits also the aid in making products environmentally friendly because of their ease of disassembly, making components of different materials easy to recycle. Traditionally, snap-fit design methodology has been disorganized and anecdotal in nature, relying greatly on the skill and the experience of the individual designer. The most popular source of designing snap-fit is the design guides from resin suppliers. However the information disseminated by these guides is usually old, obsolete and inaccurate because they based on assumptions of small deformation and linear material property. To overcome these two disadvantages, the author developed a two-dimensional, plane stress, contact finite element model (FEM) considered the nonlinear material property of polymer. Using design of experiments (DOE) and response surface methodology (RSM), this research obtained a second order response surface equation to predict the retention performance of cantilever hook with high retention angles.
With the development of Internet, there is also a demanding for a web-based snap-fit design tool that is independent to all operating systems, easily accessible and can be universally upgraded by simply updating the design tool at the server location. This ii

thesis developed a web-based design tool for three different snap-fit features: cantilever hook, post & dome, and bayonet & finger. The response surface equations obtained by using the combination between FEM, DOE and RSM were applied. Constraint management (CM) was used to make several functions available such as sensitivity analysis, correction advisor.
Optimization modules such as single objective optimization and multiple objectives optimization were applied to make the design tool more flexible and powerful.
Single objective optimization was implemented in two steps. The first step is to use
Golden Section Method to identify the search direction and the second step is to use
Broydon-Fletcher-Goldfarb-Shanno (BFGS) method to find the minimum values on this search direction. To do multiple objectives optimization, the Weighted Sum strategy was choose. Each objective was assigned a weighted value based on their importance, then combine them into a single objective optimization problem.
iii

Dedicated to my parents, my wife and kids
iv

ACKNOWLEDGMENTS
I wish to thank my adviser, Anthony Luscher, for intellectual support, encouragement, and enthusiasm which made this dissertation possible, and for his patience in correcting both my stylistic and scientific errors.
I thank Prof. Srinivasan and Prof. Houser for their patience and to be my committee members. Especially I thank Prof. Gary Kinzel to be my committee member and for providing the FORTRAN source code used in this web-based design tool.
I am grateful to Gaurav Suri for discussing with me various aspects of this dissertation. I also want to thank my officemates who helped me to handle various problems, especially Jonathan Pillai and Leo Rusli.
I want to thank my parents, my wife and kids. Without your support, I can’t finish this dissertation with a full-time job.
This research was supported by a grant from CAPCE.
v

VITA
January 11, 1976 …………………………………….. Born – Zhejiang, China
1996 ………..…………………………………………B.S. Mechanical Engineering,
China Univ. of Mining & Tech.
1999 ……………………………………… ………… M.S. Mechanical Engineering,
China Univ. of Mining & Tech.
1999 – 2002 …………………………………………..Graduate Fellow
The Ohio State University
2002 – present ………………………………………...Graduate Research Assistant
The Ohio State University
2003 – present ……………………………………….. Sr. Associate R&D Engineer
Bayer HealthCare
PUBLICATIONS
Research Publication
1. T. Ruan, Z. Wang and H. Chen, “A monitoring device for the fire of belt”, Journal of Coal Science & Engineering (China), 1998.
2. Z. Luan, Z. Wang and T. Ruan, “Balance compound gear pump (motor)”,
Transaction of Huainan Institute of Technology (Chinese), 1997.
vi

FIELDS OF STUDY
Major field: Mechanical Engineering
vii

TABLE OF CONTENTS
Abstract .............................................................................................................. ii
Dedication ......................................................................................................... iv
Acknowledgments.............................................................................................. v
Vita ..................................................................................................................... vi
List of Tables .................................................................................................... xi
List of Figures ................................................................................................. xiii
Chapter 1 Introduction ....................................................................................... 1
1.1 Introduction to Snap-Fit Features ............................................................................. 2
1.2 Motivation for Research ........................................................................................... 4
1.2.1 Snap-fit Design Approaches .............................................................................. 8
1.2.2 Web-based Snap-fit Design Tool..................................................................... 11
1.2.3 Constraint Management ................................................................................... 12
1.2.4 Thesis Objectives ............................................................................................. 13
Chapter 2 Literature Review ............................................................................ 15
2.1 Numerical Formulation for Snap-fit Features......................................................... 15
2.2 Materials Issues in Snap-fit Design ........................................................................ 20
2.3 Failure Models of Cantilever Hook ........................................................................ 22
2.4 Web-based Design Tool for Snap-fit....................................................................... 25
2.5 Constraint Management .......................................................................................... 34
Chapter 3 Finite Element Analysis of Cantilever Hook With a High Retention
Angle ................................................................................................................. 37
3.1 Combination of FEA with DOE.............................................................................. 37
3.2 Nonlinear Structural Analysis of Snap-Fits ............................................................ 39
3.2.1 Moving Contact ............................................................................................... 39
3.2.2 Geometric Nonlinearities ................................................................................. 40
viii

3.2.3 Material Nonlinearities .................................................................................... 40
3.3 Numerical Model of Cantilever Hook Feature ....................................................... 41
3.3.1 Modeling Techniques for Contact Problem..................................................... 42
3.3.2 Improvement of Solution ................................................................................. 44
3.3.3 Comparison between FEM Result and Experiment Result.............................. 46
Chapter 4 The Empirical Model for Performance of Cantilever Hook With a
High Retention Angle ....................................................................................... 49
4.1 Screening Experiment of Cantilever Hooks with High Retention Angles.............. 51
4.2 Central Composite Design ...................................................................................... 59
4.3 Model Improvement................................................................................................ 64
4.3.1 Transformation of Empirical Model ................................................................ 65
4.3.2 Remove the Influential Outlying Cases ........................................................... 66
4.4 Verification Tests..................................................................................................... 74
Chapter 5 Design Optimization and Constraint Management ...................... 78
5.1 Design Optimization ............................................................................................... 78
5.1.1 Single Objective Optimization......................................................................... 81
5.1.2 Multiple Objectives Optimization.................................................................... 82
5.2 Constraint Management .......................................................................................... 83
5.2.1 Constraint Representation................................................................................ 85
5.2.2 Constraint Management Algorithms................................................................ 89
5.3 Mathematic Formulations for Snap-fits Features ................................................. 102
Chapter 6 The Web-Based Application for Snap-Fits.................................. 108
6.1 Approaches for Web-based Application................................................................ 108
6.1.1 Java ................................................................................................................ 108
6.1.2 JavaScript....................................................................................................... 110
6.1.3 VBScript ........................................................................................................ 110
6.1.4 VB DHTML Application............................................................................... 111
6.1.5 VB IIS Application ........................................................................................ 111
6.2 Principles for VB IIS Application......................................................................... 113
6.2.1 ASP Object Model ......................................................................................... 114
6.2.2 Introduction to WebClasses ........................................................................... 116
6.3 Snap-fit Web-based Application ........................................................................... 117
6.3.1 New Specification Problem ........................................................................... 121
6.3.2 Respecification Problem ................................................................................ 121
6.3.3 Unspecification Problem................................................................................ 121
6.3.4 Reverse Specification Problem ...................................................................... 124
6.3.5 Sensitivity Analysis ....................................................................................... 125
6.3.6 Correction Advisor......................................................................................... 126
6.3.7 Single Objective Optimization....................................................................... 127
6.3.8 Multiple Objectives Optimization.................................................................. 127
ix

Chapter 7 Case Studies ................................................................................. 131
7.1 Methods to Evaluate the Performance of Cantilever Hook .................................. 131
7.2 Integrated Polymeric Lens Housing...................................................................... 132
7.3 Cantilever Hook Application of Cabinet............................................................... 136
7.4 Cantilever Hook Samples ..................................................................................... 139
7.5 Case Study Conclusions........................................................................................ 145
Chapter 8 Conclusions and Recommendations for Future Research ....... 147
8.1 Scientific and Engineering Contribution .............................................................. 147
8.2 Recommendations for Future Research................................................................ 150
Appendix A “PROCEDURE” File for MSC.MARC and MSC.PATRAN ........ 151
Appendix B Visual Basic Subroutines for Cantilever Hook........................ 153
Bibliography.................................................................................................... 162
x

LIST OF TABLES
Table 4.1: The upper limit and low limit of each parameter ............................................ 52
Table 4.2: The screening array.......................................................................................... 55
Table 4.3: The result of best subset regression................................................................. 58
Table 4.4: The most significant factors for cantilever hook ............................................. 61
Table 4.5: The central composite design for cantilever hook........................................... 62
Table 4.6: Regression analysis for retention force of cantilever hook ............................. 64
Table 4.7: Regression analysis of logarithm transformation of cantilever hook .............. 67
Table 4.8: Regression analysis of reciprocal transformation of cantilever hook.............. 68
Table 4.9: The influential outlying cases.......................................................................... 71
Table 4.10: Regression analysis for retention force after deleted influential outlying cases
........................................................................................................................................... 73
Table 4.11: Verification test between FEA result and equation results............................ 76
Table 4.12: The analysis of variance for Lack-of-Fit test................................................. 77
Table 5.1: Parameters for strength of materials design equations .................................... 80
Table 5.2: The occurrence matrix of cantilever hook ....................................................... 86
Table 5.3: The occurrence matrix after specified x
1
, x
2
, x
3
, x
4
, and x
6
............................... 93
Table 5.4: Block, input/output, level and order of cantilever hook example.................... 95
Table 5.5: The occurrence matrix of respecification problem.......................................... 96
Table 5.6: The occurrence matrix of unspecification problem ......................................... 98
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Table 5.7: The response surface factors for the post & dome feature ............................ 103
Table 5.8: Design variables for the post & dome feature ............................................... 104
Table 5.9: The scaled variables for bayonet & finger feature......................................... 106
Table 5.10: The design variables for bayonet & finger feature ...................................... 106
Table 5.11: The response surface factors for bayonet & finger feature.......................... 107
Table 6.1: The advantages and disadvantages of web-based languages......................... 113
Table 7.1: The comparison of cantilever hook samples ................................................. 145
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LIST OF FIGURES
Figure 1.1: Typical experimental force curve for insertion of snap-fit features................. 4
Figure 1.2: Typical experimental force curve for insertion of snap-fit features................. 5
Figure 1.3: Common snap-fit feature topologies ................................................................ 6
Figure 1.4: Equations for dimensioning cantilevers ........................................................... 7
Figure 1.5: Force multiplier of insertion and retention....................................................... 7
Figure 1.6: Cantilever hook and mating part ...................................................................... 9
Figure 1.7: Suri’s idealized model of cantilever hook and mating part, shown in deformed configuration. .................................................................................................................... 10
Figure 2.1: Percent engagement of cantilever hook.......................................................... 23
Figure 2.2: The different failure modes with respect to the PE ........................................ 24
Figure 2.3: The two different mechanical failure modes for cantilever hook .................. 24
Figure 2.4: The IFP snap-fit design tool ........................................................................... 28
Figure 2.5: AlledSignal Plastics’ snap-fit design guide.................................................... 29
Figure 2.6: Cantilever snap-fit design tool from Eastman chemical company................. 30
Figure 2.7: Jeff Raquest’s snap-fit calculator ................................................................... 31
Figure 2.8: GE Plastics’ snap-fit wizard........................................................................... 32
Figure 2.9: Engineers Edge’s snap-fit straight beam calculator ....................................... 32
Figure 2.10: Engineers Edge’s snap-fit tapered beam calculator ..................................... 33
Figure 2.11: Brock & Wright’s design tool for snap fits .................................................. 33
xiii

Figure 3.1: The actual and bilinear stress-strain curve for plastic material ...................... 42
Figure 3.2: Typical cantilever hook meshed example ...................................................... 43
Figure 3.3: The nonlinear stress-strain curve of Duraform Polyamide ............................ 46
Figure 3.4: The comparison results for cantilever 1 ......................................................... 47
Figure 3.5: The comparison results for cantilever 2 ......................................................... 47
Figure 3.6: The comparison results for cantilever 3 ......................................................... 48
Figure 4.1: The typical cantilever hook ............................................................................ 53
Figure 4.2: The typical bayonet & finger.......................................................................... 54
Figure 4.3: The relationship between beam thickness and retention force....................... 54
Figure 4.4: The combination of central composite design................................................ 60
Figure 4.5: Retention force vs. position............................................................................ 61
Figure 4.6: The residual and normality plots for retention force...................................... 65
Figure 4.7: Residual plot vs. fitted values for logarithm transformation.......................... 69
Figure 4.8: Residual plot vs. fitted values for reciprocal transformation ......................... 70
Figure 4.9: The residual plot after deleted outlying cases ................................................ 73
Figure 4.10: Main effects for retention force.................................................................... 75
Figure 5.1: A procedure for identifying disjoint blocks of equations............................... 87
Figure 5.2: Design decomposition algorithm.................................................................... 91
Figure 5.3: The new specification problem of cantilever hook ........................................ 94
Figure 5.4: The respecification problem of cantilever hook............................................. 96
Figure 5.5: Forward dependency algorithm...................................................................... 97
Figure 5.6: The unspecification problem of cantilever hook............................................ 99
Figure 5.7: Backward dependency algorithm ................................................................. 101
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Figure 5.8: The reverse specification problem of cantilever hook ................................. 102
Figure 5.9: A typical post & dome feature ..................................................................... 104
Figure 6.1: A typical WebClass life cycle ...................................................................... 118
Figure 6.2: The flow diagram of snap-fit web-based design tool ................................... 119
Figure 6.3: The logon page ............................................................................................. 120
Figure 6.4: The selection page ........................................................................................ 120
Figure 6.5: The design interface of new specification.................................................... 122
Figure 6.6: The design interface of respecification ........................................................ 122
Figure 6.7: The final design interface of new specification............................................ 123
Figure 6.8: The design interface of unspecification........................................................ 123
Figure 6.9: The design interface of reverse specification............................................... 124
Figure 6.10: The typical sensitivity analysis page .......................................................... 125
Figure 6.11: The typical correction advisor page ........................................................... 126
Figure 6.12. Single objective optimization for retention force of cantilever hook......... 128
Figure 6.13. Single objective optimization result for retention force of cantilever hook128
Figure 6.14: Multiple objectives optimization for retention force of cantilever hook.... 129
Figure 6.15: Multiple objectives optimization result for retention force of cantilever hook
......................................................................................................................................... 129
Figure 7.1: Integral polymeric lens housing ................................................................... 133
Figure 7.2: Dimensions of snap-fits used in integral polymeric lens housing................ 133
Figure 7.3: The curve of retention force of cantilever hook of lens housing.................. 135
Figure 7.4: The web-based design tool for integral polymeric lens housing.................. 135
Figure 7.5: Calculation of beam thickness at broken area for power supply’s snap-fit.. 136
xv

Figure 7.6: Cantilever hook of power supply for Rockwell Allen Bradly PLC controller
......................................................................................................................................... 137
Figure 7.7: The detailed dimensions of the second case study....................................... 137
Figure 7.8: The retention force curve of cabinet snap-fit ............................................... 138
Figure 7.9: The web-based design page for power supply’s cantilever hook................. 138
Figure 7.10: The detailed dimensions of cantilever hook sample a................................ 139
Figure 7.11: The detailed dimensions of cantilever hook sample b ............................... 140
Figure 7.12: The detailed dimensions of cantilever hook sample c................................ 140
Figure 7.13: The retention force curve of cantilever hook a........................................... 141
Figure 7.14: The retention force curve of cantilever hook b .......................................... 142
Figure 7.15: The retention force curve of cantilever hook c........................................... 142
Figure 7.16: The web-based design page for cantilever hook sample a ......................... 144
Figure 7.17: The web-based design page for cantilever hook sample b......................... 144
Figure 7.18: The web-based design page for cantilever hook sample c ......................... 145
xvi

CHAPTER 1
INTRODUCTION
When used in plastics, snap-fits can be a simple, quick and cost-effective method of assembling two parts. When designed properly, parts with snap-fits can be assembled and disassembled numerous times without any adverse effect on the assembly. Snap-fits also aid in making products environmentally friendly because of their ease of disassembly, making components of different materials easy to recycle. Traditionally, snap-fit design methodology has been disorganized and anecdotal in nature, relying greatly on the skill and the experience of the individual design engineer. A popular source of design knowledge for snap-fit has been the design guides from resin suppliers such as Honeywell Plastics [1], General Electric Plastics [2] and Bayer Polymer [3].
However the information disseminated by these guides is fragmentary and often inaccurate. There has been tremendous progress in the areas of more accurate retention equations, more accurate finite element models, identification of critical geometric parameters and high-performance topologies.
In the current scenario most snap-fit designs are one-of-a-kind efforts with very little, if any, leveraging between different product families. Design knowledge generated in a particular project is leveraged for use in subsequent design only through the
1

experience of the designer, or through project reports describing the design process and performance data. It will be of great value if design information for snap-fit features became available to the designers in an abstracted as opposed to application-specific forum. What is needed are design equations, response surfaces, design heuristics, evaluation metrics and the like.
It is often tedious to apply design equations for unique types of snap-fits to get parameters such as insertion and retention force. This is especially true if response surface modeling equations which can be algebraically long and tedious are used. For this reason an interface for the snap-fit design tool is advantageous. With the surge in growth and availability of the Internet and advanced programs such as Java Applet and Visual
Basic’s Internet Information Service, it was decided that the easiest and most efficient way for distributing and maintaining a snap-fit design tool was to switch from a computer-based to a web-based application. With a web browser, a designer can access the snap-fit design tool independent of computer operating systems (i.e. versions of
Windows OS, Mac OSX, Unix and Linux) from anywhere in the world. This chapter presents a brief introduction to snap-fit features, some commonly used terminologies, and then explains the motivation and goal of this research.
1.1 Introduction to Snap-Fit Features
The main performance attributes of a snap-fit which will be used in this thesis are:
• Insertion Force ( F i
) is the force that needs to be applied in the insertion direction of a snap-fit feature to engage it. Insertion force can be expressed as a
2

single maximum value or as a graph of the force versus position relative to the snap-fit feature (Figure 1.1) [4].
• Retention Force ( F r
) is the force that needs to be applied in the separation direction of a snap-fit feature to disassemble it (Figure 1.2) [4]. If designed to be a permanent assembly, disengagement occurs due to fracture, permanent deformation or loss-of-engagement between the two mating parts.
• Locking Ratio is defined to be the ratio of the maximum retention force to the maximum insertion force of a snap-fit feature.
Snap-fit are molded into plastic parts to provide attachment functionality. Unlike rivets, screws etc., which are discrete fasteners, snap-fit features are integral to the part.
They thus help reduce part count and assembly time. The motions require for assembly are also usually along a single axis as other features on the parts should remove all degrees of freedom. Snap-fits have traditionally been used in lightly loaded application such as toys and other consumer products. However, with improvement in polymer technology and the rapid development of composite materials for structural applications, snap-fit features are now used in more demanding products. As an example, automotive under-hood applications that snap-fits have found use in includes air filter housings, throttle bodies, temperature and pressure sensors, and engine intake manifolds. These assemblies are expected to withstand harsh environmental conditions like high temperatures and pressures and contaminants like oil and corrosive gases. Lens housings in high-end projection television systems, pager housings, single-use cameras, and compact disc players are other examples of consumer products in which snap-fits have
3

been successfully applied. Quite often, snap-fits are used to provide secondary joining functionality to give the primary fastening (adhesive, ultrasonic welding) joints time to form. Commonly used snap-fit topologies are show in Figure 1.3 [4]. The primary cause restricting the use of snap-fit features in a larger number of polymeric products is the lack of confidence in their design process and performance attributes.
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
-3
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Figure 1.1: Typical experimental force curve for insertion of snap-fit features
1.2 Motivation for Research
The essential goal driving this research is the need for more accurate and versatile models of the performance of snap-fit design and a web-based snap-fit design tool that is independent of operating system platforms and can be distributed and upgraded
4

universally. As mentioned earlier, the models provided by the resin supplier are usually limited in scope and often inaccurate. An analytical model developed by Suri [4] overcame some limitations of current design guides. However it is unable to provide accurate prediction for high retention angles due to convergence issues. And with the development of the Internet, there is also a demanding for a web-based snap-fit design tool that is independent of all operating systems, easily accessible and can be universally upgraded by simply updating the design tool at the server location. The immediate need for an improvement in these areas forms the motivation behind the work done in this thesis.
-0.5
-1
-1.5
-2
-2.5
1.5
1
0.5
0
-3
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Time
Figure 1.2: Typical experimental force curve for insertion of snap-fit features
5

(a) Cantilever hook feature (b) Post & dome snap-fit feature
(Axisymmetric cross-section)
(c) Bayonet & finger snap-fit feature
(Two-dimensional cross-section)
(d) Loop-hook snap-fit feature
(e) Trap type snap-fit feature (f) Hollow-core hook snap-fit feature
(Two-dimensional cross-section)
Figure 1.3: Common snap-fit feature topologies
6

Figure 1.4: Equations for dimensioning cantilevers
Figure 1.5: Force multiplier of insertion and retention
7

1.2.1 Snap-fit Design Approaches
The development of equations for the detailed sizing of snap-fit features and for predicting their response can be accomplished using analytical, numerical or experimental method. The first two approaches, in particular, have been popular in the past. Because of cost and time issues, the experimental method is usually replaced by numerical method using finite element method to simulate experiments. In the following several sections, the analytical and numerical formulations will be discussed in detail.
Simplified Strength of Materials Analytical Formulation
Design guides from major resin suppliers use a strength of materials to develop simple equations (Figure 1.4 and 1.5) for the detailed sizing of snap-fits. For example the cantilever hook is one of the most common snap-fits and its basic shape is shown in
Figure 1.6. It is typically idealized as a cantilever beam with a rigid catch at the end. The transverse force required to deflect the end of the beam ( P ) by an amount equal to the offset of the catch is determined using Euler-Bernoulli beam theory. This is related to the forces acting on the catch in the engagement direction ( F i
) to determine the insertion force for the feature (Figure 1.6). A similar approach is used to determine the retention force. Strength of materials formulations however suffer from a number of limitations.
Consideration of the equilibrium of the catch will show that the beam is subject to both an axial load and a moment, in addition to the transverse load P . The deflections due to these forces and moments cannot be found by simple superposition. The bending moment in the beam due to the axial component depends on the deflection of the end, rather than being constant. This effect has not been considered in previous snap-fit analysis [5]. The
8

assumption that the maximum deflection of the end of the beam during insertion equals the offset of the catch also introduces some inaccuracy, because the end of the beam rotates, in addition to the transverse deflection. The Euler-Bernoulli beam theory is based on small deformation and long-beam assumptions, which are commonly violated in real life. y
P
F i
Insertion Direction
Mating
Part
Retention Direction
Cantilever Hook
Catch
Figure 1.6: Cantilever hook and mating part
Applied Mechanics Equations
Suri [4] has worked on an analytical formulation that includes a full applied mechanics formulation and solution. It involved idealizing the catch as a rigid body supported on a flexible structure (Figure 1.7). A set of equations that comprehensively described the system in its deformed configuration is formulated. The equation system was iteratively solved for several such configurations to obtain a model of insertion and retention processes for snap-fits. The model showed excellent agreement with experimental results for most snap-fit geometries. However, as the retention angle approaches 90 ° it becomes impossible to find a solution to the set of equations.
9

Figure 1.7: Suri’s idealized model of cantilever hook and mating part, shown in deformed configuration.
The Numerical Formulation
The numerical formulation approach involves the development of response surface equations and main effect plot using Design of Experiment (DOE) and statistical methods. The expected significant variables to the snap-fit’s performance and response variables are first identified. The portion of the design space that is of interest is then determined. An array of experimental designs spanning the region of interest is then created using standard statistical software such as SAS ® and Minitab ® . Physical or computational experiments at each of the points in the DOE array are conducted. Because of the cost related to manufacturing physical test samples and the test time, the use of computational experiments has been more popular in the literature. Typically, finite elements models of the snap-fit are created and queried for performance attributes.
Response surface equations are fit to the values of the response variables at the experimental point. These equations can then be used to predict the performance of the snap-fit at points in the design space other than those in the experimental array. The possibility of satisfactorily extrapolating the equations to other regions of the design
10

space also exists. Main effect plots depicting the sensitivity of the response variables to the factors are also created for quick determination of the most important design variables. Such an approach has been used for the cantilever hook [5], the bayonet & finger [6, 7], the compressive hook [8] and the post & dome snap-fit features [9].
While the numerical design formulation is an effective and efficient method for generating design equations for snap-fit, it has some disadvantages. Some limitations arise from it being a purely statistical method rather than being based on a fundamental understanding of snap-fit’s performance. The response surface equations are simply mathematical equations fit to the response variable values. They are very useful and effective when the analytical formulation is very difficult to model the response surface equation and the design space has just a single active physical phenomena.
1.2.2 Web-based Snap-fit Design Tool
Using the web as a development environment is a relatively new phenomenon called web-based application. The CAD/CAM area is also facing a transformation from computer-based to web-based applications. Computer-based application means that the design software packages need to be installed and operated on a standalone computer.
Famous examples in CAD/CAM are AutoCAD ® and ProEngineer ® . Eudora ® and
Outlook ® are also examples of computer-based application. Compared to computer-based applications, web-based applications refers to applications or services that are resident on a server that is accessible using a web browser such as Internet Explorer ® or Netscape ® and is therefore accessible from anywhere in the world via the Internet and control using the Web interface that browser provides. A popular example of a Web-based application
11

would be Microsoft’s Hotmail ® that is simply a large-scale email program that runs on
Microsoft’s server computers. They’ve allowed anyone with Internet access and a browser to connect to their server and check email. So instead of using Eudora ® or
Outlook ® to check email, the client can check email by using Hotmail ® anywhere. This approach has the following advantages:
• It allows users to log onto the system from anywhere in the world as long as they have a computer, an Internet connection and a web browser.
• Applications are resident on the server instead of client’s computer so that users don’t need to worry about software distribution and updates.
• User accessibility to data is definable.
• System administration can be performed remotely.
On the other hand, there are some limitations for web-based applications. The major concern with web-based applications is that of control and security. Although most
Web-based application servers protect against hackers with some security systems, no computer on the Internet is truly hacker-proof even though some web-based applications further protect their customer’s data by encrypting it in some way. Another limitation is speed. Web-based applications usually execute quickly, but their response is noticeably slower. In other words, clicking on a button or link doesn’t result in an instantaneous reaction by the program. This is simply because of the time lag that it takes for data to travel via Internet connection from client’s computer to the application server and back.
1.2.3 Constraint Management
Constraint management is a way of planning, organizing, evaluating and controlling complex systems and it has been used in a number of fields such as
12

variational geometry based Computer-Aided Drafting systems, economical management, mechanical design, chemical process synthesis, and artificial intelligent based constraint satisfaction procedures. Most of work done in the field of constraint management is closely related to sparse matrix research since the underlying matrix representations of the designs tend to be quite sparse. Kinzel and his students [57-62] developed an interactive engineering design framework consisting of a constraint manager coupled with a friendly graphical user interface. Constraint management was implemented into this design tool and several algorithms – new specification, respecification, unspecification and reverse specification – were developed.
1.2.4 Thesis Objectives
This thesis’s objective is to make significant contributions to advancing the accuracy and effectiveness of snap-fit modeling techniques in these areas. There are two objectives for the current research.
• Development of an improved numerical formulation for modeling the performance of a cantilever hook feature with a high retention angle.
As mentioned earlier, current snap-fits design guides and Suri’s analytical model are unable to predict the performance of high-retention cantilever hooks. Nonlinear and contact finite element analysis are able to overcome this limitation. Design of experiment and response surface methodology will also allow us to generate an empirical model to predict the performance of high-retention cantilever hooks.
• Creating a web-based design tool for snap-fit features. Usually empirical models developed by statistical methods have many polynomial terms and are
13

tedious to use. A web-based, platform-independent design tool of snap-fits implemented those empirical models will be very convenient to use. Three different snap-fits, Cantilever Hook, Post & Dome and Bayonet & Finger, will be built in the design tool and constraint management will be applied. Modules such as sensitivity analysis and correction advisor will also being applied. Two different optimizations, single objective optimization and multiple objectives optimization, will be applied in the design tool.
14

CHAPTER 2
LITERATURE REVIEW
The cantilever hook is the most widely utilized snap-fit feature (Figure 1.3(a)).
Trantina [10] created a model of cantilever and imposed a displacement load on the dwell surface equal to the offset of the hook. This type of model allows the geometric and material nonlinearity of the hook to be modeled, and, therefore, produces a reasonably accurate prediction of the transverse stiffness of the hook. A serious limitation to this type of model, however, is that it cannot model hooks in retention. This is especially true for non-removable hooks with retention face angles that are close to or equal to ninety degrees.
Hotra et al.
[11] modeled retention of a compressive finger by use of contact elements but did not model the insertion process. The modeling approach taken was to have the cantilever hook modeled as an elastic structure which is incrementally pushed into a retention block which is modeled as a rigid surface.
Luscher [5, 12] applied a combination of finite element analysis and design of experiments to the cantilever hook snap-fit feature. A four-factor, two-level, orthogonal array was used to study insertion and a five-factor, two-level array was used to study
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retention. The amount of engagement was also defined as a factor to incorporate the effect of warpage and shrinkage on the performance of the feature. The finite element results were compared to experimental data. Discrepancies between the two were explained. The sensitivity of the feature to geometric variables was discussed for insertion and retention separately. An optimal hook design formulation was also presented. The design equations developed by Luscher were first order and the retention face angles investigated were 65 ° and 90 ° degrees. With such a large spread of retention angles, it was not possible to accurately capture the subtle variations in retention force between angles from 85 ° to 90 ° degrees.
The post & dome feature is a high performance snap-fit that is self-datuming and can take some shear loading in addition to retention (Figure 1.3(b)). It provides a higher locking ratio than traditional cantilever hooks, and its retention strength is less dependent on friction. Nichols [9] created two design arrays to study the post & dome snap-fit feature. The first is called the catch array; a focused array constructed to determine the optimal preload and molded-in undercut at the interface between the post & dome segments. Optimal performance is judged with respect to both the snap-fits maximum retention force and maximum locking ratio. The second array, called the macro array, is a comprehensive design of experiments array. The objective of the macro array is to generate design data for a variety of dome geometries. Design equations are provided by subsequent analysis of the finite element data using Response Surface Methods (RSM).
The designed equations provide estimates for the maximum insertion force, insertion strain, and retention force applicable for an idealized post & dome snap-fit feature. To accommodate the snap-fit’s geometric nonlinearity, the post & dome are modeled with
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three dimension 8-node brick elements via the MARC ® finite element package to generate design data. This three-dimensional contact modeling utilizing an updated
Lagrangian [13] approach provides more accurate displacement predictions than small displacement beam theory.
A bayonet & finger snap-fit feature contains a bayonet, a retention finger and a support finger (Figure 1.3(c)). Wang et al. [14] analyzed a bayonet & finger feature using a contact type finite element model in ABAQUS. High-impact polystyrene (HIPS) was modeled as an elastic-perfectly plastic material. Finite element analysis predicted a snapthrough failure model. The simulation was compared to results from experimental tests.
Wang and Gabriele [15] created a finite-element model of the bayonet & finger snap-fit feature. The model is used to simulate the insertion and retention processes of the feature as well as snap-through and buckling phenomena. A “master-slave” approach is used for contact modeling in ABAQUS. An elastic-perfect plastic material model is used to describe the plastic behavior of high-impact polystyrene (HIPS). Finite-element analysis predicts a snap-through failure mode. The simulation is compared to results from experimental tests. Reasonably accurate correlation between analysis and experiment is observed for insertion. Results for retention are not presented.
Lewis et al. [16] expanded upon the previous work. A design of experiments approach with two and three dimensional finite-element methods was used to generate approximate linear response surfaces based on feature geometry which could calculate insertion and retention forces for the bayonet & finger feature. Five dimensions were chosen as the significant design factors. Two levels were chosen for each factor. A fractional factorial experiment design was used to reduce the number of experiments
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(analyses). Sixteen different trials were performed for the study. Material properties were not considered as a factor in the experiment. Material properties for HIPS were used, with an elastic-perfect plastic material model. Insertion was modeled using a 2-D model and retention using a 3-D model. Sensitivity information was generated using a level average analysis technique. Design equations for predicting the insertion force and retention force based on part dimensions are presented in the paper. An attempt at confirmation of the validity of the equations using a sample feature design fails. The FEA analysis results for the feature show large discrepancies when compared to the design equation predictions. No experimental verification is presented.
Shen [7] first found an optimum catch geometry of bayonet using an L-9 orthogonal experimental array. Then another three-lever, five-factor experimental array was used to obtain the response surfaces. These response surfaces approximated the amount of over-engagement as well as the insertion and retention forces for values of the design factors. The calculations were done through a finite element package MARC ® . A spring was connected between the root of the bayonet and an external node to simulate the base-part stiffness of the bayonet. The sensitivity of the values of this spring constant to the feature performance was investigated.
The compressive hook (Figure 1.3(e)) is a common snap-fit used in thermoplastic automotive electrical connectors. It derives its name form its retention mechanism which uses compression to provide the locking force. Hotra et al. [17] use an approach similar to Wang et. al. [13] to model performance of compressive hook. The analysis is performed for 15% glass reinforced poly-butylene terephehalate (PBT). The material data is experimentally determined at 0.08%/sec strain rate. Bending and shearing failure
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modes are investigated separately. The simulation results are compared to experimental data and are shown to be in good agreement. The paper emphasized the need for the use of advanced analysis techniques to model the performance of complex snap-fit features.
Hotra et. al. [11] extended the above work. Poly-phenylene oxide (PPO) was also considered, beside PBT. Bending and shear were again considered in separate analyses and the lower of the two taken as the actual failure mode. Tests were performed with a hard gage as a mating part in one case, and the actual terminal as the mating part in the other case. This was done to study the influence of mating part flexibility on the performance of the feature.
Roy [18] adopted an analytical approach for predicting the performance of the compressive hook feature. The hook was modeled as a cantilever beam with variable cross-section. In particular, it was divided into three difference beam sections and
Castigliano’s theorem used to determine the load-deflection relationships. A graphical solution method was also presented. Stiffness matrices for the feature were derived using an approach similar to the finite element method. Expressions for the insertion
(assembly) force were derived based on equilibrium of the feature. For retention, buckling was recognized as the primary model of failure. The Rayleigh-Ritz method was used to derive approximate expression for column buckling loads under different end conditions. The same method was also used to estimate eigen-frequencies for longitudinal and transverse vibration models of the feature. All the results were compared to results from a finite element analysis conducted using ANSYS.
Other snap-fit features such as cylindrical snap-fit hinges and ball snap-fits, have been studied using numerical simulation (FEM) and design of experiment.
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Matuschek and Michaeli [19] used a three-level, five-factor design of experiment to model the performance of cylindrical snap-fit hinge. Each experiment is a finite element model of the cylindrical snap-fit feature with the appropriate dimensions. A quadratic response surface is generated using the results of these analyses. The sensitivity of the maximum stress in the feature and the releasing force to the geometric variables is presented graphically. An equidistant grid strategy is used to partition the design space of the feature snap-fit identify the optimal set of dimensions for a set of given design criteria.
Bader and Koch [20] were the first to use a viscoelastic material model in the finite element analysis of a ball type snap-fit feature. A “standard linear solid” viscoelastic model consisting of springs and dashpots was used. Plasticity was modeled through the introduction of an additional mechanical element that remains inactive below the yield point for the material. The dependency of the spring and dashpot parameters on strain rate was modeled using a logarithmic curve fit. Plastic damage, recovery and stress relaxation behavior were experimentally observed and phenomenological models were fit to the data. Finite element analysis of the insertion and retention processes was performed using the commercial software MARC and MENTAT. The results obtained using viscoelastic and elastic-plastic material models were compared to experimental test data.
2.2 Materials Issues in Snap-fit Design
There have been some efforts to promote the use of advanced material models for polymers in the field of snap-fit design. Most of these have, however, been limited to relatively simplistic elastic-plastic material formulations, in order to model residual strain
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in the polymer. This thesis will attempt to extend the state-of-the-art in snap-fit analysis by demonstrating improvement in the accuracy of finite element analysis predictions. A brief review of past work dealing with improved material models for snap-fit design is presented below, to emphasize the benefit of the proposed approach.
In order to evaluate the applicability and accuracy of experimental test data,
Knapp et al. [21] compare three tensile test methods. It is well known that polymer behavior is dependent upon strain rate and that conventional tensile tests subject the specimen to variable strain during the test. In this work, tests were conducted at 0.002/s and 0.02/s strain rate. The experimental stress-strain data fit to an analytical curve and the value of tangent modulus (E t
) at each strain value was determined. This material model was used in a finite element analysis and its results compared to experimental data. The authors propose that designers should use true stress-strain data gathered at strain rate values appropriate for their application. Trantina and Minichelli [22] describe a software developed for automating snap-fit finite-element analysis. Deflection limited models are analyzed. Some elementary results describing the effect of dimensions on stress and strain on snap-fits are presented. The authors use a 0.1/s strain rate for determining the material properties. The work is also summarized, in more detail, in Trantina and
Minnichelli [10, 23].
Sawyer et al. [24] investigate the applicability of commonly reported coefficient of friction (µ) values for snap-fit design. The analysis of surface friction between polymers is extremely complex [25-27]. Widely different values of µ have been reported in literature. For example, coefficients of friction (µ) ranging from 0.28 to 0.62 have been reported for unfilled nylon 6/6. In this study, friction measurements were made directly
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from polycarbonate cantilever hook lock pair contacts, and their dependence on load, sliding speed and contact geometry was investigated. An experimental setup for accomplishing this is detailed. A power law dependence of µ on normal load is suggested in the paper. Changes in the value of µ with sliding speed were found to be minimal. The contributing mechanisms to friction, viscoelastic deformation and adhesion are discussed briefly. The authors contend that adhesive contributions occur at a scale smaller than that modeled by finite elements and as such should be included in the value of µ. On the other hand, the effect of viscoelastic deformation should be captured by the finite element model of the snap-fit and as such need not be included in the value of µ.
2.3 Failure Models of Cantilever Hook
In the past cantilever hook has always been selected primary in terms of retention force and the molding tooling costs. Failure modes of cantilever hook are usually ignored even although it was already recognized and understood that it will be an important factor to the performance of cantilever hook.
Luscher [5] noted a snap-fit mode of failure called loss-of-engagement at small
Percent Engagement (PE) values (Figure 2.1). In loss-of-engagement failure the snap-fit slips past the mating part without any mechanical damage to either half of the feature. At high PE values, the snap-fit’s failure mode switched to either a shear and/or tensile material failure.
Luscher et al. [28] extended the previous work. They determined the exact dividing lines between these different modes of failure with respect to the PE (Figure
2.2). Well-defined transitions between modes of failure were observed with the cantilever
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hook. The different failure modes also observed at different levels of PE . Three distinct failure modes were observed: shear failure, tensile failure and loss of engagement. Loss of engagement was the exclusive failure mode from 20 to 66 PE while shear failure was the only mode of a failure from 70 to 100 PE . Finally, tensile failure occurred at all values over 100 PE . There was only one exception, a single shear failure point at the 40
PE level. It was considered a statistical anomaly, most likely due to a material, molding or alignment condition. Figure 2.3 shows an example from each of the two mechanical failure modes in the cantilever hook. As shown in the Figure 2.3, the direct shear failure seems to occur progressively over a certain distance. Note in this figure that the failure seems to have a “stair-step” pattern showing that the mating part is tearing out the catch starting at the retention face. It is not able to tear out the catch along a line parallel to the removal direction. The tensile failure case is different. The failure surface looks like it occurs at a single point in time.
Figure 2.1: Percent engagement of cantilever hook
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Figure 2.2: The different failure modes with respect to the PE
(a) The tensile failure mode (b) The shear failure mode
Figure 2.3: The two different mechanical failure modes for cantilever hook
It is very important to identify the transition from one failure mode to another in order to accurately calculate the retention force provided by the cantilever hook feature.
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2.4 Web-based Design Tool for Snap-fit
The Internet is changing not only our daily lives, but also the professional field of product design and new product development.
Wang et al. [29] developed a fluid simulation package integrated with the
Internet, which allow the user to perform fluid simulation on the World Wide Web
(WWW), to simulate the traditional Computational Fluid Dynamics (CFD) problem -
Transient Natural Convection in a Cavity. This simulation package was developed using two different tools: one is Microsoft ® /COM ActiveX and VBScript and the other is Java and Java Applet.
Chung and Wright [30] proposed a web-based engineering framework for an infrastructure, which integers various advanced design/manufacturing systems such as
CAD, FEM and CAM, provides relationships among product data, and coordinates with manufacturing system. Designers access this framework through web browsers and start the design with some initial parameters.
Colton and Dascanio [31] described a vision and current development in a distributed design and manufacturing environment, and emphasized how current CAD tools will evolve to facilitate the distributed design and fabrication process. They also presented the development of a set of web-based design tools for fabricating parts using a machining process via the Internet and experiments on machining 2-1/2 D and freedom parts through their Java-based design tool showed the feasibility for a networked machining service via the Internet.
Kim et al. [32] developed a web-accessible CAD tool that assists a designer in producing designs that are manufacturable on a 3-axis milling machine, which simplify
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the previous complex testing for manufacturability by focusing on 3-axis milling only and by making use of new fixturing techniques using a plastic compound or a metal alloy for reference-free part encapsulation (RFPE). They also build a Java-based front end of this design tool that can be downloaded as an applet over the Internet.
Vel ά squez et al. [33] created a system named Tool Trial System (TTS) which is capable of collating and disseminating information relating to tool trials amongst a variety of user groups. TTS provided a platform from which is possible to submit and retrieve highly specific technical tooling data on the WWW can be downloaded by remote users in the form of Java applets, through any computer with Internet connection and using conventional Java enabled browsers without the requirements of using middle tiers software or hardware between clients and server sides.
Rajagopalan et al. [34] built an Internet-based infrastructure to provide designers with access to multiple layered-manufacturing services. This system contains three primary operatives: Design Clients, Manufacturing Services and Process Brokers. The
Design Clients allows designers to submit completed designs for algorithmic decomposition, or alternately, to compose a design from primitives and library components that have been primed with some process-related information.
Manufacturing Service consists of a highly automated machine that can be used to build ceramic parts, and the associated software components for design decomposition, process planning and machine control. The Process Broker implements a number of supporting services including process selection and optimal part orientation.
Ebbesmeyer et al. [35] described a web-based tool Virtual Web Plant (VWP), a tool to integrate 3D models from various CAD plant design tools and to display them
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interactively through Internet. Using the application of object-oriented database, it is possible to define various views of the logical plant structure so that the user is able to navigate easily through both the plant structure and the project documentation. The special advantages of an objected-oriental database for the storage of the graphical data are also shown.
Huang et al. [36] described a web site, WAPIP, which has been developed specifically to support new product introduction activities. It provides databases for software vendors and researchers to register their web applications with the “wapip” search engine. It also provides facilities to support practitioners in product design and manufacture to search rapidly for the right web applications suitable for solving their problems.
Many response surface equations for snap-fit feature’s performance [5, 9, 15] have already been studied using DOE and FEM. But it is often tedious to look for design equations for unique types of snap-fits to calculate the insertion force, the retention force and the locking ratio. If found, these equations are usually long, complex, and difficult to use. For this reason, an easy using front end will be very helpful during the designing process of snap-fit features.
Oh et al. [37] created a calculator (Figure 2.4) in which the design equations of seven snap-fit features (annular snap, bayonet & finger, post & dome, cantilever hook, cantilever-hole, compressive hook, L-shaped hook and U-shaped hook) were implemented. This calculator aids in designing snap-fits to meet specific loading requirements by allowing the designer to size the feature to obtain desired estimates for maximum insertion and retention forces. This calculator was developed in JAVA ®
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language that is independent of operating system platforms and can be distributed at a company site-wide over an intranet or worldwide over the Internet. This makes it easily accessible to a user, and universal upgrades can be achieved by simply updating the software at the server location.
Figure 2.4: The IFP snap-fit design tool
Honeywell Plastics [1] developed an online Snap-Fit Design Workspace (Figure
2.5). It is a web-based application that serves as an engineering tool in snap-fit design.
The program provides a workspace for the designer to test different scenarios by adjusting various input parameters and selecting different engineering material for the snap-fit design. The following snap-types were included: five different uniform beams,
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two different tapered beams, two different U shaped beam cases, and an L shaped beam.
The cantilever beam formulas used in conventional snap-fit design poorly estimates the amount of strain at the beam/wall interface because they do not include the deformation in the wall itself. To obtain a more accurate prediction of total allowable deflection and strain for short beams, a deflection magnification factor was applied in this design workspace.
Figure 2.5: AlledSignal Plastics’ snap-fit design guide
Eastman chemical company [38] also developed an interactive online support tool
(Figure 2.6) that receives inputs from the user to generate a technical recommendation for cantilever snap-fit design. This design tool calculates the theoretical strain that occurs when a cantilever is deflected. And the maximum strain occurs on the outer layer of the
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thickness, usually at the base of the latch. Strains can also be calculated for a latch, which varies linearly in both thickness and width from the base of the tip to latch. After the users specify the material, its brand and geometry dimensions for cantilever, this design tool will predict the deflection force and outer fiber strain.
Figure 2.6: Cantilever snap-fit design tool from Eastman chemical company
Jeff Raquet [39] used Java Applet to create a snap calculator (Figure 2.7) to assist in the development of correct parameters in the design of a plastic snap-fit. Designers need to select a material and give values to several specified inputs. Then based on these
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inputs, the snap-in force, engagement force and stress at root are computed. It also provide an “optimize” option to minimize the snap-fit’s volume.
Figure 2.7: Jeff Raquest’s snap-fit calculator
GE Plastics [2] (Figure 2.8), Engineers Edge [40, 41] (Figure 2.9 and Figure 2.10) and Berkeley Manufacturing Institute of University of California - Berkeley [42] (Figure
2.11) also developed similar online snap-fit calculators. All of them used conventional design equations based on small deformation and long beam assumptions.
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Figure 2.8: GE Plastics’ snap-fit wizard
Figure 2.9: Engineers Edge’s snap-fit straight beam calculator
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Figure 2.10: Engineers Edge’s snap-fit tapered beam calculator
Figure 2.11: Brock & Wright’s design tool for snap fits
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2.5 Constraint Management
For the current research, constraint management will be an important section. So it’s necessary to review the literature of Constraint Management (CM). Constraint
Management techniques have been used in a number of fields, such as variational geometry based Computer-Aided Drafting systems, mechanical design, chemical process synthesis, and artificial intelligent (AI) based constraint satisfaction procedures. Most of the work done in the field of constraint management is closely related to sparse matrix research since the underlying matrix representations of the designs tend to be quite sparse.
Steward [43, 44] was one of the earliest investigators who examined the structure of simultaneous equations. Algorithms to partition and reorder directed graphs were shown by Christensen and Rudd [45].
Duff [46] presented an excellent survey paper on sparse matrix research that included a survey of algorithms for matching and decomposing sparse matrices. Later,
Duff and Reid [47] developed an algorithm (Harwell MC13A) to permute a square occurrence matrix to its block triangular form in order to determine simultaneous equations sets. Their algorithm was based on Tarjan’s algorithm for determining the strong components of a directed graph. In order to accomplish its purpose, MC13A required the original matrix to be converted to a zero-free diagonal (maximum transversal) form. They compared various algorithms to achieve this configuration and developed algorithm MC21A to permute a sparse, square matrix to the maximum transversal form.
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Shacham [48] presented a new method to partition simultaneous nonlinear algebraic equations into smaller irreducible subsets. Prasad and Kinzel [49] presented an algorithm to perform decomposition of incidence matrices as the number of degrees of freedoms is reduced in a mathematical model. Akin [50] refers to the occurrence matrix as a Boolean Inference Array and presents an algorithm for the detection of solvable sets of nonlinear equations that is quite similar to the work of Prasad and Kinzel. The work of
Pothen and Fan [51] is aimed at converting a sparse rectangular matrix to its block triangular form. Their two stage decomposition techniques is based on the Dulmange-
Mendelsohn decomposition algorithm.
A number of researchers have studied constraint management in the field of variational geometry. Sutherland [52] was amongst the first researchers to treat 2- dimensional drafting as a constraint satisfaction problem. Later works of Lin and Light
[53], Gossard [54], Serrano [55] added more to the mathematical basis behind CM.
Serrano used directed acyclic graphs (DAGs) to determine dependencies among variables and to identify sets of simultaneous equations. A constraint modeler was also shown that allows for interactive addition/deletion of constraints. Chung and Schussel [56] compared parametric and variational geometry environments and described a commercial package.
Srinivasan [57, 58] presented the Design Shell: an interactive engineering design framework consisting of a constraint manager coupled with a friendly graphical user interface. Sridhar [59, 60] formulated strategies to interactively handle inequality constraints in the design shell framework. Agrawal et al. [61] identified and developed various constraint management, numerical solution, and numerical optimization methodologies necessary for the performance of the shell as an interactive and effective
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design tool. Thomas [62] extended the design shell constraint management strategies towards the development of a framework for performing term designs.
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CHAPTER 3
FINITE ELEMENT ANALYSIS OF CANTILEVER HOOK WITH A HIGH
RETENTION ANGLE
This chapter describes the finite element model of cantilever hook with a high retention angle. Firstly three different nonlinearities of finite element analysis were described: Moving Contact, Geometric Nonlinearities and Material Nonlinearities. Then finite element analysis techniques applied in this thesis were explained because in a typical FEA simulation usually 80%-90% time is spent to find the best parameter settings.
Finally a two dimensional, contact FEA model was created in MSC.MARC, and the results were compared to the sample testing results with the same geometry to verify the accuracy of the FEA model.
After an accurate FEA model was obtained in this chapter, in the next chapter it was been used for the simulation of each design point and an empirical model was generated by using Design of Experiment (DOE) and Response Surface Methodology
(RSM).
3.1 Combination of FEA with DOE
Modern finite element analysis and design of experiments techniques can be combined into a powerful quality-engineering tool [63]. FEA has been widely used to
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simulate the snap-fit behaviors in the recent years and proved to be a powerful tool to get better understanding of snap-fit performance. Usually FEA users worked in the way of
“one problem-one model” interactively. That is, for a given problem (say, a cantilever hook with certain geometry and under certain working condition such as loading and boundary condition), designers usually build a fixed model, solve it and get the required results (the insertion force, the retention force, etc). This has proved to be useful as shown in many applications.
However, when using FEA to do DOE and design optimizations, the snap-fits have similar geometry but the parameters of components (dimensions and material properties) are changeable. Generally, even a simple DOE often involves more than ten runs. Working in one problem-one model way, we need to build many models interactively or manually which is time consuming.
Fortunately, in MARC FEA package, besides working in interactive mode, it also allowed to use “PROCEDURE” file (Appendix A). “PROCEDURE” file allows users to build the model in terms of parameters instead of fixed value, which in turn make it very easy to change the design. Compared to the traditional one problem-one model method, the combination between DOE and FEM has the following advantages:
• By expressing all the parameters in terms of variable the model becomes very flexible. In order to rebuild other models, designers just need to change the parameters’ value. This feature is especially suitable for DOE and design optimizations.
• Store the model in text file (generally in size of a few kb) instead of in db file (in size of a few Mb) to save space.
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• Run batch of models overnight to make full use of FEA system.
3.2 Nonlinear Structural Analysis of Snap-Fits
The reasons that the snap-fits design guides are not accurate for cantilever hook with a high retention angle are material and structural nonlinearities. Suri’s analytical mathematical model has difficulties to converge for high-retention cantilever hooks.
Nonlinear structural behavior arises from a number of causes, which can be grouped into these principal categories:
• Moving Contact
• Geometric nonlinearities
• Material nonlinearities
For cantilever hook with a high retention angle, all these three nonlinearities exist in the retention stage. The following paragraphs explained these three nonlinearities in details.
3.2.1 Moving Contact
Many common structural features exhibit nonlinear behavior that is statusdependent. Status changes might be directly related to load, or they might be determined by some external cause.
Situations in which contact occurs are common to many different snap-fit applications. Contact problems are highly nonlinear and require significant computer resource to solve. Contact problems present two significant difficulties. First, users generally do not know the regions of contact until you’ve run the problem. Depending on the loads, material, boundary conditions, and other factors, surfaces can come into and go
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out of contact with each other in a largely unpredictable and abrupt manner. For example, during the retention of cantilever hook there actually exist several contact pairs depending on contact locations. Since the deformation of local contact areas it’s very difficult to identify the point when these contact pairs change. The second reason to cause contact problem highly nonlinear is that most contact problems need to account for friction. There are several friction laws and models to choose from, and all are nonlinear.
Frictional response can be chaotic, making solution convergence difficult.
3.2.2 Geometric Nonlinearities
Small deflection and small strain analyses assume that displacements are small enough that the resulting stiffness changes are insignificant. In contrast, large strain analyses account for the stiffness changes that result from changes in an element’s shape and orientation. All design guides of snap-fits and Suri’s analytical model were based on small deformation theory. However cantilever hook with a high retention angle experiences large deformations, its changing geometric configuration can cause the structure to respond nonlinearly.
3.2.3 Material Nonlinearities
A number of material-related factors can cause your structure’s stiffness to change during the course of an analysis. Nonlinear stress-strain relationships of plastic, multilinear elastic, and hyperelastic materials will cause a structure’s stiffness to change at different load levels (and, typically, at different temperatures). Creep, viscoplasticity, and viscoelasticity will give rise to nonlinearities that can be time-, rate-, temperature-, and stress-related. Most polymeric materials used for cantilever hooks have a time-
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dependent behavior, in which a certain amount of deformation is recoverable over time.
For the purpose of numerical modeling, it was assumed that the material was timeindependent.
For all design guides of snap-fits none of these material nonlinearities were included in their design guides. All of them assumed that the maximum stress always under the yield stress that means no plastic deformation occurred during the retention. So that in order to overcome the inaccuracy caused by the ignoring of plastic deformation, it’s necessary to include the material nonlinearity in the FEA model. Several plastic material options are available in most of FEA packages such as MSC. MARC and
ANSYS to describe material plasticity behavior. These options are the Bilinear
Hardening, Multilinear Kinematic Hardening, Nonlinear Kinematic Hardening, Bilinear
Isotropic Hardening, Multilinear Isotropic Hardening and Nonlinear Isotropic Hardening.
Since in this thesis only Bilinear Hardening was used to describe the material plasticity behavior of plastics, the Bilinear Hardening option was shown in Figure 3.1. Although this is an approximation of the actual material’s curve it is an improvement over ignoring plasticity. For more details of other options, please refer to [64].
3.3 Numerical Model of Cantilever Hook Feature
In order to compute the insertion force, insertion strain, retention force and locking ratio of cantilever hook, the author created a two-dimensional, contact and nonlinear finite element model on Marc for each experiment. A mesh was generated using plane stress quadrilateral elements with full integration, which can prevent instability called hourglass mode [13] in a bending dominant problem. An example of the
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meshed cantilever hook model on Marc is illustrated in Figure 3.2. A fine mesh is required on the insertion face, dwell face and retention face of cantilever catch where are the crucial regions at engagement and disengagement.
Stress
Yield
Stress
Inelastic
Region
Elastic
Region
(a) An actual stress-stain curve
Strain
Yield
Stress
Stress
Workhardening
Slope
Inelastic
Region
Elastic
Region
(b) Bilinear hardening option
Strain
Figure 3.1: The actual and bilinear stress-strain curve for plastic material
3.3.1 Modeling Techniques for Contact Problem
The modeling of the contact conditions is an advanced technique in the finite element method. To simulate the contact condition of the finite element models, the cantilever hook was specified as a deformable contact body and the fixed displacement boundary conditions were applied to the end of the cantilever hook (Figure 3.2). The mating part was defined a rigid body and the nodes on it were tied together as a group and fixed in the vertical direction.
The analysis of surface friction between polymers is extremely complex [24-26].
The classical "laws" of friction in which friction is proportional to applied load, and is independent of area of contact, velocity, and time of contact are poorly held for most polymers. In addition the conditions of a sliding snap-fit in terms of its speed, area of
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contact, and residence time are not usually replicated in resin manufacturers tests and reported friction values. Rather than delve into this area, which is not a key focus of this research, it will be assumed that background research will be done to determine a value of coefficient of friction which is appropriate for the snap-fit in question. Therefore in these computer models an average and widely reported coefficient of coulomb friction was applied on the contact surfaces of cantilever hook and mating part. The character of the contact constraints depends on whether there is sticking or slipping. In order to avoid this distinction, an approximation is made within the MARC ® in such a way that there is always slipping. However, the friction forces decrease to zero when the amount of slip also decreases to zero. The relative sliding velocity below which this decrease comes into effect is defined by the user. Too high value for the relative sliding velocity will remove the fiction effect and too low value will run into a stick and slip situation which causes the numerical chatter.
Mating part fixed in Y direction.
Cantilever Hook
Y Cantilever hook
X
Figure 3.2: Typical cantilever hook meshed example
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The insertion and retention motions of the mating part were modeled by applying a prescribed horizon velocity to the nodes on the mating part. The reaction forces collected from these nodes were the insertion force or the retention force. We divided both insertion and retention motions into hundreds of increments and ran each increment as a quasi-static analysis. One assumption was made for the quasi-static analysis - the inertia terms are negligible.
A contact tolerance is also needed such that if the distance calculated is below this tolerance, a node is considered in contact. In general, the contact tolerance should be a small number compared to the average element size. Therefore the MARC program can automatically detect the nodes in contact and calculate both the reaction and the friction forces.
3.3.2 Improvement of Solution
Snap-fits or locks represent a difficult class of features to be modeled. Consider a cantilever hook with a 90-degree retention face, during the final portion of the insertion process, the retention face rapidly snaps back into its initial position behind the mating part. This motion is actually highly dynamic in nature and produces convergence difficulties in the finite element model. This effect is even more pronounced in terms of retention models. As a permanent hook is pulled in retention, strain energy builds up in the beam portion of the hook. When the retention face slips out past the mating part, this energy is released almost instantly and this makes convergence of the model’s solution extremely difficult. In order to get the model’s solution past this sudden release of energy, some improvements need to take to achieve convergence.
44

Time Load Increments
The rigid contact mating part is moved along horizon to simulate the insertion and retention process. Each increment moves the mating part a fraction of a millimeter. The motion of the cantilever hook and mating part will be broken up into hundreds of time load steps to accurately simulate a dynamic process. Using frequent time load steps allows the finite element analysis to capture changes in the cantilever’s shape as it is displaced. The cantilever hook becomes bended as it is displaced and the high number of increments is required to resolve this geometric distortion. This distortion needs to be accounted for to accurately predict insertion force and strain. Time load increments must be selected with respect to the distance tolerance and velocity of the contact body being moved. The displacement of a contact body cannot exceed the distance tolerance.
Element
The selection of full integral as the element type to be used in this model was driven by the effect of element type on contact problem convergence rate. Reduced order elements were considered because of the reduction in solution time these elements provide. However the model with reduced order integration elements always leads to hourglass modes preventing convergence. Another reason full order integration elements were used is that constant dilatation option was only available for this type of element.
The constant dilatation option was recommended for analysis involving a high degree of bending with some plastic deformation [13].
Optimizing the mesh
The default mesh always required some improvements to provide a mesh with good aspect ratio and no distorted elements. Modifications were sometimes required to
45

facilitate contact. The transition between the base of cantilever and the catch of cantilever frequently required a transition mesh in order to preserve good element geometry.
3.3.3 Comparison between FEM Result and Experiment Result
To verify the accuracy of cantilever’s FEM model, the comparison between
FEM’s results and experiment’s result should be done before any design of experiment’s work. Three cantilevers with different geometries were defined and the SLS (Duraform
Polyamide) is chosen as material. The experiment results of the tensile test and the comparison are shown in the following Figure 3.3 - Figure 3.6.
20
15
35
30
25
10
5
0
0 2 4 6
Strain (%)
8 10 12
Figure 3.3: The nonlinear stress-strain curve of Duraform Polyamide
46

5
4
3
7
6
2
1
0
-1
-2
-3
0
Testing Result
FEM Result
5 10 15
Position (mm)
20
Figure 3.4: The comparison results for cantilever 1
25
50
40
30
20
10
80
70
60
0
-10
-20
0
Test Result
FEM Result
2 4 6 8
Position (mm)
10
Figure 3.5: The comparison results for cantilever 2
12
47
14

20
15
10
5
0
-5
-10
-15
-20
0
Test Result
FEM Result
5 10 15 20
Position (mm)
25 30
Figure 3.6: The comparison results for cantilever 3
35
48

CHAPTER 4
THE EMPIRICAL MODEL FOR PERFORMANCE OF CANTILEVER HOOK WITH
A HIGH RETENTION ANGLE
Due to the complexity of cantilever hook design, the direct relationship between design inputs such as geometrical dimensions and design outputs such as retention force is not entirely known. Suri [4] developed an advanced analytical math model by considering the interaction of beam rotation, catch geometry and axial loading. But similar to the current design guides from resin companies such as GE Plastics, Bayer
Polymer and Honeywell Plastics, it works for cantilever hooks with insertion and retention angles less than 87 degrees. For cantilever hooks with higher angles the numerical solution can have difficulty converging to a solution. Cantilever hooks commonly have a retention angle less than 90 degree in many industrial applications for two reasons. First, although they may be designed have a 90 ° retention face, polymeric parts often experience significant shrinkage and warpage which can change this angle to some other angle in the range of 85 ° to 90 ° . With increased emphasis on design robustness, the performance of these off-nominal cases must be estimated by a predictive tool. Second, angles are often designed to be somewhat less than 90 ° for tolerance considerations. By having such an angle small tolerances along the axis of the snap can be absorbed by use of near 90 ° and a preload in the axial direction. This will ensure that
49

gaps are always eliminated and contact is always maintained between the snap-fit and the mating part.
Response surface modeling (RSM) is a methodology to create a mathematical model of a design or process. Typically some performance output is measured while several input variables are changed in a systematic way. The influences are recorded and techniques are used to create a model based on the results. Such a model can be used to improve, determine the design sensitivities, or optimize a design or process. Because there is no need for an analytical model in a RSM and manufacturing processes can be very complex, extensive applications of RSM have been in the industrial world.
DOE starts with the identification of factors or variables are likely to be important in the response surface study. This leads to an experiment designed to investigate these factors called a screening experiment. Often at the outset of a response surface study, there is a rather long list of variables that could be important in explaining the response.
The objective of a screening experiment is to reduce this list of candidate variables to a relatively vital few so that subsequent experiments will be more refined, efficient and require fewer runs or tests. Once the important independent variables are identified, the next step is to map the response surface over a particular region of interest. Since there is hopefully a reduced number of design variables this second model is usually at least a second-order surface and therefore more highly refined.
In this chapter, the above steps were applied to develop the response surface models of cantilever hook with high retention angles. First, all potentially important factors to the performance of the cantilever hook were identified based on previous design experiences. Second, a two-level, fractional factorial screening array was
50

constructed to determine the most important factors by using best subset regression
(BSR). Third, a refined array was to generate second-order response surface equations of retention force for a variety of cantilever hook geometries with high retention angles.
These surface equations were provided by subsequent analysis of the finite element data using multiple linear regression analysis. Central Composite Design (CCD), the most common response surface methodology, created the refined array. Both arrays used finite element models of the cantilever hook feature to generate design performance data used in numerical modeling.
4.1 Screening Experiment of Cantilever Hooks with High Retention Angles
After carefully investigation, the following 4 material factors were chosen to study the retention force of cantilever hook with high retention angles: Young’s modulus
(E), coefficient of friction ( µ ), yield strength and workhardening slope. The following geometry factors were also included: beam length (L), beam thickness (d
0
), taper (d
0
/d
L
), undercut (offset) (y) and retention angle ( β ) (Figure 4.1).
Some comments on eliminated design factors are in order. The first is snap-fit beam width. Beam width can be modeled with the existing 2D finite element model if elements are given a linearly varying thickness as a function of their position along the cantilever beam. Although there are cantilever hook snap-fits which are tapered in width, the majority of them are not. Therefore this model did not include b
0
as part of the study.
In the current model retention forces are found on a unit width basis and the total force is calculated as a ratio of the total snap-fit width.
51

To make this research results as useful as possible, the design factors should be as generic as possible based on previous experience of generating design equations for snapfit features. Shen [7] generated second order response equations for the performance of bayonet & finger (Figure 4.2) by using FEA and DOE. However these design equations were based on a specific dimension – the bayonet base thickness (t). The bayonet base thickness, being equal to the nominal wall thickness of the plastic part, varies as part of the design process. Unfortunately since the design equations were all based on this one dimension, they did not have wide applicability. To avoid this situation, and create a mathematical model that can be widely applied, all of the geometry should ideally be based on dimensionless geometric ratios rather than specific dimensions. This lead to the formation of two ratios: the beam length to beam thickness (L/d
0
) ratio and the undercut to beam thickness (y/d
0
) ratio.
To demonstrate that the linear scaling relationship was valid, 9 cantilever hooks with different beam thickness and other parameters fixed were simulated in Marc, and from Figure 4.3 the approximated linear relationship was found to exist. Based on the previous work and experience about snap-fit design, for the screening experiment eight genetic factors and their upper and low limits were defined in Table 4.1.
Upper
Limit
Low
Limit
4000 3625 1 90 10 1.5 2 1000
1200 580 0.6 87 5 0.8 1 100
Table 4.1: The upper limit and low limit of each parameter
52

Figure 4.1: The typical cantilever hook
53

Figure 4.2: The typical bayonet & finger
90
80
70
60
50
40
30
20
10
0
0 1 2 3 4 5 6
Thickness (mm)
7 8 9 10 11
Figure 4.3: The relationship between beam thickness and retention force
54

A 1/8 fraction factorial screening experiment created in Minitab (Table 4.2).
Run E(MPa) Y.S.(MPa) Fri. Coef. Ret. Angle B.L./Th Un.C./Th. Taper Wor. Sl. Ret. Force(N)
1 1200 580 0.6 87 5 1.5 1 100 45.28
2 1200 3625 0.6 87 10 0.8 2 100 17.30
3 4000 3625 0.1 87 5 0.8 1 100 32.90
4 1200 580 0.6 90 10 1.5 2 100 20.50
5 4000 580 0.6 87 5 0.8 2 100 64.70
6 4000 580 0.6 90 5 0.8 1 1000 175.00
7 4000 580 0.1 90 5 1.5 1 100 42.00
8 1200 3625 0.6 90 10 0.8 1 1000 48.67
9 4000 3625 0.6 90 10 1.5 2 1000 68.00
10 1200 580 0.1
11 1200 580 0.6
90
90
5
5
0.8 2 100
1.5 2 1000
54.89
23.80
12 1200 580 0.1
13 4000 3625 0.1
14 4000 3625 0.1
15 1200 3625 0.1
16 4000 3625 0.6
17 1200 3625 0.6
18 4000 580 0.1
19 1200 3625 0.1
87 10 0.8 1 100
90 10 0.8 2 100 15.80
90
90
5
5
0.8 2 1000
1.5 1 1000
7.53
18.50
12.90
87 10 1.5 1 100 14.80
87 5 0.8 2 1000 20.00
87 5 1.5 2 1000 16.23
87 10 1.5 2 1000 3.20
20 4000 3625 0.6
21 4000 3625 0.6
22 1200 3625 0.1
23 1200 580 0.1
24 4000 580 0.6
25 4000 3625 0.1
26 1200 580 0.1
27 1200 3625 0.1
90
87
5 1.5 2 100 79.36
5 1.5 1 1000 156.40
90 10 1.5 1 100 11.78
90 10 0.8 2 1000 4.80
90 10 0.8 1 100 139.00
87 10 0.8 1 1000 25.11
87
87
5
5
0.8 1 1000
1.5 2 100
10.26
4.87
28 4000 580 0.1
29 1200 3625 0.6
30 1200 580 0.6
31 4000 580 0.6
87 10 1.5 2 100 10.66
90 5 0.8 1 100 54.56
87 10 1.5 1 1000 44.56
87 10 0.8 2 1000 59.67
32 4000 580 0.1 90 10 1.5 1 1000 39.26
Table 4.2: The screening array
Best Subset Regression
Once the screening experiments were run, Best subset regression (BSR) was used to find the significant design factors. BSR is a model-building technique, which finds subsets of predictor variables that best predict responses on a dependent variable by linear (or nonlinear) regression. The best subsets regression procedure can be used to
55

select a group of likely models for further analysis. The general method is to select the smallest subset that fulfills certain statistical criteria. The reason to use a subset of variables rather than a full set is because the subset model may actually predict future responses with smaller variance than the full model.
Best subsets regression identifies the best fitting regression models that can be constructed with the predictor variables that you specify. By default, all possible subsets of the predictors are evaluated, beginning with all models containing one predictor, and then all models containing two predictors, and so on. By default, Minitab reports the two best models that can be constructed for each number of predictors.
Suppose you conduct a best subsets regression with three predictors. Minitab will report the best and second best one-predictor models, followed by the best and second best two-predictor models, followed by the full model containing all three predictors.
The following 4 statistical parameters were used as comparison criteria by the best subsets procedure to screen out the most important factors [65].
• R 2 (sample coefficient of determination): Consider a regression equation between a dependent variable y and a set of explanatory variables x ' = ( x
1
, x
2
,..., x q
): y = α + γ
1 x
1
+ γ
2 x
2
+ L + γ q x q
+ z (4.1) where α is an intercept parameter, z is a random error term assumed to be uncorrelated with the explanatory variables, and γ ' = ( γ
1
, γ
2
, ..., γ q
) is a vector of coefficients to be estimated. The usual interpretation of R 2 is as the relative amount of variance of the dependent variable y explained or accounted for by the explanatory variables x
1
, x
2
, ..., x q
. For example, if R 2 = 0.762 we say that the
56

explanatory variables "explain" 76.2 % of the variance of y . Regression subsets with large values of R 2 should be chosen.
• Adjusted R 2 : Adjusted R 2 also measures the proportion of the variation in the dependent variable accounted for by the explanatory variables. Unlike R 2 , adjusted R 2 allows for the degrees of freedom associated with the sums of the squares. Therefore, even though the residual sum of squares decreases or remains the same as new explanatory variables are added, the residual variance does not.
For this reason, adjusted R square is generally considered to be a more accurate goodness-of-fit measure than R square. Those regression subsets with large values of adjust R 2 should be choose.
• Mallow Cp : This is a measure of total squared error for a subset model containing p regressors. The total squared error is a measure of the error variance plus the bias introduced by not including important regressors in a model. Therefore, it may be used to detect if too many regressors have been deleted. We should select those regression subsets with small values of Cp . Moreover, we want to be sure that these values close to the line Cp = p , where p is number of predicators included in the model.
• s (square root of MSE): s estimates true standard deviation. Those regression subsets with small values of s should be selected.
These four different standards were calculated by Minitab statistical software and the results are shown in the Table 4.3. Only 5 of the original 8 factors were selected to be carried forward to the second array. As seen by all 4 of the standards, as more factors are considered there is no reliable gain in accuracy. The values of R 2 and adjusted R 2 are no
57

longer monotonically increasing and therefore no additional benefit is guaranteed. The 5 factors chose for future investigation were (Young Modulus, Friction Coefficient,
Retention Angle, Beam Length/Beam Thickness and Taper). Other thee non-significant factors were chosen the average of the low limit and the upper limit in the following central composite design array.
Vars
Rsq
R-sq
(adj) C-p s E(Mpa)
1 28.2 25.8 20.2 37.122
Y.S.
(Mpa)
Fri.
Coe
Ret.
Ang.
B.L./
Th
Un.C
./Th
Tape r
Non.
Sl.
X
1 17.8 15.0 27.2 39.724
X
2 46.0 42.3 10.3 32.747
X X
X X
48.8 7.1 X X X
44.9 9.4 X X X
51.8 6.2 X X X X
51.6 6.3 X X X X
54.9 5.4 X X X X X
51.9 7.1 X X X X X
55.1 6.3 X X X X X X
6 63.4 54.7 6.5 29.018
X X X X X X
7 65.1 54.9 7.4 28.943
X X X X X X X
7 64.4 54.1 7.9 29.209
X X X X X X X
8 65.7 53.8 9.0 29.284
X X X X X X X X
Table 4.3: The result of best subset regression
It should be noted that none of the work hardening factors were found to be significant. The reason for this is that for a cantilever hook snap-fit design of typical dimensions the outer fibers at the fixed end experience their highest strain during insertion rather than retention. As noted in Luscher [28], the primary mode of failure under retention forces is loss-of-engagement. In this mode the catch is partially rotated about the point of contact until the angle of retention allows a sudden release of the catch and there is no engagement (or undercut) available to prevent disassembly. Therefore in
58

our modeling of retention, the material at the catch will not experience high values of strain and will never enter the working hardening region of the material curve. The above phenomenon can also explain that the factor of ratio between beam undercut and beam thickness is not significant.
4.2 Central Composite Design
The central composite design method (CCD) is the most popular class of secondorder designs. Much of the motivation of the CCD evolves from its use in sequential experimentation. It involves the use of a two-level factorial or faction combined with axial or star points (Figure 4.4). The sequential nature of the design becomes very obvious. The factorial points represent a design with minimal variance for a first-order model or a first-order plus two-factor interaction type model. Center runs clearly provide information about the existence of curvature in the system. If curvature is found in the system, the addition of axial points allows for efficient estimation of the pure quadratic terms.
The areas of flexibility in the use of central composite design reside in the selection of α , the axial distance, and n c
, the number of center points. The choice of α depends to a great extent on the region of operability or interest. The choice of n c
often has an impact on the distribution of prediction variance in the region of interest. In this study α = p = 5 = 2 .
378 ( p is number of predicators) and 3-5 center points were chosen to reach design rotatability [66]. Design rotatability means the process possesses a reasonably stable distribution of prediction variance throughout the experimental design region. It must be clearly understood that the experimenter does not know at the outset
59

where in the design space the optimum may lie. Thus, a reasonably stable prediction variance provides insurance that the quality of a prediction of future response values is roughly the same throughout the region of interest. Results for each case were obtained using finite element models, while response surface methods and multiple linear regression analysis were used to develop design equations to predict a retention force for the cantilever hook.
1
Factorial points
+
Central points
α
Star points
Figure 4.4: The combination of central composite design
60

In the last section, five factors (Table 4.4) were identified to be the most significant factors to the performance of cantilever hook. The central composite design array (Table 4.5) was generated based on these five factors. Each design point was simulated in the nonlinear finite element model. As an example of a typical model’s results, the retention force curve of one central composite design array is shown in Figure
4.5. The data plotted is for a mating part being moved at a constant velocity and the retention force is plotted against time. From the force curve, the retention force increases with the increase of time and reaches the maximum value in the upper retention face.
Young’s
Modulus (MPa)
Friction
Coefficient
Retention
Angle
Beam Length /
Thickness
Upper Limit
Low Limit
Table 4.4: The most significant factors for cantilever hook
3
2.5
Taper
2
1.5
1
0.5
0
-0.5
-1
-1.5
0 9 10 1 2 3 4 5
Position (mm)
6 7
Figure 4.5: Retention force vs. position
61
8

Run Young Modulus
1 -1
2 1
3 -1
4 0
5 -1
6 0
7 0
8 1
9 0
10 -1
11 1
12 1
13 0
14 -1
15 -1
16 1
17 1
18 1
19 -1
20 0
21 0
22 1
23 0
24 -1
25 0
26 -1
27 -1
28 0
29 -1
30 0
31 0
32 -1
33 1
34 0
35 1
36 1
37 1
38 -1
39 1
40 -1
41 0
42 1
43 1
44 0
45 0
46 0
47 -1
48 0
49 1
50 -1
51 2.37841
52 -2.37841
Fri. Coef. Ret. Angle B.L./Th Taper Ret. Force (N)
1 1 1 15.530
-1 -1 1 2.694
-1 1 1 18.600
1 -1
0 0
-1 1 -1 3.214
1 -1 -1 15.015
-1 1 1 6.284
1 1 1 10.875
-1 1 -1 18.490
-1 -1 -1 7.780
0 0 0 9.235
0 0 0 10.970
-1 0 0 1.275
1 1 -1 6.300
0 0 0 10.380
-1 -1 1 7.934
-1 1 -1 9.244
0 0 0 10.380
1 1 1 5.436
1 0 0 17.360
-1 -1 -1 2.810
0 0
1 1 -1 12.540
1 1 -1 34.270
-1 1 1 3.142
1 1 1 31.260
-1 1 1 9.298
0 0 0 10.320
-1 1 -1 6.447
1 -1
0 0 0 10.180
0 0 0 10.300
1 -1 1 4.834
0 0 0 10.300
1 1 -1 17.150
0
0
0
0
0 0 18.570
0 0 2.147
Table 4.5: The central composite design for cantilever hook
62

The resulting data from the central composite design array was analyzed with multiple regression techniques to create the response surface equation shown below:
Retention Force = 10.05 + 3.91 E + 3.49 µ - 1.33 L/d
0
- 6.84 (d
0
/d
L
) + 1.17 (L/d
0
)^2 + 2.66 (d
0
/d
L
)^2
+ 1.18 E* µ - 1.95 E*(d
0
/d
L
) – 1.63 µ *(d
0
/d
L
)
(4.2)
This equation is applicable for any cantilever hook geometry in the explored design space, and provides an estimate of a cantilever hook’s retention force. This response surface equation also provides sensitivity data of the main design variables and variable interactions. The statistical analysis package MINITAB was used for the regression analysis of this full interaction quadratic model. All design variables and interactions between variables were considered, and terms that were not significant according to their Student’s t-test were discarded. The data was fit to a R 2 (adj) value of
0.887 (Table 4.6).
The above table shows the regression analysis of the full interaction quadratic model. The first column lists all terms of this model and the second column lists their coefficients. A general rule to judge the significance is the value of P.
If one term’s P is less than 0.05, this term is significant to the model.
The plot of residuals versus the fitted values and normal probability are shown in
Figure 4.6. The residuals plot show curvature and some outliners that will obviously affect the accuracy of the response surface. Both transformation of response and deleting outliers will be discussed in the following sections.
63

Fri. Coe
Ret. Ang
3.489
0.787
0.4762
0.4762
7.327
1.654
0.000
0.108
Fri. Coe*Fri. Coe
Ret. Ang*Ret. Ang
-0.464
-0.291
0.4096
0.4096
E(Mpa)*Fri. Coe
E(Mpa)*Ret. Ang
1.181
0.276
0.5540
0.5540
-1.133
-0.711
2.132
0.497
0.266
0.483
0.041
0.622
Fri. Coe*Ret. Ang
Fri. Coe*B.L./Th
Fri. Coe*Taper
Ret. Ang*B.L./Th
Ret. Ang*Taper
0.138
-0.383
-1.623
-0.072
-0.404
0.5540
0.5540
0.5540
0.5540
0.5540
0.249
-0.691
-2.930
-0.129
-0.729
0.805
0.495
0.006
0.898
0.472
S = 3.134 R-Sq= 93.1% R-sq(adj)= 88.7%
Table 4.6: Regression analysis for retention force of cantilever hook
4.3 Model Improvement
When a regression model is considered for an application, we can usually not be certain in advance if the model is appropriate for that application. Hence, it is important to examine the appropriateness of a fitted regression model. In this section, several statistical methods for diagnosing the appropriateness of the model are discussed. The author also considers some remedial techniques that can be helpful when the data are not in accordance with the conditions of regression model. Finally validation of the regression model was conducted by comparison between the FEA simulation results and results predicted from the regression model.
64

4.3.1 Transformation of Empirical Model
Figure 4.6: The residual and normality plots for retention force
65

Whether a linear regression function is appropriate for the data being analyzed can be studied from a residual plot against the fitted values. The residuals (Figure 4.6) show that a linear regression function may not be appropriate for the data. The residuals decrease in the middle of fitted values but increase at each end. To eliminate this residual curvature, the two most common transformations were considered: logarithm and reciprocal. Their residuals plots are shows in Figure 4.7 and Figure 4.8. Compared to the residuals plot of untransformed retention force, the residual plot of logarithm transformation shows a better pattern because the residuals show random scatter without any pattern. However the residual plot of reciprocal transformation shows a worse pattern. The value of R 2 (adj) for logarithm transformation increases to 0.941 (Table 4.7) while the value of reciprocal transformation decrease to 0.710 (Table 4.8). Therefore the logarithm transformation was selected and a response surface equation based on this transformation is shown below in Equation 4.3.
4.3.2 Remove the Influential Outlying Cases
Frequently in regression analysis applications, the data set contains some cases that are outlying or extreme; that is, the observations for these cases are well separated from the remainder of the data. These outlying cases may involve large residuals and often have dramatic effects on the fitted least squares regression function. It is therefore important to study the outlying cases carefully and decide whether they should be retained or eliminated, and if retained, whether their influence should be reduced in the fitting process and/or the regression model should be revised.
66

Fri. Coe
Ret. Ang
0.1590
0.0299
0.01185
0.01185
13.420
2.521
0.000
0.017
Fri. Coe*Fri. Coe
Ret. Ang*Ret. Ang
-0.0587
0.0005
0.01019
0.01019
E(Mpa)*Fri. Coe
E(Mpa)*Ret. Ang
0.0000
0.0000
0.01379
0.01379
-5.755
0.053
0.002
-0.003
0.000
0.958
0.998
0.998
Fri. Coe*Ret. Ang
Fri. Coe*B.L./Th
Fri. Coe*Taper
Ret. Ang*B.L./Th
Ret. Ang*Taper
-0.0033
-0.0108
0.0030
-0.0011
-0.0008
0.01379
0.01379
0.01379
0.01379
0.01379
-0.240
-0.780
0.220
-0.077
-0.055
0.812
0.441
0.827
0.939
0.956
S = 0.07798 R-Sq= 96.4% R-Sq (adj)= 94.1%
Table 4.7: Regression analysis of logarithm transformation of cantilever hook
Log (Retention Force) = 1.014 + 0.163 E + 0.159 µ + 0.030 β 2
- 0.037 L/d
0
– 0.238 (d
0
/d
L
) – 0.036 E 2
+ 0.059 µ 2 + 0.035 (L/d
0
)^2 + 0.025 (d
0
/d
L
)^2
(4.3)
In this study, three statistic terms were used to identify the outlier cases. They are
Studentized Deleted Residual, Cook’s Distance and Difference between the fitted value and the predicted value (DFFITS) [65].
Studentized Deleted Residual
The Studentized deleted residual, denoted by t i
, can be computed by the following equation.
67

t i
= e i
⎡
⎢ n −
SSE ( 1 − p − h ii
)
1
− e i
2
⎤
⎥
1 / 2
(4.4)
The detailed derivation can be referred at [65]. In this equation e i
is the residual, SSE is the sum of squares error, h ii
is the hat matrix values, p is the number of parameters and n is the number of cases.
Fri. Coe
Ret. Ang
-0.06875 0.010733
-0.00859 0.010733
-6.405
-0.800
0.000
0.430
Fri. Coe*Fri. Coe 0.05055 0.009233
Ret. Ang*Ret. Ang -0.00627 0.009233
E(Mpa)*Fri. Coe
E(Mpa)*Ret. Ang
Fri. Coe*Ret. Ang
Fri. Coe*B.L./Th
Fri. Coe*Taper
Ret. Ang*B.L./Th
Ret. Ang*Taper
0.01304 0.012487
0.00311 0.012487
0.00348 0.012487
0.00189 0.012487
-0.01941 0.012487
-0.00021 0.012487
-0.00422 0.012487
2.390 0.023
5.475 0.000
-0.679 0.502
-1.180 0.247
1.044
0.249
0.304
0.805
-0.101 0.920
0.279
0.151
-1.554
-0.017
-0.338
0.782
0.881
0.130
0.987
0.738
S = 0.07064 R-Sq= 82.4% R-Sq(adj)= 71%
Table 4.8: Regression analysis of reciprocal transformation of cantilever hook
We identify as outlying Y observations those cases whose studentized deleted residuals are large in absolute value. In addition, we can conduct a formal test by means of the Boferroni test procedure of whether the case with the largest absolute studentized
68

deleted residual is an outlier. Since we do not know in advance which case will have the largest absolute value |t i
| , we consider the family of tests to include n tests, one for each case. If the regression model is appropriate, so that no case is outlying because of a change in the model, then each studentized deleted residual will follow the t distribution with n-p-1 degrees of freedom. The appropriate Bonferroni critical value therefore is t(1-
α /2n; n-p-1).
Figure 4.7: Residual plot vs. fitted values for logarithm transformation
DFFITS and Cook’s Distance
After identifying cases that are outlying with respect to their Y values, the next step is to ascertain whether or not these outlying cases are influential. We shall consider a case to be influential if its exclusion causes major changes in the fitted regression function. In this study the author applied two measures of influence that are widely used
69

in practice – DFFITS and Cook’s distance, each based on the omission of a single case to measure its influence.
Figure 4.8: Residual plot vs. fitted values for reciprocal transformation
It can be shown that DFFITS values can be computed by using the following equation [65].
( DFFITS ) i
= t i
(
1 h ii
− h ii
) ,t i
– Studentized deleted residual
h ii
– Element of hat matrix
(4.5)
Note from the last expression that the DFFITS value for the i th case is a studentized deleted residual, increased or decreased by a factor that is a function of the leverage value for this case. As a guideline for identifying influential cases, it was suggested to consider a case influential if the absolute value of DFFITS exceeds 1 for small to medium data sets
70

and 2 p / n for large data sets where p is the number of parameters and n is the number of cases.
In contrast to the DFFITS measure, which considers the influence of the ith case on the fitted value Y
) i
for this case, Cook’s distance measure considers the influence of the ith case on all n fitted values. Cook’s distance measure, denoted by D i
, is an aggregate influence measure, showing the effect of the ith case on all n fitted values:
D i
= j n
∑
= 1
( Y
ˆ j
− Y
ˆ j ( i )
) 2
, Y
) i
- The fitted value pMSE p - The number of parameters
MSE – Mean square error
(4.6)
Note that the numerator involves similar difference as in the DFFITS measure, but here each of the n fitted values Y
) j
is compared with the corresponding fitted value
Y
) j ( i )
when the ith case is deleted in fitting the regression model. These differences are then squared and summed, so that the aggregate influence of the ith case is measured without regard to the signs of the effects. To assess the magnitude of the influence, Neter refer to the corresponding F distribution [65], namely, F(p, n-p) where p is the number of parameters and n is the number of cases.
Runs
Stu. Del. Res.
(2.957)
COOK1
(0.2658)
DFIT1
(0.6794)
Table 4.9: The influential outlying cases
71

After compared to their respective standards (values in bracket) listed in Table
4.9, 4 influential outlying cases were identified. After investigation, it was found that these 4 cases are unusual design cases whose parameters were at the edge of the design space. For example, case 13 and 34 have extreme values for the ratio between beam length and beam thickness (1.555 and 13.445). Similarly case 23 and 30 have extreme friction coefficients (0.1244 and 1.4756) (Table 4.5). For CCD sometimes it’s impossible to avoid these kinds of outliners because the star points are usually far away from the design space.
It was decided that all 4 outliers should be deleted. As a result, the value of
R 2 (adj) increased to 0.991 from 0.941 (Table 4.10). The residual plot (Figure 4.9) also becomes more scatter and more concentrated which means the residual error was decreased and the residuals become more even. After all these steps, the response surface equation [4.7] about retention force of permanent cantilever hook was developed. log(Retention Force) = 1.01 + 0.163 E + 0.131 µ + 0.0299 β - 0.0130 L/d
0
- 0.229 (d
0
+ 0.0236 (d
/d
0
L
/d
) - 0.0375 E^2 - 0.0150 µ ^2 - 0.00089 β ^2
L
)^2 + 0.00003 E* µ - 0.00003 E* β
+ 0.00019 E*L - 0.00018 E*(d
0
/d
L
- 0.0108 µ *L + 0.00303 µ *(d
0
/d
L
) - 0.00331
) - 0.00106
- 0.00076 β *(d
0
/d
L
) - 0.00523 L*(d
0
/d
L
)
β
µ * β
*L
72
(4.7)

Fri. Coe
Ret. Ang
0.1309 0.004911
0.0299 0.004221
26.664
7.077
0.000
0.000
Fri. Coe*Fri. Coe
Ret. Ang*Ret. Ang
E(Mpa)*Fri. Coe
E(Mpa)*Ret. Ang
-0.0150 0.008793
-0.0009 0.003804
0.0000 0.004911
0.0000 0.004911
-9.858 0.000
-1.710 0.098
-0.234 0.817
0.006
-0.007
0.995
0.994
Fri. Coe*Ret. Ang
Fri. Coe*B.L./Th
Fri. Coe*Taper
Ret. Ang*B.L./Th
Ret. Ang*Taper
-0.0033 0.004911
-0.0108 0.004911
0.0030 0.004911
-0.0011 0.004911
-0.0008 0.004911
-0.674
-2.189
0.617
-0.215
-0.155
0.506
0.037
0.542
0.831
0.878
S = 0.02778 R-Sq =99.5% R-Sq(adj)= 99.1%
Table 4.10: Regression analysis for retention force after deleted influential outlying cases
Figure 4.9: The residual plot after deleted outlying cases
73

4.4 Verification Tests
Discussion of Main Effect
Main effect of a given variable as defined by Yates [66] is the average difference in the level of response as one moves from the low limit to the upper limit of that variable. Main effect plots can be used to quickly determine which variables have the greatest effect on the objective function. They show which variables can be changed without adversely affecting snap-fit performance. Varying a single variable while holding all other variables at their median values produces each curve.
The different settings corresponding to each parameter are coded values so that the real significant main effect can be found. Figure 4.10 illustrates that retention force is predominately driven first by Taper, then by Young’s Modulus. Retention is fairly insensitive to Friction Coefficient, Retention Angle and the ratio between Beam Length and Beam Thickness. The possible reason to explain why Friction Coefficient and
Retention Angle are not significant factors is that this study only focused on the cantilever hook with a small range of high retention angles (87 ° -90 ° ). For cantilever hooks with high retention angles, the retention force is mainly used to overcome the bending of cantilever hook. The friction only plays a minor role during retention. Also within such a small range of retention angles, the retention force will not vary much.
Another possible reason why friction coefficient is insignificant is the contact point between mating part and catch. For insertion and retention with low retention angle, mating part slides on the catch. In this case, friction is very important. However for retention with high retention angle, mating part seems roll on the catch that explained why friction is insignificant for retention covered in this research.
74

One important conclusion is that if a designer is unable to change the material, then taper is the only variable which will have a significant effect on retention with fixed beam thickness and beam width.
Verification and Lack-of-fit Test
In order to verify this response surface equation, two methods were used to check its accuracy. First, four more design cases (Table 4.11), the first two in the design space and the last two out of it, were simulated in FEA package and their results were compared to the results calculated from the response surface equation. The last two design cases are just out of the design space.
Fri. Coe.
Ret. Angle
1.1
0.8
0.5
1.7
1.4
-2
.37
84
1
-1
.00
00
0
0
.00
00
0
1
.00
00
0
2
.37
84
1
-1 0 1
-2
.37
84
1
-1
.00
00
0
0
.00
00
0
1
.00
00
0
2
.37
84
1
-1 0
Figure 4.10: Main effects for retention force
1
-2
.37
84
1
-1
.00
00
0
0
.00
00
0
1
.00
00
0
2
.37
84
1
75

Young’s
Modulus
Friction
Coefficient
Beam Length /
Thickness
Ret.
Angle
Result
(N)
Equation
Result (N)
8.75 89.25
° 1.75 10.800 10.94
6.25 87.75
° 1.25 9.357 9.06
11.25 90.00
° 2.25 11.390 10.21
3.75 86.25
° 1.00 5.900 4.34
Table 4.11: Verification test between FEA result and equation results
For the first two verification tests, the comparison results in the design space were very accurate with a difference under 3%. Beyond the design space, the comparison is still good with a difference is about 10 %. However for the last two verification tests the difference becomes large around 25%. In generally the response surface equation works very well in the design space, and can even explore some upper outside design space with good accuracy.
The lack of fit F test can also be used to test whether the multiple regression response function is an appropriate response surface. Repeat observations in multiple regression are replicate observations on Y corresponding of levels of each of the X variables that are constant from trial to trial. Thus, with two predictor variables, repeat observations require that X
1
and X
2
each remain at given levels from trial to trial. Once the analysis of variance (ANOVA) table has been obtained (Table 4.12). Sum of Square
Error (SSE) is decomposed into pure error and lack of fit components. The pure error sum of squares (SSPE) is obtained by first calculating for each replicate group the sum of squared deviations of the Y observations around the group mean, where a replicate group has the same values for each of the X variables. The lack of fit error sum of squares
(SSLF) equals the difference SSE - SSPE. The F value of Lack-of-Fit test is denoted by the ratio between lack of fit mean squares error (MSLF) and pure error mean square
76

(MSPE). To assess the magnitude of lack-of-test, Neter refer to the corresponding F distribution [65], namely, F(1α , p-2, n-c) where p is the number of parameters and n is the number of cases.
For this case, MSLF equals 0.00097 and MSPE equals 0.00035 so that the F value of Lack-of-Fit test, 2.75 which is much less than F (0.9, 41, 1) = 62.54868, which also means that this response surface model is very accurate.
Analysis of Variance
Source DF SS MS
Regression 19 4.13642
0.21771
282.11
Residual Error
Lack of Fit
Pure Error
Total
28
19
9
47
0.02161
0.01843
0.00318
4.15802
0.00077
0.00097
0.00035
2.75 0.061
Table 4.12: The analysis of variance for Lack-of-Fit test
77

CHAPTER 5
DESIGN OPTIMIZATION AND CONSTRAINT MANAGEMENT
Design optimization and constraint management are explained in this chapter. The detailed algorithms of constraint management, design decomposition, forward dependency and backward dependency, were developed by Prof. Gary Kinzel and his students [57-62]. A cantilever hook example is used through the chapter to illustrate how design optimization and constraint management were integrated into this design tool. In the last sections of this chapter response surface equations of the cantilever hook, post & dome, and bayonet & finger snap-fit features are described.
5.1 Design Optimization
Many design problems can be characterized by m nonlinear equalities and p inequalities in n unknown design variables ( x
1
, x
2
, L , x n
) as follows: h x x x n
=
(
( g x x x n
)
)
≤
0,
0, i i = 1 , 2 , L , m (5.1)
= 1 , 2 , L , p (5.2)
In addition, there might be limit constraints on the design variables in the following form: x i l ≤ x i
≤ x i u ,
78 i = 1 , 2 , L , n (5.3)

If the inequality constraints are converted to equality constraints using p non-negative slack variables, the total number of design variables increase from n to n+p . Based on this a more general formulation is:
( x
N
)
= 0, i = 1, 2, , M (5.4) i = 1, 2, , N (5.5) x i l ≤ x i
≤ x i u x i
≥ 0, = + 1, n + L N ( Slack Variables ) (5.6)
Where N = n + p and M = m + p .
Depending on the values of M and N , we can classify a design system into one of the following three categories:
•
•
•
If
If
If
M
M
M
<
=
>
N
N
N
: Under-determined system
: Fully-determined system
: Over-determined system
Most representations of mechanical design problems are under-determined, and they exhibit ( N-M ) degrees of freedom. These degrees of freedom translate into ( N-M ) inputs variables before the design is complete. The input variables can be the design specifications e.g. load requirement, safety factor, etc. or decision variables selected by optimization algorithm, e.g. insertion force, retention force. In this thesis, we treat snapfit design as an under-determined system since the number of equations (Equations 2.7 –
5.11) (M=5) is less than the number of variables (Table 4.1) (N=13).
In many design systems these undetermined design variables are set by convenience, design heuristics or manufactory concern. The strength of materials design equations for the cantilever hook (Figure 4.1) can be expressed as:
79

Where
P =
YEBD 3
4 L 3
2
P =
6 L
τ
F
= ins arctan
⎛
⎜⎜
= P
1
µ
−
⎝
+
µ
1
µ
⎞
⎟⎟
180
π
α
α
F ret
= P
1
µ
−
+
µ β
These equations can be written in a general, homogeneous form as: f
1 f
2 f
3 f
4 f
5
=
=
=
=
= x
1 x
1 x
13 x
11 x
12
−
− x
2 x
4
3 x
4 x 3
6 x
5
3 x
4 x
3 x
7
6 x
6 x
5
2
=
=
0
0
− arctan
⎛
⎜⎜
−
− x
1 1 x
9
− x
1 1 x
9
−
1 ⎞
⎟⎟
180
π
= 0 x
9
+ x
9 tan( x
10 tan( x
8
)
) x
9
+ tan( x tan( x
8
10
)
)
=
=
0
0 x
4 x
5 x
6 x
7 x
1 x
2 x
3 x
8 x
9 x
10 x
11
Y Offset
E Modulus of Elasticity
ε Insertion Strain
β
µ
Retention
Friction Coefficient
α
F
Ins
Insertion Angle
Insertion Force x
12
F
Re t
Retention Force x
13
τ Critical
Table 5.1: Parameters for strength of materials design equations
80
(5.12)
(5.13)
(5.14)
(5.15)
(5.16)
(5.7)
(5.8)
(5.9)
(5.10)
(5.11)

5.1.1 Single Objective Optimization
Single objective optimization was used to optimize a single objective function at a time. In this thesis a single-objective optimization problem was divided into two steps.
First step is to use Golden Section Method to find the search direction, and then the next step is use Broydon-Fletcher-Goldfarb-Shanno (BFGS) method to find the optimum value. The Golden Section Method for estimating the maximum, minimum, or zero of a one-variable function is the most common interval reducing methods. It reduces the interval by the same fraction with each iteration. The intervals are derived from the
Golden Section Ratio (1.61803). This ratio has significance in aesthetics as well as mathematics. The method is simple to implement. It is indifferent to the shape and continuity properties of the function being minimized. Most important, the number of iterations to achieve a prescribed tolerance can be established before the iterations start.
BFGS method is the most popular Quasi-Newton Methods and has proved to be most effective in applications.
Step 1: Estimate an initial design x (0) . Choose a symmetric positive definite matrix H (0) as an estimate for the Hessian of the cost function. In the absence of more information, let H (0) = I . Choose a convergence parameter ε . Set k = 0 , and compute the gradient vector as c ( 0 ) = ∇ f ( x ( 0 ) )
(5.17)
Step 2: Calculate the norm of the gradient vector as c ( k ) . If c ( k ) < ε then stop the iterative process; otherwise continue.
Step 3: Solve the following linear system of equations to obtain the search direction:
81

H ( k ) d ( k ) = − c ( k )
Step 4: Compute optimum step size α k
= α to minimize f ( x ( k ) + α d ( k ) ) .
Step 5: Update the design as x ( k + 1 ) = x ( k ) + α
( k ) d ( k ) .
Step 6: Update the Hessian approximation for the cost function as
H ( k + 1 ) = H ( k ) + D ( k ) + E ( k ) where the correction matrices D ( k ) and E ( k ) are given as
D ( k ) =
( y ( k ) y ( k ) y ( k ) T
⋅ s ( k ) )
; E ( k ) =
( c c ( k ) c ( k ) T
( k ) ⋅ d ( k ) )
; with s ( k ) = α k d ( k ) (change in design) y ( k ) = c ( k + 1 ) − c ( k ) (change in gradient) c ( k + 1 ) = ∇ f ( x ( k + 1 ) )
Step 7: Set k = k+1 and go to Step 2.
(5.18)
(5.19)
(5.20)
5.1.2 Multiple Objectives Optimization
Most realistic optimization problems, particularly those in design, require the simultaneous optimization of more than one objective function. For example when a designer creates a cantilever hook design, they usually want to maximize the retention force, while minimizing the insertion force and the volume of the feature. It is unlikely that the different objectives would be optimized by the same alternative parameter choices. Hence, some trade-off between the criteria is needed to ensure a satisfactory design.
In this study a Weighted Sum strategy was applied. All the objective functions are evaluated qualitatively according to their functional importance and are assigned weights,
82

W = [ w
1
, w
2
, L , w m
] T , indicating the designer’s subjective preference. Weighted sum strategy constructs a sum of all weight-factored objectives as a preference function. The expression of the preference function is:
P ( F ( X )) = i m
∑
= 1
[ w i f i
( X )] = W T F ( X )
(5.21)
Usually,
0 ≤ w i i m
∑
= 1 w i
≤ 1
= 1
The substitute problem under weighted sum strategy is: min P ( F ( X )) = W T F ( X )
Subject to X ∈ U
Where
(5.22)
5.2 Constraint Management
A valid design case is defined by a valid set of input variables such that the remaining unknowns can be determined. Typically, each equation does not contain all of the design variables, so that only certain combinations of variables can be input. In a fully determined system, when M is equal to N , there is only one possible design case. But for the under-determined systems, the number of possible cases, A t
, can be computed as the total number of combinations of n variables taken (n-m) at a time, given as follows:
A t
= n C m
=n! / ((n-m)!m!) (5.23)
The number of valid design cases is usually less than the number of possible design cases A t
because certain combinations of variables will lead to direct redundancy
83

problems. The following simple example is used to illustrate the concept of direct redundancy.
X
3
= X
1
+X
2
X
4
= X
3
2
(5.24)
(5.25)
Since there are 4 variables in 2 equations, the number of possible design cases is
6. But the case that selects X
3
and X
4
as input variables leads to direct redundancy. In a simple constraint design problem, as the one above, direct redundancies are easy to identify. But, when the size of constraint network increases, determining of direct redundancies become a complex problem. Interactive design programs built over the constraints of a mechanical system should permit the user to input any valid combination of design variables. Constraint management can determine which variables are dependent upon the known variables and the most efficient solution scheme to solve for these dependent variables. Assuming the several variables have been already been input and other variables have been solved, an interactive constraint manager has to deal with the following situations:
• New Specification Problem : Determine if any new variables can be computed based upon the updated information.
• Respecification Problem : Determine how a change in the value of an input variable affects the dependent computed variables.
• Unspecification Problem : Determine which variables can no longer be computed because one of the previous specifications has been removed.
• Reverse Specification Problem : When a computed variable is changed to a specification, determine which of the original specifications should be released.
84

5.2.1 Constraint Representation
This section described several important terms used in constraint management algorithms of the web-based design tool: occurrence matrix, Block information, Level information and Order information.
Occurrence Matrix
The structure of a set of algebraic equations can be described in a precise manner by a Boolean matrix is called an occurrence matrix. Several different names for occurrence matrix are structural matrix, incidence matrix, dependency matrix.
The elements of an occurrence matrix A , which is a ( m × n ) matrix, are defined as follows:
A ij
= 1 ⇒ Variable x j
appears in equation f i
.
A ij
= 0 ⇒ Variable x j
doest not appears in equation f i
.
One way to identify the occurrence matrix is to examine the equations symbolically and determine if a variable is present in an equation. Alternatively, a design variable x j
is present in f i
, if for distinct values of x j
, the constraint function value is also distinct. Mathematically,
A
0
=
⎧
⎨
⎩
1 if f i
( x
1
, x
2
, L , x j − 1
, x ' j
, x j + 1
, L ,
0 x n
) ≠ f i
( x
1
, otherwise x
2
, L , x j − 1
, x j
, x j + 1
, L , x n
), x ' j
= x j
The above scheme can fail by sheer coincidence when f i
( x ' j
) = f i
( x j
) . Note that x ' j
and x j
can be the upper and lower limits for the variable. This thesis uses the symbolic approach to determine the occurrence matrix. For example, the occurrence matrix of a cantilever hook’s design equations (5.12 – 5.16) is listed in Table 5.2.
85

Variable Number, j
1 2 3 4 5 6 7 8 9 10 11 12 13
1 1 1 1 1 1 1 0 0 0 0 0 0 0
2 1 0 1 1 1 1 1 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 1 0 0 0 1
4 1 0 0 0 0 0 0 0 1 1 1 0 0
5 1 0 0 0 0 0 0 1 1 0 0 1 0
Block Information
Table 5.2: The occurrence matrix of cantilever hook
In multi-component mechanical design, it is often possible to have disjointed blocks of equations and variables. That is, one group of equations and variables may be completely independent of the others. Computationally, it is efficient to consider these independent groups separately. The block information can be determined from the occurrence matrix prior to any actual design calculations. This can be done in a variety of ways, and one possible procedure is given in Figure 5.1. The algorithm starts with the variable x
1
and finds all of the equations involving it. The other variables in these equations are then identified, and all of the equations involving these variables are identified. The new set of equations is then checked for new variables, and any additional equations that contain these new variables are identified. The procedure continues until no more new variables or equations are found. These equations and variables form the first block. The algorithm then checks if all of the variables in the original formulation have been identified. If not, the variables and equations for the second block are identified. The procedure continues until all variables and equations have been included in one of the block.
86

Using the disjoint block determination algorithm, it is apparent that all of cantilever hook’s equations and variables are belong to a single block, i.e. there are no disjoint blocks. Since there are 13 design variables and 5 equations, the total number of possible input variable combinations is 1287. As pointed out earlier, all the combinations will not be valid, due to direct redundancies.
Start with the occurrence matrix
Block = 1 k = 1
Identify L equations containing variable x k
Mark L equations
Mark x k
Add the columns of L rows of occurrence matrix to form COLSUM vector
Stack any new variables having a nonzero COLSUM
BLOCK = BLOCK + 1
Is stack empty?
N k = top(Stack)
Set k to first unmarked variables
N
Y
All variables marked?
Y
Return
Figure 5.1: A procedure for identifying disjoint blocks of equations
87

Level Information
In a typical design problem, the variables can be specified as inputs in any order.
After a sufficient number of variables are inputted, some of the outputs can be computed.
However, all of the variables that are computed at any one time may not be dependent on all of the variables that have been input. The level information indicates at what stage of the input process input variables are used and output variables are computed. The current level number remains zero until some equations can be solved.
As soon as it is detected that some variables can be solved, these output variables are given level 1. The input variables in the equations solved are also assigned level 1.
The second level includes the output variables and the new input variables (i.e. input variables with level = 0) that are required to compute the second group of outputs.
Subsequent levels are defined in a similar manner. It should be noted that for the input variables, the level number indicates the first time they were used. Even if the input variables are respecified at a later stage, the level number remains unchanged for them.
Order Information
The order information gives the order in which the equations are solved and the variables determined. If several equations are solved simultaneously, the equations and variables involved are of the same order designation. The order index is specific to a particular level, i.e. at a particular level there might be more than one order. The order information indicates the way in which equations are solved at a particular level. Since the input variables are not part of the equations solution process, their order is always set to zero.
88

5.2.2 Constraint Management Algorithms
Since this web-based design tool allows the user to input any valid combination of design variables, there is the need to determine which variables are dependent on variables already input or computed. Once the dependent variables have been identified, it is necessary to determine the most efficient sequence for computing these variables, and once the dependent variables are computed, changes in both the input and output variables must be accommodated. This indicates the need for algorithms that will determine the following:
• Design Decomposition – Determine the most efficient solution sequence for the dependent variables.
• Forward Dependency – Determine all output variables dependent on a given variable.
• Backward Dependency – Determine all input variables affecting a given output variable.
Design Decomposition Algorithm
When the value for a particular variable x k
becomes known (i.e., it is either input or computed), there is a need to identify any dependent variables that can be solved using the constraint equations. This requires an algorithm that will systematically cycle through the remaining equations to determine if any equations can be solved individually or in a set of simultaneous equations. Such an algorithm is shown in the Figure 5.2.
For any known variables, x k
, the kth column of the occurrence matrix is zeroed, i.e.,
89

A ik
= 0 , i = 1 , 2 , L , m (5.26)
This process is referred to as updating the occurrence matrix. The input process is considered complete when all variables have been either specified (input) or computed
(output), i.e., the occurrence matrix is identically zero. Figure 5.2 shows an algorithm to find the variables that have to be solved individually or using simultaneous equations.
First, the entries in each row of the occurrence matrix are summed in a vector
ROWSUM, i.e.,
(5.27)
ROWSUM i
= j n
= 1
A ij
If the ROWSUM vector is identically equal to 0, then there are no more variables remaining, and the design is complete. Then, the minimum of the ROWSUM vector is found from all the nonzero entries and assigned to k . The variable k identifies the minimum number of equations that have to be checked for simultaneous solution. If k happens to be greater than p , then nothing can be solved. Here, p is the maximum number of equations that will ever be solved at any one time. This number can be determined prior to any analyses from the original occurrence matrix by an exhaustive search that considers every possible combination and order for inputting the variable values. If k is equal to 1, there is one equation that can be solved independently. This equation is solved, the occurrence matrix is updated, and the procedure is repeated.
90

Start with the occurrence matrix
Add entries in rows to get ROWSUM vector k = min (non-zero ROWSUMs)
Is k ≤ p
?
N
Return
Is k =
Y
1
?
Y Identify equations and variables
N
Generate Active occurrence matrix
Is
Y k ≤ p
?
N
Identify number ( L ) of rows with
ROWSUM
≤ k k = k +1
Y
Is
L ≤ k
?
N
Sort the candidates L in increasing
ROWSUM
Return
Return
Return
N
Y
Get a combination of k equations
Have c L k
combination been tested?
N from L equations
Add columns of k equations and put in COLSUM
Number of nonzero entries in
COLSUM = k ?
Y
Return
Identify equations and variables
Figure 5.2: Design decomposition algorithm
91

In general, k will be greater than 1, implying that a procedure to identify simultaneous equations will be executed. To improve the efficiency of the algorithm, an active occurrence matrix ( A* ) is determined. In a typical snap-fit feature design, the occurrence matrix for the constraint representation is quite sparse. This implies that, except in rare cases, the entire occurrence matrix will not be required at each design stage, and only the submatrix corresponding to the active equations need be considered.
Here, an active equation is defined as an unsolved equation containing at least one known variable. Also, an active variable is defined as any unknown variable appearing in the active equations. As the design proceeds, the dimensions of A* will change. In the beginning, A* , will be small, and it will grow until some variables can be solved.
Typically, near the end of the process after some of the equations are solved, A* will reduce in size.
A further improvement in efficiency can be achieved by determining disjoint blocks within the active occurrence matrix ( A* ). This procedure will typically reduce the combinatorial search needed to extract solvable sets.
The same cantilever hook example was used to illustrate the design decomposition algorithm. Assume that the first input is x
3
(Young’s Modulus) has been specified. This leads to the zeroing of the 3rd column of the occurrence matrix. The
Active Occurrence Matrix ( A * ) contains just the equations f
3
with ROWSUM = 2, which implies that no equation can be solved. Next x
6
(Beam Length), x
2
(Offset), x
1
(Normal
Force) and x
4
(Base Width) are specified one by one. The updated occurrence matrix along with the ROWSUM vector now looks like the Table 5.3.
92

Variable Number, j
1 2 3 4 5 6 7 8 9 10 11 12 13 ROWSUM
1 0 0 0 0 1 0 0 0 0 0 0 0 0
2 0 0 0 0 1 0 1 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 1 0 0 0 1
1
2
2
4 0 0 0 0 0 0 0 0 1 1 1 0 0
5 0 0 0 0 0 0 0 1 1 0 0 1 0
3
3
Table 5.3: The occurrence matrix after specified x
1
, x
2
, x
3
, x
4
, and x
6
The ROWSUM value equal to 1 identifies one equation in one variable. Hence the output variable x
5
is found using equation f
1
. This leads to zeroing out of column 5. As a result, the ROWSUM value of equation f
2
changes from 2 to 1. Then the output variable x
7
is found using equation f
2
. The following web page (Figure 5.3) shows the final updated input and output variables.
The above solution strategy corresponds to the input variables x
1
, x
2
, x
3
, x
4
, x
6
, x
8
, x
9 and x
10
. The order in which the input variables are specified determines the level and order information for the variables and equations. The order that was used for the illustration of the design decomposition algorithm was [ x
3
, x
6
, x
2
, x
1
, x
4
, x
9
, x
8
, x
10
]. Let the block, input/output designation, level and order be designated as BLOCK, IO,
LEVEL, and ORDER, respectively. Then the BLOCK, IO, LEVEL, and ORDER values for the variables and equations are given in Table 5.4.
93

Figure 5.3: The new specification problem of cantilever hook
In a symbolic form, the final solution strategy can be represented as follows: [ x
3
, x
6
, x
2
, x
1
, x
4 ,
{ f
1
, x
5
}, { f
2
, x
7
}], [ x
9
, { f
3
, x
13
}], [ x
8
, { f
4
, x
12
}], [ x
10
, { f
5
, x
11
}].
Forward Dependency Algorithm
Forward dependency refers to the determination of the dependent output variables for a given input variable. This problem occurs when at any stage of the interactive design process, the user wishers to respecify (change) or unspecify (delete) the value of an input variable.
Respecification Problem
If the value of an input variable is changed, then all of the output variables that depend on this input variable must be recomputed. The dependent output variables can be identified using the algorithm presented in Figure 5.5. To determine the output variables that have to be recomputed, the block and level of the changed input variable is first
94

determined, and all of the equations that are in the same block and at the same level or higher are identified. A sub-occurrence matrix is then formed from these equations, and using only the active inputs (i.e. excluding the changed input), a modified version of the design decomposition algorithm is used to determine the output variables that are not dependent on the changed input variable. These variables need not be recomputed and they are retained at their present value. Once these variables have been identified, the remaining output variables from the initial candidate set are recomputed. The equations that were used to compute the output variables can be found and resolved in the correct order for each level.
VARIABLE OR
EQUATION
BLOCK IO LEVEL ORDER x
1 x
2 x
3 x
4 x
5 x
6 x
7 x
8 x
9 x
10 x
11 x
12 x
13 f
1 f
2 f
3 f f
4
5
1 1 1 0
1 1 1 0
1 1 1 0
1 1 1 0
1 0 1 1
1 1 1 0
1 0 1 2
1 1 3 0
1 1 2 0
1 1 4 0
1 0 4 5
1 0 3 4
1 0 2 3
1 - 1 1
1 - 1 2
1 - 2 3
1 - 3 4
1 - 4 5
Table 5.4: Block, input/output, level and order of cantilever hook example
If the value of x
3
(Young’s Modulus) was changed from 30,000 to 40,000 (Figure
5.4), then only x
5
(Base Height) and x
7
(Insertion Strain) must be recomputed since the other output variables x
11
(Insertion Force), x
12
(Retention Force), and x
13
(Critical Angle)
95

do not depend on x
3
. The solution sequence to solve x
5
and x
7
is determined using the design decomposition algorithm. At this stage, the occurrence matrix will have the x
1
, x
2
, x
3
, x
4
, x
6
, x
8
, x
9
, x
10
, x
11
, x
12
, and x
13
zero out (Table 5.5).
Variable Number, j
1 2 3 4 5 6 7 8 9 10 11 12 13 ROWSUM
1 0 0 0 0 1 0 0 0 0 0 0 0 0
2 0 0 0 0 1 0 1 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 0 0 0 0 0
4 0 0 0 0 0 0 0 0 0 0 0 0 0
5 0 0 0 0 0 0 0 0 0 0 0 0 0
Table 5.5: The occurrence matrix of respecification problem
1
2
0
0
0
Figure 5.4: The respecification problem of cantilever hook
96

Start with sub-occurrence matrix, number of equations q , active input variables p . ps = min( p , q )
Return k = 1
Update the subincidence matrix Return
N
Is k < ps ?
Y
Add entries in rows to get ROWSUM vector
Identify number ( L ) of rows with ROWSUM
≤ k
Y
Any unsolvable variables left?
N k = k +1
Y
Is L < k ?
N
Get a combination of k equations from L equations
Identify variables which can be solved for
Y Have
C L k combination been tested?
N
Add columns of k equations and put in COLSUM
Number of nonzero entries in COLSUM
= k ?
Y
Figure 5.5: Forward dependency algorithm
Unspecification Problem
After a number of input variables are specified and some of the output variables are computed, it may be desirable to unspecify or delete the value for a given input variable (unforced unspecification). Another case of input variable unspecification occurs
97

when an output variable is changed to an input status by fixing it to a specific value
(forced unspecification). Whenever an unspecification occurs, the values of all of the output variables that depend on the unspecified input variable are no longer known. The algorithm to determine all such output variables is identical to that of the respecification case. The status of the identified output variables is then changed an unknown (free) state.
As an example of unspecification, assume that x
6
(beam length) is to be unspecified. Since x
6
has Level 1, all outputs that depended on x
6
need to solve again
(Figure 5.6). A sub-occurrence matrix (Table 5.6) is formed from the corresponding tows of the original occurrence matrix with only x
1
, x
2
, x
3
, and x
4
as inputs.
Variable Number, j
1 2 3 4 5 6 7 8 9 10 11 12 13 ROWSUM
1 0 0 0 0 1 1 0 0 0 0 0 0 0
2 0 0 0 0 1 1 1 0 0 0 0 0 0
3 0 0 0 0 0 0 0 0 1 0 0 0 1
4 0 0 0 0 0 0 0 0 1 1 1 0 0
5 0 0 0 0 0 0 0 1 1 0 0 1 0
Table 5.6: The occurrence matrix of unspecification problem
Backward Dependency Algorithm
2
3
2
3
3
For a complicated problem, the designer may know typical values for the output variables, and he/she may find the computed values to be unacceptable. One option in such cases is to specify the value for one of the variable that was calculated (Reverse
Specification Problem). When this is done, the problem becomes over-determined, and it
98

is necessary to unspecify or release one of the input variables. It is therefore necessary to determine those input variables which influence the given output variable (back dependency). Usually, this will be some subgroup of the original group of input variables.
Figure 5.6: The unspecification problem of cantilever hook
To determine the backward dependency, the level and order of the output variable is determined. Only those equations at the same order or lower and those input variables at the same level or lower need to be considered. If the variables and equations that have not been used yet are designated to be at the zero level, any inactive equations will automatically be excluded from consideration. Also, even though the level designation is not unique, this procedure will always include enough equations to properly determine the dependencies. Alternatively, all of the equations that have been solved at this point
99

could be considered, but this will usually involve more calculations than if the subgroup based on the level designation is used.
A procedure for finding the backward dependency is shown in Figure 5.7. First the equations that are at the same level and order as the designed output variable are found, and the input variables in these equations are identified. If these are all of the inputs at the same level or lower, the procedure terminates. To identify all of the input variables that potentially influence the designated output variable, indirect dependencies also must be checked. To determine indirect dependencies, the other output variables in the group of equations being investigated are identified. The level and order of these variables are determined, and the input variables in the equations of the same level and order are identified. If all of the potential input variables are still not accounted for, the output variables in the new group of equations are found, and their level and order identified. The equations at this order are then checked for input variables. The procedure continues until no more input variables that are directly dependent on the designated output variable are found.
As an example of the reverse specification, assume that x
5
(base height) which was one of the output variables is designated as an input variable. This variable is in
Block 1 and has a lever and order of 1 and 1, respectively. Equation 1 has the same level and order, and the input variables appearing in equation 1 are x
1
, x
2
, x
3
, x
4
, and x
6
. Since there are no other outputs in f
2 , no new input variables were considered as candidates.
Therefore x
1
, x
2
, x
3
, x
4
, and x
6 can be unspecified when x
5
is made an input. In this example x
6
(beam length) was unspecified, then the forward dependency check discussed previously is make, and all of the output variables that are dependent on the unspecified
100

input x
6
are leased. One of these released output variables will be x
5
. A value was designated for the chosen “output” variable x
5
. x
6
(beam length) changed from an input variable to an output variable (Figure 5.8).
Start with occurrence matrix and output variable x j k = 1
Find BLOCK, LEVEL, and ORDER for x k
Find an equation for BLOCK,
LEVEL, and ORDER
Mark input variables in n equations
LEVEL = 1 and
ORDER = 1?
Y
Return
N
Identify m new output variables in n equations and rank all encountered but uninvestigated output variables in order of increasing indices ( x
1
, x i+1
, … )
All encountered output variables investigated?
Y
N
Set k equal to the lowest index of variables which have yet to be investigated
Figure 5.7: Backward dependency algorithm
101

Figure 5.8: The reverse specification problem of cantilever hook
5.3 Mathematic Formulations for Snap-fits Features
In this section, the response surface equations of three most common snap-fit features – cantilever hook, post & dome and bayonet & finger, are explained. These mathematic formulations were included in this web-based design tool and all of them were generated by using the combination of FEA, DOE and RSM.
Cantilever Hook Feature
In this web-based design tool, the traditional strength of material equations [5.7 –
5.11] were used to predict the performance of cantilever hooks with low retention angles.
102

However for retention force of high-retention cantilever hooks, the response surface equation [4.7] developed was implemented.
Post & Dome Feature
The post & dome feature, shown in Figure 5.9, is a high performance snap-fit that is self-datuming and can take some shear loading in addition to retention. Its response surface equations were developed by Nichols [9] and shown in Equations 5.28 to 5.33.
F ret
( ) = 6 [ a
0
+ a h d b ) + ( d m ) + ( ) + a
4
( bm ) + a
5
( b 2 ) ]
(5.28)
F ins
( ) = 6 [ a
0
+ a
1
+ a
2
+ ( d
) + ( d
) + ( ) + a
6
( bm ) + + ]
(5.29) e ins
= a
0
+ a
8
( m E ) + a
9
( b 2 )
+ ( d
) + ( d b ) + ( d
) + + + ( ) +
(5.30)
LR =
F ret
F ins
(5.31) d =
OD post
4 y = 0.7
t
(5.32)
(5.33)
Table 5.7 and Table 5.8 show the response surface factors and design variables of post & dome’s response surface equations.
Factors F ret
F ins
ε ins a
0
-95.930329
a
1
-0.513798
a
2
122.227241
a
3
0.115491
a
4 a
5
-21.175542
0.081995
-83.405492
1.128164
281.607631
-6.571269
-0.001944
0.022354
a
6
-4.212110
-0.213418
0.04507
-0.015966
6.94e-4
3.684e-6
0.00327
0.211387
Table 5.7: The response surface factors for the post & dome feature
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Figure 5.9: A typical post & dome feature
Variable Symbol Variable Name Range h d
Height of Dome Segments 6-12 t
β
Dome Segment Thickness
Angle of Dome Segments
0.75-2
50-75
E
µ
OD post d
Modulus of Elasticity
Friction Coefficient
Post Diameter
Dwell Surface Height
1150-3000
0-0.5
3 y Molded-in Post Undercut
Retention Force
OD post
/4
0.7t
F ret
F ins
ε ins
LR
Insertion Force
Insertion Strain
Locking Ratio
Table 5.8: Design variables for the post & dome feature
104 mm
N
N
Unit mm mm
°
N/mm 2 mm mm

Bayonet & Finger Feature
A bayonet & finger snap-fit feature contains a bayonet, a retention finger and a support finger (Figure 4.2). Its response surface equations were developed by Shen [7] and shown in Equation (5.34) to (5.45). w = t
2 d = t
1 .
0928 t f
= 0 .
5585 t
(5.34)
(5.35)
(5.36) t b
= 2 .
0 t p
= t
2 l = h + t b
+ 2
(5.37)
(5.38)
(5.39) q = q 107.5
12.5
h = h -
2
8 t = t 1.5
0.3
LR = − 232 .
7 − 882 .
2 w t
+ 1640 .
9 t w f
− 1690 .
9 ( t − d
) 2 d
+ 313 .
7 t − d d
+ 419 .
9 ( w
) 2 t
− 916 .
5 ( t w f
) 2
(5.40)
(5.41)
(5.42)
(5.43)
F ins
= a
0
+ a
1
θ + +
3
+ +
5
µ + a
6
θ
+ a
12
θ t + a
13
θ +
14
θµ +
2 + a h 2 + a t 2 + a E 2 + a
10 a ht + + µ + a t µ + a E µ
µ 2 + a
11
θ h
(5.44)
105

F ret
= + b
1
θ + +
3
+ +
5
µ + b
6
θ
+ b
12
θ t + b
13
θ +
14
θµ +
2 + b h 2 + b t 2 + b E 2 + b
10 b ht + + µ + b t µ + b E µ
µ 2 + b
11
θ h
(5.45)
Table 5.9 shows the scaled variables of bayonet & finger used in Equation 5.34-
5.45. Table 5.10 and Table 5.11 show the design variables and response surface factors of bayonet & finger’s response surface equations. t q h
E m
Dimensionless q
Dimensionless h
Dimensionless t
Dimensionless E
Dimensionless m
Table 5.9: The scaled variables for bayonet & finger feature t b l
LR
F ins
F ret
Variable Symbol Variable Name
θ Retention Finger Angle h Vertical Finger Height t
µ
Base Thickness of Bayonet
Modulus of Elasticity
Friction Coefficient w d t f
Offset Size
Gap between Fingers
Thickness of Fingers
Final Bayonet Thickness
Length of Bayonet
Locking Ratio
Insertion Force
Insertion Strain
Range
95-120
6-10
Unit
° mm
1.2-1.8 mm
1750-2625 N/mm 2
0.2-0.5
0.6-0.9
1.2-1.8
0.6-0.9 mm mm mm
0.6-0.9
10-14
Eq.6
Eq.7
Eq.8 mm mm
N
N
Table 5.10: The design variables for bayonet & finger feature
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Factors Value Factors Value a
0
0.160130
0
4.059906 a
1 b
1
-3.520562 a
2
-0.162599 b
2
-0.586661 a
3 b
3
1.398942 a
4
0.018534
4
1.044682 a
5 b
5 a
6
-0.052948 b
6
-0.107505
0.855799 a
7
0.083788
7
-0.492045 a
8 b
8
-0.640947 a
9 a
10
-0.020291 b
9
-0.015100 b
10
0.367921
1.088091 a
11
0.024895
11
0.086938 a
12 b
12 a
13 b
13
-1.011719
-0.730382 a
14
-0.022910 b
14 a
15
-0.105421 b
15
0.050770
-0.025048 a
16
-0.009982 b
16
-0.091164 a
17 b
17
0.081425 a
18
0.007982
18
0.182853 a
19
0.003338 b
19
0.185332
Table 5.11: The response surface factors for bayonet & finger feature
107

CHAPTER 6
THE WEB-BASED APPLICATION FOR SNAP-FITS
In this chapter, several approaches for web-based application such as Java,
VBScript, JavaScript, Visual Basic Dynamic hypertext markup language (VB DHTML) and Visual Basic Internet Information Server (VB IIS) application are explained and their advantages and disadvantages are compared. Next, the web-based application for this thesis, VB IIS application, will be presented in detail. A cantilever hook example will be used to explain different interfaces of the web-based design tool.
6.1.1 Java
Java is an object oriented programming (OOP) language developed by Sun
®
Microsystems in 1991, a company best known for its high-end Unix workstations. Java is used to create executable content that can be distributed through networks. Java profoundly changes the Web because it brings a richness of interactivity and information delivery not possible using previous Web software systems. Java makes it possible for programmers to create software that can be distributed across networks and run on many different kinds of computers. The resulting executable content shifts the site of activity
108

from the Web server to the Web client (the Java-enabled browser). In a word, Java has the following advantages:
• Java is Platform-Independent.
Platform-independence is one of the most significant advantages that Java has over other programming languages, particularly for systems that need to work on many different platforms.
• Java is Object-Oriented.
Like most OOP languages, Java includes a set of class libraries that provide basic data type, system input and output capabilities and other utility functions. OOP technique enables programmers to take full advantage of object-oriented methodology and its capabilities for creating flexible, modular programs and reusing code.
• Java is can be quickly understand and it easy to program.
In addition to its portability and object-orientation, one of Java’s initial design goals was to be small and simple, and therefore easier to write, easier to compile, easier to debug, and, best of all, easy to learn. Although Java looks similar to C and C++, most of the more complex parts of those languages, such as pointers and pointer arithmetic, have been exclude from Java. This makes Java simpler without sacrificing much of its power.
Java programs fall into two main groups: applets and application. For now only applets has been used in web-based application. Applets are Java programs that are downloaded over the Internet and executed by a web browser on the user’s machine. The main disadvantage of applets is that it needs Java-enabled browsers.
109

6.1.2 JavaScript
Netscape ® initially developed JavaScript under the name of LiveScript. This scripting language was intended to extend the capabilities of basic hypertext markup language (HTML) and provide a partial alternative to using a large number of Common
Gateway Interface (CGI) scripts to process form information and add dynamics to a user’s page. Although JavaScript has advantages such as ability to embed into HTML and flexibility, its disadvantages are also obvious. For example, JavaScript is environment dependent, which means some browsers don’t support it. Another disadvantage is that
JavaScript is an interpreted language that takes longer for the code to execute.
6.1.3 VBScript
VBScript, sometimes known as Visual Basic Scripting Edition, is Microsoft’s answer to JavaScript. Just as JavaScript’s syntax is loosely based on Java, VBScript’s syntax is loosely based one Microsoft’s Visual Basic, a popular programming language for Windows machines. So VBScript is the scripting version of Visual Basic. This helps to explain the subset of VB features that are supported by the VBScript interpreter.
Like JavaScript, VBScript is also an interpreted language, which means it cannot be compiled into an executable program. The interpreter reads in the script file, checks some basic syntax requirements, and then executes the commands and functions in the script. It was first used in Internet Explorer so that web page authors could run code dynamically on the remote user’s PC. Microsoft took a subset of functions from Visual
Basic for Applications, which in itself is a subset of Visual Basic, that meet the
110

functionality required while maintaining performance and keeping the interpreter as small as possible.
The VBScript interpreter is available to any application via the ActiveX Scripting standard. Active Server Pages (ASP) is an example of applications that host the Microsoft
Scripting languages to enable their functions to be used. VBScript can do many of the same things as JavaScript, and it even looks similar in some cases. The main disadvantage of VBScript is that only Microsoft Internet Explorer supports it.
6.1.4 VB DHTML Application
A VB DHTML application is a VB application that runs on the client and uses the built-in dynamic HTML (DHTML) capabilities of Microsoft’s Internet Explorer (IE) browser to provide fast response to user input. These types of applications are well suited to games, simulations, and data-input application. The power of VB then combines with the user-interface manipulation of DHTML to allow for highly interactive web application. The designer can control the page’s appearance and behavior, as well as interact with business objects and data. Instead of the traditional VB forms interface, designers can build a DHTML interface and work in a familiar browser environment.
Unfortunately, they’re limited to Internet Explorer version 4 and higher at this time.
6.1.5 VB IIS Application
These applications run on the web server and under Microsoft’s Internet
Information Server (IIS). You can write Web applications for any kind of browser (or even non-browser) client using an IIS application. IIS has the following advantages:
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• IIS applications run with multiple browsers and multiple versions of those browsers. They’ll also run on multiple platforms, including Mac and Unix machines. This makes IIS applications admirably suited for business applications used by clients who may not all be running Windows or have the latest browser.
• IIS application has a single code base. Because the application resides in its entirely on the web server, the administrator can update the entire application with a single code change. The advantages of this are hard to beat when you realize how often business rules and database requirements change.
• There are no client-side installation issues. This alone makes IIS applications worth considering.
• The application is available from any location. After an IIS application is up and running, any client computer that can connect to the server can run your application.
• A client isn’t bound to one computer. The developer can use sign-on and password or other security measures to identify clients no matter which computer they’re running on.
• All the data is centralized. Clients won’t ever lose data when they use IIS application if all data is backuped on the server. Additionally, the administrator can track use of the application down to the page level. This kind of capability is priceless when it comes to justifying the cost of an application.
112

Table 6.1 lists the advantages and disadvantages of all above-mentioned languages. In this thesis, VB IIS application was choose as the computer language for the web-based design tool.
Advantage
• Platform-independent • Needs Java-enabled browsers
Java • Object-Oriented
• Easy to learn
• Ability to embed into
JavaScript
VBScript
•
•
•
HTML
Flexibility
Ability to embed into
HTML
Flexibility
• Environment dependent
• Takes longer for the code to execute
• Only Microsoft Internet Explore supports it
• Takes longer for the code to
VB DHTML
Application
•
•
Allow for highly interactive web application
Run with multiple browsers and multiple versions of execute
• Limited to Internet Explorer version 4 and higher
• Difficult to learn.
VB IIS Application those browsers.
• Run on multiple platforms
(Windows, Unix and Mac.)
• A single code base.
• No client-side installation issues.
• Data is centralized.
Table 6.1: The advantages and disadvantages of web-based languages
After compared the advantages and disadvantages of these web-based application approaches, the VB IIS application was choose as the develop tool for this research. In this section, the Active Server Page (ASP) objects which are important to web application will first be introduced. Then WebClasses, a special type of class you interact with IIS from a Visual Basic application, will be described.
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6.2.1 ASP Object Model
The ASP object model has changed very little since version 1. The version described here is ASP 3, but almost everything in this thesis will also work with version
2. There are six objects in the ASP type library.
Server Object
There’s only one Server object for a Web server. All the applications share a single server object. In an ASP page, the Server object is used to create other object instances - this is equivalent to the Visual Basic command Set myObject = new someObject . The Server object also contains methods and properties to map virtual Web paths to physical paths. In other words, if the Web application is located in the myWeb virtual directory, the user can find out where the files for myWeb are physically located.
The Server object can also encode and decode string information for transmission to or from the HyperText Transfer Protocol (HTTP) protocol.
Application Object
There’s one Application object for each Web application. An application, to the
ASP engine, is the set of all files and subdirectories within a directory that contains a file called global.asa
. The .asa extension stands for Active Server Application. The
Application object is a container object that can hold other values. In fact, although it’s not a Dictionary object, it’s easiest to think of the Application Object as a Dictionary.
Dictionaries, like Visual Basic collections, are lists of key-value pairs.
Session Object
Each application may have many sessions, one for each user accessing the application. A session begins when a user requests any file in your Web application and
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that user’s browser does not send a valid ASP-generated SessionID cookie for that site.
The Session object is a container object like the Application object. It’s also similar to a
Dictionary object, with keys and values. The biggest difference is that each user gets a unique Session object, whereas all users of the application share the Application object.
Request Object
Browsers send a good deal of information to the server for each page request. The users don’t normally see any of this header information when he is browsing a site, but it is available at the server for applications to use. The ASP engine packages this information nicely in an object called the Request object. The Request object contains all this header information as well as information about the specific page request and any form information submitted by a user. The user can retrieve the information through the properties and collections of the Request object.
Response Object
The methods of the Response object to send a response to the client browser. The
Response object is the primary way to communicate with the client.
ScriptingContext Object
This is a wrapper object that enables an external ActiveX object to obtain references to the other ASP objects. The ScriptingContext object provides a wrapper that encloses all the other ASP objects in a single object that can be passed as a parameter.
When a page containing ActiveX object references starts, the ASP engine calls the onstartpage method for each ActiveX object on the page with a ScriptingContext object as parameter. The ActiveX objects use the ScriptingContext parameter to gain reference pointers to the Server, Application, Session, Request, and Response object. Because VB
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6 WebClasses provide these references automatically, the ScriptingContext object is not important.
ObjectContext Object
The ObjectContext object is the communications channel to Microsoft
Transaction Server (MTS). Through MTS, ActiveX objects can participate in transactions initiated by an ASP page. References to the other ASP objects can also gain through
ObjectContext object.
6.2.2 Introduction to WebClasses
As mentioned earlier, the WebClasses is the ways interact with IIS from a VB application. It works hand in hand with ASP to provide server-based request processing.
Compared to ActiveX Dynamic Link Libraries (DLLs) work with ASP pages,
WebClasses has the following benefits:
• WebClasses handle a great deal of the grunt work of connecting your VB class to the web server, parsing the variables sent by the browser, and sending new content to the browser.
• WebClass code, like a DLL, is compiled code; therefore, in certain situations, it’s considerably faster than the interpreted ASP code.
• WebClasses encourage separation between the interface code and the application logic. A WebClass acts as the connection point between the user interface (HTML) and back-end databases or middle-tier business objects.
• WebClasses let users work in a robust environment that you are familiar with and that has evolved over many years.
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In their first implementation, WebClasses don’t talk directly to the server. When an ActiveX DLL is created in Visual Basic, typically another project is also created to instantiate the objects that ActiveX DLL exposes. Similarly, a WebClass is a DLL that needs to be loaded; the objects it exposes need to be instantiated from another program.
Webclass objects are instantiated by the ASP script that controls them (actually by a library mswcrun.dll
). They receive events from the framework at the start and end of each HTTP request. Figure 6.1 shows the state transitions during the lifetime of a webclass. The
Initialize
and
BeginRequest
events are triggered first, then either
Start
or one of the standard, template or custom events. If this delegates to a template or webitem then more of these events are triggered. Finally the webclass objects and other resources are torn down in the handler for the EndRequest and Terminate requests.
The current snap-fit web-based application is developed through VB IIS application and resides on a PC with Microsoft’s Windows 2000 Standard Server in
Integral Attachment Program of the Ohio State University. A flow diagram of this webbased design tool was shown in Figure 6.2.
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Initialize Initialize
Terminate
Begin Request
Start
Generating
Begin Request
Standard event
Template event
Custom event
End Request
Standard event
Template event
Custom event
Figure 6.1: A typical WebClass life cycle
In order to secure this web-based application, every client needs a username and a password to log on this website (Figure 6.3). At the same time, in the future the administrator can identify each client by his/her username and password. Currently a file is used to keep the log-on information, which will eventually be moved to a database driven by SQL server.
After clients logged on the website, there will three different snap-fits available to choose: Cantilever Hook, Post & Dome and Bayonet & Finger (Figure 6.4).
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Units switch
Log on
Username
& password
Snap-fit feature selection
• Cantilever hook
• Post & dome
• Bayonet & finger
Design
Interface parameters
Updated
User input
Any solved outputs?
Illustration Correction advisor
Sensitivity analysis
Single objective optimization
Figure 6.2: The flow diagram of snap-fit web-based design tool
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Multiple objectives optimization

Figure 6.3: The logon page
Figure 6.4: The selection page
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Next, combined with this web-based snap-fit design tool, the 4 different problems for constraint management which is mentioned earlier will describe in detail.
6.3.1 New Specification Problem
In this stage, designers continue to input variables (in any order) and to solve for any other variables which are dependent on those already input or computed. After each input was typed, the design tool will check if there is any output can be computed based on this new input and those already computed which will avoid over-constraint. For example, in this case Modulus was chosen as the first input, but no output is available based on Modulus . So designers keep adding more inputs (beam length, offset, normal force and base width) one by one, the first two outputs are available until designers type the fifth input (base width). At the same time, the Status column will shown each variable’s status (Input, Output or N/A) (Figure 6.5).
6.3.2 Respecification Problem
Designers can change the value of the variables which has already been input. For example, if designers want to change the value of Modulus from 300000 psi to 400000 psi, designers just need input the new value for Modulus . As a result, the two output’s values will also change (Figure 6.6).
6.3.3 Unspecification Problem
Finally, all variables are inputted or computed (Figure 6.7). Designers can also unspecify the value for a variable which was previously specified as an input by clicking the corresponding Reset button. After that, the value of this input and all outputs
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Figure 6.5: The design interface of new specification
Figure 6.6: The design interface of respecification
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Figure 6.7: The final design interface of new specification
Figure 6.8: The design interface of unspecification
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dependent on this input will become blank. For this case, the Beam Length is chosen to reset. From Figure 6.8, it is found that all outputs are dependent on Beam Length .
6.3.4 Reverse Specification Problem
Finally designers can specify a value for a variable which has been computed. For instance, designers want to specify a new value for Base Height . First designers should click the corresponding Reset button. Then all inputs related to Base Height will be highlighted and designers should release one of these inputs’ statuses from Input to N/A .
Now designers can specify a new value to Base Height in order to make it as an input variable. In this case, suppose designers release the status of Beam Length and specify a new value, say 0.1, for Base Height (Figure 6.9).
Figure 6.9: The design interface of reverse specification
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6.3.5 Sensitivity Analysis
Sensitivity analysis is used to determine the input’s dependency to a specified output. For example, if designers want to see which input has the most significant impact to Insertion Strain , designer click the Sensitivity Analysis link at the bottom. The following figure (Figure 6.10) shows the typical Sensitivity Analysis page. Designers can choose any output wanted from this page. The sensitivities of inputs are ordered from the smallest to the biggest. So Offset has the most significant impact to Insertion Strain , and the value means that if the value of offset changes 1.491338E-02 the value of insertion strain will change 1 unit.
Figure 6.10: The typical sensitivity analysis page
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6.3.6 Correction Advisor
Correction Advisor is used when designers want to see how much each dependent input’s value should change if one output changes to a new value. For example, the following figure (Figure 6.11) shows the typical Correction Advisor page. If designers just want to change the value of Modulus to change the value of Base Height from
5.975205E-02 to 1, the value of Modulus should change from 300000 to 64 and the difference is 99.97867%. The page also shows that if just change one of the last three inputs, the change will violate their minimum or maximum values.
Figure 6.11: The typical correction advisor page
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There also have some other functions such as Illustration which shows each snapfit’s geometry, Variable Range which shows each variable’s range. And using Switch
Units designers can switch units between Metric units and English units. Reset button is used to clear all variable’s values.
6.3.7 Single Objective Optimization
Single-objective optimization is used when designer want to optimize one parameter of snap-fit features. First the objective parameter should be selected. Then designer should decide if he/she wants to minimization or maximization. For example, in this case designer wants to maximize the retention force. The designer can designed any parameter’s value. Each time after designer defined one parameter the design tool will calculated the corresponding outputs and check constraint management to prevent overconstraint. In this case all parameters are inputted or solved except retention angle
(Figure 6.12). Finally the optimum retention force was found and the design tool also calculated the corresponding value for retention angle (Figure 6.13).
6.3.8 Multiple Objectives Optimization
In this case, two parameters of cantilever hook want to be optimized: minimize insertion force and maximize retention force. Some parameters were inputted identified by Input status and some parameters were solved based previous inputted parameters identified by Output status before the optimization. Two free parameters were remained identified by N/A status. The weighted values of objective functions need to be selected based on designer’s decision on their importance and make sure the sum of these weighted values is 1 (Figure 6.14).
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Figure 6.12. Single objective optimization for retention force of cantilever hook
Figure 6.13. Single objective optimization result for retention force of cantilever hook
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Figure 6.14: Multiple objectives optimization for retention force of cantilever hook
Figure 6.15: Multiple objectives optimization result for retention force of cantilever hook
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After click Optimization button, the optimized values of objective functions were shown. The free remained parameters were also solved based these optimized values
(Figure 6.15).
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CHAPTER 7
CASE STUDIES
In this chapter several snap-fit applications from different industries were studied.
Results calculated by three different methods were tabulated and compared. These were strength of material solutions (Equations 5.6 to 5.11), finite element analysis and the empirical model (Equation 4.3) generated in this thesis. Comparison between these results shows that the empirical model has greater accuracy as compared to design guides and matches the result of finite element analysis. The snap-fit design guide of Bayer
Polymer [3] was used to calculate the retention forces for all case studies. Since the cantilever hooks of all case studies have uniform width, two-dimensional and plane stress finite element models were generated in either ANSYS or MARC. Finally the predicated results were generated from those empirical models generated in Chapter 4.
7.1 Methods to Evaluate the Performance of Cantilever Hook
There are several plastic snap-fits design guides [1-3] and all of them use similar strength of material design equations. In this thesis the design equations [5.6 – 5.11] of
Bayer Polymer were used to calculate retention force.
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Finite Element Simulation
Similar to the previous FEA models, the retention of cantilever hook samples were simulated by a 2D plane stress, contact and nonlinear model. All case studies were conducted by the finite element package ANSYS. Figure 3.2 shows a typical cantilever hook finite element model in ANSYS. The contact areas of cantilever hook (insertion surface, dwell surface and retention surface) have a much finer mesh to achieve accurate simulation. The end of cantilever hook was fixed and the mating part was only allowed to move in the X direction. To simplify the simulation and achieve convergence, the mating part was considered as a rigid body which means no deformation on the mating part during the contact. All other modeling techniques are same as those described in the
Chapter 3.
Web-based Design Tool
The web-based design tool developed in this thesis was used to evaluate different cantilever hook case studies. The response surface models generated in the Chapter 4 were incorporated into the web-based design tool.
7.2 Integrated Polymeric Lens Housing
This case study involves a lens system (Figure 7.1) used in a projection TV system. The integral fastening program at Ohio State had worked with this company in improving the performance of their snap-fits. In this design cantilever hook snap-fits are used to attach the two halves of a polymeric lens housing.
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(a)
Figure 7.1: Integral polymeric lens housing
(b)
Figure 7.2 shows the detailed dimensions of cantilever hook of this integrated polymeric lens housing.
Figure 7.2: Dimensions of snap-fits used in integral polymeric lens housing
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Snap-fit Design Guide Result
The following properties (Figure 4.1) were identified:
• Permissible strain ( ε ) is 15 %.
•
•
•
•
•
•
Length l of cantilever hook is 10 mm.
Thickness of root
Width at root b h is 3 mm.
is 1.5 mm.
Young’s modulus E is 1800 Mpa.
Friction coefficient µ is 0.55.
Retention angle β is 90 ° .
So that the deflection force P calculated from the snap-fit design guide of is:
P = bh 2
6
E ε l
=
1 .
5 ⋅ 3 2
6
1800 ⋅ 0 .
15
10
= 60 .
75 (N) (7.1)
Since the retention angle β is greater than the critical angle, according to the design guide it is a permanent cantilever hook which means it will not manually release.
Retention force is not based on the traditional strength of material design equations but on cross-sectional area strengths.
F ret
= 1800 ⋅ 1 .
5 ⋅ 2 = 5400 (N)
Finite Element Simulation Result
(7.2)
The graph of retention force versus time is shown in Figure 7.3. The maximum retention force was 14.03 N.
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15
12
9
X: 0.3238
Y: 14.03
6
3
0
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
Position (mm)
Figure 7.3: The curve of retention force of cantilever hook of lens housing
The Web-based Design Tool Result
Figure 7.4: The web-based design tool for integral polymeric lens housing
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It’s obvious that the retention force calculated from traditional strength of material method (5400 N) is not accurate. Compared the FEA result (14.03 N) to the empirical result (12.867 N) of Figure 7.4, it was shown that the empirical response surface model predicate the retention force very accurately for this case study.
7.3 Cantilever Hook Application of Cabinet
The second case study is a power supply case for a Rockwell Automation PLC controller (Figure 7.6). It makes use of cantilever hooks to connect the upper cover to the main case.
Snap-fit Design Guide Result
Similarly to the previous case study, the material and geometrical properties were obtained and imputed to the design equations. The cantilever hook has the same material properties as the previous case study. The only difference is the detailed dimensions shown in the Figure 7.7. Since this cantilever hook has geometric symmetry, only the half of cantilever hook was studied. Similar to the previous cantilever hook, since it has a 90 degree retention angle, the strength of material method was used to calculate the retention force.
F ret
= 1800 ⋅ 2 ⋅ 2 = 7200 (N) (7.3)
The beam thickness at broken area can be calculated as follows (Figure 7.5). h1 h2 where h1 = 2.5 – 1.5 = 1mm
h2 = 4 * 2.5 / 10 = 1mm
h1 + h2 = 2 mm
Figure 7.5: Calculation of beam thickness at broken area for power supply’s snap-fit
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(a) (b)
Figure 7.6: Cantilever hook of power supply for Rockwell Allen Bradly PLC controller
Figure 7.7: The detailed dimensions of the second case study
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Finite Element Simulation Result
14
12
X: 0.1475
Y: 12.03
10
8
6
4
2
0
-2
0 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Figure 7.8: The retention force curve of cabinet snap-fit
The Web-based Design Tool Result
1
Figure 7.9: The web-based design page for power supply’s cantilever hook
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Again since this cantilever hook has a 90-degree retention angle, the snap-fit design guide can’t predict the retention force. And the empirical model (13.517 N) from Figure
7.9 showed the good accuracy to predict the FEM retention force (12.03 N) from Figure
7.8.
7.4 Cantilever Hook Samples
In a previous study, snap-fit designs created by various rapid prototyping methods were compared to determine strength. These snap-fits were made in a mold with variable feature inserts so several geometries could be created. Three of these designs were used in this case study. These three cantilever hook samples have different geometries (Figure
7.10 to Figure 7.12) and performed their finite element analysis. The results were compared to those of the design guide and the empirical response surface models.
Figure 7.10: The detailed dimensions of cantilever hook sample a
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Figure 7.11: The detailed dimensions of cantilever hook sample b
Figure 7.12: The detailed dimensions of cantilever hook sample c
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Snap-fit Design Guide Result
For cantilever hook b, because its retention angle is 90-degree, the retention force is not based on the design equations from Bayer design guide but on cross-section strength.
For cantilever hooks a and c, since the design guide doesn’t consider the height of undercut their retention forces were the same. And the retention force based on design guide was calculated in the following:
F ret
= bh 2
6
E s
ε l
µ
1 −
+
µ tan tan
β
β
=
8 .
89 ⋅ 2 .
5 2
6
1800 ⋅ 0 .
025
10
0 .
55 + tan( 45 )
1 − 0 .
55 ⋅ tan( 45 )
= 143 N
(7.4)
Finite Element Simulation Result
After imported the meshes into a finite element analysis package, the retention force curves of each cantilever hook were obtained in the Figure 7.13 to Figure 7.15.
30
25
X: 1.503
Y: 27.89
20
15
10
5
0
-5
-10
0 1 2 3 4 5 6 7 8 9
Position (mm)
Figure 7.13: The retention force curve of cantilever hook a
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10

30
25
20
15
10
5
45
40
35
30
25
20
15
10
5
0
X: 0.2171
Y: 40.34
-5
0 0.2
0.4
0.6
0.8
1 1.2
1.4
1.6
1.8
2
Position (mm)
Figure 7.14: The retention force curve of cantilever hook b
45
40
X: 3.892
Y: 42.7
35
0
0 1 2 3 4 5
Position (mm)
6 7 8 9
Figure 7.15: The retention force curve of cantilever hook c
10
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The Web-based Design Tool Result
By using the empirical response surface model, the predicated retention forces were calculated:
• Cantilever Hook a: 29.45 N
• Cantilever Hook b: 45.27N
• Cantilever Hook c: 31.46 N
The comparison between these three different results was shown in Table 7.1. From the table, it was shown that the results of design guide usually have a higher predicated retention force compared to those of FEA and the empirical model. For cantilever hook with 90-degree retention angle (sample b) it went for infinite that was unpractical. Even using a cross-section section method, it predicted a much high retention force for sample b (Equation 7.5). Between the results of FEA and those of the empirical model, the FEA simulation showed the good accuracy for sample a and sample b. However it showed difference for sample c. That was caused by the geometries of this sample which has a uncommon large catche. As a matter of factor, the ratio of between undercut (9.65) and thickness (2.54) is 3.8. However its upper limit of response surface model defined before is 1.5. As a result the dimensions are not in the design space of the empirical model. The predictions showed a large difference.
F ret
= 1800 ⋅ ( 2 .
79 − 1 .
52 ) ⋅ 8 .
89 = 20322 N
(7.5)
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Figure 7.16: The web-based design page for cantilever hook sample a
Figure 7.17: The web-based design page for cantilever hook sample b
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Figure 7.18: The web-based design page for cantilever hook sample c
7.5 Case Study Conclusions
Sample a
Sample b
Sample c
The Results of
Design Guide
143 N
20322 N
143 N
The Results of
Finite Element
Analysis
27.89 N
40.35 N
42.69 N
The Results of
Empirical
Model
29.45 N
45.27N
31.46 N
Table 7.1: The comparison of cantilever hook samples
Difference Between
FEM Results and
Empirical Results
5.25%
10.87%
26.31%
In general the design guides from resin manufacturers were inaccurate in predicting the retention force of cantilever hook with high retention angles. The reason is because those design equations are based on the Euler-Bernoulli beam theory. As a result
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once the retention angle is greater than the critical angle, the retention forces is not based on the design equations from those design guides but on cross-section strength.
For the empirical model, if the cantilever hooks are in the design space the predicted retention force of cantilever hooks with high retention angles are very accurate
(cantilever hook sample a and b). However if the cantilever hooks move out of the design space, the predicted results are usually inaccurate (cantilever hook c). The same principle also applies on other parameters such as material properties and friction coefficient.
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CHAPTER 8
CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH
8.1 Scientific and Engineering Contribution
This thesis developed a web-based design tool for three different snap-fit features, the cantilever hook, post & dome and bayonet & finger. This web-based design tool provides an intuitive interface to help users design robust snap-fits. The users choose any snap-fit parameters as inputs, then the design tool will calculate allowable combinations of output based on current inputs.
Constraint management was also applied in the design tool. It’s the first time that such design techniques have been implemented in the snap-fit design area. Constraint management allows the use of all dimensions to drive the snap-fit design process. For example it allow users to switch parameters from inputs to outputs or vice versa. By doing this the over-constraint phenomena was prevented. Correction advisor, sensitivity analysis and optimization modules were also implemented into this design tool.
The performance of cantilever hooks with a high retention angle is difficult to predicate. The first reason is its non-linear plastic material property and the second reason is the large deformation existed in its body of cantilever hook. Other snap-fit design resources (design guides, analytical tools and linear finite element models) predicate the
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performance of cantilever hooks with low retention angles very accurately, however they always have inaccurate predictions for cantilever hooks with high retention angles. The finite element model in this thesis used the real non-linear material properties of polymer as well as geometric large deformation.
This thesis made the following contributions to the area of designing the three different plastic snap-fit parts - Cantilever Hook, Post & Dome and Bayonet & Finger:
1. This thesis developed a finite element model of the actual retention process using nonlinear material properties. This is a great improvement over previous finite element models based on linear material properties.
2. This thesis generated a second-order response surface model to predict the performance of cantilever hook with a high retention angle. It allows designers use this model to evaluate their snap-fit designs, instead of performing those complicated nonlinear finite element models.
3. This thesis proposed a different two-step method to generate the response surface model for snap-fits. The first step is a screening experiment to identify the most significant factors and the second step is to generate a second-order response surface model. The advantage of this method is to significantly reduce the number of design experiments while keeping the accuracy of response surface model at the same time.
4. This thesis used Central Composite Design to generate a response surface model for cantilever hook with a high retention angle. The Central Composite Design investigates the existence of curvature in the design space. This is another
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improvement over prior response surface models of snap-fits that assumed a linear design space.
5. This thesis provided several different diagnostic methods to check the accuracy of the response surface model. This thesis also provided several remedy methods to improve the accuracy of the response surface model. Several case studies were conducted in the different design areas for the response surface model and the results were compared to the nonlinear finite element model to verify the accuracy of the response surface model.
6. This thesis provided a web-based design tool for snap-fits. This web-based design tool allows designers an intuitive design environment and avoids the direct use of polynomial equations. Moreover by making the tool web-based, designers don’t need to be involved with maintenance issues of computer-based application such as updating the software and other software maintenance.
7. This thesis applied constraint management into the web-based design tool.
Constraint management allows designers to switch the design parameters between input parameters and output parameters on the fly. Constraint management also prevents the over-constraint phenomenon of design parameters.
8. This thesis provided two additional modules – correction advisor and sensitivity analysis - for the web-based design tools. Correction advisor allows designers to see how much each dependent input’s value should change if one output changes to a new value. Sensitivity analysis is used to determine the input’s dependency to a specified output.
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9. This thesis provided two optimization modules – single objective optimization and multiple objective optimizations - for the web-based design tool.
Optimization modules allow designers to optimize one or more design parameters at the same time. The optimized results were constrained by the design space defined by designers.
8.2 Recommendations for Future Research
Additional insight into predicting the performance of the snap-fit features can be gotten by performing additional research into the following areas:
1. Creating a new friction model to simulate the friction of polymers with pressure. Current friction models are not able to accurately describe the friction of behavior of polymers.
2. Incorporating a more accurate nonlinear material model into finite element simulations instead of a bilinear nonlinear material model.
3. Performing a deformable-to-deformable contact type finite element simulation between cantilever hook and mating part. This should improve the accuracy of the finite element analysis.
4. Incorporating other material properties of polymers into finite element simulations such as temperature dependent and viscoelasticity.
5. Further fundamental study on the bifurcating phenomenon of cantilever hook with a high retention angle (loss of engagement failure or shear failure) and developing two corresponding different response surface models.
6. Updating the web-based design tool with more user-friendly interfaces.
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APPENDIX A
“PROCEDURE” FILE FOR MSC.MARC AND MSC.PATRAN
*open_model MarcModel.mud # Read solid Model Gemoetry #
*fill_view
*show_table
*table_remove
2
*table_add
0.1
4625
*table_fit
*material_type mechanical:isotropic # Specify material property #
*material_option isotropic:plasticity:elastic_plastic *material_type plasticity
*material_table plasticity:yield_stress1
StressStrain
*update_job
*submit_job 1 # Run simulation #
*post_open_default
*history_collect 0 999999999 1 # Read history #
*history_add_nodeless
Time
Force X MatingPart # Plot retention force vs time #
*history_fit
*set_history_increment_id
50
*set_deformed on *find_solid_outline
*post_contour_bands
*post_close
*job_class mechanical
*add_post_tensor stress
*add_post_tensor strain
*add_post_tensor el_strain
*add_post_var von_mises
*save_model
*update_job
*job_submit_reset
*submit_job 1
*fill_view
*update_job
*monitor_job
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*monitor_job
*job_submit_reset
*kill_job
*submit_job 1
*post_open MarcModel_Retention.t16
*history_collect 0 999999999 1
*history_add_nodeless
Time
Force X MatingPart
*history_fit
*set_deformed on *find_solid_outline
*post_contour_bands
*post_value Equivalent Von Mises Stress
*post_skip_to
250
*post_value Comp 11 of Stress
*quit yes
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APPENDIX B
VISUAL BASIC SUBROUTINES FOR CANTILEVER HOOK
Private Sub Cantilever_Form1()
Dim TempVal As Variant
Dim iSelIn As Single
Dim iSelOut As Single
Dim ValIn As Single
Dim ValOut As Single
Dim Index As Single
Dim singMainCase As Single
Dim singSubcase As Single
Dim singIC As Single
Dim StatusAction As String
Dim numvars As Single
Dim num As Single
Dim action As String
Dim i As Single
Dim tempUnit As String 'delete it in the later
Dim UnitTran(100) As Double 'To use the transfter of units
'Define the unit's transfer
UnitTran(1) = 1.45038
UnitTran(2) = 0.03937
UnitTran(3) = 0.03937
UnitTran(4) = 0.22481
UnitTran(5) = 0.03937
UnitTran(6) = 0.03937
UnitTran(7) = 1
UnitTran(8) = 1
UnitTran(9) = 1
UnitTran(10) = 1
UnitTran(11) = 0.22481
UnitTran(12) = 1
UnitTran(13) = 0.22481
'action = Request.Form("submit") 'When the user click one of button
If Request.Form("Change") <> "" Then
End If action = Request.Form("submit")
'If action = "" Then 'judge if the user press "Enter" key instead of click "Solve" button
If action = "Reset" Then 'If the user click Reset button
For i = 1 To 13
LastTextBox(i) = ""
IndexInput(i) = 0
153

IndexSolved(i) = 0
VALS(i) = ""
Button(i) = "N/A"
Next
SensFlag = 0 'To make SensOutCnt and inCnt 0
Cantilever.WriteTemplate
Exit Sub
ElseIf action = "Switch Units" Then 'If user click the "Switch Units" button
If Request.Form("Radio") = "USUnit" Then
If Unit(1) = "KPa" Then 'to decide the previous unit
'updates the unit
Unit(1) = "psi"
Unit(2) = "in"
Unit(3) = "in"
Unit(4) = "lb"
Unit(5) = "in"
Unit(6) = "in"
Unit(11) = "lb"
Unit(13) = "lb"
'updates the unit radio
Session("RadioUnit1") = "checked"
Session("RadioUnit2") = ""
'updates the values
For i = 1 To 13
If LastTextBox(i) <> " " Then
LastTextBox(i) = Round(LastTextBox(i) * UnitTran(i))
End If
Next
Cantilever.WriteTemplate
Exit Sub
ElseIf Unit(1) = "psi" Then
'ElseIf Session("Unit1") = "psi" Then
Cantilever.WriteTemplate
Exit Sub
End If
ElseIf Request.Form("Radio") = "SIUnit" Then
If Unit(1) = "psi" Then 'to decide the previous unit
'updates the unit
Unit(1) = "KPa"
Unit(2) = "mm"
Unit(3) = "mm"
Unit(4) = "N"
Unit(5) = "mm"
Unit(6) = "mm"
Unit(11) = "N"
Unit(13) = "N"
Session("RadioUnit1") = ""
Session("RadioUnit2") = "checked"
'updates the values
For i = 1 To 13
If LastTextBox(i) <> " " Then
LastTextBox(i) = Round(LastTextBox(i) / UnitTran(i))
End If
Next
Cantilever.WriteTemplate
Exit Sub
ElseIf Unit(1) = "KPa" Then
Cantilever.WriteTemplate
Exit Sub
End If
154

End If
ElseIf action = "Solve" Or action = "" Then 'If user click "solve" or press "enter" directly
GoTo 200
Else
400 Select Case Request.Form("submit")
'If the user click one of change buttons
Case "Reset Modulus"
If IndexInput(1) = "1" Then
Index = 1
GoTo 100
Else: GoTo 20
End If
Case "Reset BeamLength"
If IndexInput(2) = "1" Then
Index = 2
GoTo 100
Else: GoTo 20
End If
Case "Reset Offset"
If IndexInput(3) = "1" Then
Index = 3
GoTo 100
Else: GoTo 20
End If
Case "Reset Normal Force"
If IndexInput(4) = "1" Then
Index = 4
GoTo 100
Else: GoTo 20
End If
Case "Reset Base Width"
If IndexInput(5) = "1" Then
Index = 5
GoTo 100
Else: GoTo 20
End If
Case "Reset Base Height"
If IndexInput(6) = "1" Then
Index = 6
GoTo 100
Else: GoTo 20
End If
Case "Reset Ins. Strain"
If IndexInput(7) = "1" Then
Index = 7
GoTo 100
Else: GoTo 20
End If
Case "Reset Fri. Coe."
If IndexInput(8) = "1" Then
Index = 8
GoTo 100
Else: GoTo 20
End If
Case "Reset Taper"
If IndexInput(9) = "1" Then
Index = 9
GoTo 100
Else: GoTo 20
End If
Case "Reset Ret. Ang."
If IndexInput(10) = "1" Then
Index = 10
GoTo 100
Else: GoTo 20
End If
Case "Reset Ret. Force"
155

If IndexInput(11) = "1" Then
Index = 11
GoTo 100
Else: GoTo 20
End If
Case "Reset Ins. Ang."
If IndexInput(12) = "1" Then
Index = 12
GoTo 100
Else: GoTo 20
End If
Case "Reset Ins. Force"
If IndexInput(13) = "1" Then
Index = 13
GoTo 100
Else: GoTo 20
End If
100 Call REMOVEINPVARIABLE(Index)
Call getValueInfo
Call updateVariables
Cantilever.WriteTemplate
Exit Sub
'Call updateVarLabelNames(maindata, frmInput)
20 Cantilever.WriteTemplate
Exit Sub
End Select
End If
200 For i = 1 To 13
TextBox(i) = Request.Form("TextBox")(i)
Next
For i = 1 To 13
If TextBox(i) <> "" Then GoTo 10 'To judge if all the textboxes are empty
Next i
Session("msg") = "You should have at least one input"
Cantilever.WriteTemplate
Exit Sub
10 i = 1
'To decide which textbox just been input
For i = 1 To maindata.numvars
If LastTextBox(i) <> TextBox(i) Then
GoTo 15
End If
Next
15 iSel = i ' Until now we know which TextBox has been just inputed
val = TextBox(i) 'we will use val to call SpreadHandler
'To see if val is in the range of minimum and maximum
If val < maindata.values.XMIN(i) Then
Session("msg") = "The value you input is too small."
LastTextBox(i) = TextBox(i)
Cantilever.WriteTemplate
Exit Sub
ElseIf val > maindata.values.XMAX(i) Then
Session("msg") = "The value you input is too big."
LastTextBox(i) = TextBox(i)
Cantilever.WriteTemplate
Exit Sub
End If
'Now we want to decide if this textbox is input or output and we will sovle it
'according to 3 different cases
If i > 13 Then
156

Session("msg") = "You already have all inputs"
Cantilever.WriteTemplate
Exit Sub
End If
If IndexType(i) = 1 Then
If IndexInput(i) = 0 And IndexSolved(i) = 0 Then
Call SPREADHANDLER(iSel, val, flag(1))
ElseIf IndexInput(i) = 1 And IndexSolved(i) = 0 Then
Call SPREADHANDLER(iSel, val, flag(1))
ElseIf IndexInput(i) = 0 And IndexSolved(i) = 1 Then
Call GETBACKDEPEND(iSel, depVars(1))
iSelIn = forwardChoice
iSelOut = backChoice
ValOut = val
ValIn = val
' Call the FORTRAN module to handle the reverse specs
' of the variables reverse dependancies
Call REVERSEHANDLER(iSelIn, ValIn, iSelOut, ValOut, flag(1))
' Call the spreadsheet handler
Call SPREADHANDLER(iSel, val, flag(1))
End If
End If
' update the labels and such
Call getValueInfo i = 1 ' To set i=1 because i will must be euqal to 1 here
'Call updateVarLabelNames(maindata)
For i = 1 To maindata.numvars
If IndexInput(i) = 1 Then
If IndexType(i) = 1 Then
LastTextBox(i) = maindata.values.VALS(i)
Button(i) = "Input"
End If
ElseIf IndexSolved(i) = 1 Then
If IndexType(i) = 1 Then
LastTextBox(i) = maindata.values.VALS(i)
Button(i) = "Output"
End If
ElseIf IndexSolved(i) = 0 And IndexInput(i) = 0 Then
End If
Next i
Cantilever.WriteTemplate
Dim temp As String
End Sub
Private Sub Cantilever_Hyperlink6()
Dim cnt As Integer
Dim i As Integer
Dim cmbText As String
'frmGraph.cmbIn.Clear cnt = 0
For i = 1 To maindata.numvars
If IndexInput(i) = 1 Then
157

'frmGraph.cmbIn.AddItem " " & maindata.varnam(i)
cmbText = maindata.varnam(i)
GraVarIn(i) = cmbText
cnt = cnt + 1
'frmGraph.setInArr cnt, i
setInArr cnt, i
End If
Next i
'frmGraph.readyToGraph = False
'frmGraph.numInps = cnt
GranumInps = cnt
'frmGraph.cmbIn.ListIndex = 0
GracmbInListIndex = 0
Call GracmbIn_Click
'frmGraph.Show vbModal
'frmGraph.update_graph
Graphing.WriteTemplate
End Sub
'The subroutine for sensitivity analysis
Private Sub Cantilever_Hyperlink7()
Static inCnt As Single
Static SensOutCnt As Single
Dim numStr
Dim spcStr
Dim cmbText As String
Dim i As Integer
Dim j As Integer
Static inArr(1 To 100) As Single
Static sensmat(1 To 100, 1 To 100) As Single
'If Request.Form("TextBox")(1) = "" Then
'End If
'If Request.Form("TextBox")(1) = "" And Request.Form("TextBox2") = "" And Request.Form("TextBox3") = "" And
Request.Form("TextBox4") = "" And Request.Form("TextBox5") = "" And Request.Form("TextBox6") = "" And
Request.Form("TextBox7") = "" And Request.Form("TextBox8") = "" And Request.Form("TextBox9") = "" And
Request.Form("TextBox10") = "" And Request.Form("TextBox11") = "" And Request.Form("TextBox12") = "" And
Request.Form("TextBox13") = "" Then
' Session("msg") = "You need at least one input and output"
' Cantilever.WriteTemplate
' Exit Sub
'End If
Call ClearVariables
Call CALCSENSMAT(inArr(1), OutArr(1), inCnt, SensOutCnt, sensmat(1, 1))
' If SensFlag = 0 Then 'If click reset and click "senstivity analsysis"
' SensOutCnt = 0
' inCnt = 0
' End If
numStr = "0123"
spcStr = " "
numInps = inCnt 'should change it maybe
numOuts = SensOutCnt
For i = 1 To SensOutCnt Step 1
Call setOutArr(i, Int(OutArr(i)))
LSet numStr = OutArr(i)
cmbText = numStr & spcStr & maindata.varnam(OutArr(i))
'frmSens.cmbOut.AddItem cmbText
SenVarOut(i) = cmbText 'give the cmbText to the Sensitivity Output Variables
158

Next i
For i = 1 To 100 Step 1
For j = 1 To 100 Step 1
Call setSensMat(i, j, sensmat(i, j))
Next j
Next i
For i = 1 To inCnt Step 1
Call setInArr(i, Int(inArr(i)))
LSet numStr = inArr(i)
cmbText = numStr & spcStr & maindata.varnam(inArr(i))
'frmSens.cmbIn.AddItem cmbText
SenVarIn(i) = cmbText 'Give the cmbText to the Senstivity Input Variables
Next i
If (inCnt = 0 Or SensOutCnt = 0) Then
'Dim msg, Style, Title, Help, Ctxt, Response, MyString
Session("msg") = "You need to have at least one Input and one Output " & _
"variable defined to use the sensitivity analysis tool."
Cantilever.WriteTemplate
Else
'frmSens.cmbIn.ListIndex = 0 cmbInListIndex = 0
Call cmbIn_Click
'frmSens.cmbOut.ListIndex = 0 cmbOutListIndex = 0
Call cmbOut_Click
'frmSens.Show vbModal
For i = 1 To SensOutCnt Step 1
Select Case i
Case "1"
Session("Choice1") = SenVarOut(i)
Case "2"
Session("Choice2") = SenVarOut(i)
Case "3"
Session("Choice3") = SenVarOut(i)
Case "4"
Session("Choice4") = SenVarOut(i)
Case "5"
Session("Choice5") = SenVarOut(i)
End Select
Next
For i = 1 To (cnt - 1) Step 1
Select Case i
Case "1"
Session("SenVar1") = SenVarIn(i)
Session("SenVal1") = SenVal(i)
Case "2"
Session("SenVar2") = SenVarIn(i)
Session("SenVal2") = SenVal(i)
Case "3"
Session("SenVar3") = SenVarIn(i)
Session("SenVal3") = SenVal(i)
Case "4"
Session("SenVar4") = SenVarIn(i)
Session("SenVal4") = SenVal(i)
Case "5"
Session("SenVar5") = SenVarIn(i)
Session("SenVal5") = SenVal(i)
'Case "6"
' Session("SenVar6") = SenVarIn(i)
' Session("SenVal6") = SenVal(i)
End Select
159

Next
'Show the first output by default
Session("Status1") = "selected"
Session("Status2") = ""
Session("Status3") = ""
Session("Status4") = ""
Session("Status5") = ""
SensitivityAnalysis.WriteTemplate
End If
End Sub
'This subroutine is for Correction Adviosr
Private Sub Cantilever_Hyperlink8()
Dim i As Integer
Dim numOut As Integer
Dim j As Integer
'frmCorr.cmbOut.Clear
'clear the values of drawdown menu
Call ClearCorrVal
Corrcnt = 0 'Number of outputs
j = 1 'The index of CorrChoices
For i = 1 To maindata.numvars
If IndexSolved(i) = 1 Then
'frmCorr.cmbOut.AddItem i & " " & maindata.varnam(i)
CorrOutChoice(j) = maindata.varnam(i)
Corrcnt = Corrcnt + 1
j = j + 1
'frmCorr.setOutArr Corrcnt, i
setOutArr Corrcnt, i
End If
Next i
Session("CorrOutStatus1") = "selected"
Session("CorrOutStatus2") = ""
Session("CorrOutStatus3") = ""
Session("CorrOutStatus4") = ""
Session("CorrOutStatus5") = ""
Session("CorrStatus1") = ""
If (Corrcnt = 0) Then
'Dim msg, Style, Title, Help, Ctxt, Response, MyString
Dim msg As String
Session("msg") = "You need to have at least one Output " & _
"variable defined to use the correction adviser."
'clear the
Call CorrAdvWriteTemplate
Exit Sub
'Style = vbOKOnly
'Title = "Sensitivity Warning" ' Define title.
'Response = MsgBox(msg, Style, Title, Help, Ctxt)
Else
'frmCorr.numOut = Corrcnt
numOut = Corrcnt
'frmCorr.cmbOut.ListIndex = 0
CorrcmbOutListIndex = 0 'Output index that choosed in drawdown menu
Call Corrcmbout_click
'frmCorr.Show vbModal
End If
Call CorrAdvWriteTemplate
160

End Sub
Private Sub Cantilever_Hyperlink9()
' This subroutine is for single objective optimization
Dim i As Integer
'Initilize the setting
SingOpMaxStatus = " checked " 'The status of max and min radio buttons
SingOPMinStatus = ""
'Make all textbox empty
SingOPTextBox1 = ""
SingOPTextBox2 = ""
SingOPTextBox3 = ""
SingOPTextBox4 = ""
SingOPTextBox5 = ""
SingOPTextBox6 = ""
SingOPTextBox7 = ""
SingOPTextBox8 = ""
SingOPTextBox9 = ""
SingOPTextBox10 = ""
SingOPTextBox11 = ""
SingOPTextBox12 = ""
SingOPTextBox13 = ""
'Set the SinMulOptFlag
SinMulOptFlag = 0
SingOp.WriteTemplate
End Sub
Private Sub Cantilever_MultOp()
'This is for Multiple Optimization
Dim i As Integer
'Set the number of objective function is 0
NumofMultObj = 0
'Initialize the minimization and maximization radio buttons
For i = 1 To 13
MultOptMinStatus(i) = " checked"
MultOptMaxStatus(i) = " "
Next i
'Initialize the default weighted values to 1
For i = 1 To 10
WeightedValue(i) = 0.1
Next i
'Set the SinMulOptFlag
SinMulOptFlag = 1
MultOp.WriteTemplate
End Sub
161

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