I A DESIGN OF EXPERIMENT APPROACH TO *".. *".. -- TOLERANCE ALLOCATION/
A Thesis Presented to
The Faculty of the Russ College of Engineering and Technology
Ohio University
In Partial Fulfillment
of the Requirement for the Degree
Master of Science
Ziaul Islam/ d
June, 1995 ACKNOWLEDGEMENTS
I would like to express my sincere gratitude to my advisor Dr. Richard J. Gerth for his advice, direction, and encouragement during my pursuit of this degree.
Working with him has been a learning experience, and I have derived a lot of inspiration fiom him.
I also wish to thank the members of the graduate committee, Dr. David
Koonce and Dr. Patrick McCuistion, for reviewing the thesis and offering suggestions for its improvement. TABLE OF CONTENTS
ACKNOWLEDGEMENTS ...... i
LIST OF FIGURES ...... iv
LIST OF TABLES ...... v
LIST OF APPENDICES ...... vi
CHAPTER
1. INTRODUCTION ...... 1
2 . LITERATURE REVIEW ...... 6
Heuristic Approaches ...... 6
Minimum Cost Optimization methods ...... 9
Lagrange Multiplier ...... 11
Linear programming ...... 13
Dynamic programming ...... 14
Non-linear programming ...... 15
Design of Experiments (DOE) ...... 19
Taguchi's Parameter Design ...... 22
Taguchi's Tolerance Design ...... 23
Tolerance Design using Taguchi's Parameter Design ...... 25
3 . CASE STUDY - A BENCH VICE ...... 27
4 . EXPERIMENTAL DESIGN SETUP AND PROCEDURE ...... 3 1 Inner and outer arrays ...... 3 1
Designing an experiment using orthogonal arrays ...... 32
Experimental variable and level selection ...... 34
Selection of orthogonal arrays and assignment of factors ...... 36
Conducting the experiment ...... 37
5 . ANALYSIS AND RESULTS ...... 39
The mean response analysis ...... 40
The S/N response analysis ...... 43
Confirmation experiment ...... 49
Geometrical interpretation ...... 54
6 . DISCUSSION AND CONCLUSIONS ...... 57
BIBLIOGRAPHY ...... 60
APPENDICES ...... 64
ABSTRACT LIST OF FIGURES
Figure
A detailed drawing of the bench vice assembly ...... ,29
Mean response graph ...... 44
S/N ratio response graph ...... 47 LIST OF TABLES
Table
2.1 . Table of cost minimization techniques ...... 1
2.2 . Table of proposed cost tolerance relationships ...... 19
4.1 . An L16 InnerlOuter Orthogonal Array Structure ...... 33
4.2 . Variables for the experiment ...... 35
4.3 . An L 16 Orthogonal Array ...... 37
5.1 . Response table ...... 41
5.2 . Mean response table ...... 42
5.3 . S/N response table ...... 46
5.4 . Mean response ANOVA table ...... 45
5.5 . S/N ratio ANOVA table ...... 48
5.6 . Confirmation experiment results ...... 53 LIST OF APPENDICES
Appendix
A . List of figures of Bench Vice component and their Geometric
Dimensioning & Tolerancing Drawings ...... 65
List of figures showing analysis of movable jaw and rod ...... 70
List of formulae and calculations ...... 72
Bench vice assembly dimensions ...... 86
Bench vice assembly simulation program ...... 87
L16 orthogonal array data ...... 88
Mean response data analysis ...... 90
ANOVA table on mean response ...... 91
S/N ratio data analysis ...... 92
ANOVA table on S/N ratio ...... 93
List of formulae used in simulation program of bench vice ...... 94
Listing of Microsoft Excel Version 4.0 macro program ...... 96 CHAPTER 1
INTRODUCTION
Manufacturing is a sigdlcant portion of the U.S. economy and manufacturing has experienced a loss in its global competitive position ( Otto and Finnie, 1993). One of the key factors contributing to the loss of market share to foreign competitors is poor product quality
(Juan, 1988; Deming, 1986). Therefore, much attention has been focused on the issue of producing quality products, which we will define as a product produced with minimum variation around a target dimensional value. If a product exhibits too much variation, it is a non-confoxming product or rejects. A reject can be defined as a part produced "out-of- specification", i.e. not within the acceptable tolerance region. Tolerancing, the method by which tolerances are assigned and their cumulative effects predicted, plays a key role in reducing rejects.
A component tolerance is "the total amount by which a specific dimension in an engineering drawing is permitted to vary [ANSI, 19821". It is the difference between the upper and lower limits of the spdcation. Tolerances reflect a designer's intentions regarding product fbnctional and behavioral requirements with corresponding implications for manufacturing and quality control. They are important not only because they spec@ product performance, but also because they have a si@cant impact on the choice of manufacturing processes, which in turn determines the final production cost. Tight tolerances can result in
excessive process cost, while loose tolerances may lead to increased rejects and assembly problems. The engineer must design high quality products and processes at low cost, by
specdjkg the allowable variation that can be tolerated without loss of component interchangeability and functionality.
The typical design problem is that the product designer has identified the allowable product performance variation and must determine the allowable component tolerances, that when combined, will result in an acceptable product.
Analysis of the effects of tolerances on product performance is diicult because it requires determining the functional relationship between the component and assembly tolerances, which is often unknown or very complicated. Thus, traditional tolerancing methods are the use of past designs, handbooks, or "rules of thumb". These methods are, often imprecise, not based on relevant data, or insuflicient to guarantee a cost effective, quality design.
Let Y, be the product or assembly response of interest which is a function of n component features Xi :
Y, = f (XI, X2, ...... 9 X) [ll
Equation [I] is known as the stackup equation I function, which may be linear or non-linear. Modern tolerancing is divided into two subareas: tolerance analysis and tolerance allocation or synthesis (Lee and Woo, 1986). Tolerance analysis means to determine the resulting assembly tolerance, Y, , when the individual component tolerances are, Xi , given; tolerance allocation means to determine the required component tolerances, X,, when the assembly tolerance, Y, , is given.
Tolerance stackup or analysis methods are worst case analyses, statistical tolerancing
(Evans, 1974; Gerth, 1992), and Monte Carlo simulation (Evans, 1975; Araj and Ermer, 1989;
Bjorke, 1989). Tolerance allocation methods include standards, uniform and proportional scaling (Chase and Greenwood, 1988), various minimum cost opGmization algorithms (Gerth,
94; Sayed and Kheir, 1985; Spotts, 1973), and Taguchi methods (Liou, et. al., 1993).
These methods have various problems including the following:
1. All current tolerancing algorithms, with the exception of Taguchi methods, require
the fimctional relationship between component and assembly features to be
known/specified.
2. Uniform and proportional scaling are iterative and do not relate tolerances to cost
or quality, and the minimum cost methods require cost information that is typically
unavailable (Gerth, 94). 3. The methods are complicated and cumbersome, and thus, typically conducted
only on "critical" dimension chains, leaving other chains unanalyzed. This can
result in excessively tight tolerance and a costly product, or loose tolerance and
fiture production problems.
4. When tolerance analysis/allocation is not conducted, it is possible to have processes
that are very capable, but are still the major sources of product variation, because
their tolerances were set too loosely (Gerth, 92).
The current thesis attempts to primarily address the first two criticisms, i.e., conducting a tolerance allocation study when the stacking function is not known, and cost tolerance information is not available.
The focus of this research is two-fold:
1. to develop a method for tolerance allocation based on design of experiments to
determine:
a) the stacking finction,
b) the sigdcant component tolerances that affect the stackup finction, and
c) the levels of the tolerances that result in the highest quality product;
2. to demonstrate the use of the method on the specific case study of a bench vice. The main advantage of this methodology is the bctional relationship need not to be known.
The next chapter will review the literature on heuristic tolerance allocation approaches, minimum cost optimization methods, and Taguchi's parameter and tolerance design. The case study bench vice will be introduced in chapter 3. The experimental design method of tolerance allocation as applied to the bench vice is described in chapter 4. The results are presented in chapter 5. The results are generalized, discussed, and conclusions regarding the method's advantages and limitations are drawn in chapter 6. CHAPTER 2
LITERATURE REVIEW
Details of past studies have been critically reviewed and are divided into three
sections. It is assumed the reader is familiar with the most common tolerance analysis
methods: worst case, statistical, and Monte Carlo tolerancing (Gerth, 1992; Bjorke, 1989).
The first section briefly explains heuristic approaches to tolerance allocation. The second
section describes minimum cost optimization methods and presents Lagrangian multiplier, linear programming, dynamic programming, and non-linear programming techniques for solving the tolerance allocation problem. The third section presents Taguchi's parameter design, tolerance design, and a new method, tolerance design using Taguchi's parameter design.
Heuristic Approaches
The important issue in tolerance allocation is to determine which combination of component tolerances is best. One method is to use non-mathematical techniques such as rules of thumb, past practice, or standards. Standards, such as IS0 Recommendation 286 and ANSI B 4.2-1978 are a set of tables through which tolerances can be identified, and hence
manufacturing processes selected (Gutrnann and Caldwell, 1987).
Standards may be used to determine the initial component tolerances, from which the
final assembly tolerance is computed by any of the tolerance analysis methods, such as worst
case tolerancing, statistical tolerancing, or Monte Carlo simulation. If the resulting assembly tolerance is too large the component tolerances may be adjusted iteratively until an acceptable
set of tolerances is found. Standards are popular because of their ease of use, and they
provide a good starting point. However, they provide no guidelines on how tolerances should
be adjusted, nor do they consider cost or quality tradeoffs between tolerances. Also, they only
spec@ sizes, no geometric tolerances.
Uniform scaling, proportional scaling, and allocation by precision factor are
the tolerance allocation methods which provide non-iterative heuristics on how to allocate the
allowable assembly tolerance among the individual components (Chase and Greenwood,
1988). Uniform scaling assigns equal tolerances to all parts. The individual component
tolerances, Toli , for linear function can be calculated as:
TOI, - CTO~, Tol; =
Where, TolA= the given assembly tolerance, Tolf, = component tolerance of supplied components (fixed),
ni = total number of components, and
nf, = number of fixed components.
Proportional scaling assumes a set of initial tolerances, Tolinitial , have been selected based on experience, process, guidelines, or standards. The individual component tolerances are then stacked according to a tolerance analysis method (worst case tolerancing, statistical tolerancing, Monte Carlo simulation) and the resulting assembly variation is compared with the specified assembly tolerance. If the calculated assembly variation is too large or too small, the individual component tolerances are scaled by a constant proportionality factor, P.
For a worst case analysis, P is given by:
and for a statistical analysis P is:
The final component tolerances, Toli are then given by:
TOli = P . The precision factor method assumes that siiarly siied parts are machined to similar precision levels. This is based on the theory that the part tolerance increases approximately with the cube root of part size:
Tali = P (Di)'" [6I where, Di = basic part size,
P = precision factor.
The precision factor, P, is calculated from the assembly tolerance, Tol~,and the part size, Di :
Finally, the component tolerances are calculated by substituting [7] into [6].
All of the above methods are simple and provide non-iterative solutions to compute component tolerances. However, their underlying assumptions ignore tolerance tradeoffs and are not based on cost or quality considerations.
Minimum cost optimization methods
In order to consider the cost tradeoff between tolerances, a number of different cost minimization algorithms to allocate component tolerances have been developed. In this type of approach, a manufacturing cost is associated with each tolerance level (cost-tolerance relationship), and the problem becomes Wing the component tolerance configuration that minimizes total cost without exceeding a specified assembly tolerance.
Subject to
(worst case)
(statistical) and Ci=f(Ti) where CT= total cost
Ci = cost of i' tolerance.
Depending on whether the stackup function is linear or non linear, and whether the cost tolerance model is discrete or continuous, Lagrange multiplier, dynamic programming, linear programming and non-linear programming have been used to solve for the optimal component tolerance set (see Table 2-1). Cost Tolerance Chain Discrete Continuous Linear Dynamic Programming Lagrange Multiplier
Linear Non-linear Programming Non-linear Programming
Table 2-1. Table of cost minimization techniques.
Laprange multiplier:
The Lagrange multiplier method provides a closed-form solution for the least-cost
component tolerance problem. The method is to take the partial derivative of the total cost function and assembly function with respect to component tolerance Ti and set it equal to zero:
~CT/dTi + h aY, /aTi = 0
Where, i = 1 ...... n,
h = Lagrange multiplier.
Let us assume a reciprocal cost fbnction as an example (Chase and Greenwood, 1988)
Ci = Ai + Bi / Ti [lo]
Where, Ai = fixed cost of i' tolerance,
Bi = proportionality constant of i' tolerance.
Equation [9] can then be written as:
O/OTi(C(Ci=A,+Bi/Ti))+ h bliiTi( CT?)=O Solving for h yields :
Expressing h in terms of TI and substituting it into [I 11, one obtains for the component tolerances:
T
Spotts (1973) applied the Lagrangian multiplier method to the minimum cost tolerance allocation problem assuming an inverse square cost tolerance function:
Ci =Ai+Bi IT:
Where, Ai = fixed cost of i' tolerance,
Bi = proportionality constant of i' tolerance.
Southerland and Roth (1975) applied the method to the general class of reciprocal powers cost functions:
Ci =Ai+Bi IT: [I51 Speckhart (1972) also used Lagrange multipliers to minimize a non-linear cost hnction subject to non-linear constraints.
Ci =A,+BieCi (Ti) where, A,, Bi , and Ci are constants.
The Lagrange multiplier method provides a closed form solution to the minimum cost tolerance allocation probkm. However, it can not treat discontinuous cost-tolerance functions, because it requires a continuous first derivative. Also, it is difEcult to apply in the case of multiple loop assemblies, where assembly functions can be described by more than one assembly function with shared dimensions. Finally, little has been done to verifjr the form of cost tolerance functions and it is, thus, dacult to generalize to a specific machine or process.
Linear Pro~rammin~:
HofEnan (1982) discussed the problem of tolerance allocation usiig linear programming (LP) techniques. LP assumes a known assembly tolerance specification and operation sequence plan.
Ostwald and Huang (1977) introduced a model of optimal tolerance selection for fbnctional dimensions assuming discrete production costs. This model was solved using linear programming with 0-1 variables. In their model, all discrete cost values, CG,weighted by binary coefficients, Xj, were summed in the system cost bction. Only one cost value for each component was permitted in a trial evaluation. A complex enumeration algorithm was used to solve the problem. The result depended on the discretization of tolerances. Although it was claimed that only a small part of the total number of combinations was required for evaluation, the 0- 1 discrete search method was found insufficient for any practical application.
Dynamic Pro~ramminp:
Dynamic programming allows one to evaluate several possible tolerance combinations based on their cost (Enrick, 1985). The method, not only considers the statistical stackup of the tolerances but also identifies the combination which results in the lowest total costs.
Dynamic programming assumes different discrete costs for each tolerance, Ti . Each tolerance and its associated cost is considered one by one in each phase of the program. Beginning with
XI , Xz , and their associated costs, all possible combinations of their stack is evaluated. Thus if two possible tolerances existed for each, there would be 4 possible combinations (branches).
Any branch that exceeds the required assembly tolerance is immediately pruned (discarded).
Also any combination that results in a looser tolerance at higher cost than another combination is also pruned. In this manner an efficient search of all combinations of tolerance cost fbnctions is possible. The limiting factors are that this method can only be applied to discrete cost tolerance relationships. It should be possible to apply it to non-linear tolerance chains, although this has not been done to date and would be an area of hreresearch.
Non- linear Pro~rammin~:
Sayed and Kheir (1985) describe a technique that assigns system element tolerances in a manner that minimizes cost. They present a continuous tolerance solution which gives an absolute minimum cost and serves as a basis for the selection of discrete tolerance values.
Similar methods have been developed by Lee and Woo (1986) for the continuous non-linear solution. They propose probabilistic optimization by associating the tolerance, Tali, with a standard deviation oi. Thus, all parameters are described by random variables and their fist and second moments:
min C,
Subject to:
where, oi>Ofori=l ...... n, with i = 1 ...... n dimension index, ci = cost of ith dimension,
6 = permissible rejection probability of the assembly,
qp,o) = multivariate probability density fbnction of mean p and variance 02,
F(x) = assembly function.
Sayed and Kheir (1985) assumed the reciprocal cost fbnction, similar to Chase and
Greenwood (1988), whereas Lee and Woo (1986) assumed the following cost fbnction:
where, ai = set up cost factor
bi = coefficient compared fiom reference cost.
Equation [18] represents the yield constraints and qp, o) represents the assembly function. The integral of the assembly probability distribution over the assembly fbnction FOC)
2 0 produces the probabiity that the assembly will meet specification. The probability must be specified to be less than 1 by selecting a value for aj, the probabiity of a permissible reject. The major difficulty lies in equation [18] which involves a multiple integral bounded by nonlinear assembly functions. This can be very time consuming to evaluate numerically and thus a fast approximation was sought. Both authors handle the problem to a "standardized coordinate system" and approximates the non-linear assembly functions F(X) 2 0 by a hyperplane. Thus a
spherical domain of acceptability is created which radius is approximated solution to the
multiple integral equation 1181. Once the non-linear assembly function has been reduced to a
linear programming problem, the cost-tolerance relationship is included to find the minimum
cost solution.
Lee et. al. (1993) developed a procedure for tolerance allocation for non-linear systems with multiple, dependent design constraints. They studied the assignment of tolerances to
dimensional mechanical assemblies as an optimization problem; the objective of which was to
minimize the (manufacturing) cost, subject to the constraints of (design) functionality and
(assembly) interchangeabity. Trigonometric functions relating to component geometries give
rise to nonlinearity in the system. Therefore, they formulated least-cost tolerance allocation
mathematically as a non-linear programming problem with mhhkhg manufacturing cost as the objective function and satismg the functional requirements and non-negativity of tolerances as the constraints:
Min C(t), subject to Y(t) 2 1 - h , and t 2 0
where, C(t) is the manufacturing cost function in terms of tolerance t,
(1 - h ) is the minimal satisfjing yield, i.e., the yield given by tolerance t should be
greater than or equal to the prespedied level 1 - h. This method works well for non-linear systems, and it allows rapid evaluation of tolerance analysis embedded in tolerance synthesis. However, care should be taken in setting up the problem formulation because no existing algorithm guarantees a globally optimal
solution unless the objective function and the constraints are of certain forms.
All of the above minimum cost optimization methods require knowing the following:
1. the hnction between component and assembly tolerance,
2. the assembly tolerance,
3. the cost-tolerance relationship,
4. tolerance analysis model.
Whether it is a linear or non-linear tolerance allocation method, it is important to have the continuous hction of the cost-tolerance relationship. This relationship is shown in Table
2-2. Usually, it is ditficult to have the functional stackup relationship (Liou, et. al., 1993), therefore, a design of experiment approach is reviewed, and presented in the next section. Model Name Cost Tolerance Method Author
Reciprocal Ci=A, +Bi/Ti Lagrange Multiplier (Chase Greenwood, 1988) Reciprocal Squared Ci = A, + Bi 1 T? Lagrange Multiplier (Spotts, 1973)
Reciprocal Power Ci = A, + Bi 1 T? Lagrange Multiplier (Southerland and Roth, 1975) C~ZB~1 T/ Nan-linear (Lee and WOO, Programming 1986); (Gerth, 1994)
Exponential Ci = A, exp[ Bi Ti] Lagrange Multiplier (Speckhart, 1972)
Empirical data Discrete points Linear (Ostwald and programming Huang, 1977)
Where, Ci = Cost of i' component tolerance A, = Set up cost factor Bi = Coefficient compared fiom reference cost Ti = individual component tolerances K = an exponent
Table 2-2. Table of Proposed Cost Tolerance Relationships
Relatively few studies have been conducted in the past to allocate tolerances using designed of experiments. Design of experiments is a discipline that enables one to systematically vary a number of independent (input) variables to evaluate their effsct on a number of dependent (output) variables or responses. As the variables are changed in the experimental process, their relationships, effects, and interactions are measured, analyzed and mapped.
Taguchi introduced a method that provides a simple way to design efficient and cost effective experiments (Taguchi and Wu,1979; Taguchi, 1986; Ross, 1988). Taguchi's method has become very popular and is widely used to optimize industrial designs and processes. The method identifies those factors (independent variables) that have a significant effect on the performance (dependent) variable by using designed experiments.
Taguchi's philosophy is based on the loss function concept. "The quality of a product is the (minimum) loss imparted by the product to society fiom the time the product is shipped"
(Elyrne and Taguchi, 1987). The idea is that loss occurs not only when the product falls outside the specifications, but also that the loss continually increases as the part deviates Mher fiom its nominal dimension (target value). Taguchi uses a simple quadratic hnction to approximate the behavior of the loss.
His quadratic loss function can be represented as follows:
L=K(X-T)~ where, L = Loss in dollars
K= Cost coefficient X= Value of quality characteristic
T= Target value (m)
Thus, the minimum cost product is the one that produces the minimum variation around the nominal target. This concept is also applied to this research. The question becomes, how to design a minimum variance product.
Taguchi views the design of a product or process as a three phase program (Byrne and
Taguchi, 1987):
1. System design
2. Parameter design
3. Tolerance design
System design is the phase when new concepts, ideas, and methods are generated to provide new or improved products to customers.
The parameter design phase is crucial to improving the uniformity of a product and can be done at no cost or even at a savings. This means certain parameters of a product or process design are set to make the performance less sensitive to causes of variation.
The tolerance design phase improves quality. Quality is improved by tightening tolerances on product or process parameters to reduce the performance variation. This is done only after parameter design. The research project assumes that the system and parameter design have already been properly conducted. However, siithe proposed tolerancing method involves parameter
design concepts, a brief description of traditional parameter design will be provided followed by a discussion of Taguchi's tolerance design and why it is not a preferred method.
-chi's Parameter Desim Paradim:
Taguchi's Parameter design evaluates alternative nominal values for selected control variables by statistical experimental design to determine the best combination of values that will
result in a product that is least sensitive to noise factors. This is called robust design and is the
key to achieving high quality without increasing product cost (Byrne and Taguchi, 1987).
The key to achieving robustness against noise is to discover and utilize interactions between controllable factors and uncontrollable (noise) factors. Thus, the strategy in parameter
design is to separate controllable versus noise (uncontrollable) factors into an inner and outer
array and to study their effect in a statistical experiment. The most important. purpose of the
outer (noise) array is to deliberately create noise during the experiment so that the controllable
factor levels which are least sensitive to noise can be identified. Dr. T;aguchi adopted
orthogonal arrays, or fdly saturated fractional factorial designs, to siipl@ the experimental
design procedure. The signal to noise ratio is used as a data transformation, and the equation for calculating the S/N ratios are based on the characteristics of the response variables being evaluated.
Taguchi's Tolerance Desien Paradim:
The objective of tolerance design is to improve quality by reducing variation in response. It is only used when efforts of parameter design have not proved adequate in reducing variation. Tighter tolerances are then required of those factors identified through parameter design to reduce performance variation. Tolerance design typically means "spending money", buying better grade materials, components, or machinery to control the range of design parameters (Byrne and Taguchi, 1987).
Taguchi's tolerance design uses the loss function concept and assumes that components are statistically independent and normally distributed. The first step of tolerance design is to determine the contribution of each noise factor to the quality loss (Phadke, 1989). In order to improve the joint economics of product cost and quality loss, one should consider the following two issues:
1. Ways of reducing the variation of the noise factors that contribute a large amount to the
quality loss
2. Ways of saving cost by allowing wider variation for the noise factors that contribute only a
small amount to the quality loss. DErrico and Zaino ( 1988) presented a modification of Taguchi's tolerance design paradigm where they assume a 3 point distribution that provides a closer approximation to the normal. They recommended a three level factorial experiment with ( pi - a (3)1°, pi , pi + oi (3)" ) as the factor levels.
Taguchi's tolerance design paradigm is better than traditional statistical tolerancing methods (DErrico and Zaino, 1988) because:
1. The method is easy to use and can be easily described to scientists and engineers.
2. The method does not require the assembly hnction to be expressed in analytical form.
3. The method requires little computation as compared to Monte Carlo simulation, and the
Taylor-series expansion method.
4. This method is comparable in accuracy to Taylor Series expansion.
However, this method has the following disadvantages:
1. When the number of components is large it requires an excessive number of experiments.
For example, to determine 12 component tolerances would require 256 experiments.
2. It is diflicult to handle nonnormal distributions.
3. The loss fbnction is diflicult to quantify. Tolerance Design using Tamchi's Parameter Design:
A new application of Taguchi's parameter design technique is utilizing the inner and outer orthogonal array concept to the tolerance specification of robot kinematic parameters (Liou et al., 1993). In order to illustrate the methodology, they use two different types of robot manipulators, a two link planar manipulator and a five-degree-of-freedom Rhino robot. Initially, they assigned a reasonable tolerance range for each joint using traditional past design practices (heuristic methods). However, they believed that there were tolerance tradeoffs between the manipulator joints, which if explored, would result in more consistent placement of the robot's end effector. The advantage of larger tolerances at the joints would be that less expensive servo controllers would be required. They used the inner orthogonal array to study speciiic alternative tolerance ranges (loose, tight), and the outer orthogonal array to provide the noise to each control factor, namely the direction that tolerance value deviates fiom a nominal position (+, -). They then measured the deviation of the end effector position from nominal, and analyzed the results using Taguchi's S/N ratio and analysis of variance (ANOVA) technique. They compared the experimental results with results from a
Monte Carlo (MC) simulation and found that both methods reveal the same trends for performance improvement. Their analysis indicated that tightening the tolerance of one factor was the most cost effective way to improve the end effector's performance and to reduce the overall cost. They concluded that parameter design was more computationally efficient than
MC methods and an effective method of tolerance allocation. Application of Taguchi's parameter design to the tolerance allocation problem appears promising. Its major advantage is that the fbnctional stackup relationship between the performance parameters and the component factors need not be known. Another advantage is that it need not rely on a loss function, which cost coefficient is often difEcult to quantq. This thesis explores the application of this approach to a hypothetical design situation of a bench vice to determine it's suitabiity for tolerance allocation compared to minimum cost approaches. CHAPTER 3
CASE STUDY - A BENCH VICE
The case study is introduced at this time to provide the reader with a concrete example that will aid in the understanding of the parameter design approach to tolerance allocation. The case study is a non fixing bench vice made fiom plastic as shown in Figure 3-1. The vice is made of six components, namely, a fiont plate, a movable jaw, an end plate, two rods, and a screw rod. Each component is manufactured separately and then assembled. Each component feature has a nominal dimension and a tolerance specified by the designer.
One of the functional characteristics of the bench vice is that the end plate and the movable jaw be parallel. This means that there should be no gap between the two plates when the vice is filly closed. But due to the variation in the dimension of the parts and overall stackup tolerance of the assembly, the two plates may not be perfectly parallel.
The goal of this thesis is to apply the parameter design concept to determine the set of component feature tolerances that will result in the lowest cost product subject to quality constraints on the product's function. Lowest cost means the largest tolerances and quality constraints on the product function means a minimum gap between the jaw and the end plate when the jaw is closed. The methodology was evaluated by "building" and "assembling" the required bench vices. Since actual construction was not feasible, a simulation program based on a 2-D mathematical model of the manufacturing and assembly processes was written.
A parametric model of the bench vice was developed (see Figure 3- 1). Points P 1 and
P3 are the end points of the end plate. Similarly, points P2 and P4 are the end points of the movable jaw. Point P5 is the center of the rod hole in the end plate and point P6 is the center
of the rod hole in the fiont plate. Points P7, P8, P9, and PI0 are at the outer diameter of the
rod where it fits into the movable jaw. The complete Geometric Dimensioning & Tolerancing
(GD&T) component drawings are given in Appendix A
In the mathematical model the following assumptions are made:
1. The analysis is restricted to 2 dimensions, i.e., there is no variation along the x-axis
(into the page) for any feature (see Figure 3-1).
2. As shown in Figure 3-1, it is assumed that point P2 is very close to point P1 and
the dierence (A) between point P2 and P 1 along the Y-axis is negligible. This
assumption is justifled because the dierence (A) is very small compared to the
length of the movable jaw.
3. The rod is straight, i.e. not curved or eccentric, although assembly variation can
cause it to be at an angle with respect to the z-axis. BENCH VICE ASSEMBLY
Y A
p3 p4 -. ----- n------P8 --- Screw Rod I :P6 I ------I P 10 Rod
____C z Movable Jaw End Plate Front Plate
Figure 3-1. A detailed drawing of the bench vice assembly 4. The center of the rod was assembled to the center of its mating holes at the front and
end plates, i.e. no clearance.
5. All part surfaces are flat, but not necessarily parallel.
Appendix B (Figure B1) shows a detailed analysis of the movable jaw and rod position. Appendix B (Figure B2) shows the complete analysis for the case of Pl = P2. For illustration purposes we consider the case of P1 = P2 and a complete mathematical formula is developed for this case. Appendix C shows the list of formulae and calculations. CHAPTER 4
EXPERIMENTAL DESIGN SETUP AND
PROCEDURE
The tolerance allocation methodology involving Taguchi's parameter design was evaluated on the case study of bench vice. Details pertaining to the experiment are presented below.
Inner and Outer Arravs :
In this experiment, the concept of inner and outer orthogonal may (OA) is used. The inner OA is used to study the effect of the controllable factors and the outer OA provides noise to each control factor. The objective of the inner OA is to determine the significance of the control factors, and to select the levels of the significant factors that optimize the performance measure. Optimal performance is one that exhibits minimum variation around a target value.
The variation is considered to be due to noise factors. The outer OA introduces noise factors in the experiment in a systematic manner. Thus, it is possible to analyze and select the proper levels of the control factors that are least sensitive to noise and will result in the minimum variance response. Each outer OA noise combination is treated as a replication of the inner OA. Thus, if an outer OA requires Lo runs for each run of an inner OA that require L, runs, the size of the experiment is L, * Lo runs. The inner array specifies the combination of tolerances ( loose =
L1, tight = L2 ) used for each control variable ( component feature ). The outer array specifies the level of noise combinations, i.e., the direction ( + = N1, - = N2 ) that the tolerances deviate fiom their nominal direction. Togethers, they define a unique product which component features are at the extremes of their tolerances. The structure of the inner and outer orthogonal array is shown in Table 4-1.
Desimin~an ex~erimentusing ortho~onalarrays:
When designing an experiment, the factors, relevant interactions, and the factor levels need to be determined. In the Taguchi method, two levels (minimum and maximum) are usually recommended, but three levels (minimum, medium and maximum) may also be used.
Interaction effkcts between factors are usually assumed to be negligible compared to main effects, but can be investigated. While using the Taguchi method, the experiment is designed by following the column assignments specified by an orthogonal array (OA). The orthogonal design employed is based on the number of factors, their levels and the number of selected interactions. The most common OKs involve two level factors, and are the Ld ( 23), Lg ( 2'1,
L16 ( 215),and L32 ( 231)orthogonal arrays. In an orthogonal array designated as La( bc ), the AN L16 INNER 1 OUTER ORWGONAL ARRAY
OUTER ARRAY
Table 4- 1. An L 16 Innerrouter orthogonal array structure letters a, b, and c represent the number of runs, the number of levels for each factor, and the number of columns in the array respectively. After an orthogonal array is selected, designing
an experiment becomes a "column assignment" task.
Experimental Variable and Level Selection:
Response Variable ( Dependent Variables ) :
The response variable is the maximum gap between the end plate and movable jaw in the closed position measured at the corners of the plates.
Factors ( Independent Variables ) :
Twelve tolerances (control factors) can affect the gap between the end plate and the movable jaw. Two tolerance levels were selected for each control factor: L1 (Loose tolerance) and L2 (Tight tolerance). A linear model was assumed, and interactions were assumed to be neghgible. The specific factor levels were selected based on standards and past experience with machining processes and their capabiities (see Table 4-2). The factor levels were generally selected to be far apart so that the factor effect would large relative to the mean squared error.
However, certain tolerances, namely, B, D, H, and K had very little difference between their loose and their tight tolerance. Thus, their perceived effect may appear to be relatively small.
(see Chapter 5). Noise Variables:
The noise factor for each tolerance is the diiection, + or - , the feature deviates from its nominal.
Table 4-2 summarizes the variables and their levels used in the experiment.
I. Response Variables (1) 1. Gap between end plate and movable jaw
II. Control Factors (12) --Levels L 1 A : Tolerance of fiont plate hole position 1 0.0 1 B : Tolerance of fiont plate hole diameter 1 0.0005 C : Tolerance of end plate hole position 1 0.01 D : Tolerance of end plate hole diameter 1 0.0005 E : Tolerance of end plate P 1 plate depth (thickness) 0.02 F : Tolerance of end plate P3 plate depth (thickness) 0.02 G : Tolerance of movable jaw hole position 0.01 H: Tolerance of movable jaw hole diameter 0.0007 I : Tolerance of movable jaw plate width 0.02 J : Tolerance of movable jaw plate depth (thickness) 0.02 K : Tolerance of rod diameter 0.0005 L : Tolerance of rod length 0.01
III. Noise Factors (12) Every control factorltolerance Positive Negative
Table 4-2. Variables for the experiment Selection of Orthogonal Arravs and Assimment of Factors:
The selection of OA depends on the following :
1. The number of factors and interactions of interest
2. The number of levels for each factor of interest
These two items determine the total degrees of fieedom required for the entire experiment. The degrees of fieedom for each factor is the number of levels minus one. The degrees of fieedom for the factors under investigation, v, assuming no interactions, is given as
v = 12 *(2-1) = 12
Thus, an OA is required that will accommodate v the total number of degrees offieedom. The total degrees of fieedom available in an 04 VLN , is equal to the number of trial minus one:
v, = N-1
In order to select the particular orthogonal array for an experiment the following inequality must be satisfied.
VIJJ 2v
Since we have 12 control factors, an L 16 OA with 15 degrees of fieedom is required.
The standard an L16 orthogonal array is shown in Table 4-3. Table 4-3. An L 16 orthogonal array
After selecting the appropriate OA, the factors and interactions can be assigned to the appropriate columns.
conduct in^ the experiment :
To conduct the experimental investigation and simulate the various bench vices, a simulation program was developed in Microsoft Excel version 4.0 (Mjcrosoft, 1992) based on a geometrical model of the vice to measure the gap between the end plate and movable jaw. An L16 orthogonal array was used for both the inner and outer array resulting in 256
experiments.
Usually complete randomization should be used during an experiment to prevent the influence of unknown and uncontrolled factors (Mead, 1988; Baker, 1990). However, sii the experiment was conducted as a computer simulation, all variables were known and
controlled, and thus, randomization was not necessary.
A macro program in Microsoft Excel Version 4.0 (Microsoft, 1992) was developed to
collect and analyze the data automatically. The detailed listing of the simulation program, formulae, and macro program is in Appendix E, K, L, respectively. CHAPTER 5
ANALYSIS AND RESULTS
In the Taguchi method, the signal to noise ratio (SM) is used as the data transformation method that consolidates the data for each control array row over the various noise levels into one value which computes both the mean and the variation present in the data.
The equations for calculating the signal to noise ratios are based on the characteristics of the response variables being evaluated; nominal the better, smaller the better, and larger the better.
In the bench vice case, the response variable, i.e., the gap between the end plate and the movable jaw is a smaller the better characteristic, and the S/N equation for this characteristic is given as:
where, n = number of repetitions
yi = individual data.
The S/N ratio for the smaller the better characteristic is essentially a measure of both the mean value (signal) and the standard deviation (noise) of the response (Montgomery,
1991). Because of the negative sign in [22], higher S/N ratios are indicative of experimental conditions that will be more robust i.e., less sensitive to variation in the noise variable
(Clausing, 1988). Thus, it can be expected that statistically sigdicant results found in the analysis using S/N ratio will be the "best set" of tolerances to produce the minimum gap between the end plate and movable jaw regardless of the other tolerance levels and their direction.
The data is presented in Appendix F. The following analysis were performed: the mean response analysis and the S/N response analysis. The mean response analysis shows which factors have the greatest impact on the mean of the response distribution. The S/N analysis show which factors have the greatest impact on the variance and mean of the response distribution.
The mean response analvsis :
The following steps are used to conduct the mean response analysis ( Peace, 1993):
1. Determine the mean for each row (experimental run). The mean for each row is shown
in Table 5-1.
2. Calculate the mean response for each control factor and develop a mean response table.
This is performed by grouping the mean responses by factor level for each column in the
array, taking the sum, and dividing by the number of responses. The absolute difference or delta between the two results (two levels) is the effect of the factor. For factor A, the calculations are as follows:
A, = ( 0.03068 + 0.03039 + ...... + 0.02325 ) / 8 = 0.0214662 inches
A, = ( 0.01999 + 0.01999 + ...... + 0.02539 ) / 8 = 0.0225062 inches
Average Gap Run No. (inches) SIN 1 0.03068 28.78188 2 0.03039 28.86940 3 0.01 198 37.04638 4 0.01 138 37.39182 5 0.02037 32.43002 6 0.02012 32.54097 7 0.02356 30.45576 8 0.02325 30.57666 9 0.01999 32.60752 10 0.01999 32.72910 11 0.02353 30.57395 12 0.02323 30.70830 13 0.02141 33.00008 14 0.02083 33.22325 15 0.02568 30.12808 16 0.02539 30.19306 0.35178 5 11.25623 C Total
Table 5- 1. Response Table The results for the other columns, is presented in Table 5-2.
Level 1 Level 2 Delta
Table 5-2. Mean Response Table
3. Construct a mean response graph for each factor based on the mean response table (see
Figure 5-1).
4. Analyze the mean response table and mean response graphs. From Table 5-2 and Figure
5-1, it can be seen that factor F (End plate P3 plate depth), Factor E (End plate P1 plate
depth), and Factor G ( movable jaw hole position 1) have the greatest effect on the gap between the end plate and movable jaw, followed by factors D (End plate hole diameter
l), B (Front plate hole diameter I), and A (Front plate hole position 1). The remaining
factor effects is negligible by comparison.
5. Determine the statistical signdicance of the factors with an ANOVA (see Table 5-4). The
conclusions fiom the graphical analysis are vesed by the mean response ANOVA.
Since our quality characteristic is smaller-the-better, we want to choose the tolerance levels fiom the response table (or response graphs) that result in smaller average response values. The recommended levels are E2 , F2, and G2. Since the effect of the other factors is small, their levels can be selected based on other considerations, such as cost. In this case we will assume that larger tolerances result in lower cost, and thus the other factors should all be set at their loose tolerance level 1.
The SIN resDonse analvsis :
The same steps are used to conduct the S/N response analysis.
1. Determine the S/N ratio for smaller-the-better case for each row (experimental run).
2. Calculate the SM response for each control factor and develop a response table. For factor
A, the calculations are as follows: Figure 5- 1. Mean Response Graph I MEAN RESPONSE ANOVA TABLE 1
Source of Variation SS dof MS F-Value P- Value Front Plate hole position 1 (A) 0.00000433 1 0.00000433 229.37 6.25E-04 Front Plate hole diameter 1 (B) 0.00000556 1 0.00000556 294.98 4.30E-04 End Plate hole position 1 (C) 0.00000086 1 0.00000086 45.35 6.69E-03 End Plate hole diameter 1 (D) 0.00001557 1 0.00001557 825.51 9.26E-05 End Plate P 1 plate depth (E) 0.00013801 1 0.00013801 73 16.95 3.52E-06 End Plate P3 plate depth (F) 0.000 13274 1 0.000 13274 7037.43 3.73E-06 Movable jaw hole position 1 (G) 0.0001 1008 1 0.000 1 1008 5836.26 4.94E-06 Movable jaw hole diameter 1 (H) 0.00000043 1 0.00000043 22.55 1.77E-02 Movable jaw plate width (I) 0.00000000 1 0.00000000 0.25 6.5 1E-01 Movable jaw plate depth (J) 0.00000000 1 0.00000000 0.22 6.72E-0 1 Rod Diameter Tolerance (K) 0.00000005 1 0.00000005 2.69 1.99E-0 1 Rod length Tolerance (L) 0.00000001 1 0.00000001 0.49 5.35E-01 Error 0.00000006 3 0.00000002 Total 0.00040770 15
Table 5-4. Mean response ANOVA table Effect ofA= I A, - A, 1 = 132.2616- 31.6454 I = 0.6162 db
The S/N response for each row is shown in Table 5-3.
Factors Level 1 Level 2 Delta
A 32.262 3 1.645 0.617 B 32.338 3 1.568 0.770 C 32.329 31.577 0.752 D 3 1.773 32.134 0.361
E 30.528 33.378 2.850
F 30.542 33.364 2.822
G 30.773 33.133 2.360 H 3 1.878 32.029 0.151 I 3 1.945 3 1.961 0.016 J 3 1.943 31.964 0.021 K 3 1.938 3 1.968 0.030 L 31.961 31.945 0.016
Table 5-3. S/N Response Table Figure 5-2. S/N ratio Response Graph
3. Construct S/N response computed for each factor, S/N ratio response graphs are
constructed for each factor as shown in Figure 5-2.
4. Analyze the S/N response table and response graph. From Table 5-3 and Figure 5-2, it can
be seen that the same factors that had the greatest effect on the mean response also have
the greatest effect on the variance of the gap, namely, E, F, G followed by B, C, A with the
others having a neghgible effect.
5. Determine the statistical si@cance of the factors with an ANOVA (see Table 5-5). The
conclusions from the graphical analysis are strongly supported by the S/N ratio ANOVA.
Since our quality characteristics is smaller-the-better, we want to choose the level fiom the response table (or response graphs) that shows smaller average response value. From the above analysis, the recommended levels are selected as E2 ,F2, and Gz, which are the same levels recommended by the mean analysis.
Confirmation Ex~erirnent:
To veethe experimental conclusion obtained above, a confirmation experiment was conducted. During experiment it was assumed that the interaction effect will be negligible because the response was hear over the narrow range of tolerances. The confirmation experiment is the final step to ver@ the interaction assumption. Sixteen bench vices were simulated on Microsoft Excel version 4.0 (Microsoft, 1992) spreadsheet under the recommended conditions A1 ,B1, C1, Dl, E2, F2, G2, HI , I1 , J1, Kl , and Li .
The predicted average gap between the end plate and movable jaw when the factors are at their recommended levels is calculated as follows: (Ross, 1988)
YE= + E~ + 6,- 2y 1231
= 0.0191 + 0.0195 + 0.01936 - 2 * 0.022
= 0.014 inches
The 99 % confidence interval for the confirmation experiment of 16 runs at the recommended levels is given by
where F0.01,1,12 = 9.33 and V, is the pooled error, i.e., qa = 0.014 * 1.281 * 1p inch 0.0127 inch < Ym < 0.0153inch
The predicted SM value can be calculated as - YE,,* = E, + F, + 6, - 2 T
= 33.378 + 33.364 + 33.133 - 2 * 31.954
= 35.967 db
From the predicted SM value, the corresponding Y value can be calculated as
YE2nn = 0.016 inches
The 99 % contidence interval for the confirmation experiment of 16 runs at the recommended levels is given by
where F0.01,1,12 = 9.33 and V, is the pooled error, i.e., The confirmation experiment result are shown in Table 5-6. Run No. Minimum gap at closure position 1 0.01243 2 0.01031 3 0.01243 4 0.01031 5 0.00104 6 0.00275 7 0.02596 8 0.02072 9 0.013 18 10 0.01 152 11 0.01318 12 0.01 152 13 0.021 89 14 0.001 88 15 0.02447 16 0.02114 Average 0.01342 S/N 36.0699
Table 5-6. Confirmation experiment results
The average and S/N ratio of the confirmation experiment results are 0.0134 inch and
36.069 db. It can be seen that both are well within their predicted values of the mean response and S/N ratio analysis. The result of the experiment and analysis can be sumas follows:
1. The three si@cant factors that influence the gap between the end plate and
movable jaw at closure position are E (End plate P1 plate depth), F (End plate P3 plate
depth) and G (Movable jaw hole position 1). The recommended levels for these three
factors are Ez, Fz, and GZ. Therefore, the sigtllficant factors E, F, and G should be set
at the tighter tolerance levels and the insigtllficant factors should be set at loose
tolerance levels as their tolerance at the chosen level does not have any significant
effect on the gap at the closure position.
2. The interaction between the control factors are not sigtllficant.
3. The interaction between the noise factor (direction of tolerance) and control factors
are not signrficant.
4. The two analysis results are compatible.
5. The confirmation experimental results under the recommended conditions Ez, Fz, and
Gz are well within the intervals for the confirmation experiment.
Geometrical Interpretation :
The above analysis indicates that tightening the tolerance of factor E (End plate P1 depth), F (End plate P3 plate depth) and G (Movable jaw hole position 1) is the most effective way to minimize the gap between the end plate and movable jaw at closure position. If necessary and feasible, the tolerances of factor D (End plate hole diameter I), B (Front plate hole diameter I), and A (Front plate hole position 1) can also be set at a tighter level to further reduce the gap between the end plate and movable jaw. Of equal importance, tolerances of
insigdcant factors such as C, H, I, J, K, and L can be widened to reduce manufacturing cost without significantly affecting the gap.
Examination of Figure 3-1 shows that the above results makes physical sense and can be further elaborated through the use of a gain or sensitivity matrix. The statistical tolerancing equation for non-linear systems is given by
2 l2 2. 0 gap =xg igap 42 1=1
where, g, is the gain of the i' tolerance to the gap, i = 1, ...... 12.
From Figure 3-1 it can be seen that the points PI, P3, P2, and P4 directly affect the gap, since they define the parallelness of the mating surfaces. Points P1 and P3 have a greater effect than P2 and P4 as the clearance between the rod and the hole in the movable jaw, reduce the effective gain in the slope of the P2 - P4 surface. Specifkally, the true position of the movable jaw has a greater impact on the gap because it effectively moves the plate along the
Y-axis increasing and decreasing the gap size when the jaw is skewed. This effect is magnified by the fact that the hole is off-center creating a longer lever arm (P7 - P4 versus P9 - P2).
Because the movable jaw surface is shorter than the length of the rod, the jaw's skewness has a greater impact than the slope of the rod (P5 - P6). Thus, the physical interpretation of the problem confirm the results obtained from mean response analysis and S/N ratio response analysis. CHAPTER 6
DISCUSSIONS AND CONCLUSIONS
A new method has been demonstrated for tolerance allocation. The method is based on Taguchi's parameter design. It is a new application of Taguchi's parameter design to tolerance design where control factors are the tolerance levels (loose and tight tolerance) and the noise factors are the direction of tolerances (positive and negative). The concept of employing inner and outer orthogonal arrays to identifjl the significant tolerances and select the optimal levels was successfdly demonstrated on a bench vice case study.
From the experiment and the data analysis result, the following conclusions can be drawn:
1. As with other systems, the Pareto principles applies: there are a few tolerances that
must be held tightly to minimize the variation in the performance parameters. The
other tolerances may be wide to reduce the manufacturing cost. This conclusion is
consistent with other tolerance allocation studies (Gerth, 94).
2. The linear model assuming negligible interactions was appropriate. This is not
surprising, given that the response varies over a tolerance range and not a choice
between nominal values which is typically much wider than the tolerances. The major advantage of using Taguchi's parameter design method for tolerance allocation is that the actual stackup hnction need not be known. This is very important because many systems are so complex that the function can not be easily determined.
For example, in a direct current motor the exact relationship between the angular deviation of the commutator and the current draw and magnetic field strength is not precisely known. All other tolerancing methodologies assume that the relationship is known.
Furthermore, the method identifies the high impact tolerances. Thus, the engineer will know where to focus engineering resources to either a) change the design to reduce the gain, or b) develop more capable processing technology to reduce the variation.
The method is easy to use and can be easily described to scientists and engineers who are familiar with Taguchi's parameter design methodology for selecting optimal nominal values. The method is comparable in accuracy to statistical tolerancing which uses a first order Taylor-Series expansion, since Taguchi's method also assumes a linear response finction based on main effects.
The major disadvantage is that the method requires an inordinate number of experiments. For example, to determine 15 tolerances requires 256 experiments, and to determine 3 1 tolerances requires 1024 experiments. Gerth (1992) analyzed a complex system using Monte Carlo analysis that contained 160 toleranqes. Systems of such complexity could not be reasonably handled by this method.
In conclusion, if only a few tolerance are to be studied, the logistics of conducting the experiment are reasonable, and the tolerance stack is too complex to be described deterministically, then this method can be a very powerfbl tolerance allocation procedure.
BIBLIOGRAPHY
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List of figures of bench Vice component and their Geometric Dimensioning & Tolerancing drawing
FRONT PLATE
.50 X 13 UNC -2B
2x234 -$l+.ol (M) 141~IC -D- I
+ .ol (s) IA IC(S)IBI I+) ( ,002 (s) ]A]
INDUSTRIAL & SYSTEMS ENGINEERING OHIO UNIVERSITY + .03 < +1Deg .X .XX +.01 .XXX ,.DO5 ALL DIMENSIONS ARE IN INCHFS J
Figure A-1 . Bench vice details of Front Plate. MOVABLE JAW
INDUSTRIAL L SYSTEMS ENGINEERING OHIO UNIVERSITY X -+ .03 ALL DIMENSIONS ARE IN INCHES Figure A-2. Bench vice details of Movable Jaw. END PLATE .$ 4.01 M A BC INDUSTRIAL & SYSTEMS ENGINEERING OHIO UNIVERSrrY .x $ .03 <+ldeg .XX $ .O1 .XXX + .005 ALL DIMENSIONS ARE IN INCHES Figure A-3. Bench vice details of End Plate. RODS 2 X .03 X 45 deg .------:-?. ------1---,--.' 2 X I$ 1.49 ( No. 25) .I90 X 24 UNC -2B INDUSTRIAL & SYSTEMS ENGINEERING OH10 UNIVERSITY .X f .03 .XX f .01 < * 1 deg .XXX f ALL DIMENSIONS ARE IN INCHES Figure A-4. Bench vice details of Rod. SCREW t$ L.290 I .025 5.055 INDUSTRIAL & SYSTEMS ENGINEERING OHIO UNTVERSIlY .X f .03 < f 1 deg .XX f .Ol .xxx+ .oos ALL DIMENSIONS ARE IN INCHES Figure A-5. Bench vice details of Screw. APPENDIX - B List of figures showing analysis of movable jaw and rod Figure B-1. Details of Movable Jaw and Rod. Y IF P1= P2 A b56' amount hdes are - mj(h) - Z WZ - P2Z Figure B-2. Drawing of Movable Jaw and Rod showing analysis of pl = p2 case. APPENDIX - C LIST OF FORMULAE AND CALCULATIONS From Appendix B, the following mathematical formulae can be derived to calculate the gap between the end plate and the movable jaw. In order to compute the gap there are two cases: 1. If P1 = P2 at the closure position and we need to calculate the gap between the points P3 and P4. 2. If P3 = P4 at the closure position and we need to calculate the gap between the points P 1 and P2. CASE I : LFP1 - p2 From Figure B2, the following geometrical relationship can be derived: Psy - PSy Slope of the rod, m56 = P& - Ps* Slope of the movable jaw formula is given as follows: mj=tan(90~+P) where, a = tan" (msa) = tan'' (mj(h) 1 mj (w)) where, mj (w) = Movable jaw plate depth at nominal + variable tolerance on plate depth at Z-axis, and mj(h) movable jaw amount of holes P7y - P2y = 11 sin ( tan-'(mj)) 111 Where, 11 = movable jaw hole position 1 nominal on Y-axis + (movable jaw hole position 1 variable tolerance on Y-axis) I2 + (movable jaw hole diameter 1 variable tolerance on Z-axis) 12 where, b5; is the intercept of the end side of the rod and is given as where, D is the movable jaw hole diameter, and bs6 = P5y - m56 P5z P~Z- P~Z = 12 cos ( tan-'(mj)) [41 Where, l2 = (movable jaw plate depth on Y-axis - the difference in height between movable jaw and end plate on Y -axis + movable jaw plate depth variable tolerance on Y-axis) - 11 Gap 1 = P4z - P3z CALCULATIONS: Center of the rod hole in the end plate on the Z-axis: P5z = End plate hole position 1 at nominal position + variable tolerance on the end plate hole position 1 on the Z- axis P5z = 0.300 + 0.00 = 0.300 inches Center of the rod hole in the end plate on the Y-axis: P5y = End plate hole position 1 at nominal position + variable tolerance on the end plate hole position 1 on the Y- axis P5y = 0.750 + 0.01 = 0.760 inches Center of the rod hole in the front plate on the Z-axis: P6z = P5z + rod length at nominal + variable tolerance of rod length on Z-axis P6z = 0.300 + 3.525 + 0.01 = 3.835 inches Center of the rod hole in the front plate on the Y-axis: P6y = Front plate hole position 1 at nominal + variable tolerance at hole position 1 of front plate on Y-axis P6y = 0.75 + (-0.005) = 0.745 inches Now, slope of the rod can be calculated as follows: Pw-P~Y= 0.745 0.760 Slope of the rod, m5, = - - Psr 3.835 - 0.300 Slope of the movable jaw can be calculated as follows: mj=tan(90-a+P) where, a = tan-' (m4 = tan" (-4.2432* 10") = -0.24312 P = tan-' (mj(h) I mj (w)) where, mj (w) = Movable jaw plate depth at nominal + variable tolerance on plate depth at Z-axis = 0.50 + 0.02 = 0.52 inches mj(h) movable jaw amount of holes and can be calculated as follows: If movable jaw hole position 1 variable tolerance on Y-axis < 0 , it is ((ABS(Movab1e jaw hole position 1 variable tolerance on Y-axis)) + (ABS(Movab1e jaw hole diameter 1 variable tolerance on Z-axis)) + ( Movable jaw hole diameter 1 variable tolerance on Z-axis)) 1 2 If movable jaw hole position 1 variable tolerance on Y-axis > 0 , it is given as {- (Movable jaw hole position 1 variable tolerance on Y-axis)) - (ABS(Movab1e jaw hole diameter 1 variable tolerance on Z-axis)) - ( Movable jaw hole diameter 1 variable tolerance on Z-axis)) / 2 In this case, it is -0.005 which means < 0 mj(h) = { ABS(-0.005) +ABS(-0.0004) + (-0.0004))/2 = (0.005+0.0004 - 0.0004) 12 mj(h) = 2.5 *10" p = tan-' (2.5* 10-~/0.52)= 0.275458 Finally, jaw slope can be calculated as follows: m, = tan ( 90 -(-0.243 12) + 0.275458) = tan ( 90.5 18578) mj = -1 10.483 degree The length of the movable jaw from P2 to P7 point is defined as 11 and can be calculated as: ll = movable jaw hole position 1 nominal on Y-axis + (movable jaw hole position 1 variable tolerance on Y-axis) I2 + (movable jaw hole diameter 1 variable tolerance on Z- axis) 12 1, = 0.5 + (-0.00512) + (0.626+(-0.0004))/2 = 0.8103 inches From geometric relationship equation [I], we know that P7y - P2y = l1 sin ( tme'(mj)) P7y - P2y = 0.8 103 sin ( tan"(- 110.483)) P7y - P2y = -0.81027 From geometric relationship equation [2], we also know that P~z- P2Z = 1' cos ( taamj)) = 0.8103 cos (tan-'(-1 10.483) P~Z- P~Z = 7.333 * 10" Now, Assuming P lz = P2 z Which means, P 1& P2 points both are at the same place on Z- axis Plz = End plate P1 plate depth at nominal + End plate P1 plate depth variable tolerance on Z- axis = 0.5 + 0.02 = 0.52 inches which means, P2z = 0.52 inches Now, P7z can be calculated as follows: IF ( movable jaw slope > 0 ) P7z = P2z + 7.333*1u3 IF ( movable jaw slope < 0 ) P7z = P2z - 7.333*104 In this case movable jaw slope < 0, so P7z = P2z - 7.333 * 10" = 0.5 1267 inches From geometric relationship equation [3], we know that P7y = m56 P7z + b5s' where, b56' is the intercept of the end side of the rod and is given as where, D is the movable jaw hole diameter = 0.626, and bS6 = P5y - m56 P~z = 0.760 - (- 4.2432 *lo4) * 0.300 = 0.761273 From equation [3], P7y = (-4.2432 * 10") 0.51267 + 1.0742758 = 1.0721 inches From equation [I] P2y =P7y+ 0.81027 = 1.0721 + 0.81027 = 1.88237 inches Let, P3z = End plate P3 plate depth at nominal + End plate P3 plate depth variable tolerance at Z -axis = 0.5 + 0.02 = 0.52 inches Now, the distance between P7 and P4 is given as 12 = (movable jaw plate depth on Y-axis - the difference in height between movable jaw and end plate on Y -axis + movable jaw plate depth variable tolerance on Y-axis) - l1 = 2.25 - 0.25 + 0.02 - 0.8103 = 1.2097 From geometric relationship equation [4], we know that P4z - P7z = 12 cos ( tan-'(mj)) now, P4z - P7z can be calculated as follows: IF slope of the jaw, mj < 0, P4z - P7z = - 12 COs ( tan-'(mj)) IF slope of the jaw, mj > 0, P4z - P7z = l2 COS ( tall-'(mj)) In this case, P4z - P7z = - (1.2097 cos ( tan-'(-1 10.483))) now, CASE I1 : In this case, the following geometric relationship can be derived: p6, - PSy Slope of the rod, m56 = Pdr - Psr Slope of the movable jaw formula is given as follows: mj=tan(90~+P) where, a = tan-' (mS6) P = tan-' (mj(h) 1 mj (w)) where, mj (w) = Movable jaw plate depth at nominal + variable tolerance on plate depth at Z-axis, and mj(h) movable jaw amount of holes P4y - P7y = 12 sin ( tan"(mj)) [71 Where, lz = (movable jaw plate depth on Y-axis - the difference in height between movable jaw and end plate on Y -axis + movable jaw plate depth variable tolerance on Y-axis) - 11 Where, l1 = movable jaw hole position 1 nominal on Y-axis + (movable jaw hole position 1 variable tolerance on Y-axis) 12 + (movable jaw hole diameter 1 variable tolerance on Z-axis) 12 P~z- P~z = l2 cos ( tan-'(mj)) 181 ~7y= m56 P~Z+ bM* 191 where, b56' is the intercept of the end side of the rod and is given as 1/20 by6 = b56+ Cos( tan-' rnd where, D is the movable jaw hole diameter P2z - P7z = ll cos ( tan-'(mj)) Gap 2 =P2z-Plz CALCULATIONS: Center of the rod hole in the end plate on the Z-axis: P5z = End plate hole position 1 at nominal position + variable tolerance on the end plate hole position 1 on the Z- axis P5z = 0.300 + 0.00 = 0.300 inches Center of the rod hole in the end plate on the Y-axis: P5y = End plate hole position 1 at nominal position + variable tolerance on the end plate hole position 1 on the Y- axis P5y = 0.750 + 0.01 = 0.760 inches Center of the rod hole in the front plate on the Z-axis: P6z = P5z + rod length at nominal + variable tolerance of rod length on Z-axis P6z = 0.300 + 3.525 + 0.01 = 3.835 inches Center of the rod hole in the front plate on the Y-axis: P6y = Front plate hole position 1 at nominal + variable tolerance at hole position 1 of front plate on Y-axis P6y = 0.75 + (-0.005) = 0.745 inches Now, slope of the rod can be calculated as follows: pVf'5~ = 0.745 0.760 Slope of the rod, m,, = - pat - pss 3.835 - 0.300 m56 = -4.2432* 10" Slope of the movable jaw can be calculated as follows: mj=tan(90~+P) where, a = tan-' (m5,j)= tan-' (-4.2432* lo5) = -0.243 12 P = tan-' (mj(h) / mj (w)) where, mj (w) = Movable jaw plate depth at nominal + variable tolerance on plate depth at Z-axis = 0.50 + 0.02 = 0.52 inches mj(h) movable jaw amount of holes and can be calculated as follows: If movable jaw hole position 1 variable tolerance on Y-axis < 0 , it is ((ABS(Movab1e jaw hole position 1 variable tolerance on Y-axis)) + (ABS(Movab1e jaw hole diameter 1 variable tolerance on Z-axis)) + ( Movable jaw hole diameter 1 variable tolerance on Z-axis)) / 2 If movable jaw hole position 1 variable tolerance on Y-axis > 0 , it is given as (- (Movable jaw hole position 1 variable tolerance on Y-axis)) - (ABS(Movab1e jaw hole diameter 1 variable tolerance on Z-axis)) - ( Movable jaw hole diameter 1 variable tolerance on Z-axis)) / 2 In this case, it is -0.005 which means < 0 mj(h) = { ABS(-0.005) +ABS(-0.0004) + (-0.0004))/2 = (0.005+0.0004 - 0.0004) 12 mj(h) = 2.5 *lo9 p = tan-' (2.5* 10"/0.52) = 0.275458 Finally, jaw slope can be calculated as follows: mj = tan ( 90 -(-0.243 12) + 0.275458) = tan ( 90.5 18578) mj = -1 10.483 degree From geometric relationship equation [7], we know that P4y - P7y = lz sin ( tan-l(mj)) Where, l2 = (movable jaw plate depth on Y-axis - the difference in height between movable jaw and end plate on Y -axis + movable jaw plate depth variable tolerance on Y-axis) - 11 Where, l1 = movable jaw hole position 1 nominal on Y-axis + (movable jaw hole position 1 variable tolerance on Y-axis) 12 + (movable jaw hole diameter 1 variable tolerance on Z-axis) I2 ll = 0.5 + (-0.00512) + (0.626+(-0.0004))/2 = 0.8103 inches l2 = 2.25 - 0.25 + 0.02 - 0.8103 = 1.2097 inches P4y - P7y = 1.2097 sin ( tan"(-1 10.483)) P4y - P7y = -1.2097 inches From geometric relationship equation [8], we know that P7z - P4z = 12 cos ( tan"(mj)) = 1.2097 cos ( tan-'(mj)) = 0.0109488 inches Now, assuming P3z = P4z Which means, P3 & P4 points both are at the same place on the Z- axis P3z = End Plate P3 plate depth at nominal + End Plate P1 Plate depth variable tolerance on Z-axis P3z = 0.5 + 0.02 = 0.52 inches which means, P4z = 0.52 inches Now, P7z can be calculated as follows: 72 = 0.52 + 0.0109488 = 0.53095 inches From geometric relationship equation [9], we know that P7y = m56 P7z + b56' where, bs6' is the intercept of the end side of the rod and is given as where, D is the movable jaw hole diameter = 0.626, and bS6 = P5y - ms6 P5z = 0.760 - (- 4.2432 * 0.300 = 0.761273 P7y = (-4.2432 * 10") 0.53095 + 1.0742758 = 1.0720229 inches From equation [7], P4y = P7y - 1.20965 = 1.0720229 + 1.20965 = - 0.137627 inches Let, Plz = End plate P1 plate depth at nominal + End plate P1 plate depth variable tolerance at Z -axis = 0.5 + 0.02 = 0.52 inches Now, 11 = 0.8103 From geometric relationship equation [lo], we know that P2z - P7z = 11 cos ( tan-'(mj)) = 0.8103 cos ( tm-'(mj)) = 7.33385 *lo" inches Now, P2z = P7z + 7.33385 = 0.53095 + 7.33385 *lo" = 0.53828 inches Gap 2 =P2z -Plz Gap 2 = 0.53828 - 0.52 Gap 2 = 0.01828 inches Physically, negative gap is not possible, therefore, the final gap is given as: Final Gap = 0.01828 inches Appendix -D Bench Vice assembly dimensions Appendix - E Bench Vice assembly simulation program vice Assembly Simulation Program 111)_ P CASE f : fF Pl=PZ CASE If : f F DP4 I I I I I I I MACRO CALCULATIONS 1 Macro fonnula for nlculatlons Appendix - L Listing of h4bosoli Excel vcrsion 4.0 macro program Macro formula for cdculating maximum values MAX =FORMULA(MAX(Ea,C4),B40) =FORMULA(MAX(ES,CS),C40) =FORMULA(MAX(E6,C6),D40) =FORMULA(MAX(E7,c7),E40) =FORMULA(MAX(Es,CfJ),F40) =FORMULA(MAX(E9,C9),G40) =FORMULA(MAX(ElO,ClO).H40) =FORMULA(MAX(Ell.Cll),I40) =FORMULA(MAX(E12,C12),J40) =FORMULA(MAX(E13,C13),K40) =FORMULA(MAX(E14,Cl4),L40) =FORMULA(MAX(ElS.C1S),M40) =RETURN() Appendix - L Listing of Microsoft Excel Version 4.0 macro program ABSTRACT This thesis presents a design of experiment approach to tolerance allocation. The purpose of this thesis is to develop a new approach to tolerance allocation problems. The new approach is based on the Taguchi method, and the objective is to determine : a) the stacking function, b) the significant component tolerances that affect the stackup function, and c) the levels of the tolerances that result in the highest quality products. A bench vice is used as a case study example to demonstrate the methodology. The methodology applies Taguchi's parameter design concept to determine the set of component feature tolerances that will result in the lowest cost product subject to quality constraints on the product's function. The parameter design concept employs inner and outer orthogonal arrays to identie the significant control factors that are least sensitive to noise. For tolerance allocation, all tolerances in the stackup function are the control variables (inner array), and each is set at its high (loose tolerance) or low (tight tolerance) level. The direction of the tolerances (+ or - ) represent the noise variables (outer array). For the bench vice case study, the gap between the end plate and the movable jaw was selected as the response variable. Twelve component features affect the gap, thus requiring an L16 inner and outer array. A simulation model was developed in Microsoft Excel version 4.0 to compute the gap between the two plates. The data were transformed according to Taguchi's smaller-the-better S/N ratio. The significant component features and tolerances were identified from a graphical analysis of the S/N ratio's and verified by ANOVA. The concept tested was found to be a valuable tool and is a novel technique for tolerance allocation. The main advantage of this methodology is that the functional stackup relationship need not to be known explicitly, as in statistical tolerancing or Monte Carlo simulation analysis. The major disadvantage is that the method requires an inordinate number of experiments.