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Publication # 911  
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A Survey of Research in the Application of Tolerance Analysis to the Design of Mechanical Assemblies
ADCATS Report No. 911
April 5, 1991
Kenneth W. Chase
Alan R. Parkinson
Mechanical Engineering Department
Brigham Young University
Provo, Utah 84602
Published in Research in Engineering Design (1991) 3:2337
Tolerance analysis is receiving renewed emphasis as industry recognizes that tolerance management is a key element in their programs for improving quality, reducing overall costs and retaining market share. The specification of tolerances is being elevated from a menial task to a legitimate engineering design function. New engineering models and sophisticated analysis tools are being developed to assist design engineers in specifying tolerances on the basis of performance requirements and manufacturing considerations.
This paper presents an overview of tolerance analysis applications to design with emphasis on recent research that is advancing the state of the art. Major topics covered are:
Interest in tolerance analysis is rapidly increasing in industry. The quest for quality has focused attention on the effects of variation on cost and performance of manufactured products. Excess cost or poor performance will eventually show up as a loss of market share. Therefore, the specification of tolerance limits on each dimension and feature of engineering drawings is considered by many to be a vital design function. Tolerance requirements have a farreaching influence that touches nearly every aspect of the manufacturing enterprise as shown in Fig. 1.
Both engineering design and manufacturing personnel are concerned about the effects of tolerances. Engineers like tight tolerances to assure fit and function of their designs. Manufacturers prefer loose tolerances which make parts easier and less expensive to produce. Therefore tolerance specifications become a critical link between engineering and manufacturing, a common meeting ground where competing requirements may be resolved.
Fig. 1. The effects of assigned tolerances are farreaching.
In the last few years, numerous companies have established comprehensive programs in quality management. Noteworthy among them are the efforts of Motorola, IBM and Xerox,who have initiated formal, corporatewide programs for improved tolerance specification, monitoring and control. Their success in reducing waste while cutting cost and development time and reclaiming lost market share has received national praise [Placek 1989a, Placek 1990, Kendrick 1991].
Another indication of the growing interest in tolerancing is the Mechanical Tolerancing Workshop sponsored by NSF and ASME in 1988 which brought together an international group of experts in tolerancing to discuss the state of the art and identify research opportunities [Paleck, 1989b, ASME 1990]. This has been followed by special theme sessions at several ASME conferences, such as the Design Technical Conference in Montreal (1989), the Design Show in Chicago (1990), and the Computers in Engineering Conference in Boston (1990).
2 Models for Tolerance Accumulation
The basis for rational tolerance specification is to create an analytical model to predict the accumulation of tolerances in a mechanical assembly. Critical clearances or fits or other resultant features of an assembly are generally controlled by the stackup or sum of several component tolerances. A number of analytical models exist with varying levels of sophistication as shown in Fig. 2.
Fig. 2. Mathematical models of tolerance accumulation.
Common models for predicting how component tolerances sum are Worst Case (WC) and Root Sum Square (RSS) as shown in Eqs. 1 and 2 [Fortini 1967].
Onedimensional assemblies:  Two or threedimensional assemblies: 
Worst Case 
(1) 
Root Sum Square 
(2) 
where Xi are the nominal component dimensions, Ti are the component tolerances, dU is the predicted assembly variation, TASM is the specified limit for dU, and f(Xi) is the assembly function describing the resulting dimension of the assembly, such as the clearance or interference. The partial derivatives ¶f/¶x_{i} represent the sensitivity of the assembly tolerance to variations in individual component dimensions. For a onedimensional tolerance stack the sensitivity is ±1.0.
The Worst Case model assumes all the component dimensions occur at their worst limit simultaneously. It is used by designers to assure that all assemblies will meet the specified assembly limit. However, as the number of parts in the assembly sum increases, the component tolerances must be greatly reduced in order to meet the assembly limit, requiring higher production costs. In the RSS model, the low probability of the worst case combination occurring is taken into account statistically, assuming a Normal or Gaussian distribution for component variations. Tolerances are commonly assumed to correspond to six standard deviations (6s or ±3s). Component tolerances may be increased significantly since they add as the root sum squared (RSS).
Statistical distributions may be used to predict the yield of an assembly, that is, the number or fraction of assemblies which are likely to lie inside the spec limits. RSS analysis generally predicts too few rejects when compared to real assembly processes. This is due to the fact that the Normal distribution is only an approximation of the true distribution, which may be flatter or skewed. The mean of the distribution may also be shifted from the midpoint of the tolerance range. To account for these uncertainties, a more general form of the RSS model is frequently used:
(3)
where Z is the number of standard deviations desired for the specified assembly tolerance and Zi describes the expected standard deviations for each component tolerance. C_{f} is a correction factor added to account for any nonideal conditions. Typical values for C_{f} range from 1.4 to 1.8 [Bender 1968, Levy 1974, Gladman 1980].
Another conservative estimate of assembly tolerances assumes the component dimensions are uniformly distributed over the specified tolerance range. In this case, the value of Zi is s3. If truncated normal distributions arise due to inspection of component parts, then choose s3<Zi<3, as described by Spotts [1983]. Spotts also suggested calculating the Worst Case and RSS assembly tolerance and simply averaging the two as a safe estimate [Spotts 1978].
2.1 Estimated Mean Shift Model
Simple RSS analysis assumes that the variation of each component dimension is symmetrically distributed about the mean or nominal dimension. However, in real processes, the mean is shifted due to setup errors or drifts due to timevarying parameters such as tool wear. Ignoring mean shifts can be very detrimental, resulting in large errors in estimates of the number of assemblies within spec limits [Spotts 1978, Evans 1975b].
Further modifications to the RSS model have been proposed to take into account mean shifts or biased distributions. Mansoor [1963] proposed that the tolerance accumulation be represented as a WC sum plus a RSS sum. A similar model by Greenwood [1987], Greenwood and Chase [1987] and Chase and Greenwood [1988] introduced an estimated mean shift factor mi (a number between 0 and 1.0) which quantifies the expected mean shift as a fraction of the specified tolerances. Eq. 4 illustrates the resulting expression.
Worst Case Sum Statistical Sum
(4)
This is a versatile model. If all the mi are set to 1.0, the result is a WC model. If all the mi are set to 0, the result is a RSS model. By selecting mi between 0. and 1.0, the resulting variation will always lie between the WC and RSS predictions. And any combination of mi may be chosen to account for the degree of uncertainty in individual process characterizations.
2.2 Motorola Six Sigma Program
A new Tolerance accumulation model that has caught the attention of industry was developed by the Motorola Corp. as a basic element of their awardwinning Six Sigma quality program which is now being adopted by other leading companies[ Placek 1989a].
The basic premise of the Six Sigma Program is: in order to achieve high quality in a complex product comprised of many components and processes, each component and process must be produced at significantly higher quality levels in order for the composite result to meet final quality standards. Stated statistically, suppose there were 1000 dimensions or other characteristics of your product, any one of which could lower the quality of the finished product. If each characteristic were produced to ±3s quality (99.73% acceptable parts or 2700 defects per million), the resultant assemblies would be only (.9973)^{1000}=.067 or 6.7% defect free. To have 99.73% defect free assemblies, you would need to produce each component to a quality of (.9973)^{.001}=.9999973 or 99.99973%, which is 2.7 defects per million.
To achieve the high quality levels required for world competition in the electronics industry, Motorola has mandated ±6s quality for all processes (.002 defects per million). However, they also recognize that shifts and drifts in the mean of the processes are expected, so they have introduced a modified process model which includes an allowance for accumulated mean shifts. The result is a net quality level of ±4.5s (3.4 defects per million).
A versatile feature of the Six Sigma model of a process is the ability to distinguish between short term and long term process capability. The process capability quantifies the spread of the process. It is defined as 6.0 times the standard deviation of the process, 6s_{i}. Over the long term, however, the mean of a process will drift due to tool wear or the set up will vary from lottolot, resulting in an apparent increase in the process capability. The resulting modified standard deviation of a component process may be estimated from Eq. 5
where (5)
UL and LL are the upper and lower tolerance limits, respectively, Cpi is the Process Capability Ratio, or the ratio of the specified tolerance range to the process capability. Variable mi is the Mean Shift Factor. When mi = 0,si describes the short term variation in the process. When 0< mi <1.0, s_{i} approximates the longterm variation. For the standard Six Sigma model, the target values of Cpi = 2 and mi = 0.25 result in tolerance limits of ±4.5si over the long term. Note that other Cpi and mi values may be selected to account for the degree of uncertainty in individual process characterizations.
The Six Sigma model for tolerance accumulation is shown in Eq. 6. It accounts for process mean shift variations by using an effective standard deviation as expressed in Eq. 5. If a value for si for the process is known from prior experience, it may be substituted in the equation.
(6)
Of course, tolerance analysis of assemblies is only one component of the complete Motorola quality management system, but the Six Sigma tolerance analysis model is a significant contribution. It is more realistic and versatile than the models commonly used for design. It should have a major impact on reducing production costs and improving quality [Harry et al. 1987, 1988, 1990].
Modeling tolerance accumulation in 2D assemblies is much more difficult than for 1D. Component dimensions are joined together as 2D vector chains or loops. The loops pass from mating parttomating part, passing through the points of contact between parts. Vector loops may be open or closed. Each open loop describes a resultant assembly clearance or interference. Closed loops describe assemblies containing some adjustable element, such as a springloaded part or fastener, which takes up the slack and assures closure.
Complex assemblies may require several open and closed loops to define assembly relationships. Each vector loop results in three scalar equations which describe the response to manufacturing variations. The resulting nonlinear system of equations must be solved simultaneously for the assembly resultants [Fortini 1967, Chase & Greenwood 1988, Marler 1988, Bjorke 1989].
Manufacturing variations propagate through an assembly by small kinematic adjustments between mating parts. Thus, kinematic constraints must be applied to the vector loops. Each kinematic constraint introduces kinematic degrees of freedom into the model, such as the location of a point of contact on a sliding plane or the relative angle between two mating parts which are joined by a pin joint. The nominal values of the kinematic variables are not known. They are not dimensioned on any engineering drawing. They must be determined by assembling an ideal assembly for which all manufactured dimensions are at their nominal values. They may then be calculated by solving the set of vector loop equations and kinematic constraints describing the system.
The variations in the kinematic variables are also unknown. Small changes in the manufactured dimensions produce small changes in the kinematic variables. Thus, the kinematic variables are dependent variables. The variation in the kinematic variables may be determined by specifying the manufacturing variations (design tolerances) and solving the vector loop equations for the resulting adjustments in the assembly.
Thus, there are two steps in solving the vector equations. First, set all the manufactured dimensions to their nominal values and solve the system of vector loop equations for the nominal values of the kinematic variables and assembly resultants. Vector equations are generally nonlinear and must be solved by iterative means. If the nominal values may be determined from, say, a precise CAD layout, this step may be omitted.
Second, linearize the equations for small variations about the nominal by Taylor's series expansion, retaining first order derivatives. Substitute the design tolerances and solve this system of linearized equations for the corresponding variation in the kinematic variables and assembly resultants. [Marler 1988, Chase et al. 1989]
Geometric tolerances are being used increasingly by aerospace and other industries to assure form and function of mechanical parts. They are distinct from size or dimensional tolerances. They control the form and orientation (flatness, roundness, perpendicularity, etc.) and location (position, concentricity, etc.) of surfaces and other features as defined in the standard, ANSI Y14.5 [ASME 1983]. Recommended practice is to establish geometric tolerances on the basis of "Maximum Material Condition" (MMC), which is essentially a Worst Case analysis, resulting in tight component tolerances. The standard does not include statistical considerations or tolerance accumulation effects [Foster 1986, Levy 1974].
However, geometric deviations provide sources of variation that can accumulate and propagate through an assembly the same as size tolerances. Depending on the number of components and the geometry, they can have a significant influence on the resultant assembly variations. Efforts are being made to analyze geometric tolerances statistically and include geometric variations in vector loop models in order to compute their effects on complex assemblies along with dimensional variations [Chun 1988, Chase et al. 1989].
Fig. 3a illustrates a simple 2D problem described by Fortini [1967]. It is a drawing of a oneway mechanical clutch.
Fig. 3. Vector loop model of a clutch assembly.
This is a common device used to transmit rotary motion in only one direction. When the outer ring of the clutch is rotated clockwise, the rollers wedge between the ring and hub, locking the two so they rotate together. In the reverse direction, the rollers just slip, so the hub does not turn. The angle F1 between the two contact points is critical to the proper operation of the clutch. If F1 is too large, the clutch will not lock; if it is too small the clutch will not unlock.
The primary objective of performing a tolerance analysis on the clutch is to determine how much the angle F1 is expected to vary due to manufacturing variations in the clutch components. The independent manufacturing variables are the hub dimension a, the cylinder radius c, and the ring diameter e. The dependent assembly resultants are the location of contact point, b, and the two angles F1 and F2.
In this example, the designer begins the construction of a vector kinematic model by creating a precise CAD model of the assembly, to which the vector model will be added as an overlay. On the CAD model, he defines the kinematic joints at the contact locations between parts (Fig. 3b). When the parts are properly constrained by the joints, they will have zero kinematic degrees of freedom.
Next, the joints are connected by vectors forming a closed vector loop and making sure all critical dimensions have been included in at least one loop (Fig. 3c). The geometry from the loops and joints is used to calculate the sensitivity of the dependent kinematic dimensions to changes in each independent dimension. This information is used for computation of the expected variation in assembly resultants and possible reallocation of tolerances.
Geometric form tolerances may be added at each contact joint (Fig. 3d). The sensitivity of the dependent assembly dimensions b, F1 and F2 to form and orientation variations may then be calculated. The predicted variation in assembly resultants may then be estimated by modifying Eq. 2 to include form variations:
(7)
where dai are the form variations and dU is the resultant assembly variation dF1.
Once an expression for dF1 has been obtained, tolerance design may proceed. By substituting reasonable values for the component tolerances Ti and form tolerances dai into Eq. 7, the designer must verify that the predicted variation in F1 will be less than design requirement T_{ASM}. Alternately, to reduce production cost, one might set dF1 equal to T_{ASM} and solve for the largest possible values for Ti.
An alternative approach to vector assembly models is to create a solid model of the assembly on a CAD system. The solid model serves as the assembly function. Small changes can be simulated and their effects will propagate realistically, provided each part is located relative to its adjacent parts and provided that kinematic adjustments are permitted.
Solid models are precise representations of assembly geometry. They are constructed from surfaces or solid primitives. The main obstacle to the use of commercial solid modelers for tolerance analysis is the lack of conventional dimension and tolerance data. This imposes a great hardship upon software application developers. The trend toward featurebased solid modelers may help to overcome this deficiency.
To obtain the sensitivities required for calculating an assembly tolerance sum, the relationship between the parameters defining the model surfaces or solid primitives and the dimensioned quantities appearing on an engineering drawing of the parts must be determined. This may be accomplished by making small changes in each of the model variables, measuring the resultant change in the component dimensions and assembly resultants as shown in Fig. 4 and computing the corresponding sensitivities. The sensitivities are used to form the linearized expressions, Eq. (8), relating the variations in the component dimensions and assembly resultants to variations in the model parameters. Finally, a linear programming problem is set up to find a set of model variations, dMj, which, when substituted into Eq. (8), satisfies the specified component and assembly tolerance limits [Turner and Wozney 1987, 1990, Turner ,Wozney and Hoh 1987].
Fig. 4. Computing tolerance sensitivities from the solid model of an assembly.
Component tolerances:  
(8)  
Assembly tolerances: 
Solid modelers are CPU intensive. Changing a single parameter for a sensitivity calculation requires regeneration of the entire CAD geometry. A detailed model of an assembly may have thousands of model parameters, resulting in a substantial wait for a complete sensitivity calculation on all but the most powerful computers. However, significant progress is being made in reducing the enormous number of sensitivity calculations by prior examination of the model to eliminate noncontributing parameters [Martino and Gabriele 1989a, 1989b]. Current research efforts by Turner include the addition of kinematic constraints [Turner 1990, Turner and Srikanth 1990, Srikanth and Turner 1990].
A number of researchers are taking an axiomatic approach to 3D tolerance representations in solid models. Requicha represents the model variations as a pair of "offset boundaries," or offset surfaces, which bound each ideal surface. The set of offset boundaries form a tolerance zone which bounds the entire part [Requicha 1983,1986]. A similar definition creates a "virtual boundary" formed by taking into account the combined effects of all applicable size and form tolerances. [Jayaraman and Srinivasan 1989, Srinivasan and Jayaraman 1989]. Several problems remain to be resolved, including: potential conflicts with existing standards, incorporation of statistical models and the lack of kinematic assembly interactions. For commentaries on these issues, the reader is referred to [ Faux 1986, Etesami 1987].
Variational geometry is another fundamental approach. It requires the formulation of analytical equations describing the geometric relationships which must be maintained in an assembly. Constraints such as perpendicular surfaces or surfaces in sliding contact are defined in terms of dimensional parameters. If the design is modified, the system of equations may be solved to adjust the free variables in keeping with the constraints. The advantages of this method are the ease of design iteration and the realistic propagation of manufacturing variations by kinematic adjustments. However, the resulting system of nonlinear equations can become very large and must be solved simultaneously. Also, geometric form and feature tolerances must still be taken into account [Light and Gossard 1982, Gossard et al. 1988, Chung and Schussel 1990].
Vector loop models of assemblies may be applied to 3D assemblies. The vectors are not confined to a plane and the kinematic conditions become more complex. The system of equations is twice as large, since each 3D vector equation yields six scalar equations. Form and orientation tolerances may be added. Their interaction with the kinematic joint axes must be carefully modeled. Recently, vector models of an assemblies have been overlaid on the corresponding 3D solid model and associated with that model such that changes in the solid model are automatically reflected in the vector model [Robison 1989].
One major advantage of 3D vector models of assemblies over solid models is that the geometry is reduced to only those parameters required to perform a tolerance analysis. Sensitivity analysis is much simpler. It is very efficient computationally and well suited to design iteration. Describing manufacturing variations with vectors and kinematics is also a medium that engineering designers are already familiar with.
5 Tolerance Analysis of Mechanisms
Since tolerance analysis of assemblies involves the creation of a kinematic model, the principal differences from the classical kinematic analysis of mechanisms are the inputs and the magnitude of the motions. A mechanism will have one or more kinematic variables as a prescribed input. The nonlinear system of kinematic equations will be solved for the remaining kinematic variables. The resulting motions will be orders of magnitude larger than those due to manufacturing variations.
For tolerance analysis of a mechanism, the dimensions of the various components will be held fixed while the nominal position and orientation of each component is determined by kinematic analysis. Then a tolerance analysis may be performed by introducing manufacturing variations while holding the mechanism stationary. This may be repeated at selected positions considered of interest or the mechanism may be incremented at regular intervals to see the results of manufacturing variations over a range of motion.
References to mechanism tolerance analysis:
Mechanism 
Analysis  Authors 
4bar function generator  tolerance estimating tolerance allocation tolerance allocation 
[Garrett & Hall 1969] [Dhande & Chakraborty 1973] [Agarwal 1981] 
4bar path generator  tolerance estimating adjustable linkages tolerance allocation 
[Baumgarton & Van der Werff 1985] [Schade 1982] [Mallik & Dhande 1987] 
Slidercrank  tolerance allocation  [Fu et al 1987, Schade 1980] 
General 2D linkages  tolerance allocation  [Fenton et al 1989] 
3D 4bar function generator  tolerance allocation tolerance allocation 
[Dhande & Chakraborty 1978] [Beohar & Rao 1980] 
Disk cams  tolerance allocation  [Rao & Gavane 1980] 
6 Nonlinear Analysis, NonNormal Distributions
The linearized models for tolerance accumulation in an assembly, as expressed in Eqs. 1 through 4, assume that the sensitivity, evaluated at the nominal, is constant over the tolerance limits. That is, if you evaluated the assembly function as you varied one parameter over its tolerance range, the slope of the function would be nearly constant . This is usually a reasonable assumption, when the tolerances are very small compared to the nominal dimensions or when there are a large enough number of components to mask the effects.
In a highly nonlinear assembly, the sensitivity may not be symmetric over this range and the distribution of the assembly resultant will be skewed or asymmetric. This can happen even though all of the component distributions are symmetric. A linearized model, however, will always yield a symmetric resultant from symmetric inputs.
Analysis methods which can treat nonlinear effects are shown in Table 1. While HasoferLind retains nonlinear effects, it is limited to Normal distributions. The relative CPU efficiency values are only estimates, based on the author's own experience. Actual values could differ substantially depending on problem complexity.
Table 1. Comparison of Tolerance Analysis Methods
Assembly Model 
Distributions 
Efficiency 

Analysis Method 
Linearized 
Nonlinear 
Normal 
NonNormal 
Relative 
Worst Case  X 
X 
NA 
NA 
1 
RSS  X 
X 
1 

HasoferLind  X 
X 
6 

Method of Moments  X 
X 
X 
10 

Integration  X 
X 
X 
60 

Monte Carlo  X 
X 
X 
X 
100 
Designers seldom have sufficient data by which to specify the distribution of the manufacturing processes. Data has not been gathered because the parts have not yet been made. Tooling has not been ordered nor certified. The processes may not have been selected. It is customary in such cases for designers to assume a Normal distribution. If there is uncertainty about the process, then a Uniform distribution may be assumed. Generally, deviations from Normal are slight and vary from batch to batch, making it difficult to predict. If the component distributions have a strong central tendency, and there are five or more components contributing to an assembly sum, the result is likely to approximate a Normal distribution regardless of the component distributions. If there are mean shifts, one of the models discussed earlier may be used.
However, in some cases, where the process has been well characterized and is known to exhibit skewness (asymmetry) or kurtosis (peakedness), it may be justifiable to estimate these parameters and apply an advanced analysis method, such as Monte Carlo or Method of Moments. This is more likely to occur after production has begun and data has been gathered on finished parts.
Monte Carlo Simulation is a powerful tool for tolerance analysis of mechanical assemblies, for both nonlinear assembly functions and nonNormal distributions. It is based on the use of a random number generator to simulate the effects of manufacturing variations on assemblies. Fig. 5 illustrates the method.
Fig. 5. Assembly tolerance analysis by Monte Carlo simulation.
The Monte Carlo method consists of the following steps:
The biggest disadvantage of the Monte Carlo method is that it requires large samples to achieve reasonable accuracy. The number of simulated assemblies must be on the order of 100,000 to 400,000 to predict the small percentage rejects of modern manufacturing processes. For example, for a ±3s assembly spec, a sample of 100,000 assemblies should yield 135 rejects at each limit, but could be 10 to 20% off [Shapiro and Gross 1981]. Design iterations get pretty tedious when 100,000 simulations must be generated for each trial design. The designer is not likely to have the patience to search for the optimum design.
References to nonlinear or nonNormal analysis include:
Method 
Authors 
Worst Case  Greenwood & Chase [1988a] 
HasoferLind  Parkinson [1982, 1985], Lee&Woo [1990], Greenwood & Chase[1988b] 
Method of Moments  Evans [1970, 1974, 1975a, 1975b], Cox [1979, 1986], Shapiro & Gross [1981], Greenwood [1987] 
Integration  Evans [1967, 1971, 1972], Sorensen [1990] 
Monte Carlo  Grossman [1976], Shapiro & Gross [1981], DeDoncker & Spencer [1987], Craig [1989], Doepker & Nies [1989], Early & Thompson [1989] 
Design improvement is the principal aim of tolerance analysis. Rather than just predict the effects of variations, the goal is to systematically select tolerances throughout an assembly to assure that design requirements will be met. The ideal assigned tolerances not only assure acceptable performance, but also assure that parts can readily be produced and assembled, resulting in high process yields and reduced costs. Several strategies for design improvement exist. A good general discussion is presented in the excellent book by Spence and Soin [1988]. Although the book is applied to electronics design, the methods are just as applicable to mechanical systems.
The main topics in design improvement are: Yield Modification, Tolerance Allocation, Sensitivity Analysis, and Probabalistic Design and Design Optimization.
The yield of an assembly process may be increased by design centering. The procedure is illustrated in Fig. 6a. The figure shows the effect of varying two design parameters, p1 and p2. The region defined by shaded boundaries is called the feasible design space and represents the limits set on the values of p1 and p2 for acceptable performance. The rectangle represents the specified production limits on p1 and p2, that is, the tolerance limits, which vary about a specified nominal value. As can be seen, the original value of the nominal places most of the rectangle outside of the acceptable performance region. The resulting design would have a low yield. By adjusting the nominal to the center of the feasible design region, nearly all of the assemblies will perform satisfactorily.
The second method of increasing yield is by variance reduction, that is, by tightening the tolerances, as shown in Fig. 6b. The new, tighter tolerances place all of the produced assemblies inside the acceptable region. Of course, tighter tolerances are more costly to produce. But the increased cost may be partially offset by the reduction in waste and rework. The optimum tolerances may be found by minimizing the overall cost of tight tolerances, waste and rework [Spence & Sion 1988].
Fig. 6. Illustration of design improvement methods.
7.2 Research in Tolerance Allocation
Tolerance allocation is a design function. It is performed early in the product development cycle, before any parts have been produced or tooling ordered. It involves first, deciding what tolerance limits to place on the critical clearances and fits for an assembly, based on performance requirements; second, creating an assembly model to identify which dimensions contribute to the final assembly dimensions; third, deciding how much of the assembly tolerance to assign to each of the contributing components in the assembly. Fig. 6c shows the tolerance on dimension 2 reduced, allowing an increase in tolerance on dimension 1.
Tolerance analysis, on the other hand, is a production function. It is performed after the parts are in production. It involves first, gathering data on the individual component variations; second, creating an assembly model to identify which dimensions contribute to the final assembly dimensions; third, applying the measured component variations to the model to predict the assembly dimension variations.
A defective assembly is one for which the component variations accumulate and exceed the specified assembly tolerance limits. The yield of an assembly process is the percent of assemblies which are not defective. In tolerance analysis, component variations are analyzed to predict how many assemblies will be in spec. If the yield is too low, rework, shimming, or parts replacement may be required. In tolerance allocation, an acceptable yield of the process is first specified and component tolerances are then selected to assure that the specified yield will be met.
Often, tolerance design is performed by repeated application of tolerance analysis, using trial values of the component tolerances. However, a number of algorithms have been proposed for assigning tolerances on a rational basis, without resorting to trial and error. Several are listed in Fig. 7.
Fig. 7. Tolerance allocation methods.
By this procedure, initial values of the component tolerances are selected, substituted into the assembly tolerance sum equation, then scaled proportionally so the sum equals the assembly tolerance limit. Initial tolerance values may be selected from charts of tolerance capabilities for specified processes, from design rules, standards, etc.[Mansoor 1963, Chase & Greenwood 1988, Bjorke 1989].
A variation on this method adds flexibility by specifying weight factors to certain component tolerances so those components will receive a greater allocation of the available tolerance [Harry & Stewart 1988].
7.2.2 Cube Root of the Nominal.
This method is based on the ruleofthumb that the difficulty in obtaining a specified tolerance increases as the cube root of the nominal size of the part . The rule is the basis for the early tolerance standards for cylindrical fits.[Fortini 1967]. The procedure is to select initial tolerance values equal to the cube root of the nominal, substitute into the assembly tolerance sum equation, then scale proportionally. The resulting tolerances will each be proportional to the cube root of their nominal size [Chase & Greenwood 1988].
This is an extension of the cube root method, with more categories of difficulty, such as: size, shape, material, process, etc., where each category refers to a property affecting the cost of producing a tolerance. The designer assigns a difficulty factor to each component tolerance based on nominal size, then assigns another factor to each component based on shape (inside dimension, outside dimension, etc.), and repeats this process for each category, writing the factors in a table. The difficulty factors for each component dimension are summed and used as weight factors in the tolerance sum equation to drive the allocation [Fortini 1967, 1985].
If an empirical function of costvstolerance (or process capability) can be obtained for each dimension in the assembly sum, then an optimization algorithm may be used to systematically search for the combination of component tolerances which results in the least overall production cost. Numerous researchers have proposed different search algorithms and different forms of empirical cost functions, as summarized in Table 2.
Table 2. Proposed CostvsTolerance Models
Cost Model  Method 
Author 

Linear  A  B T  Linear prog  Edel & Auer [1965] 
Reciprocal  A + B/T  Lagrange mult Nonlin prog 
Chase & Greenwood [1988] Parkinson [1985] 
Reciprocal Squared  A + B/T²  Lagrange mult  Spotts [1973] 
Reciprocal Power  A + B/T^{k}  Lagrange mult  Sutherland & Roth [1975] 
Multi/Recip Powers  B/T^{k}_{i}  Nonlin prog Lagrange mult Lagrange mult Nonlin prog 
Lee & Woo [1990] Bennett & Gupta [1969] Chase et al. [1990] Andersen [1990] 
Exponential  B e^{mT}  Lagrange mult Geom prog Graphical 
Speckhart [1972] Wilde & Prentice [1975] Peters [1970] 
Expon/Recip Power  B e^{mT}/T^{k}  Nonlin prog  Michael & Siddall [1981,1982] 
Piecewise Linear  A_{i}  B_{i} T_{i}  Linear prog  Bjork [1989], Patel [1980] 
Empirical Data  Discrete points  Zeroone prog Combinatorial Branch & Bound 
Ostwald & Huang [1977] Monte & Datseris [1982] Lee & Woo [1989] 
The constant coefficient A represents the fixed costs, such as tooling, setup, prior operations, etc. The B term represents the cost of producing a single component dimension to a specified tolerance T. All costs are calculated on a per part basis.
7.2.5 Minimum Cost with Process Selection.
Optimization procedures have been extended to not only find the least cost set of tolerances, but to also select the least cost process from a set of alternative processes for each dimension for the assembly. That is, the computer can decide which process is the most economical to produce each part dimension while considering the tolerances of all of the parts and their cost interactions [Ostwald & Huang 1977, Lee & Woo 1989, Chase et al. 1990].
7.2.6 Minimum Cost – Complex Assembly Models.
Optimization procedures may also be applied to complex assemblies defined by 2D or multiple vector loops, as described in the next section. Further extensions have been studied as follows:
Nonlinear assemblies:  Lee & Woo[1990] 
2D assemblies:  Sutherland & Roth[1975], Monte & Datsaris[1982], Parkinson[1985], Andersen[1990] 
Multiple loop assemblies:  Bennett and Gupta[1969], Lee and Woo[1990], Andersen [1990] 
Process mean Shifts:  Andersen[1990] 
NonNormal distributions:  Parkinson[1985] 
The third area of design improvement stems from examining the tolerance sensitivities, which are the partial derivative terms appearing in the tolerance accumulation expressions of Eqs. 1 through 4. The tolerance sensitivity tells the designer which assembly parameter variations have the greatest effect on the critical assembly features. Listing the parameters and their corresponding sensitivities in order of decreasing magnitude reveals which components to focus on for design improvement. Alternately, one could list the product of the sensitivities and their corresponding tolerances in descending order and also calculate the percent contribution made by each to the assembly resultant. Then, starting with the largest contributor, the designer could try to decrease the overall variation by tightening tolerances on the most sensitive components or decrease the overall cost by loosening the tolerance on the least sensitive components [Eaton 1975].
Sensitivity reduction is another approach in which the sensitivity itself is reduced by moving the nominal values to a less sensitive portion of feasible design space. Fig. 8 illustrates this method. In the figure, the contour lines represent lines of constant assembly performance. Closely spaced contours indicate a region of high variability in performance. By moving the nominal design from a region of high variability to a region of low variability, as shown in the figure, the design is made insensitive, or robust, to manufacturing variations. A systematic procedure for accomplishing this is the popular Taguchi method developed by the wellknown Japanese expert on quality control [Taguchi 1986, Kacker 1986, Byrne & Taguchi 1987, Taguchi et al. 1989].
Fig. 8. Sensitivity reduction by shifting the nominal values.
A fundamental element of the Taguchi method is the formulation of a Quality Loss Function which quantifies the cost of deviating from the target value of a given design parameter. The loss function is expressed as a parabola, with minimum cost at the target value and increasing with the square of the deviation. It can include the full spectrum of costs, including inferior performance, increased rework and warranty costs, dissatisfied customers and lost market share. Taguchi also includes the "loss to society", if it can be quantified [Taguchi 1986].
7.4 Probabalistic Design and Design Optimization
Probabalistic design may be considered to be an extension of tolerance analysis methods to include consideration of the variational effects of both geometric and engineering parameters on design performance. Engineering parameters, such as the limiting strength of a metal or the viscosity of a lubricant, exhibit manufacturing variations which can be characterized by statistical distributions. By applying statistical analysis to the engineering performance equations for stress, lubrication, etc., the variation in critical performance resultants can be predicted [Haugen 1980, Mischke 1989].
Design optimization is a mathematical method for improving a design by applying linear or nonlinear programming techniques to search systematically for the minimum of an objective function. The objective function is derived from the engineering model and expresses some critical performance parameter which is to be minimized, such as the weight of a structure or cost of a fluid distribution system.
Often tolerance analysis is performed after the design is essentially complete and all nominal values for the design have been determined. However, by considering the effects of tolerances during the selection of the nominal values of design variables, it is possible to develop a "robust" design that is more tolerant of variation. Developing a robust design by judicious selection of variable nominal values is an important part of the Taguchi philosophy.
Taguchi’s methodology develops a model of the design problem by direct experimentation. However, when a computer model of the design exists, an appropriate means of developing a robust design is through nonlinear optimization techniques.
These methods can be used at two levels. The first level is to select design variable values such that the design remains feasible, i.e. will still function properly, despite variations arising from tolerances. The basic approach here is to calculate the variation caused by tolerances using either a first order Taylor’s series (Eq (1)) or through Monte Carlo simulation. The transmitted variation is then subtracted from the allowable values of the constraints, causing a shift of the optimum design into the feasible regionas shown in Fig. 9. A good review of applications at this level is given by Eggert (1990).
Fig. 9. Change in optimum and decrease in feasible region
to insure
design will remain feasible to variation caused by tolerances.
The second level is to explicitly consider the variation from tolerances as an objective or constraint in the problem. At this level the designer seeks to actively control variation by either minimizing it as one of the objectives in the design problem or by constraining the design to have variation less than a value specified by the designer. Active control of variation can be computationally expensive since it requires second derivatives of model equations. A good discussion with examples of this level is given by Parkinson et al. (1990).
There is probably no other design improvement effort which can yield greater benefits for less cost than the careful analysis and assignment of tolerances. Tolerancing provides a common meeting ground for engineering and production personnel where effective communication can assure that their competing requirements are met in the most economical way, with the greatest customer satisfaction.
In the foregoing discussion, a number of research areas have been surveyed where significant progress is being made. As a result of current research, powerful new design tools are becoming available which incorporate improved methods for predicting the effects of manufacturing variations on engineering performance and production quality. The effective application of these concepts will assist manufacturing enterprises in competing in the worldwide marketplace.
This literature survey has been so broad in scope that only a few papers have been referenced in each area to permit some tutorial descriptions. We apologize for any papers which may have been passed over in the selection process. A more complete bibliography is available from the authors on request [Chase 1991].
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Andersen, C. B.(1990), "General System for Least Cost Tolerance Allocation in Mechanical Assemblies," ADCATS Report No. 902, Brigham Young University.
ASME (1990), "Research Needs and Technological Opportunities in Mechanical Tolerancing", ASME Publication No. CRTD15.
ASME (1983), "Dimensioning and Tolerancing", ANSI Y14.5M1982, ASME Publication.
Baumgarten, J. and K. Van der Werff (1985), "A Probabilistic Study Relating to Tolerancing and Path Generation Error," J. of Mechanism and Machine Theory, v 20 n 1, pp.7176.
Bender, A., Jr. (1968), "Statistical Tolerancing as it Relates to Quality Control and the Designer," SAE Paper No. 680490.
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Beohar, S. and A. Rao (1980), "Optimum Stochastic Synthesis of Four Bar Spatial Function Generators," ASME Paper No. 80DET32.
Bjorke, O. (1989), ComputerAided Tolerancing, Second Ed., ASME Press.
Byrne, D. and S. Taguchi, (1987), "The Taguchi Approach to Parameter Design," Quality Progress, pp. 1926, Dec.
Chase, K. W. (1991), "A Bibliography on Tolerance Analysis of Mechanical Assemblies," ADCATS Report No. 912, Brigham Young University.
Chase, K. W., and W. H. Greenwood (1988), "Design Issues in Mechanical Tolerance Analysis," Manufacturing Review, ASME, v 1, n 1, pp. 5059, March.
Chase, K. W., W. H. Greenwood, B. G. Loosli, and L. F. Hauglund (1990), "Least Cost Tolerance Allocation for Mechanical Assemblies with Automated Process Selection," Manufacturing Review, ASME, v 3, n 1, pp. 4959, March.
Chase, K., C. Sorensen and C. Andersen (1989), "Recent Developments in Tolerance Analysis Software for Mechanical Assemblies, Part A: TwoDimensional Modeling," Proceedings of the ASQC Western Regional Conference1989, pp. 7380.
Chun, K. (1988), "Development of TwoDimensional Tolerance Modeling Methods for CAD Systems," ADCATS Report No. 887, Brigham Young University.
Chung, J. C. H. and M. D. Schussel (1990), "Technical Evaluation of Variational and Parametric Design," Computers in Engineering1990, vol 1, ASME, pp. 28998.
Cox, N. D. (1979), "Tolerance Analysis by Computer," J. of Quality Technology, v 11, n 2, pp. 8087, April.
Cox, N. D. (1986), "How to Perform Statistical Tolerance Analysis," ASQC Series Vol 11, Basic References in Quality Control.
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DeDoncker, D. and A. Spencer (1987), "Assembly Tolerance Analysis with Simulation and Optimization Techniques," SAE Paper No. 870263.
Dhande, S. and J. Chakraborty (1973), "Analysis and Synthesis of Mechanical Error in LinkagesA Stochastic Approach," J. of Engineering for Industry, ASME, v 105, pp. 672676.
Dhande, S. and J. Chakraborty (1978), "Mechanical Error Analysis of Spatial Linkages," J. of Mechanical Design, ASME, v 100, pp. 732738.
Doepker, P. E. and D. Nies (1989), "Designing Brake Components Using Variation Simulation Modeling," Failure Prevention and Reliability  1989, ASME Publ. No. DEVol 16, pp. 131138.
Early, R. and J. Thompson (1989), "Variation Simulation ModelingVariation Analysis Using Monte Carlo Simulation," Failure Prevention and Reliability  1989, ASME Publ. No. DEVol 16, pp. 139144.
Eaton, F. (1975), "Computer Model of a ThreeDimensional Assembly," ASME Paper No. 75DE21.
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Eggert R. J., and R. W. Mayne, (1990), "Probabilistic Optimization Using Successive Surrogate Probability Density Functions, Proc. of ASME 16th Design Automation Conf., Chicago, IL, DEVol 231, p. 129, Sept.
Etesami, F. (1987), "On the Theory of Geometric Tolerancing", Computers in Engineering1987, vol 2, ASME, pp. 32734.
Evans, D. H. (1967), "An Application of Numerical Integration Techniques to Statistical Tolerancing," Technometrics, v 9 n 3, pp 441456, Aug.
Evans, D. H. (1970), "A Statistical Tolerancing Formulation," J. of Quality Technology, v 2 n 4, pp 226231, Oct.
Evans, D. H. (1971), "An Application of Numerical Integration Techniques to Statistical Tolerancing, II  A Note on the Error," Technometrics, v 13 n 2, pp 315324, May.
Evans, D. H. (1972), "An Application of Numerical Integration Techniques to Statistical Tolerancing, III  General Distributions," Technometrics, v 14 n 1, pp 2335, Feb.
Evans, D. H. (1974), "Statistical Tolerancing: State of the Art, Part 1. Background," J. of Quality Technology, v 6 n 4, pp 188195, Oct.
Evans, D. H. (1975a), "Statistical Tolerancing: State of the Art, Part 2. Methods of Estimating Moments," J. of Quality Technology, v 7 n 1, pp 112, Jan.
Evans, D. H. (1975b), "Statistical Tolerancing: State of the Art, Part 3. Shifts and Drifts," J. of Quality Technology, v 7 n 2, pp 7276, Apr.
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Fortini, E. (1985), "Dimensions and Tolerances," Mechanical Design and Systems Handbook, Second Ed., H. Rothbart editor, McGrawHill.
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Fu, J., W. Cleghorn and R. Fenton (1987), "Synthesis of Dimensional Tolerances of a Slider Crank Mechanism," 10th Applied Mechanisms Conference, Oklahoma State University.
Garrett, R. and A. Hall, Jr. (1969), "Effect of Tolerance and Clearance in Linkage Design," J. of Engineering for Industry, ASME, v 91, pp.198202, Feb.
Gladman, C. A. (1980), "Applying Probability in Tolerance Technology," Trans. Inst. Eng. Austral. Mech. Eng., v ME5, n 2, p. 82.
Gossard, D. C., R. P. Zuffante and H. Sakurai (1988), "Representing Dimensions, Tolerances, and Features in MCAE Systems," IEEE Computer Graphics & Applications, v 8 n 2, pp. 5159, Mar.
Greenwood, W. H. (1987), "A New Tolerance Analysis Method for Designers and Manufacturers," (Dissertation), ADCATS Report No. 872, Brigham Young University.
Greenwood, W. H. and K. W. Chase (1987), "A New Tolerance Analysis Method for Designers and Manufacturers," J. of Engineering for Industry, ASME, v 109, pp.112116, May.
Greenwood, W. H. and K. W. Chase (1988), "Worst Case Tolerance Analysis with Nonlinear Problems," J. of Engineering for Industry, ASME, v 110, pp.232235, Aug.
Greenwood, W. H. and K. W. Chase (1990), "Root Sum Squares Tolerance Analysis with Nonlinear Problems," J. of Engineering for Industry, ASME, v 112, pp.382384, Nov.
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Marler, J. (1988), "Nonlinear Tolerance Analysis Using the Direct Linearization Method," ADCATS Report No. 886, Brigham Young University.
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Martino, P. M., and G. A. Gabriele (1989b), "Estimating Jacobian and Constraint Matrices in Variational Geometry Systems," Failure Prevention and Reliability  1989, ASME Paper No. DEVol 16, pp. 7984.
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Michael, W., and Siddall, J. N. (1982), "The Optimal Tolerance Assignment with Less Than Full Acceptance," J. of Mechanical Design, ASME, vol. 104, pp. 855860, Oct.
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original report by Kenneth W. Chase & Alan R. Parkinson