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Publication # 941  
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General
2D Tolerance Analysis of Mechanical Assemblies
with Small Kinematic Adjustments


Kenneth W. Chase, Jinsong
Gao, and Spencer P. Magleby Department of Mechanical Engineering Brigham Young University, Provo, Utah 
Assembly tolerance analysis is a key element in industry for improving product quality and reducing overall cost. It provides a quantitative design tool for predicting the effects of manufacturing variation on performance and cost. It promotes concurrent engineering by bringing engineering requirements and manufacturing requirements together in a common model.
A new method, called the Direct Linearization Method (DLM), is presented for tolerance analysis of 2D mechanical assemblies which generalizes vector loopbased models to include small kinematic adjustments. It has a significant advantage over traditional tolerance analysis methods in that it does not require an explicit function to describe the relationship between the resultant assembly dimension(s) and manufactured component dimensions. Such an explicit assembly function may be difficult or impossible to obtain for complex 2D assemblies.
The DLM method is a convenient design tool. The models are constructed of common engineering elements: vector chains, kinematic joints, assembly datums, dimensional tolerances, geometric feature tolerances and assembly tolerance limits. It is well suited for integration with a commercial CAD system as a graphical front end. It is not computationally intensive, so it is ideally suited for iterative design.
A general formulation is derived, detailed modeling and analysis procedures are outlined and the method is applied to two example problems.
An important consideration in product design is the assignment of tolerances to individual component dimensions so the product can be produced economically and function properly. The designer may assign relatively tight tolerances to each part to ensure that the product will perform correctly, but this will generally drive manufacturing cost higher. Relaxing tolerances on each component, on the other hand, reduces costs, but can result in unacceptable loss of quality and high scrap rate, leading to customer dissatisfaction. These conflicting goals point out the need in industry for methods to rationally assign tolerances to products so that customers can be provided with high quality products at competitive market prices.
Clearly, a tool to evaluate tolerance requirements and effects would be most useful in the design stage of a product. To be useful in design, it should include the following characteristics:
To illustrate the problems associated with 2D tolerance analysis, consider
the simple assembly shown in figure 1, as described by Fortini [1967]. It is
a drawing of a oneway mechanical clutch. 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 pressure 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 F 1
is expected to vary due to manufacturing variations in the clutch component
dimensions. The independent manufacturing variables are the hub dimension
a, the cylinder radius c, and the ring radius e. The
distance b and angle F 1
are not dimensioned. They are assembly resultants which are determined by
the sizes of a, c and e when the parts are assembled.
By trigonometry, the dependent assembly resultants, distance b and
angle F1, can be
expressed as explicit functions of a, c and e.
The expression for angle F 1 may be analyzed statistically to estimate quantitatively the resulting variation in F 1 in terms the specified tolerances for a, c and e. If performance requirements are used to set engineering limits on the size of F1, the quality level and percent rejects may also be predicted.
When an explicit function of the assembly resultant is available, such as F1 in equation (1), several methods are available for performing a statistical tolerance analysis. These include:
The next section will briefly review these methods.
Establishing explicit assembly functions, such as equation (1), to describe
assembly kinematic adjustments, places a heavy burden on the designer. For
a general mechanical assembly, this relationship may be difficult or impossible
to obtain. Figure 2 shows a geometric block assembly. The resultant dimension
U1 is very difficult to express explicitly as
a function of only the independent component dimensions a, b,
c, d, e and f. It is very difficult to define
such explicit assembly functions in a generalized manner for "reallife"
mechanical assemblies. This difficulty makes the use of explicit functions
impractical in a CADbased system intended for use by mechanical designers.
The approach described in this paper solves the problem mentioned above by using implicit assembly functions with a vectorloopbased kinematic assembly model, so that less user intervention is needed for computeraided tolerance analysis of any mechanical assemblies. The next section reviews the principal methods that have been used for tolerance analysis. The following sections introduce the concepts of variation sources and assembly kinematics. The formulation of the DLM assembly tolerance analysis method is then presented, followed by specific examples.
2. Methods Available for Tolerance Analysis
This section will briefly review the methods available for nonlinear tolerance analysis when an explicit assembly function is provided which relates the resultant variables of interest to the contributing variables or dimensions in an assembly. The purpose of the review is to provide background for a discussion of a generalized method for treating implicit functions.
The linearization method is based on a first order Taylor series expansion
of the assembly function, such as equation (1). Then the variation DF1
may be estimated by a worst case or statistical model for tolerance accumulation
[Cox 1986, Shapiro & Gross 1981].
The derivatives of F1 with respect to each of the independent variables a, c and e are called the "tolerance sensitivities", and are essential to the models for accumulation, hence, the need for an explicit function is apparent.
System moments is a statistical method for expressing assembly variation in terms of the moments of the statistical distributions of the components in the assembly. The first four moments describe the mean, variance, skewness and kurtosis of the distribution, respectively. A common procedure is to determine the first four moments of the assembly variable and use these to match a distribution that can be used to describe system performance [Evans 1975a, 1975b, Cox 1979, 1986, Shapiro & Gross 1981].
Moments are obtained from a Taylor's series expansion of the assembly
function F1(xi)
about the mean, retaining higher order derivative terms, as shown in equation
4:
where mk is the kth moment, E is the expected value operator, xi are the variables a, c, and e, and m i are their mean values. Expanding the truncated series to the third and fourth power yields extremely lengthy expressions for the third and fourth moments.
Clearly, this method also relies on an explicit assembly function.
The basic idea of quadrature is to estimate the moments of the probability
density function of the assembly variable by numerical integration of a
moment generating function, as shown in equation 5:
where mk is the kth moment of the assembly distribution, w(a), w(c) and w(e) are the probability density functions for the independent variables a, c and e, and m_{a} ,m_{c} and m_{e} are their mean values. Engineering limits are then applied to the resulting assembly distribution to estimate the statistical performance of the system [Evans 1967, 1971, 1972].
The HasoferLind Reliability Index, also called Second Moment Reliability Index, was originally developed for structural engineering applications [Hasofer & Lind 1974, Ditlevsen 1979a, 1979b]. This sophisticated method has been applied to mechanical tolerance analysis [Parkinson 1978, 1982, 1983, Lee & Woo 1990]. The reliability index may be used to approximate the distance of each engineering limit from the mean of the assembly, and estimate the percent rejects. It requires only the means and covariances of the independent variables, which assumes that all the independent variables are normally distributed and independent.
The general idea of the Taguchi method is to use fractional factorial or orthogonal array experiments to estimate the assembly variation due to component variations. It may further be applied to find the nominal dimensions and tolerances which minimize a specified "loss function". The Taguchi method is applicable to both explicit and implicit assembly functions [Taguchi 1978].
The Monte Carlo simulation method evaluates individual assemblies using a random number generator to select values for each manufactured dimension, based on the type of statistical distribution assigned by the designer or determined from production data. These dimensions are combined through the assembly function to determine the value of the assembly variable for each simulated assembly. This set of values is then used to compute the first four moments of the assembly variable. Finally, the moments may be used to determine the system behavior of the assembly, such as the mean, standard deviation, and percentage of assemblies which fall outside the design specifications [Sitko 1991, Fuscaldo 1991, Craig 1989].
An explicit assembly function is required to permit substitution of random sets of component dimensions and compute the change in assembly variables for each assembly.
3. Variation Sources in Assemblies
In order to create a generalized approach for generating implicit assembly functions, the sources of variation in an assembly must be identified and categorized. With these categories in place, an engineer can use them to systematically create a model that can be used to derive the implicit functions.
There are three main sources of variation in a mechanical assembly: 1) dimensional variation, 2) geometric feature variation and 3) variation due to small kinematic adjustments which occur at assembly time. The first two are the result of the natural variations in manufacturing processes and the third is from assembly processes and procedures.
Figure 3 shows sample dimensional variations on a component. Such
variations are inevitable due to fluctuations of machining conditions, such
as tool wear, fixture errors, set up errors, material property variations,
temperature, worker skill, etc. The designer usually specifies limits for
each dimension. If the manufactured dimension falls within the specified
limits, it is considered acceptable. Since this variation will affect the
performance of the assembled product, it must be carefully controlled.
Geometric feature variations are defined by the ANSI Y14.5M1982
standard [ASME 1982]. These definitions provide additional tolerance constraints
on shape, orientation, and location of produced components. For example,
a geometric feature tolerance may be used to limit the flatness of a surface,
or the perpendicularity of one surface on a part relative to established
datums, as shown in Figure 4.
In an assembly, geometric feature variations accumulate and propagate similar to dimensional variations. Although generally smaller than dimensional variations, they may be significant in some cases, resulting from rigid body effects [Ward 1992]. A complete tolerance model of mechanical assemblies should therefore include geometric feature tolerances.
Kinematic variations are small adjustments between mating parts
which occur at assembly time in response to the dimensional variations and
geometric feature variations of the components in an assembly. For example,
if the roller in the clutch assembly is produced undersized, as shown in
figure 5, the points of contact with the hub and ring will change, causing
kinematic variables b and F1
to increase.
Usually, limiting values of kinematic variations are not marked on the mechanical drawing, but critical performance variables, such as a clearance or a location, may appear as assembly specifications. The task for the designer is to assign tolerances to each component in the assembly so that each assembly specification is met.
It is the kinematic variations which result in implicit assembly functions. Current tolerance analysis practices fail to account for this significant variation source. In a comprehensive assembly tolerance analysis model, all three variations should be included. If any of the three is overlooked or ignored, it can result in significant error. Only when a complete model is constructed, can the designer accurately estimate the resultant assembly features or kinematic variations in an assembly.
Since an assembly must adjust to accommodate the three types of variation, a model of the assembly must account for kinematics. The kinematics present in a tolerance analysis model of an assembly is different from the traditional mechanism kinematics. The input and output of the traditional mechanism are large displacements of the corresponding components, such as the rotation of the input and output cranks of a fourbar linkage. The linkage is composed of rigid bodies, so all the component dimensions remain constant, or fixed at their nominal values.
In contrast to this, the kinematic inputs of an assembly tolerance analysis model are small variations of the component dimensions around their nominal values and the outputs are the variations of assembly features, including clearances and fits critical to performance, as well as small kinematic adjustments between components.
The kinematic adjustments have a similar meaning to kinematic degrees of freedom, but the input motions do not refer to displacements of a mechanism. They actually represent differences from the nominal dimension from one assembly to the next.
The kinematic assembly equations describe constraints on the interaction between mating component parts. These constraints also serve as functions by which assembly variations may be studied. Since the assembly model is similar to a classical kinematic mechanism model, the analysis methods developed for mechanism kinematics can be applied to assembly variation analysis.
5. VectorLoopBased Assembly Models
Using the concepts presented in section 3 and 4, vectorloopbased assembly models use vectors to represent the dimensions in an assembly. Each vector represents either a component dimension or kinematically variable dimension. The vectors are arranged in chains or loops representing those dimensions which "stack" together to determine the resulting assembly dimensions. The other model elements include kinematic joints, datum reference frames, feature datums, assembly tolerance specifications, component tolerances, and geometric feature tolerances (Figure 6).
Kinematic joints describe motion constraints at the points of contact
between mating parts. The assembly tolerance specifications are the engineering
design limits on those assembly feature variations which are critical to
performance. Vector models can provide a broad spectrum of the necessary
assembly functions for tolerance analysis.
Marler [1988] and Chun [1988] defined a set of kinematic joint types to
accommodate the kinematic variations at the contact points in 2D assemblies.
Figure 7 shows the joints and datums for 2D analysis. Corresponding modeling
rules have been developed for correctly representing the kinematic degrees
of freedom in an assembly.
Larsen [1991] and Trego [1993] further developed Chun and Marler's work and automated the procedure of generating vector loop relationships for assemblies.
There are several major advantage of vector models over solid models of assemblies:
6. DLM  Linearization of Implicit Assembly Functions
The Direct Linearization Method for assembly tolerance analysis is based on the first order Taylor’s series expansion of the assembly kinematic constraint equations with respect to both the assembly variables and the manufactured variables (component dimensions) in an assembly. Linear algebra is employed to solve the resulting linearized equations for the variations of the assembly variables in terms of the variations of the manufactured components. The resulting explicit expressions may be evaluated by either a worst case or statistical tolerance accumulation model.
6.1 Assembly Kinematic Constraint
Figure 8 shows a vector loop model of an assembly in 2D. Each vector
defines the relative rotation and translation from the previous vector.
If a vector represents a component dimension, then its variation is the
specified component tolerance. If it is a kinematic variable, its variation
must be determined by solving the vector equation. A similar interpretation
holds for the relative angles. Whether a length or angle is a kinematic
variable is determined by the degrees of freedom of the corresponding kinematic
joint defined at the points of contact between mating parts.
A closed vector loop, such as that shown in figure 8, defines a kinematic closure constraint for the assembly. This means there is some adjustable element in the assembly which always permits closure. Closed loop constraints can readily be expressed as implicit assembly functions.
An open vector loop describes a gap or a stack dimension, corresponding to a critical assembly feature which is the result of the accumulation of component tolerances.
Many assembly applications are described by an implicit system of open and closed loops requiring simultaneous solution. When no adjustable elements are present, no closed loops are required, in which case, open loops may be expressed as explicit assembly functions.
Assembly tolerance limits are determined by performance requirements. Component tolerance limits are determined from process characterization studies, but may have to be modified as a result of tolerance analysis, which reveals how each component variation contributes to the overall assembly variation. Engineering design limits may be placed on any kinematic variation in a closed loop or any assembly feature variation defined by an open loop. By comparing the computed variations to the specified limits, the percent rejects and assembly quality levels may be estimated.
By summing the vector components in the global x and y directions and summing the relative rotations, a vector loop produces three scalar equations, each summing to zero, as shown in equations 4, 5 and 6.
It is significant that each vector direction, represented by the arguments of the sine and cosine functions in the above equations, is expressed as the sum of the relative angles of all the vectors preceding it in the loop. Both the manufactured and kinematic angles are relative angles. This allows rotational variations to propagate realistically through an assembly, producing rigid body rotations of stacked mating parts. This effect of individual angle variations could not be described if global angles were used in the equations.
6.2. Taylor’s Expansion of Implicit Assembly Functions
The first order Taylor’s series expansion of the closed loop assembly
equations can be written in matrix form:
where {DH}: the variations of the clearance or closure relation,
{DX}: the variations of the manufactured variables,
{DU}: the variations of the assembly variables,
[A]: the first order partial derivatives of the manufactured variables,
[B]: the first order partial derivatives of the assembly variables.
Solving equation (9) for DU gives (assuming that
[B] is a fullranked matrix):
For an open loop assembly constraint, there may also be one or more closed
loop assembly constraints which the assembly must satisfy. The strategy for
such a system of assembly constraints is to solve the closed loop constraints
first, then substitute the solution in the open loop assembly constraint.
The variations of the open loop variables may then be evaluated directly.
This procedure may be expressed mathematically as follows:
where DV: the variations of the open loop assembly variables,
[C]: the first order derivative matrix of the manufactured variables in the open loop,
[D]: the first order derivative matrix of the assembly variables in the open loop.
If [B] is fullranked, equation (11) may be written as:
6.3. Estimation of Kinematic Variations and Assembly Rejects
The estimation of the kinematic variations can be obtained from equation (10) for the closed loop constraint, or equation (12) for the open loop constraint, by a worst case or statistical tolerance accumulation model.
where i = 1,..., n, tolj is the tolerance of the jth manufactured dimension, T_{ASM} is the design specification for the ith kinematic variable and [S] is the tolerance sensitivity matrix of the assembly constraint.
For closed loop constraints
For open loop constraints
The estimation of the assembly rejects is based on the assumption that the resulting sum of component distributions is Normal or Gaussian, which is a reasonable estimate for assemblies of manufactured variables. If all the component tolerances are assumed to represent three standard deviations of the corresponding process, the estimate of the related assembly variation will be three standard deviations. Equation 14 may easily be modified to account for tolerance limits which represent a value other than three standard deviations. The mean and standard deviation of the assembly variable can be used to calculate by either integration or table the assembly rejects for a given production quantity of assemblies.
As examples to demonstrate the procedure of applying DLM assembly tolerance analysis method to real assemblies, the oneway clutch assembly and the geometric block assembly are reexamined in greater detail.
7.1. Example 1. OneWay Clutch
Figure 6 illustrated the vectorloopbased model of the oneway clutch assembly. Table 1 shows the detailed dimensions for the assembly.
Part Name  Transformation  Nominal Dimension  Tolerance(±)  Variation 
Height of hub  Rotation  90°  
Translation a  27.645  0.0125  Independent  
Position of roller  Rotation  90°  
Translation b  4.81053  ?  Kinematic  
Radius of roller  Rotation  90°  
Translation c  11.43  0.01  Independent  
Radius of roller  Rotation f1  7.01838°  ?  Kinematic 
Translation c  11.43  0.01  Independent  
Radius of ring  Rotation  180°  
Translation e  50.8  0.05  Independent  
Closing vector  Rotation f2  97.01838°  ?  Kinematic 
Translation  0 
From Figure 6, the loop equations of the assembly follow naturally as:
(17)  
(18)  
(19) 
The known independent variables in this set of equations are a, c, and e. The unknown dependent variables are b, f1 and f2. Examination of the system of equations reveals that they are nonlinear functions of f1, which must be solved simultaneously for all three dependent variables. It is not clear how one would apply the tolerance analysis methods described earlier to a system of implicit assembly functions such as this, without first solving symbolically for an explicit function of f1 in terms of a, c, and e.
Note that dimension c appears twice in equation 18. Since both vectors are produced by the same process, they will both be oversized or undersized simultaneously.
Applying the DLM method, the first order derivative matrices [A], [B] and the sensitivity matrix [S] can be obtained.
With the sensitivity matrix known, the variations of the kinematic or assembly
variables can then be calculated by applying equation (13) or (14).
Worst case:  Statistical model:  (24) 
In this assembly, dimension f1 is the one which has a specified design tolerance since its mean value and variation will affect the performance of the clutch. The design limits for f1 are set to be TASM = ± 0.6°, with a desired quality level of ±3.0 standard deviations. The number of standard deviations Z to which the design spec corresponds may be calculated from the relation:
This standard deviation number can then be used to estimate the assembly reject rate h by either standard normal distribution tables or integration or empirical methods.
The assembly rejects for a production run of 1000 assemblies can be estimated
by
Assembly Rejects  = 2 h * Number of
the Assemblies
= 2(0.002959)1000 = 5.918 
(27) 
So, there would be about six which would function improperly (three at each design limit).
7.2. Example 2. Geometric Block Assembly
The geometric block assembly requires three vector loops to completely describe the assembly relationship, even though it is only a simple threecomponent assembly. Figure 9 shows the vector loop assembly model. Table 2 gives all the dimensions for the three vector loops.
Loop Name  Part Name  Transformation  Nominal Dim  Tolerance(±) 
Loop 1  Ground  Rotation  90°  
Closing angle
90° 
Translation U3  10.0477  ?  
Block  Rotation f_{2}  74.7243°  ?  
Translation U2  8.6705  ?  
Cylinder  Rotation  90°  
Translation a  6.62  0.2  
Cylinder  Rotation f_{1}  74.7243°  ?  
Translation a  6.62  0.2  
Ground  Rotation  90°  
Translation U1  18.7181  ?  
Loop 2  Ground  Rotation  90°  
Closing angle 180° 
Translation U3  10.0477  ?  
Block  Rotation f_{2}+90°  164.7243°  ?  
Translation b  6.805  0.075  
Block  Rotation  90°  
Translation U4  2.1894  ?  
Ground  Rotation f_{3}  105.2761°  ?  
Translation d  4.06  0.15  
Ground  Rotation  90°  
Translation f  3.905  0.125  
Loop 3  Ground  Rotation  90°  
Closing angle 180° 
Translation U3  10.0477  ?  
Block  Rotation f_{2}+90°  164.7243°  ?  
Translation b  6.805  0.075  
Block  Rotation  90°  
Translation U5  27.2965  ?  
Ground  Rotation U4  105.2761°  ?  
Translation c  10.675  0.125  
Ground  Rotation  90°  
Translation e  24.22  0.35  
Ground  Rotation  0°  
Translation f  3.905  0.125 
For each vector loop, three equations having the same format as equation (17) to (19) can be obtained. Therefore, nine equations are required to describe the assembly.

(27)

The variations of the kinematic or assembly variables can then be calculated by applying equation (13) or (14).
Worst case:  Statistical model:  (30) 
In this assembly, dimension U1 is the one for which a design tolerance was specified, since its value and variation will affect the desired performance of the assembly. If the design limits for U1 are set to be TASM = ± 0.28 and the estimated variation DU1 represents 3.0 standard deviations, then the design spec corresponds to Z standard deviations, where:
Then, the predicted reject rate on each design limit is estimated from
Assembly Rejects  = 2
h · Number of the Assemblies = 2(0.002540)1000 = 5.08 per 1000 assemblies 
(31) 
The Direct Linearization Method has been presented as a comprehensive method for 2D assembly tolerance analysis. It meets many of the requirements stated in the introduction.
This paper has presented a comprehensive method for assembly tolerance modeling and analysis. It will make possible new CAD tools for engineering designers which integrate manufacturing considerations into the design process. Using this tool, designers will be able to quantitatively predict the effects of variation on performance and producibility. Design and manufacturing personnel can adopt a common engineering model for assemblies as a vehicle for resolving their often competing tolerance requirements. Tolerance analysis can become a common meeting ground where they can work together to systematically pursue cost reduction and quality improvement.
Testing of the DLM assembly tolerance analysis method has shown that it produces accurate evaluations for engineering designs [Gao 1993].
In order to simplify the procedure, only dimensional tolerances and 2D assemblies were discussed in this paper. With appropriate modifications, the DLM can also be applied to 3D assemblies, including geometrical feature tolerances and kinematic adjustments. These results will be presented in a future paper.
Major portions of this research were performed by former graduate students Jaren Marler and Ki Soo Chun, with helpful suggestions by Dr. Alan R. Parkinson of the Mechanical Engineering Department. This work was sponsored by ADCATS, the Association for the Development of ComputerAided Tolerancing Software, a consortium of twelve industrial sponsors and the Brigham Young University, including Allied Signal Aerospace, Boeing, Cummins, FMC, Ford, HP, Hughes, IBM, Motorola, Sandia Labs, Texas Instruments and the U.S. Navy.
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