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

DE-Vol. 82, Volume 1, ASME 1995
Proceedings of the ASME Design Engineering Technical Conferences
Boston, MA, Sept. 17-20, 1995, pp.353-360

Jinsong Gao
Kenneth W. Chase
Spencer P. Magleby

Mechanical Engineering Department
Brigham Young University
Provo, Utah

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Two methods for performing statistical tolerance analysis of mechanical assemblies are compared: the Direct Linearization Method (DLM), and Monte Carlo simulation. A selection of 2-D and 3-D vector models of assemblies were analyzed, including problems with closed loop assembly constraints. Closed vector loops describe the small kinematic adjustments that occur at assembly time. Open loops describe critical clearances or other assembly features. The DLM uses linearized assembly constraints and matrix algebra to estimate the variations of the assembly or kinematic variables, and to predict assembly rejects. A modified Monte Carlo simulation, employing an iterative technique for closed loop assemblies, was applied to the same problem set. The results of the comparison show that the DLM is accurate if the tolerances are relatively small compared to the nominal dimensions of the components, and the assembly functions are not highly nonlinear. Sample size is shown to have great influence on the accuracy of Monte Carlo simulation.


The linearization method and Monte Carlo simulation are the most commonly used methods for statistical tolerance analysis of mechanical assemblies, due to the versatility and speed of the linearization method and the nonlinear capability and accuracy of Monte Carlo simulation. The concern for using the linearization method is its accuracy, and so far, little information on accuracy is available to analysts. For Monte Carlo simulation, accuracy is related to the sample size, although the relationship of sample size and accuracy has not been well defined.

Traditionally, both the linearization method and Monte Carlo simulation for statistical tolerance analysis of mechanical assemblies are applied to explicit assembly functions, that is, the assembly feature or dimension must be expressed in terms of the component dimensions in the assembly (Cox 1986, Shapiro & Gross 1981, DeDoncker & Spencer 1987, Doepker & Nies 1989, Early & Thompson 1989, Fuscaldo 1991). In 2-D or 3-D space, this function is usually a nonlinear implicit function of the assembly variables. It is very difficult or impossible for a designer to establish an explicit function for "real world" assemblies. The authors have developed a generalized linearization method and modified Monte Carlo simulation for 2-D and 3-D assembly tolerance analysis using implicit assembly functions (Chase, Gao & Magleby 1995a, Gao 1993). The linearization method is called the Direct Linearization Method (DLM). These two methods will be described.

This paper applies the DLM and Monte Carlo simulation using implicit assembly functions to a set of mechanical assemblies in both 2-D and 3-D space. The results from the two methods are compared. A Monte Carlo simulation with a very large sample size is chosen as the "exact" solution. The effect of sample size on the accuracy of the Monte Carlo simulation is also investigated.

The DLM and modified Monte Carlo simulation using an implicit assembly function, were developed to take advantage of the increased use of CAD in product design. Solid modelers are used to create assembly models, and an assembly tolerance modeler is employed to include the tolerances of the components in the assembly models. Assembly tolerance analysis and allocation can then proceed on the assembly models using the DLM and Monte Carlo simulation. This section will discuss the vector-loop-based assembly tolerance modeler, and both the DLM and modified Monte Carlo simulation methods for assembly tolerance analysis.

Vector-Loop-based Assembly Tolerance Modeler

The DLM and Monte Carlo simulation were applied to the same vector loop based assembly tolerance models (Chase, Gao & Magleby 1995a, Chase, Gao & Magleby 1995b). Solid modelers do not generally contain the controlled dimension and tolerance data that are required for tolerance analysis. The vector loop assembly tolerance model extends the functions and data structure of the solid modeler to include tolerancing capabilities. It allows the designer to Create 2-D and 3-D vector assembly tolerance models graphically and add them to the solid model as objects. The vector model is stored as part of the solid model database. The model contains the complete dimension and tolerance information required for performing tolerance analysis. The complete model may then be accessed by the tolerance analysis module which will perform statistical tolerance analysis and tolerance allocation (CATS Modeler 1994).

Manufactured parts are seldom used as single parts. They are used in assemblies of parts. The dimensional variations which occur in each component part of an assembly accumulate statistically and propagate kinematically, causing the overall assembly dimensions to vary according to the number of contributing sources of variation. The resultant critical clearances and fits which affect performance are thus subject to variation due to the tolerance stackup of the component part variations.

The three major sources of variation in assemblies are included in the models:

  1. dimensional (lengths and angles)

  2. geometric feature (ANSI Y 14.5)

  3. kinematic (small internal adjustments)

The model is based on a graphically generated vector chain(s) or loop(s) representing a mechanical assembly. Each vector represents a component dimension. Complex assemblies may require solving several vector loops simultaneously. Contact between mating parts is described by kinematic joints, (planar, slider, pin joints, etc.), which assist the designer to conceptually understand the adjustability within the assembly. Kinematic constraints assure that variations propagate through the assembly in a realistic way. Figure 1 shows the kinematic joints, datums and vector loop for the one-way clutch assembly discussed in a previous paper (Chase, Gao & Magleby 1995a).

Figure 1. Assembly kinematic joints, datums and vector loop of the one-way clutch.

In the one-way clutch assembly, the manufactured dimensions are a, c and e. The dimensions b, f1 and f2 are the assembly or kinematic variables, and their values are determined by the manufactured dimensions. The assembly or kinematic variables must be able to adjust to accommodate the variations of the manufactured dimensions (Shigley 1969). f1 is the critical assembly feature since its value will affect the performance of the clutch. The goal of assembly tolerance analysis of the clutch is to find out the effect of the tolerances of a, c and e on the variation of the main assembly feature f1.

The effects of ANSI Y 14.5 geometric feature control tolerances can also be included in the model. Models for propagating geometric feature variations through an assembly depend upon the corresponding kinematic joint types. With different joint types and 14 geometric feature controls, there are a great many relationships to define, but these are already tabulated (Chase, Gao & Magleby 1995b).

The assembly constraint for the one-way clutch can be expressed in the implicit function,

[Ra][Ta][Rb][Tb][Rc1][Tc1][Rc2][Tc2][Re][Te][Rf]= [I] (1)


R represents the rotational transformation and
T represents the translational transformation.
[Rf] is the final rotation to make the vector loop close.

A set of assembly tolerance specifications have been developed so the designer can express his design intent by placing tolerance limits on critical assembly features. Such features as the gap, parallelism and flushness of a car door and its mating door frame vary as the result of tolerance accumulation of a chain of component dimensions. By analyzing the variation in the dimensional chain, the gap variation may be estimated and assembly rejects predicted.

Direct Linearization Method

The DLM quantitatively estimates the variation of all critical assembly features and predicts the percent of assemblies which will fail to meet the design specs. It uses the kinematic assembly constraints as its assembly functions. For closed vector loops, the starting and ending reference frame must be located at the same point and oriented in the same direction when every vector in the loop has been traversed (equation (1)). These constraints result in nonlinear implicit functions of the assembly variables, which are difficult to solve.

The assembly constraint equations may be linearized by a first order Taylor’s series expansion and solved by linear algebra for the assembly variations in terms of component tolerances in the assembly. The assembly variations can then be obtained statistically and the assembly rejects can be predicted with the assembly variation for a given design specification (Chase, Gao & Magleby 1995a).

The DLM can also be used for tolerance allocation with predefined schemes, such as proportional scaling. Allocation is a tolerance selection tool to assist the designer in distributing the available assembly tolerance among the components of an assembly. Tolerances may be loosened on expensive processes and tightened on others to reduce cost while assuring that the design specs will be met.

Modified Monte Carlo Simulation

The Monte Carlo simulation performs assembly tolerance analysis using a random number generator which selects values for each manufactured variable, based on the type of statistical distribution assigned by the designer. These values are combined through the assembly function to determine a series of values of assembly variables. This series is then used to find the first four moments of the assembly variable, and finally, the moments can be used to determine the behavior of the assembly, such as mean, standard deviation, and percentage of assemblies which fall outside the design specifications, or assembly reject rate. For each simulation, the value of the assembly variables must be determined numerically. The final result of Monte Carlo Simulations will be accurate, provided careful attention has been given to proper sample size and accurate generation of sample variable distributions.

Since the assembly constraint equations are nonlinear implicit functions, an iterative solution must be obtained for each simulated assembly. This takes a lot CPU time, depending on the complexity of the assembly constraints. For 3-D assemblies, there may be more equations than the number of unknowns. A least squares fit must then be applied to find an approximation.
The procedures for the Monte Carlo simulation using implicit functions can be outlined as:

    1. Generate random variates representing each of the contributing manufactured dimensions in an assembly. The variations are drawn from the specified statistical distributions. The nominal dimensions of the manufactured variables plus their simulated variations serve as the input to the nonlinear assembly system.
    2. Establish the kinematic assembly constraints. The constraints act as the assembly functions describing the unknown assembly or kinematic variables. These constraints appear in nonlinear implicit functions.
    3. Choose the appropriate nonlinear solvers and solve for all unknown assembly features (or a least square fit solution).
    4. Repeat the above steps for sufficient number of simulated assemblies to generate a statistical distribution for the assembly features of interest.
    5. Apply limits to those features of interest. Use statistics to evaluate the simulated results of the assembly or kinematic variables and their variations.

The procedure is illustrated graphically in Figure 2.

Figure 2. Monte Carlo simulation for implicit assembly constraints

Test Cases

Seven 2-D assemblies and one 3-D assembly are included in the case studies. The complexity of each assembly is a function of the number of parts and the nature of the contact between parts. Sliding contact or rotational adjustments between mating parts are described by inserting kinematic joints into the vector model at points of contact. Each kinematic joint introduces degrees of freedom or variable dimensions which are the unknowns in the assembly equations. A closed vector loop will have three unknown kinematic variables, two closed loops will have six, etc., in 2-D space. Simultaneous solution of the assembly equations is required to determine the unknown variations. A summary of the size and complexity of the assembly modeling of these examples is reflected in Table 1. Complete dimensions, descriptions and sources of the example problems are presented elsewhere (Gao 1993).


Table 1. Complexity of example Assembly models
Assembly #Parts #Joints #D vector #F vector #C loops #O loops #Unkowns
Clutch 3 5 5 4 1 0 3
Bike 3 9 9   1 0 3
 Block 3 10 12   3 0 9
 Ratchet 3 9 8   2 0 6
 Tape Reel 4 10 10 4 1 1 4
Positioner 6 7 9   1 1 4
Quick slider 4 6 5   2 1 7
3D crank 5 6 6   1 0 5


# = the number of
C_loops = closed loops
D_vector = dimension vector
O_loops = open loops
F_vector = geometric feature variables


In this section, the results of the linearized analysis or the DLM solutions will be compared to Monte Carlo simulation. A Monte Carlo simulation with a sample size of 100,000 assemblies has been established as the reference value for each sample problem. Also investigated is the effect of sample size on the accuracy of Monte Carlo estimates of the assembly variations and the predicted rejects for the assemblies.

The DLM vs. Monte Carlo Simulation

The computed standard deviation of a critical assembly feature for 10 test problems is presented in Table 2. The assembly variation is the result of tolerance stackup of the contributing component dimensions in each assembly. In Table 2, the * sign means that the result includes both dimensional and geometrical feature tolerances of the components in the assembly.

Table 2. Summary of variation results - DLM vs. simulation


Variation of Controlled Assembly Variable



Relative error %

One-way clutch




One-way clutch*




Bike crank




Geometry block




Pawl and ratchet




Quick slider




Tape reel




Tape reel *








3D crank slider




From Table 2, it can be seen that the assembly variations, as predicted by the DLM and Monte Carlo methods, are in close agreement. The DLM approximation appears to be very accurate in predicting assembly variations. For most problems, the error is less than a tenth of one percent.

Table 3 compares the predicted rejects for the linearized and simulated analyses. For most of the cases, the estimated assembly rejects on both limits by the DLM are good approximations to the simulated assembly rejects. However, the one-way clutch assembly shows a 2:1 ratio for the lower and upper limit rejects, while the linear estimation of the rejects is symmetric. This asymmetry may be caused by a highly nonlinear assembly constraint skewing the resulting distribution.

Table 3. Summary of reject results - DLM vs. simulation


Upper/Lower Rejects

Total Rejects






One-way clutch






One-way clutch*






Bike crank






Geometry block






Pawl and ratchet






Quick slider






Tape reel






Tape reel *












3D crank slider






Effect of Nonlinear Assembly Functions

Nonlinear kinematic assembly constraints can produce mean shifts and skewed distributions in the assembly or kinematic variables. Table 4 lists the distribution parameters s, b1, b2 and mean shifts for the controlled assembly variables of all the tested cases (sample size 100,000). s describes the spread,, b1 the skewness, and b2 the peakedness of the distribution. For a Normal distribution, b1 is zero and b2 is equal to 3.0.

Table 4. Summary of mean shifts and distribution parameters


Controlled Assembly Variable

Estimated s

Mean shift (in s units)


One-way clutch




One-way clutch*




Bike crank




Geometry block




Pawl and ratchet




Quick slider




Tape reel




Tape reel *








3D crank slider




Normal Distribution



In Table 4, b1 and b2 do not deviate much from a Normal distribution. The one-way clutch, however, exhibited 10 times greater mean shift and skewness than any other case. The clutch results also showed the most asymmetry of all eight test problems, as indicated by the difference in rejects at the upper and lower limits in Table 3. Variance estimates, presented in Table 2, seem unaffected by nonlinearity.

A mean shift may cause nonsymmetric assembly rejects, as illustrated in Figure 3. Using the Standard Normal Tables for area under a Normal distribution, a 0.02s shift in the mean would change the clutch upper and lower rejects from 296/296 to 315/219. The rest of the shift, to 424/201, is presumed to be due to skewness.



Effect of Sample Size on Assembly Variation

The accuracy of the simulated assembly variation depends upon the sample size. If the result of the assembly variation at sample size 100,000 is considered as the accurate solution, Table 5 lists the assembly variations at four different sample sizes for each of the examples, along with the relative error compared with the accurate solution. From the table, it can been seen that even for a small sample size, such as 1,000, the assembly variations are within 5% error, comparing to those at 100,000 sample sizes. Therefore, for assembly variations alone, small sample sizes, such as 1,000 to 5,000, can give acceptable results.

This effect of sample sizes on variation can be illustrated in Figure 4. The variation limits for the DLM are shown for comparison. From the figure, it can be seen that in order for the Monte Carlo simulation to be more accurate than the DLM in predicting assembly variation, the average sample size for this group of assemblies must be greater than 30,000.

Table 5. Estimates of assembly variation vs. sample size


Variation and % Error of Controlled Assembly Variable





One-way clutch





One-way clutch*





Bike crank





Geometry block





Pawl and ratchet





Quick slider





Tape reel





Tape reel *










3D crank slider






Figure 4. Effect of sample size on PREDICTED variation: simulation vs. the DLM

Effect of Sample Size on Assembly Rejects

The relatively small error between the assembly variation at the given sample size and the more accurate solution (at sample size 100,000) does not generally mean that the error of assembly rejects between these two cases is also small. Table 6 gives the assembly rejects for lower and upper limits, respectively, at different samples sizes. In comparing the relative error, the rejects should be proportionally extended to the number corresponding to a sample size of 100,000. For example, the rejects for the 10,000 sample size should be multiplied by 10 in order to compare with the 100,000 values.

From the tables, the conclusion can be made that even for a sample sizes of 10,000, the relative errors for many of the examples are still not acceptable. Judging from the accuracy on assembly rejects, large sample sizes, say 100,000 or even larger, are required to achieve reasonable precision.

Figure 5 shows the difference range plot of the assembly rejects for the cases tested by the Monte Carlo simulation and the DLM at various sample sizes. From the figure, it can be seen that the difference range shrinks very fast as the sample size increases. In order for the Monte Carlo simulation to be more accurate than the DLM in predicting assembly rejects for this group of assemblies, the sample size must be larger than 10,000. Otherwise, the Monte Carlo simulation may give poorer predictions about assembly rejects.

For accuracy at high quality levels, the sample size must be increased considerably, as shown in Table 7 (Shapiro and Gross, 1981).

Table 6. Estimates of assembly rejects vs. sample size (upper/Lower limit)


Rejects at Upper/Lower Limits of Controlled Assembly Variable





One-way clutch





One-way clutch*





Bike crank





Geometry block





Pawl and ratchet





Quick slider





Tape reel





Tape reel *










3D crank slider






Figure 5. Effect of sample size on TOTAL rejects: Simulation vs. the DLM

Table 7. Sample size required for Monte Carlo simulations



Error in Rejects ()





























Note: The confidence interval is 90%.


Seven 2-D and one 3-D mechanical assemblies, two of them with geometrical feature control tolerances, have been tested using Monte Carlo simulation and the Direct Linearization Method. The following conclusions can be obtained from the comparison of these methods:

    1. The DLM is accurate in estimating the assembly variations. It is also accurate in most of the cases in predicting assembly rejects, except for highly nonlinear assembly constraints.
    2. The sample size for Monte Carlo simulations has great influence in predicting assembly rejects, but has little effect on the simulated assembly variations once the sample size is larger than 1,000.
    3. Nonlinear assembly constraints may cause a mean shift in the resultant assembly or kinematic dimensions and skewness in the distribution.


This work was sponsored by ADCATS, the Association for the Development of Computer-Aided 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.


Chase, K. W., Gao, J. and Magleby, S. P., 1995a, "Generalized 2-D Tolerance Analysis of Mechanical Assemblies with Small Kinematic Adjustments," Journal of Design and Manufacturing, Vol. 5, No. 2, 1995.

Chase, K. W., Gao, J. and Magleaby, S. P., 1995b, "Including Geometric Feature Variations in Tolerance Analysis of Mechanical Assemblies," Accepted for publication by Journal of Design and Manufacturing.

Cox, N. D., 1986, "Volume 11: How to Perform Statistical Tolerance Analysis," American Society for Quality Control, Statistical Division.

DeDoncker, D. and Spencer, A., 1987, "Assembly Tolerance Analysis with Simulation and Optimization Techniques," SAE Technical Paper Series, No 870263.

Doepker, P. E. and Nies, D., 1989, "Designing Brake Components Using Variation Simulation Modeling," Failure Prevention and Reliability -1989, ASME Publ. DE-Vol. 16, pp.131-138.

Early, R. and Thompson, J., 1989, "Variation Simulation Modeling-Variation Analysis Using Monte Carlo Simulation," Failure Prevention and Reliability -1989, ASME Publ. DE-Vol. 16, pp.139-144.

Fuscaldo, J. P., 1991, "Optimizing Collet Chuck Designs Using Variation Simulation Analysis," SAE Technical Paper Series, No. 911639, Aug.

Gao, J. 1993, "Nonlinear Tolerance Analysis of Mechanical Assemblies," Ph.D. Dissertation, Brigham Young University, Provo, UT.

Shapiro, S. S., Gross, A., 1981, "Statistical Modeling Techniques," Marcel Dekker.

Shigley, J. E., 1969, "Kinematic Analysis of Mechanisms," McGraw-Hill Book Company, New York.

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Original Report by Jinsong Gao, Kenneth W. Chase, & Spencer P. Magleby