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Publication # 952  
Home : Publication's Index : #952 
DEVol. 82, Volume 1,
ASME 1995
Proceedings of the ASME Design Engineering Technical Conferences
Boston, MA, Sept. 1720, 1995, pp.353360
Jinsong Gao
Kenneth W. Chase
Spencer P. Magleby
Two methods for performing statistical tolerance analysis of mechanical assemblies are compared: the Direct Linearization Method (DLM), and Monte Carlo simulation. A selection of 2D and 3D 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 2D or 3D 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 2D and 3D 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 2D and 3D 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 vectorloopbased assembly tolerance modeler, and both the DLM and modified Monte Carlo simulation methods for assembly tolerance analysis.
VectorLoopbased 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 2D and 3D 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:
dimensional (lengths and angles)
geometric feature (ANSI Y 14.5)
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 oneway clutch assembly discussed in a previous paper (Chase, Gao & Magleby 1995a).
In the oneway clutch assembly, the manufactured dimensions are a, c and e. The dimensions b, f_{1} and f_{2} 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). f_{1} 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 f_{1}.
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 oneway clutch can be expressed in the implicit function,
where:
R represents the rotational transformation and
T represents the translational transformation.
[R_{f}] 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.
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 3D 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:
The procedure is illustrated graphically in Figure 2.
Seven 2D assemblies and one 3D 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 2D 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).
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 
Where:
# = 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
Assembly 
Variation of Controlled Assembly Variable 

DLM 
Simulation 
Relative error % 

Oneway clutch 
.654091 
.653422 
+0.10238 

Oneway clutch* 
.671709 
.672153 
0.06605 

Bike crank 
.289424 
.289697 
0.09423 

Geometry block 
.259250 
.259183 
+0.02585 

Pawl and ratchet 
.349660 
.349436 
+0.06410 

Quick slider 
.014484 
.014485 
0.00690 

Tape reel 
.025586 
.025590 
0.01563 

Tape reel * 
.032966 
.032810 
+0.47546 

Positioner 
.709387 
.710589 
0.16916 

3D crank slider 
.035078 
.035012 
+0.18850 
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 oneway 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
Assembly 
Upper/Lower Rejects 
Total Rejects  
DLM 
Simulation 
Differences 
DLM/Sim 
Difference 

Oneway clutch 
296/296 
424/201 
128/95 
592/625 
33 
Oneway clutch* 
368/368 
514/274 
146/94 
736/788 
52 
Bike crank 
185/185 
201/196 
16/11 
370/397 
27 
Geometry block 
191/191 
181/208 
10/17 
382/389 
7 
Pawl and ratchet 
503/503 
444/504 
59/1 
1006/948 
58 
Quick slider 
1917/1917 
1956/2089 
39/172 
3834/4045 
211 
Tape reel 
245/245 
257/239 
12/6 
490/496 
6 
Tape reel * 
1448/1448 
1420/1355 
28/93 
2896/2775 
121 
Positioner 
154/154 
158/172 
4/18 
308/330 
22 
3D crank slider 
515/515 
483/535 
32/20 
1030/1018 
12 
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
Assembly 
Controlled Assembly Variable 

Estimated s 
Mean shift (in s units) 
Distribution(b1/b2) 

Oneway clutch 
0.218030 
0.019750 
0.104/3.0239 
Oneway clutch* 
0.223903 
0.021554 
0.087/3.0173 
Bike crank 
0.096475 
0.000415 
0.005/3.0288 
Geometry block 
0.086417 
+0.001620 
+0.002/3.0026 
Pawl and ratchet 
0.116553 
+0.006006 
+0.014/2.9876 
Quick slider 
0.004828 
+0.011392 
+0.030/3.0046 
Tape reel 
0.008529 
0.001665 
0.009/3.0094 
Tape reel * 
0.010989 
0.002921 
0.011/3.0047 
Positioner 
0.236462 
0.004287 
0.009/3.0460 
3D crank slider 
0.011693 
+0.003848 
+0.011/3.3258 
Normal Distribution 
0.00/3.0000 
In Table 4, b1 and b2 do not deviate much from a Normal distribution. The oneway 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.
FIGURE 3. MEAN SHIFT AND ITS EFFECT ON ASSEMBLY REJECTS.
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
Assembly 
Variation and % Error of Controlled Assembly Variable 

100 
1,000 
10,000 
100,000 

Oneway clutch 
.650935(0.38%) 
.643516(1.52%) 
.654599(0.18%) 
.653422 

Oneway clutch* 
.712223(5.96%) 
.689708(2.61%) 
.677427(0.78%) 
.672153 

Bike crank 
.304315(5.05%) 
.293267(1.24%) 
.287590(0.72%) 
.289697 

Geometry block 
.272015(4.95%) 
.260079(0.35%) 
.258631(0.21%) 
.259183 

Pawl and ratchet 
.352449(0.86%) 
.356603(2.05%) 
.349877(0.13%) 
.349436 

Quick slider 
.014862(2.60%) 
.014480(0.03%) 
.014383(0.70%) 
.014485 

Tape reel 
.021282(16.8%) 
.025054(2.09%) 
.025507(0.32%) 
.025590 

Tape reel * 
.031067(5.31%) 
.033235(1.30%) 
.032509(0.92%) 
.032810 

Positioner 
.690083(2.89%) 
.699265(1.59%) 
.705424(0.72%) 
.710589 

3D crank slider 
.030339(13.3%) 
.035053(0.12%) 
.035091(0.23%) 
.035012 
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)
Assembly 
Rejects at Upper/Lower Limits of Controlled Assembly Variable 

100 
1,000 
10,000 
100,000 

Oneway clutch 
0/1 
1/2 
21/41 
201/424 

Oneway clutch* 
0/0 
3/5 
32/52 
274/514 

Bike crank 
0/0 
3/0 
19/21 
196/201 

Geometry block 
0/0 
2/1 
17/13 
208/181 

Pawl and ratchet 
2/0 
5/5 
55/35 
504/444 

Quick slider 
1/4 
22/18 
211/179 
2089/1956 

Tape reel 
0/0 
3/1 
27/22 
239/257 

Tape reel * 
1/2 
20/19 
133/136 
1355/1420 

Positioner 
0/0 
1/0 
13/16 
172/158 

3D crank slider 
0/0 
6/4 
54/42 
535/483 
Figure 5. Effect of sample size on TOTAL rejects: Simulation vs. the DLM
Table 7. Sample size required for Monte Carlo simulations
Assembly 
Rejects 
Error in Rejects (±) 

5% 
10% 
25% 

0.9900 
0.0100 
107,000 
27,000 
4,300 

0.9950 
0.0050 
215,000 
54,000 
8,600 

0.9973 
0.0027 
400,000 
100,000 
16,000 

0.9990 
0.0010 
1,081,000 
270,000 
43,000 

0.9999 
0.0001 
10,823,000 
2,706,000 
433,000 
Note: The confidence interval is 90%.
Seven 2D and one 3D 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:
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.
Chase, K. W., Gao, J. and Magleby, S. P., 1995a, "Generalized 2D 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. DEVol. 16, pp.131138.
Early, R. and Thompson, J., 1989, "Variation Simulation ModelingVariation Analysis Using Monte Carlo Simulation," Failure Prevention and Reliability 1989, ASME Publ. DEVol. 16, pp.139144.
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," McGrawHill Book Company, New York.
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Original Report by Jinsong Gao, Kenneth W. Chase, & Spencer P. Magleby