TOLERANCE ANALYSIS AND ASSIGNMENT Machinery's Handbook, 31st Edition
686
(7a) (7b)
A max A min
A A = + µ 3 σ A = − µ 3 σ A
Example: Predict the minimum and maximum assembly dimension for the stack-up diagram and dimensions given in Fig. 5. Assume manufacturing data are not available for this example and apply the RSS method. Solution: Using Equations (5) and (6), calculate a mean and standard deviation to use as model parameters for each of the contributing dimensions in Fig. 5 . Starting with B: Dim Mean (μ) Standard Deviation (σ) B 0.3755 0.000167 C 0.5882 0.000800 D 1.75 0.000333 E 0.185 0.000833 F 1.75 0.001667 G 0.9375 0.000667 Repeating this calculation for dimensions C through G produces the table values shown. Next, the assembly dimension parameters are calculated using Equations (3) and (4) : = + 0376 0375 2 0376 0375 6 = − = = 03755 . 0000167 . . . . . µ σ
μ A =
+
+ + .
=
–0.
3755
– 0.
5882
–
175 .
0
185 175 .
0
.
9375
1 0 588 .
+ + + 0 000167 0 000800 0 000333 0 000833 0 001667 0 000 2 2 2 2 2 . . . . . . + +
667 0 002167 2 = .
σ
A =
Finally, the assembly limits are calculated using Equations (7a) and (7b) :
+ ( )( ) = 0.1588 3 0 002167 0 165301 . . − ( )( ) = 0.1588 3 0 002167 0 152299 . .
A max = A min =
Therefore, assemblies are expected to measure between 0.152299 and 0.165301. Monte Carlo Simulation: This random-number-based calculation method attempts to model variation realistically. The simulation runs numerous cases and builds a probabil- ity distribution for the stack-up, which is used to predict results, including defect rates. For each case, contributing dimensions in the stack-up chain are randomly chosen according to their statistical distribution (normal, uniform, or otherwise). The dimensions are com- bined according to the stack-up equation, which may be nonlinear. Tolerance Assignment.— This is the process of distributing the “tolerance budget” among contributing dimensions of the stack-up chain to manage costs and meet target quality lev- els. As explained above, analysis produces one solution (assembly dimension) from several inputs (dimensions of contributing parts), whereas assignment attempts to produce several contributing part dimensions from one assembly dimension. There are infinite ways to assign part tolerances to achieve a desired assembly tolerance (thus, analytical solutions usually are not possible), with several methods for assigning tolerances being most common. Even Assignment: Part feature tolerances are assigned equally among assignable toler- ances to stay within the required assembly tolerance. Proportional Scaling: Preliminary tolerances are assigned and analyzed. If the assem- bly tolerance exceeds the allowed amount, assignable part feature tolerances are scaled down proportionally as needed. Weight Factors: Weight factors, determined on the basis of manufacturing difficulty, are allocated to each assignable tolerance. The assembly tolerance budget is divided across each tolerance proportional to its weight factor. Cost Minimization: A cost-versus-tolerance relationship is determined for each con- tributing dimension. The optimization algorithm varies and compares the tolerances of each dimension to arrive at the combination that minimizes total cost. Total Cost Minimization: This method loosens tolerances to reduce cost while consid- ering the cost of lower yields due to production of some number of defective parts.
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