Machinery's Handbook, 31st Edition
TOLERANCE ANALYSIS AND ASSIGNMENT 685 (2a) (2b) Example: Predict the minimum and maximum assembly dimensions for the stack-up diagram and dimensions given in Fig. 5 using the worst-case method. Solution: Using Equations (2a) and (2b) : A A B C D E F G + + + – – max min min min max max max = = – – B C D E F G + + + – – min max max max min min min
A max A min
= =
+ +
1875 1755 + . 0 . 1825 1745 + . 0 .
+ +
= =
0 – .
375 376
0 – .
5858 1749 – .
0 0
.
9395 9355
0.1722 0.1454
0 – .
– .
596 1751 – .
0 0
.
Statistical: This approach produces likely values and ranges, predicts smaller assembly variations, and permits larger tolerances to be assigned. While it does not guarantee con- formity to requirements, conformance and defect rates can be predicted. Statistical calcula- tions are appropriate when many dimensions are contributing to a stack-up chain or when it can be assumed that the contributing dimensions follow normal distributions, such as when statistical process control (SPC) is used (see STATISTICAL ANALYSIS OF MANUFAC - TURING DATA on page 136 for statistics terminology used here). The assembly dimen- sion A is modeled as a normal distribution, with assembly mean ( µ A ) and assembly standard deviation (σ A ) calculated using the general stack-up equations written in statistical form: Assembly mean ( µ A ): (3) Assembly standard deviation (σ A ): (4) Example: Use the statistical method to determine the range of assembly dimension values, between which 99 percent of assembly dimensions are expected to fall, for the stack-up dia- gram and dimensions given in Fig. 5. Manufacturing data show the contributing dimensions are well modeled as normal distributions using the parameters included in the figure. Solution: Model assembly dimension A as a normal distribution with parameters calcu- lated using Equations (3) and (4) : µ µ µ µ µ µ µ A B C D E F G = + + + – – – A B C D E F G = + + + + + 2 2 2 2 2 2 σ σ σ σ σ σ σ
+ + + 375464 588126 1 75 128 185147 1 75 748 9373 . . . 0 0 0 0 0 0 31 – . – . – .
µ A =
1595 8 = 0. 0
σ
+ + + 0 000106 0 001004 0 000203 0 000874 0 001200 0 000 2 2 2 2 2 . . . . . . + +
517 0 001879 2 = .
A =
99 percent of the area under the normal distribution curve falls within the range of µ ±2.5758σ. Upper and lower limits of this range for assembly dimension A are calculated as:
+ ( − (
)( )(
) = ) =
A upper A lower
= =
0 0 0 0
1595 8 . 1595 8 .
25758 . 25758 .
000 .
1879
0 0
164348 . 154668 .
000 .
1879
Therefore, 99 percent of assembly dimensions are expected to measure between 0.154668 and 0.164348. Root Sum of Squares (RSS): This method belongs somewhere between the worst-case and statistical methods in terms of when applicable and the desired characteristics of the result. RSS is technically a statistical method with several simplifying assumptions applied. Each contributing dimension is modeled as a normal distribution, with mean (μ) and standard deviation (σ) calculated from the maximum and minimum values. Each contributing dimension mean (μ) is calculated as: (5) Each contributing dimension standard deviation (σ) is calculated as: (6) The calculations for assembly mean ( µ A ) and assembly standard deviation (σ A ) are the same as in the statistical method; see Equations (3) and (4). The assembly dimension is then predicted to fall between the limits: Max + Min 2 − Max Min 6 = µ = σ
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