1. Tolerance Analysis Worst Case Normal Distributions Six-Sigma Mixed Uniform MAE 156A

2. 2 Worst Case Tolerance Analysis Tolerance analysis deals with the accumulation of dimensional variation within an assembly of multiple components. What is the worst case tolerance that can be expected from the final assembly? A long rod is constructed by bonding three smaller segments end-to-end. Each rod segment is cut within a specified +/- tolerance. Total Assembly Length (w/variation): Worst Case Assembly Tolerance:

3. MAE 156A 3 Statistical Tolerance Analysis Continuous random variables can be algebraically manipulated using their probability density functions.

4. MAE 156A 4 Normal Probability Distribution The Gaussian normal probability density function is given by the following equation: Probabilities associated with the normal distribution can be determined using spreadsheet and numerical analysis programs. EXCEL: Pr = normdist(b,mu,sigma,true) – normdist(a,mu,sigma,true) MATLAB: Pr = erf((b-a)/sigma/sqrt(2)/2)

5. MAE 156A 5 Yield and Reject Rate Quality control worries about rejecting parts that fall in the tails of the normal probability distribution. These parts vary significantly from the mean. reject yiel d nLUyieldreject PPM 17.512.50.6827317310 25.015.00.954545500 32.517.50.99732700

6. MAE 156A 6 Random Variable Sums The probability density function representing the sum of two random variables is found using the convolution operator. The sum of two normally distributed random variables is also normally distributed. oror

7. MAE 156A 7 Statistical Stack-Up If an assembly consists of multiple components, with dimensions governed by normal probability distributions, then the standard deviation of the assembly is: Assuming an inspection/rejection process is in place, the standard deviation of the assembly can be written in terms of component tolerances.

8. MAE 156A 8 Series Connections Tolerance analysis can be used with any component assembly that results in a summation of dimensional parameters (series connection). Worst Case: Probabilistc: T i given, find T ASM : T ASM given, find T i : Worst Case: Probabilistc:

9. MAE 156A 9 Six Sigma The "six sigma" concept maintains high quality (tight tolerance) assemblies even if the mean of the component parts shift in time. Mean shifts may occur as tooling wears, or if suppliers change. Mean shifts increase reject rate.

10. MAE 156A 10 Uniform Distribution A uniform probability distribution implies that all values between lower and upper bounds are equally likely to occur. The probability density function for a uniform random variable is given as:

11. MAE 156A 11 Uniform and Normal Combinations An assembly may have components that are best modeled as a mixed combination of normal and uniform probability distributions. The sum of many uniform random variables approaches a normal distribution.

12. MAE 156A 12 Further Reading Geng, H. (ed), Manufacturing Engineering Handbook, McGraw-Hill, 2004. http://roger.ucsd.edu:80/record=b6678812~S9 Cogorno, G.R., Geometric Dimensioning and Tolerancing for Mechanical Design, McGraw-Hill, 2006. http://roger.ucsd.edu:80/record=b6676309~S9