Shanghai Jiao Tong University 1 ME250: Statistics & Probability ME 250: Design & Manufacturing I School of Mechanical Engineering

Shanghai Jiao Tong University 2 Outline Introduction Tolerances Reliability Statistical Process Control Regression Summary

Shanghai Jiao Tong University 3 Introduction Uncertainty in Design & Manufacturing Learning Objectives Understand the elementary aspects of statistics and probability. How to represent data. Discrete and continuous distributions. The normal distribution and how to use tables of the cumulative distribution. Probability of single and multiple events. Obtaining trends from noisy data by regression. Applications in statistical process control and geometric dimensioning and tolerances. Taylor’s tool life equation and how to use it.

Shanghai Jiao Tong University 4 Example 1 Suppose a company produces 1000 engines per year and 5 engines fail on average per year. The failures can be eliminated by a 100% inspection program. If it costs $10,000 to replace an engine but $100 to inspect each one, then the costs are: Inspect: 1000 * $100 = $100,000 Replace: 5 * $10,000 = $50,000 Obviously, it is cheaper not to inspect. If 10 or more engines fail, then the inspection program is cheaper. Therefore, it is very important for the company to maintain good statistical data on its manufacturing process.

Shanghai Jiao Tong University 5 Tolerances Tolerances refer to how close a dimension or a surface finish of a manufactured part is to the desired or “nominal” dimension. The explicit specification of tolerances is critical for determining the manufacturing process(es) required to make a part and therefore its final cost. If the surface finish is not important, it may be possible to use cheaper or faster process than if the surface finish is critical.

Shanghai Jiao Tong University 6 Reliability Reliability is the likelihood that an object will perform its function or a process will yield satisfactory results. Often, the reliability of system must be predicted from an estimate of the reliability of its components.

Shanghai Jiao Tong University 7 Statistical Process Control Statistical Process Control is used to monitor processes in a way such that normal variability of the operation is accounted for, but long term degradation is observed. The process is sampled at given intervals and the statistics of the samples compared. The cost of the inspection must be balanced versus the dangers of either missing a change in the process or misinterpreting a statistical fluctuation for a change in the process.

Shanghai Jiao Tong University 8 Regression Regression allows us to determine a relationship between two or more variables, even when there is uncertainly in the measurements and the relationship may not be immediately obvious. We will use regression to establish the Taylor’s Tool Life Equation.

Shanghai Jiao Tong University 9 Terminology If the parameters of a model of a system or a process are known, given, or established from past history, then we have a probability problem and we can deduce the behavior of the system from the model. However, if the parameters are unknown and have to be estimated from the available data, then we have a statistical problem.

Shanghai Jiao Tong University 10 Representing Statistical Data Histogram Group the data into bins or intervals and show the size of each bin (interval) graphically. Cumulative distribution Show graphically how much of the data is below a given value

Shanghai Jiao Tong University 11 Example 1 Consider the following grade points from a class with 28 students: 70, 73, 74, 76, 76, 76, 76, 77, 77, 77, 78, 78,79, 79,83, 83, 84, 85, 86, 87, 89, 89, 90, 90, 92, 92, 93, 97 where the numbers have been arranged in numerical order. Grouping the results together in six intervals yields

Shanghai Jiao Tong University 12 Discrete Distributions

Shanghai Jiao Tong University 13 Variance

Shanghai Jiao Tong University 14 The solution of the example

Shanghai Jiao Tong University 15 Exercise in class (1) Given the following number: 51,53,54,56,56,56,56,57,57,57,58,59,63,63,64,65, 66,67,69,69,70,72,72,72,73,76 (a) Plot a histogram of this data using six intervals. (b) Plot the cumulative distribution also.

Shanghai Jiao Tong University 16 Exercise in class (1)

Shanghai Jiao Tong University 17 Discrete Distributions

Shanghai Jiao Tong University 18 Distributions

Shanghai Jiao Tong University 19 Continuous Distributions

Shanghai Jiao Tong University 20 Continuous Distributions

Shanghai Jiao Tong University 21 Continuous Distributions

Shanghai Jiao Tong University 22 Uniform Distributions

Shanghai Jiao Tong University 23 Normal Distributions

Shanghai Jiao Tong University 24 Normal Distributions

Shanghai Jiao Tong University 25 Normal Distributions

Shanghai Jiao Tong University 26 The standard normal distributions

Shanghai Jiao Tong University 27 The standard normal distributions

Shanghai Jiao Tong University 28 The standard normal distributions

Shanghai Jiao Tong University 29 The standard normal distributions

Shanghai Jiao Tong University 30 Example 2

Shanghai Jiao Tong University 31 Example 3

Shanghai Jiao Tong University 32 Example 4a

Shanghai Jiao Tong University 33 Example 4b

Shanghai Jiao Tong University 34 Example 4c

Shanghai Jiao Tong University 35 Example 4d

Shanghai Jiao Tong University 36 Exercise in class (2) The length of a batch of 500 steel rods are approximately normally distributed with mean 11 cm and standard deviation 1 cm. Estimate the number of rods which are longer than 10 cm.

Shanghai Jiao Tong University 37 Exercise in class (2)

Shanghai Jiao Tong University 38 Probability: Weibull Distributions

Shanghai Jiao Tong University 39 Weibull Distributions

Shanghai Jiao Tong University 40 Example 5a

Shanghai Jiao Tong University 41 Example 5b

Shanghai Jiao Tong University 42 Example 5c

Shanghai Jiao Tong University 43 Example 5d

Shanghai Jiao Tong University 44 Exercise in class (3)

Shanghai Jiao Tong University 45 Exercise in class (3)

Shanghai Jiao Tong University 46 Combining Statistics

Shanghai Jiao Tong University 47 Combining Statistics

Shanghai Jiao Tong University 48 Combining Statistics

Shanghai Jiao Tong University 49 Combining Statistics

Shanghai Jiao Tong University 50 Example 6a

Shanghai Jiao Tong University 51 Example 6b

Shanghai Jiao Tong University 52 Exercise in class (4)

Shanghai Jiao Tong University 53 Exercise in class (4)

Shanghai Jiao Tong University 54 Exercise in class (5)

Shanghai Jiao Tong University 55 Exercise in class (5) Do = Di+2W = *0.139 = to = *0.004 = Do = To =

Shanghai Jiao Tong University 56 Probability

Shanghai Jiao Tong University 57 Probability

Shanghai Jiao Tong University 58 Probability

Shanghai Jiao Tong University 59 Probability

Shanghai Jiao Tong University 60 Example 7

Shanghai Jiao Tong University 61 Example 8

Shanghai Jiao Tong University 62 Example 9

Shanghai Jiao Tong University 63 Exercise in class (6)

Shanghai Jiao Tong University 64 Exercise in class (7)

Shanghai Jiao Tong University 65 Exercise in class (7) 0.99*(1-(1-0.90) 2 )*0.99 = 0.99*0.99*0.99 = *(1-(1-0.90) n )*0.99 > 0.98; n>3.991 N=4 N 接近无穷大， (1-0.90) n =0 0.99*1*0.99=0.9801<0.99 Therefore it is impossible.

Shanghai Jiao Tong University 66 Summary Uncertainty is a natural part of any human enterprise. Statistics and probability is used in many aspects of the design and manufacturing cycle.