# mel705-15.ppt

28 May 2023
1 sur 23

### mel705-15.ppt

• 1. Fractional Factorial Designs of Experiments P M V Subbarao Professor Mechanical Engineering Department Selection of Few Significant Parameters for Experimentation…..
• 2. Full Factorial Design : Design Matrix
• 3. Full Factorial Design : Result Matrix
• 5. Sample 23 Experiment 7 9 9 9 8 3 8 3 9 + 9 + 3 + 3 6 7 + 9 + 8 + 8 8 Main Effect of X1 6 – 8 = -2
• 6. Why Fractional Factorials? Full Factorials No. of combinations  For two-levels In engineering, this is the sample size -- no. of prototypes to be built. Continuous development of Knowledge is introducing more and more factors
• 7. Why so many Treatments? “There tends to be a redundancy in full factorial designs” – redundancy in terms of an excess number of interactions that can be estimated … Fractional factorial designs exploit this redundancy …”  philosophy
• 8. How to select a subset of 4 runs from a 23=8 -run design? Many possible “fractional” designs
• 11. Third Choice Wow! Balanced design All factors occur and low and high levels same number of times; Same for interactions Columns are orthogonal. Projections …
• 12. How to select a subset full factorial design • We note that the product of any two columns is zero. • Also the column sums are zero. • Hence the three columns may be considered as vectors that form an orthogonal set. • In fact while calculating the sample variance earlier these properties were used without being spelt out.
• 13. Want to study 5 factors (1,2,3,4,5) using a 2^4 = 16-run design i.e., construct half-fraction of a 2^5 design = 2^{5-1} design
• 14. DOE - Taguchi Method • Dr. Taguchi of Nippon Telephones and Telegraph Company, Japan has developed a method based on " ORTHOGONAL ARRAY " experiments. • This gives much reduced " variance " for the experiment with " optimum settings " of control parameters. • "Orthogonal Arrays" (OA) provide a set of well balanced (minimum) experiments serve as objective functions for optimization.
• 15. Taguchi Method : When to Select a ‘larger’ OA to perform “Factorial Experiments” • We always ‘think’ about ‘reducing’ the number of experiments (to minimize the ‘resources’ – equipment, materials, manpower and time) • However, doing ALL / Factorial experiments is a good idea if – Conducting experiments is ‘cheap/quick’ but measurements are ‘expensive/take too long’ – The experimental facility will NOT be available later to conduct the ‘verification’ experiment – We do NOT wish to conduct separate experiments for studying interactions between Factors
• 16. Taguchi Method Design of Experiments • The general steps involved in the Taguchi Method are as follows: • 1. Define the process objective, or more specifically, a target value for a performance measure of the process. • 2. Determine the design parameters affecting the process. • The number of levels that the parameters should be varied at must be specified. • 3. Create orthogonal arrays for the parameter design indicating the number of and conditions for each experiment. • The selection of orthogonal arrays is based on the number of parameters and the levels of variation for each parameter, and will be expounded below. • 4. Conduct the experiments indicated in the completed array to collect data on the effect on the performance measure. • 5. Complete data analysis to determine the effect of the different parameters on the performance measure.
• 18. Determining Parameter Design Orthogonal Array • The effect of many different factors on the performance characteristic in a condensed set of experiments can be examined by using the orthogonal array experimental design proposed by Taguchi. • The main factors affecting a process that can be controlled (control Factors) should be determined. • The levels at which these parameters should be varied must be determined. • Determining what levels of a variable to test requires an in-depth understanding of the process, including the minimum, maximum, and current value of the parameter. • If the difference between the minimum and maximum value of a parameter is large, the values being tested can be further apart or more values can be tested. • If the range of a parameter is small, then less values can be tested or the values tested can be closer together. • Typically, the number of levels for all parameters in the experimental design is chosen to be the same to aid in the selection of the proper orthogonal array.
• 20. Orthogonal Array Selector Number of Factors Number of Levels
• 21. Taguchi Method : How to Select a ‘larger’ OA to perform “Factorial Experiments” Control Factors and Levels Factorial Combinations Suitable OA 2 CF / 2-levels 4 L4 3 CF / 2-levels 8 L8 4 CF / 2-levels 16 L16 5 CF / 2-levels 32 L32