eprintid: 180 rev_number: 4 eprint_status: archive userid: 6 dir: disk0/00/00/01/80 datestamp: 2008-10-13 lastmod: 2015-05-29 19:48:50 status_changed: 2009-04-08 16:55:26 type: report metadata_visibility: show item_issues_count: 0 creators_name: Lamoureux, Michael contributors_name: Akers, Benjamin contributors_name: Bohun, Sean C. contributors_name: Gibson, Peter contributors_name: Hofinger, Andreas contributors_name: Lobb, Jason contributors_name: Roberts, Malcolm title: General Statistical Design of an Experimental Problem for Harmonics ispublished: pub subjects: transport studygroups: ipsw8 companyname: Michelin Tire Michelin Tire Corporation full_text_status: public abstract: Four years ago, the Michelin Tire Corporation proposed a problem on experimental design, to improve the manufacturing process for their tires. The idea is basically to determine the effects of placements for various layers built up in the construction of a tire, to allow the design of a smooth tire with a smooth ride. A highly success solution was developed, and it has been reported that this method introduced savings of over half a million dollars in their test processes. This year, Michelin returned to the workshop with an extension to the original problem, to address specific refinements in the testing method. This report summarizes the work completed in course of the five day workshop. It was clear early in the workshop that this problem could be handled quickly by reviewing the analysis which was done in 2000, and extending those ideas to the new problems at hand. We reviewed the required Fourier techniques to describe the harmonic problem, and statistical techniques to deal with the linear model that described how to accurately measure quantities that come from real experimental measurements. The “prime method” and “good lattice points method” were reviewed and re-analysed so we could understand (and prove) why they work so well. We then looked at extending these methods and successfully found solutions to problem 1) and 2) posed by Michelin. Matlab code was written to test and verify the algorithms developed. We have some ideas on problems 3) and 4), which are also described. problem_statement: Tires are subjected to a variety of force measurements that are stored as periodic waveforms. Harmonic components of these waveforms are related to tire performance characteristics such as noise and comfort and hence the control and reduction of the amplitudes of these harmonics is an important activity of manufacturing. Technicians may choose to perform designed experiments on their production processes to understand better their impact on the resulting force harmonics. It could be advantageous to have a general design of experiment methodology which allows technicians to choose optimal designs for their studies. The original problem was posed at PIMS IPSW 2000 as follows: Develop a method that allows estimation of n harmonics on m production steps from sampled waveforms on tires. This method should be flexible, robust and easily constrained to meet operating conditions. For example we measure a waveform of force variation on a cured tire sampled at 256 equally-spaced points made with 5 products with joints at fixed angular positions. Then we change these relative angular positions, construct a new tire and measure its waveform. We would like to decompose the overall effect represented in each tire into contributions due to each product. We can do this with each harmonic of the waveform but would like to ensure good estimation of all effects for all of a specified set of harmonics. The new problem for 2004 would be to extend the previous PIMS IPSW 2000 results on Statistical Design in any of several directions that are discussed below. 1. Develop fully the method called the Good Lattice Points (GLP) method presented in 2000 so that it could be implemented in practice to allow estimation without the prime number restriction. 2. Include the fitting of a few non-harmonic frequencies with the harmonics and find good designs for these (assumes that all harmonics are not fitted) such as a frequency that passes through the signal exactly 0.62 times. The harmonics are relative to the tire circumference in the old problem, but sometimes effects such as extrusion or measuring devices put sinusoidal patterns into the overall waveform but these effects are not harmonics (the periods are not integral divisors of the tire circumference) of the tire but rather have periods that are fractional parts (like 0.62). We use multiple linear regression to estimate non-harmonic frequency effects and harmonic frequency effects simultaneously (with some correlation between the estimates). The problem is to provide an optimal design strategy for this situation given that we can provide some information like number of non-harmonic frequency effects and possible ranges for their frequencies. 3. Determine the best design for a function of the harmonics of different types such as f(type 1 harmonic 1, type 2 harmonic 1) where the experiment is performed on rotation of tire components as before and f is of a specified class or form. In this case we measure two or more type of waveforms on each tire as before. We then combine these multiple outputs into single derived output (often linearly by summation etc. but it could be non-linear). We want to ensure adequate estimation of the product effects for each of a selected set of harmonics for this derived output. 4. Expand the concept to a two dimensional Fourier transform or equivalent where the surface could be considered flat or as the surface of an inflated tire (semi-toroidal). date: 2004 date_type: published pages: 19 citation: Lamoureux, Michael (2004) General Statistical Design of an Experimental Problem for Harmonics. [Study Group Report] document_url: http://miis.maths.ox.ac.uk/miis/180/1/michelin.pdf