Probability Applications in Mechanical Design

Franklin Fisher

• Publisher: CRC Press
• Year: 2000
• Print ISBN: 9780824702601
• eBook ISBN: 9781135553159
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Civil Engineering

The authors of this text seek to clarify mechanical fatigue and design problems by applying probability and computer analysis, and further extending the uses of probability to determine mechanical reliability and achieve optimization. The work solves examples using commercially available software. It is formatted with examples and problems for use

1
Data Reduction

Data for load carrying material properties can be modelled using any probability distribution function. Statistical goodness-of-fit tests should be applied to determine if the data set could be randomly drawn from that distribution. Modelling has progressed beyond a simple two parameter (ì,ž) Gaussian distribution. This book treats the three parameter (ä, â, ã) Weibull distribution, as well as the traditional Gaussian distribution.

Many authors relegate the subject of data reduction to an appendix at the back of the book. In the opinion of the authors, the topic deserves much more attention.

I. REDUCTION OF RAW TABULATED TEST DATA OR PUBLISHED BAR CHARTS

A computer program such as SAS (Statistical Analysis System) statistical software or other compatible software is used to fit test data to a Gaussian curve.

(1.1)

where -∞≤×≤+∞≤ with

μ-is the mean for an infinite sample size
ž-is the standard deviation for an infinite sample size.
The program also fits data to a Weibull curve,

(1.2)

where γ≤×≤+∞ and g (x)=0 when x≤γ and

δ-is a scale parameter for infinite sample size
β-is a shape parameter for infinite sample size
γ-is a threshold parameter

The computer solves for the Weibull parameters as well as the Gaussian mean and standard deviation for a set of individual values from mechanical testing or published bar charts with more than one or two samples at a given value for the mid point of the bar (cell width).

Some computer software solves for only two Weibull parameters δ and β while γ is set to zero. The failure curves f(x) and g(x) are used to generate the reliability

(1.3)

(1.4)

The Weibull G(x), g(x) and Gaussian F(x) f(x) are unity curves with values from zero to one. The Weibull and Gaussian curves are used throughout this book except in chapter four where the constant failure-rate for reliability is introduced to explain the wear and tear on machinery. The computer calculations for Eqs. (1.1) and (1.2) allows the individual data to be sorted or listed as a bar chart. Sturges Rule [1.12, 1.16] presents an acceptable means of plotting data on linear coordinates, where, the data is grouped in cells of width w, over the range R, of the data.

1. Number of cells (K)=1+3.3 log₁₀ N where N is the number of individual data points. Grouped data from bar charts are already partitioned as presented in the data source, so the steps outlined for using Sturges Rule will not apply: however, the number of cells can be checked.
2. Range (R)=maximum value minus the minimum value.
3. Cell width (W)=R/K.

The number of c...

Table of contents

• COVER PAGE
• TITLE PAGE
• COPYRIGHT PAGE
• MECHANICAL ENGINEERING
• PREFACE
• LIST OF SYMBOLS
• 1: DATA REDUCTION
• 2: APPLICATION OF PROBABILITY TO MECHANICAL DESIGN
• 3: OPTIMUM DESIGN
• 4: RELIABILITY
• APPENDIX A: LINEARIZATION OF THE WEIBULL EQUATION
• APPENDIX B: MONTE CARLO CALCULATIONS
• APPENDIX C: COMPUTER OPTIMIZATION ROUTINES [3.20]
• APPENDIX D: MECHANICAL FAILURE RATES FOR NON-ELECTRONIC RELIABILITY
• APPENDIX E: STATISTICAL TABLES
• APPENDIX F: LOS ANGELES RAINFALL 1877–1997
• APPENDIX G: SOFTWARE CONSIDERATIONS