The Math Of Random Signals, Fatigue Damage, Vibration Tests

The Math of Reliability Engineering, Fatigue Damage, and its Interesting Applications in the World of Vibration Tests.

What you'll learn

How to define Fatigue Damage mathematically

How to find the probability density of the peaks of a random signal

The importance of the Gaussian distribution and why it often appears when dealing with random signals

How to find the joint probability density of two Gaussian variables

What the Fatigue Damage Spectrum is, and why it is important for Fatigue tests

How to synthesize signals with a prescribed Fatigue Damage Spectrum

Requirements

Fourier Series

Fourier Transforms

Calculus and Multivariable Calculus

Random Variables

Expected value

probability density function

Description

In the first part of this course, the mathematics of fatigue damage is addressed. The most important equations are derived, and this will serve as a guide to illustrate the main applications of the theory in the second part.This course is not only aimed at those students who are interested in vibration tests and reliability engineering, but also at those who want to see the beautiful mathematics applied to engineering. In this case the theory contains: stochastic processes, probability distributions (and how they arise in nature/real-life applications), as well as single-degree-of-freedom systems, the probability density of the maxima of a random process, and much more. In this regard, the mathematical theory of damage is quite advanced mathematically; for instance, we will have to make use of the Gamma and the Bessel functions. Anyway, the concepts will be developed intuitively throughout the course, and the connection with real-life applications will always be highlighted.Most of the results shown in this course are described in the instructor’s PhD dissertation: “Advanced Mission Synthesis Algorithms for Vibration-based Accelerated Life-testing” and the references contained therein. This work will be public as of February 2023 after a two-year “embargo”, and it will be available for consultation. If you would like to consult this dissertation and you cannot find it online, just check the attachment to the first lecture of this course, you should find the dissertation attached.This theory was exploited to create Graphical User Interfaces (GUI), which were the result of the collaboration between the instructor and companies interested in the PhD project. We will use these GUI in the course to shed light on how the equations can be implemented to generate vibratory signals.Let's briefly contextualize the theory of fatigue damage in the field of vibration tests. In real-life applications, components are often subjected to stochastic loads that might lead to a premature failure; therefore, experimental tests are needed to check the components’ resistance to environmental vibrations.The type of tests related to fatigue-life estimation aims to reproduce the entire fatigue damage experienced by the component during its operational life, but in a shorter amount of time. These tests are usually referred to as fatigue-life tests or durability tests, and the common practice is to tailor them to the specific application. The signals which are used during the tests have the same "damage potential" as the environmental conditions (to which the components are subjected).The signals are created starting from a Power Spectral Density (PSD), which is used by vibrating tables or shakers to generate a vibratory motion.In the course, a method of relating a PSD to the fatigue damage will be shown. This will help create a (Gaussian) signal to be used during tests.The Gaussian distribution appears very frequently in nature and practical applications. In this course, it was deemed necessary to shed light on the importance of this distribution, by deriving the results previously derived in another course on the Central Limit Theorem (although in a different way).However, since there are signals measured in real applications that show a deviation from the Gaussian distribution (due to the presence of peaks and bursts in the signals), the synthesis of non-Gaussian signals will also be discussed.

Overview

Section 1: The mathematics of Fatigue Damage

Lecture 1 Introduction to the course

Lecture 2 Joint Gaussian Distribution

Lecture 3 Probability Distribution of the Envelope of a Gaussian Signal (Rayleigh distrib)

Lecture 4 Mathematical Definition of Fatigue damage

Lecture 5 Formula for the Expected number of Positive Peaks

Lecture 6 Power Spectral Density, Standard Deviation of a Random Process & its Derivatives

Lecture 7 Relating the Fatigue Damage Spectrum to the Power Spectral Density

Lecture 8 Practical Explanation on How to Calculate the Fatigue Damage Spectrum

Lecture 9 Maximum Response Spectrum

Section 2: Applications of the theory of Fatigue Damage

Lecture 10 Practical use of the theory of Fatigue Damage

Lecture 11 Algorithms for the Calculation of the Power Spectral Density

Lecture 12 How to Create Non-Gaussian Signals with a Prescribed Fatigue Damage Spectrum

Lecture 13 Application: Synthesis of Non-Gaussian Signals with a Prescribed Fatigue Damage

Lecture 14 Application of the Maximum Response Spectrum

Section 3: How the distribution of a Gaussian signal or sinusoidal signal occurs in nature

Lecture 15 How the Central Limit Theorem Arises from Stochastic Processes

Lecture 16 Integral of Even Powers of Sinusoids

Lecture 17 Some Representations of the Bessel function of Order Zero

Lecture 18 Probability Density of a Sinusoid

Lecture 19 Distribution of a Sinusoid Derived Intuitively

Lecture 20 Why a Random Signal with Uniformly Distributed Phases is Gaussian

Section 4: Sine on Random Vibrations

Lecture 21 Probability Density of the Envelope of Gaussian Noise + a Deterministic Sinusoid

Lecture 22 Calculating the Damage caused by Sine on Random Signals

Section 5: Appendix on Fourier series and Fourier Transforms

Lecture 23 Introduction to Stochastic Processes

Lecture 24 Derivation of the Properties of a Linear System

Lecture 25 Properties of the Fourier Transform of a Real Time Signal

Lecture 26 Fourier Series Representation of the Output of a Linear System

Lecture 27 Another Useful Representation of the Fourier Series

Lecture 28 Recalling the Relationship Between the Fourier Series and the Fourier Transform

Section 6: Appendix on Ordinary Differential Equations

Lecture 29 Solution to 2nd order ODE (ordinary differential equations)

Section 7: Appendix on Random Variables and the Central Limit Theorem

Lecture 30 Sterling's Formula

Lecture 31 How the Binomial Distribution Converges to the Gaussian One

Reliability engineers,Mechanical engineers,Mathematicians,Engineering or math students who are interested in the mathematical modeling of Fatigue Damage,Students who want to see the connections between mathematical models and practical applications,Students who already have a good mastery of mathematics and want to learn some of its beautiful applications to Random Processes,Engineering and technical managers,Quality Engineers