# Lalanne C. Mechanical Vibrations and Shock. Fatigue Damage (Volume 4) *PDF*

Hermes Penton Science, 2002. — 352 pages. — ISBN: 1903996066.

Having established the relationships which provide the response of a linear system with one degree of freedom to a random vibration, Volume 4 is devoted to the calculation of damage fatigue. It presents the hypotheses adopted to describe the behaviour of a material subjected to fatigue, the laws of damage accumulation, together with the methods for counting the peaks of the response, used to establish a histogram when it is impossible to use the probability density of the peaks obtained with a Gaussian signal. The expressions of mean damage and of its standard deviation are established. A few cases are then examined using other hypotheses (mean not equal to zero, taking account of the fatigue limit, non linear accumulation law, etc.).

Fatigue damage to a system with one-degree-of-freedom is one of the two criteria adopted for comparing the severity of different vibratory environments, the second being the maximum response of the system.

This criterion is also used to create a specification reproducing on the equipment the same effects as all the vibrations to which it will be subjected in its useful lifetime. This volume focuses on calculation of damage caused by random vibration.

This requires:

- having the relative response of the system under vibration represented by its PSD or by its rms value. Chapter 1 establishes all the useful relationships for that calculation. In Chapter 2 is described the main characteristics of the response.

- knowledge of the fatigue behaviour of the materials, characterized by S-N curve, which gives the number of cycles to failure of a specimen, depending on the amplitude of the stress applied. In Chapter 3 are quoted the main laws used to represent the curve, emphasizing the random nature of fatigue phenomena, followed by some measured values of the variation coefficients of the numbers of cycles to failure.

- determination of the histogram of the peaks of the response stress, supposed here to be proportional to the relative displacement. When the signal is Gaussian stationary, as was seen in Volume 3 the probability density of its peaks can easily be obtained from the PSD alone of the signal. When this is not the case, the response of the given one-degree-of-freedom system must be calculated digitally, and the peaks then counted directly. Numerous methods, ranging from the simplest (counting of the peaks) to the most complex (rainflow) have been proposed and are presented, with their disadvantages, in Chapter 5.

- choice of a law of accumulation of the damage caused by all the stress cycles thus identified. In Chapter 4 are described the most common laws with their limitations.

All these data are used to estimate the damage, characterized statistically if the probability density of the peaks is available, and deterministically otherwise (Chapter 6), and its standard deviation (Chapter 7).

Finally, in Chapter 8 are provided a few elements for damage estimation from other hypotheses concerning the shape of the S-N curve, the existence of an endurance limit, the non linear accumulation of damage, the law of distribution of peaks, and the existence of a non-zero mean value.

Contents:

Average value of response of a linear system

Response of perfect bandpass filter to random vibration

PSD of response of a single-degree-of-freedom linear system

Rms value of response to white noise

Rms value of response of a linear one-degree-of-freedom system subjected to bands of random noise

Rms value of the absolute acceleration of the response

Transitory response of dynamic system under stationary random excitation

Transitory response of dynamic system under amplitude modulated white noise excitation

Moments of response of a one-degree-of-freedom linear system: irregularity factor of response

Autocorrelation function of response displacement

Average numbers of maxima and minima per second

Equivalence between transfer functions of bandpass filter and one-degree-of-freedom linear system

Introduction

Damage arising from fatigue

Characterization of endurance of materials

Factors of influence

Other representations of S-N curves

Prediction of fatigue life of complex structures

Fatigue in composite materials

Evolution of fatigue damage

Classification of various laws of accumulation

Miner's method

Modified Miner's theory

Henry's method

Modified Henry's method

H. Corten and T. Dolan's method

Other theories

General

Peak count method

Peak between mean-crossing count method

Range count method

Range-mean count method

Range-pair count method

Hayes counting method

Ordered overall range counting method

Level-crossing count method

Peak valley peak counting method

Fatigue-meter counting method

Rainflow counting method

NRL counting method (National Luchtvaart Laboratorium)

Evaluation of time spent at given level

Influence of levels of load below fatigue limit on fatigue life

Test acceleration

Presentation of fatigue curves determined by random vibration tests

Introduction

Calculation of fatigue damage due to signal versus time

Calculation of fatigue damage due to acceleration spectral density

Equivalent narrow band noise

Comparison of S-N curves established under sinusoidal and random loads

Comparison of theory and experiment

Influence of shape of power spectral density and value of irregularity factor

Effects of peak truncation

Truncation of stress peaks

Calculation of standard deviation of damage: J.S. Bendat's method

Calculation of standard deviation of damage: S.H. Crandall, W.D. Mark and G.R. Khabbaz method

Comparison of W.D. Mark and J.S. Bendat's results

Statistical S-N curves

S-N curve represented by two segments of a straight line on logarithmic scales (taking into account fatigue limit)

S-N curve defined by two segments of straight line on log-lin scales

Hypothesis of non-linear accumulation of damage

Random vibration with non zero mean: use of modified Goodman diagram

Non Gaussian distribution of instantaneous values of signal

Non-linear mechanical system

Gamma function

Definition

Properties

Approximations for arbitrary x

Incomplete gamma function

Definition

Relation between complete gamma function and incomplete gamma function

earson form of incomplete gamma function

Various integrals

Having established the relationships which provide the response of a linear system with one degree of freedom to a random vibration, Volume 4 is devoted to the calculation of damage fatigue. It presents the hypotheses adopted to describe the behaviour of a material subjected to fatigue, the laws of damage accumulation, together with the methods for counting the peaks of the response, used to establish a histogram when it is impossible to use the probability density of the peaks obtained with a Gaussian signal. The expressions of mean damage and of its standard deviation are established. A few cases are then examined using other hypotheses (mean not equal to zero, taking account of the fatigue limit, non linear accumulation law, etc.).

Fatigue damage to a system with one-degree-of-freedom is one of the two criteria adopted for comparing the severity of different vibratory environments, the second being the maximum response of the system.

This criterion is also used to create a specification reproducing on the equipment the same effects as all the vibrations to which it will be subjected in its useful lifetime. This volume focuses on calculation of damage caused by random vibration.

This requires:

- having the relative response of the system under vibration represented by its PSD or by its rms value. Chapter 1 establishes all the useful relationships for that calculation. In Chapter 2 is described the main characteristics of the response.

- knowledge of the fatigue behaviour of the materials, characterized by S-N curve, which gives the number of cycles to failure of a specimen, depending on the amplitude of the stress applied. In Chapter 3 are quoted the main laws used to represent the curve, emphasizing the random nature of fatigue phenomena, followed by some measured values of the variation coefficients of the numbers of cycles to failure.

- determination of the histogram of the peaks of the response stress, supposed here to be proportional to the relative displacement. When the signal is Gaussian stationary, as was seen in Volume 3 the probability density of its peaks can easily be obtained from the PSD alone of the signal. When this is not the case, the response of the given one-degree-of-freedom system must be calculated digitally, and the peaks then counted directly. Numerous methods, ranging from the simplest (counting of the peaks) to the most complex (rainflow) have been proposed and are presented, with their disadvantages, in Chapter 5.

- choice of a law of accumulation of the damage caused by all the stress cycles thus identified. In Chapter 4 are described the most common laws with their limitations.

All these data are used to estimate the damage, characterized statistically if the probability density of the peaks is available, and deterministically otherwise (Chapter 6), and its standard deviation (Chapter 7).

Finally, in Chapter 8 are provided a few elements for damage estimation from other hypotheses concerning the shape of the S-N curve, the existence of an endurance limit, the non linear accumulation of damage, the law of distribution of peaks, and the existence of a non-zero mean value.

Contents:

**Introduction****Response of a linear one degree-of-freedom linear system to random vibration**Average value of response of a linear system

Response of perfect bandpass filter to random vibration

PSD of response of a single-degree-of-freedom linear system

Rms value of response to white noise

Rms value of response of a linear one-degree-of-freedom system subjected to bands of random noise

Rms value of the absolute acceleration of the response

Transitory response of dynamic system under stationary random excitation

Transitory response of dynamic system under amplitude modulated white noise excitation

**Characteristics of the response of a one-degree-of-freedom linear system to random vibration**Moments of response of a one-degree-of-freedom linear system: irregularity factor of response

Autocorrelation function of response displacement

Average numbers of maxima and minima per second

Equivalence between transfer functions of bandpass filter and one-degree-of-freedom linear system

**Concepts of material fatigue**Introduction

Damage arising from fatigue

Characterization of endurance of materials

Factors of influence

Other representations of S-N curves

Prediction of fatigue life of complex structures

Fatigue in composite materials

**Accumulation of fatigue damage**Evolution of fatigue damage

Classification of various laws of accumulation

Miner's method

Modified Miner's theory

Henry's method

Modified Henry's method

H. Corten and T. Dolan's method

Other theories

**Counting methods for analysing random time history**General

Peak count method

Peak between mean-crossing count method

Range count method

Range-mean count method

Range-pair count method

Hayes counting method

Ordered overall range counting method

Level-crossing count method

Peak valley peak counting method

Fatigue-meter counting method

Rainflow counting method

NRL counting method (National Luchtvaart Laboratorium)

Evaluation of time spent at given level

Influence of levels of load below fatigue limit on fatigue life

Test acceleration

Presentation of fatigue curves determined by random vibration tests

**Damage by fatigue undergone by a one-degree-of-freedom mechanical system**Introduction

Calculation of fatigue damage due to signal versus time

Calculation of fatigue damage due to acceleration spectral density

Equivalent narrow band noise

Comparison of S-N curves established under sinusoidal and random loads

Comparison of theory and experiment

Influence of shape of power spectral density and value of irregularity factor

Effects of peak truncation

Truncation of stress peaks

**Standard deviation of fatigue damage**Calculation of standard deviation of damage: J.S. Bendat's method

Calculation of standard deviation of damage: S.H. Crandall, W.D. Mark and G.R. Khabbaz method

Comparison of W.D. Mark and J.S. Bendat's results

Statistical S-N curves

**Fatigue damage using other assumptions for calculation**S-N curve represented by two segments of a straight line on logarithmic scales (taking into account fatigue limit)

S-N curve defined by two segments of straight line on log-lin scales

Hypothesis of non-linear accumulation of damage

Random vibration with non zero mean: use of modified Goodman diagram

Non Gaussian distribution of instantaneous values of signal

Non-linear mechanical system

**Appendices**Gamma function

Definition

Properties

Approximations for arbitrary x

Incomplete gamma function

Definition

Relation between complete gamma function and incomplete gamma function

earson form of incomplete gamma function

Various integrals

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