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2nd Edition. — ISTE Ltd John Wiley & Sons, Inc., 2009. — 501 p. — ISBN: 978-1-84821-121-6 (Set of 5 Volumes), ISBN: 978-1-84821-126-1 (Volume 5).This volume focuses on specification development in accordance with the principle of tailoring. Extreme response and the fatigue damage spectra are defined for each type of stress (sinusoidal vibration, swept sine, shock, random vibration, etc.). The process for establishing a specification from the life cycle profile of the equipment which will be subject to these types of stresses is then detailed. The analysis takes account of the uncertainty factor, designed to cover uncertainties related to the real–world environment and mechanical strength, and the test factor, which takes account of the number of tests performed to demonstrate the resistance of the equipment. *The Mechanical Vibration and Shock Analysis* five–volume series has been written with both the professional engineer and the academic in mind. Christian Lalanne explores every aspect of vibration and shock, two fundamental and extremely significant areas of mechanical engineering, from both a theoretical and practical point of view. The five volumes cover all the necessary issues in this area of mechanical engineering. The theoretical analyses are placed in the context of both the real world and the laboratory, which is essential for the development of specifications.**Table of Contents**

Foreword to Series

Introduction

List of Symbols** Extreme Response Spectrum of a Sinusoidal Vibration**

The effects of vibration

Extreme response spectrum of a sinusoidal vibration

Definition

Case of a single sinusoid

Case of a periodic signal

General case

Extreme response spectrum of a swept sine vibration

Sinusoid of constant amplitude throughout the sweeping process

Swept sine composed of several constant levels**Extreme Response Spectrum of a Random Vibration**

Unspecified vibratory signal

Gaussian stationary random signal

Calculation from peak distribution

Use of the largest peak distribution law

Response spectrum defined by k times the rms response

Other ERS calculation methods

Limit of the ERS at the high frequencies

Response spectrum with up-crossing risk

Complete expression

Approximate relation

Calculation in a hypothesis of independence of threshold overshoot

Use of URS

Comparison of the various formulae

Effects of peak truncation on the acceleration time history

Extreme response spectra calculated from the time history signal

Extreme response spectra calculated from the power spectral densities

Comparison of extreme response spectra calculated from time history signals and power spectral densities

Sinusoidal vibration superimposed on a broad band random vibration

Real environment

Case of a single sinusoid superimposed to a wide band noise

Case of several sinusoidal lines superimposed on a broad band random vibration

Swept sine superimposed on a broad band random vibration

Real environment

Case of a single swept sine superimposed to a wide band noise

Case of several swept sines superimposed on a broad band random vibration

Swept narrow bands on a wide band random vibration

Real environment

Extreme response spectrum** Fatigue Damage Spectrum of a Sinusoidal Vibration**

Fatigue damage spectrum definition

Fatigue damage spectrum of a single sinusoid

Fatigue damage spectrum of a periodic signal

General expression for the damage

Fatigue damage with other assumptions on the S–N curve

Taking account of fatigue limit

Cases where the S–N curve is approximated by a straight line in log–lin scales

Comparison of the damage when the S–N curves are linear in either log–log or log–lin scales

Fatigue damage generated by a swept sine vibration on a single-degree-of-freedom linear system

General case

Linear sweep

Logarithmic sweep

Hyperbolic sweep

General expressions for fatigue damage

Reduction of test time

Fatigue damage equivalence in the case of a linear system

Method based on fatigue damage equivalence according to Basquin’s relationship

Notes on the design assumptions of the ERS and FDS**Fatigue Damage Spectrum of a Random Vibration**

Fatigue damage spectrum from the signal as function of time

Fatigue damage spectrum derived from a power spectral density

Simplified hypothesis of Rayleigh’s law

Calculation of the fatigue damage spectrum with Dirlik’s probability density

Reduction of test time

Fatigue damage equivalence in the case of a linear system

Method based on a fatigue damage equivalence according to Basquin’s relationship taking account of variation of natural damping as a function of stress level

Truncation of the peaks of the input acceleration signal

Fatigue damage spectra calculated from a signal as a function of time

Fatigue damage spectra calculated from power spectral densities

Comparison of fatigue damage spectra calculated from signals as a function of time and power spectral densities

Sinusoidal vibration superimposed on a broad band random vibration

Case of a single sinusoidal vibration superimposed on broad band random vibration

Case of several sinusoidal vibrations superimposed on a broad band random vibration

Swept sine superimposed on a broad band random vibration

Case of one swept sine superimposed on a broad band random vibration

Case of several swept sines superimposed on a broad band random vibration

Swept narrow bands on a broad band random vibration**Fatigue Damage Spectrum of a Shock**

General relationship of fatigue damage

Use of shock response spectrum in the impulse zone

Damage created by simple shocks in static zone of the response spectrum**Influence of Calculation: Conditions of ERSs and FDSs**

Variation of the ERS with amplitude and vibration duration

Variation of the FDS with amplitude and duration of vibration

Should ERSs and FDSs be drawn with a linear or logarithmic frequency step?

With how many points must ERSs and FDSs be calculated?

Difference between ERSs and FDSs calculated from a vibratory signal according to time and from its PSD

Influence of the number of PSD calculation points on ERS and FDS

Influence of the PSD statistical error on ERS and FDS

Influence of the sampling frequency during ERS and FDS calculation from a signal based on time

Influence of the peak counting method

Influence of a non-zero mean stress on FDS**Tests and Standards**

Definitions

Standard

Specification

Types of tests

Characterization test

Identification test

Evaluation test

Final adjustment/development test

Prototype test

Pre-qualification (or evaluation) test

Qualification

Qualification test

Certification

Certification test

Stress screening test

Acceptance or reception

Reception test

Qualification/acceptance test

Series test

Sampling test

Reliability test

What can be expected from a test specification?

Specification types

Specification requiring in situ testing

Specifications derived from standards

Current trend

Specifications based on real environment data

Standards specifying test tailoring

The MIL–STD 810 standard

The GAM.EG 13 standard

STANAG

The AFNOR X50–410 standard**Uncertainty Factor**

Need – definitions

Sources of uncertainty

Statistical aspect of the real environment and of material strength

Real environment

Material strength

Statistical uncertainty factor

Definitions

Calculation of uncertainty factor

Calculation of an uncertainty coefficient when the real environment is only characterized by a single value**Aging Factor**

Purpose of the aging factor

Aging functions used in reliability

Method for calculating aging factor

Influence of standard deviation of the aging law

Influence of the aging law mean**Test Factor**

Philosophy

Calculation of test factor

Normal distributions

Log–normal distributions

Weibull distributions

Choice of confidence level

Influence of the number of tests n**Specification Development**

Test tailoring

Step 1: analysis of the life cycle profile. Review of the situations

Step 2: determination of the real environmental data associated with each situation

Step 3: determination of the environment to be simulated

Need

Synopsis methods

The need for a reliable method

Synopsis method using power spectrum density envelope

Equivalence method of extreme response and fatigue damage

Synopsis of the real environment associated with an event (or sub-situation)

Synopsis of a situation

Synopsis of all life profile situations

Search for a random vibration of equal severity

Validation of duration reduction

Step 4: establishment of the test program

Application of a test factor

Choice of the test chronology

Applying this method to the example of the round robin comparative study

Taking environment into account in project management**Influence of Calculation: Conditions of Specification**

Choice of the number of points in the specification (PSD)

Influence of Q factor on specification (outside of time reduction)

Influence of Q factor on specification when duration is reduced

Validity of a specification established for Q factor equal to 10 when the real structure has another value

Advantage in the consideration of a variable Q factor for the calculation of ERSs and FDSs

Influence of the value of parameter b on the specification

Case where test duration is equal to real environment duration

Case where duration is reduced

Choice of the value of parameter b in the case of material made up of several components

Influence of temperature on parameter b and constant C

Importance of a factor of 10 between the specification FDS and the reference FDS (real environment) in a small frequency band

Validity of a specification established by reference to a 1-dof system when real structures are multi-dof systems**Other Uses of Extreme Response, Up-Crossing Risk and Fatigue Damage Spectra**

Comparisons of the severity of different vibrations

Comparisons of the relative severity of several real environments

Comparison of the severity of two standards

Comparison of earthquake severity

Swept sine excitation – random vibration transformation

Definition of a random vibration with the same severity as a series of shocks

Writing a specification only from an ERS (or a URS)

Matrix inversion method

Method by iteration

Establishment of a swept sine vibration specification

Appendices

Formulae

Bibliography

Index

Summary of Other Volumes in the Series

Foreword to Series

Introduction

List of Symbols

The effects of vibration

Extreme response spectrum of a sinusoidal vibration

Definition

Case of a single sinusoid

Case of a periodic signal

General case

Extreme response spectrum of a swept sine vibration

Sinusoid of constant amplitude throughout the sweeping process

Swept sine composed of several constant levels

Unspecified vibratory signal

Gaussian stationary random signal

Calculation from peak distribution

Use of the largest peak distribution law

Response spectrum defined by k times the rms response

Other ERS calculation methods

Limit of the ERS at the high frequencies

Response spectrum with up-crossing risk

Complete expression

Approximate relation

Calculation in a hypothesis of independence of threshold overshoot

Use of URS

Comparison of the various formulae

Effects of peak truncation on the acceleration time history

Extreme response spectra calculated from the time history signal

Extreme response spectra calculated from the power spectral densities

Comparison of extreme response spectra calculated from time history signals and power spectral densities

Sinusoidal vibration superimposed on a broad band random vibration

Real environment

Case of a single sinusoid superimposed to a wide band noise

Case of several sinusoidal lines superimposed on a broad band random vibration

Swept sine superimposed on a broad band random vibration

Real environment

Case of a single swept sine superimposed to a wide band noise

Case of several swept sines superimposed on a broad band random vibration

Swept narrow bands on a wide band random vibration

Real environment

Extreme response spectrum

Fatigue damage spectrum definition

Fatigue damage spectrum of a single sinusoid

Fatigue damage spectrum of a periodic signal

General expression for the damage

Fatigue damage with other assumptions on the S–N curve

Taking account of fatigue limit

Cases where the S–N curve is approximated by a straight line in log–lin scales

Comparison of the damage when the S–N curves are linear in either log–log or log–lin scales

Fatigue damage generated by a swept sine vibration on a single-degree-of-freedom linear system

General case

Linear sweep

Logarithmic sweep

Hyperbolic sweep

General expressions for fatigue damage

Reduction of test time

Fatigue damage equivalence in the case of a linear system

Method based on fatigue damage equivalence according to Basquin’s relationship

Notes on the design assumptions of the ERS and FDS

Fatigue damage spectrum from the signal as function of time

Fatigue damage spectrum derived from a power spectral density

Simplified hypothesis of Rayleigh’s law

Calculation of the fatigue damage spectrum with Dirlik’s probability density

Reduction of test time

Fatigue damage equivalence in the case of a linear system

Method based on a fatigue damage equivalence according to Basquin’s relationship taking account of variation of natural damping as a function of stress level

Truncation of the peaks of the input acceleration signal

Fatigue damage spectra calculated from a signal as a function of time

Fatigue damage spectra calculated from power spectral densities

Comparison of fatigue damage spectra calculated from signals as a function of time and power spectral densities

Sinusoidal vibration superimposed on a broad band random vibration

Case of a single sinusoidal vibration superimposed on broad band random vibration

Case of several sinusoidal vibrations superimposed on a broad band random vibration

Swept sine superimposed on a broad band random vibration

Case of one swept sine superimposed on a broad band random vibration

Case of several swept sines superimposed on a broad band random vibration

Swept narrow bands on a broad band random vibration

General relationship of fatigue damage

Use of shock response spectrum in the impulse zone

Damage created by simple shocks in static zone of the response spectrum

Variation of the ERS with amplitude and vibration duration

Variation of the FDS with amplitude and duration of vibration

Should ERSs and FDSs be drawn with a linear or logarithmic frequency step?

With how many points must ERSs and FDSs be calculated?

Difference between ERSs and FDSs calculated from a vibratory signal according to time and from its PSD

Influence of the number of PSD calculation points on ERS and FDS

Influence of the PSD statistical error on ERS and FDS

Influence of the sampling frequency during ERS and FDS calculation from a signal based on time

Influence of the peak counting method

Influence of a non-zero mean stress on FDS

Definitions

Standard

Specification

Types of tests

Characterization test

Identification test

Evaluation test

Final adjustment/development test

Prototype test

Pre-qualification (or evaluation) test

Qualification

Qualification test

Certification

Certification test

Stress screening test

Acceptance or reception

Reception test

Qualification/acceptance test

Series test

Sampling test

Reliability test

What can be expected from a test specification?

Specification types

Specification requiring in situ testing

Specifications derived from standards

Current trend

Specifications based on real environment data

Standards specifying test tailoring

The MIL–STD 810 standard

The GAM.EG 13 standard

STANAG

The AFNOR X50–410 standard

Need – definitions

Sources of uncertainty

Statistical aspect of the real environment and of material strength

Real environment

Material strength

Statistical uncertainty factor

Definitions

Calculation of uncertainty factor

Calculation of an uncertainty coefficient when the real environment is only characterized by a single value

Purpose of the aging factor

Aging functions used in reliability

Method for calculating aging factor

Influence of standard deviation of the aging law

Influence of the aging law mean

Philosophy

Calculation of test factor

Normal distributions

Log–normal distributions

Weibull distributions

Choice of confidence level

Influence of the number of tests n

Test tailoring

Step 1: analysis of the life cycle profile. Review of the situations

Step 2: determination of the real environmental data associated with each situation

Step 3: determination of the environment to be simulated

Need

Synopsis methods

The need for a reliable method

Synopsis method using power spectrum density envelope

Equivalence method of extreme response and fatigue damage

Synopsis of the real environment associated with an event (or sub-situation)

Synopsis of a situation

Synopsis of all life profile situations

Search for a random vibration of equal severity

Validation of duration reduction

Step 4: establishment of the test program

Application of a test factor

Choice of the test chronology

Applying this method to the example of the round robin comparative study

Taking environment into account in project management

Choice of the number of points in the specification (PSD)

Influence of Q factor on specification (outside of time reduction)

Influence of Q factor on specification when duration is reduced

Validity of a specification established for Q factor equal to 10 when the real structure has another value

Advantage in the consideration of a variable Q factor for the calculation of ERSs and FDSs

Influence of the value of parameter b on the specification

Case where test duration is equal to real environment duration

Case where duration is reduced

Choice of the value of parameter b in the case of material made up of several components

Influence of temperature on parameter b and constant C

Importance of a factor of 10 between the specification FDS and the reference FDS (real environment) in a small frequency band

Validity of a specification established by reference to a 1-dof system when real structures are multi-dof systems

Comparisons of the severity of different vibrations

Comparisons of the relative severity of several real environments

Comparison of the severity of two standards

Comparison of earthquake severity

Swept sine excitation – random vibration transformation

Definition of a random vibration with the same severity as a series of shocks

Writing a specification only from an ERS (or a URS)

Matrix inversion method

Method by iteration

Establishment of a swept sine vibration specification

Appendices

Formulae

Bibliography

Index

Summary of Other Volumes in the Series

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