Mechanical Vibrations: Applications to Equipment

By Yvon Mori

The purpose of this book is to clarify the issues related to the environment of mechanical vibrations in the material life profile. In particular, through their simulation testing laboratory, through a better understanding of the physical phenomenon, means to implement to simulate, measurements and interpretations associated results. It is aimed at development of technical consultants, quality and services primarily to those testing laboratories, as well as to all those who are faced with supply reference to the environmental test calls and particularly here, vibration tests. Furthermore it should also interest students of engineering schools in the areas of competence of their future professions affected by vibration.

• Language: English
• Publisher: Wiley
• Release date: Jan 18, 2017
• ISBN: 9781119384496

Book preview

Table of Contents

Cover

Dedication

Title

Copyright

Preface

1 Vibration Theory

1.1. Problem

1.2. Different types of mechanical signals

1.3. Theory of vibration – reminders

1.4. Concept of mechanical impedance

1.5. Electromechanical analogies

1.6. Analog and logic computer simulation

1.7. Conclusion

2 Signal Analysis

2.1. Overview

2.2. Spectral density of power

2.3. FS–Fourier Integral

3 Test Preparation

3.1. Test demand analysis and associated test specifications

3.2. Test initiation

3.3. Test fixtures

3.4. Test execution

3.5. Test reporting

4 Testing

4.1. Sine vibration tests

4.2. Vibration testing in noise or random

4.3. Specific tests

5 Equipment Applications

5.1. Vibration sources and effects

5.2. Electronic equipment

5.3. Design of electronic equipment subjected to vibrations

5.4. Study of a particular case – example of analysis of an electronic bay

6 Controlling Generators of Vibrations and Shocks

6.1. General principles

6.2. Typical configuration of the equipment used

6.3. Traceability of tests

6.4. Control in sinusoidal mode

6.5. Random control

6.6. Shock and transient control

6.7. Combined vibrations control

6.8. Control: a few essential rules

7 Metrology of Measurement and Testing Methods

7.1. Introduction to accelerometer sensors

7.2. Measurement amplifiers

7.3. Validation and verification of the testing means

7.4. Control of metrology in a testing laboratory

8 Testing Means for Vibrations

8.1. Electrodynamic exciters

8.2. Hydraulic exciters

Conclusion

Appendices

Appendix A: Fundamental Laws of Hydraulics – Application to the Study of Hydraulic Vibration Generators

A.1. Law of flow of a fluid

A.2. Notion of viscosity

A.3. Reynolds number and pressure loss in the piping

A.4. Typical force equations

A.5. Typical flow equations

A.6. Power available on the piston

A.7. Study of transfer functions

Appendix B: Study of the Basic Model with Damping

Appendix C: Natural Frequencies of 1,6 mm Thick Cards Equipped with Stiffeners

Appendix D: Resonance Frequencies of IC Cards Depending on the Mounting Conditions

Appendix E: Concept of Cepstrum

Appendix F: Tolerances on Vibration Fixtures

Appendix G: Determining Measurement Uncertainty – Example of a Calibration Method for Accelerometer Sensors

G.1. Overview

G.2. Direct measurement

G.3. Direct type B measurement method

G.4. Indirect measurement

G.5. Example of accelerometer calibration

G.6. Bibliography for this appendix

Appendix H: Example of MPE Allowances

Appendix I: List of Testing Methods in Standard Environments

Appendix J: Control Strategies – Simplified Summary

J.1. Storing the last control spectrum

J.2. Feedback properties in sinusoidal state

J.3. Feedback properties in random state

J.4. Alarms

J.5. Control during shocks

Appendix K: Mathematical Elements Involved in the Estimation of Uncertainties (Supplementary)

K.1. Calculation of variances, simplified formulas

K.2. Problem of average of averages

K.3. Problem of average of variances

K.4. Calculating the average on the fly

K.5. Calculating the variance on the fly

K.6. Testing normality of distribution, elimination of outliers

K.7. Henry line

K.8. Dixon test, detection of outliers

K.9. Calculation of covariance and correlation factor

K.10. Independence test, extract from CERESTA [ref. 6]

K.11. Table of cumulative normal distribution

K.12. χ² table: this table gives χ such that P(K > x) = p

K.13. Student distribution table

Bibliography

Index

End User License Agreement

List of Tables

3 Test Preparation

Table 3.1. Various cases of uncertainty declaration

Table 3.2. Different cases of declaration of conformity

4 Testing

Table 4.1. Responses in displacement, velocity and acceleration

Table 4.2. Result of the calculation of PSD with Vibkit 98

7 Metrology of Measurement and Testing Methods

Table 7.1. Table of the measured values

Table 7.2. Table of readings and determination of the amplitude variation

8 Testing Means for Vibrations

Table 8.1. Conventional mass measurements

List of Illustrations

1 Vibration Theory

Figure 1.1. Unit circle

Figure 1.2. Fresnel model

Figure 1.3. Sinusoidal signal

Figure 1.4. Four-variable nomogram, displacement, velocity and acceleration

Figure 1.5. Typical templates of sinusoidal vibrations

Figure 1.6. Time characteristic of displacement, speed and acceleration

Figure 1.7. Quasi-periodic signals

Figure 1.8. Quasi-sinusoidal signals

Figure 1.9. Classification of practical deterministic signals

Figure 1.10. Classification of random practical signals

Figure 1.11. Basic model with damping

Figure 1.12. Damped oscillations

Figure 1.13. Curves in phase and quadrature

Figure 1.14. Admittance circle of velocities

Figure 1.15. Admittance velocity module

Figure 1.16. Acceleration admittance module

Figure 1.17. Coefficient of transmissibility module

Figure 1.18. Amplitude responses and phase of a coupled and damped system with 2 DOF

2 Signal Analysis

Figure 2.1. Relations of functions with effective g

Figure 2.2. Model of spectrum for a half-sinusoidal excitation spectrum

Figure 2.3. PSD model

Figure 2.4. Calculation with software Vibkit

Figure 2.5. Form factor model

3 Test Preparation

Figure 3.1. Transmissibility model

Figure 3.2. Electronic cabinet. Welded/screwed frame and electronic box. Screwed frame

4 Testing

Figure 4.1. Typical configurations of shooting guns associated with aircraft classes (MILSTD 810 standard)

5 Equipment Applications

Figure 5.1. Vibration spectra of a C130 Hercules turbo-propeller transport aircraft (data of the plane’s propellers in the vertical axis)

Figure 5.2. ESTECH 2012 Conference – vibration and control strategies

Figure 5.3. Different types of shock absorbers

Figure 5.4. Example of electronic visual bay

Figure 5.5. Calculation of the response of the bay’s CoG with SYMOS

Figure 5.6. Bay FEM calculation of the mode at 61Hz and 21 g

6 Controlling Generators of Vibrations and Shocks

Figure 6.1. BO model

Figure 6.2. BF model

Figure 6.3. Classic block diagram

Figure 6.4. Test identification settings

Figure 6.5. Streamlined appearance of a reconstituted sinusoidal signal

Figure 6.6. Sampling example of a period

Figure 6.7. Example of a distorted signal

Figure 6.8. Calculation of the peak value from the average value

Figure 6.9. Block diagram of filtering

Figure 6.10. Control error model

Figure 6.11. Simplified example of a transfer function

Figure 6.12. Example of notching

Figure 6.13. Example of PSD

Figure 6.14. Example of an analog BB generator

Figure 6.15. Example of Fibonacci type LFSR

7 Metrology of Measurement and Testing Methods

Figure 7.1. Examples of accelerometer sensor types

Figure 7.2. Example of a mechanical filter

Figure 7.3. Example of transverse sensitivity of accelerometric sensors

Figure 7.4. Example of influencing factors on accelerometer sensors

Figure 7.5. Example of a calibration method for accelerometer sensors

Figure 7.6. Example of a charge amplifier

Figure 7.7. Common testing setup

Figure 7.8. Test setup for measuring the transverse

Figure 7.9. Measurement charts: max. transverse rate on the five frequency points axis OX = 3.6%

Figure 7.10. Measurement charts: max. transverse rate on the five frequency point axis OY = 9,4%

Figure 7.11. Test setup

8 Testing Means for Vibrations

Figure 8.1. Diagram of an electrodynamic exciter: fragmented view

Figure 8.2. Diagram of an electrodynamic exciter: simplified view

Figure 8.3. Ratier Forest VRF 30 electrodynamic exciter

Figure 8.4. Theoretical maximum performance for ε = 0,2

Figure 8.5. Strategies for choice of pre and post-shock on a digital console

Figure 8.6. Mechanical model of the electrodynamic exciter

Figure 8.7. Model in terms of mechanical impedances

Figure 8.8. Simplified isostatic axial model

Figure 8.9. Model loaded with a resonant system with 1 DOF

Figure 8.10. Simplified model loaded with a resonant system with 1 DOF

Figure 8.11. Case for resonant case

Figure 8.12. Case for resonant load

Figure 8.13. Empty correction factor H(p) for the resonant load

Figure 8.14. Mechanical model equivalent circuit diagram

Figure 8.15. Electrical model for assessing

Figure 8.16. SPICE model

Figure 8.17. Experimental measurement of the natural frequency of the moving component suspension and the resonance frequency of the moving coil

Figure 8.18. Theoretical curves obtained using the Spice model

Figure 8.19. Simplified block diagram of a hydraulic vibration installation

Figure 8.20. Expanded view of a hydraulic vibration cylinder

Figure 8.21. Examples of alternating nozzle servo valve

Figure 8.22. Example of MOOG paddles servo valve

Figure 8.23. Two examples of electrodynamic exciter servo valves

Figure 8.24. Usual servo valve response curves

Figure 8.25. Typical performance of a hydraulic exciter

Figure 8.26. Theoretical curve performance

Figure 8.27. Example of a feedback control card

Figure 8.28. Half-sine shock test on a hydraulic exciter

Appendix A: Fundamental Laws of Hydraulics – Application to the Study of Hydraulic Vibration Generators

Figure A.1. Element in a tube of liquid

Figure A.2. Cylinder block diagram

Figure A.3. Schematic diagram of the servo valve

Appendix B: Study of the Basic Model with Damping

Figure B.1. Basic model with damping

Figure B.2. Position of the poles

Figure B.3. Amplitude and phase of the basic system

Appendix E: Concept of Cepstrum

Figure E.1. Spectral density of the power to be analyzed

Figure E.2. Comparison of analysis by correlation and spectrum

To Monika, Arnaud and Sylvain

Mechanical Vibrations

Applications to Equipment

Yvon Mori

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First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Yvon Mori to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2016959683

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ISBN 978-1-78630-051-5

Preface

After having worked for nearly 40 years as an engineer at Thales Underwater Systems (TUS) in Sophia Antipolis, as the head of a large environmental testing laboratory, Laboratoire d’Essais et d’Evaluation en Environnement (LEEE), French for laboratory of environmental assessment and testing), I became interested in such diverse fields such as the mechanics of vibrations and shocks, climatic and electromagnetic problems of electromagnetic compatibility, as well as measurements. This activity naturally led me to be reached out to by COFRAC (COmité FRançais d’ACcréditation; French Certification Organization) as an expert assessor in area 38, which deals with mechanical and environmental tests in laboratories, among other domains, which has been taking place for a long time. This is what allowed me to see up close the training needs of the personnel working in these areas and that is why I thought it would be useful to put this experience to their service, by way of this book which presents itself in the form of a course with a pedagogical spirit from a confirmed engineer and therefore with a pragmatic and practical approach.

The purpose of this book is to clarify the issues related to the environment of mechanical vibrations in the life profile of materials. In particular, this is possible through their simulation in the testing laboratory, with a better knowledge of the physical phenomenon and the means to be implemented in order to simulate them, measure them and interpret the related results.

This book is aimed at developmental engineers of consulting firms and those involved in quality services, but most of all at those working in testing laboratories who are confronted with invitations to tender in testing environments, especially in terms of mechanical vibrations. Furthermore, it should also interest engineering students in the fields of development of their future occupations regarding vibrations.

We will thus define the phenomenon of vibration from the point of view of an environment specialist through some indispensable notions in order to know how to interpret the prescriptive documents published to date and perform some simple predictive calculations of behavior of the studied material.

Finally, I would like to warmly thank Henri Grzeskowiak and Christian Lalanne, my friends and companions for 30 years within the ASTE, CINEG/DGA and COFRAC for their effective support in the proofreading of this work and their pertinent and relevant corrections.

The book is divided into eight chapters:

- In Chapter 1, there is an introduction to the theory of vibrations, with a classic analytical description of the models used at one and two degrees of freedom (DOF), undamped and damped. A more practical approach of the types of vibratory phenomena referred to in the normative documents and specifications is proposed thereafter;

- Chapter 2 specifies the mathematical tools used to analyze these vibratory phenomena, and of course refers to the theory of signals. We will find the usual tools for the temporal description of signals as well as the frequency description Power Spectral Density (PSD, Fourier transform and series), but adapted to the techniques of vibrations. The proposed approach also describes specific applications to vibration signals encountered in testing laboratories. Indeed, the tests proposed by the normative documents involve sine scans, random tests in colored noise as well as composite tests such as the sine on noise, which has some peculiarities that are important to know;

- Chapter 3 offers a quality approach to address the problem of testing within testing laboratories. It is inspired by the normative documents on the subject, in particular regarding standard NF EN ISO / IEC 17025, which governs the accreditation of laboratories for testing and analysis. It allows for the initialization of an internationally recognized quality approach;

- Chapter 4 provides a relatively complete analytical description of the sine on noise vibration tests that are the most commonly used. Developments are oriented toward laboratory tests, with the usual problems that are encountered (scan speeds, level of limitation of noise testing, etc.). We will find most of what we know about the subject in this chapter;

- Chapter 5 provides a complete practical analysis of vibratory phenomena encountered in most engineering and design departments and offers simple and common conception rules that allow for a simple qualitative and quantitative approach to avoid most of the big problems that can be encountered in this type of environment. It also describes a practical approach through a particular example, which is the study of an electronic bay mounted on shock absorbers. This simplified example allows for the understanding of a practical approach (there are others) that resolves most of the technical problems and that relies on the now very common notion of tailoring;

- Chapter 6 provides a simplified approach to the piloting ways of testing problems, in particular, vibration exciters, in sine testing, noise and shocks, so as to educate the reader on the difficult issues typically encountered in these types of techniques;

- Chapter 7 describes succinctly the metrology and measurement aspects encountered in the use of accelerometric sensors and vibration testing means within testing laboratories;

- Chapter 8 details the operation of the two main means of vibration tests: electromagnetic generators and hydraulic vibrations. This chapter can be found in [ref. l18], but it was considered useful to reproduce it here in its entirety.

This book is developed from my professional experience and also with the help of a bibliography on the subject, which encompasses:

- specialized books in the field of shocks and vibrations;

- courses that I have followed and provided by a variety of organizations offering training on the subject, or that I have taught;

- technical reviews of manufacturers or suppliers of tests and measurements;

- articles, documentation and technical bulletins from various specialist organizations of shocks and vibrations;

- normative documents dealing with the subject;

which includes a summary that was prepared by taking all or part of some of these documents, so as to develop the themes that must be known in order to better understand the subject. Despite the amount of literature, this subject is relatively poorly perceived by development engineers and study laboratories and is generally poorly developed in its practical aspects in schools and universities.

The bibliography includes most documents that are used unanimously to date, but that are not necessarily easily accessible. This is why certain concepts or calculations are detailed and developed throughout the book so as to facilitate their comprehension.

In the appendices, information more or less developed to aid in the understanding of some concepts used throughout the document will be found.

1

Vibration Theory

This chapter is an introduction to the theory of vibrations, a term commonly used to define the study of representation models, which are used to describe the behavior of systems in a vibration regime. The scope of this theory is relatively wide and still is in constant development, mainly within the framework of dynamic analysis of structures (modal analysis or other ones), which expands beyond the framework of the course and for which there are already very good books.

Here, we will strive to develop the traditional approach of systems at 1 and 2 degree of freedom (DOF) with coupling in a free or forced regime that allows a good introduction to other, more sophisticated, techniques, and especially allows the comprehension and optimization of methods for vibration testing enforced by normative documents and carried out in testing laboratories.

1.1. Problem

1.1.1. Justification of tests in a mechanical environment

Any industrial system or equipment is subject during its lifetime to the attack of environments that may degrade its performance with the appearance of different failures. The now classic concepts of quality and reliability are therefore used to define its availability involved in all phases of its design. It seems natural and even necessary to minimize as much as possible the causes of these failures.

Therefore, the specifications of the product base are generally guaranteed by a quality assurance plan by acquiring the conviction that the specifications are kept, and this belief is maintained in time through a priori and a posteriori actions (measurements, tests, preventive maintenance, etc.).

Yet, the development of a part of this belief must be done from environmental tests. Here, the need to simulate experimentally is thus clear and thus through the appropriate tests, the constraints of operating the most representative of real conditions that the material meets during its service use. In particular in the case of equipment with strong evolution of design, and/or conditions of use, and/or technologies (new components and/or report processes, and/or materials).

For questions related to reproducibility, accessibility and cost, we will be limited here to laboratory tests, except in special cases.

Thus, we generally define several categories of tests according to the usual chronology:

- design phase: feasibility and validation of tests.

- development phase: model tests, qualification, formula, accelerated tests.

- production phase: batch testing and debugging.

- storage phase: storage testing, ageing.

- end of lifetime phase: withdrawal of service.

These tests are subject to the choice of the successful normative documents, whose development can be done through a variety of methods. Through the analysis and synthesis of data of the severity levels of the real environment, if we perfectly know the conditions of use, this results in the concept of tailorization (the latter being typically used when the stakes are high as well as the risks of an under- or overvaluation and if the material does not pass the bar of standard tests), or failing that by the adaptation of standard levels of severity usually found in the common normative documents edited by (non-exhaustive) organizations such as:

- at a national level:

- the AFNOR (Association Française de NORmalisation, French Standardization Association), NF standards.

- the UTE (Union Technique de l’Electricité, Technical Union of Electricity).

- the CCT (Comité de Coordination des Télécommunications, Coordinating Committee of Telecommunications), CCTU standards.

- the Ministry of Defense: formerly GAMEG13, AIR and SEFT standards, etc.

- at a European level:

- the CENELEC (Comité Européen de Normalisation ELECtronique, European Committee of Electronic Normalization), CECC and EN standards and currently the STANAG 4370 within the NATO area. See, in particular, Best Practices by EDSTAR (https://edstar.eda.europa.eu/best-practice-recommendations).

- at an international level:

institutional:

- ISO (International Standard Organization).

- IEC (Comité Electrotechnique International, International Electro-technical Committee).

- UIC (Union Internationale des Chemins de Fer, International Union of Railways).

sectorial:

- DO 160 or EUROCAE ED 14.

- ANSI.

- at the foreign military standards level:

national level:

- the DEF STAN 0035 (United Kingdom).

- the MIL STD 810 (United States).

- the STANAG 4370 (NATO area).

- BV Standards (Germany), etc.

international scale:

- MIL STD 810 (because it is considered a reference in all regions of the world).

1.1.2. Quality of environmental tests

- RG Aero 00011 quality of environmental tests.

- Re Aero 601-11 quality provisions in mechanical environment testing.

- Re Aero 601-12 quality provisions in climatic environment-specific tests.

1.1.3. Generating sets of vibrations

- NF E90-200: Generating sets of vibration – chart intended for the layout of characteristics.

- NF E90-210: Electrodynamic test facilities used for the generation of vibrations – methods of description of features.

- NF E90-220: Servohydraulic vibration tests resources – methods of description of features.

- NF E90-230: Auxiliary tables for vibration generators – methods of description of features.

1.1.4. Shock and vibration terminology

- ISO 2041: Vibration and shock – vocabulary.

1.1.5. Testing methods

For the non-specialized civilian field within a brand of activity, the following are the main methods of the NF or EN or CEI:

- NF/EN/CEI 60068-2-1 to 2-53 (see detailed list in Appendix I).

- the defense sector now refers to, for the NATO area, the methods of the STANAG 4370-AECTP 300 (climatic tests) and 400 (mechanical tests). For the international perimeter outside of NATO, these are the methods of MIL STD 810 complemented by those of the DEF STAN 0035 that apply;

- the civil aviation sector is represented by the DO160 methods.

In addition, for some specific areas, we can quote:

- NF EN 61373: Railway applications – rolling material – shock and vibration tests.

1.1.6. Uncertainty in measurement

- Guide for the expression of the uncertainty in measurement.

- ISO 5725-1 to 6: Exactitude (accuracy and fidelity) of the results and methods of measurement.

- FD ISO/TR 22971: Exactitude (accuracy and fidelity) of the results and methods of measurement. Practical guidelines for the use of ISO 5725-2 for the design, implementation and statistical analysis of the results of repeatability and interlaboratory reproducibility.

- NF ISO 21748: Guidelines relating to the use of repeatability, reproducibility and accuracy estimates in the evaluation of uncertainty in measurements.

ISO 21748: provides the guidelines in order to:

- evaluate the measurement uncertainties from data obtained during interlaboratory diffusivity tests conducted in accordance to ISO 5725-2.

- compare the results of an interlaboratory test with the uncertainty of a measurement obtained by applying formal principles of uncertainty propagation.

ISO 5725-3 provides additional models for the measurement of intermediate fidelity. However, although the same general approach can be applied to the use of these extended models, the assessment of uncertainty from these models is not covered in this international standard.

ISO 21748 is applicable in all areas of measurement and testing requiring the determination of an uncertainty associated with a result. It does not describe the use of repeatability data in the absence of reproducibility data.

ISO 21748 assumes that the recognized non-negligible systematic effects are corrected, either by applying a digital correction in the context of the measurement method, or by searching for and eliminating the origin of these effects.

ISO 21748 recommendations are tentative above all else. It is recognized that, even if they constitute a valid method of uncertainty evaluation, in many aspects, other appropriate methods may also be adopted. In general, it is understood that the references made in ISO 21748 toward results, methods and measurement processes, also apply to results, methods and testing processes.

1.1.7. Interlaboratory comparison and proficiency testing

- NF EN ISO/CEI 17043: Conformity assessment – general requirements regarding proficiency tests.

- NF EN DIS 13528: Statistical methods for use in proficiency testing by interlaboratory comparisons.

1.1.8. Metrology management

- NF EN ISO 10012: Systems of measurement management – requirements for processes and measuring equipment.

The following concepts are also defined:

- environmental agent: One of the physical, chemical, biological, etc., phenomena that comprises the environment which is likely to take on the characteristics or the behavior of a material or a living being;

- quality: The quality of a product can be defined as its ability to meet its operational objectives. The ISO 8402-94 standard defines quality as the set of the characteristics of an entity that has the ability to satisfy stated and implied needs. In practice, it comes in two forms: external quality, which corresponds to the satisfaction of the customer (listening to the customer) and internal quality, which corresponds to the improvement of the internal workings of the company, for example through the ISO 9000 standard or NF EN ISO/CEI 17025 (with the associated concepts of technical specifications of need [STB, Spécification techniques de besoin], quality assurance plan, specification compliance, etc.);

- reliability: Characteristic of a device expressed by the probability that it performs a required function under given conditions, for a given period of time (with the associated concepts of availability, life profile and mission, environmental and functional implementation conditions, schedule and construction of reliability, Mean time between failures (MTBF) calculation [repairable systems], Mean time to failure (MTTF) [non-repairable systems], etc.).

The environmental agent that is proposed to be mastered in this book is that of mechanical vibration.

We can note that the widespread of environmental tests have made standardization necessary, the influence of the latter on the design of tests and test facilities became fundamental and their development generally precedes the technique.

1.2. Different types of mechanical signals

1.2.1. Overview

Once the need for simulating vibrations tests by appropriate means in the laboratory is established, the interest in usual specifications of the tests used have to be specified, as well as the corresponding testing machines. In the case of vibrations, it is generally quite easy to reproduce on a machine (vibration generators), the same vibration as the one measured and recorded on site. It is, nevertheless, essential to be able to define equivalent vibrations in terms of damage afflicted on structures, so as to be able to define equivalent tests in terms of severity (amplitude, duration, type of vibration, etc.). This approach, commonly used in the development of the test specifications, is rigorously and widely described in bibliography books [ref. 11 (5 volumes)], which constitute references on the subject. It is therefore appropriate to know the different types of signals to be analyzed, knowing that a vibration can be described as an oscillation maintained around an average value that lasts long enough compared to the time constant of the excited system.

1.2.2. Mathematical prerequisites

1.2.2.1. Unit circle

It is a circle of a unit radius that is used to model usual trigonometric functions. It is shown in Figure 1.1 (to also bring closer the model of circle C(0,1) of the complex plane):

equationC01_image002.png

Figure 1.1. Unit circle

Note that the measurements on the x axis support the cosines, the y axis the sines, the x' axis the cotangents and the y' axis the tangents.

Useful trigonometric relationships:

equation

1.2.2.2. Vectors and Fresnel representation

Consider a mobile M describing the circle of center 0 and radius A C(0,A), in uniform movement and angular velocity ω, modeled by the diagram shown in Figure 1.2.

C01_image004.png

Figure 1.2. Fresnel model

At the moment where t = 0, the mobile is in M0 with an initial phase angle given by C01_image005.png . At the moment t, the mobile is in M with C01_image006.png . Therefore, we have:

equation

Thus with any sinusoidal function y(t) = A.sin (ωt+φ), we can match a vector C01_image008.png revolving around 0, with an angular velocity ω with phase at the beginning φ. This is the model of Fresnel’s rotating vector.

Consider T as the duration of a rotation and f the number of laps per second. T is thereby defined as the period of the function, f, the frequency and ω the pulse:

(1.1)

equation

1.2.2.3. Rectilinear sinusoidal movement

In this case, the elongation y(t) is a sinusoidal function of time, with:

C01_image010.png

, where we can see that the concept φ/ω here represents a delay and therefore the phase is closely related to the notion of delay and thus of time (or advance, depending on the sign φ). With ymax = peak value = maximum compared to the reference (motion [m], speed [m/s = m·s−1] and acceleration [m/s/s = m·s−2]).

Several parameters are used to describe this signal. We commonly define three amplitude values:

- peak value: the maximum value from the point of balance. It has an instant character, but it is hardly noticeable when in motion;

- peak-to-peak value: the maximum value of either side of the point of balance. It is also instantaneous, but it can be perceived in an easier manner in a motion movement. Consequently, movements are often specified peak to peak (pk/pk);

C01_image011.png

Figure 1.3. Sinusoidal signal

- average value: defined by:

(1.2) equation

This value represents the continuous value that would result in the same surface between 0 and T. It is obviously zero for a sinusoidal signal due to its symmetry compared to the time axis over a period;

- the average absolute value: usually called average value, defined by:

(1.3) equation

which helps calculating the average value on a half-period. These expressions are independent of the phase φ. So we find:

equation

, therefore:

(1.4) equation

- effective value: one of the most interesting concepts because it is directly connected to the average power contained in the signal. It is defined by the square root (x)¹/² of the quadratic value (y²(t)) average (1/T):

(1.5) equation

In order to be practical, it represents, for example, in electricity, the value of a direct current that would dissipate the same power, in a passive resistive load, than the alternating signal y(t). We can define this concept for any function. Thereby we have:

C01_image017.png

. Therefore:

(1.6) equation

We can infer the following relationships between these values:

(1.7) equation

We can also write the following relations:

equation

(1.8)

equation

(1.9) equation

In the sinusoidal case, we have:

(1.10) equation

In the general case of a distorted periodic signal harmonic or not, the effective value becomes the true effective value, or root means square (RMS) value. It is then necessary to take harmonics into account when there are some (harmonic frequencies of amplitude and phase that are integer multiples of the fundamental).

The relations therefore become:

equation

The Fourier series decomposition of the periodic signal of period T = 1/f0 is then given by:

equation

With:

equation

- n defines the harmonic rank that varies from 1 to infinite (except for the complex expression);

- if n = 1, the harmonic is called fundamental;

- cn is the amplitude of the nth harmonic;

- φn is the relative phase of the fundamental harmonic n.

a) Definitions related to the distorted sinusoidal signal

(a) The effective value or RMS:

equation

(b) The peak factor: C01_image028.png

If the peak factor is different from C01_image029.png , the signal is hence deformed by harmonics.

(c) The total distortion factor (including the continuous component):

equation

NB: We note that the sum starts at n = 2 and so it does not include the fundamental power.

(d) The harmonic rate of distortion compared to the fundamental THD (or TDH), excluding the continuous component (the most common):

equation

And for each harmonic separately: C01_image032.png .

(e) In sine vibration, the amplitude of the fundamental of the piloting signal is obtained by using a tracking filter. Its effective value a1 (also noted as F) is compared to the total effective value atot of the unfiltered global signal (also denoted NF) measured in a wide frequency range (at least five times the excitation frequency). To stay within the usual tolerance T, of a deviation between global signal and filtered signal of ±5% (of the filtered signal), given the accuracy of the measuring equipment, the ratio atot/a1 should not exceed 1.05, or the relation:

equation

For that reason, we show that the value of this ratio corresponds to a total distortion of 32% according to the definition:

equation

This happens if we neglect the share of the broad band noise compared to that relating to harmonics (hypothesis generally verified in sine scanning). Refer to section 41 of GAMEG13 and the NF INTO 60068-2-6 standard.

1.2.2.4. Kinematic parameters

Physically, the visual aspect that an oscillation takes and that is directly perceived, is that of peak to peak displacement. But in addition to this parameter, we also introduce the mathematical concepts of velocity and acceleration. This is, of course, for all types of signals, which we will define, with the specific case of the sinusoidal signal.

a) Displacement

Noted d(t) = x(t) = xmax.sinωt.

Unit = [m] often provided in [mm] in the