Mechanical Vibration and Shock Analysis, Volume 1, Sinusoidal Vibration E-Book

Christian Lalanne

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch
• Veröffentlichungsjahr: 2014

Seitenzahl: 336

Beschreibung

Everything engineers need to know about mechanical vibration and shock...in one authoritative reference work! This fully updated and revised 3rd edition addresses the entire field of mechanical vibration and shock as one of the most important types of load and stress applied to structures, machines and components in the real world. Examples include everything from the regular and predictable loads applied to turbines, motors or helicopters by the spinning of their constituent parts to the ability of buildings to withstand damage from wind loads or explosions, and the need for cars to maintain structural integrity in the event of a crash. There are detailed examinations of underlying theory, models developed for specific applications, performance of materials under test conditions and in real-world settings, and case studies and discussions of how the relationships between these affect design for actual products. Invaluable to engineers specializing in mechanical, aeronautical, civil, electrical and transportation engineering, this reference work, in five volumes is a crucial resource for the solution of shock and vibration problems. The relative and absolute response of a mechanical system with a single degree of freedom is considered for an arbitrary excitation, and its transfer function is defined in various forms. The characteristics of sinusoidal vibration are examined in the context both of the real world and of laboratory tests, and for both transient and steady state response of the one-degree-of-freedom system. Viscous damping and then non-linear damping are considered. The various types of swept sine perturbations and their properties are described and, for the one-degree-of-freedom system, the consequence of an inappropriate choice of sweep rate are considered. From the latter, rules governing the choice of suitable sweep rates are then developed.

Leseprobe

Table of Contents

Foreword to Series

Introduction

List of Symbols

Chapter 1 The Need

1.1. The need to carry out studies into vibrations and mechanical shocks

1.2. Some real environments

1.3. Measuring vibrations and shocks

1.4. Filtering

1.5. Digitizing the signal

1.6. Reconstructing the sampled signal

1.7. Characterization in the frequency domain

1.8. Elaboration of the specifications

1.9. Vibration test facilities

Chapter 2 Basic Mechanics

2.1. Basic principles of mechanics

2.2. Static effects/dynamic effects

2.3. Behavior under dynamic load (impact)

2.4. Elements of a mechanical system

2.5. Mathematical models

2.6. Setting an equation for n degrees-of-freedom lumped parameter mechanical system

Chapter 3 Response of a Linear One-Degree-of-Freedom Mechanical System to an Arbitrary Excitation

3.1. Definitions and notation

3.2. Excitation defined by force versus time

3.3. Excitation defined by acceleration

3.4. Reduced form

3.5. Solution of the differential equation of movement

3.6. Natural oscillations of a linear one-degree-of-freedom system

Chapter 4 Impulse and Step Responses

4.1. Response of a mass–spring system to a unit step function (step or indicial response)

4.2. Response of a mass–spring system to a unit impulse excitation

4.3. Use of step and impulse responses

4.4. Transfer function of a linear one-degree-of-freedom system

4.5. Measurement of transfer function

Chapter 5 Sinusoidal Vibration

5.1. Definitions

5.2. Periodic and sinusoidal vibrations in the real environment

5.3. Sinusoidal vibration tests

Chapter 6 Response of a Linear One-Degree-of-Freedom Mechanical System to a Sinusoidal Excitation

6.1. General equations of motion

6.2. Transient response

6.3. Steady state response

6.4. Responses and

6.5. Responses and

6.6. Responses and

6.7. Graphical representation of transfer functions

6.8. Definitions

Chapter 7 Non-viscous Damping

7.1. Damping observed in real structures

7.2. Linearization of non-linear hysteresis loops – equivalent viscous damping

7.3. Main types of damping

7.4. Measurement of damping of a system

7.5. Non-linear stiffness

Chapter 8 Swept Sine

8.1. Definitions

8.2. “Swept sine” vibration in the real environment

8.3. “Swept sine” vibration in tests

8.4. Origin and properties of main types of sweepings

Chapter 9 Response of a Linear One-Degree-of-Freedom System to a Swept Sine Vibration

9.1. Influence of sweep rate

9.2. Response of a linear one-degree-of-freedom system to a swept sine excitation

9.3. Choice of duration of swept sine test

9.4. Choice of amplitude

9.5. Choice of sweep mode

Appendix. Laplace Transformations

A.1. Definition

A.2. Properties

A.3. Application of Laplace transformation to the resolution of linear differential equations

A.4. Calculation of inverse transform: Mellin-Fourier integral or Bromwich transform

A.5. Laplace transforms

A.6. Generalized impedance - the transfer function

Vibration Tests: a Brief Historical Background

Bibliography

Index

Summary of Other Volumes in the Series

Summary of Volume 2 Mechanical Shock

Summary of Volume 3 Random Vibration

Summary of Volume 4 Fatigue Damage

Summary of Volume 5 Specification Development

First edition published 2002 by Hermes Penton Ltd © Hermes Penton Ltd 2002

Second edition published 2009 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. © ISTE Ltd 2009

Third edition published 2014 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:

ISTE Ltd

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John Wiley & Sons, Inc.

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© ISTE Ltd 2014

The rights of Christian Lalanne 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: 2014930266

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-84821-643-3 (Set of 5 volumes)

ISBN 978-1-84821-644-0 (Volume 1)

Foreword to Series

In the course of their lifetime simple items in everyday use such as mobile telephones, wristwatches, electronic components in cars or more specific items such as satellite equipment or flight systems in aircraft, can be subjected to various conditions of temperature and humidity, and more particularly to mechanical shock and vibrations, which form the subject of this work. They must therefore be designed in such a way that they can withstand the effects of the environmental conditions to which they are exposed without being damaged. Their design must be verified using a prototype or by calculations and/or significant laboratory testing.

Sizing, and later, testing are performed on the basis of specifications taken from national or international standards. The initial standards, drawn up in the 1940s, were blanket specifications, often extremely stringent, consisting of a sinusoidal vibration, the frequency of which was set to the resonance of the equipment. They were essentially designed to demonstrate a certain standard resistance of the equipment, with the implicit hypothesis that if the equipment survived the particular environment it would withstand, undamaged, the vibrations to which it would be subjected in service. Sometimes with a delay due to a certain conservatism, the evolution of these standards followed that of the testing facilities: the possibility of producing swept sine tests, the production of narrowband random vibrations swept over a wide range and finally the generation of wideband random vibrations. At the end of the 1970s, it was felt that there was a basic need to reduce the weight and cost of on-board equipment and to produce specifications closer to the real conditions of use. This evolution was taken into account between 1980 and 1985 concerning American standards (MIL-STD 810), French standards (GAM EG 13) or international standards (NATO), which all recommended the tailoring of tests. Current preference is to talk of the tailoring of the product to its environment in order to assert more clearly that the environment must be taken into account from the very start of the project, rather than to check the behavior of the material a posteriori. These concepts, originating with the military, are currently being increasingly echoed in the civil field.

Tailoring is based on an analysis of the life profile of the equipment, on the measurement of the environmental conditions associated with each condition of use and on the synthesis of all the data into a simple specification, which should be of the same severity as the actual environment.

This approach presupposes a proper understanding of the mechanical systems subjected to dynamic loads and knowledge of the most frequent failure modes.

Generally speaking, a good assessment of the stresses in a system subjected to vibration is possible only on the basis of a finite element model and relatively complex calculations. Such calculations can only be undertaken at a relatively advanced stage of the project once the structure has been sufficiently defined for such a model to be established.

Considerable work on the environment must be performed independently of the equipment concerned either at the very beginning of the project, at a time where there are no drawings available, or at the qualification stage, in order to define the test conditions.

In the absence of a precise and validated model of the structure, the simplest possible mechanical system is frequently used consisting of mass, stiffness and damping (a linear system with one degree of freedom), especially for:

– the comparison of the severity of several shocks (shock response spectrum) or of several vibrations (extreme response and fatigue damage spectra);

– the drafting of specifications: determining a vibration which produces the same effects on the model as the real environment, with the underlying hypothesis that the equivalent value will remain valid on the real, more complex structure;

– the calculations for pre-sizing at the start of the project;

– the establishment of rules for analysis of the vibrations (choice of the number of calculation points of a power spectral density) or for the definition of the tests (choice of the sweep rate of a swept sine test).

This explains the importance given to this simple model in this work of five volumes on “Mechanical Vibration and Shock Analysis”.

Volume 1 of this series is devoted to sinusoidal vibration. After several reminders about the main vibratory environments which can affect materials during their working life and also about the methods used to take them into account, following several fundamental mechanical concepts, the responses (relative and absolute) of a mechanical one-degree-of-freedom system to an arbitrary excitation are considered, and its transfer function in various forms are defined. By placing the properties of sinusoidal vibrations in the contexts of the real environment and of laboratory tests, the transitory and steady state response of a single-degree-of-freedom system with viscous and then with non-linear damping is evolved. The various sinusoidal modes of sweeping with their properties are described, and then, starting from the response of a one-degree-of-freedom system, the consequences of an unsuitable choice of sweep rate are shown and a rule for choice of this rate is deduced from it.

Volume 2 deals with mechanical shock. This volume presents the shock response spectrum (SRS) with its different definitions, its properties and the precautions to be taken in calculating it. The shock shapes most widely used with the usual test facilities are presented with their characteristics, with indications how to establish test specifications of the same severity as the real, measured environment. A demonstration is then given on how these specifications can be made with classic laboratory equipment: shock machines, electrodynamic exciters driven by a time signal or by a response spectrum, indicating the limits, advantages and disadvantages of each solution.

Volume 3 examines the analysis of random vibration which encompasses the vast majority of the vibrations encountered in the real environment. This volume describes the properties of the process, enabling simplification of the analysis, before presenting the analysis of the signal in the frequency domain. The definition of the power spectral density is reviewed, as well as the precautions to be taken in calculating it, together with the processes used to improve results (windowing, overlapping). A complementary third approach consists of analyzing the statistical properties of the time signal. In particular, this study makes it possible to determine the distribution law of the maxima of a random Gaussian signal and to simplify the calculations of fatigue damage by avoiding direct counting of the peaks (Volumes 4 and 5). The relationships that provide the response of a one-degree-of-freedom linear system to a random vibration are established.

Volume 4 is devoted to the calculation of damage fatigue. It presents the hypotheses adopted to describe the behavior of a material subjected to fatigue, the laws of damage accumulation and 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 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.). The main laws governing low cycle fatigue and fracture mechanics are also presented.

Volume 5 is dedicated to presenting the method of specification development according to the principle of tailoring. The extreme response and fatigue damage spectra are defined for each type of stress (sinusoidal vibrations, swept sine, shocks, random vibrations, etc.). The process for establishing a specification as from the lifecycle profile of the equipment is then detailed taking into account the uncertainty factor (uncertainties related to the dispersion of the real environment and of the mechanical strength) and the test factor (function of the number of tests performed to demonstrate the resistance of the equipment).

First and foremost, this work is intended for engineers and technicians working in design teams responsible for sizing equipment, for project teams given the task of writing the various sizing and testing specifications (validation, qualification, certification, etc.) and for laboratories in charge of defining the tests and their performance following the choice of the most suitable simulation means.

Introduction

Materials which are transported by or loaded onto land vehicles, aircraft or marine vehicles, or which are installed close to turning machines, are subject to different vibrations and mechanical shocks. These materials must be able to endure such shocks and vibrations without being damaged. To achieve this goal, the first step consists of noting the values of these environments in the specifications of the material to be developed, so that the research departments can take them into account during dimensioning. The following step is the qualification of the designed material, starting from these specifications, to experimentally demonstrate its behavior under its future conditions of use.

The specifications used for dimensioning and testing today are elaborate, starting from measurements of the real environment which the equipment will undergo (test tailoring). It is thus necessary to correctly measure the vibrations and shocks before analyzing them and to synthesize them to obtain specifications leading to reasonable qualification tests of a reasonable duration.

Taking into account vibrations and shocks thus requires us:

– to identify the future conditions of use;

– to carry out, if possible, significant measurements;

– to digitize the measured signals;

– to identify each type of vibration in order to characterize them by analysis in the frequency domain, adapted to proceed to severity comparisons between the collected measurements under various conditions, or between real environments and values provided in normative documents, or with specifications established in another context;

– to finally transform measurements into specifications.

The object of this series of five volumes is thus to describe all the mathematical tools that are currently used in the analysis of vibrations and shocks, while starting with the sinusoidal vibrations.

Sinusoidal vibrations were first used in laboratory tests to verify the ability of equipment to withstand their future vibratory environment in service without damage. Following the evolution of standards and testing facilities, these vibrations, generally speaking, are currently studied only to simulate vibratory conditions of the same nature as encountered, for example, in equipment situated close to revolving machinery (motors, transmission shafts, etc.). Nevertheless, their value lies in their simplicity, enabling the behavior of a mechanical system subjected to dynamic stress to be demonstrated, and the introduction of basic definitions.

Given that, generally speaking, the real environment is more or less random in nature, with a continuous frequency spectrum in a relatively wide range, in order to overcome the inadequacies of the initial testing facilities, testing rapidly moved to the “swept sine” type. Here the vibration applied is a sinusoid, the frequency of which varies over time according to a sinusoidal or exponential law. Despite the relatively rapid evolution of electrodynamic exciters and electrohydraulic vibration exciters, capable of generating wideband random vibrations, these swept sine standards have lasted, and are in fact still used, for example in aerospace applications. They are also widely used for measuring the dynamic characteristics of structures.

After an introductory chapter (Chapter 1) to this series, pointing out the characteristics of some important vibratory environments and the various steps necessary to arrive at the qualification of a material, we follow-up with a few brief reminders of basic mechanics (Chapter 2). Chapter 3 examines the relative and absolute response of a mechanical system with one degree of freedom subjected to a given excitation, and defines the transfer function in different forms. Chapter 4 is devoted more particularly to the response of such a system to a unit impulse or to a unit step.

The properties of sinusoidal vibrations are then presented in the context of the environment and in laboratory tests (Chapter 5). The transitory and steady state response of a system with one degree of freedom to viscous damping (Chapter 6) and to non-linear damping (Chapter 7) is then examined.

Chapter 8 defines the various sinusoidal sweeping modes, with their properties and eventual justification. Chapter 9 is devoted to the response of a system with one degree of freedom subjected to linear and exponential sweeping vibrations, to illustrate the consequences of an unsuitable choice of sweep rate, resulting in the presentation of a rule for the choice of a rate.

The major properties of the Laplace transform are reviewed in the Appendix. This provides a powerful tool for the analytical calculation of the response of a system with one degree of freedom to a given excitation. Inverse transforms particularly suitable for this application are given in a table.

List of Symbols

The list below gives the most frequent definition of the main symbols used in this book. Some of the symbols can have another meaning locally which will be defined in the text to avoid confusion.

Chapter 1

The Need

1.1. The need to carry out studies into vibrations and mechanical shocks

During their service life, many materials are subjected to vibratory environments, during their transport [OST 65], [OST 67], because they are intended to equip themselves with means of transport (airplanes, road vehicles, etc.) or because they are placed beside vibratory sources (engines, wind mills, roads, etc.). These vibratory environments (vibrations and shocks) create dynamic strains and stresses in the structures which can, for example, produce intermittent or permanent breakdowns in electrical equipment, plastic deformations or fractures by up-crossing an ultimate stress of the material (yield limit, rupture limit), optical misalignments of systems or may contribute to the fatigue and the wear of the machine elements.