Practice of Vibration Measurement: Measurement Technology and Vibration Analysis with MATLAB® 365838462X, 9783658384623

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Thomas Kuttner · Armin Rohnen

Practice of Vibration Measurement Measurement Technology and Vibration Analysis with MATLAB®

Practice of Vibration Measurement

Thomas Kuttner • Armin Rohnen

Practice of Vibration Measurement Measurement Technology and Vibration Analysis with MATLAB®

Thomas Kuttner Fakultät für Maschinenbau Universität der Bundeswehr München Neubiberg, Bayern Germany

Armin Rohnen Fakultät 03 Fachhochschule München München, Bayern Germany

ISBN 978-3-658-38462-3 ISBN 978-3-658-38463-0 https://doi.org/10.1007/978-3-658-38463-0

(eBook)

This book is a translation of the original German edition Praxis der Schwingungsmessung, by Kuttner, Thomas, published by Springer Fachmedien Wiesbaden, DE in 2019. The translation was done with the help of artificial intelligence (machine translation by the service DeepL.com). A subsequent human revision was done primarily in terms of content, so that the book will read stylistically differently from a conventional translation. Springer Nature works continuously to further the development of tools for the production of books and on the related technologies to support the authors. # The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Fachmedien Wiesbaden GmbH, part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Preface to the 2nd Edition

After the positive reception of the first edition by the readership, Springer Vieweg Verlag has suggested the publication of a new edition. The present work has undergone a fundamental revision, expansion and reorientation. Thanks to the collaboration of Armin Rohnen as a colleague and friend, the book has been expanded to include the contents in Chaps. 11, 12, 13 and 15, and signal analysis with MATLAB® has been added, as well as numerous examples from measurement practice. With this new orientation, the authors hope to be able to take into account the current developments in measurement technology and signal processing and to provide the user with a handbook for solving practical measurement tasks. Like the first edition, the book is aimed at three target groups: • Students and beginners who are coming into contact with the field of vibration measurement technology for the first time and are looking for an introduction to this interdisciplinary subject. Here you will find the basics, the measurement procedure and scripts for signal processing with MATLAB®. • Experts from practice who want to exhaust the possibilities and limits of measurement procedures and evaluation. In practice, this often involves a “horizontal” comparison of different procedures and methods with the aim of selecting the most suitable. The authors hope to have described the procedures sufficiently in breadth and depth. Almost without exception, methods are presented which are used in the practical work of the authors; the methods of signal analysis have all been practically tested in teaching and applied research. • Application specialists and technical sales staff who master one part of the measurement chain and are interested in the overall context. This supplements the requirements of the previous group with a “vertical” track, which focuses on the overall context of the signal flow. As already practiced in the first edition, the knowledge is presented from the application point of view. The design of measuring chains including practical problem-solving during their commissioning, plausibility checks and troubleshooting are presented in a practical

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manner. With this orientation, the book aims to convey solution-oriented approaches for measurement practice and to provide the user with practical solutions. In the interdisciplinary field of vibration measurement technology, the three interlocking parts are dealt with in one book: vibration technology as a sub-discipline characterized by mechanics, measurement technology as a component of electronics and signal processing as an area that draws its techniques strongly from computer science. The basic knowledge of technical mechanics, electrical engineering and measurement technology is assumed, which is taught, for example, in the first semesters of a mechanical engineering or electrical engineering course. The approach of presenting everything in one book necessarily entails a limitation to the basic methods and procedures in order not to exceed the scope of the work. The proven specialist will certainly miss one or the other procedure, for which reference must be made to further literature. Not only almost 20 years of professional practice in this field but also the support and exchange with colleagues and industry, who cannot all be mentioned by name, have flowed into this technical book. Special thanks go to Dr. Kerstin Kracht for her review of the manuscript and her valuable suggestions. The authors would like to thank Mr. Begoff from SPEKTRA in Dresden for reviewing the manuscript and making improvements to the Calibration of Accelerometers section. The present work is based on a DeepL translation provided by the publishing house. A very special thanks to Tiasa Ghosh for proof-reading the manuscript. Due to her concentrated efforts, DeepL English is transferred to humans. Finally, the authors would like to express their sincere thanks to the editorial office of Springer Vieweg Verlag for their kind guidance and permanent support. Feedback from readers is important to the authors in order to further improve this work. Do you have any suggestions, have you noticed any errors in the content, or do you have any suggestions for improving the presentation? For feedback, you can reach us at the following e-mail: [email protected] https://www.professorkuttner.de [email protected] https://schwingungsanalyse.com Munich, Germany Planegg, Germany December 2022

Thomas Kuttner Armin Rohnen

Contents

Vibrations and Its Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Subject Matter and Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Definition and State Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Classification of Vibrations According to Their Time Course . . . . . . 1.4 Classification of Vibrations According to Their Mechanism of Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Systematization of the Measurement Tasks . . . . . . . . . . . . . . . . . . . 1.6 Planning and Concept of Measuring Equipment . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 8 10 12

2

Vibrations in the Time and Frequency Domain . . . . . . . . . . . . . . . . . . 2.1 Harmonic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Phasor Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Representation in the Time Domain and Frequency Domain . . . . . . 2.3.1 Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

13 13 17 21 21 22 26 29

3

Free Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Translational Vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Rotational Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Free Damped Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

31 31 34 35 43

4

Forced Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Spring Force Excitation with Constant Force Amplitude . . . . . . . . . . 4.2 Amplitude and Phase Frequency Response . . . . . . . . . . . . . . . . . . . . 4.3 Transfer Functions and Their Inverses . . . . . . . . . . . . . . . . . . . . . . . 4.4 Nyquist Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Compilation of Various Transfer Functions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 46 49 52 54 56 65

1

1 2 3 4

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5

Vibration Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Measuring Principles for Kinematic Quantities . . . . . . . . . . . . . . . . . 5.1.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Relative Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Absolute Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Selection of the Transducer and the Measured Variable . . . . . . . . . . . 5.3 Representation in Frequency Bands . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Decibels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 67 69 70 73 75 77 80

6

Displacement Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Potentiometric Displacement Transducers . . . . . . . . . . . . . . . . . . . 6.2 Capacitive Displacement Transducers . . . . . . . . . . . . . . . . . . . . . . 6.3 Inductive Displacement Transducers . . . . . . . . . . . . . . . . . . . . . . . 6.4 Displacement Transducers According to the Eddy Current Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Magnetostrictive Displacement Transducers . . . . . . . . . . . . . . . . . . 6.6 Digital Displacement Transducers . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Fibre Optic Displacement Transducers . . . . . . . . . . . . . . . . . . . . . . 6.8 Displacement Transducers Based on the Laser Triangulation Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Videographic Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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81 81 83 84

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88 89 90 91

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93 94 98

7

Velocity Transducer (Vibration Velocity Transducer) . . . . . . . . . . . . . 7.1 Electrodynamic Velocity Transducers . . . . . . . . . . . . . . . . . . . . . . 7.2 Electromagnetic Velocity Transducers . . . . . . . . . . . . . . . . . . . . . . 7.3 Laser Doppler Vibrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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101 101 103 104 109

8

Accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Piezoelectric Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 The Piezoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Signal Conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Frequency Responses and Measuring Ranges . . . . . . . . . . 8.1.5 Factors Influencing the Measurement with Piezoelectric Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.6 Mounting of Piezoelectric Accelerometers . . . . . . . . . . . . . 8.1.7 Selection of Piezoelectric Accelerometers . . . . . . . . . . . . . 8.2 Strain Gauge Accelerometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Capacitive Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Piezoresistive Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111 111 112 114 117 124 128 131 139 142 143 144

Contents

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8.5 Accelerometer According to the Servo Principle . . . . . . . . . . . . . . . . 8.6 Calibration of Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

146 146 148

Deformation Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Strain Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Structure and Mode of Operation . . . . . . . . . . . . . . . . . . . 9.1.2 Factors Influencing the Measurement with Strain Gauges . . 9.1.3 Quarter Bridge Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Half-Bridge Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Full Bridge Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.6 Carrier Frequency and DC Voltage Amplifiers . . . . . . . . . . 9.1.7 Application and Calibration . . . . . . . . . . . . . . . . . . . . . . . 9.1.8 Measurement of Uniaxial Stress Conditions by Means of Strain Gauges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Measuring Principle of Force and Moment Transducers . . . . . . . . . . 9.3 Strain Gauge Force and Moment Transducers . . . . . . . . . . . . . . . . . 9.3.1 Force Transducer with Tension/Compression Element . . . . 9.3.2 Force Transducer with Shear Deformation Element . . . . . . 9.3.3 Force Transducer with Bending Element . . . . . . . . . . . . . . 9.3.4 Torque Transducer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.5 Selection of Strain Gauge Load Cells . . . . . . . . . . . . . . . . 9.4 Piezoelectric Force and Torque Transducers . . . . . . . . . . . . . . . . . . . 9.5 Magnetoelastic Force and Moment Transducers . . . . . . . . . . . . . . . . 9.6 Multi-Component Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Installation of Force and Moment Transducers . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

149 150 150 154 157 160 160 162 165

Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Signal Flow and Device Functions . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Measurement Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Electromagnetic Influence on the Measurement Chain . . . . . . . . . . 10.3.1 Causes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Corrective Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Calibration and Plausibility Check of the Measurement Chain . . . . . 10.4.1 Methods for Calibrating the Measurement Chain . . . . . . . 10.4.2 Characteristic, Offset and Transmission Coefficient . . . . . 10.4.3 Signal Path Tracing and Troubleshooting in Measurement Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 TEDS (Transducer Electronic Data Sheet) . . . . . . . . . . . . . . . . . . . 10.6 Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 Tasks and Function of Filters . . . . . . . . . . . . . . . . . . . . .

168 177 182 183 185 185 188 189 190 192 195 198 201

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203 203 205 210 210 213 220 220 222

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231 234 235 235

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10.6.2 Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.3 Transient Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.4 Filter Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Analogue to Digital Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.1 Digital Measurement Technology . . . . . . . . . . . . . . . . . . 10.7.2 Quantisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.3 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7.4 Technical Implementation in a/D Converters . . . . . . . . . . 10.8 Telemetric Signal Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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238 243 245 247 247 248 252 255 260 266

11

MATLAB® and Data Formats an Introduction . . . . . . . . . . . . . . . . . . 11.1 MATLAB® After Startup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 MATLAB® Help . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 MATLAB® Data Type Struct . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Creating Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5 Data Formats for Measurement and Metadata . . . . . . . . . . . . . . . . . 11.5.1 Mathworks *.mat Files . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 Audio Data Format: WAV . . . . . . . . . . . . . . . . . . . . . . . 11.5.3 Comma: Separated Values – CSV . . . . . . . . . . . . . . . . . . 11.5.4 Universal Data Format: UFF . . . . . . . . . . . . . . . . . . . . . 11.5.5 ASAM ODS Format: ATF/ATFX . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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269 270 271 271 274 279 281 283 284 288 295 296

12

Measurement with MATLAB® . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Measuring with the OnBoard Sound Card . . . . . . . . . . . . . . . . . . . . 12.1.1 Measuring with the OnBoard Sound Card and some Operating Convenience . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Data Acquisition Toolbox™ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2.1 Data Acquisition Toolbox™ and Improved Ease of Use . . . 12.2.2 Data Acquisition Toolbox™: Standard Tasks . . . . . . . . . . 12.2.3 Instrument Control Toolbox™: HP-IB . . . . . . . . . . . . . . . 12.3 Data Acquisition Toolbox™ in Conjunction with Professional Audio Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Generating Signals with the Data Acquisition Toolbox™ . . . . . . . . . 12.5 MATLAB GUI: Graphical User Interface . . . . . . . . . . . . . . . . . . . . 12.5.1 Creating a Graphical User Interface . . . . . . . . . . . . . . . . . . 12.5.2 Assign Functions to the Elements of the Graphical User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6 Measurement Process: A Variable Data Logger . . . . . . . . . . . . . . . . 12.6.1 Measurement Process: Documentation . . . . . . . . . . . . . . . 12.6.2 Measurement Process: Measurement . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299 300 302 305 308 315 320 324 329 340 341 344 351 353 356 368

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13

Raspberry Pi as a Measuring Device . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Raspberry Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Digital IO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 Voltage Measurement with the A/D Converter ads1015 . . . . . . . . . 13.4 Speed Measurement Via Interrupt at Digital IO . . . . . . . . . . . . . . . 13.5 Bridge Circuit with DC Voltage Measuring Amplifier . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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369 369 373 377 379 385 391

14

Signal Analysis Methods and Examples . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Tasks and Methods of Signal Analysis . . . . . . . . . . . . . . . . . . . . . . . 14.2 Signal Analysis in the Time Domain . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 RMS Value, Power, Mean Value and Related Quantities . . 14.2.2 Application Examples: RMS Value, Power, Mean Value and Related Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.3 Envelopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 Crest Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.5 Autocorrelation and Cross-Correlation . . . . . . . . . . . . . . . 14.2.6 1/n Octave Bandpass Filtering . . . . . . . . . . . . . . . . . . . . . 14.3 Signal Analysis in the Frequency Domain . . . . . . . . . . . . . . . . . . . . 14.3.1 Amplitude Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Counting Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4 Signal Analysis in the Frequency Domain . . . . . . . . . . . . . . . . . . . . 14.4.1 Fourier Transform: FFT or DFT? . . . . . . . . . . . . . . . . . . . 14.4.2 Fundamentals of the Discrete Fourier Transform . . . . . . . . 14.4.3 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.4 Relationships Between the DFT Parameters . . . . . . . . . . . . 14.4.5 Leakage and Window Functions . . . . . . . . . . . . . . . . . . . . 14.4.6 Triggering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.7 Averaging and Overlapping . . . . . . . . . . . . . . . . . . . . . . . 14.4.8 Spectral Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.9 Axis Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.10 Differentiating and Integrating . . . . . . . . . . . . . . . . . . . . . 14.4.11 Practical Settings for Orientation Measurements . . . . . . . . . 14.4.12 Practical Implementation in MATLAB® with Test Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4.13 Spectral Quantities in MATLAB® . . . . . . . . . . . . . . . . . . . 14.4.14 Modulation Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Transfer Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

393 393 394 394

15

Experimental Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Assumptions and Explanations of Terms . . . . . . . . . . . . . . . . . . . . . 15.2 Summary of the Analytical Principles of Modal Analysis . . . . . . . . .

398 401 407 408 412 420 420 422 429 429 431 439 440 445 456 457 463 468 471 473 476 491 494 499 511 513 515 516

xii

Contents

15.3

Operational Performance of the Experimental Modal Analysis . . . . . 15.3.1 Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Object Excitation by Means of an Impulse Hammer . . . . . 15.3.3 Object Excitation by Means of a Shaker . . . . . . . . . . . . . 15.4 Evaluation of the Experimental Modal Analysis in MATLB® . . . . . 15.4.1 Evaluation of Measurements with Impulse Hammer Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4.2 Evaluation of Measurements with Shaker Excitation . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . .

521 523 523 526 528

. . .

528 537 542

1

Vibrations and Its Measurement

Abstract

This chapter outlines the topic, shows the interdisciplinary interaction of the individual sub-areas and defines basic terms. The classification of vibrations according to the course of time and the mechanism of origin are presented. This chapter thus lays the foundations for understanding the vibration process, the measurement task and the interpretation of the measurement results. The knowledge is structured using graphical representations and illustrated with numerous practical examples. "

This chapter outlines the subject, shows the interdisciplinary interaction of the individual sub-areas and defines basic terms. The classification of vibrations according to the course of time and the mechanism of origin are presented and the possible types of measurement tasks are systematized on the basis of the transfer function. The planning and design of measuring equipment for vibration measurement is dealt with on the basis of selected criteria. This chapter thus lays the foundations for understanding the vibration process, the measurement task and the interpretation of the measurement results. With this in-depth knowledge, the user will be in a better position to analyze and plan his specific measurement tasks. For the selection of measurement technology, the criteria catalog provides a decision-making aid.

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 T. Kuttner, A. Rohnen, Practice of Vibration Measurement, https://doi.org/10.1007/978-3-658-38463-0_1

1

2

1.1

1

Vibrations and Its Measurement

The Subject Matter and Usage

Vibration measurements are an important and indispensable tool for solving a number of engineering tasks. Without prejudice to the later systematization in Sect. 1.5, some of the examples are as follows: • Vibration measurement on a machine foundation to measure the vibration effect at the installation site and comparison with permissible thresholds, • Measurement of vibrations on chassis components in order to measure the occurring load-time curves (forces and moments) and to derive load assumptions for the design, • Measurement of the dynamic spring stiffness of components with the aim of determining material and component characteristics under oscillating loads. From the different tasks it is obvious that the measuring technique used and the execution of the measurement must be adapted to the respective measuring task. Vibration measurement technology represents the intersection of three scientific disciplines: vibration engineering, measurement technology and signal processing (Fig. 1.1). The fundamentals of vibration technology provide a background for defining the measurement task, selecting the measurement principle and transducer, and further processing, evaluating and interpreting the measurement. With the knowledge of sensor and measurement technology, transducers and measurement equipment can be optimally selected and set up. This in turn requires preliminary estimates from vibration engineering. With the knowledge of vibration technology and measurement technology, the requirements can be addressed in the direction of signal processing in order to further process and evaluate the signals of the transducer in a meaningful way. Signal processing – today almost exclusively in the form of digital signal processing – provides the methods for processing the electrical quantities from the transducer. For a number of transducers, digital signal processing is a basic requirement for their function (e.g. vibrometers, magnetostrictive displacement and force measurement). For signal processing, the description of the vibration process to be measured on one hand and the measurement principle used in the transducer or its mathematical model on the other hand is indispensable information for sensible further processing of the signals and their correct interpretation. From the interdisciplinary interaction of vibration technology, measurement technology and signal processing, the fields of application can be derived, for example, in fatigue strength, experimental modal analysis and condition monitoring, each of the fields mentioned requires reliable measurement results, on the basis of which evaluation and interpretation follow. On the other hand, the mentioned fields of application also drive the development in the field of sensor technology, its circuitry and system theory as mathematical background for signal processing.

1.2

Definition and State Variables

3

Modal Analysis

Structural durability

Condition monitoring

Vibration technology

Signal processing

Metrology

Systems Theory

Sensors

Hardware design

Fig. 1.1 Interaction of the sub-areas of vibration measurement technology

1.2

Definition and State Variables

DIN 1311-1 gives the following definition for vibrations [1]: " An oscillation is a change in a state variable of a system in which, in general, the state

variable alternately increases and decreases. Special temporal changes such as impact or creep processes are also referred to as oscillations in an extended sense. Mechanical vibration processes are described by state variables in a mechanical system [2]. Mechanical state variables are: • Kinematic state variables: In the case of a translational motion, this can be the displacement. Likewise, the velocity or acceleration can be considered as kinematic quantities, which result from a time derivative of the displacement. Analogously, the state variables of a rotational oscillation are the angle, the angular velocity and the angular acceleration. • Dynamic state variables: Internal reactions can be used to describe the state of an vibrational system, e.g. normal force, shear force, bending moment or torsional moment. These quantities can also be used in normalized form as stress (normal stress or shear stress).

4

1

Vibrations and Its Measurement

• Deformational state variables: In addition to stresses, deformations (strains and shear distortions) can also be used to describe the state of a vibrational system. In order to fulfil the measurement task, it is therefore important to select a suitable state variable in the vibrating system on the one hand and a state variable that is readily accessible for measurement on the other.

1.3

Classification of Vibrations According to Their Time Course

In the following, vibrations are to be classified according to various aspects. DIN 1311-1 [1] provides an overview of the classification of vibrations according to their time history, which is presented here in modified form [2, 3] (Fig. 1.2). Here, the vibrations are divided into deterministic and stochastic vibrations. Deterministic vibrations can be described over time by a function. In the course of time, a function value x(t) is determined for each time t via a mathematical function. An example of this is material testing in a vibration testing machine, where the force-time curve or displacementtime curve should be determined as precisely as possible. In contrast, stochastic vibrations are not predictable over time. A precise separation between deterministic and stochastic vibrations depends on the respective assumptions in the observation. In the example of the vibration testing machine, the future course of the signal is only predictable until the specimen breaks or the machine switches off for other reasons. Likewise, on closer examination, the measured vibration will not turn out to be a pure sine wave, but will be superimposed by stochastic components (e.g. vibrations caused by passing vehicles, noise from the transducer, etc.). In practice, an appropriate decision must then be made regarding the course of the vibration. For the measurement of the vibration on vibration testing machines and their control, it is expedient to assume that the machine continues to run (and shutdown criteria must be provided for this case) and that the stochastic vibrations can be neglected. Deterministic oscillations can be further subdivided into periodic oscillations. Periodic oscillations have the characteristic feature that the course of the function repeats itself after a period duration T: xðtÞ = xðt þ TÞ :

ð1:1Þ

This characteristic must apply over the entire time course under consideration. The section of length T from the function progression is therefore repeated infinitely often. Thus the function course repeats itself also after an arbitrary number of periods. A constant x(t) does not count to the periodic oscillations – although a signal section with arbitrary T could be repeated. In the case x = const. The periodic increase and decrease of the state variable from the definition for oscillations is not given. Likewise, the definition does not apply to decaying oscillations (e.g. Sect. 3.3), since the zero crossings occur with a period T, but the

1.3

Classification of Vibrations According to Their Time Course

5

harmonic periodic non-harmonic amplitudemodulated deterministic

modulated frequency modulated non-periodic

Vibration

transient ergodic stationary unergodic stochastic continuous transient transient

Fig. 1.2 Classification of vibrations according to their time course

amplitudes decrease. The simplest special case of periodic oscillations are harmonic oscillations, which can be understood as sinusoidal oscillations with any phase angle. For this reason, cosine oscillations also belong to the harmonic oscillations. In acoustics, however, the term “harmonic” also denotes an integer frequency ratio of individual sine oscillations. In this context, the term harmonic oscillation is not used here. Modulated oscillations occur when a harmonic oscillation is changed by a modulating process. If the modulating process changes the amplitude, we are dealing with amplitudemodulated oscillations. If the frequency is changed by the modulating process, the oscillations are called frequency-modulated oscillations. Non-periodic oscillations are deterministic oscillations whose time course does not repeat (e.g. half-sine shock). Transient oscillations are understood as a transition between two states [1]. The future course of stochastic oscillations cannot be described by a formulaic relationship. For this reason, they are also referred to as random vibrations. The vibration process during the rolling of a vehicle wheel on an uneven road surface shall serve as an example. Since the distribution of the unevenness of the road surface is generally unpredictable, it is not possible to determine the frequency. Since the distribution of the road unevenness is generally not predictable (and the selected lane is also not predictable), this oscillation process is to be regarded as a stochastic oscillation. However, statistical statements can be made about the probability with which small and large unevennesses are rolled over. From this, for example, the mean value and the distribution function can be formed as

6

1

Vibrations and Its Measurement

characteristic values and characteristic functions. If one assumes that, for example, the mean value is invariable over time, then one speaks of a stationary stochastic oscillation. It certainly requires little imagination that this is the case if the vehicle, its speed and the road condition do not change. If, on the other hand, the road condition changes, for example, the mean value also changes. In this case, we are dealing with a transient stochastic oscillation. This classification depends on the time interval considered and the characteristic values and functions used. The time interval is of finite duration and must also be shorter than the measurement to be evaluated, i.e. it must be possible to move it over the signal as a temporal section. If the time interval is shortened (e.g. to roll over a single pebble in the road surface), the deviations inevitably become apparent and the oscillation that was considered stationary becomes transient. It is therefore crucial to specify the time interval for determining characteristic values and functions (or to apply the time interval specified in standards and regulations). A further level of consideration results if, in addition to time averaging, averaging is carried out over several similar processes. In further development of the example of uneven road surfaces, one can imagine having several similar vehicles drive over the same road section. In addition to the temporal averaging, one then obtains a so-called share average at a point in time (from the start of a fixed starting point for all vehicles). If the temporal mean value is equal to the share mean value, this is referred to as an ergodic process. This form of classification is not the only possible one. Alternative classifications can be found, for example, in [4, 5], where the vibrations are divided into stationary and non-stationary. Subsequently, a subdivision into deterministic and stochastic oscillations is made in each of the two groups. With this classification, mixed forms between stationary and non-stationary as well as deterministic and stochastic oscillations are easy to describe. "

From the point of view of measurement practice, the classification according to the time course is of great importance. The temporal course of the oscillations can provide information on the mechanism of origin. For this reason, it always makes sense to first record and assess the time course before carrying out further analyses (e.g. display of the spectrum). In measurement practice, the transducers used, the measurement technology and the signal processing methods often depend on the type and characteristics of the vibrations being investigated. This means that the approach to measurement, acquisition and evaluation depends on the vibrations to be measured and their temporal course.

1.4

1.4

Classification of Vibrations According to Their Mechanism of Origin

7

Classification of Vibrations According to Their Mechanism of Origin

If one considers vibrations according to the mechanism of origin, one obtains the following classification in Fig. 1.3 according to [1, 2]. The following statements apply to linear oscillatory systems. In autonomous oscillations, the oscillation system itself determines the frequencies of the time function. Free oscillation is one of the autonomous oscillations. Free vibration occurs when a system capable of vibration is left on its own with an initial condition. If, for example, a vehicle door is slammed shut, the body panel is deflected (initial condition) and performs free oscillations. No energy is supplied from the outside, but energy is released by the damping that is always present, so that the vibration process comes to a standstill. Selfexcited oscillations also belong to the autonomous oscillations. These are characterized by the initial state and the supply of energy. Examples are vibrations of components which are in a transient flow (wings of airplanes, masts and vehicle parts). Friction-induced vibrations also fall into the group of self-excited vibrations, such as squealing brakes or the generation of sound on a violin string. Here, the energy supply comes from the falling friction characteristic (i.e. with increasing speed, the frictional force becomes smaller). Heteronomous oscillations are characterized by frequencies caused by the external influences on the system. Thus, the system oscillates with the frequency of the external influence. Forced vibrations are caused by an external influence (excitation) on the vibrating system. An example of this is the foundation vibration of a press. The foundation is set into vibration by the excitation forces of the press. The characteristic of forced vibration is the presence of the excitation forces when the system is not vibrating. In the example, this occurs when the press is not operated on the foundation. Deflection from the equilibrium position, on the other hand, is not necessary to excite forced oscillations. The

free vibrations autonomous self-excited oscillation Vibration forced oscillation heteronomous parameter excited (rheonome) oscillation

Fig. 1.3 Classification of vibrations according to their mechanism of origin

8

1

Vibrations and Its Measurement

system does not oscillate at its natural angular frequency, but at the excitation angular frequency. During the transient oscillation process, the natural oscillations of the oscillating system can also occur. However, these decay as a result of the damping and the system is in a steady state. Parameter-excited oscillations (rheonome oscillations) are caused by the temporal change of a parameter in the oscillating system. Furthermore, a deflection around the equilibrium position is necessary. Examples of this are torsional vibrations in gears. During the rolling process of the tooth flanks, a time-dependent change in the torsional stiffness of the shaft occurs. In addition to the deflection (tensioning of the gears), the prerequisites for the occurrence of parameter-excited vibrations are thus given. "

1.5

For the solution of the measurement task it is advantageous to know the connections from the cause of the vibration to its appearance. From the appearance (e.g. time curve, frequency spectrum, etc.), conclusions can be drawn about the mechanism of origin (cause). This allows the measurement to be optimally tailored to the measurement task (e.g. transducer, signal processing). If the appearance and mechanism are known, the vibration process can be specifically influenced (e.g. increasing the conveying capacity of vibratory conveyors, reducing the effect of vibration).

Systematization of the Measurement Tasks

A system capable of oscillation is now considered, which is excited by external forces or paths (Fig. 1.4). The vibration response of the system can in turn be described by a state variable (e.g. displacement, velocity or acceleration). The link between the state variables at the input and those at the output is established by the transfer function, which contains the parameters of the oscillating system – in the simplest case mass, spring constant and damping. The transfer function is frequency-dependent and is divided into an amplitude component and a phase component. The amplitude component can be thought of as the amplification of the output relative to the input. The phase component is an expression for the time difference between the output and the input. The transfer function also links the physical units (e.g. force at the input in N and acceleration at the output in m/s2). Following a suggestion by [4], measurement tasks can be divided into three categories (basic tasks): Measurement of the Vibration Response This measurement task aims at measuring state variables at the output of the oscillating system. The vibration responses can then be compared with permissible limit values. Limit values can exist in different forms. In the simplest case, they are present in a numerical value. An example of this is the measurement of foundation vibrations in the condition monitoring of machines. Here, the vibration response at the machine foundation is

1.5

Systematization of the Measurement Tasks

Fig. 1.4 Systematization of the measurement tasks. (a) Measurement of the vibration response, (b) Measurement of the excitation, (c) Determination of the transfer function

9

a

Parameter Response

Exitation Vibratory system

b

Parameter

Excitement

Reply Vibratory system

c

Parameter

Excitement

Reply Vibratory system

measured, specified as an effective value and compared with a permissible limit value. This example also illustrates that the excitation forces and parameters of the vibrating system are not the subject of the measurement. The boundaries between vibration response and excitation are fluid here and thus also depend on the definition of the vibratory system. For example, the time course of the vibration response at the foundation can also be understood as excitation of the floor slab in a structure. Measurement of Excitation The vibration response of a system depends on the excitation (e.g. forces) and the transfer function. For the dimensioning and calculation of machines, the load assumptions as input variables of the oscillating system are of great importance. Knowledge of the load assumptions provides the designer with the prerequisite for designing machines and their components in such a way that the requirements with regard to strength and vibration response are met. The excitation variables have the advantage for the calculation of being independent of the vibrating system and its response. As an example, the measurement of the operating loads on a chassis of a motor vehicle shall serve. For the designer it is of central importance to know the temporal course of the forces and moments at the wheel for the calculation. This enables dimensioning to meet the strength requirements on the one hand and estimation of the accelerations as a vibration response on the other. In real driving operation, however, the forces and moments at the wheel contact point elude direct measurement. For this reason, e.g. the road unevenness of representative road sections is digitized and used as excitation. This approach has the advantage of being able to calculate the vibration responses at chassis parts independently of the vehicle under investigation (i.e. its transfer function).

10

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Vibrations and Its Measurement

Determination of the Transfer Function To determine the transfer function, the output and input variables are measured. The properties of the oscillatory system as a function of frequency can be calculated from this. The transfer function provides information on the behaviour of the oscillating system. Based on this, a mathematical model is used so that the parameters in the mathematical model can be determined from the transfer function. In the simplest case, the vibratory system can be assumed to consist of a mass, a spring and a damper. A number of assumptions are necessary to determine the parameters. For example, the assumption of a linear and time invariant system (LTI system: linear time invariant system) is common [2, 3]. Linearity specifies in the transfer function, for example, that the spring constant is proportional to the displacement and the damper constant follows a velocity proportional approach. Time invariance means that the oscillatory system is not subject to any time variation. This then calculates the rigid mass, the displacement-proportional spring constant and the velocity-proportional damping constant. In addition to the transfer function, there is also, for example, the representation in the time domain as an equation of motion or in the state space. From the investigation of the oscillatory systems, measures for influencing them can then be derived. If, for example, it is found that the excitation frequency coincides with a natural frequency of the oscillatory system, the natural frequency can be changed by specifically changing the spring constant or the mass (lightweight construction) or the amplitudes of the oscillation response can be reduced by increasing the damping.

1.6

Planning and Concept of Measuring Equipment

A wide range of vibration transducers and measuring devices are available to the user for solving measurement tasks. These can be roughly divided into the following groups: • Single-purpose measuring devices consist of a transducer and the associated signal processing. These devices are often used for mobile applications, e.g. for vibration monitoring. The output value is shown on a display in hand-held measuring devices. Usually these measuring devices are equipped with a computer interface or a memory function for the measured data. Alternatively, the signal processing can be carried out as an interface on a notebook and the display of the output value via a software solution. • Laboratory measuring stations are modularly constructed from components for the transducer and signal processing. The design aims to solve a variety of different and frequently changing measurement tasks. The measurement tasks are usually not designed as continuous measurements. Due to the design and the power supply, the use is limited to the laboratory environment. Signal processing is carried out via computer interfaces or as a network solution.

1.6

Planning and Concept of Measuring Equipment

11

• Mobile measuring stations consist of components for transducers and signal processing and enable frequently changing measuring tasks in vehicles, in operational use, on wind turbines, etc. The components are interchangeable, so that different transducers and the associated signal processing can be used depending on the measurement task. The measurement signals are recorded and evaluated and interpreted after the measurement. • Stationary measuring devices are designed for use at a fixed location and usually with unchanging measuring tasks. The installation is carried out, for example, by hardwiring modules on a so-called top-hat rail or as plug-in modules with a data bus system. The measuring tasks are often designed as long-term measurements. Signal processing is carried out via a PC interface or network solution. It is also possible for the modules to perform independent monitoring and control functions, e.g. in test stands. This division into four groups is not strictly separated, there are flowing transitions between the groups. For example, stationary measuring equipment can be set up modularly with high-quality measuring technology so that changing measuring tasks can be carried out and the measuring equipment thus fulfils fulfills the tasks of laboratory measuring stations. The groups of measurement equipment can be systematized if the categories mobility of measurement equipment is plotted against flexibility as the ability to perform different measurement tasks (Fig. 1.5). In the phase of planning and conceptual design, the following points must also be taken into consideration for the selection of the measurement technology [1, 3, 5–9] (Table 1.1).

high

low

Permanently installed measuring equipment (e.g. DIN rail modules)

Multipurpose laboratory measuring stations (e.g. mains-powered FFT analyser)

Single-purpose measuring device (e.g. pocket devices, transducers with USB interface)

modular, mobile measuring stations (e.g. modular configurable measuring stations with battery supply)

Mobility

low

high

Flexibility

Fig. 1.5 Flexibility and mobility as characteristics of measurement technology

12

1

Vibrations and Its Measurement

Table 1.1 Requirements for the measurement technology according to [10] Request Environment

Personnel qualification

Input and output channels Measured value processing

Art Industrial environment Lab Test bench Vehicle Temperatures Place Vibrations, shocks Engineer Metrologist Technician Operator Type and number Expandability Network and interfaces (LAN, USB, CAN . . .) Recording and display (data logger, display . . .) Computer (PC, notebook . . .)

References 1. DIN 1311-1: Schwingungen und Schwingungsfähige Systeme. Teil 1: Grundbegriffe, Einteilung (2000) 2. Magnus, K., Popp, K., Sextro, W.: Schwingungen. Springer Vieweg, Wiesbaden (2016) 3. Zollner, M.: Frequenzanalyse. Autoren-Selbstverlag (1999) 4. Holzweißig, F., Meltzer, G.: Meßtechnik in der Maschinendynamik. VEB Fachbuchverlag, Leipzig (1973) 5. Piersol, A.: Vibration data analysis. In: Piersol, A., Paez, T. (Hrsg.) Harris’ Shock and Vibration Handbook, Kap. 19. McGraw-Hill Education, New York (2009) 6. Lerch, R.: Elektrische Messtechnik. Springer, Berlin (2016) 7. Niebuhr, J., Lindner, G.: Physikalische Messtechnik mit Sensoren. Oldenbourg, München (2011) 8. Schrüfer, E., Reindl, L.M., Zagar, B.: Elektrische Messtechnik: Messung elektrischer und nichtelektrischer Größen. Carl Hanser, München (2014) 9. Trentmann, W.: PC-Messtechnik und rechnergestützte Messwertverarbeitung. In: Hoffmann, J. (Hrsg.) Handbuch der Messtechnik, S. 629–661. Carl Hanser, München (2012) 10. BMC Messsysteme: Ihre Anwendung (2014). http://www.bmcm.de/index.php/de/ihreanwendung.html. Accessed: 14 Juli 2014

2

Vibrations in the Time and Frequency Domain

Abstract

This chapter deals with the fundamentals of the description of the kinematic quantities displacement, velocity and acceleration. These quantities describe the motion processes in investigated oscillatory systems as well as in vibration transducers. Thus, these relationships are of fundamental importance for the calculation as well as for the evaluation and interpretation of measurements. On the basis of harmonic oscillation, the quantities are introduced in real and complex notation and presented in the frequency domain. This is followed by an extension to transient and stochastic oscillations. "

2.1

This chapter deals with the fundamentals of the description of the kinematic quantities displacement, velocity and acceleration. These quantities describe the motion processes in investigated oscillatory systems as well as in vibration transducers. Thus, these relationships are of fundamental importance for the calculation as well as for the evaluation and interpretation of measurements. On the basis of harmonic oscillation, the quantities are introduced in real and complex notation and presented in the frequency domain. This is followed by an extension to transient and stochastic oscillations.

Harmonic Oscillations

As an introductory example of a harmonic oscillation, the path-time curve of a wiper blade on a windscreen wiper is to be used (Fig. 2.1). In the parallel projection onto a straight line perpendicular to the projection direction, a point on the wiper blade moves linearly back # The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 T. Kuttner, A. Rohnen, Practice of Vibration Measurement, https://doi.org/10.1007/978-3-658-38463-0_2

13

14

2 Vibrations in the Time and Frequency Domain

T

1 f

Parallel projection

Deflection

xˆ

Time

Windscreen wiper

Fig. 2.1 Wiper blade of a windscreen wiper as an example of a harmonic oscillation

and forth. The displacement as a state variable increases and decreases in time, thus the definition of an oscillation is fulfilled. If the path of the point is plotted over time, a harmonic oscillation is obtained – provided the windscreen wiper is in continuous operation. The time characteristic of the displacement x of a harmonic oscillation is given by the function [1–4]. This function describes the displacement x(t) at time t (Fig. 2.1) and is also referred to as the displacement-time function. xðtÞ = x  cosð ωt þ φ0 Þ

ð2:1Þ

Many vibration processes in technology can be described sufficiently well (i.e. as a useful approximation) by harmonic oscillations. For the example of the windscreen wiper, the description via harmonic oscillations is not possible if the wiper blades move non-uniformly on the windscreen (“rubbing”). In this case, too, the description by means of a harmonic oscillation provides a starting point for further considerations. For this reason, the harmonic oscillation is of outstanding importance and will be treated as an example. Equation 2.1 includes the following quantities: • the amplitude x of the oscillation: The cosine function can take values between -1 and 1. The amplitude is the maximum displacement as seen from the value 0 of the cosine function. • the angular frequency ω: The following equation exists between the frequency f and the angular frequency ω:

2.1

Harmonic Oscillations

15

ω = 2 πf:

ð2:2Þ

The angular frequency ω of the oscillation is therefore the frequency f, which is related to the circumference of the unit circle 2π. "

For differentiation, the frequency is usually given in Hz and the angular frequency in 1/s or rad/s. Frequency f and period T of the oscillation are inversely proportional to each other: T=

1 : f

ð2:3Þ

Thus, a period of T = 0.5 s corresponds to a frequency f = 2 Hz (2 full oscillations per second). Finally, one obtains the angular frequency ω ω=

2π : T

ð2:4Þ

For a time t = 0, the zero phase angle φ0. is obtained in the argument. If this is inserted into Eq. 2.1, the result is the deflection xð0Þ = x  cosðφ0 Þ for the time t = 0, which clearly illustrates the zero phase angle. The phase angle φ can be taken as the argument of the cosine function and is calculated as follows: φ = ωt þ φ0 :

ð2:5Þ

By applying the addition theorems, Eq. 2.1 can also be transformed into the following form: xðtÞ = xC  cosðωtÞ þ xS  sinðωtÞ:

ð2:6Þ

This equation does not contain a phase angle, but it does contain the amplitude of the cosine component xC and the amplitude of the sine component xS . By squaring Eq. 2.6, the relationship to the amplitude x is derived: x2 = x2C þ x2S : Equation 2.6 also shows that the description is also possible with the sine function instead of the cosine function and is also treated in this form in part of the literature (e.g [4, 5]):

16

2 Vibrations in the Time and Frequency Domain

xðtÞ = x  cosðωt þ φ0 Þ = x  sin ωt þ φ0 þ 90 °

:

ð2:7Þ

φ0S

By inserting t = 0 and φ0 = 0, the value 1 is obtained for the cosine function. Alternatively to the notation φ0 + 90°, a zero phase angle φ0S can be used, which is related to the sine function and may only be used with it. As one can also easily convince oneself, x  cos ωt - 90 ° = x  sinðωtÞ, applies, i.e. the sine function is phase-shifted by -90° with respect to the cosine function. For the example of the windscreen wiper mentioned above, it is also very important to know the velocity-time function and the acceleration-time function in addition to the displacement-time function. The vibration speed is referred to as the velocity in order to avoid confusion with the velocity of a propagating wave. For harmonic oscillation, the time function of the velocity x_ ðtÞ is obtained by formally deriving the displacement-time function (Eq. 2.1): x_ ðtÞ = - x  sinðωt þ φ0 Þ  ω:

ð2:8Þ

Deriving Eq. 2.8 again leads to the acceleration-time function € xðtÞ : €xðtÞ = - x  cosðωt þ φ0 Þ  ω2 = - xðtÞ  ω2 :

ð2:9Þ

For the harmonic oscillation the magnitudes of the motion quantities displacement, velocity and acceleration can be converted with the angular frequency ω. The minus sign in Eqs. 2.8 and 2.9 for angle φ0 = 0 means that at the time t = 0 there is a maximum of the displacement-time function and the reversal of motion takes place to the point with the coordinates x = 0. Thus, for the amplitudes, the one-time differentiation corresponds to multiplication by -ω, and the two-time differentiation corresponds to multiplication by ω2. Likewise, an integration by division by -ω and a twofold integration corresponds to a division by -ω2. For the magnitudes of velocity and acceleration, the sine and cosine functions are replaced by the value -1: x_ = - x  sinðωt þ φ0 Þ  ω = x  ω = x  2πf

ð2:10Þ

-1

€x = - x  cosðωt þ φ0 Þ  ω2 = x  ω2 = x  4π2 f 2 : -1

ð2:11Þ

2.2

Phasor Diagram

17

Example

In the vibration test of add-on parts, an acceleration magnitude of ±20 g is to be achieved at a frequency of 30 Hz. What is the displacement amplitude at the vibration test system? The displacement amplitude is calculated by rearranging Eq. 2.11: x=

20  9:81 sm2 €x = = 0:0055m = 5:5mm: 4π2 f 2 4π2  302 s12

◄

2.2

Phasor Diagram

For the representation of oscillation processes and their solution, their mathematical treatment as Phasors as a representation of ai complex number as a rotating vector has proven itself [4–6]. This enables a clear representation and a simple solution of oscillation equations by applying the calculation rules for complex numbers. Here, one imagines the harmonic oscillation as a circular motion of a point and its parallel projection on the real axis (i.e. abscissa) (Fig. 2.2). The position of the point is detected by the tip of a rotating Phasor with the length of the amplitude x. The projection on the real axis is called the real part ReðxÞ, the perpendicular part imaginary part ImðxÞ is obtained by projection on the imaginary axis. The Phasor is underlined to indicate the complex quantity. Thus, the Phasor xðtÞ in the complex number plane describes the harmonic oscillation as follows: xðtÞ = ReðxÞ þ jImðxÞ = x  ½cosðxt þ φ0 Þ þ j sinðωt þ φ0 Þ:

ð2:12Þ

The imaginary unit j is defined as j2 = - 1. The Phasor rotates counterclockwise, likewise all angles are defined in the mathematical positive directional sense. The zero phase angle φ0 is the angle between the real axis and the Phasor at time t = 0. At time t > 0, for angular frequency ω > 0, the Phasor continues to rotate by ωt. For each defined time, real and imaginary parts can be read in the projections on the axes. The kinematic quantities considered are real and, even when using the complex calculus, have only a real part. The imaginary part exists only in the mathematical treatment. Using exponential form, Eq. 2.12 is abbreviated as: xðtÞ = x  e jðωtþφi0 Þ :

ð2:13Þ

18

2 Vibrations in the Time and Frequency Domain imaginary axis Im(x) Imaginary part

x M0 real axis Re(x)

Time

Time Real part

Fig. 2.2 Representation of a harmonic oscillation in the complex number plane

The future considerations are simplified since the magnitude of the exponential expression takes the value ejðωtþφ0 Þ = 1. Finally, in Eq. 2.13 the amplitude x can be replaced by the so-called complex amplitude. The complex amplitude x describes the initial position of the Phasor at t = 0, the term ejωt is called the time function: xðtÞ = x  e jφ0 e jωt = x  e jωt :

ð2:14Þ

The complex amplitude x contains both the amplitude x and the zero phase angle φ0. The following relationship exists between the complex amplitude and the magnitude of the Phasor as an expression for the amplitude: j xj =

ðReðxÞÞ2 þ ðImðxÞÞ2 = x:

ð2:15Þ

For the zero phase angle holds: tan φ0 =

ImðxÞ : ReðxÞ

ð2:16Þ

Finally, the advantages in computing are to be illustrated by the derivation of the speed and velocity (Fig. 2.3). By formal differentiation with respect to time one obtains:

2.2

Phasor Diagram

19

Differentiate

Integrate

Fast Deflection

Acceleration

Fig. 2.3 Differentiation and integration in Phasor representation

x_ ðtÞ = jω  xðtÞ = jω  x  e jðωtþφ0 Þ :

ð2:17Þ

This corresponds to a multiplication of the deflection by jω. Second differentiation leads to the acceleration, which is equivalent to multiplying the deflection by -ω2: €xðtÞ = ðjωÞ2  xðtÞ = - ω2  x  e jðωtþφ0 Þ :

ð2:18Þ

Example

For the location-time function xðtÞ = x  cosðωt þ φ0 Þ with the following numerical values x = 1 mm, f = 2.5 Hz and φ0 = 30° the maxima of velocity and acceleration are to be calculated. The period T = 1/f = 0.4 s is calculated from the frequency f. This period duration can be read off as the distance between the maxima in Fig. 2.4. With the frequency f the angular frequency is calculated ω = 2πf = 2π  2:5

1 1 = 15:71 : s s

Thus the magnitude of the velocity is given by €x = x  ω = 10 - 3 m  15:71

1 m = 0:016 : s s

The acceleration magnitude is calculated in the same procedure

2 Vibrations in the Time and Frequency Domain

Im(x) x

Re(x)

Re(x)

Velocity (m/s)

Integrate

1,00 0,50 0,00 -0,50 -1,00

0,02

Im(x)

0,01 0,00 -0,01 -0,02

Im(x)

Re(x)

Acceleration (m/s²)

Differentiate

Deflection (mm)

20

0,30 0,20 0,10 0,00 -0,10 -0,20 -0,30 0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

Time (s)

Fig. 2.4 Differentiation and integration in Phasor representation (schematic) and over time

€x = x  ω2 = 10 - 3 m  15:712

1 m = 0:25 2 : s2 s

These maxima can be taken from the diagram in Fig. 2.4. ◄ In the complex number plane, the derivative means that the Phasor is given the magnitude x  ω and is rotated by +90°. In the time domain, the real part of the Phasor is plotted for t = 0 (Fig. 2.4). A twofold differentiation (Fig. 2.4) corresponds in the Phasor diagram to a further rotation of the quick Phasor by +90°. The Phasor has the magnitude x  ω2 and a negative sign. The arrow in the time representation shows this situation again for t = 0. In contrast to differentiating in the time domain, the second derivative can be formed immediately with the Phasor representation and the change between cosine and sine function is omitted. For an integration the Phasor must be divided by jω, for a twofold integration the division is done by -ω2: The relations are summarized in the time representation and as a Phasor in Fig. 2.4. "

In measurement practice, multiplication or division by the angular frequency is used for harmonic oscillations when amplitudes are to be converted (e.g. measurement of the acceleration amplitude and calculation of the displacement amplitude).

2.3

Representation in the Time Domain and Frequency Domain

2.3

Representation in the Time Domain and Frequency Domain

2.3.1

Terms

21

The fundamentals of the Fourier transform go back to the historical achievement of Jean Baptiste Joseph Baron de Fourier (1768–1830), who described the decomposition of arbitrary functions into harmonic oscillations for thermodynamic compensation processes in his book “Analytical Theory of Heat” in 1822. The transfer of results from thermodynamics to signal processing may be regarded as a living example of an interdisciplinary transfer of knowledge. The representation of the time function x(t) in the usual form (time t on the abscissa, function value x on the ordinate) represents the function in the time domain. If the amplitude x or the rms value x is displayed over the frequency, this is referred to as the spectrum and the function displayed there as the spectral function. The representation of the spectral function is called representation in the frequency domain or spectral domain. The Fourier transform is the link between the time domain and the frequency domain and is to be understood as a superordinate term that includes analysis and synthesis. Analysis means the decomposition of the time function into harmonic oscillations. If a spectrum is created for a time function, it is referred to as frequency analysis or spectral analysis. Synthesis is the process of combining (superimposing) the individual harmonics to form a time function. If a spectrum is formed from a time function and the time function is reconstructed from it, this is referred to as resynthesis [1, 5–10]. Shortening, the Fourier transform is understood as the transformation from the time domain into the frequency domain. Since the signal is measured in the time domain and then transformed into the frequency domain, the term “inward transformation” is also used. The transformation from the frequency domain back into the time domain is called the “back transformation” or inverse Fourier transform. For the application of the Fourier transform, a distinction must be made according to the criteria periodic or aperiodic as well as continuous and discrete. For this there is the following assignment: 1. A periodic continuous time function is transformed into a discrete non-periodic line spectrum using the Fourier series (Sect. 2.3.2). 2. A unique continuous time function is transformed into a continuous non-periodic spectrum using the Fourier transform (Sect. 14.4). 3. A unique discrete time function is transformed into a continuous periodic spectrum using the discrete time Fourier transform (DTFT). 4. A discrete time function is transformed into a discrete periodic spectrum using the discrete Fourier transform (DFT) (Sect. 11.4). The interconnections are shown again in Fig. 2.5 as a comparison of the properties of the time and spectral functions.

22

2 Vibrations in the Time and Frequency Domain

Time range

Frequency range Analysis

periodic

Fourier series

discreetly nonperiodic

nonperiodic

Fourier transformation

continuous

periodic

Discrete Fourier Transformation (DFT, FFT)

discreetly

nonperiodic

Discrete-Time Fourier Transformation (DTFT)

Spectrum

Time function

continuous

periodic

discreetly continuous

Synthesis

Fig. 2.5 Overview of the transformations between time and frequency domain

The Fourier series and the Fourier transform have a fundamental importance for the representations in the frequency domain and are therefore dealt with in the following sections. In modern measurement practice, the time functions are available in discretetime form after sampling and are transformed into a discrete spectrum using the discrete Fourier transform (DFT) or the effective Fast Fourier Transform (FFT) algorithm. Because of its special importance for signal processing, the DFT is described in detail in Sect. 14.4.

2.3.2

Fourier Series

By means of Fourier analysis, the periodic oscillation x(t) is decomposed into a sum of harmonic partial oscillations. The periodic oscillation has the period T and the fundamental angular frequency ω = 2π/T. A harmonic in this context is a cosine function of an angular frequency kω, which is an integer multiple k of the fundamental angular frequency ω. In real notation, this sum is written as follows: x ð t Þ = x0 þ

1

xk  cosðkωt þ φ0k Þ:

ð2:19Þ

k=1

In Eq. 2.19, x0 represents the arithmetic mean (constant component). The periodic oscillation is decomposed into an infinite sum of k harmonics. In practice, one makes do with a finite number of harmonics k as an approximation for the function x(t). For k = 1, one obtains the fundamental angular frequency ω. The numerator k is the order of the partial harmonics. The kth harmonic is described by the amplitude xk and the zero phase angle

2.3

Representation in the Time Domain and Frequency Domain

23

φ0k. If the amplitudes xk are plotted against the angular frequency kω or the frequency of the harmonics kω/(2π), the spectrum is obtained. If the equivalent representation with sine and cosine function is chosen for the harmonics, we obtain for Eq. 2.19: xð t Þ = x0 þ

1

ðxkc cosðkωtÞ þ xks sinðkωtÞÞ:

ð2:20Þ

k=1

In this representation form, xkc and xks are called Fourier coefficients of kth order. The Fourier coefficients can be calculated according to the following equations: T

2 xkc = T

xðtÞ cosðkωtÞdt

ð2:21Þ

xðtÞ sinðkωtÞdt:

ð2:22Þ

0 T

2 xks = T 0

Using Eqs. 2.21 and 2.22, one then obtains for the amplitudes xk =

x2kc þ x2ks

ð2:23Þ

and the phase angle φ0k = - arctan

xks : xkc

ð2:24Þ

Thus both representations can be transferred into each other. Example

Given is the real function xðtÞ = 12 þ

5 k=1

1 k

cosðkωtÞ (modified according to [9]). For

this function, the amplitudes and the harmonics are to be specified. By a coefficient comparison with Eq. 2.21, we obtain:

24

2 Vibrations in the Time and Frequency Domain 5

xð t Þ = x0 þ

xk  cosðkωt þ φ0k Þ k=1

=

1 1 1 þ 1  cosðωtÞ þ  cosð2  ωtÞ þ  cosð3  ωtÞ 2 2 3 x1

x0

x2

x3

1 1 þ  cosð4  ωtÞ þ  cosð5  ωtÞ: 4 5 x4

x5

The five harmonic components of order k = 1 to 5 (and the DC component) are sufficient to describe the given function. Due to the chosen task, the zero phase angle φ0k is zero for the five harmonic components. The graphical application can be seen in Fig. 2.6. ◄ Since complex calculus has been shown to be useful in vibration engineering, the Fourier analysis of the function x(t) is now to be performed in the complex number plane. By applying Euler’s formula, the cosine and sine components are expressed as follows: cos kωt =

1 jkωt e þ e - jkωt 2

ð2:25Þ

sin kωt =

1 jkωt e - e - jkωt : 2j

ð2:26Þ

Substituting in Eq. 2.20 and rearranging the terms gives: xð t Þ = X 0 x0 þ

1

1

1 1 Xk ðxkc - jxks Þejkωt þ X - k ðxkc þ jxks Þe - jkωt : 2 2 k=1 k=1

ð2:27Þ

For abbreviation one introduces the complex amplitudes Xk and X - k . If one now lets the summation index k run from -1 to zero in the second sum, the minus sign in the exponential term falls away and one can write with it after summarizing the terms: xðtÞ =

1

Xk e jkωt :

k= -1

The complex amplitudes Xk can be developed as follows:

ð2:28Þ

2.3

Representation in the Time Domain and Frequency Domain

25

Time function 4

x(t)

3 2 1 0 -1

Ti m e x2

x1

x0

x3

Harmo

x4

x5

nic

Fig. 2.6 Representation of the periodic function x(t) in the time and frequency domain

T

1 Xk = T

xðtÞe - jkωt dt:

ð2:29Þ

0

In Eq. 2.27 one gets the initially paradoxical result of summing over “negative” frequencies of the harmonics. The solution of this apparent contradiction results from the consideration that the function x(t) is real and initially contains no imaginary part. As Eq. 2.25 shows using the real cosine function, this function can be thought of as the sum of two so-called conjugate complex functions. In the Phasor diagram, this corresponds to two Phasors which are mirror images of the x-axis with +ω and -ω rotating in opposite directions. The complex amplitudes Xk are then also conjugate complex to the amplitudes X - k . To identify the conjugate complex amplitudes, these are given an asterisk (*) and written: Xk = X - k :

ð2:30Þ

The following relationship exists between the real amplitude xk and the complex amplitudes Xk and Xk for k > 0: xk = 2j X k j = 2

Xk  Xk :

ð2:31Þ

The complex Fourier series consequently yields complex amplitudes that can be plotted in a frequency range from -kω to +kω. This plot is called a two-sided line spectrum, which contains half of the amplitude value at -kω and half at kω. This explains the prefactor 2 in Eq. 2.31 when converting the two-sided spectrum to real amplitudes. More common than

26

2 Vibrations in the Time and Frequency Domain

the two-sided spectrum in vibration engineering is the plot as amplitude and phase versus frequency (one-sided spectrum). "

By developing the Fourier series, a discrete line spectrum is obtained from a continuous and periodic time function x(t). Since the time function repeats after the period T, the analysis of a time interval of length T is sufficient. In the discrete spectrum the lines are in integer ratio.

2.3.3

Fourier Transformation

Now we will consider the case where there is no periodicity in the time function x(t). Here one can imagine, for example, a single rectangular pulse. This pulse has no finite period T and thus cannot be described by the Fourier series. To be able to treat this case by means of Fourier transform, one uses the auxiliary notion that the pulse repeats after infinitely long period T → 1. However, this consideration is not limited to impulses, but also applies, for example, to stationary stochastic oscillations. With infinitely long period the number of harmonics tends to infinity (k → 1) and the distance between two harmonics goes over to dω. Thus the summation in Eq. 2.28 must be replaced by an integration: 1

1 xðtÞ = 2π

XðjωÞe jωt dω:

ð2:32Þ

-1

With the transition from summation to integration, there is also a transition from the complex amplitudes Xk to the continuous complex spectral function X(jω), which is defined for all circular frequencies ω. This is calculated with the so-called Fourier integral as follows: 1

xðtÞe - jωt dt:

XðjωÞ =

ð2:33Þ

-1

The division by 2π in Eq. 2.32 is obtained by integration over dω = 2πdf. The assignment of the 1/2π term to Eqs. 2.32 or 2.33 is not handled uniformly in the literature (discussion here on, e.g., in [8, 10]). The function X(jω) is defined by -1 < ω < +1 and is called a two-sided complex continuous Fourier spectrum. The spectral function is a function of frequency and, unlike the time function x(t), is represented by a capital letter X. The representation X(jω) chosen here is intended to symbolize by a complex quantity that depends on the complex frequency argument jω. Likewise, the symbol X is possible if the

2.3

Representation in the Time Domain and Frequency Domain

27

dependence on frequency is apparent. Likewise, the equivalent representations as X(f), X(ω) for the amplitudes and Xðf Þ, XðjωÞ or XðωÞ for the complex spectral function are encountered. The complex spectral function X(jω) can be represented as real and imaginary parts. However, the usual representation is a representation as amplitude spectrum (magnitude spectrum) |X(jω)| and phase spectrum φ(ω): XðjωÞ = jXðjωÞje jφðωÞ :

ð2:34Þ

The amplitude spectrum |X(jω)| and the phase spectrum φ(ω) represent the amplitude and the phase angle over the angular frequency ω and frequency f, respectively. Since the time functions are real quantities, the symmetry relation applies to the conjugate complex quantity X ðjωÞ analogous to Eq. 2.31: X ðjωÞ = Xð- jωÞ:

ð2:35Þ

To obtain the unit of the spectral function, the time function x(t) is inserted into Eq. 2.33, e.g. in meters. The integration over the time results in the spectral function with the unit ms or m/Hz. For this reason, the amplitude spectrum is also called spectral amplitude density or spectral density. With the formulation of the spectral amplitude density, the problem in the spectral function that results from the infinitely large number of spectral lines is avoided: the lines are infinitely close, but their amplitudes approach zero. Only the quotient of amplitude and distance yields a non-zero value. However, this value must not be misinterpreted as a differential quotient (slope) [6]. Example

Given is a single rectangular pulse with an amplitudex = 1 and a pulse duration τ (Fig. 2.7a). Equation 2.34 can be used to calculate the Fourier spectrum. Here, it is only necessary to integrate within the limits from -τ/2 to +τ/2, since outside the pulse duration the amplitude is zero: þ2τ

x  e - jωt dt =

XðjωÞ = - 2τ

x e - jωt - jω

þ2τ - 2τ

=

x jωt e jω

þ2τ - 2τ

=

τ x jω2τ e - e - jω2 : jω

ð2:36Þ

After rearranging the terms, the exponential terms can be replaced by a sine function according to Eq. 2.25:

28

2 Vibrations in the Time and Frequency Domain

x(t)/

1

0 -4

-3

-2

-1

0

1

2

3

4

normalized time t/W

1,2

Amplitude

1,0 0,8 0,6 0,4 0,2 0,0 -4

-3

-2

-1

0

1

2

3

4

normalized frequency ZWS

Fig. 2.7 Frequency analysis of a single rectangular pulse. Time domain (a), frequency domain (b) τ

XðjωÞ =

τ

2x ejω2 - e - jω2 2x τ = sin ω : 2j ω ω 2

ð2:37Þ

Expanding with τ and further transforming with ω = 2πf leads to the following equation: XðjωÞ =

sin ω 2τ 2ωτ τ τ = xτsi ω : sin ω = xτ ω 2τ ωτ 2 2

ð2:38Þ

The function sin(x)/x is called the si function or slit function. The argument ωτ/2 is dimensionless and can be understood as an angular frequency ω normalized by the pulse duration τ. The real amplitude spectrum shows a curve for the rectangular pulse, which is shown in Fig. 2.7b. Since the si function itself is dimensionless, one obtains, for example, with τ = 0.005 s and x = 2mm in the expression xτ a numerical value xτ = 10 - 5 m  s (or m/ Hz). This numerical value corresponds to:

References

29

• in the frequency domain: intersection of the spectral function with the ordinate axis at ω = 0 as well as • in the time domain: area under the rectangular pulse (integral). ◄ For integer multiples of π, the amplitude spectrum of the si function has zeros. With the numerical value τ = 0.005 s, one obtains for the first zero of the si function 2πfτ/ 2 = π → f = 1/τ = 200Hz. Further zeros lie at integer multiples, i.e. 400 Hz, 600 Hz, etc. First, we will consider the role of the zeros of the si function in the excitation of an oscillatory system with a square-wave pulse. For the frequencies at the zeros of the si function, the excitation of the oscillatory system is very small. For the measurement of transfer functions (Sect. 14.5) this leads to expressions in the form of “zero by zero”; there are usually no practically usable results at these frequencies. If the time duration τ is shortened in the time domain for the same area of the pulse, the amplitude x increases. The narrower the rectangular pulse, the flatter and broader the spectrum up to the first zero. "

For the theoretical limit case of the Dirac shock, the pulse duration approaches zero (τ → 0) and the amplitude grows to infinity. For ωτ/2 ≪ 1 one obtains a practically constant course of the spectral function with XðjωÞ = xτ: This has the consequence that pulses with short pulse duration τ • require a wide frequency band for transmission and signal analysis (requirements for measurement technology Chap. 10), • excite a vibrating system in a wide frequency band (transfer function Sect. 14.5). This is used, for example, to determine the vibration response of structures by excitation with short-time square-wave pulses (e.g. hammer blow).

References 1. DIN 1311-1: Schwingungen und Schwingungsfähige Systeme. Teil 1: Grundbegriffe, Einteilung (2000) 2. Blake, R.E.: Basic Vibration Theory. In: Piersol, Paez (Hrsg.) Harris’ Shock and Vibration Handbook, S. 2.1–2.32. McGraw-Hill Education, New York (2009) 3. Gross, D., Hauger, W., Schröder, J., Wall, W.A.: Technische Mechanik 3 Kinetik. Springer Vieweg, Berlin/Heidelberg (2015) 4. Jäger, H., Mastel, R., Knaebel, M.: Technische Schwingungslehre. Springer Vieweg, Wiesbaden (2016) 5. Magnus, K., Popp, K., Sextro, W.: Schwingungen. Springer Vieweg, Wiesbaden (2016) 6. Zollner, M.: Frequenzanalyse. Autoren-Selbstverlag, Regensburg (1999) 7. Broch, J.: Mechanical Vibration and Shock Measurements. Brüel & Kjaer, Naerum (1984) 8. Butz, T.: Fouriertransformation für Fußgänger. Springer Vieweg, Wiesbaden (2012)

30

2 Vibrations in the Time and Frequency Domain

9. Heymann, J., Lingener, A.: Experimentelle Festkörpermechanik. VEB Fachbuchverlag, Leipzig (1986) 10. Hoffmann, R.: Grundlagen der Frequenzanalyse Bd. 620. expert verlag, Renningen (2011)

3

Free Vibrations

Abstract

The chapter presents the fundamentals of free vibration on the basis of translational vibration and rotational vibration. Damping is treated as component damping and frictional damping. This provides the basis for the forced oscillations and representation as a transfer function. Numerous examples and illustrations illustrate the topic and contribute to a practical understanding. "

3.1

The chapter presents the fundamentals of free vibration on the basis of translational vibration and rotational vibration. Damping is treated as component damping and frictional damping. This provides the basis for the forced oscillations and representation as a transfer function. Numerous examples and illustrations depict the topic and contribute to a practical understanding.

Translational Vibration

The oscillating system in Fig. 3.1 consists of a mass m and a massless spring with the spring constant k (unit N/m). The deflection about the static equilibrium position is described with the coordinate x. Since the mathematical description of the movement is done with one coordinate, it is called a oscillator with one degree of freedom [1–4]. By deflection of the mass, the restoring force Felast(x) = kx is exerted on the spring. The mass is acted upon by the d’Alembert inertia force FT ðxÞ = m€ x: After the mass has been cut free, the equilibrium of forces at the central plane force system at the centre of gravity is written down as follows: # The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 T. Kuttner, A. Rohnen, Practice of Vibration Measurement, https://doi.org/10.1007/978-3-658-38463-0_3

31

32

3

Fig. 3.1 Translational transducer

Free Vibrations x

m kx mx

k

- m€x - kx = 0:

ð3:1Þ

Since both the displacement and the acceleration are linearly linked to the force variables as state variables and there is no time dependence of the mass and spring constant, the system is a linear time-invariant system (LTI) (cf. Sect. 1.1). Equation 3.1 represents the differential equation of motion of the oscillatory system. For an unambiguous description of the harmonic oscillation at time t, another quantity is required in addition to the deflection, e.g. the velocity. In this case, the zero phase angle φ0 = 0 is specified. This gives a velocity x_ ð0Þ = 0 as the second quantity at time t = 0. Now, as a complex representation of the deflection, Eq. 2.14 is used. xðtÞ = xejω0 t ,

ð3:2Þ

€xðtÞ = - ω20 xe jω0 t

ð3:3Þ

k  xe jω0 t - ω20 m  xe jω0 t = 0:

ð3:4Þ

or its second derivative

into Eq. 3.1 multiplied by (-1):

In addition to the trivial solution (amplitude x = 0), another reasonable solution is obtained by equating the bracketted expression to zero. k- ω20 m xe jω0 t = 0

ð3:5Þ

As a solution, the natural angular frequency ω0 in conjunction with Eq. 3.2 describes the natural oscillation of the undamped system:

3.1

Translational Vibration

33

Natural frequency in Hz

1000

100

10

1 0,01

0,1

1

10

100

Deflection in static rest position in mm Fig. 3.2 Relationship between deflection in the static rest position and natural frequency

ω0 =

k : m

ð3:6Þ

The natural frequency f0 and the period T are obtained from Eqs. 2.4 and 2.5. Only spring stiffness and mass are included in Eq. 3.6. From this it follows directly that for a linear system with linear restitution • the natural angular frequency ω0 is independent of the initial deflection or the amplitude. It is an indication of non-linearities in the oscillatory system if the natural frequency also changes with a change in amplitude. • the natural angular frequency ω0 is independent of the static rest position of the oscillating system. A change in the spring preload (with a linear spring characteristic) therefore causes no change in the natural angular frequency. The natural frequency can be estimated from the static deflection of machine foundations and vehicles (Fig. 3.2). The deflection in the static rest position xst = mg/k is the deformation of the spring due to the load with the weight force mg. By expanding the numerator and denominator in Eq. 3.6 with the acceleration due to gravity g, the relationship to the natural frequency f0 is written as follows:

34

3

f0 =

1 2π

k g 1 = m g 2π

Free Vibrations

g : xst

ð3:7Þ

Frequently, a low natural frequency f0 is aimed for in machine foundations. This leads to the lowest possible spring constant k and thus the deflection xst in the static rest position is greater, which in turn is undesirable and often not feasible from a design point of view. On the other hand, the natural frequency can be estimated in this way from the deflection in the static rest position.

3.2

Rotational Vibrations

As an example of rotational vibrations, an oscillatory 1-degree-of-freedom system with mass moment of inertia J about the axis of rotation and a massless spring according to DIN 1311-2 [1] with the linear torsional spring constant kt shall be used (Fig. 3.3). The oscillatory system performs rotational oscillations about the fixed z-axis with coordinate φ(t). By free cutting and setting up the equilibrium conditions one obtains: - J€ φ - kt φ = 0:

ð3:8Þ

A coefficient comparison with Eq. 3.1 immediately yields the solution for the natural angular frequency ω0: ω0 =

kt : J

ð3:9Þ

There is therefore an analogy between translational and rotational oscillations, by which the results of the translational oscillator can be transferred to the rotational oscillator. For the special case of a shaft with a full circle cross section which is constant over the length l of the shaft, the torsional spring constant kt can be calculated with the shear modulus G of the material and the second order polar moment of area Ip as follows:

Fig. 3.3 Rotary transducer

t

t

3.3

Free Damped Vibrations

35

kt =

GIp : l

ð3:10Þ

The natural angular frequency ω0 is then calculated to: ω0 =

GIp : lJ

ð3:11Þ

Replacing the natural angular frequency ω0 by the period T0 according to Eq. 2.5 and rearranging the equation to J, we obtain: J=

T20 GIp : 4π2 l

ð3:12Þ

Equation 3.12 can be used to calculate the mass moment of inertia with the measured period of the oscillation T0. For this purpose, the structure to be investigated (component, system) is suspended from the torsion bar spring with a known spring constant in a torsional vibration test and set in vibration [5].

3.3

Free Damped Vibrations

In practice, always occurring motion resistances cause a decay of the vibration. This property of the vibratory system, known as damping, is the superposition of material damping, component damping of the construction, bearing damping, environmental damping by surrounding media (air, oil, water, etc.) and damping by vibration dampers, understood in the narrower sense. DIN 1311-2 [1] speaks of dissipative structural elements. According to this, damping occurs when dissipative structural elements reduce the energy sum in the energy stores of the oscillating system. As an easy-to-handle approach for the mathematical description of damping, a linear relationship between damping force Fdiss and velocity (speed) is assumed: Fdiss ðx_ Þ = dx_ :

ð3:13Þ

The damping constant d is given in Ns/m. This linear relationship with the velocity is referred to as velocity-proportional or viscous damping or, according to DIN 1311-2, as component damping and can be used to describe material, component and fluid damping. Non-linear damping influences due to e.g. turbulent flows and dry friction (Coulomb friction), on the other hand, cannot be described with Eq. 3.13. When deflected about the static equilibrium position, the balance of forces in a system capable of oscillation (Fig. 3.4) can be written down as follows:

36

3

Fig. 3.4 Translational transducer with damping

Free Vibrations

mx

m

x

kx

dx d

- m€x - dx_ - kx = 0 :

k

ð3:14Þ

By substituting the natural angular frequency of the undamped system ω20 = k=m and the attenuation constant δ = d/(2m) into Eq. 3.14, one obtains €x þ 2δx_ þ ω20 x = 0:

ð3:15Þ

For the further calculation, the damping ratio ϑ (also called Lehr’s damping ratio D) is introduced as a dimensionless quantity: ϑ=

δ : ω0

ð3:16Þ

As a solution for this second order differential equation, the function x(t) = Aeλt and its derivatives x_ ðtÞ = Aλeλt and €xðtÞ = Aλ2 eλt are substituted into Eq. 3.14: Aλ2 eλt þ 2δAλeλt þ ω20 Aeλt = 0: €x

x_

ð3:17Þ

x

By truncating Aeλt in Eq. 3.17, one obtains the characteristic equation λ2 þ 2δ λ þ ω20 = 0 :

ð3:18Þ

This quadratic equation has the solution for the two eigenvalues λ: λ1,2 = - δ ±

δ2 - ω20 :

ð3:19Þ

Depending on the numerical value of the root expression in Eq. 3.19, the solution takes different forms. DIN 1311-2 distinguishes between the following cases, which are summarised in Table 3.1. The case of the damping ratio 0 < ϑ < 1, which occurs frequently in measurement, is to be examined in more detail here. From the solution of the differential equation it can be

Real, negative

ϑ≥1

Very strong damping λ1,2 = - δ ± 2

- ω20

δ2 - ω20

δ

Real, negative λ1 = λ2 = - δ

ϑ=1

Aperiodic limiting case

λ1,2 = - δ ± j

Eigenvalues Complex

Damping ratio ϑ ϑ≪1 ϑ 0 : - m€x - kx = þ FR x_ < 0 : - m€x - kx = - FR :

ð3:28Þ

Both equations differ only by the sign of the perturbation term FR. The solution of both inhomogeneous differential equations must be done separately for the positive half oscillations with x_ > 0 and the negative half oscillations x_ < 0: FR k FR x_ < 0 : xðtÞ = A2 ½cosðω0 t þ φ0 Þ þ : k x_ > 0 : xðtÞ = A1 ½cosðω0 t þ φ0 Þ -

ð3:29Þ

The constants A1 and A2 are only valid in the considered half oscillation and are to be determined from the respective initial conditions. The natural angular frequency and period correspond to those of the undamped oscillator. At each half oscillation, the amplitude decreases by the constant amount FR/k. When the amplitude has become smaller than FR/k, the mass comes to a standstill (Fig. 3.8).

Ψ=

π

1-

pd 2 km

2πpd km

Θ = arctan

2

1pd 2 km

pd km 2

1-

δ ω0

Ψ= 1-

δ ω0

4πωδ 0

Θ = arctan

Λ = 2π ωδ0 (*)

Λ=

δ ω0

2

1-

2

δ ω0

2

1 - ϑ2

p2π ϑ

ϑ

1 - ϑ2

Ψ = p4π ϑ

Θ = arcsin ϑ (*)

1 - ϑ2

Θ = arctan p

Λ = 2π ϑ (*)

Λ=

1

δ = ϑω0

p d = 2 kmϑ

Damping ratio ϑ p ωd = ω0 1 - ϑ2

kmΛ

Λ 2π

(*)

Ψ = 2Λ

Θ = arctan

1

ϑ=

Λ 2π

Λ 4π2 þΛ2

δ= p

4π þΛ2

δ = p ω02Λ

4π2 þΛ2

p

d = p2

4π þΛ2

ωd = p2πω2 0

Logarithmic decrement Λ

Ψ = 4π tan Θ

1

Λ = 2π tan Θ

ϑ = sin Θ

δ = ω0 sin Θ

p d = 2 km sin Θ

Damping angle Θ ωd = ω0 cos Θ

4π

4π

2π

Ψ

1

Θ = arctan

Ψ 4π

Ψ 1þð4π Þ

2

2

2

Ψ 1þð4π Þ

ω0 Ψ

Ψ 1þð4π Þ

p kmΨ

Ψ 4π (*) Λ = 12 Ψ (*)

ϑ=

ϑ=

Δ=

d=

Loss factor ψ ω0 ωd = Ψ 2 1þð4π Þ

3

(*) Approximation for small attenuations

Loss factor ψ

Damping angle Θ

pd 2 km

Λ = π pdkm *

1-

2

2πωδ 0

Λ=

πpd km

δ ω0

Logarithmic decrement Λ

pd 2 km

ϑ=

ϑ=

d 2m

2

ω20 - δ

Damping ratio ϑ

ωd =

1

d 2m

δ=

-

Decay coefficient δ

k m

d = 2mδ

ωd =

2

Decay coefficient δ

1

Natural frequency of the damped oscillation ωd Damping coefficient d

Damping coefficient d

Table 3.3 Natural frequency and the damping of the free damped vibration for damping ratios 0 < ϑ < 1

42 Free Vibrations

References

43

Fig. 3.8 Translational oscillator with Coulomb friction

mx

m

x

kx

FR k

References 1. DIN 1311-2: Schwingungen und Schwingungsfähige Systeme. Teil 2: Lineare, zeitinvariante schwingungsfähige Systeme mit einem Freiheitsgrad (2002) 2. Blake, R.E.: Basic vibration theory. In: Piersol, P. (Hrsg.) Harris’ Shock and Vibration Handbook. McGraw-Hill Education, New York (2009) 3. Gross, D., Hauger, W., Schröder, J., Wall, W.A.: Technische Mechanik 3 Kinetik. Springer Vieweg, Berlin (2015) 4. Jäger, H., Mastel, R., Knaebel, M.: Technische Schwingungslehre. Springer Vieweg, Wiesbaden (2016) 5. Dresig, H., Holzweißig, F.: Maschinendynamik. Springer Vieweg, Berlin (2016) 6. Zollner, M.: Frequenzanalyse. Autoren-Selbstverlag, Regensburg (1999)

4

Forced Vibrations

Abstract

Under the external influence of periodic forces and movements, forced vibrations are excited in a oscillating system. This case, which is significant in practice, occurs on the one hand on the objects to be measured, e.g. components, vehicles, foundations, buildings, etc., and on the other hand vibration transducers themselves can also be oscillating systems. Thus, these considerations form the basis for the planning of measurement tasks and the understanding of their results. Basic terms as well as the concept of transfer functions are developed on the basis of spring force excitation. The representation in the Bode and Nyquist diagrams is explained, followed by a discussion of other types of excitation. "

Under the external influence of periodic forces and movements, forced vibrations are excited in a oscillating system. This case, which is significant in practice, occurs on the one hand on the objects to be measured, e.g. components, vehicles, foundations, buildings, etc., and on the other hand vibration transducers themselves can also be vibratory systems. Thus, these considerations form the basis for the planning of measurement tasks and the understanding of their results. Basic terms as well as the concept of transfer functions are developed on the basis of spring force excitation. The representation in the Bode and Nyquist diagrams is explained, followed by other types of excitation.

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 T. Kuttner, A. Rohnen, Practice of Vibration Measurement, https://doi.org/10.1007/978-3-658-38463-0_4

45

46

4.1

4

Forced Vibrations

Spring Force Excitation with Constant Force Amplitude

Under the action of an excitation force F, the vibration response x in the oscillatory system (Fig. 4.1) is to be investigated [1–6]. An equilibrium of forces at the center of gravity in the vertical direction leads to the inhomogeneous differential equation: - m€x - dx_ - kx = - FðtÞ:

ð4:1Þ

The excitation force is assumed to be a harmonic function with variable excitation angular frequency Ω and amplitude F. In polar form this is FðtÞ = F  ejΩt

ð4:2Þ

The zero phase angle is assumed to be zero, this simplifies the solution. The force is not dependent on the oscillatory response, so there is no feedback from the oscillatory system to the force. The vibration response x and its derivatives are assumed to be phasors with phase angle ς using Eqs. 2.14, 2.18 and 2.19: xðtÞ = x  ejðΩt - ςÞ x_ ðtÞ = jΩxejðΩt - ςÞ x€ðtÞ = - Ω2 xejðΩ - ςÞ :

ð4:3Þ

In practice, the stationary state of motion of the system is usually of interest, which occurs after the decay of all transient processes under the periodically acting excitation force. As a solution of the differential equation, therefore, a partial solution of the inhomogeneous differential equation Eq. 4.1 is sought, which is obtained by substituting Eqs. 4.2 and 4.3: - Ω2 mxejðΩt - ςÞ þ jΩdxejðΩt - ςÞ þ kx  ejðΩt - ςÞ = FejΩt :

ð4:4Þ

After summing up the terms and factoring out the complex amplitude, we get:

Fig. 4.1 Translational oscillator under force excitation

F(t) mx

m

kx

dx d

k

x(t)

4.1

Spring Force Excitation with Constant Force Amplitude

Excitation

Input

47

Vibratory system

Reply

Transmission function

Output

Fig. 4.2 For the definition of the transfer function

- Ω2 m þ k þ jΩd xejðΩt - ςÞ = FejΩt :

ð4:5Þ

Now the phasor xðtÞ is split into the time function ejΩt and complex amplitude x: xðtÞ = xejðΩt - ςÞ = xe - jς ejΩt = xejΩt :

ð4:6Þ

The complex amplitude x contains the phase angle ς. Substituting Eq. 4.6 into Eq. 4.5 lifts out the term ejΩt on the left and right sides of the equation and reads: x=F

1 : - Ω2 m þ k þ jΩd

ð4:7Þ

Thus, the complex amplitude x is proportional to the magnitude of the force F. By multiplication with the time function ejΩt both phasors x and F rotate in the complex number plane with the excitation angular frequency Ω. If we now represent the excitation force as the input and the oscillation response as the output of a system capable of oscillation, the transfer function1 HxF(jΩ) links these two quantities in the frequency domain (Fig. 4.2): HxF ðjΩÞ =

x : F

ð4:8Þ

The complex amplitudes are used for the output and input quantities, since these contain information about the amplitudes (i.e. magnitudes) and the phase shift angle. The transfer function is then also a complex quantity, for which representations as amplitude (gain) and phase components (Bode plot, Sect. 4.2) or as real and imaginary components (Nyquist plot, Sect. 4.4) are common.

1

Complex quantities can be denoted by the underscore or in the form. Both notations are used equivalently in the literature [1, 7, 8]. In this presentation, the notation is used for transfer functions when their amplitude and phase components are evaluated.

48

4

Forced Vibrations

Table 4.1 Relationship between tuning ratio and designations on the oscillating system Tuning ratio η η≪1 η≫1

tuning Highly tuned Low tuned

Excitation Subcritical Supercritical

For practical applications it is of great benefit to normalize the solution to dimensionless quantities in order to reduce the abundance of possible combinations of the parameters. For this purpose, on the one hand the damping ratio ϑ (Eq. 3.16) and on the other hand the tuning ratio η as the ratio of the excitation angular frequency Ω to the natural angular frequency of the undamped system ω0 are used: η=

Ω : ω0

ð4:9Þ

Characteristic terms related to the tuning ratio are summarized in Table 4.1. By substituting the damping ratio ϑ and the tuning ratio η, one finally obtains after factoring out the spring constant k in the denominator x=

F 1 : k ð1 - η2 Þ þ j2ϑη

ð4:10Þ

For a clearer representation, the transfer function is decomposed into the amplitude component and phase component. The amplitude component is obtained by forming the magnitude (gain): HxF ðηÞ = jHxF ðjηÞj = =

1  k

ðReðHxF ðjηÞÞÞ2 þ ðImðHxF ðjηÞÞÞ2 1 ð1 - η2 Þ2 þ ð2ϑηÞ2

:

ð4:11Þ

The amplitude frequency response can again be divided into a dimensionally dependent and a dimensionless part: 1 HxF ðηÞ = κxF  αxF ðηÞ = αxF ðηÞ: k

ð4:12Þ

The amplitude part of the transfer function αxF is itself dimensionless. The prefactor κxF, on the other hand, contains the units. For the phase component, taking into account the minus sign in the complex amplitude, the following results xe - jς :

4.2

Amplitude and Phase Frequency Response

tan ςxF ðηÞ = -

ImðHxF ðjηÞÞ 2ϑη : = ReðHxF ðjηÞÞ 1 - η2

49

ð4:13Þ

The phase shift angle ς is to be understood as the difference of the phase angles of excitation (force phasor) and vibration response (acceleration phasor). Amplitude component and phase component are real quantities. The dimensionless representation of the amplitude component of the transfer function in force excitation is called the function of amplification V1 in machine dynamics. For this reason, α1 = αxF and ς1 = ςxF are set for simplification.

4.2

Amplitude and Phase Frequency Response

The plot of the amplitude component αxF and the frequency component ςxF of the transfer function versus the excitation angular frequency Ω and the tuning ratio η are called amplitude frequency response and phase frequency response, respectively. "

The amplitude component indicates the transfer coefficient (as a factor) between output and input. The (positive) phase angle indicates the angle by which the oscillation response tracks the excitation.

The amplitude component and frequency component of the transfer function completely describe the transfer behavior. Amplitude frequency response and phase frequency response together form the Bode plot. Figures 4.3 and 4.4 show the functions αxF and ςxF over the tuning ratio η. Depending on the tuning ratio η, the shape of the transfer function is discussed as follows: • η ≪ 1 (subcritical excitation): The amplitude component tends towards the value 1: αxF → 1. Thus, the vibration response corresponds to the static deflectionx = F=k: The phase shift angle ςxF has the value 0, excitation and response have the same phase shift angle. • η ≫ 1 (supercritical excitation): The amplitude component tends towards the value 0: αxF → 0. The mass is therefore at rest. The phase shift angle ςxF has the value 180°, excitation and response are 180° out of phase. • η → 1 (approaching resonance): When approaching the resonant circuit frequency Ωr the amplitudes reach a maximum. At a damping ratio ϑ = 0, the amplitudes would theoretically grow beyond all limits. In practice, the excitation power, as well as the nonlinearities and damping actually present, limit the amplitudes. For the maximum in the amplitude frequency response, the angular resonance frequency Ωrx is obtained by setting the first derivative to zero.

50

4

Forced Vibrations

Amplitude frequency response D xF=D1

5 -

4

3

0,05 0,1 0,2 0,5 0,7 1,0

2

1

0 0,0

0,5

1,0

1,5

2,0

2,5

Tuning ratio K

Fig. 4.3 Amplitude frequency response for displacement response under force excitation

Phase frequency response ]xF=]1

180

150

120

90 -

60

30

0,05 0,1 0,2 0,5 0,7 1,0

0 0,0

0,5

1,0

1,5

2,0

Tuning ratio K

Fig. 4.4 Phase frequency response for displacement response under force excitation

2,5

4.2

Amplitude and Phase Frequency Response

Ωrx =

51

1 - 2ϑ2 ω0 :

ð4:14Þ

The amplitudes reach a maximum at the point of the natural angular frequency of the damped system ωd and only for the special case ϑ = 0 at the point of the natural angular frequency of the undamped system ω0. The maximum is called resonance rise αrx or quality factor Q and reaches the function value αrx =

1 p : 2ϑ 1 - 2ϑ2

ð4:15Þ

As the damping ratio ϑpincreases, the maximum become smaller. No maximum occurs for damping ratios ϑ ≥ 1= 2 . An evaluation of the phase frequency response shows that the phase shift angle ςxF assumes the value 90° at η = 1. As the damping ratio increases, the curves become flatter, but all intersect at this point. Thus, at the point of the phase shift angle of 90°, the natural angular frequency of the undamped system ω0 = Ω (i.e., η = 1) can be read. This is also true at higher damping ratio ϑ. At this point, the amplitude frequency response takes the following value: αxF ðη = 1Þ =

1 : 2ϑ

ð4:16Þ

From this ordinate value, it is easy to read the damping ratio ϑ. The phase shift angle ς can also be expressed as phase shifting time Δt: Δt =

ς 1 ς  : = Ω ω0 η

ð4:17Þ

In the phase frequency response, the quotient ς/η can be read directly as the increase of the phase shift angle ς over the tuning ratio η. The phase shifting time Δt is proportional to the increase ς/η. A preferably linear course of the phase frequency response – i.e. low curvature – means that the phase shift time is the same for all frequencies considered. This property is desirable, for example, in vibration transducers and is realized in certain transducers by damping the transducer system by filling it with gas or oil. In addition to the linear plot, the double logarithmic plot is commonly used for the amplitude frequency response (Fig. 4.5). This application spreads the range of small function values α.xF For the case of large tuning ratios η and small damping ratios ϑ, Eq. 4.12 can be written in a simplified form

52

4

Forced Vibrations

Amplitude frequency response DxF=D1

10 -

1

0,05 0,1 0,2 0,5 0,7 1,0

0,1

0,01 0,1

1

10

Tuning ratio K

Fig. 4.5 Logarithmic representation of the amplitude frequency response of the displacement response under force excitation

αxF ðη → 1Þ =

1 ≈ η - 2: j 1 - η2 j

By logarithmizing you get: logðαxF ðη → 1ÞÞ = log η - 2 = - 2 logðηÞ: Thus, in the double logarithmic representation, the amplitude frequency responses can be approximated by a straight line with slope of -2 for large tuning ratios η and small damping ratios ϑ (cf. Figure 4.5).

4.3

Transfer Functions and Their Inverses

As an introduction, the transfer functions were discussed in the case of spring force excitation and displacement response. In vibration measurement technology, transfer functions are very important for characterizing systems capable of vibrating. The determination of dynamic parameters in vibratory systems (e.g., mass, spring stiffness, and damping) is often done by evaluating transfer functions [1, 7–10, 12]. In the discussed case of spring force excitation and displacement response, the transfer function can be understood as the dynamic compliance (receptance):

4.3

Transfer Functions and Their Inverses

HxF ðjηÞ =

53

x : F

The reciprocal of the dynamic compliance is the dynamic stiffness: -1 HxF ðjηÞ =

F : x

The dynamic stiffness differs greatly from the static stiffness for a number of components (e.g. elastomeric bearings) and is frequency-dependent. For the vibration calculation, the dynamic stiffness is therefore often more meaningful than the static stiffness. In practice, however, it is often not the displacement response that is measured, but the vibration response as velocity or acceleration. Likewise, there are a number of transducers for recording the measured quantities whose measuring principles use the transfer function of, for example, force excitation and acceleration response. Analogous to dynamic stiffness, the admittance can be defined from the velocity response and spring force excitation: HvF ðjηÞ =

x_ : F

The corresponding inverse of the transfer function is the mechanical impedance: -1 HvF ðjηÞ = Zm =

F : x_

The mechanical impedance indicates the force in N that is necessary to move the mass at a speed of 1 m/s. The complex quantity of the mechanical impedance is assigned the formula symbol Zm in acoustics. If you take the acceleration – which is usually easy to measure – you get the accelerance: HaF ðjηÞ =

€ x : F

A great practical importance has its inverse, the dynamic mass: -1 HaF ðjηÞ =

F : € x

The dynamic mass can be thought of as the moving mass in the oscillating system.

54

4

Forced Vibrations

The parameters for describing oscillatory systems can be easily transferred from the measured transfer function to its inverse. For example, the accelerance provides the reciprocal of the dynamic mass, which can be easily converted into the mass. In comparison to the displacement response, the cases of the vibration response as velocity and acceleration are compared in Table 4.2. The naming of the individual quantities is done by two indices in the order of response, followed by excitation. The amplitude frequency responses are obtained by multiplication with the tuning ratio or its square. With velocity response, the angular resonance frequency Ωr is equal to the angular natural frequency ω0. The amplitude frequency response is more symmetrical in the vicinity of the resonant circuit frequency than when using the displacement or acceleration response. Both facts can be used to advantage by using the velocity for the measurement. In the case of the acceleration response, the plot appears mirrored at the ordinate. The discussion of the curves can therefore be transferred (qualitatively).

4.4

Nyquist Diagram

Another common way of representing transfer functions is to plot the imaginary component on the ordinate against the real component on the abscissa [10–13] as a Nyquist diagram. For a given excitation angular frequency Ω, the oscillatory response is plotted as a phasor with amplitude x in the phasor diagram (Fig. 4.12). The length of the complex phasor (magnitude) indicates amplitude component of the transfer function. The phase shift angle ς is defined between the phasors of excitation F and vibration response x. The excitation F is defined as a real quantity in Eq. 4.2, so it lies on the abscissa. The phase shift angle ς with respect to the oscillatory response x is thus fixed in the clockwise direction. This is consistent with the definition of the phase shift angle ς in Eq. 4.3 having a negative sign and is consistent with the usual perception that the oscillatory response x tracks the excitation F. With a changed excitation angular frequency Ω, a rotating phasor is obtained. If the geometric locations of the phasor tip are now connected, the locus curve is obtained. The curve starts at Ω = 0 and is traversed in a clockwise direction. In order to arrive at a normalized representation, the complex transfer function αxF can be plotted in an analogous procedure, e.g. over the tuning ratio η. The complex transfer function is obtained, for example, by converting it into Cartesian form and plotting the real part over the imaginary part: αxF ðjηÞ = αxF ðηÞ cosðςÞ þ jαxF ðηÞ sinðςÞ: ReðαxF ðjηÞÞ

ImðαxF ðjηÞÞ

HvF ðjηÞ =

HaF ðjηÞ =

Acceleration

HxF ðjηÞ =

η2 1 m ð1 - η2 Þþj2ϑη

jη p1 km ð1 - η2 Þþj2ϑη

1 1 k ð1 - η2 Þþj2ϑη

Complextransfer function

Velocity

Response Displacement

Ωrv = ω0

ςvF(η) = ς1(η) - 90° ςaF ðηÞ = ς3 ðηÞ = ς1 ðηÞ - 180 °

αvF(η) = ηα1(η) αaF(η) = α3(η) = η2α1(η)

1 k

p1 km 1 m

κxF =

κvF = κaF =

1 - 2ϑ2

Ωra = p ω0

Angular Resonance frequency Ωr p Ωrx = 1 - 2ϑ2 ω0

Phase frequency response ς(η) ςxF ðηÞ = ς1 ðηÞþ 2ϑη = arctan 1 - η2

Amplitude frequency response α(η) αxF ðηÞ = α1 ðηÞ 1 = 2 ð1 - η Þ2 þ ð2ϑηÞ2

prefactor κ

Table 4.2 Transfer functions for force excitation (Figs. 4.6, 4.7, 4.8, 4.9, 4.10 and 4.11)

4.4 Nyquist Diagram 55

56

4

Forced Vibrations

The signs result from the quadrant relations of the trigonometric functions used. This relationship is plotted for the transfer function already discussed in Fig. 4.13. The Nyquist’s curves start at η = 0 and αxF = 1 and run towards the value αxF = 1 for large tuning ratios. As the damping ratio ϑ increases, the diameter of the curves decreases. "

4.5

In the Nyquist diagram, the quantities for describing the oscillatory system can be read from the curves. Compared to the representation of the amplitude and phase frequency responses, Nyquist’s curves have the advantage that the real and imaginary parts are represented directly. However, the excitation angular frequency or the tuning ratio can no longer be read directly from the display.

Compilation of Various Transfer Functions

The different types of vibration excitation with harmonic excitation functions and their vibration responses have a great influence on the determined transfer function. In addition to the cases treated so far, other transfer functions that are important for the application will be discussed. It should be noted in advance that the solution of the vibration differential equations in the cases considered leads to five different transfer functions. Apart from the case of excitation via the damper, which is not considered here, the solutions for the transfer functions have already been developed and discussed in part. The designation for the transfer functions is not uniform in the literature and in the field of technical regulations (cf. [1–6, 11]). The systematic approach of DIN 1311-2 introduces a number of different indices for designation [1]. In the following representation, the amplitude components of the transfer functions are defined as α1 to α4 analogous to the amplification functions V1 to V4 according to [11]. Tables 4.3, 4.4 and 4.5 list some vibration responses with the associated vibrational equivalent systems, differential equations and transfer functions. This description represents a selection, further transfer functions can be derived from the excitations and vibration responses. Force Excitation (Table 4.3) Force excitation includes the case already discussed of periodic force acting directly on the mass. Examples are hydraulic and gas forces as well as electromagnetically or electrodynamically generated forces. These forces can also be introduced via the motion of a spring connected to the mass and given a displacement-time function u(t). This corresponds to the cases of the force or acceleration transducer or a so-called relative transducer, which records the oscillation paths relative to a stationary reference system.

4.5

Compilation of Various Transfer Functions

57

Table 4.3 Vibration responses for force excitation

Replacement system F(t) m

x(t)

d

(a) Designation (b) Vibration response (c) Example of application (a) Force excitation (b) Displacement (c) Force transducers (Sect. 9.2), accelerometers (Chap. 8)

Differential equationa m€ x þ dx_ þ kx = FðtÞ

Amplitude frequency response H(η) = κ  α(η) x F

= 1k  α1

k

u(t) k2 m

(a) Spring force excitation (b) Displacement (c) Relative transducer (Sect. 5.1.2)

m€ x þ dx_ þ kx = k2 uðtÞ with k = k1 + k2

(a) Source isolation (b) Base excitation force Fp (c) Foundation force for machine installation

m€x þ dx_ þ kx = FðtÞ Base excitation force: Fp = kx þ dx_

x u

=

k2 k

α1

x(t)

d

k1

F(t) m

x(t)

d

Fp F

= α2

k Fp(t)

a

The equilibrium relationships were multiplied by -1 to obtain positive signs

Base Excitation (Table 4.4) In the case of base excitation, the oscillating system is excited with a harmonic displacement-time function at the base of the system (e.g. movement at the installation site of a machine). The vibration responses of the mass are calculated as absolute vibration displacements in relation to a stationary reference system or as relative vibration displacements to the movement of the base. The relative displacement is measured in so-called absolute or seismic transducers. Unbalance Excitation (Table 4.5) If the excitation takes place with a rotating unbalance, the amplitude of the excitation force depends on the angular velocity. In this case, one speaks of an unbalance or mass force

58

4

Forced Vibrations

Table 4.4 Vibration responses for base excitation

Replacement system m

x(t)

d

k u(t)

m

xr(t)

d

k

(a) Designation (b) Vibration response (c) Example of application (a) Receiver insulation (b) Absolute displacement (c) Protection of a hard disk against vibrations from the environment

Differential equation m€x þ dðx_ - u_ Þ þ kðx- uÞ = 0

(a) – (b) Relative displacement xr (c) Absolute transducers (Sect. 5.1.3)

mð€xr þ € uÞ þ dx_ r þ kxr = 0

(a) Transit (b) Base force Fp (c) Foundation force for machine installation

m€x þ dðx_ - u_ Þ þ kðx- uÞ = 0

Amplitude frequency response H (η) = κ  α(η) x u

= α2

xr u

= α3

Fp

= k  α4

u(t)

m

x(t)

d

u

k Fp(t)

u(t)

excitation. This type of excitation deals with the technically significant cases of unbalanced mass forces on machines (e.g. combustion engines, rotors, machine tools) as well as the introduction of forces into the installation site. In summary, Table 4.6 lists the amplitude frequency responses α1 to α4 for the calculation. A further discussion of the results in relation to the vibration transducers is given in Chaps. 5, 6, 7, 8, and 9. For the derivation of conclusions on machine installation and operation, reference is made to the literature [2, 4–6, 11]. The phase frequency response for ς2 – which is associated with the amplitude frequency response α2 – is shown in Fig. 4.14 and is described by the following equation: ς2 ðηÞ = arctan

2ϑη3 : 1 - η2 þ 4ϑ2 η2

ð4:18Þ

In contrast to the phase frequency responses for ς1 and ς3 (cf. Figs. 4.4 and 4.10), the phase shift angle of 90° is only found for small damping ratios ϑ close to the tuning ratio η = 1. With higher damping ratio, smaller phase shift angles are obtained at the tuning ratio η = 1.

4.5

Compilation of Various Transfer Functions

59

Table 4.5 Vibration responses for unbalance excitation (a) Designation (b) Vibration response (c) Example of application (a) Unbalance excitation (b) Displacement (c) Motor vibrations

Replacement system F(t) :

rˆ

mu m1

Differential equation m€ x þ dx_ þ kx = FðtÞ with m = mu + m1 and F = mu r∖Ω2

Amplitude frequency response H(η) = κ  α(η) x r

=

mu m

 α3

x(t)

d

k

(a) – (b) Base force Fp (c) Introduction of engine vibrations into the vehicle structure

F(t) :

rˆ

mu m1

m€ x þ dx_ þ kx = FðtÞ FBase force: Fp =

Fp r

=

mu m

k  α4

kx þ dx_

x(t)

d

k Fp(t)

For this case, the tuning ratio η = 1 and thus the natural angular frequency of the undamped oscillator cannot be read from the phase shift angle of 90°. All transfer functions shown apply to the stationary case, i.e. to the case of excitation with harmonic excitation functions after a sufficiently long time in which the homogeneous

60

4

Forced Vibrations

Table 4.6 Compilation of the amplitude frequency responses (normalized) α1 = p Amplitude frequency response DxF=D1

5 - - - - - -

4

3

0,05 0,1 0,2 0,5 0,7 1,0

2

1

0 0,0

0,5

1,0

1,5

2,0

Amplitude frequency response D2

α2 =

1 ð1 - η2 Þ2 þð2ϑηÞ2

2,5

1 þ ð2ϑηÞ2  α1 = p

5 - - - - - -

4 3

η2

2 1 0 0,0

Amplitude frequency response DaF=D3

- - - - - -

4

3

0,05 0,1 0,2 0,5 0,7 1,0

2

1

0 1,0

Voting ratio K

1,5

2,0

2,5

Amplitude frequency response DaF=D3

α4 = η2

5

0,5

 0,05  0,1 0,2 0,5 0,7 1,0

0,5

1,0

1,5

2,0

2,5

Voting ratio K

ð1 - η2 Þ2 þð2ϑηÞ2

0,0

1þð2ϑηÞ2

ð1 - η2 Þ2 þð2ϑηÞ2

Voting ratio K

α3 = η2 α1 = p

p

η 1 þ ð2ϑηÞ2  α1 = p

2

p

1þð2ϑηÞ2

ð1 - η2 Þ2 þð2ϑηÞ2

5 - - - - - -

4

0,05 0,1 0,2 0,5 0,7 1,0

3

2

1

0 0,0

0,5

1,0

1,5

Voting ratio K

2,0

2,5

4.5

Compilation of Various Transfer Functions

61

Amplitude frequency response DvF=KD1

5 -

4

3

0,05 0,1 0,2 0,5 0,7 1,0

2

1

0 0,0

0,5

1,0

1,5

2,0

2,5

Tuning ratio K

Fig. 4.6 Amplitude frequency response for velocity response under force excitation

Phase frequency response ]vF

90

60

30

0 -

-30

-60

0,05 0,1 0,2 0,5 0,7 1,0

-90 0,0

0,5

1,0

1,5

2,0

Tuning ratio K

Fig. 4.7 Phase frequency response for fast response under force excitation

2,5

62

4

Forced Vibrations

Amplitude frequency response DvF=KD1

10 -

1

0,05 0,1 0,2 0,5 0,7 1,0

0,1

0,01 0,1

1

10

Tuning ratio K

Fig. 4.8 Logarithmic representation of the amplitude frequency response of the fast response under force excitation

Amplitude frequency response DaF=D3

5 -

4

3

0,05 0,1 0,2 0,5 0,7 1,0

2

1

0 0,0

0,5

1,0

1,5

2,0

2,5

Tuning ratio K

Fig. 4.9 Amplitude frequency response for acceleration response under force excitation

4.5

Compilation of Various Transfer Functions

63

Phase frequency response ]aF=]3

0

-30

-60

-90 -

-120

-150

0,05 0,1 0,2 0,5 0,7 1,0

-180 0,0

0,5

1,0

1,5

2,0

2,5

Tuning ratio K

Fig. 4.10 Phase frequency response for acceleration response under force excitation

Amplitude frequency response DaF=D3

10

1

-

0,1

0,05 0,1 0,2 0,5 0,7 1,0

0,01 0,1

1

10

Tuning ratio K

Fig. 4.11 Logarithmic representation of the amplitude frequency response of the acceleration response under force excitation

64

4

Fig. 4.12 Phasor of displacement response to force excitation

Forced Vibrations

Im

:, K

Fˆ ]

xˆ

Imaginary part Im(DxF(jK))

0

-1

-2

-

-3

0,2 0,5 0,7 1,0

-4 -3

-2

-1

0

1

2

3

Real part Re(DxF(jK))

Fig. 4.13 Nyquist’s curve for displacement response to force excitation

Phase frequency response ]2

180 -

150

120

0,05 0,1 0,2 0,5 0,7 1,0

90

60

30

0 0,0

0,5

1,0

1,5

Tuning ratio K

Fig. 4.14 Phase frequency response ς2

2,0

2,5

Re

References

65

solution has decayed. Transient processes, e.g. transient processes when machines are switched on and off and when transducers are subjected to shock loads, cannot be recorded.

References 1. DIN 1311-2: Schwingungen und Schwingungsfähige Systeme. Teil 2: Lineare, zeitinvariante schwingungsfähige Systeme mit einem Freiheitsgrad (2002) 2. Blake, R.E.: Basic vibration theory. In: Piersol, Paez (Hrsg.) Harris’ Shock and Vibration Handbook, S. 2.1–2.32, McGraw-Hill Education, New York (2009) 3. Gross, D., Hauger, W., Schröder, J., Wall, W.A.: Technische Mechanik 3. Kinetik. Springer Vieweg, Berlin (2015) 4. Hollburg, U.: Maschinendynamik. Oldenbourg, München (2007) 5. Jäger, H., Mastel, R., Knaebel, M.: Technische Schwingungslehre. Springer Vieweg, Wiesbaden (2016) 6. Magnus, K., Popp, K., Sextro, W.: Schwingungen. Springer Vieweg, Wiesbaden (2016) 7. Hoffmann, R.: Grundlagen der Frequenzanalyse Bd. 620. expert verlag, Renningen (2011) 8. Zollner, M.: Frequenzanalyse. Autoren-Selbstverlag, Regensburg (1999) 9. Broch, J.: Mechanical Vibration and Shock Measurements. Brüel & Kjaer, Naerum (1984) 10. Heymann, J., Lingener, A.: Experimentelle Festkörpermechanik. VEB Fachbuchverlag, Leipzig (1986) 11. Dresig, H., Holzweißig, F.: Maschinendynamik. Springer Vieweg, Berlin (2016) 12. DIN 1311-1: Schwingungen und Schwingungsfähige Systeme. Teil 1: Grundbegriffe, Einteilung (2000) 13. Butz, T.: Fouriertransformation für Fußgänger. Springer Vieweg, Wiesbaden (2012)

5

Vibration Transducer

Abstract

This chapter deals with the basic measurement principles for kinematic quantities, the resulting designs of the vibration transducers used and selection criteria for use. The basic terms of the frequency bands frequently used for the characterization of vibration transducers and measuring chains as well as the level representation are also explained. For the application, the measurement principles are systematized and their possibilities and application limits are presented. "

This chapter deals with the basic measurement principles for kinematic quantities, the resulting designs of the vibration transducers used and selection criteria for use. The basic terms of the frequency bands frequently used for the characterization of vibration transducers and measuring chains as well as the level representation are also explained. For the application, the measurement principles are systematized and their possibilities and application limits are presented.

5.1

Measuring Principles for Kinematic Quantities

5.1.1

Basics

Vibration transducers convert the input variable (vibration variable) into an – usually electrical – output variable (Fig. 5.1, Table 5.1) [1–4]. In a narrower sense, the input quantity is understood to be the vibration quantity at the location of the vibration transducer, which is used to describe the mechanical vibration, or a quantity derived from it # The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 T. Kuttner, A. Rohnen, Practice of Vibration Measurement, https://doi.org/10.1007/978-3-658-38463-0_5

67

68

5 Vibration Transducer

Vibration transducer Measurement object

Input

Conversion of measured variables

Signal processing

Signal conditioning

Transfer function Amplitude component H(f) Phase component ζ(f)

Output

Fig. 5.1 Interaction of measuring object and transducer

Table 5.1 Subdivision of input variables according to [1] Kinematic quantity Translational movement

Rotatory movement

Deformational movement

Designation Vibration displacement (deflection)

Formula symbol e.g. s(t), x(t)

Vibrational velocity (velocity)

v(t), x_ ðtÞ

Vibrational acceleration Angle (rotation angle) Angular velocity (rotational speed)

a(t), € xðtÞ β(t) β_ ðtÞ, ω(t)

Units in use m, mm, μm m/s, mm/s m/s2 rad rad/s

Angular acceleration (rotational acceleration) Strain

€βðtÞ, ω_ ðtÞ

rad/s2

ε(t)

μm/m

Torsion (twisting)

γ(t)

rad/m

[1]. In [2], a distinction is made that vibration transducers should essentially be sensitive to only one of the kinematic vibration quantities: translational motion, rotational motion or deformational motion. In some cases, vibration transducers contain not only the measured quantity conversion but also parts of the signal processing [2]. If the vibration transducer is spatially separated from the rest of the measuring equipment, the connecting line is attributed to the vibration transducer [1]. The output quantity of a vibration transducer is generally the electrical quantity at the output of the transducer. The time function of the output quantity is referred to as the output signal of the transducer. The generic term is the output quantity, which additionally includes derived quantities such as signal characteristic quantity (quantity derived by a fixed formation rule) and assessment quantity (derived quantity for assessing causes or effects of the input quantities). In general, the type of output variable depends on the transducer used and its measuring principle.

5.1

Measuring Principles for Kinematic Quantities

69

A measurement signal is only referred to after it has been represented on an information carrier (e.g. storage). Measurement signals can be named according to the input quantity (acceleration signal), according to the application area (vibration signal) or its position in the signal path (output signal) and has the dimension of the represented quantity. The specification of an acceleration must therefore be in m/s2, for example. The specification in Volts for an analog signal output is only permissible if the transfer coefficient is known [5]. The designation of a time-dependent input quantity as “input signal” is permissible [5], although this would only be correct after mapping on an information carrier. The kinematic quantities of a oscillatory system (vibrational displacement, velocity and accelerations) can be measured using the measuring principles of the relative transducer and the absolute transducer, which are explained below. The measuring principle for forces and torques is dealt with in Sect. 9.2.

5.1.2

Relative Transducer

In this case, the transducer acquires the measured quantity relative to a fixed reference system (Fig. 5.2), which is located outside the measured object [1, 6, 7]. In most cases, the transducer is located at a fixed reference point and transmits the kinematic quantity in a form-fit, force-fit or contact-free manner. The difference between the movement of the measured object and the transducer is therefore measured. The reference system can also be moved, e.g. when measuring spring travel on a vehicle between the wheel hub and the body or frame. Examples of transducers operating according to this principle are, for example, displacement transducers, which are discussed in Chap. 6. The advantage of this measuring principle is that a practically frequency-independent transfer function with amplitude component α(f) = 1 and phase shift angle ς = 0 can be realized (cf. Fig. 6.4a). Measurements are possible up to 0 Hz (quasistatic measurements). The same applies to the transfer function of velocity and acceleration. Relative transducers do not require any oscillatory system in the transducer in order to implement the measurement task. However, the transducer itself represents a vibratory system due to its mass and stiffness, which corresponds to the spring force excitation in Table 4.3. The transducer with the mass m is connected to the reference system via the spring stiffness k1, the force application to the transducer takes place via the displacement u (t) to be measured and the spring stiffness k2. The influence of damping is neglected. The aim is that the transducer does not move (x(t) = 0) and thus the measured differential displacement corresponds to the measurand u(t). For this, the product κxF multiplied with α1 must assume the value 0 in the transfer function. The following cases can be discussed here: • highly tuned transducer: Thus α1 ≈ 1, the angular natural frequency of the transducer is large due to small mass and high spring stiffness k1. This leads to a stiff coupling of the

70

5 Vibration Transducer

Fig. 5.2 Principle of the relative transducer

Relative transducer

Measurement object

Reference system

transducer to the reference frame. A low spring stiffness k2 results in a prefactor κxF = k2/(k1 + k2) ≈ 0. This design uses displacement transducers that are coupled to the (fixed) reference system with a high stiffness k1 and are frictionally connected to the measured object via a soft spring. In this case, the contactless coupling is included with k2 = 0. • Low-tuned transducer: Due to a large mass m and low spring stiffness k1 the transducer is operated above the natural frequency. Then α1 ≈ 0 applies, i.e. the transducer does not oscillate with respect to the reference system. Moreover, from the practical application, the requirement for a low spring stiffness k2 ≪ k1 has to be fulfilled. This operation is not very common in practice and corresponds, for example, to the case of a displacement transducer with a large mass and coupling with a low spring constant to the reference system (e.g. hand-held transducer, geophones) [7]. Since the transducers are coupled to the measuring object without contact or with very low moving masses, these transducers show no or only very low feedback effects on the measuring object. The disadvantage of the measuring principle is the fixed reference point, which is often difficult to realize in practice. Likewise, great attention must be paid to the coupling to the measurement object. Table 5.2 compares possible coupling types. "

Relative transducers measure relative to a fixed reference system, allow quasi-static measurements and have a frequency-independent transmission behavior.

5.1.3

Absolute Transducer

Transducers based on this measuring principle record the measured quantity relative to a stationary or uniformly moving reference system (inertial system). Such transducers are constructed as a system capable of oscillating (mass, spring, possibly damping) (Fig. 5.3) and use the inertial properties of the mass in the earth’s gravitational field. For this reason the transducers are also called seismic transducers. The transducer is attached to the

5.1

Measuring Principles for Kinematic Quantities

71

Table 5.2 Coupling types of relative transducers (modified after [7]) Coupling Rigid connection

positive-locking Measurement object

Displacement transducer

Diving anchor with rod end

Friction-locked

friction-locked Measurement object

Displacement transducer

Diving anchor with button

Non-contact

non-contact Measurement object

Displacement transducer

non-contact distance measurement

Description Principle of operation: Fixed or articulated connection of the armature with the measured object Application: Measurement of large distances Pros: Wide usable frequency range Disadvantages: No movement of the measuring object transverse to the measuring direction Play in the measuring arrangement, Attachment of fastening parts to the measuring object Principle of operation: Pressing of the armature with the object to be measured by means of a spring Application: Measurement of small distances Pros: Little preparation required, no clearance Movement of the measuring object transverse to the measuring direction possible Disadvantages: Lifting of the tip at high frequencies, High contact pressure required Working principle: No mechanical connection to the measuring object Application: Depending on the transducer measurement of small or large distances Pros: Wide frequency range Low preparation effort Movement of the measuring object transverse to the measuring direction possible without restrictions Disadvantages: Depending on the measurement method Higher demands on the measuring point (material and surface properties, calibration)

72 Fig. 5.3 Principle of the absolute transducer

5 Vibration Transducer

Absolute transducer

Measurement object

measured object, and the movement of the coupling point relative to the seismic mass in the transducer is used as the measured quantity. The oscillatory system of the transducer determines the essential measurement characteristics in connection with the measurand. When approaching the natural frequency of the transducer, the amplitude frequency response deviates from the value 1, i.e. the transmission coefficient changes depending on the frequency. A phase shift can also be detected (cf. e.g. Fig. 4.4). Both are – apart from exceptions – undesirable, since a frequency-independent transmission behavior is desired. This can be achieved in two ways: • Highly tuned transducer: Small seismic mass and high spring stiffness leads to a high natural frequency. The transducer is then operated in the frequency range of max. 20 . . . 30% of its natural frequency (Fig. 5.4a). The advantage of this concept is comparatively small, light and robust transducers. This measuring principle is used, for example, for accelerometers (Chap. 8). • Low-tuned transducer: Large seismic masses and low spring stiffnesses result in transducers with low natural frequencies, which perform their measuring task at frequencies above the natural frequency (Fig. 5.4b). The seismic mass, which is suspended on soft springs, remains at rest at high frequencies, and the vibration displacement of the transducer corresponds to the vibration displacement at the measured object. High sensitivities are possible with this measuring principle. One application for such transducers is electrodynamic transducers (Sect. 7.1). Damping of the transducer reduces the maximum in the amplitude frequency response. The transducer is thus protected against overload and the usable frequency range is extended. With a damping factor ϑ > 0.7, the maximum is suppressed and the curvature of the phase frequency response is then low. This approximately satisfies the requirement for a frequency-independent delay shift in the measurement signal (cf. Sects. 4.2 and 10.6.2). If a linear course of the phase frequency response between the coordinate origin and the phase shift angle π/2 is assumed for a highly tuned transducer with a tuning ratio η = 1, a

5.2

Selection of the Transducer and the Measured Variable

10 -

1

0,05 0,1 0,2 0,5 0,7 1,0

0,1

0,01 0,1

1

Voting ratio K

10

Amplitude frequency response DaF= D3

b

Amplitude frequency response DxF= D1

a

73

10

1

-

0,1

0,01 0,1

1

0,05 0,1 0,2 0,5 0,7 1,0

10

Voting ratio η

Fig. 5.4 Amplitude component of the transfer function with (a) high and (b) low tuning

transit time shift Δt = 1/(4fn) is obtained. With this idealized consideration, the delay shift is in the same magnitude for all frequencies and in the subcritical range depends only on the natural frequency fn of the transducer. For a transducer with a natural frequency of fn = 400 Hz, a delay of 0.6 ms is obtained. "

Absolute transducers contain an oscillating system, measure absolutely in a fixed reference system and have a frequency-dependent transmission behavior.

5.2

Selection of the Transducer and the Measured Variable

The selection of the transducer and the measured variable depends on the measurement task, and in practice often on the available transducers. The large number of measuring principles used in practice and the resulting designs result from the fact that there is no universal transducer1 for all measuring tasks. As a basic guideline, the transducers should first be selected according to the measuring range and frequency of the measurement task [3, 4, 6–9]: (a) Measuring range (absolute magnitude and dynamic range, i.e. minimummaximum, signal-to-noise ratio of the transducer), (b) Frequency (quasi-static measurement required, lower and upper limit frequency). Other criteria may be used:

1

Offered universal pick-ups cover a more or less wide range of tasks and fulfill the combination pliers theorem: A combination pliers fulfills many tasks more or less well, for the solution of special tasks one needs special tools.

74

5 Vibration Transducer

Table 5.3 Selection of transducer and kinematic quantity Kinematic quantity Displacement Speed Acceleration

Relative transducer Potentiometric, capacitive, inductive, eddy current, laser triangulation Vibrometry

Absolute transducer

Electrodynamic, electromagnetic Piezoelectric, piezoresistive, capacitive, strain gauge

(c) Dimensions and mass of the transducer, reference point of the transducer necessary and at what distance available or required, (d) linearity of the transducer (especially with large measuring ranges), time-dependent drift (especially with quasi-static measurements over a long period), (e) Power supply of the transducer, moving parts, environmental influences (temperature, fields, radiation, etc.). An overview of the selection of the transducers used according to the quantity to be measured and the measuring principle is given in Table 5.3. Deformation transducers (Chap. 9, such as strain gauges) occupy a special position. Here, it is not the displacement itself but a deformation that is measured and subsequently converted to a displacement. These transducers use the undeformed structure as a reference system to measure the strain and, with a known initial length, calculate the change in length from it. In contrast to the other measuring principles, deformation transducers cannot be used for rigid bodies, since the structure to be examined must deform in this measuring principle. The selection of the kinematic measurand can be made by comparing the amplitudes of acceleration and displacement. For a constant acceleration, the displacement amplitude is calculated as a function of the frequency by replacing the angular frequency by the frequency in the equation of motion: x=

€x €x = : ω2 ð2πf Þ2

Logarithmizing leads to a straight line equation with the slope of -2 in double logarithmic representation. lgx = - 2  lgf þ lg

€ x 4π2

:

ð5:1Þ

Figure 5.5 shows for the acceleration amplitude of 1 g that a frequency of 1 Hz requires a displacement amplitude of 248.5 mm, whereas a frequency of 10 kHz requires a

5.3

Representation in Frequency Bands

75

100 10-1

Amplitude in m

10-2 10-3 10-4 10-5 10-6 10-7 10-8 10-9 1

10

100

1000

10000

Frequency in Hz

Fig. 5.5 Displacement amplitude as a function of frequency for an acceleration amplitude 1 g

displacement amplitude of 2.485 nm. Since such small displacements are difficult to measure, the choice of displacement amplitude as a measurand is not appropriate at high frequencies. At very low frequencies, on the other hand, acceleration measurements provide only very small amplitudes for the same reason and are unsuitable as a measurand. For a number of measurement tasks that have to be performed according to technical regulations and standards, the motion quantity to be measured (displacement, velocity or acceleration) is specified. If, on the other hand, the quantity to be measured has not yet been determined, Table 5.4 provides information on the selection.

5.3

Representation in Frequency Bands

In vibration engineering and acoustics, frequency ranges are often represented as octave bands (Fig. 5.6). Here, the frequency range is divided from a frequency of 1 kHz in such a way that the center frequencies fm of the frequency bands are in a ratio of 2:1. The lower frequency limit fu and upper frequency limit fo are obtained by a geometric series:

76

5 Vibration Transducer

Table 5.4 Selection of kinematic quantities Acceleration For high frequencies

Speed (Velocity) Wide frequency range for many measurement tasks (high usable dynamic range)

Mass forces (force proportional to acceleration)

Damper forces (force proportional to speed) Determination of mechanical impedance Frequently used in condition monitoring of machines and structures and machine acoustics High sensitivity of the transducers possible

Determination of the dynamic mass Comparatively small transducers with low mass

fu=710Hz

…

250 Hz-Band

500 Hz-Band

fu=355Hz

250 Hz

500 Hz

fo=1,42kHz

1 kHz-Band

fo=710Hz

Deflection (displacement) For low frequencies Transducers need reference point Spring forces (force proportional to displacement) Measurement of relative movements and possible collisions on components

2 kHz-Band

fu=1,42kHz

1 kHz

4 kHz-Band

fo=2,82kHz

2 kHz

4 kHz

…

log f 8 kHz

Centre frequencies fm

Fig. 5.6 Octave bands in the frequency ladder

f f u = pm 2 p f o = 2  f m:

ð5:2Þ

On a logarithmic scale, one obtains frequency bands of equal width that are arranged without gaps or overlaps. A finer subdivision of the frequency bands results if each octave is divided into three third octave bands of equal width.

5.4

Decibels

5.4

77

Decibels

The used measuring range (amplitudes) of vibration transducers covers several powers of ten. With a linear application, an amplitude dynamic can be represented as a ratio of largest to smallest amplitude of 100:1. In addition to the often used logarithmic representation of the measuring quantity, the representation in decibel is common in vibration engineering and acoustics [2, 9]. " The decibels of a linear quantity (e.g. displacement, force, electric voltage, etc.) is

defined as follows: y y0

LðdBÞ = 10  lg

2

= 20  lg

y : y0

ð5:3Þ

The unit of decibel ist dB and includes a reference value y0. For the ratio y/y0 levels are given in Table 5.5. "

The RMS values (Sect. 14.2.1) are used to calculate the decibels.

Table 5.5 shows that a level of 0 dB is measured when the measured quantity and the reference quantity are equal. The logarithmic representation spreads the axis in the range of small measured values, while the axis is compressed for large measured values. Over the entire range of the representation, multiplying the amplitude by a factor of 10 corresponds to a level addition of 20 dB. When the measurement signal is multiplied by a constant factor C (e.g. gain), this causes a constant component lg(C) to be added to the value (and consequently a shift along the axis): L1 ðdBÞ = 20  lg

Cy y = 20  lg y0 y0

þ 20  lgðCÞ = L þ 20  lgðCÞ:

For the power quantities that occur less frequently in vibration engineering, the level is defined as follows:

Table 5.5 Ratio y/y0 and level y/y0 =

0.32

0.5

0.71

0.89

L[dB] =

-10

-6

-3

-1

0.94 -0.5

1

1.06

1.12

1.42

2

3.16

10

100

1000

0

0.5

1

3

6

10

20

40

60

78

5 Vibration Transducer

Table 5.6 Acceleration level for reference value a0 = 10-6 m/s2 a[m/s2] = L[dB] =

0.01 80

0.1 100

1 120

LðdBÞ = 10  lg

10 140

100 160

Y : Y0

1000 180

ð5:4Þ

If this reference value is mandatory for the determination of the measurand, it is referred to as an absolute level. (Example sound pressure level with reference quantity of p0 = 2105 Pa). If, on the other hand, the reference quantity y0 is not prescribed, this level is referred to as a relative levels. "

Relative levels are useful when comparing measurements with each other or when considering changes of a measurement signal in the measurement chain.

In vibration engineering, the reference values are defined according to [1, 5] with the use of a reference value a0 = 10-6 m/s2. Thus, for an acceleration of a = 1 m/s2 (rms value), the level L in dB is given by L = 20  lg

1 = 120 : 10 - 6

ð5:5Þ

Further numerical values are given in Table 5.6. In accordance to the acceleration level, velocity level and deflection level can be defined with the reference values v0 = 10-9 m/s and deflection level of x0 = 10-12 m. In this case, an intersection of the three levels is obtained at 159.1 Hz and an ordinate value of 120 dB for 1 m/s2 (Fig. 5.7). Another definition used is obtained with reference values for the acceleration level a0 = 10-5 m/s, velocity level v0 = 5.0510-5 m/s and deflection level x0 = 2.5510-10 m. In this case, the three straight lines intersect at 31.5 Hz and 100 dB. From these two examples, the necessity of always indicating the reference value of the level in the plot can already be seen. Example

How does the acceleration and velocity level change for a vibration with constant displacement with an increase in frequency of one octave and one decade, respectively? Since the RMS value is defined as positive, the amounts from Eqs. 2.9 and 2.10 are inserted into the level equation. If the level difference ΔL is now formed, the frequencyindependent components are dropped out. If the frequency f is now increased by a decade (factor 10) or octave (factor 2), a constant value is obtained for the level difference (Table 5.7). This level difference is therefore independent of the choice of amplitude x and frequency f. ◄

5.4

Decibels

79

220 Acceleration level re a0 =10-6 m/s2

200

Velocity level re v0 =10-9 m/s

180

Displacement level re x 0 =10 -12 m

Level L in dB

160 140 120 100 80 60 40 20 1

10

100

1000

10000

Frequenz in Hz

Fig. 5.7 Frequency dependence of the velocity and displacement level at a constant acceleration level for a = 1 m/s2 Table 5.7 Dependence of acceleration and velocity level with frequency Acceleration level RMS values of the vibration amplitudes Insert into level equation LðdBÞ = 20  lg

x x0

Decibel difference with an increase of one decade (tenfold increase in frequency) Decibel difference with an increase of one octave (doubling of the frequency)

€x = - ω  x = - ð2πf Þ  x 2

2

LðdBÞ = 20  lg = 20  lg

€ x 4π2 x0

Velocity level x_ = j ω  xj = jð2πf Þ  xj

€ x 4π2 f 2 x0

LðdBÞ = 20  lg

- 2  20  lgf

= 20  lg

ΔL(dB) = L1 - L0 = - 40  lg (10f) - (-40  lg (f)) = - 40  lg (10) = - 40 ΔL(dB) = - 40  lg (2) = - 12

x_ 2πx0

x_ 2πfx0

- 20  lgf

ΔL(dB) = L1 - L0 = - 20  lg (10f) - (20  lg (f)) = - 20  lg (10) = - 20 ΔL(dB) = 20  lg = - 6

80 "

5 Vibration Transducer

Integration: Decibel drop over the frequency -20 dB/decade (or -6 dB/ octave) Differentiation: Decibel rise above frequency +20 dB/decade (or +6 dB/ octave) Measuring systems often determine decibels from electrical quantities with a reference value of 1 Veff. The decibels are then designated dBV.

References 1. DIN 45661:2013-03 Schwingungsmesseinrichtungen – Begriffe 2. DIN 45662:1996-12 Schwingungsmesseinrichtung – Allgemeine Anforderungen und Begriffe 3. Hesse, S., Schnell, G.: Sensoren für die Prozess- und Fabrikautomation. Springer Vieweg, Wiesbaden (2018) 4. Parthier, R.: Messtechnik. Springer Vieweg, Wiesbaden (2016) 5. DIN EN ISO 1683:2015-09 Bevorzugte Bezugswerte für Pegel in der Akustik und Schwingungstechnik 6. Heymann, J., Lingener, A.: Experimentelle Festkörpermechanik. VEB Fachbuchverlag, Leipzig (1986) 7. Holzweißig, F., Meltzer, G.: Meßtechnik in der Maschinendynamik. VEB Fachbuchverlag, Leipzig (1973) 8. Sinambari, G.R., Sentpali, S.: Ingenieurakustik. Springer Vieweg, Wiesbaden (2014) 9. Kolerus, J., Wassermann, J.: Zustandsüberwachung von Maschinen. Expert Verlag, Renningen (2017)

6

Displacement Transducer

Abstract

The measurement of the vibration displacement is an important task in measurement technology. This chapter presents a selection of important measurement principles used in practice. Together with the transducers, it deals with circuit-technical particularities of the signal conditioning. The possibilities and limitations of the individual methods are shown and supplemented by sketches and sectional views. "

6.1

The measurement of the vibration displacement is an important task in measurement technology. This chapter presents a selection of important measurement principles used in practice. Together with the transducers, circuit-technical peculiarities of the signal conditioning are dealt with. The possibilities and limitations of the individual methods are shown and supplemented by drawings and sectional views.

Potentiometric Displacement Transducers

Potentiometric displacement transducers use the change of a resistance as measuring principle. This resistance path can, for example, consist of wire, conductive plastic or a sprayed-on metal layer. The input variable displacement is tapped as a variable resistance by the position of a sliding contact [1–6]. By means of a voltage divider circuit, an electrical output voltage proportional to the displacement is obtained (Fig. 6.1). The length of the resistance path corresponds to the measuring length in the case of the linear potentiometer. For angle transducers, the resistance path is circular. String potentiometers also convert the translatory movement into a rotary movement. # The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 T. Kuttner, A. Rohnen, Practice of Vibration Measurement, https://doi.org/10.1007/978-3-658-38463-0_6

81

82

6 Displacement Transducer

a

b

2 1

Path

3

4

Fig. 6.1 Principle of the linear potentiometer (a) (1 housing, 2 displacement sensor, 3 resistance track, 4 contact), circuit as voltage divider. (b) (Monika Klein, www.designbueroklein.de (a), S. Hohenbild (b))

This is done by means of a wire rope with return spring and transmission gear (Fig. 6.2). This transducer design makes it possible to achieve a large measuring range with small transducer dimensions. This allows measurements to be taken even on components that are difficult to access. Potentiometric displacement transducers are relative transducers. The slider measures the movement of the measured object; the housing of the transducer is attached to the fixed point. Due to the low moving masses, potentiometric displacement transducers are considered to have low feedback to the structure. The operating frequency range of potentiometric displacement transducers covers quasi-static (lower cut-off frequency 0 Hz) and low-frequency measurements [4]. As relative transducers, potentiometric displacement transducers behave independent of frequency, i.e. the transmission coefficient is not dependent on the frequency of the oscillating displacement. At higher frequencies, undesirable jumping of the sliding contact or oscillation of the cable can occur with cable tension transducers. Likewise, heating and wear of the resistance track may occur during operation [4]. Potentiometric displacement transducers are available for a wide measuring range from several millimeters to several meters [6] and are comparatively inexpensive. The measurement uncertainty is better than 1%, typically 1.5 ppm/K is specified as temperature drift [6]. For the circuitry, it must be noted that the input resistance of the subsequent amplifier circuit must be of such high impedance that the source is not loaded [6]. To reduce measurement deviations, the supply voltage must be stabilized, since this is included in the result [4].

6.2

Capacitive Displacement Transducers

83

Fig. 6.2 Principle of the string potentiometer (a) (1 Coil spring, 2 Pull cable, 3 Roller 4 Resistance track) and sectional view. (b) (Monika Klein, www.designbueroklein.de (a), Micro-Epsilon Messtechnik GmbH & Co. KG (b))

6.2

Capacitive Displacement Transducers

The measuring principle of capacitive displacement transducers is based on the change of the capacitance of a capacitor (Fig. 6.3). The electrodes of the transducer are connected as a plate capacitor or as a cylindrical capacitor [2–7]. The cylindrical capacitor uses the change in area as the measured quantity, whereas the plate capacitor uses the change in plate spacing as the measured quantity. By changing the area of two partially overlapping plates, it is possible to measure the angle of rotation. The capacitance C of the capacitor with air as dielectric is C=

ε0 A a

ð6:1Þ

with ε0 = 8.85 pAs/(Vm) the electric field constant, A the plate area and a the plate spacing. In addition to the area, the dielectric influences the capacitance, which makes capacitive displacement transducers sensitive to environmental influences such as splash water, but on the other hand allows, for example, the level in containers to be measured. The charge Q then results in a voltage U U=

Q Q = a : C ε0 A

ð6:2Þ

84

6 Displacement Transducer

a

b

Measuring electrode

a A

c Connection cable Transducer

Massering

c

c

shield electrode (ring-shaped)

U c

c

Measurement object

Fig. 6.3 Principle of the capacitive displacement transducer (a), sectional view of a capacitive displacement transducer (b), with ring electrode and wiring as capacitive half bridge (c). (Author (a), Micro-Epsilon Messtechnik GmbH & Co. KG (b) S. Hohenbild (c))

In order to be able to use the small voltage changes, capacitive displacement transducers are often operated in a capacitive bridge circuit. The bridge circuit requires operation with high-frequency AC voltage in order to obtain an evaluable measurement signal [6] and to keep nonlinearities low [4]. This is due to the low capacitance of the capacitor in the range of a few 10 pF to a few 100 pF. Moreover, the capacitor should have a high ohmic resistance and low polarization losses in the dielectric. Liquids in particular often do not meet these requirements and thus limit the use of these transducers. The high input impedance of the circuit requires good shielding and limits the cable lengths, otherwise interference signals are coupled in. A relative measurement uncertainty of 1 . . . 3% can be achieved [2]. Capacitive displacement transducers are relative transducers. The measuring object can be an electrode, e.g. by a ring-shaped design of the two electrodes (Fig. 6.3), while the transducer housing is attached to the fixed point. Due to the low electrostatic forces, capacitive displacement transducers operate practically without feedback [6] and exhibit a frequency-independent transmission coefficient. The operating frequency range is determined less by the transducer than by the subsequent signal processing. Capacitive displacement transducers are operated with a carrier frequency amplifier (Sect. 9.1.6), which provides for the operation of capacitive displacement transducers, or with signal processing specially adapted to the transducer and the measuring task.

6.3

Inductive Displacement Transducers

Inductive displacement transducers use the change in inductance of a coil when the magnetic flux changes [1–9]. When a current I flows through a coil, a magnetic field with flux Φ is induced:

6.3

Inductive Displacement Transducers

85

Fig. 6.4 Principle of the inductive displacement transducer. (Micro-Epsilon Messtechnik GmbH & Co. KG)

Φ=

IN : Rm

ð6:3Þ

Here N is the number of turns of the coil and Rm is the magnetic resistance of the magnetic circuit (Fig. 6.4). The magnetic resistance Rm is obtained by adding the proportion of the path of the field lines in the iron core (index Fe) and in the air (index L): Rm =

lFe δ þ : μFe  AFe μL  AL

ð6:4Þ

For simplification, the cross-sectional area in the magnetic field in the iron core and in the air gap is now assumed to be the same. For the permeabilities μ, μ = μ0 – μrel. With μrel = 1 for air and μrel = 1000 for iron, it becomes clear that the magnetic resistance is essentially determined by the air gap δ: Rm =

1 lFe δ δ þ : = μ0  A μrelFe μrelL μ0  A

ð6:5Þ

When the flux Φ changes with time, a voltage U is induced in the coil with inductance L: U= -N

dΦ dI = -L : dt dt

ð6:6Þ

If the coil is operated with an alternating current, a current I is required to build up the magnetic field. When the magnetic field collapses, the voltage U is induced with a negative sign. If the magnetic resistance Rm is assumed to be a constant (i.e. change of the air gap δ

86

6 Displacement Transducer

occurs with a much lower frequency than the frequency of the alternating current in the coil), then one obtains L

dI ∂Φ dI N dI =N  =N  dt Rm dt ∂I dt

ð6:7Þ

and finally L=

N2 : Rm

ð6:8Þ

After summing up the constants, it can be seen that the inductance L changes inversely proportional to the gap δ: L=

N 2  μ0  A K N2 = : = δ δ Rm

ð6:9Þ

If the air gap of δ is now changed by the value Δδ, the inductance changes by the value ΔL. The relative change in inductance ΔL/L is thus proportional to the relative change in displacement. ΔL K = L

1 δþΔδ

-

K=δ

1 δ

=

δ - ðδ þ ΔδÞ δ Δδ =-1= : δ þ Δδ δ þ Δδ δ þ Δδ

ð6:10Þ

Inductive displacement transducers do not exhibit linearity between output and input variables; the transmission coefficient depends on the input variable. For small displacement changes Δδ, Eq. 6.10 can be linearized. In order to detect the change in inductance, inductive pick-ups are usually operated in a bridge circuit with a carrier frequency in the kHz range. The function of a carrier frequency amplifier is explained in Sect. 9.1.6. Since interference is suppressed in this circuit, bridgeconnected inductive displacement transducers are used in harsh industrial environments and under the influence of mediums. To measure larger distances, two coils are arranged in a differential circuit (Fig. 6.5). A change in inductance occurs in both coils via a core on a non-magnetic tie rod which is moved in the coils. When the core is in the middle position, the induced voltages in the secondary windings cancel out because they have a different sign. When the core is shifted, a voltage +ΔU is induced in secondary coil 1 and a voltage -ΔU is induced in secondary coil 2 (with the same number of windings in the coil). Consequently, the differential circuit provides a voltage 2ΔU. By measuring the inductance changes between both coils, both the nonlinearities in both coils and the disturbances due to electromagnetic fields, temperature

6.3

Inductive Displacement Transducers

87

b

a

L1 L1Sek. L Prim.

UB~

UB~ L2

L2Sek.

Iron core

TF carrier amplifier

Iron core

TF carrier amplifier

Fig. 6.5 Design of inductive displacement transducers according to the LVDT principle (a) and LVIT principle (b). (S. Hohenbild)

effects, etc. compensate each other [2, 4–6]. In this circuit, measurement ranges from a few millimeters to several 100 mm are possible. In the common design of the LVDT (Linear Variable Displacement Transducer or transformer circuit), the magnetic flux is generated via a primary winding and the difference of the inductance changes is measured via the secondary windings connected in opposite directions [8]. Operated in a bridge circuit, the bridge supply voltage UB  supplies the primary winding Lprim.. The difference between the voltages induced in the secondary windings L1sek. and L2sek. is measured. Angle transducers can be realized with a coil arrangement consisting of a primary coil and two or more secondary coils at an angle of 90° or 120°. For design reasons, the primary coil is designed as a rotor coil and the secondary coils as a stator. A phase-shifted voltage is induced in the secondary coils, from which the angle to be measured is calculated. This design is called RVDT (Rotary Variable Displacement Transducer), moving coil detector or Synchro. The circuit based on the principle of the LVIT (Linear Variable Inductance Transducer or immersed armature transducer in half-bridge circuit) does not require a separate primary winding, but generates the magnetic flux in both coils with the aid of a bridge circuit operated at a carrier frequency. The change in inductance of both coils is tapped via the half-bridge circuit and processed as a half-bridge circuit. The parallel-connected capacitors in the bridge supplementary branch of the carrier frequency measuring amplifier are provided for balancing the phase [5, 6]. Inductive displacement transducers are used as relative transducers. The iron core is connected to the measured object and the coils are attached to the fixed point. For common measuring tasks, the reaction of the transducer on the measuring object is low, since the iron core has a low mass. With clamp connections, it must be ensured that the coils are not mechanically damaged and that the iron core can move freely in the coils. Inductive displacement transducers are either operated on a carrier frequency amplifier (Sect. 9.1.6)

88

6 Displacement Transducer

or have signal conditioning in the transducer. The lower limit frequency is 0 Hz, the upper limit frequency is not given by the measuring principle itself, but is determined by the operation on a carrier frequency amplifier. In practice, the carrier frequency is therefore selected to be at least five times as high as the upper limit frequency to be measured.

6.4

Displacement Transducers According to the Eddy Current Principle

Displacement transducers based on the eddy current principle consist of a coil through which a high-frequency alternating current flows [4, 10, 11]. The magnetic field of the coil generates eddy currents in the metallic measuring object, which in turn generate a magnetic field (Fig. 6.6). The DUT must be conductive, but not necessarily ferromagnetic. The magnetic field induced by the eddy currents has feedback effects on the coil, since the energy decoupling changes the impedance of the coil. The change in impedance depends on the distance and is used as a measurand. Displacement transducers based on the eddy current principle are relative transducers, whereby the transducer itself is attached to the fixed point. The feedback effects caused by the eddy currents on the measured object are very low. The measuring range covers small distances (1 mm. When the upper limit frequency is increased to 100 kHz, the measuring range is reduced to Save as there is now the possibility to save the graphical output. Several file formats are offered for this purpose. With Format .fig it is possible to open the diagram again in MATLAB® at a later time in order to be able to edit it further. The .eps format is suitable for printing at a later date. The previous instructions generate the output of Fig. 11.3. The graph window contains a Context menu which can be used to interactively edit the graphical output. Among other things, functionality for smoothing data series can be found under Tools > Basic Fitting (Fig. 11.5). Example

The measurement of the velocity curve (Fig. 11.4) during a driving test shows scatter. For further data processing, the scatter must be removed from the measurement series. The Basic Fitting context menu can now be used to try out different variants for smoothing the measurement series. You will notice that smoothing without falsifying the measured values is quite difficult. Some variants (spline, shape) show no smoothing

11.4

Creating Diagrams

277 Speed curve during a driving test

Speed in km/h

150

100

50

0 0

10

20

30

40

50

60

70

80

90

Time in s

Fig. 11.4 Speed curve of a driving test

effect, the offered balancing polynomials all lead to more or less distorted representations of the velocity curve (Fig. 11.6). Only the separation of the measurement series into acceleration and coasting data allows a satisfactory smoothing of the measurement data. The best smoothing result of the coasting data is achieved in this case by a fourth order smoothing polynomial (Fig. 11.7). With ◄

ð11:18Þ

you can draw several curves to a Y-axis in a diagram (Fig. 11.8), which can be connected via the instruction

ð11:19Þ

receives a legend without a frame. The assignment into a variable is not mandatory, but is needed for the next step. With the string . . . the statement is extended to several lines. This leads to readable statements.

278

11

MATLAB® and Data Formats an Introduction

With ð11:20Þ the legend is provided with a heading. For diagrams with several curves with unequal scaling another way must be chosen for the representation. Via the statement sequence

ð11:21Þ the curve is displayed and scaled with reference to the left Y-axis, while

ð11:22Þ performs the same with reference to the right Y-axis. Multiple XY plots with reference to the respective Y-axis can also be displayed. For this purpose, the plot instructions must be embedded in hold on/hold off. A special feature is still to be mentioned in the legend heading. The legend heading in Fig. 11.4 consists of a text and the content of a variable. It has already been shown elsewhere that this can be done using the instruction ð11:23Þ is carried out. In the case of the legend title, the issue would take place via ð11:24Þ however, lead to a two-line display of the legend title. A view into the workspace resp. the input in the Command Window

ð11:25Þ shows that the variable text is a cell array. In the legend title this leads to a two-line representation. However, if a single-line display is required, the cell array must be joined using strjoin(text). If a graphic with several diagrams (Fig. 11.9), for example an evaluation sheet, is required, this is implemented using the instruction subplot(n, m, Nr). A grid with n rows and m columns is created. The subplot(n, m, Nr) statement, which precedes the graphic output, now assigns the area element to the following plot statements. The counting

11.5

Data Formats for Measurement and Metadata

279

Fig. 11.5 MATLAB® Basic Fitting Context menu

of the area elements starts at the top left. If individual area elements are to be combined into a larger area element, the number specification is in matrix notation. subplot(3,2,[3 4]), for example, creates the diagram of the second row of (Fig. 11.5). Another way to position individual diagrams in a graphic is to use a relative position specification. The subplot(‘Position’, [X Y Width Height]) statement is used for this purpose. The position specification of X and Y refers to the origin point of the graphic, which is defined at the bottom left with X = 0 and Y = 0. The value range of the specifications is between 0 and 1 and is to be considered as a normalized length specification. The graph in line 3 of Fig. 11.5 was positioned with the instruction subplot(‘Position’, [0.2 0.1 0.6 0.2]).

11.5

Data Formats for Measurement and Metadata

The traceability required by the standard for the quality assurance system (ISO 9001) requires data records for data exchange and data storage, which additionally contain descriptive information on the sensors used, the measurement technology used and further

280

11

MATLAB® and Data Formats an Introduction

Speed curve during a driving test 160 data1 4th degree

140

Speed in km/h

120 100 80 60 40 20 0 -20

0

10

20

30

40

50

60

70

80

90

Time in s

Fig. 11.6 Velocity curve of a driving test, measurement data (data1) smoothed with a fourth order balancing polynomial Speed curve during a driving test 160

data1 4th degree

140

Speed in km/h

120 100 80 60 40 20 0 10

20

30

40

50

60

70

80

90

Time in s

Fig. 11.7 Velocity curve of a driving test, measurement data coasting (data1) smoothed with a fourth order balancing polynomial

information describing the measured object and the measurement process, so-called metadata. Optimal is the use of databases, which manage the metadata and contain references to the corresponding data sets. Ideally, these data sets contain the measurement data as well as the metadata either as an entire file or as a single file summarized in folders of the data repository. A complete file would be preferable to a summary in folders. However, this is only possible for measurement data in proprietary data formats.

11.5

Data Formats for Measurement and Metadata

281

Vibration measurement exhaust system 4

10

3

B11 [N]

1 0

0

-1

B09 [m/s 2]

2

5

-2

-5 Measurement from 20.03.2017

introduced force Acceleration response

-10

2

2.02

2.04

2.06

2.08

2.1

2.12

2.14

2.16

2.18

-3

-4 2.2

Time [s]

Fig. 11.8 Diagram with two curves, each with its own Y-axis

Missing metadata means that measurement data can only be used for a short period of time, as measurements taken longer ago can no longer be assigned to the measurement object and/or the measurement process. In addition, in the event that defective sensors or measurement technology are detected, it is not possible to provide evidence of the affected data records, which would, however, be required in accordance with ISO 9001.

11.5.1 Mathworks *.mat Files These files can be opened for MATLAB® users to read. Self-explanatory logic is required to interpret the data stored in them. The import function of third party software packages for vibration analysis can usually import *.mat files. However, data matrices without metadata are expected there. As a proprietary data format, the measurement and metadata stored in *. mat files are only suitable for data exchange to a very limited extent. If the data processing takes place exclusively via MATLAB®, then a data structure can be generated via the data format struct, which meets the requirements of the quality management and ensures the long-term usability of the measurement data. This file is read with load(filename). The variables stored in the file are transferred to the workspace. Saving variables from the workspace to a mat file is done via save(filename,variables). variables can be a single variable or several variables separated by commas. Alternatively, variables can be saved from the workspace to disk by selecting them in the workspace and right-clicking.

282

11

MATLAB® and Data Formats an Introduction RMS curve B-013 Y-direction 40

60

35

Acceleration [m/s2 ]

Acceleration [m/s2 ]

Time signal B-013 Y-direction 80

40 20 0 -20 -40

25 20 15 10 5

-60 -80

30

0

50

100

150

200

0

250

0

50

100

10

PSD [(m/s2)2/Hz]

1.5

150

200

250

Time [s]

Time [s]

Power spectral density B-013 Y-direction

-3

1

0.5

0 0

100

200

300

400

500

600

700

800

900

Frequency [Hz]

Amplitude collective B-013 Y-direction (Double)amplitudes [m/s2 ]

150

100

50

0

100

101

102

103

Total frequency

Fig. 11.9 Graph with multiple diagrams

104

105

106

1000

11.5

Data Formats for Measurement and Metadata

283

11.5.2 Audio Data Format: WAV The WAV data format is an audio data format which was originally developed from the RIFF audio format defined by Microsoft for the Windows operating system. The further development includes the extension to an arbitrary number of audio tracks, technically considered measurement channels, as well as variability in the sampling rates and higher resolutions than 16 bits-per-sample. MATLAB® supports resolutions up to 64 bits-persample. Audio data has the value range -1.0 to 1.0. This is interpreted as full-scale range or modulation. The single numerical value of the sample therefore does not correspond to the measured voltage. Although MATLAB® supports the storage of value ranges greater than -1.0 to 1.0, which would allow data storage as voltage values or in physical values, this is not supported by third-party software. For a data exchange of measurement datasets in WAV data format, a total of three files are required: the measurement dataset, a calibration dataset and a text file for the metadata. The metadata can be saved in the same way as described in Sect. 11.5.3. Via ð11:26Þ is used to import data from audio files. The audio data is stored in a column-oriented matrix. The sampling rate of the signal is transferred as further information. A time vector or associated time stamps are not available. These must be generated from the length of the audio tracks and the sampling rate itself. ð11:27Þ results in the time vector for the audio tracks read in casablanca. Since the vector starts at the numerical value 0, it is one value longer than the length of the sound track. The time interval between the individual values of the audio track is 1/fs and the vector is lineoriented. With

the vector is column-oriented and truncated by one value (see Fig. 11.10). Writing audio files is done via ð11:28Þ

284

11

MATLAB® and Data Formats an Introduction

Music box Casablanca 85

Sound pressure RMS [dB]

80 75 70 65 60 55 50 45 0

2

4

6

8

10

12

14

16

18

Time [s]

Fig. 11.10 RMS level curve of the imported audio track casablanca

11.5.3 Comma: Separated Values – CSV One of the most common variants of data exchange is via CSV files. CSV (Comma separated Values) is supported by all spreadsheet programs. This is a structured table of values in the form of a text file. Each line of the text table corresponds to one line of the value table. The individual columns are separated by a separator. The separator may not be used elsewhere. If numerical values are used with a comma (,), this must not then also be used as a separator for the columns. In these cases, the semicolon (semicolon, ‘;’) is used. The spreadsheet programs currently offer import dialogs for reading CSV files. The separator and the decimal point are defined in these dialogs. MATLAB® offers the direct reading of CSV files with the instruction M = csvread (filename). Here, the notation of the CSV file must match the source. Commas are expected as column separators, the dot as decimal separator and the column headers must not contain any other information about the data. The use of csvread for the data import of CSV files is therefore possible in exceptional cases. Double-clicking on a file with the extension .csv opens a dialog (Fig. 11.11) in MATLAB® for importing data sets. The file CSVDemodaten.csv has the following structure: Date, tester, test object, sensor 03/17/2018, Armin Rohnen; Munich University of Applied Sciences, Measurement data for CSV import, PCB-X M353B17 SN59061 B-022. Time, Data Seconds, m/s^2

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Fig. 11.11 Dialog for data import with the opened example data set CSVDemodaten.csv

5573e-05,0.12268 0.00096149,0.19735 0.0019549,0.16732 0.0029249,0.19346 ... It consists of two table areas. The first table is used for the metadata, while the second table contains the measurement data set. The import dialog of MATLAB® allows the configuration of the data import. In addition to the separator, the import range and the data format in the workspace (output type) can be set interactively. In the example, the Output Type was changed to Cell Array. The data import is performed by selecting the Import Selection. Instead of importing data directly (Import Data), you can also generate a MATLAB® function or MATLAB® script. For recurring tasks, one of the latter two variants is recommended. A commented and relatively long program code is generated, which can be shortened and optimized for further use.

ð11:29Þ defines the file name, the end of the import area and the format specification for the data import. %S stands here for the import of text. Particularly in connection with the variety in the format specification, it is helpful to first generate the required program code using an exemplary data set via the MATLAB® data import dialog. Via fileID2 = fopen(filename,′r’); the read access to the file takes place. The data import itself takes place via the statement

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Fig. 11.12 Data import of the measured values from the sample data set CSVDemodaten.csv

ð11:30Þ The file is then closed via fclose(fileID);. ð11:31Þ generates the desired data matrix and with ð11:32Þ the variables that are no longer required are removed from the workspace. To import the measurement data, proceed in the same way as for the import of the metadata (Fig. 11.12). However, the labeled column headers cannot be imported, these

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would generate error messages when using the data later. Instead of a matrix with two columns, two vectors with the names Time and Data could also be imported. The automatically generated program code shows differences to the previous one only in a few sequences. Instead of the variable endRow, startRow is required and formatSpec now has a different content. The data is available for further use in the matrix CSVDemodata. For the data export, the metadata should be available in a string matrix (matrix of character strings). For larger data matrices, the structure of a matrix for the column headers, such as 1 Time Seconds

2 Measuring point 1 m/s^2

3 Measuring point 2 m/s^2

4 Measuring point 3 N

5 Measuring point 4 m

is also advisable as a string matrix. This additionally facilitates orientation within the data matrix.

ð11:33Þ Opens the one file for CSV data export and defines as output format a string followed by the end of line. The statement sequence

ð11:34Þ

creates a CSV line from each matrix line and writes it to the CSV file. The individual elements of the matrix row are separated by commas. This allows the metadata matrix to have any structure. The metadata matrix is transferred 1:1 to the CSV file. There should be one or more free lines between metadata and measurement data in the CSV file. This is achieved by fprintf(fileID,formatSpec,“);.

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For the column headers of the data matrix, the same procedure is followed as before. For simple data matrices, as in the example data, this can also be done, for example, using ð11:35Þ be made in direct instruction. The CSV file must now be closed via fclose(fileID);. The statement ð11:36Þ finally appends the data to be exported to the prepared CSV file.

11.5.4 Universal Data Format: UFF The Universal Data Format (UFF, Universal File Format, UF Format) was originally developed by the Structural Dynamics Research Corporation (SDRC) at the University of Cincinnati in the late 1960s to early 1970s. The goal of the development was a data format for data exchange between Computer Aided Design (CAD) and Computer Aided Test (CAT) [9]. Meanwhile, the UF format for data import/export is included in many software products for vibration analysis and acoustics. However, a defacto standard, especially a defined standard as stated by the originators of the data format, cannot be assumed. The individual technical realizations and interpretations of the format descriptions are too diverse for this. For data exchange on the basis of the Universal Data Format, an agreement must therefore be reached. The original definition of the Universal File Format describes a Universal File Set, which is stored in a text file with the extension .unv. This file set contains several individual data sets in different UF file formats. Of the more than 3000 different UF file formats, two are needed to build a Universal File Set for vibration measurement and analysis: Type 151 for describing the test object and the measuring points (UFF 151) Type 58 for measurement and analysis data (UFF 58) The main feature of the Universal File Format is the structured storage of measurement and analysis data, which is mainly carried out via the Type 58 format. This is contained in Universal File Sets as often as required. Data exchange based exclusively on Type 58 is also common. These file sets often contain a single UFF 58 data set, which does not correspond to the original definition of the Universal Data Format.

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MATLAB® does not provide functions for reading and writing the Universal Data Format directly. In the MathWorks File Exchange [5], a platform for the exchange of MATLAB® code, the functions readuff and writeuff are available for download by Primoz Cermelj [4]. Data Exchange Via UFF 58 File The MATLAB® script UFF58_Writer.m from the supplementary materials performs such a data export. It can be adapted for your own purposes. With the statement ð11:37Þ the actual data export takes place. It must first be clarified whether a file may contain several UFF 58 data sets. If so, the script for creating the UffDataSet can be adopted as it is. The filesets stored in UffDataSet are stored in the file DemoUFF58.uff. The parameter replace ensures that if a file already exists, it is overwritten. The parameter add would append this to the end of the file. By definition, a UFF 58 data set contains measurement data for the ordinate from a channel and associated data for the reference variable (time, speed, etc.) of the abscissa. Both vectors must be of the same length. In the case of multi-channel measurements, this leads to the corresponding duplication of the vector for the abscissa and increases the file size, but enables the transmission of channel-specific information. A Universal Data Fileset (UffDataSet{inc}) consists of one or more data structures that describe the fileset used. Some parameters are needed to control the data storage and are therefore mandatory. This includes UffDataSet{inc}.dsType = 58 as specification of the data format and UffDataSet{inc}.binary = 0 for the specification of the data storage in binary format (=1) or not (=0). A storage in binary format reduces the file size considerably, but requires the data receiver to be able to interpret this binary format. In UffDataSet{inc}.x the data of the reference variable are stored in UffDataSet{inc}.measData the measurement data. With UffDataSet{inc}.d1 and UffDataSet{inc}.d2 two fields with 80 characters each are available as descriptive information. Overlengths of these fields are truncated to the nominal length. The same applies to the optional fields UffDataSet{inc}.ID_4 and UffDataSet{inc}. ID_5. More than 4 lines of 80 characters each of descriptive information are not available in the UFF 58 data format.

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Other mandatory parameters:

Optional fields are also provided for the description of the ordinate with the same meaning as for the abscissa. UffDataSet{inc}.ordinateAxisLabel UffDataSet{inc}.ordDataChar UffDataSet{inc}.ordinateNumUnitsLabel A third axis can also be transmitted. This takes place via

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UffDataSet{inc}.zUnitsLabel – for the description UffDataSet{inc}.zAxisValue – for the values Data Exchange/Data Archiving Via Universal File Set If an entire Universal File Set is created as a UNV file, it is possible to store more extensive metadata in addition to measurement data. For better orientation within the UffDataSet data structure, a consistent sequence of use should always be aimed for. The definition of the Universal File Format does not assume the need for longer description texts. All text fields of the UFF are limited to one line with 80 characters. However, the practice in everyday measurement and the requirements by the quality management have shown that this is not sufficient. More detailed metadata than originally intended is required for the unambiguous assignment of measurement and analysis data. This problem can be circumvented by using several type 151 (UFF 51) data sets. Thetype151recordconsistsofthefields ð11:38Þ

A content check, except in the dsType field, does not take place when the file set is created. All fields must be present but may be empty, otherwise error messages from writeuff will occur. This opens up the possibility of a freely definable metadata field structure via the . modelName and .description fields. For each metadata field a record type 151 is created in the file set. For example: ð11:39Þ as an indication of the expected number of type 151 records for the metadata. ð11:40Þ etc.

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The sensor and measurement position descriptions as well as the calibration data can also be stored via additional UFF 151 sets. In the example of the exhaust system from the additional materials, a data file results for each measurement as a universal file set with 127 individual file sets. This is available as a cell array in the MATLAB® workspace.

ð11:41Þ

The nomenclature described above enables orientation within the cell array. The first file set contains the number of metadata fields, which is also the number of associated file sets.

ð11:42Þ

Direct access to the variable is via

ð11:43Þ In this example, the metadata for the measurement and object description follow from File Set 2 to File Set 7. Between the metadata and the measuring point descriptions, another file set is positioned which contains the global information for the number of measuring channels of this measurement data.

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Via

ð11:44Þ

the direct output of this information takes place. A measuring channel/measuring point is described by 7 file sets. Now 15 * 7 further UFF 151 file sets follow. Here is the description of the first measuring channel:

ð11:45Þ

For the calculation of the index of the measurement data of the first measurement channel, the information of the description fields from the first file set, number of UFF 151 file sets for metadata, and from the corresponding file set with the information on the measurement channels is required. It makes sense to store this index data in variables for the creation of analysis routines.

ð11:46Þ

Access to the data of a measuring channel can thus be carried out in a logically structured manner.

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This creates ð11:47Þ the issue of

Some of the data is redundant to the information in the measurement channel descriptions. This is intended, however, as it allows the measurement data to be interpreted and meaningfully analyzed by outsiders. About the statement sequence

ð11:48Þ

the cell array index of the respective UFF151 file set of the measuring point description is determined. The respective index is stored in a column vector whose length corresponds to the number of measuring points. Figure 11.13 is represented by the statement sequence

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ð11:49Þ

generated.

11.5.5 ASAM ODS Format: ATF/ATFX Due to the economic crisis at the end of the 1980s and beginning of the 1990s, the automotive industry was under great pressure to cut costs and rationalize. This also affected the areas of measurement technology and test automation. The respective software tools were predominantly special and in-house developments with correspondingly proprietary and incompatible data formats. The cooperation of different areas was hindered by the different tools. Data exchange was only possible with difficulty. In 1991, the development managers of the German automotive industry decided to cooperate in this respect and Measuring point B10 Sensor Dysinet-X DA1102-005 SN3973 B-015 40 30

B10 [m/s 2]

20 10 0 -10 -20 -30 -40 2

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Fig. 11.13 X/Y plot of measuring point B10 from the demo data set

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founded the Association for Standardization of Automation and Measuring Systems (ASAM) [2]. In contrast to earlier standardization committees, in which the OEMs unilaterally decided on standards and dictated them to their suppliers, in ASAM the suppliers were included in the standard development as equal partners right from the start. This ensured that their technological know-how was incorporated and that the standards could be implemented in products and services at economically justifiable expense. On December 1, 1998, ASAM e. V. was founded in Stuttgart to legally represent the standards and ensure their dissemination. ASAM ODS (Open Data Services) [1] places the focus of the definition on sustainable storage and retrieval of test data. The standard is primarily used to enable measurement and evaluation systems to generate or process data sets in the form of ODS. The operation of the respective test systems remains unaffected by ODS. A data set that complies with the ODS standard also requires a schema file that contains the description of the data model used within the data set. In particular, when defining the standard, a great deal of attention was paid to the metadata describing the test object, the measurement process and the measurement technique used. During the same period as the development of ASAM ODS, the standards and legal regulations for quality assurance were created, which in turn were also taken into account in ASAM ODS. ASAM ODS describes the data management of measurement and analysis data, while ATF (ASAM ODS Transfer Fileformat) or ATFX (ASAM ODS Transfer Fileformat XML) describes the associated data format. The respective schema files can be used to create a database with all the necessary database tables required for data management. The structure of these files is correspondingly extensive and complex. In practice, it makes sense to define the schema of the metadata before using an ATFX-capable system. A later transfer of an old metadata schema into a newer one is only possible with considerable effort. An ASAM ODS-capable test system can read in any ATF or ATFX data set and display at least the metadata contained therein in a structured manner. The extent to which the respective test system can interpret the measurement and analysis data of the data set and process them accordingly depends on the functional scope of the test system. For example, a test system for engine performance measurements will not be able to analyze vibration measurement data. The ATFX data format is ultimately an XML document that can be processed with any XML reader/XML writer. However, metadata schemas and read/write tools are required for the effective use of this data format.

References 1. ASAM ODS. https://www.asam.net/standards/detail/ods/. Accessed: 8 Apr 2018 2. ASAM, Association for Standardization of Automation and Measuring Systems. https://www. asam.net/. Accessed: 8 Apr 2018

References

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3. Bosl, A.: Einführung in MATLAB/Simulink Berechnung, Programmierung, Simulation. Hanser, München (2018) 4. Cermelj, P: Download MATLAB-Funktionen uffwrite und uffread. https://de.mathworks.com/ matlabcentral/fileexchange/6395-uff-file-reading-and-writing?focused=8851898&tab=function. Accessed: 2 Apr 2018 5. MathWorks File Exchange. https://de.mathworks.com/matlabcentral/fileexchange/. Accessed: 10 Nov 2018 6. Dieter, W.P.: MATLAB® und Simulink® in der Ingenieurpraxis Modellbildung, Berechnung und Simulation (4. Aufl). Springer Vieweg, Wiesbaden (2014). ISBN 978-3-658-06419-8 7. Stein, U.: Programmieren mit MATLAB, Programmiersprache, Grafische Benutzeroberflächen, Anwendungen. Hanser, München. ISBN: 978-3-446-44299-3, 2017 8. Stein, U: Objektorientierte Programmierung mit MATLAB. Hanser, München. ISBN: 978-3-44644298-6, 2015 9. Universal File Formats. http://www.sdrl.uc.edu/sdrl/sdrl/sdrl/sdrl/referenceinfo/ universalfileformats/file-format-storehouse. Accessed: 2 Apr 2018

12

Measurement with MATLAB®

Abstract

Take a quick reading . . . Quickly measure something without first having to create a time-consuming measurement data acquisition, and this please “low budget”! Which measurement technician or engineer has not yet been given this order? Take a Quick Reading . . . Quickly measure something without first having to create a time-consuming measurement data acquisition, and this please “low budget”! Which measurement technician or engineer has not yet been given this order? It is obvious that the common spreadsheet programs fail at this task. A little investment in measuring equipment and measuring software is already required. For MATLAB® and the alternatives such as Octave, FreeMat, Scilab as calculation programs, signal analyzers and for the visualization of measurement series, there is a large amount of literature and help available. So why not use these tools to record the measured values? This is rudimentarily possible via the alternatives to MATLAB® (Octave, FreeMat, Scilab). However, no corresponding functions are available here for simple and quick application. The situation is different with MATLAB®, which supports almost all measurement hardware from well-known manufacturers through the data acquisition and instrument control toolbox. Whether HP-IB, VXI, USB or LAN represents the (continued)

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 T. Kuttner, A. Rohnen, Practice of Vibration Measurement, https://doi.org/10.1007/978-3-658-38463-0_12

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interface to the measured value is ultimately of secondary importance, since MATLAB® finds a way to reach it. In addition to the measurement with the sound card available in most PCs, solutions for the so-called standard measurement tasks current, voltage, temperature and force (strain gauges) are shown. Another paragraph turns to the now widely used IEPE sensor technology and the HP-IB/GPIB hardware still in use, which thanks to inexpensive adapters and MATLAB® integration can still perform extremely well. For the field of acoustic and vibration measurement, measuring with professional audio hardware is a worthwhile alternative. Here, faster and more reliable measurement results are achieved with less expensive measurement hardware than is possible with considerably more expensive measurement hardware. This chapter also provides a solution for the output of the measured voltage, which is unusual in the audio sector. In the sector “Generating and outputting signals” the methods are described, with which excitation signals can be provided synchronously to the measurement data acquisition. Additional material for this chapter can be found at http://schwingungsanalyse. com/Schwingungsanalyse/Kapitel_12.html.

12.1

Measuring with the OnBoard Sound Card

The easiest and fastest way to generate measurement data is to use the on-board sound card of the PC. Every standard PC has a sound card with a microphone input or LineIn input. This can be used for simple measurement tasks. Although the input voltage range is only 1Veff, these are usually relatively high-quality A/D converters with very high sampling rates. In addition to adapter cables to convert from the usual 3.5 mm jack to the BNC commonly used in DAQ, the audio trade provides class 1 measurement microphones which can be used directly on most PC sound cards. The measurement of static quantities is usually not possible, since the sound cards have a high-pass filter in the signal input. The statement. ð12:1Þ determines the configuration of the currently activated audio system of the PC used and returns at least the answer. ð12:2Þ Further analysis of the audio system is done by looking at the system variables.

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Measuring with the OnBoard Sound Card

301

ð12:3Þ

indicates that there is an audio system with two input channels. The access to hardware is done in object-oriented programming (cf. [1]). For the access to the audio system must be done with. ð12:4Þ an object can be created. If the optional parameter ID is omitted, an object is created for the first audio device. The parameter fs stands for the sampling rate, nBits for the number of bits and nChannels for the number of measuring channels. By specifying these parameters, the “audio recorder” is already configured. However, the system has one disadvantage. Whether the configuration made is functional is only determined during use. This can be done via the statement. ð12:5Þ already take place at an earlier point in time. For IO the parameters 1 (Input) and 0 (Output) are available. ID is replaced by the ID of the audio system variable. As a response it returns 1, for functional, and 0 for non-functional. The sequence of instructions.

ð12:6Þ

performs a 20 s measurement with the sound card as the measuring device in the configuration sampling rate = 48,000 readings/second, 24 bit resolution and two measuring channels.

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The measurement data are first temporarily stored in an internal buffer. This buffer must be read out with getaudiodata to obtain the measurement data. Each new call of record overwrites the previous buffer.

12.1.1 Measuring with the OnBoard Sound Card and some Operating Convenience More ease of use and interaction is offered by the second example for measurements with the sound card. ð12:7Þ After cleaning up the workspace, which is mandatory for this example, some variables are set “globally”. The global statement removes the encapsulation of variables and makes them available outside the respective function. Preparation of the measurement:

ð12:8Þ

As in the first example, the PC sound card is created and configured as a measuring device. In order to enable flexible starting, pausing, stopping as well as live visualization of the measurement, the recording must take place in the background and the relevant data must be processed in a cycle that is as definable as possible. With. ð12:9Þ the program function ‘AudioTimerAction’ is assigned to the call ‘Timer’. In this statement, messgeraet represents the audiorecorder object, while AudioTimerAction is the function name of the program function to be called. The function is saved in an m-file of the same name and must be located in the same folder or search path of MATLAB®. Any other location will result in an error message.

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ð12:10Þ sets the period duration for the call ‘Timer’ to one second. The program function AudioTimeAction is now called periodically after one second. This ensures that the processing of the AudioTimerAction program function is sufficiently shorter than one second. ð12:11Þ initializes a position index counter. This is required for extracting the data section to be considered. ð12:12Þ Object handles (pointers to the diagram) are provided for two diagrams, which are used to visualize the measurement data from the AudioTimerAction program function. All formatting and labeling is overwritten when the diagrams are updated and must therefore be done at the time of the update. For the periodic processing of the measurement data, the function AudioTimeAction is created in a separate m-file. ð12:13Þ The first program line of a program function must contain the instruction function followed by the function name. The statement can be extended by the required response or transfer parameters. It is not followed by a closing ‘;’. Likewise, the function is not terminated with the keyword ‘end’. Variables defined as global can only be accessed if this is defined at the start of the function. ð12:14Þ transfers the entire measurement data buffer to the variable samples. For the visualization, however, only the non-displayed range of the measurement data is required, which in this example is described by the interval lastSample+1 to lastSample + fs. In ð12:15Þ the extraction of the not yet displayed measurement data as well as the conversion into physical values takes place.

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ð12:16Þ sets the index for the extraction to a higher value according to the current processing. The respective PSD is to be visualized for the two measurement channels. The operationally fastest way to determine a frequency axis and the PSD from the measurement data is to use the instruction spectrogram, which is considered again in more detail in Sect. 14. 4.12. The function spectrogram requires as input parameters the data vector to be analysed, the window function (in the example Hanning), the number of measured values in the overlap (in the example 0), the number of lines of the two-sided spectrum and the sampling rate (in this order!). spectrogram (compare Sect. 14.4.12) returns n-spectra as answer, thus nominally a three-dimensional result. The vectors of the frequency and time axis are also returned as results. If the block size is set equal to the number of measured values, the result of the function contains only one spectrum. The parameter noverlap must be 0 for this. A Hanning window function is used. In s is the one-sided complex spectrum, f the vector of the frequency axis, t the time axis and psdx the PSD. ð12:17Þ With. ð12:18Þ the visualization of the two PSDs is now carried out.

ð12:19Þ

The remaining instructions are for diagram labeling. In the command window, the measurement can now be started with the instructions.

ð12:20Þ can be controlled.

12.2

Data Acquisition Toolbox™

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Frequency spectrum - measuring channel 1

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Fig. 12.1 Visualization of the frequency spectra as PSD of the two measuring channels used

If the recording of the measurement data is finished, there is initially no more access to the measurement data. These must be brought into the access with messdaten = getaudiodata(messgeraet). The visualization carried out is only of a temporary nature (Fig. 12.1).

12.2

Data Acquisition Toolbox™

The Data Acquisition Toolbox™ provides the functionality required to connect MATLAB® to DAQ hardware. The toolbox supports a variety of DAQ hardware, including USB, PCI, PCI Express®, PXI, and PXI Express devices. The toolbox allows you to configure data acquisition hardware and read data into MATLAB® and Simulink® for immediate analysis. It can also be used to send data over analog and digital output channels provided by the data acquisition hardware. The toolbox data acquisition software has functions to control the analog input, analog output, counter/ timer, and digital I/O subsystems of a DAQ device. You can control device-specific properties and synchronize generated data from multiple devices. One can analyze data as it is obtained or save it for later processing. It is also possible to run automatic tests and iterative updates to the test setup based on the analysis results. Simulink blocks from the toolbox allow data to be streamed live into Simulink models to verify and validate a model against the live measured data as part of a design verification process.

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In order to be able to connect MATLAB® with hardware for measurement data acquisition, measurement hardware itself as well as driver software for the respective acquisition hardware is required in addition to the license for the Toolbox (further information [2–4]). The statement. ð12:21Þ lists the detected measuring devices. Devices are detected which were connected at the time MATLAB® was started and which were able to connect to the operating system via suitable driver software. If this is the case, the following response appears, for example.

ð12:22Þ

The Web site https://de.mathworks.com/products/daq/ provides information about which data acquisition devices are supported by the MATLAB® Data Acquisition Toolbox™. If more information about the data acquisition device is required, double-clicking on the Device ID will help.

ð12:23Þ

The information should now be sufficient to perform a measurement. With. ð12:24Þ

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a session object is created for the measuring device. The configuration of the measuring device is done via the instructions. ð12:25Þ which defines the individual measuring channels and their functionality. In this example, a measurement is configured at the cDAQ base with the name cDAQ2. As measurement channel 1, the input ctr0 is connected to slot 14 as a pulse counter (EdgeCount). The other two measurement channels are located in slot 7 with the connections ai0 and ai1. The former is set up as a voltage measurement, the latter as a voltage measurement with IEPE sensor. The measurement data will be listed in this order in the later measurement data matrix. ð12:26Þ sets the sampling rate to 12,800 readings per second. Since the desired sampling rate is not always set exactly, it is worth reversing the instruction to save the actual sampling rate for later calculations, just to be on the safe side. ð12:27Þ In the further one is now only the definition of the measuring time necessary, which over. ð12:28Þ is done. ð12:29Þ performs the measurement. The time stamps of each measured value are then stored in the Time vector and the associated measurement data in the Data matrix. As already mentioned, the MATLAB® Data Acquisition Toolbox™ supports a wide range of data acquisition hardware that is available on the market as USB, PCI, PCI Express®, PXI or PXI-Express devices. For the respective support, suitable software drivers must be provided by the DAQ hardware manufacturer. Since there are other manufacturer-independent platforms for the software realization of measurement tasks in addition to MATLAB®, this is generally the case outside of special applications. At http://de.mathworks.com/hardware-support.html MathWorks® lists the supported hardware with program examples. General help on measuring with MATLAB® can be found at http://de.mathworks.com/ help/daq/.

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The Data Acquisition Toolbox™ provides methods for the processing of measurement tasks across all hardware platforms. Analog Input/Analog Output (addAnalogInputChannel, addAnalogOutputChannel) Digital Input/Digital Output (addDigitalChannel) Counter Input/Timer Input/Timer Output (addCounterInputChannel, addCounterOutputChannel) Multichannel Audio Input/Audio Output (addAudioInputChannel, addAudioOutputChannel) Periodic Waveform Generation (addFunctionGeneratorChannel) Synchronized simultaneous processes (addTriggerConnection, addClockConnection) Basically, the method library distinguishes between measurement in the foreground, which generates measured values quickly and very simply, and measurement in the background, which contributes to the solution of complex measurement tasks and enables graphical user interfaces (GUI).

12.2.1 Data Acquisition Toolbox™ and Improved Ease of Use A little more ease of use is achieved by using the methods for the measurement in the background. To avoid having to make changes in the program code for each measurement configuration, it is also useful to be able to control the specific configuration via a configuration file. Start/stop of the measurement should be done by simple inputs or keystrokes. For the control of the measurement setup as well as for the general overview, a live visualization of the measurement data and a measurement pre-run should be realized. The program code listed below is required as the main function or main program. A description of the functions used follows. ð12:30Þ carries out a configuration of the measurement. The channel assignment must be stored in the file “setuptabelle.csv” in the sequence – device, channel, type, name, sensor, EU, calibration value, unit, X coordinate, Y coordinate, Z coordinate. The individual entries must be separated by a comma and a point is required as the decimal point. This corresponds to the original definition of CSV files (comma separated values).

ð12:31Þ

forces a text input, which is used as description of the measurement.

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ð12:32Þ forces another text input, which is used as file name for saving the configuration and the measurement data. ð12:33Þ saves the configuration description of the measurement. ð12:34Þ opens the write access to the file for the measurement data. ð12:35Þ the encoder configuration is set to continuous measurement. ð12:36Þ calls the function for preparing the live visualization. The pointers to the individual diagrams are stored in levelHandle. With status = 0; a variable is preset to control the further program sequence. Here, 0 stands for the measurement run-up, 1 for the measurement and 2 for the termination of the measurement. ð12:37Þ The DAQ object provides a function call ‘DataAvailable’. The number of measured values until the function call (event) is triggered is stored in the variable NotifyWhenDataAvailableExceeds. This must be an integer value. Since the sampling rate (measured values per second) is stored in messung.fs, messung.fs/3 causes the event to be triggered every 1/3 s. The processing of the measurement data accumulated in the background is carried out by a processing function. The reference to the processing function is made via the addlistener instruction. ð12:38Þ sets a listener which reacts to the event DataAvailable. @(src, event)dispData(. . .) specifies which function should be executed in response to the event.

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ð12:39Þ starts the measurement in the background. A successful setup of the DAQ hardware and an activated listener for processing the accumulated measurement data is a prerequisite for the error-free running of the background measurement. ð12:40Þ as long as status is not set to 2, the query loop is run through endlessly. ð12:41Þ waits for a numerical input. As soon as this is done, the status is changed. This causes the infinite loop to “hang” on the numeric input.

ð12:42Þ

status == 1 stands for the execution of the measurement. In order to get from the measurement pre-run to the measurement recording, another function must be assigned to the event DataAvailable for the processing of the accumulated measurement data. This is done by deleting the previous listener and setting it again with a reference to the corresponding function. This may only be done once and is secured by setting the control variable changed = 1;.

ð12:43Þ

If status = 2 is set, the measurement is terminated. This is done by stopping the configured DAQ hardware, closing the data file and deleting the listener. ð12:44Þ creates the function ‘setup_from_file’, which must be stored in the m-file of the same name. ð12:45Þ

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to make the function (later) more flexible, the file name is assigned to a variable. This makes it possible that the file name can also be passed as a call parameter of the function.

ð12:46Þ

The upper lines of the program code were generated by manually importing a CSV file once and then slightly modified. Double-clicking on the CSV file setuptabelle.csv opens an import dialog in MATLAB®, which can be completed with “Generate Script” or “Generate Function” instead of direct execution. The generated code can then be reduced to the essentials. For the import, the options “Column Delimeters: Comma”, the import range – whereby this is irrelevant for later executions of the program code – and “Cell Array” were selected. ð12:47Þ creates the object messgeraet. ð12:48Þ sets the sampling rate to a desired value and determines the actually set sampling rate. There are often slight differences in this process. ð12:49Þ the number of measuring channels corresponds to the length of the imported data array. Therefore, a structure for the measuring point description is created from 1 to the length of the data array.

ð12:50Þ

It says.

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ð12:51Þ

configures the measuring channel. ð12:52Þ completes the function. ð12:53Þ creates the handles (pointers, access variables) to the diagrams for the live visualization. ð12:54Þ creates a graphic in a row/column logic with a fixed number of eight columns. ð12:55Þ For this, only the corresponding subplot must be created. Everything else will be overwritten at the time of the plot statement. As long as the measurement is in the “Fast forward” mode, the live visualization takes place via the.

ð12:56Þ Time stamp and measurement data are taken over in the vectors time and data. Since no further backup of the data is made, they are lost after the function run! The data is available in the usual column orientation.

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ð12:57Þ

determines the color of the display by determining the current maximum voltage value of the measured values.

ð12:58Þ

The measured voltage values are converted into the physical measured values. For this purpose it is necessary to determine whether the calibration value for mV or V is present. Since only a modulation monitoring takes place, the determined measured values are secured positive by squaring and subsequent formation of the square root. Any other mathematical treatment of the measured values is also possible as long as it can be processed within the time defined by setting the variable NotifyWhenDataAvailableExceeds. If this is not the case, the processing of the function is aborted by the output of an error message and, if necessary, subsequent error messages. Since the function is called continuously, the error messages are repeated for each function call, which leads to confusion. The causal error can be found in the first error message. By ð12:59Þ the measured data is output as a plot. During the “measurement”, the live visualization now takes place via the. ð12:60Þ This differs only in a few lines of code from the previous function. ð12:61Þ also takes over the time stamps and measurement data here.

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ð12:62Þ

sorts the time stamps and measurement data into a streamable data sequence. So that the measurement data can be stored endlessly in a streaming file, these must be resorted in the sequence timestamp, measurement value(1), measurement value(2),. . ., timestamp,. . . This is done in the upper code sequence. ð12:63Þ writes the stream to the streaming file.

ð12:64Þ

is again used for live visualization (Fig. 12.2). For further processing of the measurement data, it is first necessary to sort the data back into the usual column orientation after the measurement has been completed. ð12:65Þ The number of channels for reading in the data file is determined from the configuration file via the number of measuring points plus the time stamp.

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Fig. 12.2 Live visualization of the recorded measured values

ð12:66Þ The structured readout of the data file then takes place. However, the measurement data are now line-oriented and the associated time stamps are located in the first column. This no longer conforms to the data storage as it is done by the Data Acquisition Toolbox™.

ð12:67Þ

sorts the data into the usual column-oriented form and creates a separate time vector. At the same time, the data is converted into physical values. A calibration value in mV is assumed. ð12:68Þ stores the data in the configuration file.

12.2.2 Data Acquisition Toolbox™: Standard Tasks In the examples, National Instruments DAQ boards are used. The procedure is the same for any other supported DAQ card. In order to determine individual measured values for standard measurement tasks, an object must be created for the measurement according to the already known scheme. Depending on the procedure, the measurement channels can be distributed over several objects. The daq.createSession(. . .) instruction is used to create the object first, while addAnalogInputChannel(. . .) defines the measurement channel and configures it in a general way. Further configurations are required for a successful measurement depending on the sensor and measuring card used.

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For the measuring tasks current measurement, voltage measurement, temperature measurement and force measurement a National Instruments measuring module NI 9219 is used. The query daq.getDevices followed by a double click on the corresponding device results in the following response.

ð12:69Þ

With. ð12:70Þ an object is created, which is configured with one measuring channel each for current and voltage measurement. ð12:71Þ creates the two measuring channels. With. ð12:72Þ the Device used can be checked.

ð12:73Þ

The individual measuring channels can be assigned. ð12:74Þ can be assigned a name. The measuring range is defined by. ð12:75Þ

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Of course, the measuring card must also support the desired measuring range. If not, an error message is displayed. To check the settings made, the matrix ch is brought to the output.

ð12:76Þ

With. ð12:77Þ one measured value per measuring channel is determined. ð12:78Þ In the variation. ð12:79Þ the time of recording is determined as a time stamp for the measured values. By setting the parameter CurrentVoltage.Rate = < value > the Device is prepared for a continuous measurement. CurrentVoltage.startForeground or CurrentVoltage. startBackground performs the continuous measurement. Another object is created for the temperature measurement. ð12:80Þ An additional measuring channel is defined to configure the temperature measurement. ð12:81Þ is carrying this out. The set(ch(3)); instruction is used to output the configuration options of the measuring channel.

ð12:82Þ

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The definition of the unit and the thermocouple type is necessary to be able to use the measuring channel. ADCTimingMode is initially set to a default value when the measuring channel is created. ð12:83Þ defines the temperature measurement, which in turn is defined by. ð12:84Þ records a measured value. ð12:85Þ The used DAQ board can be controlled in the effective resolution via the parameter ADCTimingMode. Overall, this is a compromise between effective resolution and speed (sampling rate). High resolution excludes high speed. Since an influence on the measured values by the power supplies used (50 Hz or 60 Hz) cannot be completely ruled out in measurement setups, the corresponding ADCTimingMode is used for the best possible suppression. Best50HzRejection at 50 Hz mains frequency or Best60HzRejection at 60 Hz mains frequency. Force measurements are often carried out via strain gauges or with sensors which are constructed with strain gauges as a bridge circuit. In order to obtain valid measurement results, it is necessary to know the bridge type (1/1, 1/2, 1/4) and the nominal resistance value of the strain gauges. The data sheet of the sensor should be able to provide information on this. A separate object is also created for the force measurement. ð12:86Þ now creates the measuring channel again. ð12:87Þ again provides the information required for the configuration of the measuring channel.

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ð12:88Þ

defines the sensor as a 1/2-bridge with internal bridge supply at 2.5 V and a nominal strain gauge resistance value of 120 Ω. Via ð12:89Þ the timing mode is set to best 50 Hz suppression and the measuring range to 10 mV/V. Measured values are displayed as usual via. ð12:90Þ determined.

As a result, values are recorded in [mV/V] by multiplying by 1000. To obtain physical values in [N], the determined measured value must be offset with a calibration value or calibration function. Vibration sensors (acceleration, force) based on the piezoelectric effect are often used for vibration analysis. IEPE (see Sect. 8.1) has established itself as the main circuit, which requires the transducer to be supplied with a constant current of 2 mA . . . 20 mA (usual: 2 mA or 4 mA). A separate object is also created for vibration measurement. ð12:91Þ and via. ð12:92Þ a measuring channel is defined, which is.

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ð12:93Þ is configured. This device does not allow an inputSingleScan. Which makes little sense for a vibration measurement. With. ð12:94Þ a sampling rate is defined and the actual sampling rate set in the device is determined. By setting the parameters. ð12:95Þ or ð12:96Þ the duration (DurationInSeconds) or the number of measured values to be acquired (NumberOfScans) is set. Both parameters are dependent on each other via the sampling rate (Rate). ð12:97Þ performs the measurement.

12.2.3 Instrument Control Toolbox™: HP-IB This is a computer interface (bus) that was developed in the 1960s by Hewlett-Packard (HP) as HP-IB. The bus connector defined by HP is the widely used standard. However, other variants are possible and quite common. The HP-IB bus was standardized as IEEE488 and adopted by the IEC as IEC-625. For today’s PCs there are USB interfaces that allow access to HP-IB/IEEE488 bus. The bus is defined as a parallel 8-bit bus. Up to 15 devices (measuring devices, plotters, printers) can be connected. One of the 30 possible addresses must be assigned once to each connected device. The device address space allows the connection of up to 30 devices, but the specification only allows the physical connection of 15 devices per bus. For smooth communication between the PC and the connected devices, only one of the connected devices may send data at any one time. However, the data can be sent to several devices, as all other connected devices are allowed to read from the bus at the same time. With regard to the data transmission rate, it should be borne in mind that when HP defined this in the 1960s, the focus was on very secure device communication that

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precluded data loss. A 3-phase handshake (ready/data → valid/data → accepted) was defined. This means that the slowest device on the bus deliberately determines the secured data transfer rate. The defined standard only defines the transmission of data, not the commands for controlling peripheral devices. For this purpose an additional protocol or commands are required, which are usually transmitted as text to the peripheral device. The commands or the protocol must be composed by the control PC. The easiest way to test the communication between MATLAB® and an instrument connected to the HP-IB bus is to use the Instrument Control Toolbox™. This is the easiest way to test the functionality and commands. This is the easiest way to test the functionality and commands. In case of wrong commands, which can block any further communication, a reset is possible. The Instrument Control Toolbox™ opens with a window (Fig. 12.3), which is divided into three sections. Left part “1” contains a structure similar to a file browser. Here, instead of files, communication interfaces are listed, such as Serial, USB, Bluetooth, etc. If these entries are clicked with the mouse, they branch out further and the connected devices become visible. The scan button in the middle area “2” can be used to search for connected devices on the selected interface. In this area, a list of the connected devices appears at the latest after the scan, which contains more detailed information such as device name, device type and serial number. Help functions are provided via the right-hand field “3”. In this example, the device connected to the GPIB ni-Board-0 interface with address 26 is an HP8904A Opts. 02 with firmware revision 00709 A and serial number 05270. Double-click on ni-Board-0 in the right-hand window to access the devices connected to it. If you select the entry for the HP 8904 A, a multifunction synthesizer, the display changes in the middle area. Here the communication with the selected instrument can now be tested. The necessary device instructions must be taken from the respective device manual. If the communication has been tested successfully, the required MATLAB® code can be transferred from the Instrument Control Toolbox™. The MATLAB® code generated by the manual test is located in the middle area behind the Session Log tab (Fig. 12.4). ð12:98Þ checks whether a GPIB object already exists. If this is not the case, then the GPIB object is created with. ð12:99Þ is created. Otherwise the existing GPIB object is closed for access and made available for further use.

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Fig. 12.3 Test & Measurement Tool of the MATLAB Instrument Control Toolbox™®

Fig. 12.4 The MATLAB® code generated by the Test & Measurement Tool of the Instrument Control Toolbox™ during manual testing

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ð12:100Þ The GPIB object is now available and communication can take place with the measuring device addressed by it, in the example an HP8904A signal synthesizer. The communication itself is done by means of fprintf and fscanf, just like accessing files. ð12:101Þ opens the communication to the GPIB device. ð12:102Þ instructs the device to perform a reset. ð12:103Þ sets the device to configuration mode. ð12:104Þ activates an OUTPUT channel.

ð12:105Þ defines an output signal (sine) with 0° phase angle, a frequency of 0 Hz at an amplitude of 0 mV. This prepares the HP signal synthesizer for further use. In fsoll and amplitude, a target frequency and target amplitude are calculated in a higherlevel function, which are to be generated via the signal synthesizer. Since the communication to the device itself takes place via strings, the calculated numerical values must be converted into strings via num2str. Via. ð12:106Þ the GPIB device is instructed to generate a signal with the frequency fsoll in [Hz] and the amplitude amplitude in [mV].

324

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Data Acquisition Toolbox™ in Conjunction with Professional Audio Hardware

Professional audio hardware refers to sound cards and A/D converters that are intended for professional use. A special feature of this hardware is the consistent design for a high number of sample-synchronous inputs and the synchronization of several devices with each other. They thus offer the best prerequisites for metrological use in vibration analysis and acoustics. Professional audio hardware is characterized by the voltage ranges of the signal transducers commonly used in musical instruments, which are specified in dBu. Manufacturers offer devices with input ranges of 0 dBu,1 +10 dBu, +19 dBu as well as +21 dBu (and more), each with several dB (quite 15 dB) headroom. Phantom power typically provides a 48 ± 4 V supply voltage. However, this requires balanced cabling, which is implemented on the hardware side in the form of 6.3 mm threepin jack and XLR connectors. The positive pole of the supply voltage is connected to the two signal lines of the balanced cabling via two exactly paired decoupling resistors. The shielding of the line is used as the negative pole. The two decoupling resistors prevent the short circuit and limit the current. Thus, there is no voltage between the two signal lines – hence the term phantom power. Due to the symmetry, any interference has no effect on the signals. Phantom power is used for signal pickups in musical instruments, studio microphones and measuring microphones. Since professional audio hardware is designed for balanced cabling, it is necessary to adapt the coaxial cable to the asymmetry that is common in measurement cabling. For this purpose, the 6.3 mm jack sockets used on some devices are already prepared for the use of two-pole (mono) jack plugs. For the three-pole XLR connector, adapters are available which bridge PIN 3 with PIN 1. Phantom power cannot be used with coaxial cabling. In order to realize complete systems with far more than 100 recording channels, professional audio hardware has several communication and data interfaces as well as a system synchronization. External audio devices are connected to the PC via USB 2 or USB 3 or via FireWire. They thus allow a data transfer seemingly limited by the PC interface used. For the possible number of measuring channels, in audio language audio tracks, in relation to the sampling rates used, the matrix shown in Table 12.1 results. The A/D conversion itself is always done in 24 bit in professional audio. In practice, such channel numbers are not realized via a single terminal device. Several professional audio hardware are linked together. This requires at least one device, the master, which establishes the connection between PC and professional audio hardware via one of the possible PC interfaces.

1

dBu: Voltage level with the reference value 0.7746 mV.

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Table 12.1 Theoretically possible channel numbers (audio tracks) for measuring systems based on professional audio hardware in relation to the PC interface used Sampling rate Real transmission rate 192 kHz 96 kHz 48 kHz 44.1 kHz

USB 2 35 MByte/s 61 122 243 265

USB 3 275 MByte/s 478 955 1910 2079

FireWire 50 MByte/s 87 174 347 378

The master itself has its own analog inputs and outputs. It provides a signal for the sampling synchronization for the integration of further A/D converters via the “Word Clock”. The data stream is forwarded to the PC via the digital inputs of the master. Different interfaces are available for data transfer between the audio devices. AES/EBU The term AES/EBU originally stands for Audio Engineering Society/European Broadcast Union. The term is also used for the two-channel digital transmission interface specified by the AES/EBU. The interface is defined for two tracks (channels) with the sampling rates 32 kHz, 44.1 kHz, 48 kHz, 88.2 kHz, 96 kHz and 192 kHz at 16 or 24 bit. S/PDIF Electrical/Optical The term stands for Sony/Philips Digital Interface and defines a digital audio interface from the consumer electronics sector. Except for the cables/ connectors used and the interface levels, the definition largely corresponds to the AES/EBU interface. The electrical interface is a coaxial RCA cable, optical TOSLINK (TOShiba-LINK). ADAT Also the term ADAT stands causally for something else than it is common in the meantime. ADAT (Alesis Digital Audio Tape) was introduced in 1992 by the company of the same name. With digital audio tape it was possible to record multitrack audio signals digitally on S-VHS tapes. By necessity, a digital interface (ADAT interface) was required for the transfer of the audio data. The original definition of ADAT refers to 8 audio tracks in 16 bit resolution at a sampling rate of 48 kHz. A farsighted decision in the interface definition was that the actual data transfer is in 24 bit resolution, which makes it still usable. Signals with sampling rates greater than 48 kHz are transmitted using the S/MUX protocol (sample multiplexing). In this protocol, the data streams are fragmented and divided into several audio tracks. This reduces the number of actually available audio tracks to 4 at sampling rates of 96 kHz or to 2 at 192 kHz. Signal transmission is usually optical.

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TDIF (TASCAM Digital Interface) This term stands for TASCAM Digital Interface and also refers to a digital interface definition. This was developed by TASCAM for direct transfer between digital tape recorders. The transmission is done electrically via a 25-pin D-SUB cable, which may only be a few meters long. MADI This interface is standardized by the AES (Audio Engineering Society) as AES10 for multi-channel transmission of audio data. MADI stands for Multichannel Audio Digital Interface. The signal transmission is done electrically via 75 Ω coaxial cable or optically. MADI can transmit up to 64 audio tracks at a sampling rate of 48 kHz. For higher sample rates than 48 kHz, the S/MUX protocol with corresponding audio track reduction is used here as well. Depending on the audio interface equipment, a professional audio device (master) can be extended by a considerable number of audio tracks. The RME Fireface 802 used in this example has four XLR/jack inputs with phantom power (XLR) with input level + 10 dBu (XLR) or + 21 dBu (jack) at the front. At the rear, the RME Fireface 802 has eight analog inputs via TRS jacks with input level + 19 dBu. This results in 12 analog inputs, which can be extended up to 30 via the digital audio interfaces. Two ADAT and one AES/EBU interface are available for this purpose. If further devices are added via the digital audio interfaces, there must be a master in the device network which provides the synchronisation via the word clock. All other devices in the network must use the word clock as slaves. There are as many analog and digital outputs available as there are inputs. The favorable channel price of these systems with a usable frequency range of 1 Hz to 75 kHz speaks for the metrological use of professional audio hardware. However, DC voltage signals cannot be recorded with audio hardware. The Support Package for Windows Sound Cards is required to operate professional audio hardware in conjunction with the MATLAB® Data Acquisition Control Toolbox™. After installing the Support Package, professional audio hardware is available for acquiring measurement data just like any other supported measurement hardware. For the RME Fireface 802 used in this example, the query daq.getDevices gives the output (excerpt):

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ð12:107Þ This lists the internal audio devices as well as the RME Fireface 802. The sequence of the Device IDs is unfortunately not in the logical order of the inputs and outputs of the device. Which Device ID is assigned to an input and which one is assigned to an output can only be seen after a closer look at the corresponding information. Another disadvantage is that the assignment of Device ID to device channel is not permanently saved. This is a potential source of error when using professional audio hardware. Via

ð12:108Þ the object for the measuring device is created as usual and the measuring channels are defined. The (audio) measuring device object provides functionality for configuration and measurement data acquisition as before. An Audio Device always consists of two channels. This arises from the logic that they are stereo audio channels, i.e. two audio tracks. Each audio track can be defined individually as a measuring channel. addAudioInputChannel(messgeraet,‘Audio16’, 1:1) is another possibility to define a measuring channel. addAudioInputChannel(messgeraet,‘Audio16’, 1:2) on the other hand defines both audio tracks of the Audio Device as measuring channels.

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normalized measured values in the range -1 ... 1

Measurement with RME Fireface 802

0.2

0.1

0

-0.1

-0.2

-0.3 1

1.05

1.1

1.15

1.2

1.25

1.3

Time in s

Fig. 12.5 Measurement of an accelerometer signal (10 m/s2 at 60 Hz) with RME Fireface 802. The applied voltage signal produces an approx. 20% modulation of the input channel

The messgeraet.StandardSampleRates instruction lists the available sample rates. The sample rate is set via messgeraet.Rate = 47,250. If this is a value that is not supported as a sample rate, the next supported sample rate is set. Via fs = messgeraet.Rate the actually set sampling rate should be determined. Via ð12:109Þ a measurement with a duration of 30 s is carried out. The list of configured measuring channels shows that they are defined in the range - 1 . . . +1.

ð12:110Þ

The type designation audi stands for Audio Input, while audo is used for Audio Output. As shown, audio signals have no reference to the input level. Audio hardware determines a “normalized input value” between -1 and + 1, which refers to the (full) level of the input channel used (see Fig. 12.5). In order to be able to determine the applied voltage from this, the measured value must be multiplied by the maximum input level of the respective input used. However, this cannot be determined beyond doubt from the

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retrievable device information. Possible settings of the input gains are also not detected by the configured measuring system. This is another potential source of error, which is not specific to professional audio hardware. Also in other measurement systems the relation between the determined (numerical) value and the applied signal voltage or physical measured value is not always clear. Reference value measurement has become established in practical use. For this purpose, the sensor is applied to a signal calibrator and the transfer factor between the physical value of the calibrator and the modulation value is determined, which converts the measured values into physical values in the further processing of the measurements.

12.4

Generating Signals with the Data Acquisition Toolbox™

To output a DC voltage, only one signal output value is required. Signal output boards that are suitable for DC voltage output will continue to output this voltage value until a new voltage value overwrites the previous output or the device is switched off. For the output of voltage waveforms or vibration signals, the signal must be generated beforehand. This is assigned to an output buffer during subsequent output. For the mathematical signal generator, some parameters are defined first, which must correspond to system parameters for a real signal. ð12:111Þ defines the sampling rate. This must be equal to the sampling rate of the measurement. ð12:112Þ defines the time length of the generated signal. If this is longer than the output buffer allows, a streaming mechanism must be used for the output. ð12:113Þ defines the reference frequency and reference amplitude. ð12:114Þ forms the required time vector. ð12:115Þ calculates the output signal with freely defined frequency and amplitude.

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Generate signals

1.5

1

Amplitude [unit]

0.5

0

-0.5

-1

-1.5

4

4.002

4.004

4.006

4.008

4.01

4.012

4.014

4.016

4.018

4.02

Time [s]

Fig. 12.6 Generated sinusoidal signal (detail) with nominal frequency = 125 Hz and nominal amplitude = 1.345 units

ð12:116Þ For the later signal output the values are expected in a matrix with column orientation, therefore the matrix is transposed (signal section see Fig. 12.6). Another option that MATLAB® provides directly for signal generation is the chirp function. This generates a signal vector with a continuous frequency change. ð12:117Þ Without the parameter ‘logarithmic’ the vector is generated with linear frequency change, with ‘logarithmic’ it becomes a logarithmic frequency change (Fig. 12.7). To be able to solve the task of continuous signal output and simultaneous execution of a measurement, the event processing of MATLAB® is used. The data acquisition object provides a total of four events that can be linked with program code to control this task. The events DataAvailable and DataRequired are used for this purpose. Since two endless loops are actually set up here, a program control is required (Fig. 12.8), which, among other things, also terminates the event processing again. The control for the event DataAvailable has already been used to continuously store measurement data on the hard disk.

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Generate signals lin-chirp 0.09 0.08 0.07 2

0.06 0.05

4

0.04

6

0.03 0.1

8

0.02

0.05 0

10 0 500 1000 Frequency (Hz)

Time

0.01

12 1500

Generate signals log-chirp 0.11 0.1 0.09 0.08 2 4

0.07 0.06 0.05

6 0.1

0.04 0.03

8

0.05

0.02

0

10 0 500

0.01

12

1000 Frequency (Hz)

Time

1500

Fig. 12.7 Spectrograms of signals generated with linear chirp (left) and logarithmic chirp (right)

In this example (setup_and_main.m) an output signal with fixed frequency and amplitude is to be generated continuously and output without gaps. At the same time, a measurement with two measuring channels is to take place.

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Signal generator + measurement Setup - Measuring device - Output device - Events DataRequired

DataAvalaible

Generate signal

visualize data

OutputQueue

back up Control

Fig. 12.8 Schematic representation of the program design for the task “Output signal” with simultaneous measurement

First, the Data Acquisition Object is created and the measurement channels are defined. In this example, the measuring device is an RME Fireface 802.

ð12:118Þ

Analog In1 and Analog In2 are used for this purpose. An accelerometer is connected to channel 1, while channel 2 is shorted to the output channel for signal output control. Both channels are given corresponding designations. For the signal output. ð12:119Þ the Analog Out1 is defined as output channel. The first eight output channels can be found under the device “DirectSound Loudspeaker” (see 12.107). ð12:120Þ defines, as already known, the sampling rate and ensures that the actually used sampling rate is known.

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ð12:121Þ sets the target frequency and target amplitude. ð12:122Þ The output buffer is to be filled with output signal for one second. The output signal is generated in the program function generator and stored in the output buffer. ð12:123Þ Via ð12:124Þ the references to the output diagrams are provided. ð12:125Þ switches to continuous mode. ð12:126Þ defines the number of acquired measured values per measuring channel until the DataAvailable event is triggered. The value must be an integer. ð12:127Þ specifies that the program function dispData is to be called when the event DataAvailable has been triggered. This line of program code controls the measurement. ð12:128Þ specifies that the generator function is to be called when the DataRequired event is triggered. This line of program code controls the signal output. ð12:129Þ starts signal output and measurement. To stop the process, the reactions to the two events must be cancelled and the measuring device stopped again. Via

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generated signal (start) 1

Amplitude

0.5

0

-0.5

-1 0

0.02

0.04

0.06

0.08

0.1

0.98

1

Time [s] generated signal (end) 1

Amplitude

0.5

0

-0.5

-1 0.9

0.92

0.94

0.96

Time [s] Fig. 12.9 Signal sections at the beginning and at the end of the generated signal. The signal starts at time 0 with amplitude 0 and must end in such a way that the next value of the signal would be 0 again

12.4

Generating Signals with the Data Acquisition Toolbox™

335

ð12:130Þ

is used to control the program. If the number 2 is entered via the keypad, the program stops. ð12:131Þ The output signal should be continuous and without jumps. This requires a generated signal at the end of which there is an end of period (Fig. 12.9). For this purpose, the integer number of possible periods for the nominal signal length is determined and shortened by a period duration. ð12:132Þ determines the time vector which is needed for the calculation of the signal. ð12:133Þ generates the output signal and makes it available in column orientation. ð12:134Þ writes the signal into the output buffer. ð12:135Þ provides the handles (pointers) to the output diagrams. For this purpose, a subplot is created for each measuring channel. This corresponds to the number of channels in the measuring device -1, since the last channel in the measuring device is the output channel. ð12:136Þ The last required function visualizes the measurement data. A storage of the measurement data is omitted in this example. This functionality has already been described above.

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0.15

0.15

0.1

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0

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-0.1

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112.75

-0.15 112.6

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Fig. 12.10 Live visualization of the recorded measured values

ð12:137Þ takes over the time stamps and recorded measurement data. The matrix data contains the same number of columns as the number of measuring channels defined.

ð12:138Þ

visualizes the measurement data, sets the font size to 20 point and gives the diagrams a title label (Fig. 12.10). If the signal to be output is too complex for a runtime calculation, the streaming method can be used to output a previously created or otherwise generated signal. Only a few changes to the previous program code are required for this. In the main program (setup_and_main) two global variables are defined directly in the first lines of code. ð12:139Þ

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337

Instead of the signal definition and the generator function, the output signal (in the example) is read into the workspace with load(′demosignal.mat′);. In the following lines, the function for signal streaming is configured and called once. ð12:140Þ is the position pointer for output streaming. This is at the beginning of the signal at the start of the program. ð12:141Þ the signal output should again take place in 1 s blocks. ð12:142Þ calls the function for output streaming once. Also in the following lines changes were made, which serve up to the output listener of a not compellingly necessary changed live visualization. ð12:143Þ has now been set to a value that leads to a live visualization every 500 ms. ð12:144Þ now calls the dispFT function for live visualization. ð12:145Þ calls the streamer function for the signal output. This function is responsible for the blocking of the output signal in the output buffer. ð12:146Þ The position pointer pos and the output signal were defined as global variables. For the use of these, a global definition must also be made in the function. Otherwise they would be variables that are only valid within the respective function call. ð12:147Þ calculates the number of values that the output buffer is long.

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The output of the signal is to be repeated until the program is terminated by the user. Since it is also unlikely that the end of the signal will coincide with an end of the buffer, a query for the position pointer is required. ð12:148Þ if the signal piece to be transferred would go beyond the signal end, which leads to a program abort, it is necessary to jump back to the signal start. The output buffer is then filled up with signal values from the start of the signal to its nominal length. Pos contains the position pointer, which marks the value position in the signal that is to be transferred next to the output buffer. ð12:149Þ the residual signal is taken over first. ð12:150Þ determines the length of the residual signal. ð12:151Þ determines the required residual length by which the output signal must be padded by values from the signal start. ð12:152Þ Fills the output signal with values from the beginning of the signal and sets the position pointer to the next following value position in the signal.

If the end of the signal is not yet reached, a signal piece n signal values long is transferred to the output buffer starting at position pointer pos. The position pointer is then reset. ð12:153Þ With.

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ð12:154Þ the output buffer is written to the output queue. The next time the function is called, the next block of signal values is written to the output queue, starting at position pos. The magnitude spectra of the measured values are to be displayed in the live visualization.

ð12:155Þ takes over the time stamps and the measurement data. ð12:156Þ A Fourier transformation of the measurement data is performed for each measurement channel.

ð12:157Þ A vector for the frequency scaling of the diagram is calculated from the relationship between the sampling rate fs and the block length L. A Fourier transformation yields a complex two-sided spectrum as a result. The spectrum is mirrored on the frequency axis at 0 Hz. Only the positive part of the spectrum is required for a display. The value at 0 Hz is the DC part of the signal. The abs function calculates the magnitude values from a complex vector. Since the result of the Fourier transform is two-sided and the integration of the transformed signal must again yield the total energy content, the magnitudes of the one-sided magnitude spectrum are too low by a factor of 2. However, this does not apply to the DC voltage component at 0 Hz. The lines of code. ð12:158Þ determine the one-sided magnitude spectrum of the signal.

ð12:159Þ

displays the magnitude spectrum and labels the diagram.

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Fig. 12.11 Selection dialog when starting GUIDE

With.

ð12:160Þ

a measuring channel dependent scaling of the diagrams is carried out.

12.5

MATLAB GUI: Graphical User Interface

MathWorks® has discontinued the creation of graphical interfaces using the GUIDE tool with the MATLAB® 2022a release. Therefore, the creation of Graphical User Interfaces described here is only possible with older MATLAB® versions. For the creation of a Graphical User Interface only the tool appdesigner is supported.

12.5

MATLAB GUI: Graphical User Interface

341

Fig. 12.12 A new working environment after starting GUIDE

12.5.1 Creating a Graphical User Interface The GUIDE tool is used to create graphical user interfaces (GUIs). Entering guide in the Command Window opens a dialog (Fig. 12.11) for creating a new GUI. Alternatively, an existing GUI can be opened to extend it or to use it as a template for a new graphical user interface. If you select a new GUI (Blank GUI), a new working environment appears (Fig. 12.12), which is used to create the graphical user interface, the GUI. Before the functionality can be programmed, the layout of the graphical user interface must be created. Already existing GUIs can be reworked and extended. More about this later. In a first example, a data set is to be selected to be displayed and labeled in an X-Y diagram. For this purpose, an interface is created as shown in Fig. 12.13. A diagram field is required for the graphical output of the measurement data. This is created with the “axes” element. Two data sets are available for the graphical display. The selection is made via two “push buttons”. The selection of the concrete measurement data

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Fig. 12.13 Example GUI for the graphical display of measurement data

is done via a selection list, the “list box”. “Edit Text” elements are required for the labels of the title line, X-axis and Y-axis, which in turn are labeled with “Static Text”. When the layout is finished, it is saved with File → Save as and the entry of any file name. A MATLAB® script appears in the editor with the same file name as just entered. This script contains the basic structure of the MATLAB® code, which is required for the execution of the GUI. In the first lines of code there is a section which starts with. ð12:161Þ begins and ends with. ð12:162Þ ends (cf. [5]). This must be observed. In this section the GUI is started. Changes in this section cause the GUI to lose its functionality. The more important lines of code follow a little further down. There one finds now among other things with.

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Fig. 12.14 Example GUI for the graphical display of measurement data in the final version

ð12:163Þ

prepared functions, which provide the execution function (the callback) and, apart from the pushbutton functions, a creation function (CreateFcn). Although there is no real functionality in this yet, but everything has been prepared by the GUIDE, which is necessary for the later use of the graphical interface. A prepared function for the output to the diagram “axes1” is not to be found as well as the possibility to adjust the static labels. This is done via the instructions. ð12:164Þ and ð12:165Þ followed by.

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ð12:166Þ which results in the changes also being adopted. If you now start the example GUI, this will be executed. No error messages appear, but an activity of the program is also not (yet) present.

12.5.2 Assign Functions to the Elements of the Graphical User Interface The first functionality assigned to the two “push buttons” is the loading of the data sets “20160906_002.mat” and “20160906_003.mat”. It is noticeable that it would be good to introduce a status line in which each activity of the program is indicated by a status message. A manual scaling of the diagram axes would also be useful. To do this, open the layout again with the GUIDE tool and insert the missing elements (Fig. 12.14). At any point in the GUI development process, the GUIDE can be called to make changes to the layout. The newly added function calls are placed at the end of the corresponding m-file. For a better overview of the program code, the individual functions can be swapped out into independent m-files. The associated m-file is divided into several sections: • Start function – functionality before the GUI becomes visible

ð12:167Þ

In this area, instructions are inserted which are required to start the graphical user interface. Here, for example, parameters can be preset or data can be loaded from files so that they are available during the execution of the GUI. • Output to the Command Window

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ð12:168Þ

In this section, the output is in the command line. This is also usually left as it is specified by MATLAB®. • Functionality of the two buttons

ð12:169Þ

As a first step to improve the readability of the MATLAB® code, it is recommended to insert additional comment lines. In this way, the individual functions receive an explanation of their functionality. This is best done directly after the layout of the graphical interface. ð12:170Þ The functionality of the two buttons is that a data set is to be read in for display in the diagram. The data structure handles is extended by the data from the data set (measurement). This makes the data generally available. ð12:171Þ Based on the loaded measurement data, the content of the listbox and thus the selection option of the displayed measurement values is changed. This is done in the function listbox_aktualisieren. ð12:172Þ By calling the callback function of the list box, a first measured value display is made. ð12:173Þ The guidata function ensures that the changes made in the GUI and to the data records are actually made. Not all changes made actually require this.

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The MATLAB® code for the second button differs from the first only in the file name from which the measurement data is loaded.

ð12:174Þ

• Functionality of the list box The measurement data of the selected entry from the list box are to be displayed in the diagram. This function is called up when the selection in the list box is changed and after the measurement data has been loaded.

ð12:175Þ

The number of the entry is determined. This corresponds to the column number (item) of the measurement data matrix. This is ensured by the structure of the entries in the list box. ð12:176Þ The plot statement in the form plot(handle, x,y) displays the measurement data as a two-dimensional line plot. The other instructions for graphical data representation can also be applied to handle.axes1, because it simply provides the area in the GUI and access to it. ð12:177Þ causes the diagram to be covered with a grid and the global font size to be set to 14.

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ð12:178Þ The title statement creates a title on the diagram. The text of the diagram heading is taken from the field edit1 with the instruction get(handles.edit1, ‘String’). The ordinate and abscissa are labeled in the same way. ð12:179Þ For the legend in the diagram, the label text from the measuring point description is used, which is stored in the data set measurement. ð12:180Þ In order to offer a comfortable change of the axis scalings, input fields were positioned at the ends of the axes in the GUI layout. These fields are now described with the initial values. The initial values themselves are determined with the statements xlim and ylim. The result of the statement is in each case a matrix with two values, the initial value and the final value of the relevant axis.

ð12:181Þ The set statements describe the respective input fields with default values. The default value must be of type string. Therefore, a conversion of the numerical values into string is required via the instruction num2str.

ð12:182Þ

In the last section of the function a status message is output in the text field text7. The output text itself is composed beforehand from direct text input and a variable.

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ð12:183Þ

The create function of the listbox is only required in this example to set the background color to a visible value. In this code section, the filling of the list box can be placed. In the example, however, this is done in a separate function. • Functionality of text input (edit text) For the individual adjustment of the diagram output several text input fields (edit text) were placed in the layout of the GUI. Each of these elements has its own callback function and create function.

ð12:184Þ

The edit1, edit2 and edit3 input fields are used to label the diagram. Entries made here are positioned at the corresponding positions in the diagram using the title, xlabel and ylabel instructions. The set statement is used to output a status message in the status line, which is activated with the guidata statement.

ð12:185Þ

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As for any other active element in the GUI, a create function is created for the text edit fields in the automatically generated MATLAB® code. Only the functionality for the background color is needed. At this point the default values could be set. However, since in the example the values for the preassignment are only known after the GUI start, this is done at a later point in the code.

ð12:186Þ

ð12:187Þ

For the individual adjustment of the axis scaling the input fields edit4, edit5, edit6 and edit7 have been placed at the respective logical positions in the GUI. The corresponding callback functions refer to the function axes_update, which actually changes the axis scaling.

ð12:188Þ

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• Function listbox_update In the function listbox_aktualisieren the selectable entries of the listbox are created from the measuring point descriptions of the selected measurement dataset. In addition, the text entry for the diagram title and the axis labels is preset.

ð12:189Þ

A text array is created. This contains the descriptions of the individual measurement points in the same order as in the measurement dataset, which are available for selection in the display. ð12:190Þ The set statement describes the list box with the entries from the text array liste. The other set statements preassign the edit fields for the diagram title and the axis labels.

ð12:191Þ

At the end of the code sequence, a status text is output. • Function axes_update An adjustment of the axis scaling is done by changing the respective value entry at the beginning or end of the axis. The corresponding callback functions of the edit fields refer to the function axes_update, in which the actual scaling of the diagram axes takes place.

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351

ð12:192Þ

The entry values (strings) are read from the corresponding edit fields, converted into numeric values and then the scaling of the diagram is adjusted via the axis statement. ð12:193Þ At the end of the code sequence, a status message is again output.

12.6

Measurement Process: A Variable Data Logger

A variable data logger is a useful aid for rapid daily measurement. It reduces the amount of time that may be required to put together a program for the specific measurement task at hand. If the measurement task is initially reduced to the recording of measured values over a defined or variable time, or alternatively over a definable value range of a measured variable, then a flexible data logger can be created in advance using simple means. If the needs for documentation and the requirements of the quality assurance processes are included, the data logger becomes a comprehensive project. The requirements for such a data logger can be listed as follows: • • • • • • • • • • •

Documentation of the processor Documentation of the client Description of the measurement object Description of the measurement procedure Flexible configuration of the measurement technology Documentation of the measurement chain used Measurement execution in variable measurement cycles Online – Visualization Overload detection Adding remarks Standardized data formats

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Fig. 12.15 Graphical user interface of the documentation

The measurement project was created on the basis of the above requirements, which do not claim to be exhaustive. The measurement project is subject to the GPL (GNU General Public License) and may be used and modified in accordance with the GPL statutes. The measurement project consists of a documentation part (doku.m), which starts the measurement task, and the measurement part (messung.m) for performing the measurements. In order to keep the handling of the measurement project as simple as possible, it has a graphical user interface. The required parameters from the documentation are transferred to the measurement in the background.

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353

The variable design of the encoder configuration is ensured via a SETUP file in CSV format. This means that the encoder configuration, the measuring point description and the documentation of the sensors used is done outside the program code. The setup is designed in such a way that either measurement hardware or audio hardware can be used. Not solved is the problem that National Instruments hardware has to be addressed with different IDs. In the current state of the measurement project, this unfortunately leads to the fact that different device IDs have to be entered in the setup file for different computers. For the selection of the measurement cycle descriptions, the transfer from a file in CSV format was also used. The description of the program code is reduced in the further course of this chapter to those elements and areas which are necessary for the understanding of the functional sequence.

12.6.1 Measurement Process: Documentation The measurement project is started by calling up the doku.m program. The graphical user interface shown in Fig. 12.15 appears for the input and selection of the required entries. It is mandatory to enter a date, a file name and to select the setup file and the measurement cycle file. Without these entries or selections, the documentation cannot be exited in the direction of measurement. In the program code this is realized by an appropriate control with Boolean variables. For this purpose, presettings are made in the doku_OpeningFcn.

ð12:194Þ

All inputs required for documentation and measurement execution are stored in the handles.meta data structure. For the callback functions of the individual input fields (edit fields), the following results (for example). ð12:195Þ as program code. The respective activities are commented in the status field by corresponding text outputs. In order to force a reasonably reasonable date input, the input string is shortened by the blanks and checked for a minimum length of eight characters. Only if this is fulfilled, the date is taken over and the associated boolean variable is set to true.

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ð12:196Þ

The shortening by the blanks is done by a regexprep statement, which replaces all blanks (‘’) with an empty string (‘’). The object and experiment description is limited to the input of 20 lines. Entries greater than 20 lines are shortened accordingly.

ð12:197Þ

As with the date, the file name must not contain any spaces. It should also be of sufficient length and must be different from existing filenames. As before, a regexprep statement removes any existing spaces from the string, while exist(handles.meta.filename) checks whether the filename already exists.

ð12:198Þ

The selection of the CSV files for the setup and the used measuring cycles is done via the functionality of the uigetfile instruction, in which the selection can be reduced to certain file

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extensions. If a valid setup was selected, then the associated Boolean variable is set to true. The selection of the measuring cycle file is made in accordance with the setup selection.

ð12:199Þ

A push button has been created to cancel the activity. This only triggers the deletion of the currently active GUI.

ð12:200Þ

As soon as a SETUP is selected, a date has been entered and a file name exists, the documentation can be exited and the measurements can be activated. This is the case if the Boolean variables handles.setup, handles.datumok and handles.datenablage are set to true. If a measurement can be performed, the data recorded in the documentation is transferred to the following application via the setappdata instruction. A measurement data directory is created for the data storage. The metadata and the setup file used are stored in this directory. Afterwards the measurement GUI is called and the documentation GUI is closed.

ð12:201Þ

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12.6.2 Measurement Process: Measurement The measurement GUI will now open, which, in addition to the level displays for up to 32 measurement channels, also contains an area for the online visualization of a selected measurement channel. There are also some push buttons which control the individual measurements (Fig. 12.16). Before the GUI becomes visible, the required settings are made in the OpeningFcn and the transfer parameters are taken from the documentation. Since this measuring project is not intended for the execution of impulse hammer measurements, a dummy is set for the information about the impact point file, which is only used to fulfill the requirements from the setup. The counter for the measurements performed is set to the start value 1. ð12:202Þ The functionality is realized via the individual push buttons, input and selection fields and is implemented as follows: • Configuration of the measurement Due to the length and complexity of the function, it has been moved to a separate file. To control the measurement process, some variables must become globally available. This is necessary because the variable handles is not available in all areas of the code. Global settings have their pitfalls and should be limited to the necessary minimum. As far

Fig. 12.16 Graphical surface of the measurement

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as possible, data about the variable handles should be used by the functions. The handles variable itself should not be set globally, as this would prevent the program from exiting properly. It would be necessary to close MATLAB® itself. ð12:203Þ The global setting of variables must be done at the beginning of the program code of the respective function. ð12:204Þ performs the encoder configuration. This is a function which is always used in the same way in several applications. The setup is read from a file. The file name to be read in is stored in handles.meta. setupfile. The file must be a CSV file with; (semicolon) as separator. Decimal separator must be a dot. Be careful when using MS-Excel with German language setting for the creation of this file, this sets a comma as decimal separator. The file must be structured as follows: Fixed parameters are stored in the second line. These are (in this order) encoder type (ni or au for audio). This specification must consist of two characters. This is followed by numerical values for (default) sampling rate, (default) pretrigger and (default) frequency resolution.

ð12:205Þ reads the second line of the setup file and stores the values in the dataArray. The assignments for device type, sampling rate, pretrigger and frequency resolution of the signal analysis are then made.

ð12:206Þ The device type is a relevant designation for the setup of the measuring device object. In the current version of the setup, National Instruments hardware and audio hardware are supported. This leads to the instruction sequence.

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ð12:207Þ

to create the encoder object. The sampling rate, the continuous mode and the size of the data buffer are defined directly afterwards.

ð12:208Þ From line 5 onwards, an input line follows in the setup file for each measuring channel. Here the device designation is entered in column 1 (character string). Column 2 contains the channel designation (character string), column 3 the measuring range in mV (number), column 4 contains the channel configuration (character string), in column 5 the name of the measuring point (character string) is entered, column 6 contains the designation and serial number of the sensor used (character string), Column 7 contains the calibration value in mV per EU (number), in the case of an audio device the scaling factor is included here, columns 8, 9 and 10 contain the coordinates X, Y, Z of the measuring point, column 11 indicates whether the measuring channel represents a reference (value = 1) or no reference (value = 0), the last column (12) contains the information about the channel type. Here ai stands for analog input, ao for analog output, ci for counter input and co for counter output. Further channel types are possible. However, these must adhere to the specified character length of two for the nomenclature. The specifications are made as expected by the commands addAnalogInputChannel or add AudioInputChannel.

ð12:209Þ

reads the channel configuration from the setup file and stores it in dataArray. In the following program loop the measuring points are defined, documented in the transfer variable messpunkt and configured as measuring channel in the measuring device object. The transfer variable messpunkt has the structure as follows:

12.6

Measurement Process: A Variable Data Logger

359

ð12:210Þ

Depending on the contents of the signal definition, the measuring channels are created in the encoder object via the instructions addAnalogInputChannel, addAnalogOutputChannel, addCounterInputChannel, addCounterOutputChannel or addAudioInputChannel and addAudioOutputChannel. In case of an impulse hammer measurement a beat point list is read in. The file name of the beat point list must be in handles.meta.beatpoints. The structure of the impact point file is analogous to the setup file. The first column contains the detailed description of the setpoint (string), the second column contains the short description (string), while the following columns 3 to 5 contain the coordinates X, Y and Z (numerical values). The beat points are passed in the return variable beat point.

ð12:211Þ

If no beat point file exists, the return is done with. ð12:212Þ In the further course of the configuration, some parameters are now preset, which can be changed later by the user.

ð12:213Þ

The behavior of the measurement application is controlled via the variable statusMeasure, which can assume four states. The possible states are: • • • •

for measuring lead started for measurement for measurement running for measurement finished.

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At the level indicators the following instructions are given via the instruction sequence. ð12:214Þ removed the scale labels. By reading out the measuring cycles file and building up a measuring cycles matrix, a list is built up for the selection of the predefined measuring cycles.

ð12:215Þ

and made available for selection in the list box. ð12:216Þ Via handles.meta.measurementtime = 30 the measurement time is preset to 30 s. ð12:217Þ presets the matrix for overload detection with 0 = no overload. At the end of the function. ð12:218Þ the measurement is started in the preprocessing. As soon as the size of the data buffer set with NotifyWhenDataAvailableExceeds is reached, the function dispData specified in the dataListener is executed to process the acquired data. The measurement is now in the “Pre-run” status. Online visualization can be configured and overload detection is activated. • Online visualization and overload detection The online visualization and overload detection is done in the function dispData, which is called by the dataListener when the data buffer is filled. Also in this function the global variables must be defined at the beginning.

ð12:219Þ

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takes over the data buffer and rescales the measured values from [V] to [mV]. In the further processing, a processing loop is used for each existing measuring channel via. ð12:220Þ converts the measured data into EU, determines the display value and the maximum value for the level meter. Overload detection is performed by comparing the display value with the maximum value of the level indicator.

ð12:221Þ

Presettings are required for the scaling of the level display, which is done by setting the variables yminvalue, xmaxvalue, xminvalue. The assignment of measuring channel to the corresponding diagram, measuring range display as well as overload display is done via a switch/case instruction sequence.

ð12:222Þ

The level display is activated by the instruction. ð12:223Þ UPDATED. ð12:224Þ deletes the axis labels and sets the axis scaling to the required values. ð12:225Þ displays the current measuring range in the text field below the respective level display. In a text field that cannot be edited, a warning is displayed if an overload is present for this channel.

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ð12:226Þ The overload warning remains until another measuring range is selected. The measuring range can be changed by overwriting the measuring range value. The measuring range change is performed by the callback function assigned to the edit fields. The file messen.m has a total of 32 entries of this type.

ð12:227Þ

The actual change of the measuring range takes place in the function measuring range, which is generally valid by the transfer parameters wert and mp. In this function the measuring range is redefined in the variable messpunkt. For measurement techniques that allow a change of the measurement range, the corresponding code can be entered here. The return value of the function is the text for the status display. The callback function resets the overload channel in the overload matrix.

ð12:228Þ

The signal analysis for a selected measuring channel is visualized in the separate diagram field. The measuring channel is selected via the select boxes at the head of the respective level display. The parameters xminvalue, xmaxvalue and ymaxvalue must be defined for the axis scaling of the diagram. The signal analysis and its visualization is again done by a switch/case statement.

ð12:229Þ

The simplest variant of the signal analysis is the display of the time signal. Here, only the output of the measurement data of the data buffer is required.

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363

ð12:230Þ

In the second case of online visualization, the RMS values are displayed. For this purpose, the envelope function is used to determine RMS values for the window width of one millisecond. The upper envelope of the time signal is displayed. ð12:231Þ In the third case of online visualization a frequency spectrum is displayed. The parameterization of the Fourier transform is largely predefined by the sampling rate fs and the data buffer. Only the WINDOW function for the weighting of the time signal and the proportion of the displayed frequency range can be changed via check boxes. If the WINDOW function square wave is selected, then in fact no weighting of the time signal takes place (compare Sect. 14.4). To ensure the overall functionality, the analysis data are passed to the variable daten_f_ft. ð12:232Þ The window function is carried out either with Hanning or Flattop Window. The functionality is implemented via two if-queries, in which the time signal is multiplied by the amplitude-corrected WINDOW function.

ð12:233Þ

Finally, the Fourier transformation and formation of the magnitude spectrum as well as the display in the selected frequency range are performed. ð12:234Þ The selection of the analysis is done via three checkboxes. The grouping of these three checkboxes by the frame is only of an optical nature. The frame has no effect on the functionality of the grouped elements. The interaction of the checkboxes with each other must therefore be programmed into the callback function.

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ð12:235Þ

The above program code shows the functionality of the callback function for the checkbox “Time signal”. For the analyses RMS and spectrum there are two further callback functions in the file measure.m. The value of a checkbox is 1 if it is activated and 0 if it is deactivated. The instruction set (handles.rms, ‘Value’, 0) deactivates the checkbox for the signal analysis RMS, for example. By setting the respective values of checkboxes, the respective dependencies can be established. If a checkbox is activated, the associated callback function deactivates the other checkboxes belonging to this function group. Proceed in the same way for the selection of the displayed frequency range and the WINDOW function. The selection of the measuring channel for the analysis also uses the same procedure. Since a total of 32 checkboxes are to be considered here and it must be taken into account that the possible number of channels can be different from the actual number of channels, a slightly different program code results for the respective callback functions. ð12:236Þ The analysis channel is controlled via the variable analyseChan. The length of the vector messpunkt can be used to determine the overrun of the existing measuring channels. If an attempt is made to select a non-existent measuring channel for the analysis, this is simply ignored.

ð12:237Þ

To simplify the code, all checkboxes of the channel selection are first deactivated via the clearChanSelect function. Subsequently, set(handles.CH1, ‘Value’, 1) sets the currently activated checkbox as active and passes it to the variable analyseChan of the currently activated analysis channel.

12.6

Measurement Process: A Variable Data Logger

365

• perform measurements The functionality of the measurement project is controlled via the variable statusMeasure. In the function dispData, which is continuously called by the dataListener, the processing of the measurement is integrated. If the measurement is in the preliminary run, statusMeasure has the value 1. Since no measurement is carried out, no further processing of the measurement data takes place, only the visualization is carried out. A total of four states of statusMeasurement are defined: • Measurement in advance, visualization and overload detection • Measurement was started, visualization, overload detection and data recording of the first data block • Measurement running, visualization, overload detection and data recording • Measurement was finished, visualization, overload detection and data recording of the last data block This functionality is realized by a switch/case statement in the function dispData. As long as the measurement is running (statusMeasure = 3), the data blocks from the data buffer are appended to the already captured data blocks. The captured time vector is corrected by the start time of the measurement and the elapsed time since the start of the measurement is displayed in a status field. It is also checked whether the entered measurement time has been exceeded or reached. If this is the case, statusMessen is set to 1 (advance) and the status variable store, which secures the saving of the measurements, is set to true. The data blocks from the buffer are appended to the existing column-oriented data blocks using the vertcat statement.

ð12:238Þ

• Measuring cycle Predefined measuring cycles can be selected via the list box. After the selection of a measuring cycle, the abbreviation appears in the MZShort field, which becomes the file

366

12

Measurement with MATLAB®

name during data storage, and the detailed description text for the selected measuring cycle appears in the MZDescription field. Both fields are designed as edit fields and can therefore also be written to directly. • Start and stop measurement A measurement is started via the push button “MEASUREMENT START”. In the associated callback function, only statusMeasure is set to 2 and a status message is output in the status line. The actual processing for the start of the measurement is done in the function dispData. A previously performed but not saved measurement is overwritten by starting a measurement again.

ð12:239Þ

The measurement data from the data buffer are written to the temporary memory for the current measurement (aktMessung). Any existing data in aktMessung is overwritten by this. The control variable statusMessung is set to 3 (measurement running) and the saving of measurement data is blocked by store = false. Measurements are terminated when either the entered measurement time has been exceeded or the push button “MEASUREMENT END” is pressed. Both procedures set statusMeasure to 4.

ð12:240Þ

If the measurement is finished, i.e. statusMeasure = 4, then only the last data block still belonging to it is taken over from the buffer into the measurement data. The measurement is now set back to the status “Pre-run” and data storage is enabled by store = true. • Comments In order to be able to add additional information or notes to the individual measurement datasets, a remark field has been included in the GUI. The content from this field is added to the metadata.

12.6

Measurement Process: A Variable Data Logger

367

• Store measurement data By pressing the push button “STORE MEASUREMENT” the data of the last measurement is stored. If the control variable store = true, measurement data can be stored. ð12:241Þ To be able to do this without interruptions by the data acquisition running in the background, it is stopped and the dataListener is terminated.

ð12:242Þ

The measurement data, the measurement point descriptions, the overload matrix, all existing metadata as well as the description of the measurement cycle are stored in a data structure. ð12:243Þ The file name is formed from the short designation of the measuring cycle and the measuring series number (handles.messinc). The measurement data are saved in version 7.3. This also allows the storage of large amounts of data.

ð12:244Þ

Subsequently, a further saving of the measurement data is blocked by store = false and the measurement series counter is incremented by one as well as the dataListener is reactivated and the measurement data acquisition is restarted. The measurement data acquisition is now in the “Preprocessing” status again. • Exit GUI The graphical user interface must not simply be closed. This would not terminate the functions running in the background. Since the GUI itself is no longer present, no interaction would be processed. However, all other activated actions would continue to work. Therefore, via the statement sequence.

368

12

Measurement with MATLAB®

ð12:245Þ the dataListener is stopped and the measuring device is terminated. Only now the closing of the GUI window is allowed.

References 1. Stein, U.: Objektorientierte Programmierung mit MATLAB. Hanser, München. ISBN 978–3–44644298-6, (2015) 2. MATLAB: Connect to data acquisition cards, devices, and modules. https://de.mathworks.com/ products/daq/. Accessed: 29. Apr. 2018 3. MATLAB: Request hardware support. http://de.mathworks.com/hardware-support.html. Accessed: 29. Apr. 2018 4. MATLAB: Data acquisition toolbox. http://de.mathworks.com/help/daq/. Accessed: 29. Apr. 2018 5. Stein, U.: Programmieren mit MATLAB, Programmiersprache, Grafische Benutzeroberflächen, Anwendungen. Hanser, München. ISBN 978–3–446-44299-3, (2017)

Raspberry Pi as a Measuring Device

13

Abstract

Originally initiated as a school project, the Raspberry Pi has now arrived in measurement technology. This chapter describes the possibilities of using the diverse electronic interfaces of the Raspberry Pi for measurements and process control. A major advantage of the Raspberry Pi, in addition to its low price, is its communication via the LAN interface, which makes it possible to set up decentralized measurement and control stations. "

Originally initiated as a school project, the Raspberry Pi has now arrived in measurement technology. This chapter describes the possibilities of using the diverse electronic interfaces of the Raspberry Pi for measurements and process control. A major advantage of the Raspberry Pi, in addition to its low price, is its communication via the LAN interface, which makes it possible to set up decentralized measurement and control stations.

Additional material for this chapter can be found at http://schwingungsanalyse.com/ Schwingungsanalyse/Kapitel_14.html.

13.1

Raspberry Pi

The Raspberry Pi is a single-board computer developed by the British Raspberry Pi Foundation. The computer, which has a very simple design compared to conventional PCs, was developed by the foundation with the aim of getting young people more interested in programming and computer hardware, robotics, etc. The computer contains # The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 T. Kuttner, A. Rohnen, Practice of Vibration Measurement, https://doi.org/10.1007/978-3-658-38463-0_13

369

370

13 Raspberry Pi as a Measuring Device

Fig. 13.1 Raspberry Pi (lowest level) with attached experiment and measurement cards. In the middle level a 24 bit A/D converter card in the top level the presented Explorer Hat Pro

a single-chip system from Broadcom with an ARM microprocessor. The footprint of the board is about the size of a business card. By September 2016, more than ten million devices had been sold. There is a large range of accessories and software for numerous areas of application. For example, the use as a media center is widespread, as the computer can decode video data with full HD resolution (1080p) and output it via the HDMI interface. The Raspberry Pi (Fig. 13.1) offers a freely programmable interface (also known as GPIO, General Purpose Input/Output), which can be used to control devices and other electronics. The GPIO interface consists of 26 pins for Model A and Model B, and 40 pins for Model A+ and Model B+, each designed as a double-row header, of which • • • •

2 pins provide a voltage of 5 V, but can also be used to power the Raspberry Pi 2 pins provide a voltage of 3.3 V 1 pin serves as ground 4 pins, which could get a different assignment in the future, currently also connected to ground

13.1

Raspberry Pi

371

• 17 pins (Model A and B) or 26 pins (Model A+ and B+, as well as Raspberry Pi 2 Model B), which are freely programmable. They are designed for a voltage of 3.3 V. Some of them can take over special functions. • 5 pins of which can be used as SPI interface1 • 2 pins have a 1.8-kΩ pull-up resistor (to 3.3 V) and can be used as I2 C interface2 • 2 pins can be used as UART interface and provide a RS232 interface via a corresponding adaptation. With the B+ model, an official specification for expansion boards, so-called hardware attached on top (HAT), was introduced. Each HAT must have an EEPROM chip; this contains manufacturer information, the assignment of the GPIO pins and a description of the attached hardware in the form of a “device tree” section. This allows the necessary drivers for the HAT to be loaded automatically. It also determined the exact size and geometry of the HAT as well as the position of the connectors. Model A+ and Raspberry Pi 2 Model B are also compatible with these. Libraries for numerous programming languages exist for controlling the GPIOs. Control via a terminal or web interfaces is also possible [1–3]. Primarily customized Linux distributions with a graphical user interface are used as the operating system; Windows 10 also exists in a special Internet of Things version without a graphical user interface for the latest model. The boot process is done from a removable SD memory card as internal boot medium. An interface for hard drives is not present; additional mass storage can be connected via USB interface. The Raspberry Pi/MATLAB® connection requires the appropriate hardware support package from MathWorks [5]. After downloading the installation package, a menu-driven installation of the support package takes place in the local MATLAB® installation. At the end of the installation process, the Raspberry Pi is configured and an operating system for the Raspberry Pi is placed on an SD memory card. The connection between the Raspberry Pi and MATLAB® is made via the network. Ideally, the LAN connection should be used for this. It is important to ensure that the addresses of the Raspberry Pi and the MATLAB® PC are compatible with each other. Both devices must be in the same address space, which is ensured if they are in the same network segment and the address assignment is done via DHCP. This must be specified in the Raspberry Pi configuration during the installation process.

1 SPI: Serial Peripheral Interface is a bus system developed by the semiconductor manufacturer Motorola (today NXP), which has established itself as one of the standards for the synchronous serial data bus. With SPI, digital circuits can be interconnected according to the master-slave principle. A chip select line is used for unique communication. 2 I2C: Inter-Integrated Circuit (German: I-Square-C) is a bus system developed by Philips Semiconductors (today NXP), which has established itself as another standard for data exchange between circuit parts. A 7-bit addressing is used for unique communication.

372

13 Raspberry Pi as a Measuring Device

Fig. 13.2 Raspberry Pi configuration tool

Once the SD memory card has been written to, the connection between MATLAB® and the Raspberry Pi can be tested. To do this, insert the SD memory card into the Raspberry Pi, connect it to the network, switch it on and after a few seconds the Raspberry Pi/MATLAB® connection can be established. The configuration of the Raspberry Pi contains a well-known password for the user pi. This should be changed immediately to an individual password, otherwise the Raspberry can be misused in the network. With ð13:1Þ a terminal window is opened on the Raspberry. Here now with the instruction ð13:2Þ the individual configuration of the Raspberry (Fig. 13.2) can be made.

13.2

13.2

Digital IO

373

Digital IO

For first experiments with the Raspberry Pi/MATLAB® connection, it is recommended to purchase a HAT, such as the Explorer HAT Pro from Pimoroni, which is available for a few euros in the relevant trade. The HAT is simply plugged onto the 40 pin connector of the Raspberry Pi and provides additional functionality via its electronics. • • • • • • •

4 buffered 5 V tolerant inputs 4 switchable 5 V outputs (up to 500 mA in sum over all outputs) 8 capacitive keys in total 4 coloured LEDs (red, green, blue and yellow) 4 analog inputs 2 motor drivers with a maximum of 200 mA each a small breadboard for testing electronic circuits

With this, the first walking tests of the Raspberry Pi/MATLAB® connection can now be carried out. A small motor is connected to one of the two motor drivers Key 1 is to be used to turn the motor counterclockwise, key 4 to turn it clockwise and keys 2 and 3 to stop the motor and end the program. The blue LED is to indicate the counterclockwise rotation and the green LED the clockwise rotation. With ð13:3Þ the object for the Raspberry Pi/MATLAB® connection is created. Without; at the end of the instruction the following output occurs, which gives an insight into the hardware.

ð13:4Þ

Supported peripherals opens the MATLAB® help for the Raspberry Pi/MATLAB® connection. In the next step the used hardware devices have to be configured. The LEDs and the motor are controlled via GPIO pins.

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13 Raspberry Pi as a Measuring Device

ð13:5Þ defines the addresses of the required GPIO pins, which are configured in the next step.

ð13:6Þ

The preparations for switching the LEDs and controlling the motor have now been made and a short test can be performed.

ð13:7Þ

switches the LEDs on and off, while

ð13:8Þ

takes over the control of the motor. For switching the motor via the capacitive keys, detection of the keystroke is required. The capacitive keys are detected via the electronic component CAP1208. This module is connected to the I2C bus. With ð13:9Þ the electronics connected to the I2C bus are detected. The I2C bus addresses of the connected electronic components are output. In the case of the Explorer Hat Pro, the addresses are 0 × 28 and 0 × 48. The CAP1208 electronics module is hidden behind the address 0 × 28. Address 0 × 48 represents the analog-to-digital converter.

13.2

Digital IO

375

ð13:10Þ

now creates the device keys. For the communication with the CAP1208 electronic component or in general for the communication with electronic components on the I2C bus, the knowledge of the mode of operation of the respective electronic components is required. The CAP1208 electronic component stores the events at up to eight PINs as states in memory registers. A configuration register (register 0) is used, among other things, to reset the electronics module. In order to be able to detect keystrokes, the register with event detection only has to be read out continuously in an endless loop. If an event is detected, the electronic module is reset. The events are recorded in register 3. Depending on the key pressed, the register contains a value. Key 1 2 3 4

Register value (hexadecimal) 0 × 10 0 × 20 0 × 30 0 × 40

Register value (decimal) 16 32 64 128

The key pressed can therefore be determined by evaluating the register value. ð13:11Þ resets the event detection registers. Event detections that may be present are thus removed from the memory. The value 0 is written to register 0. The instruction reads in detail ð13:12Þ with the parameters • • • •

myi2cdevice – the device name register – the register number value – the value to be written dataPrecision – the data format

The data format uint8 stands for unsigned integer 8 bits (corresponds to one byte). For the actual program, an “infinite loop” is implemented via a while loop. The variable run is set to true (= true). As long as the variable run remains true, this loop is executed.

376

13 Raspberry Pi as a Measuring Device

ð13:13Þ

Read in the value from register 3 as an 8-bit integer value. ð13:14Þ resets the event detection registers. ð13:15Þ Evaluation of the read value via a switch – case – construct. According to the detected keystrokes, which store a decimal value in the event detection, the LEDs are switched via the GPIO-PINs and the motor running direction is set. At the keystrokes of key 2 and key 3 (decimal 32 and decimal 64) the motor is stopped, the LEDs are switched off and the program ends.

ð13:16Þ

13.3

13.3

Voltage Measurement with the A/D Converter ads1015

377

Voltage Measurement with the A/D Converter ads1015

The Explorer HAT Pro already used for the GPIO test has a four-channel 12-bit analog/ digital converter in the form of the ads1015. This can be used to perform simple voltage measurements. The ads1015 [6] is specified for measurements in the temperature range -40 °C to +125 °C. The sampling rates available are single measurements and 128 measurements per second (samples per second, SPS) up to 3300 SPS. It has four usable measuring ranges: 1024 V, 2048 V, 4096 V and 6144 V. An overrange is detected by the maximum measured value. The usable confidence range of the measured values is therefore 99% of the respective measuring range. The four input channels allow measurements of four channels against ground (single ended) or optionally two channels differentially. This is realized via a multiplexer. The measured values of the individual measuring channels are therefore not synchronous. Communication between the Raspberry Pi and the ads1015 takes place via the I2C interface. The respective measurement is done via the 16Bit configuration register of the ads1015 [5, page 22 ff.]. In Explorer Hat Pro, the ads1015 can be addressed via address 0 × 48. In other ads1 × 15 series applications, up to four ads1 × 15 are possible on the I2C bus. Addressing is implemented via an address selector (ADDR PIN on the electronic component). • • • •

Address 0 × 48 = ADDR connected to GND Address 0 × 49 = ADDR connected to V Address 0 × 4 A = ADDR connected to SDA Address 0 × 4B = ADDR connected to SCL ð13:17Þ

creates the object rpi and defines the AD converter. Via the statement writeRegister(myi2cdevice,register,value,dataPrecision) the configuration register is written. The parameter value is used to set the required bit pattern in the configuration register. For this example, bits 14 to 12 for the multiplexer configuration and bits 11 to 9 for the measuring range are decisive for the configuration. Details on this can be found in the ads1015 data sheet [5, page 22 ff.].

378

13 Raspberry Pi as a Measuring Device

ð13:18Þ

The value for the configuration register is calculated by the sum of base + channel + measuring range.

ð13:19Þ

calculates the value for the configuration register and writes it using the statement ð13:20Þ this into the configuration register. ð13:21Þ reads out the ADC converter value. The measured values are taken over as integer values. The measured value of the ads1015 is stored as a 12-bit value in the first three 1/2 bytes of the data register (Fig. 13.3). The last four bits or the last 1/2 byte is required for byte-wise communication, but does not contain any values and is ignored in data processing. Data transmission takes place in the sequence “second byte”, “first byte” and thus in the wrong order. ð13:22Þ reads out the ADC converter value. The measured values are taken over as integer values and converted into the hexadecimal system3 by dec2hex. hexvalue is a character string with 4 characters. Each character represents a 1/2 byte of the measured value. The fourth 1/2 byte from the data register is stored in hexvalue(2) and is not needed further. 3

In the hexadecimal system, numbers are represented in base 16 with the characters 0 to 9 and A to F. In the binary system, 4 bits are used for a hexadecimal value.

13.4

Bit 1

Speed Measurement Via Interrupt at Digital IO

Bit 2

Bit 3

Bit 4

Bit 5

first 1/2 byte

Bit 6

Bit 7

Bit 8

Bit 9

379

Bit 10

Bit 11

second 1/2 byte

third 1/2 byte

hexvalue(4)

hexvalue(1)

first byte hexvalue(3)

Bit 12

Bit 13

Bit 14

Bit 15

Bit 16

fourth 1/2 byte second byte hexvalue(2)

Fig. 13.3 Schematic representation of the ads1015 measured value register and its transfer to the variable hexvalue

ð13:23Þ then generates the correct numerical value of the A/D conversion. The result is available in conversion steps digits. With a 12Bit ADC this can take the value range 0 to 4096. The interpretation of the values is done via the diagram in Fig. 13.4. ð13:24Þ The values above 0 × 7ff0h cover the negative voltage range. This ranges from 0 V (0 × fff0h) to the negative end of the measuring range (0 × 8000 h) with reverse order. This enables a simple conversion of the negative voltage values. ð13:25Þ calculates the voltage value.

13.4

Speed Measurement Via Interrupt at Digital IO

A speed measurement is technically implemented in such a way that the period between two increments or between two pulses is recorded. The Raspberry Pi provides sufficient hardware and computing power so that a period duration measurement can be implemented with microsecond accuracy (Fig. 13.5). A Python script solves the task of period duration measurement on the Raspberry Pi. Strictly speaking, an interrupt is triggered at the time of the falling signal edge at a GPIO PIN, which writes a microsecond-accurate timestamp into a buffer. Another Phyton script, which is called periodically by MATLAB®, reads the cache and transports the timestamps to the MATLAB® -PC. The buffer is created as a FIFO file in the Linux operating system of the Raspberry Pi. This has the advantage that the period duration measurement can only store timestamps in the buffer when reading from the buffer at the same time. This ensures that the Raspberry Pi is not overloaded. However, this type of speed measurement is only useful for slow condition monitoring and slow processes (pulse rates up to 10 kHz).

380

13 Raspberry Pi as a Measuring Device

7FF0h

0010h 0000h FFF0h

...

Output Code

...

7FE0h

8010h 8000h ...

-FS 2

11

-FS 2

0

...

+FS

Input Voltage VIN

11

2

-1 +FS

11

2

-1 11

Fig. 13.4 In the value range 0 × 0000 h to 0 × 7ff0h are the voltage values from 0 V to the end of the measuring range. The value range 0 × 8000 h to 0 × fff0h, i.e., above 0 × 770 h, covers the voltage range from the negative end of the measuring range to 0 V [5, page 22] Fig. 13.5 Schematic representation of the period duration measurement on the Raspberry Pi. The control signal of an incremental sensor is applied to a transistor circuit, which switches a GPIO PIN of the Raspberry Pi to ground

Time signal with pulses

Period duration

3,3 V

GPIO-PIN Raspberry Pi Incremental sensor

13.4

Speed Measurement Via Interrupt at Digital IO

381

The Python script for the timestamps:

ð13:26Þ

represents the program header and defines the required modules. ð13:27Þ The variable target is made available globally. This contains the pointer to the opened FIFO. With the instruction ð13:28Þ defines how the GPIO PINs are numbered. These are not simply numbered from 0 to 40, but have, depending on the point of view, a different, in any case the logic deviating, PIN number. ð13:29Þ In the example PIN 18, according to board scheme, is defined as used GPIO PIN.

ð13:30Þ

This is the actual routine which processes the interrupt. It calculates the current time in microseconds since 0 o’clock and converts it into an end-of-line string (out = str(t2) + ‘\n’). os.write(target, out) then writes this value into the FIFO.

382

13 Raspberry Pi as a Measuring Device

ð13:31Þ declares PIN 18 as interrupt source. The interrupt function is called on a falling edge. To prevent bouncing behavior, a “bounce time” is set to 10 ms. ð13:32Þ The program itself runs as an endless loop. ð13:33Þ formally closes the file. The text file must be set executable using chmod 755 and is started from a shell using nohup ./ &. nohup causes the shell to close after the program starts without terminating the program. This runs a program which reacts to increments at GPIO-PIN 18 and stores values in a buffer file. The Phyton script read_n is used to read the values from the buffer file.

ð13:34Þ

This reads ten values from the buffer and outputs them to standard output. This file must also be set as executable using chmod 755. On the MATLAB® PC, the time stamps are to be periodically converted into speed values and processed further. For the demonstration, the Explorer Hat Pro is again used, which has two motor drivers. A small DC motor (Fig. 13.6) is connected to one of these drivers, which generates the pulse signal for the interrupt at the GPIO PIN via a light barrier. ð13:35Þ The Device object of the Raspberry Pi must be globally available.

13.4

Speed Measurement Via Interrupt at Digital IO

383

Fig. 13.6 Test setup for speed measurement

ð13:36Þ The two Python scripts are stored on the Raspberry Pi. This ensures that they are in the current form. The instructions

ð13:37Þ

define the GPIO-PINs for the motor driver and switch on the motor. After that

384

13 Raspberry Pi as a Measuring Device

ð13:38Þ a timer and starts it. Now via openShell(rpi) the interrupt script must be started via nohup ./intrtest.py & on the Raspberry Pi. The timer is configured using the createTimer() function.

ð13:39Þ

The timer now calls the function readNValues every 2 seconds. The function ð13:40Þ reads (10, see instructions in 14.33) timestamp values via the Python script read_n on the Raspberry Pi (rpi, see instructions 14.34). The transfer takes place in a string, which is split into its individual parts via strsplit.

ð13:41Þ now calculates speed values in min-1 from the time stamp values in microseconds. The incremental disk has 24 increments. The first time stamp value is ignored in order to completely exclude misinterpretations. The period duration of the rotational frequency is 24 times (24 increments of the incremental disk) the individual period duration, which is determined via the difference of the two time stamps. Instruction

13.5

Bridge Circuit with DC Voltage Measuring Amplifier

385

ð13:42Þ outputs the mean value of the speed values.

13.5

Bridge Circuit with DC Voltage Measuring Amplifier

For the measurement of forces and pressures, circuits with strain gauges in bridge connection are often used (see Sect. 9.1). Carrier frequency measuring amplifiers are used as the standard method for signal processing, as these have excellent long-term stability and immunity to thermal voltages, electrical and magnetic fields. DMS measuring bridges can also be operated in a DC voltage circuit. This allows simple circuits to be constructed. The bridge circuit described in [4] also permits measurement without an intermediate carrier frequency amplifier. This requires a stable, low-impedance (120 Ω) supply voltage and good resolution of the A/D converter used. The ADS1115 from Texas Instruments is a 16-bit A/D converter that achieves a measurement resolution of 0.0078 mV with an input measurement range of 0.256 V. It has I2C communication and can be configured with different addresses. This has I2C communication and can be configured to different addresses. The integrated multiplexer can switch the analog inputs to four single-ended or two differential channels. For the example, two strain gauges were glued on a bending beam (Fig. 13.8) for strain measurement and connected according to Fig. 13.7. The half bridge is completed by two resistors (R1, R2) to form a full bridge. The measuring bridge is powered via VDD of the A/D converter and thus from the 3.3 V supply voltage of the Raspberry Pi. To determine the supply voltage UE, a single voltage measurement at the beginning of the measured value acquisition is sufficient. For this purpose, the differential voltage between A0 and A1 is determined. The bridge voltage (output voltage) UA is determined by the difference measurement between A2 and A3. The measured values acquired in this way can then be converted to the voltage value. With the k of the strain gauges used and the strain ε at the respective strain gauge ε =

ΔL L0

ð13:43Þ

applies to the full bridge: UA k  ð ε1 - ε2 þ ε3 - ε4 Þ = 4 UE It is better to calibrate UA/UE with known strains or forces. First, again, the session for the Raspberry Pi must be created,

ð13:44Þ

386 Fig. 13.7 Circuit diagram of the experimental setup for the bridge circuit without carrier frequency measuring amplifier

13 Raspberry Pi as a Measuring Device VDD A0

1

1

R1

DMS 1

A2

A3

R2 A1

DMS 2 2

2 GND

Fig. 13.8 Experimental setup with ADS1115 for the bridge circuit

ð13:45Þ in order to subsequently determine the connected devices on the I2C bus.

13.5

Bridge Circuit with DC Voltage Measuring Amplifier

387

ð13:46Þ

The ads1115 can be set to four different addresses via the wiring of the ADR pin. ADR pin GND VDD SDA SCL

Address 0 × 48 0 × 49 0 × 4A 0 × 4B

Since address 0 × 48 is already occupied by an ads1015 in the example setup, the ads1115 used was wired with address 0 × 49.

ð13:47Þ

creates the device, which can be addressed directly in the following code. As with the voltage measurement with the ads1015, the configuration must be written to a register of the A/D converter. This triggers the configured voltage measurement and the A/D conversion result can be read out in register 0. In the following, the necessary parameters for the configuration are provided.single conversion is firmly defined as the “operational status”. ð13:48Þ The voltage measurement of the A/D converter always takes place between AINp and AINn. Eight different configurations of the analog inputs (A0 to A1) can be defined for the measurements via the integrated multiplexer. The values required for the configuration are stored in the matrix mux.

388

13 Raspberry Pi as a Measuring Device

ð13:49Þ

The values for the measuring range are stored in the matrix pga. The first column contains the configuration value and the second column the associated voltage range, which is required for the calculation of the determined voltage.

ð13:50Þ

In the mode, a distinction is made between single measurement and continuous measurement. ð13:51Þ The value for the configuration register cfg is now calculated from the sum of “operational status”, multiplexer setting, measuring range and the mode.

ð13:52Þ This is written to the configuration register of the ads1115. ð13:53Þ A period of 150 ms is required for writing to the configuration register and performing the A/D conversion. The execution of the code is interrupted for this time. ð13:54Þ

13.5

Bridge Circuit with DC Voltage Measuring Amplifier

389

Afterwards the A/D conversion value is available in register 0 of the ads1115 and can be read with ð13:55Þ and converted to hexadecimal. Since the bytes come in the wrong order, byte1 must be swapped with byte2 before the digital value (digits) can be determined. ð13:56Þ In the last step, the digital value (digits) is converted into the measured voltage. Please note that the digital value range above 215 is intended for negative voltage values. This range is mirrored, which allows a simple conversion.

ð13:57Þ

The class library for the ads1115 described in [7] simplifies the program code considerably. The code of the class library must be stored as ads1115.m. Only in line 48 of the code, if necessary, the address of the ads1115 must be adapted. With

ð13:58Þ

is used to configure the ads1115. The A/D converter is configured by setting the corresponding parameters. ð13:59Þ In the first step, the supply voltage of the half bridge is determined. For this purpose, the mean value of 100 measured values is formed. Since the supply voltage has a nominal value of 3.3 V, the next higher measuring range must be selected.

390

13 Raspberry Pi as a Measuring Device

ð13:60Þ

In the second step, the bridge voltage is measured. Values with a few mV are to be expected for the bridge voltage. Therefore the measuring range is set to the lowest. ð13:61Þ To obtain a time vector, the datevec function is used. This provides a numeric date vector.

ð13:62Þ

From this, a time stamp can be formed for the individual measured values.

ð13:63Þ

To obtain the measurement result in the usual mV/V, the determined voltage values of the bridge voltage were divided by the supply voltage. Via ð13:64Þ the visualization of the performed measurement takes place (Fig. 13.9). Although this circuit cannot compete with commercial measuring amplifiers in terms of long-term constancy and temperature stability, the example illustrates very clearly how useful results can already be achieved with simple means.

References

391 Measurement of strain gauge half bridges without bridge amplifier

Measured values [mV/V]

2.5

2

1.5

1

0.5

0

0

2

4

6

8

10

12

14

16

18

Time [s]

Fig. 13.9 Bridge circuit measurement result

References 1. Bartmann, E.: Die elektronische Welt mit Raspberry Pi entdecken. O’Reilly, Köln (2013) 2. Dembowski, K.: Raspberry Pi – Das Handbuch. Springer Fachmedien, Wiesbaden (2013) 3. Dembowski, K.: Raspberry Pi – Das technische Handbuch. Springer Fachmedien, Wiesbaden (2013, 2015) 4. Hoffmann, K.: Anwendung der Wheatstoneschen Brückenschaltung. Hottinger Baldwin Messtechnik GmbH, Darmstadt (1974) 5. MathWorks Hardware Support Raspberry Pi. https://de.mathworks.com/hardware-support/ raspberry-pi-matlab.html 6. Datenblatt ads101x Reihe. http://www.ti.com/lit/ds/symlink/ads1015.pdf 7. Open Source Klassenbliothek für ads1115, ADS1115 interface with raspberry pi in Matlab. https:// www.raspberrypi.org/forums/viewtopic.php?f=91&t=84491

Signal Analysis Methods and Examples

14

Abstract

By means of signal analysis, the accruing data are to be processed in such a way that their meaningful representation and interpretation is made possible. The chapter presents a selection of basic and contemporary methods for signal analysis in the time domain, frequency domain and frequency range. Special attention is given to the methods for spectral representation using Fourier transforms. The presentation is practice-oriented with numerous examples in MATLAB®. "

14.1

By means of signal analysis, the accruing data are to be processed in such a way that their meaningful representation and interpretation is made possible. The chapter presents a selection of basic and contemporary methods for signal analysis in the time domain, frequency domain and frequency range. Special attention is given to the methods for spectral representation using Fourier transforms. The presentation is practice-oriented with numerous examples in MATLAB®. Additional material for this chapter can be found at http:// schwingungsanalyse.com/Schwingungsanalyse/Kapitel_14.html.

Tasks and Methods of Signal Analysis

Large amounts of data are generated in the acquisition of vibration measurement data. The task of signal analysis is to prepare these data sets in such a way that further interpretation of the results is possible. This can include the following tasks, for example:

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 T. Kuttner, A. Rohnen, Practice of Vibration Measurement, https://doi.org/10.1007/978-3-658-38463-0_14

393

394

• • • •

14

Signal Analysis Methods and Examples

Measurement of vibrations on a machine and comparison with a limit value, Detection of the change in bearing vibrations on a machine, Determination of a stress collective on a chassis component, Determination of the natural frequency of an aggregate bearing.

Each of the tasks requires a different approach and measurement technique. For all tasks, however, it is necessary to reduce the amount of data so that statements can be made. In Example 1, the measurements of the vibration response are to be prepared in such a way that a characteristic numerical value is obtained. This is then compared with the limit value. This descriptive procedure requires that the limit value can be described with a number and often requires a processing (filtering) of the signals. The change of vibrations on machines is a task area of the condition monitoring of machines (Example 2). Here, the focus is not on the individual vibration process, but on the long-term trend between two repairs. Thus, the signal analysis must provide indicators for trends and at the same time separate the signal components that are not required before evaluation. When determining the stress collective for subsequent calculations and tests, the sequence and frequency of the stresses are not recorded. Thus, the data are condensed in such a way that they contain stresses and (sum) frequencies. The determination of characteristics of oscillatory systems – such as natural frequencies – is usually not performed in the time domain, but by evaluation in the frequency domain. In linear time-invariant systems, the time characteristic and the excitation amplitude do not play a role, so that these quantities are eliminated in the evaluation. Signal analysis is therefore generally associated with data reduction. The irrelevant data are removed so that the results can be better interpreted. These procedures are by no means to be understood merely as cosmetic embellishment of the data. On the other hand, the tools of signal analysis can be used to reveal correlations that cannot be established by sifting through the measurement signals generated. A self-contained approach to signal analysis along, for example, a decision tree is not universally possible. In the individual fields of application, different approaches have been established and have found expression in standards and regulations. In the following, a series of modern methods for evaluation is described, whereby no claim is made to completeness.

14.2

Signal Analysis in the Time Domain

14.2.1 RMS Value, Power, Mean Value and Related Quantities The effective value of an oscillation is formed according to (14.1)

14.2

Signal Analysis in the Time Domain

395

T

x=

1 T

x2 ðt Þ dt

ð14:1Þ

0

The time course of the observed oscillation x(t) is squared and then integrated over the observation period T. The mean value of the so-called interval mean value is then calculated. Subsequently, the mean value of the so-called interval mean value is calculated over the observation period and the root extraction is performed. The RMS value is a so-called single-number value and is also referred to as the “root mean square” (RMS) value. A constant value for x would also give the value |x| as the RMS value. However, this numerical value is not an RMS value of an oscillation. For this reason, only the alternating variables are considered in the following observations; the DC component (offset) is omitted. The DC component must be eliminated before the RMS value is formed, e.g. by offset correction, subtraction or high-pass filtering, otherwise the RMS value formed is incorrect. In the case of a periodic oscillation with a DC component of zero, the maximum magnitude is referred to as the peak value. In measurement practice, the effective value of the vibration is very often measured. The amplitude or the maximum value and the vibration width (peak-to-peak value, peak-topeak) are also frequently used in measurement technology. For this reason, it is always important to specify which value was used. For periodic oscillations, a period duration or an integer multiple of the period duration is integrated. In this case, the exact rms value is obtained. For stationary stochastic oscillations, integration is performed over an observation period T and this value is used as an estimate for the RMS value. The observation period must be long enough so that the statistical certainty is high enough for the measurement task. On the other hand, the observation period must be chosen short enough to capture transient changes in the vibration (machine diagnostics) and to keep the overall measurement effort reasonable. For the comparability of measurement results, the additional specification of the observation period is required. The mean value of the power is calculated according to T

1 x2 = T

x2 ðt Þ dt

ð14:2Þ

0

The mean value of the power of the measured electrical quantity can be thought of as the power of a DC voltage U converted at an ohmic resistor R in the time period T, i.e. P = U2/ R and explains the squaring of the time function. Here, the resistor R has the function of a normalization variable. Integration over time T gives the area under the curve. This is transformed into a rectangle with the same area (constant) by dividing by T. The arithmetic mean is calculated as follows

396

14

Signal Analysis Methods and Examples

Table 14.1 Conversion of characteristics of harmonic oscillations Conversion factor (row = factor × column) Amplitude Oscillation width RMS value

Amplitude (peak value) 1 2 0.707

Oscillation width (peak-to-peak value, peak-to-peak) 0.5 1 0.353

RMS value 1.414 2.828 1

T

1 x= T

xðt Þ dt

ð14:3Þ

0

The arithmetic mean value thus corresponds to the DC component of a signal. Integration over the magnitudes, on the other hand, yields the so-called rectified value, which will not be considered further here. For harmonic oscillations (amplitude x = 1 and arithmetic mean value x = 0 ), the relationships between amplitude, amplitude amplitude and rms value are shown in Table 14.1 and in Fig. 14.1. The integration takes place from the starting time t0 = 0, to which the zero point of the time counting is set, along the time axis for T > 0 in an interval up to t0 + T. In the case of a recorded measurement signal, this is conceivable until the end of the data set is reached. In the case of simultaneous acquisition and processing in real time, the future data are not yet available from the initial time t0. In this case, integration is performed over the data acquired in the past (T < 0).1 This free choice of integration limits is justified in the case of periodic or stationary oscillations. In the case of periodic oscillations, the oscillation process repeats itself with the period T. In the case of stationary oscillations, the characteristics are time-independent by definition. The floating RMS value weights the individual values in the overall result all the less, the longer they lie in the past. This temporal evaluation is carried out with the exponential function t

xr ð t Þ =

1 τ

e

- ðt - ξ Þ τ

x2 ðξÞ dξ

ð14:4Þ

ξ=0

The variable ξ serves here as an integration variable, the time evaluation is carried out via the time constant τ. A total of three time amounts for the time constant τ are defined by standardization (including DIN EN 61672-1:2014-07) and common use

1

A negative time axis could not be implemented in the presentation, so that in both cases the time axis is positive.

14.2

Signal Analysis in the Time Domain

397

1

x(t)

0.5

0

-0.5 x(t) x2(t) -1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time [s]

Fig. 14.1 Amplitude, rms value and mean value of the power of a harmonic oscillation

• Slow τ = 1 s • Fast τ = 0.125 s • Pulse τ = 0.035 s for transient and τ = 1.5 s for decay [26]. Any other time constant τ can be used depending on the application, but should be indicated on the measurement result. Special parameters are specified in the standards for a number of applications. In the standard for evaluation in the human time domain (DIN EN ISO 8041, EN ISO 5349), for example, a frequency-dependent evaluation filter Wh is prescribed for the evaluation of hand-arm vibrations, with which the accelerations in the three spatial directions ax, ay, az are to be filtered. The total vibration value is formed from the frequency-weighted accelerations by vectorial addition. aw =

a2x þ a2y þ a2z

ð14:5Þ

With this vibration total value the effective value is calculated with an averaging period T = 2000s T

ahv =

1 T

a2w ðt Þ dt

ð14:6Þ

0

For different load sections i with the time duration Ti, the daily vibration load of an eighthour working day A(8) is formed from the vibration total values a .hvi

398

14

N

1 T0

A ð 8Þ =

Signal Analysis Methods and Examples

ð14:7Þ

a2hvi T i

i=1

Example Calculation of a daily vibration load For this, the measured values 1.5 m/s2 with an exposure time of 4 h, 2 m/s2 with 2 h and 0.5 m/s2 with 9 h are available. N

1 T0

A ð 8Þ =

a2hvi T i

i=1

Að8Þ =

1 8h

1, 5

m s2

2

 4h þ

2

m s2

2

Að8Þ = 1, 55

 2h þ

0, 5

m s2

2

 9h

m s2

14.2.2 Application Examples: RMS Value, Power, Mean Value and Related Quantities The RMS value according to Eq. 14.1 of a digitized signal is calculated with the MATLAB® statement ð14:8Þ is determined. The mean value of the power according to (14.2) is determined by squaring the RMS value, since Mathworks does not offer a MATLAB® function for this purpose. ð14:9Þ is determined and for the determination of the arithmetic mean of a signal the instruction ð14:10Þ used.

14.2

Signal Analysis in the Time Domain

399 Time signal

RMS value: 0.70711 +++ Average power: 0.5 +++ DC component: -1.4727e-16

1

Amplitude

0.5

0

-0.5

-1

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Time [s]

Fig. 14.2 Section from t = 1 s to t = 2 of the generated 10 Hz signal

A synthetic signal is suitable for the first test of these functions. This offers the advantage that the signal is already known before the signal analysis and thus the applied signal analyses must provide expectable results. For the later application example, a 10 s long 10 Hz time signal is generated with the sampling rate fs = 51,200 readings/second and the amplitude = 1.

ð14:11Þ

According to the theory presented in (14.1) to (14.3), the expected result is x = 0, 707 x2 = 0, 5 x=0 From Fig. 14.2 it can be seen that theory and implementation in MATLAB® lead to equivalent results. The value of the DC component (x) with -1.472710-16 is so small that it can be equated to 0. The legend of the diagram was made via

400

14

Signal Analysis Methods and Examples

ð14:12Þ

The num2str statement converts the numeric values into strings while legend sets the legend. Example

Measurement of vibrations on two bearing blocks of a condition monitoring demonstration test rig (Fig. 14.3). ◄ At different speeds and a stepped speed ramp, the signals from the accelerometers on the left bearing pedestal were recorded at a sampling rate fs = 51,200 measured values/second. The signals are plotted as acceleration values over time and, as with the test signal, the values of the effective value (RMS), the mean value of the power (power) and the DC component of the entire signal are shown in the legend (Fig. 14.4). The evaluation of the three time signals shows that the relationships between signal amplitude, RMS value, mean value of the power as well as the DC component are no longer the same as for a sinusoidal signal. The rms values are initially surprisingly low compared to the visible signal amplitudes. Higher speed leads to higher signal amplitudes, to higher values of the RMS value and the mean value of the power. Calculating single values over the entire stepped speed rampup does not produce meaningful results. The calculated single number value is a considerable data reduction, but it represents the average value over the entire measurement period. For the description of the signal change over a time course, it is therefore necessary to divide the time signal into time segments.

2 6 1

3

5 4

Fig. 14.3 Measurement setup on the condition monitoring test stand: (1) Drive motor, (2) Torque measuring shaft, (3) Bearing pedestal left with temperature measuring point, (4) Shaft with radial runout, (5) Bearing pedestal right with temperature measuring point, (6) Magnetic particle brake

14.2

Signal Analysis in the Time Domain

401

Bearing left 1020/min

Bearing left 4020/min

10

30

0.29164 m/s 2 RMS 0.085053 (m/s 2 )2 Power

3.0315 m/s 2 RMS 9.1901 (m/s 2 )2 Power

20

0.13433 m/s 2 DC

0.111 m/s 2 DC

Amplitude [m/s2]

Amplitude [m/s2]

5

0

10 0 -10

-5 -20 -10

-30 0

5

10

15

20

0

Time [s]

5

10

15

20

Time [s]

Bearing left Speed run-up 7.4396 m/s 2 RMS

150

55.3482 (m/s 2 )2 Power 0.13309 m/s 2 DC

Amplitude [m/s2]

100 50 0 -50 -100 -150 0

20

40

60

80

100

120

Time [s]

Fig. 14.4 Representation of the recorded time signals at the bearing pedestal on the left of the condition monitoring test rig for the speeds 1020/min, 4020/min as well as a stepped speed ramp-up

14.2.3 Envelopes The moving RMS value described in Eq. 14.4 is one of the envelope curve methods. This is implemented in analog level meters and is used in digital measurement technology, among other things, for the real-time visualization of time signals. The subsequent application to long time signals is computationally intensive and therefore requires long computing times. Figure 14.5 contains the representation of the time signal (upper diagram) and the course of the moving RMS value (lower diagram) over time, with a time constant τ = 0, 125 s. Each individual value in the course of the moving RMS value is the RMS value over the - ðt - ξ Þ period τ = 0.125 s of a signal section weighted with e τ . The signal section is calculated into the “past”. For the determination of the moving RMS value, the weighting function wðξÞ = e

- ðt - ξ Þ τ

mit ξ = 0, . . . , t

ð14:13Þ

for the used time constant τ and the sampling rate fs given by the measurement. In application example this is τ = 0, 125 s and fs = 51200 Messwerte/ Sekunde. This is done by the statement sequence

402

14

Signal Analysis Methods and Examples

Time signal acceleration sensor bearing left 200 7.4396 m/s 2 RMS

150

55.3482 (m/s 2 )2 Power 0.13309 m/s 2 DC

Amplitude [m/s2]

100 50 0 -50 -100 -150 -200 0

20

40

60

80

100

120

100

120

Time [s]

Effective value acceleration sensor bearing left 25 Moving RMS value

= 0,125 s

Amplitude [m/s2]

20

15

10

5

0 0

20

40

60

80

Time [s]

Fig. 14.5 Representation of the time signal of the acceleration transducer on the bearing pedestal on the left of the condition monitoring test rig during the stepped speed run-up. The upper diagram shows the time signal over time, the lower diagram shows the moving RMS value

ð14:14Þ

Make sure that blocksize becomes an integer value. This gives the weighting function (Fig. 14.6) 6400 values in the application example. For the process-optimized calculation of the moving RMS value, the measured values are squared and multiplied by 1/fs, this maps x2(ξ) dξ from (14.14). ð14:15Þ This reduces the calculation effort, which must be carried out for each individual measured value, to t

x r ð nÞ = or in MATALB notation

ξ = 1

wðξÞ xðξÞ τ

ð14:16Þ

14.2

Signal Analysis in the Time Domain

403

Weighting function

1.2

Weighting factor

1

0.8

0.6

0.4

0.2

0 0

1000

2000

3000

4000

5000

6000

Fig. 14.6 Representation of the weighting function for the application example

ð14:17Þ n serves as a run variable and maps the number of original measured values of the signal. At the beginning of the calculation run, as long as n has not yet exceeded blocksize, the calculation is performed with a correspondingly shortened weighting function. The calculation of the moving RMS value for the application example requires several minutes of computing time and does not result in any data reduction. If, in addition, a data reduction is to take place, then this can be achieved via a modified time resolution of the moving RMS value. In the previous determination, the time resolution is 1/fs, in the application example 1/51200 s. If, for example, Δt = 1 ms is selected for the temporal resolution of the signal analysis, the amount of data is reduced by a factor of 52. Numerically correct would be a factor of 51.2. However, since no interpolation is performed for intermediate values, the actual Δt deviates slightly from 1 ms. The calculation according to (14.16) or (14.17) is then only carried out for every 52nd measured value. Figure 14.7 shows the comparison between a time resolution Δt = 1/51200 s and Δt = 0.001 s. Both courses of the moving RMS value show almost no deviations from each other. A comparative representation in a diagram would only show one signal curve visibly. Data reduction by reducing the temporal resolution is therefore permissible. However, Fig. 14.8 also shows that reducing the temporal resolution can result in a loss of information. A Δt to 0, 1 s still appears acceptable in the application example, while for Δt = 1 s the loss of information is clearly visible. This may be otherwise depending on the application. This also applies to all other methods for representing envelopes. Another method for determining the envelope of a time signal is the MATLAB® function.

404

14

Signal Analysis Methods and Examples

Effective value acceleration sensor bearing left 25

Amplitude [m/s 2]

moving RMS value

= 0,125 s,

t = 1/51200 s

20 15 10 5 0

0

20

40

60

80

100

102

Time [s]

Effective value acceleration sensor bearing left 25

Amplitude [m/s 2]

moving RMS value

= 0,125 s,

t = 0.001 s

20 15 10 5 0 0

20

40

60

80

100

102

Time [s]

Fig. 14.7 Comparison of different time resolutions of the moving RMS value. The upper diagram has a time resolution Δt = 1/51200 s, while the lower diagram with Δt = 0.001 s has a considerably coarser time resolution Effective value acceleration sensor bearing left 25 moving RMS value moving RMS value moving RMS value

Amplitude [m/s 2]

20

= 0,125 s, = 0,125 s, = 0,125 s,

t = 0.001 s t = 0.1 s t=1s

15

10

5

0

0

20

40

60

80

100

120

Time [s]

Fig. 14.8 Comparison of different time resolutions of the moving RMS value

ð14:18Þ is displayed. For each value from the data vector, a value is calculated for the upper and lower envelope according to the selected method and the specified block size. The method

14.2

Signal Analysis in the Time Domain

405

is similar to the moving RMS value, it is not calculated in the “past” and it does not weight the values. The following methods are available • rms – calculates the rms value over the number of values specified in Blocksize from the data vector • peak – determines the maximum value from the block size wide interval • analytic – determines the analytically correct envelope curve are available. Blocksize specifies the number of values to be used for calculation for the respective method. The block size must be an integer value. The larger the block size is selected, the smoother the curve progression of the envelope. Example

An airborne sound recording of an electric adjustment drive for vehicle seat adjustment is available. The recording was made during an adjustment process. The time signal is shown in Fig. 14.9. The signal was described as “whining”. The qualification of the noise as “whining” indicates a modulation of the signal, which is already visible in the time signal representation. Via ð14:19Þ the envelopes of the positive and negative signal deflections are determined. The “peak” method often provides the better usable result than the “analytic” method. This is particularly the case when measuring signals with higher sampling rates are involved, such as with fs = 48000 Messwerte/Sekunde in this application example (Fig. 14.10). ◄ Both envelopes, the upper and the lower one, can be used for further analyses and representations. Further analyses on the envelope are usually performed on the upper, the positive envelope. The envelope is available in the same data density, i.e. also in the same sampling rate as the original time signal. As a further analysis, a frequency analysis, for example, can now be performed on the entire signal. The modulation depth m, the ratio of the modulation amplitude to the carrier amplitude, is calculated from the envelope. Since the carrier amplitude cannot usually be correctly determined from a measured signal, the arithmetic mean value of the envelope is used as a substitute. The modulation amplitude is the peak value around which the arithmetic mean of the envelope fluctuates. Under this condition, the modulation depth is given by m =

x - x 100% x

ð14:20Þ

406

14

Signal Analysis Methods and Examples

Time signal electric drive

Amplitude [Pa]

0.2

0.1

0

-0.1

-0.2

-0.3 0

2

4

6

8

10

12

14

16

18

20

Time [s]

Fig. 14.9 Time signal of the electric adjustment drive evaluated as “whining” Time signal electric drive Time signal Enveloping positive Enveloping negative

Amplitude [Pa]

0.2

0.1

0

-0.1

-0.2

-0.3

0

2

4

6

8

10

12

14

16

18

20

Time [s]

Fig. 14.10 Time signal of the electric actuator evaluated as “whining” with the envelopes of the positive and negative signal deflection

This degree of modulation is also referred to as fluctuation strength or degree of fluctuation. However, the use of the term “fluctuation strength” leads to confusion with the phsychoacoustic quantity fluctuation strength, which is defined completely differently. Executed in MATLAB® instructions, the modulation depth is calculated via ð14:21Þ

14.2

Signal Analysis in the Time Domain

407

Envelope of the time signal Electric drive 0.3

Amplitude [Pa]

0.25

0.2

0.15

0.1

0.05

0 0

1

2

3

4

5

Values [-]

6

7

8

9

105

Fig. 14.11 Envelope of the electric drive time signal. The markings on the signal curve represent the determined peaks

In practical application, the problem still arises that no unique x can be determined. Instead of x, the arithmetic mean of the signal peaks found in the envelope is used. Strictly speaking, this is then a mean modulation. In the application example, this is m = 9:97%. About the MATLAB® function ð14:22Þ Figure 14.11 is created. Here the signal course of the envelope and the determined signal peaks are marked. If the function findpeaks is embedded in another function or if the result of findpeaks is assigned to a variable, then no graphical representation takes place.

14.2.4 Crest Factor The crest factor is defined as the ratio of the crest value to the rms value. It is used to characterize the impulsiveness of periodic and stochastic oscillations: CF =

j xmax j x

ð14:23Þ

The maximum magnitude value2 is divided by the RMS value. A high crest factor indicates that the signal contains a lot of pulses. For a sinusoidal oscillation, for example, this results 2

The definition and formula symbols of the maximum value of the amount according to DIN 1311-1 are used, which is defined as the largest value of the amount (first form the amount, then calculate the maximum).

408

14

Signal Analysis Methods and Examples

p in a crest factor CF = 2. The determination of the magnitude maximum value depends on the observation duration: the longer the measurement, the greater the probability of detecting an even larger magnitude maximum value. Due to the integration process, on the other hand, the RMS value is less affected by the observation duration. For signal analysis, the crest factor should be understood as a trend indicator for signals with regularly occurring pulses in the observation duration. Measured stationary noise signals show a crest factor of ≤3, since the normal distribution is truncated towards the large magnitude maximum values (e.g. by the input voltage range of the measurement technology). In rolling bearing diagnostics, a crest factor of >3.6 is considered a warning condition [5]. With the MATLAB® function ð14:24Þ the crest factor of the vector x is determined. As can be seen from Fig. 14.12, there is basically a connection that a crest factor deviating from CF = 1, 4142 indicates that the evaluated signal is not a sinusoidal oscillation. However, this is not certain in all cases. The crest factor of a square wave signal with 50% division is also CF = 1, 4142.

14.2.5 Autocorrelation and Cross-Correlation The autocorrelation function Rxx(τ) describes the similarity of the oscillation x(t) with itself and is calculated for stationary signals according to T

Rxx ðτÞ = lim

T →1

1 2T

xðt Þ xðt þ τÞ dt

ð14:25Þ

-T

The oscillation x(t) is multiplied by the oscillation x(t + τ) shifted by the time τ on the abscissa, the product is integrated in the considered time interval and normalized to the time interval of the observation period. The autocorrelation function is plotted as a function of time shift τ and has the properties: • The autocorrelation function is an even function Rxx(-τ) = Rxx(τ). A shift by τ produces the same function value as the shift by +τ. Therefore, the autocorrelation function is only plotted for τ > 0. • The global maximum is reached by the autocorrelation function for a value of τ = 0. This corresponds to the – as plausible as trivial – fact that the function is most similar to itself when it is not shifted along the time axis.

14.2

Signal Analysis in the Time Domain

409 SagezahnF = 1.732

Triangle C F = 1.7321

Sine C F = 1.4142 1

1

0.5

0.5

1.2

0

0.8

Amplitude

Amplitude

Amplitude

1

0

-0.5

-0.5

-1

-1

0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

0

1

0.2

Rectangle CF = 2.0001

1.2 1

0.8

0.8

0.6 0.4

0.6

0.8

0

0

0.4

0.6

0.8

1

Rectangle C F = 1.4142

Time signal electric drive CF = 3.9368

0.2

0.4 0.2

0.2

Time [s]

0.6

0.2

0

1

Amplitude [Pa]

1

Amplitude

Amplitude

1.2

0.4

Time [s]

Time [s]

0.1 0 -0.1 -0.2 -0.3

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0

5

10

15

20

Time [s]

Time [s]

Time [s]

Fig. 14.12 Crest factors of various signals

• For a periodic function x(t) the autocorrelation function is also periodic and has the same period T. For noise, one obtains a maximum at τ = 0 and no further periodic maxima. • For τ = 0, the square of the rms value or the average power Rxx ðτÞ = x2 is obtained in the autocorrelation function. • The Fourier transform of the autocorrelation function is the power spectral density Sxx( f ). This quantity is also called auto spectral density, auto power density or power spectral density (PSD) and describes the frequency content of the autocorrelation function. Replacing the time-shifted function x(t + τ) with a function y(t + τ) in the integrand of Eq. 14.25, we obtain the cross-correlation function Rxy(τ). T

Rxy ðτÞ = lim

T →1

1 2T

xðt Þ yðt þ τÞ dt

ð14:26Þ

-T

The cross-correlation function describes the mutual dependence of the two functions x(t) and y(t), when y(t) is shifted on the time axis with respect to x(t) by the parameter τ. Similar

410

14

Signal Analysis Methods and Examples

to the autocorrelation function, a maximum is obtained in the cross-correlation function when both functions are similar to each other. By Fourier transforming the cross correlation function, the cross power spectral density is obtained Sxy( f ). Power spectral density and cross power density are needed for calculating the transfer function and coherence, which further describe the relationship between two signals quantitatively. Fields of application for the auto- and cross-correlation function: • Determination of the period in a noisy signal: In the case of a superimposed periodic signal, the period T is found in the autocorrelation function. If, on the other hand, only noise is present, a single maximum is obtained at τ = 0. This can be applied, for example, when pressure pulsations in pipelines are masked by a strong flow noise. • Detection of echoes in a signal: The autocorrelation function can be used to determine propagation times in a signal and to identify one or more echoes. For mean-free signals, the autocorrelation function for τ → 1 decays to zero. • Identification of a common signal in noisy signals: Cross-correlation can be used to detect signal components that occur in both functions. • Determination of propagation times between two signals: From the cross-correlation function, a constant delay difference between two signals is obtained. If the function y(t) is identical to x(t) but shifted by the transit time τ, the cross-correlation function yields a maximum at +τ. If two signals from independent measurement systems are present that were started at different times, this method can be used to determine the time difference between the two signals and create a common time base. The measurement method assumes a constant propagation time or phase shift in the signal section under consideration, and is therefore suitable for determining the phase shift angle at constant frequency or speed, for example. With increasing displacement by the parameter τ the problem arises that the two functions x(t) and x(t + τ) or y(t + τ) originate from two different time periods. In the definition equation, the problem is circumvented by pushing the integration limits to infinity. With a finite signal section, the end of the interval is finally reached with increasing shift by τ. If the signal section is now continued periodically, this estimation of the future signal course does not have to agree with the actual signal course (if the observation period had been chosen longer). Thus, one does not compare two signals, but circulating (circulating) signal sections, which can lead to wrong results. This effect is called circular correlation. This occurs both with correlation functions in the time domain and with the usual evaluation in the frequency domain using auto and cross power density spectra. One remedy is, for example, to double the length of the signal section and fill the second half with zeros (zero padding). The correlation is then possible up to half the interval length of the extended signal (corresponds to the signal length of the output signal before zero padding). For stationary signals, the product in the integrand becomes smaller the more zeros are added

14.2

Signal Analysis in the Time Domain

411

left-hand bearing signal 30

Acceleration [m/s 2]

20 10 0 -10 -20 -30 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.7

0.8

0.9

1

Time [s]

Autocorrelation left bearing signal 0.5 0.4

Rxx ( )

0.3 0.2 0.1 0 -0.1 0

0.1

0.2

0.3

0.4

0.5

0.6

[s]

Fig. 14.13 Time signal (top) of the accelerometer at the bearing pedestal of the condition monitoring demo test rig and its normalized autocorrelation function (bottom)

from the second signal section. The correction is made by dividing by a correction function linearly decreasing to zero (“Bow-tie correction”) [1–3]. The choice of signal is crucial for the correlation analysis. In order to let the maxima stand out clearly, a signal without mean value is recommended. Since the signal sections cannot be shifted arbitrarily far against each other, the length of the signal section should correspond to a multiple of the assumed period duration for periodic signals. Example: Application of the Autocorrelation Function

On the condition monitoring demo test stand, a measurement is carried out with an accelerometer on the left bearing pedestal at a target speed n ≈ 4000/min. The normalized autocorrelation function (Fig. 14.13 below) shows a strict periodicity of the function with τ = 0, 0153 s. This corresponds to the actual value of the speed n = 3922/min. This cannot be seen from the time signal of the accelerometer (Fig. 14.13 above). ◄

412

14

Signal Analysis Methods and Examples

The normalized correlation functions are calculated using the MATLAB® function xcorr for the autocorrelation function in the call variant ð14:27Þ and for the cross-correlation function in the call variant ð14:28Þ xcorr returns a bipartite result from τ = - T to τ = T, so for practical application the values for the range τ > 0 only are used. The reduction is done via ð14:29Þ The value for τ = 0 is now missing, but this does not lead to any problems in practice. The formation of an envelope curve around the result of the correlation analysis often leads to an improved interpretability of the results.

14.2.6 1/n Octave Bandpass Filtering In the field of application of technical acoustics, standardization and legislation often refer to octave or third octave levels. According to the noise protection regulations (TRLV Lärm, Arbeitsstättenverordnung, etc.), octave or third-octave analyses, for example, are completely sufficient for determining the sequential components of the noise. Naturally, manufacturers of technical products and components derive regulations and technical instructions for the assessment of acoustic and mechanical vibrations from this. The starting point is the musical definition of the octave, which is defined as the interval between two tones whose frequencies are in the ratio 2 : 1 and has eight intermediate tones (see Sect. 7.3).3 Derived from this, each frequency doubling has its own bandpass filter, as described in Sect. 10.6, which allows only the frequency range of the respective octave [27] to pass from a time signal. Earlier technical realizations of octave analyzers were realized by analog bandpass filters arranged in parallel (see Fig. 14.14). Modern digital realizations of 1/n octave analyzers work according to the same principle. For the determination of the cut-off frequencies of the octave bandpass filter the following applies

3

The term octave comes from the Latin octava and means “the eighth”.

14.2

Signal Analysis in the Time Domain

413 Octave Analysis

12

10

RMS value

8

6

4

2

0 31,5

63

125

250

500

1000

2000

4000

8000

16000

Frequency [Hz]

Signal 1

0.5

Bandpass filter 31,5 Hz

moving average

Bandpass filter 63 Hz

moving average

Bandpass filter 125 Hz

moving average

Bandpass filter 250 Hz

moving average

Bandpass filter 500 Hz

moving average

Bandpass filter 1000 Hz

moving average

Bandpass filter 2000 Hz

moving average

Bandpass filter 4000 Hz

moving average

Bandpass filter 8000 Hz

moving average

Bandpass filter 16000 Hz

moving average

0

-0.5

-1

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

Fig. 14.14 Schematic diagram of an octave analyzer

f1 =

f2 2

ð14:30Þ

and f0 =

f1  f2

ð14:31Þ

with f0 the center frequency of the bandpass filter f1 the lower cut-off frequency of the bandpass filter f2 the upper cut-off frequency of the bandpass filter 1/n-octave filters for measurements are standardized according to DIN EN 61260, where e.g. cut-off frequencies f1 and f2, center frequency f0, bandwidth B and filter quality Q, but not the slopes in dB/Dec are specified. Most measurements are performed with filters and standard frequencies of series b according to DIN EN ISO 266 [28], where the frequency f0 = 1000 Hz is defined as the center frequency. Other standard frequencies are possible

414

14

Signal Analysis Methods and Examples

and, depending on the area of application, also common. For example, f0 = 50 Hz and f0 = 60 Hz are standard frequencies for shipbuilding applications due to the energy supply, and f 0 = 16 23 Hz is also a standard frequency for railways. Based on the standard frequencies, the bandpass filters required for the analysis are determined via Eqs. 14.31 and 14.30. Third octave filters correspond to 1/3 octave filters. For frequency doubling, thirdoctave filter analysis uses three filters instead of one, as in octave filter analysis. Other, even narrower frequency bands are also common. For the determination of the cut-off frequencies of the 1/n-octave filters the following applies f2 f1 = p n 2

ð14:32Þ

and f0 =

f1  f2

= f1 

p n

2

ð14:33Þ

Table 14.2 lists the common 1/n-octave filters used for vibration analysis. The 1/8 octave filter analysis is uncommon, but it maps the eight intermediate tones of the musical octave. The realization of 1/n octave analyzers involves a parallel connection of bandpass filters, as shown in Fig. 14.14. Filters, including digital filters, have the unpleasant property of transient oscillation for this application. It takes a period of time, dependent on the filter bandwidth, for the filter used to provide usable filtered signals. The settling time tr to be considered for the analysis cannot be determined exactly without knowledge of the specific filter design or filter programming. For 1/3-octave analysis with a smallest filter bandwidth of 4, 6 Hz, according to [29], tr = 0, 8 s is used for the transient response. For 1/24-octave analysis with a smallest filter bandwidth of 0, 024 Hz, this is tr = 137 s. The transient response of a filter always becomes effective when large amplitude differences, i.e. signal jumps, occur in the signal to be analyzed. This is the case at least at the beginning of the signal. Figure 14.15 shows an example of the transient response. For this purpose, a sinusoidal signal with the center frequency of the first 1/n-octave filter with the unit amplitude 1 was generated. For the 1/1-octave filtering this is f = 31, 5 Hz for the 1/24-octave filtering f = 20, 2 Hz. The signals were 1/n-octave filtered followed by enveloping. The 1/n octave filters were defined with an eighth order slope. For Fig. 14.16 the experiment was extended by generating a more complex signal. In the first 10 s, the signal exhibits an oscillation with the unit amplitude 1 at f = 20, 2 Hz. For the second 10 s, the amplitude of the signal was reduced abruptly to 0.5. In the third 10 s, the amplitude is lowered to 0 and a short pulse with amplitude 1 is set at the beginning of this segment.

14.2

Signal Analysis in the Time Domain

415

Table 14.2 Bandwidths at f0 = 1000 Hz and cut-off frequencies 1/n-octave filter Filter 1/1 octavo 1/3 octave (third) 1/6 octave 1/8 octave (unusual) 1/12 octavo 1/24 octavo

f2/f1 2.0000 1.2599 1.1225 1.0905

f1/f0 0.7071 0.8909 0.9439 0.9576

f2/f0 1.4142 1.1225 1.0595 1.0443

f1 at 1000 Hz 707 Hz 891 Hz 944 Hz 958 Hz

f2 at 1000 Hz 1414 Hz 1122 Hz 1060 Hz 1044 Hz

1.0595 1.0293

0.9715 0.9857

1.0293 1.0145

972 Hz 986 Hz

1029 Hz 1014 Hz

Bandwidth 707.1 Hz 231.6 Hz 115.6 Hz 86.7 Hz 57.7 Hz 28.9 Hz

Transient response 1/n octave filter 1.2 1/1 octave filter 1/24 octave filter

normalized amplitude

1

0.8

0.6

0.4

0.2

0 0

1

2

3

4

5

6

Time [s]

Fig. 14.15 Transient response of 1/n-octave filtering using the example of the first 1/n-octave filter for 1/1-octave filter and 1/24-octave filter in each case

In practical applications, this means that signals with highly erratic amplitude changes – e.g. signals containing pulses – 1/n-octave analyses with bandpass filters as wide as possible, such as 1/1-octave or 1/3-octave, will provide better results. For any 1/n-octave analysis, there must be sufficient signal duration for the bandpass filters to settle. Another problem to be considered when using 1/n-octave analyses is shown in Fig. 14.17. For this purpose, a 1/1 octave analysis was performed on a time signal, white noise with sampling rate fs = 64000 Hz. Subsequently, a Fourier transformation was performed for the filtered signals with f0 = 1000 Hz and f0 = 2000 Hz and the respective magnitude spectrum was plotted in normalized form. The two filter ranges under consideration do not connect to each other without gaps, but overlap in a not insignificant frequency range. Signal components will thus flow into the result both in one octave and in the adjacent octave; however, with very little weighting and therefore of secondary importance in practical use. More problematic is the area between the adjacent octaves. Here an evaluation gap arises.

416

14

Signal Analysis Methods and Examples

Time signal with abrupt amplitude changes

normalized amplitude

1

0.5

0

-0.5

-1 0

5

10

15

20

25

30

20

25

30

1/24 octave filtering 1.2

normalized amplitude

1 0.8 0.6 0.4 0.2 0

0

5

10

15

Time [s]

Fig. 14.16 Time signal with the frequency f = 20, 2 Hz and abruptly changing amplitudes (top) and signal characteristic of the first octave filter of a 1/24 octave filtering (bottom) Octave Analysis Ocatve f0 = 1000 Hz

1

normalized amplitude

Octave f0 = 2000 Hz 0.8

0.6

0.4

0.2

0 0

500

1000

1500

2000

2500

3000

3500

4000

Frequency [Hz]

Fig. 14.17 Magnitude spectra for the signal components with f0 = 1000 Hz and f0 = 2000 Hz of an octave-filtered noise signal

14.2

Signal Analysis in the Time Domain

417

1/3 octave “white noise” analysis 0.4 0.35 0.3

Level

0.25 0.2 0.15 0.1 0.05 0 0

63

200

631

1995

6310

19953

Fourier transform “white noise” 0.2

Level

0.15

0.1

0.05

0 10

1

10

2

10

3

10

4

Frequency [Hz]

Fig. 14.18 Frequency analysis of the “white noise” signal (top) 1/3-octave analysis (bottom) Fourier transform

At the frequency f2 of the left octave is the intersection to the right octave ( f1 of the right octave). Due to the slope and the definition of filter, the bandpass filter already has 1/3 (3 dB) of its filter effect at the cut-off frequencies f1 and f2. Signal components in this frequency range will appear with reduced level values in the analysis result. A larger filter slope only leads to a limited solution of the problem. Only the frequency range of the transition is reduced. A different standard frequency instead of 1000 Hz leads to a different positioning of the transition ranges and can be a problem solution. If white noise is analyzed, as shown in Fig. 14.18, then a 1/n octave analysis can be used to visualize the subjective impression of this signal. White noise is a test signal with constant power density over the entire frequency range. As noise, white noise is perceived as having a very strong treble emphasis. Since 1/n octave filters have wider bandwidths of the bandpass filters at higher frequencies, higher level values result.

418

14

Signal Analysis Methods and Examples

Example

For the airborne sound recording of an electric actuator for vehicle seat adjustment already used in Fig. 14.9, the modulation depth for the different frequency bands is determined.

ð14:34Þ

To want to perform a 1/n octave analysis via MATLAB®, all that is required is to call the function shown in Eq. 14.34. The data required are tdata, the time signal to be analyzed ntel, the divisor from 1/n as integer value filterord, the order number of the edge steepness e.g. 8 for 80 dB per decade f0, the standard frequency is usually 1000 fs, the sampling rate of the time signal The fdesign.octave instruction is used to define the filters and to determine the center frequencies via validfrequencies. In the subsequent loop, the 1/n octave filter is filtered out of the time signal for each bandpass filter. The signal filtering is done by the instruction filter, while design creates the respective filter. The function is activated via ð14:35Þ is called. The next step of the analysis is the envelope formation. For this only the instruction ð14:36Þ which determines one envelope for each of the n filtered signals. For all n envelopes, the modulation depth for the respective frequency band is determined in the next step, as already described in Eq. (14.21). The determination of the modulation depth considers the time range from the third to the 17th s of the signal. This hides the transient response of the 1/n octave filtering for the analysis.

14.2

Signal Analysis in the Time Domain

419 Time signal

60

Modulation depth [%]

50

40

30

20

10

0

0

61

194

613

1939

6131

19387

Frequency (1/3 octave) [Hz]

Fig. 14.19 Representation of the modulation degrees to the frequency ranges of a 1/3-octave filtering of the measurement signal

Figure 14.19 shows that not individual frequency ranges of the measurement signal are strongly modulated, but that almost all frequency ranges up to approx. 4000 Hz of the 1/3-ocative analysis are modulated. The display takes place via the statement ð14:37Þ Here the variable Modulation is a vector and contains the calculated modulation degrees. The abscissa axis is initially labeled 0 20 40 . . . 120, since a total of 120 modulation values are present for the 120 individual bandpass-filtered time signals of the 1/3-octave filtering. The numerically correct abscissa labeling is achieved by exchanging the corresponding labels on the abscissa axis. The values from the vector frequencies are used for this purpose.

ð14:38Þ

Since the vector contains the numerically correct frequencies, these are rounded up by round. ◄

420

14.3

14

Signal Analysis Methods and Examples

Signal Analysis in the Frequency Domain

14.3.1 Amplitude Density If the measured vibration values x(t) are divided into classes of constant width and the frequency with which certain values fall into the classes is determined, a frequency distribution is obtained [4–7]. For infinitely long measurement time and infinitely narrow width of the classes, this converges to the amplitude density p(x). The amplitude density (or probability density) indicates the probability with which a value occurs. If many similar and statistically independent events overlap, the distribution is a normal distribution (Central Limit Theorem). For a normally distributed random variable, the amplitude density p(x) is described by the normal distribution (Gaussian distribution) with the mean value μ and the standard deviation σ. pð x Þ =

1 p σ 2π

e - 2ð

Þ

1 x-μ 2 σ

ð14:39Þ

This function has the properties: • The maximum is the mean value μ • The function is mirror symmetrical to a straight line parallel to the ordinate with x = μ. • Integrating the area under the amplitude density gives the probability distribution PðxÞ =

x -1

pðxÞdx. Integration in the interval -1 to +1 yields P(x) = 1, i.e. the

function p(x) is normalized. • The standard deviation σ influences the width and height of the distribution. When the amplitude density is displayed, the frequency and sequence information is lost. For a sinusoidal oscillation and a synthetically generated, normally distributed noise, sections of the time function and the amplitude density are shown as a histogram in Fig. 14.20. The amplitude density distribution of a normally distributed noise (Fig. 14.20 top) approaches the normal distribution for a long observation period T, which is additionally drawn as a solid curve. For the sine function (Fig. 14.20 bottom), the largest values of the amplitude density are obtained at the reversal points (maxima and minima). If a normal distribution is present, the mean μ and the variance σ 2 can be estimated from the measurement. The arithmetic mean value x is an estimate for the mean value of the normal distribution μ in Eq. 14.39. The variance is calculated as the mean square deviation from the arithmetic mean value.

14.3

Signal Analysis in the Frequency Domain

421

1

0.5

0.5

0

0

-0.5

-0.5

x(t)

1

-1 0

-1 0.02

0.04 0.06 Time [s]

0.08

0.1

0

0.5

0.5

0

0

-0.5

-0.5

x(t)

2 3 Frequency

4

5 х105

1

1

-1 0

1

-1 0.02

0.04 0.06 Time [s]

0.08

0.1

0

1

2

3 Frequency

4

5

6 х105

Fig. 14.20 Time course and amplitude density – above normal distribution below sinusoidal oscillation T

σ

2

1 = T

ðxðt Þ - xÞ2 dt

ð14:40Þ

0

The standard deviation σ as the square root of the variance has two meanings: • With an arithmetic mean of zero (x = 0), the standard deviation corresponds to the rms value σ = x • In the interval from -kσ to +kσ, the area under the amplitude density curve p(x) indicates the probability of finding the percentage of samples within the bounds of ±kσ. Via

422

14

Signal Analysis Methods and Examples

ð14:41Þ

the histogram is generated in MATLAB®. By assigning it to a variable, the result of the classification can be accessed until the graph is closed. For further use of the classification, it is recommended to use

ð14:42Þ

in variables.

14.3.2 Counting Procedures Counting methods can be understood as a special notation of the amplitude density and are widely used in fatigue strength [8]. Basically, the following possibilities for counting arise: • • • • • •

Reversal point (maximum/minimum), Range is swept (span from minimum to maximum), Hysteresis loops in the stress-strain diagram are closed with repeated loading, Class boundary is crossed or crossed over, Measured variable is determined at fixed time intervals, Measured variable is determined as a function of a specified other variable (speed, angle of rotation).

All procedures have in common that • these give a frequency distribution of the amplitudes and • frequency and sequence are eliminated from the time function.

14.3

Signal Analysis in the Frequency Domain

423

A distinction is made between so-called one-parametric and two-parametric counting methods. Single-parameter counting methods record a frequency distribution as a function of one parameter (e.g. absolute value or vibration amplitude). Two-parameter counting methods provide a frequency distribution as a function of two parameters (e.g. vibration amplitude and mean value). The existing measurement signal is divided into classes of equal width and these are numbered in ascending order. A so-called reset width is defined in each class. The reset width prevents the count from being triggered if the measurement signal fluctuates around a class limit due to small amplitude changes. This can be the case, for example, with fine class division and a small reset width. In the selection of the number of classes (or class width), the expected maximum stress and the desired information compression can serve as orientation. The usual number of classes is e.g. 64. From the historical development, a number of counting methods exist, which are based on different algorithms for data reduction. Rainflow counting is presented as a representative example of contemporary counting methods, which has become a widely used method. Rainflow counting is based on the idea that raindrops flow from left to right on the time axis. For the sake of clarity, the present example is based on 8 classes. The count starts at a maximum. In Fig. 14.21, the raindrops run from maximum (a) to minimum (b) and then drip onto the roof below (i.e. to the right of it). At point d, the drop is stopped at the reversal point and results in a first half cycle (opening hysteresis loop). This half cycle is completed by a second half cycle d-e (closing hysteresis loop) to form a full cycle (a-d-e) (Table 14.3). At point b, the drops run along the inside of b to the reversal point c (first half cycle) and reunite with the drops that have dripped down at point b (second half cycle) to form the full cycle. The full cycles are now counted and displayed as frequencies in a square matrix (Fig. 14.22). The rows of the matrix indicate the maxima of each full cycle, the columns the minima. This matrix is called the full matrix. By mirroring it on the diagonal, one obtains the so-called half matrix and thus a further compression of information (Fig. 14.23). Here the standing and hanging full cycles are combined (e.g. b-c-b and o-p-o). This removes from the signal the direction in which the full cycles are traversed. This information is of no importance for applications in fatigue strength. It is also possible to carry out the count in such a way that the oscillation widths above the mean values are recorded in a full matrix (Fig. 14.24). The following information, among others, can be derived from the half-matrix: • Number of maxima and minima per class by summation. The maxima are obtained by row-wise summation, the minima by column-wise summation (Fig. 14.25). • Oscillation amplitude and mean value by diagonal summation (Fig. 14.26). On the main diagonal, the amplitude is zero. Lines parallel to the main diagonal indicate lines of equal amplitude. As the distance from the main diagonal increases, the amplitude increases. Along the main diagonal, the mean values increase from “top left” to “bottom right”. Lines perpendicular to the main diagonal indicate lines of equal mean values. • Full matrix in the form of the amplitude and mean (Fig. 14.27).

424

14

Fig. 14.21 Principle of Rainflow counting. (Data from [8], modified)

Class 8 a 7 6 5 c 4 3 b 2 1

Signal Analysis Methods and Examples

e

g

m

i

q

k f o j p h l

d

n

Table 14.3 Rainflow counting method from Fig. 14.21 Full cycle a-d-e b-c-b e-h-i f-g-f i-l-m j-k-j m-n-q o-p-p

From 8 3 8 7 8 5 8 4

According to 1 4 2 8 1 7 2 3

Mean value 5 4 5 7 5 6 5 4

Oscillation width 7 1 6 1 7 2 6 1

8 7 6 5 4 3 2 1

From

Fig. 14.22 Full matrix

1 2 3 4 5 6 7 8 to

The remaining open cycles are called residuals and are not initially included in the counting result. If the time signal is sufficiently long, it can be assumed that every opening hysteresis finds a closing hysteresis. It is also possible to derive one-parametric counting methods from the rainflow counting. The range pair count, for example, is obtained by considering only the amplitude and disregarding the information on the mean value. Results from one-parametric counting procedures are plotted as a collective. The sum frequency is plotted logarithmically on the abscissa, the ordinate contains the amplitude (e.g. force, stress, etc.) in linear order. Collectives are described by the four characteristics: Collective magnitude, collective extent, collective shape and constant mean stress or constant load ratio (as ratio of underload to overload).

14.3

Signal Analysis in the Frequency Domain

425

Maxima

1 2 3 4 5 6 7 8

Semi-Matrix

8 7 6 5 4 3 2 1

From

8 7 6 5 4 3 2 1

Full matrix

1 2 3 4 5 6 7 8

to

Minima

Fig. 14.24 Full matrix amplitude versus mean value

Oscillation width 8 7 6 5 4 3 2 1

Fig. 14.23 Derivation of the half matrix from the full matrix

1 2 3 4 5 6 7 8 Mean value

Often the two-dimensional representation in collectives is preferred, as these are more descriptive than three-dimensional Rainflow matrices. A number of conclusions and interpretations can already be derived from the collectives, such as plausibility checks of the measurements and comparisons between several measurements. Likewise, different operating conditions and their effect on the stress can be identified and estimated on the lifetime. For service life estimates, on the other hand, rainflow matrices should be used, as these include information about the mean value [8]. Application Example with MATLAB® Code

During a bad road run, accelerations were recorded on an assembly of a vehicle in the Z direction. A rainflow count is performed with this signal and the procedure is explained in MATLAB® (Fig. 14.28). For Rainflow counting, the MATLAB® function is available. ð14:43Þ which carries out the Rainflow counting. And as a function result

426

14

Class

Signal Analysis Methods and Examples

Number of maxima

Maxima

8 7 6 5 4 3 2 1

Semi-Matrix

2 1 5 1 2 3 4 5 6 7 8

Class

Minima

22 2

1

1

Number of minima

Fig. 14.25 Number of maxima and minima from the half matrix

Maxima 8 7 6 5 4 3 2 1

Semi-Matrix

1 2 3 4 5 6 7 8 Minima

Vibration amplitude increases

Mean value increases

Fig. 14.26 Oscillation amplitude and mean value from the half matrix

• c – Amplitude counts (cycle counts) • rm – Rainflow matrix from which the class frequencies and cumulative frequencies are calculated, oscillation width (rmr) x class mean (rmm) • rmm – Average values of the classes (Cycle Average) • rmr – vibration amplitudes of the classes in peak-to-peak (cycle range)

427

Maxima 8 7 6 5 4 3 2 1

Signal Analysis in the Frequency Domain

Maxima 8 7 6 5 4 3 2 1

14.3

1 2 3 4 5 6 7 8 Minima

1 2 3 4 5 6 7 8 Minima

7 6 2 1 Oscillation width

3,5 4,5 5 6 Mean value

7,5

Fig. 14.27 Derivation of the full matrix 80 60

Amplitude [m/s 2]

40 20 0 -20 -40 -60 -80 0

2

4

6

8

10

12

14

16

18

20

Time [s]

Fig. 14.28 Time signal for Rainflow counting

transmitted. The usual representation of the oscillation widths is via the cumulative frequency from the Rainflow count. In the first step (see Fig. 14.29) after calling the MATLAB® function rainflow, the formation of the class frequencies from the rainflow matrix is carried out via ð14:44Þ Since the class mean = 0 and the oscillation width = 0 do not occur, the rainflow matrix (rm) has the structure rmr-1 rows x rmm-1 columns. The statement sum(rm,2) sums the Rainflow matrix to the class frequency vector. For each oscillation width > 0, class frequency contains a value. The display (Fig. 14.29) is done via

428

14

Signal Analysis Methods and Examples

Frequency

Rainflow count Class frequency 10

4

10

3

10

2

101

10

0

10-1 0

18

38

58

78

98

118

138

Amplitudes peak-to-peak [m/s2]

Fig. 14.29 Rainflow matrix reduced to the class frequency vector, class frequencies plotted against the oscillation widths (peak-to-peak)

ð14:45Þ with subsequent exchange of the XTickLabels for the correct labeling of the abscissa axis. However, this form of representation is unusual in fatigue strength. The usual representation is that of cumulative frequencies Hi which contains the summation of the class frequencies hi of the respective class up to the highest class and thus complies with the calculation rule n

Hi =

hξ

mit i = 1 . . . n

ð14:46Þ

ξ=i

corresponds. In addition, the vibration amplitudes are plotted as ordinate values. Eq. 14.46 in MATLAB® code

ð14:47Þ

The class frequencies do not contain values for the oscillation width = 0, so the loop must be shifted by one class. The display (Fig. 14.30) is first done using the MATALB function bar, which displays the values in bar form, but as a sum frequency over oscillation widths. Afterwards all labels are made and as a last instruction the following is done with

14.4

Signal Analysis in the Frequency Domain

429

Amplitude collective 156

Vibration amplitude [m/s 2]

138 118 98 78 58 38 18 0 100

101

102

103

104

Total frequency

Fig. 14.30 Plot of amplitude (peak-to-peak) versus cumulative frequency

ð14:48Þ the exchange of abscissa and ordinate, so that the diagram is displayed as oscillation width over the cumulative frequency. In its current implementation (introduced in MATLAB® 2017b), the MATLAB® function rainflow has no configuration parameter that can be used to influence the number of classes. The calculation algorithm is based on the calculation rule presented in [31], which does not contain a definition of the number of classes. ◄

14.4

Signal Analysis in the Frequency Domain

14.4.1 Fourier Transform: FFT or DFT? As already described in Sect. 2.3.2, signals are regarded as additive superpositions of harmonic oscillations and are represented in the frequency domain as a spectrum in the form of amplitude and phase versus frequency. From amplitude and phase, fundamental and important information for the analysis of the oscillation process can be derived, some of which will be singled out: Evaluation of the amplitude spectrum: • Statements on the excitation frequencies (e.g. due to the drive) and the excited natural frequencies (e.g. due to imbalances), • Comparison of the measured spectrum with a limit spectrum,

430

14

Signal Analysis Methods and Examples

• Statements on mechanisms underlying the vibration process (condition monitoring, vibration reduction), Evaluation of the phase spectrum: • Balancing Technology, • necessary to synthesize the time signal from spectra. The purpose of the Fourier Transform (FT) is to transform discrete signals from the time domain to the frequency domain. The following illustrations are intended to give the user a guide to obtaining FT spectra and interpreting the results. The essential setting possibilities and stumbling blocks are pointed out, so that the user will (in most cases) obtain a reasonable result by reasonably chosen settings. The presentation does not claim to represent the FT in a mathematically comprehensive way. Therefore, only a minimum of mathematical tools is used (i.e. without mathematical representation of the convolution operation, the Drichlet function and the Dirac comb). Likewise, some procedures that go beyond the basics and are reserved for the expert are left untreated. For this, the reader is referred to the literature [2–7, 9–16, 24]. The term Discrete Fourier Transform (DFT) is derived from the objective of the Fourier Transform to transfer discrete signals from the time domain into the frequency domain. The second, more familiar term to users, the Fast Fourier Transform (FFT), is a computationally efficient algorithm for the Discrete Fourier Transform. There are several methods for mathematically performing the FFT. The best known and most widely used one is by James Cooley and John W. Tukey [17], who published it in 1965. A prerequisite for its applicability is that the number of discrete signal values is a power of two. This is usually not a problem, since the number of discrete signal values can be freely selected. The calculation of a Fourier transform as an FFT requires considerably less storage space and less computing capacity than that of a DFT. With today’s computing and storage capacities, considerations regarding computing times and storage space consumption play a subordinate role in the practical application of the Fourier transform. This was not the case at the beginning of digitization, in the 1960s. For historical reasons, the use of the FFT is therefore more widespread than that of the DFT. Almost all technical implementations as hardware and/or software analyzers use an FFT algorithm for the transfer of signals from the time domain to the frequency domain. This may be problematic, since the required number of values must also be available as a power of two. It is more modern to leave the decision to the performing analyzer or the performing program whether the transformation is calculated as FFT or DFT. This also leads to “smooth” frequency resolutions (. . .0, 5; 1; 1, 5; . . .Hz) if digitizing devices are used which are not adapted to the 2n laws. The transformation result of both calculation methods will not show any difference.

14.4

Signal Analysis in the Frequency Domain

431

Signal waveform x(t)

1,5

T

1,0

0,5

0,0 -30

t

-15

0

15

30

Sampling values k

0,15

N

Amplitude

N/2

0,10 f

0,05

0,00 -30

-15

0

15

30

Lines n

Fig. 14.31 Time and frequency range of the DFT

14.4.2 Fundamentals of the Discrete Fourier Transform Usually, the measurement signal in the time domain is in the form of a sampled signal, i.e. at discrete interpolation points (Fig. 14.31). The time section has the length T (block length) with N samples and a sampling interval t. The counting index k runs from zero (left interval boundary) to N - 1. The corresponding values in the time domain are denoted as x(k). The transformation is performed in blocks. The equation for the transformation from the time domain to the frequency domain is in this notation X ð nÞ =

1 N

N -1

xðkÞ e - j2π N

kn

ð14:49Þ

k=0

The counting index n runs in the frequency representation from 0 to N - 1. After transformation into the frequency domain, N discrete frequency values (lines) are then available as a discrete spectrum with the complex values X ðnÞ. The line spacing between two frequency lines of the spectrum is Δf and assigns a frequency to the counting index. As

432

14

Signal Analysis Methods and Examples

is immediately obtained by substituting n = 0 into Eq. 14.49, the 0th frequency line is real and contains the arithmetic mean x of the signal N -1

1 N

X ð 0Þ =

xðk Þ

ð14:50Þ

k=0

For the reverse transformation (inverse) the rule is written as follows xð k Þ =

N -1

kn

X ðnÞ ej2π N

ð14:51Þ

n=0

Besides this formulation, there are a number of equivalent notations, e.g. [3, 7, 13]. In particular, the prefactor before the summation sign 1/N is often assigned to the inward transformation instead of the backward transformation. A detailed discussion can be found in [1]. As already shown for the Fourier integral for continuous functions (Sect. 2.3.3), it is useful to extend the time domain by negative values on the time axis. The choice of the zero point on the time axis is arbitrary from the measurement technique anyway. In addition, the frequency range is extended by negative frequencies. This representation as a two-sided spectrum is advantageous for the mathematical implementation of the DFT. The two-sided spectrum is often used in signal processing, in vibration engineering and its applications the one-sided spectrum is preferred, in which only positive frequencies are plotted. Figure 14.31 shows a periodic, sampled square wave signal with N = 30. The zero point t = 0 is located at k = 0, i.e. the signal is symmetrical to the ordinate in the time domain. In the time domain the following then applies for the sampled values xð0Þ = 1, xð1Þ = 1, xð29Þ = 1 xð2Þ . . . xð28Þ = 0 For these three samples the sum is given by X ð nÞ = 1 = 30

1 N

N -1

xðkÞ e - j2π N

kn

k=0

1  e - j2π 30 þ 1  e - j2π 30 þ 1  e - j2π 30 0n

For the first summand you get immediately

1n

29n

14.4

Signal Analysis in the Frequency Domain

433

1  e - j2π 30 = 1 0n

Now the periodicity in the time domain is used advantageously, the penultimate sample (No. 29) corresponds to the first sample with a periodic continuation of the signal section e - j2π 30 = e - j2π 29n

- 1n 30

By combining the two minus signs, one obtains a rotating pointer in the mathematically positive sense. This forms a conjugate complex quantity with the second summand from the square bracket. Thus both summands result in a real quantity eþj2π 30 þ e - j2π 30 = 2  cos 2π 1n

1n

n 30

Two findings can already be formulated here: • By introducing a two-sided spectrum (and thus the negative frequency range), the calculation is simplified. • In this example, a purely real spectrum is obtained, i.e. it is composed only of cosine terms. The sine terms – which contain the imaginary parts – are omitted by the clever form of the choice of the zero point of the time counting. Of course, this is not possible in the measuring practice – but there the calculation is not done “by hand” either. The amplitude spectrum then takes the form j X ð nÞ j =

1 30

1 þ 2  cos 2π

n  30

For n = 0 you get j X ð 0Þ j =

1 ½1 þ 2 = 0, 1 30

The 0th frequency line corresponds to the arithmetic mean value obtained in the time domain by x = 3=30 = 0, 1. For n = 10, for example, this results in j X ð10Þ j =

1 1 1 þ 2  - Þ =0 30 2

Both values can be easily read from the spectrum. For measurement practice, the application of DFT has a number of consequences:

434

14

Signal Analysis Methods and Examples

Line Spectrum From the DFT, a discrete line spectrum is obtained, i.e. the spectrum is only defined at the supporting points (lines) n. The frequency resolution as the distance between two lines is given by Δf = 1/T. Between two grid points n and n + 1 the spectrum is not defined and contains no information. However, the practical representation of the spectrum is often done as a solid line and thus interpolates between the grid points. Periodicity in the Time Domain The N samples are continued periodically as a section of the interval of the block length T. This is shown in Fig. 14.32 for a sinusoidal signal where the signal length (observation duration) is greater than the block length. From the continuously measured and sampled signal (upper drawing), a time section with block length T is taken by the DFT and continued in both directions on the time axis (lower drawing). As Fig. 14.32, the periodic continuation of the block provides a different time signal than the continuation of the time signal in the measurement. This can be seen by comparing the upper and lower subimages. In order for the spectrum to contain the information content from the other signal sections of the measured signal, either the block length must be chosen sufficiently long or – often more expedient – averaging over several time sections is performed. No discontinuities (kinks and jumps) may occur at the interval limits of the block length. If the block length T is an integer multiple of the period of all frequencies contained in the signal, this condition is fulfilled. In general, however, this condition is not met. Window functions are used to reduce errors. Periodicity in the Frequency Domain The spectrum of the DFT is also periodic (Fig. 14.31). The periodicity arises when n/N in Eq. 14.49 becomes integer and repeats with a period of N lines in the spectrum. Thus, the spectra continue periodically and overlap additively. In the two-sided spectrum of Fig. 14.31, the lines n = 0. . .29(N - 1) of the spectrum can be thought of as continuing to the left on the frequency axis (n = - 30. . . - 1); X ð- nÞ = X ðN - nÞ holds. Likewise, the spectrum is continued to the right from the line n = 30 on the abscissa. Symmetry in the Frequency Domain The starting point for the DFT are N real values x(k). To “accommodate” their information in N complex values X ðnÞ, only half of the lines N/2 are necessary. These are contained in the 0. to N/2 - 1. spectral line. The part of the spectrum from the N/2. to the N - 1. spectral line contains the conjugate complex values X  ðnÞ and thus represents no additional information. On the other hand, since periodicity is present in the frequency domain, the conjugate complex values X  ðnÞ are on the negative frequency axis symmetrical to the ordinate X  ðnÞ = X ð- nÞ. The conjugate complex values do not have to be calculated, but are obtained by mirroring the spectrum on the ordinate axis (Fig. 14.31). This explains why

Signal waveform x(t)

14.4

Signal Analysis in the Frequency Domain

435

1 0.5 0 -0.5 -1 -64

0

64

128

192

256

320

384

448

512

576

640

384

448

512

576

640

Signal waveform x(t)

Time [ms] 1 0.5 0 -0.5 -1 -64

0

64

128

192

256

320

Sampling values [k]

Fig. 14.32 Time signal (top), its periodic continuation for the DFT (bottom)

a frequency analysis with 1024 lines in the one-sided spectrum represents a maximum of 512 lines (often 400 lines because of the limited slope of the antialiasing filter). Band Limiting and Antialiasing From the use of N/2 spectral lines it follows that the maximum frequency that can be displayed (maximum frequency) fmax must be less than half the sampling frequency fs/2. The maximum frequency fmax is calculated from the line spacing Δf = 1/T and a used line number N/2 to: f max =

N Δf 2

ð14:52Þ

The representation in the frequency domain can only represent one frequency band 0. . .fmax, so it is band-limited. To meet this condition, the signal must also be bandlimited in the time domain. Second, the sampling condition must be met, i.e. the sampling frequency must be greater than twice fmax (Sect. 10.7.3). The bandwidth limitation is implemented by low-pass filters (antialiasing filters). If the sampling condition is violated, additional lines appear in the spectrum, the so-called mirrors or alias frequencies. These cannot be removed from the spectrum by subsequent filtering. Therefore, all frequencies above the Nyquist frequency must be filtered out by a low-pass filter before the DFT. Complex Size The DFT generates the discrete, complex spectrum X ðnÞ. This can be clearly divided into an amplitude component j X ðnÞ j (amplitude or magnitude spectrum) and phase component ς(n) (phase spectrum). Likewise, a decomposition into real and imaginary parts is possible. This is illustrated in Fig. 14.33 using the example of a cosine function. x(t) = cos (2πft)

436

14

Signal Analysis Methods and Examples

1.2 1 1 0.5

Amplitude

x(t)

0.8

0

0.6

0.4 -0.5 0.2 -1 0 0

2

4

Sampling values k

6

8

0

1

2

3

4

5

6

7

Lines n

Fig. 14.33 Function x(t) = cos (2πft), sampling with N = 8. left – time course, right – two-sided amplitude spectrum

The two-sided amplitude spectrum therefore results in 8 frequency lines, whereby the 0. line contains the DC component. This results in zero for a full period. For n = 1 and n = 7, the value is 0.5. In order to obtain a statement about the frequency, the discrete line spectrum must be scaled with the line spacing Δf. For the two-sided spectrum, the transformation equation thus yields half the amplitudes (or 6 dB in level representation). By transition to the one-sided spectrum (Fig. 14.34), two lines always coincide for reasons of symmetry and their amplitudes are added together in terms of magnitude. Thus for n = 1 the value j X ð1Þ j = 1 is obtained, which is the amplitude of the cosine function. The phase spectrum ς(n) contains the value zero for all n. In order to rotate the analyzed cosine function to the (real) cosine axis, a phase angle of zero is required, the pointer is therefore real and the amplitude spectrum is mirror-symmetrical to the ordinate. The spectrum of the sine function x(t) = sin (2πft), would yield no difference for the amplitudes compared to the cosine function. Finally, the sine function also represents a real quantity, but in the DFT it gives a pointer in the direction of the imaginary axis and a pointsymmetric spectrum with respect to the ordinate. In the phase spectrum, a phase angle π/2 results for the sine function. "

Mnemonics

Real Axis: Imaginary Axis: Amplitude: Phase angle:

Cosine components (even function → mirror symmetric to ordinate), Sine components (odd function → point symmetric to ordinate), Amount of the pointer, Rotates in the mathematically positive direction of rotation from the real axis in the direction of the pointer.

14.4

Signal Analysis in the Frequency Domain

437

b

a

Phase

Amplitude

1,0

0,5



0,0 0

1

2

3

4

0

1

Lines n

2

3

4

Lines n

Fig. 14.34 Function x(t) = cos (2πft), sampling with N = 8. left – one-sided amplitude spectrum, right – phase spectrum

"

Information Content

Time domain and frequency domain are merely different ways of representing the same data. The transformation from the time domain to the frequency domain therefore does not lead to the loss of information (i.e. information-preserving transformation). For this reason, it is possible to generate the time characteristic again from the amplitude and phase spectrum by means of the inverse transformation (so-called resynthesis). This will be examined in more detail using two superimposed cosine oscillations. x1 ðt Þ = 0, 75  cosð2πft Þ þ 0, 25  cosð2π3ftÞ 3 x2 ðt Þ = 0, 75  cosð2πft Þ þ 0, 25  cos 2π3ft þ π 4

ð14:53Þ

The functions x1 and x2 differ only by the phase shift angle 3/4π in the second summand. By means of Fig. 14.35 the two functions are to be compared, which are sampled with N = 8 points in such a way that the block length is one period with the period duration T = 1/f. The time response of the two functions is clearly different, but the amplitude spectrum of both functions is the same. The amplitude spectrum shows the amplitude 0.75 for x1(t) (i.e. frequency f ) and the amplitude 0.25 for n = 3 (i.e. frequency 3f ). The difference lies in the phase spectrum, where the phase angle of zero for x1(t) and 3/4 for x2(t) can be read for n = 3. Consequently, no clear reconstruction of the time course is possible from the amplitude spectrum alone. For this, the phase spectrum is necessary (or assumptions about the phase angle must be made from the outset).

14

Signal Analysis Methods and Examples

1

1

0.5

0.5

x 2(t)

x 1(t)

438

0

0

-0.5

-0.5

-1

-1 0

2

4

6

8

0

2

4

6

8

3

4

3

4

Sampling values k

1.2

1.2

1

1

0.8

0.8

Amplitude

Amplitude

Sampling values k

0.6

0.6

0.4

0.4

0.2

0.2

0

0 0

1

2

3

0

4

1

2

Lines n

Phase

Phase

Lines n

/2

/2

0

0 0

1

2

Lines n

3

4

0

1

2

Lines n

Fig. 14.35 Comparison of the functions x1(t) and x2(t). top – time course, middle – one-sided amplitude spectrum, bottom – phase spectrum "

Only information that is already contained in the time domain can be represented in the spectrum. This is a consequence of the informationpreserving transformation. Therefore, the representation as a spectrum does not lead to any gain in information (i.e. no higher frequency resolution is possible in the spectrum than the sampled signal brings along over the finite block length). The gain lies in the representation: from the spectrum,

14.4

Signal Analysis in the Frequency Domain

439

information can be taken directly that cannot be read directly in the time domain (e.g. frequency). The conservation of information applies equally to the powers. Parseval’s theorem states that the temporal power is equal to the spectral power P =

1 N

N -1 k=0

ð xð k Þ Þ 2 =

N -1

jX ðnÞj2

ð14:54Þ

n=0

Using the introductory example, the temporal power P = (12 + 12 + 0 + ⋯ + 0 + 12)/30 = 0, 1 can be calculated for the pulse and scaled with physical units. If one now adds the squares of the 30 spectral lines, one obtains P = 0, + 0, 09852 + 0, 09422 + ⋯ + 0, 09422 + 0, 09852 = 0, 0999. In summary, the transition from the Fourier integral to the DFT can be thought of as sampling the continuous Fourier spectrum at discrete grid points and its periodic copies at intervals from N frequency lines. Both the time signal and the spectrum are then in discrete and periodic form, allowing for efficient numerical computation. The calculation requires N2 complex additions and multiplications. The FFT algorithm can reduce the number of arithmetic operations to N  log2N for the same result, thus increasing the speed of processing. This is achieved by using a line number N as a power of two (e.g. 1024, 2048, 4096, 8192 lines) and the multiple use of calculated intermediate results. The FFT is thus a particularly efficient algorithm for the implementation of the DFT, but not a special form of the DFT [9]. The Fourier transform algorithm is normally implemented in a software solution for which a number of settings and parameters must be made. The choice of settings depends on the measurement problem, and some parameters also influence each other. The choice of settings determines the results of the frequency analysis, since the FT (DFT or FFT) cannot check whether the selected settings are conducive to solving the measurement problem. By a preliminary consideration of the measurement task and recognition of the effect of the individual variables, however, reasonable settings can be found relatively easily.

14.4.3 Aliasing Compliance with the sampling theorem must ensure that the measurement signal contains only frequency components below the Nyquist frequency. As can be seen in Fig. 10.36, violation of the sampling theorem (i.e. aliasing) shifts frequency components above the Nyquist frequency into the spectrum and mirrors them on the ordinate. This is a – highly unfavorable – effect of symmetry and periodicity of the DFT spectrum.

440

14

Signal Analysis Methods and Examples

Low-pass filtering of the measurement signal before A/D conversion effectively suppresses aliasing. A violation of this condition cannot be corrected subsequently (e.g. by filtering after sampling or cutting off the higher frequencies in the FFT). In Sect. 10.7.3 it was pointed out that the sampling theorem is only fulfilled for frequencies below the Nyquist frequency f = fs/2. A frequency which corresponds to the Nyquist frequency is already sampled incorrectly. In the usual spectrum display, a line appears at f = fs/2. This apparent contradiction can be resolved by starting the sampling at zero and counting the space t from the second to the last data point [3]. If the frequency line N/2 is then plotted in the spectrum at the maximum frequency fmax, the frequency line with the counting index zero contains the DC component. "

Since low-pass filters have a limited slope in the transition region (Sect. 10. 6.2), often only part of the available number of lines is used (e.g. N′ = 400 lines from N/2 = 512 lines), thus avoiding misrepresented components in the spectrum.

In newer devices and measuring cards, a very high sampling frequency (oversampling) is selected in conjunction with a low-order analog low-pass filter (cf. Fig. 10.42). A subsequent (digital) low pass further increases the stopband attenuation. In this case, N/ 2 lines (e.g. 512 lines) are also used and displayed (Fig. 14.36).

14.4.4 Relationships Between the DFT Parameters Due to the blockwise transformation and the discretization in the time and frequency domain, the setting parameters of the DFT cannot be selected independently of each other. In the time domain, the block length T is divided into N samples with a sampling interval Δt. T = N  Δt

ð14:55Þ

The frequency range comprises up to the sampling frequency fs again N Frequency lines with the line spacing (i.e. frequency resolution) Δf f s = N  Δf

ð14:56Þ

In the spectrum, the maximum frequency fmax is to be represented, which is smaller than half the sampling frequency (Nyquist frequency). Thus results

14.4

Signal Analysis in the Frequency Domain

441

Level L

Cut-off frequency of the low-pass filter

Slope of the low pass filter Amplitude dynamics of the ADC e.g. 80 dB

N/2 e.g. 512 0

fs/2

Usable number of lines e.g. 400

N e.g. 1024 Frequency f

fs

Fig. 14.36 Influence of low-pass filtering on maximum frequency and number of lines

f max
0) and thus to a nonsensical result.

Signal Analysis in the Frequency Domain

451

20

20

0

0

-20

-20

Level in dB

Level in dB

14.4

-40

-40

-60

-60

-80

-80 -8 -7 -6 -5 -4 -3 -2 -1

0

1

2

3

4

5

6

7

8

-8 -7 -6 -5 -4 -3 -2 -1

Normalized frequency fT

0

1

2

3

4

5

6

7

8

Normalized frequency fT

Fig. 14.41 Window function in the frequency domain. Left rectangular window, right Hanning window (continuous-time window)

The solution to the problem lies in the bandwidth, since in this case the power is distributed over three frequency bands with an effective bandwidth Beff. Thus, the energetic level addition must be done according to Eq. 14.62. This again underlines the fact that despite amplitude-correct representation, the power is not correctly represented. Only via the bandwidth correction is it possible to bring both forms of representation into congruence. – With the exception of the three lines visible in the spectrum, the zeros of the window function are sampled and therefore do not appear in the spectrum.

L ¼ 10  lg

L5 L3 L4 1 10 10 þ 10 10 þ 10 10 Br

0 -6 -6 1 1010 þ 10 10 þ 10 10 ¼ 10  lg 1, 5

L ð14:66Þ L ¼ 0 dB

• Asynchronous scanning (Fig. 14.42 right): – In this extreme case, the maximum of the window function lies between two frequency lines. The amplitudes are displayed too low by the level error ΔL due to the course of the window function. This level error ΔL is called the picket fence effect and results from the fact that the spectrum is only viewed at the lines (through the “picket fence”). For the Hanning window, the level error for this case is ΔL = 1, 4 dB, as shown in Fig. 14.40 below. For the energetic level addition it has to be taken

452

14 20 Amplitude in dB

Amplitude in dB

20

Signal Analysis Methods and Examples

0 -20 -40 -60

0 -20 -40 -60

0

4

8

12

16

0

Lines n

4

8

12

16

Lines n

Fig. 14.42 Spectra with the Hanning window drawn in. Synchronous scanning (left), asynchronous scanning (right)

into account that the energy is now not only distributed on the two frequency lines – for the spectral power it has to be summed over all spectral lines. – In the spectrum, four frequency lines are sampled from the main lobe of the window function. The other frequency lines of the spectrum result from the sampling of the secondary maxima. In this case, the sampling is not done at the zeros, which would suppress the lines. In measurement practice, the cases discussed primarily represent theoretical borderline cases. Rarely will the frequency coincide exactly with a line or lie in the middle between two lines. To correct the frequency, there is a correction factor KF for sinusoidal signals, which uses the level difference ΔL (to distinguish it from the level error ΔL ) between the two frequency lines. KF = 1 -

ΔL 6dB

Δf 2

ð14:67Þ

The level can be corrected in an analog way. In the following, further frequently used window functions will be discussed. It should already be noted that there is no universal window for all measurement tasks – otherwise there would not be such a large number of window functions. To characterize the window function, the parameters sidelobe attenuation (SLA), mainlobe width (MLW) as a multiple of the line spacing, and the level error ΔL at half the line width already introduced above are used (Fig. 14.43). These definitions are widely used [7] but are not binding – hence other specifications exist [15, 18]. An ideal window function should strongly attenuate side maxima (high SLA), cause the smallest possible level error in the amplitudes (small ΔL), moreover, have a high selectivity (small MLW and small effective bandwidth BL). The parameters for the window functions are summarized in Table 14.5.

14.4

Signal Analysis in the Frequency Domain

453

20

L 0

SLA Level in dB

-20

-40

MLW -60

-80 -8 -7 -6 -5 -4 -3 -2 -1

0

1

2

3

4

5

6

7

8

Normalized frequency fT

Fig. 14.43 Definition of window parameters SLA, MLW and ΔL Table 14.5 Properties of frequently used window functions [7]

Windows Rectangle Triangle Hanning Hamming Blackman (approximated) EmperorBessel* Gauss* Flat top Exponential

SLA secondary maximum attenuation (dB) 13.3 26.5 31.5 42.7 58.1

Level error ΔL (dB) 0 1.82 1.42 1.75 1.1

Main lobe width MLW (lines) 1.62 3.24 3.37 3.38 5.87

Effective bandwidth (normalized to the line spacing Δf) BL = Beff /Δf 1 1.33 1.5 1.36 1.73

60

1.16

5.45

1.68

60 60 12.6

1.06 0.05 3.65

7.1 7.01 1.72

1.79 3.14 1.08

454

14

Signal Analysis Methods and Examples

Table 14.6 Selection of window functions Windows Rectangle Triangle Hanning Hamming Blackman (approximated) Kaiser-Bessel Gauss Flat-Top Exponential

Properties and application Standard window for synchronous signals as well as transient signals that fit completely into the window Literature, historical Standard window for asynchronous signals, low dynamics Properties similar to Hanning Better selectivity than Hanning windows Good spectral selectivity depending on parameter Used for short time FFT Very small level error, for calibration purposes with mono-frequency sinusoidal signals Damping of decaying oscillations, transient signals in weakly damped systems

The window functions marked with * have additional parameters which influence the properties. This is an exemplary representation. In [2, 13, 18] other parameters have been taken as a basis, therefore different numerical values result there The selection of the appropriate window function is an equally important and difficult part of the measurement task. On the one hand, the window decides on the interpretation or misinterpretation of the measurement – on the other hand, the frequency content of the signal would have to be known from the outset in order to determine the optimum window. Since there is no universal window, the choice of the window function must be adapted to the measurement task (Table 14.6). In case of doubt, the effect of different windows on the spectrum must be assessed and the optimum window determined. "

The selection of the window function is always a compromise between dynamics, bandwidth and level error

Frequently, the technical code or the underlying standard prescribes the window function for the measurement or recommends it in the informative part. However, the regulations often do not specify the parameters that would be required, for example, for a complete definition of the Kaiser-Bessel window. If the signals contain sine and noise, window functions with a small bandwidth are useful to keep the level error between sinusoidal signals and stochastic signals (noise) small. If, on the other hand, sinusoidal signals with approximately the same amplitude and a large frequency spacing are to be separated, window functions with high sidelobe suppression (SLA) – i.e. a high dynamic range – are preferable. For closely spaced frequency lines of approximately equal amplitude, on the other hand, a high SLA is not useful. With better sidelobe suppression (SLA), these window functions show increasingly

14.4

Signal Analysis in the Frequency Domain

455

poorer selectivity, which would, however, be precisely what is desired for this measurement problem. The maximum level error of the windows ΔL describes the deviation of the value displayed in the spectrum from the actual value. The level error of ΔL = 1, 4 dB corresponds to a relative deviation of 15% (due to the window function used alone), which is already no longer acceptable for many applications. The flat-top window, on the other hand, has a very small level error ΔL < 0, 05 dB. For this reason, this window is used for calibration purposes of measurement chains with a sinusoidal signal. A disadvantage in application is the high bandwidth and poor selectivity, which may be the wrong choice for practical applications (e.g. separation of adjacent frequencies). Therefore, the recommendation to use the flat-top window for periodic signals may be wrong in individual cases [18, 19]. For transient signals, the choice of the time zero point in the window is decisive, which can be defined via the trigger time. The window function must fade the signal section to be examined to 0 at the edges. However, with weak damping, the system oscillates for a long time. The oscillations have therefore not yet completely decayed when the end of the window is reached. In this case, the signal must be damped with an exponential window, for example. However, this leads to higher damping in the display. The window function must also completely enclose the signal section to be examined. If this condition is violated, it may happen that only a section of the signal or several signals are processed further. It is not the time signal itself which is analysed, but always the signal which is produced by multiplying the time signal by the window function. This is illustrated in Fig. 14.44 with a force pulse. For single pulses, the rectangular window can be applied if dimmed to zero at the window boundaries. It should be noted that only one pulse appears in a window at any one time. If multiple pulses are within the window, they are analyzed as one periodic signal (from multiple pulses). It is obvious that this leads to a different result. From the time history (of the un-windowed signal) it can be seen that the oscillation has decayed at both window boundaries. With the Hanning window, one obtains a time function with a smaller maximum. The reason for this is the multiplication with the used cos2 -function of the Hanning window. In this case a Hanning window gives nonsensical results. For this reason, the use of a Hanning window is prohibited for transient signals (e.g. pulse). If, for example, a pulse is time-weighted with a Hanning window, the pulse would have to lie exactly in the middle of the window and, moreover, would have to be narrow in comparison to the window length in order to capture its amplitudes correctly. "

In order to select the window functions sensibly, the unwindowed and windowed time signals must always be compared and checked for plausibility in addition to the spectra.

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14

200

Signal Analysis Methods and Examples

200 windowed Force Hanning-windows

150

Force F in N

Force F in N

150

100

50

0

100

50

0

-50

-50 0

0.2

0.4

0.6

0.8

1

0

0.2

Time in s

0.4

0.6

0.8

1

Time in s

Fig. 14.44 Force pulse of a hammer blow. Application of the rectangular window (correct) (left), application of the Hanning window (wrong) (right)

Trigger

Time

Signal source

Signal type

continuously

internal triggering

Absolute value (DC)

Single time (Single)

external triggering

Amplitude (AC)

repeated triggering (Repeat)

Signal edge (HL or LH)

Fig. 14.45 Settings for triggering

14.4.6 Triggering Triggering can be used to define the starting point of the measurement [2, 4, 5, 10, 20]. This always plays a role when a transient process (e.g. impulse) or measurements depending on a superordinate process (e.g. high-frequency vibrations on the engine depending on the ignition timing) are involved (Fig. 14.45).

14.4

Signal Analysis in the Frequency Domain

457

To perform triggering, the measurement signals are stored in a buffer and old data is continuously overwritten with new data. In this operating mode, data can be analyzed continuously (so-called free-running analysis). When a trigger time is defined, the values previously stored in the memory are used (pre-trigger). In this way, the analysis also acquires signal sections before the trigger is triggered or can fulfill the necessary windowing of the signal (see Sect. 14.4.5). The memory contents from the triggering of the trigger condition are called post-trigger. For further processing, a block length T of the measurement signal can be recorded after the trigger condition has been fulfilled once (single) or one block can be recorded after each fulfilment of the trigger condition (continuous triggering, repeat). The latter processing is a prerequisite for averaging over several measurements (Sect. 14.4.7). The trigger is triggered either internally or externally, depending on the signal source. With internal triggering, the trigger event is derived from the signal to be analyzed. This requires the specification of a trigger value and its triggering when the value exceeds (low-high edge) or falls below (high-low edge). In addition, a distinction can be made according to the signal type. Triggering on the absolute value (direct current, DC) triggers the measurement at the specified threshold. With DC triggering, slow processes (e.g. filling a container) can be used as triggering. Triggering on an amplitude (alternating current, AC), on the other hand, compares the change in the signal with the set threshold and can be understood as high-pass filtering of the signal. AC triggering is advantageous when unwanted DC components occur in the signal (e.g. due to zero drift of the transducer). External triggering requires an additional trigger signal (e.g. the pulse of a light barrier). External triggering is often used in the case of starting data recording depending on a higher-level process. This can be, for example, the ignition timing of an internal combustion engine, the end position in a cyclically repeating machining process or the (phase) angle of a rotating shaft. Repeated averaging of the acquired values in the time domain can reduce statistical fluctuations in the measurement signal and improve the signal-to-noise ratio. If the trigger signals are noisy and subject to interference, erroneous measurements may occur. In this case, it is recommended to filter the trigger signals.

14.4.7 Averaging and Overlapping Averaging From periodic signals, the line spectrum can be determined exactly by DFT in the mathematical sense, provided that the conditions of band limitation, aliasing and periodicity in the time domain are fulfilled [3, 7]. Purely periodic signals rarely occur in measurement practice, but are superimposed with noise (e.g., from the transducer). Very often, signals are present that consist exclusively of stochastic signal components (noise). Examples of this are the excitation of chassis by irregular road profiles or vibrations caused by turbulent flows.

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Table 14.7 Averaging types Averaging method Linear averaging

Implementation Each spectrum is equally weighted (root mean square).

Exponential averaging

The youngest spectra are weighted the most, older spectra get exponentially less weight.

Peak averaging

Averaging over the spectra with the highest amplitudes (i.e. no statistical method) Averaging in the time domain (“averaging out” of positive and negative amplitudes) Improvement of the signal-to-noise ratio

Time domain averaging

Application Short signal length Reduction of the statistical measurement deviation for stochastic signals Tracking of the influence of slow temporal changes on a signal (e.g. operating states, changes in parameters) Statistical measurement deviation is independent of frequency Acquisition of peak values (worst case scenario) Periodically repeating data, synchronization via trigger required Transient signals, rotating machines

If a block length T is taken from a measurement signal of a stochastic signal and analyzed by means of DFT, this initially provides a more or less good estimate for the spectrum of the signal. The one-time estimate from a block length is therefore subject to high scatter, i.e. further measurements deliver deviating spectra. The use of more measured values does not automatically lead to a better estimate. If, for example, the block length T is increased, more measured values are used, but more spectral lines N are also generated. However, this does not lead to an averaging of the spectrum, but to a reduction of the line spacing Δf in the spectrum and thus to an increase in the frequency resolution. The estimate of the mean value, on the other hand, improves if it is averaged over several blocks T. In mathematical terms, the estimate then converges towards the true mean value. The prerequisite for this is that the process is stationary (example: motor with constant speed, counter-example: Run-up of a motor). Averaging can be applied in different ways (Table 14.7) [2, 4, 5, 10, 20]. The influence of linear averaging is shown in Fig. 14.46. When averaging over 5 and 50 blocks, a smoothing in the spectrum is clearly visible. Averaging has no influence on the periodic signal components. For the practical application the question arises which averaging time TA is required. On the one hand, the estimation of the averaging time is useful for the planning of the measurement, on the other hand, this estimation gives information about the expected significance of a performed measurement. In a running measurement the averaging can be

Acceleration in m/s 2

14.4

Signal Analysis in the Frequency Domain

10

2

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459

Magnitude spectrum without averaging

-2

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5 Amount spectra linearly averaged

102

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102

Acceleration in m/s 2

10

0

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0

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1000

1500

2000

Frequency in Hz

Fig. 14.46 Influence of averaging on spectra. Top – no averaging, middle – averaging over 5 spectra, bottom – averaging over 10 spectra

observed and a stabilization of the spectrum can be waited for. An estimation of the required averaging time can be made depending on the signal. Periodic Signals For measured periodic signals, averaging takes into account the transient processes in the measurement chain and generally improves the signal-to-noise ratio. From the settling time

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of a filter (see Sect. 10.6.3), the required averaging time is 3 to 5 times the period of the smallest frequency of interest in the spectrum. If the smallest frequency of interest is equal to the frequency resolution in the spectrum Δf, this immediately results in averaging over three to five blocks of the block length T. This procedure reaches its limits with closely adjacent frequency lines within the bandwidth of the window function Beff or with amplitude-modulated signals. In this case, the modulation frequency (beat frequency [13]) rather than the smallest frequency of interest in the signal is used for the averaging time. The modulation frequency can be read from the representation in the time domain. Stochastic and Transient Signals For stochastic signals, a stationary noise signal is the basis, for the statistical description of which a standardized normal distribution with the mean value μ = 0 and the rms value x = 1 is used. In this signal, a frequency band with the effective bandwidth Beff is considered. If measurements are now made over the averaging time TA, the averaging can be regarded as an additional bandpass filter with the bandwidth 1/(2TA). The relative standard distribution is thus calculated. The relative standard deviation ε is thus calculated as follows

ε =

σ 1 =  2 x

1 Beff T A

ð14:68Þ

The relative standard deviation ε then describes the deviation of the measured mean from the true mean. With a probability of 68.3%, the true mean lies in the interval -ε. . . + ε. The true mean value is obtained for an infinitely long averaging time TA. Table 14.8 summarizes the corresponding BeffTA products for frequently used values of. ε From Eq. 14.68 it follows immediately that the averaging time is determined by the effective bandwidth, i.e. frequency resolution or smallest frequency in the spectrum. Figure 14.46 shows that for high frequencies a stable state in the spectrum has already occurred after a short averaging (middle) and that further averaging (bottom) acts on the low frequencies in the spectrum. If window functions are used, the effective bandwidth Beff must be used in Eq. 14.68. With higher frequency resolution (smaller Beff), the required averaging time TA increases, since the higher frequency resolution alone increases the required block length T. Accordingly, for a given averaging time TA (e.g. from a recorded measurement), a better estimate of the mean value is obtained for a lower frequency resolution. In other words: Simultaneous demand for high frequency resolution (small Beff) and low statistical deviation from the mean value require a long averaging time TA (large BeffTA product). In the literature, minimum values for the BeffTA product of 10 [2, 5] to 16 [13] are suggested.

14.4

Signal Analysis in the Frequency Domain

Table 14.8 Relative standard deviation ε and BeffTA -product

ε in dB 1 0.5

461 ε in % 12.2 5.9 5 1

Beff TA 17 71 100 2500

Example

What averaging time TA is necessary for an FFT analysis with N = 4096 lines, sampling frequency fs = 2048 Hz and Hanning window, if a relative standard deviation ε = 5% is not to be exceeded? With the number of lines N and the sampling frequency fs results: Δf =

f s 2048 Hz = 0, 5 Hz = 4096 N

By using the Hanning window, the following is calculated with Table 14.5 Beff = BL  Δf = 1, 5  0, 5 Hz = 0, 75 Hz For ε = 5%, a product BeffTA = 100 is read from Table 14.8. Thus, the required averaging time TA is calculated as follows TA =

100 100 1 = 133 s = Beff 0, 75 3 ◄

Overlap The window functions used fade out the time signal at the window boundaries in each block. This is necessary to reduce leakage, but parts of the signal are faded out and not taken into account (cf. Figure 14.44). As a workaround, the start of the current window can be placed before the end of the previous window, in which case the windows overlap. The overlap is given in percent, an overlap of 75% means that the windows overlap 75% and after 25% of the elapsed time the next window starts. For the representation of the signal power (square of the RMS value), the window function is squared and the so-called power weighting is obtained. If the overlapping power weighting is now summed, the ripple becomes smaller with increasing overlap (Fig. 14.47). From an overlap of 66.67% the power sum is constant, a further increase brings no improvement. As the overlap increases, multiple DFT analysis of the same overlapping signal sections occurs, so no statistical improvement occurs. Nevertheless, it may be useful to work with higher overlap

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Window function without overlap

w(N)

1.5 1 0.5 0

0

1000

2000

3000

4000

5000

6000

7000

8000

squared window function without overlap

2 w2(N)

Signal Analysis Methods and Examples

1 0 0

1000

2000

3000 4000 5000 Window function with 50% overlap

6000

7000

8000

0

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2000

3000 4000 5000 squared window function with 50% overlap

6000

7000

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3000 4000 5000 Window function with 66.7% overlap

6000

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3000 4000 5000 Window function with 75% overlap

6000

7000

8000

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8000

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8000

w(N)

1.5 1 0.5 0

w2(N)

2 1 0

w(N)

1.5 1 0.5 0

w(N)

4000

5000

squared window function with 66.7% overlap

2 1 0 0

1000

2000

0

1000

2000

w(N)

1.5 1 0.5 0

w(N)

4000

5000

squared window function with 75% overlap

2 1 0

0

1000

2000

3000

4000

5000

Fig. 14.47 Influence of increasing overlap, in each case upper drawing file: course of the window function over two window lengths, in each case lower drawing file: squared window function and power sum

(e.g. pulses that cannot be analyzed in one block). If, on the other hand, sections of a stationary signal are analyzed, the signal portions that are faded away are statistically equivalent to the signal portions that are analyzed. In this case, overlapping can be used to extract the greatest possible information content from a signal of limited length or to shorten the measurement time [13]. By overlapping, a different window function is

14.4

Signal Analysis in the Frequency Domain

463

obtained than with a window of greater length. Both the leakage effects and the effective bandwidth differ. In the superposition of the individual spectra to a sum spectrum, the power spectra are usually added. The phase component can be taken into account by including the transit time between the individual spectra as a phase angle [7].

14.4.8 Spectral Quantities The ordinate value is represented differently in the spectra. The ordinate representation used depends on the signal type. A distinction must be made here as to whether the signals are energy or power signals. • Energy signals Energy signals have a finite energy. For transient signals, such as single pulses, this condition is given. On the other hand, the energy is the integral of the instantaneous power over time. If the measurement is continued over an infinitely long time, the measured power approaches zero. • Power signals Power signals are characterized by finite power. Examples are sinusoidal oscillation or stochastic noise. The power as an averaged sum over the amplitude squares is constant. However, the energy of the signal increases with increasing measurement time and reaches the value infinite at infinitely long measurement time. The electrical measurement chain initially supplies only the electrical field quantities voltage U and current I. The transfer coefficient (Sect. 10.4.2) scales the electrical quantity (voltage U in V) in the measurement chain to the measured quantity (physical quantity). In signal processing, the quantities energy and power are expressed differently from the physical definition. The product of voltage U and current I is the power P. With an ohmic resistor R, the power can only be expressed with a field quantity – usually the voltage U – as P = U2/R. The power is then expressed as a standard value. In signal processing, an ohmic resistance of 1 Ω is now assumed as normalization, and P = U2 is then obtained. This means that the power is no longer expressed as a physical power, e.g. W, but in V2. In the representation of power signals, a further distinction must be made between periodic signals (e.g. sine) and stochastic signals (e.g. noise): Periodic Signals Periodic signals are represented as amplitude values. The associated spectrum is called amplitude spectrum or magnitude spectrum (MAG, Magnitude). For a sinusoidal signal with an amplitude of 1 V, the spectrum shows a spectral line with 1 V amplitude. Deviations are caused, among other things, by the window functions used.

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Signal Analysis Methods and Examples

It is also common to represent the signal as an effective value (RMS, Root Mean Square). For a sinusoidal signal with 1 V amplitude, a spectral line is obtained p 1 V= 2 = 0, 707 V. However, the RMS value in the spectrum is not formed by squaring and root extraction as in the time representation, but by rescaling the amplitude spectrum. Therefore, the RMS value in the spectrum formed in this way contains phase information. Squaring the RMS value gives the (normalized) power P. However, by squaring, the phase information is lost. The spectrum is called the power spectrum (PWR). In this representation, a display of P = 0, 5 V2 is obtained for the sinusoidal signal with 1 V amplitude. Squaring suppresses the small amplitudes that are also contained in the signal (e.g. 0.1 V amplitude results in a value of P = 0, 005 V2 in the display of the power spectrum). This could give the – false – impression that by using the power spectrum an improvement of the signal-to-noise ratio occurs. This effect is described in [1]. A level spectrum (see Sect. 5.4) is obtained from the RMS value or power. Usually a reference value of 1 Veff for dBV and 0, 775 V for dBu is used. If the spectrum is available as amplitudes, these must be converted into RMS values. The above sinusoidal signal with 1 V amplitude and a reference value of 1 Veff results in a level of p L = 20  lg 1 V= 2 =1 V = - 3 dB. If the spectrum is not multiplied by itself, but by the natural logarithm, the so-called cepstrum is obtained (the name is derived from the English spectrum, with the first four letters reversed). This representation emphasizes the small amplitudes. This representation is used in the condition monitoring of machines. Stochastic Signals Stochastic signals are specified as power spectral density (auto power spectrum, auto spectrum, auto spectral density or power spectral density) (PSD), since the Fourier transform of a stochastic stationary signal is not defined. The power spectral density is obtained by dividing the signal power P( f ) by the bandwidth Beff (or, in the case of the rectangular window, by the spacing of the frequency lines Δf ) in the spectrum.

Gxx ðf Þ =

Pðf Þ Beff

ð14:69Þ

The power spectral density has the unit V2/Hz (or square of the physical quantity per frequency, e.g. (m/s2)2/Hz). As a quantity that is independent of the analyzer settings, it is specified as the square root of the power spectral density as the voltage spectral density in p V= Hz. The power spectral density can be visualized in the time or frequency representation. In time representation, the signal is filtered with a narrowband filter with center frequency fm and bandwidth Beff and then its rms value xM is formed according to Eq. 14.1. The square of the rms value x2M is the power P( f ) of the signal with center

14.4

Signal Analysis in the Frequency Domain

465

frequency f = fm and bandwidth Beff. If the bandwidth of the filter is now reduced Beff → 0, the one-sided spectral power density is obtained from this boundary transition Gxx ðf Þ =

x2M ðf Þ d 2 x = df M Beff → 0 Beff

ð14:70Þ

lim

By squaring when calculating the RMS value, the phase information is lost. The one-sided power spectral density is defined only for frequencies f ≥ 0. The two-sided power density spectrum Sxx( f ) is symmetrical to the ordinate (i.e. an even function) and is formed as follows: Sxx ðf Þ = Sxx ð- f Þ =

1 G ðf Þ 2 xx

f ≠0

ð14:71Þ

The total power of the signal P, i.e. the square of the rms value x2 , is obtained by integrating all filtered signals over the frequency: 1

P = x2 =

1

Gxx ðf Þdf = 0

1

Sxx ðf Þdf = -1

1 2π

Sxx ðωÞdω

ð14:72Þ

-1

This follows from Parseval’s theorem that the same power is obtained by integration over time and in the spectrum by integration over frequency. The factor 2π 2 is a scaling factor in the transition from the frequency f to the angular frequency ω. "

The area under the curve of the power density spectrum corresponds to the timeindependent total power of the signal P.

In the frequency representation, it can be imagined as an equivalent approach that the complex Fourier spectrum X T ðf Þ is formed from the signal xT(t) over a time interval T. The signal is defined for positive values of, i.e. in the interval. The signal xT(t) is defined here for positive values of t, i.e. in the interval 0 ≤ t ≤ T. If we now sum up the squares of the amplitudes in the two-sided spectrum over an infinitely long measurement time T, we again obtain the real spectrum Sxx ðf Þ = lim

1

T →1 T

j X T ðf Þj 2

ð14:73Þ

For a given frequency, the complex amplitudes X T ðf Þ are squared, normalized to the measurement time and the limit values are calculated. The squaring again eliminates the phase information. The order of DFT and squaring must not be interchanged, since

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Signal Analysis Methods and Examples

squaring is a non-linear operation (e.g. the squares of the rms values x2 must not be formed in the time domain and then the spectrum). For a stationary stochastic signal and sufficiently long averaging, the power spectral density is time independent. If this signal is analyzed over the power spectrum (PWR) – which should be reserved for periodic signals – the power spectrum depends on the effective bandwidth of the window function used. Example

A periodic signal with a constant frequency of 40 Hz and a signal power of 1 V2 is to be represented in the power spectrum (PWR) and power spectral density (PSD). The assumptions • constant frequency, • synchronous scanning, • Beff = 0, 5 Hz und Beff = 0, 25 Hz (i.e. doubling the number of lines) hit. The result is shown in Table 14.9. The comparison (Table 14.9) between the power spectrum (PWR) and the spectral Power density (PSD) shows: • The bandwidth (line spacing) has no influence on the amplitudes in the power spectrum for periodic signals. • The value of the power spectral density depends on the bandwidth for periodic signals. • Halving the bandwidth doubles the power spectral density of periodic signals (or increases the power density level by +3 dB). A stationary stochastic signal with a frequency-independent power spectral density of 1 V2/ Hz is to be represented in the power spectrum and power spectral density. The result is shown in Table 14.10. The comparison (Table 14.10) between the power spectrum (PWR) and the power spectral density (PSD) shows: • The effective bandwidth (frequency resolution, line spacing) has no influence on the amplitudes of the power spectral density for stationary stochastic signals. • The amplitudes in the power spectrum depend on the bandwidth for stationary stochastic signals. • Halving the bandwidth halves the power of stationary stochastic signals (or reduces the power level by -3 dB). ◄

14.4

Signal Analysis in the Frequency Domain

467

Table 14.9 Influence of the spectral representation on a periodic signal Beff = 0.5 Hz P = 1 V2 Gxx = P/Beff = 1 V2/ 0.5 Hz = 2 V2/Hz

Effective bandwidth Power spectrum (PWR) Power spectral density (PSD)

Beff = 0.25 Hz P = 1 V2 Gxx = P/Beff = 1 V2 / 0.25 Hz = 4 V2/Hz

Table 14.10 Influence of the spectral representation on a stationary stochastic signal Beff = 0.5 Hz P = Gxx Beff = 1 V2/ 0.5 Hz = 0.5 V2/Hz Gxx = P/Beff = 0.5 V2/0.5 Hz =1 V2/Hz

Effective bandwidth Power spectrum (PWR) Power spectral density (PSD)

Beff = 0.25 Hz P = Gxx Beff = 1 V2/ 0.25 Hz = 0.25 V2/Hz Gxx = P/Beff = 0.25 V2/ 0.25 Hz = 1 V2/Hz

"

Use power level for periodic signals, power spectral density for stationary stochastic signals. Short-term energy signals (pulses, transient signals) are described as a real quantity via the spectral energy density E( f ) (ESD).

E ðf Þ = jX T ðf Þj2

ð14:74Þ

The signal energy W as an expression for power times time is obtained by integration of the time signal or by integration over the squares of the amplitudes in the spectrum. 1

W=

1

x ðt Þdt = 0

1

1 jX T ðf Þj df = 2π 2

2

-1

jX T ðωÞj2 dω

ð14:75Þ

-1

This is again an expression of Parseval’s theorem (temporal signal energy = spectral signal energy). If the transient signal fits completely into the block length T, the rectangular window can be applied. A correction factor for the bandwidth is not necessary because of the rectangular window used. The spectral energy density E( f ) does not include the window length T. This is useful for energy signals because the energy of transient signals is independent of the window length, but the power decreases with increasing window length. As with the power spectral density Gxx( f ), the order of DFT and squaring must not be reversed. The spectral energy density E( f ) is given in V2s/Hz or V2s2. By squaring, the phase information is lost in the energy density spectrum (phase angle for all frequencies equal to zero). The spectral energy density E( f ) is understood as the sampling of the continuous spectrum of a single pulse. The result is then an energy density. In another approach to energy signals, the transient signals are understood as a periodic sequence of e.g. pulses. In

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Signal Analysis Methods and Examples

this case, the spectrum can be plotted as a discrete amplitude spectrum of a periodic signal (e.g. in V or N ). However, this representation then depends on the period of the signal, i.e. on its window length T. Both representations can be converted into each other by squaring (to obtain the power P( f )), multiplying by the block length T and then normalizing to the effective bandwidth Beff. E ðf Þ =

Pðf Þ  T = Gxx ðf Þ  T Beff

ð14:76Þ

The influence of the block length T can be specifically exploited by filling the signal with additional zeros, the so-called zero padding. Here, additional zeros are added to or placed in front of the time signal. This increases the number of lines N in the DFT. On the other hand, the line spacing Δf or the effective bandwidth Beff is reduced. As a result, a higher frequency resolution is obtained in the spectrum. This increase of the frequency resolution is not connected with an increase of information, it concerns rather an interpolation of the spectrum at additionally inserted supporting points. The procedure is also used in the FFT algorithms to bring existing signals to the necessary length of a power of two 2N. Table 14.11 summarizes the representation of basic signal types. Difficulties in the representation always arise when different signals, such as periodic and stationary stochastic signals, are to be represented in one spectrum. While periodic signals require a scaling to e.g. the power, stationary stochastic signals require a representation as spectral power density. "

Power density spectra of periodic signals can be compared with each other if they have the same effective bandwidth Beff. The scaling is then not amplitude-correct, but this representation at least provides a comparative value.

To obtain the powers of periodic signals, their power spectral density must be multiplied by the effective bandwidth. In the case of pulses, there is also the fact that the square-wave window makes sense for pulses; in the case of asynchronous sampling, this leads to leakage effects for periodic and stationary stochastic signals. In practice, one usually helps oneself by either choosing the representation that corresponds most closely to the underlying signal or by attempting to reach the goal via two separate measurements [3, 20].

14.4.9 Axis Scaling The axis scaling in the spectra depends on the respective measurement task, on the transducers used and, if applicable, on specifications from standards and regulations. In the display of spectra, it is common to scale the frequency axis (abscissa) linearly or logarithmically.

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Signal Analysis in the Frequency Domain

469

Table 14.11 Summary of axis scaling in spectra Levels are defined only for rms values according to Eq. 5.3 and powers according to Eq. 5.4 Signal Periodic Stochastic Transient

Scaling Amplitude, peak, magnitude, MAG, RMS, RMS power, PWR, RMS2 Power spectral density, PSD Eq. 14.69 Energy density, ESD Eqs. 14.74 and 14.76

Level (Sect. 6.4) Signal level power level Power density level Energy density level

Values with index 0 are reference values according to Sect. 5.4

• Linear scaling of the frequency axis makes it easier to find harmonics. Since these are in integer ratio to each other, the frequency lines appear evenly spaced. As the maximum frequency increases, the distances between the frequency lines in the display become increasingly narrow and unclear, so that the display should be limited to about one to two powers of ten. • Logarithmic scaling of the frequency axis covers a wider frequency range, typically 3 to 4 decades, and is the standard representation in frequency bands. Since an octave is defined as a 2:1 frequency ratio (e.g., 32 and 16 Hz and 2000 and 1000 Hz are each an octave), in logarithmic representation the frequency bands appear to be the same width even though they have different absolute bandwidths. With increasing frequency, more and more lines are displayed in a narrow space, which limits the evaluability in this frequency range. The ordinate value is displayed either linear or logarithmic depending on the measurement task. • Linear scaling allows the representation of amplitude dynamics of approx. 20:1 to 50:1 (corresponding to approx. 13 to 34 dB). With an axis length of 100 mm, this ratio corresponds to a smallest ordinate section of 5 or 2 mm. Consequently, in linear representation approximately equal amplitudes can be represented, a large amplitude dynamic is not realizable. • Logarithmic or level scaling extends the display to approx. 106:1 (corresponds to 120 dB). This exploits the amplitude dynamics of transducers and A/D converters. Smaller amplitudes can also be displayed on the logarithmic scale. This is advantageous when smaller amplitudes are important for interpretation (e.g. condition monitoring). Logarithmic scaling also corresponds to the human perception of the strength of vibrations (Weber-Fechner’s law). For this reason, standards and regulations are almost exclusively based on a logarithmic evaluation scale. Figure 14.48 shows the influence of different axis scaling based on a vibration measurement on a shaker-excited exhaust system. When moving from linear to logarithmic amplitude scaling (Fig. 14.48 top, middle), it is clear that logarithmic scaling gives a more detailed representation of the smaller amplitudes. The linear representation, on the other hand, provides an overview of the maximum amplitudes in the spectrum.

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Linear scaling of the ordinate and abscissa

5

Acceleration a peak in m/s2

Signal Analysis Methods and Examples

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Acceleration apeak in m/s2

101

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0

500

1000

1500

Frequency in Hz

Acceleration apeak in m/s2

101

10

Logarithmic scaling of the ordinate and abscissa

0

10

-1

10

-2

10

-3

10

-4

101

102

103

Frequency in Hz

Fig. 14.48 Influence of scaling on the vibration measurement (acceleration amplitude apeak) on an exhaust system under shaker excitation

"

The display with linear frequency scaling and logarithmic amplitude scaling as level is often used in the evaluation of spectra. Here, the harmonics appear on the linearly scaled frequency axis as lines with equal spacing.

Similarly, so-called sidebands by amplitude modulation are easily recognized by the splitting of the harmonics into symmetrical sidelines. Logarithmic scaling of amplitudes

14.4

Signal Analysis in the Frequency Domain

471

gives a greater variety of detail. It should be noted here that the largest amplitudes are not necessarily the most authoritative indicators for monitoring the condition of the machine. Often, in this measurement task, the development of smaller amplitudes over time is the indicator of changes in the machine [5]. If instead of the linear scaling of the frequency axis a logarithmic scaling is chosen (Fig. 14.48 middle, bottom), the representation is stretched at low frequencies and compressed at higher frequencies. "

A double logarithmic representation is often used when two parameters (slope and intercept) are to be identified in the spectrum or a mathematical relationship is to be verified experimentally with a power approach.

By logarithmizing, the exponent then appears as a slope in the spectrum, and a parameter can be read from the intercept. Through a double logarithmic representation, the exponent can be read directly as the slope of a straight line.

14.4.10 Differentiating and Integrating In the spectrum, a differentiation and integration is very easily possible by using the complex calculus. A differentiation corresponds to multiplying each spectral line by jω. Integration is done by dividing with jω. This makes it possible, for example, to represent a velocity or displacement spectrum from the measured quantity acceleration. This is shown as an example in Fig. 14.49. In addition to the simple calculation rule, the method has the advantage that the DC component in the signal does not have a disturbing effect on the result of an integration. In contrast, with integration in the time domain, the DC component causes a linear increase in the integrated signal. This phenomenon is very disturbing, since the continuous increase in the signal exhausts the value range and finally leads to a numerical overflow. On the other hand, the fact that the amplitudes in the spectrum are defined as positive values has a disadvantage. With integration in the spectrum, these are divided by the angular frequency and thus remain positive amplitudes. On the other hand, integration in the time domain takes into account positive and negative amplitudes, which result in zero after integration. Thus, integration in the time domain can result in a better signal-to-noise ratio than integration in the frequency domain, which is used for time domain averaging (Sect. 14.4.7). For the evaluation of spectra, the fast spectrum is often used. This has several reasons: • The fast spectrum shows a low slope for many machines (maximum flatness criterion [6]). This property allows to exploit high amplitude dynamics in the signal. In reverse, this means: If the course of the spectrum over the frequency shows a rise over several

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14

Signal Analysis Methods and Examples

103 Acceleration in m/s2 Velocity in mm/s Deflection in m m

102

101

Amplitude

100

10-1

10

-2

10

-3

10-4

10

-5

10

-6

0

50 0

1000

1500

2000

2500

Frequency in Hz

Fig. 14.49 Integration of an acceleration spectrum to the velocity and deflection spectrum

decades, this part of the amplitude dynamics has already been used up and can no longer be used for the measurement task. • Under certain conditions, the square of the velocity is proportional to the radiated sound power Pak at a vibrating surface. Pak  x_

2

ð14:77Þ

The measurement of acceleration, e.g. by means of piezoelectric accelerometers (Sect. 8.1), is easier to achieve on surfaces than the measurement of velocity (Chap. 7). The reasons for this are a wider frequency range of the accelerometers, lower mass and thus a greater mechanical robustness of the accelerometers. By integrating the acceleration, one obtains the velocity and can continue to work with the generated velocity signal. Finally, limitations of differentiation and integration should be pointed out. The differentiation causes a roughening of the measurement signal at high frequencies. This can be explained by the multiplication with the angular frequency. The spectrum is amplified by +20 dB per decade. However, the superimposed high-frequency interference signals are also amplified with the measurement signal. It therefore does not make sense to measure a displacement signal and “convert” it into an acceleration signal by differentiating it twice, since the interference signals would also be amplified by +40 dB per decade.

14.4

Signal Analysis in the Frequency Domain

473

The physical limits of the transducers must also be taken into account during integration. Since the signal is divided by the angular frequency, the high-frequency signal components are smoothed. At the same time, the low-frequency components are strongly increased and thus measurement deviations are amplified. In this case, the frequency response of the accelerometer used must be taken into account. In the case of piezoelectric accelerometers, the frequency response becomes nonlinear at low frequencies and again leads to large measurement deviations. For this reason, the integration of acceleration signals in the low-frequency range is only possible with accelerometers whose frequency response permits this.

14.4.11 Practical Settings for Orientation Measurements The relationships explained so far have shown that the choice of settings on the analyzer is decisive for correct and usable results. These settings always depend on the respective measurement task. For this reason, there are no universally suitable specifications that apply to all applications. In summary, the approach is described how the settings on the analyzer or in the evaluation program are to be derived from the measurement task. Preparing the Measurement Task For the quality of the results and their interpretation, it is crucial to first define and delimit the measurement task in terms of content and to establish a picture of expectations: What is to be measured and what is expected? "

The more precisely the measurement task is outlined and the more concretely an expectation is formulated, the more specifically the measurement task can be solved.

Example

A motor (M) drives a centrifugal pump (P) via a gearbox (G) (Fig. 14.50). Engine speed: Gearbox: Pump: Frequency of the motor:

1500 rpm Motor side Z1 = 100 teeth, pump side Z2 = 250 teeth 8 shovels fM =

Frequency of the pump:

f P = f M  i = f M  100 250 = 10 Hz

Tooth mesh frequency in the gearbox:

f E = f M  Z 1 = 25 Hz  100 = 2, 5 kHz

Blade passing frequency in the pump:

f E = f P  Z 2 = 10 Hz  250 = 2, 5 kHz fSP = fP  ZS = 10 Hz  8 = 80 Hz

1500 60

= 25 Hz

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14

Signal Analysis Methods and Examples

Fig. 14.50 Centrifugal pump with motor and gearbox

G

P

M

Frequency lines at 10 Hz, 25 Hz, 80 Hz and 2.5 kHz are expected in the spectrum and can be assigned to assemblies. Further frequency lines are likely to occur due to e.g. sidebands of the gear meshing frequency, bearing vibrations and higher harmonics of the calculated frequencies. ◄ Conversely, it is also clear that deficiencies in planning and preparation cannot be compensated for by an additional investment of time and effort in evaluation. In the worst case, the measurement is incorrectly evaluated and misinterpreted. As a rule, such weaknesses in the preparation lead to a repetition of the measurement, in the least severe case to unattractively long measurement times and extremely large data volumes. Setting Parameters The settings listed in Table 14.12 are intended as a guide for preparing the measurement on the analyzer. Here asynchronous sampling and stationary signals are assumed. The settings give a first and in most cases usable impression of the spectra. A first attempt can be made with the settings, which can be refined in subsequent steps of evaluation and interpretation. If other settings are prescribed in the underlying standards and technical regulations, these should be preferred. These settings are based on empirical values, describe frequently occurring measurement tasks and may therefore not be suitable for the specific individual case. Test Measurement After making the settings, it is advisable to carry out a test measurement or evaluation. Here the signal is to be evaluated in the time domain and in the frequency domain: • Time range – No overload/limitation of the signal (if necessary measure with oscilloscope before the low pass), – Assessment of stability and drift of the signal,

14.4

Signal Analysis in the Frequency Domain

475

Table 14.12 Settings on the analyzer Settings Frequency range Sect. 14.4.4

Frequency resolution Sect. 14.4.4 Averaging Sect. 14.4.7

Measuring time Sects. 14.4.4 and 14.4.7

Window function Sect. 14.4.5 Axis scaling Sects. 14.4.8 and 14.4.9 Triggering Sect. 14.4.6

Explanations Lower limit (line distance) Zero or lowest speed (rolling bearings) or half of the lowest speed (plain bearings) Upper limit (maximum frequency) 3. harmonic of the highest frequency in the system under investigation (e.g. tooth mesh frequency) Separation of adjacent components, sidebands, influence of interfering signals Periodic signals: 3 . . . 5 averages Stationary stochastic signals: Eq. 14.68 Linear averaging 66.7% overlap Measurement time depends on frequency range, resolution, averaging and overlap Critical point: Signal stability or measuring time of the existing recording Periodic and stationary stochastic signal: Hanning window Pulse: Rectangular window Periodic signal: amplitude, effective value (RMS) or power (PWR) Stationary stochastic signal: power spectral density (PSD) Pulse: amplitude or energy spectral density (ESD) For test purposes: free wheel measurement Transients: internal triggering Higher-level processes: external triggering

– Modulation of the signal (estimation of the modulation frequency, modulation frequency must be greater than lower limit of the frequency), – Sufficient noise and signal-to-noise ratio (if possible, perform a measurement without useful signal), – Window function: no fading out of relevant signal sections by the window, for the rectangular window the signal must have the value zero at both window borders. • Frequency range – Assessment of the frequency content in the spectrum (matching of the lower or upper frequency limit to the shape of the spectrum), – Assessment of frequency resolution (separation of adjacent frequencies and sidebands), – Noise and signal-to-noise ratio (mains frequency and its harmonics, qualitative estimation of noise), – Evaluation first without, then with averaging (fluctuations of the individual spectra, influence of the averaging).

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Signal Analysis Methods and Examples

In case of doubt, it is advisable to first perform an analysis in a wide frequency range and low frequency resolution. On this basis, the lower and upper limits of the frequency range of interest can be narrowed down. Afterwards, the number of lines in the spectrum is increased and thus the influence of a higher frequency resolution on the spectrum is assessed.

14.4.12 Practical Implementation in MATLAB® with Test Signals Test signals are stored in the additional material to this chapter. The test signals are mainly stored as AUDIO files and can be played via any common PC sound card. In this way, FFT analyzers and measuring instruments can be tested. Almost all test signals are generated with a sampling frequency of 44.1 kHz. Only one test signal has a sampling frequency of 96 kHz (Fig. 14.51). This signal should also be playable with any PC sound card. The special thing about this signal is that this consists of two sine tones with f1 = 100 Hz and f2 = 45 kHz with the amplitude value = 0.354. A correctly performed and scaled DFT should result in the same amplitude values as in Fig. 14.51. This would require a sampling frequency of the analyzer of, again, 96 kHz. If this sampling frequency is not available, or if a different analyzer sampling frequency is used, no signal component shall be presented in any frequency band other than at 100 Hz. This test signal is used to test the aliasing filter. The transformation of a signal from the time domain to the frequency domain based on the Fourier transform is done in MATLAB® with the statement ð14:78Þ The result s contains a vector with N complex numbers. However, a DFT according to Eq. 14.45 is actually performed without the multiplier 1/N (the normalization). For line numbers N = 2n (e.g. 1024, 2048, etc.) a computationally optimized algorithm is used. To perform the Fourier transform, MATLAB® uses the FFTW [30] program library. In order to obtain a usable result of the transformation, the normalization ð14:79Þ is necessary. Now a two-sided complex spectrum is present, from which over ð14:80Þ a two-sided amplitude spectrum can be displayed (Fig. 14.52). The MATLAB® function does not require any information about the sampling frequency fs. Thus, it does not provide a frequency vector f for scaling the abscissa. About

14.4

Signal Analysis in the Frequency Domain

477

fs = 96 kHz test signal

1 0.9 0.8

Amplitude

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Frequency in Hz

10

4

Fig. 14.51 Amplitude spectrum of the test signal with sampling frequency fs = 96 kHz. The test signal contains two sine tones f1 = 100 Hz and f2 = 45 kHz. This signal is used to test the aliasing filter Fourier transform test signal: octaves 0.1 0.09 0.08

Amplitude

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 0

2

4

6

8

N

10

12

u105

Fig. 14.52 Two-sided amplitude spectrum of the test signal octaves

ð14:81Þ a one-sided amplitude spectrum with correct amplitude values and frequency axis is presented. The condition of Parseval’s theorem that temporal power equals spectral power (Eq. 14.54) is still valid. However, only N/2 lines are present in the one-sided spectrum. This results in a correction to Eq. 14.54

478

14

Signal Analysis Methods and Examples

Fourier transform test signal: octaves 0.15

Amplitude x

0.1

0.05

0 10-1

100

101

102

103

104

Frequency in Hz 1/n octave analysis Test signal: Octaves 0.14 0.12

Amplitude x

0.1 0.08 0.06 0.04 0.02 0 32

63

126

251

501

1000

1995

3981

7943

Frequency in Hz

Fig. 14.53 One-sided amplitude spectrum of the test signal octaves in amplitude-correct representation (top) and control of the amplitudes by 1/n-octave filtering of the test signal and peak formation (bottom). In both diagrams the amplitudes have a value of 0.12

P =

1 N

N -1 k=0

ð xð k Þ Þ 2 =

N 2

-1

ð2  jX ðnÞjÞ2

ð14:82Þ

n=0

The magnitude values of the one-sided spectrum are multiplied by a factor of 2 due to the lack of mirroring. To check that the procedure was correct, since the test signal consisted of nine sine tones with octave center frequencies, 1/n octave filtering was performed and then the peak values were determined (Fig. 14.53). If you want to follow the recommendations in Sect. 14.4.11, the MATLAB® function fft () is very difficult to use. A more versatile function for the Fourier transform in practical use is

14.4

Signal Analysis in the Frequency Domain

479

ð14:83Þ

If the time data vector x is longer than the number of values required for the Fourier transformation (parameter N ), the time data vector is divided into individual N-long data blocks and a Fourier transformation is performed in each case. Before performing the Fourier transform, the data blocks are multiplied by the window function w. This requires at least the rectangular window. Alternatively, the integer value N can be specified for the parameter w, this has the same effect in the segmentation as the rectangular window function. s then becomes a matrix containing the one-sided complex spectra. Similarly, p then contains the matrix of power spectral densities. From t, in this case, a value becomes a vector, which can be used to scale a time axis. The three-dimensional representation is called a spectrogram. Figure 14.54 shows the application of spectrogram to the test signal fs_44_1 kHz_Octaves.wav. The FT parameters for the spectrogram are derived from Fig. 14.37. Starting from the sampling frequency fs, which is often given or determined by means of fmax < fs/2, the desired line width or Δf is determined. This procedure has proved itself, since it is closest to the structured solution of technical problems. The maximum occurring frequency fmax and the required selectivity Δf can be determined from the problem definition. This results in the instruction sequence in Table 14.13 for the creation of the spectrogram in Fig. 14.54. Example

To determine the running smoothness, vibration measurements were carried out on a two-cylinder diesel engine using accelerometers. The measurement signal is specified with a sampling frequency fs = 44100 Hz. The frequency resolution chosen is Δf = 2 Hz. This results in a block length of T = 0, 5 s. The remaining parameters for performing the Fourier transform were selected as in Table 14.13. Slight speed fluctuations and the speed change during the measurement should therefore be visible. Figure 14.55 shows the spectrogram of the RMS acceleration amplitudes from the vibration measurement on the diesel engine. In the present form, little information can be taken from this form of representation. What is interesting about this measurement signal is first of all the frequencies it contains. For information on which frequencies are

480

14

Signal Analysis Methods and Examples

Spectrogram of test signal: Octaves

0.15

0.15

0.1

Amplitude

0.1

0.05

0.05

0 5 0

10 0

15

10

10

1

10

20

2

25

3

10

4

Frequency in Hz

10

30

Time in s

0

Fig. 14.54 Spectrogram of the test signal fs_44_1 kHz_Octaves.wav

contained in the signal, linear averaging is the quickest way to obtain an overview. The representation is done in a simple two-dimensional diagram. Two frequencies can be read from the diagram in Fig. 14.56: • 1. frequency 60 Hz first engine order, arises from the firing order and the free mass forces of first order • 2. frequency 120 Hz second motor order, arises from the free mass forces of second order In the further frequency range, no particular conspicuous features can be detected. For the frequency range up to 1000 Hz, a three-dimensional spectrogram is shown (Fig. 14.57), but with a changed angle of view on the representation. The view from “above” on the diagram shows that the frequency peaks were only caused by the excitation in the last 10 s. The motor load was increased during the measurement. The motor load was increased during the measurement and the motor speed has a stronger

14.4

Signal Analysis in the Frequency Domain

481

Table 14.13 Sequence of instructions for creating the spectrogram in Fig. 14.54 Statement df = 0.5 N = ceil(fs/df) T = N/fs overlap = ceil(2/3*N) w = hann(N) FM = sum(w)/N PM = sum(w.^2)/N B_eff = PM/(T * FM^2) [s, f, t, p] = spectrogram(x, w, overlap, N, fs) sNorm = s/N MAG = 2*abs(sNorm)/FM surf(t, f, MAG) colormap(spring) c = colorbar set(gca, ‘FontSize’, 16) title(‘Spectrogram of test signal: octaves’) xlabel(‘time in s’) ylabel(‘Frequency in Hz’) set(gca, ‘yscale’, ‘log’) zlabel(‘Amplitude’) view(120, 20) axis([0 30 0 20000 0 0.15]) caxis([0 0.15])

Explanation Determination of the frequency resolution Δf Number of lines of the bipartite spectrum, brought to an integer value with the ceil() function Block length Overlap, Sect. 14.4.7, must be an integer value Window function, Sect. 14.4.5, other window functions in the MATLAB® help under window Window mean value according to Eq. 14.68 Average power value according to Eq. 14.64 Effective bandwidth according to Eq. 14.65 Calculation of the spectrogram. If the PSDs are not required, then the tilde character (~) can be used as a placeholder. This causes that this result is not taken over Normalization of the one-sided complex spectra Calculation of the amplitude spectrum Representation of the amplitude spectrogram Definition of the color gradient, further gradients in the MATLAB help under colormap Display of a colorbar for the color scaling of the amplitude values Determining the font size Heading of the spectrogram Labeling X-axis Labeling Y-axis Conversion of the Y-axis to logarithmic division Labeling Z-axis Rotate the spectrogram for better visualization Defining the value ranges of the axis scalings Adjustment of the color scaling to the axis scaling

change between the 20th and 25th second than before and after. Both is well visible in the view from “above” (top view). The view from “above” is provided by the instruction ð14:84Þ achieved. ◄ This form of representation is used in the following for a time-varying signal. In a first step, the method is also tested on a known signal. The file fs_44_1 kHz_sweep.wav is

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14

Signal Analysis Methods and Examples

Spectrogram diesel engine with constant speed

5

4.5

4

5

4.5 3.5 4 3

Amplitude aRMS

3.5

3 2.5 2.5 2

2

1.5 1.5 1 0

0.5

1

5

0 0

1000

2000

0.5

10 3000

4000

5000

6000

15 7000

Frequency in Hz

8000

9000

10000

0

Time in s

20

Fig. 14.55 Spectrogram of the vibration measurement on the two-cylinder diesel engine Spectrogram diesel engine with constant speed

linearly averaged amplitude a RMS in m/s2

15

10

5

0 0

100

200

300

400

500

600

700

800

900

1000

Frequency in Hz

Fig. 14.56 Spectrogram of the vibration measurement on the two-cylinder diesel engine in the relevant frequency range and for determining the frequencies contained linearly averaged over the entire measurement time

14.4

Signal Analysis in the Frequency Domain

483 Spectrogram diesel engine with constant speed

Fig. 14.57 Spectrogram of the vibration measurement on the two-cylinder diesel engine in the relevant frequency range in the block from “above”

30

12

25 10

20

Time in s

8

15 6

10 4

5

2

0

0 0

50

100

150

200

Frequency in Hz

available in the additional materials. This test signal consists of a cosine with amplitude 1. Within 30 s, the signal changes from f1 = 25 Hz to f2 = 2500 Hz. Figure 14.58 contains another representation of spectrograms. Three diagrams are shown. In the diagram at the top left (large) the spectrogram is shown from “above”. The respective amplitude value (Z-axis) can be taken from the colour coding. In the two other diagrams the horizon lines are shown. In the diagram on the right, the time is shown as the ordinate and the amplitude as the abscissa. This leads to a better interpretability of this partial representation, since it represents the view from the right into the spectrogram. In this diagram, sections along a frequency line or the horizontal line, which is drawn over ð14:85Þ was displayed. Analogous to this, the lower diagram contains the view into the spectrogram from the frequency axis. In this diagram, sections along a time (line) can be displayed, or the corresponding horizontal line, which is drawn over ð14:86Þ was presented.

484

14

Signal Analysis Methods and Examples

Spectrogram Frequency sweep View from “above”

30

Horizon line View from “right”

0.1 0.09

25

0.08 0.07

20

15

0.05

Amplitude

Time in s

0.06

0.04 10

0.03 0.02

5 0.01 0

0

0.05

0.1

0.15

0

0.15

0.1

0.05

0

0

500

1000

1500 2000 Frequency in Hz

2500

3000

Fig. 14.58 Spectrogram frequency sweep in a 3-diagram representation. Diagram top left (large) shows the spectrogram from “above”. Diagram top right shows the view from the right and in this case represents the “horizontal line” over time. The bottom diagram is the view along the frequency axis and in this case shows the corresponding horizontal line

If we look at the amplitude values of the horizontal lines of the right and lower diagrams, we can see that they are only approx. 1/10 of the setpoint 1. In the representation over the frequency, a comb-like course of the amplitude line can also be seen. The reason for this is to be found in the unsuitable frequency resolution for this sweep. With a frequency change from f1 = 25 Hz to f2 = 2500 Hz during 30 s, the sweep has a slope of f ST =

f 2 - f 1 2500 Hz - 25 Hz Hz = 82, 5 = 30 s s t

ð14:87Þ

On. The DFT was performed with a Δf = 0, 5 Hz. From which a block length T = 2 s is derived. During the block length the signal undergoes a change of

14.4

Signal Analysis in the Frequency Domain

485 Horizon line View from “right”

Spectrogram Frequency sweep View from “above”

30

1 25 0.8

0.6

15

Amplitude

Time in s

20

0.4

10

0.2

5

0

0

0.5

1

1.5

0

1.5

1

0.5

0

0

500

1000

1500 Frequency in Hz

2000

2500

3000

Fig. 14.59 Analogous to Fig. 14.58 Spectrogram frequency sweep in a 3-diagram representation, but with Δf = 82, 5 Hz

Δf Signal = T  f ST = 2 s  82, 5

Hz = 165 Hz s

ð14:88Þ

This corresponds to 330 lines of the spectrum. It is true that the 66.7% overlap “saves” a little and raises the amplitude values again. Overall, however, the parameterization for this case is incorrect. For Δf, at least the value of fST = 82, 5 Hz must be adopted to obtain the true amplitude values (Fig. 14.59). Figure 14.59 shows the result of the DFT with Δf = 82, 5 Hz. The amplitude values that are to be expected from the signal with amplitude = 1 can now be seen as far as possible. However, in the range f near 0 and t near 0, too high amplitude values can now be seen. This is an effect of the combination of the signal shape (cosine), the window function used (Hanning), the overlap performed (66.7%) and the amplitude correction required as a result. The amplitude correction is simply wrong for the first spectrum of the spectrogram. The only solution that can be adopted is to dispense with the start of the signal.

486

14

Signal Analysis Methods and Examples

Spectrogram speed up/ speed down diesel engine

180

Frequency cuts

180

160

160

140

140

120

120

60

50

100

100

80

80

60

60

40

40

20

20

30

Amplitude in m/s

Time in s

2

40

20

10 66 Hz 134 Hz

0

0

50

100

150

200

250

0

0

20

40

Amplitude in m/s

0

Time slices

70

140 s 40 s

60

Amplitude in m/s

2

60 2

50 40 30 20

10 0

0

50

100 150 Frequency in Hz

200

250

Fig. 14.60 Spectrogram of the DFT of a speed run-up and speed run-down

Example

The motor from the previous example is measured during the speed run-up/run-down. This runs for 180 s. During the first 150 s, the speed ramp-up occurs from n = 700/min to n = 4000/min. According to Eq. 14.87, the result for the second motor order is fST = 0, 74 Hz/s, so that the DFT for this signal can be calculated with Δf = 2 Hz Figure 14.60 shows the spectrogram as well as frequency and time slices. The frequency slices (right diagram) show the frequency lines which appear as frequency peaks at 140 s in the time slice. The corresponding line is calculated as ð14:89Þ which is contained in the statement ð14:90Þ

14.4

Signal Analysis in the Frequency Domain

487

leads to the representation as an amplitude curve over time. The index for the time slice is determined by determining the time intervals from the vector t. In t(1,1) is the nominal time for the first spectrum t1. The time interval of all other spectra is different due to the overlap. Δt can be determined, for example, via t(5, 1) - t(4, 1). The index for the time slice is then calculated as follows Index =

t - t1 þ1 Δt

ð14:91Þ

Here, the number of spectra is determined for the time span t - t1, t minus the nominal time of the first spectrum. The index used is the next larger integer value, which will deviate slightly from the desired time slice. ◄ As explained in Sect. 14.4.7, averaging spectra improves the significance of the result. The spectrogram in Fig. 14.60 consists of 1078 spectra at a time interval of ΔtSpek = 0, 1666 s. With a frequency change of the second motor order of fST = 0, 74 Hz/s and the chosen parameterization of the DFT with Δf = 2 Hz and 66.7% overlap, a calculable number of spectra are run through until the signal has experienced a frequency change of 2 Hz. This can be used to perform averaging without any loss of information. The ratio Δf/ fST gives the time span needed for a signal change from Δf. During this period n spectra with a time interval of ΔtSpek will have a spectrum that differs only due to measurement scatter. It should be noted that ΔtSpek describes the time offset of the spectra from each other, while T describes the block length of a spectrum (Fig. 14.61). The calculation of the averaging number thus results in

n =

Δf f ST

-T þ1 Δt Spek

ð14:92Þ

The next smaller integer value is then used as the number of spectra that can be used for averaging. If n becomes smaller than 1, the parameterization of the DFT for this analysis is not correctly selected. For the spectrogram in Fig. 14.60, the possible averaging number of n = 14.2 with an averaging time of T A = T þ ðn - 1ÞΔt Spek T A = 2, 7 s

ð14:93Þ

which leads to a BeffTA -product of size 4.05 and to a relative standard deviation ε according to Eq. 14.68 of 24.8%. In the MATLAB® code the averaging is realized by a loop.

488

14

Signal Analysis Methods and Examples fSignal= ft0 + ∆f

fSignal= ft0 DFT

T

∆tSpek t=0s

t = T+n ∙ ∆tSpek Time

Fig. 14.61 Schematic diagram for determining the number of spectra that can be used for averaging a time-varying signal without loss of information

ð14:94Þ

In addition to the averaged spectra, a new time vector t2 must also be formed. The last averaging is done with less than 14 spectra, if this is the case. Another form of the calculation run is also conceivable, in which the averaging starts from the seventh. Spectrum and is carried out smoothly up to the seventh last (Fig. 14.62). In this case, a total of 14 spectra are lost at the beginning and end of the spectrogram due to averaging compared to the original DFT. The MATLAB® code for this is:

14.4

Signal Analysis in the Frequency Domain

489

moving averaged spectrogram speed up/ speed down diesel engine

180

60

160 50 140

Time in s

100 30 80

Amplitude in m/s2

40

120

20

60

40 10 20

0

0

50

100

150

200

250

0

Frequency in Hz

Fig. 14.62 Moving average spectrogram

ð14:95Þ

Another approach to averaging spectra can be followed if the measurement time is divided into discrete time intervals tStep (interpolation distance in seconds) and n averages are assigned to each time point as a spectrum. It is important to define the nominal time of a spectrum. This can be set at the beginning, in the middle and at the end of the block length T. Usually the middle is used, this corresponds to the times in the time vector t of the MATLAB® function spectrogram. An averaging is done with n spectra on the left and n spectra on the right at the time t of the middle block with block length T. Figure 14.63 shows the scheme for this type of averaging. In the MATLAB® code, the following specifications are available for this purpose

490

14

Signal Analysis Methods and Examples

2n+1 averages with 66.7% overlap

t n ∙ ∆tSpek

t0

T

TA

n ∙ ∆tSpek

∆tSpek

t1

Time

Fig. 14.63 Schematic representation of the averaging of 2n + 1 spectra at a nominal time t, which was defined in the middle of the averaging time TA

ð14:96Þ for the interpolation distance and the one-sided averaging. From this, the parameters for the stepwise calculation of the spectrogram are determined: ð14:97Þ For the calculation of the spectrogram, the instruction spectrogram is called once each along the signal via a loop at the intervals nStep and the calculated spectra are averaged. In addition, a time vector t must be calculated.

14.4

Signal Analysis in the Frequency Domain

491

ð14:98Þ

Analogous to this, averaging is carried out for signals without temporal change. The consideration of the number of spectra for the averaging is then carried out exclusively via Eq. 14.68. In addition to linear averaging, there are other possibilities for averaging or evaluating the calculated spectra. These are described in Table 14.14. The matrix of the spectra calculated by the MATLAB® function is column-oriented. This means that the first spectrum is stored in column 1, the second spectrum in column 2 and so on. Averaging or other evaluation of the spectra must therefore be performed row by row, which is indicated by the additional parameter dim = 2 in the corresponding instructions.

14.4.13 Spectral Quantities in MATLAB® As explained in Sect. 14.4.8, the ordinate value in the spectra is represented differently depending on the signal type, use case, and agreement. The implementation in MATLAB® code (Table 14.13) is not only a rescaling of the ordinate. The MATLAB® statements ð14:99Þ transform the signal x from the time domain into the frequency domain. sNorm contains the normalized one-sided complex spectrogram. In the special case that the time signal contains as many values as the desired line number N/2, the spectrogram is reduced to a spectrum. The further processing by averaging etc. is not necessary thereupon. Amplitude Spectrum, Magnitude Spectrum-MAG The linearly averaged amplitude spectrum is calculated by the instruction

ð14:100Þ

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Signal Analysis Methods and Examples

Table 14.14 List of possible MATLAB® functions for processing spectra Functionality Linear mean

Quadratic mean

Function mean(s, dim) rms(s, dim)

Description Calculates the linear mean value of the matrix s along the calculation direction specified in dim according to x =

Standard deviation Maximum value Mode

sum(s, dim) std(s,[ ], dim) max(s,[ ], dim) mode(s, dim)

n ξ=1

xðξÞ

Calculates the root mean square value of the matrix s along the calculation direction specified in dim according to X RMS = 1 n

Total

1 n

n ξ=1

xðξÞ2

Calculates the sum of the matrix s along the calculation direction specified in dim Calculates the standard deviation of the matrix s along the calculation direction specified in dim Determines the maximum value from the matrix s along the calculation direction specified in dim Determines the most frequently occurring value of the matrix s along the calculation direction specified in dim

is calculated. The result is the averaged peak values xðf Þ. Further averaging types are shown in Table 14.14. For the rms value the calculation is done via ð14:101Þ and displayed as level in dBV (LV(re 1V )) ð14:102Þ Multiplying the normalized spectrum (sNorm) by 1/FM (FM according to Eq. 14.63) performs the required amplitude correction in Eqs. 14.100, 14.101 and 14.102. Power Spectrum- PWR The averaged power spectrum is calculated according to the MATALAB instructions

ð14:103Þ

Since the order of the instructions plays a decisive role for the result, it is often more pragmatic – and easier to read – if instead of nesting the instructions within each other, they are carried out in individual steps. First, the magnitude spectrum is formed from the matrix

14.4

Signal Analysis in the Frequency Domain

493

of the normalized complex spectra s. In the next step, the rms values are calculated by the matrix of the normalized complex spectra. In the next step, the rms values are formed by p division with 2, while in the third step the squaring to the power spectrum is performed and then reduced to a spectrum by linear averaging. For correct power representation, the magnitude spectrum was multiplied by the correction value 1/FM (FM according to Eq. 14.63). Spectral energy density, auto power spectrum – ESD, APS ð14:104Þ

Power Spectral Density, Auto Power Spectrum, Auto Spectral Density, Power Spectral Density: PSD There are three possibilities for determining the PSD:

1. via the MATLAB function spectrogram and subsequent linear averaging ð14:105Þ 2. by the MATLAB® function pwelch, which already performs averaging ð14:106Þ 3. in individual instruction steps according to Eq. 14.68 ð14:107Þ As Fig. 14.64 shows, all three calculation methods lead to the same result. Cepstrum The cepstrum is calculated by the MATLAB® function.

ð14:108Þ

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14

Signal Analysis Methods and Examples

Test signal for PSD comparison, analysis with 66.7% overlap and Hanning window function

100

Amplitude m/s2

50

0

-50

-100 0

2

4

6

8

10

12

14

16

18

20

30

35

40

45

50

30

35

40

45

50

35

40

45

50

Time in s PSD from spectrogram

30

PSD in (m/s2)2/Hz

25 20 15 10 5 0 0

5

10

15

20

25

Frequency in Hz PSD from pwelch

30

PSD in (m/s2)2/Hz

25 20 15 10 5 0 0

5

10

15

20

25

Frequency in Hz PSD according to equation 15.65

30

PSD in (m/s2)2/Hz

25 20 15 10 5 0 0

5

10

15

20

25

30

Frequency in Hz

Fig. 14.64 Comparison of the calculation methods for determining the PSD from a measurement signal

14.4.14 Modulation Analysis Amplitude modulation is the addition of a low-frequency oscillation to a higher-frequency oscillation. The addition of both oscillations leads to a change in amplitude over time. Amplitude-modulated oscillations occur in machines, e.g. in gearboxes. This is triggered, for example, by shaft eccentricity.

14.4

Signal Analysis in the Frequency Domain

495

Amplitude modulated oscillation 1

Amplitude

0.5

0

-0.5

-1 1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

1.05

Time in s

Fig. 14.65 Schematic representation of the gear stage (top) and the amplitude-modulated oscillation (bottom)

Example

A gear stage (Fig. 14.65) consists of the gears Z1 = 40 teeth and Z2 = 25 teeth. For a speed of Z1 with nZ1 = 3000/min or rotational frequency fD, Z1 = 50 Hz, there is a gear meshing frequency fZ = 2000 Hz and for the speed of Z2 by the gear ratio nZ2 = 4800/ min or rotational frequency fD, Z2 = 80 Hz. By amplitude modulation in the respective rotational frequency, the gears can modulate the tooth mesh frequency. Modulation analysis determines the modulation frequency and the modulation depth of the carrier frequency, in this example the gear meshing frequency. Several steps are required to perform a modulation analysis: • Use 1/n octave filtering to divide the signal into frequency bands. • Form the envelope for each frequency band • DFT of the envelope The result is plotted as a color-scaled two-dimensional image. The abscissa is scaled with the carrier frequency, the ordinate with the modulation frequency and the color information as the modulation depth. From Fig. 14.66, it can be read that for the example, the carrier

496

14

Signal Analysis Methods and Examples

Modulation analysis

200

60

180 50

160

140

120

100

30

Modulation depth in %

Modulation frequency in Hz

40

80 20

60

40 10

20

0

0

0

63

200

631

1995

5012

(carrier) Frequency in Hz

Fig. 14.66 Modulation analysis for the gearbox example

frequency f~2000 Hz is strongly modulated with f~50 Hz. From this it can be deduced that the gear meshing frequency fZ = 2000 Hz is modulated by the rotational frequency of the gear Z1 ( fD, Z1 = 50 Hz). MATLAB® code (excerpt) In order to avoid analyzing the filter transients in the subsequent Fourier transform, the filtered signal and the time vector are shortened by 5 s at the beginning and 5 s at the end. ð14:109Þ For 1/3 octave filtering the filter parameters are defined: ð14:110Þ

14.4

Signal Analysis in the Frequency Domain

497

The signal filtering, envelope calculation and Fourier transform are performed for each frequency band within a loop:

ð14:111Þ

performs the signal filtering (cf. instruction 14.34). ð14:112Þ forms the envelope around the signal and ð14:113Þ performs the Fourier transform. ð14:114Þ forms the averaged amplitude spectrum. This is the modulation amplitude spectrum of the frequency band. ð14:115Þ closes the loop. The variable MOD contains a 6401 × 24 matrix with the modulation amplitudes. This is to be interpreted as modulation frequency x carrier frequency band. The modulation frequencies are present in the vector f from the spectrogram function calls. For the representation, the “placeholder” vector ð14:116Þ formed. About ð14:117Þ the modulation analysis is displayed graphically. At the end over

498

14

Signal Analysis Methods and Examples

Modulation analysis

200

15

180

160

140

120

Modulation depth in %

Modulation frequency in Hz

10

100

80 5

60

40

20

0

0

0

63

200

631

1995

6310

19953

(Carrier) frequency in Hz

Fig. 14.67 Modulation analysis of an electric actuator for vehicle seat adjustment

ð14:118Þ

the correct frequency values of the carrier frequency bands of the abscissa labeled. ◄

Example

Using the airborne sound recording of an electric actuator for vehicle seat adjustment already used in Fig. 14.9, a modulation analysis is performed (Fig. 14.67). From the modulation analysis it can be seen that several frequency bands are modulated with a modulation frequency fMOD = 101 Hz. The color-coded X/Y display has become established for displaying the results of the modulation analysis. In order to be able to define the range limits of the axes, it is advisable to first of all make yourself familiar with the display (Fig. 14.68 left).

Transfer Function

499

14

15

12 10

10

8 6 5

4 2

0 200

0 3 x10

4

30

2 20 1

150

30 100

10 0

0

Modulation frequency in Hz

200

Modulation analysis

15

150 10

100 5

50

Modulation depth in %

14.5

20 50

10 0

0

0 0 63 200 631 (Carrier) frequency in Hz

0

Fig. 14.68 Step-by-step editing of the graphical representation up to the final version

ð14:119Þ and gradual adjustment of the scaling (Fig. 14.68 middle) ð14:120Þ to the final display (Fig. 14.68 right) ð14:121Þ to work ahead. ◄

14.5

Transfer Function

The transfer function links the variables of output and input on a system capable of oscillation. With the transfer function, the complete description of a linear timeindependent system (LTI system) is possible. From the results, conclusions can be drawn about the system itself and its parameters. As parameters in mechanical systems, e.g. mass, spring constant, damping can be determined by the transfer function; in electrical engineering and acoustics, the gain, non-linear distortions and phase shift angle can be determined via the transfer function. If the quantities are available as a time function for the input x(t) and for the output y(t) of the oscillatory system, the following is obtained for the function h(t)

500

14

Signal Analysis Methods and Examples

1

yð t Þ =

xðt Þ hðt - τÞdτ

ð14:122Þ

-1

The function h(t) is called the impulse response function. Symbolically one writes for the executed convolution operation yðt Þ = xðt Þ  hðt Þ

ð14:123Þ

In measurement practice, however, the formulation in the frequency representation is usually preferred. H ðjωÞ =

Y ðjωÞ X ðjωÞ

ð14:124Þ

The quotient of the complex spectra of the input X( jω) and the output Y( jω) results in the complex function H( jω) as a function of the angular frequency ω. The function H( jω) is called the frequency response function (FRF). Alternatively the notation FRF ðjωÞ =

AðjωÞ EðjωÞ

bzw:

FRF =

A E

ð14:125Þ

is possible, whereby with the latter notation a confusion with the absolute value of the transfer function is possible. Figure 14.69 illustrates the relationships between signal input and signal output in a linear time-invariant system. Input and output are linked in the frequency representation by the transfer function; in the time representation by the weight function. The experimental determination of the transfer function using the example of the vertical vibrations of a motorcycle fork is shown in Fig. 14.70. The input x(t) is understood to be the excitation at the tire contact point, and the output y(t) is understood to be the vibration response. Both vibration quantities are measured in the vertical direction in this example. The signals at the input x(t) and output y(t) of the oscillatory system are acquired synchronously in time on two channels. Then the Fourier transformation of both signals is performed (Sect. 14.4.2). These are then available as complex spectra X( jω) and Y( jω) of the input and the output. The transfer function H( jω) is then formed from the spectra X( jω) and Y( jω). The transfer function H( jω) is also complex, i.e. it contains two numerical values (amplitude and phase angle or real and imaginary part) for each (circular) frequency. The amplitude and phase frequency response are usually plotted against the angular frequency ω or frequency f (bottom diagram, see Sect. 4.3). The Nyquist diagram (locus) is also used (Sect. 4.4). However, the transfer function cannot be formed simply as “output spectrum Y divided by input spectrum X” since superimposed noise would distort the transfer function. For this reason, the transfer function H must be modified for metrological detection by multiplying

14.5

Transfer Function

501

INPUT

OUTPUT

TIME REPRESENTATION

FREQUENCY DISPLAY

Frequency display

Time representation

Fig. 14.69 Relationships between input and output on a linear, time-invariant system

Input

Output Output y (t ) Input x (t )

X (j

)

H (j

)=

Y (j

X (j

) )

Amplitude share

Y (j

)

Phase component

Fig. 14.70 Block diagram for experimental determination of the transfer function

the numerator and denominator by the complex conjugate X [2, 3, 21]. The result is referred to as the transfer function H1 H 1 ðjωÞ =

Y ðjωÞ  X  ðjωÞ X ðjωÞ  X  ðjωÞ

ð14:126Þ

In the denominator we now obtain with the formal definition of the two-sided auto power spectrum SXX(ω) = X( jω)  X( jω) and in the numerator with the cross power spectrum SXY(ω) = Y( jω)  X( jω)

502

14

H 1 ðjωÞ =

Signal Analysis Methods and Examples

SXY ðjωÞ SXX ðωÞ

ð14:127Þ

The auto power spectrum SXX is real, but the cross power spectrum is complex. In the transition to the one-sided auto power spectrum4 GXX and the one-sided cross power spectrum GXY one obtains using Eq. 14.67 H 1 ðjωÞ =

GXY ðjωÞ GXX ðωÞ

ð14:128Þ

Thus, the transfer function is no longer specified as “output divided by input”, but as “cross power spectrum divided by auto power spectrum of the input”. Analogously, one can proceed by multiplying the numerator and denominator by the complex conjugate Y. Then the transfer function H2( jω) is obtained in the formulation “auto power spectrum of the output GYY divided by the cross power spectrum of the input GYX”. H 2 ðjωÞ =

Y ðjωÞ  Y  ðjωÞ S ð ωÞ G ð ωÞ = YY = YY X ðjωÞ  Y  ðjωÞ SYX ðjωÞ GYX ðjωÞ

ð14:129Þ

In the user interface of signal analyzers or the software, the selection between H1 and H2 must then be made. The difference between the transfer functions H1 and H2 becomes apparent when a disturbance signal is superimposed on the input or the output of a system capable of oscillation. The following cases are distinguished [2, 3, 21]: • Noise on the output: The system is excited with the quantity e(t), this is measured as a measurement signal x(t) without interference. On the other hand, a disturbance is additively superimposed on the output signal a(t). This disturbance can act as m(t) at the input or as n(t) at the output. It is assumed that the noise m(t) or n(t) is unrelated to the excitation e(t). In this case, both signals are referred to as “identically uncorrelated”. Due to the assumptions made (LTI system, identically uncorrelated noise), the order of arrangement of the individual elements in the signal flow is interchangeable. This disturbance can, for example, be the noise of the transducer or the measuring amplifier at the output and is measured in the noisy measuring signal y(t). • Noise on the input: A disturbance m(t) is additively superimposed on the excitation e(t) at the input of the system and is measured as a noisy measurement signal x(t). In this case, too, the input signal e(t) and noise m(t) are identically uncorrelated. The output

4

The formula symbol G is also used in control engineering for the transfer function; here it has the meaning of the one-sided power spectrum.

14.5

Transfer Function

503

signal a(t), on the other hand, is measured without interference as the measurement signal y(t). • Noise on input and output: The noise signals m(t) and n(t) are superimposed on the excitation e(t) and the output a(t), respectively. The measured quantities at the input x(t) and at the output y(t) are therefore both noisy. The noise signals m(t) and n(t) are identically uncorrelated to each other and to the input. e(t) The transfer functions for these cases are given in Fig. 14.71. In the case of noise on the input, the transfer function H1 provides an estimate for the transfer function H, which contains only a statistical error. The transfer function H2 also contains the systematic error Gnn/Gaa. For error correction, the auto power densities of the noise itself and of the non-noisy signal would have to be used, which, however, are usually not accessible by measurement. For the case of noise on the output, on the other hand, the transfer function H2 provides an estimate for the transfer function H without systematic error. "

Noise on the output: Use transfer function H1. Noise on the input: Use transfer function H2.

If both input and output are noisy, both H1 and H2 have a systematic error. Since the systematic error is greater than 1, H1 returns values that are too small, while H2 returns values that are too large. For this case, the transfer function H lies between the two limits H1 and H2. Usually H1 gives a better approximation in the transfer function H for the minima [3]. The excitation signal is much larger than the output signal, i.e. Gee > Gaa, thus the bracket expression in H1 becomes small. Maxima can be better approximated by H2 as Gaa > Gee. These uncertainties in the estimation of the transfer function H show that it is useful to eliminate interference sources to a large extent. This means to acquire the measurement signals x(t) and y(t) as far as possible without intermediate interference sources. If this is not possible, the measurement setup should be optimized so that the interference sources are assigned either to the output or to the input. In order to assess the linear dependence of the output spectrum Y with respect to the input spectrum X, the coherence function γ 2 (coherence spectrum, coherence) is used. γ 2 ð ωÞ =

jSYX ðjωÞj2 SXX ðωÞ  SYY ðωÞ

ð14:130Þ

The coherence function γ 2 is real and covers the value range between 0 and 1. The value of the coherence γ 2 states for the frequency to what extent the signal y(t) originates from the signal x(t). The coherence function thus answers the question of the causal relationship between two signals. Between the coherence function γ 2 and the experimental transfer functions H1 and H2 there is the relationship

504

14

Signal Analysis Methods and Examples

Equivalent circuit diagram

+

+

+

+

+

Fig. 14.71 Transfer function with superimposed disturbances

Transfer function

14.5

Transfer Function

505

γ2 =

H 1 ðjωÞ H 2 ðjωÞ

ð14:131Þ

Thus, the coherence function γ 2 represents, among other things, the link between the transfer functions. If the measurement signal y(t) is assigned to the output and the measurement signal x(t) to the input, the result is: • γ 2 = 1 for perfect linear dependence, i.e. the signals are fully correlated. The measurement signals at the output and at the input have a common cause in the excitation. The system is linear and time-invariant. • γ 2 = 0 for perfect linear independence, i.e. the signals are uncorrelated. The signals are therefore causally unrelated to each other. These statements apply without offset and without noise interference in the signals. Often values for the coherence of 0 < γ 2 < 1 are measured. The causes are found in the following cases: • Superimposed interference (noise) on the input and/or output (cf. Figure 14.71), • Non-linear relationship between the signals y(t) and x(t) (non-linearity in the oscillating system, e.g. rattling, friction, etc.; non-linearity in the measurement signal, e.g. filtering, overload), • Leakage due to window functions (see Sect. 14.4.5), • Transit time difference between both signals in the range of the block length (see Sect. 14.4.4). This allows a measurement chain to be optimized so that the coherence function is maximized in the frequency range of interest. In addition to the interpretation as the quotient of the transfer functions H1 and H2, the coherence can also be interpreted as the signal-to-noise ratio SNR for the respective frequency SNR =

γ2 1 - γ2

ð14:132Þ

In the fraction, the numerator is considered proportional to the output power, and the denominator is considered proportional to the noise power. "

The measured transfer function must always be considered in conjunction with the coherence function.

The transfer function does not indicate whether there is a causal relationship between output and input for this frequency. For this purpose, coherence must be used as an additional indicator. In practice, coherence is often considered sufficient for γ 2 ≥ 0, 8 to

506

14

Signal Analysis Methods and Examples

indicate a causal relationship and thus correlation of the two signals [20]. On the other hand, if the coherence is small, the transfer function is not interpreted at this frequency. For values in the coherence function of γ 2 ≤ 0, 2, a causal relationship between output and input is no longer assumed [20]. In the following, a linear oscillatory system consisting of a mass, spring and damper is considered. The dynamic mass, the spring constant and the damper constant can be calculated from the experimentally determined transfer function according to Sect. 4.3. For further information, please refer to the literature [12, 22, 23]. As a simple example, the determination of the dynamic mass on a pendulum is discussed. This procedure is practically used for the calibration of an impulse hammer. To test the measurement process, a compact mass suspended from two wires is struck with an impulse hammer. The natural frequency of this pendulum is approx. 1 Hz and is not considered. The impulse hammer has a piezoelectric force transducer (Sect. 9.4) and is used with a hard tip. When the mass is struck, the acceleration is measured by a piezoelectric accelerometer (Sect. 8.1). The measurement signals of the accelerometer and the impulse hammer (force transducer) are transformed in a signal processing into the frequency representation and the transfer function H1( f ) is formed. The entire measuring arrangement and signal processing can be seen in Fig. 14.72. If the acceleration is taken as input in m/s2 and the force as output in N, the dynamic mass mdyn can be calculated as the quotient “output divided by input”. mdyn = H ðf Þ =

F €x

ð14:133Þ

can be calculated. The dynamic mass mdyn in Eq. 14.133 represents the amplitude frequency response of the transfer function. The phase frequency response is not considered further in this example. Since the measurement is performed on a compact mass, a frequencyindependent course of the amplitude frequency response is to be expected. Input and output signals were acquired with the same setting parameters for the measurement channels. The data acquisition was performed synchronously, but without a start trigger, so that the time signals (Fig. 14.73) must first be trimmed for further analysis. For the signal analysis, a signal of block length T is required, which is dependent on the desired frequency resolution. To determine the correct transfer function, the input and output signals must contain the entire pulse hammer blow. Therefore, the time signals are truncated in such a way that a short time span before the pulse is retained, e.g. 5 ms.

14.5

Transfer Function

507

Suspension compact mass

Accelerometer

Impulse hammer

Measuring Amplifier

Signal processing Transmission function

FT Measuring Amplifier

Signal processing

12.01.2015

Fig. 14.72 Measurement setup and block diagram for determining the transfer function Impulse hammer

Accelerometer

800

800

700

700

600

600

500

Acceleration in m/s2

Force in N

900

500 400 300 200

400 300 200 100

100

0

0

-100

-100

-200 0

5

10

15

Time in s

20

25

30

0

5

10

15

20

25

Time in s

Fig. 14.73 Time signals of the force at the impulse hammer and the acceleration at the mass

30

508

14

Signal Analysis Methods and Examples

ð14:134Þ

The MATLAB® code shown in Eq. 14.121 performs the trimming (see Fig. 14.74) of the time signals. The for loop determines the index of the force vector (time signal from the force measurement) at which the force value first exceeded the threshold. The statement break ends the execution of the for-loop. Hereby inc contains the searched index. The starting point for the cutting of the signals is the determined index minus the specified pre-trigger. The end point results from the start point plus the block length N. Since the pulse-shaped signals have the value zero at both window boundaries, a rectangular window can be used (see Sect. 14.4.5). The advantage of using the rectangular window in this case is that no correction has to be made in the spectrum. The signal is interpreted as a periodic pulse train of block length T. Therefore, the energy density is not specified. For this reason, the energy density is not specified, but as a discrete spectrum in m/s2 or in N (cf. Sect. 14.4.8). The measurement results of the force and the acceleration can be seen in the time representation in Fig. 14.74 and in the frequency representation in Fig. 14.75. The time curve of the acceleration signal is characterised by an impulse followed by a highfrequency oscillation process. This post oscillation process can also be seen in the spectrum at a frequency of 15 kHz (Fig. 14.75 below) and corresponds to the natural frequency of the mounted accelerometer. The transfer function (FRF) is calculated by element-wise division of the force spectrum with the acceleration spectrum. ð14:135Þ The result is the complex transfer function FRF. For the representation as amplitude frequency response is done via ð14:136Þ or as phase frequency response

14.5

Transfer Function

509

Impulse hammer

900

700

700

600

600

500

Acceleration in m/s2

800

500

Force in N

Accelerometer

800

400 300 200

400 300 200 100

100

0

0

-100

-100

-200 0

0.5

1

1.5

2

0

0.5

Time in s

1

1.5

2

Time in s

Fig. 14.74 Time representation of the force and acceleration at the mass as used to determine the transfer function Impulse hammer

100

Force in N

10-1

10-2

10

-3

10-4 1 10

Acceleration in m/s2

10

10

10

2

10

3

10

4

Accelerometer

0

-1

10-2

10

-3

10-4 101

102

103

Frequency in Hz

Fig. 14.75 Frequency representation of force and acceleration

104

FRF in N/m/s2

510

14

10

1

10

0

Signal Analysis Methods and Examples

Transfer function

10-1

10-2

10

-3

10

1

10

2

10

3

10

4

Frequency in Hz

Fig. 14.76 Amplitude frequency response of the transfer function

ð14:137Þ The amplitude frequency response of the transfer function is shown in Fig. 14.76. In the frequency range from 20 Hz to approx. 5 kHz, the frequency response shows an almost constant, i.e. frequency-independent curve. At higher frequencies, the curve deviates increasingly from the constant. The cause can be found in the course of the acceleration and force spectrum. At higher frequencies, both the acceleration and the force spectra bend. Thus, the transfer function increasingly takes on the form of “zero divided by zero”, which leads to the deviations mentioned at higher frequencies. The coherence function (Fig. 14.77) has the value of 1 over the entire frequency range as an expression of the linear dependence. In the frequency range > 5 kHz the linear dependence is still given, although the results of the transfer function in this range are obviously wrong. As a consequence for practice, it can be inferred from the course of the transfer function that the excitation must take place with sufficient power in the entire frequency range of interest in order to obtain a sufficient response signal. This applies in particular to damping and to nonlinearities that can never be avoided in practice (backlash, friction, nonlinear spring and damper characteristics, etc.). In another example, the sound-determining resonant frequencies and their damping of a flywheel are determined. As described in detail in Chap. 15, the resonant frequencies of the damped oscillation fd (or ωd resonant circuit frequency) can be determined using the transfer function. The procedure is identical to the measurement setup in Fig. 14.72. The exact determination of the resonant frequencies via the MATLAB® function modalfit, which is also discussed in Chap. 15. From Fig. 14.78, the frequencies 1222 Hz, 3215 Hz and 3235 Hz can be taken as resonant frequencies. Checking the frequency ranges using the MATLAB® function

References

511 Coherence function

1

Coherence

2

0.8

0.6

0.4

0.2

0 101

102

103

104

Frequency in Hz

Fig. 14.77 Coherence function of the measurement signals Resonant frequencies Flywheel

10-5

FRF in N/m/s2

10-6

10

-7

10-8

10-9

10

-10

500

1000

1500

2000

2 50 0

3000

3500

4000

Frequency in Hz

Fig. 14.78 Determination of the resonant frequencies of a flywheel using the transfer function

modalfit shows 1221.2 Hz, 3214.3 Hz and 3222.1 Hz as the exact resonant frequencies. Further MATLAB® examples can be found in [25].

References 1. Butz, T.: Fouriertransformation für Fußgänger. Springer Vieweg, Wiesbaden (2012) 2. Randall, R.B.: Frequency Analysis. Bruel & Kjaer, Naerum (1987) 3. Zollner, M.: Frequenzanalyse. Autoren-Selbstverlag, Regensburg (1999) 4. Klein, U.: Schwingungsdiagnostische Beurteilung von Maschinen und Anlagen. Stahleisen, Düsseldorf (2003) 5. Kolerus, J., Wassermann, J.: Zustandsüberwachung von Maschinen. expert verlag, Renningen (2011)

512

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Signal Analysis Methods and Examples

6. Piersol, A.: Vibration data analysis. In: Piersol, S. (ed.) Paez: Harris’ Shock an Vibration Handbook. McGraw-Hill Education, New York (2009) 7. Zollner, M.: Signalverarbeitung. Autoren-Selbstverlag Regensburg (1999) 8. Jenne, S., Pötter, K., Zenner, H.: Zählverfahren und Lastannahme in der Betriebsfestigkeit. Springer, Heidelberg (2012) 9. Brigham, E.O.: FFT: Schnelle Fourier-Transformation. Oldenbourg, München (1992) 10. Broch, J.: Mechanical Vibration and Shock Measurements. Bruel & Kjaer, Naerum (1984) 11. Hoffmann, R.: Grundlagen der Frequenzanalyse, Bd. 620. expert verlag, Renningen (2011) 12. Heymann, J., Lingener, A.: Experimentelle Festkörpermechanik. VEB Fachbuchverlag, Leipzig (1986) 13. Randall, R.B.: Vibration analyzers and their use. In: Piersol, S. (ed.) Paez: Harris’ Shock an Vibration Handbook. McGraw-Hill Education, New York (2009) 14. Randall, R.B.: Vibration-Based Condition Monitoring: Industrial, Aerospace and Automotive Applications. John Wiley & Sons, Hoboken (2011) 15. Schrüfer, E., Reindl, L.M., König, A., Zagar, B.: Elektrische Messtechnik: Messung elektrischer und nichtelektrischer Größen. Hanser, München (2012) 16. Thrane, N.: The Discrete Fourier Transform and FFT Analysers. Bruel & Kjaer, Naerum (1979) 17. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19(90), 297–301 (1965) 18. Gade, S., Herlufsen, H.: Use of Weighting Functions in DFT/FFT Analysis. Bruel & Kjaer, Naerum (1987) 19. DIN 45662:1996-12 Schwingungsmesseinrichtung – Allgemeine Anforderungen und Begriffe 20. Goldman, S.: Vibration Spectrum Analysis: a Practical Approach. Industrial Press, New York (1999) 21. Herlufsen, H.: Dual Channel FFT Analysis Part I and II. Bruel & Kjaer, Naerum (1984) 22. Dresig, H., Holzweißig, F.: Maschinendynamik, 12. Aufl. Springer Vieweg, Berlin (2016) 23. Sinambari, G.R., Sentpali, S.: Ingenieurakustik. Springer Vieweg, Wiesbaden (2014) 24. Brandt, A.: Noise and Vibration Analysis: Signal Analysis and Experimental Procedures. Wiley & Sons, Ltd (2010) 25. ABRAVIBE toolbox http://www.abravibe.com/toolbox.html. Accessed: 18 Nov 2018 26. Möser, M. (ed.): Messtechnik der Akustik. Springer, Berlin (2010) 27. Elektroakustik – Bandfilter für Oktaven und Bruchteile von Oktaven – Teil 1: Anforderungen (IEC 61260-1:2014); Deutsche Fassung EN 61260-1:2014, Ausgabe 2014-10 (2014) 28. Akustik – Normfrequenzen (ISO 266:1997); Deutsche Fassung EN ISO 266:1997, Ausgabe 1997-08 (1997) 29. Peter Zeller (Hrsg.): Handbuch Fahrzeugakustik, 3. Aufl. Springer Vieweg, Wiesbaden (2018) 30. C-Bibliothek FFTW. http://www.fftw.org/. Accessed: 23 Aug 2018 31. ASTM Standard E 1049, 1985 (2011), Standard practices for cycle counting in fatigue analysis, West Conshohocken: ASTM International (2011)

Experimental Modal Analysis

15

Abstract

This chapter deals with the sequence of the individual steps of the experimental modal analysis. The focus is on the practical operational execution and implementation with the aid of MATLAB®. The theoretical principles required for this are presented in summary form with references and without further derivation. This chapter does not claim to contribute to the discussion of the theoretical foundations of modal analysis. For this purpose, reference is made in particular to the literature (Ewins DJ (2003) Modal Testing: Theory, Practice and Application, Aufl. 2. Research Studies Press, Baldock; Døssing O (1988) Structural Testing Part I: Mechanical Mobility Measurements. Brüel & Kjær, Nærum (Revision April 1988); Døssing O (1988) Structural Testing Part II: Modal Analysis and Simulation. Brüel & Kjær, Nærum (March 1988); Kokavecz J (2010) Messtechnik der Akustik, Kapitel 8, Modalanalyse. Springer, Berlin) "

This chapter deals with the sequence of the individual steps of the experimental modal analysis. The focus is on the practical operational execution and implementation with the aid of MATLAB®. The theoretical principles required for this are presented in summary form with references and without further derivation. This chapter does not claim to contribute to the discussion of the theoretical foundations of modal analysis. For this purpose, reference is made in particular to the literature [1–4]. There are differences in the notation of equations and formula designators between the notation used so far in this book and that used in the field of modal analysis. Where notations differ, they are listed in Table 15.1.

# The Author(s), under exclusive license to Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2023 T. Kuttner, A. Rohnen, Practice of Vibration Measurement, https://doi.org/10.1007/978-3-658-38463-0_15

513

514

15 Experimental Modal Analysis

Table 15.1 Comparison of different notations

Designation Mass Spring constant Damping constant Deflection Receptance, dynamic compliance Admittance, agility, mobility, mobility Acceleration, Inertance, acceleration, inertia Force

Notation according to DIN 1311 m k d x(t)

ξ

HxF

ndyn

HvF

Y

HaF

acc

F(t)

f(t)

Alternative spelling

Source

Literature [4] Literature [4] Literature [4] Literature [4] Literature [3]

Notation in exp. Modal analysis m k c x(t) HxF HvF HaF F(t)

In order to analyze vibration problems, it is necessary to know the natural frequencies, the associated vibration mode and the respective modal damping of the structure under consideration. Here, one depends on computer-aided measurements and analyses. The most important method in this field is experimental modal analysis. This is not a self-contained computational process realized by a single function in the analysis software, but a process consisting of methods of measurement and functions for analysis. The experimental modal analysis is the process with which this is determined for the considered frequency range. Scanning synchronous measurements of the structure excitation and the structure response(s) are required. Transfer functions are calculated from the measured values, which are used to identify system parameters of the object structure. Classical exciters for object excitation are the impulse hammer and the shaker. With the 2017a release, Mathworks introduced features of the MATLAB® Signal Processing Toolbox for modal analysis, which are applied to experimental modal analysis in this chapter. Additional material for this chapter can be found at http:// schwingungsanalyse.com/Schwingungsanalyse/Experimentelle_Modalanalyse. html.

15.1

15.1

Assumptions and Explanations of Terms

515

Assumptions and Explanations of Terms

A mechanical structure usually has several natural frequencies with associated vibration patterns. A mode (mode of appearance, mathematically eigenfunction ψ) is understood as a natural frequency with its oscillation pattern, the oscillation mode. From this it is derived that a structure has a first mode, second mode and so on. This form of representation lists the modes of the structure in the sequence of the natural frequency. It is common to have a description that verbally describes the vibration pattern of the mode. The subscript r is used for the mode number. Example • (first) torsional mode, for a mode whose vibration pattern indicates torsion along the principal axis of the structure with the lowest (1st) torsional natural frequency. • (second) bending mode, for a mode whose vibration pattern indicates bending of the structure in the second bending natural frequency. One of the most important assumptions for the experimental modal analysis is the orthogonality of the eigenfunctions and thus the independence of the modes from each other. For each mode, a separate closed system is assumed in the form of the single-mass oscillation system described in Chap. 3 consisting of the mass m, the spring stiffness k and the damping constant c, with one degree of freedom (Fig. 15.1). This means for the boundaries of the respective system that they have either ideally free or absolutely rigid edges. In practice, this assumption is fulfilled if the natural frequencies can be determined via the determined transfer functions. The single mass oscillating system is also called single mass oscillator or SDOF (Single Degree Of Freedom) system. Another important assumption is that they are linear time invariant systems.1 If this is not the case, the analysis shall be limited to such an extent that this assumption is fulfilled. For example, by reducing the frequency range considered or shortening the measurement time to a time interval in which the condition of time invariance can be fulfilled. The principle of superposition is assumed. The superposition of individual system responses is to be understood as the sum of the system responses without the individual system responses influencing each other. This also means that the response of the system to simultaneous excitation with several signals is identical to the sum of the system responses to the individual excitation signals.

1

Linear means that a doubling of the excitation leads to a doubling of the response. Time invariant means that the system properties do not change over time.

516

15 Experimental Modal Analysis

m1

k1

m2

c1

k2

m3

c1

k3

mn

c3

...

kn

cn

Fig. 15.1 Model for the assumption of a structure with n uncoupled modes

Proportionality is assumed. Proportionality exists, for example, when the excitation force is changed by a certain factor y and the system response also changes by the same factor y. There is reciprocity. This means that the location of the excitation can be swapped with the location of the response. This is a very important assumption to verify when performing experimental modal analysis, since the exchange of locations between excitation and response is performed very often. Locations of excitation are given index n, locations of response are given index m. There is causality. Accordingly, there is no system response without a system excitation. Therefore, the required measurements must take into account that the system response and the system excitation are recorded as a whole. The system is stable. If the system excitation is terminated, the oscillations of the system decay. The decay behaviour is determined by the damping of the system.

15.2

Summary of the Analytical Principles of Modal Analysis

The basis of the theory for the experimental modal analysis is the equation of motion of the forced oscillation of the single-mass oscillator with velocity-proportional damping, a differential equation of second order (compare Eq. 15.1). Considering the different notations and multiplying by (-1), Eq. 4.1 becomes. m€x þ c_x þ kx = F ðt Þ

ð15:1Þ

For the description of the oscillation form, deflection values are required at several points of the examined structure. In addition, it can be assumed that the real system is not a system with one degree of freedom, but a system with several degrees of freedom, which is called MDOF (Multi Degree Of Freedom) system, for which the assumption made in Fig. 15.1 applies. From the measurements carried out, the transfer function matrix discussed in [1, 4– 7] and the notation of the equation of motion results in. M€x þ C x_ þ Kx = F Herein is

ð15:2Þ

15.2

Summary of the Analytical Principles of Modal Analysis

517

• M is the matrix of the modal mass • C the damping matrix • K is the stiffness matrix A mathematically more manageable model provides the consideration in the frequency domain. By using the transfer function(s), (compare Sect. 4.3 or Table 4.2) • dynamic stiffness, H xF ðωÞ, when measuring the applied force and the deflection as system response, which is operationally less common • Mobility,H vF ðωÞ, when measuring the force applied and the velocity as the system response. • Inertia, H aF ðωÞ, when measuring the applied force and acceleration as the system response. a modal parameter model for the single-mass transducer can be derived. The transfer function H xF ðωÞ is required for this. This can be determined either directly from the Fourier transforms of the measurement or by derivation from the transfer functions H vF ðωÞ and. H aF ðωÞ. H xF ðωÞ =

X ðωÞ H vF ðωÞ H aF ðωÞ = = jω ðjωÞ2 F ð ωÞ

ð15:3Þ

The transfer function can be determined by a practical measurement (compare Sect. 14.5). In [3], the transfer function H xF ðωÞ is obtained by substituting the equation of motion in. H xF ðωÞ =

1 1 1 þ k jωc ω2 m

ð15:4Þ

is transferred. This describes mathematically the characteristic course of the transfer function of the single-mass oscillator with velocity-proportional damping (Fig. 15.2). In the frequency range up to the resonant circuit frequency of the damped oscillation ωd this is characterized by the spring and is described by the stiffness. The higher the stiffness of the system, the lower the magnitude of the transfer function. In the frequency range above the resonant frequency of the damped oscillation ωd, the angular frequency and the mass determine the course of the transfer function. The higher the angular frequency, the lower the magnitude of the transfer function. In the frequency range of the resonant circuit frequency of the damped oscillation ωd, also referred to as modal frequency (see [4]), the damping of the system is determinant for the magnitude of the transfer function. Systems with pronounced and clearly identifiable resonant peaks have low damping. Here, the resonant circuit frequency of the damped oscillation ωd can be easily determined, but the determination of the damping is more

518

15 Experimental Modal Analysis

|HxF| in m/N

1/j c

1/k

- 1/ 2 m

d

in 1/s

Fig. 15.2 Characteristic curve of the transfer function of the single-mass oscillator with velocityproportional damping

difficult due to the narrow frequency range. The opposite is true for systems with high damping. These show a flatter and broader course of the resonance peak. For the determination of the modal parameters • Modal frequency ωd(r)

→

• Vibration mode described by the deflection vector ϕ ðr Þ • modal damping δ(r) the eigenvalues from Eq. 15.1 are required. Three cases are distinguished: • The limiting case of critical damping ck p This is present at c = ck = 2 mk where the two eigenvalues are equal and become s = - c=2m. p • Aperiodic damping c ≥ 2 mk Here the deflection decays non-oscillatory exponentially and the eigenvalues become real. p • Non-critical damping c < 2 mk Here the deflection decays exponentially and the solution has conjugate complex eigenvalues.

15.2

Summary of the Analytical Principles of Modal Analysis

s= -

c þj 2m

k c m 2m

s = -

c -j 2m

k c m 2m

2

2

519

= - δ þ jωd

ð15:5Þ

= - δ - jωd

ð15:6Þ

From. s = Refsg þ Imfsg

ð15:7Þ

c 2m

ð15:8Þ

it is derived that. δ = - Refsg = and ωd = Imfsg =

k c m 2m

2

=

ω20 - δ2

ð15:9Þ

Where ω0 is the natural angular frequency of the undamped system. The eigenvalues of the system under consideration are referred to as pole positions2 and consist of the modal damping δ and the modal frequency ωd. The eigenvalues and thus the pole positions can be found in the transfer function H xF ðωÞ at ωd(r). If one performs the solution of the eigenvalues as a matrix calculation (see [4, 6]), then each individual element of the transfer function matrix can be given by. N

H xF,mn = r=1

- ω2

ϕm ð r Þ  ϕn ð r Þ þ jω2δðr Þ þ ω20 ðr Þ

ð15:10Þ

can be determined. For the consideration of the individual mode r this means in mathematical notation. H xF,mn =

- ω2

ϕm ð r Þ  ϕn ð r Þ þ þ jω2δðr Þ þ ω20 ðr Þ

N q=1 q≠r

ϕm ðqÞ  ϕn ðqÞ - ω2 þ jω2δðqÞ þ ω20 ðr Þ Bmn

2

The zeros of a characteristic polynomial are called poles.

ð15:11Þ

520

15 Experimental Modal Analysis

Due to the assumption of orthogonality, the transfer function in the interval around the r-th resonance is determined by the share of the r-th mode alone. The influence of the neighboring modes, which is described in Eq. 15.11 with Bmn, is neglected in the parameter determination by Bmn = 0. The determination of the modal parameters is trustworthy if the individual modes are clearly identifiable as resonance points in the transfer function, which is referred to as weak modal coupling. If this is not the case, strong modal coupling is present and ωd(r) and δ(r) cannot be determined unambiguously. For Eq. 15.11 as transfer function H aF,mn holds with Bmn = 0 and ω = ωd(r). H aF,mn =

ω2d ðr Þ  ϕm ðr Þ  ϕn ðr Þ - ω2d ðr Þ þ jω2δðr Þ þ ω20 ðr Þ

ð15:12Þ

With the damping ratio D. D ðr Þ =

δðr Þ ω d ðr Þ

ð15:13Þ

and ω20 = ω2d ðr Þ þ δ2 ðr Þ from Eq. 15.9, the amplitude jHaF(d(r))j becomes. j H aF ðωd ðr ÞÞ j =

ϕm ð r Þ  ϕn ð r Þ

ð15:14Þ

D4 ðr Þ þ 4  D2 ðr Þ

To determine the deflections ϕm(r), the deflection at the response point, and ϕn(r), the deflection at the excitation point, a measurement is required in which the response point m is equal to the excitation point n. This is called the driving point measurement. This measurement is called the driving-point measurement. In this case, Eq. 15.15 becomes. j H aF ðωd ðr ÞÞ j =

ϕn ð r Þ 2

ð15:15Þ

D4 ðr Þ þ 4  D2 ðr Þ

and it can be found at ϕn(r) via. ϕn ð r Þ =

j H aF,mn ðωd ðr ÞÞ j 

D4 ðr Þ þ 4  D2 ðr Þ

can be determined. The following then applies for ϕm(r).

mit m = n

ð15:16Þ

15.3

Operational Performance of the Experimental Modal Analysis

ϕm ðr Þ =

"

D4 ðr Þ þ 4D2 ðr Þ 

j H aF,mn ðωd ðr ÞÞ j ϕn ð r Þ

mit m ≠ n

521

ð15:17Þ

Mnemonic

To determine the mode shapes of modes r of a structure with the capabilities presented here using MATLAB® are required: • the amplitudes from the transfer functions j H aF,mn ðωd ðr ÞÞ j at ω = ωd(r) for the excitation response point m = n, in order to determine the deflection of the excitation point ϕn from them • the amplitudes from the transfer functions j H aF,mn ðωd ðr ÞÞ j at ω = ωd(r) for all response measurement points m ≠ n, in order to determine the deflections of the response points ϕm from them • the range around od(r) of the transfer functions j H xF,mn ðωd ðr ÞÞ j, in order to determine from this the damping D(r) and the exact resonant circuit frequency ωd(r) or alternatively the resonant frequency. fd(r) • the sign of the imaginary part of the pole s, in order to determine the direction of the deflection. This solution path requires that the underlying measurements are measurements with one excitation point n and multiple response points m. If this is not the case, the index n can be swapped with m in the equations due to reciprocity.

15.3

Operational Performance of the Experimental Modal Analysis

The measurement itself is carried out as a pulse hammer measurement (Fig. 15.3) with one or more excitation points with one or more response measuring points, which are ideally equipped with accelerometers. Alternatively, a shaker (Fig. 15.4) can be used, but then usually at one excitation point. The object of analysis must be discretized in a meaningful way for the modal analysis. Individual measurement or excitation points must be defined on the structure to be investigated – a process which initially appears to be simple. However, it must be remembered that if the measurement points are evenly distributed, individual modes cannot be detected. If, for example, excitation or measurement is performed in the vibration node of a mode, this mode cannot be determined. Shannon’s sampling theorem also applies to the determination of the vibration mode, according to which more than two measured values are required for the determination of a vibration. Applied to experimental modal analysis, this means that only those modes can be determined whose wavelength is longer than twice the minimum distance between two measurement points.

522

15 Experimental Modal Analysis

Fig. 15.3 Measurement setup of the experimental modal analysis on a flywheel. The response signal is measured at a response point m with an accelerometer on the inside of the flywheel. The excitation is performed at a total of 10 points along the flywheel with an impulse hammer. The picture shows the excitation-response point measurement Fig. 15.4 Measurement setup of the experimental modal analysis on a beam. The response signal is measured via the two accelerometers on the beam, while the excitation at one point is performed by an electrodynamic shaker, which is coupled to the beam via a force sensor

In the case of complex structures, care must be taken to ensure that all vibrations of each characteristic component are recorded. If the result of the experimental modal analysis is also used for the adjustment with a simulation model, it must be ensured that the measuring points of the measurements match the simulation points. Only then is an adjustment possible. The number of measurement points depends on the geometry of the structure, the frequency range considered and the number of modes. Accordingly, this cannot be determined unambiguously at the beginning of an analysis. However, this task can be adequately mastered through experience with similar objects and comparison with a simulation model.

15.3

Operational Performance of the Experimental Modal Analysis

523

Each individual measuring point has six degrees of freedom, of which usually only one to three translational degrees of freedom are measured. If the natural frequencies and natural mode shapes are already known (e.g. from a calculation or on the basis of similar components), a few measurement points are sufficient for metrological verification. If, however, a complex model is to be analyzed from the experimental modal analysis, a very large number of measurement points, up to several hundred, are required. The operational effort in the measurement execution is correspondingly high.

15.3.1 Storage For the quality of the results of an experimental modal analysis, the boundary conditions under which the required measurements are carried out are decisive. It makes sense to investigate the structures to be analyzed in boundary conditions that are as realistic as possible. Whenever the excitation and measurement points are difficult or impossible to reach in the installed state, idealized storage and boundary conditions are used. However, this is very often the case in practice. If the measurement serves to adjust an FEM model, the framework conditions of the FEM model must also be observed for the measurement. Often, the measurement and simulation results do not match because the boundary and transition conditions of the calculation and measurement models are different. The influence of a high structural damping and nonlinearities will strongly influence the result in the installed object state, since these are in contrast to the prerequisites of a modal analysis. In the practical implementation, an attempt is made to get as close as possible to the real boundary condition. An alternative approach is to work in the extreme opposite direction by trying to achieve the ideal free boundary conditions. For this, it must be achieved that the natural frequency of the suspension is as far away as possible from the frequency ranges to be investigated, e.g. natural frequency of the suspension at most 1/5 of the first mode of the structure to be investigated.

15.3.2 Object Excitation by Means of an Impulse Hammer The ideal impulse with a unique event at t = 0 is only a thought model. The Fourier transformation of this thought model is the infinite broadband frequency excitation. With a pulse hammer only a real pulse excitation is possible. The real time pulse causes a broadband excitation in a limited frequency range (compare Sect. 14.5). The simulated test with impulse hammer blows of different widths shown in Figs. 15.5 and 15.6 demonstrates that very narrow impulses must be taken into account when carrying

524

15 Experimental Modal Analysis Evaluation of pulse widths Pulse width

1

0.2 ms 2 ms 5 ms

Pulse height

0.8

0.6

0.4

0.2

0 0.01

0.015

0.02

0.025

0.03

0.035

Time

Fig. 15.5 Time series of simulated impulse hammer blows to evaluate the suitability of impulse hammer measurements for experimental modal analysis

out impulse hammer measurements. Pulse widths of significantly less than 1 ms are required for usable measurements. The measured time series of the force signal of an impulse hammer blow shown in Figure 15.7 has a total of 10 usable measured values for the impulse at a sampling rate fs = 51,200 Hz. Measurements of impulse hammer blows therefore require sufficiently high sampling rates. The 51,200 Hz used in this example as the sampling rate of the transducer must be regarded as borderline low. Figure 15.8 shows the usable frequency bandwidth of the measured impulse hammer blow. The frequency range up to 2 kHz (max. 2.5 kHz) is read off as usable. Up to this cut-off frequency the frequency response is sufficiently linear and the amplitude or normalized amplitude is not reduced by more than 20%, but it is not optimal. In the very low frequency range (0 to 10 Hz) this measurement is unusable. The reason for this is also to be found in the measuring technique used, since in this case it only has no influence on the measurement result above 10 Hz. The required impulse height, i.e. the strength of the impulse hammer blow, depends on the energy required to excite the structure to be examined. For this purpose, the trade offers various heavy impulse hammers. In addition, different impulse hammers can be equipped with different hammer tips. Depending on the material properties of the hammer tip, the same impulse hammer can be used for different frequency ranges. In the frequency range up to an amplitude drop of 3 . . . 5 dB, the force excitation of the impulse hammer can be used. For the example in Fig. 15.9 this means a usable frequency range up to • 4200 Hz when using a metal tip • 1400 Hz when using a plastic tip • 380 Hz when using a rubber tip

15.3

Operational Performance of the Experimental Modal Analysis

525

Spectral evaluation of pulse widths Pulse width

Pulse height (normalized)

1

0.2 ms 2 ms 5 ms

0.8

0.6

0.4

0.2

0 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Frequency [Hz]

Fig. 15.6 Frequency spectrum of simulated impulse hammer blows of different widths Measured impulse hammer blow

200

Pulse width 0.135 ms

Pulse height (N)

150

100

50

0 3.023

3.024

3.025

3.026

3.027

3.028

3.029

3.03

Pulse width [s]

Fig. 15.7 Measured pulse hammer impact with 0.135 ms pulse width

With a different impulse hammer, with a different structure to be examined and with other contact partners, these cut-off frequencies (and excitation amplitudes) will be different. It is recommended to determine this individually for the respective measuring task. The quality of the measured impulse is additionally strongly dependent on how the structure is struck with the impulse hammer. An evaluation of the impact quality is therefore mandatory and some test impacts before the measurement are recommended. The evaluation is made in pulse height and pulse width. In addition, it must be checked whether a blow with several excitation pulses has not been made by mistake (compare Fig. 15.10). For the excitation response point measurement, a measuring point is required for which the condition m = n applies. This is achieved by striking the response sensor, e.g. an

526

15 Experimental Modal Analysis Frequency spectrum of the measured impulse hammer blow

normal pulse height

1

0.8

0.6

0.4

0.2

0 0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

Page 17

Frequency [Hz]

Fig. 15.8 Amplitude spectrum of the measured impulse hammer blow Impulse hammer blows with different tips

1

Metal tip Faux fabric tip Rubber tip

0

Normalized amplitude in dB

-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 0

1000

2000

3000

4000

5000

6000

7000

Frequency in Hz

Fig. 15.9 Amplitude spectrum of different impulse hammer tips with the same attachment point

acceleration sensor, with an impulse hammer in the immediate vicinity. Of course, care must be taken to ensure that the response sensor itself is not struck by the impulse hammer blow. In the flywheel example, this was achieved by placing the accelerometer on the inside. The impulse excitation was achieved by striking the flywheel from the outside (see Fig. 15.3).

15.3.3 Object Excitation by Means of a Shaker Shakers (electrodynamic or hydraulic) allow the excitation of structures with arbitrary signals. Thanks to a wide power range, even large, thus very complex, strongly damped

15.3

Operational Performance of the Experimental Modal Analysis

527

Impulse hammer blow

100 90 80 70

Force in N

60 50 40 30 20 10 0 -10 0

0.5

1

1.5

Measured values

2

2.5 10

4

Fig. 15.10 Impulse hammer blow with several excitation pulses

and very heavy structures can be excited. It should be noted that shakers with high power values have a lower highest excitation frequency. However, this is often not problematic because heavy structures, which require high excitation powers, have low modal frequencies. This circumstance only has to be taken into account when designing the excitation signal. The excitation force is measured between the shaker and the structure under investigation, at best directly at the initiation point. This force signal is used for the calculation of the transfer functions. Alternatively, an impedance measuring head can also be used at this measuring point, which supplies the acceleration signal in addition to the force signal. Otherwise, an accelerometer must also be placed at the force application point. Accelerations are usually measured at all specified measuring points. Noise, sine signals, sine sweep (chirp) or other arbitrary signal forms can be used as excitation signals. It is also possible to use measured signals from operation (referred to as road load). Care must be taken to ensure that the shaker is securely positioned and carefully aligned. No static forces may be introduced into the structure to be examined and the introduction mechanism must not have any play under any circumstances. Both would lead to considerable deviations in the calculated transfer functions. Reactions from the test object to the coil of the shaker must be prevented as far as possible. Both are achieved by connecting the shaker to the excitation point via a thin rod (called a stinger), often designed like an expansion shaft screw.

528

15 Experimental Modal Analysis

Fig. 15.11 Measuring object flywheel

15.4

Evaluation of the Experimental Modal Analysis in MATLB®

15.4.1 Evaluation of Measurements with Impulse Hammer Excitation The procedure for evaluating the experimental modal analysis with impulse hammer excitation is to be illustrated using the example of a flywheel of an internal combustion engine (Fig. 15.11). The flywheel was excited from the outside with the impulse hammer at 10 positions, i.e. at a distance of 36 degrees each. An accelerometer was attached to the inside at position 3. This results in the excitation response point measurement, i.e. to determine the excitation deflection ϕn, the measurements from position 3 with which this example begins. Measurements were taken without any further programmed interactive assistance. The measuring technique used was simply created and the measurements were carried out individually by startForeground. Up to 16 individual impulse hammer blows per position are available for further evaluation. Figure 15.12 shows the measured time series of the impulse hammer and the acceleration response, but the diagram also shows that the individual time series have a time offset from each other due to the measurement procedure. The MATLAB® function modalfrf used to determine the transfer functions allows several measurements belonging to each other to be evaluated. For this purpose, the individual measurements are to be brought to the same measurement length. From the observation of the impulse responses, it can be seen that a decay time of 3 s is completely sufficient. In addition, the impulse hammer blow must always occur at the same time, which is defined as 2 ms. These and other data required for modalfrf are defined by parameterization.

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Evaluation of the Experimental Modal Analysis in MATLB®

529

Measured impulse hammer blows

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Fig. 15.12 Representation of the time series of the impulse hammer blows measured without triggering as well as the impulse responses by excitation at position 3

ð15:18Þ

The “trimming” of the time series is realized via a loop. This determines the data position of a threshold value overrun of the pulse hammer time series. The matrix index position stands for the measurement position, while inc stands for the number of the executed blow. Finally, the script must evaluate all blow positions with the totality of the blows.

ð15:19Þ

The threshold value for the impulse hammer blow is defined here as 50 N. The instruction *1000/kaliHammer converts the voltage measured in volts into the required physical quantity Newton. The calibration factors are usually available in mV per physical quantity. The determined data position is now corrected forward by the number of measured values of the pre-trigger, while the end is located blocksize-1 further back in the time series.

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ð15:20Þ The respective raw data of response and excitation can now be formed from the measurement time series. This is done with simultaneous conversion into physical values. ð15:21Þ For the selection of measurements suitable for modal analysis, it is suitable in the first step to consider the pulse widths in milliseconds and pulse heights in newtons of the pulse hammer time series. To determine the pulse width, MATLAB® provides the function pulsewidth(data, fs). The pulse height can be determined using max(data). ð15:22Þ The pulse heights and pulse widths determined in this way are first evaluated for plausibility and usability using an X-Y plot. Not all of the impulse hammer blows shown in Fig. 15.13 are suitable for further evaluation. A suitable pulse width seems to be in the range of 0.12 to 0.13 ms, a usable pulse height between 200 and 300 N. These values are not generally valid and differ from test object to test object. The subjective decision is confirmed by calculating the coherence (compare Sect. 14.5, Eq. 14.117) of the signals to each other. This is one of the result vectors of the MATLAB® function modalfrf. By means of. ð15:23Þ the transfer functions as well as the coherence are determined. Before this, however, the excitation and response signals must be combined (Fig. 15.15). This is done by simply joining the signals selected by pulse height and pulse width. Modalfrf is based on the Fourier transform (compare Sect. 14.4). To reduce leakage, the signals must be weighted with a window function (compare Sect. 14.4.5). However, the problem shown in Fig. 14.44 must be considered. A window function is required which forces the signal to decay to zero at the end of the signal. No weighting should be carried out at the start of the signal, as the system is at rest here and the measured values are therefore zero. The RectExpo function is provided for this purpose in the additional materials. This window function consists of a rectangular (rect) portion at the beginning of the window and a decaying portion towards the end of the window. ð15:24Þ As with all window functions, the length (window length) of the window function is required, this must be identical to the specification N or nfft in the Fourier transform and

15.4

Evaluation of the Experimental Modal Analysis in MATLB®

350

531

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Pulse height [N]

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150

100

50 0.1

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0.12

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Pulse width [ms] Fig. 15.13 X-Y plot of the pulse heights and pulse widths for the evaluated impact position

must be an integer. The parameter fs stands for the sampling rate used for the measurement performed. Rect specifies how the specification in nrect, the length of the rect component, is to be interpreted. The following parameters are available • %: as a percentage • time: as time specification in ms • n: as the number of measured values Depending on the parameter used, the number of measured values of the rect component is calculated differently. The parameter weighting specifies the weighting of the decay function. The last parameter type specifies the form of the decay function. This contains • 1 for linear decay. The parameter weighting has no function here, it only has to contain a value. • 2 for a decay curve e-t  weighting. This window function does not necessarily decay to 0 in the given window length! • 3 for a decay curve sin(x)2. weighting specifies the data range in % over which the decay function is to extend. If the addition of the rect-part and the length of the decay function is smaller than the window length, the defined window length is extended with 0-values. This window function also does not necessarily decay to 0 within the window length.

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15 Experimental Modal Analysis Excitation signal

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Fig. 15.14 History of the window function of antWindow

Figure 15.14 shows the course of the weighting using the example of the window function for the response signal. The window function for suggestion and response do not have to be identical. ð15:25Þ provides different window functions for the excitation signal and the response signals respectively. For the window function of the excitation pulse hammer signal, a fast decay to zero is recommended. This causes the signal component to be forced to zero after the excitation pulse. This means that no interfering signal components, e.g. due to the handling of the impulse hammer after the impact, are included in the calculation of the transfer function. The excitation and response signals shown in Fig. 15.15 were used to calculate the transfer function. The MATLAB® function modalfrf uses long individual segments from the blocksize signals to average the result. The evaluation of the calculated transfer function j H xF ðf Þ j (Fig. 15.16, blue line) can be done using the coherence (Fig. 15.16, red line). If this lies in the range 0.8 to 1 for the frequency ranges in question, the result is trustworthy. Dips in the coherence in the frequency ranges with transfer function values close to zero (is called antiresonance) are common. For the use of the MATLAB® function modalfrf it is to be noted that the excitation signal must be a time series of the force. No other physical quantity is provided here. Time series as acceleration, velocity or displacement are possible as response signal. The measured object acceleration is assumed in the standard setting. If this is not the case, the interpretation of the response signal can be changed by an additional parameter pair.

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Evaluation of the Experimental Modal Analysis in MATLB®

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Window function RectExpo 1

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Fig. 15.15 Composite excitation and response signal for transfer function calculation using modalfrf

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3000

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Fig. 15.16 Representation of the transfer function and the coherence in a diagram, facilitating a qualitative evaluation of the transfer function

ð15:26Þ By extending the instruction with the parameter pair ‘sensor’, ‘dis’ a response signal is now assumed as displacement. Possible values for ‘Sensor’ are ‘acc’ for the acceleration sensor (acceleration, default), ‘dis’ for the displacement and ‘vel’ for the velocity. There is another special feature to note in the blocksize parameter (referred to as window in the MATLAB® help). If this is an integer value, it is considered as the length of each signal segment. If the blocksize parameter is a vector to be considered as a window function, then both the excitation and response signals are weighted by this vector and the length of the vector is used for segmentation. However, it does not make sense to

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15 Experimental Modal Analysis

weight the excitation signal and the response signal with the same weighting vectors for impulse hammer measurements (see instruction 15.25). The parameter noverlap has a similar effect in the function modalfrf as in the function spectrogram (see Table 14.13 overlap). For impulse hammer measurements this must be 0, as they are contiguous single impulse hammer measurements. Thanks to the weakly damped structure, the frequencies of three modes can be read directly from the transfer function. The frequencies found are stored in the vector. ð15:27Þ for further analysis. Accurate modal frequencies and the associated damping ratios are required for the calculation of the vibration mode. The MATLAB® function modalfit is used for this purpose. ð15:28Þ For the calculation, this requires the transfer function frf, the associated frequency vector f, the sampling rate fs and a specification for the maximum mode number up to which the calculation is to be performed. Without the other parameters, a “standard” calculation is performed. This gives an initial overview of the existing modes and is of particular interest if no clearly identifiable peaks are to be found in the transfer functions. If peaks in the transfer function can be identified, then modal fit around the specification PhysFreq followed by a vector of frequencies is performed a default. The frequency range to be considered can be limited by appropriate parameterization. This should be used, since with impulse hammer excitations the sampling rates must always be very high and thereby the analysis frequency range lies above the frequency range of interest. Different fitting methods are offered to determine the damping: • lsce (default) stands for least-squares complex exponential method. The lsce method uses the least squares method. It calculates the impulse response of the FRF and fits it by summing the complex damped (sine) oscillations. It uses an algorithm described as Prony analysis or Prony’s method (see [8]), which was developed in 1795 by the French mathematician and engineer Gaspard Riche de Prony. • lsrf stands for least-squares rational function and comes from the MATLAB® System Identification Toolbox. This algorithm requires less data than the other two methods for determining attenuation. It is the only applicable method when a frequency vector f with non-uniform frequency spacings is present. • pp stands for peak picking method. The peak picking method assumes that exactly one structural mode exists for each significant peak in the transfer function. This represents a single-mass oscillatory system. The peak picking method provides the most trustworthy result for transfer functions with significant peaks.

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Evaluation of the Experimental Modal Analysis in MATLB®

535

Table 15.2 Comparison of results of fitting methods for determining modal frequencies and attenuations Comparison of modal frequency and attenuation lsce result lsrf result Fn Dn Fn 1221.9 0.0005 1222.0 3222.0 0.0004 3222.9 3700.3 0.0010 3222.9

Dn 0.0004750 0.0004605 0.0004605

pp-result Fn 1222.1 3218.7 3702.9

Dn 0.0005 0.0006 0.0018

In the subjective comparison by examining the transfer function in Fig. 15.16, the attenuations of the first mode (1222 Hz) and the second mode (3220 Hz) should be approximately the same, while the attenuation of the third mode (3707 Hz) must be much larger than that of the first two modes. This is confirmed by the lsce and the pp. results. With the lsrf method the third mode is not detected at all (see Table 15.2). In general, the determination of the modal parameters is subject to large scatter. The background to this is the complexity of the method, which relies on Fourier transforms. In the further progress of the analysis, it is now necessary to determine the deflection values ϕ, which are required for the description of the oscillating shape. The oscillation shape itself can only be determined after the analysis of all measured positions. From the frf determined in instruction 15.24, the magnitude maximum is determined in the span around the respective pole location (idx). ð15:29Þ idx is the index of the pole position which is given by. ð15:30Þ is determined. The span is determined with. ð15:31Þ Analytically, the maximum of the transfer function HxF ±1 index must be found around the pole location. This deviation results from the condition of integer values of blocksize in modalfrf (instruction 15.26). From. ð15:32Þ the deflection direction is determined. To calculate the deflection itself, the evaluation of the excitation response point measurement is required first. The deflection ϕn results from Eq. 15.16 as MATLAB® -statement from the transfer function H xF,nm to.

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15 Experimental Modal Analysis

ð15:33Þ and the deflection ϕm from Eq. 15.17 to. ð15:34Þ The numerical results are stored in a data structure for archiving and further use.

ð15:35Þ

Visualization can be done as a diagram for simple structures, such as Fig. 15.17, or via an animated 3D visualization. The determination of the vibration modes of the experimental modal analysis provides a qualitative result. The determined deflection amounts are a relative deflection to the excitation deflection. All variable parameters of the vibration measurement and the performed analysis and calculation steps have an influence on the numerical result. A further problem in the calculation of the oscillation form is due to the fact that the values must be taken from the resonance point. However, the determination accuracy is lowest at the resonance point itself. The numerical result is therefore subject to considerable scatter. A total of six steps are required for the evaluation of the experimental modal analysis. • Examine raw data (example script: Mod01_ImpulsRohdaten.m). • Trimming of the individual measurements (example script: Mod02_ImpulsBewerten. m). The pulse beat must always be positioned at the same time in the time series. This is achieved by triggering the data. From the evaluation of the raw data, an estimate can be made as to when the vibration of the object has decayed and therefore what the time length of the evaluation data should be. • The first selection of the pulse hammer measurements used for the evaluation is then made via the evaluation of the pulse widths and pulse heights (example script: Mod03_ImpulsAufbereiten.m). • The coherence of the impulse hammer blows used so far is determined. Only if this is sufficiently good, the further steps in the evaluation take place, if not, the first selection is refined. In the same step the FRF was calculated (example script: Mod04_FRF.m).

15.4

Evaluation of the Experimental Modal Analysis in MATLB®

537

Deflection diagram mode 1 90 1.1 120

60

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Fig. 15.17 Representation of the vibration mode of the first mode of the flywheel in a polar plot

• If the data quality is found to be sufficiently good, the last calculation step is the determination of the deflection values (example script: Mod05_BerDat.m). • In the last step the swinging shapes are shown.

15.4.2 Evaluation of Measurements with Shaker Excitation The evaluation of the experimental modal analysis with shaker excitation is basically carried out with the same working steps as for the impulse hammer excitation. As an example, an experimental modal analysis with shaker excitation was performed on an exhaust system (Fig. 15.18). In the example, the exhaust system was excited with a frequency sweep from 15 to 250 Hz over 2 s. The measurement was repeated several times. The measurement was

538

15 Experimental Modal Analysis

Fig. 15.18 CAD model of the exhaust system under consideration. Acceleration sensors were installed at the positions marked with red dots. The blue elements are absorbers, which were installed by the manufacturer of the exhaust system to reduce vibrations. These absorbers are also equipped with acceleration sensors (green dots)

repeated several times. The comparison of the measurements with each other via the coherence function showed that averaging over several measurements is not possible in this case. The sequence of the frequency sweep is very difficult to keep equal. Figure 15.19 shows the transfer functions from the measurement with shaker excitation. In contrast to the impulse hammer measurement on the flywheel, no clearly pronounced resonance points can be found. The reason for this is not the measurement method, but the damping in the structure, which leads to a higher modal coupling. With real test objects, this quality of the determined transfer functions is to be expected rather than those from the flywheel measurement. Natural frequencies can be assumed at the frequencies 30 Hz and 78 Hz. The stability diagram shown in Fig. 15.20 is now helpful, as it shows the. ð15:36Þ is generated. The input parameter for this function is the FRF matrix, in which all transfer functions from the excitation point to the respective response points are located. In the example, these are six response points. From this, a mean value is formed, which is shown as a red line in Fig. 15.20. In addition, the “stable” frequencies determined by mathematical approximations are represented by an o-sign, the “stable” frequencies and attenuations by a + sign and the instabilities by a . - sign.

Receptance [m / N]

Receptance [m / N]

Receptance [m / N]

Receptance [m / N]

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Receptance [m / N]

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Evaluation of the Experimental Modal Analysis in MATLB®

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FRF / Coherence at measuring point B11

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Fig. 15.19 Transfer functions of the measuring points on the exhaust system relevant for the experimental modal analysis

The interpretation of the diagram is carried out independently of the type of measurement via the averaged transfer functions. Poles are only present if a local peak can be seen in the transfer function. In mathematical approximations of higher order, however, very many poles are found in a transfer function, even if these produce hardly visible peaks in the line course of the transfer function. A further criterion for the presence of a pole is that the frequency and the attenuation must be the same across all model orders (Model Order in Fig. 15.20) – the mathematical approximations.

540

15 Experimental Modal Analysis Stabilization Diagram Stable in frequency Stable in frequency and damping Not stable in frequency Averaged response function

45

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-6

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Fig. 15.20 The stability diagram with the transfer function averaged over all transmission paths and markers for the stability criteria “stable in frequency”, “stable in frequency and attenuation” and “unstable in frequency”

The lines from + and o must therefore be perpendicular and cross with the peak of the transfer function. Only then is there a pronounced natural frequency in the structure under investigation. In the example, the modal frequencies 30.34 and 77.8 Hz with an average D of 2.4% and 1.9%, respectively, are taken from the instruction.

ð15:37Þ determined. The calculation of the deflections is done via two interleaved for-loops from the FRF matrix the required values are determined as described in the impulse hammer evaluation.

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Evaluation of the Experimental Modal Analysis in MATLB®

541

ð15:38Þ The exhaust system is mounted at a total of three positions. The driving point is located in front of the first bearing when viewed in the X-direction, which is located at approx. X = 200 mm. Figure 15.21 shows the vibration mode of the second mode at 77.8 Hz.

542

15 Experimental Modal Analysis Oscillating shape of the exhaust duct in Z-direction

10-7

5

Mode 2: 77,8 Hz Swing mould Mode 2

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Fig. 15.21 Illustration of the vibration shapes of the second mode on the exhaust system

References 1. Ewins, D. J.: Modal Testing: Theory, Practice and Application, Aufl. 2. Research Studies Press, Baldock (2003) 2. Døssing, O.: Structural Testing Part I: Mechanical Mobility Measurements. Brüel & Kjær, Nærum (Revision April 1988) 3. Døssing, O.: Structural Testing Part II: Modal Analysis and Simulation. Brüel & Kjær, Nærum (March 1988) 4. Kokavecz, J.: Messtechnik der Akustik, Kapitel 8. Modalanalyse. Springer, Berlin (2010) 5. Natke, H. G.: Einführung in Theorie und Praxis der Zeitreihen- und Modalanalyse – Identifikation schwingungsfähiger elastomechanischer Systeme, Aufl. 3. Vieweg + Teubner, Wiesbaden (1992) 6. Brandt, A.: Noise and Vibration Analysis: Signal Analysis and Experimental Procedures. Wiley, Chichester (2011) 7. Strohschein, D.: Experimentelle Modalanalyse und aktive Schwingungsdämpfung eines biegeelastischen Rotors. Dissertation. (2011) 8. Peter, T.: Generalized Prony Method. Dissertation,. Göttingen (2013)