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Application of the Round-Trip Theory to Electro-Dynamic Sources of Vibration

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Application of the Round-Trip Theory to Electro-Dynamic Sources of Vibration APPLICATION OF THE ROUND-TRIP THEORY TO ELECTRO-DYNAMIC SOURCES OF VIBRATION Master of Science by Research Thesis Acoustics Research Centre, The University of Salford, Greater Manchester _____________________ Author JAMES R. EVANS Supervisor PROF. ANDY MOORHOUSE A thesis submitted in partial fulfilment of the requirements for the degree of Master of Science by Research March 2021 PREFACE SECTION TITLE i Abstract ii Contents iii Symbols, Values & Constants iv List of Tables & Figures v Acknowledgements 2 i – Abstract The Round-Trip Identity, developed by Moorhouse & Elliott at The University of Salford allows the reconstruction of mechanical frequency response functions at remote locations on a test structure, without the need to physically measure at that location. This makes the identity a powerful tool in the field of modal analysis, as it may be applied to structures where measurement at a desired location may be practically difficult or impossible. This work aims to extend the applications of the Round-Trip Identity, which was originally applied to solely mechanical systems. Work by Moorhouse & Elliott in 2013 suggested that the identity could be applied to any system that is linear and time invariant. Electro-mechanical systems can be found throughout the modern world. This project investigates potential electro-mechanical extensions of the identity, which would allow application of the identity to these hybrid systems. A thorough summary of electrical network theorems and modal analysis theories is presented. Theories from both fields are subsequently combined to derive the electro-mechanical Round-Trip Identity. The first objective of this study was to verify the mechanical Round-Trip identity. Following this, an electro-mechanical version of the identity is derived. A series of increasingly complex experiments is presented that aim to verify the electro-mechanical Round-Trip Identity, the simplest case being a single degree of Freedom on a simply supported beam. The design of further experiments that would aim to verify the identity for more complex, multi degree of freedom systems is proposed. These relate to the analysis of plates and electric motors. Finally, recommendations are made for how the identity could be further developed and implemented in an industrial context. 3 ii – Contents PREFACE ....................................................................................................................................................... 2 i – Abstract ................................................................................................................................................... 3 ii – Contents ................................................................................................................................................. 4 iii - Symbols, Values & Constants ................................................................................................................ 9 iv - List of Tables & Figures ....................................................................................................................... 10 v - Acknowledgements ............................................................................................................................... 12 PART 1: RESEARCH CONTEXT, EXISTING WORKS & THEORY ................................................................. 13 Part 1 Outline ......................................................................................................................................... 13 1.1 - Introduction & Research Context ..................................................................................................... 14 1.1.1 - Research Context ....................................................................................................................... 14 1.1.2 - Thesis Aims & Objectives .......................................................................................................... 14 1.1.3 - Thesis Outline ............................................................................................................................. 14 1.2 – Review of Practical & Theoretical Background .............................................................................. 16 1.2.1 - Modal Testing ............................................................................................................................. 16 1.2.2 - Reciprocity: A Brief History & its Applications .......................................................................... 17 1.2.3 - Circuit Analysis ........................................................................................................................... 18 Kirchhoff’s Laws ................................................................................................................................. 18 Tellegen’s Theorem ............................................................................................................................ 19 Equivalent Circuits ............................................................................................................................. 19 Two-Port Analysis ............................................................................................................................... 20 1.2.4 - The Round-Trip Identity .............................................................................................................. 22 1.2.5 - Electro-Dynamic Shakers & Transduction ................................................................................ 23 Transduction & Equivalent Circuits ................................................................................................... 23 General Explanation of Electro-Dynamic Shakers ........................................................................... 24 Shakers as an Electro-Dynamic Transducer .................................................................................... 24 1.2.6 - Section Summary ....................................................................................................................... 25 1.3 - Existing Scientific Theory & Background ......................................................................................... 26 1.3.1 - Vibration Measurement Equipment & Procedures .................................................................. 26 Sensors and Monitoring .................................................................................................................... 26 4 1.3.2 - Modal Testing ............................................................................................................................. 29 General Principles & Practices .......................................................................................................... 29 Applications of Modal Testing ........................................................................................................... 29 Signal Processing & Analysis ............................................................................................................ 29 Limitations of Modal Testing ............................................................................................................. 34 The Round-Trip Identity ...................................................................................................................... 35 Geometry and Definitions .................................................................................................................. 35 1.3.3 - Circuit Analogies ........................................................................................................................ 37 Mechanical Systems .......................................................................................................................... 37 Acoustic Systems ............................................................................................................................... 40 1.3.4 - Electric Motors ........................................................................................................................... 42 General Principles .............................................................................................................................. 42 DC Motors ........................................................................................................................................... 43 Brushless DC Motors ......................................................................................................................... 44 AC Motors ........................................................................................................................................... 46 Other Notable Motor Types ............................................................................................................... 47 Motor Selection for Electro-Mechanical Tests ................................................................................. 47 1.4 - Part 1 Summary & Conclusions ....................................................................................................... 48 PART 2: DEVELOPMENT OF EXISTING THEORY & EXPERIMENTAL VERIFICATION ................................ 50 Part 2 Outline ......................................................................................................................................... 50 2.1 - Verification of The Published Round-Trip Identity ........................................................................... 51 2.1.1 - dBFA Setup ................................................................................................................................
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