Wave Propagation Approach for Structural Vibration [1st ed.] 9789811572364, 9789811572371

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Springer Tracts in Mechanical Engineering

Chongjian Wu

Wave Propagation Approach for Structural Vibration

Springer Tracts in Mechanical Engineering Series Editors Seung-Bok Choi, College of Engineering, Inha University, Incheon, Korea (Republic of) Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Yili Fu, Harbin Institute of Technology, Harbin, China Carlos Guardiola, CMT-Motores Termicos, Polytechnic University of Valencia, Valencia, Spain Jian-Qiao Sun, University of California, Merced, CA, USA Young W. Kwon, Naval Postgraduate School, Monterey, CA, USA

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Chongjian Wu

Wave Propagation Approach for Structural Vibration

123

Chongjian Wu Wuhan, PR China

ISSN 2195-9862 ISSN 2195-9870 (electronic) Springer Tracts in Mechanical Engineering ISBN 978-981-15-7236-4 ISBN 978-981-15-7237-1 (eBook) https://doi.org/10.1007/978-981-15-7237-1 Jointly published with Harbin Engineering University Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Harbin Engineering University Press. ISBN of the Co-Publisher’s edition: 978-7-5661-2199-8 © Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved by the Publishers, 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 publishers, 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 publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

We must collect and sort the phenomena details until we find the truth from the scientific impression and find the laws from various phenomena while using these laws to deduce various phenomena and explain the future in turn [1]. W. R. Hamilton

The continuous improvement of structural dynamics is driven by the pursuit of engineering refinement. Generally, they can be separated into two categories: One is analytical methods, such as modal analysis method, transfer matrix method, modal truncation/synthesis method, modal impedance synthesis method, modal flexibility synthesis method, mobility method, dynamic stiffness synthesis method, etc., which are appropriate for basic theoretical analysis. The other is numerical methods, including the finite element method, boundary element method, and statistical energy method (SEA), which are appropriate for complex engineering structure calculation and system simulation. Unless with the special experience accumulation, researchers only rely on the exhaustion of numerical calculation, leading to difficulty in revealing the general law. Therefore, the analytical and numerical methods are still in further development. This book introduces the equation derivation and the formation of the WPA intending to sort out its design. From the perspective of the structural wave, the author describes the input, propagation, and reflection of structural vibration energy and analyzes the waveform conversion. Through the simplification of the system and the clarification of the mechanism problems, designers can better find and summarize the mechanism causes of some frontier problems of such complex giant systems such as ships, connecting the model test and the real ships verification, and refining the general rule with conceptual, key, and common features while, therefore, guiding the technical research direction of the complex systems. WPA Analysis The analysis of structural dynamics is derived from the differential equation of vibration. In general, the method of separation of variables is used to establish characteristic functions: Time function is described by expðjxtÞ while space function is described by the hyperbolic function. The WPA is also derived from the

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differential equation of vibration, with double-exponential function as its foundation. This book, by a unified framework, expresses simple systems, complex systems, and hybrid power systems consistently while forming unity and regularity. The unity of the WPA is reflected in the regularity of equation derivation. The structural wave is used as the basic parameter to describe the structural vibration. For the wave being the ultimate parameter, it forms a direct causal relationship with the target control parameters, which is different from such indirect parameters as the acceleration of the structural vibration, insertion loss, or vibration level difference, although in many cases, they have similar or even identical mathematical descriptions. Therefore, in the event of differences between theoretical analysis and engineering measurement, designers should go back to the basics and rethink the wave in the structure. The regularity of the WPA comes from the exponential function. The “noumenon” of the partial derivative of an exponential function expðkn xÞ always remains unchanged. Regardless of the times for taking the derivative, its derivative always remains a constant. Such a characteristic is described by Western researchers as being similar to cutting a watermelon, meaning that no matter how you cut a solid ball, its cross -section is always round. The Chinese explanation is more interesting: It is as if you cut off part of the Monkey King, which you think is just a small piece but essentially it is another Monkey King that is identical to the original. Each instantaneous value of the solution is “forced” to coincide with the boundary conditions, enabling the WPA to meet the strict boundary conditions. The traditional analytical methods only have analytical solutions under a few constraint conditions, while the WPA is subject to relatively fewer restrictions. The convenience brought by the exponential function expðkn xÞ is genetic, which does not need to be expanded thoroughly. It is so consistent with simple harmonic motion that it demonstrates the WPA’s magic for simplifying that is seemingly tedious but essentially neglected in a hidden way. After the introduction of symbolic operators, the derivation and deduction of mathematical equations show the rhythm of harmonic motion, which enjoys an inherent logic, complex representation, and unexpected unity and regularity between the harmonic excitation and simple harmonic motion with the hypothesis of dynamics. This is the characteristic of the WPA, which does not lose the original expression of the physical phenomena. The Structural Wave There are both simple structures, such as clamp-free beams or plates and complex structures, such as automobile axles, the stiffened cylindrical shell of a submarine, and a space platform composed of various base connections. These systems can be regarded as a waveguide collection with appropriate connections at the junction. The waveguide determines the energy of propagating the wave along the length direction, so we can study the power flow from a new perspective and observe how the structural wave relates to the target control parameters, such as structural-acoustic radiation. The physical information conveyed by the structure waveguide is much more complex than the recorded control parameters. Various sources gather on the radiation surface in the form of waves, some of which are

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transformed into air-borne noise or underwater sound. The role of studying the waves and analyzing the input, propagation, and attenuation of the vibration energy becomes increasingly important for the improved design of the carrier. There are different types of waves in the structure, including longitudinal waves, shearing waves, flexural waves, and their combination. Ship designers focus more on the flexural waves since it is directly coupled with the vibration of the surrounding acoustic medium to form the most effective radiation. This reminds us that the structural units can be understood as the waveguide, for example, a finite thin rod excited in the length direction will generate a longitudinal wave; if the thin rod is excited in the transverse direction, the generated wave will propagate in a semi-infinite body as if there is only one free surface. Only after some time will the wave meet other transverse surfaces and produce reflections. For a finite structure, the particle motion is a simple superposition of the initial wave with all reflected waves at certain nodes in the waveguide. However, the synthesis is very complex with some resonance peaks disappearing, and the phenomenon of “blanking” is observed but without correct prediction. The study of structural waves is beneficial to grasp basic control. Beams and plates are common units in ships. The dimension of a beam in one direction is much larger than that in the other two directions, so the structural wave mostly propagates along the length direction while the dimension of the plate structure in two directions is much larger than that in the thickness direction, so the wave can propagate along the two directions. Plates usually contact with the surrounding fluid medium, allowing it to be the main radiator of underwater sound. To control the structural-acoustic radiation, it is necessary to store more vibration energy in the beams and less flexural waves in the plates, which is opposite to the design of the instrument! Wave Propagation and Vibration Designers pay attention to structural vibration, which enjoys a simple and intuitive expression. Essentially, wave and vibration sometimes have an identity and occasionally have a causal relationship, both of which are “languages” in two distinct and independent fields. This book tries to establish the relationship between the basic parameter waves and the target control parameters. The concepts of the traveling wave and near-field wave must first be established to study the wave propagation. The near-field waves do not propagate far away, which only exist near the “discontinuity” and decay very quickly, so they are also called the decaying waves. However, this does not mean that we should only focus on the traveling wave since the near-field wave has an important function of waveform conversion. The conversion of the flexural wave to the longitudinal wave must be accompanied by the exchange and redistribution of vibration energy. The structure waves become so complex in the giant system that the sample study of the decaying waves shows great importance, influencing the structural sound radiation uniquely. The advantage of the WPA is to establish a direct correlation between structural waves and control parameters.

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Different boundaries exist in engineering structures, so we can find a phenomenon: one of the main research contents of analytical methods is to continuously break through the influence of boundary and structural discontinuities on the analysis, such as studying the rational approximation of boundary conditions so that the mathematical solution of the system can be successfully “matched”. The WPA is less affected by the boundary so it is relatively easy to obtain the solution of a complex boundary. The establishment of the concept of universal “discontinuity” can enable researchers to better understand the giant system. The Theme and Outline This book concentrates on the propagation of structural waves in beams, plates, distributed parameter systems, and hybrid power systems. The WPA is assumed to analyze the above dynamic problems in a consistent framework, such as the “mixing effect” of floating rafts. Although we find out through practice that it is difficult to express such a seemingly simple multi-input single-output dynamic system in a centralized and clear way. Without fail, the WPA is more suitable. This book mainly studies the propagation of structural waves in a continuum, and their reflection and conversion at discontinuities such as the boundary, junction, and inflection point with a focus on their final reflection in engineering. The author aims to explore the dual integration of a giant system from “reductionism” to “holism” based on the simplification and abstraction of a giant system with mechanism research. White initiated the theory of vibrational power flow. Professor Mead conducted considerable pioneering research on the analysis of structural waves. The outstanding research conducted by Pinnington and Langley supplemented the wave propagation. Wu Chongjian and White also worked to develop the WPA as an independent analytical method. Chapter 1 of this book makes a brief introduction to the basic theory of vibration and noise, discusses the common functional analytical methods for a differential equation of vibration, which are used to analyze the vibration modes of beams and plates, and outlines the Continuous Fourier Transform and the spectral analysis to examine the sound pressure, sound power, and sound radiation efficiency. Chapter 2 describes the use of the WPA to solve differential equations, explains the same-origin relations between the WPA and the traditional analytical methods in detail, and provides two key concepts in the analysis process: One is to retain the spectrum relationship (the specific relationship between frequency and wave number) while the other is to express the space-time of the characteristic function with the double-exponential function. For these concepts, the constraints are automatically relieved when solving practical boundary value problems, even if all solutions are not strictly defined mathematically. In Chap. 3, the WPA is used to examine the plate structure and calculate the flow mode of vibration energy in the plate. Chapter 4 studies the parameter system of complex dynamics, and the unified response expression and spectrum relation are obtained by the WPA. Relying on the idea from mechanism analysis to practical engineering application and based on the finite arbitrary multi-supported beams, the specific engineering cases are analyzed. Chapter 5 analyzes the hybrid power

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system by the WPA, for example, the dynamic model of the multi-supported mast with heavy objects mounted on top and the dynamic characteristics of multi-supported beams with TMD; the interaction and mechanical coupling of structural waves and structural discontinuity are also reviewed. The applicability of the WPA in engineering is demonstrated by the analysis of the submarine mast, concluding that “there is more static stiffness and less dynamic stiffness”. The mechanism study points out that the engineering choice is to change the dynamic design of the mast instead of the high-strength material. In Chap. 6, the WPA is used to analyze the structural response under distributed force excitation. Chapter 7 explores the engineering application of MTMD in vibration isolation of the main motor. Chapter 8 makes an analysis of the floating rafts by the WPA. The force basis, as a “signal generator”, is embedded into the analytical equation, while some of the research is not demonstrated here for the reason of inconvenience. Chapter 9 describes the power flow carried by the structure waves by the WPA. The structural sound intensity shares some similarity with the (air) sound intensity but the waveguide analysis is more complicated. The early experimental research results are also provided in this chapter. The use of structural waves to describe the vibration and noise can be found in all chapters of this book. Readers are expected to understand the control parameters from the perspective of the waves and “observe” the wave process. Another analysis subject includes “universal discontinuity”. Using the discontinuity and inflection point as the starting point for analysis, researchers analyze the multiple effects of structural discontinuity on wave propagation, and understand how discontinuity can change the wave propagation and attenuation in a complex system fundamentally due to a quantitative change. Relying on the deep understanding of the simple structure, engineering designers can establish the thinking of “holism” for complex systems and better cope with increasingly complex engineering projects. The WPA is particularly suitable for dynamics analysis of beam structures, such as tricky finite quasi-periodic structure and hybrid power systems. Considering that any method has its limitations, the author has added the relevant reference documents to the corresponding chapter in as much detail as possible for theoretical development and application explanation, on which further research and discussions can be based. Yingfu Zhu Academician of the Chinese Academy of Engineering Wuhan, China

Preface

The vibration and acoustic performance of the carrier are not only linked to the comfort of the staff but are also closely linked to its comprehensive operational capability. Presently, the acoustic power radiated by an international advanced submarine is already less than 0.3 mW. Therefore, the acoustic energy radiated by a submarine in seawater is smaller than the screen-on power consumption of a smartphone. So, the vibration and noise reduction of submarines drive the structural dynamics into an era of refining development, which is considered as tackling a key cutting-edge technical issue among the core technological secrets of world powers. Behind this silent contest are the continuous breakthrough of engineering ability and basic theory! From components and equipment to systems, structural dynamics, and structure-borne noise to power flow theory, new theories are emerging and applied continuously, which is the result of the profound integration of theoretical methods and engineering practice, as well as the concentrated embodiment of decades of continuous improvement. The Wave Propagation Approach (WPA) is a supplement of the analytical method, which arises in response to the need for refinement. It provides a microscopic perspective and analysis means for new thinking. In short, WPA has the following characteristics: I. The WPA focuses on the study of structural waves. Structural inputs are transformed into forces or structural waves, which form a direct causal relationship with such target control parameters as a vibration level, quantity of vibration isolation, and acoustic radiation. In terms of the wave propagation, reflection and attenuation, and the waveform conversion of longitudinal waves and flexural waves, WPA has advantages in reducing misjudgment in combination with the power flow analysis, resulting in the easy formation of internal relations between parameters, featuring relatively simple mathematical description, which will affect the ways of thinking and design concepts. II. The WPA uses “structural discontinuity” to divide the units, which is different from the “geometric division” of the finite element method. In the complex giant system, the discontinuity is in the same category as structural damping

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and has a “universal correlation” to structural vibration. The new division method is adopted to analyze the complex systems, making the main structure and dominant characteristics clearer. III. The WPA has more relaxed constraint conditions. Many boundary conditions cannot be examined because function types cannot fulfill the mathematical matching. The WPA uses the linear superposition of the general solution of the infinite system with the particular solution of the finite system to define structural vibration, allowing a computer to complete the tedious complex matrix operations and “forcing” the instantaneous value of the solution to coincide with the boundary conditions and compatibility conditions, thereby resulting in fewer constraints and consequently leading to wider boundary adaptability. Large-scale complex projects, such as ships, aviation, aerospace, and bridge construction have experienced decades of rapid development and technological accumulation. While enjoying the results, researchers face the new stage of returning to the mechanisms, clarifying the details, and completing the increase. Designers having a good command of the basic theoretical methods, and the achievements are of great benefit to promote professional integration. The WPA may help you better understand structural waves and apply it. At present, we have entered the critical period of detail determining the success or failure of the carrier development, and structural dynamics analysis is confronted with the fundamental theoretical challenges of development: certain seemingly insignificant parameters become the main causes, with a high contribution rate to the system while certain influential factors that seem to possess great weight are either overcome or there are deviations from past understanding, leading them to make little actual contribution. Designers should continuously achieve simplicity and optimization schemes. Wave is the microscopic trace of structural vibration and transmission and it is the criterion of vibration source and mechanism formation when rising to the macro levels, so the improvement direction established and the conclusion reached are comprehensive and refined. The differences between theoretical calculations and data measured on real ships often make designers despondent and frustrated. Structural waves provide a new perspective for the interpretation of these problems, such as the phenomenon of the “blanking” or misplacing of the modal test peaks of complex systems, as explained by D. J. Ewins of the Imperial College London, UK. Combined with the power flow, the WPA, from basic units such as beams, power sources, discontinuity to coupled systems, and hybrid power systems, establishes the concept of discontinuity to help us better understand important issues, such as whether the “breathing mode” of a submarine radiates a strong voiceprint. Many papers have been published on that respect at home and abroad. The WPA adds an argument for independent judgment in advance that some radiated energy of a submarine as a complex system is consumed by numerous discontinuities instead of viscoelastic damping or other ways.

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Numerical analyses can complete the complex engineering dynamics analysis, with three issues encountered: First, the homogenization competition of commercial software is intensified while the personalized application and improvement are insufficient. Second, the needs of mechanism research are emerging constantly. There are some open loops in the simulation of complex systems while the numerical calculation is only a special one among exhaustive cases rather than a general rule. Third, the interface between analytical methods and numerical methods is becoming increasingly blurred, with the most important constructs depending on mechanism research. We need to admit that many dynamic types of research of complex systems are far from the unified theoretical analysis framework, and cognitive shortcomings persist. The author has attempted to maintain the integrity of this book as much as possible and has tried his best not to affect the extended understanding of the readers. The limitations of personal ability, shortcomings of WPA, and popular research topics’ transition may lead to unbalanced contents of the book, for example, research on longitudinal waves of rods is not yet included and some parts are limited by the content, so only the basic conclusions are given without the process. Moreover, as the WPA is still in the process of development and perfection, the author sincerely hopes that readers can get to grips with the characteristics and shortcomings of the WPA through the application of different scenarios. My email address is [email protected] and I look forward to your comments. The author would like to thank Sir Bao Yugang Foundation for the scholarship that allowed me to complete the relevant research at the Institute of Sound and Vibration Research of the University of Southampton in the UK, laying the foundation for the WPA. I would also like to extend thanks to my tutor, Prof. R. G. White, Academician Yang Shuzi, Academician Zhu Yingfu, Prof. LuoDongping, and researcher Ma Yunyi, as it was their continuous encouragement that gave me the courage to finish this monograph. I would also like to thank Prof. D. J. Mead, Prof. F. Fahy, and Dr. R. J. Pinnington for their knowledge, experience, moral character, and academic discussions from which I greatly benefited. Academician Hu Haiyan read the first draft and provided many useful suggestions. I am also thankful to Academician Yang Wei, Academician Wu Yousheng, Academician Zhang Qingjie, and Academician He Yaling for their encouragement and guidance. Xiong Jishi and Chen Zhigang assisted in the writing of Chaps. 3 and 8. Doctoral students Lei Zhiyang, Zhang Shiyang, and Yan Xiaojie conducted the programming and assisted in drawing a large number of calculation charts. Professor Li Tianyun and Associate Prof. Zhu Xiang provided many useful suggestions for the revision of the manuscript. Many seminar-style discussions were also held with Du Kun, Xu Xintong, Chen Lejia, Qiu Changlin, Wang Chunxu, and others, yielding fruitful results. An academic monograph such as this cannot be completed without the selfless help of many people, in particular, Lu Xiaohui,

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Cai Daming, Yi Jisheng, Hu Wenli, Zhang Zhipeng, Zhu Xianming, Xue Bing, Wang Yan, Yang Yuting, Fan Yongjiang, Xia Guihua, Zhang Ling, and Xue Li, whom I thank for their sincere support and guidance, as well as their selfless contribution. Wuhan, PR China August 2018

Chongjian Wu

Introduction

Based on fourth-order differential equations, the Wave Propagation Approach (WPA) continually utilizes the time-domain spatial exponential function in the equation derivation and reconstruction to create a unified framework for analyzing structural vibration. Focusing on wave propagation, reflection, attenuation, and waveform conversion combined with the power flow research, this book aims to examine the finite beam structures, periodic/quasi-periodic structures, coupling structures, and the offsetting mechanism of multiple disturbance sources from the microscopic perspective of structural waves, beginning with the basic elements such as beams and TMD. WPA is a supplementary analytical method. The readers include undergraduate students, graduate students, and engineering designers that wish to deepen their studies on structure-borne noise, power flow theory, fine control of vibration noise, etc. To read this book, one needs to have a certain basic theoretical knowledge of vibration and noise control and preferably engineering practice.

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Contents

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2 Basic Theory of WPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Challenges and Evolution of Analytical Method . . . . . . . . . . 2.2 Mathematical Description of WPA . . . . . . . . . . . . . . . . . . . 2.2.1 Development History of WPA . . . . . . . . . . . . . . . . . 2.2.2 Characteristic Function Expressed by Exponential Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Coefficients of Response Function of Point Harmonic Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Coefficients of Point Harmonic Bending Moment Response Function . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 The Basic Theory of Structure–Borne Noise . . . . . . . . . . . . . . 1.1 The Vibration Modes of Beams . . . . . . . . . . . . . . . . . . . . . 1.1.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 MATLAB Examples . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Vibration Modes of Plates . . . . . . . . . . . . . . . . . . . . . 1.2.1 Basic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Calculation Examples for Plates . . . . . . . . . . . . . . . 1.2.3 The Natural Frequencies of Plates . . . . . . . . . . . . . . 1.3 Sound Pressure, Sound Power, and Sound Radiation Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Far-Field Sound Pressure . . . . . . . . . . . . . . . . . . . . 1.3.2 The Wave Number Transform Solution . . . . . . . . . . 1.3.3 Volume Velocity and Sound Pressure . . . . . . . . . . . 1.4 Sound Power and Sound Radiation Efficiency . . . . . . . . . . 1.4.1 Basic Equations for the Radiation Mode Theory . . . 1.4.2 Examples of Beam and Plate Structures . . . . . . . . . 1.4.3 Radiation Efficiency in Terms of Radiation Modes . 1.4.4 Radiation Efficiency in Terms of Structural Modes . 1.4.5 Examples of the Calculation of Radiation Efficiency References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2.5 2.2.6 2.3 WPA 2.3.1

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . Analytical Reconstruction of Finite Beam . . . . . . . . for Analysis of Finite Simple Structures . . . . . . . . . . WPA Expressions of Displacement, Shear Force, and Bending Moment . . . . . . . . . . . . . . . . . . . . . . 2.3.2 S–S Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.3 C–C Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.4 C-F Beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.5 Comparison Between WPA and Classical Analytical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Traceability and Characteristic Analysis of WPA . . . . . . . . 2.4.1 Traceability of WPA . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Characteristics of WPA . . . . . . . . . . . . . . . . . . . . . 2.5 Introduction to the Various “Parameters” in WPA . . . . . . . 2.5.1 WPA and Mechanism Analysis . . . . . . . . . . . . . . . 2.6 Shortcomings of WPA . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Contents

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3 Analysis of Plate Structure Using WPA Method . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Bending Vibration and Wave of Uniform Plate . . . . . . . . . . 3.3 Response of Infinite Plate Under Harmonic Force (Moment) . 3.3.1 Response of an Infinite Plate Under Harmonic Force . 3.3.2 Response of Infinite Plate Under Harmonic Moment . 3.4 Wave Propagation in Infinite Plate at the Vertical Incidence of Bending Wave in Discontinuous Interface . . . . . . . . . . . . 3.4.1 Plate Simply Supported at the Middle . . . . . . . . . . . . 3.4.2 Plate Simply Supported at One End . . . . . . . . . . . . . 3.4.3 Plate Firmly Supported at One End . . . . . . . . . . . . . 3.4.4 Plate Free at One End . . . . . . . . . . . . . . . . . . . . . . . 3.5 Wave Propagation When Bending Wave of Infinite Plate Is Incident on Discontinuous Interface . . . . . . . . . . . . . . . . . 3.6 Forced Vibration of a Rectangular Plate with Both Ends Simply Supported . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Analytical Solution Example for Vibration of a Plate Using WPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 WPA Method for Solving Structure Power Flow of Plate . . . 3.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 WPA for Analyzing Complex Beam Structures . . . . . . . . . . . . . . . . 4.1 Research History and Methods of Complex Beam Structures . . . . 4.2 WPA Analysis of Elastic Coupled Beams . . . . . . . . . . . . . . . . . .

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Contents

4.2.1 4.2.2 4.2.3 4.3 Finite 4.3.1 4.3.2

Establishment of WPA Expression . . . . . . . . . . . . . . Boundary Conditions and Consistency Conditions . . . Vibration Response of Elastic Coupled Beam . . . . . . Arbitrary Multi-Supported Elastic Beam . . . . . . . . . . . Mechanical Model and WPA Expression . . . . . . . . . WPA Superposition Under Multi-Harmonic Force Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Dynamic Response and Stress of Four-Supported Mast . . . . . 4.4.1 Mechanical Model and WPA Expression . . . . . . . . . 4.4.2 Analysis of Dynamic Stress of Four-Supported Mast . 4.5 Periodic and Quasi-Periodic Structures . . . . . . . . . . . . . . . . . 4.5.1 Properties of Periodic Structure . . . . . . . . . . . . . . . . 4.5.2 Properties of Quasi-Periodic Structure . . . . . . . . . . . . 4.6 Energy Transmission Loss Due to Flexible Tubes . . . . . . . . 4.6.1 Establishment of WPA Expression . . . . . . . . . . . . . . 4.6.2 Boundary Conditions and Consistency Conditions . . . 4.6.3 Analysis of Dynamic Characteristics of Pipe Sections with Flexible Tubes . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 “Double-Stage Vibration Isolation” Device for Pipeline . . . . 4.7.1 Establishment of WPA Expression . . . . . . . . . . . . . . 4.7.2 Boundary Conditions and Consistency Conditions . . . 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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102 103 103 105 108 110 112 114 114 116

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118 121 123 124 128 129

5 WPA for Analyzing Hybrid Dynamic Systems . . . . . . . . . . . . . . . 5.1 Hybrid Power Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Continuous Elastic Beam System with Lumped Mass . . . . 5.2.1 The Mechanical Model and Derivation of the WPA Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 The Dynamic Characteristics of the Multi-support Mast with a Heavy End . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The Analysis of the Dynamic Characteristics of Multi-supported Beams with Dynamic Vibration Absorbers . . . . . . . . . . . . . . . . 5.3.1 The General Equation of WPA . . . . . . . . . . . . . . . . . . . 5.3.2 Dynamic Flow Expression . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Calculation and Discussion . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Summary of the Analysis . . . . . . . . . . . . . . . . . . . . . . . 5.4 The Analysis of Mast Retrofitting with TMD Using the WPA Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 The Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Calculation Example . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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138 139 140 142 146

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147 148 151 154 155

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Contents

6 WPA for Calculating Response Under Distributed Force Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Mechanical Model and Formula Deduction . . . . . . . . 6.3 Simple Cantilever Beam Structure . . . . . . . . . . . . . . . 6.4 Comparison Between Examples of WAP Method and Classical Analytical Method . . . . . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Discrete Distributed Tuned Mass Damper . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Velocity Impedance of MTMD . . . . . . . . . . . . . 7.3 The Vibration Absorption Characteristics of MTMD . 7.4 Analysis, Calculation, and Discussion . . . . . . . . . . . 7.4.1 The Basic Parameter Analysis . . . . . . . . . . . 7.4.2 The Comparison Between MTMD and TMD 7.4.3 The Comparison in Under/Over-Tuned States 7.4.4 Influence of the Mass Ratio . . . . . . . . . . . . . 7.5 The Actual Vibration Elimination Effect . . . . . . . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 Analysis of Raft Using WPA Method . . . . . . . . . . . . . . . . . . . . 8.1 Single-Stage and Double-Stage Vibration Isolation . . . . . . . . 8.1.1 Vibration Isolation System Model and Basic Transmission Characteristics . . . . . . . . . . . . . . . . . . . 8.1.2 Influence of Mass Ratio . . . . . . . . . . . . . . . . . . . . . . 8.2 Raft Vibration Isolation System . . . . . . . . . . . . . . . . . . . . . . 8.2.1 History of Raft Research . . . . . . . . . . . . . . . . . . . . . 8.2.2 Definition, Modeling, and Basic Characteristics of Rafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Physical Modeling and Coordination Conditions . . . . 8.2.4 Analysis of Basic Transfer Characteristics . . . . . . . . . 8.3 System Thinking and Consideration of Rafts . . . . . . . . . . . . 8.3.1 Raft Application Paradox . . . . . . . . . . . . . . . . . . . . . 8.3.2 Definition of Emergence . . . . . . . . . . . . . . . . . . . . . 8.3.3 Several Inferences . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Large Raft and Small Raft . . . . . . . . . . . . . . . . . . . . 8.4 Analysis of Rafts Using the WPA Method . . . . . . . . . . . . . . 8.4.1 Internal Coupling Force Acting on the Raft . . . . . . . . 8.4.2 Vibration Displacement of the Raft Beam . . . . . . . . . 8.4.3 Boundary Conditions and Compatibility Conditions . . 8.4.4 WPA Expression of Vibration Isolation Effect of Raft

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Contents

xxi

8.5 “Mass 8.5.1 8.5.2 8.5.3

Effect” Analysis of Raft . . . . . . . . . . . . . . . . . . . . . . . Basic Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . Comparative Study of Rigid Installation of Equipment . Impact of Equipment Location on the Effect of Vibration Isolation . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 “Mixing Effect” Analysis of Rafts . . . . . . . . . . . . . . . . . . . . . 8.6.1 Offset of Two Structural Waves . . . . . . . . . . . . . . . . . 8.6.2 Offset of Multisource Structural Waves . . . . . . . . . . . . 8.6.3 External and Internal Mixing Effects . . . . . . . . . . . . . . 8.6.4 Equal-Master Rafts and Master-Slave Rafts . . . . . . . . . 8.6.5 Impact of Raft Frame Damping . . . . . . . . . . . . . . . . . 8.7 “Tuning Effect” of Rafts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 WPA Analysis and Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1 Raft Test Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.2 Analysis of Test Results . . . . . . . . . . . . . . . . . . . . . . . 8.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 Vibration Power Flow and Experimental Investigation . 9.1 Basic Theory of Vibration Power Flow . . . . . . . . . . 9.1.1 Research Review of Power Flow . . . . . . . . . 9.1.2 Basic Characteristics of Power Flow . . . . . . . 9.1.3 Development and Focus of Power Flow . . . . 9.1.4 Input Power . . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Transmitted Power . . . . . . . . . . . . . . . . . . . . 9.2 Power Flow Test of Structure . . . . . . . . . . . . . . . . . 9.2.1 Summary of Test and Measurement Research 9.2.2 Input Power Measurement . . . . . . . . . . . . . . 9.2.3 Transmitted Power Measurement . . . . . . . . . 9.3 Testing and Measurement . . . . . . . . . . . . . . . . . . . . 9.3.1 Test Structure and Parameters . . . . . . . . . . . . 9.3.2 Test Procedure . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 Input Power Measurement . . . . . . . . . . . . . . 9.3.4 Transmitted Power Measurement . . . . . . . . . 9.4 Control Power Measurement Accuracy . . . . . . . . . . 9.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Afterword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

About the Author

Chongjian Wu born in October 1960, is a Doctoral Tutor and the Chief Technical Expert in the field of vibration and noise reduction for China State Shipbuilding Corporation Limited, the Chief Designer of key types of submarines, and a Guest Director of the Chinese Society of Theoretical and Applied Mechanics. He has once worked as a Visiting Scholar/Assistant Researcher in ISVR of the University of Southampton in the UK, enjoying the experience of in-depth research on the basic theory and engineering application of vibration and noise reduction. He has won a Special Award and the First Prize for the National Science and Technology Progress Award, respectively, and was awarded the “Ship Design Master”.

xxiii

Symbols

an ; bn ; c n ; dn A An ; Bn ; Cn ; Dn b cg cb cl D E EI F G h i; Z I j k; k1 ; k2 ; k3 ; k4 kx ; ky ; kz K [k], [K] L M; Mx n N ~po ðtÞ; ^pðxÞ ~ ^ PðtÞ; PðxÞ q r R t

The coefficients response function of an infinite structure Coefficient Coefficient related to the frequency Thickness, depth Group velocity Flexural wave velocity pffiffiffiffiffiffiffiffiffiffiffiffiffiffi Longitudinal wave velocity, EA=qS Plate stiffness, Eh3 =½12ð1  v2 Þ Young’s modulus The flexural rigidity of a beam Axial force Shear modulus, single-side frequency function The height of a beam or rod, the thickness of a plate Integer The second moment of a section, rectangular beam I ¼ bh3 =12 pffiffiffiffiffiffiffi Imaginary number, j ¼ 1 Wave number Two-dimensional wave number Spring stiffness Stiffness matrix Beam length, distance from a boundary (Bending) moment Frequency count Integer Transverse force (harmonic force), sound pressure Power/energy flow in the time domain/frequency domain Distributed load Radial coordinate External radius, the correlation function Time

xxv

xxvi

S T uðtÞ u; v; w V W x; y; z

Symbols

Cross-sectional area, cross-correlation density function Time window, temperature The response, velocity, strain, etc. Displacement response, spatial variable function Velocity response of the structure Spatial transformation window Rectangular space coordinates

Greek Alphabet a b di;j D g h v l k q r; e n UðtÞ w x; xr xc

Coefficient Loss factor, viscoelastic damping coefficient Kronecker symbol Determinant The damping loss factor of a structure Angular coordinate Poisson’s ratio Ratio Wavelength Density Stress, strain Viscous damping coefficient Time variable function Lateral contraction Angular frequency and resonance frequency Cutoff, coincidence, and critical frequency

Specific Symbols Rn P r2 ½ ; M f g; X

Stochastic noise Radiated sound power @2 Laplace operator, @x 2 þ Matrix, or M in black Vector, or X in black

@2 @y2

þ

@2 @z2

Symbols

xxvii

Subscripts a; u; m d nf ; ff þ; n r 1; 2 B; L

Input power, the power generated by shearing force and bending moment Dynamic vibration absorber, TMD (tuned mass damper) Near field, far field Negative and positive waves along the coordinates Integer, counting unit Resonance frequency Sensor, mode label Flexural wave and longitudinal wave

Superscripts   : ^  ' T H

Complex conjugate Average value Time derivative Quantity in the frequency domain (after the conversion) Quantity in the time domain Derivation of the parameters Matrix transpose Matrix conjugate transpose

Chapter 1

The Basic Theory of Structure–Borne Noise

Depth determines breadth! —Qian Xuesen

Before we discuss the WPA method, it is necessary to examine the basic theory of structural vibration noise, including basic parameters such as wave number, wavelength, and lateral displacement. The examination of continuous systems such as the bending vibration of a beam and a plate is the focus of this chapter. The vibration modes and natural frequencies of beams and plates are discussed and then the sound pressure, sound power, and sound radiation efficiency of simple structures are analyzed and discussed.

1.1 The Vibration Modes of Beams 1.1.1 Basic Equations The structural wave is the basic parameter of structural vibration and acoustic radiation, which is directly linked to the target control parameters. The theory of bending vibration of beams and plates is derived from the fourth-order differential equation [1]: ⎫   2 ∇ 2 ∇ 2 w˜ + ρ S ∂∂tw2˜ = p˜ o ⎪ ⎬ w˜ = w(x, ˜ y, z, t) (1.1) ⎪ 2 2 ∂2 ⎭ ∇ 2 = ∂∂x 2 + ∂∂y 2 + ∂z 2 where ∇2 w˜ ρ

Laplace operator; Lateral displacement of the structure; Material density of the beam;

© Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 C. Wu, Wave Propagation Approach for Structural Vibration, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-7237-1_1

1

2

1 The Basic Theory of Structure–Borne Noise

Fig. 1.1 A schematic diagram of a vibrating beam

w( x)

x 0

Lx

S Cross-sectional area of the structure; p˜ 0 (x, y, t) External excitation harmonic force The displacement w˜ is linked not only to the spatial coordinate of the particle (x, y, z) but also to the time. It is important to calculate the structural mode and modal frequency when conducting a structural analysis. The excitation features and the generation of the state matrix are based on the structure mode and modal frequency. With consideration of the Bernoulli–Euler beam as shown in Fig. 1.1, the vibration equation of Eq. (1.1) degenerates to free vibration [2]: ˜ t) ˜ t) ρ S ∂ 2 w(x, ∂ 4 w(x, · + =0 4 ∂x EI ∂t 2

(1.2)

where EI Bending stiffness of the beam; E Young’s modulus of the material; I = bh 3 /12 Cross-sectional moment of inertia of the beam, where b is the width and h is the thickness of the beam. For harmonic vibration, the displacement response can be divided into two parts: the space function and time function according to the process of separating variables: w(x, ˜ t) = w(x) · (t)

(1.3)

where w(x) Structure mode shape function; (t) Time correlation function When substituting Eq. (1.3) into Eq. (1.2) and dividing the variable for time t and space x, two ordinary differential equations are attained as ∂ 4 w(x) ρ Sω2 w(x) = 0 − ∂x4 EI

(1.4)

∂ 2 (t) + ω2 (t) = 0 ∂t 2

(1.5)

where ω is the circular frequency.

1.1 The Vibration Modes of Beams

3

Set k4 =

ρ Sω2 EI

(1.6)

Therefore, Eq. (1.4) can be rewritten as ∂ 4 w(x) − kn4 w(x) = 0 ∂x4

(1.7)

where kn is the complex wave number of the beam’s bending wave, n = 1, 2, 3, 4. For beam-type structures, the general boundary conditions are as follows: (1) Simply supported boundary condition (S-S beam): w(0, ˜ t) = 0, w(L ˜ x , t) = 0,

∂ 2 w(0,t) ˜ = ∂2x 2 ∂ w(L ˜ x ,t) ∂x2

 0 =0

(1.8)

(2) Clamped boundary condition (C-C beam): w(0, ˜ t) = 0, w(L ˜ x , t) = 0,

∂ w(0,t) ˜ = ∂x ∂ w(L ˜ x ,t) ∂x

0 =0

(1.9)

(3) Free boundary condition (F-F beam): 3 ∂ 2 w(0,t) ˜ ˜ = 0, ∂ w(0,t) =0 ∂x2 ∂x3 2 ∂ w(L ˜ x ,t) ∂ 3 w(L ˜ x ,t) = 0, = 2 3 ∂x ∂x

 0

(1.10)

The general solution to the differential equation Eq. (1.7) is w(x) = A sin(kx) + B cos(kx) + C sinh(kx) + D cosh(kx)

(1.11)

where A, B, C, and D are unknown coefficients, respectively, k 4 = ρ Sω2 /E I . Using simply supported beams as an example and substituting the displacement of the beam into the boundary conditions, it can be used to solve the unknowns A, B, C, and D. At the left end x = 0, substituting Eq. (1.11) into Eq. (1.8), we get w(0) = B + D = 0

(1.12)

∂ 2 w(0) = k 2 (−B + D) = 0 ∂x2

(1.13)

Thus, B = D = 0. At the right end x = L x , we get

4

1 The Basic Theory of Structure–Borne Noise

w(L x ) = A sin(k L x ) + C sinh(k L x ) = 0

(1.14)

∂ 2 w(L x ) = k 2 [−A sin(k L x ) + C sinh(k L x )]= 0 ∂x2

(1.15)

From Eqs. (1.14) and (1.15), we get A sin(k L x ) + C sinh(k L x ) = 0

(1.16)

Since sinh(k L x ) = 0, provide k L x = 0, and therefore, C = 0, k =

nπ , Lx

A=



2 mLx

(1.17)

where m is beam mass per unit length. When substituting Eq. (1.17) into Eq. (1.11), the nth mode shape function for a simply supported beam can be attained as  wn (x) = A sin

nπ x Lx

(1.18)

When substituting Eq. (1.17) into Eq. (1.6), the corresponding natural frequencies can be written as

 2 nπ EI · (1.19) ωn = m Lx The mode shapes are orthogonal with respect to the mass and stiffness distribution [1, 3]: L x mw j (x)wk (x)dx = μ j δ jk

(1.20)

0

L x EI 0

∂ 2 w j (x) ∂ 2 wk (x) · dx = μ j ω2j δ jk ∂x2 ∂x2  δ jk =

(1.21)

1 j =k 0 j = k

where δ jk Kronecker delta symbol; μ j Modal mass of the nth mode. The generalized mass corresponding to the mode shapes in Eq. (1.18) is m L x /2.

1.1 The Vibration Modes of Beams

5

Table 1.1 The structural mode shapes and natural frequencies Boundary conditions Structural mode shape functions

 Simply supported wn (x) = 2 ρ S L x · sin(kn x)

Natural frequencies

Clamped–clamped

cos(kn L x ) · cosh(kn L x ) − 1 = 0

wn (x) = cosh(kn x) − cos(kn x)−

kn = nπ/L x

βn [sinh(kn L x ) − sin(kn L x )] βn = Clamped-free

cosh(kn L x )−cos(kn L x ) sinh(kn L x )−sin(kn L x )

wn (x) = cosh(kn x) − cos(kn x)−

cos(kn L x ) · cosh(kn L x ) + 1 = 0

βn [sinh(kn L x ) − sin(kn L x )] βn = Clamped–simply supported

cosh(kn L x )−cos(kn L x ) sinh(kn L x )−sin(kn L x )

wn (x) = cosh(kn x) − cos(kn x)−

tan(kn L x ) · tanh(kn L x ) + 1 = 0

βn [sinh(kn L x ) − sin(kn L x )] βn =

cosh(kn L x )−cos(kn L x ) sinh(kn L x )−sin(kn L x )

As a result of the mode shapes being orthogonal to each other, the response of the beam can be expressed at any arbitrary point as a linear combination of these mode shape functions. This is known as the mode superposition method. The WPA method selects different technical paths, as shown in Sect. 2.4 of Chap. 2. w(x, ˜ t) =

∞ 

wn (x) · n (t)

(1.22)

n=1

As a result of the infinite modes in a continuous system, it is necessary to intercept a finite number of modes, such as N. In this way, the analysis of the complex system is simplified, and analytical accuracy is certain. For other boundary conditions, the structural mode shapes and natural frequencies are listed in Table 1.1.

1.1.2 MATLAB Examples Consider equations in Table 1.1 for this example. Using the MATLAB program to calculate the first 5th mode shapes of the beam structure, we can get the structure modal shape functions corresponding to different boundary conditions listed in the Table, as shown in Figs. 1.2, 1.3, 1.4 and 1.5.

6 Fig. 1.2 The first 5th mode shapes of simply supported beams

Fig. 1.3 The first 5th mode shapes of the clamped supported beams on both sides

Fig. 1.4 The first 5th mode shapes of clamp-free beams

1 The Basic Theory of Structure–Borne Noise

1.2 The Vibration Modes of Plates

7

Fig. 1.5 The first 5th mode shapes of clamped–simply supported beams

1.2 The Vibration Modes of Plates 1.2.1 Basic Equations The previous section discussed the vibration of beam-type structures. In this section, we will extend the two-dimensional structure vibration. The governing equation and mode shape for the free vibration of the anisotropic and non-damping plates will be analyzed. Equation (1.1) can also be derived [1] as ˜ y, t) + m s D∇ 4 w(x,

∂ w˜ 2 (x, y, t) ∂t 2  2

∇ 2 = ∂∂x 2 + ∂∂y 2 h3 E D = 12(1−υ 2) 2

(1.23)

(1.24)

where w(x, ˜ y, t) Lateral displacement of the plate; m s = ρh Area density of the plate, where ρ and h are the volume density of the plate and the thickness of the plate, respectively; υ Poisson’s ratio For harmonic free vibration, the w(x, ˜ y, t) can be expressed as the superposition of an infinite number of mode shape functions as follows: w(x, ˜ y, t) =

∞  ∞  m=1 n=1

with the properties

wmn (x, y)mn (t)e jωt

(1.25)

8

1 The Basic Theory of Structure–Borne Noise



L x L y mwmn (x, y)w jk (x, y) dydx = 0

Mmn m = j, n = k 0 other

(1.26)

0

where wmn the modal amplitude of the (m, n)th mode of the plate; Mmn the (m, n)th modal mass When substituting Eq. (1.25) into Eq. (1.23), this yields the following results: ∂4 ∂4 ∂4 2 D + 2 2 2 + 4 wmn (x, y) − ωmn m S wmn (x, y) = 0 ∂x4 ∂x ∂x ∂y 

(1.27)

where ωmn is the (m, n)th natural frequency. The structural mode shape functions can be randomly selected as long as they are quasi-orthogonal and both of them satisfy the boundary conditions. This is a method of dimensionality reduction; that is, the binary equation is divided into two univariate equations. The mode shape functions can be written as the product of two independent beam functions [4, 5]: wmn (x, y) = X m (x) · Yn (y)

(1.28)

The shape functions X m (x) and Yn (y) can be randomly selected only if they are quasi-orthogonal and satisfy the boundary conditions. And L x

X j (x)X k (x)dx =

o

L x

L x ∂ 2 X j (x) ∂ 2 X k (x) o

Y j (y) · Yk (y)dy =

∂x2

L x ∂ 2 Y j (y) o

0

∂x2

∂ y2

·

dx = 0 ( j = k)

∂ 2 Yk (y) dy ∂ y2

= 0 ( j = k)

(1.29)

(1.30)

From Eq. (1.25), by using the orthogonal relationship in Eqs. (1.29) and (1.30), the natural frequencies are given by [3] ωmn =



 D/m S ·

I1 I2 + 2I3 I4 + I5 I6 I2 I6

(1.31)

where I1 =

L x ∂ 4 X m (x) 0

I3 =

 Lx 0

∂x4

X m (x)dx, I2 =

∂ 2 X m (x) X m (x)dx, ∂x2

L y

[Yn (y)]2 dy

(1.32)

∂ 2 Yn (y) Yn (y)dy ∂ y2

(1.33)

0

I4 =

 Ly 0

1.2 The Vibration Modes of Plates

I5 =

 Ly 0

9

∂ 4 Yn (y) Yn (y)dy, ∂ y4

I6 =

 Lx 0

[X n (x)]2 dx

(1.34)

For simply supported boundaries, we can select the shape functions as follows: X m (x) = sin(km x) Yn (y) = sin(kn y)

(1.35)

where km Wave number in the direction-x, km = mπ/L x ; kn Wave number in the direction-y, kn = nπ/L y . For a clamped plate, the shape functions can be selected as follows:      λm x λm x λm x λm x − cos − βm sinh − sin (1.36) Lx Lx Lx Lx       λn y λn y λn y λn y Yn (y) = cosh − cos − βn sinh − sin (1.37) Ly Ly Ly Ly 

X m (x) = cosh

βm =

cosh λm − cos λm sinh λn − sin λn

where λm and λn are the roots for the equation cosh λ cos λ = 1. Notice that β Z ≈ (2Z + 1)/2 for large values of the integer Z.

1.2.2 Calculation Examples for Plates Figure 1.6 and Fig. 1.7 show the first 6th mode shapes for the simply supported plate and clamped plate, respectively.

1.2.3 The Natural Frequencies of Plates Similar to the simply supported plate, by substituting equations from (1.32) to (1.35) into Eq. (1.31), the natural frequencies are obtained as follows:  ωmn =

D mS



mπ Lx

2

 +

nπ Ly

2  (1.38)

The natural frequencies for simply supported plates are easily obtained. It is more difficult to obtain the natural frequencies for other boundary conditions, i.e.,

1 The Basic Theory of Structure–Borne Noise

Mode shape

(2,1) mode shape

(2,2) mode shape

(1,3) mode shape

Mode shape

Mode shape

(1,2) mode shape

Mode shape

Mode shape

(1,1) mode shape

(2,3) mode shape

Mode shape

10

Fig. 1.6 The first 6th mode shapes for the simply supported plate (prepared by Lei Zhiyang) (1,1) mode shape

(1,2) mode shape

(2,1) mode shape

(2,2) mode shape

(1,3) mode shape

(2,3) mode shape

Fig. 1.7 The first 6th mode shapes for the clamped plate (prepared by Zhang Shiyang)

1.2 The Vibration Modes of Plates

11

clamped √ plates. Table 1.2 shows the nondimensional frequency parameter Smn = ωmn L 2x m/D for a clamped plate with an aspect ratio of L x /L y .

1.3 Sound Pressure, Sound Power, and Sound Radiation Efficiency 1.3.1 Far-Field Sound Pressure Assume that a structure is in an infinitely rigid baffle. Take the coordinate system (x, y, z) so that its origin lies at the center of the structure and the x − y plane coincides with the plane of the structure, as shown in Fig. 1.8. The acoustic pressure can be expressed in terms of the velocity by using Rayleigh’s integral [5, 6]: p(r ) =

¨

jωρ0 2π

v(r0 ) S

exp(− jk|r − r0 |) dS |r − r0 |

(1.39)

where ρ0 the density of the acoustic medium; v(r0 ) the normal vibration velocity of the plate; S the plate area

|r − r0 | =



(x − x0 )2 + (y − y0 )2 + z 2

(1.40)

Assume that the distance |r − r0 | is large when compared to the characteristic dimension of the structure. The distance |r − r0 | in the denominator of Eq. (1.39) can be approximated by the R. A simplified expression is obtained for the far-field sound pressure: p(r ) =

jωρ0 2πR

¨ v(r0 ) exp(− jk|r − r0 |)dS

(1.41)

S

Expressing r in spherical coordinates (R, θ, ϕ) x = R sin θ cos ϕ

(1.42)

y = R sin θ sin ϕ

(1.43)

12

1 The Basic Theory of Structure–Borne Noise

√ Table 1.2 The nondimensional frequency parameter Smn = ωmn L 2x m/D for the clamped plate Mode index

Aspect ratio L x /L y

m

n

0.1

1

1

22.4419

22.6599

23.0621

23.7026

24.6480

1

2

22.6335

23.4941

25.1664

27.9146

31.9618

1

3

22.9426

24.9258

28.9480

35.5549

44.9729

1

4

23.3833

27.0804

34.6999

46.8896

63.6595

1

5

23.9682

30.0585

42.5174

61.8282

87.7430

2

1

61.7650

62.0456

62.5265

63.2269

64.1724

2

2

62.0188

63.0815

64.9312

6736720

71.4252

2

3

62.4187

64.7402

68.8605

75.0681

83.6327

2

4

62.9727

67.0788

74.4940

85.7589

101.2463

2

5

63.6847

70.1471

81.9505

99.8969

124.3611

3

1

121.0042

121.3086

121.8224

122.5554

123.5205

3

2

121.2811

122.4258

124.3715

127.1710

130.8917

3

3

121.7158

124.1918

128.4400

134.6185

142.9067

3

4

122.3153

126.6459

134.1488

145.1579

159.9984

3

5

123.0816

129.8197

141.5768

158.9428

182.4034

4

1

199.9652

200.2835

200.8179

201.5741

202.5600

4

2

200.2554

201.4498

203.4622

206.3242

210.0769

4

3

200.7104

203.2851

207.6461

213.8893

222.1323

4

4

201.3371

205.8228

213.4639

224.4720

239.0841

4

5

202.1367

209.0869

220.9688

238.1907

261.1723

5

1

298.6645

298.9925

299.5407

300.3140

301.3204

5

2

298.9633

300.1932

302.2525

305.1667

308.9730

5

3

299.4318

302.0790

306.5260

312.8465

321.1408

5

4

300.0765

304.6805

312.4425

323.5230

338.1244

5

5

300.8985

308.0184

320.0409

337.2859

360.1369

0.9

1.0

0.2

0.3

0.4

0.5

Mode index

Aspect ratio L x /L y

m

n

0.6

1

1

25.9694

27.7322

29.9888

32.7747

36.1087

1

2

37.4354

44.3734

52.7604

62.5608

73.7372

1

3

57.1935

72.1298

89.6963

109.8289

132.4831

1

4

84.8361

110.2715

139.8698

173.5726

211.3440

1

5

119.9684

158.3916

202.9253

253.4852

310.0624

2

1

65.3936

66.9248

68.8021

71.0616

73.7372

2

2

76.3107

82.4316

89.8657

98.6632

108.8499

2

3

94.7485

108.5206

124.9831

144.1270

165.9226

0.7

0.8

(continued)

1.3 Sound Pressure, Sound Power, and Sound Radiation Efficiency

13

Table 1.2 (continued) 2

4

121.1172

145.3876

174.0073

206.9087

244.0295

2

5

155.2869

192.6920

236.4647

286.3926

342.4677

3

1

124.7341

126.2152

127.9850

130.0663

132.4831

3

2

135.6092

141.4014

148.3422

156.4978

165.9226

3

3

153.4771

166.4747

182.0060

200.1383

220.9063

3

4

178.9224

202.0876

229.5672

261.3758

297.4941

3

5

212.0869

248.2572

290.9249

339.8329

395.0199

4

1

203.7856

205.2626

207.0045

209.0263

211.3440

4

2

214.7693

220.4553

227.1910

235.0316

244.0295

4

3

232.5019

245.1219

260.1032

277.5373

297.4941

4

4

257.5239

279.9771

306.5785

337.4118

372.5196

4

5

290.0529

325.2376

366.8631

414.6822

468.8186

5

1

302.5561

304.0420

305.6538

307.7926

310.0644

5

2

313.6668

319.3335

326.0299

333.7160

342.4677

5

3

331.4229

343.8862

358.6538

375.6502

395.0199

5

4

356.2963

378.3651

404.5136

434.6013

468.8186

5

5

388.4967

423.0448

464.0322

510.8738

563.9284

Fig. 1.8 A coordinated system for the baffled plate structure

z Plate structure

r (R,

r0 ( x0 , y0 ) 0

, )

y

x Rigid baffle

z = R cos θ

(1.44)

After manipulation, Eq. (1.40) can be rewritten as |r − r0 | =



  R − 2[x0 sin θ cos ϕ + y0 sin θ sin ϕ] + x02 + y02

(1.45)

For large values of R compared to x0 and y0 , the second-order term (x02 + y02 ) can be neglected and by using the first-order Taylor expansion, Eq. (1.45) can be

14

1 The Basic Theory of Structure–Borne Noise

simplified as |r − ro | = R − x0 sin θ cos ϕ − y0 sin θ sin ϕ

(1.46)

When substituting Eq. (1.46) into Eq. (1.41), the results yield a simplified expression for the Rayleigh’s integral: jωρo exp(− jk R) p(R, θ, ϕ) = 2π R ¨ ν(x0 , y0 ) exp[ jk(x0 sin θ cos ϕ + y0 sin θ sin ϕ]dS

(1.47)

S

For simply supported beams, by referring to the integral formula derived above, the solution of sound pressure can be obtained as [7] ωρ0 exp(− jk R) 2π R     Lx L y ˙ m (−1)m exp( jα) − 1 1 − exp( jβ) mπ (α/mπ )2 − 1 β m=1

p(R, θ, ϕ) =

(1.48)

For a simply supported plate jωρ0 exp(− jk R) 2π R      Lx L y ˙ mn (−1)m exp( jα) − 1 (−1)n exp( jβ) − 1 mnπ 2 (α/mπ )2 − 1 (β/nπ )2 − 1 m=1 n=1

p(R, θ, ϕ) =

(1.49) where α = k L x sin θ cos ϕ, β = k L y sin θ cos ϕ. The sound power radiated from a source is defined as the integral over a surface surrounding the source of the component of the time-averaged intensity vector normal to the surface. For harmonic excitations, the time-averaged acoustic intensity I at field point r is defined as I (r ) =

 1  Re p(r ) · u ∗ (r ) 2

(1.50)

where p(r ) Sound pressure complex amplitude; u(r ) Vector of fluid-particle velocity components In the far field, the particle velocity ν(r ) tends to become normal to the hemisphere centered on the source and its amplitude is approximated by p(r )/(ρ0 c0 ) as in the case of plane waves. Therefore, the time-averaged acoustic intensity in the far field

1.3 Sound Pressure, Sound Power, and Sound Radiation Efficiency

15

becomes I (R, θ, ϕ) =

1 | p(R, θ, ϕ)|2 2ρ0 c0

(k  1)

(1.51)

The integral over a hemisphere in the far field of the average acoustic intensity yields the total acoustic power radiated by the structure [6, 8]: 2π π/2 = 0

| p(R, θ, ϕ)|2 2 R sin θ dθ dϕ 2ρ0 c0

(1.52)

0

For plate structures, the above surface integral generally needs to be completed numerically.

1.3.2 The Wave Number Transform Solution For a two-dimensional surface described by the rectangular coordinate system as shown in Fig. 1.8, the spatial Fourier transformation and its inverse are defined as [9] +∞ +∞ F(k x , k y ) =

  f (x, y) exp jk x x + jk y y dxdy

(1.53)

−∞ −∞

+∞ +∞ f (x, y) =

  F(k x , k y ) exp − jk x x − jk y y dk x dk y

(1.54)

−∞ −∞

This is equivalent to the usual Fourier transformation from the time to the frequency domain. Here, the transform is from the spatial to the wave number domain. For planar radiators described by the Cartesian coordinate system as shown in Fig. 1.8, the Helmholtz equation describing a three-dimensional sound pressure field is given as [6, 10] 

 ∇ 2 + k 2 p(x, y, z) = 0

(1.55)

where k = ω/c0 represents the acoustic wave number. The continuity condition is  ∂ p  =0 jωρ0 V (x, y) + ∂z x,y,z=0

(1.56)

where V (x, y) is the velocity of the vibrating surface in the positive z-direction.

16

1 The Basic Theory of Structure–Borne Noise

When applying the wave number transformation to the Helmholtz equation, we get +∞ +∞ −∞ −∞

∂2 ∂2 ∂2 2 p(x, y, z) exp( jk x x + jk y y)dxdy = 0 (1.57) + + + k ∂x2 ∂ y2 ∂z 2

Equation (1.57) can be rewritten as  +∞ +∞ 2 ∂ ∂2 ∂2 ∂2 2 2 2 2 k − kx − k y + 2 · + + + k ∂z ∂x2 ∂ y2 ∂z 2 −∞ −∞    ∂2 p(x, y, z) exp jk x x + jk y y dxdy = k 2 − k x2 − k 2y + 2 P(k x , k y , z) = 0 ∂z (1.58) where +∞ +∞ P(k x , k y , z) =

  p(x, y, z) exp jk x x + jk y y dxdy

(1.59)

−∞ −∞

The general solution to Eq. (1.59) can be written in the following form:

  P(k x , k y , z) = A exp − j z k 2 − k x2 − k 2y

(1.60)

where A is the unknown parameter. Similarly, the boundary condition can be written in the wave number transformation: jωρo V (k x , k y ) +

∂ P(k x , k y , z = 0) =0 ∂z

(1.61)

The structural velocity wave number transform V (k x , k y ) is expressed as +∞ +∞   ν(x, y) exp jk x x + jk y y dxdy V (k x , k y ) =

(1.62)

−∞ −∞

The unknown parameter A can be found by substituting Eq. (1.60) into the transformed boundary condition Eq. (1.61), and we get

1.3 Sound Pressure, Sound Power, and Sound Radiation Efficiency

  ρ0 ωV k x , k y A=

k 2 − k x2 − k 2y

17

(1.63)

When substituting Eq. (1.63) into Eq. (1.60), the transformed pressure can be expressed as  

  ρ0 ωV k x , k y P kx , k y , z =

exp − jz k 2 − k x2 − k 2y k 2 − k x2 − k 2y 



(1.64)

By applying the inverse double Fourier transformation in Eq. (1.64), the sound pressure can be obtained as ρo ω p(x, y, z) = (2π )2

+∞ +∞ −∞ −∞

V (k x , k y ) exp(− jk x x − jk y y − jk z z)

dk x dk y (1.65) k 2 − k x2 − k 2y

where kZ =



k 2 − k x2 − k 2y

(1.66)

Recall Eq. (1.50); the surface particle normal velocity u(x, y, 0) is equal to the structural out-of-plane velocity ν(x, y). The surface time-averaged intensity can be written as I (x, y) =

 1  Re p(x, y, z = 0) · ν ∗ (x, y) 2

(1.67)

The sound power can be expressed as ⎡ +∞ +∞ ⎤   1 ⎣ p(x, y, z = 0)ν ∗ (x, y)dxdy ⎦ (ω) = Re 2

(1.68)

−∞ −∞

In Eqs. (1.67) and (1.68), Re is the real part of a complex value. According to the following Parseval’s formula [10]: +∞ +∞

p(x, y)w˙ ∗ (x, y)dxdy

−∞ −∞

1 = 4π 2

+∞ +∞ −∞ −∞

P(k x , k y )V ∗ (k x , k y )dk x dk y

(1.69)

18

1 The Basic Theory of Structure–Borne Noise

The sound power in Eq. (1.69) can be rewritten as ⎧ +∞ +∞ ⎫ ⎨  ⎬ 1 ∗ Re P(k , k , z = 0)V (k , k )dk dk (ω) = x y x y x y ⎭ 8π 2 ⎩

(1.70)

−∞ −∞

From Eq. (1.64), the pressure on the surface can be expressed as ˙ x , ky ) ρ0 ω(k P(k x , k y , z = 0) =

k 2 − k x2 − k 2y

(1.71)

When substituting Eq. (1.71) into Eq. (1.70), we get ⎧ +∞ +∞  ⎫ 2    ⎬ V (k x , k y ) ρo ω ⎨

(ω) = Re dk dk x y ⎭ 8π 2 ⎩ k2 − k2 − k2 −∞ −∞

x

(1.72)

y

Note that

k 2 − k x2 − k 2y is real only if k 2 ≥ k x2 − k 2y ; Eq. (1.72) can be rewritten as ρ0 ω (ω) = 8π 2

¨ k 2 ≥k x2 +k 2y

  V (k x , k y )2

dk x dk y k 2 − k x2 − k 2y

(1.73)

From Eqs. (1.72) and (1.73), it is important to note that only supersonic wave number components (the values of the wave number satisfying k 2 ≥ k x2 − k 2y ) radiate to the far field and the subsonic wave number components (the values of wave number satisfying k 2 < k x2 − k 2y ) are associated with decaying near-field waves only and do not contribute to the sound radiation in the far field.

1.3.3 Volume Velocity and Sound Pressure The spatial Fourier transform of Eq. (1.53) is now expressed in terms of the structural velocity wave number transformation. This is expressed as +∞ +∞   ν(x, y) exp jk x x + jk y y dxdy V (k x , k y ) = −∞ −∞

For a finite plate, Eq. (1.74) can be simplified as

(1.74)

1.3 Sound Pressure, Sound Power, and Sound Radiation Efficiency

¨

  ν(x, y) exp jk x x + jk y y dxdy

V (k x , k y ) =

19

(1.75)

S

where S denotes the surface of the plate structure. Recall the far-field sound pressure in Eq. (1.47): jωρo exp(− jk R) p(R, θ, ϕ) = ¨2π R ν(x0 , y0 ) exp[ jk(x0 sin θ cos ϕ + y0 sin θ sin ϕ)]dS

(1.76)

S

Compare Eqs. (1.75) and (1.76) the far-field pressure can be expressed in terms of the wave number transformation of the velocity distribution as p(R, θ, ϕ) =

  jωρo exp(− jk R)V k x , k y 2π R

(1.77)

where k x = k sin θ cos ϕ

(1.78)

k y = k sin θ sin ϕ

(1.79)

Equation (1.77) shows the fundamental relationship between far-field radiated pressure and structural wave-number information: The far-field acoustic energy radiated in the direction defined by θ and ϕ is solely determined by the single structural wave number component evaluated at the wave number (k x , k y ) defined in Eqs. (1.78) and (1.79). These values of k x and k y fulfill

k x2 + k 2y ≤ k

(1.80)

Equation (1.80) defines the supersonic region of the wave number domain; this region is associated with the radiating components of the structural vibration. The wave number components outside the supersonic region, also referred to as a subsonic wave number, only contribute to near-field radiation. The total volume velocity is defined as [4] ¨ Vν =

ν(x, y)dxdy

(1.81)

S

Consider a special case of k x = 0 and k y = 0 in Eq. (1.74). The structural velocity wave number transformation can be expressed as

20

1 The Basic Theory of Structure–Borne Noise

  V k x = 0, k y = 0 =

¨ ν(x, y)dxdy = Vν

(1.82)

S

From Eq. (1.82), it is apparent that V (k x = 0, k y = 0) is equal to the volume velocity of the structure. Thus, the far-field radiated sound pressure corresponding to the special case of k x = 0 and k y = 0 is proportional to the volume velocity of the structure. Examining Eqs. (1.78) and (1.79) shows that this special case corresponds to radiation in a direction normal to the planar structure where θ = 0 (ϕ can take any value). Therefore, if we can design a controlled system in which the modes are all non-volumetric, i.e., V (k x = 0, k y = 0) = Vy = 0, theoretically, the far-field sound pressure in the normal direction to the structure will be zero.

1.4 Sound Power and Sound Radiation Efficiency 1.4.1 Basic Equations for the Radiation Mode Theory Consider a vibrating rectangular plate of length L x and width L y in an infinite rigid baffle, as shown in Fig. 1.9. From this, we obtain [5, 6, 11] jωρ0 p(rn ) = 2π R

¨ ν(rn ) S

exp(− jk|rn − rm |) dS |rn − rm |

(1.83)

where ω k ρ0 S

Angular frequency of the sound wave; Wave number, k = ω/c0 ; Density of the air; Area of the plate;

y

Fig. 1.9 Finite element discretization of the plate Ly

S

R

S

rm

rn 0

x Lx

1.4 Sound Power and Sound Radiation Efficiency

21

r m , r n Any two random position vectors on the surface of the plate. Assuming that R = |r m − r n |, Eq. (1.83) can be rewritten as ¨

jωρ0 p(rn ) = 2π R

ν(rn )

exp(− jk R) dS R

(1.84)

S

From Eq. (1.68), the sound power radiated into the semi-infinite space above the plate is ⎡

⎤

1 ⎣ Re 2

p(rn ) · v ∗ (rn )dS ⎦

¨

=

(1.85)

S

When substituting Eq. (1.84) in Eq. (1.85), the sound power can be rewritten as ⎤

⎡

¨ ¨

=

ωρo ⎣ Re 4π

ν(rm ) S

ωρo = ν(rm ) · 4π

j exp(− jk R) ∗ · ν (rn )dSdS ⎦ R

S

¨ ¨ S

sin(k R) ∗ · ν (rn )dSdS R

(1.86)

S

Consider that the rectangular plate is divided into N elements with an equal area of S. Then, Eq. (1.86) can be approximated as a finite series: =

J J ωρ0   sin(k R) ∗ · vn S · S vm · 4π m=1 n=1 R

(1.87)

where νm and νn are the velocities at the mth and nth element, respectively. Equation (1.87) can be rewritten in matrix form:  = ν H Rν

(1.88)

where superscript H denotes the complex conjugate transpose. The (m, n) element of matrix R is Rmn =

ω2 ρ0 (S)2 sin(krmn ) · 4π co krmn

(1.89)

From Eq. (1.89), it can be found that the matrix R is purely real. R is also symmetric due to reciprocity. Since the sound power must be greater than zero (unless the surface velocity is zero), R is a positive definite. Therefore, the matrix R is real, symmetric, and positive definite. R can be diagonalized through orthogonal transformation and written as R = Q Q T , where superscript T signifies a transfer. The eigenvalues λk

22

1 The Basic Theory of Structure–Borne Noise

are real. The corresponding eigenvectors Q k are orthogonal to one another. When substituting R into Eq. (1.88), since the eigenvectors matrix Q is real, this Q T = Q H gives us  H  = v H QΛ Q H v = Q T v Λ Q T v

(1.90)

Each of these eigenvectors Q k signifies a possible velocity pattern; any surface velocity can be represented as a linear combination of Q k . Eigenvector Q k as a velocity pattern signifies a natural radiation pattern, which we will term “radiation modes” [12]. Since Q k being orthogonal to one another, the sound power of each radiation mode is entirely independent of one another. Physically, radiation modes are the base vectors orthogonal to one another in the vector space. Each base vector signifies a possible radiation pattern while every radiation mode signifies a possible radiation pattern on the surface of the radiator and it is the natural character of the radiator. Radiation modes are dependent on the radiator geometry and frequency but not on any material characteristic of the radiator itself. Each of these radiation modes has independent radiation efficacy. The main advantage of the radiation modes is to eliminate complex coupling terms in the structural modes. This makes analysis and active control structure-borne sound radiation easier. The kth radiation mode amplitudes yk can be calculated by linearly transforming the surface velocity vector v with the kth radiation mode shape. Evidently, the kth radiation mode Q k is a base vector in the vector space and yk is velocity vector v’s projection on Q k : yk = Q kT ν

(1.91)

When substituting Eq. (1.91) into Eq. (1.90), the sound power can be written as =

N 

|yk |2 λk

(1.92)

k=1

1.4.2 Examples of Beam and Plate Structures 1. Radiation Mode Shapes for Beams Firstly, the beam chosen in this study was L x = 0.5 m, L y = 0.04 m. The numerical calculation was then made by dividing the beam into 100 equal elements. Figure 1.10 shows the first 4th radiation modes of the beam when the dimensionless frequencies are equal to 0.1, 1, 5, and 10, respectively. It can be found that the radiation mode shape is related to the frequency.

0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25

23

kl=0.1 Radiation mode shapes

Radiation mode shapes

1.4 Sound Power and Sound Radiation Efficiency

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25

kl=1

0

0.1

0.2

0.3

0.4

x/Lx

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

kl=10 Radiation mode shapes

Radiation mode shapes

kl=5 0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25

0.5

x/Lx

0.6

0.7

0.8

0.9

1

0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2

0

0.1

0.2

0.3

0.4

x/Lx

0.5

0.6

x/Lx

Fig. 1.10 The first 4th radiation mode shapes for the beam structure when dimensionless frequency kl = 0.1, 1.5, and 10 (prepared by Lei Zhiyang) (solid line: 1st mode; dotted line: 2nd mode; dash–dot line: 3rd mode; dashed line: 4th mode)

2. Radiation Mode Shapes for Plates The panel chosen in this study was L x = 0.38 m, L y = 0.3 m. The numerical calculation was then made by dividing the panel into 19 × 15 equal elements. Below, we provide Figs. 1.11, 1.12, 1.13 and 1.14 to calculate the radiation mode shapes for the beam structures when kl = 0.1, 1, 5, and 10, respectively.

1.4.3 Radiation Efficiency in Terms of Radiation Modes Radiation efficiency was first defined by Wallace [8] as the ratio of the acoustic power that a structure radiates to the power radiated by a piston in the equivalent area vibrating with an amplitude equal to the spatial average mean-square velocity of the structure. The radiation efficiency is generally defined as σ =

 $ % ρ0 c0 S |v|2

where ρ0 , c0 the density of the acoustic medium and sound velocity; S$ % the total surface area of the radiator; |v|2 the spatial mean-square velocity

(1.93)

24

1 The Basic Theory of Structure–Borne Noise The 1st radiation mode

The 4th radiation mode

The 2nd radiation mode

The 5th radiation mode

The 3rd radiation mode

The 6thradiation mode

Fig. 1.11 The first 6th radiation mode shapes when kl = 0.1 (prepared by Lei Zhiyang) The 1st radiation mode

The 4th radiation mode

The 2nd radiation mode

The 5th radiation mode

The 3rd radiation mode

The 6thradiation mode

Fig. 1.12 The first 6th radiation mode shapes when kl=1 (prepared by Lei Zhiyang)

Assuming the radiator is divided into N elements with an equal area, the vector of normal velocities of these elements is denoted as v,

1.4 Sound Power and Sound Radiation Efficiency The 1st radiation mode

The 4th radiation mode

The 2nd radiation mode

The 5th radiation mode

25 The 3rd radiation mode

The 6th radiation mode

Fig. 1.13 The first 6th radiation mode shapes when kl = 5 (prepared by Lei Zhiyang) The 1st radiation mode

The 2nd radiation mode

The 3rd radiation mode

The 4th radiation mode

The 5th radiation mode

The 6thradiation mode

Fig. 1.14 The first 6th radiation mode shapes when kl = 10 (prepared by Lei Zhiyang)

26

1 The Basic Theory of Structure–Borne Noise

$

% 1 |ν|2 = 2S

¨ |v|2 ds = S

N 1  2 νHν |ν| S = 2S n=1 2N

(1.94)

where S = S/N . When introducing Eq. (1.88), we can rewrite Eq. (1.93) as σ =

2N v H Rv ρ0 c0 S v H v

(1.95)

And we further define the radiation efficiency of the kth radiation mode as σk =

2N Q kH R Q k ρ0 c0 S Q kH Q k

(1.96)

Since radiation modes are orthogonal to one another, Q kH R Q k = Q kH Q Q T Q k = Q kH λk Q k = λk Q kH Q k

(1.97)

When substituting Eq. (1.97) into Eq. (1.96), we get σk =

2N λk Q kH Q k 2N λk = H ρ0 c0 S Q k Q k ρ0 c0 S

(1.98)

Evidently, the eigenvalues λk of the radiation matrix R are proportional to the radiation efficiency σk .

1.4.4 Radiation Efficiency in Terms of Structural Modes The velocity distribution of the structure can be characterized by a series expansion ν(x, y) =

N 

wn (x, y) · Φn (x, y)

(1.99)

n=1

where wn (x, y) the nth structural mode shape; n (x, y) the modal velocity Equation (1.99) can be rewritten in the matrix form: ν = w · n

(1.100)

1.4 Sound Power and Sound Radiation Efficiency

27

where w is a real orthonormal matrix, w H = w T . Because sound power  = ν H Rν, using Eq. (1.88), we get  =  H w T Rw =  H M

(1.101)

where M = w T Rw. From Eq. (1.101), the radiation power induced by an nth structural mode can be obtained as n = Mnn |n |2

(1.102)

where Mnn is the nth diagonal element of the matrix M. The spatial mean-square velocity for the nth structural mode is $

% 1 v n |2 = 2S

¨ |wn (x, y) · Φn |2 ds =

1 2S

S

¨ |wn (x, y)|2 ds · |Φn |2 = K n · |Φn |2 S

(1.103) where Kn =

1 2S

¨ |wn (x, y)|2 ds S

When substituting Eqs. (1.102) and (1.103) into Eq. (1.93), the radiation efficiency of the nth structural mode can be expressed as σnn =

Kn Mnn ρ0 c0 S

(1.104)

Accordingly, the radiation efficiency of coupling the mth and nth structural modes can be defined as σmn =

Kn Mmn ρ0 c0 S

(1.105)

σnn and σmn can be called “self- and mutual-radiation efficiencies”, respectively.

28

1 The Basic Theory of Structure–Borne Noise

1.4.5 Examples of the Calculation of Radiation Efficiency 1. Radiation Efficiency for the Radiation Mode The radiation efficiency of the first six radiation modes for beams and plates σi (i = 1, 2, . . . , 6) are plotted in Figs. 1.15 and 1.16 as a function of nondimensional frequency kl. The radiation efficiency of each radiation mode increases when the frequency increases and finally, it reaches 1. One of the important aspects of the radiation modes is that their radiation efficacies fall off very rapidly with the increasing mode order at low frequencies. At low frequencies, it is evident that the sound power will be weakened so that the sound power of other modes can be ignored [6].

100

Radiation efficiency

Fig. 1.15 The radiation efficiency of the first six radiation modes for beam structures (prepared by Lei Zhiyang)

10-2 10-4

1st mode 2nd mode 3rd mode 4th mode 5th mode 6th mode

10-6 10-8 10-1

100

101

kl

100

Radiation efficiency

Fig. 1.16 The radiation efficiency of the first six radiation modes for plate structures (prepared by Lei Zhiyang)

10-2 10-4

1st mode 2nd mode 3rd mode 4th mode 5th mode 6th mode

10-6 10-8 10-1

100

101

kl

1.4 Sound Power and Sound Radiation Efficiency

29

2. Radiation Efficiency for Structural Modes Figures 1.17, 1.18 and 1.19 display the self- and mutual-radiation efficiency for simply supported beams. It can be found that the values of radiation efficiencies of the odd modes are much larger than even modes at low-frequency ranges. Figures 1.20, 1.21, and 1.22 display the self- and mutual-radiation efficiencies of plates. The mutual-radiation efficiency curves in each figure have all been normalized with the self-radiation efficiency of the lowest mode in that group. The degree of the modal coupling between two modes appears to decrease with their distance in the structural wave number space. The mutual-radiation efficiencies have significant effects on the sound power at low frequencies. As shown in Figs. 1.18, 1.19, 1.21, and 1.22, it can be found that the mutualradiation efficiencies can be negative at some frequencies. This means that the sound power may be over or underestimated if the effects of the modal couplings are not Fig. 1.17 The self-radiation efficiency for simply supported beams Radiation efficiency

100 10-2 10-4 1st mode 2nd mode 3rd mode 4th mode

-6

10

10-8 10-1

100

101

kl

0.35

Fig. 1.18 The odd mutual-radiation efficiency for simply supported beams

m=3

Radiation efficiency(σ1m/σ11)

0.3

m=5 m=7

0.25

m=9

0.2 0.15 0.1 0.05 0 -0.05

0

10

20

30

kl

40

50

30 0.5

Radiation efficiency (σ2m/σ22)

Fig. 1.19 The even mutual-radiation efficiency for simply supported beams

1 The Basic Theory of Structure–Borne Noise

m=4 m=6

0.4

m=8 m=10

0.3 0.2 0.1 0 -0.1

0

10

30

20

40

50

kl

100

Self-radiation efficiency

Fig. 1.20 The self-radiation efficiency for simply supported plates (prepared by Lei Zhiyang)

10-2

10-4

(1,1)mode (1,2)mode (2,2)mode

10-6

(1,3)mode (1,4)mode

10-8 10-1

100

101

Fig. 1.21 The (odd, odd) mutual-radiation efficiency for simply supported plates

Radiation efficiency (σ11,1m/σ11,11)

kl 0.5 m=3

0.4

m=5 m=7

0.3

m=9

0.2 0.1 0 -0.1 -0.2

0

10

20

30

kl

40

50

Fig. 1.22 The (even, even) mutual-radiation efficiency for simply supported plates

Radiation efficiency (σ22,2m/σ22,22)

1.4 Sound Power and Sound Radiation Efficiency

31

0.5 0.4

m=4

0.3

m=6

0.2

m=10

m=8

0.1 0 -0.1 -0.2 -0.3

0

10

20

30

40

50

kl

taken into account. This result agrees with Refs. [13, 14]. Assume a unit point force applied to a simply supported beam with a size of 500 mm × 40 mm × 5 mm at two different locations at xc = 100 mm and xc = 220 mm Figs. 1.23 and 1.24 show the sound power with and without mutual-radiation efficiencies. The effects of the mutual-radiation efficiencies tend to be more notable at a non-resonance frequency, as estimated. It should also be noted that the sound power computed without mutualradiation efficiencies can be over or underestimated depending on frequency and excitation locations. For example, when the exciting frequency is 350 Hz, the mutualradiation efficiencies contribute approximately 4 dB when xc = 220 mm and it turns out to be −2 dB when xc = 220 mm. There are many published papers and literature on the basic theory of structural acoustics. Works [6], Works [7], and other literature have been used as references for certain content in this chapter.

90 80

Radiation sound power (dB)

Fig. 1.23 The sound power with and without mutual-radiation efficiencies when xc = 100 mm

70 60 50 40

the exact sound power Without mutual-radiation mode

30 100

200

300

400

500

Frequency (Hz)

600

700

800

32 90

Radiation sound power (dB)

Fig. 1.24 The sound power with and without mutual-radiation efficiencies when xc = 220 mm

1 The Basic Theory of Structure–Borne Noise

80 70 60 50 40

the exact sound power Using structural mode without coupling terms

30 100

200

300

400

500

600

700

800

Frequency (Hz)

References 1. Hamilton WR (1940) The mathematical paper of Sir William R. Hamilton. Cambridge University Press, Cambridge 2. Hu H (2005) Mechanical vibration foundation. Beihang University Press, Beijing 3. Doyle JF (2016) Wave propagation in structures (trans: Bin W, Cunfu H, Jingpin J et al). Science Press, Beijing 4. Fuller CR, Elliott SJ, Nelson PA (1997) Active control of vibration. Academic Press, London 5. Clark RL, Saunders WR, Gibbs GP (1998) Adaptive structures: dynamics and control. Wiley, New York 6. Fahy F (2007) Sound and structural vibration/radiation, transmission and response. Academic Press, London 7. Mao Q, Pietrzko S (2016) Control of noise and structural vibration: a MATLAB-based approach (trans: Wenwei W, Zhenping W, Fei W), vol 1. Harbin Engineering University Press, Harbin 8. Wallace CE (1972) Radiation resistance of a rectangular panel. J Acoust Soc Am 51:946–952 9. Williams E (1983) A series expansion of the acoustic power radiated from planar sources. J Acoust Soc Am 73:1520–1524 10. Williams E (1999) Fourier acoustics. Academic Press, London 11. He Z (2001) Structural vibration and acoustic radiation, vol 12. Harbin Engineering University Press, Harbin 12. Elliott SJ, Johnson ME (1993) Radiation modes and the active control of sound power. J Acoust Soc Am 94(4):2194–2204 13. Li WL, Gibeling HJ (1999) Determination of the mutual radiation resistances of a rectangular plate and their impact on the radiated sound power. J Sound Vib 229:1213–1233 14. LI WL (2001) An analytical solution for the self-and mutual radiation resistances of a rectangular plate. J Sound Vib 245(1):1–16

Chapter 2

Basic Theory of WPA

John Napier Mirifici Logarithmorum Canonis Descriptio (1614).

Abstract

John Napier (Scotland) (Discoverer of logarithms, astronomer, and mathematician)

When calculating the earth’s orbit, John Napier was plagued by the vast volume of calculations required. “Seeing there is nothing so troublesome to mathematical practice, nor that doth more molest and hinder calculators, than the multiplications, divisions, square and cubical extractions of great numbers, which besides the tedious expense of time are for the most part subject to many slippery errors, I began therefore to consider in my mind by what certain and ready art I might remove those hindrances.”

The Wave Propagation Analysis method is called “WPA”, an abbreviation of “Wave Propagation Analysis” or “Wave Propagation Approach”. From the point of view of elastic waves, the difference between elastic waves in a beam and in a bar is that the bending wave in the beam is dispersive, while the longitudinal wave in the © Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 C. Wu, Wave Propagation Approach for Structural Vibration, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-7237-1_2

33

34 Fig. 2.1 Dispersive waves in a beam (Each wave line represents the transient deformation of the beam with time)

2 Basic Theory of WPA

Time

Location

bar has no dispersion effect, so the wave in the beam has no d’Alembert solution, as shown in Fig. 2.1. Due to the different traveling speeds of bending waves at different frequencies, the wave forms are distorted with time. At the same time, as the characteristic function is a fourth-order differential equation, there are two basic wave modes: the propagation wave, described by the traveling wave, and the dissipative wave, expressed by the near-field wave. After deriving the coefficients of the point response function of an infinite beam, the traveling wave and near-field wave caused by the structural boundary reflection of the beam can be reconstructed, and the general solution of the WPA method can be obtained.

2.1 Challenges and Evolution of Analytical Method The pursuit of high technical performance of carriers promotes the development of structural dynamics. From classical Newtonian mechanics, modern micromechanics [1, 2], to quantum mechanics, the theoretical methods by segment include structure-borne noise [3–9], power flow [2, 10–12], sono-elastic theory [13, 14], etc. The general methods for the exploration of new mechanisms of engineering include modeling, boundary treatment, and mathematical analysis. Some of the complex engineering involves abstraction of structural nature and model simplification, and for which basic theoretical research are used to analyze and summarize the general laws of their concepts, key points, and commonalities so as to guide the optimization of giant systems. Theoretical analysis methods can be divided into two broad categories: Analytical methods: analytical methods are numerous, including the modal analysis method, transfer matrix method, modal truncation and synthesis method, modal impedance synthesis method, admittance method, dynamic stiffness synthesis method, WPA method, and spectral analysis method, as shown in Table 2.1. Numerical methods: numerical methods are divided into the finite element method (FEM), boundary element method (BEM), and statistical energy analysis (SEA) method [13–15], e.g., the almost forgotten SAP, SAP5, Super sap, etc., and

2.1 Challenges and Evolution of Analytical Method Table 2.1 List of analytical methods

No Analytical method

35 Notes

1

Transfer function method

2

Modal analysis method

3

Transfer matrix method

4

Modal truncation and synthesis

5

Modal impedance synthesis

6

Modal flexibility synthesis

7

Admittance method

8

Four-terminal parameter method

9

Dynamic stiffness synthesis Method

10

SEA method

11

FFT method (Spectral analysis method)

Frequency domain

12

WPA …

Time domain

Approximate treatment

Typical apparent parameter method

the commonly used ANSYS, ADINA, Nastran, and other large computing programs. As a narrow specialty, Virtual Lab (originally SYSNOISE), AutoSEA, VIOLINE, and other commercial software still appear in structural acoustic analysis. There seem to be enough theoretical methods, so why do people develop new ones? In fact, many seemingly simple systems are unable to provide complete analytic solutions or else require the further simplification of boundary conditions. For example, when Kojima [16] analyzed the natural frequency and vibration modes of a finite multi-bay beam, it was necessary to assume that when the bay number of the beam exceeded 3, it had to be simply supported. Snowdon [17, 18] studied a cantilever beam with a Tuned Mass Damper (TMD) which had to satisfy xd = L; namely, the TMD had to be installed at the free end of the beam, as shown in Fig. 2.2, or else it could not be solved analytically. After improvement, the vibration response  with a TMD installed at the midpoint xd = L 2 of the beam could be obtained. Mead [19, 20] analyzed the power flow of an infinite beam with period  theory, and assumed that the point harmonic force must act at the midpoint xd = l 2 of a singlebay periodic beam. The theory was then further improved to remove this restriction [21]. In several more complicated cases, the ideas and analysis perspectives of theoretical methods are far from the unified framework of mechanism analysis, and there still exist shortcomings in theoretical cognition. For example, the “periodic” characteristic refers to the standing wave effect formed by the superposition of the propagation wave and the reflection wave. The understanding of physical mechanisms can help us to predict the “single frequency” and “repeat frequency” characteristics

36

2 Basic Theory of WPA

Fig. 2.2 Physical model of C-F beam with TMD (The positions of harmonic forces and TMD are strictly limited)

of turbine impellers and ship propellers for the reason that the periodic structure must have common characteristics in dynamics. The main difference is that for a cyclic symmetric structure, it is called a “cyclic period”, which is not as intuitive as “linear period”. Analytical perspectives also limit the discovery of new features. For example, as a new vibration isolation method, the floating raft system not only exhibits the “passive” damping dissipation in the raft similar to that of a doublestage isolation system, but also produces the “active” neutralization mechanism of multisource waves. In the engineering-oriented era, such numerical methods as FEM almost dominate engineering calculations. Commercial software is constantly updated, expanded, and upgraded [22–24], with more user-friendly interfaces and the ability to perform the dynamic analysis of very complex engineering problems. However, in the absence of mechanism analysis conclusions, the results obtained by numerical calculation can be only regarded as “individual cases”, which are special cases obtained through the method of exhaustion rather than general rules. Due to the different analytical perspectives of theoretical methods, the discovery of new characteristics is restricted. These and many other fundamental theoretical issues in engineering are the challenges that analytical methods face and the keys to their evolution [25].

2.2 Mathematical Description of WPA 2.2.1 Development History of WPA Although there are many mathematical and physical descriptions of waves in structures, such as those in the classic Structure-Borne Sound by L. Cremer and M. Heckl, they are mostly used as explanatory remarks. D.J. Mead and R.G. White et al. [19, 26–28] derived simple structural equations using exponential descriptions which they called the “Bending Wave Method” in their papers. Although this theoretical method

2.2 Mathematical Description of WPA

37

has not yet been popularized, it has already indicated possible new characteristics and seems very worthy of an extension. In 1990, following a proposal by Professor R.G. White, C.J. Wu and D.J. Mead derived the displacement response equations for a multi-supported Bernoulli–Euler beam on the basis of the bending wave method [27, 29]. They preliminarily discussed the matter and decided to use the Wave Propagation Approach (WPA) in their published papers in the future. Later, certain scholars adopted Wave Propagation Analysis, which has the same abbreviation of WPA. By using this method, C.J. Wu and R.G. White studied vibration power flow, obtained the general expression of WPA for beam-like structures, and gradually extended their research to arbitrary beams, periodic beams, quasi-periodic simply supported beams, and beams with additional TMD [2, 21]. At the same time, they carried out theoretical calculations and experiments regarding the power flow of finite periodic structures. C.J. Wu and R.G. White used conventional polycrystalline sensors to obtain experimental results that showed high agreement with the theoretical results for the first time under constrained conditions. Regarded as milestone progress in early power flow research, this research won the 1992 Achievement Award of Southampton University and was exhibited as one of two ISVR achievements in the university’s 1992 Open Day. At present, WPA research is increasingly extensive, developing from beam structures to plate and shell structures [30–35].

2.2.2 Characteristic Function Expressed by Exponential Function Taking a Bernoulli–Euler beam as an example, when subjected to space- and timedependent loads, the lateral displacement of the beam can be described by the fourthorder differential equation [36]: EI

˜ t) ˜ t) ∂ 2 w(x, ∂ 4 w(x, + ρ S = p˜ 0 (x, t) 4 2 ∂x ∂t

(2.1)

wherein w—transverse ˜ displacement and deflection of the beam; S—cross-sectional area of the beam; E I —flexural rigidity; ρ—material density. Assuming external excitation p˜ 0 = 0, the free vibration equation of the beam can be obtained, i.e., Eq. (1.2) in Chap. 1. Elastic waves move with simple harmonic motion in the structure, and the transverse displacement of the beam has the form of Eq. (1.3), which is rewritten for readability as follows:

38

2 Basic Theory of WPA

w(x, ˜ t) = w(x) · (t) The decomposition of the characteristic function is [28, 34] ⎧ ⎨ ⎩

w(x) =

4 

A n e kn x

(2.2)

n=1

(t) = e

jωt

wherein An (n = 1, 2, 3, 4) are four unknown coefficients of spatial function; there are four different kn ; and j is an imaginary unit. In this book, kn is, respectively, taken according to the order in Eq. (2.3), and the real root of kn is denoted by k and can be rewritten according to Eq. (1.6) as follows: kn = {k1 , k2 , k3 , k4 } = {k, −k, jk, − jk|n = 1, 2, 3, 4 }  k=

ρ Sω2 EI

(2.3)

1/ 4 (2.4)

In order to analyze system response, the unknown coefficient An in Eq. (2.2) must be calculated by combining boundary conditions and constraint conditions. As such, the complete solution of Eq. (2.1) is again expressed as w(x, ˜ t) =

4

 A n e kn x

· e jωt

(2.5)

n=1

Equation (2.5) refers to the general equation of WPA for the motion of beam structures. It is described by a double-exponential function, the first term of which refers to the sum of spatial correlation terms and the second term to the time correlation term. This equation can be expanded to w(x, t) =



A1 ekx + A2 e−kx A3 e jkx A4 e− jkx · e jωt

(2.6)

Equation (2.6) describes bending waves in beams, and the physical explanation corresponds to Fig. 2.3. The harmonic force generates two types of bending waves on both sides of the action point, respectively: a traveling wave and a near-field wave. The first and second terms in Eq. (2.6) refer to real terms A1 ekx and A2 e−kx , which represent the near-field wave along the negative and positive directions of the x-axis, respectively, also called “decaying waves” or “evanescent waves” in the literature. The near-field wave does not propagate and will rapidly decay, playing the role of wave mode conversion at the discontinuity points of the structure. The third and fourth terms refer to imaginary terms,A3 e jkx represents the traveling wave along the negative direction of the x-axis, and A4 e− jkx represents the traveling wave along the positive direction of the x-axis, which is also called the “propagation wave”.

2.2 Mathematical Description of WPA

39

Fig. 2.3 Traveling wave and near-field wave excited by harmonic force

The shape function of WPA has a form of the double-exponential function and retains the traditional expression of spatial function exp(kn x), as well as its simplicity.

2.2.3 Coefficients of Response Function of Point Harmonic Force Assuming that any point x = x0 on a Bernoulli–Euler beam is subject to external harmonic force p˜ 0 ; the differential equation of the transverse displacement of the infinite beam is ∂ 4 w(x, ˜ t) po δ(x − xo ) · e jωt + kn4 w(x, ˜ t) = 4 ∂x EI

(2.7)

wherein δ(x − x0 ) refers to the Dirac delta function. Due to the symmetry of the infinite beam, the excitation at x = x0 can be translated to x = 0 so as to simplify the equation. In order to avoid confusing them with the aforementioned An , the four unknown coefficients are reset. Now the general formula of the bending vibration response of the beam is

w(x, ˜ t) = Aekx + Be−kx + Ce jkx + De− jkx · e jωt

(2.8)

As the waves of the excited infinite beam propagate to both ends and there is no reflection wave, the following conditions should be satisfied: when x < 0,B = D = 0, and when x ≥ 0,A = C = 0. Thus 

w˜ − (x, t) = Ae+kx + Ce+ jkx · e jωt (x < 0) w˜ + (x, t) = Be−kx + De− jkx · e jωt (x ≥ 0)

(2.9)

wherein the subscript “ + ” indicates the positive coordinate direction and the subscript “−” indicates the negative coordinate direction.

40

2 Basic Theory of WPA

Because of the symmetry of the infinite beam, the rotational displacement at x = 0  is zero, then ∂ w(x, ˜ t) ∂ x = 0. Thus 

+k A + jkC = 0 −k B − jk D = 0

(2.10)

According to the force balance  and symmetry at x = 0, the shear force on both sides of the loading point is p0 2 and the equilibrium equation of the shear force is 

po = −E I (k 3 A − jk 3 C) 2 po = −E I (−k 3 B + jk 3 D) 2

(x = 0− ) (x = 0+ )

(2.11)

Combining Eq. (2.10) and Eq. (2.11), one can obtain jA = jB = C = D

(2.12)

Substituting Eq. (2.12) and Eq. (2.10): D=−

j po 4E I k 3

(2.13)

Thus, the excitation at x = 0 can be obtained and the forced vibration of the infinite beam can be expressed as 



w˜ − (x, t) = 4Ej pIok 3 ejkx − jekx · e jωt (x < 0)

w˜ + (x, t) = 4Ej pIok 3 e− jkx − je−kx · e jωt (x ≥ 0)

(2.14)

The piecewise response function Eq. (2.14) of the infinite beam is further written in a unified form: w(x, ˜ t) =

po −k|x| e + je− jk|x| · e jωt 3 4E I k

(2.15)

Based on the translation principle, the response function of the excited infinite beam at any point x = x0 is w(x, ˜ t) =

po −k|x−x0 | e + je− jk|x−x0 | · e jωt 3 4E I k

(2.16)

When the value of the point harmonic force is unit size, Eq. (2.16) can be further arranged in the general form:

w(x, ˜ t) = a1 e−k|x−xo | + a2 e− jk|x−xo | · e jωt a1 =

1 , 4E I k 3

a2 =

j 4E I k 3

(2.17) (2.18)

2.2 Mathematical Description of WPA

41

Sandbox Fig. 2.4 Simulation experiment of infinite beam with both ends buried in the sandbox (to measure the response function coefficients of point harmonic force)

wherein an (n = 1, 2) are the response function coefficients of the infinite beam under the point harmonic force (bending wave). In order to verify the point response function coefficients of the infinite beam structure, a testing device is set up in Fig. 2.4. When both ends of the steel beam buried in the sandbox are about 0.9 m long, the reflections of the elastic waves on both end boundaries can be ignored. Figure 2.5a shows the model of the infinite beam. When harmonic force p˜ 0 is exerted at x0 = 0, the transverse displacement response at both ends is caused by the traveling wave and near-field wave together [2]: ⎧ N  ⎪ ⎪ an e−kn (+|x|) (x ≥ 0) ⎨ w˜ + (x) = po n=1

N  ⎪ ⎪ ⎩ w˜ − (x) = po an ekn (−|x|) (x < 0)

(2.19)

n=1

The harmonic force generates N elastic waves in both the positive and negative directions of the beam coordinates, with the wave number kn . When kn is a pure real number, it indicates a decaying wave. When the k is a pure imaginary number, it indicates a traveling wave with a wavelength of λ = 2π/k.N depends on the number of degrees of freedom of the structural cross section. For beams with a uniform cross section,N = 2, k1 = k, and k2 = jk.

Fig. 2.5 Harmonic Excitation, Response, and Sign Convention of Infinite Beam

42

2 Basic Theory of WPA

The displacement response described by the piecewise function Eq. (2.19) can be further written in a unified form: w(x) ˜ = po

N 

an e−kn |x| (−∞ < x < +∞)

(2.20)

n=1

2.2.4 Coefficients of Point Harmonic Bending Moment Response Function Likewise, assuming that any point x = x0 on the beam is subjected to external harmonic bending moment M˜ 0 , the fourth-order differential equation of the transverse displacement of the infinite beam is [34] ˜ t) Mo ∂ 4 w(x, δ(x − xo ) · e jωt + kn4 w(x, ˜ t) = ∂x4 EI

(2.21)

According to the symmetry of the infinite beam, the excitation response at x = x0 can be translated to x = 0. In order to facilitate the computation, the exciting loads are first assumed as x = x0 . At this moment, the bending vibration equation of the beam is the same as in Eq. (2.7). Similarly, there are no reflection waves in the beam. Thus, the following conditions should be satisfied: when x < 0,B = D = 0, and when x ≥ 0 x ≥ 0,A = C = 0. Thus

 w˜ − (x, t) = Ce+ jkx + Ae+kx · e jωt (x < 0) (2.22) w˜ + (x, t) = De− jkx + Be−kx · e jωt (x ≥ 0) According to the central symmetry of the infinite beam and the displacement compatibility of the bending vibration at x = 0: A+C = B + D =0

(2.23)

Meanwhile, the rotational displacement at x = 0 is consistent: k A + jkC = −k B − jk D

(2.24)

Based on the moment equilibrium at x = 0, the moment equilibrium formula can be obtained as   E I (−k)2 B + (− jk)2 D − k 2 A + ( jk)2 C = M0

(2.25)

2.2 Mathematical Description of WPA

43

Combined with the compatibility of displacement in Eq. (2.23) and the consistent rotational displacement in Eq. (2.24): A = −B = −C = D

(2.26)

Substituting Eq. (2.26) into Eq. (2.25), the following can be obtained on the basis of the moment equilibrium formula: D=−

j M0 4E I k 2

(2.27)

Now the forced vibration equation of the infinite beam subjected to the harmonic bending moment at x = 0 can be expressed as 



w˜ − (x, t) = + 4Ej MI k0 2 e+ jkx − jekx · e jωt (x < 0)

w˜ + (x, t) = − 4Ej MI k0 2 e− jkx − e−kx · e jωt (x ≥ 0)

(2.28)

The above piecewise response functions of the infinite beam subject to the harmonic bending moment at x = 0 can be written in a unified form: w(x, ˜ t) = −

Mo sgn(x) − jk|x| e − e−k|x| · e jωt 2 4E I k

(2.29)

Likewise, based on the translation principle, the response function of the infinite beam excited at any point x = x0 is w(x, ˜ t) = −

Mo sgn(x − x0 ) − jk|x−x0 | − e−k|x−x0 | · e jωt e 2 4E I k

(2.30)

As observed in the above equation, a harmonic bending moment can generate two kinds of bending waves on both sides of the application point: the first term in the brackets is a traveling wave, also called a “far-field wave”, and the second term in the brackets is a near-field wave, also called a “decaying wave” or “non-propagating wave”. When subject to a unit bending moment, the general form of Eq. (2.30) can be expressed as

w(x, ˜ t) = b1 e−k|xo −x| + b2 e− jk|xo −x| · e jωt 0) , b2 = b1 = − Mo sgn(x−x 4E I k 2

Mo sgn(x−x0 ) 4E I k 2

(2.31) (2.32)

wherein bn (n = 1, 2) are the response function coefficients of the infinite beam under the point harmonic bending moment (bending wave). Similarly, the rotational displacement response θ = ∂w/∂ x is analyzed. Taking the partial derivatives of Eq. (2.19) with regard to x, the rotational displacement response caused by the harmonic force can be obtained as

44

2 Basic Theory of WPA

⎧ N  ⎪ ⎪ cn e−kn (+|x|) (x ≥ 0) ⎨ θ+ (x) = po n=1

N  ⎪ ⎪ ⎩ θ− (x) = − po cn ekn (−|x|) (x < 0)

(2.33)

n=1

Taking the partial derivatives of Eq. (2.28) with regard to x, the rotational displacement response caused by the harmonic bending moment can be obtained as θ (x) = Mo

N 

dn e−kn |x| (−∞ < x < +∞)

(2.34)

n=1

The functions described by Eq. (2.17) and Eq. (2.31) define the “infinite system point response function” of a beam with a constant cross section derived by WPA. The coefficients {an , bn , cn , dn } can be solved from the related equilibrium and coordination conditions of the force or bending moment equations. Meanwhile, it can be directly derived from these equations that the response function coefficients  have the relationship {cn , dn } ⇒ {an , bn }. As the rational displacement θ = ∂w ∂ x, the relationship between the point force and bending moment response function coefficients is cn = −kn an , namely c1 =

1 , 4E I k 2

c2 = − 4E jI k 2

(2.35)

Likewise, the corresponding bending moment response function coefficients d1 and d2 are, respectively: d1 =

1 , 4E I k

d2 =

j 4E I k

(2.36)

2.2.5 Boundary Conditions After obtaining the complete harmonic solution of the fourth-order differential equation of the beam, the different boundary conditions of the beam can be solved with the different derivatives of w, ˜ specifically: Displacement: w˜ = w(x, ˜ t) ˜ Slope: ϕ= ˜ ∂ w(x,t) ∂x ˜ ˜ +E I ∂ 2 w(x,t) Bending moment: M= ∂x2 3 ˜ Shear force: Q˜ = −E I ∂ w(x,t) ∂x3 Table 2.2 shows several typical boundary conditions and their corresponding equations. Note that each boundary of the beam corresponds to two boundary conditions which are exactly equal to the unknown number of wave reflections occurring at the end boundary, although near-field waves are not waves in the usual sense.

2.2 Mathematical Description of WPA Table 2.2 Typical boundary conditions of beam

45

Boundary condition type

Boundary condition 1 Boundary condition 2

Fixed

w(0, t) = 0

Simply supported Free

w(0, t) = 0 EI

∂w(0,t) ∂x

=0

EI

∂ 2 w(0,t) ∂x2

=0

∂ 3 w(0,t) ∂x3

=0

∂ 2 w(0,t) ∂x2

=0

EI

∂ 2 w(0,t)

=0

E I ∂ w(0,t) = ∂x3 +K w(0, t)

Linear spring

EI

Torsion spring

w(0, t) = 0

Damping

EI ∂

2 w(0,t)

Vibration isolation mass

EI ∂

2 w(0,t)

∂x2

∂x2

∂x2

3

EI ∂

3 w(0,t)

∂x3 −α ∂w(0,t) ∂x

=

=0

3 w(0,t) ∂x3 ∂w(0,t) +β ∂t

=

=0

EI ∂

=

EI ∂

3 w(0,t)

+m ∂

∂x3 2 w(0,t) ∂t 2

Note The simplest boundary condition is the simply supported boundary. The application of this table is specifically described in each chapter

2.2.6 Analytical Reconstruction of Finite Beam For a finite beam, as shown in Fig. 2.6, the excitation response function of the infinite structure is first listed and then superimposed with the traveling wave and near-field wave generated by the elastic wave reflection of the finite structure at both ends of the beam. Finally, the linear superposition reconstruction of WPA with regard to the spatial characteristic functions can be completed, as shown in Fig. 2.7. An S–S Beam with a length L is taken as an example to explain the analytical process of WPA in detail, as shown in Fig. 2.6.

Fig. 2.6 Forced wave and free wave of finite beam generated by harmonic force

46

2 Basic Theory of WPA

Infinite Structure Point Response Function

Reconstruction

WPA Finite Structure

Finite Structure Boundary Conditions

Fig. 2.7 Equation reconstruction of infinite and finite beam

Based on Eq. (2.20), point harmonic force p˜ o will generate four waves on the beam, and both beam ends will generate two free waves, respectively, due to boundary reflection. According to WPA reconstruction and the wave superposition principle, forced waves and free waves constitute the total motion equation of the beam. The transverse displacement of any point x(0 ≤ x ≤ L) on the beam can be expressed as [2, 21, 27] w(x, ˜ t) =

4

An e

kn x

+ po

n=1

2

 an e

−kn |xo −x|

· e jωt

(2.37)

n=1

Equation (2.37) is an important equation of WPA. Space and time variables are described by double-exponential functions. They constitute the basic expression framework of the characteristic functions of WPA, and will be further developed. In order to facilitate the derivation of the following formulas, the first three partial derivatives of displacement with regard to x are directly listed as 4 2

∂ w(x, ˜ t)

= A n k n e k n x + po an ( j f )kn e−kn |xo −x| ∂x n=1 n=1

(2.38)

4 2

∂ 2 w(x, ˜ t)

2 kn x = A k e + p an kn2 e−kn |xo −x| n o n ∂x2 n=1 n=1

(2.39)

4 2

∂ 3 w(x, ˜ t)

3 kn x = A n k n e + po an ( j f )kn3 e−kn |xo −x| ∂x3 n=1 n=1

(2.40)

wherein ( j f ) is the symbolic operator: ∂ |xo − x| = (jf) = ∂x



+1 i f x ≥ xo −1 i f x < xo

(2.41)

2.2 Mathematical Description of WPA

47

As the displacement and bending moment at both ends of the simply supported beam are zero, according to Table 2.2, the following can be obtained: w(x) = w(L) = 0

⎫ ⎬

∂ 2 w(0) ∂ 2 w(L) + = 0⎭ ∂x2 ∂x2

(2.42)

When w(0) = 0, it can be found from Eq. (2.37) that 4

A n = − po

n=1

2

an ( j f )e−kn xo

(2.43)

n=1

When M(0) = 0, it can be found from Eq. (2.39) that 4

An kn2 = − po

n=1

2

an kn2 e−kn xo

(2.44)

an kn2 e−kn (L−xo )

(2.45)

an kn2 e−kn (L−xo )

(2.46)

n=1

Similarly, when w(L) = 0, 4

An kn2 ekn L = − po

n=1

2

n=1

and when M(L) = 0, 4

An kn2 ekn L = − po

n=1

2

n=1

Because ( j f )2 ≡ 1 and ( j f )3 = ( j f ), the switching value ( j f ) is reduced. Equations (2.39), (2.40), (2.43), and (2.46) constitute four instantaneous equation sets for solving An which are expressed in matrix form: ⎤ 1 1 1 1 ⎢ k2 −k 2 k2 −k 2 ⎥ ⎥ S1 = ⎢ ⎣ ek L e jk L e−k L e− jk L ⎦ k 2 ek L −k 2 e jk L k 2 e−k L −k 2 e− jk L

(2.47)

X 1 = {A1 , A2 , A3 , A4 }T

(2.48)

⎡

P 1T

=

2

n=1

an ( j f )e

−kn xo

,

2

n=1

an kn2 e−kn xo ,

2

n=1

an ( j f )e

−kn (L−xo )

,

2

 an kn2 e−kn (L−xo )

n=1

(2.49)

48

2 Basic Theory of WPA

The system of linear matrix equations is denoted as S 1 X 1 = − p0 P 1

(2.50)

The vibration response, wave propagation, and stress analysis of the beam can be performed by solving Eq. (2.50).

2.3 WPA for Analysis of Finite Simple Structures 2.3.1 WPA Expressions of Displacement, Shear Force, and Bending Moment The previous section discusses the reconstruction and general expressions of WPA, while this section will present the application of WPA in simple structures. Figure 2.8 shows the response of a beam with a length L when subject to point harmonic force p˜ 0 at x = x0 : The WPA equation under point harmonic force excitation is presented in Eq. (2.37). When the beam is subjected to a harmonic bending moment at x = x0 , the transverse displacement expressed by WPA is as follows: w(x) =

4

An ekn x + ( jm)Mo

n=1

2

bn e−kn |x M −x|

(2.51)

n=1

wherein ( jm) is the symbolic operator: ∂|x − x M | = ( jm) = ∂x



+1 x ≥ x M −1 x < x M

(2.52)

Equation (2.37) and Eq. (2.51) have the four unknown coefficients An , respectively. The determination of these four unknown coefficients requires four corresponding equations which can be derived from the boundary conditions at x = 0 and x = L. When deriving these coefficients, the bending moment and shear force Fig. 2.8 Finite beam excited by harmonic force

2.3 WPA for Analysis of Finite Simple Structures

49

on both sides of the application point under harmonic force p˜ and harmonic bending moment M˜ are expressed as  4  2



∂ 2 w˜ |x M(x) = E I 2 = E I k 2 An μ2n ekn x + po an μ2n e−kn o −x| ∂x n=1 n=1

(2.53)

 4  ⎧ 2   ⎪ ∂ 2 w˜ + (x) 2 2 kn x 2 −kn |xo −x| ⎪ An μn e + Mo bn μn e ⎨ M+ (x) = E I ∂ x 2 = E I k n=1 n=1  (2.54) 2 4   ⎪ ∂ 2 w˜ − (x) |x 2 2 k x 2 −k −x| ⎪ n n o ⎩ M− (x) = E I ∂ x 2 = E I k An μn e − Mo bn μn e n=1

n=1

 4  ⎧ 2   ⎪ ∂ 3 w˜ + (x) 3 3 kn x 3 −kn |xo −x| ⎪ S (x) = E I = E I k A μ e − p a μ e ⎨ + n n o n n ∂x3 n=1 n=1  2 4   ⎪ ∂ 3 w˜ − (x) |x 3 3 k x 3 −k −x| ⎪ n n o ⎩ S− (x) = E I ∂ x 3 = E I k An μn e + po an μn e n=1

(2.55)

n=1

In the above equations, μn =

kn = { +1, −1, + jν − j|n = 1, 2, 3, 4} k

(2.56)

x0 is the point coordinate of the harmonic force, and may be positive or negative. Due to space limitations, this chapter deduces the WPA expressions of the vibration response of a finite S–S Beam, C–C Beam, and C-F Beam. The reader can deduce other constraint cases according to different boundary, constraint, and coordination conditions. Figure 2.9 shows the vibration response of a simply three-supported beam. At frequencies below 250 Hz, there are three resonance frequencies and two antiresonance frequencies. It is of great significance to try to find a physical explanation of the formation of anti-resonance frequencies from the propagation of elastic waves in the beam and their reflections at the simple supports and boundaries.

2.3.2 S–S Beam Figure 2.10 shows the S–S Beam considered in this section. At both ends of the simply supported beam, the displacement and bending moment listed in Table 2.2 are zero, thus w(0) = w(L) = 0

⎫ ⎬

∂ 2 w(0) ∂ 2 w(L) + = 0⎭ ∂x2 ∂x2

(2.57)

2 Basic Theory of WPA

Displacement response(m)

50 10

-1

10

-2

10

-3

10

-4

10

-5

10

-6

10

-7

10

-8

0

50

100

150

200

250

300

Frequency(Hz) Fig. 2.9 Vibration response of simply three-supported beam

Fig. 2.10 S–S beam

When w(0) = 0, the following can be obtained by Eq. (2.37): 4

A n = − po

n=1

2

an e−kn xo

(2.58)

an kn2 e−kn xo

(2.59)

n=1

When M(0) = 0, 4

n=1

An kn2 = − po

2

n=1

2.3 WPA for Analysis of Finite Simple Structures

51

Similarly, when w(L) = 0, 4

An e

kn L

= − po

n=1

2

an e−kn (L−xo )

(2.60)

an kn2 e−kn (L−xo )

(2.61)

n=1

and when M(L) = 0, 4

An kn2 ekn L = − po

n=1

2

n=1

Equation (2.58) and Eq. (2.61) constitute four instantaneous equation sets for solving An which can be expressed in matrix form:

⎡

1 1 1 2 2 ⎢ k2 −k k ⎢ jk L −k L ⎣ ek L e e k 2 ek L −k 2 e jk L k 2 e−k L

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

2 

−kn xo

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

an e n=1 ⎤⎧ ⎫ 2  A1 ⎪ 1 ⎪ ⎪ an kn2 e−kn xo ⎨ ⎪ ⎬ 2 ⎥ −k A 2 n=1 ⎥ = − po 2  ⎪ ⎪ e− jk L ⎦⎪ ⎪ ⎪ A3 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ an e−kn (L−xo ) ⎪ ⎪ ⎪ 2 − jk L ⎪ ⎪ −k e A4 ⎪ n=1 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪  ⎪ 2 −kn (L−xo ) ⎪ ⎪ ⎪ an k n e ⎩ ⎭ n=1

(2.62) Equation (2.62) is denoted as S1 X 1 = P 1

(2.63)

Substituting boundary conditions into the above equation, the kinematic wave and response can be solved. The vibration response of a simply supported beam is shown in Fig. 2.11:

2.3.3 C–C Beam As shown in Fig. 2.12, the displacement and rotational displacement at both ends of the C–C beam are equal to zero, thus w(0) = w(L) = 0 ∂w(0) = ∂w(L) =0 ∂x ∂x

 (2.64)

Substituting the above boundary conditions into Eq. (2.37), the following equation set can be obtained:

52

2 Basic Theory of WPA

Displacement response (m)

10

-1

10-2 10

fr1

fr2

-3

10

-

4

10-5 10

-

6

far1

10 7 -

10

-

8

0

20

40

60

80

100

120

140

160

Frequency (Hz) Fig. 2.11 The vibration response of a simply supported beam

Fig. 2.12 C–C beam

4

A n = − po

n=1 4

A n k n = − po

A n e k n L = − po

n=1

An kn e

(2.65)

2

an kn e−kn xo

(2-66)

an e−kn (L−xo )

(2-67)

an kn e−kn (L−xo )

(2.68)

n=1

n=1 4

an e−kn xo

n=1

n=1 4

2

2

n=1

kn L

= + po

2

n=1

Equations (2.65)–(2.68) can be expressed in matrix form:

2.3 WPA for Analysis of Finite Simple Structures

53

Displacement response (m)

10 0 fr1

10- 2

f r2

10- 4

10- 6

10- 8

10-10 0

fa r1

20

60

40

80

100

120

140

160

Frequency (Hz) Fig. 2.13 The vibration response of C–C beam

⎧ 2  ⎪ ⎪ ⎪ − an e−kn xo ⎪ ⎪ ⎪ n=1 ⎪ ⎪ 2 ⎪  ⎪ ⎪ an kn e−kn xo ⎨ −

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬

⎤⎧ ⎫ 1 1 1 1 ⎪ ⎪ A1 ⎪ ⎪ ⎢ k ⎥⎨ A2 ⎬ jk −k − jk n=1 ⎢ ⎥ = po 2 ⎣ ek L e jk L e−k L  ⎪ ⎪ e− jk L ⎦⎪ A3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ − an e−kn (L−xo ) ⎪ ⎪ ⎪ kL jk L −k L −jk L ⎪ ⎪ ke jke −ke − jke A4 ⎪ ⎪ n=1 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪  ⎪ −k (L−x ) o ⎪ ⎪ ⎪ an k n e n ⎩ ⎭ ⎡

(2.69)

n=1

The above equation is denoted as S2 X 2 = P 2

(2.70)

Four An coefficients corresponding to x0 can be solved by Eq. (2.70). With the given natural frequency and An values, the deflection of the corresponding point on the beam can be calculated by gradually changing the value of x according to Eq. (2.37), which is then normalized to obtain the modal shape of the beam. The vibration response of the C–C beam is shown in Fig. 2.13. Similarly, Eq. (2.70) can also be used to analyze the frequency response characteristics of the structure.

2.3.4 C-F Beam The C-F beam is fixed at x = 0 and free at x = L, as shown in Fig. 2.14. The displacement and rotational displacement at the fixed end are equal to zero, and the shear force and bending moment at the free end are also equal to zero, thus

54

2 Basic Theory of WPA

Fig. 2.14 C-F beam

∂w(0) =0 ∂x ∂ 3 w(L) 0, ∂ x 3 =



w(0) = 0, ∂ 2 w(L) ∂x2

=

(2.71) 0

Substituting the above boundary conditions into Eq. (2.37), the following equation set can be obtained: 4

2

A n = − po

n=1 4

2

A n k n = − po

an kn e−kn xo

(2.73)

an kn2 e−kn (L−xo )

(2.74)

an kn3 e−kn (L−xo )

(2.75)

n=1

An kn2 ekn L = − po

n=1 4

(2.72)

n=1

n=1 4

an e−kn xo

2

n=1

An kn3 ekn L = + po

n=1

2

n=1

Equations (2.72)–(2.75) can be expressed in matrix form: ⎤ 1 1 1 1 ⎢ k jk −k − jk ⎥ ⎥ S3 = ⎢ ⎣ kek L jke jk L −ke−k L − jke− jk L ⎦ k 3 ek L − jk 3 e jk L −k 3 e−k L jk 3 e− jk L ⎡

 P 3T = − p0

2  n=1

an e−kn xo ,

2  n=1

an kn e−kn xo ,

2  n=1

an kn2 e−kn (L−xo ) ,

2  n=1

(2.76)  −an kn3 e−kn (L−xo ) (2.77)

which can be denoted as S3 X 3 = P 3

(2.78)

2.3 WPA for Analysis of Finite Simple Structures

55

10 -2

fr2

Displacement response (m)

10 -3

fr3

fr1

10 -4 10 -5 10 -6

far1 10 -7

far2 10

-8

0

20

40

60

80

100

120

140

160

Frequency (Hz)

Fig. 2.15 The vibration response of C-F beam

Figure 2.15 shows the vibration response of the C-F beam. Similarly, Eq. (2.78) can also be used to analyze the vibration response curves of the beam.

2.3.5 Comparison Between WPA and Classical Analytical Method C-F beams and S–S beams are the two most typical structural forms of uniform straight beams, and the classical analytic expression of the natural frequencies of C-F beams and S–S beams are [2–5]  ω0 j = δ 2j

  L2 E I ρ S

(2.79)

wherein the subscript j = 1, 2, 3, ν, N represents the first natural frequencies N of the beam. For C-F and S–S beams, the coefficients δ j are, respectively:

C − F Beams : δ j = 1.8751, 4.6941, 1.8548

S − S Beams: δ j = 4.7300, 7.8532, 10.9956 After substituting the physical and geometric parameters of the beam, the natural frequencies calculated by the classical analytical method and WPA are listed in Table 2.3. As can be seen from the table, the calculated results of the two show high

56

2 Basic Theory of WPA

Table 2.3 Comparison between WPA and Classical Analytical Method [1]. Unit: Hz Calculation method

1st order 2nd order 3rd order 4th order 5th order

Classical analytical method C-F beam 13.858 WPA

86.856

243.223

476.631

787.820

S–S beam 38.906

155.624

350.150

621.490

972.640

S–S beam 38.906

155.622

350.150

622.490

972.640

Note The outside and inside diameters of the circular beam are D = 0.18 m and d = 0.16 m,  respectively, and the cantilever length is L = 3.506 m.E = 2.02 × 1011 N m,ρ = 7900kg/m3 ,β = 0.0

agreement. Meanwhile, the response results under different boundary conditions are compared and also show good consistency.

2.4 Traceability and Characteristic Analysis of WPA In recent decades, the precision design of structures reflects the significant value of basic theoretical research to improvement [42]. Designers today than at any time in the past hope to “observe” waves in structures, to analyze their propagation, reflection, attenuation, and mode conversion [2, 4]. As a supplement to the analytical method, WPA is suitable for designers to focus on structural waves and carry out mechanism research.

2.4.1 Traceability of WPA The most commonly used method for calculating the characteristic solution of Eq. (1.1) is the variable separation method, namely, finding a spatial function which can better satisfy the various boundary conditions and constraints. Spatial functions can be represented by series, trigonometric functions or hyperbolic functions, and their combinations: w(x, ˜ t) = {A cosh kx + B sinh kx + C cos kx + D sin kx} · e jωt

(2.80)

Characteristic functions with the above forms are simple and in common use.

2.4.1.1

Modal Analysis Method

Modal analysis is a typical analytical scheme that can be solved in modal coordinates. The characteristic solutions of Eq. (2.2) can be expressed as the linear superposition of the dominant modal shapes of the system, namely

2.4 Traceability and Characteristic Analysis of WPA

w(x, ˜ t) =

∞

57

wr (x) · qr (t)

(2.81)

r =1

wherein wr (x) is the shape function and qr (t) is the corresponding dominant coordinate. The modal shape function is used instead of the spatial function to improve the compatibility of boundary conditions. Mathematically, the exact solution can be approximated by the linear superposition of an infinite number of structural modes.

2.4.1.2

Modal Truncation and Synthesis Method

If the first few dominant modes of a structure are selected, such as N = 7 rather than ∞, Eq. (2.81) further degenerates into an equation of the mode truncation and synthesis methods: w(x, ˜ t) =

N

wr (x) · qr (t)

(2.82)

r =1

The finite number of modes can simplify the expression and shorten the computation time. Therefore, Eq. (2.82) is an approximation of Eq. (2.81), and the calculation accuracy is degraded to some extent, especially when the number of truncated modes is fewer. Of course, as the first few dominant modes of a structure are usually selected, calculation accuracy is guaranteed.

2.4.1.3

WPA

It can be seen from the aforementioned equation derivation that WPA, like other analytical methods, is also derived from the fourth-order differential equation of a Bernoulli–Euler beam, as shown in Fig. 2.16. The fourth-order differential equation can directly express the characteristic solution for a uniform straight beam using the double-exponential function: w(x, ˜ t) =

4

 An e

kn x

· e jωt

(2.83)

n=1

Equation (2.83) is the general equation of the bending vibration of the beam. This equation constitutes the mathematical basis of WPA, and the physical explanations of each component are as mentioned before and will not be repeated. This is not a new expression but a traditional mathematical description. Compared with Eq. (2.80), the spatial exponential function in Eq. (2.83) is not further expanded into the trigonometric function and hyperbolic function.

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2 Basic Theory of WPA

Separation of Variable

Modal Superposition

Double Exponential Function

Modal Truncation

WPA Method (exact solution)

Analytic Method (approximate solution)

Fig. 2.16 Different Analytical processes of typical analytical methods and WPA

In the mid-1980s, the mathematical expression of the double-exponential function attracted the renewed attention of researchers, and this was reflected in the research papers and monographs of Prof. D.J. Mead and Prof. L. Cremer [28, 37]. The authors discussed the uniqueness and prospects of this method in references [2] and [21]. In the linear range of structures, WPA simply sums up the harmonic force response function of an infinite structure and the response function of a finite structure, and reconstructs the general equation of the finite structure. Of course, WPA is also suitable for studying the longitudinal vibration of a rod, although this book does not include the relevant research. The above discussion is confined to the time domain. In recent years, the Fast Fourier Transform (FFT) method, also known as “spectral analysis”, has become very popular. Through the Fourier transform and inverse transform, an equation can be solved in the frequency domain, which means that spectral analysis has the advantages of frequency analysis. The general paths of spectral analysis and WPA are shown in Fig. 2.17. Fig. 2.17 Comparison between WPA and FFT methods (Time-domain analysis and frequency-domain analysis)

w ( x, t ) ⇔ wˆ ( x, ω ) WPA

FFT

2.4 Traceability and Characteristic Analysis of WPA

59

2.4.2 Characteristics of WPA Analytical methods find the general laws of science and technology from simplified and abstracted structures so as to guide complex engineering practices. They can usually provide the optimization function of a system or search for the explicit function that meets the boundary conditions in order to promote the further improvement of engineering [38]. Analytical methods are highly recommended, while WPA is supplementary to them. Such characteristics are described by Western people as cutting a watermelon. That is, no matter how you cut a solid ball, you will always get a round cross section. The explanation from China is more interesting: it is as if you cut off a small slice of Wukong (Monkey King), you think you get a small slice of meat. However, when you open your eyes, you will see another Wukong in the same size!

2.4.2.1

Exponential Function and Differential Equation

The exponential function brings the following changes to the characteristic solution of the differential equation: (1) Greater regularity of equation derivation. The exponential function is easy to write and calculate. In the past, the transformation from exponential function into trigonometric function and hyperbolic function has been regarded as a simplification, and research on sine and cosine functions are more sufficient. However, the largest disadvantage of trigonometric functions and hyperbolic functions is that when you take partial derivatives, the “ontology” is always changing between them, sin kx ⇔ cos kx or sinh kx ⇔ cosh kx, although this kind of transformation also has a rule to follow. However, the “ontology” of the exponential function always remains the same, and the equation set has greater neatness and regularity. (2) Sub-item expression of far-field wave and near-field wave. Focusing on the far- and near-field waves in a structure benefits from the sub-item expression of WPA. Once the solution values are obtained, the changes and correlation of the traveling wave and near-field wave can be studied separately. It is very interesting and valuable to investigate the effects of propagation, reflection, attenuation, and interference cancelation of these kernel parameters on the target parameters. For researchers accustomed to indirect parameters (also known as “apparent parameters”), this adds a new perspective from which to “observe” the problem. (3) The computer completes the “instantaneous value” coincidence of boundaries. In this way, the corresponding boundary, constraint, and coordination conditions are satisfied for solving the boundary problem. In the past, the characteristic function was found to satisfy the boundary condition on the whole domain, and the characteristic function was required to meet the “serial coincidence”. Now, only “matching one by one” must be satisfied, which is obviously

60

2 Basic Theory of WPA

much easier. Although the amount of required computation does increase, the boundary constraints always seem to be satisfied. 2.4.2.2

Characteristics of WPA

A computer can rapidly satisfy the “matching one by one” of boundary conditions. Before computers had simple and easy-to-use complex algebraic programs, WPA expressed by the double-exponential function seemed less than promising. Thus, the exponential function and the computer complement each other. WPA brings the following features.

2.4.2.3

First, WPA is a Method for Describing Structural Waves

Describing structural response by elastic waves is different from describing it by such indirect parameters as acceleration. Wave and target control parameters form the directness, such as floating raft isolation efficiency and structural acoustic radiation. Vibration research deepens from the apparent parameters to the kernel parameters of the structure, which can be used to study the traveling wave and near-field wave in detail. Figure 2.18 shows the variation rules of near-field waves, clarifies their internal relations from a microscopic perspective, and enables designers to interpret Fig. 2.18 Motion of finite beam subjected to harmonic excitation (The dotted line represents the motion ignoring the near-field wave. T is the period)

2.4 Traceability and Characteristic Analysis of WPA

61

engineering phenomena from the deepening level of elastic waves and discuss the dynamic characteristics of structures.

2.4.2.4

Second, WPA Uses “Discontinuity Points” for Element Meshing

Different from the “geometric division” of FEM, WPA uses “discontinuity points” as nodes for element meshing. A “discontinuity point” in a structure is defined as any “obstacle” that causes a discontinuity in the structure, and this is consistent with the concept of “functional damping” in structural noise theory. When using discontinuity points for element division, the main elements are not “finely divided” but often large elements. This treatment not only retains the main characteristics of the complex system, but can also adapt in advance to the refinement requirements in the new development stage of vibration and noise control. “Discontinuity point” and “functional damping” will become the focus of high-level applications.

2.4.2.5

Third, WPA is Suitable for More Extensive Boundary Conditions

Under many boundary conditions, analytical methods cannot obtain mathematical solutions because it is too difficult to obtain mathematical “exact matching” under certain circumstances when the characteristic functions “coincide” with the boundary. Only a finite number of boundary conditions have a good spatial agreement with vibration modes. WPA adopts the linear superposition of the general solution of the infinite system and particular solution of the finite system, and the boundary constraint is consistent with the instantaneous value of the compatible point, allowing the computer to complete the necessary complex matrix operations so as to reduce boundary constraints.

2.4.2.6

Fourth, Formulas Have Regularity in WPA

The exponential function is usually not expanded, the “ontology” of the partial derivative always remains the same and only the coefficient term “switches” regularly with the derivation order; the introduced symbol operator is equivalent to a switching value. Similar to harmonic motion, the solution of the characteristic function has regularity and rules, as well as the corresponding “rhythm”. Although the formulas of WPA appear complicated, they are all matrix sub-items with regular changes. It will be very meaningful for designers to master them.

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2.5 Introduction to the Various “Parameters” in WPA (1) Introduction to the Damping Parameter Structural damping will destroy orthogonality in the modal coordinates. In FEM, it is often assumed that structural materials have proportional damping, although they are inconsistent with reality in some cases, thereby limiting calculation accuracy. In WPA, structural damping can be simply introduced by complex stiffness, and different damping loss factors can be selected for any composite structure. E I j∗ = E I j (1 + jβ j ) ( j = 1, 2, 3, ...)

(2.84)

wherein β j is the damping loss factor of different structures. (2) Embedding of Subsystems such as TMDs External loads such as the coupling of a TMD and structure can be understood as a “discontinuity point” between the mass spring and the structure. The TMD moves harmonically with the structure, as shown in Fig. 2.19, and the applied coupling dynamic reaction is [21] ∗ · w(xd ) pd (ω) = K tot

(2.85)

wherein pd refers to the dynamic reaction on the structure exerted by the TMD and ∗ refers to the complex stiffness of the TMD. K tot For hysteretic damping, K tot =

ω2 m d K d∗ K d∗ −ω2 m d

 (2.86)

K d∗ = (1 + jβ)K d Fig. 2.19 Damping introduction of TMD through complex stiffness

Structure

mass

2.5 Introduction to the Various “Parameters” in WPA

63

For viscous damping, K tot =

ω2 m d (K d + jς ω) K d − ω2 m d + jς ω

(2.87)

wherein m d is the mass of the TMD;K tot is the equivalent stiffness of the TMD;K d∗ is the complex stiffness of the TMD;β is the hysteresis damping loss factor of the TMD;ς is the viscous damping factor; and w(xd ) is the unknown displacement response of the position at which the TMD is installed. From the derivation of the formula, it can be seen that there are no restrictions on the number, application position, or damping type of TMDs. (3) Loading of Various External Loads Various external loads or connections such as various harmonic forces or moments, as well as such additional constraints as vibration isolation mass and elastic supports, can be generalized to the “discontinuity points” in the structure. When they are added into the main structural system as external systems, their positions and numbers are also not limited, as shown in Fig. 2.20. The random function in the MATLAB program is programmed into the structural dynamic analysis program, and the initial phase and phase changes of different excitation forces are assumed, as shown in Fig. 2.21. In this way, the loading of multiple external systems under excitations of different frequencies, phases, and amplitudes can be investigated, and the cancelation mechanism of elastic waves analyzed statistically. Finally, the arithmetic mean of Eq. (2.89) is used as the evaluation parameter to realize the numerical simulation.  4 N 2



A n e kn x + an e−kn |xoi −x| · e jωi t (2.88) poi w(x, ˜ t) = n=1

n=1

i=1 T

w(x, T ) =

1 T

w(x, ˜ t)dt 0

Fig. 2.20 External loading of a complex system (Different excitation sources have different amplitudes and initial phases)

(2.89)

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2 Basic Theory of WPA

Fig. 2.21 Random signal embedded in the structure

Signal Random

Source device

Structure

wherein the superscript “−” represents the average value within the set period. Providing rich descriptions of elastic waves in structures, WPA is the development trend of structure-borne noise theory and has made great progress in active acoustic control and active vibration control. WPA does not require abstruse mathematical skills to complete dynamic analysis, making it suitable for designers to master and use.

2.5.1 WPA and Mechanism Analysis In the last decade, the application range of analytical and numerical methods in engineering has become increasingly vague. On the one hand, this reflects that the analytical method is facing more and more complex system analysis problems. On the other hand, it shows that with the emergence of a large number of basic theoretical achievements and the activity of large commercial computing software, many complex basic theoretical problems have been deeply analyzed. Interface fuzziness and local convergence are inevitable in the development of technology. Each analytical method has its own advantages and applicable scenarios. How can the mechanism analysis of complex systems be freed from the generalization of the numerical method? Designers should not only make full use of commercial software to carry out refined design, but also clearly recognize that “depth determines breadth” when facing scientific and technological innovation, and deepening research into basic theories must be relied upon. Theoretical methods are divided into two broad categories. As shown in Fig. 2.22, WPA is a supplement and extension of the analytical method which provides designers with a new perspective on elastic waves. After mastering WPA, designers can quickly clarify certain inherent mechanism problems without advanced mathematical theories, mathematical deduction skills, or even complex programming [38–40].

2.5 Introduction to the Various “Parameters” in WPA

65

Analytical Method Mechanism Analysis

WPA

Mechanism analysis

Sci. &Tech. Theoretical Methods

Numerical Method Engineering Analysis

Fig. 2.22 WPA: A supplement and extension of analytical method

2.5.1.1

Reductionism and Mechanism Analysis

Technical competition in such fields as ships, aerospace, hydraulic engineering, and bridges is like an invisible hand which promotes the deepening of theoretical methods. Mastering the inherent laws and essential characteristics of the system may lead to new breakthroughs or release the potential of the carriers. The dynamic analysis of complex systems is always a difficult engineering problem. In the process of simplifying complex systems and abstracting mechanism analysis, the general laws of conceptuality, criticality, and commonality should be summarized; for example, how to abstract complex structures into simple structures such as beams, plates, shells, or their combinations, and master the characteristics of these basic structures. Complex systems should be abstracted into simple systems in order to summarize their general rules. When focusing on the structural noise problem, the key to reductionism is to discover the core elements that affect the vibration transmission, energy flow and vibration, and acoustic radiation transformation of complex systems. This is the logic of reductionism between mechanism analysis and complex engineering.

2.5.1.2

Holism and Engineering Analysis

Mechanism analysis is a part of “reductionism”. The concept of “holism” involves summarizing the overall rules of complex systems and grasping the research directions. Mechanism analysis provides a theoretical basis for all research. Unfortunately, not all mathematical calculations can provide analytic functions or explicit optimization directions, and designers must integrate their long-term experience and make reasonable judgments according to the overall engineering so as to turn to the qualitative and quantitative optimization of theoretical and empirical complex systems. According to the concept of “holism”, designers should not only master the basic meaning of mechanism analysis, but also control the integrity and direction-sense of

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engineering design so as to better guide innovation in complex system engineering. The dialectical relationships between reductionism, holism, mechanism research, and complex engineering calculation are shown in Fig. 2.23. In light of this, the analytical method can be regarded as the pioneer of mechanism analysis. FEM and other numerical methods can first be used to carry out the engineering calculation, which is then combined with mechanism analysis to completely analyze complex engineering problems. Taking single-stage and double-stage isolation as examples, the reason why engineers can use the numerical method to carry out complex engineering calculations is that the two systems have completed mechanism analysis and established the explicit optimization function and theoretical framework of vibration isolation design. When considering complex engineering, designers can use this to guide the “correction direction” of engineering design. The diversity of analytical methods and their interweaving with numerical analysis can lead to the improved optimization of engineering design. Although numerical methods can satisfy complex engineering calculations, they are usually individual cases rather than general rules. Of course, after a large number of analysis examples, the numerical method can also summarize certain rules, but it must be based on the premise of a large number of case studies. When faced with the new mechanisms of large complex systems, it is difficult to “search” all the rules in a short time using the numerical method. However, the analytical method can obtain the general rules of the system and even such hidden objective phenomena as the various attenuation and conversion mechanisms of near-field waves. Fig. 2.23 Analytical method and numerical Method (Intertwining and evolution of reductionism and holism)

Complex Engineering Calculatioon (numerical method)

Reductionism

Holism

Mechanism Research (analytical method)

2.6 Shortcomings of WPA

67

2.6 Shortcomings of WPA Like all other analytical methods for dynamics, WPA has inherent shortcomings and deficiencies which can be summarized as follows: • It is not suitable for mechanism analysis of complex structures. Due to its adaption to extensive conditions, particularly the application to mechanism analysis, the WPA method has its unique advantages if used to analyze beam-type structures as it can be used to analyze the propagation, reflection, and attenuation of structural waves. However, for complex structures, such as stiffened plate and plate–beam combination structures, the complexity of the analysis increases exponentially due to poor adaption. • Computation amounts increase rapidly. After a “discontinuity point” is added to a structure or an external load or external system is introduced, four corresponding unknowns such as the propagation analysis of the bending wave in the beam will be added to the matrix term, increasing the order of the linear matrix equation by 4. • It is easily distorted. In the WPA matrix, for the typical S1 matrix in Eq. (2.47), some matrix elements are equal to 1, some are close to 0, and some may be very large. For example, when x = L, the ratio between matrix elements reaches the maximum, which easily leads to the distortion of the linear matrix equation during the solution process. The maximum ratio can be expressed as

max{μ} =

max {S1 }

n∈4×4

min {S1 }

= e2k L

(2.90)

n∈4×4

WPA is still in development. The theoretical mining and physical/mathematical interpretations of this method, as well as the corresponding computation cases, need to be developed and improved in their application. The continuous application and experimental correction of WPA should be carried out under various scenarios, and researchers should improve its deficiencies by identifying its shortcomings and mining its advantages.

2.7 Summary This chapter preliminarily analyzes the challenges faced by the dynamic theory and the origin, equation derivation and reconstruction, basic ideas, and characteristics of WPA. As a theoretical method developed on the basis of traditional structural dynamics, WPA can also be traced back to the fourth-order differential equation, and certain unique properties of this method for mechanism analysis are illustrated in the comparison and application of the modern analytical method. In general, WPA

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is a supplement to the analytical method which provides a new perspective on the structural dynamic analysis of elastic waves. If the velocity, acceleration, or impedance of structural vibration are called “indirect” or “apparent” parameters, the elastic wave of the structure is the “direct” or “kernel” parameter. Using kernel parameters to describe structural vibration raises theoretical research to a higher level and starts an important trend in the development of structural vibration and structural noise theory, both of which are rooted in the deep physical description of elastic waves. The influence of structural kernel parameters on the change rules of systems is of particular importance and interest. For certain more complicated engineering problems, such as structural noise, there are huge challenges. And the existing theoretical method is far from the unified framework of mechanism analysis, and shortcomings remain in theoretical cognition. On the one hand, researchers are required to carry out numerical analysis on complicated engineering. On the other hand, if we go back to the start point and clarify some basic problems, we could better master the direction of giant system abstraction, source identification, and optimized design, The author believes that the finite-line cyclic period structure, the ring cyclic period structure, such as the single frequency and repeat frequency characteristics of propellers, and the forced “modulation” characteristics of the cyclic period structure can help us reveal some frontier doubts. Mastering the structural waves enables us to master the nature of structure noise analysis. Designers are required to recognize and understand the importance of structural waves to refined design and innovation. However, the mathematical and physical equations of structural waves are too abstruse for them to conduct in-depth research. In addition, engineering books usually use indirect parameters as vibration and acceleration, and insert loss, making it difficult to establish the complete system concept. Focusing on the analysis of structural waves, the WPA method is expected to relieve engineers from upset and frustration.

References 1. Rayleigh J (1945) The theory of sound [M]. Dover Publications, New York 2. WU CJ (2002) WPA analysis of structural vibration and its application [D]. Huazhong University of Science and Technology (Chinese) 3. Ewins DJ (1984) Modal testing: theory and practice [M]. Research Studies Press, John Wily, England 4. Pinnington RJ (1991) Approximate mobilities of built up structures [M]. ISVR Technical Report 162 5. Junger MC, Feit D (1993) Sound, structures, and their interaction [M]. Acoustical Society of America, Woodbury 6. Wang WL, DU ZR (1985) Structural vibration and dynamic substructure method [M]. Fudan University Press, Shanghai (Chinese) 7. Cheng GD (1994) Introduction to optimum design of engineering structures [M]. Dalian University of Technology Press, Dalian ((Chinese)) 8. Lu BH, GUC X (1989) Complex mode synthesis approach of damped structures with lumped dampers [J].Chinese J Appl Mech 2:37–44, 127 (Chinese)

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9. He L, Shuai CG (2015) Vibration theory and engineering application [M]. Science Press, Beijing ((Chinese)) 10. Goyder HG, White RG (1980) Vibration power flow from machines into built-up structures, part I: introduction and approximate analysis of beam and plate-like foundation [J]. Journal of sound and vibration 68(1):59–75 11. Goyder HG, White RG (1980) Vibration power flow from machines into built-up structures, part II: wave propagation and power flow in beam-stiffened plates [J]. J Sound Vibration 68(1):77–96 12. Goyder HG, White RG (1980) Vibration power flow from machines into built-up structures, part III: power flow through isolation system [J]. J Sound Vibration 68(1):97–117 13. Wu YS, Ming SZ, Tian C, Sima C et al. (2016) Theory and application of coupled fluid-structure interaction of ships in waves and ocean acoustic environment [J]. J Hydrodyn 28(6):923–936 14. Ming SZ, Wu YS (2018) A three-dimensional Sono-elastic method of ships in finite depth water with experimental validation [J]. J Ocean Eng 164:238–247 15. Wang XC (2003) Finite element method [M]. Tsinghua University Press, Beijing ((Chinese)) 16. Brebbia CA, Walkers (1980) Boundary element techniques in engineering [M]. NewnesButterworth, London 17. Norton MP, Karczub DG (2003) Fundamentals of noise and vibration analysis for engineers (Second Edition) [M]. Cambridge University Press, New York 18. Kojima H, Saito H (1983) Forced vibrations of a beam with a non-linear dynamic vibration absorber [J]. J Sound Vib 88(4):559–563 19. Snowdon JC (1964)Approximate expressions for the mechanical impedance and transmissibility of beams vibrating in their transverse mode. J Acoust Soc Am 36(4):673–680 20. Snowdon JC (1968) Vibration and damped mechanical systems [M]. John Wiley & Sons, New York 21. Mead DJ (1986) A new method of analyzing wave propagation in periodic structures: applications to periodic Timoshenko beams and stiffened plates [J]. J Sound Vib 104(1):9–27 22. Mead DJ (1996) Wave propagation in continuous periodic structures: research contributions from Southampton, 1964–1995 [J]. J Sound Vib 190(3):495–542 23. Wu CJ & White RG (1995) Reduction of vibrational power in periodically supported beams by use of a neutralizer[J]. J Sound Vibration 187(2):329–338 24. Li RF, Wang JJ (1992) The principles of structural analysis program SAP5 and its application [M]. Chongqing University Press, Chongqing (Chinese) 25. Zhu YW, Wei QR, Gub D (1993) Microcomputer finite element pre- and post-processing system ViziCAD and its application [M]. Science and Technology Literature Press, Beijing (Chinese) 26. MSC/NASTRAN (1995) Dynamic Analysis [M]. The Mac Neal-Schwendler Corporation 27. Qian XS, Yu JY, Dair WA (1990) New discipline of science—the study of open complex giant system and its methodology [J]. Chinese J Nature 13(1):3–9 (Chinese) 28. White RG, Walker JG (1982) Noise and vibration [M]. Ellis Horwood Press, England 29. Wu CJ (1992) Vibration reduction characteristics of finite periodic beams with a neutralizer [R]. ISVR Academic Report, England 30. Cremer L, Heckl MA, Ungare E (1988) Structure-borne sound [M]. Springer-Verlag, Berlin 31. Wu CJ, White RG (1995) Vibrational power transmission in a finite multi-supported beam [J]. J Sound Vibration 181(1):99–114 32. Zhu X, Ye WB, Li TY (2013) The elastic critical pressure prediction of submerged cylindrical shell using wave propagation method [J]. Ocean Eng 58:22–26 33. Li TY, Xiong L, Zhu X (2014) The prediction of the elastic critical load of submerged elliptical cylindrical shell based on the vibro-acoustic model [J]. Thin-Walled Struct 84:255–262 34. Chen MX, Zhang C, Deng NQ (2014) Analysis of the low frequency vibration of a submerged cylindrical shell with endplates based on wave propagation approach [J]. J Vib Eng 27(6):842– 851 (Chinese) 35. Zhou HJ, He CC, Li WY (2015) Application of wave propagation method to vibration analysis of rod-and-beam structures with arbitrary boundary conditions [J]. Noise Vibration Control 35(2):32–35 (Chinese)

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36. Wang P, Li TY, Xiang X (2017) Free flexural vibration of a cylindrical shell horizontally immersed in shallow water using the wave propagation approach [J]. Ocean Eng 142:280–291 37. Bahrami A (2017) Free vibration, wave power transmission and reflection in multi-cracked nanorods [J]. Compos Part B: Eng 127:53–62 38. Wu CJ (2017) Scientific consciousness: an important dimension in ship innovation [J]. Chinese J Ship Res 12(4):1–5. https://doi.org/10.3969/j.issn.1673-3185.2017.04.001 39. Mead DJ (1970) Free wave propagation in periodically supported, infinite beams [J]. J Sound Vib 11(2):181–197 40. Wu CJ, Chen ZG (2018) Core value and theoretical logic of structure-borne noise, part 1: summary, value and cognitive subversion [J]. Chinese J Ship Res 13(1):1–6. https://doi.org/10. 3969/j.issn.1673-3185.2018.01.001 41. Wu CJ, Cai DM, Zhu YF (2018) Core value and theoretical logic of structure-borne noise, Part 2: blocking mass and complex huge systems [J]. Chinese J Ship Res 13(4):1–6, 32. https://doi. org/10.19693/j.issn.1673-3185.01287 42. Wu CJ, Lei ZY, Wu YS (2018) Core value and theoretical logic of structure-borne noise, part 3: WPA and mechanism analysis [J]. Chinese J Ship Res 13(5):1–9, 76. https://doi.org/10.19693/ j.issn.1673-3185.01408

Chapter 3

Analysis of Plate Structure Using WPA Method

Progress means that what has been achieved becomes the means of achieving the next goal… —B.O. Kluchevsky

3.1 Introduction When point harmonic force or acoustic wave excitation acts on the surface of a structure, the wave propagates in the structure and the structure generates vibration. However, when structural wave propagation encounters the structural boundary or discontinuous “discontinuity points” or “discontinuity lines”, some structural waves in the structure will be reflected—some will be transmitted along the boundary, and some may be converted into other forms of waves or energy. When the reflected wave and incident wave interact and superpose, standing waves will be formed at certain specific frequencies, and the standing waves are related to the structural vibration modes. From the perspective of structural waves, it is very helpful for us to analyze the structural vibration caused by external excitation as this is conducive to gaining an in-depth understanding. In addition, the establishment of the concept of structural wave propagation is also conducive to the understanding of certain specific problems. In this chapter, the WPA method is used to analyze the vibration of a plate structure under external excitation, mainly considering the bending vibration of the plate (Fig. 3.1). As foundational structures, there is a great deal of research and literature on the bending vibration of plates and shells [1–9].

3.2 Bending Vibration and Wave of Uniform Plate When a plate is subject to a vertical and time-dependent out-of-plane load p˜0 (x, y, t), the vertical displacement w(x, ˜ y, t) satisfies the fourth-order differential equation of the plate, which can be written as [1]. © Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 C. Wu, Wave Propagation Approach for Structural Vibration, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-7237-1_3

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Fig. 3.1 Response of uniform plate structure (Programmed and drawn by Xiong Jishi)

⎫ ˜ y, t) ∂ 2 w(x, ⎪ D∇ w(x, ˜ y, t) + ρh = p˜ o (x, y, t)⎪ ⎪ ⎪ ⎪ ∂t 2 ⎪ ⎪ 4 4 4 ⎬ ∂ ∂ ∂ 4 ∇ = 4 +2 2 2 + 4 ∂x ∂x ∂y ∂y ⎪ ⎪ ⎪ ⎪ 3 ⎪ Eh ⎪ ⎪ ⎭ D= 2 12(1 − ν ) 4

(3.1)

where D—Bending stiffness of plate; E—Young’s modulus; ρ—Density of plate; h—Thickness of plate; v—Poisson’s ratio of materials. In free vibration, Eq. (3.2) becomes the following bending wave equation: D∇ 4 w(x, ˜ y, t) + ρh

˜ y, t) ∂ 2 w(x, =0 ∂t 2

(3.2)

When the motion form of the wave is harmonic, w˜ can be expressed by the following formula: w(x, ˜ y, t) = w(x, y)e jωt

(3.3)

After omitting the time term e jωt , the formula (3.3) can be written as ∇ 4 w˜ + kn4 w˜ = 0

(3.4)

 where kn4 = ω2 ρh D. Equation (3.4) applies to all forms of bending wave vibration, not only to waves along the x-axis. When only the wave propagation along the x-direction is considered, the solution of the equation can be written in the form of wave propagation as

3.2 Bending Vibration and Wave of Uniform Plate

73

w(x) = Aekn x

(3.5)

 Since kn4 = ω2 ρh D, k is used to represent the positive real root and k =   4 ω2 ρh D is the bending wave number, then the four different k n are kn = {k1 , k2 , k3 , k4 } = {k, −k, jk, − jk|n = 1, 2, 3, 4 } In this way, the form containing the k-complete solution is w˜ + = A2 e−kx + A4 e− jkx



w˜ − = A1 ekx + A3 e jkx

(3.6)

where the “+” sign in the subscript indicates the wave propagating in the positive direction along the x-axis, and the “−” sign in the subscript indicates the wave propagating in the negative direction along the x-axis. The wavelength of the bending wave propagating in the x-direction is



D λ = 2π k = 2π ρhω2

1/ 4 (3.7)

The wave velocity a f of the bending wave is obtained by the product of the wavelength and the frequency, so  af = ω

1/ 2

af =

√

D ρh

1/ 4 (3.8)

ωal r

(3.9)

 where r 2 = h 2 12; r—Radius of gyration of plate section; α l —Velocity of quasi-longitudinal wave in plate [3]. al =

E ρ(1 − ν 2 )

1/ 2 (3.10)

It can be seen that the velocity of the bending wave is proportional to the 1/2 power of the frequency, so the propagation velocities of waves with different frequencies are different, and the motion of the bending wave is dispersed. Similar to the bending vibration of the beam in Chap. 2, it can be seen from observation formula (3.6) that the first two terms A1 ekx and A2 e−kx describe the negative and positive near-field waves along the plate x, respectively. They decay exponentially (also called evanescent waves) and do not transmit wave energy. Therefore, these two terms of waves are usually called attenuated waves or near-field waves.

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3 Analysis of Plate Structure Using WPA Method

These waves decay rapidly with the increase of distance from the wave source, and are usually negligible at a sufficiently long distance. The latter two terms A3 e jkx and A4 e− jkx represent bending waves propagating negatively and positively along the x coordinate of the plate, respectively. If there is no damping, the amplitude of these waves does not change with the increase of propagation distance, so these waves can propagate to the far field; thus, they are also called “far-field waves” or “propagation waves”. In addition, it can be seen from the first term or second term of Eq. (3.6) that the wave propagating along the positive direction and that propagating along the negative direction both include the propagation wave and near-field attenuation wave. Because the near-field attenuation wave attenuates rapidly with the increase in distance from the wave source, when the propagation of structural waves in the plate structure is considered, the energy carried by these waves can often be ignored at the position far away from the vibration source or boundary, and only the energy carried by the propagation waves (A3 and A4 ) can be considered. When the vibration energy carried by the bending wave in the plate is analyzed, because bending motion is the main motion, the vibration energy carried by the bending moment and shear force in the section are mainly considered. For a given section of a plate, the energy flow per unit of time can be expressed as the average value of the sum of the power carried by the bending moment (section bending moment M x x angular velocity of the section at the corresponding position) and the power carried by the shear force (shear force in the section S x x corresponding linear velocity). These two wave energies are equal in the free wave. The total energy transmitted by the bending wave with amplitude |A| along the wave propagation direction in a unit width plate can be expressed as



Π f = Dk 3 ω A2 = ω2 ρha f A2 .

(3.11)

3.3 Response of Infinite Plate Under Harmonic Force (Moment) 3.3.1 Response of an Infinite Plate Under Harmonic Force Assuming that an infinite and uniform flat plate is subject to harmonic line force p˜ 0 , the line force acts at the x = 0 position, and the range of action in the y-direction is from −∞ to ∞, as shown in Fig. 3.2a. Similar to the response of an infinite beam under harmonic force, the near-field wave and far-field wave propagate from the excitation along the x-axis into the distance. The external force can be written as

3.3 Response of Infinite Plate Under Harmonic Force (Moment) Fig. 3.2 Stress, response, and symbolic agreement of an infinite uniform plate under harmonic force. a Force excitation; b Moment excitation

p0 e j

-∞ x

-∞

M 0e j

75

t

+∞

0

t

+∞

x 0

p˜ 0 = p0 e jωt

(3.12)

In an infinite uniform plate, the displacement along both the positive and negative directions of the x-axis can be expressed as

w˜ + (x) = A2 e−kx + A4 e− jkx (x ≥ 0) . w˜ − (x) = A1 e+kx + A3 e+ jkx (x < 0)

(3.13)

In Eq. (3.13), the coefficient An (n = 1, 2, 3, 4) can be obtained by continuity and equilibrium conditions at x = 0, which satisfy A1 = A2 = − j A3 = − j A4 = −

p0 4Dk 3

(3.14)

It is worth noting that at x = 0, the amplitudes of the near-field wave and far-field wave are equal, but the phase difference between the near-field wave and excitation force is π, while the phase difference between the far-field wave and excitation force is π /2. With x = 0 as the boundary, the displacement of the plate in different regions is   −kx  P w˜ + (x) = − 4Dk + je− jkx (x ≥ 0) 3 e  +kx  P w˜ − (x) = − 4Dk + je+ jkx (x < 0) 3 e

(3.15)

From the wave solution expression of the plate, it can be seen that for the forward wave when kx = 4.36 (i.e., x = 0.732λ) and exp(−kx) is 1/100; that is to say, the nearfield wave has attenuated to 1/100 of the initial value of x = 0 at a distance exceeding 3/4 of the wavelength. This shows that the influence of the near-field wave can be ignored at a distance exceeding 3λ/4 from the wave source.

76

3 Analysis of Plate Structure Using WPA Method

3.3.2 Response of Infinite Plate Under Harmonic Moment Similarly, the response of an infinite uniform plate subjected to distributed moment M˜ 0 at x = 0 can be derived, as shown in Fig. 3.2b. M˜ 0 = M0 e jωt (Unit width)

(3.16)

The near-field wave and far-field wave propagate outward in two directions from the excitation. The displacements w+ and w− of the plate are expressed as follows: ⎫  M0  − jkx −kx (x ≥ 0) ⎪ ⎬ − e e 4Dk 2  M0  kx ⎭ w− (x) = − e − e jkx (x < 0) ⎪ 4Dk 2

w+ (x) = −

(3.17)

Similarly, at x = 0, the amplitudes of the near-field wave and far-field wave are  (0) the same. At x = 0, the displacement of the plate is 0, but the rotation angle w+ can be expressed as  w+ (0) =

Mx (1 − j) 4Dk

(3.18)

3.4 Wave Propagation in Infinite Plate at the Vertical Incidence of Bending Wave in Discontinuous Interface The structural vibration response of an infinite plate under harmonic force or harmonic torque is analyzed in Sect. 3.3. The propagation of bending waves in infinite and discontinuous plates is analyzed in the section from the perspective of structural waves. Part or all of the waves in a structure will be reflected during propagation when meeting a “discontinuous point” or “discontinuous line” and producing near-field waves on the reflecting interface. Meanwhile, the wave also propagates to the other side of the discontinuous interface to “transmit” the wave energy. It is assumed that the uniform infinite plate is discontinuous where x = 0, as shown in Fig. 3.3. The incident bending wave A˜ i in the region where x < 0 is A˜ i = Ai e j (ωt−kx)

(3.19)

Structural waves reflect and transmit in the structure as the uniform plate is discontinuous where x = 0. The reflected wave can be expressed as A˜ r = Ar e j (ωt+kx)

(3.20)

3.4 Wave Propagation in Infinite Plate … Fig. 3.3 Wave propagation in infinite and discontinuous plates

77

Discontinuous where x

Ai -∞

0

+∞

Ar

At

Anl

Anr

where Ar is the amplitude of the reflected wave. The near-field wave produced from the reflection interface is A˜ nl = Anl e j (ωt+kx)

(3.21)

where Anl is the amplitude of the reflected near-field wave. The transmitted propagation wave and near-field wave in the area where x > 0 are, respectively, A˜ t = At e j (ωt−kx)

(3.22)

A˜ nr = Anr e j (ωt−kx)

(3.23)

where At , Anr are the amplitudes of the transmitted propagation wave and near-field wave, respectively. Therefore, the displacements in the both sides of the plate at x = 0 can be describe as  

− jkx + Anr e−kx · e jωt  (x ≥ 0) w˜ + =w+ e jωt = A te  (3.24) w˜ − =w− e− jωt = Ai e− jkx + Ar e jkx + Anl ekx · e jωt (x < 0) The wave expression of the plate on the discontinuous interface will be used in the following to analyze waves on various support interfaces.

3.4.1 Plate Simply Supported at the Middle The wave characteristics of a bending wave propagating in a plate with incidence in the forward direction are analyzed considering the infinite model of a plate simply supported in the middle, as shown in Fig. 3.4. Based on continuity and equilibrium conditions, it is as follows:

78

3 Analysis of Plate Structure Using WPA Method

Ai

Fig. 3.4 Wave transmission in infinite plate simply supported in the middle

x

0

-∞

w˜ + (0) = w˜ − (0) = 0 M˜ + (0) = M˜ − (0) = 0

+∞

 (3.25)

Four amplitudes of unknown structural waves, namely Ar , Anl , At , and Anr , can be obtained by substituting the expression of the wave into the continuity condition as follows: ⎫ Ai ⎪ ⎪ Ar = − j ⎪ ⎪ 1− j⎪ ⎬ Anl = Anr = − j Ar (3.26) ⎪ ⎪ ⎪ Ai ⎪ ⎪ At = − j ⎭ 1− j

3.4.2 Plate Simply Supported at One End The wave characteristics of bending wave A˜ i propagating in the plate with incidence in the forward direction are analyzed considering a plate simply supported at one end, as shown in Fig. 3.4. Based on equilibrium conditions, it is as follows (Fig. 3.5): w(0) ˜ =0 M˜ + (0) = M˜ − (0) = 0

Ai

 (3.27)

x

0

-∞

Fig. 3.5 Wave transmission in infinite plate simply supported at one end (Where x = 0)

3.4 Wave Propagation in Infinite Plate …

79

Two unknown amplitudes Ar and Anl of the reflected waves can be obtained by substituting Eq. (3.24) into the conditional expression of equilibrium (3.27) as follows:  Ar = −Ai (3.28) Anl = 0

3.4.3 Plate Firmly Supported at One End The wave characteristics of a plate firmly supported at one end can also be solved using similar methods when a bending wave with incidence in the forward direction propagates in the plate when the reflected wave is generated at the end, as shown in Fig. 3.6. The equilibrium conditions at the point where x = 0 are as follows: ⎫ w(0) ˜ = 0⎬ ∂ w(0) ˜ = 0⎭ ∂x

(3.29)

Two unknown amplitudes of the reflected wave can be obtained when the boundary is fixed by substituting Eq. (3.24) into the equilibrium condition as follows: Ar = − j A i

 (3.30)

Anl = −(1 − j)Ai

Ai

x

0

-∞

Fig. 3.6 Wave transmission in infinite plate firmly supported at one end (Where x = 0)

80

3 Analysis of Plate Structure Using WPA Method

Fig. 3.7 Wave transmission of infinite plate free at one end (Where x = 0)

Ai

x

0

-∞

3.4.4 Plate Free at One End A near-field wave and propagation wave are generated after the wave is incident on the boundary in a plate free at one end, as shown in Fig. 3.7. According to the equilibrium condition, it is as follows: ∂ 2 w(0) ˜ ∂x2 ∂ 3 w(0) ˜ ∂x3

=0 =0

 (3.31)

Two unknown amplitudes of the reflected wave can be obtained when the boundary is free by substituting Eq. (3.24) into the equilibrium condition as follows: Ar = − j A i



Anl = (1 − j)Ai

(3.32)

3.5 Wave Propagation When Bending Wave of Infinite Plate Is Incident on Discontinuous Interface It is assumed in Sect. 3.4 that a propagation wave in the far field is vertically incident to the discontinuous interface. More generally, there is an angle between the incident bending wave and the discontinuous interface. If the angle between the propagation direction of the structural wave and the axis is α, the incident bending wave can be expressed as w˜ i = wi e jωt = Ai e j[ωt−(k cos α)x−(k sin α)y]

(3.33)

where k indicates the number of incident bending waves and ω indicates the circular frequency of the bending waves. If Eq. (3.33) is substituted into expression (3.4) with k x = k cos α and k y = k sin α, expression (3.33) can be obtained as follows: 2  2 k x + k 2y = k 4 or k 2y = k 2 − k x2

(3.34)

3.5 Wave Propagation When Bending Wave of Infinite Plate …

81

It is assumed that the bending wave is incident on the fixed boundary where x = 0 and y varies from −∞ to +∞. w˜ r is defined as the far-field wave reflected from the boundary, the reflection angle of which is equal to the incident angle. The propagation direction of the wave component in the y-direction is the same as that of the incident wave. However, the propagation direction of the wave component in the x-direction is opposite to that of the incident wave. Therefore, w˜ r = wr e jωt = Ar e j[ωt+(k cos α)x−(k sin α)y

(3.35)

The near-field wave is also generated beside the far-field wave. The number of waves in the y-direction k y of the near-field wave is as same as that of the far-field wave. However, the value of the attenuation rate is no longer equal to k x , as shown above. The near-field wave can be expressed as w˜ ne = wne e jωt = Ane ekd x+ j (ωt−k y y)

(3.36)

 2 2 kd − k 2y = k 4 can be obtained by substituting Eq. (3.36) into Eq. (3.4). Therefore,  kd = ± k 2y ± k 2 can be expressed as  kd = ±k 1 + sin2 α

(3.37)

As such, the attenuation rate of the near-field wave generated by the incident wave depends on the angle between the incident wave and the boundary. The total wave displacement produced by the incident wave is   w˜ = w˜ i + w˜ r + w˜ ne = Ai e− jkx x + Ar e jkx x + Ane ekd x · e j (ωt−k y y )

(3.38)

w˜ and w˜  are equal to 0 at the fixed boundary where x = 0. The following expressions can be obtained according to the two boundary conditions:    Ane = −Ai cos2 α − j cos α 1 + sin2 α

(3.39)

   Ar = −Ai sin2 α + j cos α 1 + sin2 α

(3.40)

The amplitudes of Ar and Ai in the expression are the same. However, the phase difference between the two depends on α. The phase difference is 90° when = 0, which conforms with the result in Sect. 3.4.3. There is no near-field wave at the simply supported boundary, while Ar = −Ai can be obtained. A reflected wave is generated without consideration of the incident angle α. The total wave movement can be expressed as

82

3 Analysis of Plate Structure Using WPA Method

  w˜ = Ai e− jkx x − e jkx x e j (ωt−k y y ) = −2 j Ai sin(k x x)e j (ωt−k y y)

(3.41)

A standing wave is generated at the x-axis and the nodal line location of the wave is mπ /k x . The wave movement features the form of a sinusoidal standing wave transmitting toward the y-axis with the wave number k y . The movement of the lateral plate of the nodal line can be ignored as the nodal line is parallel to the simply supported boundary, and the nodal line itself can be used as the simply supported boundary. Therefore, the plate discussed here can be considered as moving along the simply supported boundary within scope with the width of b. At that time, the bending wave propagates in the x-direction in the form of a sinusoidal standing wave of sin(k x x) = sin(mπ b) and in the y-direction in  the form of k = ± k 2 − k x2 to infinity. k y = 0 can be obtained when k 2 = k x2 = y  2 2 2 m π b . Therefore, the wavelength in the y-direction is infinite and there is no valid propagation along the plate. The wave with the displacement mode of sin(mπ x/b) cannot be propagated. The critical frequency is  fc =

m2π 2 b2



 ×

D ρh

21

where b—indicates the length; h—indicates the thickness; m—indicates the order number of the order m mode of the plate. The propagation frequency of the critical mode is also called the truncation frequency. The wave will not propagate if it is below the frequency. The frequency is the minimum truncation frequency when m = 1, which is equal to the natural frequency of the plate vibrating in the ground state. If the semi-infinite plate rotates elastically rather than rigidly at the boundary where x = 0, the amplitudes of the reflected wave and near-field wave can be further deduced. If the rotational stiffness per unit width on the boundary is k r , then −2 j Ai cos α Ane =  1 + sin2 α − 2Dk − j cos α kr   Ai 1 + sin2 α − 2Dk/kr + j cos α  Ar = − 1 + sin2 α − 2Dk − j cos α kr

(3.42)

(3.43)

3.6 Forced Vibration of a Rectangular Plate with Both Ends Simply Supported

83

3.6 Forced Vibration of a Rectangular Plate with Both Ends Simply Supported Considering that the two simply supported boundaries are vertical to the x-axis in terms of the rectangular plate with both ends simply supported, it is assumed that the load of a pressure wave acts on its surface as follows: p˜ o (x, y, t) = p0 e j (ωt−k p x )

(3.44)

where p0 indicates the amplitude and k p indicates the wave number of the pressure wave on the surface of the plate. This expression indicates that the resonant pressure wave of the plate propagates along the x-direction with a propagation speed of ap =

ω kp

(3.45)

where p is the sinusoidal component in the y-direction at the point where time t and location x are given. Therefore, p˜ 0 =

∞  r =1

∞

pr

   sin(2r − 1)π y = pr sin kr y y Ly r =1

(3.46)

We will only take the first term into consideration so that the analysis can be simplified, namely   p˜ o (x, y, t) = p1 sin k1y y · e j (ωt−k p x )

(3.47)

The control equation of response w(x, ˜ y, t) of the plate is D∇ 4 w˜ + ρhω2 w˜ = p1 sin(k1y y)e j (ωt−k p x)

(3.48)

The plural form D(1 + jβ) is granted through the bending rigidity of the plate. The internal damping can be considered. β is the loss factor of damping. The solution of Eq. (3.48) consists of two parts: the special solution and the general solution. The special solution is equal to the response of the infinite plate in the xdirection, which represents the propagation of the bending wave in the x-direction, and can be expressed as w p1 =

p1 sin(k1y y)e j (ωt−k p )   2 D k1y + k 2p − ρhω2

(3.49)

84

3 Analysis of Plate Structure Using WPA Method

The length of the plate is finite. The wave will reflect when the structural wave propagates to the boundary of the plate, and the structural wave reflected from all boundaries will generate the far-field wave and near-field wave. These waves can be expressed as   wC F = A1 ekn x + A2 e−kn x + A3 e jkx x + A4 e− jkx x · sin(k1y y)e jωt

(3.50)

where 2 2 , k x2 = k 2 − k1y , k4 = kn2 = k 2 +k1y

ρhω2 D

Therefore, the total displacement at any location can be expressed as w(x, ˜ y, t) =

A1 ekn x + A2 e−kn x + A3 e jkx x + A4 e− jkx x + p1 e− jk p x   · sin(k1y y) · e jωt 2 D k1y + k 2p − ρhω2 (3.51)

The solution includes four unknown numbers which can be obtained when the four boundary conditions are met. The boundary conditions that shall be met at the point where x = 0 and x = L x are ⎫ w(x, y, z) = 0⎬ ∂ 2 w(x, y, z) = 0⎭ ∂x2

(3.52)

The four expressions related to An can be obtained as follows: ⎤⎧ ⎫ A1 ⎪ 1 1 1 1 ⎪ ⎪ ⎨ ⎪ ⎬ ⎥ ⎢ kn −kn jk x − jk x ⎥ A2 ⎢ ⎣ e kn l x e−kn lx e jkx lx e− jkx lx ⎦⎪ A ⎪ ⎪ ⎩ 3⎪ ⎭ kn l x −kn l x jk x l x kn e −kn e jk x e − jk x e− jkx lx A4 ⎧ ⎫ 1 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ p1 − jk p  ×  = − jk l p x ⎪ ⎪ e 2 ⎪ ⎪ D k1y + k 2p − ρhω2 ⎩ ⎭ − jk p e− jk p lx ⎡

(3.53)

It should be noted that a large value of A may be obtained under two different conditions: (1) When the real part of the denominator at the right side is near  to 0, namely 2 2 2 2 Re[D(k1y + k p ) − ρhω ] → 0, the first is approximated as k p → k 2 − k1y = Re(k x ). Therefore, a large response will be generated when the wave number of the excitation pressure field is equal to the natural wave number of the free

3.6 Forced Vibration of a Rectangular Plate with Both Ends Simply Supported

85

bending wave movement. If the propagation speed of the pressure field in the plate is equal to the free wave speed of the corresponding bending wave at a certain frequency, the wave numbers of the two waves will be the same. (2) It occurs when the rank of the determinant at the left side has a minimum value and the frequency is equal to any natural frequency of a finite plate. The peak value of the resonance type of frequency will be obtained from the response Eq. (3.51) under the corresponding conditions. Case (1) is a “consistency condition” in which the propagation speed of the pressure field is consistent with the natural propagation speed of the bending wave. Case (2) is a simple resonance condition. In terms of the k ry of the pressure field, an equation with a form of value equivalent to that of Eq. (3.53) exists. Those values and the given p0 , k p , and ω can be obtained by solving the equations. Therefore, the displacement of any point on the plate can be obtained with the given k p and ω, and written as w(x, ˜ y, t) = Y (x, y, k p , ω) p0 e jωt

(3.54)

where Y (x, y, k p , ω) is the response function of the wave. This is the complex harmonic response of point (x, y) in terms of the given unit amplitude in the y-direction and given frequency and wave number.

3.7 Analytical Solution Example for Vibration of a Plate Using WPA An example is used to verify the accuracy of the WPA method for calculating the response of the plate structure in the section. The selected example is a rectangular plate with its surroundings simply supported of which the thickness is 0.002 m and density is 7,800 kg/m3 . The elastic modulus E is 2.1× 1011 N/m2 , Poisson’s ratio is 0.3, and the geometric dimensions of the plate meet the condition that its length L x = 1 m, its width L y = 1 m, and the harmonic force of the point is acting on point F (0.25 m, 0.25 m). The solution of the mode superposition method for a simply supported rectangular plate under harmonic force at a point is as follows [3]. ∞  ∞ sin mπ x F sin nπ y F sin mπ x sin nπ y  L Ly Lx Ly  x  u(x, y) =  2  2 !2 m=0 n=0 mπ nπ 4 D + Ly − kb Lx

(3.55)

The number of modal truncation terms is selected to meet the condition that m = n = 50 when the response calculation of the modal superposition method is conducted

86

3 Analysis of Plate Structure Using WPA Method

according to Eq. (3.55). It can be known that the calculation results have converged when this group of values is selected through analysis. The analytical solution of the normal displacement of the plate is shown in Fig. 3.8 when the frequency of the excitation force is 150 Hz, and the calculation results using the WPA method are shown in Fig. 3.9. The difference between the value calculated by the WPA method and the value calculated by Eq. (3.55) is shown in Fig. 3.10, and the order of difference is 10−7 . The displacement nephograms of the two calculation methods fit well based on a comparison between Figs. 3.8 and 3.9, and it can be seen from the error diagram in Fig. 3.10 that the solution of the WPA method and the results of mode superposition fit well. x 10

1.0

-6

4

Direction of width/m

3

0.8

2 1

0.6 0 -1

0.4 -2 -3

0.2

-4 -5

0.2

0.4

0.6

0.8

1.0

Direction of length/m Fig. 3.8 Nephogram of real part of analytical solution for displacement response of a rectangular plate under concentrated force at 150 Hz (prepared by Xiong Jishi through programming)

x 10

1.0

-6

4 3

Direction of width/m

0.8

2 1

0.6 0 -1

0.4 -2 -3

0.2 -4 -5

0.2

0.4

0.6

0.8

1.0

Direction of length/m Fig. 3.9 Nephogram of real part of displacement response of a rectangular plate under concentrated force at 150 Hz calculated by WPA method (prepared by Xiong Jishi through programming)

3.8 WPA Method for Solving Structure Power Flow of Plate

87 x 10 25

1.0

-8

Direction of width/m

20

0.8 15

0.6

10

5

0.4

0

0.2 -5

0.2

0.4

0.6

0.8

1.0

Direction of length/m Fig. 3.10 Nephogram of absolute error of real part of displacement response of a rectangular plate at 150Hz

3.8 WPA Method for Solving Structure Power Flow of Plate Based on the WPA method, it is possible to conduct the analysis of the wave propagation of a plate, thereby solving the wave equation and displacement response of the plate under forced vibration. Meanwhile, the solution and description of the energies carried by waves in the structure can also be achieved, as well as the direct reflection of those energies using visualization technology. In this section, WAP analysis is applied to further solve the power flow of structural waves in a plate on the basis of the solved displacement of the plate, and a visual display of it is performed. As for the plate and shell structure, its internal force is unit width force since it is obtained by solving the upper integral of unit stress in the direction of shell thickness. For the bending vibration of the plate, the structural power flow per unit length in the x-direction can be indicated by outer displacement w [10]: Px = D

∂(∇ 2 w) w˙ − ∂x





∂ 2w ∂ 2 w ∂ w˙ ∂ 2 w ∂ w˙ + ν − (1 − ν) ∂x2 ∂ y2 ∂ x ∂ x∂ y ∂ y

(3.56)

The power flow in the y-direction can be obtained by interchanging subscript x and subscript y of the above formula. It is necessary to consider a plate structure which is simply supported at the two short sides and free at the two long sides, as shown in Fig. 3.11; apply F = 1N resonance at the midpoint of plate surface for excitation, calculate the responses of all points of the plate, and then solve the power flow transmitted in the structure according to the power flow calculation model and compare it with the power flow calculation results of Reference [11]. In order to facilitate obtaining the distribution characteristics of the power (energy) in the plate, it is necessary to disperse the plate

88

3 Analysis of Plate Structure Using WPA Method

Fig. 3.11 Diagrammatic figure of rectangular plate

in two directions to obtain its displacements on all coordinate points, then calculate the power flow of each point and draw a power flow vector diagram and motion pattern. The power flow vector diagram and motion pattern of a rectangular plate under excitation frequency of 60 Hz are, respectively, shown in Figs. 3.12 and 3.13. It can be seen from these figures that the vector distribution of the power flow matches well with the results of Reference [11]. The energy input position and distribution as well as the flow characteristics of energies can be clearly recognized in both the vector diagram and motion pattern. The power flow vector diagram and motion pattern under excitation force frequency of 480 Hz are, respectively, shown in Figs. 3.14 and 3.15. It can be seen from these figures that the distribution characteristics of the power and motion characteristics of energy change after changes in harmonic force frequency, and an energy vortex occurs in the plate. No matter how the energy flows, the position of the excitation source and input position of energy can be clearly recognized in the lines of the figures. Therefore, carrying out the structural power flow analysis on the basis of the WPA method makes it possible to clearly analyze the propagation of structural waves in plates and shells, and identify the vibration source of the structure from the perspective of vibration energy.

3.9 Summary In this chapter, the WPA method is used to analyze the vibration response of a structure. Through classic boundary conditions, the WPA solution of plate structure displacement is found to be consistent with classic analysis and is verified to fit

3.9 Summary

89

Fig. 3.12 Power flow vector diagram of a plate under resonance excitation of 60 Hz (Programmed and drawn and by Zhu Xiang)

Fig. 3.13 Power Flow Motion Pattern of a Plate under Resonance Excitation of 60 Hz (Programmed and drawn by Zhu Xiang)

90

3 Analysis of Plate Structure Using WPA Method

Fig. 3.14 Power flow vector diagram of a plate under resonance excitation of 480 Hz (Programmed and drawn by Li Tianyun)

Fig. 3.15 Power flow motion pattern of a plate under resonance excitation of 480 Hz (Programmed and drawn by Li Tianyun)

3.9 Summary

91

the vibration response analysis of a plate structure. Moreover, a power flow study regarding a plate structure is conducted and the vector lines of power flow are drawn. The description of the energy flow in the structure can be used to recognize the vibration source and control path of the structure. Based on the WPA method, we can describe the fluctuation characteristics of the plate and easily express energy distribution. The application of the WPA method is expected to expand the analysis of plate structure under various boundary conditions in many times.

References 1. White RG, Walker JG (1982) Noise and vibration. Halsted Press, Toronto 2. THOMPSON W (1963) Acoustic power radiated by an infinite plate excited by a concentrated moment. J Acoust Soc Am 36:1488–1490 3. Junger M (1972) Sound, structures and their interaction. MIT Press, Cambridge, MA, USA 4. Timoshenko S, Woinowsky KS. Theory of plates and shells, 2nd ed. McGraw-Hill Book Company 5. Feit D (1966) Pressure radiated by a point-excited elastic plate. J Acoust Soc Am 40:1489–1494 6. Xiong J, Wu C, Xu Z et al (2010) Three-dimensional structure based on wave methods-coupling of sound filed study. Comput Eng Appl 46(98):35–37 7. Desmet W, van Hal B, Sas P et al (2002) A computationally efficient prediction technique for the steady-state dynamic analysis of coupled vibro-acoustic systems. Adv Eng Softw 33:527–540 8. Ilkhani MR, Bahrami A, Hosseini-Hashemi SH (2016) Free vibrations of thin rectangular nano-plates using wave propagation approach. Appl Math Model 40(2):1287–1299 9. Sarayi SMMJ, Bahrami A, Bahrami MN (2018) Free vibration and wave power reflection in Mindlin rectangular plates via exact wave propagation approach. Compos B Eng 144:195–205 10. Xiang Z, Tianyun L, Yaozhu Z (2017) Vibration energy flow features and damage identification of crack damage structure. Huazhong University of Science and Technology Press, Wuhan, p 9 11. Gavric L, Pavic G (1993) Finite element method for computation of structural intensity by the normal mode approach. J Sound Vib 164(1):29–43

Chapter 4

WPA for Analyzing Complex Beam Structures

Research on simplifying abstractions and analysis of complex engineering structures seem to be two extremes, and ensuring an effective connection between the two is important for designers to innovate. The simple modeling with WPA has a certain uniqueness, which is an easy theoretical method for designers to master! —Author

Beam structures can be abstracted from many engineering practices. The system needs to be simplified and abstracted in order to obtain its basic characteristics. The wave propagation and wave mode conversion at discontinuous points in the structure are studied, as shown in Fig. 4.1, and then dynamic boundary or coordination conditions, thereby realizing overall control from simplicity to complexity, and from theory to engineering practice. In recent years, research on complex beam-like structures as distributed parameter systems has achieved rapid development. The Wave Propagation Approach (WPA) studies the propagation, reflection, attenuation, and wave mode conversion of elastic waves in structures from the perspective of “elastic waves”, seeking to go deep into the structural waves, establish their relationship with target parameters such as structural vibration and acoustic radiation, and find the influencing factors and rules. This is what the WPA hopes to bring to readers.

4.1 Research History and Methods of Complex Beam Structures Complex beam is the most basic structural form of an analytical method, and many theoretical methods and calculation methods have emerged, including the admittance method, transfer matrix method, characteristic quantity correction method, Lagrange multiplier method, modal analysis method, Green’s function method, matrix analysis method, and so on [1–6]. The emergence of numerous theoretical analysis methods not only reflects the rapid development of dynamics, but also shows the different © Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 C. Wu, Wave Propagation Approach for Structural Vibration, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-7237-1_4

93

94

4 WPA for Analyzing Complex Beam Structures y t

t

0

t

/2

t

/

3 /2

t 2

/

Fig. 4.1 Wave propagation and energy exchange at discontinuous points

approaches of different theoretical methods to analyzing problems, their accuracy, and their adaptability to boundary and constraint conditions. Although extensive and in-depth theoretical researches on beam-like structures have been conducted in a large number of papers, some common difficulties exist that deserve discussion by researchers. First, the analytical method is somehow confined to strict boundary conditions. The analytical method can obtain explicit or accurate mathematical solutions only in a few cases, and only certain simplified solutions or solutions under assumed boundary conditions can be obtained [7, 8]. For example, the lateral bending motion equations of beams under simple support, fixed support, and free conditions can always produce desirable analytical solutions. However, analytical solutions cannot be obtained for more than 40% of boundary conditions. When boundary conditions are complex, either tedious equation derivation is required or there are many unknown problems. For example, if the piecewise assumed vibration mode function is used to analyze multi-supported masts [9], analytical equations need to be established piecewise; when the number of beam bays exceeds 3, the analysis becomes very difficult. Second, it is necessary to find the exact characteristic function. The analytical method strictly depends on the conformity of the characteristic function to the boundary. In complex beam structures, there are always a few cases in which the characteristic function can accurately “mesh” with the boundary conditions. Although great progress has been made [10, 11], it seems that more complex beam structures are waiting to be analyzed. It is very important for designers to master certain kinds of mechanism analysis for the active development of products. Third, an exact analying requires approximate treatment. As most boundary conditions are difficult to “match” in characteristic function, the analysis of distributed parameter systems is generally limited to single or branchless elastic

4.1 Research History and Methods of Complex Beam Structures

95

continuums. Analysis of complex systems often requires approximate treatment. See Pestel and Leckie [2], Butkovskiy [7], and other researchers for the analysis of complex systems. WPA can provide more information for the study of the connection of multi-branching elastomers by single-point, multipoint, and continuous lines. Fourth, damping parameters have several limitations. In the analysis of distributed parameter systems, it is often assumed that their main vibration modes have orthogonal characteristics. When such orthogonal characteristics are described using the characteristic function, the structural damping should be assumed as proportional damping in FEM. However, non-proportional damping will lead to nonorthogonal characteristics of the mass matrix and main vibration modes, and there is no unified conclusion on the degree of influence and verification of analytical results. The WPA [13–17] has shown some unique advantages in analyzing beam-like structures, such as finite quasi-periodic structures. Since exponential function expressions exist all the time, they decouple in the time domain, which makes up for the shortcomings of complex distributed parameter systems which are easy to model but difficult to analyze. This chapter analyzes many complex beam systems including elastic coupled beams, finite arbitrary multi-supported beams, four-supported masts, embedded flexible tubes in pipelines, etc. As two special cases of arbitrary multisupported beam systems, the finite periodic beam and quasi-periodic structure are also analyzed.

4.2 WPA Analysis of Elastic Coupled Beams The vibration analysis of coupled beams is an important part of structure-borne noise research. The typical connection types of coupled beams include linear connection, L-shaped connection, T-shaped connection, multipoint connection, and elastic coupling, as shown in Fig. 4.2. During the propagation of vibration waves in a coupled beam, wave mode conversion will take place, which is an important part of mechanism analysis and can be conveniently analyzed using the WPA. Many studies have been made on the vibration wave propagation of linear connection, L-shaped connection, T-shaped connection, and other connection modes. In this

p% 0

p% 0

p% 0

b

c

Fig. 4.2 Typical coupled beams a L-shaped coupled beam; b T-shaped coupled beam; c Multipoint coupled beam

96

4 WPA for Analyzing Complex Beam Structures

section, WPA-based vibration analysis is carried out with an elastic coupled beam as an example. See the typical system in Fig. 4.3. The system consists of two finite straight beams connected by Spring K at points x K and p˜ 0 (x, t), where Beam ➀ is at point x = x0 . Harmonic force p˜ 0 is acting on the beam, and it is often necessary to know the motion response or power flow at any point on Beam ➁. The model can also be solved using such numerical methods as the Finite Element Method (FEM), as shown in Fig. 4.3b. The purpose of the FEM division is that once the basic structure of the element is established, these simplified structures can be combined and the solutions to complex problems become very simple. This idea is also highly applicable to the WPA. See Fig. 4.3c for the WPA modeling of this model. The “discontinuous points” are used to divide the model into 5 beam sections, and each section is described by the WPA. Obviously, there will be “super elements” much larger than those in the FEM. This division mode focuses on the information at the discontinuous points. Discontinuous points are association nodes, the essence of which is that once a local coupling relation is established, the analysis matrix of the whole model can be established through this connection relation.

Fig. 4.3 Elastic coupled beam a elastic coupled beam; b FEM; c WPA

4.2 WPA Analysis of Elastic Coupled Beams

97

4.2.1 Establishment of WPA Expression For Beam ➀, assume that the lateral displacement of any point on the beam is w˜ 1 (x, t), and it can be expressed as follows: w˜ 1 (x, t) = w1 (x) · e jωt w1 (x) =

4 

An ekn x + Rk

n=1

2 

(4.1)



an e−kn |xk −x| + po

n=1

2 



an e−kn |xo −x|

(4.2)

n=1

  4 ρ1 S1 ω2 kn = E I1

(4.3)

  kn = k  , −k  , jk  , − jk  |n = 1, 2, 3, 4

(4.4)

wherein An —4 unknowns to be solved, n = 1, 2, 3, 4; an —point response function coefficients of an infinite beam, n = 1, 2; S1 —cross-sectional area of Beam ➀; E I1 —flexural rigidity; ρ1 —material density; L 1 —beam length; x0 —coordinates of point of external force action; x K —coordinates of spring joint; k  —number of bending waves in Beam ➀; R K —counteracting force of spring on Beam ➀ (unknown). For Beam ➁, the lateral displacement at any point on the beam is w˜ 2 (y, t), and it can be expressed by the following equations: w˜ 2 (y, t) = w2 (y) · e jωt w2 (y) =

4 

Bn e

kn y

n=1

+

Rk

2 

(4.5) 

an e−kn |yk −y|

(4.6)

n=1

  4 ρ2 S2 ω2 k = E I2

(4.7)

  kn = k  , −k  , jk  , − jk  |n = 1, 2, 3, 4

(4.8)

wherein S2 —cross-sectional area of Beam ➁;

98

4 WPA for Analyzing Complex Beam Structures

E I2 —flexural rigidity; ρ2 —material density; L 2 —beam length; y K —coordinates of spring joint; k  —number of bending waves in Beam ➁; R K —counteracting force of spring on Beam ➁ (unknown); the following equation can be obtained: R K = R K .

(4.9)

It is assumed that when there is no external force p˜ o acting on the system, Spring K is in a free state and its rigidity is k ∞ . The relationship between the dynamic counterforce applied to Beam ➁ displacement under loading conditions is as follows: Rk = −K [w1 (xk ) − w2 (yk )]

(4.10)

wherein spring rigidity is k ∞ . Equation (4.10) is substituted into Eqs. (4.2) and (4.6), and the following equations are obtained: w1 (x) =

4 



An ekn x − K

n=1

2 

2      an e−kn |xk −x| · w1 (xk ) − w2 (yk ) + po an e−kn |xo −x|

n=1

n=1

(4.11) w2 (y) =

4  n=1

Bn e

kn y

+

2 



an e−kn |yk −y| · K [w1 (xk ) − w2 (yk )]

(4.12)

n=1

4.2.2 Boundary Conditions and Consistency Conditions In this system, Beam ➀ is fixed at x = 0 and free at x = L. According to the boundary conditions, Eq. (2.71) is rewritten as follows (see 2.3.4): ∂w1 (0) =0 ∂x ∂ 3 w1 (L 1 ) 0, ∂ x 3 =



w1 (0) = 0, ∂ 2 w1 (L 1 ) ∂x2

=

(4.13) 0

For Beam ➁, both ends are simply supported; the following equation can be obtained from Eq. (2.57) (see 2.3.2):

4.2 WPA Analysis of Elastic Coupled Beams

99

⎫ ⎬

w2 (0) = w2 (L 2 ) = 0

∂ 2 w2 (0) ∂ 2 w2 (L 2 ) = = 0⎭ ∂x2 ∂x2

(4.14)

According to the consistency conditions of point x = xk , the following two equations can be obtained from Eqs. (4.11) and (4.12): w1 (xk ) =

4 

An e

kn xk

− K [w1 (xk ) − w2 (yk )]

2 

n=1

a n + po

n=1

w2 (yk ) =

4 

2 



an e−kn |xo −xk | (4.15)

n=1



Bn ekn yk + K [w1 (xk ) − w2 (yk )]

n=1

2 

an

(4.16)

n=1

Equations (4.15) and (4.16) are summarized as follows: 4 

An e

kn xk

− 1+K

n=1

2 

an w1 (xk ) + K

n=1

2 

an w2 (yk ) = − po

n=1

2 



an e−kn |xo −xk |

n=1

(4.17) 4  n=1



Bn ekn yk + K

2 

an · w1 (xk ) − 1 + K

n=1

2 

an w2 (yk ) = 0

(4.18)

n=1

4.2.3 Vibration Response of Elastic Coupled Beam Equations (4.11) and (4.12) contain a total of 10 unknowns. Ten equations can be established from Eqs. (4.13), (4.14), (4.17), and (4.18). They are expressed in matrix form as follows: SX = P

(4.19)

X = {A1 , ..., A4 , B1 , ..., B4 , w1 (xk ), w2 (yk )}T

(4.20)

wherein

P=

P 1T , 0, ...0,

2  n=1

T an e

−kn |xK −xo |

,0

(4.21)

100

4 WPA for Analyzing Complex Beam Structures

wherein, in Eq. (4.19), P is the system external force matrix; in Eq. (4.21), P 1 is the external force matrix of Beam ➀. See Eq. (2.40). Matrix S is as follows: ⎡

S3 ⎢0 S=⎢ ⎣ S4 0

0 S1 0 S5

S6 S7 S9,9 S10,9

⎤ −S6 −S7 ⎥ ⎥ S9,10 ⎦ S10,10

(4.22)

See S1 and S3 in Eqs. (2.39) and (2.53), respectively. Take the derivative of terms containing w1 (xk ) in Eq. (4.11) and the following equations can be obtained: 2 

K

an e

n=1

∂ K ∂x

2  n=1

   

−kn |x−xk | 

 2  ∂3  −kn |x−xk |  K a e  n  ∂ x 3 n=1

= −K

2 

an e

−kn x k

n=1

S7T

=K

,

2 

an k n e

n=1

=K

=K



an kn e−kn xk

(4.24)



an (kn )2 e−kn (L 1 −xk )

(4.25)

2 



an (kn )3 e−kn (L 1 −xk )

(4.26)

n=1 

an (kn )2 e−kn (L 1 −xk ) , −

n=1

2 



an (kn )3 e−kn (L 1 −xk )

n=1

(4.27) 2  n=1

an e

−kn yk

,

2 

 an kn e−kn (L 2 −yk ) ,

n=1

2 

 an (kn )2 e−kn yk ,

n=1







2 



 an (kn )3 e−kn (L 2 −yk )

n=1





S4T = ek1 xk , ek2 xk , ek3 xk , ek4 xk



      S5T = ek1 yk , ek2 yk , ek3 yk , ek4 yk and

(4.23)

n=1

x=L 1 2 

2 

2 

x=L 1

,



an e−kn xk

n=1

x=0

= −K

−kn x k

2  n=1

x=0

  −kn |x−xk |  an e  

 2  ∂2  −kn |x−xk |  K a e  n  ∂ x 2 n=1

S6T

=K

(4.28) (4.29) (4.30)

4.2 WPA Analysis of Elastic Coupled Beams

 S9,9 = − 1 + k

∞

2 

101

 an , S9,10 = k

n=1

S10,9 = k

∞

2 

∞

an , S10,10 = − 1 + k

an

n=1



n=1

2 

∞

2 

 an

n=1

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ (4.31)

⎪ ⎪ ⎪ ⎪ ⎪ ⎭

Substitute the structural parameters into Eq. (4.19) to obtain the displacement response of the coupled beam structure. It should be pointed out that there are no restrictions on the position of the spring and the acting point of the external exciting force in the derivation of the equation. The damping of beams can be introduced through the complex bending rigidity of Beam ➀ and Beam ➁, respectively, where the complex bending rigidity can be set as E In = E In (1 + jβn ). When n = 1, 2, β1 and β2 are the damping loss factors of Beam ➀ and Beam ➁, respectively.

4.3 Finite Arbitrary Multi-Supported Elastic Beam 4.3.1 Mechanical Model and WPA Expression The uniform elastic straight beam is considered. See Fig. 4.4 for arbitrary support, boundary conditions, constraint conditions, and external load: Harmonic force p˜ 0 acts at x = x0 . The lateral force exerted by supporting constraints on the beam is denoted as R j and the amplitude is unknown. Based on the linear assumption of structural displacement, all displacements caused by external forces p˜ 0 and R j are obtained by linear superposition. For convenience of expression, the boundary conditions are simplified to N simple supporting constraints. The lateral displacement of any point on the beam caused by the harmonic force can be obtained by the following equation:

w( x ) R1

R2

R4 ......

R3

wyt 2(,)

c

Mo

Rn

K1

po

EI , S , , L2 ,

K2

L Fig. 4.4 Simplified mathematical model of an arbitrary multi-supported elastic beam

102

4 WPA for Analyzing Complex Beam Structures

w(x, ˜ t) =

4 

A n e k n x + po

n=1

2 

an e−kn |xo −x| +

n=1

N 

Ri

2 

an e−kn |xi −x|

(4.32)

n=1

i=1

Similarly, the lateral displacement caused by harmonic moment M0 is as follows: w(x, ˜ t) =

4 

An ekn x + ( jm)Mo

n=1

2 

bn e−kn |xo −x| +

n=1

N 

Ri

i=1

2 

bn e−kn |xi −x| (4.33)

n=1

wherein bn is the coefficient of the point moment response function and ( jm) is a symbolic operator.  ( jm) =

−1, x < xo +1, x ≥ xo

(4.34)

In Eqs. (4.32) and (4.33), the first four terms are generated by the two terminal reflected waves in the beam, With 4 An and N R j --middle supports. There are N + 4 unknowns in total, and they are determined simultaneously by structural boundaries, constraints, and continuity conditions.

4.3.2 WPA Superposition Under Multi-Harmonic Force Excitation When the beam is subject to multiple harmonic forces or (and) moments, the left parts of Eqs. (4.32) and (4.33) remain unchanged, and only the parameters under corresponding conditions on the right parts of the equations need to be changed. Taking the above structure as an example, it is assumed that there are two harmonic forces p˜ 01 and p˜ 02 , and external moment M˜ 0 acting on x ∈ {x1 , x2 , x3 } of the beam at the same time; Eqs. (4.32) and (4.33) are combined as follows: w(x, ˜ t) =

4 

A n e kn x +

N 

n=1

j=1

+ po1

2  n=1

Rj

2 

an e−kn |x j −x | + ( jm)Mo

n=1

an e−kn |y2 −x| + po2

N  j=1

2 

Rj

2 

bn e−kn |y3 −x|

n=1

an e−kn |y2 −x|

n=1

(4.35) More generally, the following equation can be derived from the case of M harmonic forces:

4.3 Finite Arbitrary Multi-Supported Elastic Beam

w(x, ˜ t) =

4 

A n e kn x +

N 

n=1

( jm)Mo

N 

Rj

103

Rj

bn e

−kn |y3 −x|

+

M 

n=1

j=1

an e−kn |x j −x | +

n=1

j=1

2 

2 

poi

2 

(4.36) an e

−kn |y2 −x|

n=1

i=1

Similarly, the case of N moments can be deduced as w(x, ˜ t) =

4 

A n e kn x +

n=1

( jm)

n 

Moi

i=1

N  j=1

Rj

N  j=1

2 

Rj

2 

an e−kn |x j −x | +

n=1

bn e−kn |y3 −x| + +

n=1

M  i=1

poi

2 

(4.37) an e−kn |y2 −x|

n=1

wherein N is the number of external moments.

4.4 Dynamic Response and Stress of Four-Supported Mast 4.4.1 Mechanical Model and WPA Expression The analysis process of a finite multi-supported beam is used to study the multisupported mast. The four-supported mast is simplified to a four-supported beam, and the structure of the cantilever part varies greatly. However, due to its position on the upper part of the mast, the stiffness contribution is not large, which mainly affects the quality. See the simplified physical model in Fig. 4.5. It is assumed that all four supporting points are simply supported. After the constraint is released, four unknown supporting counterforces R j ( j = 1, 2, 3, 4) are applied to the mast beam. Under the assumption of structural linearity, Eq. (4.32) is simplified as follows:

w( x ) R1

R2

R4

R3

o

x EI , S , , L2 ,

L Fig. 4.5 Simplified mathematical model of a four-supported elastic beam

0

104

4 WPA for Analyzing Complex Beam Structures

w(x, t) =

4 

A n e k n x + po

n=1

2 

an e−kn |xo −x| +

n=1

4 

Rj

2 

an e−kn |x j −x |

(4.38)

n=1

j=1

It can be seen that the position of supporting point x j is not defined in Eq. (4.38). In addition, from the derivation process of the equation, as long as the boundary conditions and constraints are removed, the influence of these supports on the beam can be equivalent to a dynamic counterforce or dynamic counter moment at the supporting point of the beam, and the dynamic characteristics of the beam can be expressed and solved using the WPA. Equation (4.38) has a total of 8 unknowns which can be determined by the instantaneous value according to the boundary conditions and constraint conditions, and the following equation can be obtained: ⎫ w(x j ) = 0 j = 1, 2, 3, 4 ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎬ ∂ w(0) ∂ w(L) = = 0 ∂x2 ∂x2 ⎪ ⎪ ⎪ ⎪ ∂ 3 w(0) ∂ 3 w(L) ⎪ ⎭ = = 0 3 3 ∂x ∂x

(4.39)

Equation (4.39) contains 4 boundary conditions and 4 consistency conditions, and an 8 × 8 linear equation set can be listed accordingly: SX = Q

(4.40)

X = {A1 , A2 , A3 , A4 , R1 , R2 , R3 , R4 }T

(4.41)

The lateral deflection of any point on the beam can be obtained by substituting the coordinate value xc of the calculated point and the instantaneous value X of Eq. (4.40) into (4.38). In the equation, the bending rigidity is expressed in complex numbers as E In∗ = E In (1 + jβn ), and the influence of beam hysteretic damping on dynamic response can be introduced. In Eq. (4.40), the elements of matrix S are as follows: s1,n = μ2n , s2,n = μ2n · ekn L , s3,n = μ3n , s4,n = μ3n · ekn L s1,(4+ j) =

2 

an μ2n e−kn x j , s2,(4+ j) =

n=1

s3,(4+ j) =

2  n=1

2 

an μ2n e−kn (L−x j )

n=1

an μ3n e−kn x j , s4,(4+ j) =

2 

an μ3n e−kn (L−x j )

n=1

s(4+ j),1 = e−k1 x j , s(4+ j),2 = e−k2 x j , , s(4+ j),3 = e−k3 x j , s(4+ j),4 = e−k4 x j

4.4 Dynamic Response and Stress of Four-Supported Mast

s(4+m),(4+ j) =

2 

105

an e−kn |xm −x j |

n=1

The elements of matrix Q are as follows: q1 = −

2 

an μ2n e−kn xo , q2 = −

n=1

q3 = −

2 

2 

an μ2n e−kn (L−xo )

n=1

an μ3n e−kn xo , q4 = +

n=1

2 

an μ3n e−kn (L−xo )

n=1

q(4+m) = −

2 

an e−kn |xm −xo |

n=1

μn = {1, j − 1, − j}, (n, j, m = 1, 2, 3, 4)

4.4.2 Analysis of Dynamic Stress of Four-Supported Mast In the WPA, the function expression of dynamic stress is very simple. According to a basic knowledge of material mechanics, the stress equation of any point x on the cross section of a uniform beam under harmonic excitation is as follows: σ (x) =

M(x) W

(4.42)

wherein M(x)—bending moment function; W —section modulus of the beam. M(x) = E I

∂ 2 w(x, t) ∂x2

(4.43)

Take the second partial derivative of Eq. (4.32) and substitute the result into Eq. (4.43) [17] ⎫ ⎧ 4 R 2 2 ⎬    E I ⎨ σ (x) = An kn2 ekn x + po an kn2 e−kn (xo −x) + Rj an kn2 e−kn |x j −x | ⎭ W ⎩ n=1 n=1 n=1 j=1 (4.44) Equation (4.44) is the dynamic stress equation of a uniform straight beam expressed by the WPA. It can be seen that since it is a secondary derivative, the

106

4 WPA for Analyzing Complex Beam Structures

Table 4.1 Geometrical and physical parameters of the mast Category

Parameters

Geometrical and physical Parameters of the mast

Outer diameter of the beam D = 0.18m; Inner diameter of the beam d = 0.16m; Cantilever length L = 3.506 m, xc = l  E = 2.02 · 1011 N m, ρ = 7900 kg/m3 β0 = 0.0, p0 = 1000 N x3 = 4.671 m,x4 = 5.971 m L = 9.477 m,xc = L − 0.01 m βo = 0.001(loss factor of beam)

Calculation parameters of the supporting point

function in the bracket in this equation is very similar to that in Eq. (4.36), and only the sum term increases by kn2 . This shows that the WPA seems complicated but is in fact simple and regular. See Table 4.1 for the geometrical and physical parameters of the mast. The vibration characteristics are analyzed as follows: For a four-supported mast, N = 4. Substitute this value along with the relevant parameters in Table 4.1 into Eq. (4.40), the dynamic response of the four-supported mast under the unit external excitation force is calculated as shown in Fig. 4.6, and the resonance frequencies and an anti-resonance point of the first 2 orders of the mast are shown. ⎧ ⎨ 11.6089 Hz f oi = 75.8322 Hz (i = 1, 2, 3) ⎩ 147.5369 Hz The calculated natural frequency is substituted in Eq. (4.36) and the x value of the equation is gradually changed to obtain the modal shape of the corresponding order. The first-order mode frequency is f o1 = 11.6 Hz, and its modal shape is shown in

Displacement(m)

Fig. 4.6 Natural frequency and anti-resonance point of the first 2 orders of the mast

10

-2

10

-4

10

-6

10

-8

10

-10

0

20

40

60

80

100

Frequency/Hz

120

140

160

Fig. 4.7 First-order modal model of the mast a overall view; b partially enlarged view

ND Displacement/m

4.4 Dynamic Response and Stress of Four-Supported Mast

107

Mast support

Length coordinates of beam/m

ND Displacement/m

(a)

Mast Supports

Length coordinates of beam/m

(b)

Fig. 4.7. It can be seen from the partially enlarged view of Fig. 4.7b that the dynamic displacement between the 3rd and 4th bays is much larger than that between other bays, indicating that the arrangement of supporting points makes the structural stress unbalanced and far from the optimal value. The multi-supported mast takes the dynamic response at the sight glass as the target optimization function. Ideally, the cantilever of the beam is in an ideal fixed beam state, which can be regarded as the optimal extreme value of the multi-supported beam. Corresponding to the extreme value, the natural frequency of the clamp-free beam is f max = 13.86 Hz and the first-order natural frequency f 0 = 11.86 Hz is calculated. The difference between the two reflects the ultimate improvable potential under the ideal state of the system, which can be indirectly reflected as follows:  = f max − f 01 ≈ 2.25(Hz) Figure 4.8 shows the stress distribution under the external force of a multisupported clamp-free beam obtained from Eq. (4.41). The stress is greatest at the root of the multi-supported clamp-free beam.

4 WPA for Analyzing Complex Beam Structures

Stress pa

108

Lengthwise direction of beam (m) Fig. 4.8 Normal bending stress along the lengthwise direction of the mast

With other conditions unchanged, it is assumed that the loss factor of the beam is β0 , the outer diameter D and inner diameter d of the mast change synchronously within the ranges of 0.18–0.36 and 0.16–0.32 m, respectively, and the corresponding bending rigidity increases by 16 times; the 3D mesh of frequency f , cross-sectional bending rigidity E I , and displacement response w(x) are obtained, as shown in Fig. 4.9. It can be predicted from Fig. 4.10 that increasing the rigidity of the mast will have little effect on improving the dynamic response of the system. The best way is to adopt TMD, followed by adjusting the position of the supporting point. The accuracy of dynamic analysis is important, but sometimes macro analysis can give us a better direction. The modeling of this case is greatly simplified, but it does not affect the conclusion of the analysis. The WPA is simpler than other analytical methods in the analysis of any multisupported beam, and has better calculation accuracy. In the analysis of the dynamic response and dynamic stress of a structure, the WPA can introduce the different damping loss factors, supporting forms, and constraint conditions of the structure into equations. The case analysis shows that a long and thin structure such as a submarine mast has “sufficient static rigidity but insufficient dynamic rigidity”. Increasing the rigidity of the mast to prevent it from shaking is not the best way to improve the response. WPA analysis has laid a good theoretical foundation for the optimal design of mast structures.

4.5 Periodic and Quasi-Periodic Structures The train rail is a typical simple periodic structure. The seamless rail of a high-speed train is simplified as shown in Fig. 4.11. The ring-stiffened ribs are periodically distributed in aircraft cylindrical shells and submarine pressure shells which demonstrate “weak” periodicity. The energy attenuation due to the reinforcing rib can also

4.5 Periodic and Quasi-Periodic Structures x 10

109

12

2.5

Bending

2

1.5

1

0.5 20

40

60

80

(a)

100

120

140

160

Disp. Response/dB

Frequency/Hz

×1012

Displacement/m Frequency/Hz

(b)

Fig. 4.9 Equipotential line and 3D mesh of corresponding frequency and bending rigidity a equipotential line; b 3D mesh x 10

12

2.5

Bending rigidity/EI

Fig. 4.10 Equipotential line of corresponding frequency and bending rigidity (programmed and drawn by Lei Zhiyang)

2

1.5

1

0.5 20

40

60

80

100

Frequency/Hz

120

140

160

110

4 WPA for Analyzing Complex Beam Structures

Fig. 4.11 Infinite periodic structure

be regarded as mass damping in structural noise, which is collectively referred to as “universal damping” to indicate their attenuation property. An interesting topic is how and to what extent the local non-periodic “perturbation” changes in a system dominated by periodicity affect the main features of the system.

4.5.1 Properties of Periodic Structure Infinite periodic structural analysis adopts an order reduction method, namely, reducing complex problems into simple ones. The difference is that, for the periodic structure, the result is not a differential equation but a difference equation, thus completing the effective analytical analysis. For the periodic structure, there is the following expression [18]: Fn+1 = Fn eμ w˙ n−1 (x) = w˙ n (x) · eμ

(4.45)

wherein μ—periodic constant of the structure; Fn , Fn+1 —force before and after periodic bay of the structure. The complex propagation constant μ can be expressed as follows: cosh μ = 1 − 2

ω2 ωl2

(4.46)

k∞ m

(4.47)

Here, “limiting frequency” is introduced as  ωl = 2π f 1 =

wherein k ∞ is the spring rigidity; when μ is written as real and imaginary parts, the meaning of ωl is clear: μ = a + jb

(4.48)

wherein a is the attenuation constant of the periodic structure and reflects the exponential attenuation of the amplitude, and b is the “phase constant”. So, Eq. (4.46) can be written as follows:

4.5 Periodic and Quasi-Periodic Structures

111

cosh a cos b + j sinh a sin b = 1 − 2

ω2 ωl2

(4.49)

On the right side of the equation is a pure real number, so the imaginary part on the left side must be zero, i.e., the attenuation constant: a=0

(4.50a)

b = nπ n = 0, 1, 2, 3, . . .

(4.50b)

The first condition is only applicable below the limiting frequency, while the second is only applicable above the limiting frequency. In the first region which is called the “transmission area”, the phase changes only between neighboring units, which is the same as the result obtained by substituting Eq. (4.50a) into (4.49) as shown in Fig. 4.12. Conversely, attenuation constant a rapidly increases with the increase of frequency in the attenuation region, and Eq. (4.51b) can be obtained by substituting Eq. (4.50b) into (4.49):   ω 2ω2 b = arccos 1 − 2 = arcsin ωl ωl  2  2ω ω a = arccos h − 1 = 2 arcsin h ωl ωl2

(4.51a)

(4.51b)

The attenuation is very high above the limiting frequency but very low below it, so the infinite periodic structure, like a low-pass filter, shows a “low-pass” property, as shown in Fig. 4.13. Fig. 4.12 Attenuation constant and phase constant of the periodic structure

4

Constant a、b

3.5 3 2.5 2

b a

1.5 1 0.5 0 10-1

100

Frequency ratio f/f1

101

112

4 WPA for Analyzing Complex Beam Structures 30

Constant b/dB

Fig. 4.13 Pass bands and stop bands of periodic flexural vibration

20

10

0

2

3

4

Dimensionless Frequency kl

4.5.2 Properties of Quasi-Periodic Structure In a finite-length quasi-periodic structure, the periodic constant method discussed earlier has failed. The perturbation method is applicable to very small individual adjustments. However, it is also extremely cumbersome to greatly adjust the mathematical analysis. In contrast with the WPA, it is very simple to analyze finite periodic structures. It is only necessary to substitute the supporting point coordinate values x j ( j = 1, 2, 3, . . .) into Eq. (4.32). The periodic beam structure is a special case of arbitrary multi-supported beams. Figure 4.14 shows the modal shapes of a five-equal-bay periodic supporting beam. Figure 4.15 shows the tuning of TMD to the power flow of periodic structures. It can be seen from the figure that the number of bays of the periodic structure and the number of peaks of the resonance frequency of each cluster will eliminate individual modes under special circumstances. Each modal frequency is not independent but coupled to the others. Different from a cyclic symmetric structure, we call this periodic structure a “line period” while the former is called a “ring period". It is relatively easy to understand that the linear periodic structure is constrained by the coupling correlation of the standing wave effect. However, it is not so easy to understand how Fig. 4.14 Modal shapes of first 5 orders of a five-equal-bay periodic supporting beam

Fig. 4.15 Power flow of a five-equal-bay periodic supporting beam

113

Dimensionless Transmission Power

4.5 Periodic and Quasi-Periodic Structures

Without TMD

with TMD

ND Frequency (kl)

Fig. 4.16 Power flow of a 15-equal-bay periodic supporting beam (comparison between periodic and quasi-periodic, and comparison between pass band and stop band)

Dimensionless Transmission Power

the ring period of a cyclic symmetric structure causes the constraint of the same effect. A TMD creates a point of discontinuity for the structure. This point of discontinuity causes new reflections in the wave propagation. All points of discontinuity in the structure redistribute the propagation energy. When the point of discontinuity is strong enough, it will affect all the resonance peaks in the cluster, thus providing the relationship between the point of discontinuity and cluster control. When certain bays of the periodic structure are no longer equidistant, they have a limited influence on periodicity. The more numerous the bays are, the stronger the constraint and the weaker the influence will be, as shown in Fig. 4.16. Like rate gyroscope, the whole pass band is basically unchanged and only a small harmonic burr appears at its edge. It is useful to remember these seemingly trivial conclusions as they allow one to maintain clearer insight during the treatment of complex beam structures. It can be seen in Fig. 4.16 that there is no absolute passand stop bands for the finite-length periodic beam, which is different from the case

Perturbation point

ND Frequency (kl)

114

4 WPA for Analyzing Complex Beam Structures

of an infinite periodic beam. When there is an increasing number of supports, their conductive characteristics become closer to the conditions of infinite-length beams.

4.6 Energy Transmission Loss Due to Flexible Tubes In engineering practice, it is an effective measure for controlling the vibration energy transmission of the “second acoustic channel” by inserting a flexible tube into the pipeline and isolating the propagation of structural waves by the reflection of waves. In general, the longitudinal and transverse flexural rigidity of flexible tubes is much lower than that of steel pipelines. We call such a method for the isolation of vibration transmission by impedance mismatch as “soft measurement”. The model of the coupled system is shown in Fig. 4.17. It is made up of two uniform straight tubes and flexible tubes. They can be simplified as beam segments with different material characteristics, in which the flexural rigidity of flexible tube Beam ➁ is much lower than that of Beams ➀ and ➂, E I2  E I1 . Assume that Beam ➀ is subjected to harmonic force p˜ 0 at point x = x0 ; the response and power flow of the beam section at point x = xm2 on Beam ➂ are analyzed.

4.6.1 Establishment of WPA Expression Beam ➀ can be regarded as the uniform beam; it is assumed that the lateral displacement of any point on the beam is w˜ 1 (x, t): w˜ 1 (x, t) = w1 (x) · e jωt

Fig. 4.17 Coupled system of the flexible tube and elastic beam

(4.52)

4.6 Energy Transmission Loss Due to Flexible Tubes

w1 (x) =

4 

An ek1n x + po

n=1

115 2 

an e−k1n |xo −x|

(4.53)

n=1

(k1n )4 =

ρ1 S1 ω2 E I1

k1n = {k1 , −k1 , jk1 , − jk1 }

(4.54) (4.55)

wherein S1 —cross-sectional area of Beam ➀; E I1 —flexural rigidity; ρ1 —material density; k1n —number of bending waves in Beam ➀; L 1 —length of Beam ➀; x0 —coordinates of point of external force action; An —unknowns. Beam ➂ is made of the same material as pipe section ➀. Beams ➁ and ➂ are also regarded as uniform beams, and the lateral displacement at any point on them is recorded as w˜ 2 and w˜ 3 , respectively. The following equations can be obtained: w2 (x) =

4 

Bn ek2n x

(4.56)

Cn ek3n x

(4.57)

n=1

w3 (x) =

4  n=1

(k2n )4 =

ρ2 S2 ω2 E I2

(4.58)

(k3n )4 =

ρ3 S3 ω2 E I3

(4.59)

k2n = {k2 , −k2 , jk2 , − jk2 }

(4.60)

k3n = {k3 , −k3 , jk3 , − jk3 }

(4.61)

wherein S2 , S3 —cross-sectional areas of Beams ➁ and ➂; E I2 , E I3 —flexural rigidity; ρ2 , ρ3 —material density; L 2 , L 3 —length;

116

4 WPA for Analyzing Complex Beam Structures

xm1 , xm2 —evaluated coordinate points on Beams ➀ and ➂; k2n , k3n —number of bending waves in Beams ➁ and ➂; Bn , Cn —unknown coefficients of Beams ➁ and ➂.

4.6.2 Boundary Conditions and Consistency Conditions In this system, Beam ➀ is the free end at x = x1 . According to the boundary conditions, Eq. (2.48) is rewritten as follows (see 2.3.4): ∂ 2 w1 (x1 ) ∂x2

= 0,

∂ 3 w1 (x1 ) ∂x3

=0

(4.62)

At x = xa w1 (xa ) = w2 (xa )

(4.63)

∂w2 (xa ) ∂w1 (xa ) = ∂x ∂x

(4.64)

E I1

∂ 2 w1 (xa ) ∂ 2 w2 (xa ) = E I2 2 ∂x ∂x2

(4.65)

E I1

∂ 3 w1 (xa ) ∂ 3 w2 (xa ) = E I 2 ∂x3 ∂x3

(4.66)

At x = xb w2 (xb ) = w3 (xb )

(4.67)

∂w3 (xb ) ∂w2 (xb ) = ∂x ∂x

(4.68)

E I2

∂ 2 w2 (xb ) ∂ 2 w3 (xb ) = E I3 2 ∂x ∂x2

(4.69)

E I2

∂ 3 w2 (xb ) ∂ 3 w3 (xb ) = E I 3 ∂x3 ∂x3

(4.70)

At x = x2 ∂ 2 w3 (x2 ) ∂x2

= 0,

∂ 3 w3 (x2 ) ∂x3

=0

(4.71)

There are 12 equations in Eqs. (4.62) to (4.71); the corresponding unknowns An ,Bn ,Cn (n = 1, 2, 3, 4) of Eqs. (4.53), (4.56), and (4.57) can be calculated. The following is assumed:

4.6 Energy Transmission Loss Due to Flexible Tubes

u 21 =

E I2 , E I1

117

u 32 =

E I3 E I2

(4.72)

At x = x1 4 

2 An k1n + po

n=1 4 

2 

2 k1n an e−k1n xo = 0

(4.73)

3 k1n an e−k1n xo = 0

(4.74)

n=1 3 An k1n + po

n=1

2  n=1

At x = xa 4 

An e

k2n xa

−

n=1 4 

Bn e

k2n xa

= − po

4 

n=1

An k1n ek2n xa −

4 

n=1 4 

4 

n=1

(4.75)

an (−k1n )ek2n (xa −xo )

(4.76)

n=1

Bn k2n ek2n xa = − po

n=1

An (k1n )2 ek2n xa − u 21

an ek2n (xa −xo )

4  n=1

4 

Bn (k2n )2 ek2n xa = − po

n=1

4 

an (−k1n )2 ek2n (xa −xo )

n=1

(4.77) 4 

An (k1n )3 ek2n xa − u 21

n=1

4 

Bn (k2n )3 ek2n xa = − po

n=1

4 

an (−k1n )3 ek2n (xa −xo )

n=1

(4.78) At x = xb 4 

Bn ek2n xb −

n=1 4 

4 

2 k2n xa Bn k2n e

4 

n=1

At x = x2

(4.79)

Cn k3n ek3n xb =0

(4.80)

n=1

− u 32

n=1 4 

Cn ek3n xb =0

n=1

Bn k2n ek2n xa −

n=1

4 

4 

2 k2n xb Cn k3n e =0

(4.81)

3 k3n xb Cn k3n e =0

(4.82)

n=1 3 k2n xa Bn k2n e − u 32

4  n=1

118

4 WPA for Analyzing Complex Beam Structures 4 

2 k3n x2 Cn k3n e =0

(4.83)

3 k3n x2 Cn k3n e =0

(4.84)

n=1 4  n=1

Equations (4.63) to (4.84) correspond to a 12 × 12 linear equation set. SX = P

(4.85)

X = {A1 , A2 , A3 , A4 , B1 , B2 , B3 , B4 , C1 , C2 , C3 , C4 }T

(4.86)

wherein

In the derivation of the above equations, there is no restriction on the acting position of the external exciting force. The damping loss of the beams and flexible tubes can be introduced into βn (n = 1, 2, 3), respectively, through the complex flexural rigidity E In∗ = E In (1 + jβn ) of Beams ➀, ➁, and ➂. Here, βn is the loss factor of the structure.

4.6.3 Analysis of Dynamic Characteristics of Pipe Sections with Flexible Tubes In Fig. 4.17, it is assumed that pipe sections ➀ and ➂ are seamless steel pipe while pipe section ➁ is a flexible tube (rubber or fiber composite). For the convenience of calculation, the pipe section parameters are listed in Table 4.2. A typical pipeline system inserted with a rubber hose is shown in Fig. 4.17. The transmission loss of the system at different positions under the exciting force can be obtained by substituting the relevant parameters in Table 4.2 into Eq. (4.85), as shown in Fig. 4.18. It can be seen in the figure that the energy transmission loss basically increases in the frequency band above f ≥ 10 Hz. The first extreme value corresponds to the anti-resonance frequency of the system, I L = 30 dB. However, the resonance frequency of the system is around f ≈ 2, 150 Hz, and the transmission loss is greatly reduced, but because of the higher level of the entire attenuation in the region, no vibration amplification occurs around the point of resonance of the system. It is assumed that the other parameters are unchanged and only elasticity modulus E of the flexible tube is changed; the transmission loss of the system under harmonic force can be obtained as shown in Fig. 4.19. Inserting a flexible tube into the pipeline can effectively attenuate the energy transmission of bending waves along the pipeline, and the frequency band above 10 Hz shows a progressively increasing trend; the

4.6 Energy Transmission Loss Due to Flexible Tubes Table 4.2 Geometrical and physical parameters of pipe sections

119

Category

Parameters

Geometrical parameters

L 1 = 500 mm, L 2 = 300 mm, L 3 = 500 mm x1 = 0.0 mm, xa = L 1 , xb = L 2 + L 2 , x2 = L 3 + L 3 + L 3 xm1 = xa − 100 mm, xm2 = xa + 100 mm D1 = D3 = 75 mm, D2 = 80 mm, d1 = d2 = d3 = 65 mm  ρ1 = ρ3 = 7900 kg m3 ,  ρ2 = 1200 kg m3

Material parameters

β1 = β3 = 0.001, β2 = 0.15  2 Characteristic parameters E 1 = E 3 = 2.02 × 1011 N m ,  E 2 = 6.0 × 106 N m Exciting force parameters p0 = 1.0N , x0 = 0.5L 1

Transmission Loss (dB)

Anti-resonance point of the system

Frequency (Hz) Fig. 4.18 Transmission loss of harmonic force at different acting points

higher the frequency is, the more obvious the attenuation will be. The transmission loss is about 2 dB at 10 Hz and reaches 18 dB at around 1,000 Hz. Further comparative analysis shows that when the elastic modulus increases from E = 6 × 106 to E = 6 × 107 pa, that is, every time the elastic modulus increases by 10 times, the transmission loss moves down 10 dB in the whole frequency band curve above 100 Hz, but the energy leakage risk of the system increases; in particular, “amplification” will occur in the low-frequency stage f ≈ 35 Hz(see Label ➀ in the figure above). When the elastic modulus increases by another order of magnitude to E = 6×108 pa, the overall energy transmission loss of the system tends to deteriorate,

4 WPA for Analyzing Complex Beam Structures

Transmission Loss (dB)

120

Frequency (Hz) Fig. 4.19 Transmission loss of elastic rubber tube at different moduli

Fig. 4.20 Transmission loss of elastic rubber tube during change of outer diameter

Transmission Loss/dB

with an amplification of about 3dB around the specific leakage point, such as the frequency f ≈ 150 Hz. After a flexible tube is inserted into the pipeline system, the resonance frequency decreases greatly. The vibration isolation effect of the second acoustic channel needs to be matched with the floating raft. In particular, it is important to realize that the installation of flexible tubes does not always maintain effective vibration attenuation. When the attenuation increases, the risk of energy leakage also increases at certain frequency points, especially when the flexible tubes are relatively rigid. The model is always simple, and the effective vibration isolation of pipelines sometimes requires the accumulation of experience to avoid “amplification effect”. Changing the outer diameter of the flexible tube without changing the other parameters is equivalent to increasing the flexural rigidity of the flexible tube. The transmission loss curve under external force is shown in Fig. 4.20. The larger the diameter is (i.e., the greater the flexural rigidity), the smaller the amount of attenuation will be as a whole.

Frequency/Hz

Fig. 4.21 Transmission loss of elastic tube at different lengths

121

Transmission Loss (dB)

4.6 Energy Transmission Loss Due to Flexible Tubes

Frequency (Hz)

The results obtained from different flexible tube lengths allow the designer to establish the most intuitive value of transmission loss, as shown in Fig. 4.21, L 2 = {0.3, 0.15, 0.6}. A large amount of analog computation secured by the WPA was compared with the early experimental results and found to be valid within the scope of the project. During analysis, the WPA seems to be more simplified than the traditional analytical method in terms of its treatment of boundary and constraint conditions. The vibration insertion loss I L of the pipeline system is negatively correlated with the elastic modulus of the flexible tube; in other words, the greater the elastic modulus of the flexible tube, the smaller the transmission loss, which is easily understandable. However, it is not appropriate to draw the conclusion that flexible tubes with a low modulus should be selected as far as possible, because modulus selection is not only restricted by vibration isolation effect and pipeline safety. Sometimes, it is more important to avoid the “interval”, after all, the pipeline is the second acoustic channel. After the flexible tube is installed, the resonant frequency of the pipeline system moves to the low frequency as a whole, which may increase the risk of energy leakage. The low elastic modulus of the flexible tube is suitable as a common option at the beginning of the schematic design stage, but the application risk must be resolved by simple debugging. Mechanism analysis reveals basic principles. These principles enable designers to copy and transplant these conclusions from simple to relatively complex systems, and carry out a new round of theoretical and experimental verification. From the abstraction of simple systems, it is not easy to discover the universal laws of concept, criticality, and commonality in complex systems, which requires multiple-factor analysis. And the WPA method provides convenience for mechanism analysis.

4.7 “Double-Stage Vibration Isolation” Device for Pipeline In engineering practice, the vibration isolation of a single flexible tube is often not ideal. For example, the flexible tube bears high internal pressure, resulting in a sharp increase in the rigidity of the tube wall or a decrease in the length of the

122

4 WPA for Analyzing Complex Beam Structures

effective section of the flexible tube, which can in turn decrease vibration isolation efficiency. This problem will become more prominent when a flexible tube is used in combination with a floating raft. What happens if we insert two flexible tubes into a pipeline? The advantage of this method is that the modulus is not necessarily low, and the vibration isolation efficiency of the system is improved through the loss along the path. This section puts forward the new concept of “pipeline double-stage vibration isolation” based on single-stage and double-stage vibration isolation. Pipeline double-stage vibration isolation is not equal to simply adding a onestage flexible tube; it requires systematic design instead. Flexible tubes have both the attenuation mechanism for longitudinal wave transmission and the mechanism for bending waves, or a combination of the two. Here, the attenuation mechanism of bending waves is discussed. Let us assume the following: ⎫ ρ1 c1 = ρ5 c5 ⎬ ρ2 c2 = ρ4 c4 ⎭ ρ3 c3 = ρ1 c1

(4.87)

wherein Ci (i = 1, 2, 3) is the wave velocity of different structural materials. In an elastic pipeline, the complex coupled beam structural model inserted with two flexible tubes and one intermediate tube is shown in Fig. 4.22. Among them, the flexural rigidity of the flexible tubes is much lower than that of pipe sections ➀ and ➂, E I2  E I1 . It is assumed that harmonic force p˜ 0 is acting on pipe section ➀ at point x = x0 ; the response and energy flow attenuated by the flexible tube at any point on pipe section ➂ can be calculated.

Fig. 4.22 “Double-stage vibration isolation” device for pipeline

4.7 “Double-Stage Vibration Isolation” Device for Pipeline

123

4.7.1 Establishment of WPA Expression Pipe section ➀ is regarded as a uniform beam. Assume that the lateral displacement of any point on the beam is as follows: w˜ 1 (x, t) = w1 (x, t) · e jωt w1 (x) =

4 

An ek1n x + po

n=1

2 

(4.88)

an e−k1n |xo −x|

(4.89)

n=1

(k1n )4 =

ρ1 S1 ω2 E I1

(4.90)

k1n = {k1 , −k1 , jk1 , − jk1 }

(4.91)

wherein S1 —cross-sectional area of Beam ➀; E I1 —flexural rigidity; ρ1 —material density; L 1 —length; x0 —coordinates of point of external force action; An —unknowns. The material of pipe section ➄ is the same as that of pipe section ➀, and the material of pipe section ➃ is the same as that of pipe section ➁. Pipe section ➂ is special and is also regarded as a uniform beam with the lateral displacement of any point on it recorded as w˜ 2 (x, t), w˜ 3 (x, t),w˜ 4 (x, t), and w˜ 5 (x, t), respectively. The following equations can be obtained: w˜ i (x, t) = wi (x) · e jωt i = 2, 3, 4, 5 w2 (x) = w4 (x) =

4 ! n=1 4 !

Bn e

k2n x

, w3 (x) =

Dn ek4n x , w5 (x) =

n=1

(k2n )4 (k3n )4 (k4n )4 (k5n )4

4 ! n=1 4 !

Cn e

k3n x

(4.92) ⎫ ⎪ ⎪ ⎬

⎪ ⎭ E n ek5n x ⎪

(4.93)

n=1

⎫ = ρ2 S2 ω2 /E I2 ⎪ ⎪ ⎬ = ρ3 S3 ω2 /E I3 = ρ4 S4 ω2 /E I4 ⎪ ⎪ ⎭ = ρ5 S5 ω2 /E I5

kin = {ki , −ki , jki , − jki } i = 2, 3, 4, 5 n = 1, 2, 3, 4

(4.94)

(4.95)

124

4 WPA for Analyzing Complex Beam Structures

wherein S2 , S3 , S4 , S5 —cross-sectional area of each pipe section; (E I )2 , (E I )3 , (E I )4 , (E I )5 —flexural rigidity of each pipe section; ρ2 , ρ3 , ρ4 , ρ5 —material density of each pipe section; L 2 , L 3 , L 4 , L 5 —length of each pipe section; xm1 and xm2 —coordinates of measuring points on pipe sections ➀ and ➃; k1n , k2n , ...k5n —number of bending waves in each pipe section; Bn , Cn , Dn , E n —unknown coefficients of each pipe section.

4.7.2 Boundary Conditions and Consistency Conditions In this system, Beam ➀ is the free end at x = x1 . According to the boundary conditions, Eq. (2.71) is rewritten as follows (see 2.3.4): ∂ 2 w1 (x1 ) ∂x2

= 0,

∂ 3 w1 (x1 ) ∂x3

=0

(4.96)

At x = xa w1 (xa ) = w2 (xa )

(4.97)

∂w2 (xa ) ∂w1 (xa ) = ∂x ∂x

(4.98)

E I1

∂ 2 w1 (xa ) ∂ 2 w2 (xa ) = E I 2 ∂x2 ∂x2

(4.99)

E I1

∂ 3 w1 (xa ) ∂ 3 w2 (xa ) = E I2 3 ∂x ∂x3

(4.100)

At x = xb

At x = xc

w2 (xb ) = w3 (xb )

(4.101)

∂w3 (xb ) ∂w2 (xb ) = ∂x ∂x

(4.102)

E I2

∂ 2 w2 (xb ) ∂ 2 w3 (xb ) = E I 3 ∂x2 ∂x2

(4.103)

E I2

∂ 3 w2 (xb ) ∂ 3 w3 (xb ) = E I3 3 ∂x ∂x3

(4.104)

4.7 “Double-Stage Vibration Isolation” Device for Pipeline

125

w3 (xc ) = w4 (xc )

(4.105)

∂w3 (xc ) ∂w4 (xc ) = ∂x ∂x

(4.106)

E I3

∂ 2 w3 (xc ) ∂ 2 w4 (xc ) = E I 4 ∂x2 ∂x2

(4.107)

E I3

∂ 3 w3 (xc ) ∂ 3 w4 (xc ) = E I 4 ∂x3 ∂x3

(4.108)

At x = xd w4 (xd ) = w5 (xd )

(4.109)

∂w5 (xd ) ∂w4 (xd ) = ∂x ∂x

(4.110)

E I4

∂ 2 w4 (xd ) ∂ 2 w5 (xd ) = E I 5 ∂x2 ∂x2

(4.111)

E I4

∂ 3 w4 (xd ) ∂ 3 w5 (xd ) = E I5 3 ∂x ∂x3

(4.112)

At x = x2 ∂ 2 w5 (x2 ) ∂x2

= 0,

∂ 3 w5 (x2 ) ∂x3

=0

(4.113)

There is a total of 20 equations in Eqs. (4.96) to (4.113), and the corresponding unknowns An ,Bn ,Cn ,Dn ,E n in Eqs. (4.89) and (4.93) can be solved. Assume the following: u (i+1)i =

E Ii+1 E Ii

i = 1, 2, 3 , 4

(4.114)

At x = x1 , the equation set is 4 

2 An k1n + po

n=1 4 

2 

2 k1n an e−k1n xo = 0

(4.115)

3 k1n an e−k1n xo = 0

(4.116)

n=1 3 An k1n + po

n=1

At x = xa , the equation set is

2  n=1

126

4 WPA for Analyzing Complex Beam Structures 4 

An ek2n xa −

4 

n=1 4 

n=1

An k1n ek2n xa −

4 

n=1 4 

2 k2n xa An k1n e − u 21

4 

an ek2n (xa −x0 )

(4.117)

an (−k1n )ek2n (xa −x0 )

(4.118)

n=1

Bn k2n ek2n xa = − p0

n=1

n=1 4 

Bn ek2n xa = − p0 4  n=1

4 

2 k2n xa Bn k2n e = − p0

n=1

An (k1n )3 ek2n xa − u 21

4 

n=1

4 

2 k2n (xa −x0 ) an k1n e

(4.119)

n=1

Bn (k2n )3 ek2n xa = − p0

n=1

4 

an (−k1n )3 ek2n (xa −x0 )

n=1

(4.120) At x = xb , the equation set is 4 

Bn e

k2n xb

−

n=1 4 

Bn k2n ek2n xb −

4 

(4.121)

Cn k3n ek3n xb = 0

(4.122)

n=1

2 k2n xb Bn k2n e − u 32

n=1 4 

Cn ek3n xb = 0

n=1

n=1 4 

4 

4 

2 k2n xb Cn k3n e =0

(4.123)

3 k3n xb Cn k3n e =0

(4.124)

n=1 3 k2n xb Bn k2n e − u 32

n=1

4  n=1

At x = xc , the equation set is 4 

Cn e

k3n xc

−

n=1 4  n=1 4 

Cn k3n ek2n xc −

4 

n=1

(4.125)

Dn k4n ek4n xc = 0

(4.126)

n=1 4  n=1

2 k3n xc Cn k3n e − u 43

n=1 4 

Dn ek4n xc = 0

4 

2 k4n xc Dn k4n e =0

(4.127)

3 k4n xc Dn k4n e =0

(4.128)

n=1 3 k3n xc Cn k3n e − u 43

4  n=1

4.7 “Double-Stage Vibration Isolation” Device for Pipeline

127

At x = xd , the equation set is 4 

Dn e

k4n xd

−

n=1 4 

Dn k4n ek4n xd −

4 

(4.129)

E n k5n ek5n xd = 0

(4.130)

n=1

2 k4n xd Dn k4n e − u 54

n=1 4 

E n ek5n xd = 0

n=1

n=1 4 

4 

4 

2 k5n xd E n k5n e =0

(4.131)

3 k5n xd E n k5n e =0

(4.132)

n=1 3 k4n xd Dn k4n e − u 54

n=1

4  n=1

At x = x5 , the equation set is 4 

2 k5n x5 E n k5n e =0

(4.133)

3 k5n x5 E n k5n e =0

(4.134)

n=1 4  n=1

Equations (4.115) to (4.134) correspond to a 20 × 20 linear equation set. SX = Q

(4.135)

wherein A = {A1 , A2 , A3 , A4 }T B = {B1 , B2 , B3 , B4 }T C = {C1 , C2 , C3 , C4 }T D = {D1 , D2 , D3 , D4 }T E = {E 1 , E 2 , E 3 , E 4 }T X = {A, B, C, D, E}T

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(4.136)

(4.137)

In the derivation of the above equations, there is no restriction on the acting position of the external exciting force. The damping loss of the beams and flexible tubes can be, respectively, introduced into βn (n = 1, 2, 3, 4, 5) through the complex flexural rigidity E In∗ = E In (1+ jβn ) of Beams ➀–➄. Here,βn is the loss factor of the structural material. In this way, the response and vibration transmission properties of the beams can be obtained.

128

4 WPA for Analyzing Complex Beam Structures

4.8 Summary Complex beam structures are widely used in engineering. They are all typical distribution parameter systems in dynamics which are common in engineering practice. In this chapter, the elastic coupled beam, finite arbitrary multi-supported beam, four-supported mast system, periodic and quasi-periodic structures, flexible tubes embedded in pipelines, and “double-stage vibration isolation device for pipelines” are analyzed with WPA. For example, if a submarine mast is abstracted into a “slender beam structure”, the analysis shows that the submarine mast structure has “sufficient static rigidity but insufficient dynamic rigidity”. Improving the dynamic response of the mast and preventing the “fuzzy image” caused by the vibration of the mast under high-speed navigation by increasing the rigidity of the mast and optimizing the position of the supporting points and structure form have little effect. TMD is the best scheme to improve the dynamic rigidity of the mast, with good tuning effect and low-quality ratio. This is similar to the TMD effect of woodpecker brain structure. The transmission loss of the flexible tube is negatively correlated with its elastic modulus; in other words, the greater the elastic modulus of the flexible tube is, the smaller the transmission loss will be. However, the conclusion of theoretical analysis should not be simply extrapolated to general engineering applications. Modulus selection is not only restricted by vibration isolation effect and pipeline safety. Sometimes, it is more important to avoid the “low-frequency region”. The key to the second acoustic channel is matching. After the flexible tube is installed, the resonant frequency of the pipeline system moves to the low frequency as a whole, which may increase the risk of energy leakage, and the application risk must be resolved by debugging. The corresponding linear loss is safer. The periodic structure has the characteristics of “Pass Band” and “Stop Band”. The comparative analysis of the finite quasi-periodic structure and the infinite periodic structure is very meaningful, which helps us to understand the coupling in dynamics. Will the alteration of a single bay destroy the overall dynamic performance of the system? A 15-equal-bay beam has shown a “stubborn” periodicity. When the geometric parameters or material characteristics of a single bay are changed, the periodic characteristics of the system remain basically unchanged, and only the “side lobes” in the frequency response curve are changed. When a relatively large system has only small “perturbation” changes, the overall periodic characteristics of the system change little. Finally, the characteristics of using WPA to analyze complex beam structures are summarized as follows: (1) When WPA is used to study complex beam structures, the derivation of the equations and programming is very regular; the mathematical modeling is easy to complete and the physical concept is clear. (2) The WPA has certain unique advantages in the analysis of finite beam structures. This characteristic can be extended to the analysis of beam-like structures, and

4.8 Summary

129

the dynamic analysis can be carried out directly for many complex engineering problems.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Glandwell GML, Bishop RED (1960) Interior receptances of beams. J Mech Eng Sci 2(1):1–15 Pestel EC, Leckie FA (1963) Matrix methods in elasto-mechanics. McGraw-Hill, New York Lin YK (1969) Dynamics of beam-type periodic structures. J Eng Ind 91(4):1133–1141 Jacquot RG, Soedel W (1970) Vibration of elastic surface systems carrying dynamic elements. J Acoust Soc Am 47(5B):1354–1358 Dowell EH (1971) Free vibrations of a linear structure with arbitrary support conditions. J Appl Mech 38(3):595–600 Hallquist J, Sneyder VW (1973) Linear damped vibratory structures with arbitrary support conditions. ASME J Appl Mech 40:312–313 Butkoviskiy AG (1983) Structural theory of distributed system. Halsted Press, Wiley, New York Yang B (1992) Eigenvalue inclusion principles for distributed gyroscopic systems. ASME DE 37:7–12; J Appl Mech 59(3):650–656 Yang B (1992) Transfer function of constrained/combined one-dimensional continuous dynamic systems. J Sound Vib 156(3):425–443 Nicholson JM, Bergman LA (1986) Free vibration of combined dynamical systems. ASME J Appl Mech 112:1–13 Bergman LA, McFARLAND DM (1988) On the vibration of a point-supported linear distributed system. ASME J Vibr, Acoust, Stress Reliab Des 110(4):485–492 Wu CJ, White RG (1995a) Vibrational power transmission in a multi-supported beam. J Sound Vib 181(1):99–114 Wu CJ, White RG (1995b) Reduction of vibrational power in periodically supported beams by use of a neutralizer. J Sound Vib 187(2):329–338 Li TY, Zhang XM (1995) The vibration wave and power flow in a periodical simply supported curved beam. J Huazhong Univ Sci Technol 23(9):112–115 Wu CJ, Yang SZ, Luo DP, Zhu YF (1998) The dynamic response and stress analysis of multi supported elastic beam by use of WPA method. J Huazhong Univ Sci Technol 27(1):69–71 Wu CJ (2002) WPA and its application in structural vibration. Huazhong University of Science and Technology, Wuhan Mead DJ (1973) A general theory of harmonic wave propagation in linear periodic system with multiple coupling. J Sound Vib 27(3):253–260 Mead DJ, Yaman Y (1990) The harmonic response of uniform beams on multiple linear supports: a flexural wave analysis. J Sound Vib 141(3):465–484

Chapter 5

WPA for Analyzing Hybrid Dynamic Systems

A country without advanced science and technology will be easily defeated while a country without national spirit will be defeated without a fight. —Yang Shuzi

5.1 Hybrid Power Systems The system that is comprised of continuous elastomer and discrete lumped mass is called a hybrid dynamic system in dynamics. In hybrid dynamic system engineering, for instance, the bridge and fuel tank for military vehicles, heavy-duty artillery on ships, heavy-duty machine guns on aircrafts, etc., can be considered the lumped mass on elastic structures. The Tuned Mass Damper (TMD) is organized around the aircraft cabin. The main structure can also be abstracted as a hybrid dynamic system. Two simple hybrid dynamic systems are shown in Fig. 5.1. Figure 5.1a is an elastic coupling beam system. The transmission and coupling changes of the vibration power flow in the system are studied and some basic mechanical problems are discussed by simplifying the abstract model. Figure 5.1b shows a multi-support elastic beam system in which the impact of lumped mass or TMD on the system is examined from engineering applications. From the simple system to the hybrid dynamic system, the partial differential equation and ordinary differential equation must be solved simultaneously. There are a few literature pieces concerning TMD installation in the early elastic system, therefore, the theoretical analysis has experienced mathematical challenges. Until 1952, Young [1] obtained the solution of the clamp-free beams with TMD but the installation point should be limited to the end of the clamp-free beams. In 1964, Neubert [2] gave the equation of straight bar plus TMD while Snowdon [3] reviewed the TMD on the end or middle point of the uniform straight beam and challenged the multiple supports (N ≥ 3) of the uniform straight beam at a later stage. However, due to the complexity of mathematical analysis, it is difficult to obtain anlytial answer in practice. Jones [4] obtained the approximate solution of the finite straight-beam © Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 C. Wu, Wave Propagation Approach for Structural Vibration, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-7237-1_5

131

132

5 WPA for Analyzing Hybrid Dynamic Systems

Fig. 5.1 A diagram of the hybrid dynamic system

q2 y

x

L

W2

K

W1 q1

(a) Coupling beam structure

W(x)

~ po(x,t)

md *

kd

x

(b) Hybrid dynamic system

hybrid dynamic system by the presumed mode function method. Jacquot [5] used the mode truncation and mode synthesis method to expand the analytical method to the general elastic structure installation TMD. Wu Chongjian and White [6–8] extended the WPA method to the hybrid dynamic system and found that the advantages of the WPA method were particularly apparent when the number of beam supports was large, from 1992 to 1993. The WPA method was applied to analyze the dynamic characteristics of submarine mast [9], and later TMD vibration suppression was reviewed from 1993 to 1996. Wu Chongjian and Fu Tongxian [10] assumed the WPA method to study the mechanism of a floating raft, embedded multiple disturbance sources with amplitude, and the initial phase in the modeling. They also analyzed the cancelation mechanism of the structural wave in 1994. Wu Chongjian [11] in 1998 and Chen Zhigang [12] in 2005 proposed the “mass effect” and “tuning effect” of a floating raft and continued to use the WPA method in their research.

5.2 The Continuous Elastic Beam System with Lumped Mass 5.2.1 The Mechanical Model and Derivation of the WPA Formula The installation of any multi-support beam with lumped mass is a typical hybrid system. Its simplified mechanical model is shown in Fig. 5.2. It is expected that the

5.2 The Continuous Elastic Beam System with Lumped Mass

133

Fig. 5.2 The simplified mathematical model of any multi-support elastic mast with lumped mass

time correlation of displacement response is in the form of exp(jωt). This will be absent in the following expression. In Chap. 4, we infer the displacement response of the beam when there is no lumped mass on the beam and any point where x = x0 is subjected to harmonic force p˜ 0 . Here, the copy Eq. (4.32) is as follows and is recorded as w˜ b (x, t) w˜ b (x, t) =

4  n=1

An e

kn x

+ p0

2 

an e

−kn |x0 −x|

+

n=1

N  j=1

Rj

2 

an e−kn |x j −x |

(5.1)

n=1

See Chap. 2 for an . If the flexural rigidity in the wave number kn is expressed as the complex number E I ∗ = E I (1 + jβ0 ), the internal damping β0 of the beam can be introduced. The first term of Eq. (5.1) is the displacement of the reflected wave at two ends of the beam, four An are unknowns; the second term is the displacement of the external excitation force; the third term is produced by the support reaction R j ; and N support reactions are also unknowns. Then, consider the influence of the installation of lumped mass at the end of the beam on the system. When the constraint is released, the lumped mass is considered a reaction force Fm (unknown internal force) on the beam. Under the hypothesis of linear structure, according to the superposition principle of the bending wave, the transverse displacement of any multi-supported beam when it is installed with lumped mass is as follows: w(x, ˜ t) = w˜ b (x, t) + Fm

2 

an e−kn |xm −x|

(5.2)

n=1

When the beam is in harmonic motion, the internal force Fm exerted by the lumped mass is the response function of the harmonic displacement of the beam: Fm (ω) = m d ω2 w(xm ) where xm is the coordinate of the lumped mass.

(5.3)

134

5 WPA for Analyzing Hybrid Dynamic Systems

Substituting Eq. (5.3) into Eq. (5.2), an important equation is acquired as w(x, ˜ t) = w˜ b (x, t) + w(xm )

2 

an m d ω2 e−kn |xm −x|

(5.4)

n=1

The last item on the right of Eq. (5.4) is the dynamic reaction force exerted on the beam by the lumped mass; w(xm ) is used to characterize the dynamic displacement response (unknown) of any point with x = xm on the beam. To simplify the following calculation, the last item on the right of the partial derivative Eq. (5.4) of Eq. (5.4) is listed, which is the dynamic reaction force exerted on the beam by the lumped mass; w(xm ) is used to express any point with x = xm on the beam: 2  ∂ 2 w(x, ˜ t) ∂ 2 w˜ b (x, t) = + w(x ) an m d ω2 kn2 e−kn |xm −x| d ∂x2 ∂x2 n=1

(5.5)

2  ∂ 3 w(x, ˜ t) ∂ 3 w˜ b (x, t) = − w(x ) an m d ω2 kn3 ( jm)e−kn |xm −x| d ∂x3 ∂x3 n=1

(5.6)

where ( jm) is the sign operator:  ( jm) =

+1, x ≥ xm −1, x < xm

(5.7)

There is no restriction on the central mass coordinate xm in the derivation of the equation, which means it can be anywhere on the beam. There are N + 4 unknowns [1] in Eq. (5.1) and then there are N + 5 unknowns in Eq. (5.4), which can be explained by simultaneous equations of boundary conditions, constraint conditions, and continuous conditions. A simple case is considered here. When N supported beams are all simply supported beams, the above three conditions are equivalent to ⎫ ˜ t) ˜ , t) ∂ 2 w(0, ∂ 2 w(L ⎪ ⎪ = = 0⎪ ⎪ 2 2 ⎪ ∂x ∂x ⎪ ⎪ ⎪ 3 2 ⎪ ˜ t) ˜ , t) ∂ w(0, ∂ w(L ⎪ ⎪ ⎬ = = 0 3 3 ∂x ∂x w(x j ) = 0 ( j = 1, 2, 3, . . . , N ) ⎪ ⎪ ⎪ ⎪  2  ⎪ ⎪ ⎪  ⎪ 2 ⎪ an m d ω − 1 = 0 ⎪ w˜ b (xm ) + w(xm ) · ⎪ ⎭

(5.8)

n=1

Equation (5.8) corresponds to a system of linear equations with (N +4) unknowns

5.2 The Continuous Elastic Beam System with Lumped Mass

135

SX = Q



X = [A1 , . . . , A4 , R1 , R2 , . . . , R N , w(xm )]T

(5.9)

By substituting the instantaneous solution X of Eq. (5.9) into Eq. (5.4), the displacement admittance of any point on the beam can be acquired, that is, the displacement response under the unit dynamic load. The specific derivation process is abbreviated. Here, the matrix S and elements of Q are listed as follows: s1,n = μ2n , s2,n = μ2n ekn L , s3,n = μ3n , s4,n = μ3n ekn L s1,(4+ j) =

2 

an μ2n e−kn x j , s1,(5+N ) =

n=1

s2,(4+ j) =

2 

an μ2n e−kn (L−x j ) , s2,(5+N ) =

2 

2 

an μ2n m c ω2 e−kn (L−xm )

n=1 2 

an μ3n e−kn x j , s3,(5+N ) =

n=1

s4,(4+ j) =

an μ2n m c ω2 e−kn xm

n=1

n=1

s3,(4+ j) =

2 

2 

an μ3n m c ω2 e−kn xm

n=1

an μ3n e−kn (L−x j ) , s4,(5+N ) =

n=1

2 

an μ3n m c ω2 e−kn (L−xm )

n=1

s(4+ j),1 = ek1 x j , s(4+ j),2 = ek2 x j , s(4+ j),3 = ek3 x j , s(4+ j),4 = ek4 x j s(5+N ),1 = ek1 xm , s(5+N ),2 = ek2 xm , s(5+N ),3 = ek3 xm , s(5+N ),4 = ek4 xm s(4+m),(4+ j) =

2 

an e

−kn |xm −x j |

, s(5+N ),(4+ j) =

n=1

2 

an e−kn |xm −x j |

n=1

s(5+N ),(5+N ) =

2 

an m c ω 2 − 1

n=1

q1 = −

2 

an μ2n e−kn x0 , q2 = −

n=1

q3 = −

2  n=1

q4+m = −

2  n=1

2 

an μ2n e−kn (L−x0 )

n=1

an μ3n e−kn x0 , q4 = +

2 

an μ3n e−kn (L−x0 )

n=1

an e−kn |x j −x0 | , q5+N = −

2  n=1

an e−kn |xm −x0 |

136

5 WPA for Analyzing Hybrid Dynamic Systems

μn = {1, j, −1, − j}, (n = 1, 2, 3, 4), ( j = 1, 2, 3, . . . , N ), (m = 1, 2, 3, . . . , N )

5.2.2 The Dynamic Characteristics of the Multi-support Mast with a Heavy End The mast with 4 lumped-mass supports is a simplification of engineering practice. To simplify the abstract model, designers hope to investigate some regular conclusions of engineering. See Table 5.1 for the calculation parameters. The results of placing the input parameters in Table 5.1 into the MATLAB calculation program are shown in Fig. 5.3. The dotted line in the figure shows the displacement admittance at the end of the mast when no lumped mass is added to the 4 supports while the solid line is the displacement admittance after the lumped mass (m d = 25 kg) is added. When the lumped mass xm = L is installed, the peak value of the first natural frequency of the system is efficiently suppressed. Compared with that without lumped mass, the end displacement amplitude is reduced to 1.012% of the original value. Table 5.1 The input parameters of mast with 4 lumped mass supports Material performance parameters E = 2.02 × 1011 N/m2 , ρ = 7900 kg/m3 Mast geometry parameters

Mast external diameter D = 180 mm, mast internal diameter d = 160 mm, L = 9477 mm, β = 0.005

Supporting coordinate parameters x 1 = 0.2 m, x 2 = 3.726 m, x 3 = 4.67 m, x 4 = 5.971 m md = 25 kg xm = L (standard value)

External force parameters

po = 1N, x0 = L − 1.5 (m), xc = L (coordinates of the response points)

Fig. 5.3 The displacement response of the mast end with corresponding frequency and different coordinates

Displacement admittance (dB)

Lumped mass parameters

Lumped m mass Coordinates(m) Frequency (Hz)

5.2 The Continuous Elastic Beam System with Lumped Mass Table 5.2 The response ratio of the mast end corresponding to different coordinates unit: m

137

No.

xm

max(w)

max(w) max(w0 ) × 100%

1

5.9710

185.3684 × 10−5

100.00

2

6.2897

36.1248 × 10−5

25.8232

3

6.6085

14.5605 × 10−5

7.8549

4

6.9272

7.7702 ×

10−5

4.1918

5

7.2459

4.8996 × 10−5

2.6432

6

7.5646

3.4709 × 10−5

1.8724

7

7.8834

2.6894 ×

10−5

1.4508

8

8.2021

2.2425 × 10−5

1.2097

9

8.5208

1.9887 × 10−5

1.0728

10

8.8395

1.8610 ×

10−5

1.0039

11

9.1583

1.8292 × 10−5

0.9868

12

9.4770

1.8870 × 10−5

1.0180

Fig. 5.4 The displacement response of the mast end

Displacement admittance (dB)

When the lumped mass changes in the range of x m = 5.971–9.477 mm, the maximum displacement response of the mast end is acquired. The calculated value is listed in Table 5.2. In Table 5.2, w and w0 signify the displacement response of the mast end when the lumped mass is installed and not installed, respectively. They are functions of frequency f . It is evident from the table that the installation position of the lumped mass has a significant effect on the dynamic response of the mast. As shown in Fig. 5.3, the closer the lumped mass is to the supporting point, the poorer the vibration suppression effect is. Thereafter, the influence of the lumped mass on the dynamic response of the multi-supported beam is analyzed. With other conditions unchanged, assume that the lumped mass changes in the range of md = 0–10 kg. The calculated displacement response w(x, ˜ t) is shown in Fig. 5.4. Figure 5.5 is a three-dimensional diagram displaying the dynamic response of the mast under different lumped masses and frequencies.

Without mass

with mass

Frequency (H Hz)

Fig. 5.5 The displacement response of the mast end with corresponding frequency and different lumped masses

5 WPA for Analyzing Hybrid Dynamic Systems Displacement admittance (dB)

138

Lum mped masss /kg Frequancy (H Hz)

From the above analysis, it is evident that the WPA method can be assumed to analyze the hybrid dynamic system with multi-support mast and the beam with lumped mass, which is not only appropriate for any support and boundary conditions but also takes into account the damping of the beam and the lumped mass at any point. Compared with the mode synthesis method and the assumed mode function method, it indicates the convenience of mathematical treatment. (1) After the boundary constraint and additional mass are released, the lumped mass can be equivalent to a dynamic reaction/moment of the connection point on the beam. Therefore, the WPA method can be assumed to express and solve it, which is the distinctive advantage of this method. (2) The results demonstrate that the lumped mass has a significant effect on the dynamic features of the multi-support mast periscope. When it is located at xm = L, the displacement response corresponding to the first resonance frequency of the multi-support mast will be reduced by more than 90%, which is an effective engineering control method.

5.3 The Analysis of the Dynamic Characteristics of Multi-supported Beams with Dynamic Vibration Absorbers In this section, the WPA method will be assumed to analyze the dynamic features of TMD on multi-support beams, which display different dynamic characteristics from the two-degree-of-freedom system with lumped mass and TMD.

5.3 The Analysis of the Dynamic Characteristics …

139

5.3.1 The General Equation of WPA The dynamic coupling association between TMD and the multi-support elastic beam can be divided into a dynamic force, as shown in Fig. 5.6; the force Fd between TMD and the structure is exerted by the mass-spring system on the beam. With the passive harmonic motion of the beam, the dynamic reaction (internal force) forced by TMD on the beam is recorded as Fd [3] and its damping is introduced by the complex stiffness K d∗ of TMD Fd (ω) = K tot · w(xd )

(5.10)

For hysteresis damping: ⎫ ω2 m d K d∗ ⎪ ⎬ K tot = ∗ K d − ω2 m d ⎪ K ∗ = (1 + jβ)K ⎭ d

(5.11)

d

For viscous elastic damping loss factor of TMD: K tot =

ω2 m d (K d + jς ω) K d − ω2 m d + jς ω

(5.12)

where K tot K d∗ β ς w(xd )

Equivalent stiffness of TMD; Complex stiffness of TMD; Hysteresis damping loss factor of TMD; Viscous damping; Dynamic displacement response of the structure where TMD is installed, unknown

Fig. 5.6 TMD equivalent to a dynamic reaction force on the structure

Structur

Mass

140

5 WPA for Analyzing Hybrid Dynamic Systems

When the multi-support beam is subject to point harmonic force p˜0 , simple support N reaction forces R j (undetermined unknowns), and internal force Fd imposed on the beam by TMD coupling vibration, as shown in Fig. 5.1, the response function of transverse displacement of any point (0 ≤ x ≤ L) on the beam with time changes is [1, 3] w(x, ˜ t) = w(x) · e jωt w(x, ˜ t) =

4 

An ekn x + Fd

n=1

+

N  j=1

2 

an e−kn |xd −x| + F0

n=1

Rj

2 

(5.13) 4 

an e−kn |x0 −x|

n=1

an e−kn |x j −x |

(5.14)

n=1

In Eq. (5.14), the first item is the bending wave produced by the reflection wave at both ends of the beam; the second item corresponds to TMD, as shown in Eq. (5.10); the third item is associated with the external excitation force; and the last item is contributed by the reaction force of simply supported beams. There are N + 5 unknowns, 4An s, NRj s, and one w(xd ) in total, which are determined by the boundary, constraint, and continuous conditions of the structure. It is not challenging to see that there is no limit in the general Eq. (5.14) for the number of supports of beams, the position of external forces, and the installation coordinates of TMD, which is better than the method [4] introduced in the relevant literature. Comparably, the internal resistance of the beams can be introduced through the complex bending stiffness, which will not be discussed in detail.

5.3.2 Dynamic Flow Expression According to the basic concept of dynamic flow, the vibration energy that the point harmonic force operating on the structure input into the structure is [7] PS =

1 | p0 |2 Re{ϑ0 } 2

(5.15)

where ϑ0 is the origin admittance at the harmonic force on the structure. Suppose the amplitude of the excitation force is p0 = 1, and the real number in Eq. (5.15) is as follows [6, 7]:   Re{ϑ0 } = Re − jωw∗ (x0 ) = −Im ωw∗ (x0 ) where the superscript * is the conjugate of the complex number.

(5.16)

5.3 The Analysis of the Dynamic Characteristics …

141

When we place x = x0 into Eq. (5.14), we recognize that the displacement of the excitation point is w(x0 ) =

4 

An e

kn x 0

+

n=1

2 

an + Fd

n=1

2 

an e

−kn |xd −x0 |

+

n=1

N −1  j=1

Rj

2 

an e−kn |x j −x0 |

n=1

(5.17) The dynamic transmitted power by the shear force and bending moment at any point on the beam (0 ≤ x ≤ L) is shown in formulas (9.16) and (9.17), respectively, which can be written as Pu (x) =

1  1  Re S(− jω) · w∗ = Re S ∗ · jωw 2 2

(5.18)

Pm (x) =

1  1  Re M(− jω) · θ ∗ = Re M ∗ · jωθ 2 2

(5.19)

where w = A1 E 1 + A2 E 2 + A3 E 3 + A4 E 4 + a1 (E 5 + j E 6 ) + Fd a1 (E 7 + j E 8 ) + L 1  M = E I k 3 A1 E 1 − A2 E 2 + A3 E 3 − A4 E 4 + a1 (E 5 − j E 6 ) + Fd a1 (E 7 − j E 8 ) + L 2  θ = k A1 E 1 + j A2 E 2 − A3 E 3 − j A4 E 4 − a1 ( j f )(E 5 − E 6 ) − Fd a1 ( jd)(E 7 − E 8 ) − L 3  S = E I k 3 A1 E 1 − j A2 E 2 − A3 E 3 + j A4 E 4 − a1 ( j f )(E 5 + E 6 ) + Fd a1 ( jd)(E 7 + E 8 ) − L 4

Pay attention to the differences, between the above formulas and Eqs. (9.18)– (9.21) in Chap. 9. Furthermore, L 1 , L 2 , L 3 , and L 4 are, respectively: L1 =

N −1 

Ri a1 (E 1i + j E 2i )

i=1

L2 =

N −1 

Ri a1 (E 1i − j E 2i )

i=1

L3 =

N −1 

Ri a1 ( jr )(E 1i − E 2i )

i=1

L4 =

N −1 

Ri a1 ( jr )(E 1i + E 2i )

i=1

E 1 = ekx , E 2 = e jkx , E 3 = e−kx E 4 = e− jkx , E 5 = e−k|x0 −x| , E 6 = e− jk|x0 −x| E 7 = e−k|xd −x| , E 8 = e− jk|xd −x| , E 1i = e−k|xi −x| , E 2i = e− jk|xi −x|

142

5 WPA for Analyzing Hybrid Dynamic Systems

Here, ( jd), ( j f ), and ( jr ) are symbolic operators  ( jd) =  (jf) =  ( jr ) =

−1, x ≤ xd +1, x > xd

(5.20)

−1, x ≤ x0 +1, x > x0

(5.21)

−1, x ≤ xj +1, x > xj

(5.22)

By using the formulae (5.18) and (5.19), the vibration power flow transmitted along the beam can be acquired. The vibration power flow is the sum of the components of shear force power and bending moment power, that is Pa (x) = Pu (x) + Pm (x)

(5.23)

5.3.3 Calculation and Discussion

Analyze and discuss the three-equal-bay beams (suppose N = 3 and l = L N , as shown in Fig. 5.1b). After the installation of the TMD, the first 3 modal vibration modes of the beam in the simply supported state are shown in Fig. 5.7. Suppose that the point harmonic force is at x = x0 and the coordinate of TMD is x = xd , the power flow of the input structure and the transmitted power flow are shown in Fig. 5.8. Analyze the power flow of the point harmonic force input structure and the power flow transmitted along the beam. In the examination and calculation of Figs. 5.8, 5.9, 5.12, and 5.13, the dimensionless frequency kl and hysteresis damping are assumed. The mass ratio μ and kl are, respectively: Fig. 5.7 The displacement and the modal diagram of the first 3 simply supported beams of 3 equal spans

Modal 1

Modal 2

Modal 3

143

Transmitted power

5.3 The Analysis of the Dynamic Characteristics …

Dimensionless frequency /kl

Transmitted power

Fig. 5.8 The vibration power input to a 3 equal-span simply supported beams (μ = 0.2, β = 0.005, β d = 0.25, x 0 = x d = 1.5 l, Ω = 1.15)

Dimensionless frequency /kl

Fig. 5.9 Power flow in the 3 equal-span simply supported beams (μ = 0.2, β = 0.005, β d = 0.25, x 0 = x d = 1.5l, z = 2.5l, Ω = 1.15)



kl = ωρbhl E I

μ = md M

(5.24)

where M = ρbh L is the total mass of the beam. The tuning frequency ratio is described as Ω= where ωd

the natural frequency of TMD;

ωd ωm

(5.25)

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5 WPA for Analyzing Hybrid Dynamic Systems

ωm the natural frequency of the modal of the beam; Ω Tuning frequency ratio When m = 1, ω1 corresponds to the first resonance frequency of the system. The optimization objective is to reduce the maximum resonance peak corresponding to tuning in the transmission band. An efficient method is to use nonlinear mathematical programming to find the minimum and maximum values for the power flow defined by Eq. (5.23), which is written as min max P a ( Ω|β, f ) f ∈B Ω∈(0,2)

(5.26)

where B the frequency band cluster corresponding to the equal span; β the damping loss factor of TMD;

f Nonlinear digital planning frequency range

Power flow/dB

The TMD is located at the central point of the second span of the three-equal-bay beams and is tuned to the first resonance frequency of the corresponding frequency band cluster of the first span. The input structure and the vibration power flow transmitted along with the structure are shown in Figs. 5.8 and 5.9, respectively, in which the full line is the input structure and the vibration power flow is transmitted along the structure when there is no TMD, and the imaginary line suggests the power flow after the installation of TMD. Figures 5.10 and 5.11 are the three-dimensional distribution diagram and contour map of the impact of altering the vibration absorption frequency on power flow, respectively. It is evident from Figs. 5.8, 5.9, 5.10, 5.11, 5.12, and 5.13 that multiple resonance

Vibration absorption Frequency/Hz

Fig. 5.10 The impact of changing the tuning frequency on power flow (the mass ratio of vibration absorption is 5%) (programmed and drawn by Lei Zhiyang)

5.3 The Analysis of the Dynamic Characteristics …

145

150 140

• • • • • H z•

Vibration absorption frequency

130 120 110 100 90 80 70 60 50 40

20

40

60

80

100

120

140

• • • Hz•

Frequency (Hz)

Transmitted power

Fig. 5.11 The contour map of changing the tuning frequency to power flow (programmed and drawn by Lei Zhiyang) (μ = 0.2, β = 0.005, β d = 0.25, x 0 = x d = 1.5l, z = 2.5l, Ω = 1.15)

Dimensionless frequency /kl

Fig. 5.12 The comparison of the input power flow of 3 equal-span simply supported beams (the mass ratios are μ = 0.2, β = 0.005, β d = 0.40, x 0 = x d = 1.5l, z = 2.5l, Ω = 1.15)

peaks are tuned by one TMD and the inhibitory effect is significant. This “one tuning multiple” phenomenon is very interesting; it is a universal law of equal-span periodic beams and is caused by the standing wave effect. In the 5-span beam shown in Fig. 5.13, the frequency band cluster corresponding to the first span has 5 resonance peaks, all of which are suppressed at once. The examination of the numerical example shows that in generally finite equal-span beams when TMD is tuned to the first natural frequency of any resonance peak cluster, all resonance peaks of the corresponding frequency band cluster are suppressed.

5 WPA for Analyzing Hybrid Dynamic Systems

Transmitted Power

146

ND Frequency (kl)

Fig. 5.13 5 equal-span simply supported beams plus TMD power flow (the mass ratios are μ = 0.2, β = 0.005, β d = 0.40, x 0 = x d = 1.5l, z = 2.5l, Ω = 1.15)

As a result of the standing wave effect of the structure wave, the periodic features that the finite periodic structure displays originate from the strong coupling correlation, and the resonance frequency is forced to modulate. Therefore, resonance peaks appear in clusters and are tuned once. In each resonance peak cluster, the number of resonance peaks is always less than or equal to the number of the spans of finite period beams. In Figs. 5.7 and 5.8, the second resonance peak is hidden since the external excitation force operates on the second modal vibration method node of the structure, x0 = 1.5l. Another feature of the periodic beam with TMD is that its optimized tuning frequency ratio Ωopt is not always less than 1 like in discrete mass systems. When TMD and harmonic force are located at the same point on the beam, Ωopt ≥ 1, otherwise, Ωopt < 1, as shown in reference [2, 5]. However, when the periodic change is large, the constraints caused by the coupling are weakened. See Figs. 5.12 and 5.13 for more details, which are the input and transmission power flow of 3 equalspan simply supported beams and 5 equal-span simply supported beams. Figure 5.14 is a comparison between the calculated energy transfer values of the 5 equal-span, simply supported beams, the finite period structure, and the test measurement results, which are in the good anastomosis.

5.3.4 Summary of the Analysis The WPA formulae given in this section can examine general multi-support beams. The finite period structure is a special form of a multi-support beam. The analysis examples assumed are all TMD periodic beams since their characteristics are more distinct. By examining the vibration power and transmitted vibration power of the input structure, particularly the finite period structure, some new findings are

147

Transmitted Power

5.3 The Analysis of the Dynamic Characteristics …

Frequency (Hz)

Fig. 5.14 The theoretical and measured values of 5 equal-span simply supported beams plus TMD power flow (μ = 0.095, β = 0.005, βd = 0.4, x0 = 1.3l, xd = 0.5l, z = 4.5l, Ω = 1.04)

summarized as follows: only one TMD used in 5.3.4 can suppress a cluster of resonance peaks in the propagation domain corresponding to the finite period structure. The number of tuned resonance peaks is equal to the number of spans of equalspan beams. When the point harmonic force and TMD are installed on the nodes of the beam, a resonance peak is reduced. The tuning frequency should be planned to correspond to the first resonance frequency of the resonance peak cluster in the propagation domain. (1) In a two-degree-of-freedom rigid body system, the TMD tuning frequency ratio is always unidirectionally optimized and is always less than 1, namely, Ω < 1. However, in the periodic structure, the best tuning frequency ratio of TMD may be greater than 1 or less than 1, which is bidirectionally optimized. When the TMD and the external load on the beam operate at the same point, the optimal tuning frequency ratio is always greater than or equal to 1, that is, Ωopt ≥ 1, otherwise, Ωopt ≤ 1. (2) Among the multiple measures for controlling the dynamic characteristics of the mast, retrofitting TMD is most efficient.

5.4 The Analysis of Mast Retrofitting with TMD Using the WPA Method The analysis content in this section is an extension of the theoretical research in Sect. 5.3, which focuses on the practical application of a certain project, examining the changes in the dynamic characteristics of the mast after retrofitting TMD, and investigating the general methods and best solutions for mast vibration control. Figure 5.15 shows the schematic diagram of retrofitting TMD on the mast.

148

5 WPA for Analyzing Hybrid Dynamic Systems

330mm from the first support

Convex mirror

TMD

Marginal data Lined rubber Peripheral mass

Fig. 5.15 Retrofitting TMD on the mast (the mass of TMD is m < 4 kg, the actual embedded design)

5.4.1 The Physical Model Figure 5.16 displays the physical model of retrofitting TMD on the mast, which is essentially upright. The elastic mast is assumed to be a straight beam, with four simple supports at the rear end for the counterweight, forming a typical hybrid dynamic system. Figure 5.17 is a structural schematic diagram of TMD. For completeness of expression, the derivation of key equations is still retained. Assuming that any point on the mast x = x0 is subjected to harmonic force p˜ 0 and no TMD is installed, the harmonic displacement response of the mast is recorded as w˜ b (x, t) =

4  n=1

A n e k n x + p0

2  n=1

an e−kn |x0 −x| +

4  j=1

Rj

2 

an e−kn |x j −x |

(5.27)

n=1

Then, consider the impact of retrofitting TMD on the dynamic characteristics of the system. When the constraint is released, it is treated as if TMD exerted a reaction force Fd (unknown number) on the mast. Under the linear assumption, the lateral displacement of the multi-support mast with TMD retrofitted is acquired according

M Counter weight mass

Fig. 5.16 The simplified mathematical model of a multi-support elastic mast with TMD

5.4 The Analysis of Mast Retrofitting with TMD Using the WPA Method

149

Fig. 5.17 A schematic diagram of the cross section of the compound TMD structure (produced by Li Zheran) (external scheme, actually using a built-in design) Mast

Marginal data Lined rubber Peripheral mass

to the principle of superposition of bending waves: w(x, ˜ t) = w˜ b (x, t) + Fd

2 

an e−kn |xd −x|

(5.28)

n=1

With the harmonic motion of the mast, the reaction force exerted by the TMD to the mast is shown in Eq. (2-86), which is a function of the harmonic displacement of the mast connection point: ∗ pd (ω) = K tot · w(xd )

(5.29)

where ∗ K tot ∗ kd md xd

the equivalent stiffness corresponding to TMD and hysteretic damping; the complex stiffness of TMD, kd∗ = kd (1 + jβ); the mass of TMD; Installation coordinate of TMD

When we substitute Eq. (5.29) into Eq. (5.28), it can be attained as follows: w(x, ˜ t) = w˜ b (x, t) + w(xd )

2 

an K∗tot e−kn |xd −x|

(5.30)

n=1

The last item on the right side of Eq. (5.30) is the dynamic reaction force introduced by TMD and w(xd ) is the dynamic displacement (unknown amount) of the point x = xd on the mast. To facilitate the calculation, the partial derivative (5.30) to x is listed in Eq. (5.31): 2  ∂ 2 w(x, ˜ t) ∂ 2 w˜ b (x, t) ∗ 2 −kn |xd −x| = + w(x ˜ ) an K tot kn e d ∂x2 ∂x2 n=1

(5.31)

150

5 WPA for Analyzing Hybrid Dynamic Systems 2  ∂ 3 w(x, ˜ t) ∂ 3 w˜ b (x, t) ∗ 3 = − w(x ˜ d) an K tot kn ( jd)e−kn |xd −x| ∂x3 ∂x3 n=1

(5.32)

where ( jd) is a symbolic operator:  ( jd) =

+1, x ≥ xd −1, x < xd

(5.33)

It is recognized that Eq. (5.27) has a total of N +4 unknown numbers and Eq. (5.30) has a total of (N + 5) unknown numbers, in which N + 4. They are explained simultaneously by boundary, constraint, and continuous conditions. When the N supports are all simply supported, these 3 boundary conditions are equivalent to ⎫ ∂ 2 w(0, ˜ t) ˜ , t) ∂ 2 w(L ⎪ ⎪ = = 0⎪ ⎪ ⎪ ∂x2 ∂x2 ⎪ ⎪ ⎪ 3 2 ⎪ ˜ t) ˜ , t) ∂ w(0, ∂ w(L ⎪ ⎪ ⎬ = = 0 3 3 ∂x ∂x w(xi ) = 0 (i = 1, 2, 3, 4) ⎪ ⎪ ⎪ ⎪  2  ⎪ ⎪ ⎪  ⎪ ⎪ ∗ wb (xd ) + w(xd ) · an K tot − 1 = 0⎪ ⎪ ⎭

(5.34)

n=1

Equation (5.34) corresponds to an (N + 5) unknown linear system of equations SX = Q

(5.35)

where X = {A1 , . . . , A4 , R1 , R2 , . . . , R N , w(xd )}T ; the elements in the matrix S and Q are as follows: s1,(5+N ) =

2 

∗ −kn xd an μ2n K tot e , s2,(5+N ) =

n=1

s3,(5+N ) =

2 

2 

∗ −kn (L−xd ) an μ2n K tot e

n=1 ∗ −kn xd an μ3n K tot e , s4,(5+N ) =

n=1

2 

∗ −kn (L−xd ) an μ3n K tot e

n=1

s(5+N ),1 = ek1 xd , s(5+N ),2 = ek2 xd , s(5+N ),3 = ek3 xd , s(5+N ),4 = ek4 xd s(5+N ),(4+ j) =

2  n=1

an e−kn |xd −x j | , s(5+N ),(5+N ) =

2  n=1

∗ an K tot −1

5.4 The Analysis of Mast Retrofitting with TMD Using the WPA Method

q1 = −

2 

an μ2n e−kn x0 , q2 = −

n=1

q3 = −

2 

an μ2n e−kn (L−x0 )

n=1

an μ3n e−kn x0 , q4 = +

n=1

q4+m = −

2 

151

2 

2 

an μ3n e−kn (L−x0 )

n=1

an e−kn |x j −x0 | , q5+N = −

n=1

2 

an e−kn |xd −x0 |

n=1

μn = {1, j, −1, − j } ( j, n, m = 1, 2, 3, 4) When we substitute the instantaneous value solution of Eq. (5.35) into Eq. (5.30), the displacement admittance at any point on the mast can be attained, that is, the displacement response under the unit dynamic load.

5.4.2 Calculation Example

The tuning frequency ratio is defined as Ω = f d f 0n , in which f d is the tuning frequency designed for TMD, f 0n is the n th natural frequency corresponding to the 4 support mast, and the first natural frequency is f 01 = 11.61(Hz), as shown in Chap. 4, Sect. 4.4. Place the input parameters in Table 5.3 and program and calculate them to obtain Figs. 5.18, 5.19, 5.20, and 5.21. When TMD is not installed, the first 2 natural frequencies, anti-resonance frequency, and the fundamental modal vibration mode of the mast are shown in Figs. 4–7 in Sect. 4.4. Figures 5.18 and 5.19, respectively, correspond to the performance during under/over tuning. Figure 5.20 shows the response after system optimization. The highest peak value is the displacement response of the end of the 4 support mast. After retrofitting TMD, the system response of the two kinds of tuning ratios is greatly reduced. Table 5.3 The calculation parameters of a 4 support mast with TMD

Material performance parameters E = 2.02 × 1011 N m2 , ρ = 7900 kg m3 Geometrical parameters and performance parameters of the mast

Outer diameter of the pole D = 0.18 m, inner diameter of the pole d = 0.16 m, L = 9.477 m

Coordinate parameters of the bearing point

x 1 = 0.2 m, x 2 = 3.726 m, xd 3 = 4.671 m, x 4 = 5.971 m

TMD parameters

m d = 3.99 kg,xd = L − 0.001 m, βd = 0.15, f d (design parameters)

External force parameters

p0 = 1N, x 0 = L − 1.5 m, xc = L (coordinates of the response points)

5 WPA for Analyzing Hybrid Dynamic Systems

Displacement mobility (m/N)

152

Under-tu uning

Frequency y(Hz)

Displacement mobility (m/N)

Fig. 5.18 TMD suppresses the displacement response of the mast end and the under-tuning is Ω = 0.994

Over-tuning

Frequency(Hz) Fig. 5.19 TMD suppresses displacement response of the mast end and the over-tuning is Ω = 1.08

When TMD is tuned to the frequency ratio Ωopt = 1.06, the peak displacement is efficiently suppressed: The original resonance peak splits into two approximately equal small peaks. The displacement amplitude of the end of the mast is reduced from 10−3 without TMD to 10−5 with TMD, which is approximately 1% of the original displacement, corresponding to the mass of TMD of less than 4 kg. The imaginary line in Fig. 5.20 corresponds to the under-tuning condition of Ω = 1.0. Explore the relationship between the tuning frequency ratio and the vibration suppression effect. Under the situation that other conditions remain unchanged, assuming that TMD takes f = 12.34 Hz as the central tuning frequency with 12.5% of the peak cluster frequency bandwidth range, that is, when TMD corresponds to

Displacement mobility (m/N)

5.4 The Analysis of Mast Retrofitting with TMD Using the WPA Method

153

Without TMD

Frequency (Hz)

Displacement mobility

Fig. 5.20 TMD suppresses the displacement response of the mast end (the optimized tuning frequency ratio is)

Tuning frequency ratio Ω

Frequency(Hz)

Fig. 5.21 3D displacement response of the mast end corresponding to the frequency and tuning frequency ratio

the tuning frequency f d = 10.80–13.89 Hz, the 3D curve of displacement response is shown in Fig. 5.21. It is evident from the figure that the tuning frequency of TMD has a significant impact on the dynamic response of the mast. The technical measures for controlling the mast include the increase in the diameter, the increase in the rigidity, the increase in the number of supports, the shortening of the support span, the adjusting of the support form and combination, retrofitting a streamlined pod, and the increase in structural damping. As a slender cantilever structure, the effect that the mast is retrofitted with TMD to suppress the dynamic response is most significant. Theoretical analysis demonstrates that TMD can decrease the

154

5 WPA for Analyzing Hybrid Dynamic Systems

displacement amplitude of the mast end by more than 95% and solve the problem of mast “jitter” and image blurring during high-speed underwater navigation. Through comparative studies, we have reached the following important conclusions: (1) That the submarine mast has “sufficient static stiffness and insufficient dynamic stiffness” is the root cause of its high-speed jitter. Insufficient dynamic stiffness dominates the system characteristics. So, even if the modeling is simple, the system characteristics can be calculated accurately. (2) TMD is the best way to control the dynamic response of the mast at high speed. (3) “Bundle” the bearing point of the mast with the enclosure structure and use its stiffness elements to increase the stiffness of the bearing point and reduce the total weight of the system.

5.5 Summary The continuous elastomer is retrofitted with concentrated mass or TMD to shape a hybrid dynamic system. In this chapter, the WPA method was used to complete a theoretical analysis on a range of hybrid dynamic systems, such as multi-support masts with heavy objects at the ends, TMD and any multi-support beams, and periodic structures. The analytical characteristics of the WPA method were evident and several conclusions were drawn: (1) Only one TMD can suppress a group of resonance peak clusters in the propagation domain of the periodic structure. The number of resonance peaks in each group is equal to the number of spans unless the TMD is located exactly on the node. The tuning frequency should be targeted for the first resonance frequency in the peak cluster. This conclusion discloses the strong coupling relationship of the periodic structure, and the overall periodicity has strong constraints and modulation on the inherent characteristics of the subsystem. (2) In a two-degree-of-freedom rigid body system, the optimal tuning frequency ratio of TMD is always unidirectionally optimized, that is, Ωopt ≤ 1. In the periodic structure, the optimal tuning frequency ratio of TMD is two-way optimal. When TMD and the external load on the beam operate at the same point, the optimal tuning frequency ratio Ω is always greater than or equal to 1, that is, Ωopt ≥ 1, otherwise, Ωopt ≤ 1. (3) TMD is one of the efficient measures to control the dynamic response of slender beam structures and is typically more effective than other methods. The submarine mast, as a typical slender beam structure, jitters at a high speed and the root cause for image blurring is caused by possessing “sufficient static stiffness but insufficient dynamic stiffness”. (4) The analysis of finite, periodic, and quasi-periodic beams is problematic. The WPA method analyzes that this type of system has distinctive advantages and has no restrictions on structural damping, support forms, and boundary conditions.

5.5 Summary

155

The theoretical analysis of finite quasi-periodic structures has theoretical and engineering value. The exposed slight “perturbation” changes can also be used as diagnostic criteria for periodic structures.

References 1. YOUNG (1952) Proceedings of the first US National Congress of Applied Mechanics [C]:91– 96 2. Neubert VH (1964) Dynamic absorbers applied to a bar that has solid damping. J Acoust Soc Am 36(4):673–680 3. Snowdon JC (1968) Vibration and damped mechanical systems. Wiley, New York 4. Jones DIG (1967) Response and damping of a simple beam with tuned dampers. J Acoust Soc Am 42(1):50–53 5. Jacquot RG (1978) Optimal dynamic vibration absorbers for general beam systems. J Sound Vib 60(4):535–542 6. Wu CJ (1992) A new method of reduction of beam vibration by the use of neutralisers. University of Southampton, England 7. Wu CJ, White RG (1993) Reduction of vibrational power in periodic beams by use of a neutralizer. Proc Inst Acoust 15:263–270 8. Guoqun Zheng (2002) Using WPA method to study hybrid danamic dynamic system. Wuhan Ship Design Institute, Wuhan 9. Chongjian Wu (1995) Theoretical analysis and design of dynamic vibration absorber to suppress mast vibration. Ship Eng Res 2:36–40 10. Wu Chongjian, Fu Tongxian. Research and Design of Double-Layer Vibration Isolation System of Intermediate Raft Body with Elastic Properties [J]. Ship Engineering Research, 1994, (4):20– 24 11. WuChongjian. A Summary of Comparative Research on Floating Raft and Double-Layer Vibration Isolation [J]. Ship Engineering Research, 1998, (1):29–33 12. Zhigang Chen (2005) Research on “Mass Effect” in Floating Raft Vibration Isolation System [M]. China ship research and design center, Wuhan

Chapter 6

WPA for Calculating Response Under Distributed Force Excitation

The secret to being healthy for me is to think and use my brain frequently. I am interested in carrying out acoustic research. Solving scientific problems and breaking through acoustic barriers are bringing the greatest joy to my life. ——Ma Dayou

6.1 Introduction Common structures in engineering bear distributed forces, and the excitation components include distributed forces. For example, the fluid excitation on the periscope in the latent state is the distributed force [1, 2]. Wind loads of high-rise buildings, longspan bridges, high-voltage transmission structures, wave loads borne by ship hulls and various masts, etc., are or include distributed force excitations, which cannot be ignored in design. We use the WPA method to analyze the structural response problem of point harmonic force [3, 4], which is also suitable for structural dynamic analysis under distributed force excitation [5]. In this chapter, according to the superposition principle of bending waves, the dynamic response, and dynamic characteristics of cantilever beam structures under arbitrary distributed forces are theoretically studied, and the general expressions for the solution derived [6–8]. Compared with the classical algorithm, the accuracy of the method is verified. The equation derived above can be easily extended to beam structures with arbitrary supports and boundary conditions and excited by transient forces.

© Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 C. Wu, Wave Propagation Approach for Structural Vibration, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-7237-1_6

157

158

6 WPA for Calculating Response Under Distributed Force Excitation

6.2 Mechanical Model and Formula Deduction To express the continuity, the Eq. (4.32), of single-point excitation for any N supporting beams in Chap. 44 is introduced, and the time term exp( jωt) is omitted, thus it becomes w(x, ˜ t) =

4 

A n e kn x +

n=1

N  i=1

Ri

2 

an e−kn |xi −x| + po

n=1

2 

an e−kn |xo −x|

(6.1)

n=1

where, kn is the wave number, see Eq. (2.3), for details; a1 and a2 are the coefficients of the point response function of the bending wave at the origin of an infinite beam given in Chap. 2. Item 1 in Eq. (6.1), is the displacement of the reflected wave at the two ends of the beam, and 4 An are unknown numbers; Item 2 is generated by supporting reaction force Ri . The third term is the displacement term generated by external excitation force. The magnitude of N supporting reaction forces Ri is also unknown to be determined, with a total of N + 4. For the convenience of the following discussion, Eq. (6.1) is abbreviated as w(x, ˜ t) = wh (x) + po

2 

an e−kn |xo −x|

(6.2)

n=1

where wh (x) =

4  n=1

A n e kn x +

N 

Ri

i=1

2 

an e−kn |xi −x|

(6.3)

n=1

Under the excitation of distributed force, the diagram of a complex beam system is shown in Fig. 6.1. Assume that z 1 and z 2 are the upper and lower limit coordinate values of the action area of the distributed force q(x), ˜ respectively, and 0 ≤ z 1 ≤ z 2 ≤ L. The distribution force can be dispersed into m equal parts with a spacing of x.

Fig. 6.1 Complex beam structure under distributed force

6.2 Mechanical Model and Formula Deduction

159

Solve the excitation force p0i located in the coordinates xi (z 1 ≤ xi ≤ z 2 ) and interval xi + x and get the following equation: poi = q(xi )x

(6.4)

Equation (6.2) is rewritten as w(x, ˜ t) = wh (x) + q(xi )x

2 

an e−kn |xi −x|

(6.5)

n=1

According to the superposition principle of bending waves, in Eq. (6.5), q(xi )x is equivalent to a point force p0i , and the lateral displacement equation of the beam subjected to distributed force is equivalent to the sum of m external loads, i.e. w(x, ˜ t) = wh (x) +

m 

q(xi )x

2 

an e−kn |x−xi |

(6.6)

n=1

i=1

In Eq. (6.6), the limit of x is obtained, and the general expression of the lateral displacement of the beam under arbitrary support and distributed force excitation can be obtained by using the integral principle as follows: z2 w(x, ˜ t) = wh (x) +

q(z)

2 

an e−kn |x−z| dz

(6.7)

n=1

z1

The complete equation is as follows: w(x, ˜ t) =

4 

An e

kn x

n=1

+

N  i=1

Ri

2 

an e

−kn |x−xi |

z2 +

n=1

q(z) z1

2 

an e−kn |x−z| dz (6.8)

n=1

Similar to Eq. (6.1), there are only N + 4 unknowns in Eq. (6.7) and Eq. (6.8), so the two equations can also be obtained by the boundary conditions and constraint conditions of the beam. In the above conclusion, the boundary conditions and constraint conditions of the beam are not limited, and the action form, action position and magnitude of the distributed force are not specified. For the case that the beam is simultaneously acted by distributed force and concentrated load, it is not difficult to deduce the expression of the WPA method according to the superposition principle of bending waves, i.e. w(x, ˜ t) = wh (x) +

k  i=1

poi

2  n=1

an e

−kn |x−xi |

z2 +

q(z) z1

2  n=1

an e−kn |x−z| dz

(6.9)

160

6 WPA for Calculating Response Under Distributed Force Excitation

where item 2 is the quantity related to concentrated dynamic load; p j is the jth concentrated external load, j = 1, 2, ...k. For various periodic transient excitation forces, corresponding power functions can be obtained by Taylor transformation. For example, let po (x) = g(x)

(6.10)

After Taylor transformation, we get 1  g (xo )(x − xo )2 + ν 2!   1 + g (n) (xo )(x − xo )n + ν + 0 |x − xo |n n!

g(x) = g(xo ) + g  (xo )(x − xo ) +

(6.11)

where the last term 0() represents a negligible infinitesimal. Substitute Eq. (6.11) into Eq. (6.6) to obtain Eq. (6.12), which can be used to analyze the structural response under transient excitation force. z2 w(x, ˜ t) = wh (x) +

q(z) z1

2 

an e−kn |x−z| dz

(6.12)

n=1

Assume that N supports are simply supported, and the two ends of the beam are also simply supported, the corresponding boundary conditions can be obtained as follows: ⎫ w(0) = w(L) = 0 ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ∂ w(0, ˜ t) ˜ , t) ∂ w(L ⎬ = = 0 2 2 (6.13) ∂x ∂x ⎪ ⎪ ⎪ w(xi ) = 0 ⎪ ⎪ ⎭ i = 1, 2, ν, N According to the above conditions, the corresponding linear equations of N + 4 unknowns can be obtained as follows: SX = Q

(6.14)

X = {A1 , ..., A4 , R1 , R2 , R3 ..., R N }T

(6.15)

where, S——Parameter matrix; X——Unknown variable; Q——External excitation force vector.

6.2 Mechanical Model and Formula Deduction

161

According to Eq. (6.14), N + 4 unknowns can be solved, and the vibration wave propagation and displacement response of the structure can be obtained by substituting them into Eq. (6.6).

6.3 Simple Cantilever Beam Structure The straight cantilever beam mentioned in this section has one end fixed and the other end free. The beam is subjected to a uniformly distributed force q(x) = c, where c is a constant. The boundary condition of the structure is different from the Eq. (6.13), and is w(0) = 0

(6.16)

∂ w(0, ˜ t) =0 ∂x

(6.17)

∂ 2 w˜ 1 (L , t) =0 ∂x2

(6.18)

∂ 3 w˜ 1 (L , t) =0 ∂x3

(6.19)

Corresponding to the boundary conditions of Eqs. (6.16) to (6.19), the following equations set can be obtained from Eq. (6.7): 4 

An + c

n=1 4 

0

kn An + c

n=1 4 

kn2 An ekn L

+c

n=1

where

kn3 An ekn L

+c

an (−kn )e−kn z dz = 0

(6.21)

an (−kn )2 e−kn (L−z) dz = 0

(6.22)

an (−kn )3 e−kn (L−z) dz = 0

(6.23)

n=1

L  2 0

(6.20)

n=1

L  2 0

an e−kn z dz = 0

n=1

L  2 0

n=1 4 

L  2

n=1

162

6 WPA for Calculating Response Under Distributed Force Excitation

c

L  2 0

an e

−kn z

dz = c

n=1

L  2 0

an e

−kn ν L−zν

dz = −c

n=1

L  2 0

n=1

 an  −kn L e − 1 dz kn (6.24)

Assume

( jn)n = kn k1

(6.25)

( jn) = {1, j, −1, − j } (n = 1, 2, 3, 4)

(6.26)

then

Substitute Eq. (6.26) into Eq. (6.20) to (6.23), and get 4 

An − c

n=1 4 

( jn)n An − c

n=1 4 

2   an  −kn L e −1 =0 k n=1 n

(6.27)

2   an ( jn)n  −kn L e −1 =0 kn n=1

(6.28)

( jn)2n An ekn L − c

2   an ( jn)2n  −kn L e −1 =0 kn n=1

(6.29)

( jn)3n An ekn L − c

2   an ( jn)3n  −kn L e −1 =0 kn n=1

(6.30)

n=1 4  n=1

Write Eqs. (6.27) to (6.30), in matrix form and get the matrix equation shown in Eq. (6.31) SX = Q

(6.31)

The coefficient matrix of the matrix equation is as follows: ⎤ 1 1 1 1 ⎢ ( jn)1 ( jn)2 ( jn)3 ( jn)4 ⎥ ⎥ S=⎢ ⎣ ( jn)2 ek1 L ( jn)2 ek2 L ( jn)2 ek3 L ( jn)2 ek4 L ⎦ 1 2 3 4 ( jn)31 ek1 L ( jn)32 ek2 L ( jn)33 ek4 L ( jn)34 ek4 L ⎡

(6.32)

6.3 Simple Cantilever Beam Structure

Q = +c

⎧ ⎫ 2  ⎪  ⎪ an  −kn L ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 e ⎪ ⎪ ⎪ ⎪ k n ⎪ ⎪ ⎪ ⎪ n=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪  ⎪ ⎪   a ( jn) ⎪ ⎪ n n −kn L ⎪ ⎪ e − 1 ⎪ ⎪ ⎪ ⎪ ⎨ ⎬ k n n=1 2 ⎪  ⎪ ⎪ ⎪ an ( jn)2n  −kn L ⎪ ⎪ ⎪ ⎪ e − 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ k n ⎪ ⎪ n=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ 3    ⎪ ⎪ a ( jn) n ⎪ ⎪ n −kn L ⎪ ⎪ e − 1 ⎪ ⎪ ⎩ ⎭ k n n=1

163

(6.33)

6.4 Comparison Between Examples of WAP Method and Classical Analytical Method Take the cantilever beam of the above section as an example. At this time, it is fixed at one end x = 0 and the other end is the free end. The uniform distribution force on the beam is q = 1.0 N/m. The material performance parameters of the beam are as follows: E = 2.02 × 1011 N /m2 ; ρ = 7900kg/ m3 . The geometric parameters of the circular cross section beam are as follows: Outer diameter D = 0.1 m; Inner diameter d = 0.06 m; Length L = 5 m. The structural damping loss factor of the beam β = 0.001, i.e., E I ∗ = E I (1 + jβ), where E I is the bending section stiffness of the beam. According to the classical analytical method, the first 5 order natural frequency analytical expressions of the beam are as follows: ⎫ ω1 = 1.8752 a/L 2 , ω2 = 4.6942 a/L 2 ⎪ ⎪ ⎬ 2 2 2 2 ω3 = 7.855 a/L , ω4 = 10.966 a/L ⎪

⎪ ⎭ ω = 14.1372 a/L 2 , a = E I ρ S

(6.34)

5

where ω j —The first 5 natural frequencies of the beam, j = 1, 2, 3, 4, 5; E I ——Stiffness of bending section of the beam; S——Cross-sectional area of the beam; ρ——Material density. The first five natural frequencies of the cantilever beam can be calculated by substituting structural parameters into Eq. (6.34). At the same time, the WPA method is used to analyze Eq. (6.31), and various analysis and calculations can be performed by obtaining response values. The natural frequency values calculated by the classical analytical method and the WPA method are listed in Table 6.1 for comparison. It can be seen from this table

164

6 WPA for Calculating Response Under Distributed Force Excitation

Table 6.1 Comparison of calculation results between WPA method and classical analytical method. Unit: Hz

Number of order

Classical method

WPA method

1 order

20.731

20.731

2 order

129.932

129.932

3 order

363.851

363.851

4 order

713.018

713.018

5 order

1178.500

1178.500

that the calculation results of the WPA method are in accurate agreement with those of the classical analytical method. It can be seen that the WPA method is an accurate solution and has a wide range of applications. After programming and calculation, the vibration shape, natural frequency, and displacement response of the beam are obtained as shown in Fig. 6.2 and Fig. 6.3,

Normalized modal shape

order modal shape (20.7371 Hz) 2-order modal shape (129.93251 Hz) 3-order modal shape (363.8513 Hz)

Beam length direction coordinate (x)

Fig. 6.3 Displacement response curve of cantilever beam

Displacement mobility

Fig. 6.2 First 3 modes of cantilever beam

Frequency (Hz)

6.4 Comparison Between Examples of WAP Method and Classical Analytical Method

165

respectively. Figure 6.2 shows the first 3 modal shapes of the cantilever beam. The abscissa indicates the x-direction coordinates of the cantilever beam length and the ordinate indicates the modal displacement of each point on the beam. Figure 6.3 shows the frequency response curve of the beam end x = L in the frequency range of 0 ~ 250 Hz. It can be seen in the figure that the first 2 resonant frequencies (including one anti-resonance point) of the beam are 20.7 Hz and 129.9 Hz, respectively, which are in good agreement with the classical solution.

6.5 Summary According to the assumption of structural linearity and the superposition principle, through Taylor transformation, the limit of displacement function under multi-force action is obtained, and the structural dynamic equation under distributed force excitation is ascertained by using the integral principle. Taylor transformation is carried out on the distributed force function to obtain the corresponding power function, which means it is possible to deal with the general problem of a nonuniform distributed external load. In this chapter, the idea of the WPA method to deal with distributed force response is given, and the displacement vibration equation of the multi-support beam is deducted. The research shows that the WPA method is not only convenient and efficient to analyze the response of point harmonic force, but also very simple to analyze the response of distributed force excitation. The method is trustworthy, and the theoretical calculation of the cantilever beam under uniform distributed force is completely consistent with the classical calculation results. At the same time, it is not difficult to see that the WPA method can be arbitrarily expanded and extended its internal advantages as discussed in the previous chapters.

References 1. Chongjian Wu, Shuzi Y, Dongping L et al. (1999) The dynamic response and stress analysis of multi-supported elastic beam by use of WPA method [J]. J Huazhong Uni Sci Technol 27(1):69– 71 2. Chongjian Wu, Dongping L, Shuzi Y et al. (1999) Vibration analysis of multiple supported elastic mast with MTD by using WPA method [J]. Journal of Huazhong University of Science and Technology 27(2):22–24 3. Wu CJ, White RG (1995a) Vibrational power transmission in a multi-supported beam [J]. J Sound Vib 181(1):99–114 4. Wu CJ, White RG (1995b) Reduction of vibrational power in periodically supported beams by use of a neutralizer [J]. J Sound Vib 187(2):329–338 5. Chongjian Wu (2002) WPA method for structural vibration and its application[D]. Huazhong University of Science and Technology, Wuhan 6. Chongjian Wu (1995) Theoretical analysis and design of dynamic vibration absorber for suppressing mast vibration [J]. Ship Eng Res 69(2):36–40

166

6 WPA for Calculating Response Under Distributed Force Excitation

7. Chongjian W, White RG (1996) Reduction characteristics of finite periodic beams by use of a dynamic absorber [J].Vib Imp 15(4):27–31 8. Chongjian W, Xiangdong L, Yuepeng C (1998) WPA analysis method for dynamic characteristics of the mast with a heavy lifting head [J]. Noise Vib Control (6):6

Chapter 7

Discrete Distributed Tuned Mass Damper

The WPA method is a type of analytical method. It is both a new theoretical method and a new kind of thinking —The Author

7.1 Introduction Tuned Mass Damper (TMD) is a passive device [1–3] that conducts vibration energy exchange through tuning and is used to control the vibration of the main vibrating body. Its applications include controlling the low frequency vibration of bridges, ship mast jitter, auxiliary vibration energy transmission, and the noise leakage control of floating raft structures, etc. It also has a notable effect on tuning frequency points. In engineering applications, the mass ratio μ ≥ 0.3 is commonly required. Even for civilian ships and planes, arranging a large TMD is overall, a difficult design, not to mention military tanks, planes, and submarines with strict space restrictions. If a large TMD is separated into several smaller ones, a “discrete distributed” MTMD (Multiple Tuned Mass Damper) can be constructed, which can both better adapt to the vehicle space and also expand the tuning bandwidth so that the system has better engineering adaptability. Presently, there are mainly four methods to expand the tuning bandwidth of TMD: (1) increasing the damping loss factor of TMD elastic elements; (2) using soft springs; (3) using compound TMD; and (4) adopting discrete MTMD with staggered tuning frequencies. In as early as the 1960s, Snowdon [2] examined splitting TMD with mass m into three small TMD with equal mass. The calculation results showed that this method could slightly expand the tuning bandwidth of MTMD. So, how do we design an MTMD system to obtain the widest tuning frequency and higher engineering value? Sun et al. [4] carried out theoretical analysis and experimental research on multiple liquid TMD with the same tuning frequency. A similar engineering application example can be seen in the reference [5] when the main mass is a rigid body, which can better adapt to the space limitation. Igusa and Xu [6] also described that the © Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 C. Wu, Wave Propagation Approach for Structural Vibration, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-7237-1_7

167

168

7 Discrete Distributed Tuned Mass Damper

optimal distribution of MTMD is nonlinear. Ming et al. [7] applied MTMD with equal spacing and linear distribution to cable-stayed bridges to suppress bridge buffeting. Chongjian et al. [8, 9] completed research on the application of MTMD to a special marine large motor and described that the vibration suppression bandwidth could exceed 3 Hz. This chapter examine show to construct MTMD with 32 small TMD and its application in ships. The theoretical analysis shows that the tuning frequency bandwidth of MTMD can reach  ≈ 5 − 14Hz, which can better adapt to the engineering space limitation and also has a higher comprehensive performance.

7.2 The Velocity Impedance of MTMD In the large motor example shown in Fig. 7.1, vibration isolators were used to isolate vibration components above tooth frequency and yoke vibration components with relatively small MTMD tuning energy with its simplified mathematical and physical model shown in Fig. 7.2. The natural frequency of the i TMD is recorded as ωi , the damping coefficient as ξi , and the mass as m i , where i = 0, 1, 2, . . . n, i = 0 corresponds to the tuning main vibrator (motor). It is further supposed that the tuning frequencies of TMD are different and conform to the sequence characteristics ω1 < ω2 < … ωn . The velocity impedance is defined as the force amplitude necessary to generate unit harmonic velocity at the TMD footing and the impedance of the harmonic force acting at the corresponding frequency ω at the i TMD footing, which is [3] Z i (ω) = − jωm i

ωi2

ωi2 − j2ωi ξi − ω2 − j2ωi ωξi

(7.1)

Therefore, the total impedance at the N TMD feet is

Legend Description TMD Vibration isolator

Fig. 7.1 M TMD on a special marine large motor (prepared by Fan Yongjiang and Li Zheran)

7.2 The Velocity Impedance of MTMD

169

Fig. 7.2 The physical model

Z (ω) =

N  n=1

N  m i (ωi2 − j2ωi ξi ) Z i (ω) = − jω ωi2 − ω2 − j2ωi ωξi i=1

(7.2)

A dimensionless frequency interval parameter, i.e. βi =

ωi+1 − ωi ωi

(7.3)

where, ω1 < ω2 < . . . < ωn . When we define functions D(ω) and ξ(ω)  ⎫ D(ωi ) = ωi βi ⎪ ⎬ ξ(ωi ) = ξi ⎪ ⎭ i = 1, 2, . . . , n

(7.4)

where, D(ω) is the normalized mass density function of MTMD, that is, it’s mass and tuning frequency distribution in frequency coordinates. After the Eq. (7.2) is made, it becomes Z (ω) = − jω

N  D(ωi )ωi [ωi − j2ωξ(ωi )] ωi2 − ω2 − j2ωi ωξ(ωi ) i=1

(7.5)

where ωi = ωi+1 − ωi . The optimal design condition of MTMD is that the tuning frequency values corresponding to D(ω) and .. should change slowly.

170

7 Discrete Distributed Tuned Mass Damper

7.3 The Vibration Absorption Characteristics of MTMD MTMD is comprised of a plurality of TMDs and the motion equation of the main vibrator is ¨ + 2m 0 ω0 ξ0 x(t) ˙ + m 0 ω02 x(t) = F(t) + FTMD m 0 x(t)

(7.6)

where X(t)—The displacement response of the main vibrating body F(t)—The external force acting on the main vibrating body FTMD (t)—The reaction force of MTMD. When the external excitation is the unit harmonic force F(t) = e jωt , the steadystate response has the form of x(t) = x(ω)e jωt . Similarly, according to the definition of velocity impedance, the reaction force of TMD is FTMD (t) = jωx(ω)Z (ω)e jωt

(7.7)

When we substitute Eq. (7.7) into Eq. (7.6) to get the complex amplitude of the harmonic response, we get x(ω) =

1  2 2 m 0 ω0 − ω − j2ωω0 ξ0 − jωZ (ω)

(7.8)

When the external excitation force function F(t) is a stationary process G(ω) with a one-sided power spectral density, the mean square value of the response of the main vibrator is

∞ σx2

=

G(ω)|x(ω)|2 dω

(7.9)

0

The external excitation force function F(t) is approximately expressed by white noise and the input power spectral density function is expressed by a constant. Assume G(ω) = G 0 and substitute it into Eq. (7.9). This assumption is reasonable for the wideband input of power spectral density with a stable variation. According to the multivariable mass density function D(ω), we can develop ⎧ ⎫   2 √ ⎬ 4m 0 ⎨ ω 4/3 D(ω) = √ − 1 − 3ξ0 γ1 + (2 − 1) γ12 − ⎭ ω0 3π ⎩

(7.10)

where, f 0 (1 − γ1 ) ≤ f ≤ f 0 (1 + γ1 ); and πm γ1 = 4m o ξ0

 −1   π  πm 4/3 1+ 1+ 2 −1 −1 √ 4 2 3m o ξ0

(7.11)

7.3 The Vibration Absorption Characteristics of MTMD

171

The simplified results of Eqs. (7.10) and (7.11), are as follows:    2  √ 3.23m o f D(ω) = m o m 0.409 + 0.621 1 − −1 m f0  m γ1 = 0.556 m0

(7.12)

(7.13)

    where f 0 (1 − 0.556 m m 0 ) ≤ f ≤ f 0 (1 + 0.556 m m 0 ). From Eq. (7.12), it is evident that the mass density function a semi-elliptic   D(ω) is   curve with its principal axis point as ( f 0 (1 ± 0.556 m m 0 ), 0.409 m m 0 ) and   the other minor axis point as ( f, 1.03 m m 0 ). The total area of the elliptic curve envelope is the total mass of the MTMD, where the tuning frequency bandwidth of the TMD is  m (H z) (7.14)  f = 1.112 · f n m0

7.4 Analysis, Calculation, and Discussion 7.4.1 The Basic Parameter Analysis There are mainly three types of electromagnetic vibrations in the motor: Vibration caused by a motor slot path, yoke vibration, and vibration caused by teeth and poles. According to the relevant theoretical analysis and measured data verification, the amplitude of the vibration caused by teeth and tooth poles are tens or even hundreds of times different from that of yoke vibrations. Tongxian and Guohua [5] described that the electromagnetic vibration of the motor was basically determined by the yoke vibration and groove vibration of the motor, which account for the main components, as shown in Fig. 7.3. The vibration frequency caused by the tank circuit is fc = n ·

Z 60

(7.15)

where n is the motor speed in r/min; Z is the number of the tank circuit of the motor. Yoke vibrations are the natural vibration of motor stator and its frequency calculation equation is as follows:

172

7 Discrete Distributed Tuned Mass Damper

Fig. 7.3 The frequency distribution of internal and external interference forces

 Eh m m Rc2  k∞ 1 f e1 = 2π m m  p( p2 −1) Eh 3 · √ · 12m 4 ( p ≥ 2) 2 mR f e0

p

fe =

1 2π

1 = 2π

p +1

(7.16)

(7.17) (7.18)

c

where p—Order ( p ≥ 2) mm —The surface density, i.e., the mass of the average cylindrical surface of the yoke h—Height of the stator yoke E—Elastic modulus of the stator yoke composition material Rc —Average radius of the stator yoke k ∞ —Total stiffness of the vibration isolator. By substituting the parameters into the above equation, it is found that the groove vibration frequency of the motor is greater than 100 Hz while the first three natural frequencies of the yoke vibration are 14.5 Hz, 38.8 Hz, and 110 Hz, respectively. In the yoke vibration, the 2-order vibration accounts for the largest proportion, so the 2-order yoke frequency f e2 = 38.8 Hz of the motor is obtained as the main tuning frequency. MTMD is installed in the middle of the motor with N = 32, each weighing 30 kg. The structural design confirms that the spring stiffness can be adjusted in a small range and is sealed and fixed after the adjustment. The main parameters of the motor are as follows: mass m 0 = 92000 kg, damping ratio ξ = 0.007, total mass of MTMD m = 960 kg, and mass ratio μ = 0.0104. The above parameters are substituted into Eqs. (7.12) and (7.13), to attain the optimized mass density function and the calculation results are shown in Fig. 7.4.  D( f ) = 3843.73 + 5836.08 1 − 309.54



f −1 f0

 (7.19)

173

Mass density function D(f)

7.4 Analysis, Calculation, and Discussion

Tuning frequency (Hz)

Fig. 7.4 The mass density function of MTMD D(f )

The tuning frequency of each TMD is not continuous but discontinuous and its lowest natural frequency is    m = 36.5983 f min = f 0 1 − 0.556 m0

(7.20)

Other tuning frequencies values will be measured according to the following equation: 

f i+1

mi = f i (1 + βi ) = f i 1 + D( f i )

 (7.21)

Substitute f i into Eq. (7.19) to get D( f i ) and then substitute this value into Eq. (7.21) to obtain f i+1 . By comparison, the optimal tuning frequency and related parameters of MTMD can be obtained. All calculation results are listed in Table 7.1. According to Eq. (7.14), the distributed tuning frequency width of MTMD is 4.407Hz. Calculate the total impedance according to Eq. (7.2). The total mass is the same and the impedance curves of MTMD and TMD are shown in Fig. 7.5. MTMD has a wider frequency characteristic. Although the maximum impedance value is noticeably smaller than TMD, the overall vibration suppression effect of this difference on the system is positive, as shown in Fig. 7.6.

7.4.2 The Comparison Between MTMD and TMD The Eq. (7.8), is used to calculate the system response and compare it with the frequency response value of the main vibrator without TMD. To enable this comparison and regularize the calculation results, Fig. 7.5 gives the harmonic response curve

174

7 Discrete Distributed Tuned Mass Damper

Table 7.1 The distribution characteristic parameters of MTMD i

f i /Hz

Di /kg

βi

i

f i /Hz

Di /kg

βi

1

36.6427

3762.8

0.0080

17

38.9200

9467.2

0.0032

2

36.9349

6640.0

0.0045

18

38.0434

9439.6

0.0032

3

37.1017

7290.2

0.0041

19

39.1674

9392.6

0.0032

4

37.2544

7751.7

0.0039

20

39.2925

9325.3

0.0032

5

37.3986

8108.7

0.0037

21

39.4190

9236.2

0.0032

6

37.5370

8396.2

0.0036

22

39.5470

9123.1

0.0033

7

37.6711

8632.6

0.0035

23

39.6770

8983.3

0.0033

8

37.8020

8828.9

0.0034

24

39.8095

8812.9

0.0034

9

37.9304

8992.1

0.0033

25

39.9451

8606.2

0.0035

10

38.0570

9127.0

0.0033

26

40.0843

8355.1

0.0036

11

38.1821

9237.0

0.0032

27

40.2282

8047.1

0.0037

12

38.3061

9324.5

0.0032

28

40.3782

7661.2

0.0039

13

38.4293

9391.2

0.0032

29

40.5363

7158.0

0.0042

38.5521

9438.2

0.0032

30

40.7062

6444.9

0.0047

38.6746

9466.4

0.0032

31

40.8957

5135.2

0.0058

16

38.7972

9476.0

0.0032

32

40.9573

3762.8

Dimensionless displacemen

14 15

1 TMD

32TMD

Tuning frequency /Hz

Fig. 7.5 The impedance comparison between MTMD and TMD (Normalized processing)

of the main vibrator with only a single lumped mass TMD. It is evident from this figure that MTMD has a wider vibration suppression band than a single TMD with the same total mass reaching  f ≈ 5Hz and has a better tuning effect, with its maximum value being only 56% of TMD.

Dimensionless displacement

7.4 Analysis, Calculation, and Discussion

175

0 TMD

1 TMD 32TMD

Tuning frequency /Hz

Fig. 7.6 The harmonic response of the main vibrator

7.4.3 The Comparison in Under/Over-Tuned States

Dimensionless displacement

The TMD tuning frequency is lower than the optimal operating frequency point of the system, which is called under-tuning or over-tuning. MTMD is expressed by the center tuning frequency of the system. Examining the adaptability of MTMD and TMD to under/over-tuned states is an important concern in engineering applications. The tuning frequency will change due to many factors such as rubber aging, external load changes, new defects or modifications of engineering structures caused by longterm use of components and will gradually deviate from the optimal working point. It is supposed here that the tuning frequency f d = 39.8Hz, i.e., the deviation from the tuning point in +1 Hz, is in an over-tuned state. While the vibration suppression performance decreases as expected, the vibration suppression performance of MTMD is far better than that of TMD, as shown in Fig. 7.7.

0 TMD

1 TMD 32TMD

Tuning frequency(Hz)

Fig. 7.7 The response comparison of (f = 1.0 Hz) main vibrator under an over-tuned state

7 Discrete Distributed Tuned Mass Damper

Dimensionless displacement

176

0 TMD

32 TMD Mass ratio 0.0104

32 TMD Mass ratio 0.104

Tuning frequency (Hz)

Fig. 7.8 The vibration elimination bandwidth and vibration elimination effect under

7.4.4 Influence of the Mass Ratio When the total mass ratio is μ = 0.0104, the bandwidth calculated by Eq. (7.14) is 4.32 Hz. When the mass ratio is increased to 0.1043, the corresponding bandwidth is increased to 13.94 Hz. Increasing the total mass ratio of MTMD will considerably improve the effective vibration elimination bandwidth and the vibration elimination effect of the system, as shown in Fig. 7.8.

7.5 The Actual Vibration Elimination Effect To evaluate the tuning effect, 7 measuring points are organized on the propulsion motor, namely, 3 measuring points are arranged on the left middle cross section, 1 measuring point on the front,1 measuring point on the rear of the inclined foot, 1 measuring point on the bow end, and 1 measuring point on the stern end of the flat foot. The vibration acceleration at the foot of the propulsion motor before and after installing MTMD was measured under four rotational speed conditions of 45, 67, 100, and 200 r/min. The alteration of the average vibration suppression effect of the vibration acceleration with the rotation speed before and after the MTMD was installed at each measuring point, as shown in Fig. 7.9. The vibration acceleration of the propulsion motor was reduced by 53.7% on average under the four main working conditions.

Fig. 7.9 The vibration suppression effect of the main propulsion motor

177

Acceleration speed m/g 2

7.6 Summary

0.8 0.6 0.4 0.2

0

0

50

100

150

200

250

Rotational speed (rpm)

7.6 Summary This chapter demonstrates the theoretical research and engineering test results of MTMD. When the tuning frequency of MTMD is distributed according to a specific optimal design, it can better adapt to the space limitation of the vehicle and also shows better dynamic characteristics and system stability, which has a wide range of engineering application value in numerous vehicles and large bridge structures. The main features are as follows: (1) The tuning frequency band of the system is wider. The theoretical analysis shows that when MTMD is dispersed according to a specific elliptical mass, the tuning frequency width is in proportion to the square root of the total mass ratio, which is several times that of TMD. (2) The adaptability of under/over-tuned states is stronger. When the tuning system deviates from the optimal operating frequency and is in an under/over-tuned state, the vibration suppression effect of MTMD on the main vibrator is better than TMD. For a single TMD, when the tuning frequency deviates from the optimal operating point, it is equal to 10% of the deviation of the complete system, which is fatal. The MTMD, in this case, consists of 32 TMD. When one or even all of them have a deviation of 10%, they display a Gaussian probability distribution in a statistical sense, therefore, maintaining the vibration suppression and stability of the system itself. Also, the more the number is, the better the stability of the vibration suppression effect system will be. (3) Only a small mass ratio is necessary for tuning and controlling the vibration. In this case, the mass ratio was μ = 0.0104, so it is usually required to be μ ≥ 0.3 to differ by more than 20 times compared with the two-stage vibration isolation. The distribution of MTMD is endogenous and the constraint mechanism of its coupling is worthy of further study.

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7 Discrete Distributed Tuned Mass Damper

References 1. Haiyan H (2005) Foundation of mechanical vibration. Beijing University of Aeronautics and Astronautics Press, Beijing, p 7 2. Snowdon JC (1968) Vibration and damped mechanical systems. John Wiley & Sons, New York 3. Chongjian W (1995) Theoretical analysis and design of TMD for suppressing mast vibration. Ship Eng Res 69(2):36–40 4. Sun LM, Fujinoy, Pacheco B et al (1992) Modeling of tuned liquid damper (TLD). J Wind Eng Ind Aerodyn 41:1883–1894 5. Tongxian F, Guohua Z (1996) Development of vibration isolation device for propulsion motor of a submarine. Ship Sci Technol 1(44–49):65 6. Igusa T, Xu K (1994) Vibration control using multiple tuned mass dampers. J Sound Vib 175(4):491–503 7. Ming G, Gengren C, Jieming W, et al (1997) The study on the property of MTMD in controlling the buffeting response of cable-stayed bridges. Vibr Impact 16(1):1–5 8. Chongjian W (1995) Suppression characteristics of distributed dynamic vibration absorber and its design example. Ship Eng Res 71(4):22–27 9. Chongjian W, Dongping L, Shuzi Y, Yingfu Z, Yunyi M (1999) Design and application of multiple tuned mass damper for ships. J Vibr Eng 12(4):584–588

Chapter 8

Analysis of Raft Using WPA Method

U.S. intelligence services have long tracked former Soviet Union (FSU) submarines. In the mid-80s, the United States found that the radiation noise of FSU submarines had suddenly dropped tremendously, which was abnormal and was quickly reported to the US Department of Defense. According to the analysis of the Technical Department, the United States drew the unanimous conclusion: “It seems that the Soviets have mastered raft technology.” —Junior Johnson, Commander of the US Navy

“Raft” is a revolutionary form of technology that facilitates the mechanical noise control of submarines. By clarifying the evolution of vibration isolation system architecture, we cannot only analyze the transmission characteristics of the raft from a dynamic perspective, but also systematically study the reasons for the “emergence” of raft function/performance based on the system science, thereby achieving the efficient vibration isolation and overall resource conservation of carriers. This chapter starts with single-stage and double-stage vibration isolation to ensure systematic analysis.

8.1 Single-Stage and Double-Stage Vibration Isolation 8.1.1 Vibration Isolation System Model and Basic Transmission Characteristics It is necessary to review the basic characteristics of single/double-stage vibration isolation. Figures 8.1 and 8.2 show the physical model of single/double-stage vibration isolation; source device m and intermediate mass m  are simplified to singledegree-of-freedom rigid bodies. Complex models may also be used with identical basic transmission characteristics.

© Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 C. Wu, Wave Propagation Approach for Structural Vibration, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-7237-1_8

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Fig. 8.1 Physical model of single-stage vibration isolation

Fig. 8.2 Physical model of double-stage vibration isolation

The engineering design is made with reference to Professor Yan Jikuan’s Mechanical Vibration Isolation Technology [1]. The force transmission rate curve of singlestage vibration isolation is shown in Fig. 8.3. The system resonance frequency divides the frequency coordinates into three areas: the stiffness control area, damping control area, and mass control area. Isolator damping determines the amplitude and half bandwidth of the formant. Although it has a simple form, the physical mode of single/double-stage vibration isolation contains the core elements of the system and describes the essential transmission characteristics of the system. The system resonance frequency fr of single-stage vibration isolation is also called “mounted frequency” in engineering. fr =

1 2π



K m

(8.1)

8.1 Single-Stage and Double-Stage Vibration Isolation

181

2

Force transmission rate

Mounted frequency 1

0

A

B

-1

High frequency Harmonics

-2

0dB/Oct -3

Stiffness control area

-12dB/Oct

Damping Mass control area control area

fr

-4 10-1

10

0

10

1

C 102

Frequency (Hz) Fig. 8.3 Transmission characteristics of single-stage vibration isolation

where K —Isolator stiffness; m—Source device mass.

√ Within the frequency range of f ≤ f r 2, the transmission characteristics of the system are attenuated by 0 dB/Oct per octave; that is, the asymptote AB. √ Within the frequency range of f ≥ 2 f r , the transmission characteristics of the system are attenuated by 12 dB/Oct per octave; that is, the asymptote BC. The force transmission rate curve can be simplified as a fold line, with mounted frequency as the intersection. If the source device is treated as a multi-degree-of-freedom rigid body or even an elastic body, the force transmission rate curve will be superimposed with some harmonic components. In double-stage vibration isolation, the source device and intermediate mass are also assumed to be single-degree-of-freedom rigid bodies. Figure 8.4 shows the force transmission rate characteristics, with a curve and the corresponding asymptote CD added. The intersection of the three-step asymptote corresponds to the two mounted frequencies of the system f r1 and f r2 [2].   ⎫ 1 K1 m1⎪ ⎬ 2π 1  ⎪ f r1 = K 2 /m 2 ⎭ 2π

f r1 =

where K 1 —Upper isolator stiffness; m 1 —Source device mass;

(8.2)

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Force transmission rate

Mounted frequency Mounted frequency Efficient vibration isolation domain

High frequency harmonics

Frequency (Hz)

Fig. 8.4 Transmission characteristics of double-stage vibration isolation

K 2 —Lower isolator stiffness; m 2 —Intermediate mass. Certainly, the CD line in Fig. 8.4 above, is the ideal state. When corresponding to mass ratio μ → ∞, the force transmission rate curve is attenuated by 24 dB/Oct (J. C. Snowdon 1962, 1968, 1971, 1979). When corresponding to other mass ratios, the transmission rate curve of double-stage vibration isolation will lift up, with the asymptote of single-stage vibration isolation (extended line CC) as the upper limit. Generally, under discretionary parameters, the force transmission rate curve of double-stage vibration isolation is between BC and CC, the upper and lower numerical boundaries. Isolator damping suppresses the formant and determines the half bandwidth. The role of damping in engineering is somewhat exaggerated, and the engineering design is often in an “over-designed” state. The standing wave effect of the isolator and the modal frequency of the intermediate mass may lead to a risk of higher harmonic leakage in the system, which will have a significant effect on the total vibration level drop and a complex effect on energy leakage at a specific frequency. The AB segment in force transmission rate curve of double-stage vibration isolation, the transmission characteristics are attenuated by 0 dB/Oct. In the CD segment, i.e., the range of f ≥ ξ1 f r2 , the transmission characteristics follow the laws of attenuation of 24 dB/Oct (ξ1 ≈ 1.3, set constant). However, within the range of f r1 and f r2 , the laws of attenuation are 12 dB/Oct, which is identical to the BC segment in the transmission characteristics of single-stage vibration isolation with a certain level of deterioration. If the low-order vibration energy of the source device is concentrated within this range, double-stage vibration isolation may not be as good as single-stage vibration isolation.

8.1 Single-Stage and Double-Stage Vibration Isolation

183

After conducting research, the author found that the description of the transmission characteristics of a vibration isolation system with force, acceleration or power flow has a uniform shape, with differences only in the translation of the ordinate. The first two are expressed in relative quantities, and power flow is expressed in absolute values. Insertion loss is described as a relative quantity.

I L =

⎧ ⎪ ⎨

0 dB/Oct, ( f ≤ f r1

√

2)

−12 dB/Oct, ( f r1 ≤ f ≤ f r2 ) ⎪ ⎩ −24 dB/Oct, ( f ≥ ξ1 f r2 )

(8.3)

It can be seen that the performance of double-stage vibration isolation is better than that of single-stage vibration isolation due to the maximum application of the system in the CD area. In the sensitive area of this frequency band, the following deterioration of vibration isolation performance should be avoided for double-stage vibration isolation [3]. (1) The BC segment is exposed to the risk of overall lifting up and enlarging. Because of f r2 > f r and f r2 → f r1 , the area around the formant f r2 may be enlarged. Here, f r is the resonance frequency of single-stage vibration isolation. The lower limit of f r1 is restricted by the vibration intensity index of the device, so there is a possibility of harmonic deterioration within the frequency range f r2 . (2) The BC segment may be superimposed with harmonic components (commonly known as “burr”). The source device is actually a multi-degree-of-freedom rigid body or even an elastic body, with the further reduced transmission characteristics of the system. The accurate selection of system parameters according to the characteristics of the source device and system helps to avoid the degradation of double-stage vibration isolation performance. (3) The damping β1 and β2 of the upper and lower isolators is directly related to the half bandwidth and peak value of formants f r1 and f r2 , respectively, and they are denoted as β1 ⇒ f r1 and β2 ⇒ f r2 . Raft frame damping controls the higher harmonics of the CD segment. This is similar in base damping, with systematic performance in the transmission curve. (4) The addition of mass ratio μ = m 2 m 1 , compression of the BC segment and enlargement of the CD segment may generally help to improve the vibration isolation efficiency. However, it is necessary to stagger the force source frequency and BC segment. The basic transmission rate curve is the most basic description of single/doublestage vibration isolation. Designers are required to master the core points and understand the principles in order to better help the design and interpret the raft.

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8.1.2 Influence of Mass Ratio Increasing the mass of the intermediate raft can improve the mass ratio and enlarge the CD area, but as resource consumption, the intermediate raft has no other effect on the carrier. In order to ensure the vibration isolation effect, the mass ratio is generally 0.3–0.8. Assuming that the source device is 8.0 t and the mass ratio is μ = 0.5, then the mass of the intermediate raft is 4.0 t. This is a huge overall resource consumption of the carrier weight/space. It is easy to imagine how huge the overall resource consumption will be if double-stage vibration isolation is provided for all source devices. What changes will be brought by the use of the complex model if the source device and raft are elastic? First, they will not affect the basic transmission characteristics of the system unless it has a completely unrealistic design. Second, for each-degree-offreedom released by the physical model, the transmission rate curve will superimpose a side lobe harmonic component around the characteristic frequency. These are all possible frequency leakage points for the vibration energy of the raft which will impair vibration isolation performance.

8.2 Raft Vibration Isolation System Raft is the abbreviation of the Raft Vibration Isolation System which was developed and first applied to attack submarines by Americans, resulting in significantly reduced radiation noise. Figure 8.5 shows a schematic diagram of a typical raft, including the main components. If single/double-stage vibration isolation is considered a simple system, the raft is usually much more complicated. Designers are required to not only analyze the dynamic characteristics of the raft, but also research the functional properties of the system, as well as general properties of the raft. It is also necessary to explore the potential of the raft through top-level optimization from multiple perspectives according to different application scenarios.

8.2.1 History of Raft Research Raft is one of the core technologies for the mechanical noise control of carriers. In the early days, raft was mainly applied in military ships, then later in civilian ships. Military ships such as submarines and minesweepers are important application platforms for the raft. Applications of raft technology in military ships were rarely publicly reported, but their importance in engineering application was still apparent. Raft technology was first used on submarines. The former Soviet Union

8.2 Raft Vibration Isolation System

185

Fig. 8.5 Schematic diagram of typical raft

mastered and applied raft technology in the early 1980s. As a result, the underwater radiation noise of submarines was significantly reduced. Raft technology with various structures was used on K-class submarines for the concentrated vibration isolation of the auxiliary engines. Damping rafts were also used as the power units of Japanese Oyashio-class and Soryu-class conventionally-powered submarines to control vibration transmission. In the field of civilian ships, rafts were used on the fishing survey ship built by Glasgow Shipyard in Scotland, in the 1980s, to solve the problem of vibration noise. This is an early case of rafts being used in the civil industry [4]. Professor Yan [5] introduced raft technology for the first time in his New Development of Vibration Isolation and Noise Reduction Technology. On the basis of double-stage vibration isolation, the paper briefly introduces the transmission characteristics, design points, and engineering applications of raft technology. Shen and Yan [6–8] introduced and analyzed some typical examples of raft projects. Professor Shi et al. of the Naval University of Engineering [9] published Research on DoubleLayer Vibration Damping (Raft) Device for Luxury Yacht in which the two diesel generator sets used on the Lijiang Luxury Yacht constituted the first raft vibration isolation device in China. In the early days of basic theoretical research on raft technology in China, the main focus was on dynamic modeling and characteristics. The applied theoretical methods mainly included rigid body dynamics, the finite element method, mechanical impedance method, and transmission matrix. Zhu [10] proposed the modal impedance synthesis method to study raft technology, which overcame the shortcomings of the impedance synthesis method. Wu and Fu [2] studied the double-stage vibration isolation of elastic rafts to determine the influence area of viscoelastic

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8 Analysis of Raft Using WPA Method

damping, and conducted a comparative study of rafts and double-stage vibration isolation [3], parameter and structure correlation, and analog analysis. Chen et al. [11] based on the maximum entropy method, put forward a 3D optimized solution for double-stage vibration isolation which differed from the traditional conclusion about the superiority of single-stage. Liu et al. [12] studied the impact response of a multilayer vibration isolation system. Wu et al. [13] proposed three effect principles of rafts, including Mass, Tuning and Mixed Neutralized, and conducted research on the double-stage vibration isolation of a public raft body, the early prototype of emergence. Wang et al. [14] analyzed the response of a raft isolation system with the state space method. Chen et al. [15] conducted computer simulation research on rafts based on the equation of state. Lin [16] discussed the application and development of rafts on ships and looked forward to the application prospects of rafts. Shang and Li [17] analyzed the evolution of the characteristics of rafts on ships and the trends and reasons for the changes. Sheng et al. [18] established a statistical analysis model of vibration mechanics for rafts and studied the vibration response and power flow-based statistical characteristics of rafts under steady-state random force excitation. Yu et al. [19] studied the parameter optimization of rafts. From the perspective of frequency optimization, their paper takes the isolator stiffness parameter as an example to discuss the system parameter selection scheme of a multistage vibration isolation system, proposes the variable tolerance polyhedron optimization method and perturbation optimization method, and studies such hidden parameter problems as frequency selection. Feng et al. [20] studied the effect of raft frame and foundation elasticity on the vibration transmission of a complex vibration isolation system and analyzed the vibration transmission process and characteristics of an asymmetric multipoint elastic raft-flexible foundation system with the power flow calculation method based on the introduction of a substructure. Yu et al. [21] studied the power flow of rafts through testing and proposed a set of testing methods. Sun and Song [22], based on the establishment of a dynamic model of a flexible foundation multi-support elastic raft coupled with a vibration isolation system, gave the dynamic transmission equation and power flow expression of the coupled system using the comprehensive analysis method of the dynamic characteristics of the subsystem, and discussed the flexible coupling effect of mounted frequency and supporting structure, and its influence on vibration isolation. Zhang and Fu [23] analyzed the influence of the main parameters of a raft on the vibration isolation performance of the system. Li et al. [24] studied the power flow transmission characteristics of flat-plate rafts under complex excitation conditions, derived the expression of power flow transmission from elastic rafts, evaluated the effect of raft vibration isolation from the perspective of vibration energy transmission, and analyzed the influence of structural parameters such as unit layout and raft stiffness on power flow transmission. As a result, the general guidelines for the selection of structural parameters in raft design were given. Chen [25] studied the Mass Effect of rafts using the WPA method and completed the comparison of theory and experiment. Early research was very extensive and innovative. By the end of the 1990s, raft theory was sufficiently complete to support and guide engineering design. There has

8.2 Raft Vibration Isolation System

187

been an increasing number of analysis cases for the Complete Engineering Model based on the FEM method, promoting the research and application of the raft.

8.2.1.1

System Emergence Facing the Complex Requirements of Raft Technology

In the new century, the theoretical methods and engineering practice cognition of raft technology are gradually increasing, and designers regard rafts as a subsystem under the complex giant system of vehicles. At different times, hot research topics included whether the third-level raft was better than the second-level raft, whether the integration level of rafts was complex and appropriate, and whether multistage rafts or combined rafts were superior. However, the development of raft technology has reached a new crossroads: first, why are multistage rafts so rarely used in vehicles? Is it simply because we have not seen enough or are there other underlying reasons? Second, as rafts are so useful, why are they seldom used in general industries? This is the so-called “raft application paradox”. With the development of theories [13, 25, 28], we may have taken another step forward. We study rafts through systematic thinking and further reveal the confusion behind the phenomenon by studying the two cross-disciplines of dynamics and rafts, and the prominent contradiction between rafts maintaining efficient vibration isolation and the limited overall resources. This is the common requirement of the carrier, which has led to two important research topics: the top-level optimization of the raft of the carrier in open complex giant systems such as ships, and the in-depth analysis and discussion of structural waves so as to study the cancellation mechanism of force source input raft structural waves, the transmission and attenuation of structural discontinuities to waves, the energy storage and wave type conversion of the structure, and finally, the system contribution rate of these parameter changes to the target control parameters of the vehicle, thereby continuously tapping the engineering application potential. In theory, it may be difficult to argue for a broader discussion of rafts under the limited boundaries of the project at the moment, but one truth has been clarified: if the boundaries discussed are too broad, there will be no answer as to which kind of raft is best. Since the project has boundaries, the secondary raft constrained by the project can meet the short-term and future long-term planning of the project. As an integrated innovation, research on the matching of raft and system contribution rate has more practical engineering value.

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8 Analysis of Raft Using WPA Method

8.2.2 Definition, Modeling, and Basic Characteristics of Rafts Just like double-stage vibration isolation, theoretical research on rafts has also gone through a simple, complex, and then simple iterative development process. From structural dynamic analysis to systematic thinking research, from general definition to special definition, this gradual cross-integration and research subdivision often obtains twice the results with half the effort in optimizing the rafts of vehicles.

8.2.2.1

General Definition of Raft

A centralized G(G ≥ 2)-level vibration isolation system for N (N ≥ 2) independent source equipment. N in the definition represents the number of source devices and G represents the vibration isolation level. This definition can be understood as the general definition of a raft, such as “G-level raft”. When G ≥ 3, the weight and volume of the raft, vibration isolation efficiency, harmonic leakage, and vibration intensity of the source equipment should be comprehensively balanced into the system contribution rate of the vehicle. After decades of research at home and abroad, there are many definitions and explanations of rafts which we can refer to, but it is important to keep pace with the times.

8.2.2.2

Definition of Submarine Raft

A centralized double-stage vibration isolation system for N (N ≥ 2) individual source equipment [13]. G = 2 is the “Two-level raft”. Why is it necessary to subdivide submarine rafts? Obviously, because the secondary raft is becoming an application trend, not to reflect the unique refinement value. In practice, the performance of a raft is not ideal, and it should be proven afterward that it is not because multistage vibration isolation is not adopted, but because there are problems in the theoretical prediction, design, energy leakage or debugging. The two definitions of raft convey the following information of the system: (1) It is suggested to adopt a twin-stage raft. There are various types and combinations of rafts. The author does not recommend the use of G ≥ 3-level rafts. This is not only the conclusion of theoretical analysis, but also the indirect proof of the existing application of vehicles in various countries. (2) Double-stage vibration isolation is a special case of raft when N = 1 and G = 2. Compared with double-stage vibration isolation, a raft is a multisource system; the source equipment is not only a force source element, but also a mass element that does not work at a certain speed in time order. In this way, designers can actively utilize the emergence characteristics of element correlation to save

8.2 Raft Vibration Isolation System

189

weight and space resources for the vehicle, and ensure good vibration isolation efficiency. (3) Independence of source equipment. This definition is exclusive to avoid confusing partial double-stage vibration isolation with rafts. For example, diesel generator sets, which consist of two power sources linked with the same frequency, are atypical rafts. There are many options for the rafts of vehicles, but the author recommends the 2-level raft. This is not the result of compromise; long-term practice has proven that the 2-level raft is basically meet the most stringent acoustic design requirements of submarines. With the continuous reduction of the vibration level of the source equipment, the 2-level raft can give consideration to the vibration intensity, impact resistance, and smaller overall resource burden of the equipment, which is the mainstream application framework and trend of vehicles.

8.2.3 Physical Modeling and Coordination Conditions The physical model of a 2-level raft is shown in Fig. 8.6 [25]. Here N = G = 2, the source equipment is regarded as a rigid body, and the raft frame is represented by beam ➀. The foundation is simulated by beam ➁ and beam ➂. The installation points of vibration isolation for beam ➀ are marked as x21 , x11 , x12 , x22 in order and beam ➁ and beam ➂ as y21 , y22 . The source equipment is a rigid body, and the raft frame and foundation are simulated by elastic beams, forming a hybrid dynamic system. One question is the rationality of such modeling. The key to the complexity and advantages or disadvantages of modeling is whether it can correctly express the main characteristics of the system. Complex physical models seem to be more easily recognized. The above model can reflect the essential mechanism of a raft:

Fig. 8.6 Physical model of raft

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8 Analysis of Raft Using WPA Method

(1) The main features of the raft have been preserved. The model includes the main characteristics of a raft, ignoring certain secondary characteristics, and the overall characteristics of the system are clear. Further analysis and calculation reveal that if all six-degrees-of-freedom of the source equipment are released, the system transfer curve only superimposes small harmonic components at the peak cluster of the installation frequency. (2) It highlights the essential transfer characteristics of the system. After simplification and abstraction, the system retains the most core parameters, such as the modal frequencies of the beams, installation frequency of the system, structural “discontinuity points”, influence of viscoelastic damping, etc., and clarifies their influence areas, coupling attributes, and influence modes. On the other hand, complex modeling is not always positive. Too much local information often masks the main features, making it difficult to distinguish or even identify errors. (3) Simple modeling is not equal to simple analysis. When single/double-stage vibration isolation is written into the textbooks, the physical models are very simple, but they can reveal the essential laws. The reason why we can abstract and summarize the essential laws is also inseparable from the characteristics and enlightenment of different complexity models in the iterative process. Induction means summarizing new research findings and existing research results. The cycle of the simple and complex modeling of rafts is a process of deepening our understanding. The secondary factor of cognition, which may be separated by stratification, has become the main factor again, which represents a spiral rise. On the contrary, this kind of development process is more secure for the project.

8.2.4 Analysis of Basic Transfer Characteristics In the early days, we carried out a comparative study between rafts and doublestage vibration isolation. The research on double-stage vibration isolation is more mature, giving the research on rafts a relatively high starting point, while also being conducive to summarizing the similarities, differences, and related attributes of the two, and strengthening the overall cognition of raft system attributes [1, 13]. The comparative study is based on mature theories, and we can better explore the most essential dynamic characteristics of rafts and summarize their evolution laws. Attention should also be paid to the description of the force transmission rate and power flow. The most essential asymptotic relationship in system attributes remains unchanged between the force transmission rate and power flow, and between the raft and double-stage vibration isolation, except that the overall translation of transmission curve and formants become formant clusters, as shown in Fig. 8.7. When the transmission rate characteristic is expressed in the simplest form of asymptote, as shown in Fig. 8.8, the raft and double-stage vibration isolation have exactly the same essential system attributes.

8.2 Raft Vibration Isolation System

191

Transfer power flow /W

Invalid frequency domain Dangerous frequency domain Effective frequency domain High efficient domain

External

High frequency

C’

Mixing

Frequency/Hz

Transfer power flow /W

Fig. 8.7 Basic transfer characteristics of raft

A

Invalid frequency domain

B

Dangerous frequency domain Effective frequency domain High efficient domain

C

-12dB/Hz 0dB/Hz

-12dB/Hz

C’

-24dB/Hz D

0

E

Frequency/Hz

Fig. 8.8 Basic characteristics of transfer power flow of raft

(1) The transmission rate characteristic is the same as that of double-stage vibration isolation. The formants of the two systems evolve into two groups of formants S2 Si clusters which are recorded as ( f r1 , f r2 ) ⇒ ( f rS1 j , f r j ), where f rc corresponds to the center frequency of the peak clusters, j = 1, 2, . . . , N . (2) In each group of formant clusters, the number of peaks is equal to (N + 1), where N is the number of source devices (in terms of single-degree-of-freedom). However, when the equipment is rigidly installed, due to K → ∞ ⇒ f r → ∞, one resonance peak is thus reduced. (3) The raft follows the rule of three-segment line division. As with two-stage vibration isolation, the AB section is attenuated by 0 dB/Oct, BC section by 12 dB/Oct, and CD section by 24 dB/Oct. The structural wave energy input

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8 Analysis of Raft Using WPA Method

from the source equipment to the raft produces a “mixing effect”. This “external mixing” can increase the additional attenuation of the system. The dotted line indicates that the transferred power flow moves downward as a whole. (4) The damping of the upper vibration isolator is associated with the first group of formant clusters and marked as β1 j ⇒ f rS1 j ; the damping of the lower vibration isolator is associated with the second group of formant clusters and marked as β2 j ⇒ f rS2 j ; raft damping controls the half bandwidth of the formant generated by structural higher harmonics in the transfer curve, which is marked as β0 j ⇒ f rhj , where f rhj is the amplitude of higher harmonics. Attenuation caused by damping is also called “internal mixing”. In the basic theoretical research of rafts, we have adopted the two methods of complex accurate modeling and simple abstract modeling, which only reveal the essential attributes of the system. The simple abstraction at this time expounds upon the logical concept of simple modeling, which does not always bring us simplicity: (1) Rafts are similar to double-stage vibration isolation. The number of formant clusters on rafts is greater, which means that the possibility of the deterioration of transmission rate performance or harmonic leakage increases. This also shows that rafts do not always have higher vibration isolation efficiency than doublestage vibration isolation. (2) The necessity of selecting a raft in level-G ≥ 3 is weakening. It is an inevitable trend that the vibration value of the source equipment is continuously decreasing. With the development of technology, it has become opposite to the selection of raft levels. The secondary raft is not only consistent with the actual application of vehicles at home and abroad, but also represents the mainstream direction of development. (3) When the source device m 1 releases all six-degrees-of-freedom, based on the physical model of Fig. 8.6, the transfer rate curve will increase from a single formant to six. They are generated by the translation or rotation of the rigid body of the source equipment and superimposed in the high frequency region of the formant cluster f rSj or the transmission rate curve in the form of harmonics. Under the engineering boundary, the main peak shape of the raft and doublestage vibration isolation transmission rate does not change, and the peak clusters are all generated by the vertical displacement of the equipment and beams. A mathematical description is as follows:   f riS1 ∈  f r1S1 , f r2S1 , . . . , f rnS1  S2 S2 S2 f rS2 j ∈ f r1 , f r2 , . . . , f rn

(8.4)

Where superscript S1 represents the first group of formant clusters, S2 represent the second group of formant clusters, the subscript number n represents the device number and r represents the system formant. Some resonance summits in Eq. (8.4), are left blank. Therefore, there is always (i, j ≤ N ). Theoretically, the first group of formants has N + 1 formants, of

8.2 Raft Vibration Isolation System

193

which the main resonance frequency f rcS1 = f r1 in the first group of formants is estimated by Eq. (8.2). This behavior of the coupling system is caused by the standing wave effect of an elastic wave in the beam. Of course, these influences are of the second level and do not affect the main transmission rate characteristics of the system. They are parasitic on the main formant and usually “very high in absolute value, but small in relative value”. (4) From the above analysis of the raft, we can easily conclude the following logical inference: the transfer rate characteristic of the three-stage raft will add an asymptotic line DE or D E and decay according to 36 dB/Oct.

8.3 System Thinking and Consideration of Rafts In design, objects should always be considered in a slightly larger scope: chairs in rooms, rooms in residences, residences in their surrounding environment, and the surrounding environment in urban planning. —Eliel Saarinen

8.3.1 Raft Application Paradox It was mentioned in the title page of this chapter that raft technology is regarded as a revolutionary technology of submarine vibration and noise reduction. The application of raft technology is indispensable in vehicles with high acoustic requirements. However, there is a long-standing phenomenon that cannot be ignored: “Why is raft so rarely used or not used in general industries now that it is so useful?” Since it is useful but not used, it is difficult and awkward to comment on it. Another phenomenon is that three-stage vibration isolation or three-stage raft is rarely adopted by vehicles. The reason behind these confusing phenomena is not merely cost. The raft application is due to another important demand: saving overall resources.

8.3.2 Definition of Emergence Saving overall resources is a general demand of vehicles. A system with a high integration level has more emergence according to the principles of system thinking. A raft has a higher level of integration than double-stage vibration isolation, and the source device is also a mass element. Due to the internal logical relationship between these mass elements, new system functions (saving resources) have emerged and system density has increased, which is the highest pursuit of weapons platforms. The definition of emergence [26] is: “Something that shows up, appears or emerges during the operation of a system”. When a system is formed by the integration of

194

8 Analysis of Raft Using WPA Method

Fig. 8.9 Raft integrated by N double-stage vibration isolation

many entities, the interaction between these entities will bring out the functions, behavior, performance, and other intrinsic attributes of the system. The definition of emergence seems simple, but it perfectly explains how overall resources (weight) are saved through system thinking. For example, when N double-stage vibration isolation with parameters from (m 1 , k1 , m 1 , K 1 ) to (m N , k N , m N , K N ) is combined into a raft, the interaction between mass elements will bring out the system’s functions, performance, and other intrinsic attributes. Usually, raft mass m 0 is taken as the maximum of {m 1 , m 2 , . . . , m N }, and then raft body (N − 1) can be saved from the raft, as shown in Fig. 8.9. It is obvious that the more source devices and the greater abundance of emergence, the more significant the resource savings. From single/double-stage vibration isolation to raft, the system integration level is enhanced; the raft has achieved satisfactory emergence by using system complexity, namely high vibration isolation efficiency; and the consumption of weight/space resources is lower, which means that the vibration isolation efficiency per unit weight has been improved. A raft is a known-unknown. The internal logical relationships between its architecture and elements are very reasonable. The emergence of the system can save valuable weight/space resources for the vehicle. The study of raft dynamics is extensive, in-depth, and fruitful [13, 18–25]. The dynamic method can contain emergence, but it cannot guide emergence analysis independently. System thinking analyzes the order of elements and the connections between them and the raft architecture. The former evolves more functions while the latter determines the main characteristics of transmissibility so as to make the most of things. The study of rafts with system science is supplemental to dynamic analysis. It explores the more general transmissibility attributes in detail, enabling us to better master the overall attributes of rafts.

8.3 System Thinking and Consideration of Rafts

8.3.2.1

195

Case Analysis

In the following case analysis, we can see the overall amount of resources that can be saved by raft technology, what corresponding relationships it has, and the possible optimization degree. Let us start with the simple case in Fig. 8.9. Two double-stage vibration isolations are combined: source device m 1 = 8.0 t, m 2 = 5.0 t. If both are designed for doublestage vibration isolation and the mass ratio is maintained at μ = 0.5, the mass of intermediate raft body m 1 = 4.0 t and m 2 = 2.5 t, so the total mass consumption the raft. The common intermediate of the raft is m 1 + m 2 = 6.5 t. Now  reconstruct  raft body is selected as m  = max m 1 , m 2 according to the maximum mass ratio, then raft body m 2 can be saved; at the same time, the equivalent mass ratio of the source device m 2 is increased from μ2 = 0.5 to μ2 = 0.8. Something magical happens because of system emergence: the vibration isolation efficiency is improved, weight/space resources are saved and the impact resistance of the system and devices are improved. According to the principles of system science, we continue to analyze more general situations. Double-stage vibration isolation is adopted for all N source devices, or all these elements are reconstructed into a raft. The two design cases are as follows: Double-stage vibration isolation scheme: Suppose N source devices in the vehicle are designed for double-stage vibration isolation, and the mass ratio is uniformly agreed to be μn = 0.5(n = 1, 2, 3, . . . , N ). In this case, the accumulated mass of the double-stage vibration isolation raft body is M (D) =

N 

μn · m n

(8.5)

n=1

 where m n is the mass of the source device n and μn = m  mn is the equivalent mass  ratio of the source device n, m  = max m 1 , m 2 , m 3 , . . . , m n . Raft scheme: Provided that N source devices can be integrated into a raft, the most important control device is set as the main device, and the mass ratio is μ0 = 0.5. In this case, the mass of the raft frame M (R) is M (R) = μ0 · max{ m n |n = 1, 2, 3, . . . , N }

(8.6)

where M (R) is the mass of the raft frame and μ0 is the mass ratio of the source device that has the greatest mass among N source devices. The total mass difference of the two design schemes is (only some of the design elements are selected for the raft) M = M (D) − M (R) =

N  n=1

μn m n −μ0 · max{ m n |n = 1, 2, . . . , N }

(8.7)

196

8 Analysis of Raft Using WPA Method

Table 8.1 Comparison of weight consumption between double-stage vibration isolation and raft Weight of source device/t

Weight of raft body/t

Mass ratio of double-stage vibration isolation

Mass of raft/t

Mass ratio of raft

10.0

5.0

0.5

5.0

0.50

8.0

4.0

0.5

0

0.63

7.8

3.9

0.5

0

0.64

6.6

3.3

0.5

0

0.76

5.4

2.7

0.5

0

0.93

4.6

2.3

0.5

0

1.09

3.6

1.8

0.5

0

1.39

46.0

23.0

0.5

5.0

0.50–1.39

Let N = 8. It can be seen from Table 8.1 that the total weight of the source devices is 46 t. The total weight resources consumed in the double-stage vibration isolation scheme is 23 t, while that in the raft scheme is 5.0 t, just 21.7% of that in the doublestage vibration isolation scheme, thus saving 18 t of total weight resources (the additional consumption of weight and space resources by the vibration isolator which has been saved is not counted). At the same time, in the raft scheme, the equivalent mass ratio of other source devices is increased from 0.5 to 0.50–1.39 (in variety), and the impact resistance of the system is also greatly improved. Emergence endows the raft with unique characteristics that are different from those of double-stage vibration isolation, which is of great value to the vehicle.

8.3.3 Several Inferences System science explains that rafts can save resources through emergence. This provides a broader balance in terms of vibration isolator efficiency, mass ratio, impact performance, and emergence utilization for the application of rafts in carriers. Based on emergence research, we can make the following inferences. Inference 1: The more complex the system, the more sufficient its emergence. Therefore, the more equipment on the raft, the more sufficient the emergence of the system, the more significant the overall resource saving such as weight/space. The emergence of the system is as important to open complex giant systems as rafts are too quiet carriers. The combination of system science and structural dynamics provides new ideas for the design of high-density rafts on carriers. Inference 2: The raft frame will have higher mass as more equipment is provided on the raft. Therefore, the high mass ratio of the raft is genetic. The important theoretical concept of the mass ratio is established for double-stage vibration isolation. Rafts evolved from double-stage vibration isolation, which is the

8.3 System Thinking and Consideration of Rafts

197

product of system emergence and reconstruction, and an extended innovation from the concept of mass ratio. Inference 3: The whole mass of the raft is much greater than that of doublestage vibration isolation. Therefore, the impact resistance of the source device on the raft is good and genetic! The overall mass of the raft can easily reach more than five times that of double-stage vibration isolation. At the same time, after doublestage impact buffer design (approximate vibration isolation, not the best impact resistance), the source device is usually provided with better impact resistance. Inference 4: The more equipments on the raft, the higher isolation efficiency of weight ratio. It is not difficult to deduce this inference from Inference 1. This is the advantage and charm of rafts. As an integrated innovation achievement, the raft has the maximum system contribution rate in the carrier. We have been identifying the general system properties of rafts. In the crossdisciplines of structural dynamics and system science, the author and his team experienced the transformation and refocusing of research/design ideas in the process of spiraling cognition. By connecting raft elements, emergence, and architecture, we can find new optimization space and optimization paths such as harmonic leakage coefficient and interaction of force source elements.

8.3.4 Large Raft and Small Raft It is not easy to divide into “large raft” and “small raft”, simply according to their respective numbers of integrated pieces of equipment. In engineering applications, the reasonable design size of rafts involves the carrier, resources, and vibration isolation efficiency. From the above inferences, it can be concluded that the greater the equipment integration, the larger the raft frame (logically), the greater the equivalent mass ratio and overall resources saved thereby, and the better the vibration isolation efficiency and impact resistance performance. In short, the larger the raft, the better, but it must be matched with the design capacity, as shown in Fig. 8.10. Their advantages and disadvantages are analyzed as follows: (1) With the permission of the general layout of the carrier, a large raft will be provided with a high degree of integration, which can save more valuable overall weight and volume resources through the emergence of the system. (2) In general, the larger the number of source devices, the larger the raft and the larger the raft frame, so the more sufficient the external mixing effect, the more significant the transfer attenuation effect. This is an autonomous “active vibration control” behavior which illustrates the theoretical benefits of the application of large or super-large rafts. (3) The vibration energy of the raft frame is input by the source device and stored in the form of structural waves. Large rafts can store structural waves with lower frequency, i.e., long wave. These low frequency components are eliminated by the mixed offset effect or dissipated through the damping and discontinuity

198

8 Analysis of Raft Using WPA Method

Fig. 8.10 Increase in number of source devices and harmonic leakage increment

reflection. Therefore, a large raft is conducive to the storage and attenuation of low frequency structural waves. The richer the storage, the more effective the internal mixed offset effect. However, there are always two sides of things, and large rafts have many disadvantages  (1) The energy transmission of the source device is attenuated by 12 dB Oct or  24 dB Oct after passing single/double-stage vibration isolation. If the relative resonance frequency of the disturbance frequency is high, the vibration isolation quantity will be generally sufficient. If the raft is too large, there will be many resonance points and energy leakage. The seemingly small leakage or even air noise of the equipment will destroy the overall vibration isolation performance of the raft. Thus, the size of the raft depends on whether the designer can control it. (2) A more complex system will lead to worse local design, or acoustic short circuit and interference caused by pipeline processing, construction, etc., all of which may lead to the leakage of vibration energy. Even if a small local failure occurs in a large raft, the overall vibration isolation performance will deteriorate. These possible leakage points (equivalent to bugs in software) limit the upper limit of the vibration isolation efficiency of a super-large raft. (3) The “highway principle” limits the vibration isolation efficiency of a large raft. According to the theory of power flow, the vibration energy transfer of a raft follows the “highway principle” in which the vibration energy in the system always flows from the place with high energy to the place with the weakest system and escapes through the most spacious channel. Generally speaking, structural waves are regarded as automobiles which carry vibration energy, and will automatically choose to go to highways rather than country lanes. Large rafts result in a large raft frame, low natural frequency, more serious energy

8.3 System Thinking and Consideration of Rafts

199

leakage in theory, and a higher possibility of the systematic deterioration and significant reduction of the vibration isolation efficiency. Therefore, in engineering application, if the raft is larger, the new functions and performance produced by emergence will be more significant. However, the selection of raft scale and equipment quantity must be matched with current technical capacity, vibration level of source devices, and system integration, especially system concealment ability such as the interference effect between multiple sources, which requires designers to consider both efficiency and engineering control ability. It is advisable to upgrade them gradually in practice and avoid application risks through careful debugging.

8.4 Analysis of Rafts Using the WPA Method The physical model of the raft is in Fig. 8.6. The source devices m 1 and m 2 are simplified as a single degree of a free rigid body, the raft frame is simplified as uniform straight beam ➀ and the installation foundation is simulated by uniform straight beam ➁ and beam ➂, which are connected into a coupling system through isolators. This simplification retains the main characteristics of the raft and can focus on how the inherent characteristics of the core parameters of the system, such as beam ➁ and beam ➂, affect the transmission characteristics of the system. This kind of hybrid dynamic system has some advantages by using the WPA method.

8.4.1 Internal Coupling Force Acting on the Raft There are two action points x21 and x22 of k21 and k22 on beam ➀ and the vibration isolator under the raft. The point forces after the release of the constraint are recorded as R11 and R12 respectively. In the same way, R2 and R3 are set as the forces exerted by the lower vibration isolator on beam ➁ and beam ➂ respectively R11 = −R2 R12 = −R3

 (8.8)

Under the action of point force, the lateral displacement of beam ➀ at x21 is w1 (x21 ), and the lateral displacement at x22 is w1 (x22 ), so the forces of the two lower vibration isolators are  R11 = −k21 [w1 (x21 ) − w2 (y21 )] (8.9) R12 = −k22 [w1 (x22 ) − w3 (y22 )] According to Eq. (8.8), we get

200

8 Analysis of Raft Using WPA Method

Fig. 8.11 Analysis of movement and stress at installation point of source device

 R2 = k21 [w1 (x21 ) − w2 (y21 )] R3 = k22 [w1 (x22 ) − w3 (y22 )]

(8.10)

The mass of source device p˜ 01 and isolator k11 constitute a mass spring system. The source device makes a coupling motion with the raft beam ➀, and internal force p˜ 01 applied to the beam through the isolator is as shown in Fig. 8.11. The general situation should first be taken into account. The source equipment has simple harmonic force p˜ 01 , which is connected with beam ➀ at point x21 through upper vibration isolator k11 , and Fb is the force acting on the beam [27, 28] Fb = k21 {wm − w1 (x21 )}

(8.11)

For simple harmonic motion, the time-dependent term e jωt is omitted. Assuming that the displacement of source device m 1 at a certain time is wm , and the displacement at the installation point x21 of the vibration isolator on the beam is w1 (x21 ), then the stress analysis of source device m 1 is p01 − Fb = −ω2 wm m 1

(8.12)

The result of Eqs. (8.11) and (8.12), simultaneously is Fb = k11 [wm − w21 (x21 )]



Fb = p0 + ω2 wm m 1

(8.13)

The force of the source device acting on beam ➀ through the vibration isolator is Fb =

m 1 ω2 k21 k21 w(x21 ) + p01 2 k21 − m 1 ω k21 − m 1 ω2

(8.14)

8.4 Analysis of Rafts Using the WPA Method

201

 Make Q = k21 (k21 − ω2 m 1 ), then force acting on the beam will be Fb = m 1 ω2 Qw(x21 ) + Qp01 = ktot w(x21 ) + Qp01

(8.15)

where ktot —equivalent dynamic stiffness; Q—magnification factor.

ktot = ω2 m 1 Q

(8.16)

The damping of the isolator can be introduced by its complex stiffness. Damping for structural hysteresis ∗ = k21 (1 + jβ) k21

(8.17)

where β is the hysteresis damping loss factor. Magnification factor Q is Q=

1 1 − (ω/ωm )2 /(1 + jβ)

(8.18)

 where ωm2 = k21 m 1 is the inherent frequency of the un-damped single-degree-offreedom mass spring system. For viscous damping, when the source device is subject to a damping force proportional to the velocity, then the force of the vibration isolator on the beam is Fb = k21 [wm − w1 (x21 )] + c[w˙ m − w˙ 1 (x21 )]

(8.19)

where c is the viscous damping coefficient. w˙ m and w˙ 1 (x21 ) are, respectively, the partial derivatives of time, i.e., velocities w˙ m = jωwm and w˙ 1 (x21 ) = jωw1 (x21 ). The force on the source device is determined by Eq. (8.5). Combined with Eqs. (8.12) and (8.5), we get Fb = p01 + m 1 ω2 wm Fb = (k21 + jcω)[wm − w1 (x21 )]

 (8.20)

According to Eq. (8.20), the force of the source device acting on the beam through the vibration isolator under the condition of viscous damping is obtained as follows: Fb =

 (k21 + jcω)  m 1 ω2 w1 (x21 ) + p01 2 k21 + jcω − m 1 ω

(8.21)

202

8 Analysis of Raft Using WPA Method

8.4.2 Vibration Displacement of the Raft Beam For beam ➀, the lateral displacement response of any point x on it is w˜ 1 (x, t), so w˜ 1 (x, t) = w1 (x) · e jωt

(8.22)

According to the reconstruction of the WPA method [25], the following is obtained: w1 (x) =

4 

An ek1n x + R1

n=1

+ F1

2 

an e−k1n |x−x21 | + R2

n=1 2 

an e−k1n |x−x22 |

n=1

an e−k1n |x−x11 | + F2

n=1

2 

2 

an e−k1n |x−x12 |

(8.23)

n=1

(k1 )4 =

ρ1 S1 ω2 E I1

(8.24)

k1n = {k1 , jk1 , −k1 , − jk1 |n = 1, 2, 3, 4 }

(8.25)

where S 1 —cross-sectional area of beam ➀; EI 1 —flexural stiffness obtained by introducing structural damping through E I1 (1 + jη), where η is the loss factor of structural damping; ρ 1 —material volume density of the beam; L 1 —length of the beam; x11 , x12 —coordinates of connection points of isolator on raft on beam ➀; x21 , x22 —coordinates of connection points of isolator under raft on beam ➀; R1 , R2 —forces exerted by source devices k 21 and k 22 of the lower isolator on beam ➀; F1 , F2 —forces of source devices m1 and m2 on raft on beam ➀, both of which are unknown; k1 —bending wave numbers on beam ➀. For beam ➁, assuming the lateral displacement response at any point y on the beam is w˜ 2 (y, t), then w˜ 2 (y, t) = w2 (y) · e jωt w2 (y) =

4  n=1

Bn ek2n y + R2

2  n=1

bn e−k2n |y−y21 |

(8.26)

(8.27)

8.4 Analysis of Rafts Using the WPA Method

203

For beam ➂, the lateral displacement response at any point y on the beam is w˜ 3 (y, t), so w˜ 3 (y, t) = w3 (y) · e jωt w3 (y) =

4 

Bn ek3n y + R3

n=1

2 

(8.28)

bn e−k3n |y−y22 |

(8.29)

n=1

where (k2 )4 = (k3 )4 =

ρ2 S2 ω2 , E I2 ρ3 S3 ω2 , E I3

2 k2n = {k2 , jk2 , −k2 , − jk2 |n = 1, 2, 3, 4 } k3n = {k3 , jk3 , −k3 , − jk3 |n = 1, 2, 3, 4 }



where S2 , S3 —cross-sectional areas of foundation beams ➁ and ➂; E I2 , E I3 —flexural rigidities of beams ➁ and ➂; ρ2 , ρ3 —material volume densities of beams ➁ and ➂; L 2 , L 3 —lengths of beams ➁ and ➂; y21 , y22 —coordinates of installation points of two lower isolators of beam ➀; k2n , k3n —bending wave numbers of beams ➁ and ➂; R2 , R3 —reaction forces of lower isolators on beams ➁ and ➂ (unknown quantity); Bn , Cn —corresponding unknown coefficient to be determined. According to the above stress analysis of the source devices, there are two source devices m 1 and m 2 F1 = ktot11 · w1 (x11 ) + Q 1 p˜ 01 F2 = ktot12 · w1 (x12 ) + Q 2 p˜ 02

 (8.30)

where ktot11 and ktot12 are, respectively, the equivalent dynamic stiffness of isolators k11 and k12 , and the following equation can be obtained according to Eq. (8.9) ktot11 = m 1 ω2 Q 1 ktot12 = m 2 ω2 Q 2

 (8.31)

The amplification factors are, respectively Q1 = Q2 =

1 1−(ω/ ωm1 )2 (1+ jβ1 ) 1 1−(ω/ ωm2 )2 (1+ jβ2 )

 (8.32)

where p˜ 01 and p˜ 02 , are respectively, the amplitudes of the harmonic forces acting on the source devices m1 and m2 .

204

8 Analysis of Raft Using WPA Method

If m 2 is a non-power source device, let p˜ 02 = 0. Substituting Eqs. (8.9), (8.10), and (8.30) into Eqs. (8.23), (8.27), and (8.29) to obtain the following equations, we get w1 (x) =

4 

An e

k1n x

−

n=1

+

2 

4 

2    k2 j w1 (x2 j ) − w2 (y2 j ) an e−k1n |x−x2 j |

Fj

2 

an e−k1n |x−x1 j |

4 

(8.33)

n=1

Bn ek2n y + k21 [w1 (x21 ) − w2 (y21 )]

n=1

w3 (y) =

n=1

j=1

j=1

w2 (y) =

2  

2 

bn e−k2n |y−y21 |

(8.34)

cn e−k3n |y−y22 |

(8.35)

n=1

Cn ek3n y + k22 [w1 (x22 ) − w3 (y22 )]

n=1

2  n=1

8.4.3 Boundary Conditions and Compatibility Conditions In the raft model, both ends of beam ➀ are free ends. According to the boundary conditions in Chap. 2, the shear force and bending moment at the completely free end of beam ➀ are all zero  ∂ 2 w1 (0) ∂ 2 w1 (L 1 ) = = 0 2 2 ∂x ∂x (8.36) 3 ∂ 3 w1 (0) 1) = ∂ w∂1x(L =0 3 ∂x3 In the simulated raft foundation of beams ➁ and ➂, both ends are simply supported with zero displacement and bending moment and boundary conditions. Foundation beam ➁ ⎫ w2 (0) = w2 (L 2 ) = 0 ⎬ 2 2 (8.37) ∂ w2 (0) ∂ w2 (L 2 ) ⎭ = = 0 ∂x2 ∂x2 Foundation beam ➂ w3 (0) = w3 (L 3 ) = 0

⎫ ⎬

∂ 2 w3 (0) ∂ 2 w3 (L 3 ) = = 0⎭ 2 ∂x ∂x2

(8.38)

8.4 Analysis of Rafts Using the WPA Method

205

Beam ➀ is connected to beams ➁ and ➂ through the isolator, and the compatibility condition is Eq. (8.8). The compatibility deformation condition of the action point between beam ➀ and the source device is Eq. (8.30). The lateral displacement response derivative is used in the above conditions. The derivations from Eq. (8.33) are 4 2  ∂w1 (x)  = An k1n ek1n x + sign(x, x21 )k21 [w1 (x21 ) − w2 (y21 )] an k1n e−k1n |x−x21 | ∂x n=1 n=1

+ sign(x, x22 )k22 [w1 (x22 ) − w3 (y22 )]

2 

an k1n e−k1n |x−x22 |

n=1

− F1 sign(x, x11 )

2 

1 an kn1 e−kn |x−x11 |

− F2 sign(x, x12 )

n=1

2 

an kn1 e−kn |x−x12 | 1

n=1

(8.39) 4 2  ∂ 2 w1 (x)  2 k1n x 2 −k1n |x−x21 | = A k e − k (x ) − w (y )] an k1n e [w n 21 1 21 2 21 1n ∂x2 n=1 n=1

− k22 [w1 (x22 ) − w3 (y22 )]

2 

2 −k1n |x−x22 | an k1n e

n=1

+ F1

2 

2 −k1n |x−x11 | an k1n e + F2

n=1

2 

2 −k1n |x−x12 | an k1n e

(8.40)

n=1

4 2  ∂ 2 w1 (x)  3 k1n x 3 −k1n |x−x21 | = A k e + sign(x, x )k (x ) − w (y )] an k1n e [w n 1n 21 21 1 21 2 21 ∂x3 n=1 n=1

+ sign(x, x22 )k22 [w1 (x22 ) − w3 (y22 )]

2 

3 −k1n |x−x22 | an k1n e

n=1

− F1 sign(x, x11 )

2 

3 −k1n |x−x11 | an k1n e − F2 sign(x, x12 )

n=1

2 

3 −k1n |x−x12 | an k1n e

n=1

(8.41) where  sign(x, x j ) =

+1, x ≥ x j −1, x < x j

The equation is a sign operator produced in the process of deriving absolute value. The derivations from Eqs. (8.34) and (8.35) are

206

8 Analysis of Raft Using WPA Method

4 2  ∂w2 (y)  Bn k2n ek2n y − sign(y, y1 )k21 {w1 (x21 ) − w2 (y21 )} bn k2n e−k2n |y−y21 | = ∂y n=1 n=1

(8.42) 4 2  ∂ 2 w2 (y)  2 k2n y 2 −k2n |y−y21 | {w = B k e + k (x ) − w (y )} bn k2n e (8.43) n 2n 21 1 21 2 21 ∂ y2 n=1 n=1 4 2  ∂w3 (y)  = Cn k3n ek3n y − sign(y, y2 )k22 {w1 (x22 ) − w3 (y22 )} cn k3n e−k3n |y−y22 | ∂y n=1 n=1

(8.44) 4 2  ∂ 2 w3 (y)  2 k3n y 2 −k3n |y−y22 | {w = C k e + k (x ) − w (y )} cn k3n e (8.45) n 3n 22 1 22 3 22 ∂ y2 n=1 n=1

where  sign(y, y j ) =

+1, y ≥ y j −1, y < y j

Substituting the above equations into the boundary conditions and compatibility conditions, the following can be obtained for beam ➀: 4 2  ∂ 2 w1 (0)  2 2 −k1n x21 {w = A k − k (x ) − w (y )} an k1n e n 1n 21 1 21 2 21 ∂x2 n=1 n=1

− k22 {w1 (x22 ) − w3 (y22 )}

2 

an (kn1 )2 e−k1n x22

n=1

+ F1

2 

2 −k1n x11 an k1n e + F2

n=1

2 

2 −k1n x12 an k1n e =0

(8.46)

n=1

4 2  ∂ 3 w1 (0)  3 3 −k1n x21 {w = A k − k (x ) − w (y )} an k1n e n 21 1 21 2 21 1n ∂x3 n=1 n=1

− k22 {w1 (x22 ) − w3 (y22 )}

2 

3 −k1n x22 an k1n e

n=1

+ F1

2  n=1

3 −k1n x11 an k1n e

+ F2

2 

3 −k1n x12 an k1n e =0

(8.47)

n=1

4 2  ∂ 2 w1 (L 1 )  2 k1n L 1 2 −k1n |L 1 −x21 | {w = A k e − k (x ) − w (y )} an k1n e n 1n 21 1 21 2 21 ∂x2 n=1 n=1

8.4 Analysis of Rafts Using the WPA Method

− k22 {w1 (x22 ) − w3 (y22 )}

207 2 

2 −k1n |L 1 −x22 | an k1n e

n=1

+ F1

2 

2 −k1n |L 1 −x11 | an k1n e + F2

n=1

2 

2 −k1n |L 1 −x12 | an k1n e =0

(8.48)

n=1

4 2  ∂ 3 w1 (L 1 )  3 k1n L 1 3 −k1n |L 1 −x21 | {w = A k e + k (x ) − w (y )} an k1n e n 21 1 21 2 21 1n ∂x3 n=1 n=1

+ k22 {w1 (x22 ) − w3 (y22 )}

2 

3 −k1n |L 1 −x22 | an k1n e

n=1

− F1

2 

3 −k1n |L 1 −x11 | an k1n e

n=1

− F2

2 

3 −k1n |L 1 −x12 | an k1n e =0

(8.49)

n=1

For foundation beam ➁ w2 (0) =

4 

Bn + k21 {w1 (x21 ) − w2 (y21 )}

n=1

w2 (L 2 ) =

4 

Bn e

2 

(8.50)

n=1 k2n L 2

+ k21 {w1 (x21 ) − w2 (y21 )}

n=1

∂ 2 w2 (0) = ∂x2

bn e−k2n y21 = 0

2 

bn e−k2n |L 2 −y21 | = 0 (8.51)

n=1

4 

2 Bn k2n + k21 {w1 (x21 ) − w2 (y21 )}

n=1

2 

2 −k2n y21 bn k2n e =0

(8.52)

n=1

4 2  ∂ 2 w2 (L 2 )  2 k2n L 2 2 −k2n |L 2 −y21 | {w = B k e + k (x ) − w (y )} bn k2n e =0 n 2n 21 1 21 2 21 ∂x2 n=1 n=1

(8.53) Similarly, for foundation beam ➂ w3 (0) =

4  n=1

w3 (L 3 ) =

4  n=1

Cn + k22 {w1 (x22 ) − w3 (y22 )}

2 

cn e−k3n y22 = 0

(8.54)

n=1

Cn ek3n L 3 + k22 {w1 (x22 ) − w3 (y22 )}

2 

cn e−k3n |L 3 −y22 | = 0 (8.55)

n=1

4 2  ∂ 2 w3 (0)  2 2 −k3n y22 {w = C k + k (x ) − w (y )} cn k3n e =0 n 22 1 22 3 22 3n ∂x2 n=1 n=1

(8.56)

208

8 Analysis of Raft Using WPA Method

4 2  ∂ 2 w3 (L 3 )  2 k3n L 3 2 −k3n |L 3 −y22 | {w = C k e + k (x ) − w (y )} cn k3n e =0 n 3n 22 1 22 3 22 ∂x2 n=1 n=1

(8.57) Compatibility conditions of installation points of isolators under raft w1 (x21 ) =

4 

An ek1n x21 − k21 {w1 (x21 ) − w2 (y21 )}

n=1

2 

an e−k1n |x−x21 |

n=1

− k22 {w1 (x22 ) − w3 (y22 )}

2 

an e−k1n |x−x22 |

n=1

+ F1

2 

an e−k1n |x−x11 | + F2

2 

n=1

w1 (x22 ) =

4 

An e

an e−k1n |x−x12 |

(8.58)

n=1 k1n x22

− k21 {w1 (x21 ) − w2 (y21 )}

n=1

2 

an e−k1n |x−x21 |

n=1

− k22 {w1 (x22 ) − w3 (y22 )}

2 

an e−k1n |x−x22 |

n=1

+ F1

2 

an e−k1n |x−x11 | + F2

2 

n=1

an e−k1n |x−x12 |

(8.59)

n=1

The following can be obtained for beam ➀: 4  n=1

 An e

k1n x21

− 1 + k21

2 





an w1 (x21 ) + k21

n=1



− k22

2 





an e−k1n |x21 −x22 | w1 (x22 ) + k22

n=1

+ F1

2  n=1

4  n=1

An e

k1n x22

an e−k1n |x21 −x11 | + F2 2 

an e





an w1 (x22 ) + k22

n=1

+ F1

2  n=1

an w2 (y21 )

n=1 2 



an e−k1n |x21 −x22 | w3 (y22 )

an e−k1n |x21 −x12 | = 0 



−k1n |x22 −x21 |

n=1

− 1 + k22

2  n=1



2 



n=1

− k21



2 

an e−k1n |x22 −x11 | + F2

w1 (x21 ) + k21

2 



2 

(8.60)  an e

−k1n |x22 −x21 |

w2 (y21 )

n=1

an w3 (y22 )

n=1 2  n=1

an e−k1n |x22 −x12 | = 0

(8.61)

8.4 Analysis of Rafts Using the WPA Method

209

For foundation beam ➁, the displacement where y = y21 is w2 (y21 ) =

4 

Bn e

k2n y21

+ k21 {w1 (x21 ) − w2 (y21 )}

n=1

2 

bn

(8.62)

n=1

The following can be obtained: 4 

 Bn e

k2n y21

+ k21

n=1

2 





bn w1 (x21 ) − 1 + k21

n=1

2 

 bn w2 (y21 ) = 0

(8.63)

n=1

For foundation beam ➂, the displacement where y = y22 is w3 (y22 ) =

4 

Cn ek3n y22 + k22 {w1 (x22 ) − w3 (y22 )}

n=1

2 

cn

(8.64)

n=1

The following can be obtained: 4 

 Cn e

k3n y22

+ k22

n=1

2 





cn w1 (x22 ) − 1 + k22

n=1

2 

 cn w3 (y22 ) = 0

(8.65)

n=1

The device on beam ➀ is installed where x = x11 and x = x12 , and the compatibility deformation condition by the action point of the device is as follows according to Eq. (8.40): ⎫ F1 − Q 1 p1 ⎪ ⎪ w1 (x11 ) = ⎬ ktot11 F2 − Q 2 p2 ⎪ ⎪ ⎭ w1 (x12 ) = ktot12

(8.66)

The displacement response of the installation point of the source device on beam ➀ can be obtained w1 (x11 ) =

4 

An ek1n x11 − k21 {w1 (x21 ) − w2 (y21 )}

n=1

2 

an e−k1n |x11 −x21 |

n=1

− k22 {w1 (x22 ) − w3 (y22 )}

2 

an e−k1n |x11 −x22 |

n=1

+ F1

2  n=1

an + F2

2  n=1

an e−k1n |x11 −x12 |

(8.67)

210

8 Analysis of Raft Using WPA Method

w1 (x12 ) =

4 

An ek1n x12 − k21 {w1 (x21 ) − w2 (y21 )}

n=1

2 

an e−k1n |x12 −x21 |

n=1

− k22 {w1 (x22 ) − w3 (y22 )}

2 

an e−k1n |x12 −x22 |

n=1

+ F1

2 

an e−k1n |x12 −x11 | + F2

n=1

2 

an

(8.68)

n=1

Combining Eq. (8.66) with Eq. (8.68), we get 4 

An ek1n x11 − k21 {w1 (x21 ) − w2 (y21 )}

n=1

an e−k1n |x11 −x21 |

n=1

− k22 {w1 (x22 ) − w3 (y22 )}  + F1

2  n=1

4 

2 

an −

1 ktot11

2 

an e−k1n |x11 −x22 |

n=1



+ F2

2 

an e−k1n |x11 −x12 | = −

n=1

An ek1n x12 − k21 {w1 (x21 ) − w2 (y21 )}

n=1

2 

(8.69)

an e−k1n |x12 −x21 |

n=1

− k22 {w1 (x22 ) − w3 (y22 )}

2 

an e−k1n |x12 −x22 |

n=1

+ F1

Q 1 p1 ktot11

2  n=1



an e−k1n |x12 −x11 | + F2

2 

an −

n=1

1 ktot12

 =−

Q 2 p2 ktot12

(8.70)

Equations (8.46)–(8.57), (8.60), (8.61), (8.63), (8.65), (8.69), and (8.70) are written in the form of a matrix SX = P X T = {A1 , . . . A4 , B1 , . . . , B4 , C1 , . . . , C4 , w1 (x11 ), w1 (x12 ), w2 (y21 ), w3 (y22 ), F1 , F2 }   Q 1 p1 Q 2 p2 ,− P T = 0, 0, . . . , − ktot11 ktot12 where

(8.71)

8.4 Analysis of Rafts Using the WPA Method

211

X—unknown quantity matrix where Ai , Bi , and Ci are unknown coefficients to be determined; w1 (x21 ), w1 (x22 )—displacements at x21 and x22 on beam ➀; w2 (y21 )—displacement on beam ➁ where y = y21 ; w3 (y22 )—displacement on beam ➂ where y = y22 ; F 1 , F 2 —internal forces of source device acting on beam ➀ through isolators k11 and k12 ; P—external excitation force matrix; S—coefficient matrix. By substituting the relevant parameters into Eq. (8.71), for the simultaneous solution, and then substituting the solution value into Eqs. (8.33)–(8.35), the instantaneous value of displacement response at any point on beams ➀, ➁, and ➂ can be obtained. Furthermore, by substituting the calculated structural velocity and acceleration response values into Eq. (5.23), the average power flow transmitted on the beam can be solved.

8.4.4 WPA Expression of Vibration Isolation Effect of Raft There are many ways to evaluate the effect of vibration isolation, such as the transmission rate of vibration isolation, insertion loss, vibration level drop, etc. In order to facilitate the comparison with the test data, the vibration isolation effect of the raft is expressed by the vibration acceleration level drop, which is the ratio of the vibration acceleration of the foundation beam to the vibration acceleration of the source device [13]  Re(w¨ 2 ) (dB) T L = 20 log Re(w¨ m ) 

(8.72)

where w¨ 2 —real part of vibration acceleration of beam ➁; w¨ m —real part of vibration acceleration of the source device m. Assuming that the excitation of the source device is harmonic force, the acceleration response of the source equipment is also a simple harmonic parameter. From Eq. (8.12), it can be seen that the acceleration response of the source device is w¨ m = −ω2 wm =

p0 − Fb m

(8.73)

where p0 —harmonic force amplitude; Fb —internal force applied by the source device to beam ➀ through isolator;

212

8 Analysis of Raft Using WPA Method

Table 8.2 Main parameters of raft calculation model Structure

Length L (m)

Width b (m)

Section height h (m)

Material parameters

Beam ➀

1.5

0.08

0.012

Beam ➁

0.8

0.07

0.012

Beam ➂

0.8

0.07

0.012

Elasticity modulus: E = 2.02 × 1011 pa Volume density:  ρ = 7850 kg m3 Loss factor:β = 0.01

m—mass of source device.

8.5 “Mass Effect” Analysis of Raft The double-stage vibration isolation study shows that the greater the mass of the raft body, the closer the second-order mounted frequency of the system is to the first-order mounted frequency [13], as shown in Fig. 8.5. To improve vibration isolation performance, the BC region should be compressed, and the CD frequency domain should be expanded. We have drawn a similar conclusion in the study of rafts: the greater the mass of the raft frame, the closer the second group of formant clusters is to the first group of formant clusters in the system, which can compress the BC frequency domain and expand the CD frequency domain, thereby improving vibration isolation efficiency. The author proposed the “mass effect” in 1994 [3, 25]. The mass effect studies the contributions of equipment to the equivalent mass of the system during elastic or rigid installation, and the use of mass effects can further save overall resources. The WPA method is used in this chapter to analyze and discuss how to balance the equipment and the mass contribution.

8.5.1 Basic Parameters The mechanism analysis of the raft is shown in Fig. 8.6. The basic parameters of the system are shown in Tables 8.2 and 8.3. The transmissibility of the vibration acceleration response of source equipment m1 , the midpoint of beam ➂, and Eq. (8.72) are used to describe the vibration isolation effect.

8.5.2 Comparative Study of Rigid Installation of Equipment In order to study the impact of the installation method on the transmission characteristics of the system, on beam ➀, it is assumed that source equipment m 1 is installed

8.5 “Mass Effect” Analysis of Raft Table 8.3 Equipment and vibration isolator parameters in raft calculation model

213

Components etc.

Dynamic stiffness  k(N m)

Loss factor β

Upper vibration isolator

8.0 × 104

0.08

Lower vibration isolator

8.0 × 104

0.08

Quality of source equipment

m 1 = m 2 = 10 kg

Installation position

x11 = 0.4 m, x12 = 1.1 m, x21 = 0.2 m x22 = 1.3 m, y21 = y22 = 0.4 m

elastically, m 2 is installed rigidly and the mass is changed. Compared with the actual project, the model is greatly simplified, but the damping parameters of the raft frame, elasticity of the base, and vibration isolator are completely preserved. The calculation results are shown in Fig. 8.12. The selected calculated frequency range is very wide, and the overall transmissibility curve seems to be quite different from the ideal raft. It is necessary to subdivide by frequency domain and add marked line segments to help us to see clearly the basic features retained in the transmissibility curve of the raft (1) In the frequency of 18–50 Hz, the transmissibility changes with 12 dB/Oct of line segment BC as the baseline, which is shown in Fig. 8.12a. The modal frequency of the beam is low, and the contribution to the harmonic components of the transference curve after coupling is significant. In practical engineering, the modal frequency of the raft frame is relatively higher, which will only affect the high frequency region of the transmission characteristics and only contribute to the harmonic components. (2) For the frequency band above 55 Hz, the system has high vibration isolation efficiency which follows the decreasing law of 24 dB/Oct, which is shown in the marked line segment CD in Fig. 8.12b. At f ≥ 100 Hz, the transmission characteristic shows a higher resonance peak. They are not typical formants but harmonic components caused by the integral mode of the beam, causing the transmissibility characteristics to be elevated overall. In engineering practice, the mode of the raft frame will not be designed so low, which is related to the softness of the selected beam ➂. (3) The anti-resonance points f ≈ 265 Hz and 340 Hz are also one of the main modes of the beam, and the transmissibility is dominated by the modes. The antiresonance frequency of foundation beam ➂ appears as high frequency components through coupling. In the model, the rigidity of the foundation beam is much lower than in the actual project, and the damping value will not be so low, resulting in sharp peaks. It can be seen from Fig. 8.12a that rigidly installed equipment m 2 has a “frequency shift” effect to increase the mass. As the mass increases, the system’s firstorder mounted frequency moves in a low frequency direction. It is interesting that

214

8 Analysis of Raft Using WPA Method

Fig. 8.12 Impact of rigid installation equipment quality on vibration isolation effect

this change rule is quite consistent with two-degree-of-freedom dual-stage vibration isolation. The Eq. (2.188), in Ref. [1], is reproduced as follows: ⎫ ⎧     2 m1 m1 ωc2 ω22 ω22 ⎬ 1⎨ ω22 1+ 1+ + 2 −4 2 = + 2 ± m2 2⎩ m2 ω1 ω1 ⎭ ω12 ω1

(8.74)

8.5 “Mass Effect” Analysis of Raft

215

where m 1 —mass of source equipment; of raft body; m 2 —mass   2 ω1 = K 1 m 1 —natural frequency of source equipment without coupling;   ω22 = K 2 m 2 —natural frequency of raft body without coupling.  Taking the different mass ratios μ = m 2 m 1 and different stiffness ratios α =  K 2 K 1 , we can draw a nomogram of the natural frequency of the system. When the mass ratio is μ = 1, we can obtain the two installation frequencies ω0H and ω0H of the system, and the optimal stiffness ratio to minimize the frequency ratio is αopt = 1 + μ

(8.75)

The raft is modeled as a hybrid dynamic system. The raft frame and foundation are elastic beams which are coupled and connected to the much less rigid isolators. Therefore, the first natural frequency in the first mounted frequency cluster of the raft in the first group is close to the corresponding frequency of the dual-stage vibration isolation system of the rigid body. According to Eq. (8.74), we get Sj

f r1

⎫ ⎧    ⎨ 2 2 2 2⎬ m1 m1 1 ω ω ω 1+ 1+ = + 22 ∓ + 22 − 4 22 · f r1 2⎩ m2 m2 ω1 ω1 ω1 ⎭

(8.76)

In this equation, j = 1, 2, and when the “−” or “+” signs in the “∓” are, respectively, taken, the f r1S2 of the f r1S1 of the raft can be obtained, which are the first natural frequencies in the first and second frequency clusters, respectively. To make a simple conversion, the value range of the mass is m 2 = {0.1, 1.0, 5.0, 10} kg, and the maximum value is max(m 2 ) = m 1 . Thus, the mass ratio is μ = 1 and the stiffness ratio is α = 1. The following can be obtained from Eq. (8.76):  √  sj f r1 = 3 ∓ 5 · f r1 The calculation shows that when m 2 = m 1 , the effect of “frequency shift” is obvious, and when the mass of m2 is increased to 10 kg, namely when the mass of m2 is the same as that of the raft frame, the mass of the rigid installation equipment is further increased, and the promotion of the overall effect is not obvious. Similarly, the optimal stiffness ratio basically conforms to Eq. (8.75). The author conducted a comparative study of rafts and dual-stage vibration isolation, and concluded that under engineering conditions, the mass contribution of elastic installation equipment to the raft is close to zero, while the rigid installation equipment contributes nearly 100% to the mass of the system! This conclusion represents important insights for project weight loss. It is possible to improve the mass of

216

8 Analysis of Raft Using WPA Method

a raft frame by rigidly installing non-source equipment or nonoperating equipment on the raft frame, thereby achieving the purpose of system optimization and giving designers more room for development.

8.5.3 Impact of Equipment Location on the Effect of Vibration Isolation Equipment m 2 is rigidly fixed to different locations of the raft frame, which does not change the mounted frequency clusters of f S1 and f S2 of the 1- and 2-order of the system, as shown in Fig. 8.13. The double-stage raft follows the description of the three-section line ABC of double-stage vibration isolation, which is the common attribute of the two. The peculiarities of the raft are as follows: (1) Due to the rigid fixing of m2 , the two resonance peak clusters of the system degrade into single peaks; (2) The modal characteristics of the beam are clearly reproduced in the system characteristics. Moreover, the modal characteristics of the beam dominate the overall characteristics of the system through coupling because the beam selected for the test is too soft, leading to the overall deterioration of the vibration isolation performance within the frequency domain f ≥ f S2 . The area 12 dB/Oct of line B in the ABC line is raised overall (higher than the marked line); (3) When the device is at x = 1.1 m within the frequency domain f ≥ f S2 , the vibration isolation performance further deteriorates. There are two antiresonance points in the system which correspond to the two modal frequencies

Fig. 8.13 Impact of rigid installation of equipment at different positions on vibration isolation

8.5 “Mass Effect” Analysis of Raft

217

of the beam. Rafts are prone to design defects due to their large number of parameters and interrelationships. Therefore, designers need to predict and evaluate comprehensively.

8.6 “Mixing Effect” Analysis of Rafts 8.6.1 Offset of Two Structural Waves For multiple source equipment, the structural waves input into the raft frame will counteract each other, forming the so-called “mixing effect” of the raft [13, 25]. The mixing effect has the same principle as the mutual offset of elastic waves in active vibration control, as shown in Fig. 8.14, assuming that p˜ 0 is a primary source and p˜ c is a control source that has different phase angles ϕ1 and ϕ2 compared to the primary source. In Refs. [29, 30], the “active control of the acoustic radiation of 1D structural beams” is discussed. The point harmonic force p˜ 0 acts as a primary source at x = x 0 , and the vibration displacement generated by the control source p˜ c is w(x) =

∞ 

ηm wm (x) · p˜ c

(8.77)

m=1

ηm is the modal amplitude generated by the unit force, which is given by the following equation: ηm (x) =

wm (x0 ) ωm2 − ω2 + j2βm ωωm

where x 0 —position of control force; ω—exciting frequency; βm —modal damping coefficient; p˜ c —amplitude of control force. Fig. 8.14 Offset of two elastic waves in one-dimensional beam (pc = control source, p0 = primary source in active control)

(8.78)

218

8 Analysis of Raft Using WPA Method

In terms of control strategy, active vibration control has a counteracting volume velocity mode and a minimum acoustical power mode, and p˜ c adjusts amplitude and phase by adaptive methods, which were all designed by humans.

8.6.2 Offset of Multisource Structural Waves Assuming that there are N pieces of source equipment, the harmonic force is recorded as p˜ 0 j ( j = 1, 2, . . . , n), and they have different phase angles ϕ j with the primary source p˜ 01 . If all the structural waves input into the raft frame are recorded in the general form, we get w˜ Σ (x) =

n 

H˜ F∗ j · p0 j · e j (ωt+ϕ j )

(8.79)

j=1

where H˜ F∗ j —transfer function of j ( j = 1, 2, . . . , N ) exciting force; p0 j —point force amplitude; ϕ j —initial phase. According to structural dynamics, when the disturbing force of the N source equipment is input into the raft frame, the structural waves will counteract each other, then further isolated by the second-stage spring and finally output to the foundation. The mutual offset mechanism of structural waves generated by multiple force sources is called the “mixing effect” of rafts, which can generate additional attenuation and further improve the vibration isolation efficiency. The author [3, 13] once compared this process to sewage treatment, in which acid-base object replaced vibration energy: the acid-base materials “mixed first, then neutralized”. The acid-base consumption in this technological process is more effective. This is a natural offset process without adaptive intervention in the active control and without “extra” energy consumption.

8.6.2.1

Expression of Relative Amount and Absolute Amount

The power transmissibility T, insertion loss I L(A), and vibration level difference with which we are familiar in the evaluation of the vibration isolation effect are all expressions of relative amount, which are convenient for engineering surveying and have gradually been developed into test methods and measurement standards. The description of relative amount has limitations; for example, it can mask the additional attenuation caused by the mixing of force sources. Therefore, it may not be the best expression for research.

8.6 “Mixing Effect” Analysis of Rafts

219

Fig. 8.15 Internal and external mixing effects

Internal mixing Mixing effect External mixing

Due to calculation in the early days, it was found that the transmissibility curve described by the vibration power flow “paralleled down” as a whole. The energy transmitted by the system is not only due to the isolation components, but also the reduction of absolute energy.

8.6.3 External and Internal Mixing Effects The “mixing effect” of rafts can be subdivided into external and internal mixing [13], as shown in Fig. 8.15. We define the mutual and autonomous offset of the vibration energy between force source equipment as external mixing. The viscoelastic damping of the raft frame, wave reflections, and losses caused by structural discontinuity points are defined as internal mixing. Theoretically, external mixing starts from 0 Hz! Mixing is unique to multisource rafts and is also a fundamental difference from dual-stage vibration isolation. Damping in raft frames and isolators, wave reflection, attenuation, and wave type conversion caused by structural discontinuity points can all attenuate vibration energy. The mixing effect is an autonomous and random systematic optimization method which originates from the positive emergence of the logical relationship between force source elements. Its discovery and mechanism research provided a new technological approach for the development of efficient rafts.

8.6.4 Equal-Master Rafts and Master-Slave Rafts There are many ways to classify rafts [16, 22], but it is of greater practical value to classify them according to the comprehensive numbers of the source equipment. This classification guides how to make the mixing effect more sufficient and improve the vibration isolation efficiency of the raft. In the group integration of equipment, the raft is designed around the source equipment which takes the dominant position in the input vibration energy of the raft frame, and is called the “master equipment” or “class A equipment”, and that which takes the subordinate position is called the “slave equipment” or “class B equipment”. The following equation is always valid:

220

8 Analysis of Raft Using WPA Method

  Π (Ai ) Π B j (i = 1, 2, 3, . . . , M; j = 1, 2, 3, . . . , N ) where Π (A)—energy input into the system by class A equipment; Π (B)—energy input into the system by class B equipment. Thus, rafts can be divided into “master-slave rafts” and “equal-master rafts”. Obviously, the mixing effect of the equal-master raft is the most adequate and can provide the maximum additional attenuation. The physical model of a typical equalmaster raft is shown in Fig. 8.16. The performance parameters of the five pieces of source equipment are the same. Figure 8.17 shows the rafts of multiple hydraulic pumps on the “Meteor [Germany]” ship [4], according to the design and installation example of the equal-master raft. Most of the source equipment of the carrier is integrated according to the system and functions, so there are relatively more master-slave rafts. Figure 8.18 is a schematic diagram of a typical single master-slave raft, including one piece of master

Fig. 8.16 Equal-master raft

Fig. 8.17 Raft layout on the “Meteor [Germany]” ship

8.6 “Mixing Effect” Analysis of Rafts

221

Fig. 8.18 Single master-slave raft

equipment and four identical pieces of slave equipment. The master and slave equipment here are defined from a relative point of view. Reconstructing the master-slave raft as an equal-master raft is a better optimization process. The amplitude of the structural waves in the raft frame is the result of the synthesis of such parameters as the source equipment disturbing force, support stiffness, mass (or moment of inertia), and viscoelastic damping. The principle of raft optimization is to make the structural waves generated by the main disturbing force in the raft frame gain a more adequate offset mechanism. The designer fine-tuned the parameters and adjusted the master-slave raft into an equal-master raft as much as possible without having a major impact on the system. Several directional comments are as follows: (1) The equal-master raft has more isolation efficiency than the master-slave raft, thereby virtue of having a higher amount of additional vibration isolation. It is particularly important that the additional attenuation is also effective for low frequencies. (2) The study of the master-slave raft and equal-master raft reveals a principle: try to install source equipment with close comprehensive performance parameters in the same raft frame, and this is the same reason that sound insulation theory emphasizes that high-noise source equipment should be housed in an acoustic enclosure as much as possible. The same result reveals some essential similarities and objective laws of vibration isolation and sound isolation. (3) The greater the amount of raft source equipment, the more obvious the mixing effect and the more stable the data. It can also improve the transient stability and displacement response of the system. In the statistical sense, throughout

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history, the stability of an equal-master raft with three or more pieces of source equipment has been better. Fortunately, carriers, especially large naval vessels, meet the mixing conditions. The revolving speed of the source equipment, initial phase of the disturbing force, and amplitude and weight of equipment vary with the load within a small range, which conforms to Gaussian statistical distribution, and carriers also have a relatively stable number of pieces of source equipment. Through the “man-made” preset, the amplitude input into the raft frame from the source equipment is as close as possible,   ≈ p02 · · · ≈ p0 j . The simulation statistics results show that the greater namely p01 the amount of equipment, the better the dynamic stability of the raft, and having three or more pieces of source equipment can produce better mixing effects.

8.6.5 Impact of Raft Frame Damping As expected from the analysis, the structural damping of the raft frame does not change the basic transmission characteristics of the system and does not suppress the first- and second-order formant clusters of the system, as shown in Fig. 8.19. It suppresses the system peak response which corresponds to the structural mode of the raft frame and determines the amplitude and half-band width of these formants. Such peaks have higher frequencies and are within the high frequency band. Damping is represented by the composite loss factor of the viscoelastic structure. The composite loss factor of marine rafts is usually not high. Increasing the raft damping will increase the resistive part of the input power flow. Figure 8.20 shows the vibration power flow of input foundation beam ➂. The line

Tran. of acceleration (dB)

High-frequency area

m2 Elastic installation

Frequency (Hz) Fig. 8.19 Impact of raft damping on vibration isolation effect

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Vibration power flow

Line segment ABC

Line segment abc

Frequency/Hz

Fig. 8.20 Impact of raft beam damping on power flow input to foundation beam

segment corresponding to the vibration isolation system is the ABC line segment, not the abc line segment. The composite loss factors of the raft frame β = 0.01, 0.02, and 0.05 have no impact on the low frequency band of the input power flow. It is the resistant part that is in action, but the composite loss factors change the high frequency resonance peak. If it is a curved beam, the impact will be more significant. When the overall stiffness distribution of the raft frame is reasonable, the impact of damping on the high frequency band is also weakening. Raft damping can only suppress the resonance peak corresponding to the raft mode in the transfer curve. Everything has two sides, so do not generalize the application of viscoelastic damping as a “broad spectrum pill”. In fact, its control range is limited, and its overall application trend has reduced. It is interesting to analyze which properties of the system are affected by composite damping and to what extent. This suggests that we need to apply damping in a targeted manner. For example, the research on the energy storage and energy loss of rafts is highly inadequate, which is a topic worthy of everyone’s discussion, so that the application evaluation of viscoelastic damping can truly gain a system contribution rate.

8.7 “Tuning Effect” of Rafts It is almost inevitable for a raft to have a frequency point of energy leakage, which leads to the deterioration of performance. In many cases, a tuned mass damper must be applied to control it. Abbreviated as TMD, the tuned mass damper also has other English expressions, such as “dynamic absorber” and “neutralizer” [27]. It is used to absorb part of the vibration energy so that it does not escape from the raft, as shown in Fig. 8.21.

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Transmissibility (dB)

Standard double-stage vibration isolation

0

System tuning

1

10

2

10

Frequency (Hz)

Fig. 8.21 Tuning effect of equipment

We have two ways to use a TMD to control the leakage frequency point. One is to directly attach a TMD, while the other better way is to design the equipment or source equipment in the carrier as a TMD to tune its leakage frequency point. Adopting the latter method can further save overall resources, which we call the “tuning effect” of rafts. It can be seen that the complexity of the raft does not solely present negative factors for designers. It also gives designers greater design space and modification flexibility, and the use of the tuning effect can further save carrier resources, which illustrates the general benefits that the improvement of the integration level can bring to designers. In the figure, TMD tunes the second-order mounted frequency in the transfer curve, and the main peak is divided into two for greater suppression. The TMD should be designed separately, using the device/equipment as a mass block and tuning for specific peaks. It is relatively simple to apply non-source equipment or devices in a carrier, but how should the source equipment be applied? This refers to source equipment that does not work under certain working conditions, or source equipment which is subdivided into equipment that works at high speed and equipment that does not, and is expressed in system engineering as the order of time of elements. This equipment also has the TMD tuning function through installation methods and parameter adjustments. The basic knowledge of TMD and the specific application of the design are not repeated here.

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8.8 WPA Analysis and Test The WPA method was previously used to deduce the vibration response of a “beam” raft, and the transmission characteristics and their “mass, tuning and mixing” effects were theoretically analyzed. This section discusses raft tests and verification.

8.8.1 Raft Test Device The source equipment is simulated by a combined rigid body mass block, which is convenient for adjusting the mass, as shown in Fig. 8.22. They are rigidly installed on the raft beam ➀ through four vibration isolators, as shown in Fig. 8.6, and the corresponding test device is shown in Fig. 8.23. In order to be as consistent as possible with the theoretical model, straight beams and rectangular section steel are selected in the test to ensure the frequency range of the structural waves. Foundation beams➁ and ➂ are replaced by two beams with the same rectangular section, and both ends are rigidly fixed. The raft frame is installed on the foundation beam through two side by side isolators, with a total of four lower isolators. The parameters of each component of the test device are shown in Table 8.4. The three-section finite beams are consistent with the simple beam theory in material

Fig. 8.22 Raft test device

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Fig. 8.23 Combined source equipment simulation Table 8.4 Parameter list of test equipment

Beam

Vibration isolator

Equipment

Type

Dimensions Elasticity (length × width × modulus E height) (mm) (N/m2 )

Material density ρ (kg/m3 )

Beam ➀ (raft beam)

1,500 × 80 × 12

2.02 × 1011

7,850

Beam ➀ (raft beam)

1,500 × 80 × 8

2.02 × 1011

7,850

Beam ➁ = ➂ (foundation beam)

800 × 70 × 12

2.02 × 1011

7,850

Mode

Dynamic stiffness (N/m)

Rated load (N)

Damping ratio

BE-5

20,000

50

0.08 ± 0.01

BE-10

40,000

100

0.08 ± 0.01

Combination

Boundary dimensions (mm)

Mass (kg)

6 steel blocks

150 × 100 × 120

2.2 × 6 = 13.2

4 steel blocks

150 × 100 × 80

2.2 × 4 = 8.8

1 steel block

150 × 100 × 20

2.2 × 1 = 2.2

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mechanics; that is, the impacts of shear deformation and rotational inertia in the beam can be ignored, which is consistent with the theoretical model.

8.8.2 Analysis of Test Results In order to ensure that the test device is in a good and stable state, the simulator of the elastic installation source equipment is equipped with four BE-5 type vibration isolators, and foundation beam ➁ and the beam ➂ are each installed with two BE10 type vibration isolators, then connected with beam ➀. During the test, due to the need for rigid installation, the actual span of the foundation beam is 0.626 m after the two ends are fixed. The raft frame is installed in the middle of the foundation beam through the lower isolator, with its center at 0.313 m, which will naturally generate a node. It should be noted that the source equipment is connected to beam ➀ through four vibration isolators, and simplified to one point in the theoretical calculations. In order to be as close as possible to the ideal state of the beam, no holes are allowed in the beam to avoid discontinuity points. During the test, the lower ends of the isolator and the beam are glued to the beam with strong glue, and the rigid installation source equipment is glued to the beam with the same method to reduce the impact on the transmission of elastic waves in the beam. These are slightly different from the rigid assumptions, which will affect some test data. The relevant parameters and installation methods of the source equipment are changed. The specific test conditions are shown in Table 8.5 In the theoretical calculations, it was assumed that K → ∞ represented rigid installation. First, the theoretically calculated values coincide well with the experimental test values in the low frequency band f ≤ 50 Hz, as shown in Figs. 8.24 and 8.25. Due to the use of power flow expression, the starting ordinate of the transfer curve is no longer 0 value. The transfer curve retains the main characteristics of the raft, as shown in the auxiliary line in the figure. Second, the consistency of the mounted frequency is also very good. The two conclusions verify the accuracy of the WPA method and its theoretical analysis well. However, there is a big difference between beam ➀ and the engineering raft frame. The low frequency coincides well without Table 8.5 Test conditions when raft beam is 12 mm Operating conditions

1–1

1–2

1–3

1–4

1–5

1–6

1–7

Mass (kg)

–

13.2

13.2

8.8

2.2

8.8

8.8

Rigidity (N/m)

–

Rigidity

Rigidity

Position (m)

–

1.157

1.105

1–8 2.2 × 104

Rigidity

Rigidity

Rigidity

8.0

1.105

1.105

0.8

1.105

8.0 × 104 1.105

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Acceleration of Tran. (dB)

228

Test results Theoretical calculation result

Frequency (Hz)

Acceleration of Ttran. (dB)

Fig. 8.24 Comparison of theory and test results for working conditions 1–4

Test results Theoretical calculation result

Frequency (Hz) Fig. 8.25 Comparison of theory and test results for working conditions 1–7

impacts. The modal characteristics in the higher frequency band affect the form of the transmission characteristics. In the high frequency band f h ≥ 60 Hz, the vibration isolation performance deteriorates completely and the system harmonic components fully escape. This example shows that the raft frame is not good and the high frequency vibration isolation performance of the raft is poor. From the above analysis, it can be seen that the raft is not complex, but it is not easy to include all the parameters such as force source into the analysis program accurately, and it remains difficult to deal with this. Therefore, designers should maintain the balanced development of theoretical analysis and experimental debugging, summarize the main characteristics of rafts through engineering methods and promote the

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combination of theory and practice to finally form an engineering treatment method of dialectical unity. There is a big difference between the theory and test results which can be divided into two situations: one is that the model is the same, which is related to whether the modeling is reasonable or the accuracy of the theory; the other is due to the influence of such nonmainstream factors as the installation accuracy of the main source equipment, unknown error, etc. Finding out the cause of the error is an important research content of the test. Comparing the theoretical values and experimental measurement values, we can see that the frequency is much higher than the installation frequency of mechanical equipment in the high frequency section of f h ≥ 100 Hz. Because of the rich harmonic components caused by the beam mode frequency, the overall transmission characteristics of the raft are greatly increased. The reasons for the large error between the theoretical values and test values are as follows: (1) Due to the elastic installation of the source equipment, it cannot be guaranteed that there is only vertical displacement during the test. In the model test, there will be torsion, rolling, and other external displacement responses and the beam will not be subject to pure bending vibration. (2) There are many harmonic effects in testing the elastic vibration isolator. Due to the large lateral stiffness of the BE-type vibration isolator, the vibration energy escaping through the isolator cannot be ignored, and the harmonic effect is obvious. (3) Matching of vibration isolators: The impedance value of the isolator is frequency-dependent, and the smaller the isolator, the greater the dispersion of dynamic stiffness. There was no matching choice in the test, which resulted in some errors. (4) The frequency of the selected beam mode is low, and there are many beam modes in the transmission characteristics. In the model, the low-order modes of raft beam ➀ and foundation beams ➁ and ➂ are all shown in the transmission curve, and such additional low frequency modes as torsion even appear, leading to the energy leakage of the raft. In engineering practice, the overall stiffness of the raft is high, and the structural mode will not appear in this area. They may affect high frequency performance but show sparsity. The above factors are combined and the inconsistency of transmission characteristics is enlarged. Some of them are expected, but they are still within a reasonable range overall. Some of them can be solved by device debugging, for example, if the source equipment is installed in the middle of the beam, there must be an antiresonance point, which is almost impossible in an actual raft. Energy leakage will seriously affect the vibration isolation efficiency of the raft.

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8.9 Summary The simple application of rafts is not complicated, but it is not simple to achieve the ultimate effects. In this chapter, the source equipment is simplified as a single free rigid body, and the raft and base are treated as elastic beams. Based on the theoretical research of the WPA method and the previous research results, we can draw the following conclusions: (1) The double-stage raft meets the requirements of vehicle vibration isolation. Practice shows that the double-stage raft can meet the most stringent acoustic design requirements, as well as taking into account the vibration intensity, impact resistance, and reduced overall resource consumption of the source equipment. The performance of the raft is not ideal, which is often unrelated to multistage vibration isolation. The main reason is the harmonic leakage of the system, which is not related to the efficiency of the double-stage raft. (2) If the mixed damping effect is not considered, the isolation efficiency of the raft is equivalent to that of the two-stage isolation which is the optimization limit. The two-stage raft is also asymptote with “three-section lines”, 12 dB/Oct and 24 dB/Oct, and the two-stage isolation is its limit boundary. Moreover, the raft is more complex and connected, and the possible factors that lead to the deterioration of transmission characteristics due to various acoustic leakages are also increased. Our past cognition has been biased, and we may need to appropriately update our design concepts, architecture selection, and optimization paths. (3) The “mixed offset effect” can bring additional attenuation. This is the nondominant advantage of the raft. The counteraction mechanism between force sources is unique to the raft, and the “mixed offset effect” is one of the most special and important advantages as it improves isolation efficiency through additional attenuation. The “mixed offset effect” of the raft corresponds to the isolation efficiency, the “mass effect” corresponds to the overall resource saving, and the “tuning effect” has both. (4) The bigger the raft, the better. Obviously, the more equipment, the more sufficient raft systems emerge, the more significant the overall weight/space resource savings. If the raft is appropriately large and the number of pieces of equipment is appropriate, all the functions will be improved. Compared with two-stage vibration isolation, 78.3% of the total resources of seven rafts can be saved. At the same time, it can improve the vibration isolation performance and impact the resistance of the system. However, the selection of large rafts should also be balanced with the current technical capacity, equipment vibration level, and overall matching. The vibration isolation efficiency and engineering control ability should also be taken into account, and it is advisable to gradually upgrade and add refined commissioning links. From single/double-stage vibration isolation to rafts, the vibration isolation efficiency is not the only reason for selection, but should be perfectly unified with the

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231

overall weight/space resources of the carrier! From this point of view, designers should understand rafts, study rafts, apply rafts, and debug rafts.

References 1. Yan J (1985) Mechanical vibration isolation technology. Scientific and Technical Documentation Press, Shanghai 2. Wu C, Fu T (1994) Research and design of a double-layer vibration isolation system for middle raft with elastic characteristics. Ship Eng Res 4:20–24 3. Wu C (1998) A summary of the comparative study of raft isolation and double-layer isolation. Ship Eng Res 80(1):29–33 4. Kings R (1986) Some observations on the achievable properties of diesel isolation systems. In: Shipboard acoustics-proceedings ISSA’86 5. Yan J (1991) New development of vibration isolation and noise reduction technology. Noise Vib Control (5/6):11–16 6. Shen M, Yan J (1994) Engineering example of ship raft installation. Noise Vib Control 1:21–23 7. Shen M, Yan J (1994) Engineering example of ship raft installation (continued I). Noise Vib Control 3:45–48 8. Shen M, Yan J (1994) Engineering example of ship raft installation (continued II). Noise Vib Control 5:45–48 9. Shi Y, Zhu S, Lü Z et al (1995) Research on double-deck vibration absorber (raft type) for luxury Yacht. J Naval Univ Eng 2:1–5 10. Zhu H (1994) Theoretical modeling and analytical method of raft installation. Ship Sci Technol 134(2):14–19 11. Chen X, Qi H, Wu C et al (1999) Optimization design of two-stage vibration isolation system based on maximum entropy method. J Vib Shock 18(4):45–49 12. Liu Y, Shen R, Yan J (1996) Study on shock response of multilayer vibration isolation system. Noise Vib Control 3:2–8 13. Wu C, Du K, Zhang J et al (1996) Design method of raft vibration isolation system. Ship Eng Res 74(3):37–45 14. Wang C, Zhang Z, Shen R (1998) Application of state space method in response analysis of raft vibration isolation system. Noise Vib Control 5:11–16 15. Chen X, Lu F, Qi H, Wu C (1999) Computer simulation of raft vibration isolation system based on equation of state. Noise Vib Control (1):15–17, 29 16. Lin L (1999) The application and development of raft vibration isolation technology in ship. Tech Acoust 18(Suppl):1–4 17. Shang G, Li W (1999) Feature evolution of ship raft system. Ship Sci Technol 01:21–23 18. Sheng M, Wang M, Sun J (1999) A statistical analysis model for the raft isolation system. J Northwest Polytech Univ 17(4):633–637 19. Yu W (1995) Study on several problems of multilayer vibration isolation system. Shanghai Jiao Tong University, Shanghai 20. Feng D, Song K, Zhang H, Sun Y (1997) Influence of raft elasticity on vibration transfer of complex vibration isolation system. J Shandong Inst Build Mater 11(3):219–223, 253 21. Yu W, Shen R, Yan J (1995) Research on parameter optimization of multilayer vibration isolation system. Noise Vib Control 5:14–19 22. Sun L, Song K (2003) Analysis of power flow characteristics of raft vibration isolation system. Chin J Appl Mech 20(3):99–102 23. Zhang H, Fu Z (2002) Influence of main parameters of raft vibration isolation system on performance of system. J Vib Shock 19(2):4–8 24. Li X, Song K, Sun Y (2003) Study on the transferring characteristics of power flow through raft by complex excitations. Chin J Appl Mech 20(2):32–37

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25. Chen Z (2005) Study on the “Mass Effect” of raft vibration isolation systems. China Ship Development and Design Center, Wuhan 26. Crawley E, Cameron B, Selva D (2017) System architecture: strategy and product development for complex systems (Feixiang A, Trans). China Machine Press, Beijing 27. Wu CJ, White RG (1995) Reduction of vibrational power in periodically supported beams by use of a neutralizer. J Sound Vib 181(1):99–114 28. Wu C (2012) WPA analysis of structural vibration and its application. Huazhong University of Science and Technology, Wuhan 29. Mao Q, Pietrzyk S (2016) Control of noise and structural vibration: a matlab-based approach (Wu W, Wong Z, Wang F, Trans). Harbin Engineering University Press, Harbin 30. Fuller CR, Elliott SJ, Nelson PA (1997) Active control of vibration. Academic Press, London

Chapter 9

Vibration Power Flow and Experimental Investigation

ey started learning at a young age and made some achievements when their hair turned grey. Learning from books is far from enough. Only practice guarantees you a better understanding and the correct way to knowledge. ——Instructions to Ziyu about Learning on a Winter Night, Lu You, Song Dynasty

9.1 Basic Theory of Vibration Power Flow Also known as “vibration energy flow”, vibration power flow was initiated by the internationally renowned scholar R. G. White of the Institute of Sound and Vibration Research (ISVR) in the UK. The theoretical research and engineering practice of power flow provides new theoretical methods and analytical perspectives for structural dynamics design.

9.1.1 Research Review of Power Flow Power flow theory is a significant branch of structural noise theory. Many scholars have made a host of pioneering and original contributions to the theoretical development and engineering practice of power flow. White [1] proposed the concept of power flow and constructed the complete theoretical framework. Goyder and White studied the power flow of longitudinal waves, torsional waves, and bending waves in infinite beams, as well as the power flow carried by bending waves in a board in Ref. [2]. In Ref. [3], they analyzed transmission power flow in a rib-stiffened plate. In Ref. [4], they further researched transmission power flow from machine to base and provided the power flow expression of single-

© Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 C. Wu, Wave Propagation Approach for Structural Vibration, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-7237-1_9

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and double-stage vibration isolation systems excited by force source or velocity force. Mace [5] studied the reflected waves and transmitted waves generated in the discontinuous sections of a beam and established the coefficient equation. Cuschieri [6] adopted the point admittance method to analyze the power flow of an L-shaped plate and determined the resonant mode response of the coupling system. Cuschieri [7] subsequently studied the transmission characteristics of power flow in an infinite periodic structure using point admittance method and utilized the admittance function between the units to describe the coupling of subunits in the periodic structure. Grice and Pinnington [8, 9] put forward an analysis method for the vibration of a ribstiffened panel, which was computed in the form of analysis in Ref. [8] and further computed using the finite element method and compound resistance method in Ref. [9]. Seo and Hong [10] used power flow theory to predict the response of a beamplate coupling system at high frequency. Park et al. [11] researched the bending wave power flow of a finite orthotropic plate and utilized the time and local space average far-field wave energy density to deduce the energy equation. Fahy et al. [12] studied the time average power flow of a spring and damping element coupled two oscillator system under the force source of white noise. Leung and Pinnington [13] analyzed the reflection coefficient and transmission coefficient when two infinite beams are vertically connected using the WPA method, and analyzed the mutual transformation of longitudinal and bending waves. Homer and White [14–16] analyzed the propagation and reflection of structural waves in the joints of a torsion beam and studied the transmission and power flow of medium waves in a curved beam. Langley [17] simulated all parts of the framework with beam units and analyzed the vibration characteristics of beam and framework structures with the dynamic stiffness approach, as well as power flow under harmonic force or random excitation. Wu and White [18–20] used the WPA method to study the input and transmission power flow of a finite length multi-supported beam, and carried out research on periodic and quasi-periodic structures in Ref. [19], and the tuning effect of a tuned mass damper on the power flow of a coupled beam in Ref. [20]. C. J. Wu made a comprehensive summary with the universal equation established via the WPA method in Ref. [21], discussed the relationship between input power flow and point mobility, and elaborated upon engineering application cases. Pavic [22] analyzed the power flow of a cylindrical shell and provided the axial and circumferential power flow. It was found that the axial, circumferential, and radial shell movements made distinct contributions to energy. The derived equation corresponded to research on two plate movement items and one curvature item. Pavic [23] also analyzed the four forms of structural waves when a fluid-filled pipe vibrates at low frequency, and calculated and discussed the power flow. The results show that all structural waves transmit energy through both the pipe walls and the fluid. Ji et al. [24–27] cited the modal concept, transformed the power flows generated by N discrete excitation forces into N independent power flow modes, and simplified the calculation methods. They later [27] used the same method to analyze the vibration characteristics and power flow of a coupled long wave emitter and shortwave receiver at medium frequency. Wester and Mace [28–30] utilized the statistical wave method

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to analyze the wave components and power flow of a complex structure, disintegrated the structure to inter-coupled subsystems, utilized two scattering matrixes to describe the reflection and transmission of the wave components, and studied two regular and irregular coupling plates. In addition to the analysis method, many scholars combined the finite element method [31–34], spectral element method [35–39], and other numerical methods to study the power flow of structural units and complex structures. Nefske et al. [40] proposed the finite element method of power flow in 1989, and applied it to beam structures. It has been continuously developed ever since. Chinese scholars have also actively participated in research on power flow and carried out a large quantity of works covering multiple fields. Jikuany [41] used the four-end parameter method to study the vibration energy flow of an inclined vibration isolation system. Shungen et al. [42] elaborated upon the transmission of energy flow in vibration isolation systems and carried out the theoretical and experimental analysis of energy transmission in single- and double-stage vibration isolation systems with the fundamental structure of an infinite viscoelastic beam in Ref. [43]. Sun [44–46] studied the statistical energy analysis of nonconservative systems. Zhang and Zhang [47–50] combined the periodic structure theory and power flow method to explore the input and transmission power flow of an infinite periodic simply supported beam under force excitation. In Ref. [50], periodic structure theory was adopted to analyze the vibration power flow of a periodic rib-stiffened cylindrical shell. Li and Zhang [51] studied the bending and longitudinal vibration wave and power flow of an infinite periodic simply supported beam in a plane. Yi et al. [52] studied the power flow characteristics of a plate structure and beam-plate coupling structure [53]. Li et al. [54, 55] researched the power flow of a viscoelastic damping combination L-shaped plate, and used the admittance method to study the input and transmission power flow of an L-shaped reinforcing rib plate with external applied load. Xu [56] introduced power flow into the cylindrical shell-flow filed coupling system, analyzed the interaction of the structure-flow field from the perspective of energy and the power flow of free vibration and wave propagation characteristics, and studied the ribstiffened cylindrical shell-flow field coupling system. Wang and Sheng [57] carried out research on the power flow of an infinite reinforcing rib plate structure under force excitation. Wang and Sha [58] analyzed the vibration energy transmission of a rod plate structure. Liu [59] researched power flow using the SEA method. Xie and Wu [60] developed a secondary structural power flow analysis module based on Nastran and embedded it into a system solver. They then used the strong postprocessing function of MSC Patran to complete the result post-processing of power flow and vibration energy, and made it a valid tool for engineering personnel to analyze structural power flow. Zhu et al. [61] applied vibration power flow theory to the damage identification of structures. Some scholars expanded and implanted power flow theory in the medical field to study heart blood flow and pharmacology flow.

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9.1.2 Basic Characteristics of Power Flow In structural vibration analysis, the traditional research method involves a single physical quantity such as force and displacement being used to measure the degree and transmission of structural vibration, while the internal relations between physical quantities are ignored. Vibration power flow not only considers the amplitudes of force and velocity, but also the phase relationship between them, which gives the vibration power flow method the following advantages: (1) Structural vibration power flow describes the vibration energy of each point on a structure, and is not sensitive compared with phase and frequency dispersion; (2) The vibration energy of a certain point on a structure is a scalar, but vibration power flow is a vector with size and direction, which is a simple physical concept that is easy to understand. Researchers can determine the flow direction by analyzing the waveform conversion, energy storage, and flow in the structure; (3) The energy comparison of each transmission path of power flow is studied so as to establish a new judgment rule, giving the important order of energy passing through each transmission path according to the order; (4) By studying the power flow of different types of structural waves, such as bending waves, longitudinal waves, and torsional waves, the refined analysis of the vibration energy transfer mechanism is completed. The fact that structural vibration energy and structural waves are essentially related provides researchers with a new analysis perspective. For example, the spatial image of structural power flow represents more effective information than the distribution of structural vibration acceleration, becoming an important tool for research on the structural noise source identification and formation mechanism.

9.1.3 Development and Focus of Power Flow The development of power flow theory originated from the demands of engineering practice. Airborne sound was the earliest and most direct form to relate to human beings. With the development of research, researchers noticed that describing airborne sound with the energy concept could represent more practical information. Finally, the sound intensity method was developed. F. Fahy of the ISVR in the UK, and the American scholar J. Y. Chung are officially recognized as the two co-inventors of the sound intensity method. Sound intensity describes sound in the medium of air in terms of energy. Accordingly, structural sound intensity describes vibration in a solid medium in terms of energy, in which sound refers to structural noise in particular. According to different media, three types are recognized internationally: air sound, structural sound, and underwater sound. The sound intensity method has been used in industries for decades. Why did the theoretical research of power flow lag behind? Because it was more difficult

9.1 Basic Theory of Vibration Power Flow

237

to develop power flow theory. However, although power flow theory is not perfect, this does not affect the wide recognition of its theoretical and engineering value. Its research fields cover aerospace, ships, satellites, aircraft, automobiles, tanks, etc. Power flow research reached its peak in the middle and late 1980s, becoming a “trend” in the field of dynamics. It has been a hot topic for more than 20 years, forming a relatively large scientific research team, contributing to the establishment of the theoretical framework of power flow, gradually developing into an independent theoretical method and contributing to China’s strength. Energy is used to describe structural vibration, and power flow theory gives structural dynamics a new theory and control method, becoming the inevitable path from the systematic to the comprehensive to the refined process of engineering design. Power flow research should focus on the following contents: (1) Structural waves: Displacement, velocity, and acceleration are the results rather than the causes of structural vibration. Structural waves and target control parameters, such as structural sound radiation, acceleration response, stress intensity, etc., stand in a causal relationship. There are unique internal relationships between physical quantities. Therefore, we should focus on the study of structural waves and grasp a new analysis perspective. (2) The study of wave transmission and wave conversion: There is a process from the excited “component” to the final target system, including a complex transfer path, storage, attenuation, and waveform conversion. In the past, this aspect received insufficient attention. It has important added value for revealing the formation mechanism of the target control parameters and seeking the optimal control method. As we all know, the underwater radiation efficiency of a bending wave in a ship structure is the highest. For example, waveform conversion used to be of little concern, but it will be a new aspect in the dynamic optimization of complex systems, establishing a new horizon of wave flow. (3) The input of vibration energy: When research is extended from a single point to the whole system, the vibration energy input is not only related to the origin impedance and transfer impedance, but also to the whole system being studied. How much energy can the whole system input? The relationships with input subsystems are very limited in traditional analysis. Naturally, how to control the specific associated energy input through the overall structural design has very important practical value. When using vibration energy to describe the dynamic characteristics of structures, we can learn from the mature theory and laws of energy. Just as in thermodynamics, the study of power flow shows that the vibration energy of structures follows the energy laws. It always flows from the highest point of energy to the lowest point of energy, and automatically converges to the “avenue” of the system, which refers to the structural path which can most easily gather and conduct the vibration energy in the structure.

238

9 Vibration Power Flow and Experimental Investigation

9.1.4 Input Power Any point in a structure is excited by point harmonic force p˜ o , and the input power is equal to the product of the real part of the excitation force and the velocity of the corresponding point of the harmonic force on the structure. As shown in Fig. 9.1, the instantaneous input power on the structure is as follows [19]:   jωt   ˙ P˜S = Re po e jωt · Re we

(9.1)

where P˜S —Instantaneous input power of injection structure. p˜ o (x, y, t)—Amplitude of harmonic force. w(x, ˙ y, t)—Lateral displacement velocity of structure ∂w(x)/∂t. In engineering, average power is more commonly used than instantaneous power, which is represented by the following formula: ω PS = 2π

2π/ω 

  jωt   ˙ dt Re po e jωt · Re we

(9.2)

0

Substituting Eq. (9.1) into Eq. (9.2), the average power of the input structure of point harmonic force is as follows: P S (xo ) =

 1   1  Re po w˙ ∗ = Re po∗ w˙ 2 2

This can be further rewritten as

Fig. 9.1 Input power structure

(9.3)

9.1 Basic Theory of Vibration Power Flow

PS =

239

1 | po |2 Re{βo } 2

(9.4)

 where βo = V p˜ o is the origin mobility of the structure, which is more suitable for engineering applications. Assuming a finite beam of length L, apply unit harmonic force p0 = 1 at x = x0 . For convenience, the displacement response of Eq. (2.20), regarding the beam is copied as follows: w(x) =

4 

A n e k n x + po

n=1

2 

an e−kn |x−xo |

(9.5)

n=1

At this time, the real part of the structural vibration response is     Re{βo } = Re − jωw ∗ (xo ) = −Im ωw ∗ (xo )

(9.6)

where superscript * indicates the conjugate of the complex number. Substituting the known conditions into Eq. (9.5), gives us w(xo ) =

4 

A n e kn x o +

n=1

2 

an

(9.7)

n=1

 4  2   1 kn x o ˜ An e + an PS = − ω · Im 2 n=1 n=1

(9.8)

Power of the infinite uniform beam input by point harmonic force [1] PS =

ω| po |2 8E I k 3

(9.9)

Similarly, the input power of an infinite uniform beam under flexural moment is PS =

 1   1  Re M · θ˙ ∗ = Re M ∗ · θ˙ 2 2

(9.10)

More generally, the structure is subject to multiple external forces, such as nk harmonic forces and mk harmonic torque. The total power of the input structure is calculated using the following formula:  1   1  = Re pi w˙ i∗ + Re Mi θ˙i∗ 2 i=1 2 i=1 nk

P S,total

mk

(9.11)

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9 Vibration Power Flow and Experimental Investigation

where the coefficient i = 1, 2, 3, ... takes a positive integer. If structural damping is considered, Young’s modulus can be simply replaced by the complex modulus E(1 + jβ).  If and only if the damping value β is small, the complex wave number k(1 − jβ 4) is used instead of the real wave number. Here β is the damping loss factor of the structure.

9.1.5 Transmitted Power Two kinds of internal load, shear load and moment load, are induced by bending wave. Both of these internal forces conduct vibration energy in the form of structural waves. In Euler beams, the instantaneous power generated by the shear component is equal to the product of shear force and transverse velocity [19, 21]. ˜ t) ∂ w(x, ∂ 3 w(x, ˜ t) · P˜s (x, t) = E I ∂x3 ∂t

(9.12)

The instantaneous power generated by the moment component is equal to the product of the moment and angular velocity ˜ t) ∂ 2 w(x, ˜ t) ∂ 2 w(x, P˜m (x, t) = −E I · 2 ∂x ∂ x∂t

(9.13)

So, the total power is

3 ˜ t) ∂ w(x, ˜ t) ∂ 2 w(x, ˜ t) ∂ w(x, ˜ t) ∂ 2 w(x, P˜a = P˜s + P˜m = E I − · · ∂x3 ∂t ∂x2 ∂ x∂t

(9.14)

The beam moves in a simple harmonic way,w(x) ˙ = jωw(x), θ˙ = jωθ . In this way, the instantaneous power transmitted along the beam can be expressed as the sum of the transmitted power of the two internal loads, as in Eq. (9.14). They are further expressed as  3 ˜ t) ˜ t) ∂ w(x, ∂ 2 w(x, ˙ P˜a (x, t) = S w˙ − M θ˙ = E I θ w ˙ − ∂x3 ∂x2

(9.15)

Similarly, the time average of the transmitted power is more of an engineering concept. The transmitted power of the time average generated by the moment in the beam corresponding to any point on the beam (0 ≤ x ≤ L) is P u (x) =

 1   1  Re S(− jω)w ∗ = Re S ∗ jωw 2 2

(9.16)

P m (x) =

 1   1  Re M(− jω)θ ∗ = Re M ∗ jωθ 2 2

(9.17)

9.1 Basic Theory of Vibration Power Flow

241

where w = A1 E 1 + A2 E 2 + A3 E 3 + A4 E 4 + a1 (E 5 + j E 6 )

(9.18)

M = E I k 2 {A1 E 1 − A2 E 2 + A3 E 3 − A4 E 4 + a1 (E 5 − j E 6 )}

(9.19)

θ = k{A1 E 1 + j A2 E 2 − A3 E 3 − j A4 E 4 − a1 (i f )(E 5 − E 6 )}

(9.20)

S = E I k 3 {A1 E 1 − j A2 E 2 − A3 E 3 + j A4 E 4 − a1 (i f )(E 5 + E 6 )}

(9.21)

and E 1 = ekx , E 2 = e jkx , E 3 = e−kx , E 4 = e− jkx , E 5 = e−k|xo −x| , E 6 = e− jk|xo −x|

(9.22)

where (i f ) is the symbolic operator. If x < x0 , then (i f ) = −1; or (i f ) = +1. Using Eqs. (9.16) and (9.17) to calculate the total average power of the point harmonic force along the beam, which is the sum of the shear power and power components of the flexural moment in the beam, we get P a (x) = P u (x) + P m (x)

(9.23)

For comparison, the transmitted power of an infinite beam is given in Refs. [2] and [19] Pa =

ω| po |2 16E I k 3

(9.24)

Comparing Eq. (9.9) with Eq. (9.24), it can be seen that the total transmitted power is independent of distance under undamped conditions, which is exactly half of the input power of the point harmonic force.

9.2 Power Flow Test of Structure There are two purposes to carrying out power flow test research [62]: first, to verify the power flow theory, the new method must withstand the test verification; and second, to explore the engineering application of the new theory. Theory and measurement shall form a complementary pair. Scholars have made systematic and original contributions in power flow measurement theory, measuring instrument design, model design, test bench construction,

242

9 Vibration Power Flow and Experimental Investigation

theory, and test verification, which constitute important links in the development of power theory. Power test includes input power measurement, transmitted power measurement, and power measurement through vibration isolators.

9.2.1 Summary of Test and Measurement Research In the related literature, power measurement can be divided into the direct measurement method and indirect measurement method, which are further divided into the contact type and noncontact type. The input power measurement of structures is relatively easy. Most of the early measurement methods are direct measurement methods: the force sensor measures the amplitude and phase of the force, and the accelerometer picks up the origin velocity or acceleration of the excitation force, then the transient input power of the structure can be calculated directly. However, the practical operation of this method is difficult and the measurement results are not stable. In 1969, Fahy [63] proposed a cross-spectral density method for measuring input power using an impedance head on an exciter, which has become a classic method of input power measurement and a natural extension of the author’s research on the sound intensity method. Noiseux [64] initiated research on the power measurement of infinite beam and plate structures. He expressed the power of uniform mass beams and plates with the surface parameters of beams and plates. After approximate processing, the expression is simplified as an indirect measurement method which only requires two acceleration sensors. In the experiment, the power in an infinite plate was measured for the first time. However, this approximation makes the method only applicable to measurements far away from the boundaries of beams and plates, and the location of the discontinuous section. The measurement of the transmitted power is much more complicated. The Bulgarian researcher, Pavic [65] derived a general equation for measuring the power in a beam or plate. He used the differential approximation of the higher derivative of displacement in the power equation, so that only four sensors are needed to measure the transmitted power at a certain point in a beam, and nine sensors can complete the measurement in a plate. This method greatly advanced power measurement. The Belgian researcher, Verheij [66] proposed a method using cross-spectral power density in the frequency domain. By measuring the cross-spectrum function of three pairs of signals at four different points on the beam, the transmitted power measurement was completed. In this paper, force and acceleration sensors are used to measure the input power of the structure at the excitation point, and the transmitted power of flexural waves, longitudinal stretching waves, and torsional waves propagating in the beam structure is expressed as a cross-spectral density form of finite differential approximation. This paper has become a classic in the literature on measuring power using the cross-spectrum method. In addition, these two important achievements of Pavic and Verheij were completed under the guidance of their tutor White during their doctoral studies at ISVR. Due to the measurement conditions at that time, Redman and White [67] carried out two research explorations during

9.2 Power Flow Test of Structure

243

Redman’s doctoral research at ISVR: first, design a power measuring instrument; second, the amplitude of the excitation force is found to be unstable. It changes with the frequency response characteristics of the tested structure, especially in the resonance region, and it cannot maintain a constant force amplitude. C. J. Wu and White [19, 20] directly used the spectrum analyzer to obtain the input power and transmitted power of finite long periodic and quasi-periodic beam structures. Under the measurement conditions, the theoretical value was in good agreement with the measured value. Their contributions are as follows: first, the measurement is extended from a general infinite structure to a finite structure; and second, random white noise excitation is used to overcome the problem that the input amplitude of the excitation force changes with the frequency response characteristics of the tested structure, especially in the resonance region, and the measurement results are unstable. Chinese researchers Zhao [68], Li et al. [69] also conducted research on power measurement. Zhu et al. [70] carried out a study on the measurement method of the vibration energy flow of a pipe wall. Zhu et al. [71] used the power method to study impact dynamics and damage identification and carried out the structural diagnostic analysis. The measurement of power by isolators is another important aspect in the development of power theory. Pinnington and White [72] successfully measured transmitted power for the first time through a vibration isolator by using a double accelerometer and measured the power of the input system of the excitation point using a force sensor and acceleration sensor. This method is still the main method for measuring power in vibration isolation systems. Zhou [73] studied the application of the strain gauge method in power measurement. Finally, two points are made: The direct measurement method uses a filter system which has a fast measurement speed and ample information, making it suitable for general-purpose measurement research and particularly suitable for occasions that require the expression of octave bandwidth and decibel number. The indirect measurement method requires an FFT spectrum analyzer to measure and analyze the relevant data; the advantage of this is that random error analysis can be carried out during the measurement, while its disadvantage is that the analysis time is relatively long. The power measurement method can be divided into the contact type and noncontact type based on whether the sensor is in contact with the tested structure. Most of the work related to vibration power measurement are based on contact methods such as dual sensor methods [74]. In the noncontact type, the measuring probe does not need to make contact with the structure surface, so there is no interference to the vibration measurement, such as laser Doppler vibration measurement [75, 76], acoustic holography [77], electronic speckle interferometry [78], and holographic interferometry [79]. The contact measurement method, the source method of power measurement, is practical. Noncontact measurement represents the future development, but the practical engineering measurement method is still uncertain.

244

9 Vibration Power Flow and Experimental Investigation

9.2.2 Input Power Measurement The direct measurement method, which adopts the excitation force and origin response, is simple and direct, but it is difficult to implement in engineering. The indirect measurement method transforms the force and structural response into origin velocity mobility or acceleration mobility to complete the indirect measurement.

9.2.2.1

Direct Measurement Method

Applying harmonic force at any point on the structure and inputting the time average power Pin  of the structure equal to the real part of the product of the point harmonic force and structural speed response at the excitation point, we get Pin  =

1 1 Re{ po · V ∗ } = Re{ po∗ · V } 2 2

(9.25)

where p0 —Simple harmonic excitation force applied p0∗ —Conjugation ofp0 V —Speed response of structure V ∗ —Conjugation of V. The above equation is the physical expression of direct measurement. In actual operation, the output amplitude of the excitation force is difficult to keep constant in the resonance region of the structure. Therefore, it is not easy to obtain accurate and stable measurement results.

9.2.2.2

Indirect Measurement Method

Using the indirect measurement method, the mathematical expression is as follows [14]: Pin  =

| po |2 Im{A11 } 1 | po |2 Re{M11 } = 2 2ω

(9.26)

where M11 —Mechanical mobility at action point of harmonic force on structure (Inertance); A11 —Acceleration mobility at action point of harmonic force on structure. The indirect measurement method completes the spectrum analysis according to the FFT transformation of two signals. The correlation function is introduced here, which describes the time average relationship between two-time domain signals. The cross-correlation function between the harmonic force and corresponding point velocity on the structure is defined as [80]

9.2 Power Flow Test of Structure

245

1 R f v (τ ) = lim T →∞ T

T p˜ o (t) · V (t + τ )dt

(9.27)

0

In Eq. (9.27), the intensity component of average power shall be arrived at by 1 Pin  = lim T →∞ T

T p˜ o (t) · V (t + τ )dt = R f v (0)

(9.28)

0

The velocity distribution of the product of harmonic force p˜ 0 and velocity V is equal to the FFT transform of the cross-correlation function, which defines the correlation density function between the excitation force and the structural velocity response 1 S f v (ω) = 2π T

+∞ p˜ o (τ ) exp(− jωτ )dτ

(9.29)

−∞

The function is complex and represents the average phase relationship between p˜ 0 and V . R(τ ) and S(ω) in Eqs. (9.28) and (9.29), form an FFT transform pair, namely +∞ S f v (ω) exp(− jωτ )dω R f v (τ ) =

(9.30)

−∞

and +∞ S f v (ω)dω Pin  = R f v (0) =

(9.31)

−∞

In this case, S f v (ω) represents the frequency distribution of the average intensity. The cross-spectrum function has the following characteristics: 



Re S f v (ω) = +Re S f v (−ω)



 Im S f v (ω) = −Im S f v (−ω)

(9.32)

Spectral function S(ω) is defined as a function in the full frequency (positive and negative axis) coordinates, while it is also represented by a reverse phase vector pair. For the convenience of use, the spectral density function is often defined as unilateral frequency function

246

9 Vibration Power Flow and Experimental Investigation

⎫ G f v (ω) = 2S f v (ω) ω > 0 ⎬ G f v (ω) = S f v (ω) ω = 0 ⎭ ω=

EI

3 ¨ ¨ ¨    4 a3 a2 − a4 a2 − a3 a1

(9.46)

where is the distance between two adjacent sensors. In the frequency domain, the corresponding power expression of Eq. (9.46) is Ps  =

  EI Im 4G a3 a2 − G a4 a2 − G a3 a1 3 (ω )

(9.47)

where G ai a j is the cross-spectral density of acceleration at the i point and j points on the structure. The one-dimensional flexural wave measurement of the beam structure is shown in Fig. 9.2. In the actual measurement, the linear array of four acceleration sensors is used to measure the transmitted power. Among them, the spacing is also the test content.

250

9 Vibration Power Flow and Experimental Investigation Point of interest

Fig. 9.2 Arrangement of accelerometers for one-dimensional measurement of flexural wave intensity

9.3 Testing and Measurement 9.3.1 Test Structure and Parameters The test beams are made of low carbon steel with cross sections of 50 × 60 mm and lengths of 2,100 mm and 4,100 mm, respectively. The former is designed as a simply supported multi-span beam; the latter is designed as an embedded section with 1,000 mm embedded in the sandbox at both ends to simulate an infinite beam with no reflecting end, equivalent to a working length of 2,100 mm. Such a design can be used to examine the following four situations: (1) simply supported beams; (2) finite periodic beams; (3) finite multi-span beams; (4) infinite periodic beams. The test bench is shown in Fig. 9.3. See Table 9.1, for the data parameters of the beams.

9.3.2 Test Procedure The input power and transmitted power of the structure are measured by the programmable HP3566A/67A spectrum analyzer (B&K Precision) and excited by an electric vibration exciter. The accelerometer is a B&K4344 (with a mass of 2.2 g) from Denmark. During the experiment, three kinds of excitation methods were tried (1) Steady-state sine sweep mode (2) Short pulse chirp mode (3) Random excitation mode During the test, the short pulse chirp mode is a self-contained excitation mode of the analyzer. The schematic diagram of testing equipment and facilities is shown in Fig. 9.3. The linear array of the accelerometer shown in Fig. 9.2, is used for measurement.

9.3 Testing and Measurement

251

Oscilloscope Multi-supported beam

B&K 4344

Exciter

Power Amp.

B&K 2635 电荷放大器 电荷放大器 电荷放大器 电荷放大器 电荷放大器

Charge Amp.

Signal generator

ch.1

HP 3566A/67A

FFT Analyzer

Ch.6

——Input Power ——Transmitted Power

Post-processing

PC Computer

Maths Program

Fig. 9.3 Instrumentation for input power and power transmission measurement of simply supported finite periodic beam

Table 9.1 Physical quantities and parameters of test

Classification

Parameters

Length

L = 2100 mm, L = 4100 mm

Width

b = 50 mm

Height

h = 6.0 mm

Material density

ρ = 7900 kg/m3

Loss factor

β = 0.05

Young’s modulus

E = 207 × 109 N/m2

9.3.3 Input Power Measurement In the test, three excitation modes including continuous random, steady-state sine sweep, and short pulse chirp are used to study the excitation structure in the frequency range. The cross-spectral density between the measured force (channel 1) and origin

252

9 Vibration Power Flow and Experimental Investigation

Freq. Response (m/s2 Hz)

acceleration signal (channel 2) is monitored and displayed during the test. In the programmer of the spectrum analyzer, the self-programmed INPUT.MAT is loaded and the input power calculated according to Eq. (9.35). The results are in the frequency domain and the unit is W/Hz. The random excitation sampling data is averaged 50 times to smooth the measurement curve and eliminate random errors. The structural frequency response of the excitation point is shown in Fig. 9.4. By comparing Fig. 9.4a, b, it can be seen that the low frequency response under random excitation is cleaner than that under sine sweep excitation. There is little difference between the two, but the gap will be enlarged in power measurement. In the relatively high frequency band, they are both accompanied by some undesirable small peaks. Many tests show that these tiny peaks, which are in the support and

(a)

Freq. Response (m/s2 Hz)

Frequency (Hz)

Frequency (Hz)

(b)

Fig. 9.4 Original acceleration inertance of three-equal-bay simply supported beam. a Random excitation, averaging 25 times; b sine sweep excitation

9.3 Testing and Measurement

253

Input Power Density (N2/Hz)

boundary constraints, are caused by the additional flexural moment due to abnormal friction, and the more friction points, the more serious the impact. The power density of the input structure is shown in Fig. 9.5. The corresponding measurement results of random white noise excitation are shown in Fig. 9.5a. This is the average result of 50 measured data samples, which is much better than the average result of two data samples. Figure 9.5b is the measurement result of sine sweep excitation. The excitation force is normalized in the solution, but the input power is not ideal because it is unstable in the resonance region and fluctuates greatly with frequency. The curve appears to be much smoother than that under random excitation, but the statistical variance of the latter is actually much higher than that of the former, which is caused by the instability of the input excitation force.

（a）

Input Power Density(N2/Hz)

Frequency (Hz)

Frequency (Hz)

(b)

Fig. 9.5 Input power of a three-equal-bay simply supported beam at center of beam. a Random excitation (averaging 50 times); b sine sweep excitation

254

9 Vibration Power Flow and Experimental Investigation

Coherence

The measurement of input power is relatively simple. We can see that when measuring power, the number of sensors increases from two to six, and the advantages of the random excitation method in measuring power are gradually revealed. Figure 9.6 shows the measured coherence function of structural origin. In the low frequency band, the coherence coefficient of some frequency points is not ideal. When using normal variables and taking sine excitation with a step frequency interval of 5.01 Hz, the system correlation is greatly improved, as shown in Fig. 9.6b. The power of the input simply supported beam is shown in Figs. 9.7 and 9.8, and the solid line corresponds to the theoretical prediction value of loss factor β = 0.05 and excitation force p0 ≡ 1N; the measured data (dotted line) also normalizes the excitation force to unit force. They are drawn on the same map, and in good agreement in most cases.

(a)

Normal Variables

Frequency (Hz)

Frequency (Hz)

(b)

Fig. 9.6 Coherence function of force and origin of periodic beam with three bays. a Random white noise excitation (averaging 50 times); b sine sweep excitation (resolution 5.01 Hz)

255

Input Power (Nm s)

9.3 Testing and Measurement

(a)

Input Power (Nm s)

Frequency (Hz)

Frequency (Hz)

(b)

Fig. 9.7 Input power to simply supported beam. The data is normalized (i.e., p0 = 1N), random excitation, averaging 50 times,  = 50 mm (predicted values and measured values, dotted line). a 0–200 Hz; b 0–400 Hz

Note that the measured value between resonance frequencies always deviates from the theoretical estimate. By analyzing the test process, it is found that they are mainly caused by the additional moment due to the friction at the support, as well as the nonideal state of the beam itself, such as distortion and internal stress. Of course, the phase error between the acceleration sensor and the analyzer is also one of the reasons for the error. In particular, the experimental results are improved greatly after the sensor is matched and selected. These unnecessary additional friction disturbances are rectified by adjusting the support centering, adding lubricating oil, etc., and the effect is obvious. Unfortunately, it is too late to manufacture a new model.

9 Vibration Power Flow and Experimental Investigation

Input Power (W/Hz)

256

(a)

Input Power (W/Hz)

Frequency (Hz)

Frequency (Hz)

(b)

Fig. 9.8 Input power to simply supported periodic beam. As above, p0 = 1N, averaging 50 times,  = 25 mm (predicted values and measured values, dotted line). a 0–350 Hz, random excitation, resolution 1.00 Hz; b 0–350 Hz, sine excitation, resolution 5.01 Hz

In order to test the influence of the excitation methods, two of the three excitation methods are selected quickly, as shown in Fig. 9.8a, b. From the comparison, we can see that the measurement result of random excitation is the best, better than that of sine sweep excitation, with higher resolution and faster measurement speed.

9.3 Testing and Measurement

257

9.3.4 Transmitted Power Measurement In this chapter, taking a finite length multi-span beam as an example and based on the theoretical calculation of the WPA method, power test measurement is carried out. The measurement uses the indirect method and is implemented using a multichannel FFT spectrum analyzer. To measure the power transmitted along the beam, first specify the second acceleration sensor in the accelerometer line array as the reference channel (that is, the total channel 4 in the FFT). Monitor and demonstrate the crossspectral density between the fourth and second acceleration sensors and the crossspectral density between the third and second acceleration sensors. Next, change the reference channel to the first sensor in the sensor linear array (i.e., the total channel 3) and analyze the cross-spectral density between the third and first sensors. In the programmer of the spectrum analyzer, the self-programming solver PFLOW.MAT is loaded, and the vibration transmitted power in the frequency domain along the structure is calculated according to Eq. (9.47). Finally, the measurement results are normalized to the corresponding results of p0 = 1N. Change the reference channel to channel 1 (force signal) again and display the input power density. The operation of the first power measurement was a bit complicated, limited by the capabilities of the HP3566A/67A analyzers at the time, although they were among the most advanced FFT spectrum analyzers in the world at the time. It should be noted that the amplitude of the excitation force shall not be changed during the above measurement operation or else greater error will be caused. The average number of sampled data is the same as the previous measurement of the input power, and the 50 times of smooth processing shall be maintained. Sensor spacing selection, including the three cases of 10, 25, and 50 mm, and the best spacing of opt = 25mm shall be confirmed through the test. The internal measurement error of the analysis system caused by the finite difference method shall be corrected by the following equation [19, 62]: Ps  = Ps 

(k )3 2 sin(k ) − sin(2k )

(9.48)

or 1 Ps  = Ps  1−(k ) 2 /4 (k ≤ 0.8)

(9.49)

where Ps  corresponds to the measured value osf the transmitted power. When the k value is high, such as k = 0.8 and = 50 mm, the error of the corresponding frequency f ≈ 350 Hz is 20%. When = 50 mm is excited randomly, the measured power is as shown in Fig. 9.9. It can be seen that the measured value in the first propagation domain of the vibration power of the periodic structure is highly consistent with the theoretical prediction value. When the sine sweep method is used for excitation, the measurement error increases greatly; the reader

9 Vibration Power Flow and Experimental Investigation

Input Power (W/Hz)

258

（a）

Power Flow (W/Hz)

Frequency (Hz)

Frequency (Hz)

（b）

 Fig. 9.9 Power transmission in periodic beam with three bay random excitation at x = 5L 6. As above, p0 = 1N, averaging 50 times,  = 25 mm, predicted values and measured values. a 0–300 Hz; b 0–750 Hz

can independently verify this experiment. According to Eqs. (9.48) and (9.49), the spacing of sensors should be as small as possible. However, the relative error between the test results of the three different spacings is not large in this chapter. The mathematical description of power measurement is more complex than that of the sound intensity method, and there are more factors that cause an error, including sensor accuracy, phase error, and matching consistency. It cannot be as accurate as the measurement of structural acceleration. In the same measurement system, the structural response measurement accuracy is much higher than power measurement, especially transmitted power measurement. The measured value is a combination of multi-sensor measurement and multi-data calculation. Any small error will be multiplied in the measurement system. The relative inaccuracy of the power measurement

9.3 Testing and Measurement

259

results is endogenous. In this paper, crystal sensors are used for measurement in the experiment, and the overall agreement is good. It was exhibited at the school’s Open ‘92 as a scientific research achievement.

9.4 Control Power Measurement Accuracy Compared with the air sound intensity method, the measurement accuracy of structural sound intensity is much lower at present. The reason for the relatively large measurement error: first, there are more types of structural waves in the structure such as longitudinal waves, torsional waves and flexural waves, etc. In addition to the existence of the strong boundary of the structure, it is difficult to ensure that only the expected pure wave appears in the test. Any small, possible interference such as structural shape, beam, plate, or shell structural defects may be amplified in power measurement; second, more sensors are used in the measurement of structural power, and the multichannel signals overlap, further enlarging the measurement system error; and third, the strong coupling between the structural waves far exceeds that of the air medium. Power measurement was carried out in strict accordance with the operating procedures, and the test results are generally in good agreement. If amorphous sensors are used, the accuracy of the test device will be higher and the measurement accuracy will be further improved. In order to ensure measurement accuracy, the measurement of power shall be controlled as follows: (1) Strict measurement of system calibration: In addition to the requirements of conventional measurement systems, special attention shall also be paid to the matching of the group of acceleration sensors, especially when they are arranged in groups in a linear array; this was very difficult in the past, but now that the quality of sensors has improved, the consistency guarantee is much better. (2) The boundary, support design, installation, and commissioning of the test structure shall be strictly controlled to minimize the error of the test device itself. The ideal test structure with high processing accuracy and good conditions shall be selected to ensure that there is no distortion, no prestress, and accurate dimensions. (3) The measurement accuracy of the input power is much higher than that of the transmitted power. Since only two sensors are used, it is not only easy to achieve, but it also has a lower phase deviation. (4) Random white noise excitation can be used as a standard measurement. It shows a relatively higher consistency with the theoretical prediction value and fastest measurement speed. Sine steady-state excitation, affected by the unsteady output of the excitation force at the resonance frequency in the sweep process, produces results unworthy of confidence, especially in the measurement of transmitted power.

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(5) The main reason affecting measurement accuracy is the phase error of several acceleration sensors in the measurement system. The experimental results show that the measurement error is greatly affected by the “external load” caused by the extra friction at the support points. In addition, the effects of the torsion and initial prestress of the test beam cannot be ignored. It has been proven by trial and error that the “extra” small formant which deviates from the rule of the main curve is generated by the extra torque caused by the friction at the support points.

9.5 Summary Power theory is required for deepening structural noise theory. This method has clear and advanced physical concepts, as well as no obstacles to the theoretical part and input power measurement, but transmitted power measurement is still difficult. Not only does it lag far behind the theoretical analysis, it also encounters great development obstacles. Simple structural measurement is better suited to power theory verification. In this chapter, input power and transmitted power are measured in simple beam structures. We can see that the difficulty of transmitted power measurement is not comparable to (airborne) sound intensity measurement. From beams to plates, the number of sensors will increase significantly, and a complex measurement of the engineering structure will be required, involving plate thickness, boundaries, structural inhomogeneity, and so on. These obstacles cannot be solved by new sensors and measuring instruments alone; theoretical and engineering breakthroughs are also required. However, the defects of power measurement do not affect its theoretical and engineering value. Engineering measurement can be subdivided into three types: input power measurement, transmitted power measurement, and vibration isolation system power measurement. Input power measurement is easy to achieve, and the measurement accuracy is high because the phase deviation is small. Transmitted power measurement is the most difficult. The third type is approximately equivalent to the first. To our relief, if the first and third types of measurement solutions are developed more quickly, the practical application of complex engineering can certainly go further. For a period of time, we can establish an engineering correlation method, and develop input power measurement with theoretical and measurement parameters, and transmitted power with theoretical calculation values for comprehensive judgment. As the sound intensity method has been used in engineering for many years, it can be used for reference directly to develop such high-precision measurement as laser Doppler, acoustic holography, electronic speckle pattern interferometry, etc. In addition, if structural defects are the main causes of power measurement error, then the application of power theory to structural diagnosis [71] will be highly expected.

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Afterword

To control the flood, one can neither inflexibly stick to the ancient systems and regulations, nor follow the words of others hastily since the terrain extends from low to high and the water flows fast and slowly with a deep and shallow swamp and a winding and straight river. One cannot understand its real situation without observation and measurement and cannot fully understand its details without consultation and asking for opinions. Therefore, it is essential to painstakingly climb mountains and wade through rivers in person.

Flood Control Must Be Undertaken Personally Writing a book is like building a house during which one always hopes to make it perfect. However, with the limitation of my own knowledge, I wrote this book just like building a house by new methods, in which no model structure can be referred to. Inevitably, something wrong and improper may be found in this book, to which I look forward to your criticism. There are both satisfactions and imperfections when I finally finished the book, but more importantly, I hope the readers will feel free to point out the shortcomings of this book. This book centers on the complex structure of Bernoulli–Euler beams and the author has long been engaged in the research of complex structures. A beam, as a basic structure, is far from a Bernoulli–Euler beam. Though, they have something in common in terms of essence and abstraction, such as a profound understanding of the waveguide and structural discontinuity of structural connection, which offers far-reaching enlightenment for the analysis of complex structures. This book deduces the WPA equation and makes an analysis from the foundation beam to the complex hybrid power system, then to the engineering masts, floating rafts, etc., which gives a relatively complete mathematical equation and model test. In the framework of the theoretical analysis, WPA is simple, feasible, and clear in concept to analyze

© Harbin Engineering University Press and Springer Nature Singapore Pte Ltd. 2021 C. Wu, Wave Propagation Approach for Structural Vibration, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-15-7237-1

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structural damping, the coupling effect, and power flow. Several analytic examples show that WPA has good consistency with a distinctive perspective. Outstanding achievements have been made in structural dynamics, with perfect mathematical expression. It is a pity that the mathematical analysis of complex systems is not always “engaged”, with some necessary system features disappearing. We are full of doubts in the face of the frontier problems in our work. It is also an engineering method to go back to the basic, find the essential features of the basic structure, and establish the theory, model, and real ship criterion in combination with experience. Therefore, the secrets hidden in the large system will be revealed through deductive induction and logical thinking. This book aims to make WPA become a mechanism analysis tool that designers are willing to master, using structural waves to think and establishing logical relationships with the target control parameters of structural vibration and sound radiation. The content not included in this book is the examination of longitudinal waves in the rod, studying the waveform conversion between longitudinal waves in the rod, and flexural waves in the connecting structure, which needs to be further supplemented. Chapters 4 and 5 analyzes the complex systems. WPA uses “discontinuity” to divide the units and its efficiency lies in that the analysis is only related to the connection of “discontinuity” or the “line of discontinuity”, so the unknown is fewer. There is a lot of literature studying the super large units. The WPA’s characteristic is to elaborate on the important structure in the system with a super large unit. If necessary, it can also focus on the transmission of structural waves in nodes. The number of “discontinuities” in a giant system changes qualitatively due to quantitative changes, and some local structural characteristics have subtle relations with the giant system if the giant system will disappear or be enlarged. Qian Xuesen, while combining the research and practices of complex giant systems, warned us that “the complex giant system enjoys a different situation, so sometimes it is difficult, costly, and timeconsuming to establish a mathematical description.” WPA is suitable for mechanism analysis and it is hoped to play a role in the essential analysis of complex systems and effectively connect with experiments rather than relying solely on mathematical analysis. Chapter 8 is about the progress in the research of floating rafts and the “mixing effect”, which will be supplemented later. In Chap. 9, many new methods are adopted for the power flow measurement. The input of power flow in complex systems does not have many problems, but the engineering measurement of the power flow transmission still faces challenges. The main content that is not discussed in this book is the radiation of structures in different media. It is of great importance and much interest to analyze a series of problems related to wave transmission, attenuation, conversion, and sound radiation. The use of the WPA to analyze these problems needs to be deepened mathematically, for example, the “super large unit” will be implanted into the commercial software to cover engineering applications. On the existing basis, the WPA has two directions worth studying: one is to study the correlation between vibration transmission, energy change, and structure waves in coupled structure combining the power flow; the other

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is to deepen the analysis of waveform conversion of longitudinal wave and flexural wave in structure. In the end, the author intends to point out that WPA is not perfect as an independent theoretical method that needs profound theoretical interpretation, application expansion, and verification, and the author hopes that better consistency between the theoretical research and engineering practice can be achieved. However, you may find that some descriptions are omitted due to the constraints and limitations, resulting in suddenness and abruptness for which the author felt regretted!