Asymptotic Modal Analysis of Structural and Acoustical Systems (Synthesis Lectures on Mechanical Engineering) 1681739879, 9781681739878

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Series ISSN: 2573-3168

SUNG • ET AL

Synthesis Lectures on Mechanical Engineering Asymptotic Modal Analysis of Structural and Acoustical Systems

This book describes the Asymptotic Modal Analysis (AMA) method to predict the high-frequency vibroacoustic response of structural and acoustical systems. The AMA method is based on taking the asymptotic limit of Classical Modal Analysis (CMA) as the number of modes in the structural system or acoustical system becomes large in a certain frequency bandwidth. While CMA requires both the computation of individual modes and a modal summation, AMA evaluates the averaged modal response only at a center frequency of the bandwidth and does not sum the individual contributions from each mode to obtain a final result. It is similar to Statistical Energy Analysis (SEA) in this respect. However, while SEA is limited to obtaining spatial averages or mean values (as it is a statistical method), AMA is derived systematically from CMA and can provide spatial information as well as estimates of the accuracy of the solution for a particular number of modes. A principal goal is to present the state-of-the-art of AMA and suggest where further developments may be possible. A short review of the CMA method as applied to structural and acoustical systems subjected to random excitation is first presented. Then the development of AMA is presented for an individual structural system and an individual acoustic cavity system, as well as a combined structural-acoustic system. The extension of AMA for treating coupled or multi-component systems is then described, followed by its application to nonlinear systems. Finally, the AMA method is summarized and potential further developments are discussed.

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ABOUT SYNTHESIS This volume is a printed version of a work that appears in the Synthesis Digital Library of Engineering and Computer Science. Synthesis lectures provide concise original presentations of important research and development topics, published quickly in digital and print formats. For more information, visit our website: http://store.morganclaypool.com

ASYMPTOTIC MODAL ANALYSIS OF STRUCTURAL AND ACOUSTICAL SYSTEMS

Shung H. Sung, SHS Consulting, LLC Dean R. Culver, U.S. Army Research Laboratory Donald J. Nefske, DJN Consulting, LLC Earl H. Dowell, Duke University

Synthesis Lectures on Mechanical Engineering

Asymptotic Modal Analysis of Structural and Acoustical Systems

Synthesis Lectures on Mechanical Engineering Synthesis Lectures on Mechanical Engineering series publishes 60–150 page publications pertaining to this diverse discipline of mechanical engineering. The series presents Lectures written for an audience of researchers, industry engineers, undergraduate and graduate students. Additional Synthesis series will be developed covering key areas within mechanical engineering. Asymptotic Modal Analysis of Structural and Acoustical Systems Shung H. Sung, Dean R. Culver, Donald J. Nefske, and Earl H. Dowell 2020

The Engineering Dynamics Course Companion, Part 2: Rigid Bodies: Kinematics and Kinetics Edward Diehl 2020

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Introduction to Deep Learning for Engineers: Using Python on Google Cloud Platform Tariq M. Arif 2020

iii

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Essential Engineering Thermodynamics: A Student’s Guide Yumin Zhang 2018

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Engineering Dynamics Cho W.S. To 2018

Solving Practical Engineering Problems in Engineering Mechanics: Dynamics Sayavur Bakhtiyarov 2018

Solving Practical Engineering Mechanics Problems: Kinematics Sayavur I. Bakhtiyarov 2018

C Programming and Numerical Analysis: An Introduction Seiichi Nomura 2018

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Unmanned Aircraft Design: A Review of Fundamentals Mohammad Sadraey 2017

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Resistance Spot Welding: Fundamentals and Applications for the Automotive Industry Menachem Kimchi and David H. Phillips 2017

MEMS Barometers Toward Vertical Position Detection: Background Theory, System Prototyping, and Measurement Analysis Dimosthenis E. Bolanakis 2017

Engineering Finite Element Analysis Ramana M. Pidaparti 2017

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Asymptotic Modal Analysis of Structural and Acoustical Systems Shung H. Sung, Dean R. Culver, Donald J. Nefske, and Earl H. Dowell www.morganclaypool.com

ISBN: 9781681739878

paperback

ISBN: 9781681739885

ebook

ISBN: 9781681739892

hardcover

DOI 10.2200/S01053ED1V01Y202009MEC032

A Publication in the Morgan & Claypool Publishers series SYNTHESIS LECTURES ON MECHANICAL ENGINEERING Lecture #32 Series ISSN Print 2573-3168

Electronic 2573-3176

Asymptotic Modal Analysis of Structural and Acoustical Systems

Shung H. Sung SHS Consulting, LLC

Dean R. Culver U.S. Army Research Laboratory

Donald J. Nefske DJN Consulting, LLC

Earl H. Dowell Duke University

SYNTHESIS LECTURES ON MECHANICAL ENGINEERING #32

M C &

Morgan

& cLaypool publishers

ABSTRACT This book describes the Asymptotic Modal Analysis (AMA) method to predict the high-frequency vibroacoustic response of structural and acoustical systems. The AMA method is based on taking the asymptotic limit of Classical Modal Analysis (CMA) as the number of modes in the structural system or acoustical system becomes large in a certain frequency bandwidth. While CMA requires both the computation of individual modes and a modal summation, AMA evaluates the averaged modal response only at a center frequency of the bandwidth and does not sum the individual contributions from each mode to obtain a final result. It is similar to Statistical Energy Analysis (SEA) in this respect. However, while SEA is limited to obtaining spatial averages or mean values (as it is a statistical method), AMA is derived systematically from CMA and can provide spatial information as well as estimates of the accuracy of the solution for a particular number of modes. A principal goal is to present the state-of-theart of AMA and suggest where further developments may be possible. A short review of the CMA method as applied to structural and acoustical systems subjected to random excitation is first presented. Then the development of AMA is presented for an individual structural system and an individual acoustic cavity system, as well as a combined structural-acoustic system. The extension of AMA for treating coupled or multi-component systems is then described, followed by its application to nonlinear systems. Finally, the AMA method is summarized and potential further developments are discussed.

KEYWORDS vibration, sound pressure, modal analysis, classical modal analysis, asymptotic modal analysis, random vibration, Lagrange equation, nonlinear dynamics, high frequency response, vibroacoustic

ix

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 1.2 1.3

2

Classical Modal Analysis with Random Excitations . . . . . . . . . . . . . 7 2.1 2.2

2.3

2.4

2.5 2.6

3

Overview of Vibroacoustic Modal Analysis . . . . . . . . . . . . . . . . 1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Structural System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.1 Lagrange Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2.2 Classical Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.3 Random Excitation Response . . . . . . . . . . . . . . . . . . . 12 Acoustic Cavity System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.1 Classical Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.2 Random Excitation by Sound Sources . . . . . . . . . . . . . 16 2.3.3 Random Excitation by Wall Panels . . . . . . . . . . . . . . . 17 Coupled Structural-Acoustic System . . . . . . . . . . . . . . . . . . . . 18 2.4.1 Classical Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . 19 2.4.2 Random Excitation Response . . . . . . . . . . . . . . . . . . . 20 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Asymptotic Modal Analysis of Structural Systems . . . . . . . . . . . . . 25 3.1 3.2 3.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 AMA Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Frequency Band Response . . . . . . . . . . . . . . . . . . . . . .

25 25 27 28 30

x

3.4 3.5 3.6

4

31 32 36 37

Asymptotic Modal Analysis of Acoustic Cavity Systems . . . . . . . . 39 4.1 4.2 4.3

4.4

4.5 4.6

5

3.3.3 Asymptotic Limit of Classical Modal Analysis . . . . . . AMA Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Asymptotic Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Sound Source Excitation . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Wall Panel Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . AMA Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Rectangular Enclosure . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Rectangular Cavity-Plate System . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39 39 41 42 44 48 48 51 53 53

Asymptotic Modal Analysis of Coupled Systems . . . . . . . . . . . . . . 55 5.1 5.2

5.3

5.4

5.5

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Lagrange Multiplier Method . . . . . . . . . . . . . . . . . . . . 5.2.2 Equations-of-Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Frequency Response Functions . . . . . . . . . . . . . . . . . . 5.2.4 Coupled Natural Frequencies and Damping . . . . . . . . 5.2.5 Random Excitation and Mean-Square Response . . . . Discrete Classical Modal Analysis . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Reduced-Order Model . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Random Excitation and Mean-Square Response . . . . Asymptotic Modal Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Modal Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Frequency Reduction and Frequency Band Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Displacement Distribution . . . . . . . . . . . . . . . . . . . . . .

55 55 56 57 58 59 60 60 61 62 64 64 66 69 69

xi

5.6 5.7

6

Asymptotic Modal Analysis of Nonlinear Systems . . . . . . . . . . . . . 75 6.1 6.2 6.3

6.4 6.5 6.6 6.7

6.8 6.9

7

5.5.2 Frequency Band Response . . . . . . . . . . . . . . . . . . . . . . 69 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Modal Analysis Formulation . . . . . . . . . . . . . . . . . . . CMA Transient Solution Method . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Eigenvalue Analysis and Coupled Natural Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Time Marching Method . . . . . . . . . . . . . . . . . . . . . . . CMA Frequency Response Method . . . . . . . . . . . . . . . . . . . . . Dominance-Reduced CMA Method . . . . . . . . . . . . . . . . . . . . Asymptotic Modal Analysis Method . . . . . . . . . . . . . . . . . . . . Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Nonlinear Response Solution . . . . . . . . . . . . . . . . . . . . 6.7.2 Method Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.3 Runtime Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 77 79 82 82 84 87 89 89 90 91 93 94

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 7.1 7.2

7.3 7.4

State of the Art in Predicting the Response of Systems with High Modal Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Two-Component System Example . . . . . . . . . . . . . . . . . . . . . . 98 7.2.1 System Natural Modes and Frequencies . . . . . . . . . . . 99 7.2.2 Forced System Response . . . . . . . . . . . . . . . . . . . . . . 101 Future Developments in AMA . . . . . . . . . . . . . . . . . . . . . . . . 103 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

Authors’ Biographies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

xiii

Preface In this short book, Classical Modal Analysis (CMA) is first reviewed for application to structural and acoustical, e.g., plates and acoustic cavities, as well as dynamical systems that can be considered as comprising interconnected components, e.g., interconnected plates. Also treated are combined structural and acoustical systems. Ultimately, at higher frequencies, we consider Asymptotic Modal Analysis (AMA) as the asymptotic limit of CMA when a large number of resonant modes are excited in a frequency bandwidth, and we illustrate the challenges and opportunities that analyzing this type of system presents to the analyst and engineer. The topics covered are shown in the Table of Contents, and it is believed that the book will be useful to a range of readers including those encountering theoretical modal analysis for the first time, as well as those who have been using modal analysis for many years and are interested in the recent developments, particularly at higher frequencies. An experienced reader may use the initial CMA chapter as a review and reference source, but we hope each chapter is sufficiently self-contained that a reader may begin the reading of the book at almost any Chapter of special interest. A word is appropriate about the relationship of modal analysis to Finite Element Analysis (FEA) and Statistical Energy Analysis (SEA). As many readers will know, SEA (and subsequently AMA) was developed in significant part because of the difficulty of using FEA when a large number of modes are needed to describe the dynamics of a system. Yet, one of the great virtues of FEA is that it allows one to determine the modes of complex systems which are beyond the reach of classical eigenvalue/eigenfunction analysis for simpler geometries. But one may most usefully think of FEA, SEA, and indeed AMA as complementary approaches. And, indeed, when using FEA, one may wish to consider whether the system dynamics have effectively entered the domain of SEA and/or AMA. Thus, a knowledge and understanding of AMA and SEA will help the FEA analyst as well.

xiv

PREFACE

Finally, as will be evident from the chapter references, the use of CMA, its extension to higher frequency ranges, and ultimately to AMA, has been a research effort of the authors that has also stimulated them to think harder and more deeply about these fascinating topics. We hope that the reader will share this deep interest which has led the present authors to compile this book that summarize the extension of CMA capabilities for higher frequency applications. Shung H. Sung, Dean R. Culver, Donald J. Nefske, and Earl H. Dowell October 2020

1

CHAPTER

1

Introduction 1.1

OVERVIEW OF VIBROACOUSTIC MODAL ANALYSIS

Modal analysis is one of the pervasive and powerful methods for analyzing the dynamics of complex systems including the vibroacoustic responses of structures and fluid flow fields. Other well-known applications include electromagnetic field analysis and quantum mechanics, which can be viewed as a form of modal analysis. In the present book, we treat elastic structures and acoustical fields, but other systems where modal analysis can be used may be treated successfully using similar methods. This book starts with a review of Classical Modal Analysis (CMA) of linear systems under random excitations. Chapter 2 is a brief discussion that brings out the essential ideas needed for the remainder of the book. There are many fine texts on classical modal analysis of linear systems and on random vibration theory and among those the authors would mention are the books by Meirovitch [1] and Newland [2]. When the excitation of the system is random, then modal analysis takes a certain special form and the concepts of correlation functions in time and power spectra in the frequency domain are found to be powerful in deriving the asymptotic modal response. In CMA one first determines the eigenmodes of the system and then determines the response of the system to a prescribed excitation. If the system is linear, then each eigenmode can be treated individually and the total system response is determined by summing the results of each mode. For CMA to work well in practice, the number of modes that needs to be considered must be relatively small. Often, the principal difficulty is not summing the modes per se, but rather accurately determining a very large number of eigenmodes, especially at higher frequencies. However, there are cases when the number of modes that are responding becomes so large that the computational requirements of CMA becomes impractical. In these cases, alternative methods such as Statistical Energy Analysis

2

1. INTRODUCTION

(SEA) and Energy Finite Element Analysis (EFEA) have been developed [3]. As an alternative to these established methods, it is conceptually interesting and useful to consider what happens in a certain limit as the number of modes becomes very large. Thus, one considers the limit as the number of responding modes becomes large in a finite frequency bandwidth and this approach is referred to here as Asymptotic Modal Analysis (AMA). The discussion of the development of AMA is one of the principal goals of the present volume. Chapter 3 presents the development of AMA for treating an individual structural system, and Chapter 4 presents the development of AMA for treating individual acoustic cavity systems as well as structural-acoustic systems which consist of a single component. The extension of AMA for treating coupled or multi-component systems is described in Chapter 5 using the ideas from component mode analysis. Component mode analysis considers a system which consists of two or more components. The eigenmodes that represent the response of each component are identified, and the dynamics of the system is analyzed in terms of these component modes. There are several alternative approaches to component mode analysis, mainly distinguished by the modes chosen for each component and how the connections between two components are modeled. A well-known text on component mode analysis is that of Craig [4]. Another treatment by one of the present authors is in Dowell and Tang [5]. In the approach of Dowell and Tang, each component is assumed to have a free end at the interface connection between the components and a constraint condition is imposed to connect the components. This is the approach followed here. In Chapter 5, the prototype example of two parallel plates connected by a rigid spring illustrates the development of AMA for treating coupled system in the high frequency range. Chapter 6 considers the extension of AMA to nonlinear systems. When nonlinear elements are present, all modes are coupled. In principle, nonlinear normal modes may be identified that are the nonlinear counterpart of the eigenmodes of linear systems, as discussed in [5]. In general, these nonlinear normal modes are also coupled and their response cannot be determined individually, but must be determined simultaneously. Thus, in practice, the eigenmodes of the corresponding linear system are often used, but because of the nonlinear elements, the modes are indeed coupled and solution of the equations of motion must be solved simultaneously. However, as discussed here, if

1.2. BACKGROUND

the nonlinear elements are concentrated in a single component, this can be taken advantage of using the ideas of component mode analysis. Finally, Chapter 7 summarizes the status of the AMA methodology in predicting the vibroacoustic response of structural and acoustic systems subjected to stationary random excitations, as well as where one might productively press forward to advance the methodology. This discussion is primarily from an AMA perspective, but alternative approaches are also considered.

1.2

BACKGROUND

The development of AMA for treating the case of a very large number of modes was inspired by the ideas of statistical mechanics used to develop SEA, as described by Lyon and De Jong [6]. Indeed, one of the original motivations for developing AMA was to determine the relationship between modal analysis and SEA. In both AMA and SEA, it is assumed that the number of modes is sufficiently large in a certain frequency bandwidth and that the modal frequencies and damping can be represented by the center frequency of the bandwidth. It is in that sense that the limit is asymptotic, i.e., the bandwidth must be large enough to include a large number of resonant modes (typically at least 10–100 depending on the accuracy required), but at the same time the bandwidth must be small enough that the center frequency properties are representative of those throughout the bandwidth. For most dynamical systems these two competing claims on the bandwidth requirements are most likely to be satisfied at high frequencies and high mode numbers. In practice, one often uses the most appropriate of the above methods to analyze complex dynamical systems. For the lower modes at the lower resonant frequencies where such frequencies are well spaced in the frequency spectrum and it is practical by analytical or numerical means (using finite element analysis), the CMA method often works well even for a nonlinear systems. On the other hand, if there are many modes in a relatively narrow frequency band, SEA or AMA may be useful. As mentioned previously, often it is the accurate computation of a high number of modes that is more challenging than the summation of modes. Both SEA and AMA avoid the necessity of computing the individual modes of the dynamical system. In that spirit, yet another strategy is to use a reduced order model based on an approximate set of modes that are good enough.

3

4

1. INTRODUCTION

Both SEA and AMA assume that the modes underlying these methods do not depend sensitively on the geometric shape or boundary conditions of the system. Finally, we note that in a recent paper by Newland and Sharman [7], the authors offer the following comment on the experimental observations by Crandall and colleagues [8] that Chladni-type patterns can be found even in plates excited by random vibration: “the computer modeling of how intensified lanes [corresponding to the Chladni-type patterns] of vibrational response develop is complicated and does not appear to have been exploited by anyone yet.” In Chapter 3, computer modeling using AMA gives support and insight into these experimental observations. However, outstanding questions remain about SEA and AMA and both still provide a rich opportunity for further advances. The authors hope that the present volume will prove helpful in that process.

1.3

REFERENCES

[1] L. Meirovitch, Analytical Methods in Vibration, Macmillan, NY, 1967. 1 [2] D. E. Newland, An Introduction to Random Vibrations, Spectral and Wavelet Analysis, Dover Publications, NY, 1993. 1 [3] S. A. Hambric, S. H. Sung, and D. J. Nefske, Engineering Vibroacoustic Analysis – Method and Applications, John Wiley & Sons, NY, 2016. DOI: 10.1002/9781118693988. 2 [4] R. R. Craig, Structural Dynamics: An Introduction to Computer Methods, Wiley, NY, 1981. DOI: 10.1115/1.3139698. 2 [5] E. H. Dowell and D. Tang, Dynamics of Very High Dimensional Systems, World Scientific, Singapore, 2003. DOI: 10.1142/5346. 2 [6] R. H. Lyon and R. G. De Jong, Theory and Application of Statistical Energy Analysis, Butterworth-Heinemann, Boston, 1995. DOI: 10.1016/C2009-0-26747-X. 3 [7] D. E. Newland and L. H. Sharman, Creating visual images with sound and vibration, 26th International Congress of Sound and Vibration, Montreal, Canada, 2019. 4

1.3. REFERENCES

[8] S. H. Crandall, Random vibration in one- and two-dimensional structures, Developments in Statistics, 2:1–82, P. R. Krishnaiah Ed., Elsevier B. V., 1979. 4

5

7

CHAPTER

2

Classical Modal Analysis with Random Excitations 2.1

INTRODUCTION

This chapter reviews the CMA method to predict the random response of individual structural and acoustic systems, and of coupled structural-acoustic systems. The CMA method is based on the modal expansion solution using the normal modes of the individual systems. The random vibration theory based on stationary transient excitations is then applied in the CMA method to predict the random vibration response of the individual structural and acoustic systems, and of coupled structural-acoustic systems. Section 2.2 develops the methodology to treat individual structural systems using the Lagrange method to develop the governing equations of motion. Section 2.3 develops the methodology to treat individual acoustic cavity systems based on the uncoupled rigid-wall modes. Finally, Section 2.4 develops the methodology to treat coupled structural-acoustic systems based on the uncoupled structural modes and the uncoupled, rigid-wall cavity modes. The development of AMA to treat individual and coupled systems, as well as nonlinear systems, follows in the subsequent chapters.

2.2

STRUCTURAL SYSTEM

2.2.1

LAGRANGE METHOD

The governing equations-of-motion of a structural system can be obtained using the Lagrange method [1]. Figure 2.1 shows a structural system of area AS with displacement field w .r; t / and external force F .rj ; t / applied at location rj . The Lagrangian L of the system is expressed in terms of the generalized displacements (modal degrees of freedom) qk and generalized velocities qP k as L .qk ; qP k ; t /  T .qk ; qP k ; t /

U .qk ; t / ;

k D 1; 2; : : : ;

(2.1)

8

2. CLASSICAL MODAL ANALYSIS WITH RANDOM EXCITATIONS

w(r,t)

F(rj,t)

AS

rj

Figure 2.1: Elastic structure of area As subject to an applied external dynamic
load F rj ; t at location rj resulting in a vibration displacement response w .r; t /. where T is the kinetic energy and U is the potential energy. The Euler–Lagrange or simply Lagrange equation is then derived by applying Hamilton’s principle to obtain [1]:   d @L @L D Qk ; k D 1; 2; : : : : (2.2) dt @qP k @qk Here, Qk represents the generalized (modal) forces acting on the system which consists of any external generalized (modal) force FkE and any generalized (modal) damping force Dk as Qk D FkE

(2.3)

Dk :

The generalized (modal) damping force Dk in Eq. (2.3) can be expressed as viscous damping using the Rayleigh dissipation function R so that @R 1X Dk D (2.4) D ck qP k where R D ck qP k2 ; @qP k 2 k

where ck is the generalized (modal) damping constant. Substituting Eqs. (2.3) and (2.4) into Eq. (2.2) then gives the Lagrange equation as   d @L @L C ck qP k D FkE ; k D 1; 2; : : : : (2.5) dt @qP k @qk The external generalized (modal) force FkE is related to the applied force F illustrated in Fig. 2.1 as FkE D Fj


@wj ; @qk

(2.6)

2.2. STRUCTURAL SYSTEM




where Fj D F .rj ; t/ is the external point force at rj and wj D w rj ; t is the displacement at rj . Summing up all multiple external point forces acting on the plate at multiple points rj where j D 1; 2; : : : ; L, Eq. (2.6) can then be expressed as L X @wj Fj
; (2.7) FkE D @qk j D1 where L is the total number of point forces. In the case of distributed external pressure normal loading p E .r; t/ acting on the plate, Eq. (2.7) can be modified as Z Z @.w
b n/ @w FkE D p E dA D p E dA; (2.8) @qk @qk AS

AS

where AS is the surface area, b n is the surface normal vector, and w is the structural surface normal displacement. 2.2.2

CLASSICAL MODAL ANALYSIS

In many technical applications, the flexible structural element may be represented as a plate (shells, curved plates, beams, or combinations). The CMA method will therefore be illustrated here for a flat plate. For an isotropic plate with normal displacement w .x; y; t /, the kinetic energy T and the potential energy U can be expressed as [2]: Z 1 T D s hwP 2 dA (2.9a) 2 AS

1 U D 2

 2 2 Z "  2 2 @ w @ w @2 w @2 w D C C D 2D @x 2 @y 2 @x 2 @y 2

AS

C 2.1

/D



2

@ w @x@y

2 #

(2.9b) dA:

Here s is the structural mass density, AS is the surface area, h is the plate thickness, D is the bending rigidity of the plate, and  is Poisson’s ratio. The displacement response of the plate can be expressed by the superposition of the normal displacement modes 'm as X w.r; t/ D qm .t/'m .r/; (2.10) m

9

10

2. CLASSICAL MODAL ANALYSIS WITH RANDOM EXCITATIONS

where qm are the modal displacement coefficients and r D .x; y/. The Lagrange method is then applied by substituting Eqs. (2.9a), (2.9b), and (2.10) into Eqs. (2.1) and (2.5) to obtain XZ
s h'l 'm qR m C Dr 4 'l 'm qm dA C cm qP m D FmE ; (2.11) l A S

where r 4 is the biharmonic operator. The structural modes 'm are taken as in-vacuo modes that satisfy the eigenvalue equation
2 'm D 0; (2.12) Dr 4 s h!m where !m are the structural modal frequencies. The orthogonality of the mode shapes gives 8 Z