Dynamic analysis of payloads and structures with intermediate modal density

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(1)UNIVERSIDAD POLITÉCNICA DE MADRID ESCUELA TÉCNICA SUPERIOR DE INGENIEROS AERONÁUTICOS. DYNAMIC ANALYSIS OF PAYLOADS AND STRUCTURES WITH INTERMEDIATE MODAL DENSITY TESIS DOCTORAL. Elena Roibás Millán Ingeniera Aeronáutica. 2014.

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(3) DEPARTAMENTO DE VEHÍCULOS AEROESPACIALES ESCUELA TÉCNICA SUPERIOR DE INGENIEROS AERONÁUTICOS. DYNAMIC ANALYSIS OF PAYLOADS AND STRUCTURES WITH INTERMEDIATE MODAL DENSITY Elena Roibás Millán Ingeniera Aeronáutica. Director. D. Francisco Simón Hidalgo Doctor en Ciencias Físicas. 2014.

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(5) Tribunal nombrado por el Magfco. Y Excmo. Sr. Rector de la Universidad Politécnica de Madrid. El día de de 20 . Presidente: Vocal: Vocal: Vocal: Secretario: Suplente: Suplente:. Realizado el acto de defensa y lectura de la Tesis el día de 20 en la E.T.S.I./Facultad. de. Calicación EL PRESIDENTE. LOS VOCALES. EL SECRETARIO.

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(7) To all individuals and organizations who, directly or indirectly, have enriched my life profoundly along this journey.

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(9) Acknowledgements Foremost, I would like to express my sincere gratitude to my doctoral advisor, Dr. Francisco Simón, for his continuous support of this PhD research, his thoughtful guidance and critical comments. He has always found the time to propose consistently excellent improvements and ideas. The completion of this work could not have been possible without his assistance. My deepest acknowledge to Dr. Jesús López Díez, in memoriam, for providing me the opportunity to embark on this project. I gratefully acknowledge to the European Space Agency to support this PhD Thesis within the ESA's Networking and Partnering Initiative (NPI). Very special thanks to the sta at the Mechanical Department and, in particular, to Dr. Julián Santiago Prowald for his helpful comments during the development of this work. I would like to thank to the Universidad Politécnica de Madrid and the E.T.S.I. Aeronáuticos. Thanks also to the people of the Space Vehicles Department. Specially thanks to Marcos Chimeno, for being always available for my questions. I am indebted to him for his help. I am also very grateful to the international experts, for their favourable reports to this dissertation, and to EADS CASA Espacio, for providing me the opportunity to carry out a research stay. Getting through my dissertation required more than academic support, and I have many, many people to thank for listening to and, at times, having to tolerate me over the past three years. Thanks to my family and friends, I cannot begin to express my gratitude and appreciation for their love and friendship. Specially, I would like to thank my parents for the support, love and encouragement they provided me through my entire life. This work is for and because of you. To my three sisters, for being a source of constant and unconditional love. Irene, I could not ask for a better sister and friend. To my wonderful husband, Alex, for his practical and emotional support. Your love has been my greatest strength. Without you, I would be lost.. E. Roibás Madrid, 2014.

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(11) Abstract Over the last years an increasing need of novel prediction techniques for vibroacoustic analysis of space structures has arisen. Current numerical techniques are able to predict with enough accuracy the vibro-acoustic behaviour of systems with low and high modal densities. However, space structures are, in general, very complex and they present a range of frequencies in which a mixed behaviour exist. In such cases, the full system is composed of some sub-structures which has low modal density, while others present high modal density. This frequency range is known as the mid-frequency range and to develop methods for accurately describe the vibro-acoustic response in this frequency range is the scope of this dissertation. For the structures under study, the aforementioned low and high modal densities correspond with the low and high frequency ranges, respectively. For the low frequency range, deterministic techniques as the Finite Element Method (FEM) are used while, for the high frequency range statistical techniques, as the Statistical Energy Analysis (SEA), are considered as more appropriate. In the mid-frequency range, where a mixed vibro-acoustic behaviour is expected, any of these numerical method can not be used with enough condence level. As a consequence, it is usual to obtain an undetermined gap between low and high frequencies in the vibro-acoustic response function. This dissertation proposes two dierent solutions to the mid-frequency range problem. The rst one, named as The Subsystem based High Frequency Limit (SHFL) procedure, proposes a multi-hybrid procedure in which each sub-structure of the full system is modelled with the appropriate modelling technique, depending on the frequency of study. With this purpose, the concept of high frequency limit of a sub-structure is introduced, marking out the limit above which a substructure has enough modal density to be modelled by SEA. For a certain analysis frequency, if it is lower than the high frequency limit of the sub-structure, the sub-structure is modelled through FEM and, if the frequency of analysis is higher than the high frequency limit, the sub-structure is modelled by SEA. The procedure leads to a number of hybrid models required to cover the medium frequency range, which is dened as the frequency range between the lowest substructure high frequency limit and the highest one. Using this procedure, the mid-frequency range can be dene specically so that, as a consequence, an improvement in the continuity of the vibro-acoustic response function is achieved, closing the undetermined gap between the low and high frequency ranges..

(12) The second proposed mid-frequency solution is the Hybrid Sub-structuring method based on Component Mode Synthesis (HS-CMS). The method adopts a partition scheme based on classifying the system modal basis into global and local sets of modes. This classication is performed by using a Component Mode Synthesis, in particular a Craig-Bampton transformation, in order to express the system modal base into the modal bases associated with each sub-structure. Then, each sub-structure modal base is classied into global and local set, st ones associated with the long wavelength motion and second ones with the short wavelength motion. The high frequency limit of each sub-structure is used as frequency frontier between both sets of modes. From this classication, the equations of motion associated with global modes are derived, which include the interaction of local modes by means of corrections in the dynamic stiness matrix and the force vector of the global problem. The local equations of motion are solved through SEA, where again interactions with global modes are included through the inclusion of an additional input power into the SEA model. The method has been tested for the calculation of the response function of structures subjected to structural and acoustic loads. Both methods have been rstly tested in simple structures to establish their basis and main characteristics. Methods are also veried in space structures, as satellites and antenna reectors, providing good results as it is concluded from the comparison with experimental results obtained in both, acoustic and structural load tests. This dissertation opens a wide eld of research through which further studies could be performed to obtain ecient and accurate methodologies to appropriately reproduce the vibro-acoustic behaviour of complex systems in the mid-frequency range..

(13) Resumen La necesidad de desarrollar técnicas para predecir la respuesta vibroacústica de estructuras espaciales ha ido ganando importancia en los últimos años. Las técnicas numéricas existentes en la actualidad son capaces de predecir de forma able el comportamiento vibroacústico de sistemas con altas o bajas densidades modales. Sin embargo, ambos rangos no siempre solapan lo que hace que sea necesario el desarrollo de métodos especícos para este rango, conocido como densidad modal media. Es en este rango, conocido también como media frecuencia, donde se centra la presente Tesis doctoral, debido a la carencia de métodos especícos para el cálculo de la respuesta vibroacústica. Para las estructuras estudiadas en este trabajo, los mencionados rangos de baja y alta densidad modal se corresponden, en general, con los rangos de baja y alta frecuencia, respectivamente. Los métodos numéricos que permiten obtener la respuesta vibroacústica para estos rangos de frecuencia están bien especicados. Para el rango de baja frecuencia se emplean técnicas deterministas, como el método de los Elementos Finitos, mientras que, para el rango de alta frecuencia las técnicas estadísticas son más utilizadas, como el Análisis Estadístico de la Energía. En el rango de medias frecuencias ninguno de estos métodos numéricos puede ser usado con suciente precisión y, como consecuencia a falta de propuestas más especícas se han desarrollado métodos híbridos que combinan el uso de métodos de baja y alta frecuencia, intentando que cada uno supla las deciencias del otro en este rango medio. Este trabajo propone dos soluciones diferentes para resolver el problema de la media frecuencia. El primero de ellos, denominado SHFL (del inglés Subsystem based High Frequency Limit procedure), propone un procedimiento multihíbrido en el cuál cada subestructura del sistema completo se modela empleando una técnica numérica diferente, dependiendo del rango de frecuencias de estudio. Con este propósito se introduce el concepto de límite de alta frecuencia de una subestructura, que marca el límite a partir del cual dicha subestructura tiene una densidad modal lo sucientemente alta como para ser modelada utilizando Análisis Estadístico de la Energía. Si la frecuencia de análisis es menor que el límite de alta frecuencia de la subestructura, ésta se modela utilizando Elementos Finitos. Mediante este método, el rango de media frecuencia se puede denir de una forma precisa, estando comprendido entre el menor y el mayor de los límites de alta frecuencia de las subestructuras que componen el sistema completo. Los resultados obtenidos mediante la aplicación de este método evidencian una mejora.

(14) en la continuidad de la respuesta vibroacústica, mostrando una transición suave entre los rangos de baja y alta frecuencia. El segundo método propuesto se denomina HS-CMS (del inglés Hybrid Substructuring method based on Component Mode Synthesis). Este método se basa en la clasicación de la base modal de las subestructuras en conjuntos de modos globales (que afectan a todo o a varias partes del sistema) o locales (que afectan a una única subestructura), utilizando un método de Síntesis Modal de Componentes. De este modo es posible situar espacialmente los modos del sistema completo y estudiar el comportamiento del mismo desde el punto de vista de las subestructuras. De nuevo se emplea el concepto de límite de alta frecuencia de una subestructura para realizar la clasicación global/local de los modos en la misma. Mediante dicha clasicación se derivan las ecuaciones globales del movimiento, gobernadas por los modos globales, y en las que la inuencia del conjunto de modos locales se introduce mediante modicaciones en las mismas (en su matriz dinámica de rigidez y en el vector de fuerzas). Las ecuaciones locales se resuelven empleando Análisis Estadístico de Energías. Sin embargo, este último será un modelo híbrido, en el cual se introduce la potencia adicional aportada por la presencia de los modos globales. El método ha sido probado para el cálculo de la respuesta de estructuras sometidas tanto a cargas estructurales como acústicas. Ambos métodos han sido probados inicialmente en estructuras sencillas para establecer las bases e hipótesis de aplicación. Posteriormente, se han aplicado a estructuras espaciales, como satélites y reectores de antenas, mostrando buenos resultados, como se concluye de la comparación de las simulaciones y los datos experimentales medidos en ensayos, tanto estructurales como acústicos. Este trabajo abre un amplio campo de investigación a partir del cual es posible obtener metodologías precisas y ecientes para reproducir el comportamiento vibroacústico de sistemas en el rango de la media frecuencia..

(15) Contents. Contents. i. List of Figures. vi. List of Tables. xv. List of Symbols. xviii. I Introduction and State of the Art. 1. 1 Introduction. 3. 1.1. Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3. 1.2. Vibro-acoustic analysis: strengths and limitations . . . . . . . . . . . . . . . .. 4. 1.3. The research goal: Objectives and justication . . . . . . . . . . . . . . . . .. 7. 1.4. Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 2 State of the art on numerical methods for vibro-acoustics of space structures 13 2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. 2.2. Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 2.2.1. 15. Formulation of the elastic structural domain . . . . . . . . . . . . . . . i.

(16) 2.2.2. 2.3. Formulation of the uid domain . . . . . . . . . . . . . . . . . . . . . .. 17. 2.2.2.1. The homogeneous acoustic wave equation . . . . . . . . . . .. 17. 2.2.2.2. The acoustic velocity potential . . . . . . . . . . . . . . . . .. 18. 2.2.3. Fluid-structure coupling . . . . . . . . . . . . . . . . . . . . . . . . . .. 19. 2.2.4. Solution methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 20. Methods for the analysis in the low frequency range . . . . . . . . . . . . . . .. 21. 2.3.1. Introduction to low-frequency methods . . . . . . . . . . . . . . . . . .. 21. 2.3.2. Finite element method . . . . . . . . . . . . . . . . . . . . . . . . . . .. 22. 2.3.2.1. Structural domain . . . . . . . . . . . . . . . . . . . . . . . .. 22. 2.3.2.2. Finite uid domain. Acoustic cavities . . . . . . . . . . . . .. 24. 2.3.2.3. The coupled structure-acoustic system . . . . . . . . . . . . .. 26. Boundary Element Method . . . . . . . . . . . . . . . . . . . . . . . .. 28. 2.3.3.1. Fast multipole method . . . . . . . . . . . . . . . . . . . . . .. 29. 2.3.3.2. Innite Element Method. . . . . . . . . . . . . . . . . . . . .. 29. . . . . . . . . . . . . . . . . . . . . . . .. 30. 2.3.3. 2.3.4. 2.3.4.1. Modal interaction analysis. . . . . . . . . . . . . . . . . . . .. 30. 2.3.4.2. Joint acceptance analysis . . . . . . . . . . . . . . . . . . . .. 31. 2.3.4.3. Transmissibility Method: Admittance modelling . . . . . . .. 31. Condensation and sub-structuring techniques . . . . . . . . . . . . . .. 32. 2.3.5.1. Dynamic Model Reduction Methods . . . . . . . . . . . . . .. 32. 2.3.5.2. Component Mode Synthesis . . . . . . . . . . . . . . . . . . .. 35. Methods for the analysis in the high frequency range . . . . . . . . . . . . . .. 37. 2.4.1. Introduction to high-frequency methods . . . . . . . . . . . . . . . . .. 37. 2.4.2. Statistical energy analysis . . . . . . . . . . . . . . . . . . . . . . . . .. 38. 2.4.2.1. Basic hypothesis of the SEA method . . . . . . . . . . . . . .. 38. 2.4.2.2. Formulation of the SEA method . . . . . . . . . . . . . . . .. 40. 2.4.2.3. Characteristics of the analysis by SEA . . . . . . . . . . . . .. 42. 2.4.2.4. Coupling loss factor determination . . . . . . . . . . . . . . .. 43. Other methods for high frequency estimations . . . . . . . . . . . . . .. 45. 2.4.3.1. Spectral Finite Element Method . . . . . . . . . . . . . . . .. 45. 2.4.3.2. Energy Finite Element Method . . . . . . . . . . . . . . . . .. 45. 2.3.5. 2.4. 2.4.3. ii. Other low frequency method.

(17) 2.4.3.3 2.5. 2.6. Energy Boundary Element Method . . . . . . . . . . . . . . .. 46. Methods for the analysis in the medium frequency range . . . . . . . . . . . .. 47. 2.5.1. Introduction. The mid-frequency problem . . . . . . . . . . . . . . . .. 47. 2.5.2. Resolution of the mid-frequency problem . . . . . . . . . . . . . . . . .. 49. 2.5.2.1. Variational Theory of Complex rays . . . . . . . . . . . . . .. 49. 2.5.2.2. Fuzzy structure theory . . . . . . . . . . . . . . . . . . . . . .. 50. 2.5.2.3. A Hybrid Finite Element/Statistical Energy Analysis method. 51. 2.5.2.4. A Hybrid method for structural analysis of complex systems. 54. Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54. II New mid-frequency solution methods. 57. 3 A Subsystem based High Frequency Limit procedure. 59. 3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 59. 3.2. The Subsystem High Frequency Limit procedure (SHFL) . . . . . . . . . . . .. 63. 3.2.1. Subsystems High Frequency Limit determination . . . . . . . . . . . .. 65. 3.2.1.1. Statistical Energy Analysis applicability conditions . . . . . .. 66. The mode count as high frequency criterion . . . . . . . . . . . . . . .. 67. 3.2.2.1. Relation between mode count and boundary conditions . . .. 69. 3.3. Application to a reference case . . . . . . . . . . . . . . . . . . . . . . . . . .. 76. 3.4. Application to a typical case . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82. 3.4.1. Description of the case: satellite structure . . . . . . . . . . . . . . . .. 82. 3.4.2. Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 82. 3.4.2.1. Structural load test . . . . . . . . . . . . . . . . . . . . . . .. 82. 3.4.2.2. Acoustic test . . . . . . . . . . . . . . . . . . . . . . . . . . .. 84. 3.4.3. Numerical models description . . . . . . . . . . . . . . . . . . . . . . .. 85. 3.4.4. Numerical models classication . . . . . . . . . . . . . . . . . . . . . .. 85. Discussion of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 88. 3.5.1. Numerical models reliability. Comparison with experimental results . .. 88. 3.5.2. Numerical results obtained with the SHFL procedure . . . . . . . . . .. 91. Response of the SWARM satellite through the SHFL procedure . . . . . . . .. 95. 3.2.2. 3.5. 3.6. iii.

(18) 3.7. 3.6.1. Structure description . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 95. 3.6.2. Sub-structures High Frequency Limit determination . . . . . . . . . .. 98. 3.6.3. Vibro-acoustic models classication . . . . . . . . . . . . . . . . . . . . 101. 3.6.4. Vibro-acoustic analysis results . . . . . . . . . . . . . . . . . . . . . . . 102. 3.6.5. Comparison of numerical models with experimental results from acoustic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106. 4 Hybrid Sub-structuring method based on Component Mode Synthesis 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107. 4.2. Mathematical Formulation of the HS-CMS method . . . . . . . . . . . . . . . 109 4.2.1. Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.1.1. 4.2.2. 4.2.3. The Craig-Bampton transformation . . . . . . . . . . . . . . 109. Hybrid formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.2.2.1. Sub-structures denition. 4.2.2.2. Eigenvalue problem and wave partitioning . . . . . . . . . . . 111. 4.2.2.3. Hybrid partitioning in the complete system . . . . . . . . . . 112. 4.2.2.4. Global equations of motion . . . . . . . . . . . . . . . . . . . 116. 4.2.2.5. Local equations of motion . . . . . . . . . . . . . . . . . . . . 117. The acoustic excitation. . . . . . . . . . . . . . . . . . . . . 110. . . . . . . . . . . . . . . . . . . . . . . . . . . 119. 4.2.3.1. Rain on the roof excitation . . . . . . . . . . . . . . . . . . . 119. 4.2.3.2. Equivalent ROF excitation . . . . . . . . . . . . . . . . . . . 120. 4.2.3.3. Application to a diuse eld excitation . . . . . . . . . . . . 123. 4.2.3.4. Radiation eciency and coupling with external air . . . . . . 124. 4.3. Implementation of the method. 4.4. Application of the hybrid method to elastic beams . . . . . . . . . . . . . . . 131. . . . . . . . . . . . . . . . . . . . . . . . . . . 127. 4.4.1. Structure denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131. 4.4.2. Eigenvalue problem and wave partitioning . . . . . . . . . . . . . . . . 132. 4.4.3 iv. 107. 4.4.2.1. Subsystems wavelength and global modes selection . . . . . . 132. 4.4.2.2. Hybrid partition in the complete system . . . . . . . . . . . . 134. Global response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135.

(19) 4.4.4. 4.4.5 4.5. Input power from the external force . . . . . . . . . . . . . . 138. 4.4.4.2. Input power from the global modes. 4.4.4.3. Coupling loss factor of the system . . . . . . . . . . . . . . . 140. . . . . . . . . . . . . . . 139. Total response of the two coupled beams . . . . . . . . . . . . . . . . . 141 Specimen description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 4.5.1.1. Panel A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143. 4.5.1.2. Panel E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144. 4.5.1.3. L-shaped panels: Panel A + Panel E conguration . . . . . . 144. 4.5.2. Structural load test description . . . . . . . . . . . . . . . . . . . . . . 145. 4.5.3. Dynamic response of the L-shaped structure . . . . . . . . . . . . . . . 147 4.5.3.1. Mode count and modal density calculations for sandwich panels147. 4.5.3.2. Global response . . . . . . . . . . . . . . . . . . . . . . . . . 151. 4.5.3.3. Local response . . . . . . . . . . . . . . . . . . . . . . . . . . 154. 4.5.3.4. Mean response of the panels . . . . . . . . . . . . . . . . . . 155. 4.5.3.5. Comparison of numerical models with experimental results from the structural load test . . . . . . . . . . . . . . . . . . 158. Vibro-acoustic response of Large Antenna Reector through HS-CMS method 161 4.6.1. Specimen description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161. 4.6.2. Acoustic test description . . . . . . . . . . . . . . . . . . . . . . . . . . 162. 4.6.3. Vibro-acoustic response . . . . . . . . . . . . . . . . . . . . . . . . . . 165. 4.6.4. 4.6.5 4.7. 4.4.4.1. Dynamic response of a L-shaped structure by the HS-CMS method . . . . . . 143 4.5.1. 4.6. Local response of the beams system . . . . . . . . . . . . . . . . . . . 138. 4.6.3.1. FEM Model of the Structure . . . . . . . . . . . . . . . . . . 165. 4.6.3.2. Acoustic load . . . . . . . . . . . . . . . . . . . . . . . . . . . 165. Vibro-acoustic response of the structure . . . . . . . . . . . . . . . . . 168 4.6.4.1. Global response . . . . . . . . . . . . . . . . . . . . . . . . . 169. 4.6.4.2. Global response comparison with acoustic test results . . . . 171. 4.6.4.3. Local response . . . . . . . . . . . . . . . . . . . . . . . . . . 177. Total response comparison with acoustic test results . . . . . . . . . . 180. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 v.

(20) 5 Comparison between new mid-frequency methods. 185. 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185. 5.2. System description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187. 5.3. Dynamic response through the HS-CMS method . . . . . . . . . . . . . . . . 188 5.3.1. Sub-structures wavelength and global modes selection . . . . . . . . . 188. 5.3.2. Global response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190. 5.3.3. Local response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193. 5.3.4. Validation study: comparison with Monte Carlo results . . . . . . . . . 197. 5.4. Dynamic response of the four coupled beams through the SHFL procedure . . 200. 5.5. Comparison between methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 202. III Conclusions and recommendations for further work. 207. 6 Conclusions and recommendations for further work. 209. 6.1. Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209. 6.2. Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212. References. vi. 215.

(21) List of Figures. Applicability ranges for the proposed mid-frequency methods: The Subsystems based High Frequency Limit procedure (SHFL) and The Hybrid Sub-structuring method based on Component Mode Synthesis (HS-CMS). . . . . . . . . . . . . . . . . . . .. 9. 2.1. Sketch of the structure-uid problem. . . . . . . . . . . . . . . . . . . . . . . . .. 15. 2.2. Direct and reverberant elds decomposition for subsystem k . . . . . . . . . . . . .. 52. 3.1. Application of the modal density criterion for a typical satellite structure. Red vertical dashed lines represent the high frequency limits of the subsystem: f1,lim , f2,lim and. 1.1. f3,lim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 65. 3.2. Accumulated mode count for the aluminium rod. . . . . . . . . . . . . . . . . . .. 71. 3.3. Modal density for the aluminium rod. . . . . . . . . . . . . . . . . . . . . . . . .. 71. 3.4. Accumulated mode count for bending motion in the beam. . . . . . . . . . . . . .. 72. 3.5. Modal density for bending motion in the beam. . . . . . . . . . . . . . . . . . . .. 73. 3.6. Modal density for the plate bending motion. . . . . . . . . . . . . . . . . . . . . .. 74. 3.7. Accumulated mode count for the plate bending motion. . . . . . . . . . . . . . . .. 75. 3.8. The Ariane 5 upper part and location of the SYLDA structure [Arianespace, 2009] .. 76. 3.9. Structural response of the cylinder, reproduced following the two models dened in Ref. [Mid-Frequency D28]: the SYLDA FE model and the SYLDA hybrid model. . .. 77. 3.10 Mode count (Nm ) and modal overlap (M ) for the subsystems of the SYLDA structure. 78 3.11 Frequency response of the cylinder of the SYLDA structure using the SHFL procedure. 79 vii.

(22) 3.12 Frequency response of the upper cone using the SHFL procedure. . . . . . . . . . .. 79. 3.13 Frequency response of the lower cone using the SHFL procedure. . . . . . . . . . .. 80. 3.14 Frequency response of the upper cone calculated using the SHFL procedure versus. frequency response calculated by means of a full FE model of the SYLDA structure.. 80. 3.15 Mock up of the satellite (left) and subsystems of the numerical model (right). . . . .. 82. 3.16 Structural and acoustic test; a detail of the accelerometers and shaker used in the test campaign. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 83. 3.17 Acoustic load measured by the far eld microphones. . . . . . . . . . . . . . . . .. 84. 3.18 Number of structural modes in each frequency band for the elements of the complex structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 86. 3.19 Dierences between FE/BE model and FE/SIF model, to study the application range of the Semi Innite Fluids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 87. 3.20 Structural test results: Experimental and numerical responses of accelerometers placed on a lateral face, shown in narrow band and in 1/3rd octaves. . . . . . . . . . . . .. 88. 3.21 Structural test results: Experimental and numerical responses of accelerometers placed on the upper platform, shown in narrow band and in 1/3rd octaves. . . . . . . . . .. 89. 3.22 Structural test results: Experimental and numerical responses of accelerometers placed on a solar array, shown in narrow band and in 1/3rd octaves. . . . . . . . . . . . .. 89. 3.23 Acoustic test results: Experimental and numerical responses of accelerometers placed. on the upper platform, shown in 1/3rd octaves. Black dashed line: Acoustic test results. 90. 3.24 Acoustic test results: Experimental and numerical responses of accelerometers placed. on a lateral face, shown in 1/3rd octaves. Black dashed line: Acoustic test results. .. 90. 3.25 Upper platform response to a diuse acoustic eld of a constant pressure level (1 Pa). 91 3.26 Lower platform response to a diuse acoustic eld of a constant pressure level (1 Pa). 91 3.27 Lateral face response to a diuse acoustic eld of 1 Pa pressure level. . . . . . . . .. 92. 3.28 Adapter cone response to a diuse acoustic eld of 1 Pa pressure level. . . . . . . .. 92. 3.29 Solar array response to a diuse acoustic eld of 1 Pa pressure level. On the top graph,. response using a single hybrid model for mid-frequencies; on the bottom graph, the proposed SHFL procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3.30 SWARM spacecraft structure.. 93. Courtesy of ESA. .. . . . . . . . . . . . . . . . . . .. 95. 3.31 Mode count as a function of the frequency for dierent boundary conditions for the Solar Panel PY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 99. 3.32 Mode count for the SWARM sub-structures (Part I). . . . . . . . . . . . . . . . . 100 viii.

(23) 3.33 Mode count for the SWARM sub-structures (Part II). . . . . . . . . . . . . . . . . 100 3.34 Acceleration spectral density of the Solar Array PY (equal response is obtained in solar array MY, due to symmetry). . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.35 Acceleration spectral density of the subsystem Top Panel MX.. . . . . . . . . . . . 102. 3.36 Acceleration spectral density of the subsystem Rib 1, inside the SWARM body. . . . 103 3.37 Acceleration spectral density of the deployable boom assembly. . . . . . . . . . . . 103 3.38 Comparison of acceleration spectral density calculated with the SHFL procedure and. the acoustic test results in the Top Panel MX. Black line, acoustic text; Green line, the SHFL procedure; Blue line, FE/BE model; and red line, FE/SIF model. . . . . 105. 3.39 Comparison of acceleration spectral density calculated with the SHFL procedure and. the acoustic test results in one rib inside the satellite body. Black line, acoustic text; Green line, the SHFL procedure; Blue line, FE/BE model; and red line, FE/SIF model. 105. 4.1. Global modal coordinates calculated at high frequencies. . . . . . . . . . . . . . . 118. 4.2. Flowchart of the hybrid method for its implementation in Matlab code. . . . . . . . 129. 4.3. System of coupled elastic beams. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131. 4.4. Wavelength as a function of the mode count for both beams. . . . . . . . . . . . . 134. 4.5. Mode count for both beams. Black dashed lines represent the critical mode count, Nc (5 modes within a frequency band). . . . . . . . . . . . . . . . . . . . . . . . 134. 4.6. Mean velocity of beam 1 depending on the number of global modes selected and it comparison with a conventional FEM response (black line). Note that all velocity values below 500 Hz are indistinguishable. . . . . . . . . . . . . . . . . . . . . . 136. 4.7. Mean velocity of beam 2 depending on the number of global modes selected and it comparison with a conventional FEM response (black line). Note that values of Nc = 8 (red line) and FE model are indistinguishable up to 2,000 Hz. Dierences between FE model and Nc = 5 (blue line) are shown only for frequencies higher than 1,600 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137. 4.8. Input power from local force injected to the SEA subsystem beam 1 (note that input power from local force is zero for beam 2). . . . . . . . . . . . . . . . . . . . . . . 139. 4.9. First values of global modal coordinates of beam 1, calculated at high frequencies. . 139. 4.10 Input power from the global modes for beams 1 and 2 in 1/3 octave band. . . . . . 140 4.11 Total input power injected into the SEA model of the system of beams beams, including contribution of local force and global modes. . . . . . . . . . . . . . . . . . 140 ix.

(24) 4.12 Mean velocity of the sub-structures beam 1 (upper graph) and beam 2 (lower graph).. Green lines, global responses in narrow band; dark blue lines with markers, global responses averaged in 1/3rd octave bands; red lines with markers, local responses averaged in 1/3rd octave bands; dashed blue lines with markers, global responses in 1/3rd octave bands; red dashed lines with markers, local responses in 1/3rd octave bands; black dashed lines with markers, conventional SEA. Red vertical dashed lines represent the high frequency limit of the beams: f1, and f2, . . . . . . . . . . 142 lim. lim. 4.13 Finite element model of Panel A. Red triangles stand for the structure inserts. . . . 143 4.14 Finite element model of Panel E. Red triangles stand for the structure inserts. . . . 145 4.15 Finite element model of the L-shaped specimen. . . . . . . . . . . . . . . . . . . . 145 4.16 Test facilities and specimen set up. Specimen hanging conditions. . . . . . . . . . . 146 4.17 Excitation points in Panel A (left) and in Panel E (right) and positions of the accel-. erometers [ESA/ESTEC, 2006]:(◦) location of the accelerometers for the response of the plate, (•) position of the accelerometer and cell force in the dierent excitation points, and (⊗) location of the joint with the L brackets. . . . . . . . . . . . . . . 146. 4.18 The wavenumber plane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.19 Modal density for both sub-structures of the L-shaped structure. . . . . . . . . . . 150 4.20 Mode count for the sub-structures of the system. Lines with markers, mode count. values calculated through a FEM model with xed boundary conditions; continuous lines, mode count values obtained through the modal density. Black dot-dashed line represents the critical mode count, Nc = 5 . . . . . . . . . . . . . . . . . . . . . . 150. 4.21 Global response of Panel A due to a point load excitation in Panel A. Comparison. with FEM results. Red vertical dashed line represents the high frequency limit of the Panel, fA, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 lim. 4.22 Global response of Panel E due to a point load excitation in Panel A. Comparison. with FEM results. Red vertical dashed line represents the high frequency limit of the Panel, fE, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 lim. 4.23 Global response of Panel A due to a point load excitation in Panel E. Comparison. with FEM results. Red vertical dashed line represents the high frequency limit of the Panel, fA, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 lim. 4.24 Global response of Panel E due to a point load excitation in Panel E. Comparison. with FEM results. Red vertical dashed line represents the high frequency limit of the Panel, fE, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 lim. 4.25 Coupling loss factors for the L-shaped structure: 4, for ηAE ; •, for ηEA . . . . . . . 154 4.26 Input power coming from global modes. Red vertical dashed lines: fA,lim and fE,lim x. 155.

(25) 4.27 Acceleration spectral density in Panel A due to a point load excitation in Panel A.. Blue line, global response in narrow band; blue line with markers, global response in 1/3rd octave bands; green line with markers, local response in 1/3rd octave bands. Red vertical dashed line represents the high frequency limit of the Panel, fA, . . . 156 lim. 4.28 Acceleration spectral density in Panel E due to a point load excitation in Panel A.. Blue line, global response in narrow band; blue line with markers, global response in 1/3rd octave bands; green line with markers, local response in 1/3rd octave bands. Red vertical dashed line represents the high frequency limit of the Panel, fE, . . . 156 lim. 4.29 Acceleration spectral density in Panel A due to a point load excitation in Panel E.. Blue line, global response in narrow band; blue line with markers, global response in 1/3rd octave bands; green line with markers, local response in 1/3rd octave bands. Red vertical dashed line represents the high frequency limit of the Panel, fA, . . . 157 lim. 4.30 Acceleration spectral density in Panel E due to a point load excitation in Panel E.. Blue line, global response in narrow band; blue line with markers, global response in 1/3rd octave bands; green line with markers, local response in 1/3rd octave bands. Red vertical dashed line represents the high frequency limit of the Panel, fE, . . . 157 lim. 4.31 Comparison between numerical model and experimental data for point excitation A.1. Response of accelerometers A5 in Panel A. Red vertical dashed line: fA,lim . . . . . 158 4.32 Comparison between numerical model and experimental data for point excitation A.1. Response of accelerometers A2 in Panel E. Red vertical dashed line: fE,lim . . . . . 158 4.33 Comparison between numerical model and experimental data for point excitation E.2. Response of accelerometers A8 in Panel A. Red vertical dashed line: fA,lim . . . . . 159 4.34 Comparison between numerical model and experimental data for point excitation E.2. Response of accelerometers A4 in Panel E. Red vertical dashed line: fE,lim . . . . . 159 4.35 Specimen under study. Location of the rib and cleats. . . . . . . . . . . . . . . . . 161 4.36 Reverberation room geometry.. . . . . . . . . . . . . . . . . . . . . . . . . . . . 162. 4.37 (A): Specimen in vertical orientation. Position of the microphones in the chamber; (B): Specimen in vertical orientation, side view. . . . . . . . . . . . . . . . . . . . 163 4.38 (A): Specimen in vertical orientation, front view (concave side); (B): Specimen in vertical orientation, rear view (convex side). . . . . . . . . . . . . . . . . . . . . . 163 4.39 Readings of the microphones in the acoustic test in 1/3rd octave bands. On the left, near eld microphones; on the right, far eld microphones. . . . . . . . . . . . . . 164 4.40 Position of accelerometers in the dish (on the left) and in the rib (on the right).. . . 164. 4.41 Specimen FEM. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.42 PSD of pressure measured during the acoustic test. . . . . . . . . . . . . . . . . . 165 xi.

(26) 4.43 Modal radiation eciencies of the rib (grey lines) and its comparison with radiation eciency calculated following section 4.2.3.4 (solid blue line). . . . . . . . . . . . . 166 4.44 Modal radiation eciencies of the dish (grey lines) and its comparison with radiation eciency calculated following section 4.2.3.4 (solid blue line). . . . . . . . . . . . . 166 4.45 Nodal forces randomly distributed in the space. . . . . . . . . . . . . . . . . . . . 167 4.46 Modal density of the sub-structures of the system. . . . . . . . . . . . . . . . . . . 168 4.47 Mode count of the sub-structures of the system. Lines with markers represent the. mode count values calculated through a FEM model with xed boundary conditions; continuous lines, mode count values obtained through the modal density. Black dotdashed line represents the critical mode count Nc = 5 . . . . . . . . . . . . . . . . 169. 4.48 Vibro-acoustic response of the dish in narrow band ∆f = 1.25 Hz. Green line, FEM. model of the system; Blue line, global response. Red vertical dashed line, f. Dish,lim. . 170. 4.49 Vibro-acoustic response of the rib in narrow band ∆f = 1.25 Hz. Green line, FEM model of the system; Blue line, global response. . . . . . . . . . . . . . . . . . . . 170 4.50 Vibro-acoustic global response of the dish (left graph) and the rib (right graph), in. 1/3rd octave bands. Green line, FEM model of the system; Blue line, global response. Red vertical dashed line, fDish,lim . . . . . . . . . . . . . . . . . . . . . . . . . . 171. 4.51 Vibro-acoustic global response and acoustic test results of accelerometer 4023, located in the dish, in narrow band ∆f = 1.25 Hz. Red vertical dashed line, fDish,lim . . . . 172 4.52 Vibro-acoustic global response and acoustic test results of accelerometer 5029, located in the dish, in narrow band ∆f = 1.25 Hz. Red vertical dashed line, fDish,lim . . . . 172 4.53 Vibro-acoustic global response and acoustic test results of accelerometer 6032, located in the dish, in narrow band ∆f = 1.25 Hz. Red vertical dashed line, fDish,lim . . . . 173 4.54 Vibro-acoustic global response and acoustic test results of accelerometer 7209, located in the dish, in narrow band ∆f = 1.25 Hz. Red vertical dashed line, fDish,lim . . . . 173 4.55 Vibro-acoustic global response and acoustic test results of accelerometer 7226, located in the dish, in narrow band ∆f = 1.25 Hz. Red vertical dashed line, fDish,lim . . . . 173 4.56 Vibro-acoustic global response and acoustic test results of accelerometer 7233, located in the dish, in narrow band ∆f = 1.25 Hz. Red vertical dashed line, fDish,lim . . . . 174 4.57 Vibro-acoustic global response and acoustic test results of accelerometer 7275, located in the dish, in narrow band ∆f = 1.25 Hz. Red vertical dashed line, fDish,lim . . . . 174 4.58 Vibro-acoustic global response and acoustic test results of accelerometer 7286, located in the dish, in narrow band ∆f = 1.25 Hz. Red vertical dashed line, fDish,lim . . . . 174 4.59 Vibro-acoustic global response and acoustic test results of accelerometer 7556-Y, located in the rib, in narrow band ∆f = 1.25 Hz. . . . . . . . . . . . . . . . . . . . . 175 xii.

(27) 4.60 Vibro-acoustic global response and acoustic test results of accelerometer 7556-Z, located in the rib, in narrow band ∆f = 1.25 Hz. . . . . . . . . . . . . . . . . . . . . 175 4.61 Vibro-acoustic global response and acoustic test results of accelerometer 4023 and. 5029, located in the dish, in 1/3rd octave bands. Red vertical dashed line, f. Dish,lim. . 175. 4.62 Vibro-acoustic global response and acoustic test results of accelerometer 6032 and. 7209, located in the dish, in 1/3rd octave bands. Red vertical dashed line, f. Dish,lim. . 176. 4.63 Vibro-acoustic global response and acoustic test results of accelerometer 7226 and. 7233, located in the dish, in 1/3rd octave bands. Red vertical dashed line, f. Dish,lim. . 176. 4.64 Vibro-acoustic global response and acoustic test results of accelerometer 7275 and. 7286, located in the dish, in 1/3rd octave bands. Red vertical dashed line, f. Dish,lim. . 176. 4.65 Vibro-acoustic global response and acoustic test results of accelerometer 7556-Y and 7556-Z, located in the rib, in 1/3rd octave bands. . . . . . . . . . . . . . . . . . . 177 4.66 Input power into the Dish SEA model. Green line, total input power; Blue line, input. power due to the local force; and red line, additional input power coming from the presence of global modes. Red vertical dashed line, f . . . . . . . . . . . . 178 Dish,lim. 4.67 Input power into the Rib SEA model. Green line, total input power; Blue line,. input power due to the local force; Red line, additional input power coming from the presence of global modes. Red vertical dashed line, f . . . . . . . . . . . . . 178 Rib,lim. 4.68 Coupling loss factors for the sub-structures. Comparison between results using V-. SEA module [ESI, 2011] and results obtained by means of punctual junctions [Lyon, 1975; Lyon and De Jong, 1995]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 179. 4.69 Total response of the dish. Green dashed line, total response in narrow band; red line. with markers, total response in 1/3rd octave band; and blue line with markers, local response. Red vertical dashed line: f . . . . . . . . . . . . . . . . . . . . . 179 Dish,lim. 4.70 Total response of the rib. Green dashed line, total response in narrow band; red line. with markers, total response in 1/3rd octave band; and blue line with markers, local response. Red vertical dashed line: f . . . . . . . . . . . . . . . . . . . . . . 180 Rib,lim. 4.71 Total response of the Dish and its comparison with acoustic test results. . . . . . . 180 4.72 Total response of the Rib and its comparison with acoustic test results. . . . . . . . 181 5.1. Schematic of the four coupled beams, with an axial load P applied to beam 1. . . . 187. 5.2. Wavelength as a function of the subsystems mode count. Notice that values for beams 1 and 4 are identical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189. 5.3. Mean velocity in the beam 1. Green line, results of the global response; Blue line, its comparison with a FE model of the system. . . . . . . . . . . . . . . . . . . . . . 190 xiii.

(28) 5.4. Mean velocity in the beam 2. Green line, results of the global response; Blue line, its comparison with a FE model of the system. . . . . . . . . . . . . . . . . . . . . . 191. 5.5. Mean velocity in the beam 3. Green line, results of the global response; Blue line, its comparison with a FE model of the system. . . . . . . . . . . . . . . . . . . . . . 191. 5.6. Mean velocity in the beam 4. Green line, results of the global response; Blue line, its comparison with a FE model of the system. . . . . . . . . . . . . . . . . . . . . . 192. 5.7. Values of mode count and modal overlap for the beams. Red vertical dashed lines represent the high frequency limits for each beam: f1,lim , f2,lim , f3,lim and f4,lim . . 192. 5.8. Values of global modal coordinates, ξg for the beam 1 (on the top) and beams 2, 3, and 4 (on the bottom). Continuous lines represent the values calculated with the global equations. Dot-dash lines represent the calculated values for high frequency. . 194. 5.9. G Power input from global modes into the SEA model. Each values of Pin is calculated for frequencies above the high frequency limits of each subsystems. Red vertical dashed line: High frequency limit for each sub-structure, fi, . . . . . . . . . . . . 194 lim. 5.10 Mean velocity of the beam 1. Blue line, global response (narrow band); Green line,. global response (1/3rd octave bands); Red line, local response. Red vertical dashed line: f1, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 lim. 5.11 Mean velocity of the beam 2. Blue line, global response (narrow band); Green line,. global response (1/3rd octave bands); Red line, local response. Red vertical dashed line: f2, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 lim. 5.12 Mean velocity of the beam 3. Blue line, global response (narrow band); Green line,. global response (1/3rd octave bands); Red line, local response. Red vertical dashed line: f3, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 lim. 5.13 Mean velocity of the beam 4. Blue line, global response (narrow band); Green line,. global response (1/3rd octave bands); Red line, local response. Red vertical dashed line: f4, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 lim. 5.14 Mean velocity in beam 1. Light gray lines, each Monte Carlo realization; bold gray. line, Monte Carlo ensemble average in narrow bands; black line with markers, Monte Carlo ensemble average averaged in 1/3rd octave bands; solid blue line, global response; red line with markers, global response averaged in 1/3rd octave bands; green line with markers, local response. . . . . . . . . . . . . . . . . . . . . . . . . . . 197. 5.15 Mean velocity in beam 2. Light gray lines, each Monte Carlo realization; bold gray. line, Monte Carlo ensemble average in narrow bands; black line with markers, Monte Carlo ensemble average averaged in 1/3rd octave bands; solid blue line, global response; red line with markers, global response averaged in 1/3rd octave bands; green line with markers, local response. . . . . . . . . . . . . . . . . . . . . . . . . . . 198. xiv.

(29) 5.16 Mean velocity in beam 3. Light gray lines, each Monte Carlo realization; bold gray. line, Monte Carlo ensemble average in narrow bands; black line with markers, Monte Carlo ensemble average averaged in 1/3rd octave bands; solid blue line, global response; red line with markers, global response averaged in 1/3rd octave bands; green line with markers, local response. . . . . . . . . . . . . . . . . . . . . . . . . . . 198. 5.17 Mean velocity in beam 4. Light gray lines, each Monte Carlo realization; bold gray. line, Monte Carlo ensemble average in narrow bands; black line with markers, Monte Carlo ensemble average averaged in 1/3rd octave bands; solid blue line, global response; red line with markers, global response averaged in 1/3rd octave bands; green line with markers, local response. . . . . . . . . . . . . . . . . . . . . . . . . . . 199. 5.18 Mean velocity of the beam 1 using the SHFL procedure.. . . . . . . . . . . . . . . 200. 5.19 Mean velocity of the beam 2 using the SHFL procedure.. . . . . . . . . . . . . . . 201. 5.20 Mean velocity of the beam 3 using the SHFL procedure.. . . . . . . . . . . . . . . 201. 5.21 Mean velocity of the beam 4 using the SHFL procedure.. . . . . . . . . . . . . . . 202. 5.22 Mean velocity in the beam 1. Red dashed line, the HS-CMS method; Blue line, the SHFL procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 5.23 Velocity in the beam 2. Red dashed line, the HS-CMS method; Blue line, the SHFL procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.24 Velocity in the beam 3. Red dashed line, the HS-CMS method; Blue line, the SHFL procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 5.25 Velocity in the beam 4. Red dashed line, the HS-CMS method; Blue line, the SHFL procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204. xv.

(30) xvi.

(31) List of Tables. 3.1. Natural frequencies of the longitudinal vibrations of a rod. . . . . . . . . . . . . .. 70. 3.2. Accumulated mode count and modal density for a rod in longitudinal vibrations. . .. 70. 3.3. Natural frequencies, modal density and accumulated mode count for bending motion in a beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 72. 3.4. Modal density for a plate vibrating in bending. . . . . . . . . . . . . . . . . . . .. 74. 3.5. Accumulated mode count for a plate vibrating in bending. . . . . . . . . . . . . . .. 74. 3.6. Computation time for the SHFL procedure. . . . . . . . . . . . . . . . . . . . . .. 81. 3.7. Comparison between SHFL and full FEM model computation times. . . . . . . . .. 81. 3.8. Numerical models required to predict the vibro-acoustic response of the satellite. . .. 87. 3.9. Comparison between computation cost associated with models FE/BE and FE/SIF.. 88. 3.10 Process time (total CPU time) associated with the models needed to apply the SHFL procedure and the single-hybrid procedure. . . . . . . . . . . . . . . . . . . . . .. 94. 3.11 Sub-structures of the SWARM satellite (Part I). . . . . . . . . . . . . . . . . . . .. 96. 3.12 Sub-structures of the SWARM satellite (Part II). . . . . . . . . . . . . . . . . . .. 97. 3.13 Sub-structures of the SWARM satellite (Part III). . . . . . . . . . . . . . . . . . .. 98. 3.14 Hybrid numerical models classication.. . . . . . . . . . . . . . . . . . . . . . . . 101. 4.1. Geometrical and physical properties of the beams. . . . . . . . . . . . . . . . . . . 131. 4.2. Computational eort to perform the analysis between 10-2,000 Hz (∆f = 1 Hz), compared with a FE model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 xvii.

(32) 4.3. Design parameters of the skins material of Panel A and Panel E. . . . . . . . . . . 144. 4.4. Design parameters of the core material of Panel A. . . . . . . . . . . . . . . . . . 144. 4.5. Design parameters of the core material of Panel E. . . . . . . . . . . . . . . . . . 144. 4.6. Position of excitation and measurement points in Panel A and Panel E. . . . . . . . 147. 4.7. Dish and Rib sandwich conguration. . . . . . . . . . . . . . . . . . . . . . . . . 162. 5.1. Geometrical and physical properties of the four coupled beams. . . . . . . . . . . . 187. 5.2. First values of the bending wavelength, associated with the interior modes of each beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188. 5.3. Frequency ranges in which global and local response are dened for each sub-structure. 191. 5.4. Numerical models used to apply the SHFL procedure. . . . . . . . . . . . . . . . . 200. 5.5. Computation times required by the HS-CMS method. . . . . . . . . . . . . . . . . 204. 5.6. Computation times required by the SHFL procedure. . . . . . . . . . . . . . . . . 204. xviii.

(33) List of Symbols. Latin letters A B. C c0 cB C eq cg cL cS D. D DS E Ei. ES f fc f~ F~ flim G h. Area Bending stiness Damping matrix Sound velocity Velocity of bending waves Equivalent spatial coherence function Group wave propagation velocity Velocity of longitudinal waves Velocity of shear waves Bending stiness Dynamic stiness matrix Constitutive matrix Modulus of elasticity Subsystem energy Green-Lagrange strain tensor Frequency Critical frequency Volumetric force Applied forces High frequency limit Green function Thickness xix.

(34) i I Ii. I IS J. K k kB L m M. M mi ni ~n ng nl N. N Na Nc Nm Ns p P Pin q. q Re r rg S SF Sp Spp Svv. SS t. T xx. Imaginary unit Intensity In Unitary matrix Inertia forces Joint Acceptance Function Stiness matrix Wavenumber Bending wavenumber Length Mass Modal overlap Mass matrix Normalized attenuation factor Modal density of subsystem i Normal unitary vector Number of global modes Number of local modes Number of subsystems Finite element shape functions Accumulated mode count Critical model count Mode count Shear rigidity Pressure Power Input power Degree of freedom / Nodal displacement in FEM Vector of degrees of freedom Real part of a complex number Position vector Radius of gyration Surface PSD of nodal forces PSD of nodal pressure Power spectral density Velocity spectral density Cauchy stress tensor Time Kinetic energy.

(35) T u v V w Wrad Y Z i, j , n. Transformation matrices Displacement Velocity Volume Normal surface displacement Radiated sound power Admittance/Mobility Impedance Counters. Greek letters αi δ ∆ ∆ω ε ηi ηij ηi,tot λ λB λL λmf µ ν ξ ξg ξl ρ σ σrad τij τ φ Φ φa ψ. Random angles for DAF excitation Dirac delta Divergence operator Bandwidth Strain Internal dissipation loss factor of SEA subsystem i Coupling loss factor from SEA subsystem i to subsystem j Total loss factor Wavelength Bending wavelength Limit wavelength Mode wavelength Shear modulus Poisson's ratio Modal coordinate Global modal coordinate Local modal coordinate Density Cauchy stress tensor Radiation eciency Transmission coecient Traction vector Normal mode Matrix of modal shapes Acoustic velocity potential Acoustic pressure mode shape xxi.

(36) Ψ ∂Ω ω Ω. Craig-Bampton transformation matrix Boundary of the domain Ω Circular frequency Domain. Acronyms and Abbreviations ASD BEM CLF CMS CPU dof DAF DLF FEM FFT FMM FRF GDR HS-CMS IEM ILS IRS JAA MIA ODE PDE PSD SEA SHFL SIF SYLDA. xxii. Acceleration Spectral Density Boundary Element Method Coupling Loss Factor Component Mode Synthesis Central Processing Unit Degree of freedom Diuse acoustic eld Damping Loss Factor Finite Element Method Fast Fourier Transformation Fast Multipole Method Frequency Response Function Generalised Dynamic Reduction Hybrid Sub-structuring based on Component Mode Synthesis Innite Element Method Internal Loss Factor Improved Reduced System Joint Acceptance Analysis Modal Interaction Analysis Ordinary Dierential Equation Partial Dierential Equation Power Spectral Density Statistical Energy Analysis Subsystems High Frequency Limit Semi Innite Fluid Systeme de Lancement Double d'Ariane 5.

(37) Part I. Introduction and State of the Art. 1.

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(39) CHAPTER 1 Introduction. 1.1 Background ver the last years, the evolution of the space systems, has led to a situation in which. O. a successful space mission requires to consider the interaction between vibrations and vibro-acoustic eects. One of the primary sources of structural vibrations and internal loads during the lift-o phase is these acoustic loads that, in addition to shock loads, belong to the main design loads in spacecraft structures. Acoustic loads are due to several causes as, for example, the noise generated during ignition of rocket engines, which manifests itself to the launch vehicle, the payload and the launch pad, in the form of air-borne acoustics and structure-borne vibrations. An early knowledge of the random vibration environment is required during the development phases of a spacecraft, especially in the process of designing and testing the equipment.. The problem of predicting the vibro-acoustic behaviour of a system is mainly characterized by the interaction between two dierent domains, the uid and the structure. Whenever an elastic structure is in contact with a uid, the structural vibrations and the acoustic pressure eld in the uid are inuenced by the mutual vibro-acoustic coupling interaction. The force loading the structure, caused by the acoustic pressure along the uid-structure interface, inuences the structural vibrations. At the same time, the acoustic pressure eld in the uid is also sensitive to the structural vibrations along the uid-structure interface. The strength of this vibro-acoustic coupling interaction is largely dependent on the geometry of the structure and the uid domain, and on the uid and structural material properties, as well as on the frequency of the dynamic disturbances. 3.

(40) Chapter. 1.. There are a number of dierent approaches that have been developed to allow estimation of structures response to acoustic loads and the estimation of transmitted sound or vibration. The fundamental factors on which the selection of an analysis method should be based are the type of response information required, the levels of accuracy that are acceptable, the nature of the system under study and the time and resources available. The vibro-acoustic analysis of a complex vibro-acoustic system is dicult for two reasons: rst, the system may require many degrees of freedom to describe the response and, second, the response may be sensitive to small imperfections in the system. Both of these points become increasingly severe as the excitation frequency increases, due to the decreasing wavelength of the deformation. This research will deepen in the knowledge of numerical methodologies, providing state of the art insight on the technologies, a better understanding of the software tools, an advance in the numerical algorithms and, in addition, it will consolidate analysis methods and procedures for the vibro-acoustic analysis. The methodologies formulated will be also tested on experimental data corresponding to representative spacecraft structures. This work belongs to the Networking and Partnering Initiative (NPI) of the European Space Agency, entitled Dynamic Analysis of Payloads and Structures with Intermediate Modal Density [NPI, Statement of Work, 2011]. This activity allows to increase cooperation between the Agency, European universities, research institutes and the private sector, by means of the co-funding of PhD Thesis.. 1.2 Vibro-acoustic analysis: strengths and limitations Ideally, modelling techniques for vibro-acoustic studies of space structures should be applicable over the whole frequency range of interest, which is typically the frequency range from 10 Hz up to 10 kHz. In practice, however, specic methods are applicable only within a limited frequency range. Often it is required to comply with noise specications in a complex system of acoustical and structural subsystems, with several broadband sources of noise and vibrations. In such systems, noise problems need to be identied early in the design stage so that treatments can be eectively integrated into the design. The vibro-acoustic analysis of systems is often partitioned in three ranges: low, high and medium modal density ranges. In practice, in space structures, the number of existing modes makes equivalent this modal density ranges with the low, high and medium frequency ranges. If the system has low modal density, in the low frequency range, the response of the system is usually described in terms of modes, which are relatively few in number. This fact causes that modal analysis procedures, as Finite Element Method (FEM) [Zienkiewicz et al., 4.

(41) 1.2. VIBRO-ACOUSTIC ANALYSIS: STRENGTHS AND LIMITATIONS 2005], can be used to predict the acoustic environment by identifying the structural modes of the surfaces and acoustical modes of the interior space. Since the structure is modelled in a spatially discrete manner, acoustic forcing requires also to be represented in a similar way. However, the representation of the acoustic eld by discrete forces becomes more dicult as frequency increases. As a consequence, when a Finite Element Method is used to model the structure, the acoustic eld modelling is better performed by means of the Boundary Element Method (BEM). The main advantage of BEM is that only the boundary of a component needs to be discretized and, in general, this leads to a reduced number of degrees of freedom that are needed by a single FEM. However, the matrices involved tend to be fully populated rather than banded, and the computational eort to assemble all the equations can be signicant. Nevertheless, as the frequency range of interest becomes higher, the number of modes in the analysis and, therefore, the size of the numerical models increase rapidly. Thereby, the computational cost may become very expensive making the use of deterministic methods limited to the low frequency range (i.e. the characteristic dimensions of the considered problem is much shorter than the free eld wavelength). Using special techniques, as Component Mode Synthesis [Craig, 1981], and increasing computer power may enable an ecient analysis of a system when a limited number of modes are being excited. When the response of a system is modelled at higher frequencies, the analysis must take into account the large number of acoustic and structural modes contributing to the dynamic response. At such a high frequencies, it is appropriate to select and analysis method that requires relatively few degrees of freedom and provides statistical information. Statistical Energy Analysis represents a eld of study in which statistical descriptions of a system are employed in order to simplify the analysis of complicated structural-acoustic problems. At higher frequencies there are so many modes and the vibrations can be better described in terms of waves, which propagate through the system. The system is considered to be built up of a group of subsystems, and the response is given in terms of subsystem properties and the physical relations among their parts. Methods as Statistical energy Analysis (SEA) [Fahy, 1970; Lyon, 1975; Lyon and De Jong, 1995] provide prediction procedures that are appropriate for high frequencies. By carrying out a statistical description of the power ow among the modes corresponding to the dierent parts of the whole system, and by considering the energy of these parts as variable, the dynamic equations can be formulated straightforwardly. Once the energies of the subsystems are determined, other secondary parameters as displacement, velocities or pressure can be calculated. The method involves relatively few degrees of freedom and, therefore, parameter studies can be carried out with little computational eort. Furthermore, SEA allows to make response predictions at frequencies for which very few other analysis methods are available. However, as a drawback, by using this statistical method, it is not 5.

(42) Chapter. 1.. possible to obtain results at individual locations and single frequencies. This analysis method provides space and frequency averages of the target quantities. Between the low frequency range, where the whole system can be described deterministically, and the high frequency range, where statistical methods are used, a mid-frequency range exists where none of the methods described above can be stated as appropriate. Often, for a given frequency band, certain number of subsystems have high modal density overlapping each other and fullling the requirements for the application of a statistical description, while some other parts of the system still present a deterministic behaviour. In the full system, this undetermined range is called the medium modal density range, where the density of modes in the frequency band is not either enough high to carry out an statistical analysis nor enough low to use a deterministic method. For complex structures, this medium modal density is shown for mid-frequencies and, in general, it is not clearly determined. Broadly, current approaches can be grouped into three types, although there is a signicant overlap in methods and philosophies between the types.. • Deterministic response prediction methods • Energetic response prediction methods • Hybrid and mixed methods. Deterministic methods Deterministic methods are based on the solution of a set of deterministic equations of motion. These methods are based on calculating primary magnitudes as displacement, velocity or acceleration. Modal analysis procedures will enable ecient analysis of systems in which a limited number of modes are being excited. However, for vibroacoustic problems, a large number of acoustic and structural modes may be contributing to the dynamic response and modal synthesis can become computationally unfeasible.. Energy methods The number of degrees of freedom required by deterministic methods can become impracticable at high frequencies. In such cases carrying out a detailed prediction of the structure response is not possible. Energy methods use vibrational energy to formulate the dynamic equations providing great simplications to the analysis. The methods involve relatively few degrees of freedom and it is possible to perform parameter studies with little computational eort. However, energy methods does not provide detail about the variation of the average response in discrete positions of the system.. Hybrid and mixed methods One mid-frequency issue of real practical importance is the case where sti, low mode-count components are coupled to exible, high mode-count components. In other words, elements with low modal density coexist with elements with 6.

(43) 1.3. THE RESEARCH GOAL: OBJECTIVES AND JUSTIFICATION high modal density. The problem is that both types of component are well modelled by, for example, either FEM or SEA respectively, but the structure as a whole is not well modelled by one single approach. The size of a full FE model is prohibitively large, while some components have so few modes that the SEA assumptions do not apply. This has motivated the search for new techniques that can be classied into two dierent philosophies: (i) Techniques which combine FE and SEA models of dierent regions of the structure, where FE and statistical elements are linked by hybrid junctions [Cotoni et al., 2007; Langley and Cordioli, 2009; Shorter and Langley, 2005a]. (ii) Techniques that include information from one methodology into another. Several hybrid methods have been developed in the last years. Methods belonging to this group are: the theory developed by Langley and Bremner [1999], based on partitioning the system response into a global part, solved using deterministic methods, and a local part, incorporated towards their dynamic stiness matrices; The theory of fuzzy structures [Lyon, 1995; Soize, 1993], that proposes that the master structure is modelled deterministically and the inuence of fuzzy attachments on the master structure is modelled as an increase of inertia and damping, etc.. 1.3 The research goal: Objectives and justication Throughout the dierent sections of this introductory chapter, an overview of the strengths and limitations arising from the vibro-acoustic analysis of structures has been presented. For the resolution of vibro-acoustic problems, dierent types of existing methodologies will be studied along this dissertation. From the study of the state of the art in vibro-acoustic modelling techniques, it has been concluded that the available methodologies cover with enough accuracy the low and high modal density ranges. Nevertheless, an opportunity for improvement exists in the eld of medium frequency modelling. The proposed Ph.D research is focused in the range of mid-frequencies due to the lack of knowledge in ecient prediction techniques for these specic cases, so that is justied in the strong need of novel prediction techniques that allow to narrow the currently existing mid-frequency twilight zone. Given the lack of a clearly denition of the mid-frequency range, one of the rst issues to deal with is to propose criteria to mark out these limits and their characteristics, to develop appropriate modelling techniques. In view of the properties of the mid-frequency dynamic behaviour of vibro-acoustic systems, the prediction techniques should meet the following requirements: (i) The prediction techniques must be applicable for general, real-life vibro-acoustic engineering applications. 7.

(44) Chapter. 1.. (ii) In contrast with SEA models, the prediction techniques should be able to provide detailed information on the spatial distribution of the mid-frequency response within the various components of the considered vibro-acoustic system. (iii)) The prediction techniques should provide continuous responses in the whole frequency range. (iv) Many engineering structures, specially built-up structures such as satellites and other space structures, have a particular mid-frequency dynamic behaviour: parts of the structure consist of sti and strongly connected components that still exhibit long wavelength dynamic deformations in the mid-frequency range, while other parts already exhibit a highly resonant behaviour with short wavelength deformations. The prediction techniques should enable a hybrid modelling approach iby which the long wavelength components are modelled deterministically, while the short wavelength components are described in a probabilistic manner. The study is focused on hybrid methodologies, in order to develop guidelines and analysis procedures providing higher computational eciency and better spatial resolution. As Fig. 1.1 shows, vibro-acoustic methods could be classied according to the heterogeneity of the system and the frequency range of analysis. This PhD work proposes two methods in order to solve the vibro-acoustic problem: The Subsystems based High Frequency Limit procedure (SHFL) and the Hybrid Sub-structuring method based on Component Mode Synthesis (HS-CMS). Their proposed applicability ranges are shown in Fig. 1.1. 1- The Subsystems based High Frequency Limit procedure (SHFL) proposes guidelines to perform a vibro-acoustic analysis, focusing in the improvement of the system response in the medium frequency range. The procedure is based on a combination of FEM/BEM and SEA techniques in dierent frequency ranges, being the key point of the method the denition of the frontier between both techniques. Therefore, the concept of subsystem high frequency limit is presented, which marks the frequency frontier above which a certain subsystem could be appropriately modelled by SEA. By this method, for a certain frequency of study, the subsystems which have a high frequency limit lower than the frequency of study are modelled through FEM, while the rest of subsystems are modelled by SEA. As the frequency of study increases, more subsystems must be modelled through SEA and less through FEM. The opposite happens as the frequency of study decreases. As a consequence, as set of hybrid models are needed to appropriately reproduce the system behaviour in the mid-frequency range. As a consequence, a denition of the mid-frequency range is proposed, based on the subsystems behaviour rather than in the full systems one. The mid-frequency range 8.

(45) 1.3. THE RESEARCH GOAL: OBJECTIVES AND JUSTIFICATION. Figure 1.1: Applicability ranges for the proposed mid-frequency methods: The. Subsystems based. procedure (SHFL) and The Hybrid Sub-structuring method based on Component Synthesis (HS-CMS).. High Frequency Limit Mode. limits are as follows: (i) The minimum frequency corresponds with the lowest high frequency limit of all the subsystems by which the whole system is composed. Under this frequency, the procedure leads to a pure FEM model. (ii) The maximum frequency corresponds with the highest high frequency limit of all the subsystems by which the full system is composed. Above this frequency, the procedure leads to a pure SEA model. Throughout this procedure, a signicant improvement in the response continuity is obtained, without increasing the associated computational cost. The procedure is tested in space structures and veried with experimental results. One advantage of the procedure is that the necessary numerical methods for its implementation, as the hybrid junctions between FEM and SEA subsystems [Shorter and Langley, 2005a], are currently available in commercial software, so that the procedure could be already used for industrial applications. As the gure shows, the method is recommended for heterogeneous systems, being the obtained responses more continuous as the heterogeneity of the system rises. The procedure is dened for the mid-frequency range, tending to a pure FEM for low frequencies and a SEA model for high frequencies. 9.

(46) Chapter. 1.. 2- The Hybrid Sub-structuring method based on Component Mode Synthesis (HS-CMS) is founded on the hybrid method developed by Langley and Bremner [1999]. The method uses some aspects of this existent hybrid methodology, imposing some conditions and modications that enable its direct application for vibro-acoustic problems. In particular, the classication scheme of Ref. [Langley and Bremner, 1999] is adopted, proposing to classify the system response into a global and a local part, modelling each one of them by deterministic and statistic approaches, respectively. The proposed method is based on classifying the sub-structures modes into global and local sets, by means of a Component Mode Synthesis approach. The Craig-Bampton transformation is used, in order to express the system modal base into the interior modes (modal shapes with xed boundary) of each sub-structure and into the constraint modes, related with the motion of the sub-structures boundaries. The sub-structures modal bases obtained are partitioned into a global and a local set of modes. Global modes are related with the sub-structure long wavelength motion, while local modes are related with the short one. Constraint modes are always included into the global set, since they are related with the boundary motion, and sub-structure interior modes are global or local depending on their wavelengths (or associated eigenfrequencies). In particular, an interior mode will be global if its associated wavelength (or associated eigen-frequency) is higher than a limit wavelength dened by the sub-structures high frequency limit. Thus, the concept of sub-structure high frequency limit is used again, marking out the frequency frontier between global and local behaviour. The equations of motion in the Craig-Bampton modal space are derived, resulting classied into a global and a local set of equations. The global response is calculated from the equations, where corrections, coming from the presence of local modes, are introduced through additional terms in the dynamic stiness matrix of the global problem and through an additional force vector. The local equations of motion are solved using SEA, where interactions between global and local modes are taken into account through the introduction of an additional power driven into the SEA model. The sub-structures vibro-acoustic responses are obtained by juxtaposition of a global and a local response, where the interactions between global and local types of modes are taken into account. The HS-CMS method is implemented using Matlab code [Matlab, 2012] and tested in simple structures to stablish the basis of the method. The method is also tested in real structures made of composite materials, showing a good agreement with experimental results. The method has been proved to be valid for complex structures, whatever the heterogeneity of the system is.. 10.