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Statistical Description of Submarine Hull Vibration:

Statistical Description of Submarine Hull Vibration: 2 3 M. Blakemore', J. Woodhouse and D.J.W. Hardie , . 'Topexpress Ltd.• 511 Coldhams Lane, Cherry Hinton, Cambridge CB1 3iS. UK. 2Engineering Department, University of Cambridge, Trumpington St. Cambridge CB2 9PZ, UK. 3Defence Evaluation and Research Agency, Winfrith Technology Centre, Dorchester, Dorset. DT2 8Xi, UK Received 6 February, 1999 INTRODUCTION Vibrational noise in a submarine hull close to any source comprises a whole gamut of modes. Some of these modes spread and travel along the hull. Small amplitude modes that readily propagate ultimately dominate the vibrational field. sufficiently far away. The self-noise problem faced when fitting hull­ mounted sonar arrays is not only one of achieving some general low noise levels over a band but also in reducing particular coherent components that may be confusing to the signal processing. No complete theory of the structural acoustics of pressure hulls exists. Any theoretical approach requires a degree of simplification. retaining the essential physics for the frequencies of interest. Here we review the basic ideas of theoretical models developed to predict the vibrational character of submarine hulls. Particular attention is paid to considering the self-noise field around fluid loaded ribbed cylinders. This provides the theoretical framework in which to study submarine pressure hull dynamics. What is seen is that the regular periodic nature of the ribbed cylinders produces vibration Pass and Stop bands with regards to frequency. Waves within a Pass band continue propagating through the hull with little attenuation and can become highly coherent due to regular frame spacing in the hull. The wave modes consist of in-plane (e.g. compressional waves), out-of-plane motions (i.e. flexural waves). These are modified to a degree by dynamic interaction with the rib motion. There are various types of structural vibration model of submerged hulls available. The fundamental group are deterministic based on idealised f1uid­ loaded cylindrical structures. The solutions are expressed in closed form, usually via some series expansion. These models can also be used to consider how certain crucial dynamical aspects (e.g. dispersion relations) behave under change in parameters such as shell radius or thickness. This aids our basic qualitative and quantitative understanding of design issues [I]. To complement these closed-form models there are numerical treatments based on finite element methods which provide quantitative information on specific design configurations. Recently we have developed a novel statistical treatment based on the deterministic theories which is very useful for quick assessment of behaviour over a wide frequency band. Structural detail is incorporated in only an implicit fashion since it follows on from the deterministic treatment of a hull with its shell dimensions and material properties. The method takes account of the periodicity inherent in the deterministic models with internal ribs. Further the Journal of Low Frequency Noise, 167 vibration WIt! Actil'/' Control Vlil. 18 No. -4 1C)1.)C) STATISTICAL DESCRIPTION OF SUBMARINE HULL VIBRATION treatment can account for the non-reverberant nature of the hull shell flexural vibration which loses energy by radiation into the surrounding fluid. The latter idea stems from the treatment of power flowing along quasi-one dimensional structures such as long cylindrical hulls. The method adopted is termed Statistical Power Flow [2, 3]. DETERMINISTIC MODELS The deterministic models which have been used so successfully so far, are mostly based on a long cylindrical elastic shell. Surrounding the shell is an acoustic fluid of infinite extent. The shell is stiffened by T-shaped ribs. The model must account for the dynamic effect of ribs which may be identical and equally spaced or be of different design and arranged in an arbitrary fashion. It is important that the theory adopted must be sufficiently sophisticated to describe the rib dynamics in a comprehensive way. The motion of the ribs can be quite complicated, especially at high frequencies, and the characteristics of the Pass bands are dependent on the interaction between the shell modes and the ribs. The cylindrical nature of the structure allows for the ready decomposition of the vibrational field into a set of angular order modes. The fluid-loading is formally incorporated by a variational approach using an additional complex kinetic term which encompasses both the mass loading and radiation damping due to the presence of the fluid. To solve for the fluid-loaded modes a non­ linear eigen-solution is necessary. This gives each mode a complex frequency, group velocity. radiation loss factor shape etc. A useful tool for assimilating the vibrational characteristics of a particular hull design is the frequency vs angular order dispersion diagram. This can be readily derived from experimental data or from a theoretical model. Comparing predictions with measurements for a typical configuration, excellent agreement is seen. A major benefit, provided by these ribbed cylinder models is in the capacity to examine that character of the various modes that are present. For flexural wave modes three distinct groups are apparent, denoted "L", "R" and "C", corresponding to modes on the left, right and central regions of the dispersion diagram (figure 1). All these modes have the potential to cause problems for a hull-mounted sonar array and are of different character [11. Investigations reveal that the "C" modes, in particular, readily travel past rib­ like structures. STATISTICAL POWER FLOW APPROACH Assimilating the results from a typical study performed using the deterministic fluid-loaded ribbed cylinder models is an involved task. Very many cases each involving a large number of modes are considered. It has long been known that statistical treatments of vibrating structures can be formulated in various guises. Of the most successful is that of Statistical Energy Analysis (SEA). Many refinements of SEA have been proposed but, until now none have been able to consider heavily fluid-loaded structures such as submarine hulls in any consistent way. SEA begins by decomposing the structure into a number of discrete subsystem. Within each subsystem the response is assumed to be homogeneous in a temporal and spatial sense. A subsystem usually corresponds to a recognisable structural components (e.g. a cover plate or a main pipe). The subsystems are examined in tum and described by simple expressions which represent the modal distribution and energies, the dispersion relations, for a general system supporting wave motion. The subsystems are then coupled via 168 STATISTICAL DESCRIPTION OF SUBMARINE HULL VIBRATION simple linear equations which describe the rate of energy exchange and balance between subsystems. From these equations a spatial average of response levels within each individual subsystem can be derived. An accurate representation of the inter-subsystem couplings is crucial. However their derivation can be difficult and recourse to experimental methods is common. Coupling factors can be estimated for simple idealised structures (e.g. plates etc.) and simple theoretical expressions can be derived. Their suitability for a given problem, however, is not always certain. Ribbed Cylindrical ShellSection. "R" m ode Di spersion Dia gram Wavenumber Figure 1. Schematic of Wave Modes around a Pressure Hull Applying SEA in a simplistic manner to describe the response of a submerged pressure hull is liable to run into difficulties. SEA deals with energy or the intensity of a response. Consequently all information regarding the relative phase between modes confined within a subsystem is averaged over or 169 STATISTICAL DESCRIPTION OF SUBMARINE HULL VIBRATION lost. There are no preferential modes within a subsystem, save for those selected by a particular type of forcing mechanism. Ignoring all phase information, a key assumption in standard SEA, removes the Pass and Stop band nature of the spatially periodic pressure hull from the SEA model. The fundamental issue governing the transmission of vibration along the hull, as seen in other deterministic theoretical models and in measured data is lost. Further, the SEA assumption of homogeneity of levels throughout the subsystem is incorrect. Radiation damping due to the presence of the surrounding fluid, reduces the vibration levels as waves progress along the hull. Finally, the coupling due to the acoustic fluid is not local. The fluid accounts for the influence of one region of the shell on others (and vice-versa). This phenomenon is not readily described by standard SEA. SEA still has attractions. It can cope with highly modally dense systems, which is the case for submarine hulls at frequencies of interest to flank array sonars. We have proposed a novel method based on SEA concepts to describe submarine pressure hull dynamics [2]. The first stage is to decompose the whole hull into convenient regions or "chunks". In this we are following broadly the path of a standard SEA procedure. Submarine chunks are sections of hull that are in some constructional sense separate, such as the adjacent sections separated by a bulkhead or perhaps the main machinery compartment (figure 2). The next and novel approach is to choose a set of subsystems that are germane to the pressure hull dynamics. Here we choose the mode families briefly mentioned above. Four families are considered, the in-plane modes which are essentially shear and compressional waves within the steel shell itself and the three flexural wave modes; "L", "C" and "R". Regarding each mode families as an individual subsystem has tremendous advantage. They are only very weakly coupled to one another so that the dependence on coupling factors is reduced. The dispersion relations for these modes are derived using the formulae from original deterministic theory of fluid-loaded ribbed cylinders. This embodies the main advantage of the approach which lies in the inherent periodicity of the dispersion relations. Pass band and Stop bands are retained. The derived modal behaviour includes the influence of fluid, albeit in a restricted sense, over a limited region of shell, the chunk. The influence of fluid-loading from chunk to chunk is performed only locally at their interface. Thus this theory does not have the full long-range fluid coupling that is present in the deterministic model and in this sense it is lacking. We are considering further refinements to address this omission. However, in a broad context, this deficiency should not affect results unduly. There is the issue of homogeneity of levels within a chunk. By including a further refinement to the method using notions from Power Flow theory dissipation within a chunk can be imposed. This is introduced by a term which imposes a length scale for the decay of energy as we progress axially along the hull. This length can be predicted from an involved set of calculations or can be readily gleaned from measured data. Our new method, of which a more detailed exposition of the theory is in [3] results in a simple set of equations for the amplitudes of vibrational "energy waves" propagating from the left and right for each modal subsystem. These exhibit exponential decay along the length of the hull chunks in the manner of heat flow along a conducting rod. Finally, the influence of internal structural complexity, in so far as it influences the vibration in the cylindrical shell, can be included in the formulation. This is done by introducing a "mixing term" which accounts for an average coupling between subsystems. The magnitude of this term can be estimated again from measurement, ideally during hull 170 STATISTICAL DESCRIPTION OF SUBMARINE HULL VIBRATION construction. Using this technique it is possible to estimate the vibrational response to a broadband excitation within parts of a submarine hull. Input Power Chunk I Chunk2 I' Decay to Left 00000 0·0 0' ..... ..... ........... ... 111·········· ..111·············· ~~aytORight ... Power Flow Schematic of Statistical Power Flow Model of a Pressure Hull Figure 2 DISCUSSION Deterministic models based on a fluid-loaded ribbed cylinder have been developed. Great success has been achieved in using these models to predict the response of model scale submerged hulls and in full scale hulls under construction. These models have enabled us to achieve a detailed understanding of the various mode families present in the dynamics of submarine pressure hulls. It is important to note that these deterministic models can give dynamic response results that exhibit greater sensitivity to the initial conditions and design configuration than is seen in reality. 171 STATISTICAL DESCRIPTION OF SUBMARINE HULL VIBRATION A statistical approach for modelling the transmission of vibration along a realistic submarine pressure hull has been developed from a synthesis of various ideas. These notions come from a SEA and Power Flow interpretation of the original deterministic models of fluid-loaded ribbed cylinders. Standard SEA is not adequate to deal with the special geometry and fluid-loading conditions seen in pressure hulls. Our method needs further testing and refinement but already shows promise as a useful aid to submarine hull designers. A synthesis of a deterministic model of the array site and baffle with a refined version of the statistical model of the rest of the hull, including the excitation regions, seems to be a promising future development. This should provide the designer with a predictive tool that is detailed enough to address specific design questions concerning the array site and its vicinity. The concern over spatially coherent noise components can still be addressed. However, the anomalously strong spatial phase dependency of the response field is mitigated by the "smeared" statistical description of the further regions of the hull which includes irregularities and internal structures that deviate from the perfect cylindrical symmetry. This would form the basis of a model which offers the prospect for a "whole vessel noise prediction", encompassing the local detailed Pass and Stop band dynamic behaviour with a global approach for the surrounding structure, including all its actual inherent complexity. This could be described by a small number of "mixing terms" which account for the diffuse angular order couplings (due to decks etc.) and could be derived experimentally Juring hull constructions ("in build"). REFERENCES I D.J. W. Hardie, M. Blakemore and J. Power, "The Influence of Hull Vibration on Array Performance", Proc. loA, Julv 1996, Vol.J8(5), /01-110. 2 M. Blakemore, J. Woodhouse and D.J.W. Hardie, "Statistical Power Flow Method for Submarine Self-Noise", Proc. loA. December 1996, Vol.18( /0) 85-94. 3 M. Blakemore, J. Woodhouse and D.J.W. Hardie, "Statistical Power Flow Analysis of an Imperfect Ribbed Cylinder", Journal (?l Sound and Vibration (submitted for publication).; © British Crown Copyright I 999/DERA. Published with the permission of the Controller HMSO. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png "Journal of Low Frequency Noise, Vibration and Active Control" SAGE

Statistical Description of Submarine Hull Vibration:

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Abstract

2 3 M. Blakemore', J. Woodhouse and D.J.W. Hardie , . 'Topexpress Ltd.• 511 Coldhams Lane, Cherry Hinton, Cambridge CB1 3iS. UK. 2Engineering Department, University of Cambridge, Trumpington St. Cambridge CB2 9PZ, UK. 3Defence Evaluation and Research Agency, Winfrith Technology Centre, Dorchester, Dorset. DT2 8Xi, UK Received 6 February, 1999 INTRODUCTION Vibrational noise in a submarine hull close to any source comprises a whole gamut of modes. Some of these modes spread and travel along the hull. Small amplitude modes that readily propagate ultimately dominate the vibrational field. sufficiently far away. The self-noise problem faced when fitting hull­ mounted sonar arrays is not only one of achieving some general low noise levels over a band but also in reducing particular coherent components that may be confusing to the signal processing. No complete theory of the structural acoustics of pressure hulls exists. Any theoretical approach requires a degree of simplification. retaining the essential physics for the frequencies of interest. Here we review the basic ideas of theoretical models developed to predict the vibrational character of submarine hulls. Particular attention is paid to considering the self-noise field around fluid loaded ribbed cylinders. This provides the theoretical framework in which to study submarine pressure hull dynamics. What is seen is that the regular periodic nature of the ribbed cylinders produces vibration Pass and Stop bands with regards to frequency. Waves within a Pass band continue propagating through the hull with little attenuation and can become highly coherent due to regular frame spacing in the hull. The wave modes consist of in-plane (e.g. compressional waves), out-of-plane motions (i.e. flexural waves). These are modified to a degree by dynamic interaction with the rib motion. There are various types of structural vibration model of submerged hulls available. The fundamental group are deterministic based on idealised f1uid­ loaded cylindrical structures. The solutions are expressed in closed form, usually via some series expansion. These models can also be used to consider how certain crucial dynamical aspects (e.g. dispersion relations) behave under change in parameters such as shell radius or thickness. This aids our basic qualitative and quantitative understanding of design issues [I]. To complement these closed-form models there are numerical treatments based on finite element methods which provide quantitative information on specific design configurations. Recently we have developed a novel statistical treatment based on the deterministic theories which is very useful for quick assessment of behaviour over a wide frequency band. Structural detail is incorporated in only an implicit fashion since it follows on from the deterministic treatment of a hull with its shell dimensions and material properties. The method takes account of the periodicity inherent in the deterministic models with internal ribs. Further the Journal of Low Frequency Noise, 167 vibration WIt! Actil'/' Control Vlil. 18 No. -4 1C)1.)C) STATISTICAL DESCRIPTION OF SUBMARINE HULL VIBRATION treatment can account for the non-reverberant nature of the hull shell flexural vibration which loses energy by radiation into the surrounding fluid. The latter idea stems from the treatment of power flowing along quasi-one dimensional structures such as long cylindrical hulls. The method adopted is termed Statistical Power Flow [2, 3]. DETERMINISTIC MODELS The deterministic models which have been used so successfully so far, are mostly based on a long cylindrical elastic shell. Surrounding the shell is an acoustic fluid of infinite extent. The shell is stiffened by T-shaped ribs. The model must account for the dynamic effect of ribs which may be identical and equally spaced or be of different design and arranged in an arbitrary fashion. It is important that the theory adopted must be sufficiently sophisticated to describe the rib dynamics in a comprehensive way. The motion of the ribs can be quite complicated, especially at high frequencies, and the characteristics of the Pass bands are dependent on the interaction between the shell modes and the ribs. The cylindrical nature of the structure allows for the ready decomposition of the vibrational field into a set of angular order modes. The fluid-loading is formally incorporated by a variational approach using an additional complex kinetic term which encompasses both the mass loading and radiation damping due to the presence of the fluid. To solve for the fluid-loaded modes a non­ linear eigen-solution is necessary. This gives each mode a complex frequency, group velocity. radiation loss factor shape etc. A useful tool for assimilating the vibrational characteristics of a particular hull design is the frequency vs angular order dispersion diagram. This can be readily derived from experimental data or from a theoretical model. Comparing predictions with measurements for a typical configuration, excellent agreement is seen. A major benefit, provided by these ribbed cylinder models is in the capacity to examine that character of the various modes that are present. For flexural wave modes three distinct groups are apparent, denoted "L", "R" and "C", corresponding to modes on the left, right and central regions of the dispersion diagram (figure 1). All these modes have the potential to cause problems for a hull-mounted sonar array and are of different character [11. Investigations reveal that the "C" modes, in particular, readily travel past rib­ like structures. STATISTICAL POWER FLOW APPROACH Assimilating the results from a typical study performed using the deterministic fluid-loaded ribbed cylinder models is an involved task. Very many cases each involving a large number of modes are considered. It has long been known that statistical treatments of vibrating structures can be formulated in various guises. Of the most successful is that of Statistical Energy Analysis (SEA). Many refinements of SEA have been proposed but, until now none have been able to consider heavily fluid-loaded structures such as submarine hulls in any consistent way. SEA begins by decomposing the structure into a number of discrete subsystem. Within each subsystem the response is assumed to be homogeneous in a temporal and spatial sense. A subsystem usually corresponds to a recognisable structural components (e.g. a cover plate or a main pipe). The subsystems are examined in tum and described by simple expressions which represent the modal distribution and energies, the dispersion relations, for a general system supporting wave motion. The subsystems are then coupled via 168 STATISTICAL DESCRIPTION OF SUBMARINE HULL VIBRATION simple linear equations which describe the rate of energy exchange and balance between subsystems. From these equations a spatial average of response levels within each individual subsystem can be derived. An accurate representation of the inter-subsystem couplings is crucial. However their derivation can be difficult and recourse to experimental methods is common. Coupling factors can be estimated for simple idealised structures (e.g. plates etc.) and simple theoretical expressions can be derived. Their suitability for a given problem, however, is not always certain. Ribbed Cylindrical ShellSection. "R" m ode Di spersion Dia gram Wavenumber Figure 1. Schematic of Wave Modes around a Pressure Hull Applying SEA in a simplistic manner to describe the response of a submerged pressure hull is liable to run into difficulties. SEA deals with energy or the intensity of a response. Consequently all information regarding the relative phase between modes confined within a subsystem is averaged over or 169 STATISTICAL DESCRIPTION OF SUBMARINE HULL VIBRATION lost. There are no preferential modes within a subsystem, save for those selected by a particular type of forcing mechanism. Ignoring all phase information, a key assumption in standard SEA, removes the Pass and Stop band nature of the spatially periodic pressure hull from the SEA model. The fundamental issue governing the transmission of vibration along the hull, as seen in other deterministic theoretical models and in measured data is lost. Further, the SEA assumption of homogeneity of levels throughout the subsystem is incorrect. Radiation damping due to the presence of the surrounding fluid, reduces the vibration levels as waves progress along the hull. Finally, the coupling due to the acoustic fluid is not local. The fluid accounts for the influence of one region of the shell on others (and vice-versa). This phenomenon is not readily described by standard SEA. SEA still has attractions. It can cope with highly modally dense systems, which is the case for submarine hulls at frequencies of interest to flank array sonars. We have proposed a novel method based on SEA concepts to describe submarine pressure hull dynamics [2]. The first stage is to decompose the whole hull into convenient regions or "chunks". In this we are following broadly the path of a standard SEA procedure. Submarine chunks are sections of hull that are in some constructional sense separate, such as the adjacent sections separated by a bulkhead or perhaps the main machinery compartment (figure 2). The next and novel approach is to choose a set of subsystems that are germane to the pressure hull dynamics. Here we choose the mode families briefly mentioned above. Four families are considered, the in-plane modes which are essentially shear and compressional waves within the steel shell itself and the three flexural wave modes; "L", "C" and "R". Regarding each mode families as an individual subsystem has tremendous advantage. They are only very weakly coupled to one another so that the dependence on coupling factors is reduced. The dispersion relations for these modes are derived using the formulae from original deterministic theory of fluid-loaded ribbed cylinders. This embodies the main advantage of the approach which lies in the inherent periodicity of the dispersion relations. Pass band and Stop bands are retained. The derived modal behaviour includes the influence of fluid, albeit in a restricted sense, over a limited region of shell, the chunk. The influence of fluid-loading from chunk to chunk is performed only locally at their interface. Thus this theory does not have the full long-range fluid coupling that is present in the deterministic model and in this sense it is lacking. We are considering further refinements to address this omission. However, in a broad context, this deficiency should not affect results unduly. There is the issue of homogeneity of levels within a chunk. By including a further refinement to the method using notions from Power Flow theory dissipation within a chunk can be imposed. This is introduced by a term which imposes a length scale for the decay of energy as we progress axially along the hull. This length can be predicted from an involved set of calculations or can be readily gleaned from measured data. Our new method, of which a more detailed exposition of the theory is in [3] results in a simple set of equations for the amplitudes of vibrational "energy waves" propagating from the left and right for each modal subsystem. These exhibit exponential decay along the length of the hull chunks in the manner of heat flow along a conducting rod. Finally, the influence of internal structural complexity, in so far as it influences the vibration in the cylindrical shell, can be included in the formulation. This is done by introducing a "mixing term" which accounts for an average coupling between subsystems. The magnitude of this term can be estimated again from measurement, ideally during hull 170 STATISTICAL DESCRIPTION OF SUBMARINE HULL VIBRATION construction. Using this technique it is possible to estimate the vibrational response to a broadband excitation within parts of a submarine hull. Input Power Chunk I Chunk2 I' Decay to Left 00000 0·0 0' ..... ..... ........... ... 111·········· ..111·············· ~~aytORight ... Power Flow Schematic of Statistical Power Flow Model of a Pressure Hull Figure 2 DISCUSSION Deterministic models based on a fluid-loaded ribbed cylinder have been developed. Great success has been achieved in using these models to predict the response of model scale submerged hulls and in full scale hulls under construction. These models have enabled us to achieve a detailed understanding of the various mode families present in the dynamics of submarine pressure hulls. It is important to note that these deterministic models can give dynamic response results that exhibit greater sensitivity to the initial conditions and design configuration than is seen in reality. 171 STATISTICAL DESCRIPTION OF SUBMARINE HULL VIBRATION A statistical approach for modelling the transmission of vibration along a realistic submarine pressure hull has been developed from a synthesis of various ideas. These notions come from a SEA and Power Flow interpretation of the original deterministic models of fluid-loaded ribbed cylinders. Standard SEA is not adequate to deal with the special geometry and fluid-loading conditions seen in pressure hulls. Our method needs further testing and refinement but already shows promise as a useful aid to submarine hull designers. A synthesis of a deterministic model of the array site and baffle with a refined version of the statistical model of the rest of the hull, including the excitation regions, seems to be a promising future development. This should provide the designer with a predictive tool that is detailed enough to address specific design questions concerning the array site and its vicinity. The concern over spatially coherent noise components can still be addressed. However, the anomalously strong spatial phase dependency of the response field is mitigated by the "smeared" statistical description of the further regions of the hull which includes irregularities and internal structures that deviate from the perfect cylindrical symmetry. This would form the basis of a model which offers the prospect for a "whole vessel noise prediction", encompassing the local detailed Pass and Stop band dynamic behaviour with a global approach for the surrounding structure, including all its actual inherent complexity. This could be described by a small number of "mixing terms" which account for the diffuse angular order couplings (due to decks etc.) and could be derived experimentally Juring hull constructions ("in build"). REFERENCES I D.J. W. Hardie, M. Blakemore and J. Power, "The Influence of Hull Vibration on Array Performance", Proc. loA, Julv 1996, Vol.J8(5), /01-110. 2 M. Blakemore, J. Woodhouse and D.J.W. Hardie, "Statistical Power Flow Method for Submarine Self-Noise", Proc. loA. December 1996, Vol.18( /0) 85-94. 3 M. Blakemore, J. Woodhouse and D.J.W. Hardie, "Statistical Power Flow Analysis of an Imperfect Ribbed Cylinder", Journal (?l Sound and Vibration (submitted for publication).; © British Crown Copyright I 999/DERA. Published with the permission of the Controller HMSO.

Journal

"Journal of Low Frequency Noise, Vibration and Active Control"SAGE

Published: Aug 1, 2016

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