Passive and Active Control of the Sound Radiated by a Submerged Vessel due to Propeller Forces

Transcription

1 Passive and Active Control of the Sound Radiated by a Submerged Vessel due to Propeller Forces by Sascha Merz A thesis presented to the University of New South Wales in fulfilment of the thesis for the degree of doctor of philosophy in Mechanical Engineering Sydney, New South Wales, Australia, 2010

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3 Originality Statement I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project s design and conception or in style, presentation and linguistic expression is acknowledged. Signature... Date... iii

4 Copyright Statement I hereby grant to the University of New South Wales or its agents the right to archive and to make available my thesis or dissertation in whole or part in the University libraries in all forms of media, now or hereafter known, subject to the provisions of the Copyright Act I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertation. I also authorise University Microfilms to use the abstract of my thesis in Dissertations Abstract International (this is applicable to doctoral theses only). I have either used no substantial portions of copyright material in my thesis or I have obtained permission to use copyright material; where permission has not been granted I have applied/will apply for a partial restriction of the digital copy of my thesis or dissertation. Signature... Date... Authenticity Statement I certify that the library deposit digital copy is a direct equivalent of the final officially approved version of my thesis. No emendation of content has occurred and if there are any minor variations in formatting, they are the result of the conversion to digital format. Signature... Date... iv

5 Abstract An important cause of sound radiation from a submarine in the low frequency range is fluctuating forces at the propeller. The forces arise from the operation of the propeller in a non-uniform wake and are transmitted from the propeller to the submarine hull through both the shaft and the fluid. The sound radiated from the submarine is due to the combination of sound radiation caused by hull and propeller vibrations as well as dipole sound radiation from the propeller. To improve the stealth of a submarine, the radiated sound power can be reduced using passive and active noise and vibration control mechanisms. In this thesis, dynamic models of a submarine hull and propeller/shafting system are developed. To reduce the radiated sound fields, passive control is introduced using a hydraulic vibration attenuation device known as a resonance changer, which is implemented in the propeller/shafting system. Active control techniques are implemented using either tuned actuators or a control moment. To initially obtain the structural and acoustic responses of a submarine hull, an analytical model and fully coupled finite element/boundary element model are developed, for a simplified physical model of the hull. The submerged body under axial excitation is modelled as a ring-stiffened cylindrical shell with finite rigid end closures and separated by bulkheads into a number of compartments. Lumped masses are located at each end to maintain a condition of neutral buoyancy. In the low frequency range, only the axial hull modes in accordion motion and axial vibration of the propeller/shafting system are examined, which gives rise to an axisymmetric case. The frequency responses, axial and radial responses of the cylinder and the radiated sound pressure from both the analytical and computational models are compared. A dynamic model of the propeller/shafting system developed computationally and including the resonance changer is then coupled to the FE/BE model of the hull which is subject to both structural excitation from the propeller/shafting system and acoustic excitation from the propeller. The influence of tailcone properties on the structural and acoustic responses of the submarine are investigated. v

6 Passive control is implemented to attenuate the hull responses using a resonance changer. It is demonstrated that the performance of the resonance changer is negatively influenced at frequencies above the fundamental axial resonance of the hull by the effect of forces transmitted through the fluid. When the resonance changer is optimised to minimise excitation of the hull via the propeller shaft, the increased axial movement of the propeller results in an additional sound field that excites the submarine hull in a similar manner to the fluid forces that arise directly from the hydrodynamic mechanism. Cost functions that represent the submarine radiated sound power are developed, where the virtual stiffness, damping and mass of the resonance changer were chosen as design parameters. The minima of the cost functions are found by applying gradient based optimisation techniques. The adjoint operator is employed to calculate the sensitivity of the cost function to the design parameters. The influence of sound radiation due to propeller vibration on the optimisation of the resonance changer is investigated. The influence of the reduction in amplitude for higher harmonics of the blade passing frequency on the control performance is also examined. Different active control strategies are investigated, in which active control is applied to the propeller/shafting system and/or to the submarine hull. Active vibration control and discrete structural acoustic sensing based on the far field radiated sound power were considered in the development of the cost functions. In addition, the performance of a of a combined passive and active control system is investigated. Significant reduction of radiated sound power is achieved when an active control system using tuned actuators is combined with a resonance changer. The structural responses of a model scale, free flooded submarine tailcone are investigated computationally and experimentally. The tailcone is represented by a thin-walled conical shell attached to a stiff plate. The stiff plate represents the pressure hull end plate of the submarine and is subject to axial excitation correlated to propeller forces. Good agreement between the computational and experimental results are found for the tailcone in air as well as for the submerged tailcone. vi

7 Acknowledgements I would like to express my deep gratitude to my supervisor Dr Nicole Kessissoglou for giving me the opportunity to realise this important part of my life. The amount of heart and soul that she put in support can not be expressed through words and her patience seems unlimited. I am deeply indebted to thank my co-supervisor Dr Roger Kinns for his fantastic on-going assistance. Much of the energy that has been required to realise this work was drawn from his motivation and indispensable ideas. I would also like to express my special thanks to Dr Steffen Marburg. His invaluable guidance contributed to a major part of the work that is presented in this thesis. Special thanks have to be expressed to the people who had to share the office with me. The social and technical communication were essential for getting fresh impetus to keep up with new ideas. I need to express deep thankfulness to my family and friends for enduring all the highs and lows with me. I wish to acknowledge the people at the DSTO Maritime Platforms Division for making this research project available and for providing technical and scientific assistance. I also wish to acknowledge the UNSW CFD research group for providing essential technical equipment. Lastly, it is with my pleasure to thank the people at the Technische Universität Dresden for their support during my academic exchange. vii

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9 Contents List of Figures List of Tables List of Symbols List of Abbreviations xvii xviii xxxii xxxiii 1 Introduction Topic of Research Thesis Overview Literature Review Hull excitation by propeller forces Analytical vibro-acoustic modelling of a submarine hull Dynamic modelling of the propeller/shafting system Coupled finite element/boundary element methods Optimisation Active noise and vibration control Contribution to Research Structural/Acoustic Modelling of a Submarine Introduction Simplified Physical Models of a Submarine Analytical Model of a Submarine Hull Motion of the cylindrical hull Motion of the bulkheads Dynamic response of the submarine hull ix

10 Contents Radiated sound field Computational Model of a Submarine Sound field radiated by the propeller Coupled FE/BE modelling Sound power far field approximation Conditioning of the BE matrices Structural and Acoustic Responses of a Submarine Introduction Analytical and Computational Results for the Hull Axial responses at the cylinder ends Axial and radial displacements Radiation directivity patterns Radiation modes of the submarine hull Influence of Tailcone and Propeller/Shafting System Properties Dynamic behaviour of the propeller/shafting system Influence of the tailcone properties on the structural and acoustic responses of the submarine hull Structural and acoustic responses for the coupled model under structural and acoustic excitation Conclusions Optimum Passive and Active Control of a Submarine Introduction Optimisation and Sensitivity Analysis of Resonance Changer Parameters Sensitivity analysis Optimisation Results Active Control of a Submarine x

11 Contents Active control systems Computation of actuator forces Results Conclusions Conclusions and Recommendations for Future Work Summary of Conclusions Recommendations for Future Work Bibliography 124 A Structural Responses of a Tailcone 138 A.1 Introduction A.2 Computational Analysis A.3 Description of the Experimental Rig and Equipment A.4 Experimental Procedure A.5 Computational and Experimental Results A.6 Conclusions B Structural Finite Element Modelling 153 B.1 Axisymmetric Kirchhoff-Love Shell Elements B.2 Axisymmetric Reissner-Mindlin Shell Elements B.3 One-Dimensional Finite Elements C Acoustic Boundary Element Modelling 168 C.1 Symmetry and Anti-Symmetry at a Plane of Infinite Extent C.2 Special BE Formulation for Axisymmetric Problems D Gauss-Legendre Integration 180 E Distributed Adaptive Integration on a Grid 182 Publications arising from this Thesis 186 xi

12 List of Figures 1.1 Wake of a torpedo Stern of a torpedo Non-cavitating noise of a submarine propeller Physical model of a submarine Propeller/shafting system Schematic diagram of the propeller/shafting system [1] Resonance changer Geometrical parameters for a thin-walled cylinder of length l c Cylinder-plate junction of the submarine model, where i = 1, Cylinder with semi-infinite baffles Rigid disc in a closed-back baffle Dipole directivity pattern Cylindrical shell with interior ring stiffeners FE meshes for the submarine pressure hull BE meshes for the submarine pressure hull Condition numbers for the BE matrices Axial response of the hull at z = Axial response of the hull at z = l h Axial displacement at the first axial resonance Axial displacement at the second axial resonance Axial displacement at the third axial resonance Radial displacement at the first axial resonance Radial displacement at the second axial resonance Radial displacement at the third axial resonance xii

13 List of Figures 3.9 Radiated sound pressure at the first axial mode Radiated sound pressure at the second axial mode Radiated sound pressure at the third axial mode Most efficient radiation modes at the first hull axial resonant frequency Most efficient radiation modes at the second hull axial resonant frequency Most efficient radiation modes at the third hull axial resonant frequency FE model for the propeller/shafting system, where n i denotes FE node i. Nodes n 1 to n 4 have only translatory axial degrees of freedom. Node n 5 has only one virtual translatory degree of freedom which is defined as the difference between the axial displacements of nodes n 3 and n 4. Nodes n 6 to n 9 belong to the foundation, which is only partially shown. The nodes of the foundation have both radial and axial translatory degrees of freedom and one rotational degree of freedom around the circumferential coordinate Real and imaginary parts of the radiation impedance for a rigid disc subject to an axial force at its centre using semi-analytical and numerical models Force transmissibility of the propeller/shafting system Axial propeller mobility amplitude Locations of the excitations to investigate the influence of the tailcone properties on the structural and acoustic responses of the submarine Point mobility of the stern end plate using different representations of the free-flooded tailcone Mobility of the stern end plate due to dipole excitation near the tailcone end, using different representations of the free-flooded tailcone Sound power level due to structural excitation of the stern end plate, using different representations of the free-flooded tailcone xiii

14 List of Figures 3.23 Sound power level due to dipole excitation near the tailcone end, using different representations of the free-flooded tailcone Locations of the excitations to investigate the influence of the PSS on the structural and acoustic responses of the submarine Net force on the water corresponding to the dipole due to propeller vibration Mobility of the stern end plate ignoring acoustic excitation Mobility of the stern end plate including acoustic excitation (dipole due to non-uniform wake and dipole due to propeller vibration) Sound power level due to structural excitation Sound power level, where structural and acoustic excitation of the hull are considered. The dipole due to propeller vibration is not taken into account Sound power level, where structural and acoustic excitation of the hull are considered. The dipole due to the operation of the propeller in a non-uniform wake is not taken into account Sound power level, where structural and acoustic excitation of the hull are considered. The acoustic excitation is due to the dipole caused directly by the operation of the propeller in a non-uniform wake and due to the dipole caused by propeller vibration Weighting functions for the force used in the three cost functions Sound power level with no RC, with an RC tuned using Goodwin s method [2] and using the optimum RC parameters for the three cost functions Cost function (a) Radiated sound power at the maximum and minimum values of cost function (a) Sensitivity of cost function (a) with respect to the virtual damping of the RC xiv

15 List of Figures 4.6 Sensitivity of cost function (a) with respect to RC virtual stiffness Cost function (b) Radiated sound power for the minimum and maximum values of cost function (b) Sensitivity of cost function (b) with respect to RC damping Sensitivity of cost function (b) with respect to RC stiffness Cost function (c) Radiated sound power for the minimum and maximum values of cost function (c) Sensitivity of cost function (c) with respect to RC damping Sensitivity of cost function (c) with respect to RC stiffness Inertial actuator Propeller/shafting system with an inertial actuator Active control system using inertial actuators AVC stiffener to generate a control moment Multi-channel control system Active control system using a control moment Control system components Performance of the system using tuned actuators Forces for the tuned actuators using AVC Performance of the system using a control moment Equivalent force to generate the control moment using AVC Performance of the system using tuned actuators and an RC Forces for the tuned actuators when an RC is present using AVC Performance of the system using tuned actuators and AVC Performance of the system using a control moment and AVC Performance of the system using tuned actuators and an RC A.1 Geometry of the cone (dimensions are given in millimetres) A.2 Drive point impedance amplitudes of the cone xv

16 List of Figures A.3 Displacement patterns for the cone in vacuo A.4 Displacement patterns for the cone in water A.5 Experimental set-up A.6 Dewetron portable data acquisition and analysis system A.7 Arrangement of the rig for measurements in water A.8 Excitation of the cone using a modal hammer A.9 Sealed accelerometer A.10 Drive point impedance amplitudes of the cone in air A.11 Drive point impedance amplitudes of the cone in water A.12 In-plane displacement at the first axial resonant frequency A.13 In-plane displacement at the second axial resonant frequency A.14 In-plane displacement at the third axial resonant frequency A.15 In-plane displacement at the fourth axial resonant frequency B.1 Coordinate system and geometry for the axisymmetric FEM formulation B.2 Axisymmetric Kirchhoff-Love shell element B.3 Interpolation functions for the Kirchhoff-Love element B.4 Axisymmetric quadratic Reissner-Mindlin shell element B.5 Interpolation functions for the Reissner-Mindlin element B.6 One-dimensional elements C.1 Exterior Ω fe and interior Ω fi acoustic domains with boundary Γ f and normal vectors pointing towards the exterior domain n e and pointing towards the interior domain n i C.2 Scattering problem C.3 Symmetry and anti-symmetry using an image C.4 Coordinate system and geometry for the axisymmetric BEM formulation C.5 Subdivision of an axisymmetric boundary element into panels C.6 Interpolation functions for a linear discontinuous line element xvi

17 List of Figures E.1 Adaptive numerical integration on a grid E.2 Distributed adaptive numerical integration xvii

18 List of Tables 3.1 Data for the propeller/shafting system Data for the submarine hull Optimisation results for cost function (a) Optimisation results for cost function (b) Optimisation results for cost function (c) A.1 Locations for the measurement D.1 Some Gauss-Legendre quadrature rules xviii

19 List of Symbols Greek symbols α i Coefficient for modified Simpson rule, where i = 1, 2, 3 β β f Γ c Γ f Γ f,l Dimensionless shell thickness parameter Wave number factor for the damping parameter Axisymmetric boundary Boundary of the acoustic domain Boundary of the acoustic domain at element l γ i Parameter for modified Donnell-Mushtari equations, where i = 1, 2, 3, 4 γ (k 1) i Γ int Γ l,j Γ s Γ s,l γ sn δ ij Adjustment parameter for the moving asymptote i at the outer iteration point k 1 Fluid/structure interface Boundary at the element l that corresponds to collocation point j Structural surface Structural surface for an element Shear strain between the directions along and normal to the shell midsurface Kronecker symbol δ (k 1,k 2 ) MMA adjustment parameter for the iteration point (k 1,k 2 ) Γ s ǫ ǫ ǫ 0 ǫ θ ǫ s ρ s Structural surface Strain Strain vector Strictly positive small number Strain in θ-direction Strain in s-direction Structural density xix

20 List of Symbols ρ f ρ f ρ r ρ s η η s θ Θ θ θ ij ϑ θ i ϑ i Density of the fluid Structural density for the foundation Mass density of the resonance changer fluid Structural density for the shaft Intrinsic element coordinate Structural loss factor Rotation matrix Transformation matrix Tangential coordinate Element of the rotation matrix θ Vector of design parameters Tangential coordinate at node i Design parameter i ϑ (k 1) i Design parameter i at outer iteration k 1 ϑ (k 1) ϑ (k 1,k 2 ) θ R κ Λ λ Λ f λ f λ i Optimum solution for the MMA subproblem at the outer iteration Optimum solution for the MMA subproblem of the inner iteration Angle for field point vector Coefficient for plate sound radiation Surface in the acoustic field Wave type parameter for the characteristic equation Field surface Wave length of the fluid Particular wave type parameter for the characteristic equation µ Dynamic viscosity of the resonance changer fluid ξ Ξ θ Ξ s Π Intrinsic element coordinate Moment per unit length around tangential direction normal to the angular coordinate Moment per unit length around angular direction Approximate solution for the radiated sound power xx

21 List of Symbols π Π ζ ζ a ζ m τ σ σ σ θ σ θ σ (k 1) i σ s σ s τ I τ sn ν ν f Φ φ f φ θ ˆφ θ,i χ χ θ χ s ω Ψ ψ ψ (k 1,k 2 ) Ratio between circumference and diameter of a circle Exact solution for the radiated sound power Distance of the collocations from the element centre Damping ratio for an inertial actuator Intrinsic coordinate for panel m Coefficient to limit the control forces Stress Stress vector Force per unit length in θ-direction Normal stress in circumferential direction Moving asymptote for design parameter i Force per unit length in s-direction Normal stress along the s-coordinate Tolerance for integral error estimate Shear stress between the directions along and normal to the shell midsurface Poisson s ratio Poisson s ratio for the foundation Integration matrix Global interpolation functions for the fluid variables Rotation about the tangential coordinate direction Nodal rotation for element node i Test function Rotational strain in θ-direction Rotational strain in s-direction Circular frequency Differential operator matrix Conical angle for a Kirchhoff-Love shell element MMA adjustment parameter for the inner iterations xxi

22 List of Symbols ω r Ω f Ω fc Ω fe Ω fi Ω s Ω s,l Ring frequency Acoustic domain Complementary domain Exterior acoustic domain Interior acoustic domain Structural domain Structural domain for an element Roman symbols 1 g First element node for a linear discontinuous element 1 v First collocation point for a linear discontinuous element 2 g Second element node for a linear discontinuous element 2 v Second collocation point for a linear discontinuous element a A a 0 A 0 A 1 a c a f a p A rs A r A s b B B b 0 b a Lower limit for integral Structural system matrix Lower limit for root integral Cross-sectional area of the resonance changer cylinder Cross-sectional area of the resonance changer pipe Cylinder radius Minor radius of the foundation Radius of the plate Cross sectional area of a ring stiffener Cross-sectional area for rod element Cross-sectional area of the shaft Upper limit for integral Matrix that relates the strain to the nodal displacements Bulk modulus of the resonance changer fluid Upper limit for root integral Susceptance for rigid disc xxii

23 List of Symbols b f B l b l b rs c c a c b c c s c f C l c pl c r C sc c se C sf D d d b d f Major radius of the foundation Matrix that relates the strain to the nodal displacements for the element l Matrix that relates the strain to the nodal displacements for the rod element l Stiffener spacing Coefficient for representation formula Damping coefficient for an inertial actuator Damping of the thrust bearing Coefficient for complementary representation formula Structural velocity of sound Velocity of sound in the fluid Elemental damping matrix Velocity of sound for in-plane waves in a plate Virtual damping of the resonance changer Constant damping matrix Damping coefficient for a spring-damper element Frequency dependent damping matrix Elasticity matrix Vector of disturbance signals Directivity for plate in infinite baffle Directivity for freely suspended disc D i Displacement coefficients for plate motion, where i = 1, 2, 3 D p E E e e 0 e 1 Plate flexural rigidity Complete elliptic integral of the second kind Young s modulus Vector of error signals Error sensor signal for the stern end plate Error sensor signal for the propeller xxiii

24 List of Symbols E f E s F f ˆf f a f cz f cz,0 f cz,lh f cz,i f cz,i+1 f cr f cz,i f cz,i+1 f Γs f h f p f p,i f pr f pr f pr,i f pz f pz,i f se f se,opt g g a G Γf G Γf,prop Young s modulus for the foundation Young s modulus for the shaft Cost function for the subproblem Force amplitude Nodal vector of discrete forces Force for an inertial actuator Longitudinal cylinder force Axial force of the cylinder at the stern end plate Axial force of the cylinder at the bow end plate Axial force for first cylinder at junction i Axial force for second cylinder at junction i Radial cylinder force Radial force for first cylinder at junction i Radial force for second cylinder at junction i Continuous surface traction vector Force acting on the hull Force acting on the propeller Force vectors at discrete points Radial force for plate Vector of primary excitations Radial force for plate at junction i Transverse plate force Axial force for plate at junction i Vector of secondary excitations Optimum vector of secondary excitations Green s function Conductance for rigid disc Single layer potential BEM influence matrix Matrix to take into account propeller vibration in the coupled system xxiv

25 List of Symbols g i,j G pr G se G Ωf H Γf h c h f H (1) i H i H (0) i H (1) i h i,j h p H Ωf I I i Entry for a single layer potential BE matrix Matrix of primary paths Matrix of secondary paths Single layer potential BEM transfer matrix Double layer potential BEM influence matrix Shell thickness Thickness of the foundation Hankel function of the first kind and order i Struve function of order i Incomplete cubic Hermitian interpolation function for element node i Complete cubic Hermitian interpolation function for element node i Entry for a double layer potential BE matrix Plate thickness Double layer potential BEM transfer matrix Integral Unity matrix Index I 0 Modified Bessel function of the first kind and order 0 I s I t J J j j J i K k k k 1 Simpson s rule approximation Trapezium rule approximation Jacobi matrix Cost function Imaginary unit Index Bessel function of the first kind and order i Complete elliptic integral of the first kind Vector for the elliptic formulation of the axisymmetric BEM Modulus for the elliptic integral Iteration index for outer iterations xxv

26 List of Symbols k 2 k a k b k c k f K G K G,1 K G,1,m K G,2 K H K H,1 K H,1,m K H,2 K i K l k pb k pl k r K s k se l L Iteration index for inner iterations Spring coefficient for an inertial actuator Stiffness of the thrust bearing Wave number of the structure Wave number of the fluid First integral over the circumference for the axisymmetric BE formulation Singular part of the first integral over the circumference for the axisymmetric BE formulation Singular part of the first integral over the circumference for the axisymmetric BE formulation for panel m Non-singular part of the first integral over the circumference for the axisymmetric BE formulation Second integral over the circumference for the axisymmetric BE formulation Singular part of the second integral over the circumference for the axisymmetric BE formulation Singular part of the second integral over the circumference for the axisymmetric BE formulation for panel m Non-singular part of the second integral over the circumference for the axisymmetric BE formulation Modified Hankel function of order i Elemental stiffness matrix Plate bending wave number In-plane wave number for a plate Virtual stiffness of the resonance changer Structural stiffness matrix Stiffness coefficient for a spring-damper element Element index Length of the resonance changer pipe xxvi

27 List of Symbols l c L f L fs l h l j L l L s l s l se L sf m M m 0 m lh m a m b M c M c,i M c,i+1 m eq m f M l m me M p m p m pf M p,i Cylinder length Surface density matrix Matrix that relates acoustic normal velocity in the BE collocation points to structural displacements in the FE nodes Pressure hull length Length of pressure hull segment j Length of a line element Domain correlation matrix Length of the shaft Effective length of the shaft Matrix that relates the acoustic pressure at the BE collocation points to the forces at the FE nodes Panel for the axisymmetric BE formulation Control moment Lumped mass for stern side end plate Lumped mass for bow side end plate Moving mass of an interial actuator Dynamic mass of the thrust bearing Moment for cylinder Moment for first cylinder at junction i Moment for second cylinder at junction i Distributed mass Fluid mass loading Elemental mass matrix Mass for a point mass element Moment for plate Propeller mass Propeller fluid-loading Moment for the plate at junction i xxvii

28 List of Symbols m r M s n n n e N f N i n i N i,g N i,v N j n l n m N s N s,l N Γs,p N Γs,v N v n v N v,l n v,n n v,r n v,s n v,z Virtual mass of the resonance changer Structural mass matrix Coordinate normal to shell direction Outward normal vector Normal vector pointing towards the exterior domain Global matrix of fluid interpolation functions Interpolation function for node i Normal vector pointing towards the interior domain Interpolation function for node i of a linear discontinuous element Interpolation function for collocation point i of a linear discontinuous element Interpolation function for collocation point j Number of elements Number of panels for the axisymmetric BE formulation Global matrix of fluid interpolation functions Elemental matrix of nodal interpolation functions for the coordinates Permutation matrix for nodal forces Matrix of displacement functions that relate the displacements on the structural surface to the nodal displacements Matrix of interpolation functions for the field variables Vector of interpolation functions for rod element Elemental matrix of nodal interpolation functions for the field variables Elemental vector of interpolation functions for the field variables related to the direction normal to the shell mid-surface Elemental vector of interpolation functions for the field variables in radial direction Elemental vector of interpolation functions for the field variables related to the direction along to the shell mid-surface Elemental vector of interpolation functions for the field variables in axial direction xxviii

29 List of Symbols P p ˆp p P p Γf p inc ˆp inc ˆp inc,prop p inc,ωf p Λ P n p Ωf Q Q q (k 1,k 2 ) i r R r R r a r f ˆr i r (k 1,k 2 ) i ˆr l r m r P r P Point in the acoustic domain Acoustic pressure Vector of pressures at the collocation points Acoustic pressure in time domain Collocation Point Acoustic pressure due to radiation from the boundary Acoustic pressure due the incident field Vector of pressures at the collocation points due to the incident field Vector of pressures at the collocation points due to propeller vibration Acoustic pressure due the incident field at the field points Pressure in the integration points of the surface Λ Legendre polynomials Vector of field pressures at a set of discrete points Effort matrix Source point Coefficient for MMA that changes for inner and outer iterations Radial coordinate Distance to field point Coordinate vector Distance to field point projected to the generator for an axisymmetric problem Resistance for rigid disc Fluid radiation damping Radial coordinate at node i Coefficient for MMA that changes for inner and outer iterations Elemental vector of nodal coordinates Radial coordinate at the mid-surface Field point coordinates Collocation point coordinates xxix

30 List of Symbols r Q S s T t u ü ˆü û U z u c,0 u c,i U z,i U r,i u c,i+1 U r,i,j u c,lh u Γs û l û l,n u n û n,i u p u p,i u p,i Source point coordinates Vibro-acoustic system matrix Coordinate in shell direction Acoustic transfer matrix Time Displacement vector Acceleration vector Vector of nodal accelerations Vector of nodal displacements Displacement amplitude for general solution in axial direction Axial displacement of the cylinder at the stern end plate Axial cylinder displacement for first cylinder at junction i Displacement amplitude for general solution in axial direction for a one wave type Displacement amplitude for general solution in radial direction for one wave type Axial cylinder displacement for second cylinder at junction i Displacement amplitude for general solution in radial direction for one wave type for segment j Axial displacement of the cylinder at the bow end plate Displacement vector at surface Vector of nodal displacements for element l Elemental vector of local nodal displacements Displacement normal to the shell mid-surface Nodal displacement in direction normal to the shell mid-surface for element node i Displacement vectors at discrete points Radial plate displacement at junction i Displacement vector at discrete point i xxx

31 List of Symbols u r ũ r û r,i u s û s,i u z û z,i v V ˆv v a v f v h v n v n v n v n,i v n,r,i v n,z,i v p v s v s v s v s,i v s,r,i v s,z,i U r w c,i Displacement in radial direction Spatial Fourier transform Nodal displacement in radial direction for element node i Displacement in shell direction Nodal displacement in direction along the shell mid-surface for element node i Displacement in axial direction Nodal displacement in axial direction for element node i Normal velocity Volume of the resonance changer reservoir Vector of normal velocities at the collocation points Velocity of the moving mass for an inertial actuator Normal surface velocity of the acoustic domain Axial velocity of the stern end plate Velocity normal to the surface Unit vector normal to the mid-surface Vector normal to the mid-surface Unit vector normal to the mid-surface at node i Radial unit vector component normal to the mid-surface at node i Axial unit vector component normal to the mid-surface at node i Axial velocity of the propeller Normal surface velocity of the structural domain Unit vector in s direction Vector in s direction Unit vector in s direction at node i Radial unit vector component along the mid-surface at node i Axial unit vector component along the mid-surface at node i Displacement amplitude for general solution in radial direction Radial cylinder displacement for first cylinder at junction i xxxi

32 List of Symbols w c,i+1 w i w p,i x x a x i z i z j,a z j,b y Y i z z z a z f z i ẑ i z m z M Radial cylinder displacement for second cylinder at junction i Weighting factor for integration point i Axial plate displacement at junction i Solution vector for coupled problem Reactance for rigid disc Function argument for integration point i Axial coordinate for junction i Axial coordinate, where cylinder segment j starts Axial coordinate, where cylinder segment j ends Excitation vector for coupled problem Bessel function of the second kind and order i Axial coordinate Intermediate solution for adjoint problem Radiation impedance Characteristic impedance of the fluid Axial coordinate of bulkhead i Axial coordinate at node i Axial coordinate at the mid-surface Axial coordinate for the control moment xxxii

33 List of Abbreviations AVC Active vibration control BE Boundary element bpf Blade passing frequency DSAS Discrete structural acoustic sensing FE Finite element GCMMA Globally convergent method of moving asymptotes IA Inertial actuator KL Kirchhoff-Love L-BFGS-B Bounded limited memory Broyden-Fletcher-Goldfarb-Shanno algorithm LP Linear programming MMA Method of moving asymptotes MPI Message passing interface NLP Non-linear programming PSS Propeller/shafting system RC Resonance changer RM Reissner-Mindlin xxxiii

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35 Chapter 1 Introduction 1.1 Topic of Research The reduction of noise emitted by submarines has long been a key topic in naval research. This arises from water being a very good sound transmitter, allowing detection of submarines by passive sonar from hostile sensors over large distances. In addition, detection techniques are constantly improving. In particular, evaluation of underwater noise by computers allows for the detection and allocation of noise sources more reliably. In order to find ways to reduce the noise radiated by a submarine, its sound sources need to be identified. Noise emitted from sources internal to the hull and from the propeller can be distinguished. These internal sources include on-board machinery such as diesel engines and generators, fluid systems and exhaust systems, as well as activity by the crew. Propeller noise can be attributed to cavitation, flow noise and blade vibration, which are strongly influenced by spatial and temporal variations of velocity in the wake, as well as by the rotation of blade thickness and thrust. The prevalence of the sound sources depends on the frequency band, speed and depth. At large distances from the hull, lower frequencies dominate since absorption increases with frequency. Tonal components are more distinctive than random noise, but the combination of tonal components at different frequencies with broadband noise provides important information about the identity and speed of the submarine. At high speeds, the propeller is the most significant source of tonal 1

36 1 Introduction and broadband noise. High levels of radiated noise can occur at hull resonances that correspond to the acoustically efficient structural modes. Vibration correlated to such modes is excited through the water and the propeller shaft by fluctuating propeller forces that arise from operation of the propeller in a non-uniform wake. The overall sound signature is a combination of tonals and broadband random noise. Since submarines usually operate at large depths, cavitation is suppressed by high water pressure. Propeller blades are sickle shaped to reduce net fluctuating forces due to imperfections in the incident wake field. Flow noise is moderated by travelling at low speeds. Transmission of structure-borne noise from the on-board machinery through the submarine hull to the surrounding fluid can be reduced by appropriate mounts. For the reduction of axial hull vibrations that are induced by propeller forces, passive and active control of vibration can be employed. In this thesis, a passive control mechanism known as a resonance changer (RC), which acts as a hydraulic vibration absorber, is implemented in the propeller/shafting system (PSS). It is shown that careful selection of the RC virtual mass, stiffness and damping parameters is required as a change of the PSS dynamics can increase propeller axial vibration, leading to a net increase in overall radiated noise. Active control techniques are also implemented in this work, in which additional force or moment excitations are introduced to the PSS and/or the submarine hull in order to minimise hull vibration or sound radiation. The objective of this thesis is to investigate passive and active control mechanisms to reduce noise radiated from a submarine due to axial propeller forces that are transmitted to the hull via both the PSS and the fluid. Simulations are conducted using a numerical vibro-acoustic model of a submarine in an infinite fluid, where the complex structural and acoustic interaction between hull and propeller is taken into account. The numerical model has been validated by comparing results with those obtained for an analytical model. As hull excitation from the fluid mainly occurs at the tailcone of the submarine near the propeller, various tailcone configurations are investigated. The effect of a resonance changer on the structural and 2

37 1.2 Thesis Overview acoustic responses of the submarine is then compared for models with and without consideration of hull excitation through the fluid. Optimum design parameters for the RC are determined for minimisation of the overall radiated sound power. Active control systems using control forces or a control moment are implemented to minimise propeller force induced vibration of the submarine hull, or the overall radiated sound power of the submarine which is due to sound radiated from the hull and sound radiated from the propeller. 1.2 Thesis Overview The work presented in this thesis is divided into five main chapters, followed by conclusions. In the remainder of Chapter 1, a literature review on previous work related to submarine hull excitation by propeller forces and the dynamic modelling of a submarine hull and its propeller/shafting system is given, followed by an overview of coupled finite/boundary element methods, optimisation and control. Chapter 2 presents the analytical and numerical models that are used to to describe the structural and acoustic responses of a submarine hull and propeller/shafting system. A modular approach to model the propeller/shafting system is discussed, where the resonance changer is represented by a spring-mass-damper system. An axisymmetric vibro-acoustic model of a simplified submarine hull under axial excitation, comprising a fluid loaded, ring stiffened cylindrical shell with two internal bulkheads and rigid end plates, is presented first. Superposition of the individual responses from the end plates and the cylindrical surface is used to give an approximation of the structure-borne radiated sound pressure. In the second part of Chapter 2, a numerical model of a more detailed submarine structure is presented. Detailed formulations of the finite element (FE) method to model the structure and of the boundary element (BE) method to model the fluid are described, as well as the coupling between the FE/BE models. Chapter 3 presents results for the structural and acoustic responses of a submarine due to propeller forces. Results for a simplified physical model of the hull 3

38 1 Introduction consisting of a finite cylindrical hull under axial excitation are initially presented. For the simplified physical hull model, the frequency response functions for axial motion of the stern and bow end plates, as well as the axial and radial modeshapes of the cylindrical shell obtained both analytically and from a coupled FE/BE model are compared. Radiation directivity patterns are also given for the first three axial hull resonances. Results from a more detailed fully coupled FE/BE model of a submarine hull and its propeller/shafting system are then presented. The submarine is excited by propeller forces that are transmitted to the hull via both the shaft and the fluid in the vicinity of the propulsion system. The radiated sound power is contributed from the structure borne noise radiated by the hull and sound radiated directly from the propeller. The structural and acoustic responses of the submarine for different tailcone representations, including a rigid tailcone, a stiff tailcone in vacuo, a flexible tailcone in vacuo and a flexible free flooded tailcone, are investigated. The responses of the submarine with and without a resonance changer are presented. Chapter 4 presents passive and active control strategies applied to a submarine using FE/BE simulations. For passive control, optimum design parameters of the resonance changer are determined, where the overall radiated sound power due to sound radiation from both the hull and the propeller are taken into account to define the cost function. An optimisation technique based on the method of moving asymptotes is presented to determine the virtual mass, spring and damping parameters of the resonance changer, in order to minimise the radiated sound power. A sensitivity analysis of the radiated sound power with respect to the resonance changer parameters is presented. The performance of different active control systems is investigated, where active vibration control or discrete structural acoustic control was combined with either a control system featuring tuned actuators or a control moment. In Chapter 5, the work conducted in this thesis is summarised. Conclusions are drawn from the results presented in Chapters 3 to 5. Suggestions are given for subsequent work on the research topic of this thesis. 4

39 1.3 Literature Review In Appendix A, the structural responses of a submerged tailcone are investigated using numerical models as well as an experimental rig. Appendix B summarises the theory for the structural finite elements that are used in this thesis. Boundary element formulations which were employed in this work are discussed in Appendix C. Schemes for numerical integration which are required for both finite and boundary element methods are given in Appendix D. Appendix E presents algorithms for adaptive numerical integration that were used in this work to obtain optimisation cost functions. 1.3 Literature Review Reduction of noise radiated by a submarine due to propeller forces is based on a wide field of research subject areas. This includes general topics such as hydrodynamics, structural dynamics, acoustics, optimisation and control theory as well as numerical and experimental methods. There is little non-classified literature on the subject area as a whole due it strategic sensitivity. However, literature can be found on many subtopics that are more or less related to the subject area. The following literature review is organised in six sections. In each section a subtopic of the subject area is discussed, where both general literature and literature related to submarine research are considered. In the first section, literature concerned with the origin of the problem is discussed, namely propeller forces which cause excitation of the submarine hull. In order to understand the physical system which is subject to the excitation mechanism, the second section presents literature on the analytical dynamic modelling of a submarine hull. As a significant part of the hull excitation from the propeller occurs via the propeller/shafting system (PSS), literature on the dynamic modelling of the PSS including dynamic vibration absorbers implemented in the PSS is presented in the second section. In order to model respectively the submarine and the surrounding fluid domain, finite elements (FEs) and boundary elements (BEs) are used in this work. Literature on FEs and BEs is discussed in the fourth section. To minimise the overall sound radiated by the submarine, 5

40 1 Introduction design parameters for the dynamic vibration absorber implemented in the PSS are optimised in this work. Hence, the fifth section is dedicated to literature on optimisation techniques. In addition to reduction of sound radiated by a submarine using passive components, active control of sound and vibration is a topic considered in this thesis. Literature on active control of sound and vibration is presented in the sixth section Hull excitation by propeller forces At operational cruise speed, the sound radiated by a submarine is mainly due to propeller forces [3]. The propeller forces are caused by the operation of the propeller in a non-uniform wake, as shown in Fig. 1.1, which results in tonal noise in the frequency range up to 100Hz [4 6]. The non-uniformity of the wake is due to asymmetry in the hull or protrusions of control surfaces, as shown in Fig As the propeller blades rotate through areas of different water velocity, fluctuations in thrust are generated at the blade passing frequency (bpf, number of blades multiplied by the propeller rotational speed) and its multiples [4]. Fluctuating forces of similar order also occur in the vertical and transverse directions. The observed tonal components arise primarily from this spatial variation in the wake field, combined with the usually smaller effects of the rotation of blade thrust and blade thickness [5]. These tonal components are complemented by random components due to turbulence in the wake flow at entry to the propeller and also turbulence generated by flow over the blades. The spectrum of a non-cavitating propeller is shown in Fig The variation in thrust causes structural excitation of the hull through the propeller/shafting system, resulting in vibration of the hull and the propeller [7 10]. An early model to predict the structural response of submarines due to propeller forces, where only structural excitation is considered, was developed by Greenspon [7]. The rotation of the propeller results in a fluctuating pressure field in addition to the fluctuating structural force [6, 11 14]. The pressure field is due to both 6

41 1.3 Literature Review U U a = r R = 1.0 Figure 1.1: Wake of a torpedo [5] Figure 1.2: Stern of a torpedo Sound pressure level Discrete tones Frequency Broadband noise Figure 1.3: Non-cavitating noise of a submarine propeller [4] 7

42 1 Introduction the passage of the blades through a spatially varying velocity field and the rotation of blade thickness and thrust. The rotation of blade thickness represents moving displacement of water with respect to a fixed point in the acoustic field. The rotation of thrust represents moving forces on the water with respect to a fixed point in the acoustic field. The pressure field due to rotating blade thickness and thrust has been investigated by Breslin and Tsakonas [11]. The relation of the pressure field to blade sweep was investigated by Amiet [12]. The pressure field in the immediate vicinity of the propeller due to non-uniform inflow has been found to be highly complex as it reflects the distribution and variation with time of fluctuating forces over the whole of the propeller disc [14]. Breslin and Anderson [6] showed that the pressure field due to rotating blade thickness and thrust decays rapidly, leaving a simpler field with well-defined characteristics further from the propeller. A numerical study on the combined propeller pressure field due to rotating blade thickness and thrust and due to a non-uniform wake in the absence of cavitation was investigated by Seol et al. [13]. It was found that the net fluctuating forces on the propeller tend to govern the field at more than a few diameters from the propeller. This field has the characteristics associated with acoustic dipoles in the axial and radial directions. Each dipole has a near field decaying as 1/r 2 and an acoustic far field decaying as 1/r [5]. The transition between the near and far field components occurs within a few propeller diameters at frequencies of interest, becoming closer to the propeller as frequency increases. Simplification of the hull excitation mechanism to a combination of axial thrust and the associated axial dipole allows fundamental aspects of the propeller/hull interaction to be explored and understood; both the structural and the fluid forces have the potential to excite breathing modes of an axisymmetric hull. The simplified model for hull excitation also avoids the need to specify details of a propeller design and wake field that would be required to define the complete pressure field. The sound radiation from the propeller also contributes to excitation of the hull, via the combination of the hydrodynamic near field and the acoustic far field. In 8

43 1.3 Literature Review early research, only the near field was taken into account and blade loading was considered as an important factor regarding excitation of nearby boundaries [15 17]. Tsakonas and Breslin [18] found that the axial propeller clearance and slenderness of the vessel stern are important factors for hull excitation by the propeller pressure field. The influence of propeller dynamics on the excitation of ship hulls due to the propeller field was investigated by Tsushima and Sevik [10]. It was often assumed that the excitation of submarine hulls by the propeller pressure field is negligible, as earlier research predicted a contribution from the propeller pressure field to hull excitation that is only 6-8% of the contribution from the structural force, where the Laplace equation was used to model the fluid [8]. Recent work using the Helmholtz equation has shown that the contribution of fluid forces to the overall excitation is between 10% and 50% of the response due to excitation through the shaft [9]. This is because the transition from the hydrodynamic near field to the acoustic far field occurs close to the propeller at frequencies of practical interest. Kinns and Bloor [19, 20] investigated propeller forces on a cruise liner hull. Fixed fluctuating volumes to represent cavitation and fixed fluctuating dipoles to represent the propeller pressure field due to a non-uniform wake were considered. Rath Spivack et al. [21,22] developed methods to determine the forces from the propeller pressure field on axisymmetric floating and submerged vessels. Fixed fluctuating monopoles and dipoles were used to respectively represent cavitation and the propeller pressure field due to a non-uniform wake. Steady rotating monopoles and rotating dipoles were used to respectively represent blade thickness and blade loading. Both far and near field contributions were taken into account Analytical vibro-acoustic modelling of a submarine hull A submarine is a highly complex structure. In order to completely understand and model the dynamic behaviour of a particular submarine, details about the design of the submarine type are required. However, this information is generally classified and therefore assumptions and simplifications need to be considered in order to develop general yet meaningful submarine models. A historic description 9

44 1 Introduction of different submarines with some technical details is given in Ref. [23]. The basic design concepts are common for most submarines. The fundamental component of all submarines is the pressure hull. It is required to provide atmospheric conditions for the crew at high water depths by withstanding the water pressure. It is essentially a cylindrical steel shell that can be modelled using thin shell theory [24 29]. As the shell is reinforced by ring stiffeners, the additional dynamic properties of the rings need to be taken into account. This can be accomplished by smearing them over the cylindrical shell surface if the structural wave length is large compared to the stiffener spacing, which results in orthotropic material behaviour for the cylindrical shell [30]. Often semi-numerical approaches are used in order to take into account the stiffeners as discrete elements [31 33]. Ruotolo [34] compared different thin shell theories for the dynamic analysis of stiffened cylinders. A comparison with results from finite element models showed good agreement for most of the thin shell theories. The pressure hull of a submarine is often subdivided into compartments by bulkheads. The bulkheads can be represented by circular plates [35]. The dynamic behaviour of cylindrical shells caused by joint discontinuities such as cylinder/plate junctions has been investigated by several researchers [36 38]. Domes or flat circular plates are used as end closures for the pressure hull. They might also be treated as rigid plates, as they are stiff compared to other components of the submarine [39]. The influence of external water pressure on the dynamics of a ring-stiffened circular cylinder has been investigated by Ross et al. [40]. However, changes in hydrodynamic pressure on the cylindrical shell do not cause the overall dynamic behaviour to change significantly at operational depths [41]. As the density of the water is similar to the density of steel, the fluid loading effect has to be considered for the vibro-acoustic modelling of the submarine hull. This can be accomplished by adding a distributed mass to represent the entrained water and by adding damping due to sound radiation. Fluid loading coefficients for finite cylindrical shells have been determined by Sandman [42]. The effect of external fluid loading on the dynamic behaviour of thin conical shells has been investigated 10

45 1.3 Literature Review by Caresta and Kessissoglou [43], in which the conical shell was modelled as an assembly of short cylindrical shells. In order to calculate the sound field due to vibration of a shell, the boundary integral representation for the Helmholtz equation originally derived by Kirchhoff needs to be solved [44]. However, this is not possible analytically for arbitrary bodies, as the boundary integral equation can not be solved in a closed form for complex geometries. An approximation to determine the sound pressure due to a vibrating structure is given by the Rayleigh integral, when the contribution of the pressure on the radiating surface to the field point pressure is negligible [45]. Junger and Feit [46] presented methods for the calculation of the sound power radiated by simple geometries such as cylinders and circular plates. The dynamic behaviour of the radiating structure was also addressed. More recently, Junger [47] compared basic concepts for acoustic fluid-structure interaction that were developed over the past decades. A comprehensive work on sound/structure interaction for geometrically simple structures was presented by Fahy [45]. Analytical models to describe the structural and acoustic responses of a submarine hull modelled as a fluid loaded cylindrical shell with bulkheads and conical end closures have been presented by Caresta [48] Dynamic modelling of the propeller/shafting system Most submarines have a single propeller wake propulsion. It is mounted on a slender propeller shaft, where the aftmost journal bearing is located at the stern end of the tail cone. The shaft passes through the stern end plate of the pressure hull. The thrust is transferred from the shaft to the foundation through the thrust bearing. A resonance changer (RC) can be implemented between the thrust bearing and the foundation, when a Michel thrust bearing is used. The foundation is a shell-like structure mounted at the stern end plate. The thrust bearing, resonance changer and foundation form the thrust block. As a simplification, the propeller and shaft can be envisaged as a spring-mass system having one natural frequency, where the propeller is the mass and the shaft is the spring. However, other components such 11

46 1 Introduction as the thrust bearing contribute significantly to the overall dynamic behaviour of the propeller/shafting system [49,50]. The dynamic behaviour of axially excited propeller/shafting systems of ships has been investigated analytically by Kane et al. [49] using a series of spring-massdamper systems and experimentally by Pan et al. [50]. The propeller can be treated as a lumped mass, where the fluid loading was modelled as an additional virtual mass. The virtual propeller mass due to fluid loading has been experimentally determined for marine propellers by Burrill and Robson [51]. Another way is to treat the propeller as a freely suspended rigid disc, where mass loading coefficients can be approximated from the radiation impedance, if the fluid wavelength is large compared to the disc diameter [45, 52]. The radiation impedance for freely suspended discs with arbitrary ratios of fluid wavelength and disc diameter is discussed by Mellow and Kärkkäinen [53]. Stiffness, damping and mass coefficients to model the thrust bearing as a spring-mass-damper system have been presented by Schwanecke [54]. A resonance changer attenuates axial vibration of the propeller/shafting system, thereby reducing the transmission of axial forces from the propeller to the hull. The RC, initially derived from a thrust-meter, is a hydraulic vibration absorber that can be represented by a virtual spring-mass-damper system [2]. It detunes the natural frequencies of the propeller/shafting system and introduces additional reactive force and damping. For best performance of the RC, its optimum virtual stiffness, damping and mass parameters need to be found. Early models to find optimum RC parameters treated the submarine hull as a rigid termination [2] or one-dimensional rod model [55]. In recent work, a simplified physical model of a submarine was developed to include the hull impedance, hull resonances and radiated sound due to shaft excitation [1]. Results were also obtained for a series of resonance changers with variable parameters. However, excitation of the submarine hull from the dipole field radiated by the propeller was not considered. Another problem arising from the use of a resonance changer to minimise excitation of a submarine hull is that the axial movement of the propeller can be increased [1]. This causes additional sound radiation from the propeller. 12

47 1.3 Literature Review Coupled finite element/boundary element methods Whilst analytical vibro-acoustic models are computationally very efficient, their drawback is that only basic geometries can be used to represent the structure. Furthermore, the Helmholtz equation can only be solved analytically for very simple geometries. This requires approximations such as the superposition of the pressure fields from different parts of the model which involves neglecting their interaction. Numerical approaches to solve the structural and acoustic problems in the low frequency range that allow for more complex geometries are the finite element (FE) and boundary element (BE) methods [56 62]. The structure as well as the fluid can be modelled using either the FE or the BE method. For the FE method, the domain is subdivided into elements, where geometrical as well as field quantities are elementwise interpolated by polynomials. The individual elements are then combined to a more complex system which gives rise to a sparse linear system of equations. The usually symmetric system matrix is obtained by numerical integration of the material properties over the elements [63]. The FE method and its various applications has been described in detail [56, 57]. Finite element procedures with a focus on non-linear structural problems are discussed by Bathe [58]. Cook [59] summarises the most common FE methods and elements. For the BE method, only the the boundary of the domain needs to be subdivided into elements, where the geometry and the unknown quantities on the boundary are described by polynomials in a similar manner as for the FE method. The BE method gives rise to a system of linear equations, however with dense and usually non-symmetric matrices. A difficulty of the BE method is that singular or even hypersingular integrals need to be solved for computation of the system matrices. Special techniques need to be applied to address this issue such as coordinate transformations or special quadrature rules [64, 65]. The quantities in the domain are then found in a second step by integration of the solution from the first step over the boundary. With regards to structure/fluid coupling, it is often convenient to use one method for both the structural and fluid domains. However, for the ex- 13

48 1 Introduction terior radiation/scattering problem, it is generally simpler to use the BE method as only the fluid/structure interface needs to be meshed. The fluid domain needs to be meshed up to an ellipsoidal boundary of infinite elements at some distance from the structure when using finite elements to model the fluid [62]. Using the BE method for the structural domain requires definition of subdomains if different materials are used, where for each subdomain interface a coupling procedure has to be performed. Using the FE method allows more flexibility as the material parameters are an element property rather than a mesh property. In addition, discrete finite elements can be used to physically simplify the model and increase computational efficiency. Therefore the combination of the FE method for the structure and the BE method for the fluid is a common approach to solve coupled scattering/radiation problems [62, 66 73]. Application of the boundary element method for acoustics is described by Brebbia [60] and Wu [61]. More recent work on FE and BE acoustics is presented by Marburg and Nolte [62]. For the finite element representation of a submarine hull, the appropriate element type needs to be chosen. Both the cylindrical shell as well as the bulkheads represent thin-walled structures and hence shell elements are used. In this work, the excitation of the submarine hull is assumed to be axially introduced to the centre of the stern end plate. Using the assumption that the pressure hull is an axisymmetric structure, special axisymmetric elements are used to reduce computational cost. In general, two types of shell elements exist, namely Kirchhoff-Love elements and Reissner-Mindlin elements. The major difference between the element types is that shear stress normal to the shell surface is neglected for the Kirchhoff- Love elements [57]. An axisymmetric shell element was presented by Zienkiewicz et al. [74], using axisymmetric thin shell theory [24]. An element based on the Reissner-Mindlin assumptions was presented by Ahmad et al. [75]. The Reissner- Mindlin assumptions are simpler to treat when it comes to FE implementation [57]. When the direct BE method is used to solve the pressure based Helmholtz equation for the acoustic domain, the Helmholtz equation needs to be transformed into either the Helmholtz boundary integral equation or the hypersingular normal 14

49 1.3 Literature Review derivative Helmholtz boundary integral equation [60 62]. For the transformation of the Helmholtz equation into the Helmholtz boundary integral equation, the solution of the Helmholtz equation for a point source needs to be found, which is known as Green s function. The hypersingular Helmholtz boundary integral equation is obtained by taking the normal derivative of the Helmholtz boundary integral equation at the field point with respect to the direction to the source point [61]. The direct BE method can then be employed to solve both the exterior radiation/scattering problems as well as the interior/cavity problem. In contrast to the acoustic FE method, the acoustic BE method has the advantage in that the Sommerfeld radiation condition is implicitly fulfilled for application to exterior problems. This means that the acoustic pressure goes to zero as the distance from the source point on the boundary goes to infinity. A shortcoming of the BE method for the exterior problem is that the integral equations break down at the eigenvalues of the corresponding interior Dirichlet or Neumann problems. However, several methods exist to overcome this problem. Examples are the Burton-Miller approach [76], where a linear combination of both equations is used, and the CHIEF method proposed by Schenck [77]. A distinction can be made between continuous and discontinuous boundary elements [62]. For continuous elements, the unknown variables are forced to match at the element boundaries, as required for finite elements. This principle was initially common as its implementation is simple. However, there is no requirement for continuity of the unknown variables across the borders of boundary elements and for many applications, discontinuous elements yield higher accuracy than continuous elements at comparable computational effort [78]. Special acoustic BE formulations have been derived for axisymmetric problems. Seybert et al. [79] proposed a method in which elliptic integrals were used to address the singularities. However, their approach is only valid for axisymmetric boundary conditions. Acoustic BE formulations for axisymmetric problems with arbitrary boundary conditions can be established by using Fourier transforms [80 83]. The boundary element method can be used to obtain acoustic radiation modes of a structure, where the eigenvalues correspond to the radiation efficiencies of the 15

50 1 Introduction correlated modes [84]. The acoustic radiation efficiency of a structure is defined as the ratio between the sound power per unit area radiated by the structure and the sound power per unit area radiated by a reference model. The reference model is given by a uniformly vibrating piston which has the same average mean square velocity as the structure and a radius that is much larger than the fluid wavelength. Hence, the radiation efficiency can take values larger than unity [45]. The radiation modes are not correlated to the structural properties and dynamics, but only depend on the geometry of the radiating structure, the fluid properties and the fluid wave number. Strong coupling of the acoustic BE and the structural FE methods is achieved by imposing that (i) the normal velocity of the structure equals the normal velocity of the fluid and (ii) the normal distributed surface load of the structure equals the acoustic surface pressure at the structure/fluid interface. In terms of implementation it is convenient to use matching BE and FE meshes. This means that the coordinates of the interpolation nodes for the FE mesh match the coordinates of the interpolation nodes for the BE mesh. Condition (i) is then simply achieved by enforcing continuity at the nodes [67 71] or by enforcing continuity of the average velocity over an element to element surface [66]. Condition (ii) is obtained by element-pair-wise integration. However, the interpolation functions for some shell elements are usually of a higher order than the interpolation functions for the BEs of the same geometry. Apart from convenience in programming and meshing, matching meshes are not always to be preferred. Often the mesh of the structural problem needs to be much finer than the mesh of the acoustic problem since the wave number of fluids is generally much smaller than the wave numbers of solids. However, the computational effort for solving the BE problem increases excessively with the element number. For non-matching grids, the continuity condition (i) cannot be imposed directly. A solution is to relax the continuity condition and to introduce weak coupling, where the pressure interpolation functions for the BEs are used as test functions. This approach is similar to the mortar element method that was originally developed for FE domain decomposition [85, 86], but has also been 16

51 1.3 Literature Review formulated for FE domain coupling of structural/acoustic and acoustic/acoustic problems [87]. Extensive research on FE/BE coupling using the mortar element method has been conducted [72,73] Optimisation Mathematical optimisation is the process of finding the maximum or minimum for a real valued function of multiple parameters. Optimisation plays an important role in a lot of different areas, for example, in engineering, economics, operations research and game theory [88 91]. The function to be maximised or minimised is called a cost function or objective function. Sometimes constraints can be implied such as bounds on the function parameters and equality or inequality conditions on values for additional functions that depend on the parameters. The most common optimisation problems can be categorised into linear programs (LP) [88] and nonlinear programs (NLP) [89]. For linear programming problems, the values of the cost function as well as the additional constraint functions depend linearly on the parameters. A nonlinear programming problem arises if the cost function value and/or at least one of values for the constraint functions does not linearly depend on the parameters. Many optimisation applications are subject to a convex optimisation problem. This means that a local minimum for the cost function is always a global minimum. For convex optimisation problems, methods have been found to unambiguously determine the optimum parameters that correspond to the global minimum. A commonly used globally convergent method to find the optimum solutions for LP problems, as a subclass of convex problems, is the simplex algorithm [88]. For higher order convex optimisation problems, Lagrange multipliers can be used to reduce an optimisation problem with constraints to an optimisation problem with no constraints, if the cost function and the constraint functions are differentiable [89]. The reduced optimisation problem can then be solved using an appropriate method [91]. There is no mathematically proven method to find the global optimum for nonconvex NLP problems deterministically [90]. Even if a local optimum is found, 17

52 1 Introduction there is no certainty that this also represents the global optimum. Methods to find local optimum parameters for NLP problems with no constraint equations can be categorised into gradient and non-gradient based methods. If the first order gradient of the cost function is available, the gradient, conjugate gradient and quasi-newton methods can be used. If the cost function is twice differentiable, the Newton method is applicable [91]. General optimisation methods that can be applied to arbitrary NLP problems with constraint equations that are not differentiable are heuristic and can often be related to natural processes. The most basic algorithm is hill climbing [92]. More advanced approaches are simulated annealing and evolutionary algorithms [93,94]. For numerical applications, the evaluation of the cost function is often computationally expensive. To address this problem, an approach known as sequential programming was proposed by Schmit and Farshi [95] in which an NLP problem is approximated by a linear subproblem at an initial set of parameters and the subproblem is solved by an appropriate method. The optimum solution for the approximating subproblem then represents the next iteration point. The iteration is stopped when some convergence criteria are fulfilled. It has to be noted that this approach can only be applied if the cost function is differentiable, as the subproblem is constructed using the gradient of the cost function. Extended concepts of this approach using more complex function approximations have been investigated by Houten [96]. A formulation of sequential programming with improved convergence was proposed by Svanberg [97], where the cost function is approximated by moving asymptotes. The resulting subproblems are not linear but convex and can be solved by an appropriate gradient based method such as quasi-newton algorithms [98]. As the approximation concept for the subproblem is local, the process can easily get trapped in a local minimum. This can be avoided by introducing another level of iterations, in which the inner iterations are utilised to verify the solution of the subproblem that is based on the outer iterations [99]. Further improvement of this approach was achieved by Bruyneel et al. [100], where two successive design points are used to form the subproblems. 18

53 1.3 Literature Review Structural optimisation was initially applied to find optimum geometries for minimum structural weight [101]. A comprehensive work on structural optimisation was published by Haftka et al. [90], where most of the aforementioned techniques and their application to structural optimisation are discussed. Research on optimisation techniques related to numerical structural-acoustic problems has been summarised recently by Marburg [102]. The aim of structural-acoustic optimisation is to find optimum structural design parameters that minimise some cost function related to sound radiated by the structure. The cost function is often given as the integral of some acoustic property such as the sound pressure for interior problems or the radiated sound power for radiation problems over the frequency range of interest [103]. As numerical models of vibro-acoustic problems require significant computational effort, the use of gradient based methods can accelerate the optimisation process. Formulations to obtain analytically the gradient or sensitivity of the radiated sound power for plates with respect to structural design parameters have been investigated by Salagame et al. [104]. The proposed method can also be applied to numerical models, resulting in a semi-analytical formulation [105]. There is very little literature on structural-acoustic optimisation related to submarines available in the public domain. Optimum design parameters for the resonance changer have been found by Dylejko for an analytical submarine model [1]. Different cost functions to represent structural and acoustic responses of the submarine hull to propeller excitation were minimised using a genetic algorithm [94]. The submarine hull was represented by a rod [55] or a thin-walled, ring-stiffened cylinder with bulkheads [1] Active noise and vibration control Active control of noise and vibration is generally based on the minimisation of a cost function by introducing a secondary excitation to the vibro-acoustic system [106, 107]. The cost function is obtained by evaluation of the complex signal from a sensing system which is formed by at least one discrete physical sensor. The 19

54 1 Introduction cost function is often defined as the sum of the squares of the sensor signals. This approach is known as the least mean square (LMS) algorithm [108]. The signal strength of the error sensors is anti-proportional to the performance of the active control system. In order to minimise the sound radiated by a structure, active noise control (ANC), active vibration control (AVC) or active structural acoustic control (ASAC) can be used [107]. For ANC, anti-noise, which means noise of equal amplitude but opposite phase of the undesired noise, is generated using loudspeakers, where the error sensors are represented by microphones [109]. ANC finds applications for which interior noise needs to be minimised [110]. For AVC and ASAC, actuators are used to introduce a secondary excitation to the vibrating structure, where the secondary excitating force is of equal magnitude but opposite phase of the primary excitating force [106]. In practical applications, multi-channel systems are commonly used, where many error sensors and actuators or loudspeakers form the control system. A cost function for such a multi-channel system is given by Fuller et al. [106], where the relative contribution of the individual error sensor signals to the cost function is determined by a weighting matrix. Using active control, it is desirable to use actuator forces that are as small as practical in order to reduce cost, power, space and weight requirements of the control system. One method of achieving small optimum control forces is to use added masses and springs that are tuned separately to each major resonance of the hull and shafting system. These masses and springs can also have a significant effect on hull vibration and sound radiation when the active control system is not operating, because they act as passive vibration absorbers. Gündel [111] used AVC to minimise vibration of an aircraft fuselage, where actuators were attached to the frames of the fuselage. The actuators featured passive elements that were tuned to the first three harmonics of the blade passing frequency. Active vibration control of low frequency hull vibration and radiated noise for a submarine has been presented recently by Forrest [112] and Pan et al. [113, 114]. They used a method derived from work conducted by Young [115], in which a control 20

55 1.4 Contribution to Research moment is generated on the radiating surface using a stiffener. Pan et al. [113,114] achieved reduction of the radiated sound power by up to two-thirds compared to the sound power radiated by the uncontrolled model. Lewis and Allaire [116] implemented active control of the axial shaft vibrations in the propeller/shafting system of a ship using an electromagnetic thrust bearing. AVC has also been applied by Baz et al. [117] to minimise axisymmetric vibrations of cylindrical shells using active constrained layers with piezoelectric elements. Both AVC and ASAC of cylinders using piezoelectric actuators have been investigated analytically [118] and experimentally [119] using piezoelectric elements to generate the secondary forces. Laplante et al. [120] employed active constrained layers with piezoelectric elements to minimise the radiated sound of a submerged cylinder, where hydrophones were used to obtain the error signals. As the use of hydrophones is impracticable for submarines, discrete structural acoustic sensing (DSAS) can be applied, where the radiated sound field is estimated by the structural response at the wet surface [121]. DSAS has been applied to minimise the sound radiated by a cylinder in the time domain by Maillard et al. [122]. Feedback control is suitable for systems in which the excitation is random, as only the signal from the error sensors is used to determine the secondary excitation. Feedforward control may only be used if the excitation is deterministic, for example, when a reciprocating machine of known rotational speed represents the primary excitation. The optimum secondary excitation can then be determined as a fixed system property and adapted based on tachometer data of the reciprocating machine. In this case, error sensors are not necessary, but they may be used to monitor the performance of the system or to tune the controller [106]. 1.4 Contribution to Research The work presented in this thesis contributes to naval research as well as the field of numerical acoustics. The contribution to the naval field is related to the reduction 21

56 1 Introduction of noise radiated by a submarine. A fully coupled structural/acoustic model of the submarine was developed for optimisation of the resonance changer parameters as well as for design of an active control system. This means that the complex fluidstructure interaction between the hull and the propeller that has been neglected in previous work on resonance changer optimisation is taken into account. The stiffeners of the cylindrical pressure hull were modelled using discrete elements instead of stiffener smearing. The radiated sound field is contributed by sound radiation from the submarine hull as well as sound radiation from the propeller. In addition, the stern end of the submarine was modelled as a cone rather than a flat circular plate, because the structural and geometric properties of the submarine stern may have significant influence on the vibro-acoustic interaction between the propeller and hull. Using active control, a novel approach to submarines is investigated in which inertial actuators have been implemented. The inertial actuators were tuned to the low frequency axial resonances of the submarine in an attempt to reduce the required control forces. There is no literature on the implementation and application of axisymmetric discontinuous boundary elements that employ elliptic integrals. Such elements are used in this work. Little literature exists on the application of an axisymmetric coupled finite element/boundary element method, where non-matching meshes are employed. In this work, non-matching meshes were used to minimise computational cost. The non-conformity difficulty involved with non-matching meshes was addressed by using mortar elements at the structure/fluid interface. It is shown that the mortar coupling matrix becomes diagonal when discontinuous elements are employed. This implies that the saddle point problem vanishes, which arises in previous mortar formulations for boundary element/finite element coupling. Very little research exists on the application of the method of moving asymptotes to acoustic structure/fluid interaction problems. In this work, the method of moving asymptotes was applied to minimise the radiated far field sound power of the coupled acoustic BE/structural FE system. The sensitivity of the radiated far field sound power which is required for the method of moving asymptotes is obtained in a semi- 22

57 1.4 Contribution to Research analytical way. This is realised by employing the adjoint operator, where acoustic sources in the fluid domain were considered. The approach presented in previous work does not allow for additional acoustic sources in the fluid domain, as only the radiating surface of the structure is used to calculate the radiated sound power. 23

58 Chapter 2 Structural/Acoustic Modelling of a Submarine 2.1 Introduction In this chapter, analytical and numerical models to describe the structural and acoustic responses of a submarine hull are developed. The excitation mechanism for the submarine is the operation of the propeller in a non-uniform wake, resulting in a time-harmonic fluctuating force at the propeller blade passing frequency and its multiples. Only the steady state response for time harmonic excitation is taken into account, where the time dependence is given by e jωt. j is the imaginary unit, ω is the circular frequency and t is the time. Under the assumption that the propeller diameter is half the hull diameter and the tip speed is limited to 40 m /s to avoid cavitation, the maximum fundamental blade passing frequency is approximately 25 Hz for a medium size submarine with a 7-bladed propeller. As the first four harmonics of blade passing frequency are taken into account, the frequency range between 1 and 100 Hz is considered. The pressure hull of a submarine is represented by a cylindrical steel shell and can be treated as an axisymmetric structure [1]. A modern single screw submarine has the propeller shaft along the axis of the cylindrical pressure hull. Hence, the propeller/shafting system is modelled as a one-dimensional dynamic problem [49]. This results in axisymmetric boundary conditions on the pressure hull, allowing for special formulations of both the structural and the acoustic domains. An analytical model is used to validate a simplified fully coupled FE/BE numerical model. 24

59 2.2 Simplified Physical Models of a Submarine Tail cone Pressure hull Propeller Foundation Resonance changer Bulkhead Propeller shaft Thrust bearing Ring stiffeners Rigid end plate / lumped mass Figure 2.1: Physical model of a submarine A more detailed numerical model of the submarine hull, its excitation and the radiated sound field is then developed. The computational models extend the analytical model by (i) using a modular approach to take into account the dynamics of the propeller/shafting system, (ii) development of a semi-analytical model of the propeller to consider the complex interaction between the propeller pressure field and the hull, and (iii) using a tailcone instead of a rigid end plate to represent the stern. 2.2 Simplified Physical Models of a Submarine A real submarine has the pressure hull as its main structure with external attachments, such as buoyancy tanks, that are of relatively light construction. The pressure hull was modelled as a thin-walled cylinder with evenly spaced ring stiffeners of rectangular cross-section, and with two evenly spaced bulkheads which were modelled as circular plates. The end plates of the pressure hull have been treated as rigid as they are relatively stiff. The on-board machinery and remaining internal structure were considered as a distributed mass of the cylindrical shell. The distributed mass was chosen in such a way that neutral buoyancy of the submarine is guaranteed [1]. Lumped masses were added to both end plates to represent the water in ballast tanks and free-flooded structures. The elongated tailcone was considered explicitly in the numerical models since the dipole excitation of the hull originates at the propeller hub. A schematic diagram of the submarine is shown in Fig

60 2 Structural/Acoustic Modelling of a Submarine Shaft Thrust Bearing Foundation Propeller Resonance Changer Figure 2.2: Propeller/shafting system The propeller/shafting system was only considered in the FE/BE models. The basic elements of the propeller/shafting system are the propeller, shaft, thrust bearing, resonance changer (RC) and foundation, as shown in Fig A schematic representation of the propeller/shafting system is shown in Fig. 2.3, where the propeller force and velocity amplitude are given by f p and v p, respectively. The hull drive point force and velocity are denoted by f h and v h. The propeller is represented by a lumped mass m p that also includes the added mass effect of the water. The propeller dimensions for calculating the propeller mass and the added mass of water are chosen by assuming that the propeller volume is 1 /1000 of the volume displaced by the pressure hull [22]. The propeller diameter is assumed to be half the pressure hull diameter. The propeller shaft was modelled as a simple rod with an effective length l se and an overall length l s, where the overhang was represented by another lumped mass. The shaft properties are also defined by its cross-sectional area A s, Young s modulus E s and density ρ s. The thrust bearing was Propeller Shaft l s Thrust bearing c b Resonance changer c r Foundation m p l se m b k r a f b f A s, E s, ρ s k b m r E f, ρ f, ν f, h f f p v p f h v h Figure 2.3: Schematic diagram of the propeller/shafting system [1] 26

61 2.2 Simplified Physical Models of a Submarine V A 1 L A 0 Figure 2.4: Resonance changer assumed to act as a spring-mass-damper system with mass m b, damping coefficient c b and spring constant k b. For the present model, the thrust bearing is attached to a single RC that has been reduced to a spring-mass-damper system according to Goodwin [2]. The RC incorporates a hydraulic cylinder that is connected to a reservoir via a pipe, as shown in Fig The virtual mass, damping and stiffness are calculated using [2] m r = ρ ra 2 0L, c r = 8πµL A2 0, k r = A2 0B A 1 V A 2 1 (2.1) where ρ r is the density of the hydraulic medium, µ is the dynamic viscosity and B is the bulk modulus of the oil in the RC. V is the volume of the reservoir, A 1 is the cross-sectional area of the pipe, L is the pipe length and A 0 is the cross-sectional area of the cylinder. In a real submarine, the foundation of the propeller/shafting system is a complex shell-like structure. In this work, the foundation is represented as a truncated cone with end radii a f and b f. The Young s modulus, density, Poisson s ratio and thickness of the foundation are given by E f, ρ f, ν f and h f, respectively. 27

62 2 Structural/Acoustic Modelling of a Submarine 2.3 Analytical Model of a Submarine Hull In order to confirm the computational FE/BE models that are used in this work, results from the FE/BE model are compared with results obtained from an analytical model, where just the submarine hull is considered. The analytical model employs thin-shell theory to model the cylindrical pressure hull. The bulkheads that subdivide the pressure hull are represented by circular plates. The overall dynamic response of the submarine is obtained by dynamic coupling of the individual substructures and by assuming appropriate boundary conditions at the cylinder ends. The overall radiated sound field is obtained by superposition of sound radiated from the cylindrical hull and sound radiated from the end plates Motion of the cylindrical hull The equations of motion for the cylinder of the submarine hull were derived using the Donnell-Mushtari equations, but contain additional terms representing several influencing factors including the effects of ring stiffeners, fluid loading and a distributed mass. Tangential motion of the shell was not considered as it does not couple well with fluid motion and hence does not significantly contribute to sound radiation. Under axisymmetric motion, the dynamic problem can be described in terms of the radial coordinate r, the axial coordinate z, the radial displacement u r and the axial displacement u z. The equations according to Fig. 2.5 are given by [25,46] where 2 u z z γ c 2 s ν u z a c z + γ 2 2 u z t 2 + ν a c u r z = 0 (2.2) u r a 2 c + β 2 a 2 4 u r c z + γ u r t + γ c 2 s c 2 s u r t = 0 (2.3) γ 1 = 1 + A rs b rs h c + m eq ρ s h c, γ 2 = 1 + A rs(1 ν 2 ) b rs h c γ 3 = 1 + A rs b rs h c + m f ρ s h c, γ 4 = r f ρ s h c (2.4) 28

63 2.3 Analytical Model of a Submarine Hull The cylinder has shell thickness h c and radius a c. The structural speed of sound for longitudinal waves in the cylinder is given by c s = E/ρ s (1 ν 2 ), where E, ρ s and ν are respectively Young s modulus, the structural mass density and Poisson s ratio. The dimensionless shell thickness parameter β is given by β = h c / 12a c. m eq is the equivalent distributed mass of the internal structure and on-board equipment, m f is the mass fluid loading parameter and r f is the damping fluid loading parameter. The effect of ring stiffeners is taken into account using stiffener smearing [30], where the dynamic properties of the stiffeners are characterised by their cross sectional area A rs and axial spacing b rs. The fluid loading parameters are given by [46] for k c k f and m f = ρ fa c K 0 (β f ) β f K 1 (β f ), r f = 0 (2.5) J 0 (β f )J 1 (β f ) + Y 0 (β f )Y 1 (β f ) m f = ρ f a c H (1) β f 1 (β f ) 2, r f = πβ 2 f 2ρ f c f k f a c H (1) 1 (β f ) 2 (2.6) for k c < k f, where β f = a c k 2 c k 2 f, k c is the wave number for longitudinal waves in the cylinder, k f is the fluid wave number, ρ f is the mass density of the fluid and c f is the velocity of sound in the fluid. H (1) i, K i, J i and Y i represent respectively the Hankel functions of the first kind, the modified Hankel functions, the Bessel functions of the first kind and the Bessel functions of the second kind of order i [123]. General solutions for the differential equations of motion are given by u r (z,t) = U r e jωt+λz/ac (2.7) u z (z,t) = U z e jωt+λz/ac (2.8) where λ represents a wave type, U z is the axial complex displacement amplitude and U r is the radial complex displacement amplitude. Substitution of equations (2.7) and (2.8) back into the equations of motion (equations (2.2) and (2.3)) leads to a system of linear homogeneous equations in terms of U z and U r. The determinant of 29

64 2 Structural/Acoustic Modelling of a Submarine l c r, u r z, u z a c h c Figure 2.5: Geometrical parameters for a thin-walled cylinder of length l c the coefficient matrix has to be zero for a non-trivial solution, which leads to the characteristic equation β 2 λ 6 +γ 1 β 2 ω 2 rλ 4 +(γ 2 (γ 3 jγ 4 /ω)ω 2 r ν 2 )λ 2 γ 1 (γ 3 jγ 4 /ω)ω 4 r +γ 1 γ 2 ω 2 r = 0 (2.9) ω r = ωa c /c s is the ring frequency. The characteristic equation is a polynomial of the third order in terms of λ 2. Thus three pairs of the wave type parameter λ can be obtained for a given frequency, where one pair represents wave motion in the positive and negative directions. As the fluid loading is a function of the structural wave number k c, an iterative procedure is required to find the roots. The complex amplitudes U z,i and U r,i that correspond to the particular solution λ i can be found from the boundary conditions. The complete solution for shell motion is then given by u r (z) = u z (z) = 6 U r,i e λ iz/a c (2.10) i=1 6 U z,i e λ iz/a c (2.11) i= Motion of the bulkheads The bulkheads are modelled as circular plates and result in an increase of the stiffness around the region of the plate-cylinder junctions. The motions of the cylinder 30

65 2.3 Analytical Model of a Submarine Hull will generate bending and in-plane waves in the circular plates. For axisymmetric motion, the equations of motion for bending and in-plane waves in a circular plate are given by [38] 4 u z + k 4 pbu z = 0 (2.12) 2 u r r + 1 u r 2 r r 1 2 u r = 0 (2.13) c 2 pl t 2 where u z and u r are respectively the bending and in-plane displacement of the plate. k pb = 4 ρ s ω 2 h p /D p is the plate bending wave number, where D p = Eh 3 p/12(1 ν 2 ) is the flexural plate rigidity, h p is the thickness of the plate and k pl = ω/c pl is the in-plane wavenumber. 4 is the square of the Laplacian operator for axisymmetric motion. The solutions of the differential equations are given by [38] u z = D 1 J 0 (k pb r) + D 2 I 0 (k pb r) (2.14) u r = D 3 k pl J 1 (k pl r) (2.15) where I 0 is the modified Bessel function of the first kind of order 0, which in this case is given by I 0 = J 0 (jk pl r) [123]. D 1, D 2 and D 3 are plate displacement coefficients which are determined from the boundary conditions. r is the radial coordinate from the centre of the plate Dynamic response of the submarine hull To determine the overall dynamic response of the submarine hull, continuity and equilibrium equations at the cylindrical shell/circular plate junctions are required, as well as the boundary conditions. Figure 2.6 shows a cylinder-plate junction of the submarine model. f cz, f cr and M c denote the longitudinal force, transverse force and bending moment of the cylinder per unit length of circumference. f pz, f pr and M p are respectively the transverse force, radial force and bending moment per unit length of the outer radius of the circular plate. For the sign convention used in Fig. 2.6, the force and moment equations for a cylinder-plate junction are given by [38] 31

66 2 Structural/Acoustic Modelling of a Submarine f cz = Eh ( c uz 1 ν 2 z + ν ) u r a c (2.16) Eh 3 c 3 u r f cr = (2.17) 12(1 ν 2 ) z 3 M c = Eh3 c 2 u r (2.18) 12(1 ν 2 ) z ( 2 3 u z f pz = D p r u z 3 a c r 1 ) u z (2.19) a 2 c r f pr = Eh ( ) p ur 1 ν 2 r + u ν r (2.20) a c ( 2 u z M p = D p r + ν ) u z (2.21) 2 a c r The pressure hull is represented by a finite cylindrical shell with rigid end plates. The boundary conditions at the ends of the cylinder corresponding to z = 0 and z = l h are given by u r = 0 (2.22) u r z = 0 (2.23) where l h is the length of the pressure hull. The lumped masses m 0 and m lh represent the main ballast tank and casing. In addition, they maintain a condition of neutral buoyancy together with the equivalent distributed mass m eq and the mass of the pressure hull. The entrained mass of water at the end plates is obtained by using the reactive part of the normalised radiation impedance for an unbaffled piston and is given by [53] m ep = 2ρ f a 3 c (2.24) Together with the exciting axial harmonic force, the equations for the boundary conditions are [1] f h 2πa c f cz,0 = (m 0 + m ep ) 2 u c,0 t 2 at z = 0 (2.25) 2πa c f cz,lh = (m lh + m ep ) 2 u c,lh t 2 at z = l h (2.26) where f h is the sinusoidal force applied to the stern hull end plate. 32

67 2.3 Analytical Model of a Submarine Hull w c,i+1 u c,i+1 f cz,i+1 f cz,i+1 z M c,i u p,i f pz,i w p,i f pr,i M p,i M c,i+1 w c,i u c,i f cz,i f cz,i z i Figure 2.6: Cylinder-plate junction of the submarine model, where i = 1, 2 [55]. Equations for continuity of displacement and slope as well as the equations for equilibrium of forces and moments, applied at each cylinder-plate junction, are given by [1] u c,i = u c,i+1 (2.27) w c,i = w c,i+1 (2.28) u c,i = w p,i (2.29) w c,i z w c,i z = w c,i+1 z = w p,i r (2.30) (2.31) f cz,i + f pr,i f cz,i+1 = 0 (2.32) f cz,i f cz,i+1 f pr,i = 0 (2.33) M c,i M p,i M c,i+1 = 0 (2.34) where i = 1, 2. Equations (2.27) to (2.34) represent a system of 24 linear homogeneous equations in terms of the unknown coefficients U r,i, U z,i, i = 1, 2,..., 6 33

68 2 Structural/Acoustic Modelling of a Submarine for each of the three cylinder segments, and D 1, D 2, D 3 for each bulkhead. After determining the unknown coefficients by solving the system of equations at each frequency, the overall dynamic behaviour of the shell can be obtained by substituting the displacement coefficients back into the general solutions for the cylindrical shell and circular plates, given by equations (2.10), (2.11), (2.14) and (2.15) Radiated sound field In the simplified analytical model, the sound field radiated by the submarine is computed by superposition of the sound field radiated by a baffled cylinder and two rigid fluctuating discs. It is assumed that the mutual influence of the cylinder and the two end plates on the overall radiated sound field is negligible. A formulation for sound radiation from a cylindrical hull with two semi-infinite baffles is given by Junger and Feit [46]. The pressure field is obtained by application of a spatial Fourier transform to the Helmholtz equation in cylindrical coordinates. In the far-field and for axisymmetric motion, the pressure is given by [1] p(r,θ R ) = jρ fω 2 e jk frũ r (k f cos θ R ) πk f R sinθ R H (2) 1 (k f a c sin(θ R ) (2.35) where R is the distance from the centre of the cylinder to the field point, θ R is the angle according to Fig. 2.7, ũ r is the spatial Fourier transform of the radial displacement of the submarine hull and k f is the fluid wave number. The spatial R p(r,θ R ) baffle θ R baffle Figure 2.7: Cylinder with semi-infinite baffles 34

69 2.3 Analytical Model of a Submarine Hull baffle p(r,θ R ) side of the disc not exposed to the fluid θ R side of the disc exposed to the fluid Figure 2.8: Rigid disc in a closed-back baffle Fourier transform is obtained by the sum of the transforms of the individual cylinder segments by ũ r = 6 i=1 3 j=1 U r,i,j e λ il j /2a c e z j,a(λ i /a c+jk f cos θ R ) e z j,b(λ i /a c+jk f cos θ R ) λ i /a c + jk f cos θ R (2.36) where z j,a and z j,b respectively denote the axial start and end coordinates for a shell segment and l j is the axial length of the shell segment. The sound radiation from the end plates is obtained by treating each plate as an axially fluctuating unbaffled piston. The piston can be modelled as a rigid disc, where only one side of the disc is exposed to the fluid and the other side of the disc is baffled. The sound radiation for a disc in such a closed-back baffle is obtained by summing half the sound radiation for a disc in an infinite baffle to half the sound radiation for a freely suspended disc [53]. According to Fig. 2.8, the pressure is obtained by [53] p(r,θ R ) = jk f z f g(r)πa 2 p [d b (θ R ) + z a d f (θ R )] v n (2.37) where the characteristic impedance of the fluid is given by z f = ρ f c f and v n is the axial velocity of the plate. g is Green s function, which for wave propagation in a 35

70 2 Structural/Acoustic Modelling of a Submarine three dimensional space is defined as g(r) = e jk fr /4πR (2.38) The directivities for a disc in an infinite baffle and for an ideal freely suspended disc are respectively obtained by d b (θ R ) = 2J 1(κ/2 sin θ R ) κ/2 sin θ R (2.39) and d f = d b cos(θ R ) (2.40) where κ = 2k f a p. The acoustic radiation impedance of a freely suspended disc is given by [53] z a = 1 g a + jb a (2.41) where g a and b a are respectively the acoustic radiation conductance and susceptance, and are given by g a = 1 + 2J 1(κ) κ 2J 0 (κ) π (J 1 (κ)h 0 (κ) J 0 (κ)h 1 (κ)) (2.42) b a = κh 1 (κ) ( ) 2 (2.43) κ 2J 1(κ) 2 κ + (H1 (κ)) 2 H 0 and H 1 are the Struve functions of the zeroth and first order, respectively [123]. 2.4 Computational Model of a Submarine The computational FE/BE submarine model includes both the propeller/shafting system and the submarine hull. The acoustic properties of the propeller are modelled using a semi-analytical approach, which is described in the following section. 36

71 2.4 Computational Model of a Submarine The structural properties of the propeller/shafting system and the submarine hull are modelled using finite elements. The acoustic properties of the submarine hull are modelled using boundary elements Sound field radiated by the propeller The propeller sound field is dominated by sound radiation due to (i) the hydrodynamic mechanism that arises through the propeller operating in a non-uniform wake and (ii) the axial fluctuation of the propeller due to vibration of the shafting system. Hence, the sound field due to rotating thrust and blade-thickness which results in rotating monopoles and dipoles has been neglected in the proceeding analysis. The sound radiation originates from the propeller blades as multiple dipoles. The dipoles can be simplified to a single dipole located at the propeller hub, because the wavelength is large relative to the propeller diameter and the propeller is small relative to the submarine. A derivation of the dipole field pressure due to a fluctuating force is provided by Ross [5]. The directivity pattern of the dipole is governed by cosθ R, where θ R is the angle between the field point vector with respect to the source and the force direction. The dipole amplitude is directly proportional to the structural force. The radial variation of the amplitude follows 1/R 2 in the near field and 1/R in the far field, where R is the distance of the field point from the source. The transition is a function of the wavelength λ f and occurs at λ f /2π. A polar diagram of a dipole is shown in Fig For (i), corresponding to the sound radiation due to the hydrodynamic mechanism, the sound radiation due to the force on the propeller hub is ( p(r,θ R ) = jk f fg(r) 1 j ) cos θ R (2.44) k f R where k f is the wave number, g(r) is the free space Green s function and f is the amplitude of the exciting force. θ R denotes the angle between the submarine axis and the vector pointing from the propeller hub to the field point, at which the pressure is evaluated. 37

72 2 Structural/Acoustic Modelling of a Submarine For (ii), corresponding to propeller vibration, the propeller was simplified as a rigid circular disc. The problem reduces to a dipole according to Eq. (2.44) for small values of k f a p, where a p is the disc radius. The axial velocity v p of the propeller can be related to the equivalent force by f = 2πa 2 pz f z a v p (2.45) where z f is the characteristic impedance of the fluid and z a = r a + jx a, where [53] r a = 8(k fa p ) 4 and x 27π 2 a = 4k fa p 3π (2.46) are respectively the normalised radiation resistance and reactance. The added mass of water for the propeller due to fluid loading can be found by substitution of jx a into Eq. (2.45) as the radiation impedance and is given by [45] m pf = 8 3 a3 pρ f (2.47) where ρ f is the mass density of the fluid. The radiation damping due to fluid loading is neglected for the model presented here Figure 2.9: Dipole directivity pattern 38

73 2.4 Computational Model of a Submarine Coupled FE/BE modelling For the computational models, finite elements have been used to represent the structural domain. The dynamic finite element formulation has been derived by assuming linear elasticity as discussed in Appendix B. The cylindrical pressure hull, the bulkheads and the foundation of the propeller shafting system were represented by axisymmetric shell elements based on Kirchhoff-Love theory or Reissner-Mindlin theory as described in Appendices B.1 and B.2, respectively. Ring stiffeners with a rectangular cross-section as shown in Fig were considered using shell elements. The FE meshes have been generated using ANSYS 11. Fig. 2.11(a) shows detail of the FE mesh at the cone and Fig. 2.11(b) shows detail of the FE mesh of a bulkhead. Other parts of the mesh have similar structure. The end plate is rigid and therefore its grid resolution has no influence on the results. As the joints, especially at the bulkheads and end plates, lead to evanescent near-field waves with a small wavelength, a fine FE mesh is required for convergence of the results in terms of the natural frequencies and displacements. The one-dimensional finite elements presented in Appendix B.3 were used to model the dynamics of the propeller/shafting system as only axial excitation of the submarine hull was taken into account. Rod elements were used to model the propeller shaft. The lumped masses at the end plates, the propeller structural and fluid loading masses, the virtual mass of the thrust bearing and the virtual mass of the resonance changer were modelled using point mass elements. The virtual stiffness and damping of both the thrust bearing and the resonance changer have been modelled using spring-damper elements. Using finite elements, the structural domain is represented by the following system of equations Aû = N T Γ s,vf Γs dγ s + ˆf (2.48) Γ s where A is the system matrix for the structure, û is the nodal displacement vector, N Γs,v is the global matrix of interpolation functions at the structural surface Γ s, 39

74 2 Structural/Acoustic Modelling of a Submarine Figure 2.10: Cylindrical shell with interior ring stiffeners r z Axis of rotation (a) FE mesh at tailcone r z Axis of rotation (b) FE mesh at a bulkhead Figure 2.11: FE meshes for the submarine pressure hull 40

75 2.4 Computational Model of a Submarine f Γs is the surface traction vector and ˆf is the vector of nodal forces. The system matrix A was generated using ANSYS 11. The direct boundary element method described in Appendix C has been used to represent the interior and exterior fluid domains by solving the Helmholtz equation. A special BE formulation for axisymmetric problems as given in Appendix C.2 was employed to reduce computational cost. Fig. 2.12(a) shows the BE mesh for the exterior domain and Fig. 2.12(b) shows the BE mesh of the cone internal water. The BE meshes were generated by ANSYS 11. r z Axis of rotation (a) BE mesh for the exterior domain r z Axis of rotation (b) BE mesh for the tailcone internal water Figure 2.12: BE meshes for the submarine pressure hull Application of the boundary element method results in the following system of equations G Γfˆv + H Γfˆp = ˆp inc (2.49) where G Γf and H Γf are the BEM system matrices, ˆv is the vector of normal velocities at the collocation points and ˆp is the vector of pressures at the collocation points. The vector ˆp inc contains the pressure in the collocation points due to the dipole correlated to the operation of the propeller in a non-uniform wake and is computed using equation (2.44). For the interior fluid domain, the right hand side of equation (2.49) becomes zero. 41

76 2 Structural/Acoustic Modelling of a Submarine As the dipole pressure ˆp inc,prop due to axial propeller fluctuation depends on the axial surface normal velocity of the propeller, it can be expressed in terms of ˆv ˆp inc,prop = G Γf,propˆv (2.50) The sparse matrix G Γf,prop is computed using equations (2.44) and (2.45) and subtracted from matrix G Γf. The pressure at a set of points in the acoustic domain is obtained by p Ωf = G Ωfˆv + H Ωfˆp + p inc,ωf (2.51) where G Ωf and H Ωf are matrices that arise from integration of Green s function and its normal derivative over the boundary. p inc,ωf is a vector of pressure values at the field points due to the incident field. For computation of the BE matrices, a software implemented in SciPy and C++ was used. Strong coupling of the acoustic BE and the structural FE methods is achieved by imposing that (i) the normal velocity of the structure equals the normal velocity of the fluid and (ii) the normal distributed surface load of the structure equals the acoustic surface pressure at the structure/fluid interface. For non-conforming meshes at the coupling interface, condition (i) can not be considered in a strong sense. Therefore, an approach similar to that presented in Ref. [85] is used. The pressure can be interpreted as a Lagrange multiplier and continuity of the surface normal velocity is only established in a weak sense [105] φ f v f dγ int = φ f v s dγ int (2.52) Γ int Γ int where Γ int is the fluid/structure interface. The test function φ f corresponds to the global interpolation function for the fluid domain variables. v s and v f are the surface normal velocities of the structural and fluid domains, respectively. Equation (2.52) can be expressed in matrix form as 42

77 2.4 Computational Model of a Submarine L fˆv = L s û, (2.53) where L f = Γ int N T f N f dγ int and L s = jω Γ int N T f n T N s dγ int = L T sf. Equations (2.49), (2.48) and (2.53) can be combined to form the saddle point problem given by A L sf 0 û ˆf 0 H Γf G Γf ˆp = ˆp inc L s 0 L f ˆv 0 (2.54) When cone internal water is considered using boundary elements, the matrices G Γf and H Γf are combinations of the matrices G Γf and H Γf for the exterior radiation problem and the interior cavity problem. In addition, the vector ˆp inc has zero entries for collocation points correlated to the interior domain. When discontinuous boundary elements are used, the matrix L f becomes a diagonal matrix and the last row of equation (2.54) can be explicitly solved for û. The reduced system of equations can then be written as A L sf û ˆf = G Γf L fs H Γf ˆp }{{}}{{} S x ˆp inc (2.55) where L fs = L 1 f L T sf. The system of equations given by equation (2.55) was solved using SciPy routines and UMFPACK. When the solution vector of equation (2.55) is known, the pressure vector p Ωf can be found by combining equations (2.51) and (2.53) p Ωf = Tx + p inc,ωf (2.56) with the acoustic transfer matrix ] T = [G Ωf L fs H Ωf (2.57) 43

78 2 Structural/Acoustic Modelling of a Submarine Sound power far field approximation The complex radiated sound power through a surface Λ is given by [61] Π = 1 2 pv dλ (2.58) Λ where p is the acoustic pressure of the fluid and v is the normal velocity of a fluid particle at the surface. If the surface Λ is spherical and in the far-field with respect to the sound sources, equation (2.58) simplifies to [5] Π 1 2ρ f c f Λ pp dλ (2.59) When Λ is subdivided into polygons and the pressure is expressed as a piecewise linear approximation, then equation (2.59) can be rewritten as Π = p H ΛΦp Λ (2.60) where p Λ is the vector of pressures in the vertices of the polygons and the diagonal matrix Φ describes the geometry of Λ and the fluid properties. The sound power radiated from the submarine is computed by setting p Λ = p Ωf, where p Ωf is the vector of sound pressure values due to sound radiation from the submarine at the integration points for the sphere Conditioning of the BE matrices The fluid domain has been modelled using the direct boundary element method, where the pressure based Helmholtz integral equation given by equation (C.7) is considered. For both exterior radiation problems and interior cavity problems, the double layer potential matrix H Γf in equation (C.15) is identical. This implies that the matrix becomes ill-conditioned at the resonant frequencies of the interior Dirichlet problem, where the pressure on the boundary is prescribed as zero. An 44

79 2.4 Computational Model of a Submarine axisymmetric boundary element model for a cylinder of 45m length and 3.25 m radius has been checked for irregular frequencies. The condition numbers for both the single layer potential matrix G Γf and the double layer potential matrix H Γf are shown in Fig for the frequency range of interest. The condition number for the single layer potential matrix is less than ten for the whole frequency range. The condition number for the double layer potential matrix H Γf is around 10 3 which is considerably higher than for the double layer matrix. However this is still sufficient for a double precision machine with a numerical accuracy of around As a resonant frequency of the interior Dirichlet problem is assumed to involve condition numbers higher than for the double layer potential matrix, it can be concluded that no resonance frequencies for the interior Dirichlet problem occurs at the frequency range of interest. Hence, no special care has to be taken as suggested by Refs. [77] and [76] Single layer potential matrix Double layer potential matrix 1000 Condition number Frequency (Hz) Figure 2.13: Condition numbers for the BE matrices 45

80 Chapter 3 Structural and Acoustic Responses of a Submarine due to Propeller Forces 3.1 Introduction In this chapter, analytical and computational results for the structural and acoustic responses of a submarine excited by propeller forces are presented. Analytical and numerical results are initially compared for a cylindrical hull model only, under axial excitation at the stern end plate. In the analytical model, fluid loading was taken into account using additional terms in the equations for the cylindrical pressure hull. The frequency response function for the stern and bow end plate displacements due to a unity force excitation of the stern end plate was computed in order to identify the hull axial resonances. The vibration patterns of the hull and the sound pressure directivities at the resonance frequencies have been investigated. Results for a more detailed FE/BE model are then presented, in which the hull is under excitation from both forces transmitted through the propeller/shafting system and acoustic excitation of the hull. The overall radiated sound power is due to structure-borne sound from the hull and sound radiated directly from the propeller. The influence of the tailcone properties and the effect of a resonance changer on the hull structural response and radiated sound power are presented. 3.2 Analytical and Computational Results for the Hull Results are presented for a ring-stiffened steel cylinder of diameter 6.5 m, hull plate thickness 40 mm, length 45 m, and with two evenly spaced bulkheads of thickness 46

81 3.2 Analytical and Computational Results for the Hull 40 mm. The stiffeners have a cross-sectional area of m 2 and are 0.5 m apart. The cylinder is submerged in water (density of 1000 kg /m 3 ). A neutrally buoyant condition is maintained by using a distributed mass of 1000 kg /m 2, and with lumped masses of 200 tonnes at each end. Internal structural damping is included in the analysis by using a structural loss factor of In the FE model, the ring stiffeners were modelled with a rectangular cross-section with axial and radial edges of length 0.08 m and 0.15 m respectively. The spacing between the rings is 0.5m. In the analytical model, the stiffeners were smeared over the shell to result in an increase in the hoop stiffness. A primary axial force of 1 N was applied to one end of the finite cylindrical shell. Only the natural frequencies of the accordion modes were obtained corresponding to axisymmetric motion of the hull. The numerical models use either Reissner-Mindlin or Kirchhoff-Love shell theory to model the cylindrical hull and the bulkheads. The fluid domain is represented by boundary elements, where sufficient conditioning of the BE matrices is considered. The tailcone has not been considered in the models Axial responses at the cylinder ends Figures 3.1 and 3.2 show the axial displacement at z = 0 and z = l h, respectively, as a function of frequency. The results for the analytical model, the numerical model using Reissner-Mindlin (RM) shell theory and the numerical model using Kirchhoff-Love (KL) shell theory are shown. It can be observed that the pressure hull follows rigid body motion at very low frequencies. The first three axial modes for the accordion modes occur at frequencies of around 19, 40 and 64 Hz for the numerical model using RM theory, and at frequencies of around 19, 40 and 60 Hz for the numerical model using KL theory as well as for the analytical model. The small peaks at approximately 9 and 36 Hz are caused by resonance of the bulkheads Axial and radial displacements The axial and radial displacements for the first three axial hull modes as a function of axial position along the hull (from z = 0 to l h ) are given in Figs. 3.3 to 3.8. In 47

82 3 Structural and Acoustic Responses of a Submarine Displacement magnitude (db re m) Analytical Numerical RM Numerical KL Frequency (Hz) Figure 3.1: Axial response of the hull at z = 0 Displacement magnitude (db re m) Analytical Numerical RM Numerical KL Frequency (Hz) Figure 3.2: Axial response of the hull at z = l h 48

83 3.2 Analytical and Computational Results for the Hull each figure, the results obtained analytically and numerically using RM and KL shell theory are compared. Figures 3.3 to 3.5 show the axial displacement for the first three axial modes, in which the phase relationship between the ends of the pressure hull can clearly be observed. At the first and third axial resonant frequencies in Figs. 3.3 and 3.5 respectively, the end plates of the cylinder are vibrating out of phase with each other. At the second axial mode (Fig. 3.4), the end plates of the cylinder are vibrating in phase. Good qualitative agreement between the analytical and computational results of the structural responses is obtained for the first and second hull axial resonances. However, the amplitudes differ by up to 25% for the third axial resonance, as the amplitude of the axial displacement is significantly larger for the analytical model, as shown in Fig This is attributed to the fact that radiation damping of the end plates was neglected in the presented model due to mathematical complexity. 3e-09 2e-09 Axial displacement (m) 1e e-09-2e-09 Analytical Numerical RM -3e Numerical KL Axial position (m) Figure 3.3: Axial displacement at the first axial resonance 49

84 3 Structural and Acoustic Responses of a Submarine 4e-10 3e-10 Axial displacement (m) 2e-10 1e e-10-2e-10 Analytical Numerical RM -3e Numerical KL Axial position (m) Figure 3.4: Axial displacement at the second axial resonance Axial displacement (m) 1e-10 8e-11 6e-11 4e-11 2e e-11-4e-11-6e-11-8e-11 Analytical Numerical RM Numerical KL -1e Axial position (m) Figure 3.5: Axial displacement at the third axial resonance 50

85 3.2 Analytical and Computational Results for the Hull 0-2e-11 Radial displacement (m) -4e-11-6e-11-8e-11-1e e e-10 Analytical Numerical RM Numericaal KL Axial position (m) Figure 3.6: Radial displacement at the first axial resonance Figures 3.6 to 3.8 present the radial response of the pressure hull for the first three axial modes. In these figures, the localised effect of the bulkheads on the radial displacement is shown. In addition, the effect of the individual ring stiffeners which were modelled as discrete elements in the computational results of the radial displacement is clearly observed. This effect is not observed in the analytical results of the radial displacement as the stiffeners are modelled using orthotropic shell properties. Comparison of the axial and radial displacements shows that when the axial response is a maximum, the radial response is a minimum, and vice versa, due to the Poisson effect which causes coupling between the axial and radial motion Radiation directivity patterns The total far-field pressure radiated by the hull consists of contributions due to axial motion of the end plates and radial motion of the cylindrical shell. Figures 3.9 to 3.11 present the structurally radiated pressure for the first three axial modes, at a distance R of 1000 m from the geometric centre of the submarine hull. The direction of the radiation directivity corresponds to Fig. 2.7, where the bow of the 51

86 3 Structural and Acoustic Responses of a Submarine 4e-11 3e-11 Radial displacement (m) 2e-11 1e e-11-2e-11 Analytical -3e-11-4e Numerical RM Numerical KL Axial position (m) Figure 3.7: Radial displacement at the second axial resonance 1e-11 8e-12 6e-12 Radial displacement (m) 4e-12 2e e-12-4e-12-6e-12-8e-12-1e Analytical Numerical RM Numerical KL Axial position (m) Figure 3.8: Radial displacement at the third axial resonance 52

87 3.2 Analytical and Computational Results for the Hull submarine is at 0, the stern is at 180 and the exciting force is applied at the stern. As the frequency increases, the radiation directivity increases in complexity which is characterised by a greater number of lobes as well as a loss of symmetry. The radiation directivity at the first axial mode is shown in Fig At 90, the sound pressure obtained from the analytical model is predicted to be about 15% higher than the sound pressure obtained from the numerical models. At 0 and 180, the highest predicted sound pressure is 100% greater than the lowest predicted sound pressure. The highest sound pressure is obtained from the model using RM theory, whereas the lowest sound pressure is obtained from the analytical model. The radiation directivity at the second axial mode is shown in Fig The sound pressure predicted by the numerical models is about 40% higher than the sound pressure obtained from the analytical model. Good agreement can be observed for the radiation directivities at the third hull axial resonance shown in Fig The effect of damping in reducing the pressure amplitude increases as the frequency increases, which is observed in the results. Pressure (Pa) 90 Analytical Numerical RM Numerical KL e Figure 3.9: Radiated sound pressure at the first axial mode 53

88 3 Structural and Acoustic Responses of a Submarine Pressure (Pa) 90 Analytical Numerical RM Numerical KL e Figure 3.10: Radiated sound pressure at the second axial mode Pressure (Pa) 90 Analytical Numerical RM Numerical KL e Figure 3.11: Radiated sound pressure at the third axial mode 54

89 3.2 Analytical and Computational Results for the Hull Radiation modes of the submarine hull The radiation efficiency of a vibrating structure is defined as [61] σ r = Π ρ f c f S 0 v n 2 (3.1) where S 0 denotes the surface area of the radiating structure and the the time and space averaged surface normal velocity is given by vn 2 = 1 v n 2 ds 0 = 2S 0 S 0 1 2S 0 ˆv H L fˆv (3.2) The radiated sound power can be approximately computed using the Rayleigh integral Π ˆv H R { G H Ω f ΦG Ωf } ˆv (3.3) Using equations (3.2) and (3.3) in equation (3.1) and reordering yields [ 2 R { ] } G H Ω ρ f c f ΦG Ωf σr L f ˆv = 0 (3.4) f Equation (3.4) represents a general eigenvalue problem, where the eigenvectors are the radiation modes and the eigenvalues are the corresponding radiation efficiencies [84]. The three most efficient radiation modes of the submarine hull at the first, second and third hull axial resonant frequencies are investigated, where the resonant frequencies obtained from the model using Reissner-Mindlin shell elements have been used. The three most efficient radiation modes for the first hull axial resonant frequency at 19.3 Hz are shown in Fig. 3.12, where the generator for the axisymmetric cylindrical hull is represented by the thin line and the modeshape is represented by the thick line. For radiation mode 1, no nodes occur on the shell surface and all nodal normal velocities have the same sign. At the first axial resonance, mode 1 has the highest radiation efficiency of Radiation mode 1 is comparable to the 55

90 3 Structural and Acoustic Responses of a Submarine radial displacement of the hull at the first axial resonance shown in Fig. 3.6, which implies that the submarine hull is an efficient sound radiator at 19.3 Hz. Radiation mode 2 has one node and the second highest radiation efficiency of Radiation mode 3 has two nodes and a radiation efficiency of , which is low compared to the radiation efficiencies of modes 1 and 2. The radiation modes at the second hull axial resonant frequency at 39.8 Hz are shown in Fig Mode 1 has the highest radiation efficiency of 0.745, however mode 2 has a radiation efficiency of which is similar to the radiation efficiency of mode 1. Radiation mode 2 is comparable to the radial displacement of the hull at the second hull axial resonance shown in Fig Hence, the submarine hull is also an efficient sound radiator at 39.8 Hz. The radiation modes at the third hull axial resonance frequency at 65.3 Hz are shown in Fig Mode 3 has the highest radiation efficiency of 1.1. Modes 1 and 2 have radiation efficiencies of and 1.05 respectively, which are of similar order as the radiation efficiency for mode 3. In contrast to the first and second hull axial resonance frequency, radiation mode 2 has 3 nodes instead of 1 node for the third axial resonance frequency. Radiation mode 3 can be compared to the radial displacement of the hull at the third hull axial resonance shown in Fig This indicates that the submarine hall is also an efficient sound radiator at 65.3 Hz. (a) Mode 1, σ r = 0.35 (b) Mode 2, σ r = (c) Mode 3, σ r = Figure 3.12: Most efficient radiation modes at the first hull axial resonant frequency 56

91 3.2 Analytical and Computational Results for the Hull (a) Mode 1, σ r = (b) Mode 2, σ r = (c) Mode 3, σ r = Figure 3.13: Most efficient radiation modes at the second hull axial resonant frequency (a) Mode 1, σ r = (b) Mode 2, σ r = 1.05 (c) Mode 3, σ r = 1.1 Figure 3.14: Most efficient radiation modes at the third hull axial resonant frequency 57

92 3 Structural and Acoustic Responses of a Submarine 3.3 Influence of Tailcone and Propeller/Shafting System Properties on the Structural and Acoustic Responses Computations were performed on fully-coupled numerical models, where the fluid domain is represented by direct discontinuous boundary elements and the structural domain is represented by finite elements. Reissner-Mindlin shell elements were used to model shell structures such as the cylindrical pressure hull, the bulkheads and the foundation. Rod elements were used to model the propeller shaft. The thrust bearing and the resonance changer are represented by spring-damper and mass elements. For some of the presented models, the internal fluid of the cone has been simplified to a lumped mass. A two-field problem has been formulated and convergence was achieved with a grid as shown in Fig. 2.12(a). In this case, there are at least 20 elements per fluid wavelength present. For the models where the cone internal fluid is modelled using boundary elements, a three-field problem has been formulated and a much smaller element size is required to achieve convergence. Results for a model scale free-flooded tailcone obtained from the computer programs used in this thesis are compared to experimental results in Appendix A for validation purposes. The mesh for the cone internal fluid is shown in Fig. 2.12(b). The same element size was used for the exterior fluid domain at the cone for the three-field model. The model parameters are given in Tables 3.1 and 3.2 for the propeller/shafting system and the submarine hull, respectively. The majority of the properties for the propeller-shafting system, including the resonance changer, are from Ref. [55] Dynamic behaviour of the propeller/shafting system Results for the dynamic response of the propeller/shafting system are presented with and without a resonance changer, with emphasis on the force transmissibility and propeller vibration. The FE model for the propeller/shafting system is depicted in Fig The propeller was acoustically represented by a rigid fluctuating disc. The analytical model given by equations (2.44) to (2.47) was 58

93 3.3 Influence of Tailcone and Propeller/Shafting System Properties Table 3.1: Data for the propeller/shafting system Parameter Value Unit Propeller structural mass 10,000 kg Propeller added mass of water 11,443 kg Shaft Young s modulus 200 GPa Shaft Poisson s ratio 0.3 Shaft density 7,800 kg/m 3 Shaft cross-sect. area m 2 Shaft length 10.5 m Effective shaft length 9 m Bearing mass 200 kg Bearing stiffness 20,000 MN/m Bearing damping 300,000 kg/s Resonance changer mass 1,000 kg Resonance changer stiffness 169 MN/m Resonance changer damping kg/s Foundation major radius 1.25 m Foundation minor radius 0.52 m Foundation half angle 15 deg Foundation thickness 10 mm Foundation Young s modulus 200 GPa Foundation density 7,800 kg/m 3 59

94 3 Structural and Acoustic Responses of a Submarine Table 3.2: Data for the submarine hull Parameter Value Unit Cylinder length 45.0 m Cylinder radius 3.25 m Shell thickness 0.04 m Stiffener cross-sectional area m 2 Stiffener spacing 0.5 m Young s modulus of structure without foundation 210 GPa Young s modulus of foundation 200 GPa Poisson ratio of structure 0.3 Density of structure 7,800 kg/m 3 Structural loss factor 0.02 Added mass 796 kg/m 2 Stern lumped mass (two-field problem) kg Stern lumped mass (three-field problem) kg Bow lumped mass kg Cone half angle 24 deg Cone length m Cone smaller radius 0.3 m Density of fluid 1,000 kg/m 3 Speed of sound 1,500 m/s 60

95 3.3 Influence of Tailcone and Propeller/Shafting System Properties Mass element m r n 5 r m p m b k b, c b k r, c r A s, ρ s n 3 n 6 n 7 n 8 n 9 z n 1 n 2 n 4 Shell element Rod element Spring-damper element Figure 3.15: FE model for the propeller/shafting system, where n i denotes FE node i. Nodes n 1 to n 4 have only translatory axial degrees of freedom. Node n 5 has only one virtual translatory degree of freedom which is defined as the difference between the axial displacements of nodes n 3 and n 4. Nodes n 6 to n 9 belong to the foundation, which is only partially shown. The nodes of the foundation have both radial and axial translatory degrees of freedom and one rotational degree of freedom around the circumferential coordinate. 61

96 3 Structural and Acoustic Responses of a Submarine used to represent the propeller. When the dipole at the propeller hub due to the operation of the propeller in a non-uniform wake is taken into account, a semianalytical model for the propeller is required. Otherwise, the fluctuating rigid disc can be modelled using boundary elements. The semi-analytical model was derived for a fluctuating rigid disc in a free field. However, the submarine hull close to the propeller influences the radiation impedance for the propeller. The real and imaginary components of the radiation impedance for a rigid disc using the analytical model, the boundary element model in a free field and the boundary element model near the submarine hull are given in Fig The reactance and resistance impedances are both greater for the numerical models than for the semi-analytical model. This is attributed to the disc having finite thickness for the numerical models. The presence of the submarine hull does not have a significant influence on the reactance which corresponds to the mass loading effect. However, the hull axial resonances lead to a slight local increase in resistance for the disc in addition to other diffraction and coupling effects at higher frequencies. In the subsequent analysis, the propeller is modelled using the semi-analytical model for a rigid disc. The resistance is not taken into account as it is small compared to the reactance. The axial force on the propeller is proportional to the square of propeller rotational frequency. In Ref. [55], the maximum of the ω 2 -weighted force transmissibility in the frequency range of interest was considered as the cost function, when optimising the RC virtual stiffness, damping and mass parameters for a simplified submarine/shafting system model of dimensions similar to the models in this work. Acoustic excitation of the submarine hull was not taken into account. Optimisation of the weighted force transmissibility was chosen under the assumption that it corresponds to the radiated sound power from the hull. Due to the frequency weighting, lower force transmissibility of the optimised system was obtained at the higher frequency range. The RC parameters from Ref. [55] have been employed in this work as the structural and geometric parameters of the submarine models presented here and in Ref. [55] are similar. 62

97 3.3 Influence of Tailcone and Propeller/Shafting System Properties Semi-analytical solution BE solution in free field BE solution with submarine hull Resistance Frequency (Hz) 9e+06 8e+06 7e+06 Semi-analytical solution BE solution in free field BE solution with submarine hull 6e+06 Reactance 5e+06 4e+06 3e+06 2e+06 1e Frequency (Hz) Figure 3.16: Real and imaginary parts of the radiation impedance for a rigid disc subject to an axial force at its centre using semi-analytical and numerical models 63

98 3 Structural and Acoustic Responses of a Submarine The drive point impedance is examined here in order to investigate its influence on the force transmissibility of the propeller/shafting system. The force transmissibility of the structurally excited propeller/shafting system with a rigid termination is compared to the force transmissibility of the propeller/shafting system that is coupled to the simplified physical model of the fluid loaded submarine hull. The results are shown in Fig The local peaks at 21 Hz and 43 Hz are the first and second hull axial resonant frequencies. The RC reduces the maximum force transmissibility and detunes the fundamental resonant frequency of the propeller/shafting system from 37 Hz to 12 Hz. This is much lower than the first axial eigenfrequency for the shaft of about 250 Hz. In addition, the RC leads to a significant decrease of the force transmissibility above the fundamental resonant frequency. It is evident that the 180 phase shift due to the resonant frequency of the propeller/shafting system is not sustained in the case of the model with the RC, but reduces gradually to a 58 phase shift. This is assumed to play an important role for the combined dipole and structural excitation, as cancellation or reinforcement effects may occur, depending on their relative phase. The difference between the model with a rigid termination and the model with the simplified physical model of the submarine are the local peaks at the hull resonant frequencies. Comparing the results with a rigid termination and a physical model of the hull shows that the effect of the drive point impedance on the force transmission is very small for the first hull axial resonant frequency (21 Hz) and and negligible for the second and higher hull resonant frequencies. Clearly, the natural resonances of the submarine hull at 21 Hz and 43 Hz have much less influence on the model with the RC, showing that the propeller/shafting system with the RC is less strongly coupled to the submarine hull. The axial mobility of the propeller with and without the RC is shown in Fig. 3.18, where the simplified physical model of the submarine has been used. The maximum vibration of the propeller occurs at the fundamental resonant frequency of the propeller/shafting system, leading to increased sound radiation from the propeller. 64

99 3.3 Influence of Tailcone and Propeller/Shafting System Properties 100 No RC Force transmissibility RC RC, rigid termination Frequency (Hz) No RC RC RC, rigid termination Phase (degrees) Frequency (Hz) Figure 3.17: Force transmissibility of the propeller/shafting system 65

100 3 Structural and Acoustic Responses of a Submarine Influence of the tailcone properties on the structural and acoustic responses of the submarine hull For structural and acoustic optimisation problems, computationally cost efficient models are required. However, for the free-flooded cone, a three-field problem is present rather than a two-field problem, leading to significantly larger and denser system matrices. An investigation of different tailcone models is therefore of interest. A first approximation to the tailcone is a rigid cone with no internal water, where the mass effect of the water is considered as a lumped mass at the pressure hull end plate in order to maintain the global dynamic behaviour of the hull. However, the stiffness of the tailcone has an influence on the sound radiation as well as the sensitivity of the structure to excitation from a nearby sound source. Therefore, additional models were developed that represent a link between the model with internal fluid loading and the model with a rigid tailcone. The range of examined models includes: a) a flexible tailcone with internal water; b) a flexible tailcone with the same structural parameters as a) but with a lumped mass representation of the internal water, where the lumped mass is attached to the stern end plate; c) a flexible tailcone with the lumped mass representation of the water as used for b), but with an increased Young s modulus; d) a rigid tailcone with the lumped mass representation of the internal water as used for b) and c). The mobility of the stern end plate is used to compare the models, where a unity exciting force is applied to (i) the stern end plate or a dipole to represent a unity net force on the water is applied at (ii) near the end of the tailcone, as shown in Fig Results obtained using (i) allow assessment of the global behaviour of 66

101 3.3 Influence of Tailcone and Propeller/Shafting System Properties 1e-05 No RC RC 1e-06 Mobility (m/ns) 1e-07 1e-08 1e Frequency (Hz) Figure 3.18: Axial propeller mobility amplitude 0.46 m (i) (ii) Figure 3.19: Locations of the excitations to investigate the influence of the tailcone properties on the structural and acoustic responses of the submarine 67

102 3 Structural and Acoustic Responses of a Submarine the model due to excitation via the shaft, whereas results obtained using (ii) are relevant for excitation by the propeller pressure field. Figure 3.20 shows the point mobility of the hull stern end plate. The lumped mass used to simulate the effect of the cone internal water was chosen such that the simplified model yields the best match with the model that includes internal water. The resonant frequencies and amplitude for the drive point mobility were chosen as criteria for the comparison of the models. A lumped mass that is about half the mass of the internal water was found to be the best approximation. In the case of the flexible cone without internal water, the stiffness was increased through the Young s modulus. The global behaviour of the structure is seen to be only weakly dependent on assumptions about the tailcone properties for the models with no internal water. However, the structural mobility of the hull end plate changes for the model with internal water as local resonances of the cone occur within the investigated frequency range. The first four hull axial resonances can be identified at 21, 43, 70 and 99 Hz for tailcone models b) to d). For model a), the first axial resonance of the cone at 39 Hz shifts the first resonance of the hull down by about 2 Hz and the second resonance up by about 4 Hz. Figure 3.21 shows the transfer mobility of the hull stern end plate for a dipole applied at 0.46 m distance from the end of the tailcone. The results for model a) and models b) to d) now differ significantly for frequencies above 20 Hz. The results for models b) to d) diverge above about 43 Hz. It can be seen that an increase in flexibility leads to stronger responses for models b) to d). Using model a), cone resonances also occur at 39, 64, 81 and 96 Hz. It can be concluded that the cone properties will not have a crucial impact on the structural response of the submarine hull for structural excitation. The structural response will, however, be underestimated for dipole excitation, where significant forces are applied near the end of the tailcone. The influence of the assumed cone properties on radiated sound power is more significant, when a dipole is applied at the propeller end of the cone. Figures 3.22 and 3.23 show the sound power level for the different representations of the cone, 68

103 3.3 Influence of Tailcone and Propeller/Shafting System Properties 1e-06 1e-07 Mobility (m/ns) 1e-08 1e-09 a) Flexible cone / internal water b) Flexible cone / lumped mass c) Stiffened flexible cone / lumped mass 1e d) Rigid cone / lumped mass Frequency (Hz) Figure 3.20: Point mobility of the stern end plate using different representations of the free-flooded tailcone Mobility (m/ns) 1e-06 1e-07 1e-08 1e-09 a) Flexible cone / internal water b) Flexible cone / lumped mass c) Stiffened flexible cone / lumped mass d) Rigid cone / lumped mass 1e Frequency (Hz) Figure 3.21: Mobility of the stern end plate due to dipole excitation near the tailcone end, using different representations of the free-flooded tailcone 69

104 3 Structural and Acoustic Responses of a Submarine when an axial force is applied at the stern end plate and when a dipole is applied at 0.46 m distance from the tailcone end, respectively. In Fig. 3.22, the results are all similar apart from the shift of the second hull axial resonant frequency. When a dipole is applied near the propeller end of the cone (Fig. 3.23), the assumed cone flexibility now has a significant influence on the radiated sound power at frequencies above about 25 Hz. The rigid cone has been used in the complete FE/BE model that includes both the submarine hull and the propeller/shafting system as hull excitation via the shaft is dominant and the focus is on the influence of propeller/shafting system properties on the structural and acoustic responses of the submarine hull. Fig shows that excitation due to the dipole is likely to be underestimated at frequencies above about 30 Hz, particularly at the cone resonances Structural and acoustic responses for the coupled model under structural and acoustic excitation The structural and acoustic responses of the complete submarine model, corresponding to the hull coupled to the propeller/shafting system, are presented here, where the tailcone was modelled as a rigid structure. Results are shown with and without the use of a resonance changer. The RC parameters used here are from Ref. [55], in which a simplified analytical submarine model was used and excitation of the submarine hull via the fluid was not taken into account [55]. In the model presented here, both structural excitation through the propeller/shafting system and acoustic excitation of the submarine hull are considered. The acoustic excitation is due to dipole sound radiation caused by operation of the propeller in the non-uniform wake as well as propeller vibration. Both the structural force and the dipols are applied at the propeller hub. The propeller hub is located at a distance of 0.46 m from the stern as depicted in Fig The acoustic response in the far field is a combination of sound radiated from the submarine hull due to structural and acoustic excitation, sound radiated directly from the propeller as well as sound radiated from the propeller and diffracted by the submarine hull. 70

105 3.3 Influence of Tailcone and Propeller/Shafting System Properties Sound power level (db re W) for 1 N force a) Flexible cone / internal water b) Flexible cone / lumped mass c) Stiffened flexible cone d) Rigid cone Frequency (Hz) Figure 3.22: Sound power level due to structural excitation of the stern end plate, using different representations of the free-flooded tailcone Sound power level (db re W) for 1 N force a) Flexible cone / internal water b) Flexible cone / lumped mass c) Stiffened flexible cone d) Rigid cone Frequency (Hz) Figure 3.23: Sound power level due to dipole excitation near the tailcone end, using different representations of the free-flooded tailcone 71

106 3 Structural and Acoustic Responses of a Submarine In order to represent excitation of the system due to the operation of the propeller in a non-uniform wake, a unity force has been applied to the propeller hub as well as the corresponding dipole. This dipole is equal in magnitude but opposite in phase with respect to the structural force. The dipole field due to propeller vibration has also been taken into account. The net force on the water due to the dipole that corresponds to propeller vibration shown in Fig is implicitly computed in the coupled models. It can be observed that the force has its maximum at the fundamental resonance of the propeller/shafting system. Presence of an RC results in a smaller maximum force for the investigated frequency range. Figure 3.26 shows the mobility of the hull stern end plate, due to structural excitation at the propeller hub. For comparison, results are also presented for models where linear finite shell elements based on Kirchhoff-Love (KL) assumptions have been used for the structure [56]. Sysnoise was utilised to obtain the solution for the coupled model using boundary elements for the fluid. The frequencies are slightly shifted downwards for the models using Sysnoise/KL. The reason is that the Reissner-Mindlin elements are stiffer than the Kirchhoff-Love elements. It is evident that vibration of the hull is significantly reduced by the RC for frequencies greater than 20 Hz. However, when considering acoustic excitation, the effect of the RC on the end plate mobility is decreased, as shown in Fig The acoustic far field of the dipole plays an important role in the excitation of the submarine hull at higher frequencies, since the radius of transition from the near to the far field decreases with an increase of frequency. The sound power was computed by integration of the intensity over a sphere of 1000 m radius, with the centre on the axis of the submarine and 22.5 m forward in the axial direction from the stern end plate. For models with no dipole sources present, the sound power was obtained by integration of the intensity over the submarine surface. Four cases have been examined. For each case, results with and without an RC are presented. For the first case, only structural excitation of the submarine hull through the propeller/shafting system has been considered. For the second case, excitation by the dipole field due directly to operation of the propeller 72

107 3.3 Influence of Tailcone and Propeller/Shafting System Properties 0.46 m PSS End plate Force Dipoles Figure 3.24: Locations of the excitations to investigate the influence of the PSS on the structural and acoustic responses of the submarine 10 No RC RC Force (N) Frequency (Hz) Figure 3.25: Net force on the water corresponding to the dipole due to propeller vibration. 73

108 3 Structural and Acoustic Responses of a Submarine 1e-06 No RC RC Mobility (m/ns) 1e-07 1e-08 1e-09 No RC (Sysnoise/KL) RC (Sysnoise/KL) 1e Frequency (Hz) Figure 3.26: Mobility of the stern end plate ignoring acoustic excitation 1e-06 No RC RC 1e-07 Mobility (m/ns) 1e-08 1e-09 1e Frequency (Hz) Figure 3.27: Mobility of the stern end plate including acoustic excitation (dipole due to non-uniform wake and dipole due to propeller vibration) 74

109 3.3 Influence of Tailcone and Propeller/Shafting System Properties in a non-uniform wake is taken into account in addition to structural excitation via the propeller shaft. For the third case, excitation by the dipole field caused by propeller vibration is taken into account in addition to structural excitation via the propeller shaft. For the fourth case, structural excitation via the propeller shaft and acoustic excitation by both the dipole field due to operation of the propeller in a non-uniform wake and the dipole field caused by propeller vibration have been considered. Figure 3.28 presents the results for the first case and shows that implementation of an RC in the propeller/shafting system leads to a significant reduction of overall radiated sound power above 25 Hz. Results are also given for the model using linear finite elements and Sysnoise/KL. Increased sound radiation at frequencies above 83 Hz can be identified for the models using Sysnoise/KL, where the downward shift of the fourth axial resonant frequency of the hull due to the less stiff Kirchhoff-Love elements becomes evident. The peak at 37 Hz due to the fundamental resonance of the propeller/shafting system shifts to 12 Hz when an RC is implemented, leading to an increase in radiated sound power at lower frequencies. For the second case in Fig. 3.29, the sound radiation due to both structural and dipole excitation without the RC is greater than in Fig. 3.28, particularly at higher frequencies. This is attributed to excitation of the hull by the dipole field. Figure 3.29 shows that excitation of the hull via the fluid has limited the effect of the RC to frequencies below 60 Hz. For the third case in Fig. 3.30, a reduction in overall radiated sound power is predicted for frequencies above about 17 Hz, when an RC is introduced. However, the beneficial effect of the RC is not as significant as for the case in Fig. 3.28, where only structural excitation is taken into account. Figure 3.31 shows the sound power level for the fourth case, in which both the dipole due to propeller vibration as well as the dipole due to the operation of the propeller in a non-uniform wake are taken into account. Implementation of an RC does not lead to a decrease of sound radiation at higher frequencies due to dominance of dipole excitation. In addition, the dipole due to propeller vibration has a smaller magnitude at higher frequencies, when an RC is implemented. Therefore, 75

110 3 Structural and Acoustic Responses of a Submarine Sound power level (db re W) for 1 N force No RC RC No RC (Sysnoise/KL) RC (Sysnoise/KL) Frequency (Hz) Figure 3.28: Sound power level due to structural excitation Sound power level (db re W) for 1 N force No RC 80 RC Frequency (Hz) Figure 3.29: Sound power level, where structural and acoustic excitation of the hull are considered. The dipole due to propeller vibration is not taken into account. 76

111 3.4 Conclusions a cancellation effect of the two dipoles is more significant for the model without an RC. The dipole field due to propeller vibration causes an increase in radiated sound power at the fundamental resonance of the propeller/shafting system, when an RC is implemented. 3.4 Conclusions The structural and acoustic responses for a fluid loaded submarine pressure hull have been presented. The hull was axially excited from the propeller/shafting system, resulting in excitation of the hull axial modes. The total radiated sound pressure was contributed by axial motion of the cylinder end closures and radial displacement of the pressure hull flank. Results were observed for the first three axial resonant frequencies of the hull circumferential breathing modes. Good agreement between the models was obtained for the frequency responses at each end of the hull, for the axial displacement along the hull length, and for the radiation directivity patterns corresponding to the first three axial modes. Discrepancies between the analytical and computational models for the amplitude levels of the displacements and pressure were observed, which are attributed to the differing underlying assumptions used for the modelling techniques, associated with the fluid loading and the shell assumptions. The radiation modes with the three highest radiation efficiencies have been investigated for the computational models. The boundary element method was used to obtain the radiation impedance of the submarine hull. The first three hull axial resonant frequencies were considered for the analysis. Some of the radiation modes were found to be comparable to the radial displacement patterns at the hull axial resonances. This implies that the submarine hull is an efficient sound radiator at its axial resonant frequencies. Axial excitation of a submarine hull due to propeller forces, where propeller forces transmitted to the hull via both the propeller/shafting system and the fluid are considered, has been investigated. The tailcone of the submarine was represented as 77

112 3 Structural and Acoustic Responses of a Submarine Sound power level (db re W) for 1 N force No RC 80 RC Frequency (Hz) Figure 3.30: Sound power level, where structural and acoustic excitation of the hull are considered. The dipole due to the operation of the propeller in a non-uniform wake is not taken into account. Sound power level (db re W) for 1 N force No RC 80 RC Frequency (Hz) Figure 3.31: Sound power level, where structural and acoustic excitation of the hull are considered. The acoustic excitation is due to the dipole caused directly by the operation of the propeller in a non-uniform wake and due to the dipole caused by propeller vibration. 78

113 3.4 Conclusions a rigid structure connected to the pressure hull. This was shown to be a reasonable approximation to a flexible free flooded cone at low frequencies, when excitation through the propeller/shafting system is dominat, but the effects on overall sound radiation of forces transmitted through the fluid to the tailcone and hull structure are likely to be underestimated. The performance of a hydraulic vibration attenuation system known as a resonance changer in the propeller/shafting system has been examined. It was found that the performance of the RC is significantly affected by the influence of dipole fields due to hydrodynamic forces, which are always present, and propeller vibration. The two dipole fields may reinforce or partially cancel each other, depending on frequency and parameter selection. The RC can provide significant reduction in overall radiated sound power at low frequencies, including propeller blade passing frequency, but will have a much more limited effect at higher frequencies where the dipole fields tend to be the dominant cause of radiated sound power. 79

114 Chapter 4 Optimum Passive and Active Control of a Submarine 4.1 Introduction In this chapter, passive and active control of the noise radiated by a submarine due to propeller forces is investigated. Work on this subject area has previously been conducted by Dylejko [1]. In the work presented by Dylejko in Ref. [1] as well as in this work, passive control is realised by implementation of a resonance changer (RC) in the propeller/shafting system (PSS). Dylejko used an analytical representation of the submarine with flat rigid end plates at the stern and at the bow. Optimum RC parameters were found by application of genetic algorithms, where the sound radiated from the hull and sound radiated due to propeller vibration were considered in an uncoupled fashion. In addition, Dylejko presented a preliminary theoretical study on active control of PSS vibration. In this work, the submarine is represented by a fully coupled finite element / boundary element model, where the complex interaction between the submarine hull and the propeller pressure field is taken into account. The propeller pressure field is due to both operation of the propeller in a non-uniform wake and propeller vibration. The stern of the submarine is represented by a rigid tailcone instead of a flat rigid end plate, where the tailcone internal water is considered as a lumped mass. Optimum parameters for the RC were found using gradient based optimisation techniques. Different approaches are investigated for active control, where either a control moment is applied to the cylindrical pressure hull or inertial actuators are implemented in the propeller/shafting system and at the stern end plate. The model data as given in 80

115 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters Tables 3.1 and 3.2 was used for optimisation of the RC parameters as well as for the active control simulations. 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters Numerical models of the sound power radiated by a submarine are presented, where the sensitivity of the weighted sound power over the relevant frequency range to resonance changer design parameters has been computed. The sensitivity is obtained in a semi-analytical way by employing the adjoint operator [102]. Optimum parameters are found for the virtual stiffness, damping and mass of the RC by applying the globally convergent method of moving asymptotes [99] Sensitivity analysis Let ϑ be a vector of the RC virtual parameters. The sensitivity of the sound power radiated by the submarine to the RC parameters ϑ is then obtained by differentiation of equation (2.60) and is given by Π ϑ = 2pH ΛΦ p Λ ϑ (4.1) The sensitivity of the pressure at the integration points with respect to the RC parameters is obtained by differentiation of equation (2.56) and is given by p Λ ϑ = T x ϑ (4.2) In order to obtain an expression for the sensitivity of the vector x with respect to the RC parameters, equation (2.55) has to be differentiated which yields S x ϑ + S ϑ x = y ϑ (4.3) 81

116 4 Optimum Passive and Active Control of a Submarine Equation (4.3) can then be reordered such that an expression for the sensitivity of the vector x with respect to the RC parameters is obtained by ( x y ϑ = S 1 ϑ S ) ϑ x (4.4) In order to compute the sensitivity of the sound power, equations (4.1) to (4.3) can be combined in an adjoint operator formulation [102] ( Π y ϑ = 2pH ΛΦTS 1 ϑ S ) ϑ x (4.5) Let b T l = 2p H Λ ΦT and zt = b T l S 1, then the sensitivity of the sound power can be found for any set of parameters ϑ, as long as the solution of the system of equations S T z = b l is known. This means that for an arbitrary number of structural design parameters, only two systems of equations have to be solved Optimisation For optimisation of the resonance changer parameters, the following cost function has been defined to represent the radiated sound power over the frequency range of interest [102] J = 1 ω ω Π(ω)dω (4.6) The gradient of the cost function can be obtained by differentiating equation (4.6) with respect to the RC parameters ϑ and is given by J ϑ = 1 ω ω Π(ω) dω (4.7) ϑ The non-linear problem of minimising the radiated sound power can be written as minimise J(ϑ) subject to ϑ ϑ ϑ (4.8) 82

117 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters where ϑ and ϑ are the lower and upper bounds for the design parameters, respectively. As the first derivatives of the cost function J with respect to the design parameters ϑ are explicitly available, an appropriate family of methods to find local minima are the quasi-newton algorithms [90]. An example of such an algorithm that is applicable to equation (4.8) is the limited memory Broyden Fletcher Goldfarb Shanno algorithm with parameter bounds (L-BFGS-B) [98]. However, applying the L-BFGS-B directly to equation (4.8) can require a large number of computationally expensive cost function evaluations. In addition, the process can get easily trapped in a numerically related local minimum. In order to reduce the number of required evaluations of J and J, an iterative algorithm can be applied, ϑ where the problem is locally approximated by an explicit subproblem minimise F(ϑ) (k 1) subject to ϑ (k 1) ϑ ϑ (k 1) (4.9) for an outer iteration point k 1. The subproblem is solved using a gradient based method such as the L-BFGS-B. The optimum parameters for the subproblem represent the next iteration point and the formulation for the next subproblem is modified based on data from previous iterations. The iteration is stopped when certain convergence criteria are fulfilled. An example for this approach is the method of moving asymptotes (MMA), where asymptotes are used to approximate the cost function [97]. For the algorithm used in this paper, inner iterations k 2 are conducted in addition to the outer iterations k 1. This approach is called the globally convergent method of moving asymptotes (GCMMA) [99]. The cost function is approximated near the iteration points using F(ϑ) (k 1,k 2 ) = ( n i=1 q (k 1,k 2 ) i ϑ (k 1) i ϑ i + σ (k 1) i + r (k 1,k 2 ) i ϑ i ϑ (k 1) i + σ (k 1) i ) q(k 1,k 2 ) i + r (k 1,k 2 ) i σ (k 1) i + J(ϑ (k 1) ) (4.10) 83

118 4 Optimum Passive and Active Control of a Submarine where n represents the number of parameters, i is the index for a parameter, ϑ (k 1) represents the optimum solution from the last outer iteration step and σ (k 1) i moving asymptotes. The parameter bounds for a subproblem are given by are the ( ) ( ) max ϑ i,ϑ (k 1) i 0.9σ (k 1) i ϑ i min ϑ i,ϑ (k 1) i + 0.9σ (k 1) i (4.11) The asymptotes are moved after each outer iteration. If the process oscillates, the asymptotes are moved closer to the iteration point to make the approximation more conservative. In contrast, if the process is slow, the asymptotes are moved away from the iteration point. This is accomplished by letting σ (k 1) i = 1 ( ) ϑi ϑ i 2 if k 1 = 1, 2 σ (k 1) i = γ (k 1) i σ (k 1 1) i if k 1 > 2 (4.12) where [99] γ (k 1) i 0.7 if 1.2 if 1.0 if ( ( ϑ (k 1) i ϑ (k 1 1) i ϑ (k 1) i ϑ (k 1 1) i ( ϑ (k 1) i ϑ (k 1 1) i ) ( ) ( ) ϑ (k 1 1) i ϑ (k 1 2) i ϑ (k 1 1) i ϑ (k 1 2) i ) ( ϑ (k 1 1) i ϑ (k 1 2) i < 0 ) > 0 ) = 0 (4.13) The coefficients q (k 1,k 2 ) i and r (k 1,k 2 ) i are given by ( q (k 1,k 2 ) i = ( r (k 1,k 2 ) i = σ (k 1) i σ (k 1) i ) { 2 max 0, J ( i ϑ i ) { 2 max 0, J ( i ϑ i ) } ϑ (k 1) + ψ(k 1,k 2 ) σ (k 1) i 4 ) } ϑ (k 1) + ψ(k 1,k 2 ) σ (k 1) i 4 (4.14) (4.15) where the parameter ψ (k 1,k 2 ) is adjusted for the inner iteration in order to achieve ( ) global convergence. This is accomplished by increasing ψ (k 1,k 2 ) until J ϑ (k 1,k 2 ) ( ) is smaller than F ϑ (k 1,k 2 ), where ϑ (k 1,k 2 ) denotes the optimum solution for the subproblem of the inner iteration. For the first iteration point at an inner iteration 84

119 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters loop, that is k 2 = 0, the parameter ψ (k 1,k 2 ) is chosen to be 1 for k 1 = 1 and [99] ) ψ (k 1,k 2 ) = max (0.1ψ (k 1 1, ˆk 2 ),ǫ 0 (4.16) for k 1 > 1, where ˆk 2 is the number of inner iterations for the outer iteration point k 1 1 and 0 < ǫ 0 1. For k 2 > 0, the parameter ψ (k 1,k 2 ) is given by ψ (k 1,k 2 ) = min ( 10,ψ (k 1,k 2 1) 1.1 (ψ (k 1,k 2 1) + δ (k 1,k 2 1) ) ) if δ (k 1,k 2 ) > 0 ψ (k 1,k 2 ) = ψ (k 1,k 2 1) if δ (k 1,k 2 1) 0 (4.17) where δ (k 1,k 2 ) = Subsequently, when J [ ( ) ( )] J ϑ (k 1,k 2 ) F ϑ (k 1,k 2 ) 1 2 ( ) ϑ (k 1,k 2 ) next outer iteration point ϑ (k 1). ( i σ (k 1) i ( ϑ i ϑ (k 1) i ) 2 ( ) 2 ϑ i ϑ (k 1) i ) 2 (4.18) ( ) is smaller than F ϑ (k 1,k 2 ), ϑ (k 1,k 2 ) becomes the Results Results are presented for optimisation of the RC virtual damping, stiffness and mass parameters using different cost functions. The sensitivities of the cost functions to the virtual damping and stiffness parameters for an optimum virtual mass is investigated. For efficient generation of the results, the calculation of the cost function has been parallelised with respect to the frequency using the Message Passing Interface (MPI) as described in Appendix B. Integration over the frequency range was implemented in an adaptive manner by comparing results for the Simpson rule to results for the trapezium rule. A minimum number of 210 integration points was used. System matrices that are independent of the design parameters have been precomputed and stored in a database at a step size of 0.1 Hz. By taking into account physical feasibility as described in Ref. [1], the RC virtual damping was varied between to kg/s. Ranges from to 85

120 4 Optimum Passive and Active Control of a Submarine N/m and from 1 to 20 tonnes were chosen for the RC virtual stiffness and mass, respectively. The optimisation was conducted using the GCMMA, where eight different initial parameter sets were used. The initial parameter sets were obtained by dividing each dimension of the three-dimensional parameter space by three, where the intersections of the dividing borders form the initial parameter sets. The iterations were stopped when the cost function values differed by less than 1 20 W between two subsequent iterations. An optimisation run for a single initial parameter set required an average of 40 minutes on a computer cluster comprising of six Pentium 4 CPUs at 3 GHz. Three different cost functions have been investigated. For cost function (a), sound radiation due to propeller vibration was neglected. A single axial exciting force and the corresponding dipoles at the propeller hub that ranges from 1 to 100 Hz were considered. The force was weighted with (ω/ ω) 2, as it increases proportionally with the square of the radian frequency. For cost function (b), the exciting force was weighted as for cost function (a) where sound radiation due to propeller vibration has been taken into account. For cost function (c), sound radiation due to propeller vibration was taken into account. Furthermore it has been assumed that the exciting force is a superposition of the first four harmonics of bpf and that the relative force amplitude is smaller for higher harmonics of bpf. Under the assumption that the propeller diameter is half the hull diameter and the tip speed is limited to 40 m /s to avoid cavitation, for a 7-bladed propeller the maximum fundamental bpf is approximately 25 Hz. The force amplitude for higher harmonics of bpf was assumed to be 1/n times the force amplitude for bpf at a given shaft speed for the n th harmonic of bpf. The cost function is computed by superposition of the energy contributions from the individual harmonics of bpf. The superposition can be considered implicitly during integration over the frequency range. The resulting weighting factors for the exciting force are shown in Fig. 4.1 for the three cost functions. The optimisation results are given in Tables 4.1, 4.2 and 4.3 for cost functions (a), (b) and (c), respectively. For cost function (a), six out of the eight sets of initial parameters lead to a common minimum with a function value of W (for three digits of 86

121 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters Weighting factor for exciting force Weighting for cost functions (a) and (b) Weighting for cost function (c) Frequency (Hz) Figure 4.1: Weighting functions for the force used in the three cost functions 87

122 4 Optimum Passive and Active Control of a Submarine Table 4.1: Optimisation results for cost function (a) Parameters c r ( kg /s) k r ( N /m) m r (tonnes) Function value J (W) Outer iterations Initial Optimum

123 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters Table 4.2: Optimisation results for cost function (b) Parameters c r ( kg /s) k r ( N /m) m r (tonnes) Function value J (W) Outer iterations Initial Optimum

124 4 Optimum Passive and Active Control of a Submarine Table 4.3: Optimisation results for cost function (c) Parameters c r ( kg /s) k r ( N /m) m r (tonnes) Function value J (W) Outer iterations Initial Optimum

125 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters accuracy). The optimum RC parameters for the lowest function value of W are given by c r = kg/s, k r = N/m and m r = 1 tonne. For cost function (b), seven out of the eight sets of initial parameters lead to a common minimum with a function value of around W. The optimum parameters are given by c r = kg/s, k r = N/m and m r = 1 tonne. For cost function (c), all eight sets of initial parameters lead to a common minimum with a function value of around W. The optimum parameters for this function value are given by c r = kg/s, k r = N/m and m r = 1 tonne. All of the cost functions result in the lower limit of 1 tonne for the optimum RC virtual mass. In order to compare the performance of the optimum RC parameters from the three cost functions, the radiated sound power for the submarine model with and without the use of an RC are compared in Figure 4.2, where structural excitation from the propeller, dipole excitation due to the operation of the propeller in a nonuniform wake and dipole excitation due to propeller vibration have been considered. The exciting force was weighted by (ω/ ω) 2, where ω = 99 Hz. The peaks in the radiated sound power at around 20, 45 and 70 Hz represent the first three axial hull resonances. The maximum sound radiation occurs at the fundamental propeller/shafting system resonance which occurs at 37.3 Hz. Results were also obtained for the sound power level using RC parameters according to Goodwin [2], in which the natural frequency of the RC is tuned to match the natural frequency of the propeller/shafting system. The RC virtual mass was chosen to be at the lower parameter bound of m r = 1 tonne, as proposed by Goodwin. In addition, the lower limit of m r = 1 tonne for the RC virtual mass was obtained during the optimisation process, as well as in Ref. [55]. The RC virtual stiffness is obtained using [2] k r = 4π 2 fpsm 2 r (4.19) where f ps = 37.3 Hz is the fundamental resonance frequency of the propeller/shafting system. This yields an RC virtual stiffness of k r = N/m. The RC virtual damping parameter is calculated using [2] 91

126 4 Optimum Passive and Active Control of a Submarine Sound power level (db re W) No RC RC tuned using Goodwin s method RC optimised using cost function (a) RC optimised using cost function (b) RC optimised using cost function (c) Frequency (Hz) Figure 4.2: Sound power level with no RC, with an RC tuned using Goodwin s method [2] and using the optimum RC parameters for the three cost functions c r = mr k r q r 4 ( 6 + q r + ) 8q r + qr 2 (4.20) where q r = 4π 2 fpsm 2 p k r. This yields an RC virtual damping of c r = kg/s. Using Goodwin s method, a reduction of the sound power level can be observed for frequencies between 10 and 75 Hz. For the majority of the frequency range, the radiated sound power for a submarine model with no RC is significantly higher than for submarine model with an RC that has been optimised using any of the cost functions. The curves for cost functions (a) and (b) are similar, but the radiated sound power for cost function (b) is slightly lower for frequencies above about 70 Hz. This is attributed to the fact that the sound radiation in the high frequency range is strongly correlated to propeller vibration which is accounted for in cost function (b). A higher RC virtual damping obtained using cost function (b) therefore leads to a decrease of radiated sound power at higher frequencies. For cost function 92

127 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters (c), the sound radiation has been significantly decreased in the low frequency range and slightly increased in the high frequency range. This means that sound radiation from the propeller does not have a significant influence on the optimisation using cost function (c) due to the assumption that the amplitude for higher harmonics of bpf is only a fraction of the amplitude for the fundamental bpf. This leads to a very compliant propeller/shafting system due to the small RC virtual stiffness. Results for the cost functions and its sensitivities to the RC virtual damping and stiffness are presented. The variation of the cost functions and its sensitivities over the parameter space spanned by the RC virtual damping and the RC virtual stiffness are examined. An analysis of the cost function and sensitivity dependency on the RC virtual mass has been omitted as all investigated cost functions lead to the same RC virtual mass of 1 tonne. This value has also been found previously by Dylejko [1]. For the sensitivity analyses presented here, the optimum RC virtual mass parameter of 1 tonne has been used. The variation of cost function (a) over the parameter space is shown in Fig Low values for the cost function are obtained by using small values for the RC virtual stiffness k r. In this case, the propeller/shafting system becomes more flexible and uncoupled from the hull. The RC virtual damping c r has only a notable influence for higher values of k r, when the coupling between the propeller/shafting system and the hull is strong. In this case, an increase of the damping leads to a decrease of the cost function values. The radiated sound power for the maximum and minimum values of the cost function in Fig. 4.3 is presented in Fig It can be seen that for the maximum cost function values, the fundamental propeller/shafting system resonance is detuned to around 27 Hz, but peak sound radiation still occurs. For the minimum cost function value, the fundamental propeller/shafting system resonance is detuned to about 14 Hz. Peak sound radiation occurs at higher frequencies above about 80 Hz, where no decrease in radiated sound power can be observed. 93

128 4 Optimum Passive and Active Control of a Submarine J (W) 2.5e e e e e e e e e e+09 kr ( N /m) 9.0e e e e e e+05 c r ( kg /s) 8.0e e+06 Figure 4.3: Cost function (a) 40 Sound power level (db re W) Maximum at c r = kg/s, k r = N/m Minimum at c r = kg/s, k r = N/m Frequency (Hz) Figure 4.4: Radiated sound power at the maximum and minimum values of cost function (a) 94

129 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters The sensitivities of cost function (a) with respect to the RC virtual damping and stiffness are shown in Figs. 4.5 and 4.6, respectively. It can be seen that the minimum cost function value is stable with respect to the RC virtual damping c r. However, moderate changes of the RC virtual stiffness k r may lead to an increase of sound radiation. The variation of cost function (b) over the parameter space is shown in Fig. 4.7, where an increase of the RC virtual damping leads to lower values for J. Two distinct local maxima of the cost function can be identified. The first local maximum occurs at the upper limit for k r and the lower limit for c r. The second local maximum occurs at the lower limit for both the RC virtual stiffness k r and damping c r. The variation of sound power with frequency is shown in Fig. 4.8 for the corresponding RC parameters. For the first local maximum, the cost function is dominated by sound radiation at the fundamental propeller/shafting system resonance. In this case, the fundamental resonance of the propeller/shafting system has been decreased by 10 Hz to 27 Hz when compared to the configuration with no RC. For the second local maximum, the cost function is dominated by the sound power due to propeller vibration in the high frequency range as a decrease of the values for c r and k r involves an increase of the propeller/shafting system axial flexibility. For the minimum cost function value, the fundamental hull resonance occurs at around 15 Hz. Due to the frequency weighting, the contribution of the radiated sound power to the cost function is small at this frequency. The sensitivity of cost function (b) with respect to the virtual damping and the virtual stiffness of the resonance changer is shown in Figs. 4.9 and 4.10, respectively. The plots are similar to the results for cost function (a), when sound radiation due to propeller vibration has been neglected. The major difference occurs for the sensitivity of the cost function with respect to the virtual damping, which is increased for low values of k r. It can be seen in Fig. 4.7 that the first maximum of the cost function at the lower limits of the RC parameters is primarily sensitive to the RC stiffness, whereas the second maximum of the cost function at the lower limit of the RC virtual damping and the upper limit of the RC virtual stiffness is primar- 95

130 4 Optimum Passive and Active Control of a Submarine J (Ws /kg) cr 2.0e e e e e e e e e e+09 kr ( N /m) 9.0e e e e e e+05 c r ( kg /s) 8.0e e+06 Figure 4.5: Sensitivity of cost function (a) with respect to the virtual damping of the RC J (WN /m) kr 8.0e e e e e e e e e e e+09 kr ( N /m) 9.0e e e e e e+05 c r ( kg /s) 8.0e e+06 Figure 4.6: Sensitivity of cost function (a) with respect to the virtual stiffness of the RC 96

131 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters J (W) 1.2e e e e e e e e e+09 kr ( N /m) 9.0e e e e e e+05 c r ( kg /s) 8.0e e+06 Figure 4.7: Cost function (b) 40 Sound power level (db re W) Maximum at c r = kg/s, k r = N/m Maximum at c r = kg/s, k r = N/m Minimum at c r = kg/s, k r = N/m Frequency (Hz) Figure 4.8: Radiated sound power for the minimum and maximum values of cost function (b) 97

132 4 Optimum Passive and Active Control of a Submarine J (Ws /kg) cr 0.0e e e e e e e e e+09 kr ( N /m) 9.0e e e e e e+05 c r ( kg /s) 8.0e e+06 Figure 4.9: Sensitivity of cost function (b) with respect to RC damping J (WN /m) kr 5.0e e e e e e e e e+09 kr ( N /m) 9.0e e e e e e+05 c r ( kg /s) 8.0e e+06 Figure 4.10: Sensitivity of cost function (b) with respect to RC stiffness 98

133 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters ily sensitive to the RC virtual damping. It can be concluded that an increase in RC virtual stiffness reduces axial propeller vibration in the higher frequency range. An increase in RC virtual damping will primarily lower sound radiation at the propeller/shafting system fundamental resonance. Cost function (c), where the decrease in amplitude for higher harmonics of bpf was considered, is shown in Fig The values for J are much lower than in Fig. 4.7, as the influence of sound radiation at higher frequencies on the cost function has been reduced. There is one global maximum due to the propeller/shafting system resonance. Sound radiation from the propeller in the high frequency range does not have a significant impact. As k r increases, the propeller/shafting system fundamental resonance increases and the overall sound radiation becomes larger since the exciting force is weighted with the square of the frequency. However, as the propeller/shafting system fundamental resonance becomes higher than the upper frequency limit for the first harmonic of bpf, the value for J drops. This can be predicted from the weighting function applied to the exciting force for cost function (c) shown in Fig The radiated sound power over the investigated frequency range for the weighted exciting force is shown in Fig For the maximum value of J, the fundamental resonance of the propeller/shafting system can be identified near 25 Hz. At the curve for the minimum value of J, the fundamental resonance of the propeller/shafting system can not be identified and the maximum sound radiation is due to the first axial hull resonance. The sound radiation in the high frequency range is slightly higher for the RC configuration that corresponds to a minimum for J. However this does not have a significant influence on J, as J is dominated by contributions that can be attributed to the first harmonic of bpf. The sensitivity of the cost function to the RC virtual damping and stiffness is shown in Figs and 4.14, respectively, where a bpf weighted exciting force has been used. The RC parameters have considerable influence on the cost function at the global maximum. For a large part of the parameter space, the cost function is relatively stable and a wide variety of RC parameter configurations is available to provide efficient reduction of the radiated sound power. 99

134 4 Optimum Passive and Active Control of a Submarine 4.0e e-11 J (W) 2.0e e e e e+09 kr ( N /m) 9.0e e e e e e+05 c r ( kg /s) 8.0e e+06 Figure 4.11: Cost function (c) 40 Sound power level (db re W) Maximum at c r = kg/s, k r = N/m Minimum at c r = kg/s, k r = N/m Frequency (Hz) Figure 4.12: Radiated sound power for the minimum and maximum values of cost function (c) 100

135 4.2 Optimisation and Sensitivity Analysis of Resonance Changer Parameters J (Ws /kg) cr 2.0e e e e e e e e e e+09 kr ( N /m) 9.0e e e e e e+05 c r ( kg /s) 8.0e e+06 Figure 4.13: Sensitivity of cost function (c) with respect to RC damping J (WN /m) kr 1.0e e e e e e e e e+09 kr ( N /m) 9.0e e e e e e+05 c r ( kg /s) 8.0e e+06 Figure 4.14: Sensitivity of cost function (c) with respect to RC stiffness 101

136 4 Optimum Passive and Active Control of a Submarine 4.3 Active Control of a Submarine Estimates of the fluctuating propeller forces at each speed and frequency as well as constraints on the size of the control force amplitudes are utilised in order to select appropriate actuators for practical design of an active vibration control system for a submarine. The performance of different active control strategies as well as the combination of a passive/active control system is presented for the case of axial excitation of a submarine hull due to propeller forces. The frequency range between 1 and 100 Hz was considered in order to address the first four harmonics of bpf. Since the fluctuating force at each multiple of bpf is at least approximately proportional to bpf 2, then the fluctuating force at half maximum speed will be only 25% of its maximum value, while the frequency is halved. At bpf, the order of magnitude of the fluctuating force might be 1 kn rms for a hull with the parameters used in this paper, but the force magnitudes are likely to be significantly smaller at higher multiples of bpf. As the excitation is related to bpf, feedforward control has been implemented. Excitation of the hull via the propeller/shafting system and excitation of the submarine hull via the fluid were taken into account. In addition, the sound field directly radiated from the propeller has been considered for the computation of the overall radiated sound power. The secondary excitation was applied using actuators with tuned passive elements or by a moment generated by means of a torsionally stiff ring. A moment control system can be implemented using stack actuators that apply a force between a ring stiffener of T-shaped cross section and the cylindrical pressure hull of the submarine. As most submarines feature ring stiffeners of T-shaped cross section, a moment control system can easily be implemented in many submarines. For both systems, paired point forces of opposite direction need to be applied, where electromagnets are used for the tuned actuators and piezoelectric stacks are used to generate the moment. The passive elements of the actuators were tuned such that their natural frequencies are similar to the natural frequencies of the pressure hull or to the fundamental resonance of the 102

137 4.3 Active Control of a Submarine propeller/shafting system, in order to reduce the required control force at peak sound radiation. The performances of AVC (Active Vibration Control) and frequency domain DSAS (Discrete Structural Acoustic Sensing) are investigated. In order to evaluate the performance of the implemented active control systems, the computational submarine model presented in section 2.4 has been used. The performance of a combined passive/active control system using a resonance changer implemented in the propeller/shafting system as the passive control mechanism and the tuned actuators is also presented Active control systems In order to efficiently address peak sound radiation at resonances of the hull and propeller/shafting system, inertial actuators (IAs) with additional passive elements are used [111]. Each actuator features a mass m a that is linked to a base by a spring of stiffness k a and a damper of damping coefficient c a, as shown in Fig A pair of control forces f a of opposite direction to the mass and base is introduced by an electromagnet. The base is fixed to the stern end plate in order to address vibration correlated to the hull resonances, or to the thrust collar to address vibration correlated to the fundamental resonance of the propeller/shafting system, as shown in Fig The configuration of an active control system for a submarine using inertial actuators is shown in Fig In order to take into account the axisymmetric nature of the problem, a set of actuators that are tuned to a certain frequency are evenly distributed on the end plate over a virtual concentric ring k a c a m a f a v h v a Figure 4.15: Inertial actuator 103

138 4 Optimum Passive and Active Control of a Submarine Propeller Shaft l s Thrust bearing c b Resonance changer c r Foundation m p l se m b k r a f b f f p v p A s, E s, ρ s m a c a k b m r E f, ρ f, ν f, h f fh vh k a f a Figure 4.16: Propeller/shafting system with an inertial actuator End plate Inertial actuators Propeller shaft Axially freely suspended mass Electromagnet Motor Thrust bearing / resonance changer Thrust collar Auxilliary AVC collar Figure 4.17: Active control system using inertial actuators 104

139 4.3 Active Control of a Submarine around the propeller shaft. Three sets of actuators are attached to the end plate, where each set of actuators is tuned to respectively control the first, second and third axial resonances of the submarine hull. The actuator used to address the fundamental resonance of the propeller/shafting system is attached to the thrust collar. This could be accomplished by using a freely suspended mass that interacts with an auxilliary shaft collar through an electromagnet only, where the magnet can simultaneously act as spring, damper and force [116]. Using another control strategy similar to that proposed by Pan et al. [113,114], a control moment is applied to the cylindrical pressure hull. This can be accomplished by implementing piezoelectric stack actuators between a simply supported, torsionally stiff ring and the cylindrical shell, as shown in Fig Pan et al. [114] concluded that the proposed control system is most efficient when the control moment is applied close to the source of excitation. Hence, the control moment is applied near the stern end plate in this work, as shown in Fig The advantage of such a system is that the overall dynamics of the submarine are not changed significantly. It is also possible to use one of the ring stiffeners of the pressure hull to support the stack actuators. However, this lowers the efficiency of the control system as the ring stiffener is welded to the pressure hull, which leads to a partial cancellation of the moment through the weld Computation of actuator forces Since excitation correlated to bpf is deterministic, feedforward control is used. A multi-channel system as shown in Fig is considered, where the error sensor signals are given by e = G pr f pr + G se f se (4.21) f pr and f se are vectors of the primary and secondary excitation spectra, respectively. G pr and G se are matrices of the primary and secondary path spectra, respectively. As the spectra f pr, G pr and G se are known, it is possible to find the optimum spectra for the secondary excitation f se in order to minimise a cost func- 105

140 4 Optimum Passive and Active Control of a Submarine Cylindrical shell f pr G pr d Piezo stack Torsionally stiff ring f se G se + + e Figure 4.18: AVC stiffener to generate a control moment Figure 4.19: system Multi-channel control Special AVC stiffener End plate Ring stiffeners Propeller shaft Motor Thrust bearing / resonance changer Thrust collar Figure 4.20: Active control system using a control moment 106

141 4.3 Active Control of a Submarine tion obtained from e. A cost function similar to that proposed by Fuller et al. [106] is given by J = e H Qe + f H seτif se (4.22) where I is the unity matrix, Q is a Hermitian, not necessarily diagonal, matrix that determines the weighting of the individual error sensor signals and τ 0. As the force of the actuators have a physical limit, τ has to be found iteratively such that this limit is not exceeded [111]. Using equation (4.21) and setting J f se = 0, it can be shown that the optimum secondary force f se,opt is given by f se,opt = [ G H seqg se + τi ] 1 G H seqg pr f pr (4.23) For DSAS, the cost function can be defined as the estimated radiated sound power as given in equation (2.60), where the weighting matrix Q = Φ. In order to estimate the far-field pressure p Λ, the contribution from the pressure at the radiating surface is neglected such that p Λ = G Ωf e + p inc,ωf. The error signals e represent the normal velocities at the radiating surface of the pressure hull. The optimum secondary force is then given by f se,opt = [ (G Ωf G se ) H ΦG Ωf G se + τi ] 1 (GΩf G se ) H ΦG Ωf G pr f pr (4.24) Results Results are compared for two active control excitation system configurations shown in Fig For the first excitation system, a set of three tuned actuators at the stern end plate and one tuned actuator at the propeller shaft was used. For the second excitation system, a control moment M as been applied to the cylindrical hull by means of a simply supported torsionally stiff ring at a distance z M from the stern end plate. Results are also presented for a combined active and passive control system, where a resonance changer is combined with the system using tuned IAs. In the numerical model, a hypothetical allowance of m a = 1 tonne has been included for the mass of each tuned actuator associated with the hull axial resonances 107

142 4 Optimum Passive and Active Control of a Submarine e n M e 1 e 2 IA 4 IA 3 IA 2 IA 1 e 0 z M Figure 4.21: Control system components and the propeller/shafting system resonance. This is sufficient to have a significant effect on these principal resonances with a damping factor of ζ a = 0.02 (loss factor = 0.04) when the control system is turned off, owing to the passive vibration absorber effect. If this benefit is not needed, then the added mass can be reduced to much smaller values. In practice, it is possible to obtain an actuator force of 1 kn peak using a commercially available electromagnet shaker such as the Brüel & Kjær Type 4828 shaker which has a total mass of 80 kg and a total moving mass of only 1.3 kg; the larger mass of the magnet itself can be used to form part of the added mass. The damping parameter for the actuators is given by c a = 2ζ a ka m a, where the stiffness coefficients k a were chosen such that the natural frequency of a tuned actuator equals one of the hull resonances or the propeller/shafting system resonance. The moment was applied by implementing an additional simply supported steel ring according to Fig at z M = 0.25 m, where the cross section has a radial dimension of 0.2 m, an axial dimension of 0.25 m and a thickness of 5 mm. In order to demonstrate the potential of the active control system with small actuator forces, the force magnitude was initially restricted to 10% of the fluctuating propeller force at each frequency. This is a severe restriction, because electrodynamic shakers, for example, can give almost constant force amplitude over a wide frequency range from less than a few Hz to a khz or more, while the required forces reduce with speed and also with the multiple of bpf. It is also demonstrated how further improvements in performance can be achieved by allowing the actuator force to be 108

143 4.3 Active Control of a Submarine increased to 30% and then 100% of the propeller force at each frequency. For the control moment, the equivalent stack actuator force per circumferential metre to generate the moment has been considered. AVC and DSAS have been used to obtain a cost function respectively based on axial vibration of the stern end plate and far-field radiated sound power. For AVC, a single sensor e 0 to measure the velocity of the stern end plate was used. For DSAS, one sensor e 1 to measure the axial velocity of the propeller and 43 sensors (e 2 to e 44 ) to measure the normal velocities of the hull were used in order to estimate the overall radiated sound power, using linear boundary elements for the hull and a rigid disc approximation for the propeller. The number of sensors for DSAS was chosen such that there are at least ten boundary elements per fluid wave length. The performance of AVC has been compared to the performance of DSAS using either tuned actuators, a control moment or tuned actuators combined with a passive resonance changer. All frequency response functions were obtained for unity force excitation of the propeller and the correlating dipole excitation. The results using AVC and DSAS are initially compared where the control force was limited to 10% of the primary force. Figure 4.22 shows the radiated sound power levels of the submarine when (i) no control system is implemented, (ii) the tuned actuators are implemented but no control force is applied, (iii) the tuned actuators are implemented and active control is used, and (iv) a control moment is applied. The peaks at around 20, 44 and 69 Hz represent the first three axial resonances of the hull and the peak at around 37 Hz represents the fundamental resonance of the propeller/shafting system. It can be seen that the system using tuned actuators reduces the peak sound radiation at the aforementioned resonances in passive mode by 2 5 db, if, for example, the active system fails. The tuned actuators work particularly well in active mode at the hull resonances, where a reduction of the radiated sound power by up to 25 db is achieved. The performance at the propeller/shafting system resonance is less significant. DSAS can only deliver a slight improvement over AVC for frequencies above 64Hz. This means that hull 109

144 4 Optimum Passive and Active Control of a Submarine vibration correlated to axial fluctuation of the end plate couples well to sound radiation. Figure 4.23 presents the corresponding control forces for the shakers. It can be seen that for the first and third axial hull resonances, less than 10% of the primary force is necessary for optimum control, whereas for the second hull resonance and the PSS resonance, the system needs to be driven to its limit. The performance of the system using a control moment is shown in Fig It can be concluded that the system is not viable for frequencies below around 40 Hz. However, for frequencies between 50 and 70 Hz, moment control works particularly well. This is due to good coupling between the rotational displacement of the cylindrical shell around the circumferential coordinate and the axial resonances in this frequency range. As for the tuned actuators, DSAS does not deliver a significant improvement over AVC. This confirms that axial vibration of the end plate due to excitation from the propeller/shafting system couples well to the overall sound radiation of the submarine. The equivalent control force per metre to generate the control moment is shown in Fig The system is driven to its limit up to 48 Hz and only performs above about 40 Hz. The dip at around 78 Hz corresponds to the first resonance of the simply supported torsionally stiff ring that is required to generate the control moment. The influence of a resonance changer (RC) on the performance of the control system is shown in Fig. 4.26, where optimum parameters for the RC corresponding to m r = 1 tonne, k r = N/m and c r = kg/s were used. The RC detunes the propeller/shafting system resonance from 38 Hz to 18 Hz, which is very close to the first hull resonance at around 20 Hz. An improvement over the case when just tuned actuators are implemented is achieved for frequencies between 25 and 60 Hz. No significant improvement is obtained when DSAS is used instead of AVC. Figure 4.27 presents the corresponding force amplitudes of the control actuators using AVC for each of the hull and propeller/shafting system axial resonances. A comparison between Figs and 4.27 shows that less control amplitude is required for the shakers when implementing a resonance changer for additional passive vibration control of the propeller/shafting system. It can be seen 110

145 4.3 Active Control of a Submarine Sound power level (db re W) No control system Tuned actuators / passive Tuned actuators / AVC Tuned actuators / DSAS Frequency (Hz) Figure 4.22: Performance of the system using tuned actuators fs/fp st hull resonance 2 nd hull resonance 3 rd hull resonance 1e PSS resonance Frequency (Hz) Figure 4.23: Forces for the tuned actuators using AVC 111

146 4 Optimum Passive and Active Control of a Submarine Sound power level (db re W) No control system Control moment / AVC Control moment /DSAS Frequency (Hz) Figure 4.24: Performance of the system using a control moment fs/fp e Frequency (Hz) Figure 4.25: Equivalent force per metre to generate the control moment using AVC 112

147 4.3 Active Control of a Submarine from Fig that optimum control is achieved for the frequency range between 40 and 100 Hz when a resonance changer is implemented. It has been shown that for AVC, using a control that is limited to 10% of the primary exciting force at the propeller hub, a significant reduction of radiated sound power can be achieved when a resonance changer is implemented in the propeller/shafting system. The increase of performance of the control system using AVC is investigated when the force limit is raised to both 30% and 100% of the primary force. Figure 4.28 shows the performance of the control system using the tuned actuators for 10%, 30% and 100% of the primary force. An increase of the force limit to 30% leads to an improvement in performance at the first hull axial resonance and the propeller/shafting system resonance, due to the tuning of the actuators to these resonances. An increase of the control force to 100% of the primary force provides significant attenuation for frequencies above 14 Hz. The dip in the radiated sound power at around 53 Hz is due to cancellation of the dipole due to the hydrodynamic mechanism by the dipole due to propeller vibration. The performance of an AVC system using a control moment, when the limit of the equivalent control force per circumferential metre to generate the control moment is limited to 10%, 30% and 100% of the primary force, is shown in Fig An increase of the equivalent control force to 30% of the primary exciting force only has an effect on the radiated sound power around the second hull axial resonance. If the equivalent control force is increased to 100% of the primary exciting force, some improvement in the attenuation can be observed at the first hull axial resonance. No attenuation is achieved at the propeller/shafting system resonance. Figure 4.30 compares the control performance using tuned actuators and an RC for the cases when (i) using AVC and the control force limited to 30% of the primary force, (ii) using AVC and the control force is 100%, (iii) using DSAS and control force is 30% and (iv) using DSAS and control force is 100%. Comparison of Figs and 4.30 shows that when the control force limit is increased from 10% to 30%, a significant reduction of radiated sound power over the majority of the frequency range is achieved. A further increase of the control force limit to 113

148 4 Optimum Passive and Active Control of a Submarine Sound power level (db re W) No control system RC only AVC and RC DSAS and RC Frequency (Hz) Figure 4.26: Performance of the system using tuned actuators and an RC fs/fp st hull resonance 2 nd hull resonance 3 rd hull resonance 1e PSS resonance Frequency (Hz) Figure 4.27: Forces for the tuned actuators when an RC is present using AVC 114

149 4.3 Active Control of a Submarine Sound power level (db re W) No control system 10% 30% 100% Frequency (Hz) Figure 4.28: Performance of the system using tuned actuators and AVC Sound power level (db re W) No control system 10% 30% 100% Frequency (Hz) Figure 4.29: Performance of the system using a control moment and AVC 115

150 4 Optimum Passive and Active Control of a Submarine Sound power level (db re W) No control system AVC (30%) and RC AVC (100%) and RC DSAS (30%) and RC DSAS (100%) and RC Frequency (Hz) Figure 4.30: Performance of the system using tuned actuators and an RC 100% of the primary exciting force only shows slight improvements at frequencies near the first axial hull resonance. A further improvement over all investigated configurations using AVC can be achieved if DSAS is used for either a control force limit of 30% or a control force limit of 100% with respect to the primary force. This can be attributed to the fact that due to the RC, the propeller and shaft are almost dynamically uncoupled from the hull. The actuator at the auxiliary collar will then excite the shaft in such a way that the dipole due to propeller vibration cancels the dipole due to the hydrodynamic mechanism. The remaining excitation of the stern end plate from the foundation is then cancelled by the actuators at the end plate such that sound radiation from the hull is minimised. It can be concluded that optimum control of hull vibration will not result in optimum reduction of radiated sound due to the contribution of propeller vibration to the overall radiated sound power. 116

151 4.4 Conclusions 4.4 Conclusions Optimum design parameters for a passive vibration attenuation device known as a resonance changer have been computed. The objective was to minimise the overall radiated sound power due to propeller forces in the low frequency range. The overall radiated sound power is due to both sound radiated from the hull as well as sound radiated from the propeller. Cost functions have been obtained by integration of the frequency-weighted radiated sound power over the frequency range of interest. In order to use gradient based optimisation, the sensitivity of the cost function to the resonance changer parameters was also computed using an adjoint operator formulation. The globally convergent method of moving asymptotes was applied in conjunction with the L-BFGS-B method, in order to find the optimum virtual damping, stiffness and mass parameters of the resonance changer. With respect to the parameter space, eight equally distributed initial parameter sets were used, where at least six optimisation runs resulted in a common minimum. The influence of sound radiation due to propeller vibration on the optimisation has been investigated. In addition, the influence of the reduction in amplitude for higher harmonics of the blade passing frequency on the optimisation has been examined. It was shown that inclusion of sound radiation due to propeller vibration leads to a higher RC virtual damping parameter which reduces axial vibration of the propeller/shafting system, and therefore sound radiation due to propeller vibration. When the reduction in amplitude for higher harmonics of bpf is considered, the sound radiation due to propeller vibration becomes insignificant which leads to a very resilient configuration of the RC with a low RC virtual stiffness and damping. The variation of the cost functions and its sensitivities to the RC virtual damping and stiffness over the parameter space has been visualised. The minima and maxima of the cost functions have been analysed. For the global minima of all cost functions, sound radiation at the fundamental resonance of the propeller/shafting system does not have a significant influence on the cost function value. 117

152 4 Optimum Passive and Active Control of a Submarine The performance of different active control systems implemented in a submarine has been investigated. The control systems are used to reduce vibration and submarine radiated noise caused by propeller forces. Two control systems were implemented. The first system employs tuned actuators at the thrust collar and the stern end plate of the hull. The second system generates a control moment by means of a torsionally stiff ring at the cylindrical pressure hull near the stern end plate. A limitation of the control force and moment amplitudes has been considered. The system using tuned actuators performs better than the system using a control moment. The performance of the individual systems as well as a combination of the system using tuned actuators and a resonance changer have been investigated using AVC and DSAS. A further improvement is achieved when the passive control system corresponding to the RC is used in conjunction with the active control system. In this case, a control force that is limited to 30% of the primary exciting force guarantees optimum control for 90% of the frequency range. DSAS does not provide a significant benefit compared to AVC. However, it leads to a significant reduction in radiated sound power when the system using tuned actuators is combined with an RC. 118

153 Chapter 5 Conclusions and Recommendations for Future Work 5.1 Summary of Conclusions Analytical and computational models for the low frequency dynamic and acoustic responses of a submarine have been developed. The physical system models of the submarine, which consists of the propeller, shaft, thrust bearing and foundation hull was modelled using a modular approach. The model for the propeller/shafting system also included a hydraulic vibration attenuation device known as a resonance changer. The submarine was considered to be under axial excitation by fluctuating forces due to propeller rotation in a non-uniform wake. This results in excitation of the hull by forces transmitted along the propeller shaft and also by the pressure field surrounding the propeller. Full coupling between the water and the pressure hull has been considered. Radiated sound due to hull and propeller vibration, and sound radiated directly from the propeller, have been taken into account. The propeller itself was modelled as a rigid disc. Analytical and computational results for the submarine hull in the absence of the propeller/shafting system were initially presented and compared. The radiated sound pressure was due to axial motion of the cylinder end closures and radial displacement of the pressure hull flank. Results for the first three axial resonant frequencies of the hull circumferential breathing modes were observed. Good agreement between the analytical and computational results was obtained for the frequency responses at each end of the hull, for the axial displacement along the hull 119

154 5 Conclusions and Recommendations for Future Work length, and for the radiation directivity patterns corresponding to the first three axial modes. Discrepancies between the results for the amplitude levels of the displacements and pressure are attributed to the differing underlying assumptions used for the modelling techniques, associated with the fluid loading and the shell assumptions. The radiation modes with the three highest radiation efficiencies have been investigated for the boundary element domain of the computational models. The first three hull axial resonant frequencies were considered for the analysis. Some of the radiation modes were found to be comparable to the radial displacement patterns at the hull axial resonances. This implies that the submarine hull is an efficient sound radiator at its axial resonant frequencies. Axial excitation of a submarine hull due to propeller forces was then implemented, where propeller forces transmitted to the hull via both the propeller/shafting system and the fluid are considered. The tailcone of the submarine was represented as a rigid structure connected to the pressure hull. This was shown to be a reasonable approximation to a flexible free flooded cone at low frequencies, when excitation through the propeller/shafting system is dominat, but the effects on overall sound radiation of forces transmitted through the fluid to the tailcone and hull structure are likely to be underestimated. The performance of a hydraulic vibration attenuation system known as a resonance changer in the propeller/shafting system has been examined. It was found that the performance of the RC is significantly affected by the influence of dipole fields due to hydrodynamic forces, which are always present, and propeller vibration. The two dipole fields may reinforce or partially cancel each other, depending on frequency and parameter selection. The RC can provide significant reduction in the overall radiated sound power at low frequencies, including the propeller blade passing frequency, but will have a much more limited effect at higher frequencies where the dipole fields tend to be the dominant cause of radiated sound power. Optimum design parameters for the resonance changer to minimise the overall radiated sound power have been computed. Cost functions were developed by inte- 120

155 5.1 Summary of Conclusions gration of the frequency-weighted radiated sound power over the frequency range of interest. In order to use gradient based optimisation, the sensitivity of the cost function to the resonance changer parameters was also computed using an adjoint operator formulation. The globally convergent method of moving asymptotes was applied in conjunction with the L-BFGS-B method, in order to find the optimum virtual damping, stiffness and mass parameters of the resonance changer. With respect to the parameter space, eight equally distributed initial parameter sets were used, where at least six optimisation runs resulted in a common minimum. The influence of sound radiation due to propeller vibration on the optimisation has been investigated. In addition, the influence of the reduction in amplitude for higher harmonics of the blade passing frequency on the optimisation has been examined. It was shown that inclusion of sound radiation due to propeller vibration leads to a higher RC virtual damping parameter, which reduces axial vibration of the propeller/shafting system and therefore sound radiation due to propeller vibration. When the reduction in amplitude for higher harmonics of bpf is considered, the sound radiation due to propeller vibration becomes insignificant which leads to a very resilient configuration of the RC with a low RC virtual stiffness and damping. The variation of the cost functions and its sensitivities to the RC virtual damping and stiffness over the parameter space was presented. The minima and maxima of the cost functions have been examined. For the global minima of all cost functions, the fundamental resonance of the propeller/shafting system occurs below the fundamental hull resonance and does not have a significant influence on the cost function value. The performance of different active control systems implemented in a submarine to reduce vibration and radiated noise caused by propeller forces has been investigated. Two control systems were implemented. The first system employed tuned actuators at the thrust collar and the stern end plate of the hull. The second system generated a control moment by means of a torsionally stiff ring at the cylindrical pressure hull near the stern end plate. A limitation of the control force and moment amplitudes has been considered. The system using tuned actuators performed better than the system using a control moment. The performance of the 121

156 5 Conclusions and Recommendations for Future Work individual systems as well as a combination of the system using tuned actuators and a resonance changer have been investigated using two active control techniques. Active vibration control (AVC) was implemented to minimise axial vibration of the stern end plate and discrete structural acoustic sensing (DSAS) was implemented to minimise the estimated overall radiated sound power of the submarine. A further improvement in noise reduction was achieved when the passive control system corresponding to the RC is used in conjunction with the active control system. In this case, a control force that is limited to 30% of the primary exciting force guarantees optimum control for 90% of the frequency range. DSAS does not provide a significant benefit compared to AVC. However, it leads to a significant reduction in radiated sound power when the system using tuned actuators is combined with an RC. 5.2 Recommendations for Future Work In future work on this thesis, a more realistic model of the submarine hull can be developed to take into account non-axisymmetric motion. A more realistic model would include complex submarine internal structures such as on-board machinery, deep frames and tanks in detail. In addition, the submarine bow can be represented by a hemisphere rather than a flat plate. The propeller can be modelled as a flexible structure to take into account the effects of blade resonances, for example. Using a non-axisymmetric model, the blades can be represented by individual elements to investigate miss-tuning effects. The hydrodynamic excitation mechanism associated with the operation of the propeller in a non-uniform wake can be investigated further in presence of a submarine tailcone with control surfaces using CFD models or virtual source theory. The Ffowcs-Williams-Hawkings analogy can be used to identify the equivalent acoustic sources. The propeller shape could be optimised to reduce both weight and sound radiation. Tailcone internal structures such as stiffener frames and the ballast tank can be taken into account in future models. The interaction of the propeller pressure field with the tailcone can be modelled 122

157 5.2 Recommendations for Future Work for a non-axisymmetric submarine, where tailcone internal fluid is considered. This would also allow investigation of the significance of vertical excitation of the tailcone through the journal bearing. The tailcone could be optimised to minimise hull excitation by the propeller through the fluid. The effect of anechoic cladding and constrained layer dampers for the tailcone on excitation of the submarine hull through the propeller pressure field could be investigated. Finally, active control could be applied to address vibration of the pressure hull associated with the bending modes. The active control systems proposed in this thesis could be investigated experimentally. 123

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172 Appendix A Structural Responses of a Tailcone A.1 Introduction In order to validate the computational model of the free flooded tailcone used in Chapter 3, a model scale tailcone has been computationally and experimentally investigated. The tailcone comprises of thin conical shell with a thick base plate at the major cone radius and a thin end plate at the minor cone radius. The base plate represents the end closure of the submarine pressure hull. The cone is submerged in a water tank. The base plate is aligned with the water surface such that no fluid is present at the submarine side of the base plate, as it corresponds to the dry interior of the pressure hull. The base plate is axially excited at its centre with respect to the cone to represent excitation through the foundation of the propeller/shafting system. The structural responses for the computational cone model are initially examined, where the drive point impedance is used to identify the axial resonances. The vibration patterns obtained computationally and experimentally are compared for the first four axial resonances. A.2 Computational Analysis A pure finite element model and a coupled finite element/boundary element model have been developed to compute the structural responses of the truncated cone to an exciting force, which is applied axially at the centre of the base end plate. The pure finite element model has been used to obtain the structural responses of the cone in vacuo. Axisymmetric Reissner-Mindlin elements as described in Appendix B.2 have been used. A coupled FE/BE model was then developed to 138

173 A.2 Computational Analysis Hole for flooding Circumferential locations for accelerometers z Hole for flooding Base plate tapped holes for eye bolts Tapped hole for shaker Figure A.1: Geometry of the cone (dimensions are given in millimetres) obtain the structural responses of the free flooded cone, where both the interior and exterior water have been considered by two independent direct boundary element domains. For both the interior and the exterior boundary element domains, the axisymmetric boundary element formulation described in Appendix C.2 was used. The boundary element domains were both coupled to the finite element domain using the approach described in section The water surface of the tank is taken into account in the computational models using the method of images, as described in Appendix C.1. The speed of sound and the mass density of the fluid are respectively 1500 m /s and 1000 kg /m 3. The geometry of the cone is given in Fig. A.1. The flooding holes and the tapped holes have not been considered in the computational model. The base plate has a thickness of 10 mm. The conical shell and the small end plate have a thickness of 2 mm. Steel has been used for the cone, where Young s modulus is 210 GPa, Poisson s ratio is 0.3 and the mass density is 7800 kg /m 3. A constant damping factor of 0.02 was considered for the structure. The drive point responses for the cone in vacuo and in water are shown in Fig. A.2, where the frequency range from 5 to 4000 Hz has been considered. The first four axial resonances are investigated. For the cone in vacuo, the first four axial reso- 139

174 A Structural Responses of a Tailcone 0.01 In air In water Impedance amplitude ( m /Ns) e-05 1e Frequency (Hz) Figure A.2: Drive point impedance amplitudes of the cone nances occur at 275, 1257, 2780 and 3310 Hz. For the free flooded cone, the first four axial resonances can be identified at 635, 873, 1316 and 1508 Hz. The displacement patterns for the cone in vacuo and in water are respectively shown in Figs. A.3 and A.4. For the cone in vacuo, the displacement patterns are dominated by out-of-plane vibration of the base plate. For the free flooded cone, motion of the base plate and the conical shell are coupled by the interior water. This leads to displacement magnitudes of similar order for both the end plate and the conical shell. A.3 Description of the Experimental Rig and Equipment The tailcone shown in Fig. A.1 was used in the experiments. The tailcone was manufactured by welding a truncated conical shell to the thick base end plate and the thin small end plate. The conical shell required another weld along the axial direction as it was manufactured by rolling a flat sheet of metal. The cone model was then painted to prevent it from rusting. 140

175 A.3 Description of the Experimental Rig and Equipment (a) First axial resonance (b) Second axial resonance (c) Third axial resonance (d) Fourth axial resonance Figure A.3: Displacement patterns for the cone in vacuo (a) First axial resonance (b) Second axial resonance (c) Third axial resonance (d) Fourth axial resonance Figure A.4: Displacement patterns for the cone in water 141

176 A Structural Responses of a Tailcone Axial force Bungee cord Force transducer Accelerometer Cone Data acquisition system Accelerometer Reverberant water tank Figure A.5: Experimental set-up The arrangement of the experimental rig in the water tank is schematically shown in Fig. A.5, where the cone was mounted using a crane. For measurements in air, the cone was taken out of the water. The equipment and instrumentation comprised of: Water tank of length 10 m width 10 m and depth 6 m with reverberant, nonrectangular walls Brüel & Kjær 4292 accelerometer calibrator that gives an acceleration of 10 m /s 2 at a frequency of Hz Modal hammer PCB 086C05 with a sensitivity of 1 mv /lbf. PVC tip for measurements in water, metal tip for measurements in air PCB 302A02 accelerometer with a sensitivity of 1 mvs2 /m to measure the drive point acceleration in air PCB 352C66 accelerometer with a sensitivity of 10 mvs2 /m to measure the acceleration of the conical shell in air PCB 303A02 accelerometer with a sensitivity of 1 mvs2 /m for all acceleration measurements in water Dewetron 2021 data acquisition system 142

177 A.4 Experimental Procedure Figure A.6: Dewetron portable data acquisition and analysis system The Dewetron data acquisition system is shown in Fig. A.6. The Dewetron data acquisition system software DeweFRF was employed to compute the frequency response function for the drive point impedance and the vibration pattern along the conical surface. For the measurements in water, the cone was located such that the distances between the cone and the walls of the water tank are as large as possible in order to reduce re-excitation of the cone by sound reflected at the walls. Hence, the cone had to be accessed by a bridge as shown in Fig. A.7. A.4 Experimental Procedure Prior to the measurements, the accelerometers were calibrated. The structural responses of the cone in air were initially obtained. The drive point impedance was determined by an accelerometer placed next to the excitation stud. The tangential 143

178 A Structural Responses of a Tailcone Figure A.7: Arrangement of the rig for measurements in water 144

179 A.5 Computational and Experimental Results acceleration of the conical shell was measured at 20 points along the cone axis. As the structural wave length of the conical shell is smaller near the large end plate, the spacing between the accelerometers needs to be smaller near the large end plate. Hence, a geometric series with a scale factor of 1.1 was used for the accelerometer locations to increase the spacing with distance from the large end plate. In order to provide ample space to mount the accelerometers, a spacing of 10mm between the edges and the outer accelerometers was considered. The measurement points are given in Table A.1. Table A.1: Locations for the measurement i z (mm) Measurements were taken at two lines along z. All accelerometers were attached to the cone using beeswax. The excitation was introduced using a modal hammer as shown in Fig. A.8. An average of ten measurements was used for each point. For measurements in water, the PCB 303A02 accelerometer was sealed using beeswax as shown in Fig. A.9. A.5 Computational and Experimental Results Results for the axisymmetric structural responses obtained computationally and experimentally were compared for the cone in air and in water. Discrepancies were expected in the results due to simplifications in the numerical models. In addition, manufacturing imperfections, welds, holes and the mounting approach introduce non-axisymmetric components to the experimental results. The rigid body motion frequency for the experimental model is slightly greater than 0 Hz due to the stiffness of the bungee cords which were required for mounting. For the computational model, the rigid body motion frequency is 0Hz as no mounts have been considered. 145

180 A Structural Responses of a Tailcone Figure A.8: Excitation of the cone using a modal hammer Figure A.9: Sealed accelerometer 146