Journai of Ship Researcii, Voi 55, No 3, September 2011, pp 149-162 Journal of Ship Research Reduction of Hull-Radiated Noise Using Vibroacoustic Optimization of the Propulsion System Mauro Caresta and Nicole J Kessissoglou School of Mechanical and Manufacturing Engineering, The University of New South Wales, Sydney, Australia Vibration modes of a submarine are excited by fluctuating forces generated at the propeiier and transmitted to the huii via the propeiier-shafting system The iow frequency vibrationai modes of the huii can result in significant sound radiation This work investigates reduction of the far-fieid radiated sound pressure from a submarine using a resonance changer implemented in the propulsion system as well as design modifications to the propeiier-shafting system attachment to the hull The submarine hull is modeled as a fluid-loaded ring-stiffened cyiindricai sheii with truncated conical end caps The propeller-shafting system is modeled in a modular approach using a combination of mass-spring-damper eiements, beams, and sheiis The stern end piate of the hull, to which the foundation of the propeller-shafting system is attached, is modeied as a circular plate coupied to an annular plate The connection radius of the foundation to the stern end plate is shown to have a great infiuence on the structural and acoustic responses and is optimized in a given frequency range to reduce the radiated noise Optimum connection radii for a range of cost functions based on the maximum radiated sound pressure are obtained for both simple support and clamped attachments of the foundation to the huii stern end plate A hydraulic vibration attenuation device known as a resonance changer is implemented in the dynamic model of the propeiier-shafting system A combined genetic and pattern search aigorithm was used to find the optimum virtual mass, stiffness, and damping parameters of the resonance changer The use of a resonance changer in conjunction with an optimized connection radius is shown to give a significant reduction in the iow frequency structure-borne radiated sound Keywords: vibrations; noise; propuision; ship motions; loads Introduction ROTATION OF a submarine propeller in a spatially nonuniform wake results in fluctuating forces at the propeller blade passing frequency (Ross 1976) This low frequency harmonic excitation is transmitted to the submarine hull by the propeller-shafting system (Kane & McGoldrick 1949, Rigby 1948, Schwanecke 1979) Early work to reduce the transmission of axial vibrations to the hull include increasing the number of propeller blades (Rigby 1948), modifying the hydrodynamic stiffness and damping of the thrust bearings (Schwanecke 1979), implementation of a hydrauManuscript received at SNAME headquarters February 28, 2010: revised manuscript received October 2010 SEPTEMBER 2011 lie vibration absorber in the propeller-shafting system (Goodwin 1960), and application of active magnetic feedback control to reduce the axiai vibrations of a submarine shaft (Parkins & Homer 1989) Goodwin (1960) examined reduction of axial vibration transmitted from the propeller to a submerged hull using a resonance changer that acts as a hydrauiic vibration absorber, using a simplified spring-mass model of the propeiier-shafting system with a rigid termination The resonance changer is designed as a hydraulic cylinder connected to a reservoir via a pipe Goodwin developed expressions to descdt)e the virtual mass, stiffness, and damping of the resonance changer in terms of its dimensions and properties of the oil contained in the reservoir In recent work on the resonance changer, a dynamic model of a submarine hull in axisymmetric motion was coupled with a dynamic model of a 0022-4502/11/5503-0149$00.00/0 JOURNAL OF SHIP RESEARCH 149 propeller-shafting system (Dylejko 2007) Optimum resonance changer parameters were obtained from minimization of the hull drive-point velocity and structure-bome radiated sound pressure The radiated sound power with and without the use of a resonance changer has also been investigated using an axisymmetric fully coupled finite element/boundary element (FE/BE) model of a submarine, in which the hull was excited by structural forces transmitted through the propeller-shafting system and acoustic excitation of the hull via the fluid in the vicinity of the propeller (Merz et al 2009) The structural and acoustic responses of a submarine hull have been presented previously by the authors (Caresta & Kessissoglou 2009, 2010) In Caresta and Kessissoglou (2009), the hull was modeled as a fluid-loaded cylindrical shell with internal bulkheads and ring stiffeners and closed at each end by circular plates The far-field radiated sound pressure was approximated using a model in which the cylinder was extended by two semi-infinite rigid baffles The effect of the various complicating effects such as the bulkheads, stiffeners, and fluid loading on the vibroacoustic responses of the finite cylindrical shell was examined in detail In a later paper (Caresta & Kessissoglou 2010), the authors presented a similar model of a finite fluid-loaded cylindrical shell that was closed at each end by truncated conical shells Harmonic excitation of the submerged vessel in both the axial and radial directions was considered The forced response of the entire vessel was calculated by solving the cylindrical shell equations with a wave solution and the conical shells equations using a power series solution, taking into account the interaction with the external fluid loading Once the radial displacement of the whole structure was obtained, the surface pressure was calculated by discretizing the surface Using a direct boundary element method (DBEM) approach, the sound radiation was then calculated by solving the Helmholtz integral in the far field The contribution of the conical end closures on the radiated sound pressure was observed The results obtained from this semianalytical model were compared with results obtained from a fully coupled finite element/boundary element model and was shown to give reliable results in the low frequency range In this paper, a dynamic model of the propeller-shafting system is coupled with the hull dynamic model presented previously by the authors (Caresta & Kessissoglou 2009) While previous work in Dylejko (2007) and Merz et al (2009) modeled the connection between the foundation of the propeller-shafting system and the pressure hull using a rigid end plate, here a more realistic flexible plate is used The foundation of the propeller-shafting system is coupled to the hull by means of the stem end plate, which is modeled as a circular plate coupled to an annular plate Two types of connection between the foundation of the propeller-shafting system and the hull stem end plate are considered, corresponding to simply supported and clamped boundary conditions The results presented here examine the influence of the flexibility of the end plate, different types of connection, and the radius of the connection location on the vibroacoustic responses of the submarine The use of a resonance changer implemented in the propeller-shafting system in conjunction with the flexible end plate to attenuate the structural and acoustic hull responses is presented In Merz et al (2009), the resonance changer parameters were optimized using gradient-based techniques, since genetic algorithms are not viable for coupled FE/BE models because of their high computational cost In this work, a semianalytical model is used, and the virtual 150 SEPTEMBER 2011 mass, stiffness, and damping parameters of the resonance changer are optimized with a new approach by combining genetic and pattern search algorithms The flexible stem end plate is shown to have a significant influence on the structural and acoustic responses of the submarine, due to the change in force transmissibility between the propeller-shafting system and the hull The connection radius is then optimized by minimizing the far-field radiated sound pressure in a wide frequency range or at discrete frequencies The use of a resonance changer implemented in the propeller-shafting system is investigated initially considering a rigid attachment to the hull, as done in Dylejko (2007) and Merz et al (2009), and then using the attachment at the optimum connection radius The resonance changer acts as a dynamic vibration absorber and introduces an extra degree of freedom in the propeller-shafting system The parameters of the resonance changer are tuned to a single frequency It is shown that the flexibility of the end plate and attachment of the propeller-shafting system to the hull at the optimum connection radius, combined with the use of a resonance changer, results in very good reduction of the radiated sound pressure over a broad frequency range Dynamic model of the submarine In this paper, a dynamic model of the propeller-shafting system is coupled with the hull dynamic model presented in Caresta and Kessissoglou (2010) for axisymmetric motion only The low frequency dynamic model of a submarine hull is approximated: The main pressure hull is modeled as a finite cylindrical shell with ring stiffeners, intemal bulkheads, and end caps The end caps are modeled as truncated conical shells that are closed at each end by circular plates The entire structure is submerged in a heavy fiuid A schematic diagram of the submarine model is shown in Fig The propeller-shafting system is located at the stem side of the submarine The propulsion forces generated by the fluctuating forces at the propeller are transmitted to a thrust bearing located along the main shaft The thrust bearing is connected to the foundation, which in turn is attached to the stem end plate A schematic diagram of the propeller-shafting system is shown in Fig, The flexible end plate is modeled as a circular plate coupled to an annular plate, where the annular plate is attached to the cylindrical hull 2.1 Cylindrical shell The fluctuating propeller forces, arising from its rotation through a spatially nonuniform wake field, are transmitted through the propeller-shafting system and result in axial excitation of the hull A detailed dynamic model of the submarine hull under axial and radial harmonic excitation was previously presented by the authors (Caresta & Kessissoglou 2010), This model is briefiy reviewed here for axisymmetric motion and then coupled to a Cylindrical shell Truncaled aine -^I.V) (21 Shaliing system Stiffeners Fig Knd piales Diagram of the submarine hull JOURNAL OF SHIP RESEARCH Conical shell Shaft Cylindrical shell the radiation damping Furthermore in the low frequency range, the axial wave number is supersonic and the fiuid introduces mainly a damping effect Hence at low frequencies, the results from the fiuid-structure interaction problem for an infinite cylindrical shell can be used to estimate the fiuid loading for a finite cylindrical shell The external pressure p can be written in terms of an acoustic impedance Z by (Junger & Feit 1986) Circular plate Propeller p = Zw = Annular plate Foundation (rigid) Thrust Bearing Fig Diagram of the propelier-shafting system dynamic model of the propeller-shafting system Flügge equations of motion were used to model the cylindrical shell T-shaped ring stiffeners are included in the hull model using smeared theory, in which the mass and stiffness properties of the rings are averaged on the surface of the hull (Hoppmann 1958) The smeared theory approximation is accurate at low frequencies where the structural wave numbers are much larger than the stiffener spacing The Flügge equations of motion for axisymmetric motion of a ringstiffened fluid-loaded cylindrical shell are given by (Caresta & Kessissoglou 2010) (4) -w Pf is the density of the fluid, to is the angular frequency, anáj is the imaginary unit, k and /.>, are respectively, the axial and the acoustic wave numbers Wo is the zero-order Hankel function of the first kind, and H'Q is its derivative with respect to the argument The validity of the approximation for the fiuid loading is shown in Caresta and Kessissoglou (2010), where structural and acoustic responses were compared with results from a fully coupled FE/BE model Results showed that for a large submarine hull in the low frequency range, an infinite fiuid-loaded shell model gives reliable results; hence a fully coupled model is not necessary In addition, the analytical method presented here is computationally faster than a fully coupled FE/BE model, thus providing an advantage for a vibroacoustic optimization routine 2.2 Circular and annular plates àu VOM ^ ^ y ^" vdu (1) = (2) The end plates and bulkheads were modeled as thin circular plates in both in-plane and bending motion The stem end plate is modeled as an internal circular plate coupled to an annular plate For the annular plate, w^ and w^ are, respectively, the axial and radial displacements For axisymmetric motion, the equations of motion for the annular plate are given by (Leissa 1993a) M and w are the axial and radial components of the cylindrical shell displacement in terms of the axial coordinate v, which originates r2 r dr ) D^ dt^ ~ ' dr\dr r) at the stem side of the main cylindrical hull, a is the mean radius of the shell, and h is the shell thickness CL = [Ê/p(l - v')]"^ is the longitudinal wave speed £, p, and v are, respectively, the Young's modulus, density, and Poisson's ratio of the cylinder r is the plate radius, and h¡, is the plate thickness The coefficients ß, 7, d(,, and d^ are given in Appendix A in accordance with Caresta and Kessissoglou (2010) The axial and radial displacements for the cylindrical shell can respectively be is the flexural rigidity, and written as (Leissa 1993b) u(x,t) = ', w{x, = =1 (3) C, = Ui/W, is an amplitude ratio and U¡, W¡ are the wave amplitude coefficients of the axial and radial displacements, respectively In equation (2), p is the external pressure from the surrounding water The fluid-structure interaction problem can only be analytically solved for infinite cylindrical shells, in which the axial modes are uncoupled as in the in vacuo case For a finite shell, coupling between axial modes occurs and the acoustic impedance has both self and mutual terms This aspect makes the problem analytically nondeterminate However, a finite cylindrical shell can be approximated by extending the cylinder by two semi-infinite rigid baffles (Junger & Feit 1986) Junger and Feit (1986) showed that mutual reactances are generally negligible Mutual resistances are negligible for supersonic modes and even for slow modes when structural damping is sufficient to dominate SEPTEMBER 2011 r ^ dt'^ ~ (5) is the longitudinal wave speed, where £a, Pa> and Va are the Young's modulus, density, and Poisson's ratio, respectively General solutions for the axial and radial displacements of the annular plate are, respectively, given by (Leissa 1993a) H'a(r,i) = (6) (7) and ¿aL = are the wave numbers for the bending and in-plane waves J^), [Q, YQ, and KQ are the zero-order Bessel and modified Bessel functions of the first and second kind (Abramowitz & Stegun 1972) The coefficients /4, (/ = : 4) and ß, (( = : 2) are determined JOURNAL OF SHIP RESEARCH 151 from the boundary conditions For a full circular plate, similar expressions for the axial Wp and radial «p displacements as given by equations (6) and (7) for an annular plate can be used, where the coefficients A3, A4, and ¿2 are set to zero 2.3 Conical end caps The equations of motion for the fluid-loaded conical shells are given in terms of u^ and w^ that are, respectively, the orthogonal components of the displacement in the axial and radial directions The axial position, x^, is measured along the cone's generator starting at the middle length, and M\, is directly outward from the shell surface Fluid loading was taken into account by dividing the conical shells into narrow strips that were considered to be locally cylindrical The equations of motion to describe the dynamic response of a conical shell under fluid loading are given by 'a sm a cos a R Vc COS a dUc R — d^u ç) ] • l [X,) (11) (8) 2.4 Propeller-shafting system cos- a sm a cos a ^0 z = [«cl (Xc) where Uci(Xc) and WdiXc), (( = I : 6), are base functions arising from the power series solution (Caresta & Kessissoglou 2008) v^ is a vector of six unknown coefflcients that are determined from the boundary conditions Ve cos a ñWc í—«c H R by matching terms of the same order for the axial position x^ The recurrence relations allow the unknown constants of the power series expansion to be expressed by only eight coefflcients that can be determined from the boundary conditions of the conical shell A mathematical procedure to describe the vibration of a truncated conical shell in vacuo using the power series approach is initially presented by Tong (1993) for shallow shell theory This approach has been modified by the authors to consider a truncated conical shell with fluid loading (Caresta & Kessissoglou 2008) The axial and radial conical shell displacements can be then expressed as "c Î;:^—1 Pc = (9) where sin a d dx¡ ' R dx¡: a is the semivertex angle of the cone R is the radius of the cone at location Xc The propeller-shafting system consists of the propeller, shaft, thrust bearing, and foundation and is modeled in a modular approach using a combination of spring-mass-damper elements and beam/shell systems, as described in Merz et al (2009) Mpr is the mass of the propeller, which is modeled as a lumped mass at the end of the shaft, as shown in Fig The shaft is modeled as a rod in longitudinal vibration The connection of the thrust bearing on the shaft is located at x^t = L^i Hence, the shaft dynamic response is obtained by separating the shaft in two sections The motion is described by the displacements «,, and u^2 along the Xs\ and Ys2 coordinates, respectively The equation of motion for the shaft in longitudinal vibration is given by i is the longitudinal wave speed E^, pc, /¡c, and v^ are, respectively, the Young's modulus, density, thickness, and Poisson's ratio of the conical shell Similar to the cylindrical shell, the external pressure p^ on a conical shell due to the surrounding water can be written in terms of an acoustic impedance Z^ by Pc=Z,w, (10) The impedance Z^ is similar to that given by equation (4), with the mean radius of the cylindrical shell, a, replaced by the mean radius of the conical shell, /?(, The validity of the fluid-loading approximation for a conical shell in the low frequency range is presented in Caresta and Kessissoglou (2008), in which results for the structural responses of a large truncated cone with different boundary conditions obtained analytically are compared with those from a fully coupled FE/BE model At low frequencies, the conical shells behave almost rigidly and the axisymmetric motion is supersonic The effect of the fluid loading is mainly a radiation damping, and its effect is small compared with the structural damping At higher frequencies or using a cone with a larger semivertex angle, the approximation for the fluid loading could lead to errors The axial-dependent component of the orthogonal conical shell displacements are expanded with a power series Substituting the power series solutions into the equations of motion, two linear algebraic recurrence equations are developed 152 SEPTEMBER 2011 (13) ^ is the longitudinal wave speed E^ and p» are the Young's modulus and density of the shaft The general solution for the longitudinal displacement for the two sections / of the shaft is given by «„(AS,,Í) = {A,ie-^'''" + ß,-e'**)e-^"', / = 1,2 Fig (14) Displacements and coordinate system for the propeller-shafting system JOURNAL OF SHIP RESEARCH where k^ = W/CSL is the axial wave number of the shaft The thrust bearing dynamics can be modeled as a single degree of freedom system of mass Mt,, stiffness K^,, and damping coefficient Cb The foundation is modeled as a rigid cone which function is to transmit the force to the end plate /?ap is the connection radius between the foundation and the plate Also shown in Fig is a resonance changer that is a hydraulic device located between the thrust bearing and the foundation The resonance changer is modeled as a single degree of freedom system of virtual lumped parameters connected in parallel (Goodwin 1960), denoted by mass M^, stiffness Kr, and damping coefficient C^ Its motion is described by coordinate ll^, In the absence of a resonance changer, «b = Wp 4)^ = dwjdxç for the conical shell To take into account the change of curvature between the cylinder and the cone, the following notation was introduced «c = "c cos a — w'c sin a, H'C = w, cos a + Uç sin a /Vrc = Wjccosa — Vv.c sina, V'^c = Kt,c cos a-I-A^^ ^ sin a (16) At junction (2) in Fig 1, the continuity conditions between the cone, annular plate, and cylindrical shell are given by U = Uç=W^, H'= H'c = Ma, cf) = (t)c = -(}>a N, + /V,,c - Af,,a = 0,M,- (17) M,,c + M,,a = 0, V^ - V,,e - /V,,a = (18) 2.5 Boundary and continuity conditions for the hull The dynamic response of the submarine structure is expressed in terms of W¡ (i = : 6) for each section of the hull, Aj (/ = : 4) and B, (/' = : 2) for each circular plate, x^ for each piece of frustum of cone, and A^,, B^, (/ = : 2) for the shaft The dynamic response is calculated by assembling the force, moment, displacement, and slope continuity conditions at each junction of the hull (corresponding to junctions to in Fig 1), as well as the boundary conditions of the hull (junctions and 6) The positive directions of the forces, moments, displacements and slopes are shown in Fig The membrane force N^, bending moment My, transverse shearing force Qy, and the Kelvin-Kirchhoff shear force V^ for the cylindrical shell, conical shells, and circular plates can be found in Caresta & Kessissoglou (2010), where the forces and moments are given per unit length The slopes are given by g,,.; Fig Modes I (a) and II (b) of collapse and (c) tripping of an in-plane compressed stiffened panel (adapted from Hughes 1983) where I^, is the moment of inertia of the stiffener (only) about an axis through the centroid of the stiffener and parallel to the web, /sp is the polar moment of inertia of the stiffener (only) about the center of rotation, d is the distance from the shear center to the point of attachment, t is the plating thickness, a is the panel length, G is the shear modulus,,/ is the torsion constant, E is the elastic modulus, m is the failure mode, and Q is a correction factor defined as; ¿»k-r = T-^E (13) Analogously, the plate failure stress crp ¡j becomes a portion of the plate yielding stress r-0.1 (14) Moreover, when the load eccentricity is considered, an additional nondimensional parameter must be introduced as (Hughes 1983); (7) (16) in which fw is the stiffener web thickness The failure modes considered for the tripping stress determination are the first five (i.e., w = 1, 2, 3, 4, 5), Higher modes are unnecessary because they would provide higher stress values For example Fig, l(c) shows the second failure mode (w = 2), The tripping stress is the minimum stress associated with each failure mode Once the effect of buckling has been taken into account, the failure stress o-pj will be the minimum stress between the tripping CTa,T and the yielding stress (Ty: where Ap is the load eccentricity acting on the section of platingstiffener system, h is the distance from the midplane of the plating to the centroid of the stiffener, A^ is the sectional area of the stiffener, A„ = A^ + b'^,^t is the total transformed area, A = As + h'^t is the total nontransformed area, yp^^ is the distance between the neutral axis of the transformed section and the plating, and Pu is the transformed radius of gyration defined as; (8) (17) According to Hughes (1983), once the strength ratio /?i (associated with Mode I of collapse) is obtained, the maximum stress (failure value) that can be reached by the cross section at the farthest point from the centroid is; where /,r is the moment of inertia of the stiffened plate about an axis through the centroid and parallel to the plate The effect of this eccentricity combined with the section slendemess is taken into consideration by the strength ratio R^ (associated with Mode II of collapse) Thus, the maximum stress that can be reached by the section is given by; (jfi = min(CTaxaY) O'a,u.l = CTF.I (9) R[ 2.1.2 Mode collapse II: compression failure of the plating In the second mode, the collapse occurs as a result of the compression failure of the plating In this case, the combination of in-plane compression and positive bending moment induces compression on the plating as shown in Fig l(b) Because of the compression of the welded plating, which has a complex inelastic failure behavior, the relation between the applied stress and the average strain becomes highly nonlinear The procedure introduced involves the consideration of the secant elastic modulus f^, which is the slope of the line joining the origin and the point of plate failure in the stress-strain diagram, so that Es = TE {E is the original value of the elastic modulus) in which (Hughes 1983); T = 0,25 {2 210 SEPTEMBER 2011 10.4 (10) (18) O'a,u,U — Hence, once the failure stress is evaluated for each stiffenerplating system, the relation between strain and curvature can be easily obtained using the stress-strain relationship (Hughes 1983); for Mode I (19) for Mode II (20) where eg „ is the stiffened plate ultimate strain, o is the curvature of the whole hull girder corresponding to the first stiffened plate JOURNAL OF SHIP RESEARCH failure (component failure), A'p is the number of stiffened plates, y is the distance between the centroid of the stiffened plate and the neutral axis of the whole hull, and / refers to the /th stiffened plate under consideration However, it is also necessary to check for a possible failure occurring by tensile yielding of the stiffened plate; thus the curvature equation becomes: niin ey (21) and hence the force in the element (T¡JAJ, where Zj and Aj are the centroid and the cross-sectional area of the jth element, respectively, and NAi and K, are the neutral axis position and the curvature at the /th iteration, respectively Determine the new neutral axis position NA, by checking the longitudinal force equilibrium over the whole transverse section Hence, adjust NA, until the force is F, = 'Í,cr¡jAj = Calculate the corresponding moment by summing the force contributions of all the element as M, = SCT,, AJ(ZJ — NA,) (^ j This novel optimization method is able to find the ultimate moment in a limited number of steps, lower than the classic incremental method In fact, by applying this method, the convergence to the ultimate moment value is much faster, so that much = Elo^o (22) computational time and resources can be saved This procedure is in which /() is the inertia of the whole box girder having all the included in the developed MATLAB program panels intact and having the transformed plating area (Ap „ = eg ^ /) 2.3 Corrosion model being used if the panel fails according to Mode II of collapse where ey is the yielding strain of the stiffened plates Finally, the first failure moment of the whole structural system, represented by the hull girder, is: 2.2 Ultimate failure evaluation The ultimate failure moment of the hull girder is calculated by using the optimization-based method proposed by Okasha and Frangopol (2010b) In this method, the ultimate moment is obtained by adopting an optimization-based procedure instead of a classic incremental curvature method Briefly, the moment-curvature relationship can be treated as an implicit function For a given curvature value K, a corresponding value of bending moment AÍ(K) can be computed by following the procedure proposed by the IACS (2008) guidelines Therefore, by using an optimization search algorithm, it is possible to find the curvature value that maximizes the associated bending moment M(K) The result obtained is the ultimate failure moment of the whole structure The main steps of the optimized iterative method are summarized as (Okasha & Frangopol 2010b): Step Divide the hull girder midship transverse section into structural elements, stiffened plates (system components), and hard comers Step Derive the stress-strain curves for all structural elements Step Provide the first trial of the curvature Step Optimize the function A/(K) by using an optimization algorithm As previously stated, the main purpose of this study is the evaluation of the time-variant redundancy index of ship structures A time horizon of 30 years is used as the target period Over time, the resistance of the hull section changes due to corrosion attack that reduces the thickness of plates and stiffeners; consequently, the global section modulus decreases The assumed corrosion model is defined as follows (Akpan et al 2002): r(t) = Cí(t - tof' (25) in which r{t) is the thickness loss (mm), /o 's the corrosion initiation time depending on coating life (years), C[ is the annual corrosion rate (mm/years), C2 is a constant usually set to unity, and t is the time expressed in years The annual corrosion rate and the initiation corrosion time are treated as random variables Table summarizes type of distribution, mean, and coefficient of variation (COV) of the assumed annual corrosion rate with respect to the location of the stiffened plates (Akpan et al 2002) Corrosion initiation time is described by a log-normal distribution with mean of years and coefficient of variation of 0.4 Load effects The hull is mainly subjected to two types of bending moments, due to still water and wave-induced Sagging bending moment is induced when the deck of the ship is in compression due to waves The optimization process is subjected to constraints; the curva- located at the extreme points of the hull; contrarily hogging bendture is bounded by lower and upper limits A realistic lower bound ing moment is caused by the compression of the keel (the bottom is the yielding curvature, while a reasonable upper bound can be part of the ship) due to waves positioned under the midship set as three times the yielding curvature, as specified by the IACS guidelines (IACS 2008) Therefore, the considered constraints are set up as: Table K < 3Ky and K > Ky for sagging (23) K > —3Ky and K < — Ky for hogging (24) The first trial curvature value (step 3) can be set as the yielding curvature value (lower bound) In step optimization of the objective M(K) is performed where each time the A/(K) objective is calculated by performing the following steps: For each jth element calculate the strain Sy = K,[...]... may print, download, or email articles for individual use Journal of Ship Research, Vol 55, No 3, September 2011, pp 163-184 Time Domain Prediction of Added Resistance of Ships Fuat Kara Energy Technology Centre, Cranfield University, United Kingdom The prediction of the added resistance of the ships that can be computed from quadratic product of the first-order quantities is presented using the near-field... distance between the shell midsurface and '^e centroid of a nng / is the area moment of inertia of the stiffener about its centroid and m^^ is the equivalent distributed rnass on the cylindrical shell to taike into account the onboard equipment and the ballast tanks JOURNAL OF SHIP RESEARCH Copyright of Journal of Ship Research is the property of Society of Naval Architects & Marine Engineers and its content... Investigations on the hydrodynamic stiffness and damping of thrust bearings in ships Transactions of the Institute of Marine Engineers, 91, 68-77 SKELTON, E A., AND JAMES, J H 1997 Theoretical Acoustics of Underwater Structures, Imperial College Press, London JOURNAL OF SHIP RESEARCH 161 TONO, L 1993 Free vibration of orthotropic conical shells International Journal of Engineering Science, 31, 719-733 Tso, Y K.,... formulas of Hess and Smith (1964) For linear system in a later time of motion of the body surface when small values of r the integrals are done exactly For intermediate the impulse is applied at one instant of time The body boundary Fig 2 The memory part of the transient free surface wave Green function G(ò, p.) SEPTEMBER 2011 JOURNAL OF SHIP RESEARCH 167 condition corresponding to an impulsive velocity of. .. The Journal of the Acoustical Society of America, 30, 77-82 JUNGER, M C., AND FEIT, D 1986 Sound, Structures, and Their Interaction, MtT Press Cambridge, MA KANE, J R., ANU MCGOLDRICK, R T 1949 Longitudinal vibrations of marine propulsion shafting systems Transactions of the Society of Naval Architects and Marine Engineers, 57, 193-252 LEISSA, A W 1993a Vibration of Plates, American Institute of Physics,... HORNER, D 1989 Active magnetic control of oscillatory axial shaft vibrations in ship shaft transmission systems Part 1 : System natural frequencies and laboratory scale model Journal ofTribology Trans- actions, 32, \10-\lằ, RiGBY, C P 1948 Longitudinal vibration of marine propeller shafting Transactions of the Institute of Marine Engineers, 60, 67-78 Ross, D 1976 Mechanics of Underwater Sound, Pergamon, New... Variation of cost function J25.50.75 with connection radius, for hard and soft connections between the foundation of the propellershafting system and the hull stem end plate The optimum radius for each connection is shown by a solid marker SEPTEMBER 2011 Jo-m /0-40 /25.50,75 J25 Optimum connection radius Soft connection Hard connection 0.87 m 1.27 m 0.88 m 1.48 m 0.79 m 1.44 m 0.87 m 2.02 m JOURNAL OF SHIP. .. frequency of the linear system, ò is the angle of the wave propagation direction with the positive x direction, k is the wave number and is related to the absolute frequency w in the case of infinite depth by k = (li^lg, and in = v cos (ò) + y sin (ò) Figures 13 and 14 show convergence of nondimensional exciting force impulse response functions as a function of nondimensionai JOURNAL OF SHIP RESEARCH... amplitude or proportional to the square of the wave amplitude The solution of the second-order problem results in mean forces, and forces JOURNAL OF SHIP RESEARCH 160 140 Asymlotic Continuation 120 -mj-WAVE 100 80 20 0 -20 -40 0.5 2 2.5 3.5 t*sqrt(g/L) Fig 26 Comparison of the heave impulse response function of Wigleylll hull at Fn = 0.3 between the solution of integral equation and asymptotic continuation... first line of equation (48) (b) Pressure due to the quadratic first-order velocitythe second line of equation (48) (c) Pressure due to the product of gradient of first-order pressure and first-order motionthe third line of equation (48) (d) Pressure due to the product of first-order pressure and first-order rotational motionthe fourth line of equation (48) (e) Total added resistance the sum of (a), (b),