Tài liệu Journal of ship research, tập 55, số 03, 2011

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 1. 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 3. 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. 2. 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. 1. 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, 2. 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. 1 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. 2 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 2 ^ ^ y ^" vdu (1) = 0 (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, 0 = 1 =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, (/ = 1 : 4) and ß, (( = 1 : 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 — 1 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 = 0 (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. 3. 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. 3 (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. 3 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 = 0 (18) 2.5. Boundary and continuity conditions for the hull The dynamic response of the submarine structure is expressed in terms of W¡ (i = 1 : 6) for each section of the hull, Aj (/ = 1 : 4) and B, (/' = 1 : 2) for each circular plate, x^. for each piece of frustum of cone, and A^,, B^, (/ = 1 : 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 2 to 5 in Fig. 1), as well as the boundary conditions of the hull (junctions 1 and 6). The positive directions of the forces, moments, displacements and slopes are shown in Fig. 4. 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 p, ^v.p. M^p, A'r.p). At the cylindrical shell/ circular plate junctions corresponding to junctions (3) and (4) in Fig. 1, similar expressions for the continuity conditions are used in which the conical shell terms are omitted. Likewise, for the boundary conditions at the free ends of the truncated cones corresponding to junctions (1) and (6) in Fig. 1, similar expressions for the continuity conditions between the conical shells and circular plates are used in which the cylindrical shell terms are omitted. The continuity equations between the propeller-shafting system and the hull in the absence of a resonance changer are initially presented. The boundary and continuity conditions for the shaft of cross-sectional area A^ are given by =0,^,1=0 (19) (20) (21) - M'-)] = = O, r = /?a Fig. 4 Positive direction of forces, moments, dispiacements, and slopes for the cyiindricai sineii, conicai siieii, and circuiar piates SEPTEiVIBER 2011 The shaft is attached to the power system by means of a flexible joint, resulting in the free end boundary condition given by equation (20). In equation (19), «s is the shaft acceleration. The propeller is modeled as a rigid disc of radius a^,, immersed in water. The mass load of the fluid can be calculated from the radiation impedance and is given by M„ = S/3a^^pf (Fahy 1985). The mass of water M^ is added to the propeller mass Mp^, resulting in Mpr = Mp^ + M« (Merz et al. 2009). Equation (21) describes the continuity of axial force at the junction of the thrust bearing along the propeller shaft. At the attachment location between the foundation of the propeller-shafting system and the hull stem end plate (/• = /?ap)- two different types of connections are considered corresponding to a "soft" connection and a "hard" connection. The soft connection implies that only an axial force is transmitted from the foundation; that is, the connection between the foundation and end plate is a simple support. This connection can be realized by means of an attachment that would be rigid only under axial motion. In the case of a hard connection, the foundation is clamped to the plate. JOURNAL OF SHiP RESEARCH 153 At r = /?ap, the boundary conditions for a soft connection are given by (22) = 0, integral with a direct boundary element method. The far field is defined in polar coordinates (R^, (^r) with the origin set at the geometric center of the hull. The sound pressure is given by (Skelton & James 1997) 1 H'a(r) = Wp(r) = «si(x,i), x,i = Ls, (23) Nr,a(r) - Air,p(r) = 0, M,,a(r) - W,,p(r) = 0 (24) u,(r) = «p(r), c^ai/-) = ct)p(r) (25) Equations (22) to (25) represent the continuity of displacement, slope, force, and bending moment between the circular and annular plates. For a hard connection, the boundary conditions at r = /?ap given by equations (24) and (25) are substituted by Ma(r) = «p(r) = 0, a(/-) = c|>p(r) = 0 (26) When a resonance changer is introduced in the propeller-shafting system, equations (21) and (22), respectively, become --(Kb-ß )[«s - «b] = 0,r = (27) - «b] = 0 {N,.,(r) - N.,p (28) where KRQ = f^r — J^C^ — w'^Mp An extra equation is also introduced for the resonance changer displacement u^. cosß, Y(ro, zo) = p(ro, dZn (31) sin , cos (32) The surface of the hull is represented in Cartesian coordinates (;•, Zr) where z^ is in the axial direction with its origin set at the geometric center of the hull, (r, Zr) is the node location on the hull surface SQ. oir = kf cos <^r< 7r = ^f sin (|)r, and ßr is the slope of the hull surface, if is the speed of sound in the fluid, /Q is the derivative with respect to the argument of the zero-order Bessel function Jo. Once the radial displacement M'N(/'O,Z()) is known at each node location on the hull boundary, the shell surface pressure p(rQ,Zo) at each node on the shell surface can then be calculated by p = D W N , where D is the fluid matrix and p, WN are, respectively, the vectors of the surface pressure and displacement (Caresta & Kessissoglou 2010). The integral in equation (31) is evaluated numerically using an adaptive Gauss-Kronrod quadrature, by separately considering the contribution of each section of the submarine corresponding to the conical and cylindrical shells. 4. Results (Kb -jtí)Cb)\uh - «si(j^si)] +KYi,c[ub - H'p('-)] +Mbüb = 0 (29) The boundary and continuity equations for the entire hull and between the hull and propeller-shafting system are arranged in matrix form Bx = 0, where x is the vector of unknown coefficients. The vanishing of both the real and imaginary parts of the determinant of B gives the natural frequencies of the system. The location of the natural frequencies can be conveniently checked from the local minima of the absolute value of the determinant, because of the complex nature of the matrix B and its determinant. The steady-state response of the hull under harmonic axial force excitation from the propeller can be calculated using a direct method in which the force is considered as part of the boundary conditions. Under a harmonic axial force, the boundary condition of the shaft corresponding to equation (19) becomes _ Ai ¿¿3, (x,^ ) = x,, = 0 (30) The boundary and continuity equations can be arranged in matrix form Bx = F, where F is the force vector with only one nonzero element corresponding to the force amplitude Fo. From x = B~ ' F , the unknown coefficients of the various plate and shell displacements can be obtained. 3. Far-field sound pressure A detailed acoustic model of a submarine was previously presented by the authors (Caresta & Kessissoglou 2010). The radiated sound pressure was calculated by solving the Helmholtz 154 SEPTEMBER 2011 Results are presented for a submarine and its propeller-shafting system using the data presented in Table 1. To account for the onboard equipment and ballast tanks in the cylindrical section of the hull, a distributed mass on the shell of Weq = 1500 kg/m* was used (Tso & Jenkins 2003). All the structures are made of steel. Structural damping was introduced using a complex Young's modulus £.^ = £(1 —jr\), where TI = 0.02 is the structural loss factor. A unity axial harmonic force from the propeller was used to excite the hull. In a real submarine, the harmonic excitation from the propeller is tonal at the blade passing frequency. Superharmonics with smaller amplitude would also appear in the spectrum of the propeller force. 4.1. Effect of the connection radius on the structural and acoustic responses The resonance changer is initially not included in the following results. For a hard connection in which the foundation is considered clamped to the stem end plate, the frequency response function (FRF) of the axial displacement at junction 2 is shown in Fig. 5, for different values of the connection radius Äap. The first two axial resonances of the hull are located at around 22 and 44 Hz. The amplitudes at resonances are affected by the damping effect of the fluid loading and become smoother as the frequency increases. The lowest frequency peak is due to the resonance of the end plate and corresponds to large deformation of the annular plate. As the connection radius becomes larger, this resonance shifts to higher frequencies, increasing from 2.8 Hz for Äap = 0.5 m to 14.7 Hz for R^p = 2.5 m. Furthermore, for larger values of /?ap, the deformation of the inner circular plate relative to the annular plate increases, as shown in Fig. 6. Other peaks in JOURNAL OF SHIP RESEARCH Tabie 1 Parameters of the submarine huli and propeiier-shafting system «^ = 0.5m,/=2.8Hz Stiffeners Cylindrical shell Bulkheads and end plates End cones Propeller-shafting system Thrust bearing Shaft Properties for steel Properties for exterior fluid Type Cross section Moment of inertia Torsion constant Spacing h Thickness h Length ¿ Radius Thickness /ip Semi-vertex angle a Thickness h^ Small radius Propeller mass A/p, Mass of water displaced M„ Mass Mh Stiffness Kf, Damping coefficient Cb Length of shaft section 1 L^i Length of shaft section 1 L,2 Radius Density p Poisson ratio v Young's modulus E Density pf Speed of sound C[ T section 509 X 10"' m^ 2.3764 X 1 0 ' m" 9.8567 X 10 ^m" 0.5 m 0.04 m 45 m 3,25 m 0.04 m 18» 0.014 m 0.50 m 10" kg 11.443 X 10'kg (Merz et al. 2009) 200 kg 2 X 10'" N/m 3 X 10'kg/s 9.0 m 1.5 m 0.15 m 7,800 kg/m-' 0,3 2.1 X 10" 1,000 kg/m' 1,500 m/s the FRFs at 9 and 36 Hz are due to the bulkhead resonances and are unaffected by the location of the connection radius. In general, as the connection radius increases and approaches the hull radius (^ap -^ «). higher amplitudes of the FRFs are observed. This occurs because force is transmitted to the hull more directly, without being filtered by the transmissibility of the end plate. The dynamic behavior of the end plate at its second resonance is more complex, as shown in Fig. 7. For increasing values of R^p, the resonant frequency increases and then decreases. The decrease in the resonant frequency occurs when the connection radius approaches the antinodes of the plate deformation, resulting in a greater structural response. R__^=1.0m,/=3.8Hz R^=l.5m./=5.5Hz - - R =2.0m,/=8.8Hz ap R^ = 2.5m,/=14.7Hz Fig. 6 Operating deformation shape of the stern end plate at its first resonance. Location of the connection radius is shown by a cross. The undeformed plate is also shown as a dashed bold line. As the connection radius increases, the deformation of the inner circular plate relative to the annular plate also increases The complexity of the change in resonance location as the connection radius increases from a small value of R¡,p = 0,5 m to the maximum value of R.^p = a can be observed in Fig. 8. This figure presents a contour plot of the frequency response function in terms of the connection radius and frequency. The resonances and antiresonances are shown by white and black lines, respectively. The left white branch presents the increase of the fundamental plate resonance as both the connection radius and frequency increases. At around R^p = 2,5 m, the plate resonance is interrupted by the intersection of an antiresonance that increases with frequency as the connection radius becomes smaller. The other two white branches correspond to higher resonances of the end plate. The maximum radiated sound pressure is defined as /'max = „ max p{R) In the far field at /f = 1000 m, the maximum sound pressure level (SPL) for different values of the connection radius /?ap ranging from 0.5 to 3,0 m is shown in Figs. 9 and 10 for hard and soft connections, respectively. The main hull axial resonances occur at around 22, 44, and 70 Hz for all values of the connection radius, except for values around R^p = 2,5 m due to the interaction of the hull with the end plate vibration. The second and third hull axial resonances are less evident due to the structural and radiation damping. The small peaks visible at 9 and 36 Hz are due to the 3 R = I.Oin /=l.5iii 2 - - *„^»2.0™ R^^ = 2.5 m R = 3.0 m E ' ji 0 ap = l,0m./=4i).I Hz 2 -1 = 2.0 m ./= 26.3 Hz R ap 30 40 50 Frequency [Hz] -O^m i'-?74H ap -7 = 2.5 m ./= 19.5 Hz -3 Fig. 5 Frequency response function of the axial displacement for the cylinder at x = 0 for different values of the connection radius. The lowest peak is due to the resonance of the end plate and corresponds to large deformation of the annular plate. As the connection radius increases, this resonance shifts to higher frequencies SEPTEIUIBER2011 -2.5 Fig. 7 -2 -15 -1 -0.5 Displacements w , w^ [ml 0 0.5 ^ ^Q-^ Operating deformation shape of the stern end plate at its second resonance JOURNAL OF SHiP RESEARCH 155 p ap = 0.5 R = 1.0 R = 1.5 D '•5 Rap = 2.0 R = 2.5 ap - - - % = 3.0 30 40 50 Frequency [Hz] 30 40 50 Frequency [Hz] lu Fig. 8 60 70 80 Contour piot of the frequency response as a function of the connection radius and frequency out-of-plane vibration of the bulkheads and end plates. The bulkhead resonances do not significantly contribute to the sound radiation and are not considered further. The resonance of the propeller-shafting system occurs at around 48 Hz and is very close to the second axial resonance of the hull. The propeller-shafting system resonance falls in the low frequency range because of the large mass of the propeller which, when summed to the mass of the water displaced by the propeller, becomes around 20 tons (Mpr = 20 ton). The sound radiation increases considerably as the connection radius becomes larger, especially in the medium frequency range, and is attributed to the increase in the structural response. For a hard connection (Fig. 9), the radial motion is constrained at the junction, resuhing in an increase of plate rigidity. Figure 10 shows that for a soft connection, the SPL is lower in value at certain frequencies, which occurs because the junction only affects the axial motion of the end plate. 4.2. Optimization of the connection radius 4.2.1. Maximum radiated sound pressure. It is evident from the results presented in the previous section that the value of the connection radius has a significant infiuence on the structural and Fig. 10 iVIaximum far fieid sound pressure ievel for different vaiues of the connection radius, for a soft connection between the foundation of the propeiier-shafting system and the huii stern end piafe. The junction oniy affects the axiai motion of the end piate, resulting in a iower SPL at certain frequencies acoustic responses of the hull. This is shown by a considerable shift in the natural frequencies with a related increase or decrease of the structural and acoustic responses in the entire frequency spectrum. The connection radius can thus be optimized to minimize the radiate sound pressure. In this section, the optimum value for the connection radius /i^p is found by minimizing the total maximum sound pressure in the frequency range A / = [0 —/max]> Since the axial force on a propeller is approximately proportional to the square of the propeller rotational speed (Goodwin 1960), the sound pressure is conveniently weighted by if/Afy^, where/is the discrete frequency and A/is the frequency bandwidth considered. The weighted cost function to be minimized is defined as (33) The cost function given by equation (33) has units of pressure. The overall maximum radiated sound for two frequency ranges defined by/max = Hz and/max = Hz are given in Figs. 11 and 12, respectively, for both soft and hard connections. A coarse increment for the radius of 0.1 m was used. The numerical integration was performed using the trajjezoidal method. The cost function was also minimized at one or several discrete frequencies. — B — Hard connection — 9 — Soft connection 30 40 50 Frequency [Hz] Flg. 9 Maximum far-field sound pressure ievei for different vaiues of the connection radius, for a hard connection between the foundation of the propeiier-shafting system and the huil stern end piate. The radial motion is constrained at the connecting junction resuiting in an increase of plate rigidity. As the connection radius increases, the SPL aiso increases and is attributed to the increase in the structural response 156 SEPTEMBER 2011 1.5 Connection radius/? 2.5 3.25 [m] ap Fig. 11 Variation of cost function Jo_8o with connection radius, for hard and soft connections between the foundation of the propeiier-shafting system and the huii stern end piate. The optimum radius for each connection is shown by a solid marker JOURNAL OF SHIP RESEARCH 10" 10" 10 9-Q nr — B — Hard connection - O — Soft connection 10' 1.5 2 Connection radius R^ [m] 0.5 2.5 3.25 Fig. 12 Variation of cost function Jo_^o with connection radius, for hard and soft connections between the foundation of the propeller-shafting system and the hull stern end plate. The optimum radius for each connection is shown by a solid marker 10" 0.5 1.5 2 Connection radius R 3.25 2.5 [mj Fig. 14 Variation of cost function J25 with connection radius, for hard and soft connections between the foundation of the propeller-shafting system and the hull stern end plate. The optimum radius for each connection is shown by a solid marker In Fig. 13, Pmax is minimized at the fundamental hpf imd its hull, especially at higher frequencies. Figure 16 shows the force n harmonics, scaled by l/n. In Fig. 14, the maximum radiated transmissibility between the propeller and the stem end plate at sound pressure F^ax is minimized at the fundamental propeller the hull junction, determined by 7", = NyJFo where Ny_„ is the hpf of 25 Hz. The optimum value for the connection radius for membrane axial force of the annular plate and F» is the amplitude the various cost functions are highlighted in Figs. 11 to 14 with a of the harmonic axial force generated at the propeller. Similar solid marker and summarized in Table 2 after refinement using a trends are observed in the results for the frequency response of resolution for the radius of 0.01 m. For Jo-s.0 in Fig. 11, the soft the axial displacement and the force transmissibility. It can be and hard connections give similar trends, but the lower values are shown that using the force transmissibility or the axial velocity at given by a soft connection. It is also observed that minimization of the cylinder/cone junction as cost functions does not result in the the cost function for the full frequency range (7()_8o) and at the hpf optima connection radii found by minimization of the far-field and its superharmonics (./25.50.75) results in nearly identical values radiated sound. This occurs because the optimization does not for the optimum connection radius due to minimization of the cost take into account the radiation efficiency of the excited structural modes. A plot of maximum sound pressure level as a function of functions over a broader frequency range. frequency for the optimum values of the connection radius /?ap 4.2.2. Frequency response function and force transmissibility. using a soft connection between the foundation of the propellerThe frequency response function of the axial displacement at the shafting system and the hull stem end plate is presented in Fig. 17. connection between the cylindrical hull and stem end plate The maximum SPL for a rigid connection is also shown. As expected, the optimum connection radius of R.^p = 0.88 provides is presented in Fig. 15, for the optimum values of/?ap with a soft the best overall reduction in maximum radiated sound pressure as connection and for a rigid connection to the hull (^ap = ^)- A this radius was obtained from minimization of the cost function significantly lower structural response is observed as the connecover a broader frequency range. tion radius moves away further from the outer periphery of the 4.2.3. Radiated sound power. Results similar to those pre.sented in section 4.2.1 using the far-field sound pressure as a cost function can be obtained by minimizing the radiated sound power, which has been estimated at the hull surface. The sound power can be expressed as an integral over the surface of the structure So (Skelton & James 1997) 10" — B — Hard connection — 9 — Soft connection (34) Table 2 10" 0.5 1.5 2 Connection radius 2.5 3.25 [m] Fig. 13 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 RESEARCH 157 20 30 40 50 Frequency [Hz] Fig. 15 Frequency response function of the axiai displacement for the cylinder at x = 0 for optimum vaiues of the radius using a soft connection between the foundation of the propeiier-shafting system and the huil stern end plate. The FRF for a rigid connection is aiso shown. Over the majority of the frequency range, the structural response decreases as the connection radius decreases 40 Frequency [Hz] 60 80 Fig. 17 Maximum sound pressure levei for the optimum vaiues of the connection radius using a soft connection between the foundation of the propeiier-shafting system and the huil stern end piate. The maximum SPL for a rigid connection is aiso shown respectively. The weighted cost function to be minimized in terms of the radiated sound power becomes (36) WQ is the surface normal velocity and the asterisk * denotes the complex conjugate, po is the surface pressure and can be expressed in terms of an acoustic impedance Zac = Po/^o- Equation (34) can be rewritten as (35) In equation (35), the radiated sound power is proportional to the real part of the acoustic impedance and is responsible for the effect of damping on the shell because of the fluid loading. In addition, its imaginary part contributes to the power retained by the hull, resulting in a mass effect. The acoustic impedance for the cylindrical and conical shells are given by equations (4) and (10), Results for the variation of the cost function with connection radius are shown in Fig. 18 for a hard connection between the foundation of the propeller-shafting system and the hull stem end plate. The optimum value for the connection radius for the various cost functions are highlighted with a solid marker. It can be seen that the general trend and values of the optimum connection radii for minimization of the radiated sound power at the hull surface are very similar to those obtained by minimizing the far-field maximum sound pressure, since these quantities are directly related. However, minimizing the radiated sound power provides an advantage in that it does not require solving the Helmholtz integral in the far field. 4.3. Acoustic transfer function Optimization of the resonance changer parameters requires calculation of the sound pressure several times, which becomes computationally very time consuming. It is therefore useful to use an acoustic transfer function to obtain the maximum sound pressure for a specific value of the connection radius. The acoustic transfer 10 20 30 40 50 Frequency [Hz] 60 70 80 Fig. 16 Force transmissibility for optimum vaiues of the connection radius using a soft connection between the foundation of the propellershafting system and the huii stern end piate. The force fransmissibiiity for a rigid connection is also shown 158 SEPTEIUIBER2011 1.5 2 Connection radius R Fig. 18 Variation of the cost functions Ju with Rap for a hard connection between the foundation of the propeiier-shafting system and the hull stern end plate. The optimum radius for each cost function is shown by a solid marker JOURNAL OF SHiP RESEARCH Table 3 xlO' .^0-80 .^0-40 •'25.50.75 hi 10 20 30 40 50 Frequency [Hz] 70 80 Fig. 19 Acoustic transfer function Hp.« for nap e [0.5 - 1.6] m with a soft connection between the foundation of the propeller-shafting system and the hull stern end plate function is defined as the ratio between the maximum pressure /"max and the radial displacement at some location x on the cylindrical hull surface. The location along the hull surface is x^ — 4>i,/3, where «t is the conjugate golden ratio given by 4) = (1 + \ß)l2 - 1 « 0.618 (Dunlap 1997). The golden ratio is an irrational mathematical constant, which when multiplied by the length of a section of the cylindrical hull, results in a location at which a large number of structural modes can be observed. The acoustic transfer function is given by //p « = Fmax^'C*^*)The acoustic transfer function for different values of the connection radius ranging from 0.5 to 1.6 m with steps of 0.1 m is shown in Fig. 19, for a soft connection between the foundation and the hull stern end plate. The peaks in Fig. 19 correspond to the frequencies where high radiation efficiency occurs. The acoustic transfer function has the advantage of calculating the maximum sound pressure at a much faster rate than directly solving the Helmholtz integral and is used in the optimization of the resonance changer parameters. 4.4. Optimization of the resonance changer with a rigid connection (Ägp = a) A resonance changer is implemented in the propeller-shafting system to reduce the transmission of axial vibration from the propeller to the hull. The various cost functions of the maximum weighted sound pressure 7^/ are initially minimized with respect to the N = 3 resonance changer parameters, corresponding to (Cr, A'r, Afr), using a rigid connection of the shafting system foundation to the hull; that is, /?ap = a. Taking into account the physical feasibility of the system, the lower and upper bounds for the resonance changer mass, stiffness, and damping parameters are given by (Goodwin 1960) Cr e [5.0 x lO'^ - 1.1 x 10*] kg/s, K, e [1.5 X Kf - 1.5 X 10"] N/m, and Mr € [1 x 10^ - 20 x 10^] kg. The parameters belong to a bounded space DRC e K . The minimum value of the cost function was obtained using the genetic algorithm and direct search toolbox of Matlab, using the following procedure. The space DRC is divided in 6'^ subspaces DRC such that 6« [N/m] CJkg/s] :, 5,000 5,000 5,000 5,176.1 9.1048 2.4480 9.9060 3.7784 Mr [kg] X 10' X 10' X 10' X 10" 1,000 1.000 1,000 1,513 The center points of these spaces are used as the starting points for a generalized pattem search algorithm to find 6'^ local minima. The local minima are then used as part of the initial population for a genetic algorithm. A total population of 8"^ points is used, the remaining 8^^ - 6'^ points are randomly created. For the genetic algorithm the following default parameters were used: crossover fraction = 0.8, elite count = 2, migration fraction = 0.2, generations = 100. Both pattem search and genetic algorithms use the augmented Lagrangian pattem search algorithm (Lewis & Torczon 2002, Conn et al. 1991, 1999). The maximum sound pressure was calculated using the acoustic transfer function Wpw, which greatly reduced the computational time. The optimum resonance changer values are summarized in Table 3. It can be seen that with the exception of minimizing the SPL at the hpf of 25 Hz only, the optimum parameters are achieved with the lowest feasible values of Cr and M^. Similar to optimization of the connection radius, optimization of the resonance changer parameters leads to similar values obtained for the optimum parameters from minimization of the cost function for the full frequency range (Jo-so) and at the t>pf and its harmonics (Í25,50.75)4.5. Optimization of the resonance changer with the optimum connection radius The resonance changer acts as a vibration absorber and is located between the propeller shaft and the hull. Optimization of the resonance changer is realized using the optimum value of the connection radius for a flexible stem end plate. Using the optimum /?ap values for a soft connection presented in Table 2, the optimum resonance changer values obtained from minimizing the cost function for different frequency ranges are given in Table 4. The natural frequency of the resonance changer for each cost function is also provided. For the four frequency-weighted cost functions, the weighted maximum SPLs are shown in Figs. 20 to 23 with and without the use of a resonance changer, for both a rigid connection of the shafting system to the hull (/?ap = a) and using the optimized Tabie 4 Optimum vaiues for {R^p, C,, K,, M,) and resonance changer naturai frequency C, [kg/s] .'0-80 0.87 .^0-40 •'25.50.75 /25 SEPTEMBER 2011 Optimum values for (Cr, K„ M,) with flap = 0.88 1.48 5,000 5,000 5,000 5,176.1 iCr [N/m] 8.8435 2.4144 9.7085 8.6820 X X X X 10 10' 10' 10' Mr [kg] [Hz] 1,000 1,O(K) 1,000 3,497.5 47.3 24.7 49.6 25 JOURNAL OF SHIP RESEARCH 159 50 ^ K — ^ O • I CQ - without RC R =a CL -50 ap - without RC R =0.87 m -50 . . _ with RC, R = a -100 1 without RC, R = 0.88 tn ap With RC, R =a - with RC, R 10 20 - withRC, R =0.88 m 30 40 50 Frequency [Hz] 60 70 80 10 Fig. 20 Maximum sound pressure levels as a result of minimizing Jo-80- The weighted maximum SPLs are presented with and without the use of a resonance changer, for both a rigid connection of the foundation to the hull and using the optimized connection radius with a flexible end plate connection radius with a fiexible end plate. It can be observed that regardless of the cost function, the use of a resonance changer greatly reduces the SPLs. Minimization of ,/o_8o results in good reduction of the SPLs over the entire frequency range, while minimization of JÍ)^Q enhances the performance in the range up to 40 Hz, Minimization of J25.50.75 with the resonance changer results in two antiresonances at exactly 50 and 75 Hz. In the absence of the resonance changer, only a single antiresonance at 75 Hz occurs. The resonance changer introduces an extra degree of freedom in the system. By carefully tuning the resonance changer it is possible to get two antiresonances, that is, two zeros in the transfer function between the propeller and the hull. Minimizing the sound radiated at only the single propeller hpf of 25 Hz results in a significant reduction of 40 dB at this frequency due to the introduction of an antiresonance. The optimum resonance changer parameters, the connection radius and the corresponding sound pressure levels for the case of minimizing /Q-SO and 725,50,75 50 *1 1 ap -100 1 20 30 40 50 Frequency [Hz] 80 are almost identical, providing the best parameters for the design of the propeller-shafting system. When the resonance changer is used with a rigid connection of the propeller-shafting system to the hull, the location of the antiresonances coincide with the natural frequencies of the resonance changer shown in Table 4. This occurs because at these frequencies, most of the energy from the propeller forces is absorbed by the motion of the resonance changer. Figures 20 to 23 show that optimization of both the connection radius and the resonance changer results in a significant reduction in the radiated .sound pressure. The main results of this work are summarized by Fig. 24 where the cost functions for optimization of the connection radius only, the optimization of the resonance changer only and the combined optimization of both the connection radius and resonance changer parameters are normalized respect to the maximum value of the cost function in the frequency range considered for each case. It is shown that the resonance changer reduces the cost functions by ! y 0 • 0 - • ^ — without RC, R =a ap without RC,Ä = 1.27 m I 70 Fig. 22 Maximum sound pressure levels as a result of minimizing ^25.50,75- The weighted maximum SPLs are presented with and without the use of a resonance changer, for both a rigid connection of the foundation to the hull and using the optimized connection radius with a flexible end plate 50 -50 60 ap si with RC,R =a Â0 /—^^^-^ .,-_---'- —^^ without RC, R -a without RC, R - 1.48 m -50 with RC, R =a • ap ap 48 m ap -100 10 20 30 40 50 Frequency [Hz] 60 70 80 Fig. 21 Maximum sound pressure levels as a result of minimizing Jo-40- The weighted maximum SPLs are presented with and without the use of a resonance changer, for both a rigid connection of the foundation to the hull and using the optimized connection radius with a flexible end plate 160 SEPTEMBER 2011 -100 1 10 20 30 1 40 50 Frequency [Hz] i__ 60 70 80 Fig. 23 Maximum sound pressure levels as a result of minimizing J25. The weighted maximum SPLs are presented with and without the use of a resonance changer, for both a rigid connection of the foundation to the hull and using the optimized connection radius with a flexible end plate JOURNAL OF SHiP RESEARCH I I opt. R I 0.8 I "^ ap I opt. RC I 0.6 I 0.4 I 0.2 0 opt. R + opt. RC h 11 transfer function that was minimized using a combined genetic and pattern search algorithm. Using a resonance changer in conjunction with a flexible connection of the propeller-shafting system to the hull can introduce two antiresonances in the hull response at design frequencies, thereby resulting in a significant reduction in the radiated sound pressure levels in both narrow and broad frequency ranges. References ''25,50.75 Fig. 24 Cost functions variation with the use of a resonance changer and a fiexible connection more than a half with respect to the minimization of only the connection radius. Optimization of both the connection radius and the resonance changer results in a significant improvement in control performance. ABRAMOwrrz, M., AND STEGUN, L A. 1972 Handbook CONN, A. R., GOULD, N . I. M., AND TOINT, P. L. 5. Conclusions SEPTEMBER 2011 Mathematical 1991 A globally conver- gent augmented Lagrangian algorithm for optimization with general constraints and simple bounds, SIAM Journal on Numerical Analysis, 28, 545-572. CONN, A. R., GOULD, N. t. M.. ANDTOINT. P. L. A dynamic model of a propeller-shafting system coupled to a .submarine hull through a flexible end plate has been presented. The submarine hull was modeled as a fiuid-loaded ring-stiffened cylindrical shell with truncated conical end caps. The propellershafting system was modeled in a modular approach using a combination of mass-spring-damper elements, beams, and shells. A hydraulic vibration attenuation device known as a resonance changer was also included in the dynamic model of the propeller-shafting system. The foundation of the propeller-shafting system was coupled to the hull using the stem end plate, which was modeled as a circular plate coupled to an annular plate. The various cylindrical shell, conical shell, and circular plate motions were coupled together by applying the continuity conditions at each junction. The steady-state response of the hull under harmonic force excitation from the propeller was calculated using a direct method in which the external force was considered as part of the boundary conditions. An acoustic model to describe the structure-borne radiated sound pressure from the submarine was calculated by solving the Helmholtz integral with a direct boundary element method. Both soft and hard connections between the foundation of the propeller-shafting system and the hull stem end plate were considered, which respectively correspond to a simple support and clamped boundary condition. The connection radius was shown to influence the structural and acoustic responses of the submarine and was optimized in order to reduce the radiated noise. Cost functions based on the maximum radiated sound pressure for both discrete frequencies and a specific frequency range were defined. The best results were obtained for a soft connection of the foundation to the pressure hull, due to the transfer of only an axial force between propellershafting system and hull. The use of a resonance changer in conjunction with an optimized connection radius was investigated, where the presence of a resonance changer introduces an extra degree of freedom in the propeller shafting system. The resonance changer parameters were optimized using an acoustic of Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York. CARESTA, M., ANO KESSISSOGLOU, N . J. 2008 Vibratioti of fluid loaded conical shells. The Journal of the Acoustical Society of America, 124, 2068-2077. CARESTA, M . , AND KESSISSOGLOU, N . J. 2009 Structural and acoustic responses of a fluid toaded cylindrical hull with .structural discontinuities. Applied Acou.itics, 70, 954-963. CARESTA, M., AND KESSISSCMLOU, N. J. 2010 Acoustic signature of a submarine hull under harmonic excitation. Applied Acou.itics, 71, 17-31. 1999 A globally cotivergent augmented Lagrangian barrier algorithm for optimization with general inequality constraints and simple bounds. Mathematics of Computation, 66, 261-288. DuNLAP, R. A. 1997 The Golden Ratio and Fibonacci Numbers, World Scientific, Singapore. DYLEJKO, P. G. 2007 Optimum Resonance Changer for Submerged Vessel Signature Reduction, Ph.D. Thesis. The University of New South Wales, Sydney, Australia. FAHY. F. J. 1985 Sound and Structural Vibration, Academic Press, London. GOODWIN, A. J. H. 1960 The design of a resonance changer to overcome excessive axial vibration of propeller shafting. Transactions of the Institute of Marine Engineers, 72, 37-63. HoppMANN, W. H. It. 1958 Some characteristics of the flexural vibrations of orthogonally stiffened cylindrical shells. 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, New York. LEISSA, A. W. 1993b Vibration of Shells, American Institute of Physics, New York. LEWIS, R. M., AND TORCZON, V. 2002 A globally convergent augmented Lagrangian pattern search algorithm for optimization with general constraints and simple bounds, SIAM Journal on Optimization, 12, 1075-1089. MERZ, S., KINNS, R., AND KESSISSOGLOU, N. J. 2009 Structural and acoustic responses of a submarine hull due to propeller forces. Journal of Sound and Vibration, 325, 266-286. PARKINS, D . W., AND 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-\l», 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 York. SCHWANECKE, H. 1979 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., AND JENKINS, C. J. 2003 Low Frequency Hull Radiation Boise. Defence Science and Technology Organisation, UK, Report No. Dstl/ TR05660. ^^ The coefficients in the Flügge equations of motion given by equations (1) to (2) are given by Caresta and Kessissoglou (2010): P_ 162 ^ ''^ ' _ 1 I jJL_L^!£î d _ i + (^ + ^X . _ 1 + 3 T | fe/i p/i ' fl^ ' a^ SEPTEMBER 2011 . (j _ y2\^ |JL = "" (1 — v^)Az ,\ = "^" El , 'i\ = "'^ Eh^ ,D =— r— 12(i—v) (A2) ^ ' The ring stiffeners have cross sectional area A. h is the stiffener ^P'»'^'"^' ^nd r, is the 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 may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users 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 method based on the direct pressure integration over floating body in time domain. The transient wave-body interaction of the first-order radiation and diffraction problems are solved as the impulsive velocity of the floating body by the use of a threedimensional panel method with Neumann-Kelvin method. These radiation and diffraction forces are the input for the solution of the equation of the motion that is solved by the use of the time marching scheme. The exact initial-boundary-value problem is linearized about a uniform flow, and recast as an integral equation using the transient free-surface Green function. A Wigley III hull form with forward speed is used for the numerical prediction of the different parameters. The calculated mean second-order added resistance and unsteady first-order impulse-response functions, hydrodynamics coefficients, exciting forces, and response amplitude operators are compared with experimental results. Keywords: resistance (general) 1. Introduction THE EXTRA POWER REQUIRED to maintain the service speed in a seaway needs to be quantified at the design stage of the vessel. This extra power requirement is the added resistance of the ship due to the responses of the vessel to a wave system. The resistance in a seaway for a ship traveling at a given speed will usually be greater than the calm water resistance due to added resistance in waves. A ship can experience a 15% to 30% resistance increase in a seaway (Strom-Tejsen et al. 1973), where the added resistance is the main reason for this increase. If a ship is designed to achieve a given speed in a seaway, then its propulsion capacity must include a margin for added resistance. Extra power requirement from added resistance will also reduce cavitation inception speed, which can be particularly important for naval vessels. Hence, the accurate prediction of the added resistance is very important for the design of both commercial and naval ships, since it affects economic performance of the vessels. It is well known that oscillating body in waves transmits the energy to the sea. It is this energy due to the damping of the oscillatory motion that increases the resistance. The effect of the Manuscript received at SNAME headquarters February 28, 2010; revised manuscript received October 3, 2010. SEPTEMBER 2011 hydrodynamic damping due to heave and pitch motion that are the dominant motions for added resistance is much bigger than the viscous damping. This implies that the prediction of the added resistance is an inviscid problem, and the potential formulations can be applied and assumed to give accurate predictions. It can be expected that the biggest contribution due to the radiation problem to the added resistance will be in the region of the resonance frequency of heave and pitch motion. The diffraction-induced added resistance will be dominated by high incident wave frequencies where the floating body motions are smail. It is assumed that the added resistance in a seaway is considered independent from calm water resistance (Kara et al. 2005), which is due to the forward speed of the floating body, and these two resistances are added to each others to get the total resistance. The added resistance is the longitudinal component of the mean second-order wave forces in the case of nonzero forward speed. This second-order force is proportional to the square of the wave amplitude and hence is nonlinear. The second-order forces on a ship due to the diffraction of the waves on a fixed body and due to relative first-order motions of the floating body were pioneered by Havelock (1940, 1942). The added resistance can be computed from quadratic products of the first-order quantities. Three different methods can be in general used for the prediction of this second-order force. The first one is the method of radiated energy 0022-4502/11 /5503-0163$00.00/0 JOURNAL OF SHiP RESEARCH 163 (Gerritsma & Beukelman 1972), which is based on the determination of the radiated energy of the damping waves during one period of oscillation. The main advantage of this method is that it does not require solving any hydrodynamic boundary conditions and only geometric data as input are required. The second one is the near-field method (Pinkster 1976, 1980), which is based on the direct pressure integration of the quadratic pressure over the instantaneously wetted surface. This method gives individual forces on the body surface as the time average of the integrated pressure. This method can be used for both single and multihull problems. The third one is the far-field method (Maruo 1960, Newman 1967), which is based on the momentum-conversation principles and applied over entire fluid volume. This method is advantageous in terms of accuracy and computational efficiency. However, only the horizontal force and vertical moment on a single body can be obtained, and it is not applicable for multiple body interaction. Kim and Yue (1990) developed the more general complete second-order sum- and difference-frequency approximation for fixed or freely floating axisymmetric bodies in the presence of bichromatic incident waves. The semianalytical formulation for second-order diffraction by a vertical cylinder in bichromatic waves is studied by Eatock Taylor and Huang (1997). There are two popular Green function methods for the solutions of the second-order forces both in frequency and in time domain; the wave Green function that satisfies the free-surface boundary condition and condition at infinity automatically, while Rankine source Green function does not satisfy these boundary conditions. Hence, in the case of former Green function only the body surface needs to be discretized using quadrilateral or triangular element. Both body surface and some part of the free surface needs to be discretized to satisfy the free surface and radiation condition numerically in the case of Rankine source Green function. Ferreira (1997) used a hybrid method in which the impulse response functions is obtained using transient free-surface wave Green function, and then the frequency domain velocity potential and fluid velocities are obtained by the Fourier transform of these impulse response functions for the prediction of the second-order steady forces. Choi et al. (2000) used higher-order boundary element with frequency domain wave Green function including second-order potential effect for the evaluation of second-order nonlinear forces. Hermans (2005) used time domain Rankine .source Green function and asymptotic approximation in which problem is linearized with respect to double-body potential for the prediction of added resistance. Kashiwagi et al. (2005) used higher-order boundary element method with frequency domain wave Green function to predict the second-order hydrodynamic interactions for side-by-side vessels. Fang and Chen (2006) used a method based on the second-order steady-state method, and three-dimensional pulsating source distribution approximation is applied for the prediction of added resistance. Recently, Joncquez et al. (2009) used a three-dimensional Rankine source Green function time domain higher-order boundary element method to predict the added resistance using Neumann-Kelvin and doublebody linearization together with both near-field and far-field approximations. A different approach from previous studies for the prediction of the mean second-order forces and unsteady first-order radiation and diffraction forces is used in the present paper. The numerical solutions of these forces are studied directly in the time domain using Neumann-Kelvin approximation method. The initial bound164 SEPTEMBER 2011 ary value problem is transformed from the volume to boundary integral equation on the fluid boundary applying the Green's theorem over the transient free-surface wave Green function (Kara & Vassalos. 2003, 2007). Then, the exact initial boundary value problem is linearized using the free stream as a basis flow, replaced by the boundary integral equation. The resultant boundary integral equation is discretized using quadrilateral elements over which the value of the potential is assumed to be constant and solved using the trapezoidal rule to integrate the memory or convolution part in time. The free-surface and body boundary conditions are linearized on the discretized collocation points over each quadrilateral element to obtain algebraic equation. 2. Exact Initial boundary value problem Two right-handed coordinate systems are used to define the fluid action. A Cartesian coordinate system Xo = (.Vo, Vo, ZQ) is fixed in space. Positive .Vp-direction is toward the bow, positive ZQdirection points upward, and the zo = 0 plane (or A'O.VO plane) is coincident with calm water. The body is translating through an incident wave field with velocity U, while it undergoes oscillatory motion about its instantaneous body surface position. The other Cartesian coordinate system x = (.v, y, z) is fixed to the body and has the same orientation with the space-fixed coordinate system XQ = (jcoi yo. ^o)- The origin of the space-fixed coordinate system Xo = {xç,, yo, Zo) is located on the calm water, while the body-fixed coordinate system x = (.v, y, z) is located at the center of the xy plane. At time t = 0, the two coordinate systems are coincident. The solution domain consists of the fluid bounded by the free surface Sf{t), the body surface S^it), and the boundary surface at infinity S^ as shown in Fig. I. The assumptions need to be made in order to solve the physical problem. If the fluid is unbounded, except for the submerged portion of the body hull and free surface, ideal (inviscid and incompressible), and the flow is irrotational (no fluid separation and lifting effect). The principle of mass conservation dictates that the total di.sturbance velocity potential <î>(x(), t) that is harmonic in the fluid domain is govemed by the Laplace equation everywhere in the fluid domain as V'(xo,O = 0, and the disturbance flow velocity field V(xo,0 may be described as the gradient of the potential (x(),/) (e.g., V(xo,r) = V is the acceleration due to gravity, and Pa,n, is the atmospheric pressure, which is used as a reference pressure and assumed to be constant (i,e,, zero). The boundary conditions must be defined for the problem. The boundary conditions at the free surface can be defined in terms of a kinematic and a dynamic boundary condition. Since the free surface is a material surface, the kinematic boundary condition is defined in terms of substantial derivatives (or Eulerian time derivatives) on Z() = Ç(AO, yo, t), which is the unknown free-surface elevation. The dynamic free-surface boundary condition occurs when the fluid pressure equals the atmospheric pressure on the free surface. Neglecting the surface tension effect and using BemouUi's equation [equation ( 1)]. the dynamic free-surface boundary condition is given as ô<ï> 1 + VO • V*-I-;?Z(i = 0 on zo = ot L (2) The dynamic boundary condition equation (2) may be used to determine the unknown free-surface elevation 1 /â4> = 20 = - - ( — (3) and using the substantial derivative in equation (3), Ç(A'O, >'O, ') can be eliminated and the combined free-surface boundary condition can be obtained as 2VO • V df- dt 1— V ^ • V(V4* • VO) -j- P O - := 0 on zo = 2 c .() (4) On .solid boundaries, the no-flux boundary conditions are used. The fluid viscosity is not included. Thus, on the submerged part of body surface, the normal component of the flow velocity is equal to the normal component of the body surface velocity at the same location and may be written as - - = Vsiio onSb(/) an (5) where the normal vector ño is pointing out of the fluid domain and into the body surface, Vs(xo,f) is the velocity of the point Xo on the body surface, and 5b(/) is the exact position of the body surface. Two initial conditions are required, since the free-surface condition equation (4) is second order; tp = (p, = 0 on Zo = 0 / < 0 for the radiation problem and (p = ip, = 0 on ZQ = 0 r < —oo for the diffraction problem. Since an initial boundary value problem is being solved, the gradient of the velocity potential must vanish (Vip —» 0 when xo —> cx3) at a spatial infinity for all finite time. This kind of formulation is the exact description of the physical problem of a body starting at rest and reaching a uniform speed in the presence of an incident wave field. The more detailed discussion of the initial boundary value problem is presented by Wehausen and Lai tone (1960), 2.1. Linearized initial value problem It is assumed that the fluid disturbances due to steady forward motion and unsteady oscillations of the body surface are small and SEPTEi\/IBER 2011 may be separated into individual parts for the linearized problem. In addition to the separation of the fluid disturbance into steady and unsteady parts, the free-surface boundary condition, body boundary condition, and BemouUi's equation may be linearized. For the linear problem, the body-fixed coordinate sy,stem x = {x, y, z), which has the same orientation as the space-fixed coordinate system Xo = (jco, yo< ^o) ^nd travels along the .Vo direction with a constant speed U is used. In the steady problem, the body starts its motion at rest and then suddenly takes a constant velocity U parallel to free surface. After some oscillation all transients are allowed to decay to zero for the steady problem that gives rise to the calculation of the steady resistance, sinkage force, and trim moment. Then the unsteady problem, which consists of radiation and diffraction problems, is solved, when the body is in its equilibrium position. Because of the small disturbance of the fluid, the total velocity potential produced by the presence of the floating body in the fluid domain may be separated into three different parts x, 0 = 9basls (X) + ^steady {^) + fli (6) The steady problem is the combination of ipbasis (x) and tpsteady (x) potentials due to the steady translation of the fioating body at forward speed U. The incident potential ip|(x, /) is produced when the steadily translating fioating body meets with an incident wave field. If the incident wave is reflected by the floating body, the resultant potential is the scattering potential ip3(x,/) and comprises the diffraction potential. The solution of the incident wave potential and diffraction potential is called diffraction problem. L When the steadily translating floating body is forced to oscillate in any of its rigid body mode k, the floating body produces the radiation potential ipjt(x,i), the solution of which comprises the radiation problem. This kind of decomposition is given by Haskind (1953) and gives rise to the linearization of the goveming equations, which are the free-surface condition, body boundary condition, and BemouUi's equation. Physically, this kind of decomposition equation (6) ignores the interaction of the waves produced by the individual components. In the moving coordinate system (body-fixed coordinate system), the fluid velocities consist of the free stream and the undisturbed incident wave components in the far field and may be written as VO -^ -U\ -I-Vtp, X 00 (7) The basis flow ipba.sis (x) is taken as the free stream potential far from the body, and it is assumed its contribution is much bigger than the remaining potentials, which are the nomial components of the incident wave velocity on the body. The traditional selection for the basis flow is the double body flow and free stream flow. The latter is used in the present paper and may be written as 'Pbasi.,(x) = -Ux (8) This kind of selection of the basis flow gives the Neumann-Kelvin linearization of the pressure, the free surface, and the body boundary condition and eliminates the interaction between the various potentials except for the interaction of the steady flow with the body boundary conditions. For the free-surface boundary condition, the Eulerian description of the flow is used. Thus, no overtuming and breaking waves are allowed to exist. Using the JOURNAL OF SHiP RESEARCH 165