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 s
Trang 3Journai 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 fre- quency 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 lie vibration absorber in the propeller-shafting system (Goodwin
1960), and application of active magnetic feedback control toROTATION OF a submarine propeller in a spatially nonuniform reduce the axiai vibrations of a submarine shaft (Parkins & Homerwake results in fluctuating forces at the propeller blade passing 1989) Goodwin (1960) examined reduction of axial vibrationfrequency (Ross 1976) This low frequency harmonic excitation is transmitted from the propeller to a submerged hull using a reso-transmitted to the submarine hull by the propeller-shafting system nance changer that acts as a hydrauiic vibration absorber, using a(Kane & McGoldrick 1949, Rigby 1948, Schwanecke 1979) simplified spring-mass model of the propeiier-shafting systemEarly work to reduce the transmission of axial vibrations to the with a rigid termination The resonance changer is designed as ahull include increasing the number of propeller blades (Rigby hydraulic cylinder connected to a reservoir via a pipe Goodwin1948), modifying the hydrodynamic stiffness and damping of the developed expressions to descdt)e the virtual mass, stiffness, andthrust bearings (Schwanecke 1979), implementation of a hydrau- damping of the resonance changer in terms of its dimensions and
properties of the oil contained in the reservoir In recent work onManuscript received at SNAME headquarters February 28, 2010: revised the resonance changer, a dynamic model of a submarine hull inmanuscript received October 3 2010 axisymmetric motion was coupled with a dynamic model of a
Trang 4propeller-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
ex-ternal 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
connec-tion locaconnec-tion 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
mass, stiffness, and damping parameters of the resonance changerare optimized with a new approach by combining genetic andpattern search algorithms The flexible stem end plate is shown
to have a significant influence on the structural and acousticresponses of the submarine, due to the change in force transmissi-bility between the propeller-shafting system and the hull Theconnection radius is then optimized by minimizing the far-fieldradiated sound pressure in a wide frequency range or at discretefrequencies The use of a resonance changer implemented in thepropeller-shafting system is investigated initially considering arigid attachment to the hull, as done in Dylejko (2007) and Merz
et al (2009), and then using the attachment at the optimum nection radius The resonance changer acts as a dynamic vibrationabsorber and introduces an extra degree of freedom in the propel-ler-shafting system The parameters of the resonance changer aretuned to a single frequency It is shown that the flexibility of theend plate and attachment of the propeller-shafting system to thehull at the optimum connection radius, combined with the use of aresonance changer, results in very good reduction of the radiatedsound pressure over a broad frequency range
con-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 andKessissoglou (2010) for axisymmetric motion only The low fre-quency dynamic model of a submarine hull is approximated: Themain pressure hull is modeled as a finite cylindrical shell with ringstiffeners, intemal bulkheads, and end caps The end caps aremodeled as truncated conical shells that are closed at each end bycircular 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 thesubmarine The propulsion forces generated by the fluctuatingforces at the propeller are transmitted to a thrust bearing locatedalong the main shaft The thrust bearing is connected to the foun-dation, which in turn is attached to the stem end plate A sche-matic diagram of the propeller-shafting system is shown in Fig, 2.The flexible end plate is modeled as a circular plate coupled to anannular plate, where the annular plate is attached to the cylindricalhull
2.1 Cylindrical shell
The fluctuating propeller forces, arising from its rotationthrough a spatially nonuniform wake field, are transmittedthrough the propeller-shafting system and result in axial excitation
of the hull A detailed dynamic model of the submarine hull underaxial and radial harmonic excitation was previously presented bythe authors (Caresta & Kessissoglou 2010), This model is briefiyreviewed here for axisymmetric motion and then coupled to a
(21 Shaliing system Stiffeners
Fig 1 Diagram of the submarine hull
-^I.V)
Knd piales
150 SEPTEMBER 2011
Trang 5Fig 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
ring-stiffened fluid-loaded cylindrical shell are given by (Caresta &
M and w are the axial and radial components of the cylindrical shell
displacement in terms of the axial coordinate v, which originates
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
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
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
analyt-ically 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
imped-ance has both self and mutual terms This aspect makes the
prob-lem 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
the radiation damping Furthermore in the low frequency range,the axial wave number is supersonic and the fiuid introducesmainly a damping effect Hence at low frequencies, the resultsfrom the fiuid-structure interaction problem for an infinite cylin-drical 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)
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
acous-tic 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 acousticresponses were compared with results from a fully coupled FE/BEmodel Results showed that for a large submarine hull in the lowfrequency range, an infinite fiuid-loaded shell model gives reli-able results; hence a fully coupled model is not necessary Inaddition, the analytical method presented here is computationallyfaster than a fully coupled FE/BE model, thus providing an advan-tage for a vibroacoustic optimization routine
2.2 Circular and annular plates
The end plates and bulkheads were modeled as thin circularplates in both in-plane and bending motion The stem end plate ismodeled 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)
r2 r dr ) D^ dt^ ~ ' dr\dr r ) r ^ dt'^ ~
(5)
r is the plate radius, and h¡, is the plate thickness.
is the flexural rigidity, and
is the longitudinal wave speed, where £a, Pa> and Va are theYoung's modulus, density, and Poisson's ratio, respectively Gen-
eral 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
func-tions of the first and second kind (Abramowitz & Stegun 1972).The coefficients /4, (/ = 1 : 4) and ß, (( = 1 : 2) are determined
Trang 6from 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
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
by matching terms of the same order for the axial position x^ The
recurrence relations allow the unknown constants of the powerseries expansion to be expressed by only eight coefflcients thatcan be determined from the boundary conditions of the conicalshell A mathematical procedure to describe the vibration of atruncated conical shell in vacuo using the power series approach
is initially presented by Tong (1993) for shallow shell theory Thisapproach has been modified by the authors to consider a truncatedconical shell with fluid loading (Caresta & Kessissoglou 2008).The axial and radial conical shell displacements can be thenexpressed as
= [«cl (Xc) ç) ] • l [X,)
(11)
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 fromthe boundary conditions
2.4 Propeller-shafting system
The propeller-shafting system consists of the propeller, shaft,thrust bearing, and foundation and is modeled in a modularapproach using a combination of spring-mass-damper elementsand beam/shell systems, as described in Merz et al (2009) Mpr isthe mass of the propeller, which is modeled as a lumped mass atthe end of the shaft, as shown in Fig 3 The shaft is modeled as arod 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 theshaft in longitudinal vibration is given by
i
(13)
^ is the longitudinal wave speed E^ and p» are the
Young's modulus and density of the shaft The general solutionfor the longitudinal displacement for the two sections / of the shaft
is given by
«„(AS,,Í) = {A,ie-^'''" + ß,-e'**)e-^"', / = 1,2 (14)
Fig 3 Displacements and coordinate system for the propeller-shafting
system
Trang 7where 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^,
stiff-ness Kr, and damping coefficient C^ Its motion is described by
coordinate ll^, In the absence of a resonance changer, «b = Wp.
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,
displace-ment, and slope continuity conditions at each junction of the hull
(corresponding to junctions 2 to 5 in Fig 1), as well as the
bound-ary conditions of the hull (junctions 1 and 6) The positive
direc-tions 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 <j) = dw/dx
for the cylindrical shell, <()a = dwjdr for the annular plate, and
4)^ = dwjdxç for the conical shell To take into account the change
of curvature between the cylinder and the cone, the followingnotation was introduced
«c = "c cos a — w'c sin a, H'C = w, cos a + Uç sin a (15)
/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 thecone, annular plate, and cylindrical shell are given by
U = Uç=W^, H'= H'c = Ma, cf) = (t)c = -(}>a (17)
N, + /V,,c - Af,,a = 0,M,- M,,c + M,,a = 0, V^ - V,,e - /V,,a = 0
(18)Similar equations are used at junction (5), in which the displace-ment, slope, force, and moment terms associated with the annular
plate (»a, H.,, 4)a, /Vi a, M.v.a, ^r,a) a ^ replaced with those for a full circular plate (Up, Wp, 4>p, ^v.p M^p, A'r.p) At the cylindrical shell/
circular plate junctions corresponding to junctions (3) and (4) inFig 1, similar expressions for the continuity conditions are used inwhich the conical shell terms are omitted Likewise, for theboundary conditions at the free ends of the truncated conescorresponding to junctions (1) and (6) in Fig 1, similar expres-sions for the continuity conditions between the conical shells andcircular plates are used in which the cylindrical shell terms areomitted
The continuity equations between the propeller-shafting systemand the hull in the absence of a resonance changer are initiallypresented The boundary and continuity conditions for the shaft of
cross-sectional area A^ are given by
Fig 4 Positive direction of forces, moments, dispiacements, and slopes
for the cyiindricai sineii, conicai siieii, and circuiar piates
The shaft is attached to the power system by means of a flexiblejoint, resulting in the free end boundary condition given byequation (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 thrustbearing along the propeller shaft
At the attachment location between the foundation of the peller-shafting system and the hull stem end plate (/• = /?ap)- twodifferent types of connections are considered corresponding to a
pro-"soft" connection and a "hard" connection The soft connectionimplies that only an axial force is transmitted from the foundation;that is, the connection between the foundation and end plate is asimple support This connection can be realized by means of anattachment that would be rigid only under axial motion In thecase of a hard connection, the foundation is clamped to the plate
Trang 8At r = /?ap, the boundary conditions for a soft connection are
given by
(22)
= 0,
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
an-nular 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, <t>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 0,r =
)[«s - «b] =
{N,.,(r) - N.,p - «b] = 0
(27)(28)
where KRQ = f^r — J^C^ — w'^Mp An extra equation is also
introduced for the resonance changer displacement u^.
(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
sys-tem 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
calcu-lated 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
displace-ments 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
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)
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 tive with respect to the argument of the zero-order Bessel function
deriva-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 evaluatednumerically using an adaptive Gauss-Kronrod quadrature, by sep-arately considering the contribution of each section of the subma-rine corresponding to the conical and cylindrical shells
complex Young's modulus £.^ = £(1 —jr\), where TI = 0.02 is the
structural loss factor A unity axial harmonic force from the peller was used to excite the hull In a real submarine, the har-monic excitation from the propeller is tonal at the blade passingfrequency Superharmonics with smaller amplitude would alsoappear in the spectrum of the propeller force
pro-4.1 Effect of the connection radius on the structural and
acoustic responsesThe resonance changer is initially not included in the followingresults For a hard connection in which the foundation is consid-ered clamped to the stem end plate, the frequency response func-tion (FRF) of the axial displacement at junction 2 is shown inFig 5, for different values of the connection radius Äap The firsttwo axial resonances of the hull are located at around 22 and
44 Hz The amplitudes at resonances are affected by the dampingeffect of the fluid loading and become smoother as the frequencyincreases The lowest frequency peak is due to the resonance ofthe end plate and corresponds to large deformation of the an-nular plate As the connection radius becomes larger, this reso-nance 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 tothe annular plate increases, as shown in Fig 6 Other peaks in
154 SEPTEMBER 2011 JOURNAL OF SHIP RESEARCH
Trang 9Tabie 1 Parameters of the submarine huli and propeiier-shafting
Spacing h Thickness h
Length ¿
Radius
Thickness /ipSemi-vertex angle a
Thickness h^
Small radiusPropeller 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
RadiusDensity p
Poisson ratio v Young's modulus E
45 m3,25 m0.04 m
18»
0.014 m0.50 m10" kg11.443 X 10'kg(Merz et al 2009)
200 kg
2 X 10'" N/m
3 X 10'kg/s9.0 m1.5 m0.15 m7,800 kg/m-'0,32.1 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 = I.Oin /=l.5iii
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
«^ = 0.5m,/=2.8Hz 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 Theundeformed plate is also shown as a dashed bold line As the connectionradius 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 resonancesand antiresonances are shown by white and black lines, respec-tively The left white branch presents the increase of the funda-mental 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 increaseswith frequency as the connection radius becomes smaller Theother two white branches correspond to higher resonances of theend 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 rangingfrom 0.5 to 3,0 m is shown in Figs 9 and 10 for hard and softconnections, respectively The main hull axial resonances occur ataround 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 axialresonances are less evident due to the structural and radiationdamping The small peaks visible at 9 and 36 Hz are due to the
3 2
E '
ji 0
2 -1 -7
- 3
ap
R
ap ap
Trang 10'•5
ap R R
R ap R ap
Fig 8 Contour piot of the frequency response as a function of the
connection radius and frequency
30 40 50 Frequency [Hz]
Fig 10 iVIaximum far fieid sound pressure ievel for different vaiues of
the connection radius, for a soft connection between the foundation ofthe propeiier-shafting system and the huii stern end piafe The junctiononiy affects the axiai motion of the end piate, resulting in a iower SPL at
certain frequencies
out-of-plane vibration of the bulkheads and end plates The
bulk-head 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
acoustic responses of the hull This is shown by a considerableshift in the natural frequencies with a related increase or decrease
of the structural and acoustic responses in the entire frequencyspectrum The connection radius can thus be optimized to mini-mize the radiate sound pressure In this section, the optimum valuefor the connection radius /i^p is found by minimizing the totalmaximum 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 rangesdefined by/max = Hz and/max = Hz are given in Figs 11 and 12,respectively, for both soft and hard connections A coarse incre-ment for the radius of 0.1 m was used The numerical integrationwas performed using the trajjezoidal method The cost functionwas 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
1.5 Connection radius/? [m]
ap
Fig 11 Variation of cost function Jo_8o with connection radius, for hard
and soft connections between the foundation of the propeiier-shaftingsystem and the huii stern end piate The optimum radius for each con-
nection is shown by a solid marker
Trang 11Fig 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
con-nection is shown by a solid marker
10 9-Q
Fig 14 Variation of cost function J25 with connection radius, for hard
and soft connections between the foundation of the propeller-shaftingsystem and the hull stern end plate The optimum radius for each con-
nection is shown by a solid marker
In Fig 13, Pmax is minimized at the fundamental hpf imd its
n harmonics, scaled by l/n In Fig 14, the maximum radiated
sound pressure F^ax is minimized at the fundamental propeller
hpf of 25 Hz The optimum value for the connection radius for
the various cost functions are highlighted in Figs 11 to 14 with a
solid marker and summarized in Table 2 after refinement using a
resolution for the radius of 0.01 m For Jo-s.0 in Fig 11, the soft
and hard connections give similar trends, but the lower values are
given by a soft connection It is also observed that minimization of
the cost function for the full frequency range (7()_8o) and at the hpf
and its superharmonics (./25.50.75) results in nearly identical values
for the optimum connection radius due to minimization of the cost
functions over a broader frequency range
4.2.2 Frequency response function and force transmissibility.
The frequency response function of the axial displacement at the
connection between the cylindrical hull and stem end plate
is presented in Fig 15, for the optimum values of/?ap with a soft
connection and for a rigid connection to the hull (^ap = ^)- A
significantly lower structural response is observed as the
connec-tion radius moves away further from the outer periphery of the
Fig 13 Variation of cost function J25.50.75 with connection radius, for
hard and soft connections between the foundation of the
propeller-shafting system and the hull stem end plate The optimum radius for
each connection is shown by a solid marker
hull, especially at higher frequencies Figure 16 shows the force
transmissibility between the propeller and the stem end plate at
the hull junction, determined by 7", = NyJFo where Ny_„ is the
membrane axial force of the annular plate and F» is the amplitude
of the harmonic axial force generated at the propeller Similartrends are observed in the results for the frequency response ofthe axial displacement and the force transmissibility It can beshown that using the force transmissibility or the axial velocity atthe cylinder/cone junction as cost functions does not result in theoptima connection radii found by minimization of the far-fieldradiated sound This occurs because the optimization does nottake into account the radiation efficiency of the excited structuralmodes A plot of maximum sound pressure level as a function offrequency for the optimum values of the connection radius /?apusing a soft connection between the foundation of the propeller-shafting system and the hull stem end plate is presented in Fig 17.The maximum SPL for a rigid connection is also shown As
expected, the optimum connection radius of R.^p = 0.88 provides
the best overall reduction in maximum radiated sound pressure asthis radius was obtained from minimization of the cost functionover a broader frequency range
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 tion can be obtained by minimizing the radiated sound power,which has been estimated at the hull surface The sound powercan be expressed as an integral over the surface of the structure So(Skelton & James 1997)
Hard connection0.79 m1.44 m0.87 m2.02 m
Trang 1230 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
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-
Equa-tion (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),
20 40 60Frequency [Hz]
80
Frequency [Hz]
Fig 16 Force transmissibility for optimum vaiues of the connection
radius using a soft connection between the foundation of the
propeller-shafting system and the huii stern end piate The force fransmissibiiity
for a rigid connection is also shown
Fig 17 Maximum sound pressure levei for the optimum vaiues of the
connection radius using a soft connection between the foundation of thepropeiier-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)Results for the variation of the cost function with connectionradius are shown in Fig 18 for a hard connection between thefoundation of the propeller-shafting system and the hull stem endplate The optimum value for the connection radius for the variouscost functions are highlighted with a solid marker It can be seenthat the general trend and values of the optimum connection radiifor minimization of the radiated sound power at the hull surfaceare very similar to those obtained by minimizing the far-fieldmaximum sound pressure, since these quantities are directlyrelated However, minimizing the radiated sound power provides
an advantage in that it does not require solving the Helmholtzintegral in the far field
4.3 Acoustic transfer function
Optimization of the resonance changer parameters requires culation of the sound pressure several times, which becomes com-putationally very time consuming It is therefore useful to use anacoustic transfer function to obtain the maximum sound pressurefor a specific value of the connection radius The acoustic transfer
cal-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 hullstern end plate The optimum radius for each cost function is shown by
a solid marker
158 SEPTEIUIBER2011 JOURNAL OF SHiP RESEARCH
Trang 13xlO' Table 3 Optimum values for (Cr, K„ M,) with flap =
Frequency [Hz]
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
connec-tion 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
fre-quencies 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
reso-nance 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
pro-peller 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«
.^0-80 ^0-40
•'25.50.75
hi
5,0005,0005,0005,176.1
9.1048 X 10'2.4480 X 10'9.9060 X 10'3.7784 X 10"
1,0001.0001,0001,513
The center points of these spaces are used as the starting pointsfor a generalized pattem search algorithm to find 6'^ local min-ima The local minima are then used as part of the initial popu-lation for a genetic algorithm A total population of 8"^ points isused, the remaining 8^^ - 6'^ points are randomly created Forthe genetic algorithm the following default parameters wereused: crossover fraction = 0.8, elite count = 2, migration frac-tion = 0.2, generations = 100 Both pattem search and geneticalgorithms use the augmented Lagrangian pattem search algo-rithm (Lewis & Torczon 2002, Conn et al 1991, 1999) Themaximum sound pressure was calculated using the acoustictransfer function Wpw, which greatly reduced the computationaltime The optimum resonance changer values are summarized inTable 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 theresonance changer parameters leads to similar values obtainedfor the optimum parameters from minimization of the cost func-
tion for the full frequency range (Jo-so) and at the t>pf and its
For the four frequency-weighted cost functions, the weightedmaximum SPLs are shown in Figs 20 to 23 with and without theuse 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] iCr [N/m] Mr [kg] [Hz]
.'0-80 ^0-40
•'25.50.75 / 2 5
0.87 0.88 1.48
5,000 5,000 5,000 5,176.1
8.8435 X 2.4144 X 9.7085 X 8.6820 X
1010'10'10'
1,000 1,O(K) 1,000 3,497.5
47.3
24.7 49.6
25
Trang 14Fig 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
Fig 22 Maximum sound pressure levels as a result of minimizing
^25.50,75- The weighted maximum SPLs are presented with and withoutthe use of a resonance changer, for both a rigid connection of the foun-dation 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
Mini-mizing 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
are almost identical, providing the best parameters for the design
of the propeller-shafting system When the resonance changer isused with a rigid connection of the propeller-shafting system tothe hull, the location of the antiresonances coincide with the natu-ral frequencies of the resonance changer shown in Table 4 Thisoccurs because at these frequencies, most of the energy from thepropeller forces is absorbed by the motion of the resonancechanger Figures 20 to 23 show that optimization of both theconnection radius and the resonance changer results in a signifi-cant reduction in the radiated sound pressure
The main results of this work are summarized by Fig 24 wherethe cost functions for optimization of the connection radius only,the optimization of the resonance changer only and the combinedoptimization of both the connection radius and resonance changerparameters are normalized respect to the maximum value of thecost function in the frequency range considered for each case It isshown that the resonance changer reduces the cost functions by
ap
Frequency [Hz]
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
with RC, R =a ap ap
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 thehull and using the optimized connection radius with a flexible end plate
Trang 15Fig 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
5 Conclusions
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
propeller-shafting 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
propel-ler-shafting system The foundation of the propelpropel-ler-shafting
sys-tem was coupled to the hull using the ssys-tem 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
har-monic 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
corre-spond 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
maxi-mum 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
propeller-shafting system and hull The use of a resonance changer in
conjunction with an optimized connection radius was
investi-gated, 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
transfer function that was minimized using a combined geneticand pattern search algorithm Using a resonance changer inconjunction with a flexible connection of the propeller-shaftingsystem to the hull can introduce two antiresonances in the hullresponse at design frequencies, thereby resulting in a significantreduction in the radiated sound pressure levels in both narrowand broad frequency ranges
References
ABRAMOwrrz, M., AND STEGUN, L A 1972 Handbook of Mathematical 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 acousticresponses 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
subma-rine hull under harmonic excitation Applied Acou.itics, 71, 17-31.
CONN, A R., GOULD, N I M., AND TOINT, P L 1991 A globally gent augmented Lagrangian algorithm for optimization with general con-
conver-straints and simple bounds, SIAM Journal on Numerical Analysis, 28,
545-572
CONN, A R., GOULD, N t M ANDTOINT P L 1999 A globally cotivergent
augmented Lagrangian barrier algorithm for optimization with general
inequality constraints and simple bounds Mathematics of Computation, 66,
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,
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
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.
Trang 16TONO, L 1993 Free vibration of orthotropic conical shells International (j _ y2\^ (1 — v^)Az El Eh^ Journal of Engineering Science, 31, 719-733 |JL = , \ = , 'i\ = , D = — r—
Tso, Y K., AND JENKINS, C J 2003 Low Frequency Hull Radiation Boise "" "^" "'^ 1 2 ( i — v )
Defence Science and Technology Organisation, UK, Report No Dstl/ (A2)
TR05660 ^ '
^ ^ The ring stiffeners have cross sectional area A h is the stiffener
The coefficients in the Flügge equations of motion given by ^P'»'^'"^' ^nd r, is the distance between the shell midsurface andequations (1) to (2) are given by Caresta and Kessissoglou (2010): '^e centroid of a nng / is the area moment of inertia of the
stiffener about its centroid and m^^ is the equivalent distributed
P _ ^ _ 1 I jJL_L^!£î d _ i + (^ + ^X _ 1 + 3 T | rnass on the cylindrical shell to taike into account the onboard ''^ ' fe/i p/i ' fl^ ' a^ equipment and the ballast tanks.
162 SEPTEMBER 2011 JOURNAL OF SHIP RESEARCH
Trang 17Copyright 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.
Trang 18Journal 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 fromquadratic product of the first-order quantities is presented using the near-fieldmethod based on the direct pressure integration over floating body in time domain
The transient wave-body interaction of the first-order radiation and diffraction lems are solved as the impulsive velocity of the floating body by the use of a three-dimensional panel method with Neumann-Kelvin method These radiation anddiffraction forces are the input for the solution of the equation of the motion that issolved by the use of the time marching scheme The exact initial-boundary-valueproblem is linearized about a uniform flow, and recast as an integral equation using
prob-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 calculatedmean second-order added resistance and unsteady first-order impulse-responsefunctions, hydrodynamics coefficients, exciting forces, and response amplitudeoperators 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.
hydrodynamic damping due to heave and pitch motion that are thedominant motions for added resistance is much bigger than theviscous damping This implies that the prediction of the addedresistance is an inviscid problem, and the potential formulationscan be applied and assumed to give accurate predictions It can beexpected that the biggest contribution due to the radiation problem
to the added resistance will be in the region of the resonancefrequency of heave and pitch motion The diffraction-inducedadded resistance will be dominated by high incident wave fre-quencies where the floating body motions are smail It is assumedthat the added resistance in a seaway is considered independentfrom calm water resistance (Kara et al 2005), which is due to theforward speed of the floating body, and these two resistances areadded to each others to get the total resistance
The added resistance is the longitudinal component of the meansecond-order wave forces in the case of nonzero forward speed.This second-order force is proportional to the square of the waveamplitude and hence is nonlinear The second-order forces on aship due to the diffraction of the waves on a fixed body and due torelative first-order motions of the floating body were pioneered byHavelock (1940, 1942) The added resistance can be computedfrom quadratic products of the first-order quantities Three differ-ent methods can be in general used for the prediction of thissecond-order force The first one is the method of radiated energy
Trang 19(Gerritsma & Beukelman 1972), which is based on the
determina-tion 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
approxima-tion for fixed or freely floating axisymmetric bodies in the
presence of bichromatic incident waves The semianalytical
for-mulation 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
sur-face needs to be discretized using quadrilateral or triangular
ele-ment Both body surface and some part of the free surface needs
to be discretized to satisfy the free surface and radiation
condit-ion numerically in the case of Rankine source Green functcondit-ion
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
ele-ment 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
double-body 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
bound-ary value problem is transformed from the volume to boundbound-aryintegral equation on the fluid boundary applying the Green's the-orem over the transient free-surface wave Green function (Kara &Vassalos 2003, 2007) Then, the exact initial boundary valueproblem is linearized using the free stream as a basis flow,replaced by the boundary integral equation The resultant bound-ary integral equation is discretized using quadrilateral elementsover which the value of the potential is assumed to be constant andsolved using the trapezoidal rule to integrate the memory or con-volution part in time The free-surface and body boundary condi-tions are linearized on the discretized collocation points over eachquadrilateral element to obtain algebraic equation
2 Exact Initial boundary value problem
Two right-handed coordinate systems are used to define thefluid action A Cartesian coordinate system Xo = (.Vo, Vo, ZQ) isfixed in space Positive Vp-direction is toward the bow, positive ZQ-direction points upward, and the zo = 0 plane (or A'O.VO plane) iscoincident 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 submergedportion of the body hull and free surface, ideal (inviscid andincompressible), and the flow is irrotational (no fluid separationand 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'<I>(xo,O = 0, and the disturbance flowvelocity field V(xo,0 may be described as the gradient of thepotential <I>(x(),/) (e.g., V(xo,r) = V<J)(xo,r)) The fluid pressure
field, p(xo, /), is then defined from BemouUi's equation
+Patm(xo) (1)
Fig 1 Coordinate system and surface of the problem
Trang 20where p is the fiuid density, f> 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
pres-sure equals the atmospheric prespres-sure on the free surface Neglecting
the surface tension effect and using BemouUi's equation
[equa-tion ( 1)] the dynamic free-surface boundary condi[equa-tion is given as
(2)
ô<ï> 1
+ V O • V*-I-;?Z(i = 0 on zo =
ot L
The dynamic boundary condition equation (2) may be used to
determine the unknown free-surface elevation
1 /â4>
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 1— V ^ • V(V4* • VO) -j- P O - := 0 on zo = L
df dt 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(/)
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
con-dition 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
may be separated into individual parts for the linearized problem
In addition to the separation of the fluid disturbance into steadyand unsteady parts, the free-surface boundary condition, bodyboundary 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 thesteady problem that gives rise to the calculation of the steadyresistance, sinkage force, and trim moment Then the unsteadyproblem, which consists of radiation and diffraction problems, issolved, when the body is in its equilibrium position Because ofthe small disturbance of the fluid, the total velocity potentialproduced by the presence of the floating body in the fluid domainmay be separated into three different parts
= 9basls (X) + ^steady {^) + fli x, 0
(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 wavefield If the incident wave is reflected by the floating body, theresultant potential is the scattering potential ip3(x,/) and com-prises the diffraction potential The solution of the incident wavepotential and diffraction potential is called diffraction problem.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 theradiation problem This kind of decomposition is given byHaskind (1953) and gives rise to the linearization of the govemingequations, which are the free-surface condition, body boundarycondition, and BemouUi's equation Physically, this kind ofdecomposition equation (6) ignores the interaction of the wavesproduced by the individual components
In the moving coordinate system (body-fixed coordinate tem), the fluid velocities consist of the free stream and theundisturbed incident wave components in the far field and may bewritten as
sys-V O -^ -U\ -I-sys-Vtp, X 0 0 (7)
The basis flow ipba.sis (x) is taken as the free stream potential farfrom the body, and it is assumed its contribution is much biggerthan the remaining potentials, which are the nomial components
of the incident wave velocity on the body The traditional tion for the basis flow is the double body flow and free streamflow The latter is used in the present paper and may be written as
selec-'Pbasi.,(x) = -Ux (8)
This kind of selection of the basis flow gives the Neumann-Kelvinlinearization of the pressure, the free surface, and the body bound-ary condition and eliminates the interaction between the variouspotentials except for the interaction of the steady flow with thebody boundary conditions For the free-surface boundary condi-tion, the Eulerian description of the flow is used Thus, noovertuming and breaking waves are allowed to exist Using the
JOURNAL OF SHiP RESEARCH 165
Trang 21linearized potential equation [equation (6)] in the free-surface
boundary condition equation (4), the linearized free-surface
con-dition about the mean positions of the floating body in the moving
coordinate system may be written as
where <f> is used for all the perturbation potentials Using the
linearized potential equation (6) in the body boundary condition
equation (5), the linearized body boundary condition about the
mean positions of the floating body in the moving coordinate
system may be written as
'''Pstieady
andn
= Un\ on
dn on
r
where Sb is the mean position of the floating body,
generalized unit normal vector and may be written as
2, «3) = ñ, = r x R, r = (x,y,z)
(10)
(11)(12)
is the
(13)
where xi^ is the amplitude of the unsteady motion in six degrees of
freedom Vi, V2, x^ are the linear translational amplitudes, and x^,
Xf;, X(, are the linear rotational amplitudes along the x, y, and z
directions, respectively
The m* — terms in the body boundary condition for the
radia-tion problem equaradia-tion (12) implies that the steady and unsteady
potentials are coupled through the presence of these /n^ - terms
that are the gradient of the steady velocities in the normal
direc-tion and are given as
(wi|,m2, W3) = —(ff • V)V<I>, (m4,m5, Wfi) = —(n • V) (r x V<I>)
(14)where V $ is the fluid velocity vector due to steady translation of
the body and is given by Eq.(8) as
V* = V(-Ux)
For the Neumann-Kelvin linearization, the gradient of the steady
velocity mi¡ — terms reduces to
Equations (10), (11), and (12) represent the steady problem body
boundary condition, diffraction problem body boundary
condi-tion, and radiation problem body boundary condicondi-tion,
respec-tively The latter results developed by Timman and Newman
(1962) from the linearization of the complete normal body
bound-ary condition on the instantaneous body to the mean underwater
body 5b
The linearized Bernoulli's equation for the fluid pressure field
can be written in the body fixed coordinate system as
(16)The corresponding first-order wave elevation at a point z = 0 plane
is obtained from the dynamic free-surface condition
(16a)
z=0
3 Solution of boundary integral equation
The initial boundary value problem consisting of initialcondition, free-surface, and body boundary condition may berepresented as an integral equation using a transient free surfacewave Green function (Wehausen & Laitone 1960) ApplyingGreen's theorem over the transient free surface wave Greenfunction derives the integral equation It is possible to show thattransient free surface wave Green function satisfies the initialboundary value problem without a body (Finkelstein 1957) Inthe case of the prediction of fluid velocities on the body surface,which is the case for the second-order force calculations, sourceformulation over potential formulation is preferred to avoid tak-ing the spatial derivatives of potential numerically The sourceformulation can be derived by the use of the flow in the regioninterior to the body specified by the scalar potential cp' Theintegral equation for ip' is the same as for ip while normal vector
is defined in opposite direction The equations for <f' and (p can
be added The source strength is defined as CT = ipj^ — ip„ andsource formulation is obtained by choosing ip' = (p on the bodysurface Integrating Green's theorem in terms of time from —00
to 00 using the properties of transient-free surface wave Greenfunction and potential theory, the integral equation for thesource strength on the body surface may be written as in Kara(2000)
(17)
and potential on the body surface
where
G(\,é,,t,T) = 2 j dky/kgsin[y/kg(t — j)]e''''-'^^^jQ(kR) is the
memory part of the transient free-surface wave Greenfunction
X = [x(t),y(t),z(t)] is the field point.
Trang 22, Ti(í), Ç(f)] is the source point.
r = ^{x - if+{y - "<\f
field and source point - if the distance between
r'= J[x-i,f+{y-T\f+{z-\-t,f is the distance between
field point and image point over free surface
70 is the Bessel function of zero order
The memory part of the transient free-surface wave Green
function G(.v,^,i, T) represents the potential at the field point
X = (x, y, z) and time t due to an impulsive disturbance at
source point f = (^, T], Q and time T.
The ;• and /•' represent the Rankine part of the source potential
The integral equation for the source strength equation (17) is
first solved, and then this source strength is used in the potential
formulation equation (18) to find potential and fluid velocities at
any point in the fluid domain The solution of the integral equation
[equation (17)] is done using time marching scheme The form of
the equation is consistent for both the radiation and the diffraction
potentials so that the same approach may be used for all
poten-tials Since the transient free surface wave Green function
G{x,Ê,,t, T) satisfies free-surface boundary condition and condition
at infinity automatically, in this case only the underwater surface
of the body needs to be discretized using quadrilateral/triangular
elements The resultant boundary integral equation [equation (17)]
is discretized using quadrilateral elements over which the value of
the source strength is assumed to be constant and solved using the
trapezoidal rule to integrate the memory or convolution part in
time This discretization reduces the continuous singularity
distri-bution to a finite number of unknown source strengths The
inte-gral equation [equation (18)] is satisfied at collocation points
located at the null points of each panel This gives a system of
algebraic equations that are solved for the unknown source
strengths At each time step, the new value of the source strength
is determined on each quadrilateral panel
The evaluation of the Rankine source type terms (e.g., \/r, X//)
in equation ( 17) is analytically integrated over quadrilateral panels
using the method and formulas of Hess and Smith (1964) For
small values of r the integrals are done exactly For intermediate
values of /- a multipole expansion is used For large values of r asimple monopole expansion is used The surface and line integralsover each quadrilateral element involving the wave term of thetransient free surface wave Green function G(.v,f,/ T) are solvedanalytically (Liapis 1986, Beck & Liapis 1987, King 1987,Newman 1990) and then integrated numerically using a coordi-nate mapping onto a standard region and Gaussian quadrature Forsurface elements the arbitrary quadrilateral element is firstmapped into a unit square Then, a two-dimensional Gaussianquadrature formula of any desired order is used to numericallyevaluate the integrals The line integral is evaluated by sub-
dividing T{t) into a series of straight line segments The source
strength a(x, f) on a line segment is assumed equal to the sourcestrength of the panel below it
The evaluation of the memory part of the transient free-surfacewave Green function G(.v.|./ T) and its derivatives with an effi-cient and accurate method is one the most important elements inthis problem Depending on the values of x, | , and /, five differentmethods are used to evaluate G(.v,f, f, T); power series expansion,
an asymptotic expansion, a Filon integration quadrature, Bessellfunction, and asymptotic expansion of complex error function.Figure 2 shows the memory part of transient free-surface wave
is calculated by the appropriate time lag from the instant of thecorresponding impulse In the case of free surface, the linearsystem has a memory, meaning how the free surface affects thelinear system in a later time of motion of the body surface whenthe impulse is applied at one instant of time The body boundary
SEPTEMBER 2011
Fig 2 The memory part of the transient free surface wave Green function G(ß, p.)
JOURNAL OF SHIP RESEARCH 167
Trang 23condition corresponding to an impulsive velocity of floating body
using equation (13) can be written as
(19)
where 8(r) is Dirac delta function and H{t) is Heaviside unit step
function Thus, it is natural to divide the radiation potential as
impulsive and transient parts
, t) = i|;,,( x, t)H{t) (20)
where the instantaneous potential vl/u.(x) represents the
instanta-neous fluid response to the motion of the body If the body moves
and suddenly stops, the entire fluid motion associated with the
i|in(x) potential stops The time-independent impulsive potential
»|j2t(x) represents the potential due to the steady displacements In
other words, if the body is given a unit impulsive velocity in the
kth mode, the floating body will have a unit displacement in that
mode The time-dependent memory potential x,t(x, i) represents
the transient potential, which results from the effect of the free
surface In the case of the transient problem, all motions die out
after a reasonable time and all displacements approach zero
asymptotically In other words, the transient potential Xi(x, 0
is the velocity potential of the motion that results from the
impulse of the floating body velocity at time t = 0 The
time-independent impulsive potentials »|<u(x) and <^2k{^) provide initial
conditions on the potentials that describe the transient motion
Xi(x, t) The motion of the floating body is considered a sequence
of in.stantaneous motions For each impulse, there is an
imme-diate fluid response due to the incompressibility of the fluid
and the free surface results in an extended response, which is
lasting longer than the impulse itself The generalized
displace-ments x{t) and velocity x(t) are the inputs and the velocity
poten-tial X; (x, ') is the impulse response functions for the total velocity
potential
The general radiation potential for an arbitrary forced motion in
the kth direction corresponding to the impulsive velocity of a
float-ing body may be expressed in terms of a convolution integral as
(21)
It may be shown that the generalized radiation potential <I>((x, t)
satisfies the free surface boundary condition, body boundary
con-dition, and condition at infinity for all time t The integral
equa-tions that must be solved to determine v|/n(x), >|J2A(X), and x<(x, t)
are found by applying integral equation [equation (18)] on the
body surface and substituting equation (20) Gathering terms
pro-portional to 8(0 and H(t) gives integral equations for il/n(x) and
feix), respectively The remaining terms yield an integral
equa-tion for Xi(x.i) The details of the derivaequa-tion can be found in
Liapis (1986) and in Kara (2000)
The transient response of the floating body is required in the
radiation problem For each radiation problem, the steadily
trans-lating floating body is moved impulsively in mode k, and the force
on the floating body in mode j (i.e., corresponding radiation
impulse-response function) is calculated For the generalized
radi-ation force Fjicit) acting on the body in the yth direction because of
an arbitrary motion in the klh mode is determined (Cummins 1962)
The coefficient a^ is the time and frequency independent
con-stant, it depends on the body geometry and is related to added
mass The coefficients hß and c,; are the time and frequency
independent constants and depend on the body geometry, and
forward speed The coefficients />;<., Cjk are related to damping
and hydrostatic restoring coefficient, respectively The memory
coefficient KjiA^t) is the time dependent part, depends on body
geometry, forward speed, and time, and it contains the memoryeffect of the fluid response The convolution integral on the right-hand side of equation (22), whose kernel is a product of the
radiation impulse response function Kß(t) and velocity of the
floating body x<(/), is a consequence of the radiated wave ofthe floating body When this wave is generated, it affects thefloating body at each successive time step
Figures 3 and 4, which are the results of our in-house directtime domain program of ITU-WAVE, show convergence ofnondimensional radiation force impulse response functions as afunction of nondimensional time by the use of equation (26) inheave and pitch modes in terms of panel numbers for a Wigleylll
hull form at F„ = 0.3, respectively The results are converged both
heave and pitch modes for the panel numbers of 256 over halfbody for the present calculation The subsequent results areobtained using 256 panels over half body with nondimensionaltime step size of 0.05
4.1 Frequency response function for the radiation problem
In equation (22), the time domain force coefficients are related tothe frequency domain force coefficients If the motion of the body
is considered as a time harmonic motion e.g., v(/) = e'"''' / > 0 at
frequency Wg, then the force in the frequency domain in complexform may be written as
Fß{t) = {(úlAjt{b)e) - ;'ü)e.ßy^(ü)e)}?""'••' (27) Using time harmonic motion x(t) = c'"'' / > 0 in the time domain
force expression equation (22) and equating reai and imaginarypart of equation (27) and equation (22), the impulse-responsefunctions are related to the more familiar frequency responsefunctions (i.e., the added-mass and damping coefficients) through
a Fourier transform
Trang 24COS(Ü),T) (29)
where the coefficients /í/<.(We) and ß,i(ü)e) are the
frequency-dependent added-mass and damping coefficients, respectively
Figures 5 through 12 show the nondimensionai added-mass,
damping coefficients, and cross-coupling of added-mass and
damping coefficients as a function of nondimensionai frequency
for a Wigleylll hull form at F„ = 0.3 in heave and pitch modes.
Figures 5 through 12 are obtained by Fourier transform of results
of Figs 3 and 4 according to equation (28) and equation (29) for
the added-mass and damping coefficients, respectively The
experimental results, which are compared with our ITU-WAVE
numerical results, for Wigleylll hull form are taken from Joumee
(1992) It should be noted that even though there are oscillations at
larger times in the impulse response functions, such as in heave
mode of Wigleylll hull form in Fig 27, which is the expanded
5 Diffraction problem
The general diffracted wave potential due to an arbitrary dent wave on the body fixed coordinate system may be deter-mined in temis of the convolution integral as in the radiationproblem
inci-(30)
where (ps(x, r) and <Pi(x, i) are the impulse-response functions forscattering and incident wave potentials, respectively The diffrac-tion problem, that of finding the velocity potential for the case
of the floating body fixed to its mean position in the presence of
an incident wave, may be solved to find the transient excitingforce When the diffraction problem is forced by an impulsivewave elevation, the computed transient forces may be related to
Trang 2712 Nondimensional pitch-heave coupling damping coefficient of Wigleylil hull at F„ = 0.3
impulse-response functions For the generalized diffraction force
f/D(/) acting on the body in the jlh direction may be solved to
determine the transient exciting forces in the case of the presence
of given an arbitrary, known, incident wave elevation on the
body-fixed coordinate system (King 1987, Kara 2000, Korsmeyer &
where Kjs(t) and Kß(t) are the impulse response function for
diffraction and Froude-Krylov forces, respectively The kernels
and Kß(t) are of the form that corresponds to a
time-invariant linear system since the reference point of the waves isflxed with respect to the moving floating body The excitation ofthe floating body is provided by io('), the arbitrary wave elevation
in the body-fixed coordinate system Kß{t) is found by direct
integration of the time derivative of the impulse-response-function
of the incident wave potential (p](i) over the floating body surface.The scattering perturbation potential ips(O represents the dif-fracted wave potential due to an impulsive incident wave
Kjsit) is the impulse response function on the floating body
which is found from solving the diffraction problem using tion ( i l ) and (17) forced by the incident wave potential which isknown and given as
equa-(fiiix, 11 — — e ' ' e w^J
where the encounter frequency is given as We = w — Uok cos (ß),
Ü) is the absolute frequency of the linear system, ß 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 (ß) + y sin (ß).
Figures 13 and 14 show convergence of nondimensional excitingforce impulse response functions as a function of nondimensionai
Trang 28Fig 14 Nondimensional pitch exciting force impulse-response function for Wigley 111 hull at Fn = 0.3 and ß = 180 deg
time step for a Wigleylll hull form at Fn = 0.3 and ß = 180 deg in
heave and pitch modes in terms of panel numbers, respectively
The results are converged for the panel numbers of 256 over half
body for the present calculation The subsequent results are
obtained using 256 panels over half body with nondimensional
time step size of 0.05
It is assumed that the incident wave potential is a unidirectional
wave system that contains all frequencies, and it describes a wave
elevation which is the Dirac delta function 8(0 in time, when it is
viewed from the origin of the body-fixed coordinate system The
impulse is the wave elevation at the body-fixed coordinate system
at time í = 0, and the response is the fluid velocity or pressure due
to this incident wave eievation at the origin Unlike the radiation
impulse response function, the diffraction impulse response
func-tions are nonzero at time í < 0 as can be seen from Figs 13 and
14 This is the resuit of the dispersion of the free surface waves
Figures 15 and 16 (which are the breakdowns of Figs 13 and
14) show the Froude-Krylov, diffraction, and total exciting force
impulse response functions of the heave and pitch modes of
motion for Wigleylll hull at Fn = 0.3 and ß = 180 deg,
respec-tively Froude-Krylov exciting force is even about / = 0 for heave
and pitch modes This is the result of the impulsive wave pressure
which is an even function in time about t = 0 The diffraction
force impulse response functions shown in Figs 15 and 16 do notdisplay the same nice symmetry properties as the Froude-Krylovforces Because of the memory effects of the diffracted wavesystem, the results are neither odd nor even
The free surface elevation due to the incident wave Ço(0 at thebody-fixed coordinate system is given as
Trang 29Fig 16 Pitch Froude-Krylov, diffraction, and total exciting force Impulse-response functions of Wigleylll hull at Fn = 0.3 and ß = 180 deg
shown in Fig 17 Hence some disturbance is experienced before
t = 0 due to the dispensed waves and requires that the impulse
response functions have values at times less than zero In the case
of U = 0 and ß = TT; at times other than / = 0, the waves are
dispersed over just one-half of the free surface For any time t < 0,
the waves are only in the x > 0 half-space, while for t > 0, the
waves are only in the x < 0 half-space In the former, the waves
are coalescing to the impulse, and in the latter they are dispersing
from the impulse as shown in Fig 17
The computation of the free-surface impulse response function
equation (36) is carried out via an extension of algorithms
com-monly used for the calculation of the complex error function
(Gautschi 1969)
5.1 Frequency response function for the diffraction problem
Similar to the radiation problem the time domain exciting force
is related to the frequency domain exciting force via Fourier
trans-forms If the motion of the wave is considered as time harmonic
io(O = e"^' at the encounter frequency of Ue, then the exciting
force in the frequency domain in complex form may be written as
(37)
Using the time harmonic wave elevation Ço(') = *"""'' in equation(31 ) and equating real and imaginary parts of equation (37) and
equation (31), the impulse-response function are related to the
more familiar frequency response functions through a Fouriertransform
(38)
where X,(ü)e) is the complex exciting force The real part of theequation (38) is the amplitude of the exciting force, while imagi-nary part of equation (38) is the phase angle of the exciting force
in the frequency domain Thus, exciting force in the frequencydomain can be determined from the Fourier transform of theimpulse response functions for the Froude-Krylov and diffractionforces, based on a wave of impulsive elevation
Figures 18 through 21 present the nondimensionai amplitudeand phase angle of the exciting force versus nondimensionai
frequency kL for Wigleylll hull form at Fn = 0.3 and
Trang 30Fig 18 Nondimensional heave exciting force amplitude for Wigleylll hull at Fn = 0,3 and ß = 180 deg
ß = 180 deg in heave and pitch modes of motion The
fre-quency domain results in Figs, 18 through 21 are obtained
from the time domain results by taking the Fourier transform
of the results of Figs 13 and 14 by the use of the equation
(38), The experimental results are taken from Joumee (1992),
Figures 22 and 23 (which are the breakdowns of the Figs, 18
and 20) show the amplitude of Froude-Krylov, diffraction, and
total exciting forces in frequency domain for heave and pitch
modes of motions
6 Equation of motion
It is assumed that the weight, sinkage force and trim moment of
the ships are balanced by the hydrostatic pressure while the steady
resistance of the floating body is balanced through propulsion
The extemal exciting forces including Froude-Krylov and ing forces are balanced by the inertia and radiations forces Thememory effects on the free surface both radiation and excitingforces are taken into account by the use of the convolution inte-grals In the context of the linear theory, the equation of motion ofany floating rigid body may be written in a form which is essen-tially proposed by Cummins (1962)
Trang 31Fig, 19 Heave exciting force phase angle for Wigleylll hull at Fn = 0.3 and ß = 180 deg
Fig 21 Pitch exciting force phase angle for Wigleylll hull at F„ = 0.3 and ß = 180 deg
Trang 32Fig 22 Heave Froude-Krylov, diffraction and total exciting force amplitude for Wigleylll huii at Fn = 0.3 and ß = 180 deg
10 kL
Fig 23 Heave Froude-Krylov, diffraction and total exciting force amplitude for Wigleylll hull at Fn = 0.3 and ß = 180 deg
The displacement of the floating bodies from its mean position in
each of its six rigid-body modes of motion is given Xy^U) in
equa-tion (39), and the overdots indicate differentiaequa-tion with respect to
time The inertia matrix of the floating body is My,t, and linearized
hydrostatic restoring force coefficients are given by Cj^.
The radiation impulse response functions are composed of the
hydrodynamic coefficients and the kernel of the convolution on
the left-hand side of equation (40) A radiation impulse response
function is the force on the body in the jlh direction due to an
impulsive velocity in the kth direction, with the coefficients ay^,
hj)^, and Cj^ accounting for the instantaneous forces proportional to
the acceleration, velocity, and displacement, respectively, and the
memory function A'y^(O accounting for the free surface effects
thai persi.st after the motion occurs For the radiation problem
the term "memory function" is used to distinguish this portion
of the impulse-response function from the instantaneous force
components outside of the convolution on the left-hand side
of equation (39) The term Kß)(t) on the right-hand side of
equa-tion (39) are the components of the exciting force and moment's
impulse response functions due to the incident wave eievation
) defined at a prescribed reference point in the body-fixed
coordinate system Here, the kernel Kji,(t) is the diffraction
impulse response function; the force on the body in the yth tion due to a unidirectional impulsive wave elevation with a head-ing angle of ß Once the restoring matrix, inertia matrix, and fluidforces, for example, radiation and diffraction forces are known,the equation of motion of floating body equation (39) may besolved using the fourth-order Runge-Kutta method Figures 24and 25 show the heave and pitch response amplitude operatorsthat are obtained by the time marching of the equation (39) foreach encounter frequency, respectively The experimental results,which are compared with our ITU-WAVE numerical results, areobtained from Journée ( 1992)
direc-The decay of the forward speed impulse response functions intime is different from that of zero speed impulse response func-tions due to the resonance at the critical reduced frequency T =
f^cU/g = 1/4 The impulse acting on the floating body generates
energy due to the presence of the wave system This energy at thegroup velocity of wave components propagates away from thefloating body at zero-forward speed, while in the case of forwardspeed, this energy remains in the vicinity of the floating bodysince the group velocity of the wave component is approximately
SEPTEMBER 2011 JOURNAL OF SHIP RESEARCH 177
Trang 33Fig 25 Pitch response amplitude operator (RAO) for Wigleylll hull at Fn = 0.3 and ß = 180 deg
equal to the speed of the floating body For the long simulation
of the floating bodies, it is very important to avoid the
computa-tion of transient free-surface wave Green funccomputa-tion, which results
in the prediction of the impulse-response function for each
mode, at each time step In the present paper, the computation
of the impulse response functions are truncated at the
non-dimensional time step of \5\Jg/L and the asymptotic values of
each impulse response functions are approximated (Bingham
1994) a s í — 0 0
Kjk{t) « 00 -I- - [a\ cos(a)c/) -I- 02 sin(ü)c/)] (40)
The constants in equation (40) can be determined by a least
squares fit Figures 26 and 27 show comparison between a very
long calculation of the heave impulse response function and
asymptotic continuation results
The solution of the time domain discretized integral equations
demonstrates an oscillation over longer time as shown in
expanded view of the heave impulse-response function in Fig 27
The oscillatory error at large time is apparently the result of the
integral equation method of solution and not numerical
inaccura-cies The oscillatory error in the time domain discretized integralequations is the equivalence of he irregular frequencies in thefrequency domain This oscillation persists indefinitely in timefor the zero forward speed case, while its amplitude decreaseswhen forward speed is increased The oscillation amplitude both
at zero and forward speed cases can be reduced by increasingpanel numbers and by decreasing the time step size
7 Pressure integration method for second-order
or proportional to the square of the wave amplitude The solution
of the second-order problem results in mean forces, and forces
Trang 34oscillating with difference frequency and sum frequencies in
addi-tion to the linear soluaddi-tion
The fluid pressure is integrated over the hull to obtain the global
hydrodynamic forces at each time step These wave loads will
determine the subsequent motion of the body with equation (41)
Therefore, an accurate and complete description of the pressure is
essential in properly simulating the response of a body The
big-gest effects of the global hydrodynamic loads may be obtained by
integrating the linear Bemoulli pressure equation (16) over that
portion of the body that lies below the undisturbed mean free
surface However, important small-amplitude contributions to the
global force come from the quadratic Bemoulli term and by
accounting for the relative wave elevation about the body
The instantaneous forces and moments are given over the
float-ing body as
(41)
where 5b is the exact wetted surface of the body and p = p(\o, t)
is the fluid pressure on the body surface at each time step and
Xo = (XQ, yo Zo) is the local position vector in instantaneous
position The normal vectors «o, needs to be evaluated neously as a function of time Since it is more convenient toevaluate the forces and moments in the mean positions of thefloating body for the computational purposes, the integral overinstantaneous body surface 5b in equation (41) needs to be trans-formed to the mean position of the floating body 5b
instanta-This means that the evaluation of the wave-induced order forces can be separated into two parts:
second-• Integration of fluid pressure up to the mean free surface.Since the instantaneous position 5b is displaced and rotatedwith respect to mean position ,Sb, the instantaneous pressure
p(xo, 0 and normal vectors «o, need to be expressed in terms
of their values on mean position 5b It is assumed that the
pressure p(\Q,t) on the instantaneous position 5b can be
obtained in terms of a Taylor expansion with respect to meanposition 5b
Pressure integration from the undisturbed free surface to theactual wave elevation The integration over instantaneous
position 5b applied up to the wave elevation ZQ = ^(.(o, yo 0.
Trang 35Fig 28 Relationship between earth-fixed (dashed line) and body-fixed
(solid line) coordinate systems
but the integration over mean position Sb goes up to Zo = 0,
which is equivalent to ^.^ + 3'ai — ^«2 on instantaneous
surface Sb (see Fig, 28), If it is assumed that ASb be the part
of instantaneous position Sb between Ç3 -I- >'ai — va2 and
wave elevation ZQ = Ç(xo, >'o, 0- The integral over A5(, can
be written as
- p dSp(xo, t)noi = -p\dl
(43)
where | = (4,, ^2, ki) = (^i- -^2, X3) and a = (a,, «2, «3) =
(X4, A'5, X(,) is the translational and rotational first-order
fioating body motions, respectively The second-order
forces over mean position come from the integral of the
second-order pressure on the mean position and can be
written as
(44)
where the second-order pressure 77'^'(xo,i) can be written as
(45)
The second-order pressure p''^'(xo,0 in equation (45) is
derived from the sum of second-order potential in the
Bemoulli equation [equation (1)] and the interaction
between the floating body motion and the gradient of the
first-order pressure The second-order potential is
neglected in the present paper since it will not contribute
to the prediction of the mean second-order added
resis-tance (Pinkster 1976, 1980, Kim & Yue 1990)
The position vector and normal vector in the inertial
(earth-fixed) coordinate system can be expressed in terms of body-fixed
coordinate system as
(46)
where x and fï are the position and normal vectors on the
body-fixed coordinate system, respectively The H is the transformation
matrix with the adoption of roU-pitch-yaw sequence of rotation(Ogilvie 1983) and is given as
taneous pressure p from equation (42) with the consideration
of the Bemoulli equation [equation (1)] are substituted in tion (44) The final expression for the second-order force intime domain neglecting the second-order hydrostatic force (sinceits contribution to added-resistance prediction is zero) can bewritten as
Figure 29 shows the achievement of steady state of each ponent of the added resistance that is given in equation (48) at theresonance frequency and sum of these components for a Wigleylllhull form at Fn = 0,3 and ß = 180 deg The Wigleylll hull form inthe present mean second-order calculation is free of heave andpitch motions The mean second-order forces f , ' ' ( / ) over a time
com-range T is given as
T
(49)
The averaging time T must be much larger than the characteristics
period of the incident wave Figure 30 shows the mean addedresistance of Wigleylll hull form at fn = 0.3 and ß = 180 degfor a range of frequencies The experimental results, which are
Trang 36Fig 29 Achieving steady-state of the added-resistance components at the resonance frequency for a Wigleylll hull form at Fn = 0.3 and ß = 180
deg (a) Relative wave elevation along the waterline—the first line of equation (48) (b) Pressure due to the quadratic first-order velocity—the secondline of equation (48) (c) Pressure due to the product of gradient of first-order pressure and first-order motion—the third line of equation (48).(d) Pressure due to the product of first-order pressure and first-order rotational motion—the fourth line of equation (48) (e) Total added resistance—
the sum of (a), (b), (c), and (d)
Trang 37Fig 31 Nondimensional mean added resistance components for a range of nondimensional frequencies for Wigieylll hull form at Fn = 0.3 and
ß = 180 deg (a) Relative wave elevation along the waterline—the first iine of equation (48) (b) Pressure due to the quadratic first-order velocity—the second line of equation (48) (c) Pressure due to the product of gradient of first-order pressure and first-order motion—the third line of equation (48) (d) Pressure due to the product of first-order pressure and first-order rotational motion—the fourth line of equation (48) (e) Total added resistance—
the sum of (a), (b), (c), and (d)
compared with our ITU-WAVE numerical results, are taken from
Journée (1992) In order to avoid the transient effects, only the last
half of the time domain results are taken into account for the
prediction of the mean added resistance using equation (49)
Figure 31 shows each component of mean added resistance
The local quantities in equation (48) such as the fluid velocities
V 9 ' " ( x i) and wave elevations i(x,/) can be decomposed as in
integrated quantities (e.g., radiation and diffraction forces) and
related impulse response function can be defined for these local
fluid quantities A general radiation fluid velocity over floating
body surface at point x and time t can be written in terms of the
gradient of the general radiation potentials at that point and time
using equation (21) as
(50)where the instantaneous potential v|;i^(x), the time-independent
impulsive potential i|j2i((x), and the time-dependent memory
potential have the same meaning as in equation (21) Vxt(x, i) isthe radiation velocity impulse response function A general dif-fraction fluid velocity at point x and time / can be written usingequation (30) as
(51)
where V(PD(X t) is the diffraction velocity impulse response
func-tion Similar to the radiation and diffraction fluid velocities, ation and diffraction wave elevation, which is computed from thevalue of the potential and its gradient at the null points of thepanels bordering the free-surface along the waterline of the float-ing body, and wave elevation impulse response function at the
radi-point X and time I along the body waterline can be defined
(52)
Trang 38A computer code (ITU-WAVE) with the boundary-integral
equation method and Neumann-Kelvin linearization was
devei-ojjed for the prediction of the three-dimensional transient
wave-body interaction of the first-order unsteady hydrodynamic forces
including radiation, diffraction, and Froude-Krylov forces and the
second-order steady forces
Numerical experience has shown that the computational
accu-racy of the quadratic pressure forces is generally not as good as
that of the first-order forces Since the evaluation of first-order
fluid velocity is less accurate than the pressure on the floating
body surface, the accurate prediction of these forces requires more
refined discretization of the floating body that increases
computa-tional time significantly Because the prediction of the quadratic
pressure is sensitive to the distribution of the panels in the vicinity
of the sharp comer, the computational results have inaccuracy
around these regions
Results were presented to demonstrate the convergence of the
developed computer code for the impulse-response functions,
added-mass and damping coefficients, exciting forces, response
amplitude operators, and the second-order mean drift forces (e.g.,
added resistance) The calculations are shown to be in satisfactory
agreement with the experimental results
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