FORM NL- 3 1EV. IC-7 ENDORSED BY AN AND NES(O.
THIS FOR.\I .IAY B: REPP
'
35-P744. Title and Subtitle
On the Theory of Coupled Ship Motion and Vibration
5. Report I)ate
September 1980
6.
7. Author( s)
Hisaakj. Maeda 8 Perform ing c)rgr. 'toNo. 232 n Rept.
'9. P 1orrnn&Oraniz.zLun Name and Address
Dept. ot Naval Arch. & Marine Engineering, The Univ. of Michigan,
Ann Arbor, Michigan 48109
IO. Procct/T,is/Wotk Lt No.
11. Contr.ict/Gra,t No.
7-3806
12. Sportsortn Organtzatiori Name and Address
The American Bureau of Shipping, 65 Broadway Street, New York, NY 10006
and
MA RAD
13. Type of Report & Pemod
Covered
Technical
14.
15. Supplementary Notes
16. Abstracts
A forward speed, short wave, linear ship dynamics theory is developed. Particular attention is paid to vibration response of the ship hull.
17. ,ev urti'; ;tnJ I'o:urtiçrtt A;;.iyis. 1 lo, Dc scriprurs *
Forward speed
Linear ship dynamics Main hull vibration Wave induced vibration
I 7b. Ident;fters/OpenEnded Terms
'17c. COSATI Field Group
1$. Arì isr;hty Srsternenz
National Technical
Springfield, Virìnia
d for Release
19. Security Cliss (lhsEçCFASftjp
21. Nc'. of P.tes
70
Informatiop Service
DOCUMNTATE
DATUM.
Bibliotheek van de
A1eIinq Scheepsbow. en Scheeptaartkue
Techr.isde io:ho, Delft
K5c3 23
ON THE TFORY OF COUPLED SHIP MOTIONS
AND VIBRATIONS
Prepared by: Hisaaki Maeda
September 1980
Co-sponsored by: The American Bureau of Shipping
and U. S. DEPARTMENT OF COMMERCE, Maritime Administration
Office of Commercial Development
No. 232 September 1980
This report has been prepared for the purpose of collecting and developing the theoretical background for a project on ship "springing" in the Department of Naval Architecture and Marine Engineering of The University of Michigan. The purpose of the project is to determine the physical mechanism(s) causing ship springing, especially with respect to Great Lakes bulk carriers, and further to develop procedures for predicting quantitatively the hydrodynamic forces that cause springing.
In this report, "springing" means the vibratory response of a ship to the nonimpulsive hydrodynamic forces associated with ship operations in ambient waves. This definition eliminates ship vibration resulting from slamming, as well as vibration or
"shuddering" caused by various transient conditions such as pass-ing over a shoal. We shall generally restrict the study even further by considering just the vibratory response of the ship acting as a free-free beam; field observations on Great Lakes bulk carriers suggest that such overall response of the ship is the most serious dynamic response to the presence of waves.
The above restrictions do not imply that other causes and modes of vibration are unimportant. The restrictions are intro-duced only to provide bounds on the scope of the study being uncertaken.
In this report, the theory is developed in general form for motions/vibrations in the vertical plane of an elastic ship. The
term "motions" is usually used to describe the rigid-body response of a ship in waves, and "vibrations" refers to the dynamic defor-mation of the elastic body. Because of special conditions relating to very large Great Lakes bulk carriers, "motions" and "vibrations" cannot be completely separated in the usual way.
The theory herein is based mainly on Bishop et al [3]*, with important references to Chertock [4] [5] and BessIo [1] . Some improvements are made by
introducing four terms to represent structural and cargo
damping (2-2),
making clear the characteristics of hydrodynamic forces (4-3, 4-4),
investigating the equations for the coupled motions and
viirations (6-2, 6-3), and
Cd) introducing the three-dimensional (3-D) correction factors to tue sectional forces derived
by
tne strip method (4-7).There is an extension of Chertockts results in (g6-4).
This report was prepared with the encouragement of Professor T. Francis Ogilvie, Chairman of the Department of Naval Architec-ture and Marine Engineering, The University of Michigan. It is a pleasure for the author to thank Professor Ogilvie for his encour-agement and warm support. He also acknowledges his appreciation to Professor Robert F. Beck and Dr. Armin W. Troesch for their helpful discussions.
1. INTRODUCTION i
1-1. Ordinary Ship Viurations 1
1-2. Motions of the Rigid Floating Ship 2
1-3. Springing 3
1-4. Mode Shapes 5
Generalized Equations for Ship Motions/Vibrations 6
The Coordinate System 7
2. VIBRATION OF A BEAM 8
2-1. The Assumption of a Timoshenko Beam 8
2-2. The Linear Equations for a Vibrating Beam 8
2-3. The Mode Shapes in Air (Dry Modes) 12
2-4. Orthogonality Relationship for the Dry Modes 12
3. GENERALIZED EQUATIONS OF SHIP MOTIONS/VIBRATIONS 15 3-l. Expansion Theorem for the Dry Modes 15 3-2. Derivation of tne Generalized Equations of
Motions/Vibrations 16
4. GENERALIZED HYDRODYNAMIC FORCES 18
4-l. Generalized Excitation 18
4-2. Linear tlydrodynamic Forces 19
4-3. The Reciprocity Theorem for the Radiation Forces
(Dry Modes) 21
4-4. Khaskind Relationships (Dry Modes) 34
4-5. Generalized Radiation Forces in Vertical Modes 36
4-6. Generalized Wave xcitation in Vertical Modes 43
4-7. 3-D Correction Factors on Sectional Forces 52
4-8. Generalized Restorina Forces 52
5. SOLUTION OF LINEAR EQUATIONS OF SHIP MOTIONS/VIBRATIONS 54
5-1. Method of Solution 54
5-2. Response Functions 56
Scheepehydromechanica
Archief
CONTENTS Mekelweg 2,2628 CD Deift
E
. 1ELATIONSHIP TWEEN DRY AND WET IIODES 58
6-1.
The Dry Modes
586-2.
The Wet lIodes
596-3.
The Difficultj in tue Case of a
Frequency-Dependent Coefficient Matrix
616-4.
Wet Modes Corresponding to a Syrrìmetric
Frequency-Independent Inertia Matrix
647.
CONCLUDING REMARKS 67l-l. Damping and Added-Mass Coefficients for Various Ship Vibration and Motion Cases 4
l-2. Coordinate System 7
2-l. Element of a Timoshenko Beam 9
4-l. Element of the Hull Surface 21 5-1. Block Diagram for Computation of the Solution 57 6-l. Location of the Center of Gravity 59
1.
INTRODUCTIONl-l. Ordinary Ship Vibrations [22]
[23]
In the usual theory of ship vibrations in the vertical plane, there are three big assumptions that are made with regard to hydro-dynamic force, namely, (i) the natural frequencies of the modes are so high that the hydrodynamic forces are independent of fre-quency, (ii) there is no interference between rigid-body motions (heave and pitch) and vibrations, and (iii) the effects of ship speed on hydrodynamic forces are negligible.
Moreover, the assumptions associated with the ship hull are that (i) the ship hull can be represented by the equivalent Timo-shenko beam and (ii) the equations for the motion of the hull are linear. These assumptions are believed to be reasonable even if nonlinear hydrodynamic forces are taken into account.
When we refer to "ordinary ship vibrations," we shall be implying that the three assumptions for hydrodynamic forces are valid.
Chertock [4] [5] treated the ship vibration problem this way, and he ingeniously derived the relationships between the "wet" mode shapes (the mode shapes for the hull in water) and the hydro-dynamic forces. His theory was proved experimentally by Chertock
et a
[6]
[7].
The theory for ordinary vertical ship vibrations was essen-tially completed by Ohtaka
et al
[22] [23].When a strip method is used in computing hydrodynamic forces, a 3-D correction factor, the so-called "J-factor," is used to
account for end and other 3-D effects, as developed by Lewis [12] and Kumai [11]
-1-iJonhydrodynamic damping, such as structural (material) da ing, cargo damping, etc., is one of the uncertain factors in t theory. Kumai [9) has investigated material damping.
i-2. Motions of the Rigid Floating Ship
Motions of a floating rigid ship are usually separated in those in the vertical plane (heave and pitch) and those in the horizontal plane (sway, yaw and roll). As long as the ship is symmetric with respect to the centerplane, it can be assumed t there is no interaction between vertical motions and horizonta motions [24).
Hydrodynamic forces depend on the frequency of the motion because of memory effects of the free surface. Moreover, ther are strong couplings between heave and pitch and among sway, y and roll, and there is heavy damping of all of these modes thr the generation and radiation of waves. For these reasons, a s of orthogonal modes cannot be defined, and so modal analysis c not be used for predicting rigid-body ship motions. We can sp of pitch and heave, for example, as two "modes" of motion, but they are not modes in the strict sense generally implied in vi tion theory. These two "modes" are strongly coupled and their characteristics are strongly frequency dependent because of th presence of the free surface, and they cannot be decoupled thr
some kind of linear transformation.
Nevertheless, there exist some useful reciprocity relatio ships among these modes. Furthermore, the Khaskind-Hanaoka-Ne relationships can be used to compute the corresponding general
forces even when the ship has forward speed [14] [16) [18) [27).
A strip method is usually used for computing hydrodynamic forces, without any 3-D correction corresponding to the J-fact
mp-he to hat i s, e aw ough et an-eak bra-e ough n-wman i z ed ors
-3-used in ship vibration theory.
Only the linear damping due to surface waves is taken into account, except in the case of ship roll, when viscous damping is introduced in terms of an equivalent linear damping.
l-3. Springing
The hydrodynamic forces involved in the springing of a 1000-ft Great Lakes bulk carrier seem to be influenced by the presence of the free surface to an extent that they cannot be evaluated by the methods of ordinary ship-vibration theory, although the same is not true for the vibrationof giant ocean-going tankers. This is the result of the extremely low frequency of springing of a
Great Lakes ship [8].
The M/V Stewart J. Cort may be considered as typical of the new large bulk carriers operating on the Great Lakes. Its length overall is 1000 ft, with L/B 10 and B/d 3.5 ( L is length,
B is beam and d is draft) . These ships are severely draft limited, and the longer they become the more the flexibility of the hull is increased. The springing frequency (the natural fre-quency of the two-noded vibration) turns out to be about 0.33 Hz. We can check the frequency dependence of hydrodynamic forces at such a frequency by looking at the graphs of the 2-D added-mass coefficient and the wave-amplitude ratio (effectively the damping coefficient) for heave motion of a rectangular box corresponding to the parallel midsection of the ship. See Fig. l-l. The non-dimensional springing frequency of the Cort is w2B/2g 6 . The
usual rigid-body ship motions occur in the range w2B/2g 0.2 to
5 (mainly concentrated near i ). The natural frequency for
two-noded vibration of a 20,000 DWT oceangoing container ship caused by shudder (panting) is w2B/2g 50, and the vibration caused by
1.4 1.0 Added Mass: a(x,w) = -ITB2C s 1.0
pg2-2
BT Wave Damping: b(x,w) AH 3.6 T S.J. Cort Springing S-I
Springing of 300,000 DWT Tanker Figure 1.1Damping and Added-Mass Coefficients for Various Ship Vibration and
Motion Cases
T
e
30
2-noded vibration by slamming of 20,000 DWT container
50 4 Blades 100 rpm propeller excitation o 5 lo 15 20 g 2
-5-a four-bl-5-aded propeller oper-5-ating -5-at loo rpm occurs at w2B/2g 2900. The springing frequency of the Cort is thus seen to be in the range in which there are strong frequency effects on added mass and damping coefficient, although the springing frequency is higher than the frequencies of rigid-body ship motions. In the usual
problems of ship vibrations, it is seen that the hydrodynamic forces are essentially frequency independent.
We conclude that in order to analyze the springing of Great Lakes bulk carriers the frequency dependence of the coefficient matrix of the equations of motion has to be taken into account. We use the unified dynamic analysis of ship responses to waves, which combines rigid-body motions and ordinary ship vibrations. Bessho [li and Bishop et al [3] have provided the basic theory.
In this report, the concept and the theory are developed in detail.
l-4. Mode Shapes
The mode shapes for ship vibration can be discussed in terms of the mode shapes in air (the "dry modes") or the mode shapes in water (the "wet modes") [2].
The dry modes for ship vibration are defined as clearly as for structures on land. The dry modes satisfy an orthogonality condition.
On the contrary, the wet modes can be defined precisely only for the case of high-frequency vibrations, that is, for the case in which there are no frequency-dependent free-surface effects on the hydrodynamic forces. Furthermore, the wet modes depend not only on the ship hull but also on ship speed. If the hydrodynamic forces are functions of frequency of ship vibration, the modes do
not satisfy an orthogonality relationship, and so we are not abi to apply an expansion theorem to the wet modes.
Since the hydrodynamic forces acting in the springing of a
1000-ft Great Lakes bulk carrier seem to be frequency dependent, we shall use the dry modes to generalize the equations of motion
for such ships [81.
i-5. Generalized Equations for Ship Motions/Vibrations
The analysis of rigid-body ship motions and hull vibrations is unified here into a single set of equations [3] [151. In the following sections, we shall refer to this as the motions/vibra-tions problem.
The motions and vibrations of the hull are described in terms of the dry modes. All hydrodynamic forces act as excitation of these modes.
The vibration of the ship hull itself can be assumed to fol a linear law. Hydrodynamic forces may be linear or nonlinear. E when they are nonlinear, the equations can be transformed into
equivalent linear equations, as is usually done in treating roll damping in ship-motions problems.
It is reasonable to separate the generalized equations for Ship motions/vibrations into equations for the vertical-plane problem and equations for the horizontal-plane problem. This
requires that local vibrations should be negligible. Then the springing problem is treated in terms of the generalized equatio for ship motions/vibrations in the vertical plane.
e
low ven
-7-l-6. The Coordinate System
Fig. 2-2 shows the coordinate system that will be used. The
origin is located at the mean position of the intersection of the
undisturbed free surface, the centerplane of the ship and the
midship section. In forward-speed problems, the water has a
mean velocity of magnitude U in the positive x direction.
Incident waves propagate in the direction measured with respect
to the positive x axis.
z
U (flow velocity)
Figure 1-2. Coordinate System
2-I. The Assumption of a Timoshenko Beam
A ship hull can be represented by a Timoshenko beam. This is valid at least up to the five-noded vibration. However, it does not work well for eight-noded vibration and beyond, since the influence of local vibration turns out to be large then [23]
It seems improbable that one needs to consider more than the seven-noded vibration for the analysis of springing [3], and so we shall represent the ship hull as a Timoshenko beam. Then the most important problem is how to get the equivalent Timoshenko beam corresponding to a ship hull, that is, how to obtain the
dis-tributed weight, w , the rotary moment of inertia, 'y , the
bending rigidity, EI , and the shear rigidity kAG
2-2. The Linear Equations foraVìbrating Beam
We can derive the linear equations for a beam vibrating in water in the presence of a free surface by Hamilton's principle [1] [22], but here the infinitesimal deformation method is used because it is easier to understand the physical meaning of the phenomena.
Let us consider the Timoshenko beam, a part of which is shown in Fig. 2-1. The vertical displacement at any point x and time t is denoted by z(x,t) and the vertical force per unit length, including hydrodynamic forces, by f(x,t) . The system parameters are:
m(x) , mass per unit length of the beam,
Iy(X) sectional mass moment of inertia about the y axis,
EI(x) , bending rigidity, where E is Young's
modulus of elasticity and 1(x) the
-8-
-9-sectional area moment of inertia about an
axis parallel to the y axis, passing through the center of the cross-sectional area,
kGA(x) shear rigidity, where G is the shear modulus,
A is the cross-sectional area, and k is the shear rigidity factor.
The angle of rotation due to bending and the angle of distortion
due to shear are denoted as O(x,t) and y(x,t) , respectively.
The shear force and the bending moment are denoted by Q(x,t) and M(x,t) , respectively. Sign conventions are shown in the figure.
M(x,t)
Figure 2-1. Element of a Timoshenko Beam The following equations and conditions constitute the problem for the vibrating
beam:
1)
The
force equation of motion, in the vertical direction: m(x) .a2z(x,t) +c
Cx) zïx z(x,t)at2
S (2-l) =- {Q(x,t)
3Q(x,t)
} + Q(x,t) +f(x,t)x
p.
Xwhere C Cx) denotes the internal damping (due to cargo and friction) per unit length;
2) The moment equation with respect to the axis parallel
to the y axis, passing through the center of the cross-sectional area:
e2 (x,t)
I(X)
X tz + Cm(X)ABO,t)
= - {M(x,t) M(x,t) lx} + M(x,t) + Q(x,t)x
Bx
where Cm(x) is a damping coefficient similar to Cs(x)
The compatibility relation: Bz(x,t)
- O(x,t) - y(x,t) ,
3X
The moment-curvature relation:
(2-2) (2-3) M(x,t) - EI(x){ O(x,t) + (x) íe(xlt))} (2-4 x t
where (x) is the material damping per unit length with regard
to the moment;
The shear stress-strain relation:
Q(x,t) = kGA(x){ 1(x,t) + a(x) B1(x,t) } (2-5)
where a(x) is the material damping per unit length with regard to the shear force;
The boundary conditions at the ends of a free-free beam: Q(x,t) = M(x,t) = O at x = -ZF A (2-6)
where
-F
and 9A are the x coordinates of the bow and stern respectively.The above equations become, in a simpler notation,
mz.tt+C5zt =-Q+f
(2-S)Iyett + cmet - + Q (2-9)
= - EI(e + 0xt (2-11)
Q = kGA(y + cy.) ; (2-12)
Q = M O at X
= F ' (2-13)
These equations can be combined and rewritten in two groups: the governing equations for z and O
mztt + 0st - {kGA(e +z+Ot +z,t)}
= f , (2-14)+ CmOt = {EI(e+Ot)}
- kGA(O +z+Ot+ctzt)
(2-15) with the conditions at the ends,
EI(e + 8xt = O at x =
' (2-16)
kGA(O+z+cOt+czt)
= O at x = Fi A (2-17)
the equations for y , M and Q
y = - z - O , (2-18)
M = - EI(O + , (2-19)
Q = kGA(y + cyt) (2-20)
In these equations, (2-14) through (2-20), there are five unknowns, z , O , y , M and Q , corresponding to the five independent equations, (2-14), (2-15) and (2-18) through (2-20).
The boundary conditions, (2-16) and (2-17), give the natural frequencies as eigenvalues for the equations (2-14) and (2-15).
These equations represent the most general Timoshenko beam, because all of the non-hydrodynamic damping (even liquid-cargo damping) is taken into account.
The simplification of the above equations should be done only after checking the contribution of each term in the equations.
2-3. The 1ode Shapes in Air (Dry Modes)
The dry-mode eigenvalues and eigensolutions can be found by neglecting all terms in the above equations representing damping and excitation, that is, by using
mztt -
kGA(e+z)}
= 0 , (2-21)-
EI0}
+ kGA( +z)
= O , (2-22)with the corresponding boundary conditions, EIO = O at x =
F ' A (2-23)
kGA(Û+Z)
= O at x = F £A . (2-24)The last four equations give the natural frequencies, r r =
2, 3, ... , and the dry-mode shapes, Zr(x) and ®r(x) . (The natural frequencies for the first two modes will be found to be zero, since there are no restoring forces for such modes.) The solutions can be expressed in the form
iUirt
Zr(X,t) = Zr(X) e , (2-25)
Or(x,t) Or(x) elwrt lWrt F.(X) e
the last being an obvious extension.
2-4. Orthogonality Relationship for the Dry Modes
The solutions Zr and er are separable in time and space. Denoting two distinct solutions of the eigenvalue problem by Zr er and z
,
, respectively, we can write
[AI - mZr -
kGA(Or+Z)}'
= O (2-28)[B] - WrIyOr -
EIG}t + kGA(Or+Z.)
= O (2-29) < x < ZA , where the prime denotes differentiation with(2-26)
respect to x , and also
- w2mZ - {kGA(e + Z')}' = 0
s s s s
- wIG5 - fEIe}' + kGA(D5+ Z)
= O13
-also in -ZF < x < ZA . Note that m , kGA , EI
, Zr Z
are all functions of x only.
Now combine the last four equations and integrate as follows:
2.
J{ [A] x Z5 +
[B] x
-
{ [C] x Zr + [D] x Or}] dx ZA = (w- w) J
(mZsZr + lyDsOr) dxfZA[kGA(Or+Z)}IZs
- {kGA(®s+Z)}'Zr
F- kGA(3+z')O
+ kGA(D5+Z)®rJ
dx +J
[{EIG}'Sr - {EID}'Gs] dx -2.F = 0 . (2-32)The second and third integrals on the right-hand side can be inte-grated by parts and, with the use of the boundary conditions, they are both found to equal zero. Hence, Equation (2-32) reduces to
ç2.A
(w -w)
J (mZsZr+ 'y°s0r
dx
= O .
(2-33)
Recalling that the eigenvalue problem is homogeneous, we can normalize the natural modes by writing
ç2.A
J(mZsZr + lyoser) dx
=assrs
(2-34)where rs is the Kronecker delta and a5 is the generalized
mass of the ship hull in the s-th mode. The orthogonality relation-ship is expressed by Equation (2-34).
(2-30)
Another type of orthogonality relationship is obtained by the following integration: [AI x
+ [B] X
dx =- w
J
(mZrZs+ 'y®r®s) dx - J[{kGA(®
+z)}'z5
-kGA(er+
Z)OsJ
dx- J
{EIe}'e5
dx
= o (2-35)
Performing an integration oy parts, using the boundary conditions, and substituting from Equation (2-18), we obtain
J(EIG + kGArs) dx
=4arrrs
(2-36)Equation (2-36) expresses the orthogonality relationship as weil as Equation (2-34).
As might be expected, we conclude from Equation (2-36) that the natural frequencies associated with the rigid-body modes
(heave and pitch) are zero. The linear function of x represent-ing the pitch mode can be so chosen that the heave and pitch modes are mutually orthogonal.
15
-3. GENERALIZED EQUATIONS OF SHIP MOTIONS/VIBRATIONS
3-l. Expansion Theorem for the Dry Modes
Solutions of the homogeneous problem, as expressed in (2-25)-(2-27), satisfy the orthogonality relationship, (2-34), or, alter-natively, (2-36). These solutions also form complete sets, and so any function that satisfies the boundary conditions (2-23) and
(2-24) (or the condition on y(x,t) obtained by combining these conditions with (2-18)) can be represented by an absolutely and uniformly convergent* series in terms of the dry modes, as follows:
z(x,t) = Zr(x)r(t) = {Zr(x)}T{r(t)} , (3l) r= O S(x,t) = ør(x)r(t) = Or(x)}T{r(t)} , (32) r=O y(x,t) = Fr(x)r(t) = {Fr(x)}T{r(t)} , (33) where denotes the column matrix of the eigenvector and
}T
denotes the corresponding transposed (row) matrix. The subscripts will be interpreted henceforth as follows:
Subscript Mode of Motion/Vibration
O Heave
i Pitch
2 Springing (two-Noded)
n Springing (n-Noded)
From the orthogonality relations, (2-34) and (2-36), one obtains easily that
*It is assumed that the functions under consideration represent physically realizable solutions. In particular, they cannot be discontinuous.
ZA r(t) = (i/arr) J {m(x)zr(x)z(x,t) + Iy(x)®r(x)O(x,t)dx = (l/warr) J
-for r = 0, 1,2 ...Also,
Fr(x)= -
4(x)
3-2. Derivation of the Generalized Equations of Motions/Vibrations
Using the orthogonality relations, (2-34) or (2-36), we can easily derive the generalization of the governing equations, (2-14) and (2-15). We combine the latter and integrate as follows:
ZA J{(2_14)xZr + (2_l5)xOr} dx ZA = J
{mzZ
+ IyûttOr}dx + J{CtZ
+ CmOt®r}dx ZA - J{{kGA(e+zx)Jxzr - [kGA(e+z) -
(EIOx)xIOr}dx-Z
rA
-
j {[kGAc(et+zxt)]xzr -[kGAa(Ot+zxt) -
(EIext)x]®r}dx-ZF ZA
= J
f(x,t)Zr(x) dx . (3-7)
-ZF
Performing appropriate integrations by parts and taking into
account Equations (3-1)-(3-6) and the boundary conditions, (2-16)-(2-17), we obtain the generalized equations of ship motions/vibra-tions:
arrr(t) +
{c5 +Cm ++rss(t) + arrçr(t)
rs rs rs s=O r, = J Zr(X)f(X,t)dX E Fr(t) , (38)-where Fr(t) is the generalized force for the r-tiTi mode and
(3-4)
+ kGAFr} dx
,it may be noted that
- ør(X) .
(3-5)
I wìere -arr r(t) + L
-
17 -arr= f
m(x)Zr(x)Zr(x) + Iy®r(x)Gr(x)}dx (39) =J
Cs(x)Zr(x)Zs(x) dx , (3-10) = J Cm(x)er(x)es(x) dx , (3-11) -rZA rs = I kGA(x)(x)Fr(x)Fs(x) dx , (3-12) rs = JEI(x)(x)ø(x)®(x) dx
. (3-13)Note that all of the generalized damping coefficients CS rs
C5
' rs ' 3rs , are symmetric with respect to r and sThe matrix form of the generalized equation (3-8) is
r'm -s
rs '-rs rs rs r(t)} + Wrarr2
N
{r(t)}
= IFr(t)} , (3-14)
denotes a diagonal matrix. The subscript r iden-tifies the equations in the same way defined in Section 3-1, that
is, r = O denotes the heave force equation, etc. However, each mode is coupled to the others through the generalized hydrodynamic
4-1. Generalized Excitation
The following may all be considered as parts of the generalized excitation for ship motions/vibrations:
(a) hydrodynamic excitations
wave excitation radiation forces wave impact
(b) propeller-induced excitation
shaft bearing forces
distributed liull-surface forces (c) excitation by the main engine
Cd) excitation by auxiliaries
Those included in (b), (c) and (d) are usually at higher frequen-cies than the lowest natural frequenfrequen-cies of the hull vibrating as a beam, and so they relate more to local vibrations of the ship hull. Wave impact causes whipping similar to springing vibration, but the mechanism is quite different, and so we shall not consider it further here. For our study of the springing problem, we shall assume that it is enough to consider only wave excitation and
radiation forces.
besides the linear hydrodynamic forces in which the free-surface effect is taken into account, Japanese investigators [10]
[26] claim that we must consider nonlinear hydrodynamic forces
as well in analyzing the excitation of springing. Nonlinear hydrodynamic forces may occur, for example, when a bulbous bow is exposed above the free surface during ship motions; such forces contain components at harmonics of the frequency of encounter, and such harmonics may excite springing vibration. Japanese authors call this "selective resonance," meaning that the ship responds in springing selectively to waves with frequencies such that the
frequency of encounter is a submultiple of the springing frequency.
19
-This kind, of selective resonance has been observed experimentally in full-scale tests [25] and in model tests [10] when running a large oceangoing tanker with a big bulbous bow in a light-weight condition in a high sea state. The two-noded vibration in these tests was distinctly different from that caused by wave impact.
It should be noted that these Japanese investigators do not claim that nonlinear hydrodynamic forces are the only excitation of springing. They explain that, in mild sea states in which ship motions are small, springing is caused by linear hydrodynamic
forces, while nonlinear hydrodynamic forces are important in caus-ing sprcaus-ingcaus-ing of a ship runncaus-ing in a severe sea state, in which ship motions are large [26]. In this work we will consider only the linear effects of the hydrodynamic forces.
4-2. Linear Hydrodynamic Forces
If the linearity assumption holds good, the hydrodynamic forces can be divided into three parts:
wave excitation,
hydrodynamic forces due to ship motions/vibrations, and hydrostatic restoring forces due to motions/vibrations. Restoring forces, (c), can be predicted precisely from the ship lines. But it is not easy to calculate the hydrodynamic forces,
(a) and (b), since they must be obtained from the solutions of 3-D boundary-value problems. This difficulty exists even if the fluid is inviscid and the flow irrotational. The two kinds of hydrodynamic forces can, however, be found separately, because of the linearity of the problem.
Fortunately, large Great Lakes bulk carriers typically have L/B 10 , and so they are appropriate for the application of
slender-body theory, which is essentially equivalent to strip theory. Then the accuracy of the prediction of hydrodynamic
forces depends mostly on the accuracy of the solutions of 2-D boundary-value problems (except for end effects and possibly certain forward-speed effects)
There has been no theory for predicting 2-D hydrodynamic forces covering the range from very low to very high frequencies. The convergence of the usual multipole expansions turns out to be slow at higher frequency, and the singularity-distribution method shows an infinite number of singular frequencies at which it fails. Recently, two different kinds of theory were developed to remove these difficulties. One is a multipole-expansion method in which
the expansion converges rapidly at higher frequencies [13]. The other is a singularity-distribution method in which an extra
singularity is added at the origin to remove the singular frequen-cies [20]. The latter result can also be achieved by imposing an additional boundary condition on the free surface
inside
thebody [21].
Let us consider harmonic oscillations in regular waves, with the time dependence expressed by the factor elwet , where We
is the frequency of encounter. The generalized force exciting the k-noded vibration, as given in (3-8), can be divided into three terms: ZA ZA ZA
Izk(x)f(x,t)dx
= [J Zk(x)fE(x)dx + J ZA + Jzk(x)fS(x)dx]xe1t
, (4-l) -ZFfE(x) is the [complex] amplitude of the vertical component of wave excitation at x ,
f R(x) is the radiation hydrodynamic force at x (due to
motion/vibration of the hull), and
f(x)
is the hydrostatic force at xThese three terms will be discussed in detail later. where
21
-4-3. The Reciprocity Theorem. for the Radiation Forces (Dry Modes) The combination of Vorus' expression [281 for the generalized hydrodynarnic forces in the vibration modes and Newman's derivation [26] of the reciprocity theorem for rigid-body motions yields the reciprocity theorem for the motions/vibrations of an elastic body.
N
Figure 4-1. Element of the Hull Surface
Generalized Excitation [28]
In order to evaluate the generalized excitation for use in Equation (3-8) or (3-14), we must define the sectional hydrodynamic force (the force per unit length at a section). First we note that the usual force on the ship can be written
F(t)
=- IÍpndS
J Is r =-
Jdx
Jp n.
d ,(4-2)
F Cwhere C is the contour around the girth of the ship at
x ,
S is the wetted area of the hull, p is the pressure in the fluid, and n is the unit vector normal to the hull, directed into the f luid.* The force per unit length is then given by
*TIe
two lines in (4-2) are not precisely equal to each other. If is the usual slenderness parameter, the error is given by a
cross-section being considered.
(See Figure 4-1 for the definition of coordinates.) It must be emphasized that (f,f5,f6) represents a pure couple acting on the section at x ; it is not the moment about some
fixed
axiscaused by the pressure acting on the given section. We can write (n1,n2,n3) = n (4-6) (nL.,nS,n6) = - nxr' and then f1(x,t) -
Jp
ni di , (4-7)for i = l,...,6 . Again it should be noted that ni for i =
4,5,6 is not the same as the similarly denoted quantities in many versions of ship-motion theory. (If r' is replaced by r with the origin at the center of gravity of the body, these
quan-tities are identical to those frequently defined.)
Surge, sway and heave force components are given by
Jdx
f1(x,t) , i = 1,2,3, (4-8)
respectively. Roll moment about the neutral axis is given by the same integral with i = 4 (if the neutral axis is straight).
(f1,f21f3)
= -
I n di . (4-3)jc
Similarly, the moment per unit length caused by the pressure dis-tribution
at the same section
can be written(ff5,f6)
=
J p(flx')
di , (4-4)C
where
r' = r - r0 , (4-5)
r = [vector] position relative to the origin of coordinates, and
23
-Pitch and yaw moment are, respectively,
rA
jdx
f5(x,t) - Jdxxf3(x,t) , (4-9)
rA
rA
Jdx f6(x,t)
+ J
dxx f2(x,t) . (4-10)The first integral in each of the last two expressions is gener-ally negligible, and so we shall not consider f5 or f5 further. Such quantities would represent additional terms on the right-hand side of Equation (2-2).
The subscript i on identifies the direction in which the force or moment acts. To obtain the generalized force exciting any mode of oscillation, we must multiply by an appropriate weighting function and integrate the product over the length of the
hull, as in (3-8) or (3-14) , where the weighting function is the mode shape, Zk(x)
As a generalization of Section 3, let the modes shapes be given by ljik(x) . We shall follow this convention on the
subscripts: The first subscript (here i ) identifies the
orien-tation of the motion or the force under consideration and the second subscript ( k ) tells how many nodes there are in the mode
under consideration. See Table 4-l. If, for example, we take
TABLE 4-1
1 2 3 4
O surge sway heave roll
1 1-noded yaw pitch 1-noded
2 2-noded 2-noded 2-noded 2-noded
3 3-noded 3-noded
-axial lateral vertical torsional vibration springing springing vibration
i = 3 , we are considering vertical force or motion at each sec-tion;
i30
, which equals a constant, say C , is the mode shapefor heave; = C1x the mode shape for pitch; 32 = C2Z2(x) the mode shape for two-noded vibration in the vertical direction
(that is, springing); etc.
The modes listed in the first column of Table 4-1, for which i = i , will not be considered further, since they are not likely
to be excited by waves. They may be of importance in the study of propeller-induced vibrations, especially for submarines, but such problems are not covered in this report. Therefore we shall limit
i to the values 2 , 3 , 4
The analyses for modes with i = 2 and for modes with i = 3 are exactly parallel. The only difference is in the orientation: i = 2 denotes lateral (horizontal) force or motion, and i = 3 denotes vertical force or motion.
The case i = 4 is rather different, but it can still be handled similarly. Because we are considering beamlike bodies and counting nodes longitudinally, a one-noded vibration is possible corresponding to i = 4 ; it is the simplest torsional vibration.* If we want to carry this case through the entire analysis, of course we shall have to generalize the beam analysis of Section 2.
For any hydrodynamic pressure field, given by p(x,y,z,t) ,
we can compute the generalized force corresponding to the (i,k)
*In a precise analysis, torsional and lateral motions are coupled. It is then still possible to define a doubly-infinite set of modes, although they cannot be separated into distinct lateral and rotary motions. It is not necessary for us to consider such a
sophisti-cated analysis, however, since our principal concern will be for the vertical modes, which can always be separated from the anti-symmetric modes for a body with the symmetry characteristic of a ship.
25
-mode by using (3-8), with f(x,t) replaced by f1(x,t) from (4-7) and Zj(x) generalized by ik(x) . Thus we obtain
rA
Fjk(t) = j Pikf(x,t) d
(4-12) -jj5k(
n1pdS
. (4-13)As already noted, i can take the values 2, 3, 4, and k = 0, 1, 2
...
The physical phenomena causing the pressure field may be quite general. For example, p may be caused by incident waves or by the ship vibrating in any of its rigid-body or flex-ural modes. In the latter case, if the ship is vibrating in the mode designated by the indices (j,,Q)*
, Fik(t) represents a
"coupling" force between the (j,.Q) and (i,k) mocies.
The product
jki
in (4-13) has an important physical interpretation. Suppose that the Ship is vibrating in one mode, say the (i,k) mode; let the instantaneous displacement of an element of the hull surface by given by . Thecomponent of hull velocity normal to the hull is then
njkjk(t)
But ik =
ikei , where ej is a unit vector appropriatelydirected for the i class of modes. Thus
iki
= '1"ik = normal velocity component of a pointon the hull when = i .(4-14)
are used in the same sense, respectively, as (i,k)
See Table (4-1).
This result is rather obvious for translational vibrations. To extend it to roll/twist motions C i = 4 ), let the angle of
rota-tion about tile x axis be given by
X(x,t) = E(x)ç(t) (cf.
(4-1)). Then the normal velocity component is given byis the component of that is
pkkr'nO
, where ne nperpen-dicular to i and r' . The quantity r'n0 is precisely the
same as n wnen k(t)
, by
= i
Velocity Potentials and Corresponding Pressures [191 [24]
The hydrodynamic problem is formulated in the reference frame shown in Figure 1-2 (p. 7). Let U denote the speed of the stream that moves past the ship in the positive x direction, the total velocity potential, p the total pressure, g the
accelera-tion of gravity, and p the density of the water. It is assumed
that satisfies the Laplace equation and the usual boundary
conditions (linearized on the free surface). Then the hydrodynamic pressure is derived from the Bernoulli equation:
Now divide the potential into three parts, as follows: iwt
(x,y,z,t) = Ux + 5(x,y,z) + U(x,y,z)e . (4-16)
The term p(x,y,z) represents the fluid disturbance by the ship
moving in steady motion in otherwise calm water; (x,y,z)
represents the unsteady fluid disturbance, including the effects of incident waves, diffraction of those waves, and oscillations of the ship. The time-dependence factor, e1Wt , will generally be omitted.
substituting (4-16) into (4-15), we obtain for the pressure:
E = - { g z + U +
4
Vq5.V5}
-. {iwU + U+ Vc.Vc}
iwt e Bx p+ E + 4 vv +
g z t p -lU2
2 -4 vq.v
ue
2iwt (4-17)The last term in (4-17), representing nonlinear effects at a harmonic of the fundamental motion, will be neglected. Now define the steady and unsteady pressure components:
4
tJ = + +
iWjjZ(X,Y,Z)
j=2 =0
where now denotes the amplitude of (t) , that is,
iwt
=
je
The incident wave is described by
where
1(x,y,z) =
- e
igh Kz - iK(x cos+ ysin)
, (4-23)W0 27 -=
-
{ g z + u + , (4-18) p= -
{iwU + v'V}
, (4-19) where y =UI + V5
, (4-20)and the time factor has now been suppressed.
We now further subdivide the unsteady-motion potential into components:
h = amplitude of incident waves,
= radian frequency of incident waves in a reference frame fixed to the fluid at
infinity,
= angle of propagation of incident waves, measured with respect to the +x axis,
K =
w/g
= 2rr/X , the wave number.(4-21)
(4-22)
(4-24)
The frequency w is often denoted by We , suggesting "frequency
of encounter;" w0 and We are related as follows:
The term D in (4-21) represents the diffraction waves caused by the presence of the rigid motionless ship in the given incident
waves, and iwc92(x,y,z)
is the potential for the flow caused by motion/vibration in the 2,-noded mode of j type (see Table4-l), after suppression of the time factor.
Corresponding to (4-21), we can define pressure components:
4 Pu =
PI
+ PD + iWjjPj9 j=2 2=0 wherePI
= - p(iwp1
+ v71}
= - p{iWÇl
+VVD}
= - P{iw2
+ v.72}
It will sometimes be convenient to define
= I + D
= PI + Pfl
representing the combined incident and diffraction waves.
.2eciprocity Theorem for the Generalized Radiation Forces
For any of the components of pressure included in (4-26), can compute the generalized force corresponding to the (i,k)
mode by using (4-13). For the present, we consider only the pressure components associated with ship motions/vibrations, given by (4-29). To be specific, let the ship be oscillating in the (j,Q) mode and compute the resulting generalized force act
it
in the (i,k) mode. Call this force
2Fk2e
(so that Fik,j2 represents the amplitude of force due to unit amplitude(4-2 (4-2 (4-2 (4-2 (4-3 (4-3 Oa) Ob) e ing
iwt iwt
=
nik1wike
which is equivalent to (4-14).
Now, a slight generalization from Timman and Newman [27] (see Vorus [28]) gives the body boundary condition,
(4-33)
=
-ik +
(i
X v±) } , (4-34)29
-of motion). From (4-13), (4-26) and (4-29), we obtain
Fik,ji = iwp
jJikniiwj
+ dS (4-31)We now briefly sketch the derivation of the reciprocity relationships first obtained by Tirnman and Newman [27] and later generalized by Vorus [28]. These relate the various force compo-nents acting on an oscillating ship with, respectively, forward and backward motion. To indicate explicitly that (4-31) refers to the forward-speed case, we add a + sign to F , that is,
FIk,j is the generalized force in the (i,k) mode caused by
oscillation in the (j,.Q) mode, the ship moving in its forward
direction with speed U (Note that "forward" means in the direction of negative x .) The steady velocity vector, y
and the potential must also be identified according to whether they apply to forward or reverse motion, and so we write
them, respectively, as v+ and . More generally, we can
write the force for either forward or backward motion as follows:
Fk,j
= 1WJJ
ikni{iw + dS . (4-32)s
The physical interpretation of has already been noted; see (4-14). This relationship holds whether the stream moves past in the forward or reverse direction. When the ship vibrates in the (i,k) mode with the generalized deflection given by
jket
the component of velocity normal to the hull isto be satisfied on the mean position of the hull. We also have tne free-surface condition:
B 2 ± ik
(iw ± U-) ik + g
Bz =
We use (4-34) to express fljk , substituting into (4-14) for use in (4-32). Since (4-14) holds regardless of the direction of the incident stream, we can use either sign in (4-34). We
choose it to be opposite to the stream direction in the problem, as follows:
ik
i
rl±k
= B n V x (aikX v)
n iw = 1)jkfliThen the force is given by (4-32):
+ ik
FIkjQ
= - w2p JJ dS S O on z = O . (4-35) -IJS V x (j x v) dS + 1WP JJ dS (437)The second integral on the right-hand side can be transformed with standard vector identities, and Stokes' theorem applied, which yields + +
ii
FIkjj
= - w2p Jj Bn p dS S + S dS - iwp O t(ik xv) di
. (438) CThe last integral is a line integral around the intersection of the undisturbed hull and the undisturbed free surface, t being a
unit vector tangential to the curve.
31
-In (4-38), we make assumptions that are consistent with our approach elsewhere: We neglect products of the steady-motion
perturbation
velocity and the oscillatory velocity of the fluid.Since
v
= ±Ui +
the integrand of the second integral contains the quantity
(v +v)
= (V+V)
0 , (4-39)which is a negligible product by the above assumption. (If the hull were symmetrical fore and aft and moving through an infinite fluid, (4-39) would be an equality.) Similarly, the integrand of the third integral on the right-hand side of (4-38) is small by the saine standard, since t is almost parallel to v , differing
by an amount that is proportional to the steady perturbation velocity of the fluid. Thus we assume that
All that remains on the right-hand side of (4-38) is the first integral, and so we have the following result:
+ rr tk ±
FIkj
=
-jj
Bn4ji
dSS
(4-41)
In (4-36) and (4-37), we chose with the superscript sign opposite to that of . If we had not done so, we could
not have used (4-39) to eliminate the second term on the right-hand side of (4-38), and the simple symmetrical result in (4-41) could not have been obtained. This seemingly arbitrary choice
is also essential in some of the steps that follow.
Next we use Green's theorem on the right-hand side of (4-41) to obtain the actual reciprocity theorem. When the quantity
+
+
--
ik ri dSI ik +
-Bn q' - ik
is integrated over a surface completely enclosing a region of the fluid, the value of the integral is zero. We take the bounding surface as the hull surface, the plane z = O , and a closing
surface at infinity. Since both potentials satisfy a radiation condition, the integral over the third surface is null. On the plane z = O , the operator B/Bn is equivalent to B/Bz , and
we can substitute from (4-35). Then the integral with respect to x can be carried out, leaving the following result:
J
1 + .
2ik
- - Q - U
+ +
U2}]dy
The surface F is the undisturbed free surface, and the contour for the line integral is the intersection of the hull and the free surface, both in their undisturbed positions. According to the hypothesis of Newman [16], the right-hand side of (4-42) vanishes, and so we have
+ j}ç (i Bq dS = jj Bn ik dS S S or, from (4-41),
FTkjz
= F,jk
(4-42) (4-4 3) (4-44)Recall that Ftkj
is the generalized force in the (i,k) mode,the ship going forward, caused by oscillation of unit amplitude in the (j,2)
mode, and Fjk is the generalized
force in the(j,Z) mode, the ship going backward, caused by oscillation of
unit amplitude in the (i,k) mode.
In terms of actual physical quantities, if the ship oscillates in the (j,.Q) mode, we have
but For example,
Fk2j
F,2k
+ Fjk,j]ç 33 -iwt= normal velocity component of point
on hull;
ijj(x,y,z)eWt
= corresponding velocity potential, forforward or backward motion jut
= resulting generalized force in (i,k) mode, ship moving forward.
For the first, see (4-33); the second comes from (4-21); the third was given in the definitions for (4-31).
As an example, consider the case
21,O
=F0,21
(4-45)The left-hand side gives the yaw moment caused by roll, and the right-hand side gives the roll moment caused by yaw. These are equal except that the direction of ship motion is opposite in
the two cases.
The following relationships should be noted:
=
Fk,ik
±k,i9 =
Fj,jk
as well as the following: + Fjk,jZ F19,jk + -L
-ik,j2. jk,i - T:,-rLk,29,-A special case of (4-47) will be especially important in this study, namely, (4-46) (4-47) (4-48) (4-49) (4-50) (4-51)
+
F3k,39.
=F39.,gk
(4-52)For k = 9. = O , this says that the heave force due to heave motion
is the same whether the ship moves forward or backward. For k O ,
Z = i , it gives the reciprocity relationship between heave and
pitch. And so on.
The derivation above follows that of Timman and Newman [27], as already noted, and it depends in a crucial way on the neglect of interactions between the steady perturbation of the incident
stream and the unsteady flow field. This was the basis for
accepting (4-39) and (4-40). However, in a systematic perturba-tion soluperturba-tion of the hydrodynamics problem, as developed, for example, by Ogilvie and Tuck [l3, such interactions are clearly not negligible. Nevertheless, these authors found that the
coupling forces between ship heave and pitch satisfy an equation equivalent to (4-44). One may reasonably expect the same to be true for the couplings among the vertical motions and the vertical vibration modes of a ship. Thus the basic assumption made in
(4-39) and (4-40) appears not to be necessary, although no simple derivation of the reciprocity relationships has been reported that is not based on such an assumption.
4-4. Khaskind Relationships (Dry Modes)
- + iwt
We introduce Elke as the generalized wave excitation ir the (i,k) mode. Note that we must now interpret w as the
fre-quency of encounter, we (see (4-25)). From (4-13) and the development, from (4-17) through (4-30), we can express the wave excitation as follows:
Etk =
-
iki P
dS= p
35
-The superscript + signs denote that this calculation relates to
the case of the ship in forward motion.
We can apply the same procedure used to derive (4-41), which yields
Elk = ik + dS (4-54)
The diffraction potential, , satisfies the same radiation con-dition as . Moreover, these potentials all satisfy the
free-surface condition (4-35). So we can apply Green's theorem to derive the equivalent of (4-43), that is,
+ cp.
-ik
ds =
Jfik
dS . (4-55)On the hull, the boundary condition for q is
and so (4-54) reduces to --11 .+ ik
Ek =
iwp jj { ik } dS S Similarly we findEk
= iwpIf
-
ik +ik ----} dSis
(4-56) (4-57) (4-58)Equations (4-57) and (4-58) are the so-called Khaskind-Hanaoka-Newman relationships, generalized for the arbitrary (i,k) mode.
According to the consistent slender-body theory, the free-surface condition in the near field differs from (4-35) by the addition of an extra term. However, McCreight [14] [19] proved that the extra term has no effect on the excitation forces and moments for the rigid-body modes. We may expect that McCreight's method can be used to refine the derivation of the Khaskind rela-tionships for the flexural modes as well.
4-5. Generalized Radiation Forces in the Vertical Modes
We now consider only the cases in which every element of the hull oscillates in the vertical plane. Thus i =
j
= 3 . Sincewe limit the further analysis in this way, there is no need con-tinually to repeat such indices, and so we now write
and so on.
Our purpose is to obtain an expression for that can be readily calculated with acceptable accuracy. We already have two basic expressions for , namely, (4-32) and (4-41). We can
start with either, but we shall choose the former.
Before starting this program, we make some new definitions and develop a useful relationship.
First we define two sets of quantities, k and , which
will serve the same roles as k and mk in the work of Ogilvie and Tuck [18]: for 'P3j.(x) for for c3(x,y,z) for
Fk,3Z
where= flk
= n V x (ak X v±)Then the boundary condition, (4-34), can be written
+
=
iW\)k+Upk
We also define two components of the potential q
(4-6 0) (4-61) (4-62) (4-63) (4-59) J
37
-on the body. It will be assumed that and satisfy iden-tical free-surface and radiation conditions. (We shall consider this matter further presently.)
Table 4-2 gives expressions for ak vk and in terms of quantities already defined and used; for the second and third of these quantities, the expressions can be derived from (4-60) and (4-61). In the table, k is a unit vector in the z
direc-tion.
Table 4-2
A useful theorem of Tuckts (see Ogilvie and Tuck) can be generalized to encompass the flexural modes. In our present nota-tion, this theorem is as follows:
JJ{
(vV)
Up} dS
= t ( Xv)
, (4-66)where is any continuously differentiable scalar function and t is a unit vector tangent to C , which is the intersection of
the hull surface, S , in its undisturbed position with the plane
z = O . This can be proven readily by noting that k k O 1 k k -xk k(x)k n3 -xn3 k(x)n3 -n Bv±/Bz Un3+ n.Bv±/Bz t + ±t-Jkfl3 -knBv/Bz + Bn = (4-64) + = ± (4-65)
X V)
( X
v)
=X (. X
v)
= Uj
; (4-67)x V) [ (
x vi)]
= (x V). (
xv) +
vV
= Up +
vV
. (4-68)One substitutes from (4-68) into the left-hand side of (4-66) and then uses Stokes' theorem to obtain the right-hand side of (4-66).
For the cases under consideration, in which cxj is parallel
to the z axis, the integrand of the line integral in (4-66) is
identically zero provided that the hull is wall-sided (vertical) at the waterline. (When this condition is satisfied, the three vectors t
,
cxj and all lie in a plane.) So we have for
our purposes the result
JJ
vj (vV) + Up} dS
= 0 (4-69)This method of proving Tuck's theorem can also be used to derive (4-41), the reciprocity relationship, from (4-32). One
substitutes the last term of (4-68) into (4-32) and uses Stokes' theorem on the result. Condition (4-39) is equivalent to requir-ing that
U(u+ij;)
oa fact that is also needed in this alternative derivation of the reciprocity relationship.
We now return to the main purpose of this section, deriving a suitable expression for . We substitute from (4-69) into
(4-32):
=
iwp JJ
-
U}
dSs
39 -with and +
ri
+= - w2p
JJ -g--
s dS + iwpU + + - pU2 jj _Ji 'i' ds sThe Strip-Theory Assumption
In order to simplify this further, we must now make a major new assumption, namely, that and can be approximated by their strip-theory equivalents. For k > O , we then have
(see Table 4-2)
+
=
+ t +
= ± Pk(x)0 + Jk(x);
satisfying the free-surface conditions =
0,
w20
-+ =g-5--
O,
+ + } dS + = 1Jw2k+iwuv_u_u2±k}
asis
From (4-64) and (4-65), this is further transformed:
(4-70)
(4-71)
(4-72)
(4-73a)
(4-73b)
on z = O . Note that is independent of the direction of
motion, in fact, independent of the value of U , and thus so is
On the other hand, still depends on U and the direc-tion of modirec-tion through the body boundary condidirec-tion, (4-65). It is further assumed that and satisfy identical outgoing-wave radiation conditions. Of course, the introduction of strip theory implies that these potentials all satisfy the 2-D Laplace equation.
The first term on the right-hand side of (4-70) can be rewritten
rr I j-s.
ii
j dS = k(x)(x)00 dS
s = - Jk Cx) (x) [a (x) +L
b Cx) ]dx (4-74) 1W where a(x) +L
b(x) = - p (4-75)and the line integral is taken around the girth of the ship at the section x . The quantities a(x) and b(x) are the added-mass and damping coefficients per unit length in the heave mode of motion. For modes of motion other than heave, these quantities must be weighted by the appropriate mode shapes, i9,(x) . The net
effect on the k mode is obtained by further weighting them by
'Pk(x)
The second term on the right-hand side of (4-70) can be rewritten and then transfoLlued by Green's theorem (in two dimen-sions) dS Bn Bn (1 B') , + = Jis P(x)'ç] -
(x)o[±k(x)--- +
dS = ±Jk(x)(x)
- (x)(x)] [a(x) +Lb(x)I
dx. (4-76)The third term on the right-hand side of manipulated as follows: BtY] Bn T dS = J s s =
'Is
["
(x)(x)00
+ +± )k(X)(X)
° + -(4-77) (4-70) can be[±0 +(x)}
dSkoo
(x)(x)Q-j dS
o Bn11
-
41-To simplify this further, we invoke that assumption discussed in connection with (4-39) on page 31, namely that we can neglect
interactions of the steady perturbation flow and the unsteady flow. The third and fourth terms in (4-77) contain products of
and , the latter involving the steady perturbation flow,
as can be seen from (4-65) and Table 4-2:
+ +
±
U --j:j- = U
= - n
and so we neglect these terms. The second term of (4-77) is a product of two terms, both involving perturbations of the steady stream, and this is even more negligible.* Thus (4-77)reduces to
Ç
JJ
(x)(x)00
dS dS n L.s
li
i=
-Jk(x)i(x) [a(x) +b(x)] dxlui . (4-78)
-Substituting all of these results back into (4-70), we have
FL.
=Í{w2k(x)lp(x)
iwU[k(x)j(x) -(x)(x)]
+ u2(x)(x) }{a(x) +-,-b(x) }
dx . (4-79)Although this was derived from (4-31), it satisfies the reciprocity relationship, (4-52) (as, of course, it must). A change in the direction of motion changes the sign before the second term, but an
interchange of k and L. interchanges the two terms in square
brackets, leaving the net result unchanged.
The term containing U2 is lacking in the analysis of Ogilvie and Tuck [181, since it is formally of higher order than the other terms (order being measured in terms of , the [small]
slender-ness parameter). Aside from such formal mathematical considerations, *This derivation is very similar to that of Salvesen, Tuck and
Faltinsen [24], who actually assume that p0 = O . In the notation of Ogilvie and Tuck [18], p0 = m3 . Salvesen, Tuck and Faltinsen
+
w2 [A00
+L.3
-£ 0,u.,t,
-
00Then the added-mass and damping coefficients for heave are, respectively, A00 = j a(x) dx = j b(x) dx
this term is usually not of much significance numerically. Further-more, experiments have shown that
F,1+Ft,0
is approximately inde-pendent of U , which can be true only if the U2 term in (4-79) isnegligible.
One may probably expect the same result for F2+Fk
although this has not yet been demonstrated either theoretically or experimentally.
For k = 2. = 0 , since i (x) = O , we have
= w2 J{a(x) + b(x)} dx (4-80)
ihich gives the strip-theory added-mass and damping coefficients for a heaving ship. To be precise, set
(4-81)
(4-82)
(4-83)
The heave equation of motion is obtained from (3-14) by setting
r O . If the only motion is the heave oscillation, the
right-hand side of (3-14) will contain the hydrodynamic radiation force
F,00e1Wt
, which can be transferred to the left-hand side (thatis, subtracted from both sides), and then A00 combines with a00 and B00 combines with the damping coefficient
(C110+C0+ci00+00)
More generally, we can express F2.
= w2[Akj + (4-84)
43 -= S S
St
[k(x)4(x) -i(x)U(x)]b(x)} dx ;
(4-85) = J. { TI2 t ± U (x)4
(x) - i (x) i (x) la (x) j dx . (4-86)These are our final expressions for the added-mass and damping coefficients. If the mode shapes are all known and if the added mass and damping per unit length (in heave) are known, these quantities can be calculated without difficulty.
4-6. Generalized Wave Excitation in the Vertical Modes
The generalized wave excitation in the k-noded vertical oscillation is, from (4-57),
ri
+i
.L}dS
Ej = iwp
Jj
{ ---swhere we now omit the subscript i , which is hereafter equal to 3 , and we consider the force only for forward ship motion. We
assume that can be evaluated by strip theory, as given by (4-63) and (4-71), (4-72). Then we have
4 =
p JJ {{ip(x)
+s
- -
[iwk(x)0 +U(-i(x)0 +ik(x))l} dS
=
JJ{iwik(X) -U(x){-2
0i}
dS+ pU
JJk4-5#
--}
dSThe last integral represents interactions between the perturbation of the steady flow and the incident unsteady flow field, which we
For nearly head seas,
-iKysin
-e
l-iKysin
and we assume that the second term here is small. Furthermore, is odd with respect to y , and so we also neglect the term
including n2 . Noting that n3
= , we are left with
pigh
[iwk(x) -U(x)]
eIKXC0S
now neglect. Thus, substituting from (4-23), we havep igh 1
dx [ik
Cx) -
U (x) Je1 cos
wo jxJ
di e - iKySlfl
-o (Kn3 - iKn2
Sfl
)]
C dz eKzv, [l-K0}
(4-87) (4-88)Finally, we suppose that we can include the effect of the factor
e in a.mean-value sense by introducing a factor e such that
Jdi ev0[l -K0}
= eJdi
[l-K0]
(4-89)We note that
Jdi
o = - B(x),(4-90)
where B(x) is the beam at the section x . Thus we obtain our
final result:
igh 1iA
-- j
dX[iwbk(X)_U(x)Ie_1K0elJ
x{-pB(x) + K{a(x)+Lb(x)}}
1W (4-91)
where (4-75) has been used.
The term containing B(x) is frequently called the "Froude-Krylov force." It gives the generalized force that would exist
45
-on the hull surface if the hull did not modify the pressure field of the incident waves.
The remaining part of E can be considered as the generalized force that is caused by the diffraction of the incident waves. It has a particularly interesting interpretation. Let
= e-iKx cos
We note that the following manipulation is possible:
K e_
xcos
=i01(x)
wo
=
i(w-KUcos
)p1(x)=
iw1(x) +
U4(x)
where we have used (4-24) and (4-25). Then the part of (4-91) representinq the generalized diffraction force can be expressed:
= - h í:2kI
- iwU[kj-I] +U2} e
x{a(x)
r b(x)} dx (494)
If we neglect the x dependence of e , we can rewrite this
expression
(cf.
(4-79) and (4-84)):EiD = - w2{AkI+-L-
BkI}he
(4-95)where AkI and BkI are defined exactly as in (4-85) and (4-86), but with ip(x) replaced with
Equation (4-95) justifies the so-called "relative-motion hypo-thesis" for the forward-speed case. If we had assumed that there were a mode of motion given by the shape function (4-92), the
amplitude of motion in this mode being given by - h , we could have used (4-84), with (4-85) and (4-86), to obtain the generalized diffraction force in the k mode. It was necessary to assume that the x dependence of e could be neglected, but there were no other extra assumptions that were not already implied in (4-91).
(4-92)
4lternatíve Derivation of Formula for E . The preceding derivation, like that for in Section 4-5, is based essen-tially on the approach of Ogilvie and Tuck [181. Derivations have been developed by other investigators using different approaches. For the radiation forces, all of these methods appear to give more-or-less the saine results. The only significant difference is that some investigators obtain extra contributions to A and B]
(see (4-85) arnd (4-86)), the extra terms representing "end effects." These arise if a(x) and b(x) do not vanish at the ends of the ship,* but they appear not to be significant, at least not in the ordinary ship-motions problem, which is fortunate, since a substan-tial contribution of this kind from the ends of the hull would
violate the slender-body concept on which all strip theories are based.
For the wave-excitation forces, these other derivations yield formulas that are apparently quite different from those derived above. We present here a typical derivation of such kind, so that the differences and similarities may be noted explicitly.
We start this time with the expression for
4
given in (4-53). As in the above derivation, we suppress the index since i = 3 always. If one can assumet thatUi + V Ui (4-53) can be written + + + Ek - + EkD (4-96) (4-97)
*Such end effects can also be derived by the Ogilvie-Tuck approach, as shown by Salvesen, Tuck and Faltinsen [24].
§Bishop et al [31 obtain the extra end effects, for example. An interesting new derivation by Newman [171 does not predict the end effects.
tThe effect is essentially equivalent to that of our previous assump-tion that interacassump-tions between oscillaassump-tions and perturbations of the steady flow can be neglected.
where
(4-98)
47
-E1
=JJ
sko{j
+ u dSwhich is the generalized incident-wave force (Froude-Krylov force),
+ + +
and EkD is given by the same formula with
qj replaced by D
The latter has already been introduced; see (4-94).
We introduce
4
from (4-23) and note that {iw +u_}4
[iw -
iKUcos}4
=(Cf. (4-93).) Then, from (4-98), we have
+
iw01
E1 = P0
JJ
k os
A_iKxcosdx
IKz_iKysin
= - pghJ
ke j*Integrate the term containing i(x)B(x) by parts.
(4-99)
(4-100)
C
As in the derivation of (4-88) - (4-91), we assume that (a) the ship is laterally symmetrical, (b) the y dependence of the inner integrand can be neglected, and (C) eKZ can be replaced in a mean-value sense over the contour C(x) by e ; also, we use (4-90) again. This all yields
-pgh
J
eD1
COS B (x) k (x) dx . (4-101) *It may be noted that this can be obtained from that part of (4-91) containing B(x) if it is assumed that
Cx)
cos
-
i K cos B (x) cos 'x
(4-102) that is, the rate of change of B(x)e1 is small compared with the rate of change of COS This is quite reasonable if
the waves are short compared with ship length.
