Jarosław Artyszuk.pdf

PRACE NAUKOWE POLITECHNIKI WARSZAWSKIEJ z. 112

Transport

2016

Jarosaw Artyszuk Maritime University of Szczecin, Faculty of Navigation, Poland

ON SCALING OF SHIP SEAKEEPING The manuscript delivered: April 2016

Summary: The widely accepted linear model of ship motions in waves is considered in frequencydomain, where the steady-state harmonic motions as response to harmonic waves are of interest. The solution of motion transfer functions is presented in dimensionless form with special concern about the dimensionless wave frequency as independent variable. Though some facts are known within ship model testing methodology, this detailed and careful derivation is aimed to enhance in-depth understanding of the seakeeping characteristics of a ship. It is also expected that a transfer functions data exchange between one full-scale ship and another of close geometry yet of different size will be more encouraged. Keywords: ship motion, wave, seakeeping, dimensionless approach, transfer function, RAO

1. INTRODUCTION The ship behaviour in waves, traditionally termed seakeeping, is crucial both for safety and efficiency/effectiveness of operation. Through proper design and good operation, the ship motions shall be minimized or allowed for in various decision-making processes, like e.g. in ship under-keel-clearance management of ports (approach speed limiting, entrance permission issuing). For analysing ship oscillatory motions due to water wave action, it is common/acceptable to use in many cases the linear model of ship dynamics with frequency-dependent coefficients and the frequency-domain (spectral) analysis. The latter is necessary for getting the motion response to irregular/random wave. The irregular wave input is described herein by the wave energy spectral density function (briefly the wave spectrum), that is defined and parameterised for many nautical areas. The ship motion response is modelled via the so-called response amplitude operators (of complex number definition, abbrev. RAOs), also referred to as transfer functions. Both wave spectrum and ship transfer function are functions of frequency, being the wave absolute or encounter (relative) frequency. Simply speaking, the wave spectrum is multiplied by the transfer function of a particular (elementary or combined) motion to obtain this ship's motion spectrum, i.e. the real ship response. Since the transfer functions consider both amplitude ratios and phase lags of a ship's motions versus wave, they are dimensionless and may be successfully used independent of the size of geometrically similar ships. This is especially important in physical scale model

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testing. However, the actual wave absolute or encounter frequency constituting a (horizontal axis) basis for the measured/computed transfer function shall be made dimensionless – multiplied by the factor of  (with accuracy up to the constant), where L is model or ship length, thus arriving at unit length – or directly recalculated to a new length by the factor of  / , where L1 and L2 are original and target length, accordingly. Hence, the lower length, the higher frequency shall be assigned to a given value of transfer function. That is why the international standards (ITTC - International Towing Tank Conference), see also [2], recommend using the dimensionless frequency for presenting data either on transfer functions, or on the resulting motion spectra. For a ship and wave, the dimensionless frequency is 'informatively' equivalent to ship length-wavelength ratio L/O(or the inverse), or more precisely to a square root of this ratio. The lower is model/ship length, the shorter shall be the waves to provide the same ordinate of transfer function. Whenever we talk below in the paper on geometrically similar ships, we mean either scaled models with respect to their full-scale ship (o full/exact similarity), or full-scale ships of the same type, ensuring close hull form shapes/fullness coefficients and ratios of main dimensions, yet of different size/length (o partial, practical or approximate similarity). However, the ratios of main dimensions (length, beam, draught) often vary with a ship's length to get somewhat optimal hull. Therefore, applying dimensionless results from other ships shall be considered with caution, but in case of lack of another better data, absolutely encouraged within the required accuracy. There are many ways to explain the appearance, role and benefits of the dimensionless frequency as abscissa in transfer functions, commonly defined as  , where Z – wave absolute frequency, g – gravity acceleration. The plots of transfer functions are then identical, independent of an object's (model/ship) size. The approaches may differ in the level of abstraction, from a high (qualitative, general) to a low (quantitative, detailed) one. Only some of the first category are listed in the next paragraph. First, having a look at the dimensionless frequency equivalent – the aforementioned ratio L/O– one assures identical relative longitudinal geometry of the ship-wave system. Second, if a model/ship has a natural frequency of its oscillatory motions in calm water, proportional to   (see later in the paper's main body), then all surface wave phenomena and resulting object's motions are in the same proportion to this natural frequency. Third and the last, the motion frequency for angular motion essentially means its angular velocity. When we are considering roll or pitch motions, we can establish points of interest for studying linear velocities, e.g. B/2 or L/2 respectively, where B is ship's beam. The former radius means ship's side, the latter denotes ship's bow or stern. If we now construct the Froude number based on such a reference linear dimension and a linear velocity (as product of angular velocity and radius): FnL / 2

ZL 2 gL 2

Z

L 2g

, FnB / 2

ZB 2 gB 2

Z

B 2g

then we can see that dimensionless frequency is essentially the Froude number for angular motion. Preserving identity of Froude numbers is beyond a dispute. Some authors, e.g. [2],

On scaling of ship seakeeping

11

attempt to derive the formula for dimensionless frequency from 'a ratio' of Froude and Strouhal dimensionless numbers. This seems very complicated and partly confusing. The main objective of the paper is to provide a concise insight into the mechanism of transfer function generation in its fully dimensionless definition. This is a methodical/knowledge- systematizing work aiming at improving the self-confidence of prospective user to such extrapolation or scaling. Good understanding of underlying principles shall advance the art of handling/exchanging transfer functions data between ships of different size. The analysis will be performed at the low level of abstraction in that we are starting from the well-known basic motion equations and take advantage of definite dimensionless force coefficients. Such an approach with transformation of formulas is always the most convincing, if any dimensions disappear, thus providing a clear view of the phenomenon interior. As far as possible, we try to avoid a direct application of dimensional analysis, though a bit of the latter was already incorporated in the assumed coefficients.

2. LINEAR MODEL OF SHIP MOTIONS IN WAVES A ship's general movement, in 6DOFs (degrees-of-freedom), is described in an earth-fixed right-hand reference frame O0x0y0z0, where linear and angular position of the ship-fixed reference frame Mxyz is being determined, see Fig. 1. earth-fixed system x0

O0

y0

MS (midship section)

z0

x xG

M

x

WP (water-plane)

zG G y

z

y

CP (centre-plane)

z

KP (keel-plane)

Fig. 1. Reference systems

However, for studying ship oscillatory motions (disturbances or perturbations) in waves, another 'hidden' inertial frame is usually put behind the latter one, to be marked M'x'y'z' (not shown in Fig. 1), that goes with the ship's constant forward speed v (in x-axis direction). This speed will be an average speed, when the ship starts oscillations. The frame M'x'y'z' serves to establish linear (x, y, z) and angular (M, T, \) displacements of Mxyz. The angular displacements are the rotation angles around corresponding axes. For

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simplicity of final equations, the corresponding ship motion equations (providing evolution of linear ad angular velocities) are often formulated in an additional frame Gxyz (and analogically 'Gx'y'z'), that is Mxyz but parallel shifted to the ship's centre-of-gravity G. A drawback of introducing Gxyz is that, in the case of G variation due to loading condition, the equation hydrodynamic coefficients (hull-constant, but G-sensitive) shall be adjusted. Let's further assume, not impairing the generality of our next derivations, that G coincides with M (xG=zG=0, Fig. 1) Taking into account the usual ship's symmetry against the centre-plane, the general 6DOF linear model can be decomposed into two well-known uncoupled sets of three equations, corresponding to the so-called symmetrical (DOFs: 1, 3, 5) and asymmetrical motions (DOFs: 2,4,6): ªm1  k11 m13 m15 º ª xº ª n11 » « m1  k 33 m35 » « z»  «n31 « m31 « » « «¬ m51 m53 I y 1  k 55 »¼ «¬T»¼ «¬n51

n13 n33 n53

n15 º ª x º ª0 0 n35 » « z »  «0 b33 »« » « n55 »¼ «¬T»¼ «¬0 b53

ª xº ª x º ª xº >M jk @sym «« z»»  >N jk @sym «« z »»  >B jk @sym «« z »» «¬T»¼ «¬T»¼ «¬T »¼

m24 ªm1  k 22 « m I x 1  k 44 42 «  Dzx  m64 «¬ m62

º ª y º ªn22  Dxz  m46 » «M»  «n42 »« » « I z 1  k 66 »¼ «¬\»¼ «¬n62 m26

n24 n44 n64

jk

ª FxWV 1 º » « « FzWV 1 » «¬ M yWV 1 »¼

ª FxWV 1 º » « « FzWV 1 » «¬ M yWV 1 »¼

n26 º ª y º ª0 0 n46 » «M »  «0 b44 »« » « n66 »¼ «¬\ »¼ «¬0 0

ª y º ª y º ª yº «M»  >N @ «M »  >B @ «M » jk asym « » jk asym « » asym « » ¬«\¼» ¬«\ ¼» ¬«\ ¼»

>M @

0 ºª xº b35 » « z » »« » b55 »¼ «¬T »¼

ª F yWV 1 º «M » « xWV 1 » ¬« M zWV 1 ¼»

(1)

(1a)

0º ª y º 0» «M » »« » 0»¼ «¬\ »¼

ª FyWV 1 º «M » « xWV 1 » «¬ M zWV 1 »¼

(2)

(2a)

where: m– ship's mass; Ix, Iy, Iz– ship's inertia moments; Dxz, Dzx– ship's deviation moments (equal); kii– dimensionless coefficients of basic added masses, mij– other added masses, nij– damping coefficients, bij– restoring coefficients, where i, j{1, 2, ..., 6}. Terms to the right of equals sign in (1) to (2a) are components of the first-order ('WV1') wave excitations, of force or moment dimension. The variation in elevation with time of the harmonic (regular) wave is written by:

[

[ 0 cos Zt

(3)

where: [0– wave amplitude, Z– wave (absolute) frequency, t– time.

Because of linearity assumed in the wave pressure, the wave exciting generalized force F (in terms of force or moment, hereafter briefly called the force) is also harmonic, but of the amplitude F0, proportional to the wave amplitude, of the encounter frequency ZE, and of the phase lag MF versus the wave:

On scaling of ship seakeeping

F

F t

F0 cosZ E t  M F

13

(4)

which can be rearranged to:

F

F0 >cos M F cos ZE t  sin M F sin ZE t @ F C cos ZE t  F S sin ZE t

(5)

where the cosine- and sine-amplitudes, FC and FS, are introduced. This expression represents each component of the right-hand side vectors in (1) and (2).

The transformation of the wave frequency Z into its encounter frequency ZE is the direct effect of the ship-wave relative speed:

ZE

Z  kv cos J WV

(6)

where: k– wave number (=2S/OO– wavelength); v– the ship's constant forward speed (note: the derivative of x in (1), marked  , means the oscillation around this speed); JWV– wave direction (incidence angle) in the ship's horizontal plane Mxy from the range (180q,180q². In the latter, the convention is as follows: 0q– stern wave, 90q– beam wave from portside, r180q– head wave, i.e. running from ahead to aft.

Under the harmonic exciting forces, due to the linearity of differential equations (1) and (2), the resulting motions in the steady-state phase (after suppressing the transient 'memory effects') are also harmonic of the frequency of excitations, herein of the encounter frequency. The steady-state harmonic motions also directly satisfy the differential equations. The amplitude and phase lag of each motion versus the wave elevation can also be written in the analogical way to (4) and (5). Moreover, the motion amplitude is proportional to the wave amplitude, therefore allowing an introduction of a very convenient ratio of these quantities - just called a motion transfer function, in contrast to a force transfer function, which is defined with regard to the 'wave-wave force' mechanism. In other engineering fields there exist various names for 'transfer functions'. In general, the transfer function consists of not only the amplitude, but of the phase information as well. In this case, the transfer function is represented symbolically by complex numbers. The phase angle data is necessary while combining/superimposing elementary motions (of linear or angular nature) to get the motion of arbitrary interested point around the ship. In our approach, by analogy to (5), the motion sine- and cosine-amplitudes, UC and US, are introduced, that can be next easily converted to the (usual) amplitude U0 and phase angle MU:

U

U t U 0 cosZE t  MU U C cos ZE t  U S sin ZE t

(7)

where

U0

U  U C 2

S 2

, tan MU

US UC

(8)

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This approach is fully equivalent to a direct application of complex number formulation, as common in this case. In other words, our linear system is subject to a combined action of cosine and sine excitations of the form (5), that can be split and input separately to the system. The 'independent' steady-state harmonic motions arising from each of these two components can then be superimposed (added) to get the total resulting motion in the form of (7). Although we keep in (8) both US and UC dimensional at this moment, their ratio (and the produced phase angle, too) is directly dimensionless and identical for geometrically similar ships.

3. CASE OF ASYMMETRICAL MOTIONS Let's concentrate hereafter purely on the asymmetrical motions (2). The treatment of the other type of motions (1) is quite similar, thus leading to the same general conclusions from the viewpoint of the paper's objective. Computing derivatives of (7), substituting to (2), and grouping the sine- and cosine-related terms finally yields:

> Z >M @ 2 E

jk asym

 >B jk @asym

ªU yS º ªU yC º « » « C» ˜ «U M »  Z E >N jk @asym «U MS » «U\S » «U\C » ¬ ¼ ¬ ¼

@

ªU yC º « »  Z E >N jk @asym «U MC »   Z E2 >M jk @asym  >B jk @asym «U\C » ¬ ¼

>

ªU yS º « » ˜ «U MS » «U\S » ¬ ¼

@

C ª FyWV 1 º « C » M « xWV 1 » C » « M zWV 1¼ ¬

S ª FyWV 1 º « S » M « xWV 1 » S » « M zWV 1¼ ¬

(9)

(10)

where subscripts denote particular elementary motions.

Both the equations (9) and (10) must be solved together for the 6 unknown sine- and cosine-amplitudes, leading to a set of 6 linear equations with 'some' symmetry of the main matrix:

>

ª 2 «  Z E >M jk @asym  >B jk @asym «  Z E >N jk @asym «¬ 3x3

>

@

@

3x3

> Z >N @ @ E

> Z >M @ 2 E

jk asym 3x3

jk asym

 >B jk @asym

ªU yC º « C» UM º « C» « » ˜ U\ » » «U S » » « y» 3x3 ¼ « S » UM « S» «¬U\ »¼

@

C ª FyWV 1 º « C » M xWV 1» « C » « M zWV 1 « S » F « yWV 1 » S » « M xWV 1 « S » ¬« M zWV 1 ¼»

(11)

However, the analytical solution of (11), due to the size of the matrix (6x6), is complicated and not practical for our purpose.

On scaling of ship seakeeping

15

The dimensional character of the elements of matrix [Mjk]asym can be defined with the following dimensionless coefficients (see e.g. [3]):

>M @

jk asym

ª m' 22 m « Lm' 42 « «¬ Lm' 62

Lm' 24 L2 m' 44 L2 m' 64

Lm' 26 º L2 m' 46 » » L2 m' 66 »¼

(12)

where ship's length (L)-related non-dimensionalizing scheme has been adopted. The m'ij coefficients, where i, j {2, 4, 6}, are dimensionless. The original element of [Mjk]asym has been thus divided either by the ship's mass or the dimensional part of its static or inertia moment (mL, mL2, accordingly), where applicable.

Especially if we evaluate ships of varying breadth B, and identical other dimensions, it is sometimes more useful to apply in the above the breadth value, or even the half of it. Moreover, it is not necessary to use the same non-dimensionalizing term for all elements of the matrix. Such a selection only improves 'the parametric performance' of the resulting dimensionless coefficients from the viewpoint of particular purpose. However, for geometrically similar hulls (geosims), there is no matter what linear dimension is chosen and all the received dimensionless coefficients are identical for (independent of) any size of geometrically similar ship. For damping matrix [Njk]asym, using also the L-related method, one may similarly write [3]:

>N @

jk asym

m

ª n' 22 g« Ln' 42 L« «¬ Ln' 62

Ln' 24 L2 n' 44 L2 n' 64

Ln' 26 º L2 n' 46 » » L2 n' 66 »¼

(13)

where the square root expression in front of the matrix represents a certain, conventional natural (undamped) oscillation frequency in water due to buoyancy/hydrostatics, the same for all degrees of freedom. Of course, the sway (y) and yaw (\) motions, not subject to direct hydrostatic forces, undergo natural oscillations due to inherent coupling with the roll (M) motion. Expressing the damping forces by means of natural frequency is also usual for studying vibration problems of any mass-damping-spring mechanical systems. The dimensionless coefficients in (13) are constant for geometrically similar ships.

It shall be mentioned here that the matrices (12) and (13) are not symmetrical for nonzero forward speed v, what probably would facilitate an analytical solution of (11). The buoyancy/restoring force terms in [Bjk]asym and [Bjk]sym can be intermediately and finally formulated like follows (refer also to [1]): b33

UgAWP

b44

Ug

cWP 2 L L B

mg ˜ GM

(14a)

(14b)

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b55

b35

UgI yWP

Ug

cWP 2 r ' L4 L yWP B

 UgAWP ˜ LCF

b53

 Ug

(14c)

cWP LCF '˜L3 L B

(14d)

where: U– water density, AWP– ship's water-plane area (cWP– its fullness coefficient), GM– metacentric height, IyWP– water-plane are inertia moment around y-axis (r'yWP– dimensionless gyration radius), LCF– longitudinal centre of floatation (i.e. water-plane centroid's x-abscissa, LCF'– its dimensionless value).

The coefficients (14a-c) lead to the following natural frequencies in pure oscillations relevant to a particular degree-of-freedom:

Zz 0

ZM 0

ZT 0

1 rxc

L B ˜ B T ˜g cB 1  k33 L cWP

cWP g ˜ cB 1  k33 T 1

1  k 44 c ryWP ryc

˜

GM ˜ g B2

1 L rxc B

c ryWP ryc

cWP g ˜ c B 1  k 55 T

1

1  k 44

˜

(15) GM ˜ g L2

L B ˜ B T ˜g c B 1  k 55 L

(16)

cWP

(17)

where: T– ship's draught, cB– block coefficient (displaced volume versus LBT), r'x and r'y– ship's dimensionless gyration radii (in L units) versus corresponding axes.

It shall be noted that the natural frequencies (or corresponding natural periods) for heave (z) and pitch (T) are similar in magnitude for most ships. We may write all the three frequencies (15)-(17) as proportional to the term:

Z0

g L

(18)

serving as dimensioning (or non-dimensionalizing) factor in (13). Since it does not provide real value of natural frequency for elementary motions, the dimensionless damping coefficients n'ij, shown in (13), shall be properly interpreted, or scaled using (15) to (17), when a direct comparison to the classical 1DOF vibration (in ship's case– motion) equation is attempted. Let's now define the wave absolute and encounter frequencies in the units of Z0. For the former, one can read:

On scaling of ship seakeeping

Z Z0

Zc

Z

L g

17

(19)

that we will referred to as the dimensionless wave frequency. From the so-called dispersion relationship (a part of the water wave theory), one can deduce the wave number k (or wavelength) for a given frequency. In deep water conditions, this gives:

Z

kg

or k

Z2 g

(20)

Considering the definition and role of the Froude number in the free-surface flows (with wave-making action):

v gL

FnL

(21)

and (20), we can rewrite the encounter frequency expression (6) like follows (see also [5]):

ZE

Z

Z2 FnL gL cos J WV g

Z 1  Z cFnL cos J WV

(22)

that results in the dimensionless wave encounter frequency Z'E:

ZE Z0

Z Ec

f1 Z c Z c1  Z cFnL cos J WV

(23)

The dimensionless frequency Z' per (19), by means of (20), also resolves to:

Zc

2S

L

O

(24)

On account of (14b), (16), and (18), one can write down the b44 constant with its dimensionless value b'44 as:

b44

mL2

g b'44 L

mL2Z02b'44

The right-hand side wave forces in (2) may be non-dimensionalized according to:

(25)

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ª FyWV 1 º «M » « xWV 1 » ¬« M zWV 1 ¼»

ª1 c º « L FyWV 1 » c 1» mg[0 « M xWV » « c 1» M « zWV ¼ ¬

(26a)

cc 1 º ª FyWV cc 1 » mgk[0 « L ˜ M xWV « » cc 1 ¼» ¬« L ˜ M zWV

(26b)

º ª 1 c « L FyWV 1 » cc 1 » mg[0 «kL ˜ M xWV » « cc 1 » « kL ˜ M zWV ¼ ¬

(26c)

or ª FyWV 1 º «M » « xWV 1 » ¬« M zWV 1 ¼»

or (through mixing the above) ª FyWV 1 º «M » « xWV 1 » ¬« M zWV 1 ¼»

i.e. being expressed in the units of the buoyancy/gravity force or its conventional moment [3]. This is rather natural (especially (26a)), if we look at partially hydrostatic nature of waves. The right-hand side dimensionless coefficients of (26a-c) are also independent of the length of geometrically similar ships and can be partly understood as the wave force transfer functions. They can also be defined in the same way with respect to the cosine and sine parts as appearing in (11), leading to the notation with additional 'C' and 'S' superscripts.

4. DERIVATION OF TRANSFER FUNCTIONS Inserting (12), (13), and (26a) to (11), after some transformations and according to the remarks in the preceding paragraph, gives: º ª ª m' 22 Lm' 24 Lm' 26 º Ln' 24 Ln' 26 º ª n' 22 » ªU C º « 2 « » 2 2 « » » « yC » «  Z E2 « Lm' 42 L2 §¨ m' 44 b' 44 Z0 ·¸ L2 m' 46 »  Z E Z0 Ln' 42 L n' 44 L n' 46 2 ¸ ¨ « » » «U M » « ZE ¹ « » 2 2 © «¬ Ln' 62 L n' 64 L n' 66 »¼ » «U C » « « Lm' L2 m' 64 L2 m' 66 »¼ 62 ¬ »˜« \ » « ª m' 22 Lm' 24 Lm' 26 º » «U yS » « n Ln Ln ' ' ' ª º 22 24 26 « »» « S » « § Z2 · U «  Z E Z0 « Ln' 42 L2 n' 44 L2 n' 46 »  Z E2 « Lm' 42 L2 ¨¨ m' 44 b' 44 02 ¸¸ L2 m' 46 » » « MS » « » Z « » » ¬«U\ ¼» « E ¹ 2 2 © ¬« Ln' 62 L n' 64 L n' 66 ¼» « Lm' « L2 m' 64 L2 m' 66 »¼ »¼ 62 ¬ ¬

ª 1 cC º « L FyWV 1 » « M cC » 1 » « xWV cC 1 » « M zWV g[ 0 « 1 S » c 1» « FyWV L » « S c M « xWV 1 » S » « M zWV ¬ c 1¼

(27)

On scaling of ship seakeeping

19

When both sides of (27) are divided by Z02 and left-hand-multiplied by the diagonal matrix (corresponding to multiplying the rows of the affected matrix/vector by certain factor) ªL «0 « «0 « «0 «0 « ¬0

0 0 0 0 0º 0 0 0 0» » 1 0 0 0» » 0 L 0 0» 0 0 1 0» » 0 0 0 1¼

1 0 0 0 0

then the resulting matrix to the left side of (27) can be decomposed to the righ-handproduct: m' 24 m' 26 º ª ªm' 22 ªn ' 22 n ' 24 « » « § 1 · «  f 12 Z c «m' 42 ¨¨ m' 44 b' 44 2 ¸¸ m' 46 »  f 1 Z c «n ' 42 n ' 44 « f 1 Z c ¹ « » « © ¬«n ' 62 n ' 64 « m' 64 m' 66 ¼» ¬«m' 62 « ªm' 22 m' 24 « ªn ' 22 n ' 24 n ' 26 º « «  f 1 Z c «n ' 42 n ' 44 n ' 46 »  f 12 Z c «m' 42 m' 44 b' 44 « « » « « «¬n ' 62 n ' 64 n ' 66 »¼ m' 64 «¬m' 62 ¬«

n ' 26 º n ' 46 » » n ' 66 ¼» 1 f 12 Z c

º » ªL 0 » « 2 » «0 L » «0 0 »˜« m' 26 º » « 0 0 »» « 0 0 m' 46 » » « »» ¬ 0 0 m' 66 »¼ ¼»

0 0 L2 0 0 0

0 0 0 0 0 0 L 0 0 L2 0 0

0º 0» » 0» » 0» 0» » L2 ¼

(28)

The second matrix in (28) can be next combined with the vector of the motion amplitudes. Hence, after utilizing (18) and denoting the left matrix in (28) with [Aij(Z')], we get: ª LU yC º « 2 C» «L UM » « 2 C» >Aij Z c @˜ « LLUU\S » y » « « L2U MS » « 2 S» «¬ L U\ »¼

cC 1 º cC 1 º ª FyWV ª FyWV « C » « C » c 1» c 1» « M xWV « M xWV « M zWV cC 1 » cC 1 » g[ 0 « M zWV » [0 L« S » 2 « S c 1» c 1» Z0 « FyWV « FyWV S « » « M xWV cS 1 » c 1 M xWV « S » « S » c 1 »¼ c 1 »¼ «¬ M zWV «¬ M zWV

(29)

Therefore, the solution of the wanted motion transfer functions, marked with 'prime', in the case of (26a) takes the form of: ª U cyC º « C» « LU Mc » « LU\cC » « S » « U cy » « LU Mc S » « S» «¬ LU\c »¼

cC 1 º ª FyWV « C » c 1» « M xWV C 1 « M zWV >Aij Z c @ « ccS 1 »» « FyWV 1 » « M xWV cS 1 » « S » c 1 ¼» ¬« M zWV

(30)

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where C ªU McC º 1 ªU y º « S» , « S» [ 0 ¬U y ¼ ¬U Mc ¼

ªU cyC º « S» ¬U cy ¼

C ªU\cC º 1 ªU M º « S» , « S» [ 0 ¬U M ¼ ¬U\c ¼

C 1 ªU\ º « S» [ 0 ¬U\ ¼

(31)

The transfer functions of angular motions in (31), however, are still dimensional, strictly L-dependent. If we implement another definitions for them: ªU Mc*C º « *S » ¬U Mc ¼

C ªU\c*C º 1 ªU M º « S » , « *S » [ 0 / L ¬U M ¼ ¬U\c ¼

C 1 ªU\ º « S» [ 0 / L ¬U\ ¼

(32)

then we will reach a fully dimensionless solution: ª U cyC º « *C » «U Mc » «U\c*C » « S» « U cy » «U Mc*S » « *S » ¬«U\c ¼»

cC 1 º ª FyWV « C » c 1» « M xWV C 1 « M zWV >Aij Z c @ « ccS 1 »» « FyWV 1 » « M xWV cS 1 » « S » c 1 ¼» ¬« M zWV

(33)

Relating the angular amplitudes to the dimensionless wave amplitude as expressed in in the ship's length units (L), see (32), is not usual in seakeeping-oriented ship hydrodynamics. However, it is physically sound and equivalent to the more convenient, and more frequently used wave slope angle – k[0. The latter is equal to 2S[0/O i.e. proportional to the wave amplitude-wavelength ratio. The wave slope angle can be considered a dimensionless wave amplitude, where the wave amplitude is denominated in wavelength (O) units. To arrive at the latter solution, with angular amplitudes vs. wave slope angle, equation (30) can be transformed to: ª1 0 «0 1 « kL « «0 0 « «0 0 «0 0 « « «0 0 ¬

0

0

0

0

0

0

1 0 kL 0 1

0

0

0 1 0 kL

0

0

0

0º C 0 »» ª U cy º « C» » « LU Mc » 0 » « LU cC » \ ». 0 » « U cyS » » « 0 » « LU Mc S » »« S» 1 » ¬« LU\c ¼» kL »¼

ª1 0 «0 1 « kL « «0 0 « «0 0 «0 0 « « «0 0 ¬

0

0

0

0

0

0

1 0 kL 0 1

0

0

0 1 0 kL

0

0

0

0º cC 1 º ª FyWV 0 »» « C » c 1» » « M xWV 0» cC » 1 « M zWV »>Aij Z c @ « S 1 » 0» c 1» « FyWV « M xWV 0» cS 1 » » « S » c 1 ¼» 1» ¬« M zWV kL »¼

(34)

On scaling of ship seakeeping

ªU cyC º «U cC » « M » « k » «U\cC » » « « kS » « U cy » «U cS » « M » « k » « U\c S » » « ¬ k ¼

ª1 0 «0 1 « Z' 2 « «0 0 « «0 0 «0 0 « « «0 0 ¬

0

0

0

0

0

0

0

0

1 Z' 2 0 0

1

0

0

0 1 0 Z' 2 0

0 º cC 1 º ª FyWV 0 »» « C » c 1» » « M xWV 0 » cC » 1 « M zWV »>Aij Z c @ « S 1 » 0 » c 1» « FyWV « M xWV 0 » cS 1 » » « S » c 1 ¼» 1 » ¬« M zWV 2» Z' ¼

cC 1 º ª FyWV « C » c 1» « M xWV C 1 « M zWV >Dij Z c @>Aij Z c @ « ccS 1 »» « FyWV 1 » « M xWV cS 1 » « S » c 1 ¼» ¬« M zWV

21

(35)

and finally ªU cyC º « C» «U Mcc » «U\ccC » « S» «U cy » «U Mcc S » « S» ¬«U\cc ¼»

where:

cC 1 º ª FyWV « C » c 1» « M xWV C « M zWV >Cij Z c @« ccS 1 »» « FyWV 1 » « M xWV cS 1 » « S » c 1 »¼ «¬ M zWV

(36)

>C Z c @ >D Z c @˜ >A Z c @

1

ij

ij

ij

(37)

and ªU MccC º « S» ¬U Mcc ¼

C ªU\ccC º 1 ªU M º « S» , « S» k[ 0 ¬U M ¼ ¬U\cc ¼

C 1 ªU\ º « S» k[ 0 ¬U\ ¼

(38)

Assuming different non-dimensionalization schemes for wave exciting forces, like exemplified in (26b) or (26c), would lead more or less directly to the same results.

5. FREQUENCY-DEPENDENCE OF COEFFICIENTS AND OTHER ISSUES Till this point, we have treated all the dimensionless matrix elements of [Mjk]asym (strictly their added masses) and [Njk]asym (damping coefficients), (12) and (13), as constants. The same has also been the case for the wave dimensionless coefficients (26a), or in its expanded version – the 6-element vector to the right of equation (36).

22

| *

However, the dimensionless added masses and damping coefficients are oscillation frequency-dependent, herein equal to the encounter frequency ZE. This dependence is also true for the dimensionless amplitudes of the wave forces (refer e.g. to [4], [5]). Particularly in that, the diffractional (hydrodynamic) part of the wave force amplitude, similarly to water inertia forces (added masses) or damping forces, exhibits a similar physical nature. Hence, it is computed, within the widely adopted strip approach, with the same encounter frequency-dependent two-dimensional (sectional) added mass and damping potential coefficients. The same dimensionless encounter frequency relationship is also valid for the Froude-Krylov (hydrostatic) part of the wave force amplitude. In addition in our equations, if we subtract the residual dimensionless terms originating from ship's inertia, thus arriving at pure added mass coefficients, the functions m'ij (ZE) can be interrelated to the damping coefficients n'ij (Z(). The aforementioned sectional damping coefficients, and the corresponding twodimensional ship's cross-sectional oscillations, are also subject to the same frequency nondimensionalizing, as stated in comments to (13). However, the linear dimension of a crosssection (proportional to a ship's length L) is usually applied in this context. Therefore, in the considered frequency-dependence, instead of ZE, we may assume its dimensionless definition Z'E in order to obtain the most universal (independent of ship's size) description. That is why we should really write:

>C Z c, Z c @ >D Z c @˜ >A Z c, Z c @

1

ij

E

ij

ij

E

(39)

and cC 1 º ª F yWV « C » c 1» « M xWV « M zWV cC 1 » « S » c 1» « F yWV « M xWV cS 1 » « S » c 1 ¼» ¬« M zWV

>F c @ >F cZ c @ j

j

(40)

E

The both expressions (39) and (40), in the light of (23), can thus be written as sole functions of either Z' or Z'E, leading to the fully dimensionless description of ship seakeeping. Since all the three basic frequencies – Z', ZE, and Z (or four, if we add the less frequently used Z'E) – are interconnected to each other by means of (19) and (23), we may also express and present the charts of our transfer functions:

>U c

y

C

U Mcc C

U\cc C

U cy S

U Mcc S

U\cc S

@

T

as function of just one independent variable that can be each of these frequencies. The selection of a given frequency affects a shape of transfer functions. A subsequent change to another frequency type will engage a horizontal proportional scaling of their

On scaling of ship seakeeping

23

charts (if we move from any dimensional frequency, Z or ZE, to its dimensionless value, annotated with 'prime', or vice versa), or a complete horizontal transformation. The latter will happen according to (22), if we are changing over between the wave absolute frequency and its encounter frequency, either in the dimensional or dimensionless case. The vertical scale of transfer function charts is always preserved, since their ordinates are just the motion dimensionless amplitudes (in units of wave amplitude or wave slope). Using the dimensional encounter frequency ZE, or its dimensionless equivalent Z'E, while presenting/storing the transfer functions, is generally being avoided due to numerical difficulties involved in handling of such datasets [4], [2]. The wave absolute frequency corresponding to particular encounter frequency is ambiguous, see (22), though the opposite conversion is totally unique. In certain conditions of a ship-wave relative movement, mostly connected with stern/quartering waves, up to three different wave absolute frequencies may contribute to the same value of the encounter frequency. It shall also be kept in mind that a 'transfer' of transfer functions from one ship size to another is valid only for the same wave incidence angle JWV. At first hand, this belongs to geometrical similarity of ship-wave mutual arrangement, that is particularly obvious at zero forward speed. But more precisely, the wave direction is not only accommodated in the encounter frequency (22), where someone might try to manipulate with forward speed or wave absolute frequency to obtain the same encounter frequency. Namely, the amplitudes of wave exciting forces are in general explicitly dependent on the wave incidence angle, independently from forward speed. However, the necessity to retain also the absolute (dimensional) forward speed in the data exchange on transfer functions, similarly to the above problem of wave incidence angle, is not the case. A change to another ship size (length) shall automatically involve recalculating the forward speed according to Froude number identity, hence through multiplication by the factor of  / , where L1 and L2 are original and target length, respectively.

6. CONCLUSIONS In the paper, we have demonstrated a disappearance of dimensional terms in a few stages. First, under assumption of constant coefficient matrices/vectors, the dimensional encounter frequency arisen from derivatives of motions has been reduced with the help of damping and restoring forces being expressed in natural frequency units. An essential step has been also to define the transfer functions for angular motions versus the dimensionless wave amplitude (as either [0/L or [0/O). Finally, the encounter frequency-dependence of equation coefficients has been discussed, that really resolves to the impact of dimensionless encounter frequency, and subsequently to dimensionless absolute frequency itself. If a transfer function is presented versus dimensionless frequency Z' (roughly corresponding to a reduction of ship's length to unit length), as shall be by default, then we can rather better compare various hydrodynamic designs for seakeeping performance. However, the horizontal scaling of such a transfer function plot to the actual ship's length,

24

| *

via multiplication by   , always means its shift towards lower dimensional wave frequency. Also, the higher ship's length, the more on lower frequencies is lying the transfer function. Hence, the ship's actual motion spectrum (and its related probabilistic indices), as a combination of the actual transfer function and the actual wave spectrum, does not necessarily solely rely on the fully dimensionless transfer function (in other words, on 'pure' hydrodynamics). This is also a matter of ship's size and frequency range of the wave spectrum. To supplement the outcome of this study, the next research shall concentrate on a sensitivity of transfer functions to different hull form shape parameters (fullness coefficients, ratios of main dimensions). This will provide limits of accuracy for transfer function data exchange among ships, since a variation in ship's length often implies more or less significant changes in those parameters.

References 1. Artyszuk J.: Consideration on dynamic modelling of ship squat. In: Safety of Marine Transport. Marine Navigation and Safety of Sea Transportation (eds. Weintrit A., Neumann T.), CRC Press, Boca Raton, 2015. 2. Dudziak J.: Ship Theory. Ed. 2, Foundation for Shipbuilding Industry and Maritime Economy Promotion, Gdansk, 2008 (in Polish). 3. Journee J.M.J.: User Manual of SEAWAY. Rel. 4.19, Report no. 1212a. Delft University of Technology, Ship Hydromechanics Lab., Delft, 2001. 4. Journee J.M.J., Adegeest L.J.M.: Theoretical Manual of Strip Theory Program 'SEAWAY for Windows'. Report no. 1370, rev. Dec 14th, Delft University of Technology/ Ship Hydromechanics Lab./AMARCON, Delft, 2003. 5. Pawlowski M.: Linear Model of Ship Motions in Irregular Wave. Technical Report no. 41, Polish Register of Ships (PRS), May, Gdansk, 2001(in Polish).

6 3 /6) /465 369] ^ 1 13_ 4 ` / Streszczenie: H   

              

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