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(1)

The Relationship of the Frequency and

Time Domain with respect to

Ship-motions

Ir. J.C. Moulijn

(2)

Abstract

In this report the relationship of the frequency and time domain 'expressions for the hydromechanical reaction force, that act on the hull of an oscillating ship, is presented

The report is based on the theory developed by Cummins and Ogilvie in the ar1y sixties.

The boundary value problem for unsteady shipmotion is derived and linearized. Solution forms to' this 'boundary value problem are proposed in the frequency and the

time domain., The hydromechanical reaction force on the ship hull is expressed in terms

of these solutions. Some theory on the response of linear systems and the relatiònship of the fre4uency and time domain is presented. This theory is applied to the two expressions for the hydromechanical reaction force. A relatively easy way to find the constants and the impulse response function (retardation function),, that appear in the

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Contents

Abstract

3

Contents

5

i

introduction

7

2 Boundary value problem

9

3 The frequency domain formulation

11

4 intermezzo: Impulse response functions

14

5 The time domain formulation

18

6 Relationship of the frequency and time domain forces

22

7 Concluding remarks

25

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i

Introduction

We are very much used to formulating the equations of motion of a ship in the frequency domain. The. hydromechanical reaction force due to harmonic notion of the ship, which

is the hydrostatic plus the hydrodynamic reaction force, is expressed in terms

propor-tional to displacement, velocity and acceleration of the vessel. The component of the hydromechanical reaction force that is in phase with the harmonic motion of the ship is distributed over the displacement and the acceleration term, while the quadrature component of the hydromechanical force contributes to the velocity term:

= -

+ Bjk(w),k + CjkT/k} (1)

According to Cummins [1] this representation of the hydromechanical forces is far from

reality. When the hydromechanical reaction forces are set equal to the inertial forces a set of seemingly normal second order differential equations appears. However, the constants

Ak

and B3k are dependent on the frequency of oscillation. The response of

the ship to an arbitrarily shaped forcing function cannot be calculated bymeans of these equations.

Nevertheless the frequency domain expression of the hydromechanical reaction force

has proved to be very useful. When a statistical instead of a deterministical description

of the shipmotions is used it suffices very good as long as the problem is linear. A major

advantage of the frequency domain formulation is the more easily solvable boundary value problem; all time derivatives (/Ot) in the boundary conditions red'uce to iw, so

the problem does not depend on time anymore. Difficulties arising with time stepping algorithm are thus circumvented. Therefore it is not surprising that the frequency domain formulation has got (and will get) much attention.

When non-linear features are playing an important role, the frequency domain

for-mulation cannot be applied, because it is based on the assumption of linearity. Although

the shipmotions problem is non-linear, good results can be obtained when usinga linear formulation. Of course one should always be careful if linearization is allowed.

For linear systems the frequency and time domain are related by a Fourier transform.

When the response of a system is known in the frequency domain, the time domain response can be constructed. This can be useful when external non-linear forces are acting on a vessel. One might think of non-linear mooring forces, or of non-linear cushion dynamics of cushion supported vessels. The hydromechanical reaction forces can be expressed in the current arbitrary motion and the arbitrary motions of the past. Such a time domain filter can be constructed by means of the inverse Fourier transform of the frequency domain response function. Then the non-linear external forces can be

added to the hydromechanicat forces, and the motions of the ship can be solved in the time domain.

(5)

This report does not contain any new items, but it represents the :theory, that was

developed by Cummins [1] in 1962. Expressions fOr the hydromechanical. reaction force

will be derived in the frequency and in the time domain At the end the frequency and

time domain will be linked together as Ogiivie [2J did in 1964.

At first, in the second section, the boundary value problem. will be formulated and linearized. This linearized boundary value problem is the bases for both frequency and time domain formulation In the third section a frequency domain formulation for the linearized boundary value problem will be proposed The hydromechanical reaction forces will be expressed in terms of the frequency domain solution The fourth section

treats some theory of the response of, linear systems in the time dOmain and of the

inter-relation of the frequency and time domain In the fifth section a time domain solution

form for the boundary value problem is proposed. The hydromechanical reacion force is

expressed in this solution. Finally, in the sixth section, the frequency and time domain expression for the hydromechanical reaction force are linked. A relatively easy way to find the constants and the impulse response function that appear in the time domain

(6)

2

Boundary aIue problem

The problem will be formulated in a coordinate system that travels, at a constant speed U in positive x-axis' direction. The origin, of the system. is fixed in: the undisturbed free surface above the mean center of gravity of a vessel sailing at the same mean forward speed U. The z-axis points upwards. The vessel has got a small oscillating displacement frúm its mean position. The displacement is expressed in the coordinate system as (m 13) being the translation, in x-, y- and z-direction, while the rotation

of the vessel is expressed as (ifl, f15, f16) being the rotation around the x-, y and z-axis

respectively.

Because the fluid is assumed to be in-viscid. and non-rotational, a velocity potentiàl 'J! can be. introduced. The fluid velocity is equal to V'l!. Conservation of mass leads to

the governing differential equation for 'I':

(2)

Two boundary conditions hold on the 'free surface ((x, y, t): the kinematic free surface condition,

( ± VW . V)(z - 'C(x y, t)) = O

on z = (

(3)

the dynamic free surface condition,

These conditions can be combined to:

onz=(

(5)

The boundary condition on the hull reads:

VI =

. on (B) (6)

where is the displacement vector of the .hull?s surface, and' ff is' the unit vector normal

to the surface pointing into the fluid domain. At infinity a radiation condition, ensuring that the energy flux carried by the disturbance due to the vessels motion propagates

away from the vessel, must be fulfilled.

The body boundary condition is linear itself, but it holds on the unknown instan-taneóus position of the. body. The boundary condition 'of the free surface is non-linear.

(7)

Moreover it holds on the instantaneous free surface Both boundary conditions will be linearized to a mean position.

The velocity potential 'I' is divided into a steady potential and an unsteady

po-tent jal ,, which is assumed tobe small when compared to 0 The steady potential is the solution to the steady wave problem This non-linear problem has been solved by Raven [3]. Nakos [4] approximates by the double body potential, thus neglect-ing some interaction of the unsteady potential with the steady waves The results of Nakos are still very good. Frequently is approximated by the undisturbed flOw (cF = Ux) This approximation is known to give somewhat poor results with respect

to heave-pitch coupling forces As the choice of cF does not make any essential difference

for the remainder of this report, we will proceed with the most simple one: cF = Ux. When terms of Order 2 are neglected, the free surface boundary conditions for

'

canj be linearized to:

gçoz + çott - 2Uço + U2ço

=0

on z = 0 (7)

The body boundary conditions can be written as:

V

=

+ rnk) on (B) (8)

for the unsteady potential, where

(iii,712,773)i =

=

=

=

m5,rn6)T

=

(0, Un3, rU

This linearization scheme is known as the Neumann-Kelvin linearization. The notation of the boundary condition on the body (equation 8) was introduced by Ogilvie and Tuck [5]. The terms llkThk account for the interaction between the unsteady and bases flow, and are usually referred tú as the rn-terms.

Now the boundary value problem has completely been fórmulated and linearized. This problem can he solved in either, the: frequency or the time domain.

(8)

3

The frequency domain formulation

When the. problem is solved in the frequency domain, all, unsteady motions, velocity

potentials, and forces are assumed to be. harmonic. They are 'related. to the time domain

by a Fourier transform. The unsteady potential ça is split up into separate potentials due to oscillatory motion in the six degrees of freedom (iii, , ), the potential of the incident waves, and a potential describing the diffraction of the incident waves This

decomposition of ça is cptured in. the following expression:

ça {[A(çaj

+ D) ±

kçak]e (9)'

where

A = the (complex) amplitude of the incident wave (p.j. = the incident wave potential

=

ee_c03ß8mn

wo

'where w w0 kU cos, k. = .w/g, and w0 is the incident

wave frequency as viewed from the stationary frame Ç°D = the diffracted wave potential

= the complex amplitude of oscilatory motion

inkthdirectión,k=1,...,6

Ç°k = the complex potential due to oscilatory motion

in kt direction, k =

Plugging (9) into the boundary conditions (7) and (8) and separately balancing each

mode, leads to the following, boundary value problems:

= O

-

w2çak'

- 2ïwU- +

U2 ÇO

=0

onz=0

71= iWilk ± 7k on (B)

Radiation Condition (10)

for k= 1,.. .. ,6., and

g

W2ça,D_2jWUp+U2a;2D

.0

onz=O

VçaD .il= Vça1 on (B)

(9)

for ÇOD. These boundary value problems can be solved by means of a rankine panel

method (see Moulijn [6] or Nakos [4]).

When the boundary vahe probleris have 'been solved, the: pressure distribution on

the hull.can. be calcuiated:.using Bernou11i.s equation:

on(B)

(12)

Linearizing and taylòrizing to the vessels mean position of this equation leads to:

p - pa = - pyz - p(çQt - U) - p(. V)(gz,)

on (.)

(13)

The forces acting on the vessel follow from:

= - ff(p - Pa) n. dS

j = 1,. .

. ,6 (14)

where the subscript j denotes the direction of forcing. The integral of the steady pressure

term in (13) is balanced by the gravitational force. The hydromechanical force in th

direction is written as:

.{([w2Ajk(w) - iwBk(w) Gkk

±

AX:(w))et}

j

1, . ..

6

(15)

where

- U.:-)nids

Bk

ff(iwk

-

U0;)midS

(C1, C2,C3)T

-off V(gz)ndS

(C4,C5,C6)T

pff (r® V)(gz)ndS

(10)

expresses the hydromechanicai reaction force in th direction (response) due to harmonic

oscillatory motion in kt direction (input) is defined as:

Hk(w) = w2Ak(w)

- iwBk(w) -

(16)

This complex harmonic response function will he related to the time domain formulation

(11)

4

Intermezzo: Impulse response functions

In the previous section the response (hydromechanical forces acting on the hull) of a linear system was expressed in terms of input variables (the ship motions and the incident waves). The system was a linear system because the boundary value problem

and the expression for the pressure were fully linearized. The characteristics of the linear

system were captured by complex harmonic response functions. The application of a harmonic response function is, however, restricted to the frequency domain. In some

cases we want to calculate the response of a system in the time domain, for instance if the input is not harmonic, or if non-linear external forces play an important role (one might think of non-linear mooring forces, or non-linear cushion dynamics ofa cushion

supported vessel).

The response of a linear system in the time domain can be expressed by means ofan

impulse response function. An impulse response function gives the quantified response of the system to a unit impulsive input at t = O.

The unit impulse function 8(t) is often defined as:

00

6(t) = O if t O , and

f 6(t) dt = 1

(17)

-00

Some times it is more convenient to define 8(t) as limto 6(t) = 8(t), where (see also Figure 1)

8(t) = O

if t [O, Lt] ,

and 8(t) =

if t E [O, ¿st] (18)

A physical example of an impulse is a hammer stroke: a very large force acting during a very short time interval. According to mathematicians (Papoulis [7]) these definitions are not accurate, but for us they wifi suffice.

(12)

Figure 2: The response of a damped mass-spring system to an impulsive force The impulse response function h(t) gives the response of a system to an impulsive input. This could be the motion of a mass after an impulsive force (hammer stroke). If the system is stable the impulse response function approaches zero after some time. Figure 2 presents the response of a damped mass-spring system to an impulsive force.

Because of the linearity of the system the response to several impulses (acting at different moments) can be superimposed. This means that if

h(t) is

the response to

8(t), the response

r(t) to a continuous input x(t) can be approximated by (see also

Figure 3):

n

r(t)

t) x(t) i.t

(19)

i=O

or in the limit for i.t + O:

r(t)

=f

h(tt')x(t')dt'

(20)

The integral is called a convolution integral. It presents "the memory" of the system.

(13)

When t - t' is substituted by 'r, and we realize that h(t) = O if .t < O, the convolution

integral can he written as:

00

r(t).

f h(r)'x(t

-T)

dr

(21)

-00

The impulse response function is related to the cómplex harmonic response function by a Fourier transform. Assume the input x(t) to be harmonic:

x(t) =

(22)

Then r(t) can be written as:

= H(w)x(t), (23)

or as:.

00 00

r(t)

_fh(r)e(t_T)dr

=f h(r)eTdr . x(t)

(24)

When (23) and (24') are compared H(w) and: h(t). appear to be a Fourier transform pair:

H(w)

=

f h(t)etdt

h(t)

=

_f H(w)edw

(25)

This Fourier transform. exists when the following integral exists.

f Ih(t)I dt (26)

For stable systems it generally does.

Because the real part of H(w) 'is an even friction of w, and the imaginary part of H(w) is an odd function of w, h(t) can also be expressed as:

(14)

These relations imply a relationship between the real and imaginary part of.H(a). This

relationship is given by the "Kramers-Kronig relationships" or "Hubert transforms" (see

for instance Papoulis [7j:.

= _ìiz

=

ì

(29,)

These relations can be used to express {H(w)} in £.'{H(w)} or vice versa.

A special casearises when the system has "no memory"; the system.on1y. responses. during the impulsiveinput, so hEt) = A S (t):

00

H(w)= J AS.(t)eT.td'

=.

(30)

and

r(t) =J AS(r)x(

- r) dr = A. x(t)

(31)

H(w) appears to .be constant and real. The convolution integral vanishes, and H can

be applied directly in the time domain.

A more elabôrate discussion on this subject can be found in Price and Bishop [8] or

(15)

5

The time domain formulation

In this section. the hydrómechanial reactiOn force will be expressed. in a time domain solution of the boundary: value problem of, Section. 2. The hydromechanical excitation

force, which is the force due to the incident waves and the diffraction of incident waves,

is not concerned in this section.

The linear boundary value problem from SectIon 2 can be solved directly in the time domain. However, this' is time consuming and still in' a development stage. We chose to exploit the linearity Of the problem once more; we will follow the approach

of Cummins [1] In this approach the potential is expressed in terms of a convolution

integrai over the potential caused by an impulsive displacement Cummins first discusses the more simple case' without forward speed'. We will go off the deep end' at once.

The, linearized boundary value problemas formulated. inSection 2 is recapitulated::,

gçp +'t' - 2Ucpt .+U2çO

=

....

Von= >(nk±c'7kmk)

k=1 '

Radiation' Condition (32),

A solution of the following form is proposed:

(x, y, Z, t) = {Oflk(x, y, Z)7k,(t)+ 9nk(X,Y, Z»lk(t) +

k=i

fi9,(x, y, z,, t - T)7k(T) dr +

f9mk(X, Y z, t - r)llk(r) di-} ' (33)

-co

The potentials Ok(x,y,z), Omk(X,'y,Z), i9, the boundary conditions:

on z = O

on (B)

VOkfmk

VOmkilmk

on(E).

°hkO

°mk°

onz='O'

(16)

t99nk. 829nk a219k

2'2nk

g

+

2U

=0

onz=0

ôz 9x

ô19nk a2l9rnk 2 19mk

g

+

2U

+U

=0

onz=0

i9z

at

8xOt Ox2

Radiation Cóndition

And t9k(x,y, z, t) andl9mk(X,y,z, t) have to fulfill the initial conditions:

19nk0

mk0

fort<0

Ol9mk

-

00mk

for t .4-0

Fiom substitution oftheseconditions in (33) it follòws that will fulfi11 its boundary

conditions (32);

The.physicai is çhosçn to be

the unit impulse function 5(t):

(Pk(X, y, z,t) = Oflk(x, y, z)8(t) ± 9mk(X,Y, z) + 19 k(X, y, z, t) +

J19mk(XY,Z,t_T)dT

fort 0

(34) The potential °nk gives the direct response of the water to unit velocity of the ship in kt direction without free surface effects. The potential °mk gives the direct response of the

water to unit displacement in kth direction without free surface effects. The boundary

condition for O and k on z = O is fulfilled when (B) is reflected in the surface z = 0,

and the body boundary conditión is taken negative on this refiectión.

The potential 19nk describes t:he decaying disturbance of the free surface after a unit

impulsive velocity in kth direction. The potential t9mk describes the decaying disturbance

of the free surface after a unit impulsive displacement in kth direction.

The potentials 9nk and k ensure that the body boundary condition is fulfilled. The

potentials 19nk and l9mk take care of the free, surface boundary condition. The potentials

having subscript m are associated with the rn-terms of the bo4y boundary condition for ço. They account for the interaction between the unsteady and the bases flow. If the ship has no forward speed the rn-terms are zero, so then these potentials vanish.

Because we want to express the hydromechanical' fòrces acting on the vessels hull in terms of one convolution integral containing only we will integrate the second convolution integral of (33) by parts First emk(x, y, z, t - 'r') is defined as

y, z, t - y) =_

(17)

Now (x, y, z, t) can be. written as:

(p(x,y,,z,t). =

{9flk(x,y,z)k(t) +

[Omk(X,yj z) + emk(:7 y, z, O)]?lk(t) +

f[i9(x, y., zt r) - Omk(X, y, z,t - r):jik(r) dr}

(36)

-00

The potentials Omk(X y,z) + 9mk(X, y, z, O) are a correction to the steady pótential för the vessel having a constant unit displacement (1k)

As in Section 3 the pressure is written as:

p - p0 = -

pgz - p(çog - U) - p(. V):(gz,)

on (.)

(37)

The unsteady part of...thepressure (.the..second and third term.) is now expressed. interms.: of the solutiom as .propose& iequation. (33)::: .

Pu. =p

{ O,(x, y,z)ijk(t) +

k=1

o

(Omic.(x, y, z)

U-9k(x., y, Z))llk(t) +

_U(Omk(X,

y, z) + emk(x, y, z,0))ijk(.t) +

f[(n(x, y, z, t - r) - ek(X,

y., z,

t - r)

U-(9flk(x,y,z,.tr) 9mk(x,y,z,tT))]k(r)dr

+

(dk.V)(gz)} (38)

The unsteady forces acting on the vessel follow from:

1i=_ffPunïdS

j=i,....,6

(39)

so can be written as:

J.

(18)

where ff1 Ç'! k'j1,L'j2, j3)

=

Ç'! j6)

ffV(gz)ndS

pfj(® V)(gz)mdS

In literature hk(t) is frequently called retardation function. ak.

jO(x, y,,

z)n dS

Jj(Omk(X, y,, z) - y,z))nj dS

Cjk

=

jfUL(ek(x,

y, z) ± emk(x, y, Z,

o))nds +

Ck

B

a

-

r)

_pJf[ä(19nk(X,y,Z,t - T) - emk(x,Y,z,t -

r)) +

U(flk(,y,t

-

r)

emk(x,y,z,t -

r))]ndS

(19)

6

Relationship of the frequency and time domain

forces

In the previous: sections the boundary value problem 'for' a ship sailing in waves was

formulated and linearized. Solution forms for the boundary value problem were proposed

in both the frequency and time domain Expressions for the hydromecharncal forces acting on the vessels hull were found in hoth the frequency or the time domain. In the

fourth sectiOn the. impulse response function was introduced. Impulse response functions were used extensively 'in 'the time dòmain formulation.

The goal. of this section is to link the expressions for the. hydromechanical forces in

the frequency and the time domain T.his as done by Ogi'lvie '[2]. The.boundary. value

problem can be 'solved in the frequency. domain by means' of a 3D panel. method or a

more simplified (and faster) solution.scheme like the strip theory. Time -domain solvers

are: still .j a development stage,. and the computational burden is very large. W'hen

the link. between, the frequency and the time'.domain- is known, time domain'fi1ters can

be constructed from the frequency domain information, so we do not need to solve the boundary value problem in the time domain.

The expressions in .the frequency and the time domain are related. by a Fourier transform. in Section 4 the relationship of a complex harmonic response function and

an impulse response function was deri'ved:

00

H(w)

f h(t)etdt

h(t)

=

H(w) tdw (41)

For the calculation of the impulse response function H(w) must be known on the entire

frequency range.

At the end' of Section 3 we found H(w) to be:'

Hk(w) = w2Ak(w) iwBk(w)

Ck

(42)

When w * oo,

H3k(w) * oo .too, so the inverse Fourier transform does not. exist.

In Section 4 we found that a' constant and real harmonic response function can be applied directly in the time domain (the system does not possess any "memory"). The

(20)

where

Hk(w) = iw(Ak(w)

-

Ak(oo)) + (Bk(w)

-

Bk(oo)) (44)

Now Hk appears to be at

last

0(1/w2) if w

-

00.

However, Ak(w) is 0(11w2) if w * O, so the the inverse Fourier transform still does not exist. This is caused by the definition of the coefficients Ak, Bk and Gk:

When w tends to zero, çok(x, y, z, w) approaches a real non-zero function. k(X, y, z, O)

is a steady potential giving a correction to 4 for the vessel having a constant unit displacement (i = 1). Actually çok(x, y, z, O,) is equal to Omk(X, y, z) + emk(x, ., ,O)

from Section 5.. Even though k(X, y, z, O) can be small it causes to be infinite

when w

=

O. This, steady part of Ç°k contributes to the coefficient Ak while it should contribute to G'k. When Ak, Bk. and, are defined as:.

Ak(w) ff[iw.cok.(w) -, u---(cok(w) çok(0))]nj dS.

Bk(w)

=

_!-ff[iwçok(w')

U--.(çok(w) çok(0))]n dS

¡1*

ffU-(0) n'dS ± C3k

is 0(1) when w -+ O. Note that Bk(w)

=

Bk(w) because k(0) is a real function.

Now Fj can be written as:

=

{Ak(00)k.(t) + Bjk(00)hik(t) + CJk(t) + Hk(w)k(t)'} (45) where Hlk(w) = iw(A;,(w) Ak(oo)) + (Bk(w) B;k(00)) (46) ASk. =

Bjk=

(.C1,C2,C3)T =

if,

I-' Ç1 \T LLij4,'-'j5 '-'j6')

-off V(gz)ndS

_pff(® V)(gz)ndS

U)ndS

-

f(iwwk

U?)njdS

(21)

This form of F can be transformed directly to the time domain. It will get exactly the same 'appearance as the time domain expression för Fj which was derived in Section 5:

-

+f h(t-

Y)1k.(r)dr}

j=1,...,6

(47) where ak

=

Ak(oo) b,,

=

B'k(oo) Cjk

=

Ck. hk(t) =

-f Hk(w)e'dw.

In this expression 'ij. is not;necessarily harmonic.

The coefficients ak and bk can be calculated by sòlving the potentials °nk and °rnk

(see Sections 5). The coefficient. jk = can be, calculated most easily in the frequency domain by solving (x., y, ,'O).. The impulse respcnse function hk(.t) can be found from the inverse Fourier transform of H7k'(w).. Therefore H7,(w) must be known on the .entir.e frequency range. The limit values for w -+ O can be approximated by a value very close

to zero. The frequency range from O tosome' frequency w cani be calculated by solving the potentials pk. for a number of '.frequencies 'The asymptotic behavior of 11(w) or:

w

+00

is' not known. Van Oortmersen [9] gives some analytical results for zero. fora±d speed. He also suggests. a method .of triai and error:

In practice hk (t) is calculated from an inverse cosine Fo.irier transform of the real part of Hk(w)' (see equation (27)). If the sine Fourier transform of the thus calculated

hk(t)

fits el1 to the irnginary. .part of Hk(w), the high frequency extrapolation of

{Hk(w).} is accurate. Fortunately hk(t) appears to be not very sensitive to the high frequency extrap'1ation of {Hk(w)} as long as w* is large enough.

(22)

7

Concluding remarks

The boundary value problem of a vessel sailing ata constant mean forward' speed in a seaway has been, formulated and linearized.. Solution forms for this boundary value problem have been proposed for eIther the frequency or the time domain, The hy-dromechanical reaction force was expressed in these slutions. In the fourth section some theory of the 'response of linear systems has been treated. The relationship of the time and frequency domain was derived. Finally the frequency and time domain

expression for the hydromechanical reaction force were linked.

It it possible to solve the shipmotions in the time domain without solving the lin-earized boundary value problem in the time domain. The time domain expression for

the hydromechanical reaction force has the following form:

= -

{ajk(t)+ bk7k(t) + ck71k(t) + f h(t - T)k(T) dr}

-00

j=1,...,6

(48)

The frequency and time independent coefficients aJk, b3k and Cjk can be calculated

directly, while. the impulse response function (or retardation finction) hk1(t) follòws

from the inverse Fourier transform of the frequency dependent part of the' frequency domain response.

(23)

References

W.. E. 'Cummins. The impulse-response function and ship motions. Schiffstechnik,

9.(47):101i09, 1962..

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