**Journal of Sound and Vibration (1979) 62(3), 353-363 **

AN INVESTIGATION INTO THE LINEAR THEORY OF SHIP RESPONSE TO WAVES

R. E. D. BISHOP AND W. **G. **PRICE

**Department of Mechanical Engineering, University College London, London WClE 7JE, England ****(Received 29 July 1978) **

The response of an elastic ship to waves is examined in general terms by using the theory of linear non-conservative systems which suffer sinusoidal excitation. This is made possible by the conventional use of constant hydrodynamic coefficients (which can only be evaluated once the encounter frequency has been specified). A more general approach is formulated which does not depend on such coefficients and is not restricted to use in sinusoidal motions. Ship responses are expressed in modal forms in which the nature of resonance is made self-evident.

1. INTRODUCTION

Practical problems of ship motion and hull stressing in waves are becoming recognized as being due to resonances. The observable phenomena in a random sea are in the nature of narrow band responses. Any theoretical treatment of the problem requires that suitable co-ordinates be employed and, unless empirical (and possibly quite unwarranted) assump- tions are to be made, the only dependable set so far used have been the principal co-ordinates of the dry hull. The resonances have thus involved coupling.

While this coupling is in no sense a serious drawback, many discussers of work in this
field plainly feel that each resonance should be associated with a single co-ordinate. If
such were possible, the “principal co-ordinates” associated with the * wet *modes should be
used, so to speak. Although the idea is unquestionably an attractive one, the question of
what this implies in dynamical terms has remained obscure. It is necessary to discover, for
instance, what sort of modal responses are performed. The purpose of this paper is to
examine the matter.

A known technique in the theory of non-conservative linear systems is adapted to give the response to sinusoidal excitation in a new form. The importance of the new result rests not merely on the technical significance of the general problem (in the theory of machines, aeronautics, vehicle dynamics, as well as in ship theory), but also on ,the familiarity of the form that the result takes. In particular, it is possible to identify “magnification factors”.

In the theory referred to, the equations of motion only have meaning if the excitation is sinusoidal. This is because the familiar notion of “fluid derivatives” is employed (though to be sure frequency dependence of those derivatives is admitted). The theory has therefore to be extended if it is to apply to transient excitation-as would be demanded, for instance, by ship “slamming’‘-and so a general solution is found for this latter case.

* 2. *THE EQUATION OF MOTION

It is convenient to start in quite general terms without special reference to ships. The equation of motion of a linear non-conservative system subject to sinusoidal excitation is

**Aij + & + Cq = @e’“‘. ** _{(1) }

If n is the number of degrees of freedom, the n x 1 vector q represents the response at the * n *
generalized co-ordinates while the n x 1 vector Cp gives the corresponding amplitudes of
generalized excitation. The IZ x n matrices A, B and C are all real but, with a non-conser-
vative system, they possess neither of the familiar features of symmetry and positive
definiteness. A steady state solution of equation (1) has been given by Fawzy and Bishop
(see references [l-3]) but we shall make certain changes in the presentation and then show
how the solution can be thrown into an alternative and more revealing form.

3. FREE VIBRATION

If the trial solution q = de’* is introduced into the homogeneous equation Aq+&+Cq=O

it is found that A can take one of the eigenvalues A,, I,, . . . , A,, given by

IAL + BIZ + C( = 0. _{(2) }

Each root 1, is associated with an eigenvector &) such that

(L,2A) + Q3 + C)a”’ = 0. _{(3) }

Each root I, has, in addition, an associated eigenrow n“’ which satisfies the equation
d”(;i,ZA + &B + C) = 0 (r = 1,2,. . . ,2n). _{(4) }
As one would expect, the sets of *2n *eigenvectors &) and 2n eigenrows A(‘) are related to
each other by a form of orthogonality. Fawzy and Bishop have shown (see reference [l])
that, provided I, # 1, for I # s,

z@)[@, + I,)A + B]a’*’ = 0, @(cl R.A - C)a”’ = 0. _{* L } _{(51 }
How the eigenvectors and eigenrows shall be normalized is a matter of choice. In a
private communication, Professor Dr Eng. S. Miwa [4] of Aoyama-Gakuin University,
Tokyo, has suggested that a slight modification of the choice made by Fawzy and Bishop
[l] confers certain advantages and it is his suggestion that is developed henceforth. In
accordance with Miwa’s proposal, let

~“‘(23, A _{r } _{+ B)u”’ = l/I. I } (6)

and so

#) (A2A - C)&’ = _{r } 1. _{(7) }

Let

, A = diag@,, i2,. , A2,J.

We now find that the orthogonality relations reduce to

AllAZ + LlAZA + lZBZ = A-‘, *AllAr;A * *- LlCZ = I, *

where I is the unit matrix of order *2n. *

(839) When they are normalized in this way the eigenvectors and eigenrows satisfy an important relationship which is comparable with one found previously by Fawzy and Bishop [2]

**THEORY OF SHIP RESPONSE TO WAVES ** 355
and which will now be derived. The matrices A, Z and iZ may be partitioned:

A= A, 0

### [ 1

### 0 4

### )

### z

### =

### [zJCb],

### n

### =

### [3

### lz,

### =tl

### '

where the submatrices are all square of order n x n and may be assumed to be non-singular. If these matrices are substituted into equation (9) it is found that

*A,ll,A&A, * *- l&C& = I, AJl,AZ,A, * *- ll,CZb = I, *
*A&,AZ,A, * *- ll,C& = 0, A&lbAZaA, - ll,CZ,, = 0. *

When the first and third of these equations are postmultiplied by ll, and lZb respectively and added, their sum is

*A,lZ,A(ZaAJl,, * *+ &A&,) * *- l&C&_& * *+ Z&l,) = !Za. *

Similarly the second and fourth give

*A,lZ,A(Z,A,ZT, * *+ ZaAJtJ * *- lZ,C(X;,lI, + I;&) * *= lib *

When these last two equations are premultiplied by ll; ’ and n; ’ respectively and sub- tracted, they produce the result

*(l?,‘AJ?, * *- lI, ‘AJZb)A(ZaA,ll,, + ZbAbLJb) = 0. *

*This suggests that A is null, or Hi ‘AJZa = Ll; ‘A,,Hb or ZaA,lla + ZbAbLfb = 0. But *
nullity of A is a trivial case and the second possibility is untenable because the partitioning
*of ll and A is arbitrary. Therefore the third possibility is true and so *

ZAll *= 0. * *(10) *

If this last result is introduced into equation (9) when the latter has been postmultiplied
*by Ail, it gives *

*AllAZA’ll * *= All. *

*Therefore either AlLlEA * *= I, in which case ZZCZ = 0 so the result is not tenable, or *
*AZA’ll * *= I. * _{(11) }

4. FORCED VIBRATION

The general steady state solution of equation (1) may be derived by adapting the method shown by Fawzy and Bishop [l]. We shall merely check by substitution that the result is

*where Jz = (io1 - A)-‘. *
Differentiation shows that

*4 = ioB2AlZ9eiat * *= X[(ioI - A)Ja + Ad2]All@eimr *
*= Z(I + A12)AlZ@eioz = ZAJZAIZ@eia. *

Further,

4 = -02ZfiAfl@ei”* *= E[(i2021 - A’) + A2]12AlZ@eiot *

*= Z[(ioI + A)(ioI - A)f2 + A’Q]AZZ@e’“* = I;[(ioI + A)1 + A2f2]AlZ@ei”* *
*= (ZA2L! + ZA2t2An)@eimt. *

356

Consider the matrix lZ(Aij + & + Cq). On substituting the expressions for q. in and 4 we find that it is equal to

(llAE4 + IlAZA’R + ZZBZM2 + lZCZQ)AZl!@e’“’

*= (nAZA + nAI;A% * *+ ZIBZAS1+ AZlAZAQ - It2)AlMkiat *

*= [HAZA + (AllAX + t7AZA + nBZ)AR * *- U2]AlZ@eiwt *
*= nAxA2nqiei~t * *= n@eio*. *

That is to say, the solution (12) satisfies the equation of motion and is a valid particular integral.

Since the matrix I2 is diagonal it is readily inverted and the rth element in the diagonal of the inverse is l/(io - IZJ. It follows that the response q can be expressed in the alternative form

*(13) *

*The matrix &r(‘) is square of order n x n and it is identified with the rth eigenvalue ,I,.. *
When 5 is complex, it too is complcs in general. The matrix in square brackets in equation
*(13) is the n x n receptance matrix cr and we now see that each element of it (i.e., each *
receptance) consists of 2n terms added together.

5. LIGHTLY DAMPED SYSTEMS

Complex eigenvalues of a system occur in conjugate pairs because the matrices A, B and C
are all real. We shall assume henceforth that all roots are of this sort-as they normally will
be if the possibility of serious resonances exists. For a conjugate pair of roots, say ,I, =
*p,* iv,@ = 1,2 ,..., n), a conjugate pair of products &c(~) = Us) + iv’“’ (s = 1,2, . . . , n) *
may be formed.

The series summation of equation (13) can now be carried out for conjugate pairs of eigenvalues: that is,

a= f
*=l
=,t,
J 6(r’11w _{” }
r=
io - &

**4 **

(ps + ivJ(U’“) + iv(“)) + (pLs - ivJ(U(“’ - iv’“)) s=l io - (CL, + iv,) io - (pLs - ivJ I 2[ - a,ZU(“) + io(~~U’“’ - v,V’“))]

Qf - o2 - 2ip,w ’

where Qf = pz + vi = ,I A,/‘. It follows that

I,

**a= **

**41 **

s= I

-2U(s)+i~[~U(s)-~V(s)]}~~[( -$]+y}j, (14)

where i, = -~,lsL,.

The numerator of the sth term in equation (14) may be written as

R(@ + i(o/Q,)!P or - 2{U’“’ + i(o/GJ [SsU(‘) + J(1 - [~)V’“‘]},

**THEORY OF SHIP RFS’ONSE TO WAVES **

may be expressed in polar form and so the expression for a becomes

357

where

0, = tan- ‘{(X,~/Q,)/[l - (cWJ21). The complex square matrix

R(“) + i(w/l(l )S@) _{S }

cannot, of itself, display any special behaviour near resonance. As w/sZS merely passes through the value unity, By contrast the multiplier

(16)

o + 51,, the factor

G, = e-iBS/J([l - 02/Q~]” + 4~~o”/s2,‘} (17)

is very well known in linear dynamics and provides all the familiar features of resonance. In particular G, accounts for the characteristic “resonance loops” in polar receptance diagrams.

6. APPLICATION TO SHIPS

A ship under way in sinusoidal waves has an equation of motion of the type (1). The system matrices, denoted by d, .@, Cg are all sums of two matrices, one of structural origin and one of hydrodynamic origin.

The system matrices are determined by the ship’s structural form and properties, by the hydrodynamic features, by the structural and hydrodynamic theories used, by the operating conditions of speed and heading and by the choice of generalized co-ordinates q. It has proved particularly useful to identify q in equation (1) with the vector of principal co- ordinates of the dry hull; this we shall do here, adopting the symbol p. Before proceeding, however, we may observe that the foregoing theory would permit fresh types of co-ordinates to be defined, were it so desired. Although there is no obstacle in theory, the practicalities of adopting a system whereby resonance in a wet mode corresponds to resonance at a single co-ordinate are by no means clear.

In ship dynamics it is necessary to identify w with the encounter frequency we and it is convenient to write the excitation in the form Se-‘“*‘, where the n x 1 vector B is dependent on w,. The equation of motion for a ship is therefore

*d(0,yfi + 93(w,jp + V(W,)P = 9(0,)e-i? *

*(18) *

where, in general, the real n x n system matrices &, S?, ‘S are we-dependent for a given speed and heading in a long crested sinusoidal sea of unit amplitude of elevation. The (n x 1) matrix B is complex and it, too, depends on w, and on the operating conditions. The theoretical background of equation (18) has been summarized by Bishop, Price and Tam [S].

Equation (18) may be solved for the steady state response by analogy with the previous simpler case. If the equation is written as Dp = EemiaJ, then

P = D-l~e-im.f E ase-iOOt.

358 **R. E. D. BISHOP AND W. G. PRICE **

pitch and n - 1 symmetric distortion modes, then

*The real square n x n matrices @(o,) * and .Y’“‘(w,) (for s = 0, 1,2, . . n) depend on the
system matrices &(w,), a(~,) and U(w,). In other words, the eigenvectors u”) and eigenrows
n’“’ have to be determined for a given encounter frequency o, in the homogeneous equation

dbe)p +

### ab,jp +

vh,)p = 0. (20)But for a practical ship the dependence of the system matrices on we is significant only for small values of oe (i.e., in the region of 52, and fil only). The encounter frequency dependence is virtually nil for all the frequencies of resonant distortions.

7. THE SYSTEM MATRICES d(w,). I(cuc.), V?(o),,)

Although the equation of motion (18) has found very wide acceptance in the literature, it suffers from certain defects. In the first place the system matrices a(c),), a(~,) and %?(o~) are not known until the frequency of the excitation is specified. (This drawback is commonly overcome by assuming that the system matrices correspond to an infinite en- counter frequency.) Secondly there is no logical way of dealing with non-sinusoidal excita- tion, such as occurs when a ship slams. Thirdly there is a logical inconsistency in assuming that the homogeneous equation (20) governs free motion in the absence of excitation; free motion would require a process of successive approximation to determine oe under the assumption of “nearly sinusoidal” motion.

Evidently a fresh approach to the theory is desirable if transient motions are to be ad- mitted. This will be developed in the following section and it will be based on a device that will be examined first in the light of sinusoidal excitation.

Cummins [6] and, in a series of papers, Bishop, Burcher and Price [7-91 have expressed
the equation of motion of a rigid ship in water in terms of convolution integrals. If the
technique is adapted to the present problem of hydroelastic ship response to waves, it
suggests that
f
ab + bb + cp +
’ s
h(z)p(t - z) dr = E(t). (21)
**0 **

*The n x n system matrices a, b and c relate to the dry hull, being those of inertia, damping *
and stiffness respectively. The quantity p(t) describes the displacement and distortion of the
*hull, being the n x 1 matrix of principal co-ordinates, so that a and c are diagonal and b *
is merely symmetric, while all three system matrices are positive definite or semi-definite.
*The n x n matrix h(r) is that of “impulsive receptances” or “unit impulse responses” such *
that h(r) = 0 for z < 0 and r > E. The object of employing this equation is to make provision
for “fluid memory” effects while retaining linearity of the equation of motion.

Consider a sinusoidal forcing function S(t) = ge-io*t giving a steady state solution p(t) = pe-ioat. Equation (21) may be expressed as

**THEORY OF SHIP R@SPONSE TO WAVES ** 359

The integral term will be recognized as the inverse Fourier transform

for which H(Q = s m h(+Per dz, -CC h(t) = & f 03

H(o,)e- iru=r do,. -C0

The function H(o,) is complex and can be written as H(o,) = HR(w,) + iH’(o,). Equa- tion (22) can therefore be written as

[ -c$(a - HR(~,)/c$) - iw,(b - H’(o,)/o,) + c&e-‘“a’ = Eeeicoet, or

d(u,)ii(t) +

### 9qo,jp(t)

### +

**Q(t) = Eeeimef,**(23) where the encounter frequency dependent system matrices are

d(u,) **= a - ** HR(o,)/oz, W(w,) = b - H’(o,)/w,.

Equation (23) is the equation we used previously, but we now have an alternative inter- pretation of the system matrices. It is still necessary that both the excitation and response vary with the encounter frequency 0,.

8. TRANSIENT EXCITATION

Equation (21) provides a safe starting point for developing a linear theory of transient
motion. Instead of the Fourier transform, however, we now apply the Laplace. transform.
The equation is first multiplied by e-“’ and then integrated with respect to t between the
limits * t = 0 *and t = co. This gives

m m

a

**s **

e-&ji dt + b
0

**s **

0
e-““pdt + eJ: e-“‘pdt + [: eCU{jih(r)p(i - r)dz}di

=

**s **

**om **

B(t)e-"'dt. (24)
To quote from the standard theory of Laplace transforms,

s

m e-&‘)dt = Ip(l) - po, f

m

e-&fi dt = J,‘p(n) - Ip, - po,

0 0

where p. and ‘p. are the values of p(t) and b(t) at t = 0 respectively and

p(l) = s

m p(t)e-“‘dt

0

is the Laplace transform of p(t). Again from standard theory, m

**s **

**IS **

t
e -At
0 0

If these results are substituted into equation (24) it is found that

360 where

R. E. D. BISHOP AND W. G. PRICE

F&) = a& + AP,) + bp,, a(n) = s

om S(t)e-A’ dt.

Thus we may express the equation of motion in the algebraic form D(&o) = F,(A) + 3(A).

That is to say

Ip(A) = D- “(l)[F,(l) + &4.)-j, where I is the unit matrix of order n and the inverse matrix is

D-‘(A) = adj D(A)/det D(A). (25)

To find the solution p(t) of the equations of motion it is necessary to invert p(A). This may be accomplished by means of the general result

1 s

**y+im **

**p(t) ** **= 2ni ** **y_im ** e”‘@(l) dA, (26)

where the integration is performed along the straight line joining the points Y - ice and Y + ice and Y is a number sb chosen that all the poles of e^‘Ip(A) lie to the left of the line. With this general solution we are now in a position to examine the special cases that we previously discussed, viz. free motion and steady forced sinusoidal motion.

9. FREE MOTION

If the trial solution p(t) = pe”’ is introduced into the homogeneous equation

s
f
aii(t) + bp(t) + cp(t) + h(z)p(t - z)dz = 0,
**0 **
it is found that
or
[A2a + Ab + c + h(A)]p = 0.

The eigenvalues I,, A,, . . . , I,, (say) are the roots of the characteristic equation

That is to say

lA2a + Lb + c + h(A)z)l = 0. (27)

det D(A) = (A - A,)@ - A,). . . (A - AZ,“) = 0.

The number ofroots 2m depends upon the form of the function h(l). Ifh(l) = 0, then m = n and the number of eigenvalues is the same as that found previously. But notice particularly that this is not the case discussed previously since the system matrices are a, b and c rather than &(o,), &?(o,) and %‘(oJ.

**THJZORY OF SHIP RESPONSE TO WAVES ** 361
T # s, then,

**z@)[Afa + &b + c + b(A,)] = 0, ** **[AFa + I,b + c + b(2,)]a”’ = 0. **

**If the first result is postmultiplied by a”) and the second is premultiplied by x(‘), and they **

are subtracted it is found that, for I, # A,,

### w -

### h(4)

64, + **&)a + b + ** **lr _ n, **

### 1

&) _ 0**- **

### .

_{(284 }

Again if the first result is postmultiplied by Q(“) and the second premultiplied by 1,rP and they-are subtracted it is found that

76”) **I&a - c + &h(A) - 4b(4) **

**A, - As **

### 1

**b(v) _ o **

**- **

### *

Ifs = I, however, we may write as discussed previously

### PW

**n(‘)[2A,a + b + ~%(l2,)/~~,]a(” = l/l,, ** (294

n’“[+ - c + ~@h(A,)/c%,) - **h@,)]a”’ = 1. ** **(2W **

Equations (28) and (29) are the orthogonality relations for the new modes.

10. SINUSOIDAL EXCITATION

If the excitation is of the form B(t) = Eemio”, it follows that S(J) =

s

m Ee-iG’e-“‘e-hddt = A.

**0 ** **e **

The Laplace transform of the equation of motion is thus

whence

**[A’a + Ab + c + h@)]p(l) = F,(A) + S/(A + io,), **

**IdA) = [A2a + Ab + c + h(l)]- ** ’ (’ + ipy:f’ + ’ .
I?
That is to say,

**adj **D(n)Fo@)elt dr2

**1 **

**s **

**y+io, ** _{adj D(L)Ee”’ dr2 }

+ 2ni **y-io,(' ** **- ** **A,)(2 ** **- ** **i,)...(A ** **- ** **l.,,)(J. ** **+ ** io,)’

By applying the Heaviside expansion theorem, as discussed by Frazer, Duncan and Collar [lo], or by using the residue theorem we find that

denominator. If the substitution 2 = Ar is then made in the reduced expression, the response reduces to

### Pm =

adj D( - io,)Ze - ‘Q_{det D( - iWe) }

r=l where c = (1 -

### A,) a4 W

### z

r det D(L) F (n) +**0**

**(A**

**+**

**ime)**

**A=I,’**

in which ill # -iw,.

Since the ship-wave system is stable, all the roots of il,, AZ,. . . , A,,,, of the characteristic equation have negative real parts. Therefore all the terms associated with the constants CI are zero since eAr’ + 0 as t + co and the steady state periodic solution is

p(t) = adj D( - iw,)ZeiUe’/det D( - iw,). (30) Since

det DC-iw,) = (A, + io,)(n, + iw,) . . . (A,~ + ime),

equation (30) may be expanded in partial fractions to produce a solution of the form

(31)

where the A, are the appropriate complex constants. On physical grounds it is to be ex- pected that the roots 1,, A,, . . . , AZ,,, will occur in conjugate pairs and so the modal sum may be recast in the same manner as before.

It will be seen that, if h(2) = 0, the number of roots reduces to *2n *and equation (31)
reduces to the form of equation (13) with A, = ~,u%(‘).

**Il. CONCLUSIONS **

Significant responses of a ship to waves-of bodily motion or distortion-are observed as narrow band random processes. If it is assumed that linear theory can represent the responses adequately, their study reduces to that of resonant response to sinusoidal waves. In such a theory fluid actions associated with the response are usually specified in terms of hydrodynamic coefficients in the equations of motion. These coefficients depend on en- counter frequency in a complicated way.

By following this established line of approach, it is shown how responses can be expressed in modal form. The modes concerned are “wet modes”, though the responses in them are expressed in terms ofthe principal co-ordinates of the dry hull. Although the modal responses are of a straightforward form familiar in the theory of linear non-conservative systems, they do not suggest that transformation to a fresh set of generalized co-ordinates-“wet principal co-ordinates” so to speak-would be of much advantage. Orthogonality con- ditions that the wet modes satisfy have been established.

Encounter frequency dependence of the hydrodynamic coefficients is a consequence of “fluid memory effects”. That is to say, it reflects the fact that the fluid actions at any instant associated with hull motions are not determined solely by the motions prevailing at that instant. Linearity of the theory can be preserved by presenting those fluid actions by con-

THEORY OF **SHIP RESPONSE TO WAVES ** _{363 }

volution integrals instead of the more familiar hydrodynamic equations. Instead of a set of second order equations with (encounter frequency determined) constant coefficients, one has then to deal with a set of integro-differential equations. The relationship that exists between these two approaches has been developed in terms of Fourier integrals.

The second approach, with use of convolution integrals, has the inherent advantage that equations of motion can be derived which govern all motions of the ship. It is no longer necessary to require sinusoidal excitation with a prescribed encounter frequency.

By using Laplace transform techniques, this second method is also developed and modal solutions are found, The number of the new wet modes is no longer necessarily equal to the number of dry modes admitted in the theory, but may be greater; the number depends on the nature of the unit impulse response functions used in the convolution integrals. Orthogonality relations are found for the new wet modes. So far as the authors are aware, however, there is no simple way of employing these modes directly in an analysis of forced motion; the difficulty lies in the form of the orthogonality equations, which contain the Laplace transforms of the unit impulse response functions. A modal solution can neverthe- less be found for sinusoidal excitation by using the Laplace transform method.

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3. I. FAWZY and R. E. D. BISHOP 1977 * Journal of Sound and Vibration 55,475-485. * On the nature
of resonance in non-conservative systems.

4. S. MIWA 1977 Private communication.

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* Meeting Paper W8. *A unified dynamic analysis of ship response to waves.

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**BISHOP, R.**K. BURCHER and W. G. PRJCE 1973

**Proceedings of the Royal Society, London*** A332, 25-35. *The uses of functional analysis in ship dynamics.

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* A332, 37-49. * Application of functional analysis to oscillatory ship model testing.

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113-128. Fifth annual Fairey lecture: On the linear representation of fluid forces and moments in unsteady flow.

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