331

Motions of a Floating Elastic Plate in Waves

by Mikio Takaki * , Member Xiechong Gu * * , Member

Summary

This paper deals with three-dimensional responses of a very large floating structure in head and

quartering seas. First we estimate three-dimensional dry-eigenmodes of a completely free plate by employing a finite element method. Then, hydrodynamic pressures exerted on the plate are evaluated by combining a pressure distribution method and the dry-eigenmodes. Wave-induced motions of the

plate are finally determined by solving a set of equations of wave-induced motion in the principal coordinates. Present modal analysis method is verified through comparisons of calculated amplitudes of vertical motions and instantaneous shapes of a plate model in head seas with available measured data. They show reasonable agreements. Next, we apply this method to the large floating module of 300m x 60m, and investigate the motions of it and the disturbance waves around it in head and

quartering seas. Finally, the response amplitude operators of the first 10 modes in all heading waves with various wave lengths are shown.

1. Introduction

Recently, a number of studies have been made for

predicting the behavior of wave-induced motions of a

very large floating structure 1-9) . A mat-like floating

structure has been proposed to be a candidate of a

floating airport. This kind of structure has dimensions

of a few kilometers long, some hundred meters wide,

while its depth is around ten meters. Therefore, this

large length to depth ratio makes it very flexible, and

the natural frequencies of elastic vibrations are

significantly reduced. In addition to that, the incident

wave length becomes relatively short, and the elastic

vibrations are remarkably induced by wave excitations.

Thereby, the elasticity of this sort of structure also

must be taken into account. So far, this kind of study

has mainly dealt with the motions of a long body like a

beam except for the one due to Mamidipudi &

Webster 5 ) in head seas. They have proposed the method

of evaluating three-dimensional motions of the mat-

like floating structure by using the discretized panel

method. However, there are very few studies dealing

with the motions of the mat-like structure in quartering

seas.

In this paper, we develop the estimating method

dealing with three-dimensional motions of the mat-like

structure in quartering seas by using the modal method.

The present analysis method is verified through compar-

isons with available measured date). Then we apply

this method to the large floating module of 300m x 60m,

and investigate the motions of it and the disturbance

waves around it in head and quartering seas. Finally,

the response amplitude operators of each mode in all

heading waves with various wave lengths are shown.

2. Formulation

We hereafter focus our attention on a floating elastic

plate of which draft is very small comparing with its length and width. This allows us to employ a pressure distribution method 10 ) to evaluate the hydrodynamic

pressures on the plate. Thereby horizontal motions (surge, sway and yaw) are excluded in this study.

2. 1 Boundary value problem

A Cartesian coordinate system (x, y, z) with upward

z-axis is used, and the origin o is set on the center of the

plate as shown in Fig.1. A long-crested regular wave

with a small amplitude incomes from the direction of

angle x. We assume that the fluid is incompressible and

inviscid, and the flow is irrotational. The velocity

potential ƒ³ (x, y, z, t) exists in the fluid domain ƒ¶, and

can be expressed as the sum of the incident potential ƒÓƒ§,

the diffraction potential ƒÓD and the radiation potential

φR as follows

( 1 )

* Faculty of Engineering, Hiroshima University

* * Graduate School, Hiroshima University

Received 10th July 1996 Read at the Autumn meeting 14, 15th Nov. 1996

332 Journal of The Society of Naval Architects of Janan, Vol. 180

where

ω ; incident wave frequency,

k ; wave number (=ω2/g),

ζA ; incident wave amplitude,

ｇ ; gravitational acceleration.

We define pressure function p(x,y) and the vertical

motion ζ(x,y) without the term of time dependency as

follows

( 2 )

where ƒÏ is the fluid density, and Sh is the mean wetted

body surface at z = 0.

Potential function ƒÓ(x, y, z), which is ƒÓD or ƒÓR, is

governed by the three-dimensional Laplace equation

[L]. It has to satisfy a linearized free surface condition

[F], a hull surface condition [H], a bottom condition

[B], and a radiation condition [R]. They can be expres-

sed as follows ;

( 3 )

where Sf is the mean free surface z = 0, and subscript

denotes partial differentiation.

The radiation potential ƒÓR and the diffraction poten-

tial ƒÓD have to satisfy the following hull surface condi-

tions respectively

( 4 )

By introducing a Green function G( x, y, z; x', y', z')

which satisfies (3) except for [H],φ( x, y, z) can be

evaluated by the following integration

( 5 )

Substituting ( 5 ) into the hull surface condition[H]in ( 3 ), we get the following integral equation

( 6 )

where the Green function G( x, y, 0; ｘ', y', 0) is the same

as the one used by Yamashita10). We apply a constant panel method to solve this

equation for the evaluation of the pressures p(x, y). First, we divide Sh into panels, and the centroids of the

panels are selected as the collocation points. Then the pressure Pj on the j-th panel is set to be a constant. We can, therefore, get the following simultaneous equations

( 7 )

where C'iｊ is evaluated by the following integration

( 8 )

δij is the Kronecker delta

( 9 )

The equation (7) can be rewritten by using the tensor

expression in the form

( 10 )

We can get the pressure pj from (7) by using the

coeffictent Cij, the inverse of C'ij,as follows

( 11 )

We can get the corresponding pressure function PD and PR by substituting (4) into (11). Thereby we get the hydrodynamic pressure on i-th panel in the form

( 12 )

We can see from the above equation (12) that the

hydrodynamic pressures on the plate consist of two

parts. One is the wave pressures on the fixed rigid plate

and the other is the radiation pressure related to the

deformation of the plate. Once the matrix Co is calcu-

lated, the pressures can be determined without calculat-

ing the velocity potential.

2. 2 Eigenvalue problem

We employ a finite element method to obtain eigen-

modes of a completely free plate. By assuming that a

plate is orthotropic11), we have the following equation

( 13 )

where Mx, My and Mxy are bending moments and twist-

ing moment, w(x, y) the deflections and Dx, Dy, D1, Dxy

the rigidities of the plate. We divide the plate into the

rectangular elements with 4-nodes of which they are

three degrees of freedom of vertical displacement w and

the rotation Įx, Įy at each nodal point. The slope with

respect to x and y-direction are expressed in the form

( 14 )

( 15 )

A polynomial with 12 parameters due to Zienkiewicz 13) is chosen as an interpolation function of the

deflection of the element

Fig. 1 Coordinate system

Motions of a Floatinw Elastic Plate in Waves 333

( 16 )

A typical element with length 2a and width 2b has been depicted in Fig. 2 where the coefficients at (i= 1, 12) depend linearly on a nodal displacement vector Oe of an element in the form

( 17 )

The deflection w has the linear relationship with ƒÂe as

follows

( 18 )

where N is the known shape function which is a row

vector of 12 dimensions. By applying the principle of

virtual work to the element, a stiffness matrix Ke of

the element, a corresponding mass matrix Me, and a

nodal force vector fe which is equivalent to the constant

load q on the element can be derived as follows

( 19 )

( 20 )

( 21 )

Matrices in these expressions, L, K1, K2, K3, K4 and Mg

can be obtained from the refereces[12], [13]. Further-

more, ƒÏs represents mass per unit area of the plate. We

assemble all the element's mass matrix, stiffness matrix

and nodal force vector, and get the motion equation of

the plate as follows

( 22 )

where K and M are the stiffness matrix and the mass

matrix respectively, and ƒÂ is the nodal displacement

vector of the plate, and I is the nodal force vector due

to the piecewise constant load q. We can also get the

equation of the eigenvalues with respect to the

nontrivial solutions dj in the form

( 23 )

The natural frequency wj of j-th eigenmode dj; can be obtained. The second term in (23) shoud not be recog-

nized as the summation convention. We introduce a

matrix ăij

( 24 )

And we rewrite (23) with the use of the tensor expres-sion in the form

( 25 )

The eigenmodes are usually normalized with respect

to the mass matrix as follows

( 26 )

( 27 )

Some eigenmodes of an isotropic square plate are

evaluated by solving the above eigenvalue problem as

an example. Timoshenko12) has represented the natural

frequency wj of the j-th mode in the following non-

dimensional form

( 28 )

where a0 and D are the side length and the bending

rigidity of the plate respectively. Table 1 shows our

numerical values of a; for the lowest 3 modes. They are

in good agreement with the Timoshenko's results12).

2. 3 Wave-body interactions

To calculate the wave-induced motions of a plate, the

finite elements are managed to coincide with the panels,

and deflection w(x, y) of the plate corresponds to the

vertical motion Ā(x, y). Then the nodal force vector f

due to the piecewise constant hydrodynamic pressures

(12) can be expressed as

( 29 )

where the matrix Fij is transformed from matrix Cij in

(11) by using transformation (21). The motion ƒÂi due

to wave-body interactions are also harmonic, and can

be written as

( 30 )

where the displacement vector ui can be expressed as

the superposition of the eigenmodes of the plate as

follows

( 31 )

The following equation can be obtained by substitut-ing (29), (30) and (31) to (22) for determining the unknown principal coordinate qj

( 32 )

where wij is the deflection of j-th mode at the centroids

of i-th panel.

Multiplying the above equation with dim, and takingFig. 2 A typical element

Table 1 a; of 3 lowest modes of a completely free

square plate

334 Journal of The Society of Naval Architects of Japan. Vol. 180

into account the orthogonality in (26) and (27), we can

get a set of equations as follows( 33 )

Thus, qj can be determined by solving the above equa-

tions, and then the wave-induced motions (30) , (31) are

obtained.

3. Numerical results

3. 1 Validation

Murai et al9) carried out an experiment by using a

plate model. The principal particulars of it are shown

in Table 2, where Poisson's ratio of the plate is added by

the authors. The plate model is divided into panels (40 x 10) in our calculation, and the first 20 eigenmodes are used for determining the wave-induced motions. The natural frequency with two nodal line eigenmode by our calculation is 0.6990Hz. The displacements and their phase angles at the points 1, 2, 3, 4 are shown together with the measured data in Fig. 3, in which the

phase angles are the values relative to the point 1. They show good agreements. The instantaneous shape of the

plate is recorded by measuring the vertical displace-ments at point 1, 2, 3, 4 in the experiment. The calcu-lated results agree well with measured points as shown in Fig. 4.

3. 2 Motions performance of a module Next we investigate the motions of a large floating

module (300m x 60m)16). The principal particulars of the module are shown in Table 3. We divide the module into panels (60 x 12) , and the first 60 eigenmodes are evaluated for determining the motions. Fig. 5 shows the nodal lines of the first 10 eigenmodes and 16-, 17-, 20-, 21-, 22-th mode. Table 4 shows the corresponding natural frequencies. The first 3 modes correspond to the rigid modes (heave, roll and pitch) , and the 4-th mode is a bending mode of 2 nodal lines which is the first

Table 2 Principal particulars of the elastic plate model

Fig. 3 Comparison of amplitudes of vertical dis-

placement at locations of point 1, 2, 3, 4, and their phase angles relative to point 1 with measured date)

Table 3 Principal particulars of the large floating

module

Fig. 4 Comparison of instantaneous shapes of the

plate model with measured date)

Motions of a Floating Elastic Plate in Waves 335

elastic mode. While the 6-, 8-and 10-th mode are

twisting modes which have a very important role in

oblique seas. The symmetric modes with 16-, 17-, 20-,

21-, 22-th are specially shown here to emphasize that

their nodal lines are different from those of the beam . If

they are excited in head seas, the method by the beam

model fails to predict them.

It is worth to pay more attention to the value of

natural frequency of the first elastic mode, which is here

denoted as 14. The value of I:1 is equal to 0 .43H, as

shown in Table 4. If a floating airport of 2000m x 300m

is constructed by connecting this kind of modules, the

value of A is greatly reduced to 0 .00961/, and thereby it

will be very flexible. However, a natural frequency of

a conventional ship is much larger than the one of the

floating module. For example, a destroyer has the

natural frequency of 2.12Hz (13 .3rad/s)15). A character

number, f = f4•ãL, can be selected to express the

flexibility of a plate from the law of similarity4) . If the

plate of 2m x 0.3m is used as the experimental model of

Fig. 5 Several eigenmodes of the module

Table 4 Natural frequencies of some eigenmodes of the module

Fig. 6 Motions of the module at wave length ratio λ/L = 0.25 in head seas

Fig. 7 Displacement of centerline of the module at

wave length ratio ă/L = 0.25 in head seas

336 Journal of The Society of Naval Architects of Japan, Vol. 180

the floating airport of 2000m x 300m, the value of f4 for

the model is 0.30Hz due to the law of similarity. It is

much less than that of the plate model mentioned in

section 3.1.

Fig. 6 shows the motions of the module at the wave

length ratio ă/L = 0.25 in head seas. The deformations

of the centerline are small as shown in Fig. 7. The

disturbance wave profiles around the module which

Fig. 8 Disturbance wave profile around the module

at wave length ratio ă/L = 0.25 in head seas

Fig. 9 Motions of the module at wave length ratioλ/L = 0.25 in quartering seas of 135。

Fig. 10 Displacement of centerline of the module at

wave length ratio ă/L = 0.25 in qartering seas

ｏf 135。

Fig. 11 Disturbance wave profile around the module at wave length ratio A/L=0.25 in quarteringseas of 135。

Motions of a Floating Elastic Plate in Waves 337

include the diffraction and the radiation waves are shown in Fig. 8. The motions in quartering seas of the same wave length ratio are calculated, and are shown in Fig. 9. We can see that the motions are no more symmetric with respect to the centerline. The deforma- tions of the centerline are as small as that in head seas. Furthermore, we can see from the results in Fig. 10 that the amplitude of the point around .r/L= 0.3 is small. In

particular, the amplitude of the point xIL= 0.333, ylL=-0 .033 is almost a nodal point because the non-dimen-

sional amplitude is the value Z/ĀA = 0.0029. The distur-

bance wave profiles around the module in quartering

seas are shown in Fig. 11, and the hydrodynamic pres-

sures are given in Fig. 12. Fig. 13 shows the effects of

the elasticity on heaving, rolling and pitching ampli-

tudes at the wave length ratio ă/L = 0.25 in various wave

headings in which the elastic deformations are obtained

by using 60 modes. There is no significant discrepancy

between them.

It is difficult to describe the wave-induced motion

performance without using modal method because the

plate has the infinite degrees of freedom. The charac-teristics of the motions can be expressed by using the

responses of a few lower modes which are included in

the motions of the plate. We show the response ampli-

Fig. 12 Pressures on the bottom of the module at

wave length ratio ,ă/L = 0.25 in quartering

seas of 135•B

Fig. 13 Effect of elastisity on the heaving, the pitch-

ing and the rolling amplitudes at wave

length ratio ă/L = 0.25

338 Tournal of The Society of Naval Architects of Tapan, Vol. 180

Fig. 14 Response amplitude operators of the first 10 modes of the module

Motions of a Floating Elastic Plate in Waves 339

tude operators15) (RAOS) of the first 10 lowest modes in Fig. 14, in which we put the maximum vertical displace- ment the unite amplitude for each mode. We can see that only heaving mode appears in a very long wave(L/λ=0.01), where the module follows the incident

wave, and the other modes do not appear at all. Of

course, the twisting mode does not appear in head seas.

The significant responses of each mode only appear at

the limit range of the wave headings in a constant wave

length. The limit range mentioned above becomes

narrow with the reduction of wave length. Finally the

heaving mode (the 1 st mode) appears in only beam seas

of the short wave length (L/ă = 6 ). The significant

resonant phenomena do not appear in this numerical

results.

4. Conclusions

In this paper we developed the plate modal method

which combines a pressure distribution method in

hydrodynamics with a finite element method in elastic-

ity. It makes the prediction of wave-induced motions of

the very large floating structure possible in any heading

waves. The main conclusions obtained in this study

can be summarized as follows ;

( 1 ) The accuracy of present method is verified by

the comparisons of calculated vertical displace-

ments and instantaneous shapes of a plate model of

2 m x 0.5 m in head seas with the available mea-

sured data, which show reasonable agreements.

( 2 ) The involvement of twisting modes makes the

prediction of motion in oblique seas possible. Even in head seas, prediction by present method will be

more reasonable than that by the beam modal

method due to the effects of higher symmetric

modes.

( 3 ) We　 propose　 the　 character　 number　 f　related　 to

the　nat∬al　 frequency　 of　the　 first　elastic　 mode.　 The

values　 of f indicate the plate flexibility.

( 4 ) It is made clear that the twisting modes appear

in the motions of the large module of 300m x 60m in

quartering seas, and the significant resonance phe-nomena of it scarcely appear in waves.

We have carried out the computations with Cray J932/24-8194 of the Information Processing Center, Hiroshima University and HP 9000-730 at the Ocean & Space Laboratory, Hiroshima University.

Finally the authors would like to express their grati-tude to Dr. Xin Lin for his valuable discussions during this study.

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