The hydroelastic nature of vibration problems in ship structures, ISSC80, Committee II.4, - Ship Vibrations

(1)

h

TECHNISCHE HOGESCHOOL DELFT

AFDELING DER SCHEEPSBOUW- EN SCHEEPVAARTKUNDE DELFI' UNIVERSITY OF TECHNOLOGY

Department of Shipbuilding and Shipping

Ship Structures Laboratory

Report No.

SSL 233

THE HYDROELASTIC NATURE OF VIBRATION PROBLEMS

IN SHIP STRUCTURES. INTRODUCTORY REMARKS.

by

R. Wereldsma

Contribution to the I. S.S. C. -Committee 11.4 - 'Ship Vibrations'.

(2)

THE HYDROELASTIC NATURE OF VIBRATION PROBLEMS IN SHIP STRUCTURES. INTRODUCTORY REMARKS.

By R. Wereldsma.

(Conribütion to the i.S.S.C.-Committee II4 - 'Ship Vibrations').

1. Introduction

Vibration problems aboard ships do exist for a long time. In most cases they can be categorized under forced vibrations, to beanalyzed with classical linear mechanics. In general they obey the equation:

lMl{} IcI{} IKl{x} {F} (1)

where: _1Ml = mass matrix

ICI = damping matrix IKl stiffness matrix

{F} excitation vector

{ x} displacement vector.

The description of this 'dry' mechanical problem has to be completed with the effect of water, for the case a sailing ship is to be considered /1/.

When the hull of the ship moves in the water, reaction pressures of the medium will be generated, in general proportional to the acceleration _{of the motion,} the speed of the motion and to the position of the hull boundary. _{In general} all these forces are dependent on the global shape of the motion pattern and the geometry of the hull.

So, a local pressure,, introduced at a certain location of the hull will depend on the local acceleration, speed and displacement amplitude and similar quan-tities of adjacent and all other locations of the hull boundary. Therefore, when a discretized description is made,all nodes of the discretized model will show mutual couplings, in other words the _{system matrices describing.} the hydrodynamic effects will not have elements equal to zero, although for practical reasons a band matrix approximation _{may be applied.}

For the wet ship equation (1) needs to be completed as follows:

lMIO)

Iciw

+ IKI{x} {F} _{- IMl{) - ICIW - IKI{x}} (2)

where: _1Ml added mass matrix

ICI hydrodynamic damping matrix 1X1 = buoyancy matrix.

(3)

These 3 matrices describe the hydrodynamic effects to betaken into considera-tion. Elements of these matrices may depend on environmental conditions, such as e.g. forward speed.

Since we have now a combination of mechanical elasticity and hydrodynamic effects this equation governs a problem of hydroelastic nature, and defini-tions and allocadefini-tions of different areas to be studied may be deduced from this equation /1/.

In order to allocate more precizely the various hydroelastic systems _a separa-tion of ship mosepara-tions and defiecsepara-tions is necessary..

Therefore a coordinate transformation to 'dry hull' natural coordinates will

1be made.

'The result can be presented as follows:

11M1111

_i

L

IMij

lCll!lCl2l

_{mÌ

r1K11'

For the interpretation of equation (3) three cases will be considered, i.e.: the 'rigid body modes' (with index in (motions));

the 'elastic modes' (with index f (flexions)); e) special conditions.

a. 'Rigid body modes'

For the so-called 'rigid body motions' of the ship the equation for pitch and heave reads from eqûation (3):

lMiil1m}

f

₊

_-l2'f

+

IK11I{P}

r

-

_{lMiiftPm} +}

-

i21f} -

ICiil{P}_

JC12l{iif)

-

1i'm,

11C12'KPf} (14)

From this equation for the rigid body motions _{} we see that due to the} mechanical damping. _ICI and the hydrodynamic terms

_{1Ml, ICI}

and _IK.I there exists a coupling with the deflection nodes (2., 3 and higher node vibratory modes) of the elastic ship.

This means that pure rigid body motions do not exist, and as a consequence. the 'rigid body' motion problem of the ship belongs to the hydroelastic domain. Serious interaction effects can only be expected when ships longer

For.the further analysis we restrict ourselves to symmetrical modes, i.e.. heave, pitch and vertical bending.

Pf

(4)

-3-than 300 m are to be considered or more generally when the excitation frequency/natural frequency ratio is larger than (see par. 2).

For a practical analysis the damping matrices _ICI and _ICI are diagonalized i.e.. damping. coefficients are assumed to be proportional to either the mass or the stiffness distribution or a combination of them (proportional damping).. In that case the coupling between the rigid body modes and the elastic modes of the dry ship are through the addd mass and added stiff-ness matrices alone and still the so-called 'rigid body modes' do not exist. The ship motion domain therefore is in a wider sense also afflicted with hydroelasticity. .

Elastic modes

For the elastic, deflector modes equation (3) reads:

IM22l1

+ lC2iIPm} + _1C221{lPf} ₊ f

K2{ti} =

{Ff}

- lM21l{}

- lM2l{f} - lC2iIm}

lC22l{f} _-

I2ilm} - lK22I{

(5)

Similarly as in case a) there exists a coupling between the elastic modes and the rigid body modes even if diagonalisation of the _{ICI and} _ICI matrices is. carried out.

The analysis of the ship deflections as generated during the operation of a sailing ship is a hydroelastic prdblem.

Special conditions

As already stated, the magnitude of the hydrodynamic effects, represented by the elements of the _1Ml,

_{ICI and IKI}

matrices may be influenced by environmental conditions, such as the forward speed of the ship, nearby boundaries in the fluid (restricted waters)., or in the case of a profile (propeller blade, rudder) the relative speed of advance.. The value of the elements of the mentioned matrices (or in control engineering terms the amplification of the feedback loop) controls the dynamic characteristics of the system (natural frequency, damping or dynamic instability).

In this case there is a strong interaction between the hydrodynamic and elastic characteristics of the system (in the case of dynamic instability flutter phenomena are manifested) certainly belonging to the domain of hydroelasticity.

Examples are: airplane wing flutter, moving flag, propeller blade singing, possible rudder flutter.

In Table I a review is given of the.various timedependent phenomena and their classification.

(5)

Table I. Review of ship hydromechanics. Hydromechanics

Hydrostatics Hydrodynamics

Rigid body,

time independent phenomena

Rigid body,

time dependent phenomena

Elastic body,

time dependent phenomena

Draft Trim Emersion Heeling Buoyancy

Still water bending moment Bow-stern-wave bending moment Static stability

Ship motions (and deflections) Manouvring

Course stability Dynamic stability Wave bending moment

Modal vibrations and deflections Hull deflection

Binding moment due to springing, whipping, slmrning

Dynamic flexible body stability (flutter)

(6)

2. The 'wet hull hydroelasticity'

By means of an example it will be shòwn how the flexibility of the hull will influence its bending moment loading, as compared to the regular rigid body analysis. In order to make the analysis accessable for hand calculations, simplifications in the calcuiatin model will be introduced, such as:: - sagging-hogging will be represented by the 2-noded deflection mode only

(3rd natural coordinate);

- higher modes of deflection will not be considered;

- the shape of the ship will be simplified to a rectangular beam;

hydrodynamics will be simplified to quasi-static Archimedes forces and mass effects (damping neglected).

The model of the ship is shown in Fig. 1.

o o o

O O

o o

s2 where: *1 s22

M = distributed mass

[kg/mi

EI = bending stiffness

Ñm2]

b

= distributed elastic support

(boyancy) [N/rn]

Fig.1 Simplified model of a ship

Taking into account the mentioned simplifications, equation (3) reads for this model:

M000

O O M11 O O O is generalized mass

Í1ft0

M00 M01 M0 i = F1 - M10 M11 M12

[2j

LF2 M20 M21 M22

'O2

equals the heave motion equals the pitch motion.

quals the 2-noded deflection (sagging-hogging), (Fig. 2) is generalized bending stiffness

(6) OO 0l 10 S20 11 S21 12 S22 p2J

(7)

For

harmonic excitation

( ₂₂

2T

M22) ( J

2w

i

-6

M00,M01,MO2 is generalized added mass

is generalized excitation forces S00,S02,S02is generalized buoyancy.

Fig.2 Twonoded deflection of.the huti girder

Focussing attention on the flexibility of the system (q)2)and neglecting the coupling effects with the other two coordinates the equation for the 2-noded deflection reads:

S22.q)2 i- F2 - M22.2 -

₍₇₎

and a linear system we may write:

"2 - s22 q)2

where w is the frequency of encounter or excitation and.M2 equals the total mass of the ship including the added mass.. After sorne manipulation follows:

+ w2 - w2)

(8)

s

2 22

where: w22 equals the natural frequency of the 2-noded vibration of the

M . .

22 ship (inclusive the added mass) s

2 22

w

,w

M22

Equation (8) reflects the dynamic bending response of the ship when the effect of the buoyancy on the deflection is taken into consideration.

For the case this effect is not taken into consideration which means the

(8)

(r )

2D

i

-7

which reflects the dynamic bending response of the flexible dry ship (without buoyancy effects of the deflection).

In order to look for the relative effect of this hydroelastic mechanism as compared to the non-elastic case we consider the ratio:

R (*2)D -

2W

reduction of deflection of ship due to the buoyancy effect

- - deflection of the ship without buoyancy effect

+ ww) - (A)

(10)

For the case we look for the. static deflection

_{(w -}

0) we have:

Rs _{- w22} -t-w2 w

For a practical case of a larger ship we can state w22 1 Hz.

is in the order of the natural frequencies of ship motions, i.e. - Hz.

Rs - corresponding to appr. L%,, which means about '% reductibn of the bending moment, when the effect of the buoyancy of the deflection is taken into consideration.

For very large ships the natural frequency w22 may be even smaller and a stronger reduction is to be expected.

For a ship having a hinge midships w22 - O the reduction factor equals 100%, which means that all the deflection is counteracted by the buoyancy.

For the case the loading is not static but time dependent, w 0, (as is the case with a wave loading), very large deflections may occur as follows from formulas (8) and (9) and the reduction factor may have large values as well, depending on the valUe of w. In these cases however the applied simplifica-tions (neglecting the damping) are not allowed and a more sophisticated analysis is necessary (springing), starting from formula (3), beingthe sub-ject of the next paragraph.

3. Hydroelasticity and hull vibration

(9)

For this subject again the elasticity of the hull will assumed to be represent-ed by the two-nodrepresent-ed deflection solely and the complete equation for motions

(9)

and deflections in natural coordinates (based on the 'dry ship') reads: M00 O O C00 C01 CO2 o M11 0 1-

C10 C11

C12 O 0 M22 '2 2O

2l

22 M00 M01 MO2 M10 M11 M12 M20 M21 M22

From this equation can be concluded that separation of motions and deflections is only possible under certain simplifying assunptions, i.e. neglection of the coupling terms, introduced in the mechanical damping _ICI and the hydroelastic effects represented by the _1Ml, _ICI and _IS] matrices.

Doing so, we obtain for the two-noded deflection:

+ C2211)2 i S1 - r2 - M22112 - C22i2 - S221p2 (13)

Similar equations can be formulated for the higher modés., again under the same assumptions.

T2 equals the excitation force and may be composed of:

- high frequency waves having a stationary nature (springing); - interrupted high frequency waves (whipping);

- impulsive forces (slamming);

propeller shaft forces (hull vibrations); - propeller hull pressures (hull vibrations).

Whether or not higher modes need to be considered depends on the ratio of the natural frequency and the excitation frequency, because due to resonance phenomena the relative importance of a certain mode may be amplified. So for springing and whipping the 2-noded mode may suffice for the analysis.

For slamming the first 5 modes e.g. are of importance /2/ and for propeller vibrations all the modes having natural frequencies equal at or smaller than say 3 times blade: rate are of importance.

So the complexity of the analysis will increase with the increasing frequency content of the excitation force.

Practical applications for springing and whipping are generally made by a further modification of equation (13) as follows:

(M22 +

M22 + 1- S22t1)2 (114) 00 11

coo

C10 C20 coi C11 C21 co2 C12 C22 11)0 11)1

.

--s00 s01 s1,0 s11 20 s21

s2

s12 s22 iP2J (12)

(10)

Negiection of S22 against S22 is made and C22 is a damping coefficient derived

from practical experience (modal damping).

For harmonic excitation and a linear response we obtain:

{-w2 (M22 + M22) i- iw22 i- S22}ì2

F2 1

"2_1

C

j;--5

_. ₂ 22 12 ₂₂

{l- (_L)

1-iw---22

where w22 equals the natural frequency of the 2-noded deflection

₂

of the

ship in water. Equation (15) is the regular second order linear system

equa-tion, relating the deflection

against the excitation F2. Damping c22 and

excitation F2 are most of the time subject of research.

For the analysis of propeller generated vibrations more modes need to be

considered simultaneously and the application of the finite element method

is a requisite. Again the hydrodynamic effects play aniftiportant role such

as added îiass M22,, M33 etc.. Hydrodynamic damping is generally neglected

(C22, C33 ...

O) and the effect of buoyancy S22, S33 ... etc. is. also

ne-glected. The excitation r2, r3 etc.. is subject of intensivé research

theore-tically as well as experimentally. In particular for the measurement ôn the.

model of the hull pressure fluctuations a serious hydroelastic problem

is

present and recognized for a longer time but not so easy to solve /3/.

For the solution of the ship vibration prediction problem it is questionable

whether the finite element method is the appropriate tool, because the

vibra-tion analysis of those details required for an evaluavibra-tion of the habitability

aboard ships requires a very large number of nodal points in the

F.E.-approach, that may lead to an impractical. problem /4/. The transfer of the

distributed pressure, generated by the propeller and exciting, via the

stiff-ened panels in the afterbody of the ship, the main structure of the ship,. is

a problem of similar, but now hydroelastic nature and of large complexity,

because very many relatively small, one side wetted, panels are involved with

their hydroelastic response characteristics. The resolvability of the finite

element method, to catch these small details is doubtful.. (It is comparable

to the problem to determine the bending moment in the stem of a tree that is

loaded by the wind through its leaves and branches).

.

(11)

10

-General remark

The hydroelastic effects mentioned in the preceding paragraphs and indicated as added mass, damping and buoyancy have one general property: they do not depend on environmental circumstances. T]ey do depend on hull shape, draft, freqiency and may be non-linear (although in many cases linearisation is allowed), but they are not influenóed by the forward speed e.g.

These coefficients modify the dynamic characteristics of the system under consideration, i.e. natural frequency and damping but do not give rise to instability and sontaneous oscillations. Sometimes the forward speed has been considered /5/, but results show only small corrections on the values without.forward speed.

For the case vorticity is shed by the structure due to the forward speed (in the case of an oscillating pröfile, or a vibrating tube of an offshore structure), not only added mass effects but also forward speed dependent damping and coupling effects will be introduced, and these effects combined with the mechanical properties such as mass, damping and elasticity, may give rise to instability of the system under consideration (wingflutter, propeller singing, rudder vibrations),.

5. Rudder dynamics as an example of hydroelasticity

For the preliminary analysis of the rudder dynamics use is made of the two-dimensional instationary airfoil theory. The results of this theory are given in the first chapter of the àppendix of this paper, aid consist of hydro-dynamic coefficients controlled by frèquency and forward speed.

In chapter II of the appendix the elasticlly supported rudder has been dealt with, and the final equation governing the rudder dynamics is given as equa-tion (A6) and ispresented in the format of equaequa-tion (2). Soluequa-tion of this homogeneous equation leads to the condition that may introduce flutter.

The complexity of the equation requires a computerised solution of a numeric-al example not given here.

Remarks on euat ion A6

The damping and stiffness coefficients of hydrodynamic origin .do not only depend on the shape and dimensions of the body under consideration and the density of the medium, but also on environmental conditions such as the forward sp.eed and the frequency of operation. This in contradistinction with the systems of par. 2 and 3 where the effect of forward speed and frequency was not so pronounced and neglected.

(12)

Therefore, when a forced vibration analysis of the rudder is made, e.g. due to the excitation of the propeller wake vorticity, in order to analze the rudderstick bearing reaction forces in connection with a hull vibration analysis, it is necessary to take the speed and frequency dependent hydro-dynamic coefficients in consideration. if the system happens to be highly Undamped due tO hydroelastic effects, a serious vibration source is intro-duced. (a similar conclusion can be drawn for all other appendages nearby the propeller such as e.g. the shroud).

Final remarks

Hydroelastic phenomena are as a mater of fact introduced fi all wet mass-elastic vibration- and other time dependent-problems aboard ships.

The hydroelastic analysis as formulated in 71/ needs to be developed as a regular tool for marine technologists in order to be able to handle problems. in advanced ship designs and unknown future developments in the field of marine technology, in order to meet the demands and standards of safety and

habitability that will bé set fr the near future.

Complex hydroelastic instability problems as outlined in par. 5 and the appendix are unlikely to be encountered aboard ships. Analysis of e.g. the rudder dynamics is based on ideal circumstances not existing in reality. The mechanical support of the rudder or shroud and the flow conditions are so complex that a theoretical analysis needs to be verified with an experim-ent.. It might be necessary in that. case to build an elastic scaled model so that uncertainties in the environmental conditions are as much a possible eliminated. It is recommended to investigate the probability of occurrence

- of unstable or highly resonant systems aboard ships, before making thoroùgh

studies.

References

/1/ Bishop, R.E.D. and Price, W.G.: 'Hydroelasticity of ships'.

Cambridge University Press, 1979. Cambridge, Londön, U.K. /2/ Belik, O., Bishop, R.E.D. and Price, W.C.:

'On scaling the oscillatory characteristics of ship models'. R.I.N.A., 1980.

/3/ Wereldsma, R,: .

(13)

12

-on an elastic structure of a shipmodel'.

Paper to be presented at the Conference on Advances in Propeller Research and Design, Gdansk, February 1981.

¡'t/ Wereldsma, R.:

'Today's difficulties in ship vibration prediction and a possible solution'.

Report No. 226a of the Ship Structures Laboratory of the Deift University of Technology5 The Netherlands May 1980.

/5/ Grim, 0.:

'Hydrodynamische Masse bei lokalen Schwingungen, insbesondere bei Schwingungen im Bereich des Maschinenraums'.

Schiff und Hafen, Heft 11, 1975.

/6/ Bisplinghoff., R.L., Ashley, H. and Haifman, R.L.: 'Aeroelasticity'.

Addinson-Wesley Publ. Comp., Inc., 1957. ¡7/ Fung, Y.C.:

'An introduction to the theory of aeroelasticityt John Wiley & Sons., Inc.,, New York, 1955.

(14)

APPENDIX: RUDDER VIBRATION ANALYSIS

I. Results and interpretation of instationary two-dimensional profile theory. See Figure Ai.

b

\rnid chord

Jo(

¼JM

Fig. Al Definition of displacements

and forces

Lift and Moment (L and M), (Fig. Al), are the forces to be exerted by the support on the profile (midchord point) to perform the forced displacements h and , and are measured per unit of spanlength.

From the incompressible hydrodynamic theory the following force and moment terms are derived and given in table Al (see /6/ and /7/).

Table Al. Two-dimensional instationary profile forces.

The matrix equation., representing. these hydrodynamic effects, reads:

rpb2

r2pubc(.k)

irpub2l

c(k))1Jl

+

L o

.pb4J[J

L_!rpub2c(,k) irpub[l - c(k)}][J + ro

2,pu2bc(k)1h

LI

. (Al)

[o

-iru2b2c(kj[cxJ [MJ L M Remarks + irpb2li + pb

non-+ 1Tpub2 .

-

npu2b2 circulatory

- npub2fi contribution 2pubc(kTh + pub2{1 - c(k)} circulatory

+ 2ïrpu2bc(k)a

+ iîpu2b2{l - c(k)}ct distribution + ïrpu'b2c(k) +

irpub3{l

(15)

-I

w.b

u

A2

-c(k) is the 'Theodorsen function' taking into consideration the èffects of the traling vorticity and is a function of k,the reduced frequency, defined as:

w = radial frequency of oscillation b = chord length

u = speed of advance.

Remarks: 1) The reduced frequency k is not a frequency having a dimension but a dimensionless measure for the. wavelength of the oscillatory flow-disturbance.

2) c(k) is a complex function shown in Fig. A2. The. real part of the force is in phase with the speed. of displacement (F and ) and the imaginaire part is in phase with the acceleration of displacement

(fi and

ei).

3) When the effect of trailing vorticity is neglected (quasi-steady approach), c(k)

i.

(A2)

Im

o

01 - 0.2

- 0.3

koo

Q.5

80

/,

0.40

₁

0.05

0.20

0.10 Fig. A2 Complex Theodorsen function

C (k)

For the quasi-steady approach the equations. simplify to.:

1pb2 ₊

_1p1b

2P111fl

+

ro

2rrpu2blJh'L = _ILl _(A3)

L°

.pbiIuJ LpUb2 °

_iJ

Lo -rPu2b2J[aJ jMJ

and can be. interpreted as follows:

Tpb2 equals the added mass for a translatory motion fi

. pb4 _{equals the added moment of inertia for a rotational motion ä}

2rrpub equals the lift force. due to a unit transverse speed Fi of the. profile and acts at the chord point, so generating a moment -irub2

(16)

A3

--ïrpub2 equals the moment due to lift force generated by a unit transverse speed Fi

2Trpu2b _{the steady lift produced by a profile with a unit angle of attack} _a.

This lift force acts at the chord point.

This steady lift is also recognized in the well-known steady profile theory: L _CL. pu2.(chord length),

where CL approaches 2n.

-lrpu2b2 equals the moment generated by. the steady lift acting through chord point.

II. The hydroelastic system

As an application of the mentioned results an idealized rudder is examined. The rudder itself is assumed to be a rigid body supported elastically by the stick in lateral and translational direction.

The centre of gravity is assumed tobe in the middle of the rudder chord, (see Fig. A3).

Point P of the rudder stick is a distance a in front of the midchord point, the latter point coinciding the mechanical centre of gravity. In this point P

a lateral supporting spring with stiffness S1 and a torsional spring with stiffness S2 are located. Further M1 is the mass of the rudder per unitof span and I the moment of inertia per unit of span, aound the C.G.

In order to apply the results of paragraph I a coordinate transformation to the 'rudder coordinates' is necessary. The new coordinates refer to point P and are indicated by y1 and y2 (see Fig. AL1). _{The forces around point P are}

S (side force) and T (torque.).

Ya

id chord

Fig.A3 Rudder stock with

Fi.g.A4 Coordinate transformation

(17)

W2[ MT Li-aMT

Ihi

[1

lai

[o

al f

u]1,Y2 +aMT

_iSil,

+ +a2MT1-ITJ

L2i

Alt

-The coordinate transformation reads:

Transforming equation (Al) by post- and pre-muitiplication of the coefficient

matrices by the transformation matrix and the transposed transformation matrix

respectively,results in the hydrodynamic part of the hydroelastic equation:

The mechanical terms to be added to this equation read:

[M1 M1a

+ F

oJy1l

Js

[Mia

_Mia2+IiJl2j

Lo

sjy2f

lT

The complete hydroelastic equation for the rudder problem as defined in

Fig. A3 can now, for harmonic oscillations,, be formulated as follows:

iwirpub

It 2c(k)

+2ac(k)+b{1-i-c(k)}

1Í1Ì

+

Li-(2a_í)c(k)

(+2a2ç(k)-i-ab+b2(l_c(k)))J[y2J

rs1

-t-2Trpu2bc(k)

1f1T

=

ÍÌ

(A6)

[o

_{rtrpu2(2ab_b2)c(k)+S2jly2J}

where: MT is the total mass per unit span,, i.e. M1 +irpb2

is the total moment of inertia per unit span, i.e.

L1-l- -

ph4.

This set of equations needs to be solved for a rudder vibration analysis.

The homogeneous set of equations (eigen value problem)., cannot be solved

straight forward for every frequency.

The value of c(k) is frequency dependent because of the different reduced

frequencies.

From formula (A2) föllows wb

_{k.,u,, so that the coefficient for the first term}

of equation (AS) equals -w2

k22 and for the middle term of equation (A6)

iwirpub can be rewritten as: ilTpu2k, so that ali the coefficient matrices

re

only dependent on k and u.

For a selected forwardspeed u the eigen value solution can be found in terms

(A5)

+ pu2 [o

2bc(k)

f L

Js

(Ait)

[_o

(2ab-b2)c(kJ]:Y2J

{aL + Mj

jT

1IPb2[

a

_1fr1ÌÇIb

r2c(k)

2ac(k)-s-b{li-c(k)}

1JTl

[a

a21b2J

(18)

A5

-of k. Also the damping coéfficient can be evaluated for that particular value of k (and implicitly w). For an increasing forwardspeed u the change in eigen frequency and the change in damping coefficient for that frequency can be analyzed.

Arriving at the point for zero damping gives us the flutter speed and the frequency of oscillation.

Fòr a forced vibration analysis the nonhomogeneous equation needs to be solved. Excitation forces S and T, e.g. generated by the propeller wake need to be determined in the first place.(two-dimensional flutter theory: 'Sears function' The transfer from these forces to rudder vibrations and rudder stick bearing forces is controlled by the equation (A6). The transfer characteristics may be seriously influenòed by the hydroelastic behaviour. In particular the

damp-ing may be effected. Therefore it is recommended when a rudder vibration analysis has to be performed to take the hydroelastic effect into