Experimental analysis of ship’s springing through modal techniques

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CONTHIBUTION TO THE DISCUSSION OF CO.!ITEREFCT I .2

by R. Wcreldsia, Ship Structures Laboratory,

DeDartment of Shipbuilding nd ShiDping,

Deift Universlty of Techno1cgy

TCHNSCiE UV1STEff

Laboratorium voör

Scheephyc1romechanica

&rchiot

Mekelweg 2, 2628 CD Deft

FacO15-1813

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o

R. WERELDSI4A (The Netherlands)

EXPEiIMBNTAL ANALYSIS OF SHIPS' SPRINGING THROUGH MODAL SUMMARY

An analysis of the shìp motion and bending rroblem is given. For this analysis the "Normal Mode Method" is apiied. It is prorosed to use a segmented model oscillator to determine the hydrodynainic coefficients controlling the springing phenomenon of ships.

LIST OF SYMBOLS

C = hydrodynamic damping

[c] = hydrodynmic damping matrix

F = forces in ship coordinates

i imaginary unit

M mechanical mass

M

= hydrodynanic mass

[M) _[MI mechanical mass matrix [M) = hydrodynamical mass matrix

S = hydrodynamic buoyancy

S mechanical stiffness

[s) = hydrodynamic buoyancy matrix [s) [s) mechanical stiffness matrix

[T) = transformation matrix

X displacement in ship coordinates

r, Ï generalized forces

= generalized displacements w = circular frequency

INTRODUCTION

Vertical ship motions (heave and pitch), vertical ship bending

(sagging and hogging) and the t\ro-noded vertical girder vibrations can be dealt with in a general way by means of the 'Normal Mode Method" ¡1/.. This method is based on a coordinate transformation to natural

coordin-ates, coinciding with the deflections of the structure when it is freely vibrating. In this paper the attention is focussed on the design of a vertical model oscillatcr, which makes it possible to determine

exrerim-entally the generalized hydrodynaiììc

coefficients for the modal deflec-tions of the ship's girder, a necessary information for stringing anal-ysis. As a first step linearity and frequency independency of the

hydre-dynamic effects is assumed.

DESCRIPTION OF THE PROBL IN "SHIP COORDINATES"

For this analysis the ship will be approximated by a lumped mass and stiffness distribution, According an aruroach by Bishop et al. 12/ the problem vili 'be b'oken down in two parts i.e. the pure mechanical problem ("dry ship") and the hydrodynamical problem, being a generai-*

Report No. 206, Ship Structures Laboratory, Delft University of

Teci-noio.

The expression "Dry Ship" for the mechanical problem has been intro-duced by P.E.D. Bishop.

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ized description of the effects of waves and water.

In Fig. i a simolified discretization of the vertical displacements is shown. The approximation has in this case 6 degrees of freedom i.e. the rigid body motion heave and 5 different vertical deflection possibil-ities.

A similar description is possible for the rotary displacements which includes the pitch as a rigid body motion and bending flexibility. From this lumDed parameter approximation the set of equations, governing the static and dynamics of the "dry ship" can be written in abbreviated form as follows:

+ [](i) = {F} (i)

The stiffness matrix

Is)

includes the elastic stiffness against shear and rotary deformation. The displacement vector {x}- includes also the rotations of the elements. The iass matrix [] is diagonalised. {F)- equals the forcing action, generated by the waves.

Since the ship operates in water all its sections will experience forces due to accelerations -[k)-, velocity {} and displacement {x}. These forces are generally described by "hydrodynar'ic coefficients".

Part of the wave forces [F] is consumed by the surrounding water to accelerate the water (hydrodynamic mass), to overcome frictional ef-fects (hydrodynamic damping) or to supplement loss in buoyancy due to a change in displacement (hydrcdynamic spring effects).

For the complete description of the problem (i.e. the problem of the "vet ship") the set of equations (i) has to be completed with the hydrodynamic effect as follows:

I){x}

+ [J} = {y} - [

7){}

-

E

){k}

-

E

J){x}

(2) Having knowledge of all the elements of the matrices and the forcing function {F_J., the equations can be solved., resulting in vertical ship motions, vertical girder bending and, if applicable, in vertical girder vibrations. The mentioned motions and deflections have strong mechanic and hydrodynamic interactions and make the understanding and the insight in the hydro-mechanics difficult. Therefore transformation to "Natural Coordinates" will be performed.

DESCRIPTION IN "NATURkL COORDINATES"

To transform the "Ship Coordinates" to "Natural Coordinates" the arbitrary girder deflection (being a function of the longitudinal ship axis) needs to be broken down into a series of standard deflectioms, co-inciding with the free body motions and the multinode vibration shares. These standard deflections, nornalised in orte or another way, rossess the property of orthogonality, when they are mass weighted ¡1/. This transformation reads in matrix form:

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e X0 Xl X2 X3 Xli T00 T01 T02 T03 T0 '10 '11 '12 T13 a1 - 20 '21 22 23 T30 T31 T32 T33 T3 Tlio T1 T12 T13 T

Li-noded

(bending) 3-noded (bending) 2-noded (bending) 1-noded (pitch) 0-noded (heave) 0-noded (heave) -1-noded (pitch) 2-noded (bending) 3-noded (bending) -14-noded (bending) -or:

[TJT.[F

= T00 T10 T20 T30 T10 T01 T11 T21 T31 T11 T02 T12 T22 T32 T142 T T T T T 03 13 23 33 14 O14 T114 T214 T31 T141 or {x} = [T){. F0 F1 p F3 type of deflection Fo r 2

r3

r)4 (3) ( ) )

The columns of the transformation matrix [T) represent the modal deflections of the girder, in a normalised version. The vector {c)- repre-sents the strength of the various modal deflections. The vector x- ecuals the resulting displacement in ship coordinates.

o

For the forcing vector .{F} a similar break-dowm into a series of modal distributions is necessary. This break-dorn is based on the fact that a distributed force proportional to the mass distribution multiDlied with the modal deflection causes a beam deflection eoual to that modal deflection. The amplitude of this distributed force F}- ecuals the gener-alised force and fits with the "natural coordinate" system.

The relation 'bétween {F} and {F reads as follows:

Now the rows of the transformation matr*x represent the modal de-flections in a normalised version. So the vector of the generalised forces rJ- can he obtained by premultiplicaticn of the fprce vector (F} with the transposed coordinate transformation matrix TJ . See / 1/.

When transformations (3) and (1f) are carried out on eauation (2) the problem has been converted to the new modal coordinates and the eQua-tion reads:

IT)T[SJ[TJ{,I}

+ [T)T[)fT]{fl =

=

r}

[T1T[J}[:) -

_[T]'[C]jT]}

- [T]T[J[TJ{

₍₎

Further reduction gives:

1S)tIJ + [M)} = .(FJ. - -

- [s){}

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' mechanics wave ' hydrodynamics

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Equation

(6)

describes the ship motion and girder deflection problem in a suitable way. This because tie

[s]

and [1} matrices are diagonalised so that the mechanics of the hull (motions and deflections of the "dry ship") can be independently dealt with. The coupling between motions and deflections is caused by the off-diagonal terms of the [M]

_[c]

_[s]

matrices, representing the hydrodynamic properties of the "wet hull".

SEPARATION OF MOTIONS AND DEFLECTIONS

For the case attention is focussed on the springing problem, i.e. the resonance henomenon of the 2-nodecl. girder deflection, only three principal modes will be considered, i.e. heave (ip ), pitch (p ) and the

two-noded mode (IP2). O i

Equation

(6)

reads now in expanded way; -rM O 0-1

1OO

O1lO

l-00 I LO O

O+ O

M11 O

I'

22- i '2 - L o o M22]

rr

M M M 1 0 00 01 02 M Âf 1

20 2112

Ir

. _M _M J. 2. - '2O 22 22 ¿ 2

0s _IC00

Coi CO2.

<

-

Ic20

C11 C12

[c20

C22 C22 2 -s00

s0

s02 10 12 i2

_s20 s22

s22_

Ship motions can now be separated from ship deflections as follows: heave:

N

= r

- M

-

MO2i2 -

C000 C01;1

-000

0

000

01 - S00p0 - S01l1 Pitch:

= r1 -

M10Ç0 - -

M122 -

C10

- Cl11 - Ci22 +

- - S1ip1 - S222

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Springing:

s

+i=r

-M"'7i .-M

-C

-C i+

222

2

200

211

222

200

221

222

- s20r,0 - S22IP2

₍₎

A systematic extension to more modal coordinates is necessary for ship strength considerations.

All three considered deflections are mutually coupled by hydro-dynamic effects. Well known coefficients can "be recognized easily, such as:

total mass of "dry ship" = added mass for heave

ro

= total vertical wave force

IP

ra

'-4

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(I)

= hydrodynamic heave damping

S00 displacement force due to unit draft increase

= total moment of inertia measured through centre of gravity of the "dry ship"

Mn

= added moment of inertia measured through centre of gravity r1 = total pitching moment of wave force measured through the

centre of gravity

C hydrodynamic pitch damping

S = pitch moment of displacement forces duc to unit change of pitch angle

S22 = generalized bending stiffness of the ship structure M22 = generalized mass of the ship structure for the 2-noded

deflection

F2 = wave generated saggin-hogging load distribution

1122 = added mass for the 2-noded girder vibration

-C22 = hydrodynamic damping of the 2-noded girder vibration

S additional bending stiffness due to change in displacement 22

caused by sagging-hogging deflection. E)TERIÌ4ENTAL SOLUTION OF THE SPRINGING EQUATION

In order to solve the springing equation (io) and to find

2' being

the amplitude of the 2-noded deflection, it is necessary to know all the coefficients of equation (io). S22 and M22 are pure mechanical quantities. These can be calculated by means of energy theorems, S22 being the

gener-alized stiffness and 1122 the genergener-alized mass. Finite Element Procedures enable us to deal with complex structures, such as ships. The right hand side of the equation covers all the hydrodynamic effects, and it is the purpose of this paper to describe experiments to measure the matrix

elem-ents of this part of the equation.

a. Measurencnt of the enera1ized force and the ship motion interactlons

For this purpose a free model test in head waves with a speci-ally designed model makes it possible to measure the sum:

F2 -

M221 -

C211 S200

-This sum equals the total girder loading for the 2-noded de-flection caused by the waves and the additional waterforces due to forward speed, heave and pitch. A detailed description is given in /3/.

The remaining terms of equation (io), all connected to girder deflections, are ?122) C22 and S22 and for the case of strength cal-culations higher order interactions. These coefficients are to be determined experimentally with a separate experiment.

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b.

Experimcntl determination of

hyrodynarnic

coefficient.s for elastic

shiD

deflections

Hydrodynamic coefficients connected to ship deformations can 'be determined experimentally with a special sectionalized vertical ship model oscillator.

With

this oscillator the ship is forced to move in one mode so that

all

the other modes are zero.

In this paragraph an analysis will 'be given of an oscillator with 6 sections. The analysis will be limited to three modes;

and In Fig. 2 a schematic arrangement of the oscillator is given. For each section the hydrodynamic effects are indicated in ship coordinates.

Each section performs a vertical sinusoidal motion x, relative amplitudes of which are made in accordance with the modal deflection curve. The output of the force pick-ups represents the total force as generated 'by the surrounding water and caused 'by the motion x of the section. (Measures are taken to eliminate parasitic mechanical mass acceleration forces).

The measured force is broken down into three components) i.e.:

roportional to acceleration (M) mass proportional to speed (C) damping proportional to displacement (S) stiffness. For harmonic motions holds:

F0 =x0S0 +

iwx0C0 -

2x0M0

or F0 x0 (S0 -F iuC0

- w2M0)

If the motion of the oscillator is adjusted to

2 we have:

x =T

0 02 2 X1 = T12 X2 = T2 X3 = T32 X11 = T X5 = T52

Equation (12) refers to the coordinate transformation. The forces F0 can now be expressed as:

F0 T02 '2 O + iwC0 -F1 = T12 .

ij

(S,

₊ _iwCi t I t t T I I I t I

Fr =

T52 2 6 1LC5

(12)

(13)

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o

or in matrix notation:

Í0

= [TY[SJ[T)10

After multiplication ve obtain:

[ s00 s02

s0jo

1 = lO 22 s12

₁₀

- 2O 22 22 +

fc

c

c cro'

00 01 0 I C C 10 21 12 J Lc20 C22 22 2 +

[)[c[1]

-The following set of equations is obtained:

LU2O '2 i 2

2-(i6)

S0 0 0 0 0 0 0 S1 0 0 0 0 O

O S20 0

0 0 0 0 S3 0 0 0 0 0 0 S4 0

00000s

5-..-o O 2' + iw _co o o o o o o C2 O O O O o o C2 0 0 0 o o o C3 O O o o o o C4 o ._O O O O O C5

[T]o

-o v2 - w2 M0 O O O O O o M2 O O O O o o ₁₄₂₀ 0 [Tjo 10 (i1) f' o o o M3 0 O O O 0

¡14 0

h!) J

-0 0 O O 0 T00 T01 T02 T10 T11 T12 vhere [T)= T20 T21 T22 T31 T32 T30 T10

T11 T2

T50 T51 T52 I

-

w2 I 10

0002

21 p

'j

' 02 Y "12 "12 (15)

n L

2-noded mode - 1-noded mode - _{0-noded mode}

Introducing generalized hydrodynarnic forcea

4}

[T]T.{ g i ve s

ÍF'o

Ii

J F2 F3

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G)

= S02'2 + lui C2 - w2MO22

= 22'2 + iw C22 (17)

F2 = S22i2 + c22g2

When the signals are split in "in phase" and "quadrature"

com-ponents the following equations are obtained for r2.

(r ) = (S - (,)2M ) 2 in phase 22 22 2

(F )

2 ouadr.

So, for various measurements, with different frequencies w, sufficient information is obtained (i.e. (' ). and (P

2 an pnase 2 quadr. together with

2 as input) to solve equation (18) and to determine

S22,

C22 and M22, being the unknown coefficients of the springing equation (io).

From equation

₍₁₇₎

follow also other terms of the coefficient matrices and general information will become available when the

sec-tional oscillator is adjusted to P1 and/or higher order deflec-tions.

PROPOSED INSTRUMENTATION OF THE SECTIONALIZED OSCILLATOR

The fundamentals of the instrumentation are shown in the diagram of Fig. 3. This instrumentation makes an on-line conversion from ship coordinates to natural coordinates possible and the output of the in-struments reads immediately the generalized hydrodynamic forces

'

P1

and P2 of equation (18). By adjustment of the deflection of the oscil-lator to other generalized displacements all unknown matrix terms (hydro-dynamic coefficients) of equation (7) can be determined.

FINAL REMARKS

In order to obtain a closed solution of the ship springing problem It makes sense to transform the ship coordinates to generalized modal coordinates /2/.

The coordinate transformation is also favourable when dealing with irregular wave loading and to make a statistical description of the ship strength problem possible /l/.

Ship motions and ships' girder deflections belong to one system of equations, having strong interactions. A simultaneous treatment is logical.

The sectionalized vertical ship model oscillator is a multidimen-sional extension of the regular rigid model oscillator as used in seakeeping laboratories. This extension is necessary to make a rational treatment of the strength problem for vertical bending possible.

(18) =

(10)

o

REFEREN CES

/1/ W.C. Hurty and M.F. Rubinstein, Dynamics of structures, Prentice Hall, 1961f.

/21

R.E.D. Bishop ad W.G. Price, On the relationship between 'Dry Modes and 'Wet Modes' in the theory of ship response, Journal of Sound and Vibration, 1976,

pp. 151-164.

131

R. Wereldsma d G. Moeyes, Wave and structural load experiments for elastic ships, 11th Symposium on Naval Hydrodynamics, London,

April 1976.

/1/ R. Wereldsma Normal mode approach for ship strength experiments, a proposal, Proceedings of the Symposium "The Dynamics of Marine Vehicles and Structures in Waves", London, April 1974.

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e

t

LIST OF FIGURES

Figure 1. Lumped parameter approximation of the vertical bending of a ship.

Figure 2. Schematic arrangement of vertical senented oscillator. Figure 3. Instrumentation of oscillator.

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6 FORCE PICK UPS s s s s Jxo ix3 J Fo JFi ,1F2 413 JE4 M0 M1 ¡'13 M4 C0 C _C C3 C4 s0 S1 S3 54 M1 M2 H3 M4 M5 HYDRODYNAMIC MASS 01 C2 C3 04 C5 _{HYDRODYNAMIC DAMPING} So _Si S2 S3 54 55 _{HYORODYRAHIC STIFFNESS}

FIG.? SCHEMATIC ARRANGEMENT OF VERTICAL SEGMENTED OSCILLATOR.

6 STIFF DEAN SECTIONS

7 ADJUSTARLE CAl-I MECHANISMS 7 ELASTIC HINGES S SEALINGS

o

F101 U)MPED PARAMETER APPROXIMATION OF IHE VERTICAL BENDING 01 A SKIF

Mr MECUANICAL MASS S MECHANICAL STIFFNESS X DISPLACEMENT Fr WAVE FORCES H NYOR MASS C r IIYDR. DAMPING SrI-IYDR BOYANCY

TORI VINO SHAFT

6 MODEL SECTIONS 0F SEDMENTED M3CEL

I

MECHAUICJL PART OF OSCILLATOR

DEFLECTION ADJUSTED TO 2 _{DRIVING SHAFT}

o Q Ç' cl ₉ ___J N ENASE/OQADRATURE REFERENCE EENCRATCR 6 FORCE H PICK UPS 6 STRAINOt.50 INSTRUMENTS L WEIGHTING FUNCT IONS AND S u M MAEION OF 6 WEIGHTED SIGNALS

i

_-

_t_ .__-i_-.

-or SO MO2 M12 M32

--}s32 JI2 012 J22 C27 J5r032 X IN Ri _{SQ III)} OLD. 330 EQ (16> O UAD x N PN EQ (16) -1 GUAO.

---

fX ti _E9(T> 3 I .u.,O.

GENERALIZED FO;CES SOLUTION OF EQIATION (To) ACcoRDING EQ(17( (UGT 6ELO1JGII.C. TO

INSTRU-P163 INSTRUMENIATION OF OSCILLATOR MEUlAI ION) DRIVING GEAR

C)

4X4 jxo 410 DISPLACEMENTS MEASURED FORCES X2 J F2