Experimental analysis of ships springing through modal techniques

CONTRIBUTIO TO TXS DISCUSSION OF CO1i1ITTEEREPORT 1.2

by R. Wereldsma, Ship Structures Laboratory,

Department of Shipbuilding and Shipping, Deift University of Technology.

* **

R. WERELDSIsIA (The Netherlands)

EXPERIMENTAL ANALYSIS OF SHIPS' SPRINGING THROUGH MODAL TECH1'TQUES SUMMARY

An analysis of the ship motion and bending problem is given. For this analysis the "Normal Mode Method" is applied. It is proposed to use a segmented model oscillator to determine the hydrodynamic coefficients controlling the springing phenomenon of ships.

LIST OF SYMBOLS

C = hydrodynamic damping

[C] = hydrodynamic damping matrix

F = forces in ship coordinates

i = imaginary unit

M . mechanical mass

M

= hydrodynanic mass

IM] [M] = mechanical mass matrix

EM] = 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,

¡ = generalized forces

4' generalized displacements

w = circular frequency INTRODUCTION

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

(sagging and hogging) and the two-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 trarifòrmation 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 oscillator, which makes it possible to determine

experim-entally the generalized hydrodynamic coefficients for the modal deflec-tions of the ship's girder) a necessary information for springing anal-ysis. As a first step linearity and frequency independency of the

hydro-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 approach by Bishop et al. /2/ the problem will be oken down in two parts i.e. the pure mechanical problem ("dry ship") and the hydrodynamical problem, being a

general-Report No. 206., Ship Structures Laboratory, Deift University of

Tech-nology.

The expression "Dry Ship" for the mechanical problem has been

o

ized description of the effects of waves and water.

In Fig. i a simplified discretization of the vertical displacements

is sho'm. 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 lody motion and bending flexibility. From this lumped parameter approximation the set of equations, governing the static and dynamics of the "dry ship" can be written in abbreviated form

as follows:

[]{x} + [](J = {F}

(i)

The stiffness matrix [s] includes the elastic stiffness against shear and rotary deformation. The displacement vector {x}- includes also the rotations of the elements. The hass matrix [M] is diagonalised. .(FJ equals

the forcing action,, generated by the waves.

Since the ship operates in water all its sections will experience

forces due to accelerations {)., velocity {.k} and displacement .(x}.

These forces are generally described by "hydrodynamic 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 (hydrodynamic spring effects).

For the complete description of the problem (i.e. the problem of the "wet ship") the set of equations (i) has to be completed with the

hydrodynamic effects as follows:

+ [i]{i} {F} _{- I}

-

[]{} - []{x)

(2)

Having knowledge of all the elements of the matrices and the forcing function {F]-, the equations can be solved, resulting ïn vertical ship motions, vertical girder bending and,' if applicable,in vertical girder

vibrations. The mentioned motions and deflecfions htrngmechan'ic and hydrodynamic interactions and make the understanding and the insight in the hydro-mcchanics difficu]t. Therefore a transformation to "Natural Coordinates" will be performed.

DESCRIPPION IN "NATURAL 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 deflections, co-inciding with the free 'body motions and the muitinode vibration shapes.

These standard deflections, norjrialised in one or another way, possess

the property of orthogonality, when they are mass weighted ¡il. This transformation reads in matrix form:

b

14-noded (bending)

3.-poded (bending)

2-noded (bending) trpe of deflection

1-noded (pitch) 0-noded (heave)

The

columns

of the transformation matrix [T) represent the mOdal deflections of the girder, in a normalised version. The vector

repre-sents the strength of the various modal deflections. The vector lx) equals

the resulting displacement in ship coordinates.

For the forcing vector (F} a similar break-down into a series of modal distributions is necessary. This break-down is based on the fact that a distributed force proportional to the mass distribution multiplied with the modal deflection causes a beam deflection equal to that modal deflection. The amplitude of this distributed force _.(rJ- equals the gener-alised force

and fits with

the "natural coordinate" system.

The relation between {F} and {FJ. reads as follows:

0-noded (heave) -

T00 T10 T20 T30 T140

1-noded (pitch) -*

T01 T11 T21 T31 T141

2-noded (bending) -'-

T02 T12 T22 T32

T142 3-noded (bending) - T03

T13 T23 T33

T143

Ji-poded (bending)

_-' T T114 T214 T314 T141 F0 F1 F2 F3 F14 s

(3)

or:

(14)

Now the rows of the transformation matrix represent the modal de-flections in a normalised version. So the vector of the generalised

forces r} can be obtained by premultiplication of the frce vector .fF}

with the transposed coordinate transformation matrix T} . See /1/.

When transformations (3) and (Ji) are carried out on equation (2)

the problem has been converted to the new modal coordinates and the

equa-tion reads:

+

.[T)T[I][T]).

=

r}

[T]T[][Tj}

[T]T[)[T]{}

-

[T]T[S1[T]{

(5)

Further reduction gives:

[]}

= r} _{- [Mil} - [c]{) - [s]{p}

(6)

L_mechanics__i wave ' hydrodynamics

forces

T00 T01 T02 T03

_0C

x1

x2

x3

. = T1,0 T T12 T13 T114

T20T21 T22 T23 T214

T30 T31 T32 T3 T314

i

or

.(xJ. =

x

T140

_ii

T142 T14,3 T

ò

Equation

(6)

describes the ship motion and girder deflection problem

in a suitable 'way. This because the

[s)

and [M] 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],

[cl

a [s]

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

SEPARATION OF MOTIONS ARD DEFLECTIONS

For the case attention is focussed on the springing problem, i.e. the resonance phenomenon of the 2-noded girder deflection, only three

principal modes will be considered, i.e. heave (ip0), pitch (ip,1) and the

two-noded mode

(ip2).

Equation

(6)

reads now in expanded way:

--

o o

o

- M00

O 0'

o o

o

O M11 O P1 22 L 2 J O M2 ₂

--=1

rol

M00 MO2 02

10

c00 C01 CO2

10

s0

s02

F1

h-

M22 22

I

-

c10 C11 C12 - S10 S12

r2J

M20 M22 22

12

c20

C22

C22 12

s20

s21

s22

(7)

Ship motions can now be separated from ship deflecions as follows:

Heave:

M000 = r0 -

M000 -

M01f1 -

MO22 - C000

CO2,1 -

CO22+

-

50011)0 -

oii -

S02i1)2

₍₈₎

Pitch:

= r1 -

M20&0 - M111 - M222 - C100 -

C12i1

-- slOipo --

S11ip1 Springing: +

_M11

= r2 - M20f0 -

M221 - M221)2 - C200 - C22i1)1 -

C222

+

-20O

- 2l11 52211)2

(io)

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:

M00 total mass of "dry ship" M00 = added mass for heave

r0

total vertical 'wave force

C00 hydrodynamic heave damping

S00 displacement force due to unit draft increase

M11 total moment of inertia measured through centre of gravity of the "dry ship"

M11 = added moment of inertia measured through centre of gravity

r1 = total pitching. moment of wave force measured through the

centre of gravity

C11 = hydrodynamic pitch damping

S11 = pitch moment of displacement forces due 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 wavegenerated ságging-hogging load distribution M22 = added mass for the 2-noded girder vibration

C22 = hydrodynamic damping of the 2-noded girder vibration

S22 = additional bending stiffness due to change in displacement

caused by sagging-hogging deflection. E)ERIMENTAL 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,

22 being the

gener-alized stiffness and 1422 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

eiern-ents of this part of the equation.

a. Measurement of the gener]ize6 force theshiípotton-interactions

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

F2 - M20 - M211 - - C211 - - S2,1

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 M22, 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.

b. Experimental determination of hydrodynamic coefficients for elastic ship deflections

Hydrodynarnic 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

2'

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;

*2' 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 proportional to acceleration proportional to speed

proportional to displacement For harmonic motions holds:

F0 x0S0 +

iwx0C0 - w2x0M0

or _xô (s0 + icC0 - w2M0)

If the motion of the oscillator is adjusted to *2 we have: X0 = T0 = T12 x2 = T22 X3 = T3 '

x1 = T2

X5 = T52

Equation (12) refers to the coordinate transformation. The forces F05 can now be expressed as:

F0 = T02 + iwC0 - w2M0) F1 = T12 (5

+ iC

w2M)

(13)

F5 = T52 (S5 + iwC5 - w2M5)

into three components, i.e.: (M) mass

(C) damping (s) stiffness.

h or in matrix notation:. F0'

S00

0 0 0 0

C00

0 0 0 O F1

O S10

O O O

O C1O

O 0 0

loi

F2

00 S20 00

_[T}O+iu

00 C20 00

F

00OSOO

I

00000.0

I 3 F 0 0 0

0 S40

O O O O

C40

F

000005

O0O00C

--

5 '

--

5-

-- - _5. -where [T)= o o o o o M1 O O O O o O M2-0 O O o o o M3 O O O O O O M4 O O O O O O M5.. T00 T01 T02 T10 T11 T12 T20 T21 T22 T30 T31 T32 Tho T11

T2

T0 T51

T52

t

t

-'2--noded mode_{1-noded mode}

0-noded mode

{} =

[T]T[s][T){o _{+ iw} [T]T[C)[T]{0 _{} - w} [T]T[M)LT]{0 } (15) "2 - --- 2 -- '2 --

--After multiplication we obtain:

- [r01 s00 s01 s0 o - C00 C01 C0 o - M M01

M01

riF

= 1O ll O + iw c10 c11 C12 o

-

w2 M10 M _M O

L2J.

₂₀ 2l 22 '2J C20 C21 C22 _'21 M22 %2 (16)

The following set of equations is obtained:

b

= S02(2 +

it

-

_w2o2,*2 s12*2 + iw

- W2M*

= + ) C22i2 - w2M22*2

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

com-ponents the following equations are Obtained for r2.

2in phase = (s22 - 2M22)ip2

(18)

"2quadr. = wC22ii2

So, for various measurements, with different frequencies w, sufficient information is obtained (i.e. (1

_).

and (r

2 in phase 2 quadr.

together with *2 as input) to solve equation (18) and to determine

22 C22 and M22, being the unknown coefficients of the springing equation (1:0).

From equation (i') follow also other terms of the coefficient matrices and generalinformation will become available when the

sec-tional oscillator is adjusted to , q and/or higher order

deflec-tions.

PROPOSED INSTRUIVNTATiON 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 r0, i

and F2 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 ("i) 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 //.

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 oscillâtor 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.

BEFEREN CES

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

Hall, 19611.

/2/ R.E.D. Bishop and W.G.. Price, On the relationship between 'Dry Modest and 'Wet Modes' in the theory of ship response, Journal of Sound and Vibration,

_{1976, pp.}

_157-161Ï.

/3/ R. Wereidsina and 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 _19111.

*

LIST OF FIGURES

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

ship.

Figure 2. Schematic arrangement of vertical segmented oscillator.

7 .7 r-. FORCE PICK UPS 6 SIRAINGAGE INSTRUM ENTS £ WEIGHTING FUNCT IONS. AÑO

SUM MAT ION OF

6 WEIGHTED SIGNALS

DRIVING GEAR

MECHANICAL PART OF OSCILLATOR

iDEFLECTION

FtG.1 LUMPED PARAMETER APPROXIMATION OF

THE VERTICAL BENDING OF A SHIP.

6 STIFF BEAM SECTIONS

6 FORCE PICK UPS 7 ADJUSTADLE CAM MECHANISMS 7-ELASTIC HINGES S SEALIHOS

M3S

M Jx3 4xL 4F3 Mj C3 C 53 143 C3

FIG.2 SCHEMATIC ARRANGEMENT OF VERTICAL SEGMENTED OSCILLAIOR.

ADJUSTED TO 'V2

FIG.3 INSTRUMENTATION OF OSCILLATOR

M' C' s' Mr MECHANICAL MASS S r MECHANICAL STIFFNESS X DISPLACEMENT Fr WAVE FORCES Mr HYDR. MASS C r HYOR. DAMPING SHYI7R. BOYANCY M5 i DRIVING SHAFT 6 MODEL SECTIONS OF SEGMENTED MODEL DISPLACEMENTS MEASURED FORCES HYDRODY14A1IIC MASS C5 _{HYDRODYNAMIC DANPÍPG} 1 DRIVING SHAFT IN PHASE/QUADRATUNE REFERENCE GE1JERATOR QUAD.

GENERALIZED FORCES SOLUTION OF EQUATION (19)

ACCORDING Ea(17) (NOT BELONGING TO

INSTRU.-M EH 1AT ION