Springing of ships in waves

3 JUNI 1979

lab.

y.

Scheepsbouwkunde

ARCHIEF

_{Tech&sche Hogesdìool}

Def t

SPRINGING OF SHIPS IN WAVES

SPRINGING OF SHIPS IN WAVES

Pröefschrlft ter verkrijgi ng van de graad van doctor in de technischê wetenschappen

aan de Technische Hògeschool Deift op gezag van de rector magnificus prof. dr.ir. F.J. Kievits

voor een commissie aangewezen door het college vari dekanen te verdedigen op

donderdag 9november1978 te 14.00 uurdoor

Frans Frederuk van Gunsteren

scheepsbouwkundig ingenieur geboren te Wassenaar

it proefschrift is goedgekeurd door de promotor PROF. IR. J. GERRITSMA

Aan Julia,

CONTENTS

INTRODUCTION 1

THEORY OF SPRINGING 10

2.1 General 10

2.2 The seaway and ship response as stationary randôI

processes 10

2.3 Structural response to stationary r&ndom loadiñg 15

2.3.1 Structural idealization 15

2.3.2 Free vibrations 19

2.3.3 Normal mode methòd 27

2.3.4 Nónproportional damping 30

2.3.5 Response to harmonic loading 35

2.3.6 Frequency response method 40

2.4 Modified springing calculation method 43

NONSTATIONABY RANDOM EXCITATION AND RESPONSE 45

3.]. General 45

3.2 Time doínain computer simulation 46

3.3 Spectral structure of nonstationary data 50

3.4 Explanation of simulation results 52

WAVE EXCITATION 63

4.1 The seaway 63

4.1.1 Wave spectra 63

4.1.2 Wave grops 74

4.2 Wave exciting forces 75

4.2.1 VIbratory wavé excitation 75

4.2.2 Second order wave excitation 78

4.2.3 Whipping 79

DAMPING AND ADDED MASS 80

5.1 Damping 80

5.1.1 General 80

5.1.2 Hydrodynamic.da1ping 82

5.1.3 Structural damping 8

5.1.4 Order of magnitude of springing damping 86

5.2 Added mass 88

SPRINGING MEASUREMENTS 91

6.1. Full scale measurements 91

-6.2 Model experiments

97

7. CONCLÙSIOÑS

99

APPENDIX A : Egòdic processes ₁₀₂

APPEND:EX ¿ Ñonstationary spectra ₁₀₇

APPENDIX C : Response of lineär second order system

113

APPEÑDIX D Spectrum of amplitude modulated oscillations ₁₁₈

REFERENCES ₁₂₀ LIST OF SYMBOLS ₁₃₁ SUMMARY _L33 SANENVATTING ₁₃₅ DANKWOORD ₁₃₇ DE SCHRIJVER ₁₃₈

CHAPTER 1

Introduction

Wave induced main hull ship vib±atiôns are reported in a number of cases where thè. natural- frequency of the 2node vertical vibration of the ship equals the, frequency of encounteòf the

waves or a Iulti-p-1e of this frêuency.

Noticable cöntinuous 2-node vibrations, ecited by waves of

small length were reported in 1960

_Ill

for a 20.000-tdw bulk

carrier, in 1963 ₂₁ for a 47.000 tdw tanker. arid in 196 ₁₃₁

for some tankers.

In these cases a small change in speed or heading r.eul.ted into the eLimination of the vibrätion, which ïs typical for resonant

wave/ship phenomena. Therefore, other resonan.t 'phenomena such

as sh-ip/machinery and ship/propeller interactiöñs are excluded.

Moreover, the natural frequency of the 2-node vibration is too

small for sigñïficant interact-ion, with rnachiney an propel-ler.

In the last decade many report's have been made (see chapter 6)

of the continuous 2-rthde vertical vibration in moderate _sea.

states, especially for larger ships of 200 metres and thore in

length. _{The continuous wave induced 2-node vibrat-ion has been}

observed more frequently on ships f larger scale.

With increasing ship length the 2-nöde natural frequency is reduced and resonaht conditions occur for wäves of increasing wave lehth and wave energy.

The same effect occurs for increased ship speeds of foe: i±istance

container ships-.

The occurrence of signifidant continuous _{wave excited 2-node}

vibrations has only become apparent by the increase in size of

the ships and to a lesser extent by the- increase of ship's. speeds.

Large increase of the size of tankers and bulk carriers has taken place mainly between 1960 and 1970.

More and more sh-ips with lengths over 200 metres entered service and showed the continuous 2.-node vibration in moderate seas. Only since the 1960's concern about the influence of continuous

wave induced vibrätions on midship bending moments and stressés

has been expressed, espécially because not only the natural frequency decreases with ship size but the damping too.. Most

concern was exhibitéd for Great Lakes bulk carriers when these

very slender ships in the early 1960's increäsed to a length

over 300 metres

_141.-Also

iñ the mid 1960's a new denomination

has been introduced _{131 41} Sl in seakéeing ±èséarch

terminology : springing ,whiòh-dnotes the continuoús wave - induced 2-node vertical vibation. In general, the exciting

forces for springing aré active along the éntire hüll.

It is interesting to note that thïs phenomenon became clearly apparent when the ships increased in size over äbout .200 metres in length and that until that..time only the impulsive wave

induced 2-node vertical vibration, called whippïng was

investigated in seakeeping research. The exciting forces for

whipping are-only active on Certain parts of the ship's hull.

Whipping occurs in rough sèas- when, ship motions. are heavy and impulsive wave loads are- caused .by the large :reiative 'motions

and velocities between ship and wavé.

Impulsive or transient 'wave -loads result from slamming, from wave impacts on structurai thexnbers, from bow. flare and from

shipping green seas 6 . .

Ships of all sizes can suffer from whipping -if the sea state and ship's-speed a±e high enough.

In general, merChant ships reduce speed voluntarily in a heavy

seaway in order to prevent whipping and hull damage due to large

wave-impact förces. Navy ships, which are sensitive for whipping

due to their low hull stiffness, because of their mission,should

BEEGI-ILY 1972 INTERVAL.2-2 SECTION luME 01:32 COST MAY2 ,73 INTERVAL 42 SECTION LTIME 4:35 - COOT MAY2'73 INTERVAL SECTIONO TIME 507 BEEGIILT 1972 -INTERVAL 2-2 SECTION STIME 01:54 CORT I-IVY 2;73 INTERVAL 42 SECTION ITIME 4:44 A CORT MAY 273 INTERVAL 42 SECTION STRIE 5-09 BEEGHLY 972 INTERVAL 27 - SECTIONOTIME 6:17 CORTMAY 273 INTERVAL 43 SECTIONWTIME 525 - - BEEOHLV 1972 - -INTERVAL 2-7 SECTION O,TIÑE 6:32 COST MAY 2:73 INTERVAL 42 SECTION TIME 4-57

Figure

1.1 Examples of filteréd and unfiltered stress

recordings on two Great Lakes bulk carriers 181

resear.ch for navy ships for a long time.

This thesis is only concerned with springing.

It is noted that only the 2-node vertical vibration is

considered, because in practice stresses as a result of higher vertical modes and also horizontal and torsional modes are small in comparison with the 2-node vertica], mode stresses, although

displacements of higher vertical modes may be substantial _71.

An example of stress recordings, when springing wäs present, is given in figures 1.1 and 1.2. Figure 1.2 shows also typical springing stress spectra with sharp peaks at the 2-node natural

frequency.

Even before springing became more apparent in the 1960's on seagoing añd Great Lakes vessels with increasing size, Belgova

9 studied springing phenomena of inland water ships by means

of full scale observations, model tests and theoretical

investigations. Belgova io! already matched the normal mode method and beam analogy for the determination of the structural response with the simplified striptheory for the determination

of the generalized excitation in regular waves. About a decade

later two theories were reported by Van Gunsteren 1101

liii

and

Göodman 1121, both based on the saine principles for the

calculation of springing in regular waves. The modified

strip-theory 1131 (Gerritsma and Beukelinan) was used for the

calculation of the wave forces along the ship and the wave force contribution to the 2-node vibration excitation (participation

factor). The beam analogy and normal mode method were applied.

Both theories proposed the application of the superposition principle and the frequency response method for the calculation

of springing in irregular waves as given by St.Denis and

Pierson 14! in 1953. The observed springing phenomena could be

qualitatively explained by the theories of Van Gunsteren and

Goodman.

Since then, various efforts have been made to improve the springing calculation method and to verify certain underlying

assumptions:

- The method öf calculation of wave excitations by short waves has been slightly improved by Wahab and Vink 1151.

A A

£

_{'y -} -'

--i_

iiki.

&

SMALL SPRINGING STRESSES

EQUAL CONTRIBUTIONS OF 2 SPRINGIN'G ANO HÖGGING/

SAGGING STRESSES

o

3-Figure

1.2 Examples of midship longitudinal deckstress

recordings on -a tanker and a$sociated power spectra

1831 IOI

4 HOGGING/SAGGING STRESSES i i h I r

r,,1iflPFr

Id _{3 SMALL COMPARED TO} J SPRINGING STRESSES

The effect. of nonlinear bow flare induced wave forces has been

examined by Van Gunsteren 161.

- Moeyes 1171 has shown by means of experiments with a segmented

captive thodel in regular waves 1171, that the striptheory is not appropriate for the calculation of the wave excitation by

short waves.

- A simplified springing bending moment calculation method has

been given by Kaplan 1181.

-

The hydroelastic effects, 'such as the influence of buoancy

forces on the springing response, has been incorporated by

Van Gunsteren 1191 in his theory.

- Linearity of springing with wave height has been shown by

model test results at Webb Institute 1201 an by Kagawa et

al 1211.

-

Full scale measurements including the accúrate measurement of

the mode shape during springing have shown stationary nodes and mode shape so. that it can be assumed that the damping distribution is more or less. proportional to the mass distribution and the normal 'mode method can rightly be

applied.

Ail these springing investigations have not resulted in an experimentally verified calculation method, because still some major parameters, such as the wave exciting forces and the

damping, cannot' be estimated and because application of the frequency response method for springing in irregular waves has not been justified scientifically.

The last subject is addressed by this thesis and is the néw and. substantial part pf the thesis.

The applicability of the frequency response method for the calculation of springing in irregular waves has been taken for granted and has not been investigated before, probably because it has been justified in all other areas of seakeeping research. Chapter 3 shows that it is not justified for springing.

upon this aspect in his discussion of reference 1121 whiòh is

quoted as follows:

"In thè USSR a nuitiber of investigations have been carried out

concerning the problem of overall rigidity criteria and in

particular that of wave vibrat-ion

221 1231 1241 1251

1261.

The object of these investigations was the solution of

practical problems linked with the development of

regulations on overall strength and fatigue life of

high-tensile steel hulls of large ships.

In the above mentionéd papers two aspects of wave vibration

are considered:

vibration caused by wave impact

- vibrat-ion due to wa7e profile change along the

ship's hull

(in linear and non-linear formulation).

These researches have shown that dynamic bending Irtome1t

values and the number of impact vibrations decrease with the

reduction of overall hull rigidity.

From this point of view reduction of hull rigidity can be

öonsidered as a favourable factor.

The wave resonance

vibration is a more complicated aspect.

Under otherwise

equal conditions the reductipn of midship moment of.inert-ia

e.g; 1.6-1.7 times may lead to än increase of wave resonance

vibration amplitude by a factor of three.

But even that extended amplitude remains small enough for

ships with a normal ratio of mai-n dimensiôns (LIE < 14).

Moreover, a slight change in ships' speed or heading can

result in discontinuance of resistance and in practically

complete cessation of vibration.

The méntioned data were confirmed by full scale tests

on

the tanker Sof-a (L = 214 m, I

156J2000m2cm2).

It should be noted that resonance vibration could be

recorded ori board this tanker only in rather moderate seas

(Beaufort 3-4)

(Peak-to-peak value of midship vibration

stresses was 100 kg/cm).

Nô reonance vibration occurred

iñ seas of greater intensity.

At

e same time accordingto

the spectral method used by the author the resonance

vibration intênsity should on average increase with increase

in wave intensïty (growth of spectrum ordinates at fixed

frequency wz).

A question arises as to the caused of this discrepancy and whether' the spectral method is applicable in essence to the

investigation

of

resonance vibratioi.

A requisite condition of resonanòe vibration development is the stability in time of dynamic vibration characteristics

of the system (ship) an a relatively durable irifluetice àf

exciting forces., acting at a resonance rate. In an actual ituation of a sea voyage when the ship is moving in

irregular waves the realisation of these terms will occur

seldom enough. 'he occurrence of resonance vibration is practically possiblè only in a wave regime where the average period of the wave spectrum is -neat to the period of the

2-node natural frequency of the ship's hull. In these

conditions the pitchïng and rolling of the ship are usually absent (due to the- relatively small wave lengths) and in the wave spectrum waves with periods capable of generating

resonance vibration prevail.. These conditions are not taken into consideration by the spectrum theory.

Òf particular importance in estimating wave vibration aré the spectrum ordiñàtes in the region of apparent wave frequencies close to natural 2-node frequency of ships'

hull.

These regions of- high intensity wave regimes are located in

the high frequency part- of the wave spectrum, analyt-ical representation of which is- rather inaccurate.

-The above considerations suggest the necésity of amplify-iñg theoretical and experimental researches in the field of wave

vibration. These reséar5hes aré important -not only for the purpose of spècifyirig the excitatión forces (diffraction component, distributior along the s-hip length of relatively short waves) and defining the process of- vib±ätion damping

(damping rates) but also for the improvément of the method of spectrum transformations."

Moreover, Maicimadji touched not -only upon the applicability of the fréquency respör-ise methOd, but also on the difference between

impulsive arid continuous excitation (pinging and .whippinq),

non-rlineàr clontinuous excitation and the decrease of vibrations

with increàsing flelbility, which subjects haVe been treated by

Vari Gunstèren ih earlier work 161 1191 11781.

the other main problem. areas wave excitation and damping

-have been treated in chapters 4 and 5. No striking conclusions

are drawn, except that ho progress since 197Ó has beeñ made

apart from the test. results fOr wave forces on a resrre.

modél in short reguià± waves as reported by Moeyes _1171.

Chapters also gives a simnlified méthod for the calculation of added mass in springing calculations as developed earlier by Van

Günsteren 1841

liii.

1h òhapter 6 the springing measurements are dscused with

respect to their contributio to the improvement of t:he

springing calculation method and thé vérficatiön of àssuxnptiotis

underlying this method. No valid comparison of theOry and full

scale experiments is epOrtd due to làck of precise info nation

on waves, damping and uncertainties in the preseht sprinqing calculation méthod.

CHAPTER 2 -

-Theory of springing

2.1 General

This chapter gives a review of the theory of springing with speôial emphasis on the justification of assumptions used in the present springing response calculation method developed earlier by Van Gunstéren liii and Goodman 1121..

Theseassumptions are:

- linear-ity of springing with. wave height

-stationary random wave loading and thus. appiicàtion. of

uper-position principle and frequency response method

- application of normal mode mthod (proportional damping) -' only vertical 2-node vibration is relevant for springing

stresses.

2.2 The eaway and ship response as stationary random processes

St.Denis and Pierson ¡141 have challenged their quotation of a sayiñg by Lord Rayleigh - "The basic law of the seaway is the apparent lack of any law" - by a presentation, to naval

architects of a method fo the statistical description of the

seaway and the responses of a ship in it.

The method presented in 193 is a syntaxis of available

scientific results of oceanography (1948)

_1361,

of harmonie

statistical propertie of noise (1944/1945)

j27j,

the

statistical distribution of wave amplitudes 39 , of at that

time available knowledge of motion respor of a ship in regular

waVes (1950) 1401.

The validity of th-is method for the prediction of ship motions

in -irregular longitudinal waves has been

successfuY

investigated by model experiments 1411 131 and full scale tests

42l 1431

From thesè studies it is concluded, that the theory of randoth

processes can be adequately applied to quas-i linear dynamic

systems such as the longitudinal motions of a ship in a seaway.

These techniques have also been applied for the analysis of structural dynamics of aircraft subject to gust and buffeting

loads 1441 I4l 1461. The relevant conclusions of the method of

St.Denis and Pierson together with the rsults of subsequent

stüd-iés of the response of ships in a seaway 1471 1481 491 311 1501 are given in, the ñext section.

There-fore, the following assumptions are made:

- The surface of the waves is assumed to have a statioiay/

ergodic Gaus-iañ distr.butiOn

- A linear relation exists between wave surface elevation, wave loads, Înoion response and vibratory hull response.

The assumption of linearity is- generally justiied for springing

except for exceptional circu$tances discússed in chapter 4.

The assumptiôñ of ergodicit' of the wave excitátion and ship

response is subject of further discussion in chapters 3

and 4 iñ connection with the question i-f the f reqûency reponse

method can be applied to weàkly damped systems subject to quai-' and non-stationary random loading.

2.2.1 Energy spectrun

it has been shown 1141 1411 1311 I5l that the ea urface and

the ship's response in moderate sea states can adequately be described in à statistical sense by. its enèrgy spectrum, which gives the disttibution of energy as a function of frequency. The measuremeñt of sea waves shows that the mean energy per unit

area sea surface varies slowly with time. For a period of some

30 inthutes, conditions in thé öpèn sea can be considered

stationary, while in most cases a record of minimum 20 minutes contains just ènough-infòrmation for the purpose of spectral

anàlysis. Under the assumption that the process is ergodic, i.e. the time and ensemble averages give the sa-me result, the power or energy spectrum of the wàves ànd the ship response to the

waves, is obtained as presehted by Gérritsma ₁₄₇₁ and De Jong,

311 1521 as follows.

According to Rice 1271 the Gaussian random variable x(t) can

be given by thé rañdom phase model:

E c cos (u t + ) (1)

n n n

n=1

where the frequencies u are continuously distributed between '0

and , the phase angles

n are independent stochastic variables

with uniform distribution between O and 2ïr, a-nd c are the

n

amplitudes of the infinite number of harmonics.

Without decreasing the generality the meàn of x(t) may be

assuiiied to be zero.

Sorne propertiés of ergodic processes are summarized in Appendix

A.

The mean square valué can be obtained by substitution of equation (1) into eqiation (A.12) and using orthogonality

relations between

cos wt and

sin wt

E[x2(t)] ½ E c (2)

n=1

This value, which is also called the mean energy of the function

x(t), is independent of the phase angles

_Crl

Thé distribution of the inéan energy ½ c as a function of frequency u, is given by the power spectral density function

whidh is

determined

by:

= ½ c (3)

he continuous spectral density S(w) of the signa x(t) is in

thi représentation replaced by a discrete spectrum, which has

only values for w1, w, ...

N

= E. ½ c2 (w-w) (4)

n=1

where & dèlta function

It öan be shown by applicatioñ of thé central. lim,t theqrçm 1271, that equation (1) converges to a normal distributed process with average value zero and variance

.

S, dw (5)

when the diòreté sectrtiin

in équation (3) apprqaches to a

còntiñuous one f ór N ähd

w-O.

This definition of the spectral density

function or energy

spôtrim S(w)

i coñistèht with the following more general

expression of the mean square value of x(t)

T

hm

1

f

x2(t) dt

(6)

T+° 2T-T

Application of Parseval's thèòreth to Foui'ier tahsfoms iiès:

7

I(t)

2

d.t =

.!

7 _FF(w),I2 dw

()

-=

2iî-=

-where F(w), the

öuier

transfärm of

x(t),

is give1 by

f

x(t)e dt or T F w) =

um

f

x(t)eTt

dt

T

-T

13

c

½

f S(w) dw =

in which

S (w) dw

X

Substitution of equations (9) and (7) into the mean sqüare value

(6) gives c 11m

!_

.L J

IF(w) 12 d } (10) T-- 2T 2.ir or

S() =

-it j____

I

i - T-c 2ir T

This definition of the spectral density function S(w) is

generally applied in the field Of random data analysis

I

28

I of

seäkeeping research 1521 ₁₄₁₁ and in other engineering fields

I4I.

The power spectral density function or energy spectrum, S(w),

can be obtained f roiTi a recOrd x(t) of an ergdic random process

by its auto correlation function which is defined by

R(t) E { x(t) x(t

+ t)

} (13)

or

T

R(T)

= 11m i f x(t) x(t

+ t)

dt (14)

-T-

2T_T

Substitütion of equation (1) i-nto equation (17) and application

of Fourier tranforms gives the following relation between the

energy spectrum S.(w) and the. auto correlàtion function R(w):

1301 1291

I'I l5I:

S(w) = 1 f

R(T)

-iWT

(15)

Since

R(T)

is an even function, where

the enérgy spectrum can be written as thé Fourier transform of

the auto corfeiàtiön funötiön as follows:

S (w) = 2 f R

(T)ewT dT

X X

or

S, (w) e 2 f R (T) COS wtdt

52!

II

54! 41!.

Inversely, the aútb correlatiòn funtiön can be expressed by

R (T) =

L S(w) cos wtdw

2.3 Structural response to stàtioñary random loading

2.3.1 Structural idealizatioñ

The dynamic behavloùr of the ship's girder can be analyzed by

idealization of he ship's structúre uing generalized

coordinàtes or discrete coordinates.

The ship's structure is often discretized In a number of fi.nite elements, t'hih dépends on the required accuracy of the analysis and the geometry of struóture.

For the lower modes of vibration of the ship's girder a ónédimeñsionäl discretizatiOn, corresponding with the consideration of the girder âs a bêám, is jústiied.

More elâboràte disáretizatiori in two and three dimensions are required for the analysis of propeller excitéd vIbrations of

the aftérbody an deckhouses. These vibratioñs are outside thè

scope of this studr althöugh they aÍre coupled. The coupling effects are neglected in thé treatment of springing., since the natural frequencies of the aftérbôdy and dékhouse modes are

much higher than the natural frequency of the lower thain hull

girder modes. Òn the othér haiìd, the coupling effects should be taken into account for the analysis of afterbody, shaft and

deckhouse vibrations..

It has been shown

Il7I.

for a thoroughly investigated ship, that

a threedimerisional discretization does not increase the accuracy of the analysis of the lower modes of hull girder vibrations. In the present study the main hull girder is treatèd as a Timoshenko beam, in which the transverse section of the beam, originally plane, remains plane and normal to the longitudinal fibres of the beam ¿íter bending.

Analysis of the structural dynamics of the beam is complicated by the fact that the mass is continuously distributed along its

length. The displacements and accelerations must be defined for each point, since the inertia forces result from structural displacements, which in turn are influenced by the magnitudes of

the inert-là forcés. This closed cycle of cause and effect can be attacked directly only by formulating the problem in terms of partial differential equations because the position along the length as well as in time must be taken as independent

variables.

However, if the mass of the beam were concentrated in a series of discrete points or lumps, as shown in figure 2.1, the

analytical problem would be very much simplified, because the inertia fOrces could be developed only at these mass points.. In this case alsO the displacements and accelerations need only to be defined at the discrete points..

The procedure for defining the mass properties of a structure by a lumped mass system is to assume that the entire mass is

concentrated in the points at which the translational displacements are defined.

The structure is divided in segments; the nodes serve as

connection points.. The point masses are located at the nodes. Figure 2.1 illustrates the procedure for a beam type structure. The mass of each segment is assumed to be concentrated in point masses at each of its nodes, whereas -the distribution of the

segment mass to these points are determined by statics as

stiffnes segment Xi a coñtinuôus beam segrnent[ ej rn1 mass q Kt ILt

m= mass/unit length

m1= discretized macs k AG.

shear stiffness

o K1 bending joint oq1=shear joint L findtin1+rn24rn5

J.Lax k14Kk

rn*dxmLirL.

5m rn0 m1

,,iIÍrr_Tn4.4

ij

X?1

Figure 2.1

Discretizatiön of lumped parameter system

liii

1341

11711

In case of ship girder vibrations, the mass due to the hydro-synamic mass forces, caüsed by the vibration, are taken into

account. The coefficients of these forces are often referred to as "added mass". beòause the hydrodynamic mass is added to the structural mass for the dynamic analyis of free and forced

vibrations. Various aspects of added mass, such as influence of vibration mode, shipform, speed and motiôns on added mass, are

discussed in chapter 5. If the masses were not concentrated in

points, but had a finite rotational inertia (beam depth) then -the rotational displacements should be considered.

However, these rotational displacements which would double the number of required degrees of freedom of the discretized system, will not be taken into account in consideration of their small

influence compared to bending and shear for the lower natural

modes. It has been -shown by Timoshenko 11741 and others ₁₂₉₁

1351 that for a prismatic beam the effect ofrotatory inertia

on the natua1 frequency of the lower modes is small compared to the influence of shear (25%) and very small compared to the

influence of bending. The rotatory inertia cörrection terms fçr the natural vibration frequency, are very small for the lower modes, and slender beams, but increase with increasing mode number and decreasing beam slendernes (increasing, depth).

In practical springing situations, only the lower modes of shi-p

girder vibrations are relevant. Moreover,.only. slender ships

with a small natural frequency of the low hull girder modes are sensitive fór springing.

Consequently, the influence of rotatory inertia can be neglected for the calculation of the lower modes of free vibrations of slender ships and is neglected in the present study of springing.

The lumped-mass idealization provides a simple procedure of limiting the number of degrees of freedom tFïat must be

considered in dynamic analyses. The lumping procedure is' most

effective for systems where a large portion of the mass is

actually concentrated in a few discrete points-. In cases,

whére the mass 'is quite uniformly distributed-throughout the structure, the lumped mass discretization still may be used,

19

but an alternáti'ie approach to limit the degrees of freedom may

be preferable. This approach assumes that the deflected shape of the beam can be expressed as the sum of a seriés of specified displacement pattérns, whiäh become the di-p1ädement coordinates

of the beam. -w(x) = c1(x)pi + c2(x)pz or N w(x) E (20) 1=1

where the functions (x) are the N shapé, in whiôh thé beam

may bend, and are called generalized displacements and p1

determines the amplitudes of the respective fuñctiön (x) and

are referred to as distributed or generalized coordinates. The

number of shape patterns, or shape functions téptesents the number of degrees of freedom considered in the idealization of

the (beam) structure. The accuracy of the solütion will

increase with N, the number of shape functions. Älthough, a better accuracy can genealiy be achieved from a given nüihber. of

degrees of freedom by application of the shape f-unction

-rather-than by the lumpéd-mass äppròadh, it is reòognized, that greater

computational. effór is requited for eàh deg±ee of f-±eedoi when

the generalized coordinates are applied. Both methods can be

ùsed for the analysis of the structural dynamics of the ship's

girder.

If a structural system with constraints has a set of m

côordinates and if there exist among these coordinates r

-equations of constraint, then n = m - r independent

coordinates exist and the displacements and förcès may be

-completely defined by these n coordinates, which are cal-led

independent coordinates or-generalized coordinates (qi q.

n is also the number of degrees of freedom.

2.3.2 Free vibrations

The elasticity of multi-mass-systems cancovenientiy be

described by influence numbers. The influence number is

defined by: deflection in point, i, cuse by a unit forse in

poiñt

ï.

These influences numbers are calculated from the

stiffriesses of any system, discretized or not. The equations of motions can then simply be written as linear deflection

equations, instead of second order equations o motion, whiòh is

called the inverse method. Consider now the ship structure as a

long beam, discretized in n masses, m (i i ...n), which

can Ìnove in the plane through one of the two principal axes,

perpendicular to each other. The moving system can then be

substituted by a static system, which is loaded in the points, x., by the inertia forces:

- m. d2 w

-:1.

dt,2

in which the deflection, w, in point x is given by:

n

w.. m. d2 w .

i . i] (21)

dt2

This is a set öf n (i 1, ... n) linear, second order

differential equations in t, corresponding with n masses.

-According to the theory of vibrations, it is assi.imed that the

masses m perfQrm harmonic motions .with the same frequency and

different amplitudes. w. = y. Sin w t 2. J. so:

-

w2Y sin w t dt2

Inserting (22) in (21) gives n siinultaneo.ts equatiçns:

(m. .

. y. - 'i) O (ii, 2,

J=i iJ J

It is obseied that the àddèd iras

isfrequn

dépendép.t. This difficulty can be ovêxcorne by an iterative approach: using the

asmptotid alue of the added mass as a first apprbKithtiän. The flexibilityinfluenáê coeffiâieñts do riot exist fbi the

m conttaiñt

baiü büt can b

determined by pìacïn

fictituous

cOnstraints at the énds of the beàin acöordinq to thé

following

procedure given

by Koch

34 _. With these constraints the

fOrce-deflection relationships for the points i and j

can be.

established taking the effects of bending, shear and hydrostatic

bûoancy into account. Thén the équations (23) äre only à1id,

if the forces of support are taken into äccount. The set of

equations is hén: n 2 X X. X. y. = E mw a. .y; +

n+i

I

i

=i J Yo ₊ n+i -n+i

where the displaòeménts of the fictituous supports, w0 and

are' i'iiutrated in figue 2. i

and défiried

the conditiOns Of

equiiïbrium of the inertia fOrces.

n E m.w2

i=9

(x1 - x) y

= O

(24)

(25) (27) 21

and

n+]. E

mw2 X.

; = O (.26) i=1 J J J

The n equation, resulting from substitution Of (25) arid (26) into (24), have nntrivial sOlutiOns for wi if the detehiriant of the coefficients of w. is zero.

The igenvalues are obtained as outlined by KOch 34 I and

summarized for springing by Van Gunsteren ii

_I.

The eigenvalues àre giveri in notm1izéd fot by:

= deflection of normal môde r in mass poit...

The noal modes are normàliied by the following condition:

E

m. y2.

_i

ri

A détailed descripi.iono

the nercl procedure fOr the

discretiz-ation of the mass, the determination of the influence coefficients and the solution of the eigenvalue problem is given.

by De Vries 11.7.11 who gives alsO an approximate cOrrection of the natural frequency for the influence of rotatory inertia. A very important feature of normal modes is their orthognality:

m.y.y.=0 ifrs

₍₂₈₎

for which proof is referred to reference

_1341.

It is noted, that the influence of the hydrostatic buoyancy forces, due to the deflection of the beam, should be taken into

account in thé frée vibration analysis. In conträst with the

statement by Wereldsma 1731, that it is possible for the

elastic deformation to neglect the terms related to thé

hydraulic restoring forces, V

Gunsteren

11,91 has shown that

these forces can reduce the springing bending mqinent

of Grat

Lakes carriers by more than 10 percent and should be t'akei into

account.

One method of accounting for the influence of vibratory deflection induced buoyancy forces is:

- to neglect its influence on the normal mode shape - to determine its influence on the natural frequency by

application of the law of Southwell _{11! 11101}

- to account for -its influence on the response by including these forces in the vibration spring factor i-n the normal mode equation for the damped response to dynamic loading as.

indicated by Van Gunsteren in reference 1191.

Another way of taking thee fQrçes into account, is to account

for thém in the calculation of the free vibration

-characteristics of the ship in still water, whée the ship is

supported elastically by the surrounding water. This can be

achieved by extending the supports to all mass-élements and application of equations (22) (23) and (24) as suggested by Koch

1341.

Bishop et al

_-l154I

11721 prefers to consider the free

vibration chàractéristics of the 'dry" ship seperately and deal

buoyancy forces as applied forces. However, the free vibration behaviour (in still water) provides the basic information for

the wave-excited and damped vibration. This is the reason the

norma]. mode method is an. effective tool to determine the

complicated wave excited vibrations in a simple manner using the

free vibration characteristics. Therefore, it is logical to

calculate the physicalJy more real "wet" modes of vibration in in stili water instead of the hypothetical "dry" modes, bécause the influence of vibratory deflection on the wave-exciting

forces is small and can .be neglected. This observation is in

accordance with a statement by Wereldsma 11731. Thus, there are

no relevant coupling effects between free vibrations in still water and incident wave excitation and the springing response

can best be determined using the eigenvalues of the free

vibrations in still water, which is analogue to the striptheory approach 1131 for the determination of longitudinal rigid body motions like pitch and heave.

If all hydrodynaiic forces including ri.gid body induced förces would be considered in the free vibration analysis, then the

lowest modes of the transverse free vibrations would be the

translatory and angulär rigid body modes. In naval architecture

the vertical rigid body translatory and angular modes are called heave and pitch respectively.

In the last décades, extensive research has been carried out on the prediction of the ship's rigid body motïons like pitclì and heave the wave induced bending moments in the rigid hull girder

and the distribution of loads on the r4gid hull girder. The

distribütion of waves induced loads over length of the rigid

hull girder is relevant for thé springing response.. This wave

induced load distribution including the effects of rigid body

motions like pitch and heave are urther disçussed ir chapter 4.

Although for the load distribution the rigid body modes haie to be considered, structúral response of the rigid body modes and the lower vertical distortion modes, like the 2-node and 3-node vibration, can be treated completely separately due to the orthogonality of all thé natural modes.

The choice of a separate treatmeñt of rigid body modes and

deflection modes is mainly based on practical considerations. In the present study of springing only the lower modes of

deflection of the ship's girder are considered. Consequently

only the buoyancy forces due to the deflection of the girder are taken Into account in the determination of the flexibility

influence coefficients..

Distributed parameter system

The eigenvalues, with associated normal modes and nattiräl frequencies can be determined for a distributed parameter system -in a similar way as for a lumped parameter system.

For a system with n generalized coordinates (q, q ... q) and

known mass and stiffness matrices m and k the equations of

motion are

1ml {q} + kl {q} = O . (29)

Since

q = -w2q

These n algebraic equations in q with w2 unknown represent the eigenvalue problem.

Nontr±vial solutions give the eigenvàlúe 11w2 with corresponding natural frequency w and elgenvector {qJ, which represent a

particular natural modè.

For an n degree of freedom system thèré äre h such eigenvaluès

añd eigen-etos. It can 6e shown tha the eigenvectors satisfy the orthogonality relationships

{q}T

IkI

= p

} If W (30)

} 1ml {q} = O

It is noted that two modes with the sainé natüral frequency are not necessarily orthogonal.

The eigenvalues aré orthonormal if they satisfy equations (30) and the generalized mass Mr is unity:

25

The orthogoriallty relationship (30) shows that the eigenvedtors

are not dynamically coupled. Now the generalized coordinatê q,

which are staticàliy uncoupled, can be transförthed into normal

coordinates ì _{which are also dynäiali} uncoupled, be means of

thé linear transfOrmation.

{q}

= Iii

{y} (32)

in which the square transformation matrix il ïsconstructed

from the eigenvectôrs Of the system

{ï} = X]

{q}

Ç33)

where is defined by

2 T

} (34)

Further to the expression (20) for the displacement coor4i1ate

of a distribute parameter systeth, the displacement Qf a beam

can be expressed in normal coordintes and associated normal

môde shapes:

w(x, t

= i!l i

(x) (t) (35).

where w(x, t) is the

transverse

deflection of point x at time t

and

is the displacement confgat-ion of the i-th natural

mode.

It is noted that herewith thé dlsplàcethent is separated in .x

and t dependent terms.

It can be derivéd from equations (30) ana (31) that the normal

mode shapes r(x) satisfy the folow-ig orthogonality

cOnditiOns

£

f m (x) _(x) 5(x) O if r s (36)

f m (x)

o

(X) Cx)

r s r

ifr=s

(37)

För furthe± details of the free vibfation analysis arid the various methods to determine the eigenvalues and normal modes

reference is made to textbooks on structural dynamics 1291 135!.

Coupling effects

As discussed in the Introduction, the vertical natural

vibrations of the ship's girder are uncoupled from the horizontal and torsional modes due to a plane of symmetry at its centreline. These uncoupled modes are rather exceptions than a rule in

general dynamics of structure. Fortunately, for springing only

the lòwer modes, of the vertical hull girder vibration will be

considered.

The consideration of coupling of natural modes has now been confined to vertical lower modes of vibration.

Two coupling effects are, distinguished: - structural coupling effects

- hydrodynamic coupling effects.

Structural coupling effects do not exist theoretïcally, because the orthogonality of the natural modes are all uncoupled.

However, in practice, not only beam type hull girder vibrations will take place, but also local vibrations, vibrations of sub-structures lIke supersub-structures.

These substructure vibrations could be coTupled with the hull

vibrations. Their natural vibrations- have been calculated

in-cluding the coupling effects usïng various models for the

idealization of the hull and substructure of the ship 179

11801. The coupling effects between substructuré and hull

structure are neglected för the free vibration analysis for springing, because the influence of the substructure vibration on the natural vibration of the lower vertical modes of the hull

girder is small. On th other hand, the influence of the lower vertical hull modés on the natural vibrations of the substructure

cannot be neglected 11801. The effect of the substructure vibrations on the damping of the forced waves induced on vertical vibrations of the hull girder might be relevànt

according to Betts et al

_1941!

The hydrpdynamic coupling of the lower vertical modes exists theoretically, because the hydrodynamic mass is taken into account in the free vibration analysis as "added" mass and because this added mass is dependent on the frequency and mode of each natural vibration.

However, in practice, the variation of the added mass with the frequency and mode number is small for the lower vert-ical modes

of hull vibration. The determihation of the different added

mass distributions over the ship's lêngth for each natural

frequency and mode £ possibe.

The hydròdynamnic vibration induced buoyancy forces have no effect on the coupling of the lower natural vertical mpdes,

because they are independént öf frequency arid modeshape.

The coupling effects of the hydrodynamic damping forces are

dis-cussed later in the context of the forced hull 'g-irder vibration,

but are not relevant for natural mode coupling, since damping forces are not considered in free vibration analysis.

It may be concluded, that for the purpose of calculation of springing, where only the vertical lower modes of vibration of the hull girder are considered, the natural modes are uncoupled. This property appears to be important for the application of

normal mode superposition. This method is called the "normal

mode method" and is efficient for the solution of the multi-degree-of-freedom forced vibratiön with damping.

2.3.3 Normal mode method

When the displacements are expressed in terms of normal modes, the differential equations can be decoupled due to the

ortho-gonality of the normal modes. Then the n simultaneous

differential equations of motion of a structural system with n degrees of freedom can be described by n independent

differential equations The feasibility of the use of eigen

values and nOrmal modes in strüctùraidnàinics has been hôizn

clearly by Ìôh ₁₃₄₁ in i929, whereas the mètho fôr thi use

has läter been denoted as the normal-mode methOd.

Details of this method are given in various publications l9I

1351. ExtensiOh by Vãfi Guñtèen lui Of KoOh's thei5 fr

damped ibrations becâuse it ï'nipòrtänce för pflngiñg has

been given in referénce ].0j

lfi[.

-.

It appeár that both f Or the luiped paràmeér systémã wèll as

for the distributed parameter system , the eqüãtioñof motion

for a mOde òf vibratiòñ Of á béã±h, ubjeèt to liñéär vsdOùs damping aild éxterñalditribütéd forces, is the same: a linear secoñd ordèr différential equation.

For a lumped parameter shïp's girder, with exterñälfòrces Q1

and viscous dampIng foi-ces , acting on the mases n points x,

the eqätion Of thótioñ or thè r-th iödéof ibrätiòni

according to Van Gunsteren liii 1191:

n rl .

+ C

ri + w2 = Q r r r r r . (38) where

= rth mode vibration response parameter, implicitly given

by n w. = E n -y i r ri r i in which n n = E m. w. y r . i i lei

w1 = instantanèòu displacement df the i-th mas element

ri = r-th mode nhäpé displacement at i-th Iñass element,

Cr = r-th mode vibration damping coeffIcient, qien by the ratiò between the viscous damping coefficient Cri

and mass m.at mass i

c = Cri (41.)

r

m.

-ri

It is not&d that the. distribution of the viscous damping is supposed to -be proportional to the distribution of the mass over

thé length of the ship. -

-Nonpropörtional dampIng,. when the -damping is not proportional to

the mass, stiffness or a linear combination, is discussed later.

The- normal mode method cari ònly be applied for proportiohal damping.

For the distributed parameter system the equation of motion öf the .r-th mode is similar to equation (38) for the lumped system, except for the excitation, and reads:

N

+Wr

=r

r r r r

r in which

Mr is the generalized mass, given by equatiôn (37)

Nr is the generalized fötcé

In case of a distributed force having an intensity p(x, t) t

point x and time- t, the generalized force can be written as

Nr = ¡p ('

r (x) dx

This expression is obtained fìom the fact that the virtual- work by the distributed fOrce is eual to the virtual work by the

generalized force. - -

-If the force fúctioh p(x, t) is separable in x and t, then

.p(x,.t)=r

p(x) f(t)

.-

. -

(44)

where

p(x) = force distribution

function

f (t)

=

time dependent force or loading function

p0

= dimension lessmaximuxnarnplïtude of the intégrated

L

_forcé

_-.

_-

-.

t-For the séparable force fùnctibn., the

b-th generalïzed forcé is

Nr =

Po r

f(t)

-. (45)

where

rr P

rX

..

(46)

and is called the participation factör.

This factor rr is a measure for the extent to which the r-th normal mode participates in the total generalized load on the

beam.

Siibstitution of equation (45) into equation (42) gives:

PF

2

or

cr

_r

₊

_r

_{r -}

f(t)

r

2.3.4 Nonproportional damping

The normal mode method can oniy be applied if the damping is

proportional. to the mass, the stiffnéss or a li-near cöxnbinatioñ.

The equations of motioñ can be- uncoupled when the damping is proportional as given by equatioñ (41).

It can beshown thatfdr proportional dampingali points of the

structure ûibrate with the saine frequency and phase. However,

fòr nonproportorial damping the points of the structure vibrate

witb

the same frequency but different. phase angles.. The effect of the différences in phase angle on the rnoioti is illustrated

in fiqure 2.2. LEGEND Tr PERIOD OF VIBRATION

OO- tr

T

t r --.

T .-x X

t;+T

t r T

Figure 2.2

Example of sweeping of a beam

It is noted that the girder has xo fixed mode.

A system with propo±±ional linear viscous damping vibrates free-ly ïn a set df uncoupled modes, which haie the same shape as the normal modes of the undamped system.

The amplitudes o these free vibrations decay exponentially in

time and uniformly over the irder. The modes of these

vibrations have stationary nodal points.

On the othét hand, a system with nonproportional linear viscous dampIng vibrates freely in a set of uncoupled modes, where all pòints have an exponentially damped motion at the same

frequency, but have different phase angles. These modes are

hot lie the Ìuiódes ö the ündthnped System and are charàctéflzed by nonstationary "flodes" This phenomenon is called "sweeping"

Referénce is made to Hurty and Rubins.tein 291 for the sOlution

of the equations of motiOn for proportional compared to

non-proportional damping For the purpose of analysis of pringing

the damping is assumed to be propOrtional to the mass, stiffness

or a linear combinatioñ. This assumption is justified, because

sweeping has not been measured and the major part of the. damping distribution at very low frequencies., i.e. structural damping, is proportional to the mass distribution.

According. to chaptèr 5 the hydrodynamic damping i very small

compàrOd to the structural damping for springing vibrations. Full scale measurements along the length of the M.V. "Stewart J.

Cor' ¡.7

I (sée figure 2.) do not indicate the presence of

"sweeping', which is expected to be present in. case Of nönproportional damping.

ODO GREAT LAKES BULK CARRIER

LONGITUDINAL AXIS .

-Figure 2.3

Measured 2node vertical hull deflection profile.171

These measurements are conclusive in this rèspect, because the hydrodynainic damping distribution is extremely concentrated at the aft ed of the ship where the ,shipsections have an extremely high readth-draf t ratio. See figure 2.4.

-t-s.j caer

Bodyplan of.J.SCort

Io

0 4' 2 /2 / 20 24' 22 12 40

2r2T/ar/ '','ge

Figure 2.4

Calculated longitudinal distribution of sectional hydrodynam.ic damping coefficient for two Great Lakes

bulk carriers 19J 33 12 21. -t 8 1. 1. 8 12 Sodyplan of 1000 design

Although the sweeping effects suçh as moving nodal points and motion phase differences between various poiñts of the hull girder have not been observed by full scale measurements of continuous springing, it is not êxcluded that sweeping may occur

during transient hull girder vibrations (slaxnining, whipping).

During the HNLMS "Tydeman" measurement

I 1721 the wide afterbody

was subjéct to afterbody slanning in following waves and produced sweeping type noviñq hüll vibìation.

When dealing With transient response effects and impulsive types of loadings, also higher mOdes of vibration and the coupling of these modes by the influence Of nonproportional

damping should be considered and. can b calculated by solving the coupled equations of motion using complex modes of the

equations of motion 1701 1711 and application of a more

elaborate idealization of the hull girder than given by thé beam

model.

This treatment of transieñt vibrations is oùtslde the scope of

the study. It

Is

assumed that the influence of-nonportional damping is small for the purpose of analysis of springing.

Comparison of equations (3g) and (4), for the r-mode of a lumped parameter system and distr-ibutéd system, shows only a difference in right hand terpi for the external forces.

Equation (47) is also applicable for lumped parameter systems for the following expressions of the generalized mass Mn generalized fòtce Nr and pàrticipàtion factor Fr:

The generalized mass

n

M = m. y y for = s (48)

r

i=0 i ri si

The r-th generalized force Nr is obtàined from the same virtual work considerations as with the distributed force function and

is

n

= E Ql

If- the forces or Q (x1 t)} are separable ïn x and t, then

(51)

35

Q (x1, ₌

p (x) f

(t) (50)

where Po is the peak value Of Q (xi, t) and p (xi) is a

dimensionless force distribution function and f (t) is a time

function.

Then the participation factor is deftned by

n

Fr

p(x)

ri

1=0

It is concluded that equation (42) represents the équation of mOtion of the r-th thode of vibratioñ for a lumped- as well as

a distributed parameter systeme Since the external forces on a

ship are deterininéd by mean Of thé st±iptheory and the

excitation is calculated for longitudinal distributed sections. of the ship, the lumped parameter system is most suitable for

further use in the analysis of springing.

The equation fOr the 2node index 2 vertical vibratory response

112 to the excitation is given by

N2

112 + C2 112 + U) _{fl? =} (52,)

whére N2 and M2 ïs giien by equation (45) (48), (49), -(5Q) and

(51)

2.3.5 Response to harmonic loading

The solution of equation (47) öan be obtainéd é.g. by

applicatioti- of Laplace transfOrm -to the equation and is given in Appendix C. The ésúlt is given by equation (c.14):

--. p t '-½c .(tX)-O:r [ j e sin{wdr(t_A)}f(X)dXJ (-53) r o dr

whére B1 = r r drt) cos and or

= damped natural frequency

wdr -½c t. r wd

c2

_r

A = (duinmy)variable

Tbe response for zero initial cond4tiofls, is given by

substitutioñ of flr(0) = = O into equation (53):

r

t

flr(t) o r H(t-A) f ÇA)dA where ½c (t-A)

H (t-A)

= -

e sin w (t-A)

r w dr

It is rioted from equations (c.1O), (c.11) and (c.15) of appendix

C, that H(t-A) is the impuls respons function and gives the

response the r-th mode in the time domain to a unit impuls

Hence equation (56) is the convolution integral which gives the

general response flr(t) f Or thé r-th mode of the system described

by eqüation (47).

Equations (54) show the iñfluence Of iñitial conditions.

The respoñse to simple harmonic excitation is found by

sub-stitution

of

f(t) = cos (wt)

Sin

_ar

(54)

f(X) = cos (wX)

into equation (53).

The result is after some algebra-ic treatment:

i +

n±(t) =

½ Mr ¼ C2 ₊ dr + -w) w

-w

+ dr -- cos wt ¼ c + (wd - w)2 steady state ½ c - terms P F

_{or e}

_½Crt Lr Wd si-n wt ¼ c2 + (wd + w)2

½c

-+ r sin

()tJ+

¼ c. + dr - w)2 + ½ ½crt ½ C Sin

wt+

_{(wd+w) òos} M _wd ¼ ç. _dr + w)2 - -transient - terms

½Cr sin.w.t + (wrw) cos wdrt]

¼ c

_(w -, w)2 (B3 BL.) + Bi + B2 terms for initial öonditions (58) For zero initial conditions, flr(0)

= r«(0) =

O,

and

after

rearrangement of steady state

and

transient terms, the response

of the r-th mode to a cosine excitation is given by:

in which

Porr

Rr cos (wt +

-2E-

e_rt wr

RCOS

drt + dr

M _wdr R = r static 1

- cw

r = arctg

w2

-- ½ cr (w + _w2)

=arctg

{ ü)at ( - w2) r

The first terni on the right-hand side of equation (59) represents

the steady state pare of the response. the other terms on the

right-hand side of this equation represent the transient part of

the response.

The second term on the righthand

sid

is

the transient part o

the respòriê tO àn harmonic cosine typé excitation, in Oase of zero initial conditïohs.

The importance of this transient term in case of random excitation is discússed later.

It is noted, that Rr has a physical rneaning. It répresents the magnification factOr, which gives the maximum amplitude of the response

r in the r-th natural mode resulting from the loads

being applied dynamically as òompared with the same loads

applied statically:

max

- magnification factor

(59)

The teady state response is given by

or

-

g

r

since the second term of equation approaches zero with increasing time t.

Equation (63) can be written in terms of the complex frequency

response H(w), which reptesents the output ifkput ratio of a linear system excited by an harmonic input with frequency w. Substitutiohof the input

iwt

f(t) = e

into equation (47) gives the output

p _r. M; (w

-

w2 icw) Hence: i r w2 - w2 + ic w r r

Separation of amplitude and phase by

Hr(U))

IHrI e

r

Rr cos (wt

+ Ïp)

(63)

gives he relation with équation (63):

The frequency response is not only useful for determination of the response to harionid excitation but also to randoiü

excitation, when transient effects may be neglected and the system is linear.

Then the response to random exc4tation is supposed to be the sum

of the response to the harmonic components whiCh constitute the

randOm excitation

Springing is supposed t be linear. Nonlineat response is not

f(t) (64)

(6-5)

(66)

further studied. However, the other condition for the

application of the frequency response method is further analysed

in the. next chapter: the influenòe of transient effécts.

But, first the frequency response method is further sitsnrized.

2.3.6 Frequency response method

The mean square or variance of the response öf a single degree of freedom lïnear system, e.g. the r-th mode of vIbration, súbject to random loadïng can be determined by application of the frequency response method as follows using the results of

appendix C.

The mean square of the response flr(t) is according to equation

(11) and (12):

2f S

(w)dw nr where (w)

hm

1 i

F r'2

nr T- 2t T F»w) =_: flr(t)

et

dt

Substitution of equation (c.20) into euätions (67), and (68) gives the mean square value of the response n(t) of the linear single degree of freedom system, given by equation (47), to a random ergödic excitation

PF

_{o r} M r EIfl(t) I

= 4

o1r

'r

Sf(w)dw (70) where

equation (65)

-Sf(u) = spectral density function of the excitation f(t), defined by equation (13).

A convenient approximation for lightly damped systems can be

made by approximation of the power pectra1 density of the

excitation, Sf(w), by its discrete value at Wr

For a lightly damped system, where cr«l,the magnification

f-actör is sharply. peaked and excitation powr spectral dy

only coñtributes to the response for a small bandwidth of

frequencies- around w. Therefore the excitation power spectral

density function Sf(w) in this bandwidth is assumed to be constant

Sf(w) = Sf(w) (71)

Substitution of equation (71) into, equation (70) gives

E [(t)] =

Ir Sf(w) IH(W)I2dW -,.

7)

For a lightly damped single degree of- freedom system as given by

equation (47) , the- following properties can be derived:

The, half-powe±-.point bandwidth Bi,, as defined by equon (74),

is

B c

r r

The half-power point bandwidth Br is defined by

Br = lw-i -

w2l-where

-IH(wi)I2 lH(w2)12 = ½IH(w)l2

lHw')l = peak value of the frequency response function

The maximum value of the òmplex frequency response is

f

_H(W)12dW

I H(()p) 12 B (76)

Substitution Of equation (73) and (75) into (76) and (76) into

(72), gives the approximate mean square résponse cnr òf the

r-th modé f a lightly damped single degree of freedom system,

to random loading Porr f(t), as given by equation (47).:

pr

E[rijt)] o.r

S(u) ¼

(77)

For springing the applicàtïon of this very simple approximate formula is justified since springing vibrations are very lightly

damped.

The results are ñot only applicable for a sinqlé degree of

freedom system but also to a multi-degree-of-freedom system when the system is given by normal coordinates

r' because the

equatiotis of motion fOr such a system are decoupled.

It is recalled that the displacement of the beam may be written in terms of the normal coordinates fox the distributed and lumped parameter system by equations (35) and (39) respectively.

The mean square value of the response w(x, t) for the

distributed system described by equation (47) can then be

approximated by: n

p2r2

E[w2(x, t)] = E

c2() .0

r Sf.(w) ¼

-r0

o

M2

cr (78)

The approximation is possible by disrearding phase relations

between mOde responses and crOss product terms.. These two

assumptions are justified for lightly damped systems. Fu±ther

details are given by Hurty àñd Rubinstein

1291.

In a similar way the méan square of the response w for a lumped

parameter system to. stationary random loading can be Obtained by

Hr(Wp) = r

(75)

mode superposition, as is described by Van Gunsteren liii.

2.4 Modified springing calculation method

Van Gunsteren ill has given a flow chart of the calculation of

springing. In general, this scheme for the method of calculation is still válid ã±d is represented in figure 2.5

W, Od d r OCt P

St,F!ntSO 0,.trIbUtÌOfl,

Ost,., , sed mass

Prob.b I, ty c.tc,u(.tP0n

¶

Figure 2.5 Flow chart of the calculation of springiñg

4,3 Se, nP0,,P - N.tu.Ifr.Ou.nc y

S.ct!0OSI

act!., t,on curt.

Sb,, ,peea S.ctPonIIadde a

w

In general, it is concluded from calculations, ill and full scale measurements

I 831 1 7:1, that springing is dominated by the

weakly damped 2-node verticalinain hull vibration mode. Thus, the mean square of the springing response is given by

equation. (77), provided the freqi.iency response method is valid.

From the next chapter it is clear that this condition is _not

satisfied for weakly damped systems such as th 2-node vertical main hull vïbration mode (springing).

When the frequency response method is not valid, _{e.g. for}

damping ratio's of less than 3 percent and for nonstationary excitations with respect to short time durations, the above calculation method can be slightly modified in order to take the transient effects into account. This can be done by means of application of a correction factor to the mean square of the springing response.

For the time being, untill further results of research _{on the}

influence of the type of the excitation on G(=G) have been

obtained, the factor Gr as given in figure 3.2 is applied as

correction factor: 2 2 e r s where 2

e =mean square of the springing response including

2

nonstationary and transient effects

=mean square of the springing responsé acòording to the frequency response method

Gr =springing response correction factor for. the r-th mode of vibration

can be expressed as the mean square of the displacement

springing response E{w2(x,t)} or mean square of the normalized

springing response E{2(t)}.

E{w2(x,t)} is given by equations (78) añd (35).

The correction factor Gr can be determined from figure 3.2.

CHAPTER 3

Nonstationary random loading and response

3.1

General

The résporie. system to stationary random ioäding can be determined straight forward by application of the frequenáy response method, discussed in sectiOn 2.3.6.

However, the loading may also be nonstationary. This sectipn

investigates the statistical propertiès of the response of a single degree of f reedöm system to honstationary random loading. Multidegree of freedom system respOne to this loading càn be Obtained by mode superposition under conditions mentioned in

section 2.3.6, e.g. independence of the modes. Therefore the

generality of the discussion is not restricted by the òonsideration of a single degree of freedom system.

1h practice, the excitation of a ship by waves can be

non-s.tàtioha7 düe to:

-- the energy contained in the waves is dependeht of wind force

ànd swell; variability of the wave spectrum due to chaiges in wind arid swell are taken into àccount by long-term

ditr-ibùtiohs

quiçk changes of the energy in the waves over small periods.

This last aspect of nonstationarity of the wave spectrum over small periods of time or small number of wave oscillations is

the subject of this chapter.

First of all the question should be addressed if there is a differeñce in respone of a lightly damped system to stationary

or nonstationary excitation. If there would be no difference

the practical importance of the short time rionstationarity is

small.

By means of computersimulation it has been investigated if there is a difference between the response calculated with the

assumption of stationary excitation and the exact response.

The results are given in the next section and indicate a large

influence fOr small damping ratios, which are typical or

spinging.

In section 3.4 the computer simulation results are explained _y

analytical .rèsults using spectral descriptions of nonstationary data, which are summarized in section 3.3.

3.2 Time domain computer simulation

The response of a single degree of freedom with mass m, damping coefficient c and spring constant k, to random loading f(t) has

been simulated on a digital coxrputer.

The equation of motion is given by

mx + cx + kx = g(t) (80) or x +

+ ux =

g(t) (81) in which = /c damping atio Cc e cr4tical damping = 2 m

= natural frequency

k = spring constant

= mass

c e visôoùs damping coefficient

Equation (81) is sImilar to the equation for single mode springing given by equation (47) by substItutioñ of

(.) ü)

r x

= 2w

X

f() =

g(t)

The excitation g(t) has been generated by means of passing 8192

random white noise data, spaced half a second, through a haping

filtèr of t-ransfer fuhction

u(

\

A p

-

r-Fi

(p+a)

A proper choice of k and r can approximate the Pierson-Mosçowitz form of the spectrum.

The energy spectrum of the considered excitation sample is given

in figure 3.1.

The geñeratioñ in the timé dòman of the exòitation g(t) is

developed by 'Pasveer I 72 I,' based on the pr-inc-iplés of reference

1651.

The e,cact response x(t) of the mechahical system té this

excitation g(t) is. obtaihed by Pasvêer 172 using numerical

methOds.

-(82)

n 06 0.2 0,2 Wr Iw010 51r /Wp 1? Wr/Wp

WrNATURAL FREOUEY OF r-th MODE

wp. FREQUENCY AT PEAK OF EXCITATION

-SPECTRUM

DÁMPINE- COEFFICIENT ----. PERCENT OF CRITICAL DAMPING

Figure 3.2

Computersimulation results

0.5 10 15

CIRCULAR FREQUEÑCY W - RAP/SEC

Fiure 3.1

Ene±gy spectrum of excitation used in the

computer-simulation IO 08 0.6 I 'n I. 6 9 10