Second order non linear effects in marine systems

Novembre- '1986 'ECOND0,141:!ERT-NON,LINEAR.EFFECTS", -IN ,n*FHOE sy;TEnns , y.

:Scheeijsbou4,0*

Tealini,itheliogoschrool

VALE

SECOND ORDER NON-LINEAR EFFECTS

IN MARINE SYSTEMS by A. PITTALUGA M. DOGLIANI Technical Bullettin N. 98 Genova, November 1986

-Tr: The,, traditional _{approach to the study of: theil--.behaviour of} ships and'offshore systems at _{sea is based on the hypothesis that}

"- the Isystetii _is- linear. Even if this approach - it::.-SatiSfaCtory for a great , number of practical applications, _{it is considerably limited}

With respect to the study _{of ,certain. phenomena because of their}

. non-linearity.

Examples,: are the capsizing ships or

Compliant, offshore, structures:

t ;

The. definition of ,a phenomena aS noti linear: _{is iunsatisfá.ctory} because, it makes not clear which type it belongs _to r

Therefore; _{it is _necessary to distinguish the _different kinds of} :non-linear ,.systems and the casuistry they can model.

This paper deals-, with ,Volterra system,:-,in particular second _order systems, Which are defined by an -input/output function of the

kind'. -'' .-N..),.. _{.(t)ac (-tZii} ,-41

f

vista) .

-Where x(t) and y(t) are the ,input and output respectively of the

system.-- _

slow motions of

":-cirldra.

The theoretical characteristics of these, systems, the technique

by 7.1which it is possible _{to model a non-linear ' differential system by}

using a quadratic model and the _{charactetistitS17 Of the response of} such a; model- subjected to stochastic _{excitation are 'described.}

Finally: the possibility of applying, this _{.technique to typical,}

I - INTRODUCTION

It almost ,always happens., in technolo.gy. that the appearance of an .innovative .method is followed by -a period of enthusiasm.

during which more and more fields of application are found, a

period. of ...reflection during which people go on finding problems to

which .. the new .,,method,..-CANNO'T be applied; and by . a :re-analysis phase \-during _-which the .old fields Of study are resumed or further extensions of the new method_ are ..'StUdied. .

The study of the behaviour of ships and. Marine. systems at

.

sea has not overcame this taw; the appearance,. in the sixties, of

methods ..-based on respOnse-,. operators and spectral' density function was 'followed by a -flourishing ".development of aPplications and; at 7

the end. of the seventies, by a growing interest in non-linear

phenomena, those , to which method .CANNOT be _applied because of

the no longer 'vali&-hypotesis of linearity on which the 'method is

based.

As M. St'. -D,enis ,has -pointed. ,out, dealing. with non-linear

phenomena means, however,- acting like a monkey which divides the uniyerse, into bananas , and .non`-bananas: -it does not produce any _progress - about knowledge' of ,non bananas .' In .fact, very different

kinds of non-linearities exist; for example the effect of

-a-non-linear ,,damping over an otherwise-_ linear sistem, the impulsive effects' of an istanta.neous change of 'state; the Mathieu unstable

osCillations of a parametric excited system.'"

If we want to pass from negative knowledge (What does not

happen) to positive knowledge, we have to model every type of

phenomena by Using an appropriate non-linear systems which has its .ovin: main.- properties.- ,In this report, we 'deal with the Volterra

systems whose response_ y(t), to a continuo's -input x(t) is continuous

and may be , rappresented- by a convergent Volterra series of, the kind:

1(t).

_J

where the termSjii(t,...), known as Volterra kernels, depend on the properties of 'the system and not on the excitation. The assumption of steadiness (the assumption that the Volterra .kernels do not vary

with respect to time) is implicit. The extension to non-steady

The above-mentioned written definition _{is generally applied to} systems provided with state _{differential equations of the analytic}

kind. Furthermore it is meaningful only if the _{'series converges}

quite _{rapidly .because the computational effort connected with the}

utilization of a slowly convergent series is completely. _{beyond of the}

capabilities of the present generation of computers.

When it can be applied, however, it retains the elegance and incisivity of the linear method allowing _{a detailed analysis of the} problem.

In _{this work, we have restricted the "analysis to the steady}

second order Volterra systems, the ones represented by _{the two first} terms of the series, and we refer to systems _{of one degree of} freedom only, in order to simplify the formulation.

In the following, the steady systems of one degree of freedom are defined as "Volterra systems".

In chapter 3, _{Volterra systems of the second order and their}

properties are defined and. studied; in chapters _{4 and 6 the case of}

stochastic excitation is treated and, in chapter 7; we deal with

some pratical applications to marine systems.

In chapter 2 some preliminary recalls of the theory of linear systems are made in order to introduce the reader to the subject and just to show the analogies between the _{different approaches to} the linear systems and the Volterra _{systems of superior order.}

Furthermore, in chapter 5 an illustration of a very simple

physical system is given.

2 - RECALLS ON LINEAR SYSTEMS

A linear system is characterized by the' equation:

1.(t)=ja.h (t) (t. v. (2.1)

where y( t ) _{is the response of the system}

x(t) is the excitation to which the _{_system is submitted}

h(t) is the impulse response function which entirely

characterizes the system. 3

The Fourier transform of hit) is called the response operator

ON.

.. Out

14,1,4 h(4)c d

and it characterizes the response of the system subm-itted, to a sinusoidal excitation.

By performing experiments with pure sinusoidal excitation it

is then possible to characterize the system using its response

operator and by means of a Fourier antitransform it is possible to

find the impulse response function h(t).

Obviously following equation holds true:

= (-1 (",) (,-,) (2.2)

where capital letters denote the Fourier transforms of the relating

functions.

If the excitation is deterministic, equations (2.1) and (2.2) entirely solve the system.

(

If the excitation is a stochastic process, the response is

stochastic too and the equation (2.1) makes it possible to find the

characteristics of the response, among which:

the mean:

SD

(t). E[(t)]

h(z) E (t t)] d

and the autocorrelation function:

t,,, ta) E r te.2)] _{h (.4) h (co (t4 --C2) CI d r,,}

the last one allows the Fourier transform:

C° itj4L4 - i (.02 ka

FR% (W4 ,44-12)--=

if

r (t4 -2 e. e

d '

t)

ci...ta,

-co )

If, as well as being a stochastic process the excitation is also stationary, the mean is constant

E Dc. to] Y-t

-and the au t oco rrel a vi o fUnc.:tfonw depe rid

7 :

In this ,case obviously,. the autoc.orretation symmetr.fc.-.

As ap- consequence:, the regpOnSe process and the

foIIowing . equations%, hold: Its may _ -ti-ansorin- -r-3:(

an:cif-the fbllowing equation, :bolas

-(w)

:where: the . symbol ( .)* means the complex and - the .:EFourier t-r a n stotinv - which is also

symmetric, R

In the case' of stationary prOcesses it is possible to define an

aiitoiddVa ,z,

_

. i

proved_ that, if. R

(c4

) , then:, -, [ )4} *4(t-a ' " :.1.1 7 It evident- that:_ 24"4-1,ii; 2) , function is

1t is..then.: pOsSible:'

- 'autocoVariariCe. function

-where4, (co) is ,-The .:funCtiOn,,., Which- MeanS. that the 1:2x(w) contains an impulsive term at zero frequerice:

the equation:

_

Cu.+) ex (6J5. -4114 Co)

to the ,rf',ict

that t(o)

6 2 -is the ' variance ' Of: the

process an is symmetric, a: ,:power _spectral, density

-

-function is usually written as follows:. .

Aai-

- Jr

:

' - defined- lin

integral is the va,t-frice of the proCess.

;

SECOND ORDER. VO.L.TERIZA- SYSTEMS-,, 5111314ITTED TO DETERMINISTIC

EXCITAT.10N::.

-.

,We define second .order systems as the ones characterized b

(t).al oz-),)e (t-

r

- 11(0 \

where -the- terms n----:.(r)

Which -identify ,the system..

.,defihe..,the',Fourier- 7translorni...of the

6

.11,<

.stioh: a -way that its

4 tz)

an.d h (t:1,;) are the .:Volterra _kernels

-

-It is easy to .see that the kernel h(2)(t.1-1t-2) is -.symmetric,

because by replacing h(;) With

, , the equation .._ (3.1)

These kernels may either be obtained in an analytic way,

from the differential. state equations of the system. or in an

experimental one, by means of adequate tests.

There are several analityc methods; an exhaustive study has been made by W. J. Rugh /1/ in chapter 3. An example of this type

of method, based upon a perturbation technique, is reported in

appendix 1.

There are also many methods with regards to the experimental

way; they are dealt in Rugh's book /1/, chapter 7. A method based on bimodal excitation is given in appendix 2.

The _{system's representation} in the frequencies domain is a

little more complicated. It is possible to prove /1/ that the

following equation holds true:

co

>(w) =Law)X ( G) 4 4

j

1( - 9-, 4) X e) x (9) de

-ao

where capital letters indicate the Fourier transforms of the functions represented by small letters.

This equation is given to show a peculiarity of the Volterra

systems, the "pollution" of frequencies.

Unlike the linear systems, the harmonic component of the response at w frequency does not only depends on the corresponding

exciting frequency but also on the non-linear contributions of all

the other excitant frequencies.

In particular, if the excitation is a zero mean, X(0) = 0, then generally the response mean is not zero:

00

Y(e)=

2Zttt

Yele)x(-1:9x(e)

-00

Before going on to examin stochastic excitation, it would be

appropriate to underline a peculiarity of all the Volterra systems:

the uniqueness of the response.

Given an excitation x(t), one and only one response y(t) of

the system exists and this is limited if the kernels are integrable

(hypothesis implicitly made in this paper).

Therefore it jS better to test the. system'

,.creating a Volterra

or example, the ,parametric

As a consequence the.,Volterca _{systems-. are stable _arid do. not}

present; bifurcation phenomena

where the non-linear terms are supposed to be small, may be

succesfully modelled by ;means of a, _second order Volterra system, only if the excitation is far from the instability Mathieu limits.

When the excitation x(t) - is a, stochastic process, ,the response

y(t ) is _obviously stochastic too. ,

'- Thus it is possiblet o define the

...:-.. _i.

n-th order autocorrelation-_functions of the response as

Atj (t) + c () 4_ b4 (4)

(t

- ,

- ,

SECOND ORDER VOLTERRA SYSTEMS SUBJECTED

EXCITATION

Eg(,) 1(4 n)J (4.1)

.1-,Let,';s consider, -::forti the ,sake of semplicity the 'pure, _quadratic

, ,. system,:... ,,... -00 , , .,,. - , - _f:-..4.0

1r _

g-If.

,-- ticcsral lk._(t,,c,a ) le c_ .-i.2.) ,A,c,t4t2. {4.2)

. .

. T.he:;extenstah Of the. bRiorriiou s. systems w be ,;''developed later

By Substituting the equation (4,.2) in the equation .1) we

. obtain':

14(CI,r0rx

7 excited -before (C4,ct )11(c3A).--

,t2)

rmil!c.t.i.('-rcY.-_- C ce;, TQ TOCKAST IC on. 7

It is to be noted that the n-th order autocorrelation function

of the response depends on the excitation's autocorrelation function having order 2n.

For instance:

00

(Z)

'IA 1-i(t) ri) Ck- - ri ri rz.

-co

00

a) (.1)

el)a (r,,r, ) (Eli; )

r9

-CO

If, in addition, the excitation is stationary, so is the

response. As a consequence, the n-th order autocorrelation function

depends on n-1 time intervals. In particular, the response mean is constant:

14

j.0

0:1 (rAs -C2) r-y"Irt- 1-4)ot

ri

dtL

Furthermore, if the excitation is also zero mean gaussian,

c4, (2), (I) (t)

rif L4.11 r4) rit e-4)4"-r< tt) (*L- b4)₄

cz.)

4.r% t-4) _Lt2.-.t0 and

00

rit)(t, to =if h tr,ri) h(r r )r(2)(e. ) r-(23(

r)d-r

_{4 *}

1 31 4 '4 4 al A 00 . . 4-11 f (CA ( r3f r4) r2)(t., - 1/4'2 r;:1)(r-i-t _i .c4 3- 4-24c, 00 , ) (t3, rX2)(t'i- tz *C4 et)(44-tz-t

by using the symmetry of the kernels we obtain:

2 ifcc,

f h (ti, 2-2 ) h(r3,z4 ) r,n-e4 -ti 4. ti-t 2) vintl-rz 4 f-1- t-0 cir4

ri

where the second term on the right is the autocovariance function of

the response.

By using the gaussian properties of the excitation it is then

possible to obtain every order of the autocorrelation functions of

the response. These normally do not satisfy the gaussian properties; for this reason the response distribution is not gaussian.

The above mentioned considerations may be made, of course

also in the frequency domain but the results is more difficult to obtain. Thus, we compute only the mean and the power spectrum of

the response, without entirelly justifying them.

Taking account of the equation:

420 i to., r

j

rx( t)= Qx(c...,,, ) e 4 4.'4

27-i and using the

C;-)

44, roi (4,2ra

(c.,1 La-9a)

we obtain: freo

5--- jj Wttit2)1 gif R

jc,,i)jc.:. itm4r4e-ic.i2ra_go(G-si+coa) 4 CA-$2 4 Wi 41 ri cirL.

4: :2if R (c..44) So(446+(u2)/ZirmirL

e..-".".`" 'T*

ex x

iit d r., dire at., awl

-co

-4--ifix(")41 H ("41,Lua) g.Cc4,*(,,a)

27C in addition it is: 40 = [r;43)(x) -d Thus : ( r Gil 10 ao rjx(ctil) (4J4,-(lie)ci 27r -*0 020 ar, - 6,34 ( C-4 t-) -L (a")((o)o ifi v)(C,r2)''')(4-3,z4) -L4j 2, c 644, z 27t oo ; t 3-ez...z)eco2

-± j ZxCLot)a otc.-32 ci r., dra 4 r3c1 r4 cit . 2fr

- oo o

- t'o-hros -itu.e..

'

i'lit211Cw4) gli(-'2)Fcs,rs) t e ci ;vizi.

j

V) ( r34

-co -00 -00

cw2 r3

azt

/ 0°- i (Ls,. wi ic.)

if Qfc(444) ett(ak) W(434,4-12) 44 (-CUL 1'641) 2 ir go (441 1-6.3) d CUE 470' :241t cx,Qic (0.34 ) - H ( (0.3,1 ) ( Cul 4-61 6.41) cl co 4 Remembering that: Q(c.0) = (-(4) H (44-1.1,Lua) H (4.):,100 - 1-4*(-(010... yr_ and calling _ (4.)4.CA-34 )

we reach the usual notation

gr

--(2)

(Cei) ek GY) (CO + gr) ,)41 0,3 +at)

kur)ar

-co

often 14 Wz) depends only on the difference _{c02- ca)-1 ,} that

means

(Q-21,w2)

H (-"'L

= (j-j2

)

2

in this case:

V.:)(45) _{fc13,(442A (4J4a) I H(a" 1)11cAr} using to the spectral density functions, we obtain:

5 (Ca) Z.1 S)e(1/(1)$/t (-""?1) _{11("191244e4 r}

-00

( 1 4.'Oki-4 (a"- ,1)12.- su( e-G.1)1H

11

u112?e%

In _{the case of two-terms systems, defined by equation (3.1),} it is then possible to write:

where v' t) ..is the and y( t)

by .theequatIon_

r

i-.4)

-The .0:eyelol.. ..tnt is :difficult and .gives rise kind.: .7 (ei.01) rka (t-4 . in Which': .100i it) (t.4 z) . --(-Ci4) (4 ) (4 4 ) (4-,'z) -rr`14 particUlar:

4()

It) to,

Li.(ret-,.,z44

(4) . . iyo 47r re, d..rA .i4 '

()

ta,r1) ric (E-4- ,)

Equations striiiliaf).-tO/ the previous ones' may be..;.ritten_ in the

frequency domain, where the cross "spectrum of the processes

y1 and

y appears.

It should be noted that the term exciting process ,.is.,-g-aussian, thus the linear

of the response are 'untorrelated.

EXAmpLE--Just

to illustrate the equations deVeloped in the ;previous

.:chapters, let us consider the two prisms in ,fig's._ 1. and one . with al. rectangular basis, the Other with a. triangular one',.each having

mass M; unitary displacement in the 'initial. condition, 'IC, and draft .

(4f 4)

) 6- _{C§;:- ta) 4 1r}

_zero if the_ :and -quadratic, terms

h. The prisms are .float frig. W....ith .the water :plane p-ay.411::61?- tO the

genaratrixes. The added -mass 'is... independent _{of both frequency and} draft and is taken into '7acCrOi.itit in the ...Trials -A4,; _{the damping iS, L..} .

. ..., ... _,

_

. . . .

Let's (.--..?nsider the _{heave :metion.:---y(f) only and call- _x(t=)'. itS.}

.--...7 .

. excitation.

is: The state equation' of the L system (rectangular basis prism)

and.. of -- the ..the NsYstern, (trtankulatbasis _prism),

assumi.ng:

By making:_

we.robtain foi7. _Sysfem

()

2.4-00 (4t) 4

and for the N.Syste.rii-..

c) 4"Joi(j

and neglecting the the N system: - L-I6 C4) ( 2fic+e-6)1 (4 2. 4(45 42G-Jo g (4 ) _4, or higher. ,*(4) - k.E011. Ei(0-7act.)3 J r ' we obtain for

It _{is clear that, when , the N, system approaches ,the} L system 'and that the _{solution of the latter coincides With the first}

order solutiOn4.., Of the former _{Thus we -study the N -SYstern}

only,-knowing that its first orde`r. .solution; cbiriCid-es.. with _{the 'exact'}

-The firstorde.r Voiterr,A kernel is: from i,Vhich 4

()

t443( )

t

r Thus: tic4

r.)

( - 2-4)

-where the Second orde-i.. VOiterra kernel i

(r1, r1)t (ryiiei

L -r) r)- de 4

(2) ) , ( .

(A) (4) (4) (4)

-(ti)

rz -(r2)'4

+ 2 cue...va s

Th-erefore, in the frequency domain:

(A) e II, Cr) e

dt

4-,* (4-$

Ge.

A (.0 ' - I 4-4 r4 _ ( (AJ2 ezdr4 d n . (r4( ;).L- e (Al ' , (41 ' CO (4) , (.1 H (41-(02) II 041) H (4-)a)-1-1 ('-',1)-_-14 Cw 14

-If how, the stochastic exciting process x(t) .-zero meangaussian one,its autocorrelation function

,

.where? A and T

a-rec.:nstants

whose - _dim-enSi:ons

are -ELT] and [Tj respectively:, also the linear component of the response ha-s zero rile:in% 'and the .-autoCorrelation _function.

(2) '

r

(t)

--\ wherealsT the

mean:-and the autoc'orrefation fUnctiOn,

4:10(

r-1 Lt.

rid Or) _)L1Itt

r)

A2- e 47 1'

r a ci

-4, 2

1

tr i

I

_--.7--.2 4 VI (r4;r2) 4 c2

Thus, in the frequency domain:

A-(ax (G=5)

whereas the mean is: 1/4J

-In figures to 12, relating to _{the N system,, the following} items are shown :

1

i

i (r'AS., tI) iv, (.64O;) 4

quadratic component of the response has a hon zero

C's0

(c...,Tr

4-A

As stated before, due to the fact _{that the exciting process is}

gaussian , _{the linear and quadratic components of the response are}

uncoreelated_., and therefore the _{spectral density function of the}

response is.

dre

- the first arid second order Volter.ra kernels _-,

It is,- therefore, : possible to

distribution 99- (9) of the process' by

-(115)2

,the first .1 d second order_ response operators.

- the autocorrelat ion iunction =:an(.1 the spectrum o

process

= 0.05

4. 01 rad/§ec

9 sec

16

the_ ,aut0Correlat ion func,ticp and spectrum ,

respon_se

-the autocOrrelation 'function and spectrum of , the second ...order ,response

_ &Valuated-, analytically or numerically, with the following

,

given to the parameters:

e e'xciting

e %first order

values

-,RESPONSE -DISTRIBUTION IN A SECOND ORDER VOLTERRA .SYSTEM

The response being stationary, the nth order autocorrelation

-function, evaluated for zero value of all its variables, is equal to

the n-th moment of the probability distribution ,because of the steadiness of the_ response:

firid the i-nitial- . probability

Using a Gram-Charlier .series

Or which 0295 .m2

- In figures l3 to 15 the following items, 'obtained by ineans of

-a computer simulation, are shown:

a realisation of the exciting stochastic process its lineal'. response.

-- .

!where i,2. ;:)..4:3 .. ,,a4 -..., .... are -. the. cent ral ..moments,': which - are

. ea, sily, obtained

from '3 -,

rt.,_z.ny 171_A- , moments,. --_-:.fir- is the

.., ...variance 'of the process- (t.) and

j

i is the normalized variable

4 (I -5")//gt

To _evaluate- -The distribution the extremes df

(E) in , the knowledge of the joint distribution of the

process and of its derivative is needed.

'

Let ug write the ,equation (3.1) as:

ix 039 cua- 4.-

V)("(,bq5.,

from which

().r .c2)..

c(6.)cte- 4-) t1 Ck- -Gsz) lc(6-2) citGi Giqz

(2.) where h means CZ) .11 C(,2)2 r-iPe The 1-th

process and of its derivative is defined.-1Dy

and m-th order cross

(elm) et41.0 El( 4) (t-t)Zj(t-41) 17 X.( 1574) (trz correlation function '4(

Let us consider, again fOr the sake -of simplicity, _{a pure}

The (14-m)-th order moment. of the _{joint -distribution of y(t)} and Y(t) may be evaluated as follows:

(gem)

'711 ,(°/.--/

W1 ft

\

-Knowledge of the moments allow's us to iexpres's _the

charac-teristic function of_ the , joint -distribution by 'a Mc. Laurin series

and, from these, to _obtain the joint _{probability distribution /2/.}

The joint distribution p(Y,9), being known, _{the probability that}

(5,_'s,)-(t,,,,. ie.o..

quadraticsystiem:co

;00, _.(/)

. Cl) 1.00

r)( (N-r4) 4.0 ' _{)...1C(414'176iji _i) )4N4ia- .1.2t444.4)1 a C - - - 41 r2{4Z4A4}

-a given extreme (t.) will not be exceeded In the inIeryal

may be obtained as follows. .

First of'., all one must Compute the mean- number of times that

the process y(t) crosses the threshold u from below:

E [kir (1-4 =

where:

fe'if(1T Ica) c'o

More, if we assume that the crosses of the threshold are

independent (hypothesis correct when ), the distribution of

the crosses is a Poisson one and the probability, of not crossing the

threshold in 0, T _{is exP}

' The probability that a certain extreme will not be exteedeed

---4

in 0,T is, therefore:

1)(1-) P`e (-T) e

where 1;y(9T) is the probability of being below the threshold_ at the beginning of the interval.

Therefore, the derivative of POT) is the probability

dis-tribution of the extremes of 9T in the interval o, T .

7 - MARINE SYSTEMS APPLICATION

A. field of application - of the quadratic Volterra systems in

marine engineering is that of low frequency resonant systems, all those whose oscillation periods are greater than 50-100 seconds; At

these frequencies there is,, very little hydrodynamic damping, and

the linear excitation, with the 'same frequency as sea waves, is

almost zero. The consequence is, that the response is dominated by the non-linear, low frequency excitation which may arise from the presence of quadratic terms in the state equations of the system. A typical example is given by the. slow oscillations of moored marine

systems, in which- the quadratic" term arises mainly from the

relative motion between the free profile. of the

sea and the hull

surface of the system, in addition to the quadratic terms- in the

PC'irii)cii :AT

1°1

equations relative to the velocity -potentials. _{These problems are}

'dealt in references /3/ and /4/.

Another field of application is to be found in those systems

whose 'response has a _{non-zero mean even if subjected " to a zero}

mean _excitation. The _{easiest 'example is the - surface of the sea}

itself; it is known that the crest of very steep waves is much , higher and narrower than the trough.

An example in the naval field is given by the wave bending moment which may present considerable differenCieS between _hogging and sagging values. The _{reason is to be found in the flare as well} as the non-symmetry of the wave profile and the _{non-linearities of} the hydrodynamic potentials.

A quadratic model of this phenomenon _{has been proposed by}

Jensen and Pedersen /5/ with satisfactory _results.

. Another example' is the added resistance of ships. It is known, infact, that a ship's resistance_ is higher in _{a rough sea}

than in calm water. This increase in the _{resistance is due to the}

mean longitudinal wave force which is not .zero because of quadratic effects.

The _application to _{"emisymmetric" phenomena, such as the}

non-,linear roll of ships, requires the _{use of odd order Volterra}

systems .(third or greater). This leads _{to a significant increase of} the computational onus, therefore it _{is probably more convenient to} use _other methods,- like- the pure perturbation technique /6/ or

numeric simulation in the -time 'domain.'

8 - CONCLUSIONS

In the _{present_ report the main properties Of second order} Volterra systems are *summerized and _{-their applicability to the study}

of some typical non-linear marine _{systems_ is discussed.}

- In preparing this, _ our intention was not

to produce something original and we therefore apologize any readers to whom this method is already known_.

The application to the ma.rine field does not seem to have

been - Viidepread- the method -Der-haps .merits; his report seem's- to Us tO be an useful'. sy_rithe:.;iS:

"Arclii.tects; VoI. 121,.

,and therefor,e

. ,.., ... .., . . ., .

...h 1 s. c ont-1,..c.,,t.ion we - would like to,' merit ion '.tne, paper; by ,, A

,

Na'eSjs' ./-7,'',, 5imili'4,r,..i.in O:Ontent, ,publb_lished in; 'he Journal of Ship .--..Research iii'Decerriber,, -1.-985,-.3...fii1Z:..11 'reached:. Italy _while the.':'.preent,_..

.:. _ ..._ . 1" -, ,...,

- .

- " . .. ..._ , .

wOrk wa.s: -beirig-ptihted. . - ,

.

'...-,

/1/. Rugh:"'r,W.I., ."'Ncifr linea system: theory:. The.' Volterra/Wiener

app.r6ac'h--; J6lii-C;1-logkin'S; 1981' -/2/ Lin ".PrObabilistic theory

McGrad-H11I, 1967 y

structural dyridMicS'

....

/3/. Pinkster J.A.---;.--1-HUI)Sdians R.I-E,..kr, '.'The .low irk:pi:enc.)/ motions of

:

-semisubrnersible on waii-eS!' ,. ProCeedings -of-. BOSS 82,:'1.- MIT,

Cambridge, . Massachusetts, -082..

.

, . .

' /4/ RobertsJ.8.,

. . "Non' linear analysis of slow drift. oscillations of.. ., . .

-:: moored vessels in randOM,,seast, I.-..- .Shlp Researdh.. VOL 25, _N..'

'-. 2, 1981 ''.., :- - . - ...

/5/ 'Jensen J.J:, Pedersen, P.T., "Wave induced bending momehts, - ships - A quadratic theqry' -171.1e Royal Institution Of Nava.1._,

-/6/ .:Ferro; G. ,..,"..'Caret-ti. -.'..Metodi per '1' ahalisi 'del

CompartaMento-ti.Pn-. lineare delle nayi in Confirso";,;-R...I.NA., ,R,e7P:Ort N..

220, 1985

/7/ Naeas A., Statistical analysis of second. order 'response of 'Marine structures", J. Ship 'Research, Vol.. -29, N. 4, 1985

20

-L

Nick)]F

Dcto,(4)/

.

Appendix 1

ANALYTIC DERIVATION OF SECOND ORDER VOLTERRA KERNELS BASED ON THE PERTURBATION METHOD

As is well known, the perturbation technique is based _{on the}

assumption that the response y () can be developed in a Taylor series around a perturbative parameter

()

EA4,

(t)

(A1.1)

by putting _{this equation in the state equations of the system, and} equalizing the terms containing the same power of E , recursive

equations may be obtained of the kind:

4 -4 Ct.)] (A1.2)

where L is a linear operator and FL is a finite (not necessarily

linear) function.

In many cases, if h(V) _{is the impulse response function of the}

linear L system it _{is possible to write the first equation of (A1.2)} as: L

[yi(o)r [:*(4)]

t.b) from which: (A1.3)

v1(r);(4-r)dt

ti(r),c

dr

-00 _-00 and: L

(0]

1[10C t k)3

ti(r)0c(i-t)cie]

from which:

xWz

In(t) _r) (A1.4)

It _{is almost always possible, by means of some rearrangement,} to express 2(t) by a Volterra kernel series.

For instance, in the very simple case that:

I

aucip

FA CO a 1C (4) lii(4) ".: _{ICC+) h(e) ICti- -r)}

de

21

00 by cha.--Ting the equation is obtained: Whereas It is clear that,. in variable Yz-41

The 'example- refers to t e system:

(t) (4.)

141) (ri.-ri)

of integ-ratiOn. the - following

if the amplitude o the excitation u(t) is -small; we may state.:

this case,

z,

z)(c4ira)

6 n(ti)t 1(4- r4)

22

Obviously the system is a second order one if the - kernels of

Order higher than the second which may arise from the (Al 4) are hegligeble, therefore the perturbed - series (al .1) converges at the

second term. (.k) E -rc

e?

c)

0(63)

4- 62i ct) esiz ct) - tt) Ev4(t) 4-, -tJt(6)

("

with reference to we et: i1 .(.40 . ti Cr.%) ,(k-r, vi(ri, ,cC-'z'.., - 1-1) 621:ez 4-r-,

EXPERMINtAL DERIVATION OF _{SECOND ORDER VOLTERRA- KERNELS BY}

THE BIMODAL EXCITATION

_

_ This -meth<5d-,cOnsst's in subjetting.-- t esystem-to excit-ation,

of the kin'd:

:Appendix 2

i

C:2-5 0'11 ₎ 4.

The -fresponse i1I be:

_Pp

1.1 'Cc, 1 Ai _{(-Gs (C-E)1 4.-A} ,

;1)

jtc.,,r1)1Ai

.44,(t-r-4)1c.'s

4.21 _{cu2 (t--e.4)2 LA3ECua(V-r,111}

A1A s[6.4 (k,r4)) Lai tcy,7.(e,-,..ra)3

4. AtiA Er-.3% - 1.4 _dcid'!.%

by means of some rearrangements

3'('1/4 ) z (42$ ( 441 4() )₄ 4 UssLc(-44.1 4 C4/3), (24-60 Id ECC141' ":")

- ea

. .' C3 _(A2.3)

-yv(9. Ih'eT. e.:orripOhehit':-.,at,.!:.ttie. _{.fietlyehty cal -1.:ot} it ,may:!...be written as:

74.2-

et-(c-ic, cu let); i(44tri-,ahta

-) ,s

''4)'

₄

a

The, last twe,(=iiite&a.,1s: re. equal 1-..leca-iise.: t

properties of h _theiefore.:.

=

11('''cz.,,T,)-4

stv. _

2)) s

I u.(cuir.4 -4- z) d Are .4'

ZrA)rL) A 4 LA) t (co.. r: `"tr4) 4-:

-tbtCcLPl4cuDtJ Ctts' -r coll-r4 )}

d14-r

-This means that :

*.=

,

L):

e'-:Fo,ur.iii--,trahsfo,r-h.. of 1the kernel.liat&A7

24

e symmetry

(42.-5,

With the -exception of the finite number of

frequen-,cies(cof,1/4) for which (c,2/4-4)2) equalize one of the other frequencies

which -appear in the (A2.3) (*) , the term $2.(t)can be isolated by

mean of an harmonic analysis of the response signal of the system.

By completing IMOvw2)and CY44,(L). taking the .limit in maT- cases, or

g

by using other methods, it is then possible to compute h( a".1,t2.) By an analogous procedure, it is possible to demonstrate that

The term ii1)(t) may be obtained by analyzing the component at the

frequency W./ (or ci)2 ).

The disadvantage of this method is that it is impossible to

directly verify the order of the system. In other words, the first

and tsecond order terms may be affected by unrecognized higher

order .effects. If we are not sure that the system is of second

order, further enquiries will be necessary, see Rugh /1/.

(*) For instance if cOi =Oa , the component at the frequency coii-wzin

the equation (A2.3) covers the one at frequency 2G01 and

the y'12 as defined in the equation (A2.4), can not be

isolated.

t US -tal<ing.4: 0,-:1" ' _ ; . : ,

cons'idei-. the f.driction-s':

60 h(co c,r2-t.) ,..RerheMbeting that : Apjehthx 3 r-: -e. ( S.,0 (ra one mayobiain f. YE. ri.=7 26 "rt7r ( - - s-1) d one Obtains: .. ( _)ti 6,172_ z )clz I obtains: w , one -obi , ll (t) ' . . ) tl (7(4 .-) .2c.k - r.1) ...c

The-two -

:-.)-;C , formulAtions of fi,(t) -. 0Fq'',... . eiqui.v:alent..

. Thus F(L) may be obtained by their mean,. : obtaining', in "thi.s:,.:.

,

4y, .,a

: kernel - 1.4:rhi'C.Y; ;',i's ,symmetric and consistent .1.4.ithhe , adopted

,::,..

. .

simbold: .: .. .

-.

After this, 2.(t).:. ;4(0 - (4)

.c(t)

. and:

1(4).:::_ffee[J°°Id(r)4 (Li-r)

)dc

- i (t)1,1( - 14(rs)

* S-0(rz-ti)jx( E-74)1e k-1-2) dr.( Girt

making:

kt4in.) in(r) (r4- Ll(t-2--c) aft- ((c',4) (rz) Z4- rt) +

(t), -G., PROOF 2 PROOF 3 (a) ti (c11u.2) We obtain: (4) H 4- h (eA) -(1) ( -

_t

_)ci

_dr:

0-01* 6.;rc

shi (wo 14-12

t)cir

CAJ 0

(A-% 1)411- ₀

Remembering that e`c-t'2'.cos(co.erf-Lsin(we) , we obtain:

( ) c

c to) z. a _{2..c.AsCcor)seu (wo cir}

(4- 1.16

I r

Z

C)

(c.o

j't) az]

By applying _{the prostaferesi formulas and developing the}

integrals, we have:

&Jo?. YG.,02 -6-#0 4 2

cow.11

(04.13-t-2)t 4-

I g2 J1 cJ

ri(Gui,uut) (4),,u...) +la ( cAN, (1.0 4- r4(4-4,t0a)

Where, using the saffie s-imbuts as in .proofs 1. and 2, it co ti(coliwz), (r) h

ti(r -r)

Q --CO o. -1.C4-11(ri-e) ti(.)) d-r., Vi(r,-t

J

fcb%36(r) A) (41) e 1-4 (G"(4) (4) ci41 (.-#2 -Cl)

(4)dc

44 ")(04.0 CcaJz)

C4

4AI c'2

wi")

It 'rt) ="r,) r4 Cz_{ctri d Tz J.1} _ c4_944 (o/ )2,1

f

6 (t,) (4) -00 (4) Cl)

-

'4'2) 1,4 -( 6)4

In- -the,_ same wa.y:

(L, Thus: c-t) ((04 c

H (w21

+ CA-4) 'jjeo -_ cotr z e dri ) _ t't (4) (CY4) 4

(:L.4)

14(r2) (r4-rt) e ( (4-'4 +-1"/.) 't4 r4) e . dri

Ii0i

ti(r4) -zt) e -28 4 t cbrid

dr

_ e c ez ' e ct riar G41 CO 14(ch-1,0- W(.16°2) W (C4,1)-

(24d

By substituting the expression from proof 2 in the place

-of 1.4 (1) . we have:

(7)A L2 )

1.1 4"1 I CaJj) H c 4J1 C4j1) 4- A

"

&Pa cal

Q.e 11)( c.A) =

ft (

N -

_fiN)

vv, 1-1 (2)_{(6-'d ,14,2}

) =FLCFI

2r)

) Where: 2. (-

7/

- (44+_0)414.4 _{eo...,44.02,..4,E(444441Z)21-1g144641} --L(too2 -W21) 24 4 f (AJac.J2.2] 49,2 - tua)2 Fi 42

f c

(o-s,4.wt.) c44174.$1-( ca.1/4,2-G-hz _{(c4.0.21-(4.022 )4- 124,02G-hicu,,z(c.L2-24...1 4.,,t2 )4}

4" 4 S4 0-'oel 0-24 Cuz (144i _cJcs?)

Ps _{2s (4203641a jz [an (4,J. wit)}

4. Ca..94( Cia02- Gail) _{Cue, (4-11 (..4,03}

Thi way, by relation of It t =1+5, r2 exchange

must be _{underlined that, F(w1)being equal to El (wziebi ,}

e H(2) and kr) H (2) are invariable with respect to an

of c..)1 with ck.)2

s conclusion might have been reached. in a more general noting that t/(23("6--.1:60 e-144)-1Z-"le-c1-)zPr--2' _{is invariable in}

to an exchange of indiCc6, l)eLause of the simmetry

--.- SECOND ORDER 'VOLTERRA KERNEL

FIRST. ORDER RESPONSE OPERATOR

(1)

Im- (43) IMMAGINART PART

,(1)

Re H (to) REAL

PART-. 33 (1) Im H (1) Re H ) rad/tec)

6(a) :SECOND ORPt44 RESPONSE cOPERAT,61,

(REAJ PART) ,

coif:rad/sec Re H(2)(011 . .4

4

P-4-4 4 w '

4141

\N-q

4top.,,- \'.1&4W .

4

0 to ,44I Pror

r-___

...

...,,,,

Iipp A

top.

' v1. I %

S

/04*".

"

40,40 4.,

44 .401.-4--0.Z4,11,

4,,Alhas.,

4 a 0 °' 11

4-

4 1 .-44; Tj+1 s t 41 ff0W leftiel. Ags'

zr4

4

.4p,wiw

1 r

0.1

Z.

..

0 . 1 4 44111111,A16..

.4,

4..1.

1 11, 0 . 3

'.4'

.*

0.2

Fig. 6(b) SECOND' ORDER RESPONSE OPERATOR

(REAL PART)

-40 -20 - _0t[sec]

.

Fig. _{AUTOCORRELATION:FUNt7ON OF THE FIRST ORDER}

1'1

40

_

I--600

--0.050

11 - AUTOCORRELATION .FUNCTION OF THE SECOND ORDER RESPONSE_

0 . 10 0 . 20

POWER SPECTRUM OF THE SECOND ORDER RESPONSE

17,

itaisec

0

-6.

60--1 .1

1:8'00.7.?

Ti.

rjt

;

c,E1:3

10:JO 1500-:25

35O400 49.-00 _{50i.7i0 .550?}

BT _sep

BTONput4I-Siudia--46iSeparatOri filtrO per acgue Oteo'Se_di:sentina

e:su-miLlratori di'-contenUto oleos° nel-Leacgue di:-Scarito _{daLle nevi}

PT 69-- FEBBRAI0,1980 / 1116'", TTI

Un -archivio di- Coeffitienti idrodinamicii.di_pr_essione _e-.di.forZa

per cilindri bscillanti in superfitie:tibera,.

- .

41-file7of'hydrodYnaMic.'01ressure and force

coefficients-forcylinders

oscillating in a.free surface

ST 70 - LUGLIO 1986 / CAMPPTPLP,

EA1A5SIND,-M1lRCHESL._NALQINT1N-=s-7LinAlgoritMo,per'il calcolo.deLle superfici lavate deLle.nayi-_petrolliere

(COW.)

An algorithm for washed surfaces _{computation in-,tariker-s-:-(C6de Oil}

Washing)

BT 71 OTTOBRE 1900 / SPINELLI

Convenzione _{MARPOL '73, come emehdafa-:dal prOtocolL0,-78.}

Interpretazione_{. _ . normatiya=contrOlji--RINR-Oer riLascib.}

- di dichiarazioni di corrispondenzalaCie.horme deLla:con'venzione

_

BT 72 - DICEM3RE-1980 /PITTALUGP,

I:carichi

d'ondadi, progettO e La, Ltio correlazione,

- - ,

Design Wave' Loads-and their correlation.

BT JS - OVEMBRE 1981 / FERRO

Metodi e Problemi neWanalisi

_{deWaffidabiLita'}

navaLi . _

BT 73 7.DICEMBRE 1980 4HCARETTI.':

Confronto_ _{teorico-sperimentale. .degLiHoperatori}, , . _{risposta-f-deLLa}

pressione .tP-Onda tareha .

toMpariSor-rbe'tieieen'theoretical _{and..-experiMehtplesponseampLitude}

pOergtorS'OnhuLCs...,

BT 74 - FEBBRAIO 1981 / PITTALUGP,

Una ProCedura per !..a :0_alutazione del.,La.robustezza.,trasverSale

navi per iL-t.r,atOdr:tb:.di. Merci alla

procedure _{for '.thet,--iransverse strendthassessment:of_targe}

..carrier ships

Methods-and'probleMs in retiabjLj.t _{nalysis of ship structures}

BT 76 - CUCEMBRE 1981 / BISAGNO, MARCAESI, _VALENTIN

GIPSY - Un'post-procesSor-Oer

_{l'ahaLisia4ceLenentifiniti}

GIPSY - A poStOrOcessOr'for finite _{eleMent_analysis_}

_ _ .

BT _{77::'--GENNAINA98242FFJJ}

ApplicabiUita' deLle tecniche affidabilistiche

_alla

_{progettazione}

navaLe

Applicability of reliabiLity

_{concept_to ship,design}

. .

ST 78 - FEBBRAIO 1982 / SELVAGGI

ProbLemi'di progettaZione _{per na,yi}

Liguefa-tti"--4-6asSa 'teMperatura'

strut ture

L

pc) L L

_{.1rEpc=r-11.t7.a.}

-Design problems for ships carryng Low temperature _{liquefied gases} BT 79 - APRILE 1982 / SPINELLI

Convenzione MPRPOL 1973, come emendata dal protocoLlo 1978

- interpretazione della normativa controlLi RINP per il riLascio di dichiarazioni di corrispondenza aLle norme delta convenzione

Cseconda ediZione)

BT 80 - 14136I0 1962 / FERRO, ZILIOTTO

Applicazione di una _{procedura diretta per il calcoto dei carichi}

d'onda per Le analisi di robustezza trasversale

AppLication of a direct procedure to the assessment _{of wave loads}

for the transverse strength analysis of ships

BT 81 - PPRILE 1983 / MICILLO

Applicazione del procedimenii speciaLi di saldatura nelle costruzioni

navali

BT 82 - NOVEMBRE 1983 / MPRCHESI, ZILIOTTO

Comportamento post-critico di panneLli nervati-1confronti _tra

risuLtati numerici e prove sperimentali 83 - DICEMBRE 1983 /-PLIMENTO

I materiali per _La costruZione degLi sCafi

Note sulle caratteristiche e Prove del materiali second° La normativa del RINA

Materials for huLL structures

Review of the properties and tests Of the materials _accordingly.

with RINP requirements BT 84 - GENNPIO 1904 / FERRO

Advances in the calculation of the maxima of ship _responses

Paper presented at the Euromech Colloquium 165, reliability _theory

of structural engineerind systems, JUne 15-17, 1982, Engineering

Academy of Denmark - reprinted from Dialog 5-82

BT 86 - GENNPIO 1904 / FERRO, CERVETTO.

Reliability of marine structures under dynamic Loadings Paper presented at the International Workshop on stochastic Methods in structural mechanics, June 9-12, 1983, University of Pavia - reprinted from the proceedings_

BT 86 - GENNAIO 1984 / ROBIN], ZILIOTTO

Wave torsional moments in ships with -Large hatch openings

Paper presented at the VI Italian - Polish seminar - Genoa,

November 1983.

BT 87 - MPRZO 1984 / SPINELLI

Convenzione MARPOL 73, come emendata dal_prOtocoLLo 78

- Interpretazione deLla'normativa ContrOtli RINA per il riLascio di dichiarazioni di corrispondenza aLle norme della convenzione

Cterza edizione)

BT 88 - 11166I0 1984 / PPSINI

La saldatura subacquea -;*Stato dell'arte

BT 89 - GIUGNO 1984 / FERRO

Stochastic model's 'for Low-frequency, springing and impact Loads on ships

_

BT 90 7 0041410 190$ liTTPILU(11, BISFIGNO

_

-Mode rie tecnici7e= di ,-.#641-"i:gir deL cOMOOr tamektb, in mare':

BT 94'-''.,7.M1=166-10, 1985 CFAZZL/L0 . .

Pandra'inica sui fondamenti teorici deLle fliec-Canide deLLa frattura

.-BT 99,. ...I.LUGLIO 1989 :/ 1 .CERVETT m i'-'' FERRO_{. -,_} . . .

9f-f-idabiLilta!, e r idbi:idariz'a ;:ne L Le f orcd-ioni .' offshore

.5_,Nisteni- reL,iab-il: i tc,-: of offshore foundations

'BT -96 ..4 'APRILE 1986,- -I- CLBIENto . _.

, . -

-Niezi di sat_ vataggib 7'.Le' n-uove norme .-" de L-La convenzi_cne.:SOLPS,'_-. 1974. , .. . - C833

.

: , ,:-:-., . . . ,

-_61 97:14_,-Setterretire- 1986/ - Pi t taciog.a:. . ... ' _. . ...

imiLiaritieS and'differenteS...in- t1:5ih,:WL--.e:d'itleamt,, theorie.... '''-',

-,. .

DT "98 . ..- NoveMbre : 1986-i Pitt'a louga -r- Ii3gLitik. 7._.

. ,

54Ond ,di...ci .i- -146.64 n:146,4i-- effects , in marine- yatetri'S

.-., .. , _.

, _ . . .., ...

--. .

-.Prospettive deLLa Or Oge't ezipfle aff-id6.61 I. is tica -.del. Le. sit rt..it tiire marine

. ._ .

HT 91 GENNRIO -1985 1-.604v Ei ZILIOTTO.

Addl. iCazione deLLaneccanica del_ La; f r at.tur a neLLe veri.fithe a f at ica.

-IC C3 FREE C3FRIt7.- 3 lEt 1 1'9' US (S.

RP 204 -::SENNAIO '1980 :S. FANO:WU-1d.. Za,1TTO '

i n un !7' p o df.E.1 1EE:1 1 e , e: d e.(4 1 i

.

:L ED C:1 :I Hi;cZ,.. a I: C: :L t-liyaff

,

RP 205 (-*) :FEBBRAIO 1980 .1 FPINr.f.IULLI,' .TEDESCHI. , ZILIOTTO

D et Er f7.1 1 ...D. t

t

12 t e eL, a

dE g.:I. el ene i 3.pondeaz -.

de.1..1 t C.T! a f.71 1",; h y d Li S.,

RP .SFTTEMBRE. 1986 /

Desrr -.-.seippl 12nn d a su I 1 e se:zmi

trasversaI car ena

RP 2-07 - 'SETTEMBREH1990 / FA.NC FULL t ZILlOTTO

'Alcune consideraZzIIrisu 1 ana.kisi-.. deiXe sOi 1 eda. tazion i:n serbatoi

1 i ndri c 'dri ontai da. 3e11 e '208 OTTOBRE71980. /. FERRO

-_,Aff.idabd,1 it a (7, d lie condscenz e e. pr ospettive.

-dei le cerc.nT

RP 209. - OTTO8RE 1980 1 FERRO

daratt eri c h e. s tat darlcc _d pannel 1 i

ner vatl

RP 210- -:.OTTOBRE:1980 .FERRO. FITTALUBA.

sampli +.idato' per e 1 ung ter ffy.i.ne ded , di si dshing.

RP 211 _{-*AGOSTO1'081.} .FITTALUGA, ,SCI.ACCA, ZILIOTTO

A:itune. note. sufIe .w.'f...braz 1 eFSi.onai Oegl 1 bOri porta E.1 ica

.RP 212. 7-- AGOSTO 1981.: .1. FERRO, FITTALU6A

u.en ..d.(72 et:I E p :I or! iflet

sul cal cold, della ri spost

a

navi-RP 2.13 -DICEMEIRE, 1981. 1-FERRO.

Ii 1u ey:n z.a deld a A...:4.-dhfaa band a -st. a Ii st r i buz i one d 1 p !.t. chi

un process° stOtasticz

RP 214 (*) - ,MAR20 1982. / ALBERTI.4 ZILLOTTO

Ahal i si del cdmport amentO Litt i.Arale: di navi porta.contenitrrl

Sug get te i on e

Rp', 215 - eIutNo 1982 -/ CAREtTI...

] Nen-1 i near., fr-cluency doma.i'n analysis. moti..on7... and lo.Ad

of

Ships i rirregui sr- waves

RP 216 - APR ILE 1983 CAZZULD

Un programia - 'ver ifi c a. gl °tale delIe temperature die:L LO

o--RP -217 7. AGOSTO 198'3 ALBERT BERRINO

spErlMentale di Un metodc cia ci:.:c:l.

del cdmportabeto

-.RP 21E1 ( te;'): ' - MA. RZO

It§.84...:.../i..:Fkkka-... - ,...-.::- - ,,.,. , - ----

-.-F(D Llncl-arri4n-Ea l's.'' -.:.c.) i 7 : 01- 6 c....6 d Li,- iE...,

structures ' . ..

221 _{MAGOP 1.9g5,,,, CASC1ATI,}

'DIRE-TTOFiE RESPONSABILE .CARLO CASTELLI r..,CENITROSTAMPA R.I.NA. EDITOR-E- R NA_ VIA CORSICA 12 - - TEL. 53851 AUTOFIIZZAZIONE "TRIBUNALE JOIGENOVA -N° 27i732del.:10 aprile 1973