Prediction of the damping controlled response of offshore structures to random wave excitation

Dt

ARCHEF

INTRODUCT!ON

This aper introduces a simple procedure for estimating

the dynarrc response of a structure at each of its

nat'jral frquencies to the random cxci tction uf ocean waves. The ,rincipaT advantage o the proposed method is that the explicit calculation of wave forces has been eliminated !:om the analysis. This is made possible by a direct iic.tion of the reciprocity re!atiors for ocean waves. originally established by Haskind' and describ-ed by Newman,2 ii a form that is easy te impiement.

Brieiy stated, for many structures it is possible to

derive a simpl expression for tie wave force spectrum ir terr'.s of the raaiation damping and the prescribed

wave ampIide sjectrurn. In general, such substitt-tian is of l:ttl use tecause the radiation damping

coef-ficient ray be ecily difficult to find. However, the

substitutir)r leads to a very useful rsult when the

dynamicaliy ampiified response at a natural frequency is of connern. In surh cases it is slio'i that, contr2ry to

popular bcief, the response is not inverse!y

propoc-tioaf o the tote! damping but is, in fact, proportional to the ra.io of the racation damping to the tote damping. This i. beuauce the radiation damping and the wave ex-iting forces are not indepenoent quantities Therefore, in the absence uf a reliable estimate of either the total damping o i.he ratio of the radiation component to the iotal, an upoer hound estimate of thc response sll may

MiS Jour' al

/

Predkton of th Dmpng-Controed

iesponse of Offshore Structures

to Random Wave Excitation

J. Kir Varidiver

SPE-AI,4E, Massachusetts Institute 'f Tc-cfiricIogy

A method s presented for predcting the damprnq.ccntrolled response of a struLIuC at a known natural frequency to renciom wave forces. The principal advantage of the proposed method over those in current use is that tne explicit calculation of wave forces is not raquired in the analysis. Thi is ac-complished by application of tie principal of reciprocity: flat the linear wave force cpcirurn for a

particular vibration mode is proportional to the radiation wave-rìaking) dampig of that mode. Several example calculatiens are presented incii.Jing the prediction Oi the hve resronse of a

tension-leg platform. The directional distribution of the wave spectrum is included in the analysis.

Lab.

V.

kun d

Technische Hc9eschool

Deift

be achieved because the ratio is, at most, one. The dem-onstration cf th existenne cf thib upper bound is one of the key contribuions of this r)r1or.

Linear wa"e theory is acsurned; therefcre, excitatkjn caused by drag forces is n:t considered. However, for many structures drag e- :ita.ion is negligible except for

very large wave events, n tht desiì process externe

events are modeL. deterrnnisticalIy by means of a pre-scribed design wave and not stochasfical!y as is done

here. In mary circumstances linear viave forces will

dominate, arid the results show here will be ppIicable Although dregcxciting fuCC are not included,

damp-ing resultinr from hydroaynarnic raç is Included. Wa"e

diffraction effects are extremciy difficult to calcula.

eis analysis includes effec but never requires ex-plicit evaluation ci them.

It has been recoçriized tnat directftnei spreading of the

ways spectrum ¡s an importait considerarion in the

estimation of dyarric response. In this paper such cf-fects are accounted tor in Liosed-form exp'essions. The

evaluation of the expressions tquires knov.'iedge of

estimates of the variation of the modal exciting force with wave incidence angle. However, only the relative variation of the modal exciting force as a percent of that

Evaluation Á t.e wave force absolute terms still is not requirea.

There are numerous applications of present interest. For example, the fatigue analysis of a tension-leg platform must include an estimate of tee amplified reponses at

the natural frequencies of the structure in heave, pitch, and roll. This method quickly provides that response

estimate. An example calculation foi the heave

re-sponse of a tension-leg platform is included.Two addi-tional examples ara provided, which exploit simplifica-tions that frequently may be useful. The first is the case where the wa'e exciting force is incependent of inci-dence angle, as would be true when considering the

heave response of a structure v,th a vertical axis of

symmetry iCe oecorìd example iliustrates the sirnplifi-cations obtained wen Inn wave spectrum is distributed broadly in incidence angle.

The techniques applied in tus paper ar-a new to the field of ocean engineerinq. Hnwever, they are not without precedent and have ¶oLnd extensive application n the

ieIds of acoustics and vibration.3

LtNEAR OSCILLATOR MODEL

A structure in the ocean may have a l:ge numberof

natural frequencies, alihough at only a few is the dynamic response to wave excitation likely to he impor-tant. It is onvenicnt for the purpose of this paper to assume that by using the techniques of modalanalysis each of the responding natural modes ma'/ be modeled as an independert single-degree-of-freedom resonator. The general requirements foi this are that the vibration of the structure behave in linear fashion and that the damping be small. The motivation for using modai analy-sis is that it is far simpler mathematically to anayze a few independent single-degree-cf-freedom models than

one large, coupled, multidegreeotfreedom system. Reference 4 presents a thorough dir.r-.ussion of modal analysic, and. Reference 5 demonsh ates its applicoticn to offshore structures. tri some cases the technique of modal analysis does not eliminate aU of the damping-related coupling terms between modes. Fcr the response predictions considered

n this paper this

generahy is not a problem. Supporting discussion is presented ater.

This paper will be presented in terms f the response of a simple sïnle-degree-of-' eedc.n resonatorexcited by ocean wave forces. The reults shocid beinterprated in

the largor context of modal analysis: that the total

response of a SiruCtt. can be obtal roo by a

superposi-tion of the individua responses of the modes of

in-terest. Although it will not always be stated cxplicitly, the coefficients and vriabIes of the single-degree-of-freedom system ust be expressed in terms of the

a-propriate modal quantities for the specific natural mode being modeled.

The equation of motion fr the sing Ic-degree-of-freedom 32

resonator excited by ocean waves will contain terms corresponding to hydrodynamic forces a well as urely

mechanical ones, such as structura? stiffness. The

hydrodynamic exciting forces usually will be a function of the relative acce'eration, velocity, and displacement between the water particles and the generalized

coordi-nates that cpreeent the motion of the structure. For structures tCat hahave in a linear fashion, these quanti-tie may be expressed separately. Thus, the loads on tee resonator resulting troni its motion n an otherwise calm ocean may be added to the forces exerted on the resonator when held rigidly in placo arid loaded by the paaage of ocean waves. This may be exoressed

mathe-niatically as follows, where th coefficients are often functions of frequency.

(rri+m,1).'± (Ri+Rrad+Rv)S+(Ks+Khy)

f(

+g() +h(i),

(I)

wh.re the reonse quantities are: ri = modal mass cf structure,

mocal added mass of water,

R, linear internal structural modal damping, not related to the presence of the fluid,

R_{nid =} radiation srwave-mal'ing damping of the mode

(a unear frequency depend'nt term that may be expressed by potential ]ow theory),

R = viscous fluid modal damping (due to the

as-sumption of light damping, it is assumd that an equivalent linearization will he adequate),

s = appropriatè nocmal cocrdinate obiaincd by model analysis for this particular mode, K5

structural

modal

stiffness,

and

Kh hydrostatic modal stiffness that arises from changes in displacement of a body en the free surface.

On We right-hand side ppear the excitation quantities

!hat are functions of iie water particle acCeleration,

velocity, anc1 displacement ) ,, and r, respectively.

g() = drag torce excitation term that is ssum-ed smail comparssum-ed witl, the other two

terms and s drnpped, arid

fti),h(,i) =

hydrodynamic nodal forces that norilally

woud be calculated from potentia

flow theory by iotegrating th pressure c',er the surface of the rody; these are,

ir

act, the inertial cou hydrostatic

forces exerted by passing waves. The exciting forces appearing on the right-hanu side are the rrcdal forces that would be exerted on the body ¡fit wore held rigidly n place. These torces r.cludeall linear

diffrctinn efeìs. A principal conclusion of this paper

is that these forces need not be ivaluated explicitly to obtain an estimate of the me-an square rcsponse of a patUcular vibration mode.

For the dssumption of 'Nave force linearity to be valid, thc, ratio of wave amplitude to structural member d1m-eter must be on the order of one or less. For circular menbers in oscillating flow this corresponds to a ruel-engan Carpenter number of less than 2 The results presented in this paper are not valid for structures com-posed of small members excom-posed to large waves. Hoa-ever, high-cycle

low-stress fadgue consicrations

require estimates of response in low sea state., where tor even relatively small members the forces ar essen-ually linear. In such circumstances the rerults presented here will be useful.

Equatinn i is of the form of a simple singie-decree-of-freedom oscillator, as simplified by

m..,s + RT i + Ks = F(s),

(2)

w h e re:

= total viiva! mass,

R-r total damping, K total stiffness, and

F(t)

modal exciting force.

The undamped natural frequency and the damping ratio are given by these familiar expressions:

w0

= 'JK/m,

R

Mi'S Journal

m( i-.and K generally may not be assumed inde-pendent of frequency. However, i the following analy-sis, the fiequency range of interest is confined to a nar-row band about the natural frequency. Within this band we assume that mIL, and K do not vary. However, the f re-quency dependence of Rrmay not be disregarded so easiv. The radiation damping portion of RTis strongly frequency dependent. Because the behavior of an oscil-Iato at r'sonance is damping controlled, the nature of the dampino must be well-understood before simplify-ing assumptions are made.

flECIPROCITY RELATIONS

The evaluation of hydrodynarnic forces on a body in er' incident wave system is difficult. lt is necessary to know

not only the hydrodynarnic pressure in the incident

wave system but also the effects on this pressure fielu due to th presence of the body. The innident pressure field is relatively easy to evaluate, but the diffraction ef-fects usua1ly are extremely diifiCit to obtain. Haeind' and Newman have presented expression for the

excit-¡ng feces and moments on a fixed body that do not

requio knowledge of the diffraction effects but depend instead on the velocity potential for forced oscillations of the body in calm water. n other words, there is a dirct relationship between the radiation damping on a body that is forced to oscillate in calm water and the

foree exerted on that body when it is held fixed in

incident waves.

Ncwman evaluated the expressions for an arbitrary

three-dimensional bony eiLher on the surface or sub-merced in terms of the six generalized coordinates and forces relating to the six rigid-body degrees of freedom. In genera!, one would desire the relation between the modal radiation damping coefficien. and the modal ex-citing force. The modal exex-citing force and, therefore, the modal radiation damping may be obtained by a linear

transformation from the six generalized forces in accordance with the method of modal analysis.

The HaskindlNewman relation is sted here in terms of the modal quantities necessary in the remainder of this discussion:

w3 ç271F(w,ß)12

dfi,

(5)

:; _{) IA(w,j3)!2}

here:

Rrad(W) = radiation damping coefficient for the nat-ural mode of interest,

F(w,fi) = modal exioting force exerted on the fixed

body by a systP. of plano deepwater

waves of frequency w and amplitude A (w,13), incide-it on the body at an angleß; F(w,ß)andA(w,ß) both have

an e1w1 time-dependent form which will not be explicitfy written cut,

p=

densityofwater, sed

g=

acceleration of gravity.

Equatio 5 states that the modal radiation damping

coefficient is proportional to the integral of the square of ihe modal exciting torce, integrated over all angles of

incidence.

Vugts6 experimentally confirmed the validity of these results in a sanes c mode! test published in 1968. In general, for ari arbitrary body the wove fo ces wili de-pend on the shape of the body and the ancle of inci-dence of the waves. For this analysis it is useful to have a shape function defined as

F(w, $3)

=

A(w,fi)

(6)

F is a measure o the modal force pe unit wave ampli-tude as a function of wave frequency and incidence angle. A mean square value of F computed over all nei dence angles io given simcly by

(1F12)

=

2IF(,ß)i2

dß.

Theiefore, from Equation 5.

Rd

(w) may e exprc'sec1 in terms of the mean sc'tare value ')t

w3

Rrud(w)

= 2pg3

(II'I>ß.

(8)

Equation 6 m'y be rewiitten as

F(w,ß)

=

/w,L?) r(w,13).

(9)

This is the modal wave force due to the incidence of

regular waves of a single frequency and incidence angle. Again, the time-dependent e' term is implied and not explicitly written. Because only linear

pro-cesses are being consicered, superposition of waves cf

many frequencies and incidence angles results in a

modal wave force spectrum of this forrn

When possible the modal force spectrum may be s!mp?i-fled further by integrating this expression over all inci-dence angles

2i

SF(W) =

_{Sa(w,?)IF(w,)I2di3.}

(11)

lt is desirable to normalize this expression with respect to the simple wave amplitude spectrum and the mean square value of the shape function. The resulting non-dimensional normalized modal force is designated by the symbol C,, which for any given structure, sea state, and natural fretuency is a constant expressed as:

S,,(w,ß) I(,f) I2dß

s,,(w)<lrl2)

(12) Usig C1, the modal wave force spectrum may he

ex-pressed as

SF(w) = C1 S,,(w)< IF2!> (13) C is a measure of the influence of dirsetional

spreac!-ing of the seas of angular dependence of the shape

function. As will be shown, C1 = whenever the seas are distributeri broadly ¡n c4irection or the modal force is insensitive to chancos ri incidence angk. Three exam-pIes at the end 'f l'e paper show how to obtain C.

From Equation 8 the mear sqmre value of rmay be ex-pressed in terms of the raaUon damping. Substitution

into Equation 13 results in 2pg

SF(w) = C1 S,1(w) _1rad(°') (14) This is a result of considerable utility. The wave f'rce

spectrum has been expressed in terms of the simple wave amplitude spectrum and the radiation damping. This result le:ds to useful expressics for the response of the rescna'or.

RESPONSE OF A SINGLE.DEUREEOF-FREEDOM RESONATOR TO RANDOM EXCITATION Through the use of modal analysis, the total sructurI

vibration has been expressed in terris of a set of

inde-pendent single-degree-of-freedom osciIlats, ora for

each vibration irode. lf the displact.rnent ot one of these oscillators id denoted by s, the displacement response

spectrum to the modal wave force sptrum SF(w) s

given by S5(w) ₌

5r(w)lH5)2,

_.

_{- ..(15)}

i-: (w) 12 = 1/K2 L

(i

w2)2()

w0 (.3

The modal wave force spectrum and, consepjntly, the modal radiatiomi damping. R,a(w)- var' with frequency. The total damping ratio , through its dependence on Rrad(W), also is a frequency dependent term. The re-mainder of this section is devoted to presenting a

sim-ple but accurate pxpression for the response of the

resonator that arises from the dampig-contnlled reso-nant peak that is centered on the natural frequency. From random vibration theory, th mean square or a pro-cess is given by the integral of the spectrum over all

frequencies. Therefore, the mean squa9 disnlacemenL is gi"en by

- S5(w)dw SF(w)IH(w)I2dw,

(.) /

--(17) whee, for engineering purposes, only positive frequen-cias are allowed.

f the force soecrurn is a constant, S0, over all frequen-flies, the mean square displacement is simply

(s2)

S1mIHs(w)I2dw.

(18)

For light constant damping (i.e., 0.15 )the value of this integral is approximated closely by this expression, which may be found in the text by Lyon:

7rS0 irS0

= = (19)

4m12w

_2Rrmw02

34

_v.lSn.1

Sj:(w,/3) S,, (w,ß) II'.< ,ß)12 (IO) _{where H5 (w) is the complex frequency response of tie} resonator arid may be found in any vibrations text.4

where Rr = R ± R + Rr,çj ,

the total dampig of

the resonator. The largest contribution to this integral cornes from the damping-controlled peak in IJ-I )

l,

which is confined to a narrow band of freque.uics about

te atural frequency In fact, 64% can be attributed to the small band in frequency,w0 ± known as

the haf-power bandwidth, u 2w0 . The mean

square response to S0 in the half-power band ma; be exprsser1 as (l +

= S

I1f(o)2uu

'.. ¿w O

-So R

-

64°7ø (21) (s2; <s2>

If the limits of integration in Equation 20 arc doubled to include twc halt-power bandwidths, '-o,

± 2w0 , then

80% of the total dynamic response will be inciuded:

0.4irS0

(22)

RTml,wO2

An accurate estimate of the mean quare response of a lightly damped resonator excited by ocean waves may

be obtained in a half-power bandwidth. This may be

done by ass:rning that the values of the wave force

Spectrum and the radia'ion damping at the natural

fre-quezcv of the resonator, w0 , mpresent acceptable averages ovo the band &' . This assumption povides a sirple bu reasonably accurate estimate of the

damping-cotrc!leJ dynamic response ri the half-power bend,

2

S(c0)

KS >

-

₂

R7-(w0 )no0

The error introduced by this approximation is related directly to the width of the halt-power hand

w = 2w0

and, therefore, to the total damDin . For very low

damping ( 0.05) the error is negligible. This was confirmed b a numerical integration of Equation 2'.) over the half-power band for a variety of cases in whch the wave force spectrum and radiation damping were allowed u vary wi:h frequency in a realistic fashion. The worst case resjlts indicate that the error introduced by using the apprcxirnation of Equation 22 was less than

2% for = 0.05. This error will increase with an

in-crease in the tota damping . However,for any specific ariplicalion the frequency dependence of the wve force

spectrum S1-w) aid tie total damping ratio may be estimated in the neighborhood of the natnral frequency w. The actual error may he accounted to by evaluating the ratio between the expressions provided in Equatkns 23 and 20. Such a procedure would allo'; the extension

of the smpie results of Equation 20 lo include tota!

clamping values as high as 10 or 15%.

In the case of very low totei damping (0.05)the

assumption of constant force spectrum and total damp-ing my be increased to include a gre2ter portion of the damping-controlled peak. For example, Equation 22 may

bu used to provide an estimate of the

darï.ping-controlled response in a region which ¡s two half-power barvjwidths wide. For = 0.05 the worst case error ¡ri-creases to only 6%, and approximately 80% of the tota! dynamic response is contained ¡ the prediction given by

2 0.4-zr SF(W

<S >2& ° (24)

RT(wo )m,,w0

To simplify the presentation in the remainder of the

paper, response estimates will be made for the region defined by a sinçule hail-power bandwidth using

Equa-tion 23. It is implied that other estimates using broader bands, such as Equation 24, also may be used, bough larger errors will results.

ELIMINAT!ON OF EXPLICIT CALCULATiON OF WAVE FORCES

The reciprocity relation was used to derive an expres-sion for the modal wave force spectrum in terms of the radiation damping (equation 14). This expression may be substituted in F-quation 23 to obtain an expression for the mean square response ir the half-power band, which does not renuire explicit calculation of the wave force spectrum:

j_Pg3i,1(wo)

Rradwo)

Rr(w)

(25)

The most important feature revealed by this expression is tha.t the damping-contro!led response of a resonator excited by linear ocean wave forces is dependent on the ratio of the radiation to total damping evaluated at the natural frequency, w0. It is often easier to estimate the

ratio Rrad(wo )/R(w0) than

t is to evaluateRQ (')

Firthermoro, because tris ratio can never e:ceed one, an upper bound estimate still may be achieved without

ai; knowledge of the ratio. This upper bound is

ind-pendent of dampirug. The widely held belief that 'hie response of a structure al a natural frequency increases without hound as the dampung is decreased is simply not true when tre excitation is provined by linear wave forces. This is a consequence of the rec3procty relation stated in Equation 5. lt is impossible to reduce the radia-tion damping without aiso reducing the exciting force-3, thus resulting in a bounded response.

The uneva!uate'l constant c'1 is dependent on the shape

of the structure and the direconaiity of the wave

spec-trum. The following three examples will evaluate C.

These examples were selected because they may be ex-tende directly to a large variety of ocean structures.

SAMPLE RESPONSE CALCULATit)NS Example 1: Heave Resporse

of an Oceanographic Mooring

The resuts of this example apply to any structure for which it may be argued that the modal force is inde-pendent of wave incidence angel

Consider the simple oceanographic mooring shown in Figure 1. Ii consists of submerged spherical fleat and a tripod elastic tether. The 'indemped natural frequency in heava is given by Ecjatien 3, where K is a linear stiff-ness coefficient for smail vertical motions. The modal force for vibration ir' the vertical dftection 3 simply the generalized force in the vertical direction on the float. Furthermore, because the float has a vertical axis of symmetry, the heave exciting force is independent of the angle of incidence of the waves; therefore, F(w,3) is a function of w only.

(f

jT

(l)S' -/ /

---/

. .

-

--Fiqure 1. Oceanographic Mooring.

The modal exciting force is derived from Equation 1.

Because r is independnt of ß, it may be moved

out-side of the integra'. Note that in this case the magnitudé uquared of r and its means square with respect to ß must be equal:

lI't2 =

. (26)

Therefore,

=

(lrl2>4S(w,ß)dß

₍₂₇₎

= (lFl2)ßS(w).

(28)

because the integration of the directional vave

spec-turn over all incidence angles iesu!ts in he simple wave amplitude spectrum.

This result, when substituted into Equation 12, revecs

that C =

i It follows from Equation 14 that the heave exciting force spectrum is given by

'pg3

Sf(w)

= _Rrad(w),

where _{Rraci (w)} is the modal radiation damping cf the axi symmetr1c float for heave motions.

The. hesve response spectrum is as presented in qtla. tion 15, and the mean square response in the smail half-powc banc about the natural frequercy is from Equa-tion 25.

2pg3 Rrad(Wo)

Sy,("t0) X

RT(w0)

(30) This estimate of the heave response of the buoy is ap-propriate within the half-power band

w = 2w0,

pro-videi the system is reasonably linesr, the tOtal damping is small, and tnc assumptions and limitations of modal analysis are satisfied.

In this prediction of the heave response.of a moorg, no mention was made of the dependence on the depth of submergence. This is implicit in the ratioRrad(wo)/RT (w0), Newman Ehows that the radiation damping coeffi-cient decreases as where k is the wave number

of radiated waves. In the lirrit

that the depth of

submergence í - , then R0d(w0O, and the ratio also goes to zero. Thus the response of the buoy is

reuicted correctly t be zero at deths below th

reìori

of signifiant wave excitation.

The specific results shoWn n Equations 29 and 30 for this example are generally applicable to a broad range ot

structresthat is, wheiever th

modal exciting force is independent of wave incidence argle As shown next, these results also apply whenever tne waves in the fre-cuer'cy hand of interest can be assumed to have ra'dorn

incer.ce angle.

Exarnv!e 2: Random incidence Waves

When the inoidqt wave spectrum h,' Jistributed equally over all cidence angles, the resits sho'vn in

Equa-tions 29 and 30 apply. This

is _{relatively easy to}

dernonstrte, even for structures with complicated or

rjnkonwn shape functions. For waves of completely ran-(29)

dom incidence angle, the directional wave spectrum and the simple amplitude spectrum are relatea in this way

S (,fi) = S ()

(31)

2

This may he substituted into Equation 11, the genesal expression for the force spectrum:

2i-SF(w)

. 2

=

Sr(04 - Ç

li'(3) I2d,

(32)

2ir 'O

where the angular inc1eperdent wave spectrum has been moved outside of the integral. The integra' now is re-duced to that which defines the mean squar of F wth

respect to f3.

Therefore, S(i)

Sr(W) <iri2.

which leads to the onclusion that C1 = 1. This

im-rnediateiy leads to the same expression for the response in the half-power bandwidth as found n Equa-tion 30 of the previous example. In fact, for the result shown in Equation 30 to be v&id, tt is necessary that unly the waves whose frequencies lie within the hale-power band be randomly incident. Waves outside of the band need not be so randomly orented. As d practical matter, the high-frequency components of a seaway

tend to be more confused in direction than the

low-frequency waves. Therefore, the validity of the

acsump-tion of randomly incident waves may be more

ap-propriate tnan ordinarily supposed, depending on the natural frequency of the structure, geographic location, and prevailing weather.

This result applies to an arbitrary shape function. Any structural symmetries will reduce thc range o ang!e over which the waves must be rardomty incident For

eamp!e, it can be shown that for a structt"e with two

orthogonal vertical planos of symmetry, such c a steel

jacket platform with a rectangular ayou. of its pmary

leo, the waves in the half-power band need oly be ran-dcrnly incident over a semicircle (i.e., 180') ior Equa-tions 28, 29, and 30 to hold. The result ir oht be used to

predict the mean square response of te two lowest

flexural modes.

For many structures these simplifying assumptions may be justified, and the simple result for the mea-i square response within the hr-powe bandwidth as

shown in Equation 30 may be applied.

However, at times such assumptions may not be ac-ceptable, and it may be necessary to measure or

estimate ,f3) and to incorporate a directional

ae

spechum S (w,ß) Such a procedure i followed in the f mai exanipie.

Exaiip!e 3: The Response of a Tensiori-Leg PIa$orm to Random Wave Excitation

An important corcern in contemporary d3nign of all

MTS Journal

platforms is fatigue. The prediction of the fatigue life is

a process that must include the anticipated vave

statistics and response statistics of the structure. Numerous authors hve reported on difficulties

er.-countered in estimating the response at the resonant frequencies of the structure and have noted that the response prediction for the frequc;icy band about ros-onanc-e is critically dependent on the value of damping ihat is selected. This method is directed specifically at predicting the response in the resonant band and puts the role of rlamping in the proper perspecîive. Knowl-edge of the totei damping is not sufficient. lt is impor-tant to know the wa in whh the damping is distributed among radiation and all other sources.

Consider the hypothetical square tenson-leg platform shown ri Figue 2. At the preliminary design stage it

would be usetul to have an estimate of the response of the structure Io a prescribed sea state at its natural f re-quencies in heave, pitch, and moli. In the following

exam-ple only the heave 'esponse will be estimated. The

response in the roll and pitch modes would be carried

out in a very smilar fashion, as las been shown in a

thesis supervised by ihe author.7 The primary purpose of this example is to illustrate the method one might use

to take the geometry of the structure and the

direc-tionality of the wave spectrum into consideration.

Figure 2. Tensio;'-leg Platform.

Thc influence of both the directionality of the wave spectrum and the geomer of the structure has been

compressed into the unown costant

C1 shown in

Equation 25, the prediction of the mean square displace-ment response n the half-power bandwidth. C was de-fined in Equation 12, which is shown here where he in-tegral form of the nean £quaie of F(w,13)has been used to replace the ( > nritation.

37

.---,.

'-'

_..

-tF(w,)

2/

cl -

(33)

S,(w)-1

lF(,B)Vd

27 Jo

The directional wave spectrum is prescribed and here is assumed to be a cosine scivarc.d istrubution about some reference ariqie _ß0.

Sw,ß)

= --So)

cos2(L3-L0) ,

(34)

which is

valid for - 7/2

- í3

7/2 and zero

elsewhcre. lt is noted that

.(3 I- r/2

S,(w)

S,(w,3)r'3.

(35)

By substituting into Ecu'tion 33 the expression fc'

common term

S (w)

S (w3) and noting that the

cancels out, this is

btined:

,

-

Cl-cos2

3-0)I1'(w,L)t2d13

Ç

'r' -

2

jl3u_i/

X 1 .2x

-

tF(w,3)i2dß

27 JO (36) The problem has rc-duced to the need for an estimate of the angular dependence oí

IÇ(j3)

I-This task is simpl-tied because an expresson valid for aU frequecies, w,lS not necessary. An estimate. valid at only the raturai f

re-quency of interest, w0,is sufficient. Ir' Figure 3, plane progressive deapwater waves of ur,t amplitude and f re-quency,w,are shcwn approaching the tension-leg plat-form at an anglc .The magnitude of Uie heave torce

ex-L

Figure 3. Regular Waves Incident on the Tension.!eg

Platlorm.

,;-:-- \

O.'- )

38

v.lSn.1

erted on a single cxiall symmeric eg is i;dependeìt of incidence angle and may be expres5ed as F0 (w) I. The macnitude of the force exerted on the entire

stric-ture will depend primarily on the relalive phases of the four indvidual leg forces and on any leg interaction ef-fects. Th interaction effects are ascumed small

corn-pared with the phase effects and are ignored. The magnitude of the total heave force accounting for phase effects is given by = 41F0(w0)I cos(cos@' X

I

/ cos(, sinß). (37) X

where L is the leg spacing and A is the vave length

corresponding to a frequencyw0.Substitution or this ex-presion listo Equation 36 yields this result for L'1:

,

=[cßofr12

_{cos2(í-ß0) cos2 -__cosß)}

(XL

[Jß0-1-12 X

(2

2/XL

cos2

(-_-sinß) dßj

cos

2/XL

\

i

cos --sinß) d131. .. (38)

This expression was integrated numericslly for ali com-binations of heave natural peiod and !eg spac.ing rang-ing from i to 4 seconds and loo to 300 rt (30.5 to 91.5 rn). To ftl % accuracy, C1 = I for all directions ot

in-cidence, 30,of the cosine squared v.'ave spectrum. The cosine squared distribution was sufficiently broad to smooth out the effects of varying wave force phases on the four legs. This unexpected but simple conclusion allows the use cf tOe simple result of the previous two examples. The mean square heave response in the damping-controlled halfpower band is given by Equa-tian SO.

For the arge legs of a tension-leg platform the rdi.ion damping will likely be the greatest contributor to the total damping. Consequently a.conservative but

rea-sonable upper bound estirnete for the ratio c the rada-tion to total damping is 1, and Equarada-tion 30 rcduces io

2pg3S1(w0)

(39)

ni ,w0

An example calculation where:

rn = 22.000 tons (20,300 «g), the virtual mass of

the tension-leg platform r. heave, = 2.1 radians/s, whichcorresncads to a heave

period of 3 seconds, ana

0204 t2-s (189 x 10_2 m2-s), calculated for a 30-knott (15-mIs) Pierson-Moskowitz

spectm,

Yields a root mean square heave amplitude cf

The heave response is insignificant. However, to arrive at (hat conclL(sion by any other means would have L,aen much more difficult.

Damping-Induced Coupling

'At this point it is appropriate to discuss a routir;'ly ig-nored source of error for all response prediction

tech-niqi.ies. Th error arises because of the c(;u)inq be-iween otharwise independent vibration modes

tht is

introduced through damping.

Dampig-induced cc.'Jpling makes it possible for viura-tion energy to be transferred between modes. or the response prediction analysis presentd in this paper, such coupling generally is not sigilficant for the follow-ing reasons.

First, for most ocean structures of interest only a few modes have ow enough natural frequencies to be ex-cted by the wave spectrum. With a few notable excep-tions the natural frequencies of these modes tend to be

well separated. Due to the assumption of tight total

modal damping, the response of each mode is domi-nated by the damping-controlled peak centered on the natural frequency As long as no two natural frequen-cies are so close together that their response peaks overlap, then the energy transfer by damping-induced coupling between any two modes will be insignificant.

n certain typcs of structures, concident natural

fre-quencies do occur. Two common examples are the low-est end-on an broadside flexural natural frequercies of steel jacket structures and the pitch and roll natural tre quencies of a square tension-leg platform as described in the previous example. In both cases, however, the rec-tariqlar or square geometries or the structures provide symmetries in the motion of each mode that rsults in

negligibly small damping-related coupling.

Foi' example, the response of the pitch mode of the

tension-cg platform will result in port.starboard sym-metry o aciated waves. As a consequence, the radiated waves will generate no roll-exciting moment. Therefore, aven though the iesponse peaks overlap, no coupling results from the radiation component of tIte totai

damp-ing. Smiliar arguments may be applied to the other

darnpng components. Small asymmetries that do occur

in structures give rise to small coupling tcrm, which

often may be negiected.

ENGINEERING IMPLEMENT,TION OF THESE RESULTS

Ti-e implementation of new theoretical rSjliS often re-quirc3 alteration of accepted engineering practice. The theoretical importance of the ratio of radiation to total damping has not been recognized previously.

Experi-mental techniques and numerical tools for efficient

evaluation of the ratio re unavailable. Experience will reveal whmch applications are basi

suited to the

methods described here. A comparison with present practice is used to hghlight promising features of the new techniques.

Present practice la dvnai ic response prediction

re-quires estimation of the magn;tude of the wave

amplitude to wave force transfer function as a function of wave frequency and wa'e incidence angie. This

func-tion is denoted by ir (u,,ß) lin this paper. For

com-parison the method proposed here does not require an ah.olute measure of IF (,ß) Ibut only ìs relative varie-tion with repect to that at an arbi«ary incidence angle. Thus. t is easier to estimate and less sensitive io

selec-tion of, for example, the exact value of the inertia coeff i-Clttnt.

Pre3ent practice also requires an independent estimate of the total damping oc the structural natural mode of in-terest. The recent works of Ruhi aad Berdahl 8 and Vari. diver and Campbell reveal that the published results for measured modal damping or exishng structures has

been generally inaccurate. Therefore the ability to

estimate the totei damping for new designs has been hampered by a lack of accurate empirical data.

The method proposed also requires knowledge of the model demping but in the rather unique form of the ratio of the radiation component to 'etal damping. In some cases this may be easier to stiniate as it depends only on relative quantitos. Furthermore, it is helpful that in the absence of reliable knowledge of the damping, an upper bound ori the ratio may be used until further infor-matior becomes available.

A number of possibilities exi:t for th

engineering

estimation of Rrad/RT

ad II' ((Aß) I. Consider the

esiirnation of ir I first. i cases of special

sym-mctry or diffuse seas as in Examples i and 2, IF (w,) I need not be estimated at eli. For more complex

struc-tures andor drectional' concentrated wave spectra,

the relative variations of F (w,j3) I with ß may be esti-mated by one of the f!owing methods. The first is by carefully considered engineering approximation as il-lustrated in Exan pe 3. The secona is by a static, not dynamic, finite element time domain model of the struc-ture, in which unidirectional regular waves at the natural frequency of irìter:st are passed by the structure troni

many differen incidence angles. Thereby,l F (.f3 lis

obtained for each value of incoenL.e angle. The thrd method i. by a relatively simple clatie, noi dynanic.

model test. In such a model test unidirectional regular

waves would be passed by the model held rigidly in

place by a sufficient nurrer of load cells to determine

the modal exciting fOrce. In sequential runs the

in-cictermce angle of the model vouJ be variert to ilevelcp IF(u,0,ß)LSuch a test requires geometric similarity of

the model and Fcde scaling, but not dynamic imih-tude of the mass and stftiness distribution. Estimation of modal exciting forces for ri;d body modes such as heave and pitcn of a tension-leg platform are extracted easily from load cell data. For stractural modes that ex-hibit deformation of the structure at locations exposed

to significant wave forces,_{estimation of rnL1a! exciting} forces from load rel data recu:ires additional corrcctk"i for mode shapes and, _{ther&ore, s more difficuit.} A number of techniques exist for the esJmation_OfRrwj/ RT. For simple structures, such as a freestanding cis-son, Rrad may be ca1oulated_{analytically from potr.tial}

flow theory. Fo _{more complex structures such as}

tension-leg ptatftrms, a diffraction theory wave foi_ce program may be used. Coupled_{with estimates of the}

re-maining sources of damping, the analysis leads to an estimate ol

_Rrad/RT.

Another method for obtaining _{Rrad is by direct} applica-liOfl of the principie of reciprocity. Data for ti' I

may he obtained by modei test or finite element simula-tion as described _{previously. The mean}

square of trat data with respect to incidence annie, <t r

()

2

may be used directly in Equation 8 to obtain an_estimate

Of

R,ad/RT also may be hindr.ast_{from full-scalerespons} data. To do this '.'ould require sinultaneous measure-ment o the dynamic_{response and the directional}

wave spectrum, plus an independent estimate of the direc-tional dependence nf ti' _{(w,) i. Given sunh data,}

the

constant C could be _{estimated ad Equat:on25 could}

be solved for _{Rfd/RT. This procedure has}

been at-tempted and reported for the lowest flexural modes _of two separate pile-supported platforms The fIrst _{was a} small four-leg jacket in 70_{fi (21.3 m) of water. The} hind-cast value of _R,Od/RT

₌

iO _{The second estimate}

was obtained for an eight-log production platform in 325 ít (99 m) of water. That_{estimate was}

RrCd/RT O.OS.

"

Both hindcast estimates_{required numerous approxima-.} tions and small data sets; therefore, rather large cori-fidence bounds were impiied. Field experiments_{to yield} accurate estimates of _{í,3II/RT are possible hut have} not been conducied.

CO N C LUS! O NS

A method has been _{presented tor predicting the}

damp-ing-corirolled dynamic response of an offshore struc-ture. The method is appiicable to a wide variety_{of st;} tic-tures and depends only_{on the assumptions of}

linearity of wave forces and _{structiral response. Furthemnre, it} requires that the total structural damping be _small. There arc three principal_{ccncusions to be drawn. First,} tne linear wave ¶o _{ce spectrum on a srqcture}

may be ex-pressed in ierns oì the radiation damping of the struc-ture. This is a consequence of the principle_o reciproci-ty for ocean wavc forces, which has been known for many years but has oct been applied to many common ocean engineering problems.

Second, through the _{use of the above result, a method} for estirnaring the damping-controlled response of a structural natural node _{has been presented that does} not require explicit calculation of the modal_{wave torce} spectrum.

40

Finally, the role of damping in the estimation ofdyne response is placed n the proper perspective. _Lii wave forces and modal _{damping are not indepenr.} quanì;ties. Therefore, it is not the total damping I

vibration mode that governs the response to_wave

citation but, in fact, _{the ratio of the radistion}

_{L t.}

damping. Since this ratio_{has an upper bound ci 1,} response has au upper bound

independent of the e value of the damping.

NOMENCLATURE

'k

A (w.ß) _{plane prorc-seive waves of amplit'.}

A, frequency w, and incider

Q _angleß

C7 _{constant dependent ori S}

,,fi) a

F (w, f3)

linear wave forces_{on fixed uody} modal wave force_{on nxed otructu} modal wave force _{on fixed structu}

du _{to wavesA(wL)}

acceleration of gravity

drag exciting ¶erc;e on. fixed body frequency rer-ponse c'f liiear I;econi

order single-dogree-of-freedo

s'1 stem

K = total modal stiffness

= structural and hydrostatic _contribt

lions to the modal stiffness leg spacing on tension-leg platform m _{modal structural rnQss}

= modal added ries

,n1 _{total mc'lal virtual}

mass

R _{linear nonhydrodynarnic damping} Rrad,R,,d (w) = linear adiation damping

RT total linearized damning

R, = inearized viscous _hydrodynamic

damping

s = moda displacement _coordinate

mean square disniacement

_{in th}

band w

S(wt3) = directional

_{modal wave force sper}

trum

= constant force spectrum = modai displacement _spectrum

= wave amplitud spectrum

directional wave spenfrum = wave incidence angle

modal vave force _{F(w.í3) per uni} wave amplitude

<1fl2)ß

mean square of _{F(w,f3)with respec} to fi'

= half-power bandwidth = total modal damping ratio

= water particle displacement,_velocity and acceleration

V.

1' i.

=

F(t) =

X = w = =

11=

wavo length with frequency w0 density of water

wave frequency

natura! frequency of the mode denotes magnitude of

ACKNOWLEDGMENTS

This research was spoisored by grants frani the

Na-tional Sciencè Foundation, Henry L. Donerty Foci-tion, aid the Massachusetts Institute of Technoioy

Sea G'nt -rogram. This research is. cotinuing with the sponsorship of the USGS Branch of Marine Oil and Gas Operations. Particciar acknowlec4iment is extended to Walter E. Schott Ill. Bruce Dunwody, anu Floif Dietrich, whose technical contributions were invaluable in the preparation of this paper.

R E F ER E N CES

Haslcind. M.D.: "The Exciting Forces and Wetting of Ships in Waves," Izvestia Akademii N'k S.S.S.R., Otde!enie Tekhnicheskikh lVauk (1957) No. 7, 65.79 (English translatior available as David Taylor Model Basin rars!ation No. 307, Marr.h l'd62).

Nev'man, J.N.: "The Excitinç Forces on Fixed

Bodies in Waves,' J. of Ship Research, SNAME (Dec. 1962 6, No. 3.

Lyon, R.H : Stati3tical Energy Analysis of Dynamical S3's rerns: Theory and Applications, M IT Press, Cam-hidge. MA (1975).

Meirovitch, L.: Elements of Vibration Analysis, McGraw-Hill Inc., New York City (1975).

Berge, B. and Perizien, J.: 'Three Dimensional Stochastic Re.por'se of Offshore Towers to Wave Forces," Proc., Sixth Offshore Technology Con-fercace, Houston (May 1974) 173-190.

G. Vrgts, J.H.: "The Hydrodynamic Coefficients for

Swaying. Heaving :tnd Rolling Cylinders in a Free Surface." International Shipbc.lding Progress (1 9t)

15.

7. Dietrich, PA.: "The Effects of Wave Spreading or:

the Exciting Forces on a Tensi,.i Leg Platform," MS thesis, Massachusetts Inst. of Technclogy, Dept. of Ocean Eng ineering, Cambridge, MA (May 1979).

R. RuhI, JA. and Berdahi, P.M.: "Forced Vibration Tests of a Dewater Platform," Proc., 11th

Off-shcre Tecfo!ogy Conferece, Houston (May 1979)

134 '.1 354.

Vandicer, u.K. and Campbell, RB.: 'Estimation of Natural Frequencies and Damping Ratios of Three Similar Offshore Platforms Using Maximum Entropy Spectrai Analysis,' paper presentod at the ASCE Spring Convention. Boston april 6, 1979.

Schott, WE. Ill: "Response and Radiation of Struc-tures ir an Irregular Seaway," MS thcsis, Massaciu-setts Inst. of Technology, Dept. of Ocean Engineer-ing, Carnbrdge, MA (June 1977).

Note: Paper orig7aliy writtcc for t!.e Society of PetroIcm Engineers of AlMs i979, puOlished toruary 1980.