Applied Ocecm Research 15 (1993) 169-181
Extreme values of first- and second-order
wave-induced vessel motions
S. McWilliam & R.S. Langley
Department of Aeronautics, Umversity of Soutliampton, Highfield, Soutliampton, UK, S09 5NH {Received 26 January 1993; accepted 23 March 1993)
A recently developed expansion for the joint probability density function of two variables has been used to develop a means of determining the mean up-crossing rate of a combined first- and second-order random process. The first term in the series corresponds to the assumption that the response displacement and velocity processes are statistically independent, and further terms depend upon the joint displacement-velocity cumulants. The extreme statistics are then determined using the Poisson assumption of up-crossings. The accuracy of the method is examined by comparing the results with time domain simulation, and it is found that the displacement and velocity are, to a good approximation, statistically independent for the examples studied.
Key words: slow drift, dynamic response, extreme values, random waves.
I N T R O D U C T I O N
The behaviour of compliant offshore stnictures is often dominated by large-amplitude slow d r i f t motions which are a resonant response to non-linear low-frequency second-order wave forces.' Although these second-order forces tend to produce the major part of the response, linear first-order wave forces can contribute significantly to the extreme response,^ and should therefore be incorporated within the design analysis.
I n a recent publication,^ the nature of the combined first- and second-order response statistics of a linearly moored vessel were investigated. The present paper expands upon the techniques developed there and discusses the manner in which the theory can be used to analyse the extreme response statistics.
Few attempts have been made in the literature to account for the non-Gaussian nature of the combined first- and second-order response process when predicting extreme values. I n principle, the asymptotic approach developed by Naess'* may be used to predict the combined extreme, although this method may be complicated and difficult to apply in practice.^ As a compromise, Naess^ has investigated the use of empirical formulae to estimate the combined first- and second-order extreme values. However, the approximate nature of these relations restricts their use to the preliminary design stage.
A n approximate method, based on the assumption
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© 1993 Elsevier Science Publishers Ltd.
that the displacement and velocity processes are statistically independent, has also been proposed as a means o f predicting the extreme values.^'^ However, the uncertainty imposed by this approximation has pre-vented the method f r o m being adopted as a reliable design procedure.
In this paper a recently developed expansion f o r the joint probabiUty density funcdon of two variables^ is used to develop a means of determining the mean up-crossing rate of the combined first- and second-order response. The first term in the resulting expression corresponds to the assumpdon that the displacement and velocity processes are statistically independent, and further terms are dependent upon knowledge of the joint displacement-velocity cumulants. Although the joint displacement-velocity cumulants may be evaluated using the analysis developed by Longuet-Higgins, the calculadon tends to be too numerically intensive f o r practical purposes, especially i f a number o f additional terms are required. Consequendy, an approximate method is developed which reduces the computadon time significantly, at the expense of accuracy. A further reduction in computer time is attained by assuming that the velocity stadsdcs are Gaussian. Finally, the extreme response statistics are calculated using the Poisson assumpdon of independent peaks.^'^
The applicability of the present method is investigated with reference to the extreme response of a semi-submerged cylinder subjected to unidirecdonal random seas, and the results are compared with time domain
simulations.^ For the examples considered, the theore-tical results give good agreement with simulation; sufficient accuracy is provided by the first term in the series, which corresponds to the assumpdon that the displacement and velocity are stadsdcally independent.
C O M B I N E D FIRST- A N D SECOND-ORDER W A V E - I N D U C E D RESPONSE
Before considering the prediction of the extreme response, the combined first- and second-order response stadsdcs of a linearly moored vessel are summarized.^
Assuming that the response, x{t) may be modelled as a linear single degree of freedom system, the equation of motion is given by:
Mx + Bx + Kx = F{t) (1)
where M, B and K are respectively the mass, damping and stiffness o f the system and F{t) is the random wave force which includes both first- and second-order effects. Following the detailed analysis given in Ref 3, the combined first- and second-order response of a linearly moored vessel may be written as
complex Hermitian matrix'" whose nmXh. entry is H„„„ and u is the vector whose nth entry is M„exp(/w„/).
By defining a vector X such that X = R u , where R"^ is a matrix whose rows are the eigenvectors of H , eqn (8) may be rewritten as
x{t)^Y.Pj^j+>^j\^j\ ( 9 )
where real parts are assumed, Pj is the jth. entry o f the vector R * G , Xj is the yth eigenvalue of H , and Xj is the Jth entry of a random vector X . Using eqns (5)-(7), the entries of X can be shown''^ to sadsfy the following properdes: E[\Xj\'] = \ E[XiXj] = 0 £ [ z , A 7 ] = 0, ; V J (10) (11) (12) x(/) = R e ^ G „ M „ e x p ( / w „ / ) « = 1
Further, the probability density function (pdf) of this combined response may be written in the f o r m of a series whose first term corresponds to the assumption that the first- and second-order components are statistically independent. Full details of this series have been given in Ref. 3; i f it is assumed that the first term provides an adequate representation of the response statistics, then
+ Re H„„AÜ:„ exp[/(c^„ - uj,„)t] (2) P^"") " 5 S^''^
M
n=im—\
where G„ and H,„„ are defined in terms of the single-sided wave spectrum ^^^(w) which has been dlscretized into N strips o f width dw, the first-order force transfer function (i?,f - ;'i?,f), and the second-order force transfer function {T,l„ - iT,',,,), as follows:
G„ = {Rl', - iRl,)[-ujlM+ iüj„B+K]-^ 1 "J M
- E
j=M+\ (13) ^[2S,^^{uj„)doj\ ^nm ir^nm ^"^nm) (3)where $ is the cumulative normal distribution function, (Tl is the first-order rms response and Xj is the j ' t h eigenvalue of the matrix whose « H 7 t h entry is H,„„. Further, the eigenvalues have been ordered such that Xj,
j = 1,2... ,M are positive and A,-, j = M -|- 1 , . . . , A?" are
negative, and jij is given by
[ - K - ^,„fM+ i{Lo„ - Lu,„)B + K]-^
V[2S^^{u>„)du;Wl2S,,{oj„,)du;] (4)
Further, ü„ are complex Gaussian random variables which possess the following properties:
m<n\'] = 1 A-=l (14) E[ü„ü„,] = 0 E[ll„Ü*,] = O, 77 7^ 777 (5) (6) (V) Equation (2) may be conveniendy written in matrix form as
x{t) = R e G ^ u + R e u ^ H u * (8)
These relationships will be used in the following sections to determine the extreme response statistics.
E X T R E M E V A L U E P R E D I C T I O N
The following extreme value prediction method is based on the assumption that successive up-crossings of a specified level are independent and constitute a Poisson process.^
Under this assumption that probability P{b,t) that the response, x{t) will cross a level x = b at least once during a period t is given by
Extreme vahies of wave-induced vessel motions 171 where is the mean rate of crossing b with positive
slope, which is given by
xp{b,x)dx (16)
where p{x, x) is the joint displacement-velocity prob-ability density function.
Equation (15) is an approximate formula which tends to yield a conservative estimate o f the exceedance probability; more refined approaches which allow f o r the stadstical dependence o f successive up-crossings are usually limited in application to Gaussian processes;" however, Naess'"^ has considered the effects o f correla-tion on the extreme slow-drift response.
The method o f determining the mean up-crossing rate for the present application is based on a series expansion of the joint probability density funcdon (jpdf) of the displacement and velocity o f response, as detailed in the next section.
M E A N UP-CROSSING R A T E
Recently,^ it has been shown that a general expression for the j p d f o f two variables can be written as a series whose first term corresponds to the assumption o f stadstical independence. Using this expression, p{x,x) may be rewritten as'^
p{x,x) =p{x)p{x)
OO OO ff'pjx) &"p{x)
""' dx" dx"' UX' ux (17)
n=\m=\
where A,„„ are related to the joint displacement-velocity cumulants.^ The first few A,,,,, terms are given by
Au=K,+,[xx] (18) Aix ^\K2+i[x^x\ A22=\K2+2[X^X^\1-K,^,\xxf (19) (20) (21) where Ki+j[x'F] denotes the yth joint displacement-velocity cumulant.
Subsdtudng eqn (17) into eqn (16) gives
ut =p{b) xpix)dx oo OO roo ^ ^ ( - l ) " + ' % , „ / ' ) ( 6 ) / xp^'H^dx ti=\ m = l CO oo =pib)I, + ^ ^ ( - l ) " + ' " ^ , „ „ p ( " ) ( è ) / „ , (22) « = 1 » i = 1
where p''"\b) represents the «th derivative o f p{x)
evaluated at level b, p''"'\x) is the 777th derivative of
p{x), and the integrals /„, are defined accordingly.
A direct consequence of eqn (22) is that the mean up-crossing rate may be obtained f r o m a knowledge of the individual displacement and velocity distributions i n addition to the joint cumulants. The displacement pdf, under the assumption of independent first- and second-order components, is given i n closed f o r m by eqn (13). Appendix A shows that the 71th derivative o f the pdf may similarly be developed analytically. Thus, i t remains necessary to determine the nature of the velocity statistics and the joint cumulants before eqn (22) can be used to evaluate the mean up-crossing rate,
V E L O C I T Y STATISTICS
The velocity, x{t) of the combined response is determined directly f r o m eqn (2) as
A'
X t ) ^G„ii„iu!„ exp{iu)„t)
n = l
N N
H„,„ii„iXni{oJ„ - 0J,„) exp[/(w„ - uj,„)t]
(23) where the definitions given i n eqns (3)-(7) remain valid. Rewriting this equation in matrix f o r m gives
i(0 = P ^ u - f u ^ Q u * (24) where P is a vector whose 7 7 t h entry is ioj„G,„ Q is a
complex Hermitian matrix whose « 7 7 7 t h entry is ;(w„ - uj,i,)H„„„ and u is the vector whose 7 7 t h entry is 77„exp(7w„0.
By diagonalizing the matrix Q and assuming that the first- and second-order components o f the velocity are statistically independent, i t may be deduced that the velocity pdf may be written in the same f o r m as eqn (13), such that
X
O-l.vJ
XJ 0"l.v X
(25) where tJi^. is the first-order rms velocity response and is the 7 t h eigenvalue of the matrix whose 7 7 m t h entry is /(w„ - uj,„)H„,„. Further, the eigenvalues have been ordered such that j = 1 , . . . , M are positive and
= M + I,... ,N are negative, and is given by
Although it may be argued that the first- and second-order velocity components are not statistically indepen-dent, i t will be shown that this assumption provides adequate accuracy when determining the mean up-crossing rate f r o m eqns (16) and (17), where an integration over velocity is performed.
To determine the mean up-crossing rate using eqn (22) the following integral must be evaluated:
&"p{x)
: dx" -dx m = 0 , 1 , 2 . . . (27)
Equation (27) is considered in Appendix A 2 f o r the velocity pdf given by eqn (25), and a pure Gaussian process. Under the assumption that the first- and second-order velocity components are statistically independent, /„, is given by eqn (A2.6), such that
, f ^ ( - l ^ x y ^ I f . I t ; " j=M+i I V I V ^ ^ e x p t e U
^ V ( 2 - ) + 2 |
OH i - (28)A comparison with an equivalent Gaussian distribution will be presented in the numerical examples section.
J O I N T D I S P L A C E M E N T - V E L O C I T Y C U M U L A N T S
The joint displacement-velocity cumulants are calcu-lated using a technique first developed by Longuet-Higgins to determine the effect of non-hnearities on the statistical distribution o f sea waves.^ Although the Longuet-Higgins analysis is based upon a general non-linear combination scheme, the order of the analysis may be reduced so as to include the first- and second-order contributions only.
To allow use o f the Longuet-Higgins analysis, eqns (8) and (24) are rewritten in an expanded matrix form as R e H I m H x = ( R e G ' , - I m G ^ ) Y - } - Y M | Y I - I m H R e H / X = (Re P ^ , - I m P"^) Y + Y ^ R e Q I m Q ' - I m Q Re Q (29) (30)
where Y is a vector whose entries are independent Gaussian random variables each having a mean squared value o f one-half These equations may then be rewritten, using the same convention as the original Longuet-Higgins equations, as 2N 2N 2N >^ = T . ^ j Y j + Y.Y.''j'^YjY, (31) ; = i j=ik=\ 2N 2N 2N \ ^ \ ^ i= 1 k= 1 (32)
Following the detailed analysis given by Longuet-Higgins, it may be shown that the first few j o i n t cumulants are given by
^ 1 + 1 = E ^ " . / - T / + E E ^ '^i^^jk (33) j k K2+ 1 [X^X] = E E 9 "Fk^Jk + E E ^Fk^jl , n. —J'-K IJK • / J —] IK-^jk j k ^ j k + J2J2J2''Jk'^knij (34) j k I Kl + 2[xx^] = E EöT/'^'taj/i + E E "^Fkljk j k J k (35)
+ EEEww"/y
j k I K2+2[X^X^] = E E E "^"Fkajllkl J k I+ EEE^>wQ^w
J k I+EEE"-'">w
+ E E E E 3ajVcaw7/m7,»/ (36) j k I mFurther, these equations are equally valid f o r the pure second-order case with the first-order contributions set to zero.
Noting that the response is assumed to be strongly stationary, it may be deduced that K„j^i[x"x\ = 0 f o r all positive integer values of n. Hence, the j o i n t cumulants given by eqns (33) and (34) may be ignored.
Further details o f the derivation of the cumulants have been given by Longuet-Higgins^ and H u . ' ' ' Although it is not a difficult task to generate the equations for the higher-order cumulants, i t can be a numerically intensive operation to calculate their values, especially as the limits on the summations are typically i n the range 200-400.
Even i f the symmetry o f the matrices is taken into account, a substantial amount o f computation is still required. However, a more significant reduction in time can be obtained by using eqn (9) to define the
Extreme values of wave-induced vessel motions 173 displacement. Rewriting eqn (9) in the same expanded
f o r m as eqn (31) gives
A-= ( R e / 3 ^ , - I m / 3 ' ^ ) Z + Z'^ A 0
0 A (37)
where y l is a diagonal matrix containing the eigenvalues of the matrix H , and Z is a vector whose entries are independent Gaussian random variables having a mean square value of one-half
Rewriting the velocity in terms of Z and expanding the resulting expression in the same form as eqn (37) gives I T T 3 * T I m P ' ^ R * ' ^ ) Z - K Z ^ i - = ( R e P ^ R / ReR*QR'^ I m R * Q R ' ^ \ X Z V- I m R * Q R ^ R e R * Q R ' ^ / (38)
A direct consequence of using eqns (37) and (38) to redefine eqns (31) and (32) is that ajj^ = 0 whenever
j k. Hence, the first two non-zero joint cumulant
equations may be rewritten as
+ J2J2^Jk^kjajj (39) j k k
+ EEE"y"^w/
; k I (40)This scheme reduces the computation o f Ki+j[x'x^] considerably when i > j, and an analogous scheme i n which the velocity equation is rewritten so that the second-order velocity term is diagonalized will reduce the computation needed to determine the cumulants whenever ; < j.
Even when these schemes are implemented the computational effort required may sdll be excessive. As this may prove to be an influential factor when performing the analysis, an approximate technique is introduced in the following section, which permits a quicker solution at the expense of accuracy.
A P P R O X I M A T E J O I N T D I S P L A C E M E N T -V E L O C I T Y C U M U L A N T S
The approximadon used adopts the methods of diagonalization described above but neglects the second-order velocity contribution when determining the cumulants. The first two non-zero cumulants, given
by eqns (39) and (40), may then be written as
Ki+2[xx^] = Y^^lFj-ajj
K2+2[X'X'] J2^jljajjajj
(41)
(42)
It may be noted that eqns (41) and (42) contain considerably fewer summadons than eqns (39) and (40), indicating a less numerically intense calculation.
A general expression f o r the joint cumulants, under this assumption, may be derived f r o m the displacement-velocity characteristic funcdon M ( ö i , Ö 2 ) , which may be shown to be given by
M{ei, Ö2) = E[e\p {iOix) exp {i92x)] 2N
nc
exp4(1 - iajjO,) (43) The joint cumulants are related to the log characterisdc function by the following equation;'''
1 / 5"+'"
de'ld9% l n M ( ö i , Ö 2 ) =«2=0 (44)
Hence, by substituting eqn (43) into eqn (44), it may be shown that the non-zero j o i n t cumulants are, in general, given by
Kn+2[X"X''] Y ^ « ! n 2 (45)
where n is a positive integer and the expression is observed to be independent of the first-order displace-ment process.
The validity and accuracy of this approximation are investigated in the numerical example section.
S U M M A R Y OF C A L C U L A T I O N P R O C E D U R E The mean up-crossing rate of a general level b is given by eqn (22); this result may be used in conjunction with eqn (15) to estimate the rehability of the system. Equation (22) contains three main elements: the response pdf p{b) and its derivatives, the coefficients
Ai„„ and the integrals /„,. Each of these elements is
discussed below.
(1) The response pdf p{b) and its derivatives may be calculated i n a variety of different ways;^'^''^ however, for the present study, the procedure used in Ref. 3 is adopted. Consequently, eqns (13) and ( A l . 5 ) provide a means of obtaining a good approximation f o r the p d f and its derivatives, where the first- and second-order components are assumed to be statistically independent. However, an improved estimate for p{b) may be
Table 1. Linear force transfer function for a semi-submerged horizontal cylinder in beam seas, radius d 1-25 M 8 1-12 0-89 0-84 0-79 0-76 0-69 K/Pgd^ 0'0495 O'O 0-0543 0-0085 0-057 0-0168 0-0621 0-0432 0-0631 0-0463 0-0645 0-0490 0-0662 0-0498 0-0705 0-0459 calculated using additional terms in the series
expan-sion;^ this will be referred to below as the 'refined p d f . Alternatively, the numerical method of Naess and Johnsen'^ may be used, although the original equations must be modified to calculate the derivatives o f the pdf.
(2) The coefficients A„,„ depend upon the joint cumulants of the displacement and velocity. I f the response displacement and velocity are assumed to be independent, then only the first term in eqn (22) is included and the coefficients A,,,,, are not needed; this is
referred to below as the 'Indep.' approximation. The inclusion of further terms in the series is denoted by 'Indep. -I- correction', the joint cumulants being required. For the pure second-order case, eqns. (39) and (40) are used to determine the first two joint cumulants, and the higher-order joint cumulants are assumed to be negligible. Consequently, the correction term corresponds to setting the limit on the summations in eqn (22) to two. For all other cases, that is, CT, ^ 0-0, the joint cumulants are calculated using the approximate analysis presented above (eqn (45)). The correction term for these cases corresponds to setting the limit on the summations in eqn (22) to four.
(3) I n accordance with the assumption that the first-and second-order response are statistically independent, the integral /„, may be evaluated analytically, as described in Appendix 2. I n the following, it is shown that results of acceptable accuracy may be obtained by assuming that the velocity statistics are Gaussian, in which case ƒ„, reduces to the simple resuhs (A2.8) and
(A2.9), where only the mean squared velocity, CT^., is required.
Numerical results based on eqns (15) and (22) are presented in the following section.
N U M E R I C A L E X A M P L E S
As an example, the above analysis is used to investigate the sway response of a 2-D semi-circular structure subjected to the action of unidirectional random waves. The mooring is modelled as a linear single degree o f freedom system with an equation of motion of the f o r m
X + 2Pu>„x + uj\x = F/M (46)
where (3 = 0-05 is the damping coefficient, w „ = O-IO is the undamped natural frequency and M = 3-21 X 10^ kg/m is the mass plus added mass per unit length. F represents the combined first- and second-order random force per unit length, the first- and second-order transfer functions o f which are given in Tables 1 and 2 as calculated by Kato et al? and Faltinsen and L0ken,'^ respectively.
The radius of the cyhnder is chosen to be 10 m and the transfer function data is linearly interpolated within the given frequency bounds, using 120 frequency compo-nents to allow sufficient resolution over the resonant response peak.^ Further, the force transfer function data are linearly scaled so that different ratios of the standard
Table 2. Quadratic force transfer functions for a semi-submerged horizontal cylinder in beam seas, radius d o„{d/g)
1/2
1-25 1-18 1-12 1-06 0-95 0-89 0-84 0-65
(a) Real transfer function T^^Jpg
1-25 0-308 0-285 0-259 0-250 0-250 0-240 0-233 0-256 1-18 0-285 0-314 0-308 0-292 0-277 0-246 0-234 0-254 1-12 0-239 0-308 0-338 0-340 0-324 0-267 0-234 0-247 1-06 0-250 0-292 0-340 0-368 0-367 0-301 0-245 0-243 0-95 0-250 0-277 0-324 0-367 0-383 0-329 0-257 0-241 0-89 0-240 0-246 0-267 0-301 0-329 0-303 0-227 0-195 0-84 0-233 0-234 0-234 0-245 0-257 0-227 0-147 0-105 0-65 0-256 0-234 0-247 0-243 0-241 0-195 0-105 0-051
(b) Imaginary transfer function T'„„,/pg
1-25 0-000 0-043 0-059 0-061 0-059 0 069 0-112 0-160 1-18 -0-043 0-000 0 030 0-038 0-032 0-028 0-066 0-112 1-12 -0-059 0-030 0-000 0-015 0-013 0-004 0-041 0-087 1-06 -0-061 -0-038 -0-015 0-000 0-000 -0-006 0-033 0-082 0-95 -0-059 -0-032 -0-013 0-000 0-000 -0-004 0-040 0-094 0-89 -0-069 -0-023 -0-004 0-006 0-004 0-000 0-056 0-129 0-84 -0-110 -0-066 -0-041 -0-033 -0-047 -0-056 0-000 0-090 0-65 -0-160 -0-112 -0-087 -0-082 -0-094 -0-129 -0-090 0-000
Extreme vahies of wave-induced vessel motions 175
Table 3. Calculated values of /Q
0"2/o-l k Naess' method (integrated numerically) Gaussian (eqn (A2.8))
First and second independent (eqn (28)) 1- 0 2- 0 5-0 Low frequency 0-2405 0-1550 0-7642 X 10"' 0-2389 0-1541 0-7610 X 10"' 0-2691 X 10"' 0-2389 0-1541 0-7599 X 10"' 0-2755 X 10"'
deviation of the second-order response to the first-order response (ffa/o"!) can be investigated. I n accordance with previous work,-' the ratios investigated are 1-0, 2-0 and 5-0.
The incident sea-state is specified in terms of an International Ship Structures Congress (ISSC) spectrum with significant wave height H, = 2-0m and a mean period = 5-5 s:
exp
691
(47) Before considering the response crossing rate, a number of results concerning the calculation of the coefiicients ƒ,„ are presented. Equations (28), (A2.8) and (A2.9) have been used to calculate values of IQ and h, and the results are shown in Tables 3 and 4, where a comparison has been made with the value obtained w i t h the numerical method of Naess and lohnsen,'^ using a numerical integradon scheme. Equation (28) relates to the assumption that the first- and second-order velocity responses are statistically independent, and eqns (A2.8) and (A2.9) corresponds to an equivalent Gaussian process. I t can be seen that the results compare well, even f o r the pure second-order case, suggesting that the velocity statistics are near Gaussian. This assertion is not surprising, as the main contribution to the integrand arises f r o m the low response level, which, for a lightly damped oscillator, is known to be near Gaussian. Moreover, it may be argued that the low-frequency response is so slowly varying that i t makes httle or no difference to the total velocity contribution f o r the case of the combined first- and second-order response. From a practical point of view, the near-Gaussian nature of the velocity statistics provides a suitable means of calculating the coefRcient ƒ„, without recourse to the complicated eigen approach required to obtain eqn (28),
thus reducing the computational effort required to obtain an estimate of the mean up-crossing rate. Throughout the following, eqns (A2.8) and (A2.9) will be used to calculate ƒ,„.
I n what follows, all figures are plotted on a log scale, the response is normalized to have zero mean and unit variance, and the mean up-crossing rate has been standardized with respect to the mean up-crossing rate of the mean level. I n all graphs a comparison is made with the result which would be obtained for a completely Gaussian response, in which case
exp b - M.V (48)
where and cr^ represent the displacement and velocity rms values, respectively, and y.^ represents the mean response.
Figure 1 shows the mean up-crossing rate f o r the pure second-order response (exact pdf),'^ where a compar-ison has been made with numerical simulations which have been performed using the fast Fourier transform (FFT) technique described in Ref. 9. I t can be seen that the first term of the series (denoted by Tndep.') underestimates the crossing rate, as mentioned by Naess. The addition of the second term of the series (denoted by 'Indep. + correction') provides an improvement on the first term, and gives an adequate representation of the distribution. I t should be noted that, for this case, the exact joint displacement-velocity cumulants, calculated using eqns (39) and (40), are used to calculate further terms in the expression for the mean up-crossing rate. For all other examples the displacment-velocity joint cumulants are calculated approximately using eqn (45).
Figure 2 shows the mean up-crossing rate f o r the ratio o-j/cTi = 1-0, where a refined solution f o r the pdf has been used.^ Although the theoretical result tends to
Table 4. Calculated values of I2 h Naess' method (integrated numerically) Gaussian (eqn (A2.9))
First and second independent (eqn (28)) 1- 0 2- 0 5-0 Low frequency 0- 6616 1- 0269 2- 1019 0- 6662 1- 0329 2- 0913 5-7771 0- 6662 1- 0331 2- 1006 6-2023
l.OE-06 I ' ' ' ' L 1 1 0 1 2 3 4 5 6 7
(Response-mean)/rms
Fig. 1. Mean up-crossing rate, pure second order (exact pdf).
underestimate the true response, i t may be seen that the agreement is reasonable for the assumption that the displacement and velocity are statistically independent. However, a slight improvement on this result may be obtained by using a more refined expression f o r the mean up-crossing rate, suggesting that the displacement and velocity are, to a good approximation, statistically
0 1 2 3 4 5 6 7 (Response-mean)/rms
Fig. 2. Mean up-crossing rate, 0-2/0-1 = 1-0 (first- and second-order response pdf refined).
l.OE+OO
l.OE-Ol t
. l.OE-06 I ' ' ' 1 1 - 1 I
0 1 2 3 4 5 6 7 (Response-mean Vrms
Fig. 3. Mean up-crossing rate, 0 2 / 0 - 1 = 2-0 (first- and second-order response pdf assumed to be statistically independent).
independent. Although in principle, the proposed method should agree with simulation when the refined expressions for the pdf and mean up-crossing rate are used, small differences between the simulation and theoretical curves may be seen in Fig. 2. These differences may be attributed to the following two factors: (1) the series expansion of the p d f and mean up-crossing rate are truncated after a finite number of terms; (2) /„, and the joint displacement-velocity cumulants are calculated approximately.
For (Xi/oi = 2 - 0 it can be seen in Fig. 3 that a
reasonably good estimate f o r the mean up-crossing rate may be achieved by assuming that both the first- and second-order response,' and the displacement and velocity are statistically independent, where using a refined expression for the pdf has been found to make negligible difference. However, a slight improvement may be attained by using a more refined solution for the mean up-crossing rate.
Figure 4 shows the mean up-crossing rate f o r
oi/^i = 5-0, where the first- and second-order response
are assumed to be statistically independent. As shown i n Ref 3, this provides an excellent representation for the displacement pdf, hence there is no need to refine this expression f o r the p d f I t may be noted that the first term of the series, corresponding to the assumption that the response displacement and velocity processes are statistically independent, gives good agreement with numerical simulation. Further, a more refined solution for the mean up-crossing rate may be attained by including additional terms in eqn (22).
Extreme vahies of wave-induced vessel motions 111 l.OE+OO p n CE CL 3 l.OE-01 l.OE-02 1.0E-03 l.OE-04 l.OE-05 •l.OE-06 0 2 3 4 5 (Response-mean Vrms
Fig. 4. Mean up-crossing rate, 0-2/^^1 = 5-0 (first- and
second-order response pdf assumed to be statistically independent).
I t may be concluded that a reasonably good estimate of the mean up-crossing rate may be attained using the assumption that the displacement and velocity are statistically, independent, provided that a good estimate of the combined response pdf is used. This may be explained by noting that the displacement and velocity processes are statistically independent whenever the response is Gaussian. For a hghtly damped oscillator i t is known that the response statistics approach a Gaussian distribution, even though the tails are non-Gaussian. I t may therefore be reasoned that i f the damping is low then a good estimate f o r the mean up-crossing rate may be attained by assuming that the displacement and velocity are statistically independent. Furthermore, for the example considered here i t may be argued that this assumption should provide the best estimate when ai/crx = FO, as the distribution should be more Gaussian. The above reasoning may be verified by considering the joint displacement-velocity cumulants, as given in Table 5, where the smaller values indicate that the displacement-velocity statistics are more
Gaussian, and therefore more likely to be statistically independent.
Because of the excessive computer time required to obtain the joint displacement-velocity cumulants using eqns (39) and (40), it was necessary to determine the joint cumulants using the approximate method described above, where the second-order velocity contribution is neglected. Table 5 compares the values obtained for the first two non-zero joint cumulants
Kx+2[xx^] and K2+2[x^x\ where eqns (39) and (40) have
been used to calculate the exact joint cumulants and eqn (45) has been used to calculate the approximate joint cumulants. I t may be seen that the approximate joint cumulants provide a better estimate whenever the response contains a significant first-order contribution (i.e. 0 - 2 / 0 " ! = 1-0 or 2-0). This is not surprising, as the
approximating assumptions are more hkely to be satisfied under these conditions.
The response level b which has a probabihty of exceedance a in a time t can be evaluated f r o m eqn (15). Results are presented here for the case a 0-01 and / = 3 h, which yields the condition f o r • Given the above results for f ^ , the response level b which satisfies this condition may readily be calculated. Results for the present example are shown in Table 6. The first column contains the results for the first-order response alone. The second column contains results f o r the combined first- and second-order response derived under the assumption that the total response is Gaussian. The last entry in this column shows the result which would be obtained for the pure second-order response, again under the Gaussian assumption. The third column contains results, as calculated using the present (non-Gaussian) techniques. The final entry in this column shows the extreme second-order response which is predicted using this method. The final column relates to an empirical formula discussed below. The mean second-order response has been omitted in all cases, and the figures in parentheses relate to the actual response for the case = 1-Om.
I t may be seen that the combined extreme is always much greater than the first-order extreme, and the Gaussian first- and second-order extreme. This is not surprising, as the response contains a significant second-order contribution which dominates the tails of the distribution. Additionally, the combined extreme is always less than the sum o f the first- and
second-Table 5. Calculated joint displacement-velocity cumulants
0-2/0-1 K,+2[XX'']I{CJA) \xH^]l(olal)
0-2/0-1
Actual Approx. Actual Approx.
1-0 0-4255 X 10"' 0-4422 X 10"' 0-1578 X 10"' 0-1569 X 10"' 2-0 0-5388 X 10"' 0-5364 X 10"' 0-3224 X 10"' 0-2401 X 10"' 5-0 0-6014 X 10"' 0-4629 X 10"' 0-8394 X 10"' 0-2269 X 10"'
Low frequency 0-6570 X 10"' N/A' 0-2626 N / A
Table 6. Extreme vahies based on the probability of failure, a = O OL r = 3 h
0-2/0-1 First-order First- and second-order Present Empirical
Gaussian Gaussian method formulae
(eqn (49)) 1-0 4-9oi 4-80-1+2 5-801+2 6-O0-1+2 (4-9 m) (6-8 m) (8-2 m) (8-5 m ) ' 2-0 4-90-1 4-701+2 6-2o-,+2 6-40-1+2 (4-9 m) (10-5m) (13-9 m) (14-3 m) 5-0 4-9cri 4-601+2 6-10-1+2 6-601+2 (4-9 m) (23-5 m) (31-1 m) (33-4 m)
Low frequency N/A 4-4ff2 6-O0-2 N/A
order extreme. This is to be expected, as the first-and second-order extreme values will not occur simultaneously.
The extreme value which has a specified return period is considered next, where on average the extreme value is exceeded once during the specified return period. This value, which is close to the 'mean value of the maximum', corresponds to a mean up-crossing rate o f IjT, where T is the duration of the storm. Although it may be convenient to assess the extreme values in terms o f the return period, it should be noted that the probability that this value w i l l be exceeded within the return period is 0-632. Hence, the concept of the return period is not ideally suited to the prediction of extremes. However,- this method is used extensively throughout the offshore industry to gain an insight into the extreme response at the prehminary design stage.
For the present example, a return period of 3 h is studied, corresponding to a mean up-crossing rate of 9-3 X 10"Vs. The results are given i n Table 7, where the present theory is compared with an equivalent Gaussian process and pure first-order response. As above, the quoted values correspond to a zero mean response and the values in parentheses correspond to o"! = 1-Om.
In accordance with the extreme values presented above, the combined first- and second-order extreme values are non-Gaussian in nature. However, i t is worth nodng that the values are substantially more Gaussian than those obtained earlier. This is to be expected, as the extreme values based on the return period are much lower than f o r a 1% probability of failure, and therefore lie in a region where the
probability density function shows closer agreement with a Gaussian distribution.^
Naess^ has suggested that the maximum value of the combined first- and second-order response may be calculated using the following empirical formulae:
^ 1 + 2 = M.V +
q^{X\
+ xl)
(49)where X^, X2 and X 1 + 2 represent the maximum first-order, order and combined first- and second-order value, respectively, and ^ is a 'correction factor' which accounts f o r the effect o f the coupling between first- and second-order response. Although this empiri-cal formula was originally suggested for the case of almost equal first- and second-order response com-ponents, the foffowing investigation examines the accuracy of eqn (49) for the examples considered above. I n accordance with Naess,^ q takes values i n the range 1-0-1-2; a value of 1-1 is used below.
W i t h the extreme values obtained using the present analysis, the empirical response maxima are calculated using eqn (49). The results are given in Tables 6 and 7, where it may be seen that the empirical formula gives good agreement with the proposed aniaysis, the resuhs being shghtly conservative. However, i t should be emphasized that a reliable estimate o f the second-order maxima is required before eqn (49) may be used. A n existing design formula for the 'mean value o f the maximum' second-order response is CT2ln(-^)j where N is the number o f second-order cycles.^ For the examples considered here this formula yields a value o f 5-15cr2, where it may be noted that the formula significantly overestimates the second-order value. Although this may be accurate enough at the prehminary design Table 7. Extreme values based on a return period of 3 h
02/01 First-order First- and second-order Present Empirical
Gaussian Gaussian method formulae
(eqn (49))
1-0 3-801 3-7oi+2 4-10-1+2 4-10-1+2
(3-8 m) (5-2 m) (5-8 m) (5-8 m)
2-0 3-801 3-601+2 4-I01+2 4-I01+2
(3-8 m) (8-0 m) (9-2 m) (9-2 m)
5-0 3-80-1 3-4oi+2 4-0O1+2 4-I01+2
(3-8 m) (17-3m) (20-4 m) (20-8 m)
Extreme values of wave-induced vessel motions 179 stage, a more rigorous approach based on the present
analysis is more appropriate at the detailed design stage.
C O N C L U S I O N S
It has been shown that the mean up-crossing rate of the combined first- and second-order response may be expanded as a series, in which the first term corre-sponds to the assumpdon that the displacement and velocity are stadsdcally independent and subsequent terms depend upon the joint displacement-velocity cumulants. The examples considered here indicate that the first term of the series provides a good estimate of the mean up-crossing rate provided that a good estimate of the combined response pdf is used.
Although the independence assumpdon is considered to be accurate enough f o r most purposes, a refined estimate of the mean up-crossing rate may be calculated using additional terms in the series. However, an excessive amount o f computation may be required to calculate the joint displacement-velocity cumulants, especially f o r the case of spreading seas, where the size o f t h e Kac-Siegert matrix is greatly increased.^ To reduce the computation needed, an approximate method of calculating the joint cumulants has been presented, i n which the second-order velocity contribution is neglected.
The extreme response is determined directly f r o m the mean up-crossing rate using the Poisson assumption o f independent up-crossings. The accuracy of an existing design formula is validated against calculated theoretical values using the present analysis, where the design formula calculates the combined first- and second-order extreme value f r o m the individual first-order and second-order extreme values. However, the dependence of the formula on the second-order extreme value restricts its applicability. I n addition, an empirical formula f o r the expected second-order extreme is found, for the example considered here, to overestimate significantly the extreme value based on the return period.
Finally, in accordance with R e f 3, the present method may be used to predict the extreme response o f a hnearly moored vessel subjected to wind gusts, and combined wind and wave forces.
REFERENCES
1. Pinkster, J.A. Low frequency second order wave exciting forces on floating structures. Publ. 650, Netherlands Ship Model Basin, Wageningen,
2. Kato, S., Kinoshita, T. & Takase, S. Statistical theory of total second order responses of moored vessels in random seas. Appl. Ocean Res., 12(1) (1990) 2-13.
3. Langley, R.S. & McWilliam, S. A statistical analysis of first and second order vessel motions induced by waves and wind gusts. Appl. Ocean Res., 15(1) (1993) 13-23. 4. Naess, A. Statistical analysis of second-order response of
marine structures. J. Ship Res. 29(4) (1985) 270-84. 5. Naess, A. Prediction of extremes of combined first order
and slow-drift motions of offshore structures. Appl. Ocean
Res. 11(2) (1989) 100-10.
6. Langley, R.S. A statistical aniaysis of low frequency second-order forces and motions. Appl. Ocean Res. 9(3) (1987) 163-70.
7. Longuet-Higgins, M.S. The effect of non-linearities on statistical distributions in the theory of sea waves. / . Fluid
Mech. 17(3) (1963) 459-80.
8. Rice, S.0. Mathematical analysis of random noise. In: Selected Papers on Noise and Stochastic Processes, ed. N . Wax, Dover, New York, 1954, pp. 133-295.
9. Langley, R.S. On the time domain simulation of second order wave forces and induced responses. Appl. Ocean Res.
8(3) (1986) 134-43.
10. Wilkinson, J.H. The Algebraic Eigenvalue Problem. Oxford University Press, Oxford, 1965.
11. Vanmarcke, E.H. On the distribution of the first passage time for normal stationary random processes. / . Appl.
Mech. 42 (1975) 215-20.
12. Naess, A. Effects of correriation on extreme slow-drift response. Offshore Mechanics and Arctic Engineering
Conference, {OMAE), The Hague, 1989, pp. 465-74.
13. Hu, S-L.J. Probabilistic independence of joint cumulants.
/ . Eng. Mech. 117(3) (1990) 640-53.
14. Lin, Y.K. Probabihstic Theory of Structural Dynamics. McGraw-Hill, New York, 1967, p. 37.
15. Naess, A. & Johnsen, J.M. An efficient numerical method for calculating the statistical distribution of combined first-order and wave-drift response. Conference on Offshore
Mechaincs and Arctic Engitwering {OMAE), Stavangar,
Norway, 1991, Vol. 11, pp. 59-70.
16. Faltinsen, O.D. & L0ken, A.E. Slow drift oscillations of a ship in irregular waves. Appl. Ocean Res. 1(1) (1979) 21-31.
17. Naess, A. The statistical distribution of second-order slowly varying forces and motions. Appl. Ocean Res. 8(2) (1986) 110-18.
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A C K N O W L E D G E M E N T S
This work forms part o f the research programme 'Behaviour o f Fixed and Compliant Offshore Struc-tures' sponsored by SERC through the Marine Technology Directorate and jointly funded with: Admiralty Research Establishment, Aber Engineering, A M O C O Production Company, Brown and Root, BP Exploration, HSE, E l f U K and Statoil.
A P P E N D I X 1: D E T E R M I N A T I O N OF p^"\.x)
The purpose o f this Appendix is to evaluate an expression f o r the nth derivative o f p{x), where p{x) is the p d f evaluated using the assumption that the combined first- and second-order response are statisti-cally independent. Hence the «th derivative of p{x),
P^"\x), may be written as
( A l . 1 ) where p{xx) and p{x - x , ) ' ^ are given by
1 P{xx) - e x p ( - ^ ( A l . 2 ) M p{x-xi) = Y,Y^^P j=i N ( X - X i ) A, Xj > 0 and x > Xi J=M+l ( X - X , ) Ay XJ < 0 and x < x, ( A l . 3 ) and the continuity o[p^"\x) is assumed.
Substituting ( A l . 2 ) and ( A l . 3 ) into eqn ( A l . 1 ) gives lij ( - 1 ) " / X
e x p f
-^ t f A;.'+' 7(27r)a, X exp — exp - d x j VAyV \^ 2C7\) + T ^ J ^ e x p T - ^ |A,.|A;' v'(27r)CT,X exp I — exp - dxi ( A l . 4 )
where the integral has been considered in two parts to account for the nature of the second-order term. Evaluating the integral gives
M I i \ « p W ( x ) = 5 : / X ax ; = I \ ^1 2A,V X O
-Ay
(Tl ^ ( - l ) ' V y X o\ • A / 2 A ? (Tl X ) 0-1. ( A l . 5 ) where $ is the cumulative normal distribution function.A P P E N D I X 2: D E T E R M I N A T I O N OF /„,
The purpose of this section is to evaluate the integral I„,
analytically:
'OO
Jo (A2.1)
where p{x) is the velocity p d f
Assuming that the first- and second-order response are statistically independent, /„, may be written as
>oo
x;7("')(x) dx . 0
(A2.2)
where p^"'\x) is written i n a f o r m analogous to eqn ( A l . 5 ) such that
X (Ty
X O
2 ^ \t\pn exp ^ + 9 . 2 -^1
(A2.3) where and are defined in the main text.
Before /„, can be derived, it is convenient to note the integrals xexp I - - K 1 - $
+ O e x p ( - i ^
XUy
4^
dx = (A2.4) and xexp -fy-exp 0 <r: dx =(AA)
W(27r)
^ 2 j (A2.5)Using eqns (A2.2), (A2.4) and (A2.5), eqn (A2.3) may be evaluated such that
M
.(zi
)'"xi
Z-j tm+lX U j e x p
V{2n) ^ 2 , (-i )"'x.-\C.\tm J=M+l I V i S / X i6
expf
1$
^{2n) 2 (A2.6)Extreme values of wave-induced vessel motions 181 Assuming that the velocity statistics are Gaussian with
zero mean, p{x) may be written as
p{x) 1
V'(27r)a, (A2.7)
where al is the mean square velocity response.
I t is a relatively easy task to evaluate /„, f o r small values of m. For example, it may be shown that
r - -7 - ^ (A2.8)
However, the analysis becomes tedious f o r large values of w. For such cases, it is convenient to write /„, in terms of the Hermite polynomial.'^ Using such a
represen-tadon it is possible to rewrite I„, as
\m~2 1
dl
V(27r) < . ' - ' where'^ -1)"(2«)! / / „ , _ 2 ( 0 ) , m = 2,3, H2M • ^ ^ 2 „ + l ( 0 ) = 0 2"n\and « is a positive integer. More specifically
I2 '
