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Three-dimensional Fourier analysis of drag force for compliant offshore structures
Three-dimensional Fourier analysis of drag force for compliant offshore structures JAMES HAMILTON Vickers Offshore (P&D) Ltd, Craven House, Barrow-in-Furness, Cumbria L A 1 4 1AF, U K
In this paper a representation is obtained of the first Fourier component of the drag F given by: F ~ Q { Q - Q } 1/2 The relative velocity Q is assumed to consist of a mean current plus a single Fourier component, and may come from all directions during the course of a cycle. The expression is shown to agree well with the usual two-dimensional expressions, (when the relative velocity is always in one direction) and also with the exact expression in the three-dimensional case. A good representation of the drag in this manner is important if the quasi-linear spectral approach is to be used in the evaluation of the response of a three-dimensional compliant structure to ocean waves.
INTRODUCTION The importance of drag on a typical offshore structure compared with the other forces, for example, arising from pressure gradients and hydrodynamic inertia, increases with the wave slope. Thus the correct modelling of drag is particularly important for calculating the responses to the structures design wave that is the worst wave which the structure is likely to encounter during its lifetime.. Model tests and time domain integration may provide the only true confidence that a given design may survive its design wave, however, their expense is likely to restrict their use considerably. In the design process the requirement to test different platform designs, and to search for the worst loadings (which may not occur at the design wave) probably mean that the number of program runs in which drag plays an important (if not the dominant) part is more likely to be limited by time and money than by anything else. There is thus ample justification for obtaining at least the capability of a more accurate representation of drag within the framework of the linear, spectral approach. The spectral approach in general has important uses in the statistical analysis of the responses 1. For these purposes, strictly speaking we must be able to write a linear governing equation for the platform orientations X say, where typically
x = { x ~ , r~, z., ~, o, q.}~
(I)
are independent of X or X. Given the assumed form (2) for the governing equation, the response to a superposition of wave spectral components may be obtained simply by superimposing the responses to the individual components. In reality the drag force on a cylindrical element of the platform actually takes the form: F ~ = ½PCo D idli( u i - vi)lui - vll
(3)
where u i is the particle velocity normal to the element (diameter D i length d/i) and vi is the velocity of the element normal to its axis due to the platform motion, (ui - vi) is therefore linearly dependent on X (for small angular displacements). The first consequence of equation (3) is that if Ui and vi possess only one Fourier component, F~,. however, would possess many. Thus even if F ° were independent of X, (2) could not be solved simply by fourier methods. The second consequence of course is the dependence of F ° on X in a highly non-linear manner. The author is of the opinion that except in the limiting case of small response velocities compared with the total water particle velocity when we may write: (U - - V)I(U - - V)[ = (U - - VXU" U - - 2U" V) I/2 "]- O( V2)
=(u-v-u~)lu]
+O(V2)
(4)
%f X~, Y~, Z c are the locations of the centre of gravity of the platform and tp, 0, ff the angular rotations. For example the linearised Morison equation 2'3 for X takes the form: M X + C X + K X = F(t)
(2)
.where F(t) is the force-moments vector and M, C, K and F 0141-1187/80/040147~)7 $2.00 © 1980 CML Publications
a sensible linearization does not exist for many degree-offreedom systems. The approach of Borgman 4 in which the best linear approximation to QIQI is found given the distribution of Q, is not applicable when Q depends on more than one independent variable. It would be necessary to specify the phase relationship between for example -~o and (o a priori or assume that they were uncoupled, both extremely doubtful hypothesis.
Applied Ocean Research, 1980, Vol. 2, No. 4
147
Fourier analysis of drag force for offshore structures: James Hamilton jl
~r2 1/ Comooc~nt in, phase w i t h q
2 lO00~'~'d
2000 i
3000 7 6
~x,c 1
5
T1 • torln(~3
2
Component in phase with -W
same magnitude as Q~ whilst the second harmonics on the other hand are ignored, in order that 'tow out' may be studied. Note that in order that equation (6) be useful, it is not necessary that the drag itself have small second and higher harmonics merely that the response which may be dominated by the linear forces have small higher harmonics. The usual formulations of the drag for a multidegree of freedom system do not include the effect of a mean current; however, Borgman 4 (or Wu s) quote the best linear fit for aone degree of freedom system.
(vertical partical vek:x;ity)
QIQI ~ (a 2 - mZ)2@(m/a) - 2maq~(m/p) + - 2OO0
Figure 1. Convergence of iterative method of response evaluatiOn for a TBP in its 50 year worst wave. Damping matrix C = O. *, No drag; wave height, + 17 m; wave period, 20 sec. CD=0.8; O, aft tethers; e,forward tethers. Tether tension variation, T = ~ (T 1 + iT2)exp( - iogt) This non-linear dependence of the drag force on a compliant structure such as a TBP does not fortunately preclude us from using the spectral approach to study its responses. It is merely necessary to be able to estimate the first Fourier component of the drag F? given the wave particle velocity and the platform response. By first assuming a particular response X 0 (say) it is possible to calculate successive estimates for the responses X 1, X 2 , X 3 etc. which converge to give the actual non-linear response. In the Appendix we give further details of this approach, which is necessary if the effect of drag on complaint as opposed to fixed structures is to be calculated. In Fig. 1 may be found an example of the convergence which may be obtained in the case of a TBP moored in 200 m of water when subject to its design wave. It will be seen that the inclusion of drag forces on this particular TBP has a considerable effect on the tether tensions. This in spite of the fact that the drag forces are still smaller in total size than the inertial forces and that the platform is not operating near its surge resonance (which is about 60 sec). It is not appropriate to enter into an extended discussion of TBP dynamics in this paper, we merely wish to point out the effect of drag can be both unexpectedly large in crucial areas and highly non-linear. (Note that the first correction points '1' are a considerable distance from the final points '7'.) The building block which enables us to calculate these responses for a multi-degree of freedom system, however, is absent. There does not seem to be an adequate formulation of the drag force in this case and it is with this problem that this paper is primarily concerned.
We consider the problem of calculating the harmonics of (5)
(7)
where q~(x)= ~
1
exp ( - x2/2)
42.
a
f o
and Q = Q,fi (say) is normally distributed, mean m and standard deviation a. Evidently extension of this formulation to a many degree of system where both m and a are vectors is not straightforward. Wu 5 appears to use an empirical extension of equation (7), which in the case of m = 0 (zero mean current) reduces to a form equivalent to: QIQI=Q~IQ~Ii
+Q,IQ,~
+Q~IQ~Ik
(8)
This formulation of the many degree of freedom drag forces is not frame invarient as is equation (5) and is therefore not physically reasonable. The other formulation commonly used (for example in N M I wave) consists of obtaining the Fourier component of for example sin cotlsin ~otl by carrying out the Fourier transform directly4. Thus
al=~f
vlvlexp( - io~t)dt
(9)
0
and drag = ½PCoDl ~,~a 1 exp (iogt)
(10)
where v is the relative velocity component. To extend to three dimensions, however, with the inclusion of current effects, it would first be necessary to obtain: Q- Q = A + ~ { B exp (iogt)+ C exp (2i~ot)}
FOURIER REPRESENTATION OF THE DRAG
QIQ[ = Q[Q" Q]l/z
{4m~(m/a) - 4atp(m/a)}Q
(11)
where A,B and C are obtained by subsituting equation (6) into the left hand side of equation (11) and equating terms (all exponentials being converted to positive frequencies). Thus
where the relative velocity Q given by: A = Q o ' Q o +½QI" Q~ Q = Qo + , ~ Q 1 exp (io~t)
(6) B=2Qo.Q 1
is composed of a mean Qo and a fluctuating component Q1 (here,~denotes the real part). We allow Q0 to be of the
148
Applied Ocean Research, 1980, Vol. 2, No. 4
C =~Q1 "Q1
(12)
Fourier analysis of drag force for offshore structures: James Hamilton where '*' denotes the complex conjugate. The carrying out of the step equivalent to equation (9)
~2 = ~ '
[-Ql:_Or
(23)
"Q~'+ L 4x/2
2x/o
°f
al =~-
Q{Q. Q}l/2exp(- kot)dt
(13)
Since q is the positive root we must take:
O
would seem however to involve an intractable integral (hence probably the use of 8). It is this problem which leads us to suggest a different method. The nub of the problem would seem to be in finding the square root q (say) of {Q" Q}. To find an approximation to this we return to the original definition and seek e,/3, 7; (/3, 7 complex), such that q = ~ +,~{/3e '°'' + 7e 2''~'}
(14)
and q2 has Fourier components A, B, C at the zeroth first and second frequencies. Thus substituting equation (14) into the lefthand side of equation (11) and equating the coefficients of the components we obtain three simultaneous equations: (15)
2~/3+ 2/3*7 = B
(16)
27~ + 1//2 = C
(17)
Note that this process is not quite the same as finding the Fourier components of {Q.Q} 1/2 since q2 contains terms in e 31°t and e 41°t (18)
where D=/37 and E=½y 2. However, these terms (in D and E) if they were to arise in {Q .Q} would result from the presence of a second harmonic component in Q. It is therefore consistent with our single component approach to neglect these terms. Similarly we could compensate for their presence by including terms in q of 6 and e q = 0c+ ~ [/3ei°'' + 7e 21°t + 6e 31°t 8e*i°t]
(19)
These terms have no effect (except indirectly through their influence on ~,/3 and 7) on the e i'°' term in the drag. Below we compare with the single degree of freedom case. The solution of equations (15), (16) and (17) proceeds as follows. Treating/3 as a known variable, from equations (15) and (17) ~2 = ~(A - ½/3/3*)+ {~A - ½/3/3,)2 _
~c-½/32XC*-½/3"2)}1/2 7 = ( c - ½/32)/(2~)
(20) (21)
Note from equation (16) we also have:
/3= (~/3- 7/3")/2(g2 - 77")
(24)
Hence 7
+ 1
-1/2
(25)
-2{Q1. Q~},/2
Intuitively since {Q.Q}I/2 is positive definite we would expect to take the ' + ' sign in equations (24) and (25) and hence in equation (20). We may confirm this in the case of a mean current only. Thus if Q1 = 0 and hence B, C,/3 and 7 are also zero then (if~2
1"~2 ") 1/2
(26)
q=a=i~9-+~I
~2 +½/3/3, +½77* =A
q2 ___A + , ~ [ B e i'°' + Ce 2u°' + De 3i°t + Ee 4i°']
1 )1/2 ~=½{QI'Q*} w2" 1 _ ~ j ~
(22)
Clearly the' + ' sign must be taken in equation (20) to yield the correct root. Except in the case/3 = 0 when a solution of equations (15), (16) and (17) may be obtained as shown above, it does not appear possible to obtain an elementary solution of these equations. As an example in the special case when Q1 may be written in the form: Q1 = 0 1 exp (i0) where 01 and 0 are real, we may remove the complex parts of fl and 7 simply by changing the time origin in equation (6) by an amount - 0. In this simple case we may eliminate ot and 7 to obtain a quartic in/32
[,2
{4A/32 - 2fl* + 3 ~ - - 2C/32 +/3"
]}2 -
B 2 VB 2
~-[_T-2C/32 +/3"]=o
Even this equation, however, has no simple roots, thus in general we must adopt a numerical approach. The procedure proposed (which has been extensively tested) is to proceed iteratively first assuming/3 = 0, then equation (20) (with t h e ' + ' sign) and (21) may be used to find estimates for ct and y. A corrected value for/3 can then be obtained using (22) and the procedure repeated until the iteration converges. The convergence is immediate for a zero current and about 10 iterations are sufficient in general to obtain accuracy to i%. Note that where an exact representation of {Q. Q}1/2 is possible with only 3 Fourier terms, this procedure appears to yield it, thus if Q0 is greater than and parallel to Q1, Q never changes sign and ct= {Qo'Q0} 1/2, fl={Q1 "Q1} 1/2, 7 = 0
(27)
We consider first the use of zero current when Qo = B = / 3 = 0 and hence from equation (18)
can be confirmed as a solution of equations (15), (16) and (17). In Fig. 2 may be found comparisons of the Fourier series representation of pQ] (the first three terms) and the
Applied Ocean Research, 1980, Vol. 2, No. 4
149
Fourier analysis of drag Jorce Jor ofjshore structures: James Hamilton 1 ~.
Q=coswt
These expressions are on a less firm foundation than equation (31) as additional terms 6 and e can contribute directly to their values. However, in Fig. 3 we plot the first two terms.
.~-
(36)
Q{Q.Q}I,'2 = D o + D 1cos ogt
"hi2
0
'
for the Fourier series and the present theory. It can be seen that both are reasonably close together. In Fig. 4 is plotted the full, three term expression (30) and the corresponding Fourier series. Again the present theory seem to reproduce the peak values with greater accuracy than the simple Fourier transform. To illustrate the capability of the present procedure when Q0 and Q1 are not parallel we plot in Fig. 5 a polar plot of the drag and equation (30) for such a case. Also plotted is the expression (the formulae to which Sony and Wu approximate)
rr tat
O=1/3÷2/3cos~t
n/2
wt
Figure 2. Comparison ofIQl( ( - - - - - ) and present theory (
QxtQxl
..... ) with Fourier series )
+ QylQyl
[37)
It can be seen that this expression, at least in its 'raw' form does not provide good agreement with the frame invariant form.
present theory (equation (20)), for a single component relative velocity. Q = c o s ~oti
(28) 1~
and Q = (~ + 2 cos ~ot)i
(29)
It can be seen that the agreement in both cases is fairly close, with little qualitatively to choose between equation (14) and the Fourier series. It will be noted, however, that equation (14) seems to reproduce the peak values with more accuracy. Granted the adequacy of equation (14) we may obtain the first harmonic of Q { Q ' Q}1/2, Dx (say), where QIQI = Do + ~
Q=1/3+2/3~t
Dlei'~' + , ~ D2 e2i'~'
6
.12
x ~ ~
~t
(30) -1
Figure 3. Comparison of Q I Q I ( - ) with first two terms of Fourier series ( - - - ) and present theory ( )
from
D,
=
(31)
flQ0 + ~Q, +½7Q~
If Q is represented by equation (28) we may compare our result.
ID,I =½(1 + 1/x/2)l/z +~(1 ) 1+ l/xf2
~
1/2
= 0.8446
~
Q11/3+2/3cos~at
(32)
with the classical result from Fourier series: ID{I = 8/3rt = 0.8488
(33)
a less than ½% error. In addition to D 1 we may also obtain estimates for D O and Dz again assuming the adequacy of equation (14):
150
Do = ~Qo +½fl*Q1
(34)
D2 = 7Qo + ½flQ~
(35)
Applied Ocean Research, 1980, Vol. 2, No. 4
Figure 4. Comparison of QIQI (--) with first three terms of Fourier series ( --) and present theory ( )
Fourier analysis of dra9 jbrce for offshore structures: James Hamilton 3n
O
-~
4ff
ii/ .0"5
~11~ J~
\'N
-
•\1', 0.5
0.5
'
'
f" / ~ tf-- -
"
x
an objection, the argument being that the method is justified by the results embodied in Figs. 2-5 together with the agreement in the special cases discussed above. Also these results would be used to calculate single harmonic responses of semi-submersibles for example when evidently second harmonics would also be present in the motion. Most small parameter expansions would be inapplicable in this case, thus it was considered more important to obtain the best overall accuracy irrespective of the size of a small parameter.
/"ln• CONCLUSIONS
2
Figure 5. Polar plot of dra9 force (Dxi + Dyj) when the mean and oscillatory components of the relative velocity (Q) are not parallel. , QIQI; - - - - ~ QxlQxli + QylQy[J; , present theory, Q = (1+ 4acostot)i _ sino~tj
HIGHER HARMONICS It will have been noticed that the above procedure in addition to providing estimates for the mean force Do and the prime harmonic D~ also provides (equation (35)) an estimate for the second harmonic of the Drag force. In what sense is this term valid? We may argue that if the response X is actually very closely approximated by: X = X o + ~ X l e i~°' (say) then the expression (36) for the drag forces will be fairly accurate, We may therefore use D 2 to verify that the response has actually a small second harmonic component. Taudin 6 appears to show second harmonics and may do something similar; however, he gives no indication of how the drag forces are evaluated. DISCUSSION Some justification is perhaps required for suggesting a method of evaluating the first Fourier component of the drag which is not apparently the first term in a small parameter expansion. There is in fact a small parameter for this analysis, and it will measure the extent to which q = x / Q ~ Q can be represented by a time series including up to only second harmonics (e and 6 small in equation (25)). Special cases where this is exactly true are: (i) nearly uniform magnitude Q. For example, a fixed pipe beneath a sinusoidal wave. This would not be correctly represented by the usual theory. (ii) the relative velocity Q is of the form: Q = (a + bei°")r where ]a] > Ib] and r is a constant vector. Note that if Q possesses more than the first two harmonics A, B and C in equations (15), (16) and (17) and the expressions for Do, D1 and D 2 may have to be appropriately modified. However, it must be admitted that the absence of a small parameter in the usual sense was not initially seen as
We have shown that the approximate method suggested here for evaluati.ng the first Fourier component of the drag force has the following properties: (i) in the case of a one-dimensional relative velocity and in the absence of a mean current the method is within ½% of the exact (8/3rc) expression; (ii) the formula is also exact if the mean current is greater than and parallel to the oscillatory component of the relative velocity ;(iii) good agreement with the exact expression for the drag (not the Fourier components) when the phases of the relative velocity components are not equal. This latter case can occur if the structure oscillates wholly or partly at right angles to the direction from which the waves are incident. The method is not susceptible to linearization; however, in the absence of a comparable formulation it is recommended that the method be seriously considered in the analysis of the three-dimensional dynamic response of structures in waves. A suitable title for the method might be 'the approximate square root method'.
REFERENCES 1 Ochi, M. K. and Wang, S. Prediction of extreme wave induced loads on ocean structures, BOSS 76, Conf. Trondheim, 1976 2 Hallom, M. G., Heaf, N. J. and Wootton, L. R. Dynamics of marine structures, Report UR8 CIRA Underwater Engineering Group, London, 1977 3 Paulling, J, R. Wave induced forces and motions of tubular structures, Proc. 8th ONR Symposium on Naval Hydrodynamics, Pasadena 1970, p. 1052 4 Borgman, L. E. Ocean wave simulation for engineering design, J. Waterways Harbours Div. ASCE 1969, 95 (WW4), 129 5 Wu, S. C. The effects of current on dynamic response of offshore platforms, Proc. Offshore Technol. Conf. Houston, 1976, OTC 2540 6 Taudin, P. Dynamic response of flexible offshore structures to regular waves, Proc. Offshore Technol. Conf., Houston, 1978, OTC 3160 7 Malhotra, A. K. and Penzien, J. Nondeterministic analysis of offshore structures, J. Eng. Mech. Div. ASCE 1970, 96, 985 8 Burke, B. G. The analysis of motions of semi-submersible drilling vessels in waves, Proc. Offshore Technol. Conf. Houston, 1969, OTC 1024
APPENDIX Iterative method In the case when u and v are the same order of magnitude an iterative upproach to the solution of (2) would seem unavoidable. In the iterative approach 5,7,a, the drag is not associated with a stiffness matrix C but is evaluated given an assumed value of v. This is added into the r.h.s, of equation (2) and a new estimate is calculated.
Applied Ocean Research, 1980, Vol. 2, No. 4 151
Fourier analysis of drag force for offshore structures: James Hamilton
Xo=0 [ - t o 2 M + K ] X j = F ( t , Xj_O,
and convergence can be forced with the correct choice of C and X o. j=1,2,3
(A1) Evaluation of a dissipation matrix
where to is the frequency of the oscillation in radians per second. This procedure continues until X j and X j_ 1 are sufficiently close in value. Note that it is possible to speed convergence by introducing a damping matrix to either side of equation (A1). (See, however, Taudin 6 for a more extensive discussion of iterative methods.)
The dissipation matrix can, of course, be evaluated using conventional methods in which typically the drag force F is assumed to behave as: F(t, X~) ~. 8 tQ11[(ul - X~)i +
[ - coZM + itoC + K ] X j = F(t, X j_ 1) + i~CXj_ 1
This technique is applicable to all methods of evaluating the response (assumed to be of the form exp (itot)) to a single component wave field.
(A6)
(Uz- ~'~)i-I
(A2)
for a two degree of freedom system (that is i~noring rotational velocities). The dependence of IQII on Xo, Yois ignored, thus for a small change in the response dX~, d Y~ the perturbed drag is given by:
r/=~q0 exp (itoot + kox + loy + ¢Po) An interesting feature of the iterative method when carried out using equation (5) is that within the framework of the other linearizing assumptions (normally small wave height and angular velocities) it is possible to deduce information about the stability of the resulting solution. Thus if we write 6 X j = X j - X ~ , then close to X~
dF ~ 8 I Q , l ( - d X j - d
Ycj}
(A7)
It would seem preferable from the point of view of speeding convergence to evaluate the damping matrix (which just attempts to model the dependence o f F on )¢c) using the expression for D 1 developed above. Thus replacing QI by Q1 +dQ1 in equation (18) we obtain:
[ _ t o 2 M + K ] f X j _c~F(t, X ~ ) ~ . d a --~Q~dQ1 1 *
and hence
+~Q1-dQ*
dB = 2Qo' dQ1 [ -- t o 2 M +
K](bXj +1 - 3X2)
dF(t, X~) OX
(3Xj- 6Xj_ 1)
d C = Q1 "dQ1
(A3)
In general with 6 degrees of freedom, 21 independent results such as equation (A1) are necessary before t3F/aX is specified unambiguously. However, in situations where only one mode of vibration is predominant we will have: (6Xj +a - fiX) = R ( f X j - 3 X j_ 1)
where R is a complex constant and the iterates X j will describe a spiral towards the final solution X~ (see Fig. 1). By observing the direction of rotation of the spiral, the sign of R may be determined and hence the stability or otherwise of the mode stimulated inferred.
c~F(t,X~) 3X
(A8)
The perturbation values d~, dfl, d7 may then be obtained by substitution into equations (21), (22) and (23):
[
2~ 2fl
½fl* 2ct
½fl 27
½7* ½fl*
2fl*
27*
2ct
0
27
fl
0
2~
27*
0
fl*
0
L
2a.l
dT*
dC*
(A9) Giving finally the first harmonic of the drag: dD~ =~dQ~ +½7dQ* +
= ( - t o 2 M + K)R
= - icoC* (say)
Q0dfl+ Qld~ +1Q'd7 (A4)
where C* is the damping matrix near to X = X'. Note that for convergence of the iteration process we require ]RI < 1 and this imposes a limit on how close to a resonance (det(-to2M+K)=0) the iteration will converge. Close to a resonance equation (A2) must be used giving:
It is not necessary to solve equation (A9) explicitly since all but one of the terms in equation (A10) do not contribute to the damping matrix. Thus using equations (A8) and (A9), equation (A10) can evidently be written in the form: dQ1 =~dQ1 +~TdQ* + 61 [dQ x" 62] + 6 s [dQ*" 64]
OF(t, X~,) + i t o C = ( _ t o 2 M +itoC + K) R OX ito(C - C*)
152 Applied Ocean Research, 1980, Vol. 2, No. 4
(A5)
(AI0)
(A 11)
where the vectors fix, 62, 63, 64 are ultimately dependent on Qo, Q1 and Q3. The actual perturbed drag force will be given by:
Fourier analysis of drag force for offshore structures: James Hamilton dF =
RIdD lei'°'
(A 12)
Now dQt is related directly to the perturbed velocities for example
RldQ1 ei~°t= dXj + d ~ + dZ, k +
_u(U'~v) lul ~u" uj having been neglected as stated above. When the mean current Q0 is zero: I "~1/2
angular velocity terms
(A13)
ct=½(l+~2)
IQ,I
=0.6533 IQll Hence there is no difficulty in interpreting the first term of (All) namely ~tdQ~ as a damping term. The same is not true of the other terms. The second and fourth terms depend on dQ* which is not directly related to any physical response velocity. It is possible for dQ* to appear in dQ1 because the effect of dQ1 on dD 1 depends on its phase relative to the phase of Q~. The third (and fourth) terms are more in the nature of a force in a specified direction (61 or 63) whose expected values (for arbitrary directions of dQ~ relative to 52 and 64) are zero. Thus the most important contribution to the damping matrix as it is usually understood comes from the term in ,. In the case of a large mean current at = IQol which is in accordance with equation (4), the term:
(A14)
This compares with 8/3nlQxl=0.8488 IQll which is normally used. This value, however, aside from the objections raised above for a many degree of freedom system is suited more to the representation of D1 when large values of dQ1 are expected. Thus at dQ 1 = - Q l d F = - F, predicting a zero drag force as expected. Equation (A14), however, does not imply a zero drag force at zero relative velocity. It is presumably, more accurate for dQx small. The conventional expression should therefore be used for a non-iterative solution; however, for an iterative solution we should write for example dF=ct{ - d,~'c,i - d Ycj}
(A15)
Applied Ocean Research, 1980, Vol. 2, No. 4 153
In this paper a representation is obtained of the first Fourier component of the drag F given by: F ~ Q { Q - Q } 1/2 The relative velocity Q is assumed to consist of a mean current plus a single Fourier component, and may come from all directions during the course of a cycle. The expression is shown to agree well with the usual two-dimensional expressions, (when the relative velocity is always in one direction) and also with the exact expression in the three-dimensional case. A good representation of the drag in this manner is important if the quasi-linear spectral approach is to be used in the evaluation of the response of a three-dimensional compliant structure to ocean waves.
INTRODUCTION The importance of drag on a typical offshore structure compared with the other forces, for example, arising from pressure gradients and hydrodynamic inertia, increases with the wave slope. Thus the correct modelling of drag is particularly important for calculating the responses to the structures design wave that is the worst wave which the structure is likely to encounter during its lifetime.. Model tests and time domain integration may provide the only true confidence that a given design may survive its design wave, however, their expense is likely to restrict their use considerably. In the design process the requirement to test different platform designs, and to search for the worst loadings (which may not occur at the design wave) probably mean that the number of program runs in which drag plays an important (if not the dominant) part is more likely to be limited by time and money than by anything else. There is thus ample justification for obtaining at least the capability of a more accurate representation of drag within the framework of the linear, spectral approach. The spectral approach in general has important uses in the statistical analysis of the responses 1. For these purposes, strictly speaking we must be able to write a linear governing equation for the platform orientations X say, where typically
x = { x ~ , r~, z., ~, o, q.}~
(I)
are independent of X or X. Given the assumed form (2) for the governing equation, the response to a superposition of wave spectral components may be obtained simply by superimposing the responses to the individual components. In reality the drag force on a cylindrical element of the platform actually takes the form: F ~ = ½PCo D idli( u i - vi)lui - vll
(3)
where u i is the particle velocity normal to the element (diameter D i length d/i) and vi is the velocity of the element normal to its axis due to the platform motion, (ui - vi) is therefore linearly dependent on X (for small angular displacements). The first consequence of equation (3) is that if Ui and vi possess only one Fourier component, F~,. however, would possess many. Thus even if F ° were independent of X, (2) could not be solved simply by fourier methods. The second consequence of course is the dependence of F ° on X in a highly non-linear manner. The author is of the opinion that except in the limiting case of small response velocities compared with the total water particle velocity when we may write: (U - - V)I(U - - V)[ = (U - - VXU" U - - 2U" V) I/2 "]- O( V2)
=(u-v-u~)lu]
+O(V2)
(4)
%f X~, Y~, Z c are the locations of the centre of gravity of the platform and tp, 0, ff the angular rotations. For example the linearised Morison equation 2'3 for X takes the form: M X + C X + K X = F(t)
(2)
.where F(t) is the force-moments vector and M, C, K and F 0141-1187/80/040147~)7 $2.00 © 1980 CML Publications
a sensible linearization does not exist for many degree-offreedom systems. The approach of Borgman 4 in which the best linear approximation to QIQI is found given the distribution of Q, is not applicable when Q depends on more than one independent variable. It would be necessary to specify the phase relationship between for example -~o and (o a priori or assume that they were uncoupled, both extremely doubtful hypothesis.
Applied Ocean Research, 1980, Vol. 2, No. 4
147
Fourier analysis of drag force for offshore structures: James Hamilton jl
~r2 1/ Comooc~nt in, phase w i t h q
2 lO00~'~'d
2000 i
3000 7 6
~x,c 1
5
T1 • torln(~3
2
Component in phase with -W
same magnitude as Q~ whilst the second harmonics on the other hand are ignored, in order that 'tow out' may be studied. Note that in order that equation (6) be useful, it is not necessary that the drag itself have small second and higher harmonics merely that the response which may be dominated by the linear forces have small higher harmonics. The usual formulations of the drag for a multidegree of freedom system do not include the effect of a mean current; however, Borgman 4 (or Wu s) quote the best linear fit for aone degree of freedom system.
(vertical partical vek:x;ity)
QIQI ~ (a 2 - mZ)2@(m/a) - 2maq~(m/p) + - 2OO0
Figure 1. Convergence of iterative method of response evaluatiOn for a TBP in its 50 year worst wave. Damping matrix C = O. *, No drag; wave height, + 17 m; wave period, 20 sec. CD=0.8; O, aft tethers; e,forward tethers. Tether tension variation, T = ~ (T 1 + iT2)exp( - iogt) This non-linear dependence of the drag force on a compliant structure such as a TBP does not fortunately preclude us from using the spectral approach to study its responses. It is merely necessary to be able to estimate the first Fourier component of the drag F? given the wave particle velocity and the platform response. By first assuming a particular response X 0 (say) it is possible to calculate successive estimates for the responses X 1, X 2 , X 3 etc. which converge to give the actual non-linear response. In the Appendix we give further details of this approach, which is necessary if the effect of drag on complaint as opposed to fixed structures is to be calculated. In Fig. 1 may be found an example of the convergence which may be obtained in the case of a TBP moored in 200 m of water when subject to its design wave. It will be seen that the inclusion of drag forces on this particular TBP has a considerable effect on the tether tensions. This in spite of the fact that the drag forces are still smaller in total size than the inertial forces and that the platform is not operating near its surge resonance (which is about 60 sec). It is not appropriate to enter into an extended discussion of TBP dynamics in this paper, we merely wish to point out the effect of drag can be both unexpectedly large in crucial areas and highly non-linear. (Note that the first correction points '1' are a considerable distance from the final points '7'.) The building block which enables us to calculate these responses for a multi-degree of freedom system, however, is absent. There does not seem to be an adequate formulation of the drag force in this case and it is with this problem that this paper is primarily concerned.
We consider the problem of calculating the harmonics of (5)
(7)
where q~(x)= ~
1
exp ( - x2/2)
42.
a
f o
and Q = Q,fi (say) is normally distributed, mean m and standard deviation a. Evidently extension of this formulation to a many degree of system where both m and a are vectors is not straightforward. Wu 5 appears to use an empirical extension of equation (7), which in the case of m = 0 (zero mean current) reduces to a form equivalent to: QIQI=Q~IQ~Ii
+Q,IQ,~
+Q~IQ~Ik
(8)
This formulation of the many degree of freedom drag forces is not frame invarient as is equation (5) and is therefore not physically reasonable. The other formulation commonly used (for example in N M I wave) consists of obtaining the Fourier component of for example sin cotlsin ~otl by carrying out the Fourier transform directly4. Thus
al=~f
vlvlexp( - io~t)dt
(9)
0
and drag = ½PCoDl ~,~a 1 exp (iogt)
(10)
where v is the relative velocity component. To extend to three dimensions, however, with the inclusion of current effects, it would first be necessary to obtain: Q- Q = A + ~ { B exp (iogt)+ C exp (2i~ot)}
FOURIER REPRESENTATION OF THE DRAG
QIQ[ = Q[Q" Q]l/z
{4m~(m/a) - 4atp(m/a)}Q
(11)
where A,B and C are obtained by subsituting equation (6) into the left hand side of equation (11) and equating terms (all exponentials being converted to positive frequencies). Thus
where the relative velocity Q given by: A = Q o ' Q o +½QI" Q~ Q = Qo + , ~ Q 1 exp (io~t)
(6) B=2Qo.Q 1
is composed of a mean Qo and a fluctuating component Q1 (here,~denotes the real part). We allow Q0 to be of the
148
Applied Ocean Research, 1980, Vol. 2, No. 4
C =~Q1 "Q1
(12)
Fourier analysis of drag force for offshore structures: James Hamilton where '*' denotes the complex conjugate. The carrying out of the step equivalent to equation (9)
~2 = ~ '
[-Ql:_Or
(23)
"Q~'+ L 4x/2
2x/o
°f
al =~-
Q{Q. Q}l/2exp(- kot)dt
(13)
Since q is the positive root we must take:
O
would seem however to involve an intractable integral (hence probably the use of 8). It is this problem which leads us to suggest a different method. The nub of the problem would seem to be in finding the square root q (say) of {Q" Q}. To find an approximation to this we return to the original definition and seek e,/3, 7; (/3, 7 complex), such that q = ~ +,~{/3e '°'' + 7e 2''~'}
(14)
and q2 has Fourier components A, B, C at the zeroth first and second frequencies. Thus substituting equation (14) into the lefthand side of equation (11) and equating the coefficients of the components we obtain three simultaneous equations: (15)
2~/3+ 2/3*7 = B
(16)
27~ + 1//2 = C
(17)
Note that this process is not quite the same as finding the Fourier components of {Q.Q} 1/2 since q2 contains terms in e 31°t and e 41°t (18)
where D=/37 and E=½y 2. However, these terms (in D and E) if they were to arise in {Q .Q} would result from the presence of a second harmonic component in Q. It is therefore consistent with our single component approach to neglect these terms. Similarly we could compensate for their presence by including terms in q of 6 and e q = 0c+ ~ [/3ei°'' + 7e 21°t + 6e 31°t 8e*i°t]
(19)
These terms have no effect (except indirectly through their influence on ~,/3 and 7) on the e i'°' term in the drag. Below we compare with the single degree of freedom case. The solution of equations (15), (16) and (17) proceeds as follows. Treating/3 as a known variable, from equations (15) and (17) ~2 = ~(A - ½/3/3*)+ {~A - ½/3/3,)2 _
~c-½/32XC*-½/3"2)}1/2 7 = ( c - ½/32)/(2~)
(20) (21)
Note from equation (16) we also have:
/3= (~/3- 7/3")/2(g2 - 77")
(24)
Hence 7
+ 1
-1/2
(25)
-2{Q1. Q~},/2
Intuitively since {Q.Q}I/2 is positive definite we would expect to take the ' + ' sign in equations (24) and (25) and hence in equation (20). We may confirm this in the case of a mean current only. Thus if Q1 = 0 and hence B, C,/3 and 7 are also zero then (if~2
1"~2 ") 1/2
(26)
q=a=i~9-+~I
~2 +½/3/3, +½77* =A
q2 ___A + , ~ [ B e i'°' + Ce 2u°' + De 3i°t + Ee 4i°']
1 )1/2 ~=½{QI'Q*} w2" 1 _ ~ j ~
(22)
Clearly the' + ' sign must be taken in equation (20) to yield the correct root. Except in the case/3 = 0 when a solution of equations (15), (16) and (17) may be obtained as shown above, it does not appear possible to obtain an elementary solution of these equations. As an example in the special case when Q1 may be written in the form: Q1 = 0 1 exp (i0) where 01 and 0 are real, we may remove the complex parts of fl and 7 simply by changing the time origin in equation (6) by an amount - 0. In this simple case we may eliminate ot and 7 to obtain a quartic in/32
[,2
{4A/32 - 2fl* + 3 ~ - - 2C/32 +/3"
]}2 -
B 2 VB 2
~-[_T-2C/32 +/3"]=o
Even this equation, however, has no simple roots, thus in general we must adopt a numerical approach. The procedure proposed (which has been extensively tested) is to proceed iteratively first assuming/3 = 0, then equation (20) (with t h e ' + ' sign) and (21) may be used to find estimates for ct and y. A corrected value for/3 can then be obtained using (22) and the procedure repeated until the iteration converges. The convergence is immediate for a zero current and about 10 iterations are sufficient in general to obtain accuracy to i%. Note that where an exact representation of {Q. Q}1/2 is possible with only 3 Fourier terms, this procedure appears to yield it, thus if Q0 is greater than and parallel to Q1, Q never changes sign and ct= {Qo'Q0} 1/2, fl={Q1 "Q1} 1/2, 7 = 0
(27)
We consider first the use of zero current when Qo = B = / 3 = 0 and hence from equation (18)
can be confirmed as a solution of equations (15), (16) and (17). In Fig. 2 may be found comparisons of the Fourier series representation of pQ] (the first three terms) and the
Applied Ocean Research, 1980, Vol. 2, No. 4
149
Fourier analysis of drag Jorce Jor ofjshore structures: James Hamilton 1 ~.
Q=coswt
These expressions are on a less firm foundation than equation (31) as additional terms 6 and e can contribute directly to their values. However, in Fig. 3 we plot the first two terms.
.~-
(36)
Q{Q.Q}I,'2 = D o + D 1cos ogt
"hi2
0
'
for the Fourier series and the present theory. It can be seen that both are reasonably close together. In Fig. 4 is plotted the full, three term expression (30) and the corresponding Fourier series. Again the present theory seem to reproduce the peak values with greater accuracy than the simple Fourier transform. To illustrate the capability of the present procedure when Q0 and Q1 are not parallel we plot in Fig. 5 a polar plot of the drag and equation (30) for such a case. Also plotted is the expression (the formulae to which Sony and Wu approximate)
rr tat
O=1/3÷2/3cos~t
n/2
wt
Figure 2. Comparison ofIQl( ( - - - - - ) and present theory (
QxtQxl
..... ) with Fourier series )
+ QylQyl
[37)
It can be seen that this expression, at least in its 'raw' form does not provide good agreement with the frame invariant form.
present theory (equation (20)), for a single component relative velocity. Q = c o s ~oti
(28) 1~
and Q = (~ + 2 cos ~ot)i
(29)
It can be seen that the agreement in both cases is fairly close, with little qualitatively to choose between equation (14) and the Fourier series. It will be noted, however, that equation (14) seems to reproduce the peak values with more accuracy. Granted the adequacy of equation (14) we may obtain the first harmonic of Q { Q ' Q}1/2, Dx (say), where QIQI = Do + ~
Q=1/3+2/3~t
Dlei'~' + , ~ D2 e2i'~'
6
.12
x ~ ~
~t
(30) -1
Figure 3. Comparison of Q I Q I ( - ) with first two terms of Fourier series ( - - - ) and present theory ( )
from
D,
=
(31)
flQ0 + ~Q, +½7Q~
If Q is represented by equation (28) we may compare our result.
ID,I =½(1 + 1/x/2)l/z +~(1 ) 1+ l/xf2
~
1/2
= 0.8446
~
Q11/3+2/3cos~at
(32)
with the classical result from Fourier series: ID{I = 8/3rt = 0.8488
(33)
a less than ½% error. In addition to D 1 we may also obtain estimates for D O and Dz again assuming the adequacy of equation (14):
150
Do = ~Qo +½fl*Q1
(34)
D2 = 7Qo + ½flQ~
(35)
Applied Ocean Research, 1980, Vol. 2, No. 4
Figure 4. Comparison of QIQI (--) with first three terms of Fourier series ( --) and present theory ( )
Fourier analysis of dra9 jbrce for offshore structures: James Hamilton 3n
O
-~
4ff
ii/ .0"5
~11~ J~
\'N
-
•\1', 0.5
0.5
'
'
f" / ~ tf-- -
"
x
an objection, the argument being that the method is justified by the results embodied in Figs. 2-5 together with the agreement in the special cases discussed above. Also these results would be used to calculate single harmonic responses of semi-submersibles for example when evidently second harmonics would also be present in the motion. Most small parameter expansions would be inapplicable in this case, thus it was considered more important to obtain the best overall accuracy irrespective of the size of a small parameter.
/"ln• CONCLUSIONS
2
Figure 5. Polar plot of dra9 force (Dxi + Dyj) when the mean and oscillatory components of the relative velocity (Q) are not parallel. , QIQI; - - - - ~ QxlQxli + QylQy[J; , present theory, Q = (1+ 4acostot)i _ sino~tj
HIGHER HARMONICS It will have been noticed that the above procedure in addition to providing estimates for the mean force Do and the prime harmonic D~ also provides (equation (35)) an estimate for the second harmonic of the Drag force. In what sense is this term valid? We may argue that if the response X is actually very closely approximated by: X = X o + ~ X l e i~°' (say) then the expression (36) for the drag forces will be fairly accurate, We may therefore use D 2 to verify that the response has actually a small second harmonic component. Taudin 6 appears to show second harmonics and may do something similar; however, he gives no indication of how the drag forces are evaluated. DISCUSSION Some justification is perhaps required for suggesting a method of evaluating the first Fourier component of the drag which is not apparently the first term in a small parameter expansion. There is in fact a small parameter for this analysis, and it will measure the extent to which q = x / Q ~ Q can be represented by a time series including up to only second harmonics (e and 6 small in equation (25)). Special cases where this is exactly true are: (i) nearly uniform magnitude Q. For example, a fixed pipe beneath a sinusoidal wave. This would not be correctly represented by the usual theory. (ii) the relative velocity Q is of the form: Q = (a + bei°")r where ]a] > Ib] and r is a constant vector. Note that if Q possesses more than the first two harmonics A, B and C in equations (15), (16) and (17) and the expressions for Do, D1 and D 2 may have to be appropriately modified. However, it must be admitted that the absence of a small parameter in the usual sense was not initially seen as
We have shown that the approximate method suggested here for evaluati.ng the first Fourier component of the drag force has the following properties: (i) in the case of a one-dimensional relative velocity and in the absence of a mean current the method is within ½% of the exact (8/3rc) expression; (ii) the formula is also exact if the mean current is greater than and parallel to the oscillatory component of the relative velocity ;(iii) good agreement with the exact expression for the drag (not the Fourier components) when the phases of the relative velocity components are not equal. This latter case can occur if the structure oscillates wholly or partly at right angles to the direction from which the waves are incident. The method is not susceptible to linearization; however, in the absence of a comparable formulation it is recommended that the method be seriously considered in the analysis of the three-dimensional dynamic response of structures in waves. A suitable title for the method might be 'the approximate square root method'.
REFERENCES 1 Ochi, M. K. and Wang, S. Prediction of extreme wave induced loads on ocean structures, BOSS 76, Conf. Trondheim, 1976 2 Hallom, M. G., Heaf, N. J. and Wootton, L. R. Dynamics of marine structures, Report UR8 CIRA Underwater Engineering Group, London, 1977 3 Paulling, J, R. Wave induced forces and motions of tubular structures, Proc. 8th ONR Symposium on Naval Hydrodynamics, Pasadena 1970, p. 1052 4 Borgman, L. E. Ocean wave simulation for engineering design, J. Waterways Harbours Div. ASCE 1969, 95 (WW4), 129 5 Wu, S. C. The effects of current on dynamic response of offshore platforms, Proc. Offshore Technol. Conf. Houston, 1976, OTC 2540 6 Taudin, P. Dynamic response of flexible offshore structures to regular waves, Proc. Offshore Technol. Conf., Houston, 1978, OTC 3160 7 Malhotra, A. K. and Penzien, J. Nondeterministic analysis of offshore structures, J. Eng. Mech. Div. ASCE 1970, 96, 985 8 Burke, B. G. The analysis of motions of semi-submersible drilling vessels in waves, Proc. Offshore Technol. Conf. Houston, 1969, OTC 1024
APPENDIX Iterative method In the case when u and v are the same order of magnitude an iterative upproach to the solution of (2) would seem unavoidable. In the iterative approach 5,7,a, the drag is not associated with a stiffness matrix C but is evaluated given an assumed value of v. This is added into the r.h.s, of equation (2) and a new estimate is calculated.
Applied Ocean Research, 1980, Vol. 2, No. 4 151
Fourier analysis of drag force for offshore structures: James Hamilton
Xo=0 [ - t o 2 M + K ] X j = F ( t , Xj_O,
and convergence can be forced with the correct choice of C and X o. j=1,2,3
(A1) Evaluation of a dissipation matrix
where to is the frequency of the oscillation in radians per second. This procedure continues until X j and X j_ 1 are sufficiently close in value. Note that it is possible to speed convergence by introducing a damping matrix to either side of equation (A1). (See, however, Taudin 6 for a more extensive discussion of iterative methods.)
The dissipation matrix can, of course, be evaluated using conventional methods in which typically the drag force F is assumed to behave as: F(t, X~) ~. 8 tQ11[(ul - X~)i +
[ - coZM + itoC + K ] X j = F(t, X j_ 1) + i~CXj_ 1
This technique is applicable to all methods of evaluating the response (assumed to be of the form exp (itot)) to a single component wave field.
(A6)
(Uz- ~'~)i-I
(A2)
for a two degree of freedom system (that is i~noring rotational velocities). The dependence of IQII on Xo, Yois ignored, thus for a small change in the response dX~, d Y~ the perturbed drag is given by:
r/=~q0 exp (itoot + kox + loy + ¢Po) An interesting feature of the iterative method when carried out using equation (5) is that within the framework of the other linearizing assumptions (normally small wave height and angular velocities) it is possible to deduce information about the stability of the resulting solution. Thus if we write 6 X j = X j - X ~ , then close to X~
dF ~ 8 I Q , l ( - d X j - d
Ycj}
(A7)
It would seem preferable from the point of view of speeding convergence to evaluate the damping matrix (which just attempts to model the dependence o f F on )¢c) using the expression for D 1 developed above. Thus replacing QI by Q1 +dQ1 in equation (18) we obtain:
[ _ t o 2 M + K ] f X j _c~F(t, X ~ ) ~ . d a --~Q~dQ1 1 *
and hence
+~Q1-dQ*
dB = 2Qo' dQ1 [ -- t o 2 M +
K](bXj +1 - 3X2)
dF(t, X~) OX
(3Xj- 6Xj_ 1)
d C = Q1 "dQ1
(A3)
In general with 6 degrees of freedom, 21 independent results such as equation (A1) are necessary before t3F/aX is specified unambiguously. However, in situations where only one mode of vibration is predominant we will have: (6Xj +a - fiX) = R ( f X j - 3 X j_ 1)
where R is a complex constant and the iterates X j will describe a spiral towards the final solution X~ (see Fig. 1). By observing the direction of rotation of the spiral, the sign of R may be determined and hence the stability or otherwise of the mode stimulated inferred.
c~F(t,X~) 3X
(A8)
The perturbation values d~, dfl, d7 may then be obtained by substitution into equations (21), (22) and (23):
[
2~ 2fl
½fl* 2ct
½fl 27
½7* ½fl*
2fl*
27*
2ct
0
27
fl
0
2~
27*
0
fl*
0
L
2a.l
dT*
dC*
(A9) Giving finally the first harmonic of the drag: dD~ =~dQ~ +½7dQ* +
= ( - t o 2 M + K)R
= - icoC* (say)
Q0dfl+ Qld~ +1Q'd7 (A4)
where C* is the damping matrix near to X = X'. Note that for convergence of the iteration process we require ]RI < 1 and this imposes a limit on how close to a resonance (det(-to2M+K)=0) the iteration will converge. Close to a resonance equation (A2) must be used giving:
It is not necessary to solve equation (A9) explicitly since all but one of the terms in equation (A10) do not contribute to the damping matrix. Thus using equations (A8) and (A9), equation (A10) can evidently be written in the form: dQ1 =~dQ1 +~TdQ* + 61 [dQ x" 62] + 6 s [dQ*" 64]
OF(t, X~,) + i t o C = ( _ t o 2 M +itoC + K) R OX ito(C - C*)
152 Applied Ocean Research, 1980, Vol. 2, No. 4
(A5)
(AI0)
(A 11)
where the vectors fix, 62, 63, 64 are ultimately dependent on Qo, Q1 and Q3. The actual perturbed drag force will be given by:
Fourier analysis of drag force for offshore structures: James Hamilton dF =
RIdD lei'°'
(A 12)
Now dQt is related directly to the perturbed velocities for example
RldQ1 ei~°t= dXj + d ~ + dZ, k +
_u(U'~v) lul ~u" uj having been neglected as stated above. When the mean current Q0 is zero: I "~1/2
angular velocity terms
(A13)
ct=½(l+~2)
IQ,I
=0.6533 IQll Hence there is no difficulty in interpreting the first term of (All) namely ~tdQ~ as a damping term. The same is not true of the other terms. The second and fourth terms depend on dQ* which is not directly related to any physical response velocity. It is possible for dQ* to appear in dQ1 because the effect of dQ1 on dD 1 depends on its phase relative to the phase of Q~. The third (and fourth) terms are more in the nature of a force in a specified direction (61 or 63) whose expected values (for arbitrary directions of dQ~ relative to 52 and 64) are zero. Thus the most important contribution to the damping matrix as it is usually understood comes from the term in ,. In the case of a large mean current at = IQol which is in accordance with equation (4), the term:
(A14)
This compares with 8/3nlQxl=0.8488 IQll which is normally used. This value, however, aside from the objections raised above for a many degree of freedom system is suited more to the representation of D1 when large values of dQ1 are expected. Thus at dQ 1 = - Q l d F = - F, predicting a zero drag force as expected. Equation (A14), however, does not imply a zero drag force at zero relative velocity. It is presumably, more accurate for dQx small. The conventional expression should therefore be used for a non-iterative solution; however, for an iterative solution we should write for example dF=ct{ - d,~'c,i - d Ycj}
(A15)
Applied Ocean Research, 1980, Vol. 2, No. 4 153