Proceedings of the Eleventh (2001) hlternational Offshore and Polar Engineering Conference
Stavanger, Norway, June 17-22, 2001
Copyright © 2001 by The International Society of Offshore and Polar Engineers
ISBN 1-880653-51-6 (SeO; ISBN 1-880653-52-4 (Vol. I); ISSN 1098-6189 (SeO
Effect of Ramp Duration on the Dynamic Response of Spars
Iftekhar Anam and Jose M. Roesset
Texas A&M University
College Station, TX, USA
ABSTRACT
conditions to get the total solution. This is often a more efficient method
of solution, particularly for linear systems. However, even the steady
state f-d approach is not an exact representation of many real prob|ems.
Among other limitations, the f-d method can only provide solutions to
'!i_n.earized' problems or reproduce nolmineari_fies_ (e.g., those involving
forces) only up to a certain order using a perturbation approach.
Therefore, the t-d approach is still widely used in solving dynamic
problems and is often preferred because of the complexities it can
handle. A common approach adopted by several researchers, studying the
response of flexible offshore s m a ~ e s is to ~ e a ramp fi~ction (wbAch
increases gradually from 0 to 1) at the initial stage of the t-d simulation to
avoid initial transient effects that may give unreasonably large responses.
Ramp functions are also used in wave model basins testing offshore
structures to simulate the real sea situations and to get steadT state
r~°ponses "~i~,hin~e limited time of the experiment.
Larger ramp durations are generally believed to W e steadier structural
responses. But many studies on flexible offshore stmct~es like spars
(with surge natural periods of around 300 seconds) have been conducted
to simulate iaboratolT conditions, taking very small ramp durations; e.g.,
only 50 seconds proto~pe scale in Ran etal. 1995, Mekha et al. 1995,
Cao and Zhang 1996, In ._someo~_es._~.the d y n ~ c re._spo~e within the
ramp duration was also considered as part of the 'recorded' results,
which is not the correct numerical approach, nor even representative of
the way the data are recorded in the tests. Such anomalies show a lack of
clarity about the possible effect of ramp duration on the results of t-d
sim~ations.
Mekha and Roesset (1998) showed the effect of ramp duration on
the response of spars with some numerical results, concluding that a
'sufficiently large' ramp was necessary to obtain steady state
response. Tl-fis paper tries to explain tl~ese res~dts a~alytically m~d
support the main conclusions numerically. The pro'pose of this work is
to develop a better understanding of the effect of the ramp ftmction in
the numerical analyses using the t-d approach. The analytical work in
the first part of the paper is followed by some numerical results for a
monochromatic wave excitation acting on a spar platform. By
considering this very simple excitation and only a linear response one
can show clearly the transient effects with the frequency spectrum of
the response.
To e.....
':~"m,,~e
~+
the free -dbrafion terms and to simulate actual
field conditions, it is common practice in wave model basins to
multiply the excitation by a ramp function and to disregard the
portion of the response corresponding to the application of the
ramp. The effect of ramp duration on the dynamic response of a
spar platform subjected to irregular waves was shown
numerically in a previous paper° This topic is inves_ti_gated again
in this paper both analytically and numerically, using a simp|e
monochromatic excitation and a finear system. It is shown that
free vibration will result from discontinuities in the forced
response both at the time of application of the rarap and wilen it
is removed_. Both discontinuities decrease in Lmportance as the
duration of the ramp increases. To reach the steady state
condition one can thus increase the duration of the ramp and/or
neglect the initial part of the response unti! a certain time after
the end of the ramp. For normal, low values of damping the ramp
duration is the predominant factor.
K E Y W O R D S : Spar, ramp, free vibration, steady state, time domain,
frequency domain.
INTRODUCTION
In the numerical analysis of dynamic systems, the more direct
approach is to fommiate the problem ha the time domain and to solve
the governing differential equations numerically using a time-step
inte~ation scheme. The time domain method (called t-d here)
accounts for the initial conditions of the system (e.g., displacements
and velocities) and represents properly in general the nonlinearities of
the system. However this method can result in unrealistic transient
effects, which may not represent the real-life situations. This is
particularly true for flexible offshore structures (i.e., structures having
large natural periods of vibration) subjected to ocean wave loads that
develop ~adually. The large transient effects in t-d solutions due to a
sudden application of fully developed sea-conditions can therefore be
artificial.
To solve this problem, two approaches can be taken. The first is to use
a combined time-frequency domain approach that converts the steady
state frequency domain-(f-d) results to t-d and applies appropriate init/al
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01-JSC-308
I. Anam
EFFECT OF RAMP FUNCTION
.'. The steady response just after the ramp is applied, is
x(0+) = 2: Pi {cos(qgi0)/ki0 - 0.5 cos(tpi_)/ki_- 0.5 cos(qoi+)/ki+}/2
while just before, i.e., initially, it was x(0-) = 0
(5)
Similarly, just before the ramp is withdrawn, the forced response is,
x(Tr'/Wr) = ~ Pi {cos(oiTt:/~r+q)i0)/ki0+ 0.5 COS(r.Oi~//gr+~i.)/ kit- 0.5
cos(oin/~ +9i+)/k~+}/2
while just after withdrawal, i.e., it is
x(Tg+/~i~r)= ~2 Pi cos(f-°i~/UIr+(Di0)/ki0
(6)
Equations (5) and (6) show that the forced responses go through
sudden discontinuities both at the beginning and at the end of the ramp
durafior,, as was logically concluded earlier.
To see the effect of ramp duration on these d_iscx~ntimfifies,one may
consider the case when the ramp duration becomes very large;
i.e., Tr --~ oe ~ ~r--~ 0.
• Clearly, both ki. and ki+--~ ki0;also q0i. and q~i+-+ q0i0,in this case.
" x(0 +) -+ X Pi {cos(~0)/ki0 - 0.5 cos(qoi0)/ki0- 0.5 cos@pi0)/ki0}/2
= 0, which is x(0);
and x(xT~0 -~ 2 Pi {cos(~iTrJ~Jr+CPi0)/kio+ 0.5 COS(OiTr,/'UJr+q~iO)f~O+0.5
COS(OiT~//~Jr-q-(1)i0)/rki0}/2
= 2 Pi cos(°)iTg/~Jr+~i0)/ki0, which is x(x+/~0
(7)
Equation (7) gives an improved insight on how long the ramp should
be in order for it to become 'sufficiently' iarge. This ~411always remain a
qu~,ditafive statement, bm tim essential fact is that the transient effects will
be diminished iXthe ~scontku,,fifi~ of the motions on eider side of ~ e
ramp duration can be reduced. So, the ramp duration has to be
'sufficiently' large compared to (half of) the longest period which
contributes 'significantly' to the displacement.
For flexible offshore structures like spar platforms, the smallest
ffequen.cies (i.e., longest periods) coombuting si~ifica_ntly to the
structural motions are the surge natural frequencies. Therefore,
'sufficiently' large ramp duration for surge has to be 'sufficiently' large
compared to halt" of the surge natural period, it will not need to be that
large for the pitch resportse of a typical spa,-.
On the other hand, the longest 'significantly-excited' period of motion
for rind structures is within the range of the wave periods themselves,
which is much larger than the structures' natural periods of vibration.
Theretbre, the 'adequate' ramp durations tbr rind structures should be
much smaller. Tiffs is part of the reason why the work done ruth ramp
functions h ~ focused on flexAbte smacmres like spars and Tension Leg
Platforms (TLPs), and not on rind structures like gravity-based
structures or steel-jacket plaffom~s.
The ramp function is applied to enrage a continuous ~ d gradual
transition of wave loads from an initial zero value to a fully developed
stage. It is applied as a multiplying factor to the wave-load and should
therefore increase from zero to one. A ramp function often used in the
literatm-e is
framp = (1" cos(nt/T~))/2, for 0 < t < Tr
(1)
where Tr is the ramp duration.
It is a cosine function which does increase from zero to one, with
the additional fact that it has zero slopes both at the beginning and at
th,, end of the ~,,~,,~,~,l If it res~ted in continui~, of steady state
response, there would be no transient effects and the results from the
t-d solution would represent the steady state. However, the continuity
of the force does not result in similar continuity of the steady state
response due to it. The continuity of the total response (steady +
~ausient) will be maintained however, wlfich requires a discontinui~ ~
in the transient response as well.
The application of the ramp results m transient effects in two ways.
First, it will still produce a transient response when applied initially
on a structure at rest; then it will create an additional transient effect
when it is withdrawn. The decay of these two transient effects will
depend on the damping of the system and the time they are given to
die out. Clearly, the Fast transient effect will decay during the entire
period of appfication of the ramp; the~,e~or., a longer rmnp ,.m help its
decay. The second transient effect will only beg~n after the ramp is
withdrawn; therefore its decay will depend on the time given after the
withdrawal of the ramp. Thus, qualitatively it will take a 'sufficiently'
long ramp as well as a 'waiting time' after its withdrawal for the
transient effects to die out stffficieml_y and for the response to become
more ' steady'.
These concepts are easy to visualize from basic concepts of
structural dynamics. However it remains to be seen if the ramp
dm-ation has any effect on the discontinuities introduced in the forced
responses. As is shown later, a large ramp does decrease them too.
Thus, the effect of ramp in decreasing transient response is twofold.
This is now shown analytically.
For simplicity, a single-degree-of-freedom (SDOF) system is
investigated here. The main conclusions can still be presented in
general terms. The force from i~egular waves is the stun of its
different Fourier components acting at different frequencies (~i), each
having different magnitudes (Pc) and phases (q~c),
The ramp function within the ramp duration is
= (I- cos(x'ffTr))/2
= 0.5 (1- COS~rt), where Igr = ~/Wr
(2)
Here, w,. = x/Tr = 0.5 (2x/T0 = Half the natural frequency
corresponding to the ramp-period.
.'. During the ramp duration, the dynamic force
F(t) = (1- cos ~rt)/2 22 Pi cos(Oit+g)i)
= {E Pi cos(°it+(pi)- 0.5 ~2 Pi COS(Oit-~rt+(Pi)- 0.5 ~] Pi
cos(toit+Wrt+9~}/2
(3)
Thus, the force is reduced to a summation of three sets of cosines. The
forced pm-t of flae dynamic response (i.e., particttlm" solution of equations
of motion_) can be easily found to be,
x(t) = Z Pi {cos(c0it+~i0)/ki0 - 0.5 cos(oit-_m_~t+gi3/ki.- 0.5
cos(oit+~t+(pi+)/ki+}/2
(4)
where ki0 = ~/{(k-oi~m)2~ o i c);}, the dynamic stiffness at frequency oi
ki-- Dynamic stiffness at frequency ( o i - ~ ) , and ki+ = Dynamic
stiffness at frequency (Oi+'rJr).
Also, q0i0= tan-1 {oi c/(k-oi2m) }, the phase angle at frequency oi
The phase angles q)i.and (Pi+are defined likewise.
N~TvIERI CAL RESULTS
The main conclusion from the theoretical work of the previous
section was that a longer ramp helps to suppress transient effects, and
thereby should provide responses closer to the steady state (forced
vibration) solutions. Another aspect was the "waiting period' after the
ramp is withckawn; a bigger waiting period would help also to
minimize transient effects. These conclusions are now studied
numerically for a spar platform with a draft of 198 meters, a diameter
of 40.5 meters and long periods of vibration [332 seconds in surge and
66 seconds in pitch]. This particular spar operates at a water depth of
3i8.5 meters and has a mass = 259xi06 kg, spring stiffness = 188
kN/m and radius of gyration = 62.3 meters.
A detailed work has akeady been conducted showing the possible
effect of ramp duration on the dynamic response of spar platforms
(Mekha and Roesset 1998) where the above conclusions were shown
to be true even for random waves. This section deals with the shnpler
case of a monochromatic wave (called 1'v~,lI here) with mnplimde 3.0
m~ers and period !4 seconds. This way, it is possible to show the
ramp's effect purely with the frequency specmun for a linear
428
01-JSC-308
I. Anam
2
5
formulation. The response should converge to a single peak at the
wave-frequency and be zero everywhere else. Another point of
interest is the difference between the 'adequate' ramp duration for
surge and pitch motions, with their very different natural periods.
According to the analytical conclusions of the previous sections, the
adequate ramp durations for the two motions (to suppress transient
response) should be different. The results of the t-d analyses with
ramps of different duration are compared to those of a steady state f-d
solution.
The frequency-spec~a obtained from the t-d and f-d simulations are
shown in Figs. 1-12. For the four ramps chosen here, the results for
surge mad pitch are shown alternately between Figs. 1-10. Finally, the
f-d results are shown in Figs. 11 and 12.
First, a ramp duration of 20 seconds was used. This is very small
compared to the surge natural period (332 s~onds), and also with
respect to the pitch natural period (66 seconds). It results in transient
response amplitudes of 0.341 m for surge (Fig. 1) and 0.265 ° for pitch
(Fig. 2) which are about half of the amplitudes at the excitation
frequency. Obviously, these components would not be there if the
response were steady. However, the responses at the wave-frequency
(0.696 m, 0.60! °) agree very well with the f-d solution (0.712 m,
0.615 °) [Figs. 11 and 12].
The ramp was increased next to 50 seconds (a ramp value
sometimes used in the literature). This decreases the transient effects,
mad the corresponding amplitudes are now 0.233 m (Fig. 3) and
0.017 ° (Fig. 4). This ramp, being small compared to tile surge period,
cannot suppress the transient surge response adequately. But now the
transient pitch response is diminished to almost zero; therefore, it is
adequate for pitch. In all these cases, the 'waiting' period after the
ramp is taken as zero.
To demonstrate the effect of the waiting period, the next simulation
was performed with a long waiting period of 950 seconds. The
transient respotkse amplitudes decrease slightly as a result, this time to
0.186 m (Fig. 5) and 0.011 ° (Fig. 6) respectively. This decrease is
caused by the decay of the responses due to damping.
The ramp duration was then increased to 400 sec, with a waiting
period of 600 sec. This way the same 'initial' (ramp + waiting) period
of 1O00 sec (= 400 + 600) is maintained. But the transient responses
are much decreased this time. The transient surge amplitude is now
0.082 m (Fig. 7), while there is no transient pitch amplitude (in
degrees) even up to three decimal points (Fig. 8).
Finally, the ramp duration was increased to 900 sec, with a waiting
period of 100 sec again maintaining the same 'initial' period of 1000
sec (= 900 + 100). The transient responses are almost completely
removed this time. The transient surge amplitude is only 0.015 m
(Fig. 9), while there is obviously no observable transient pitch
amplitude (Fig. 10).
This not only shows the steadying effect of ramp duration on the
dynamic responses, but also the trade-off between the rennp duration
and the waiting period. In this case, the ramp duration turns out to be
more important. However, increased damping would favor the waiting
period. No matter how big a transient response is induced in the ramp
period, the waiting period should be able to suppress it more
effectively in that case. However, the effect of damping ratio on the
required waiting period is beyond the scope of this work and has not
studied here.
Although here the forces are calculated using a "linearized' form of
Morison's equation, the conclusions are valid for any dynamic
problem. Similar results could be obtained here by any combination
of loads equal to the resultant forces calculated in this case, and with
representative damping ratios.
CONCLUSIONS
The main conclusions from this work can be stmunarized as:
1. Application of a ramp function results in discontinuities in the
steady state response of a structure subjected to dynamic loads. These
discontinuities occur both at the beginning and the end of the ramp
duration.
2. TILe discontinuities cause transient responses, both at the time the
ramp is applied and when it is withdrawn.
3. A 'sufficiently' long ramp helps to decay the initial transient effect.
A 'stffficiently' long 'waiting time' helps to decay the second transient
effect. Therefore, in order to get the steady state response, the t-d
27
simulation should wait for awhiie
to allow this second transient effect to
die out.
4. A large ramp also helps to reduce the discontinuities themselves.
That is, the longer the ramp duration,, the smaller the transient effect
'induced" by the ramp.
5. For flexible structures with natural frequencies much smaller than
the wave frequencies, different modes of vibration need different ramp
durations to reach steady state. It depends on their respective natural
periods of vibration.
6. A large 'waiting period' can offset the large transient effect caused
by a small ramp. The trade-off between ramp duration and waiting
period favored a longer ramp in this case. But in general, this may
depend on the amount of damping.
It must be mentioned here that this work mainly concentrates on the
effect of ramp duration on the free vibration of flexible structures.
Some other effects of the ramp duration; e.g., increasing numerical
stability of the t-d algorit~-n, are not studied here.
ACKNOWLEDGEMENTS
The financial support provided by the Offshore Teclmology
Research Center (OTRC) through the National Science Foundation
Grant # CDR 8721512 is gratefully recognized.
REFERENCES
Cao, P and Zhang, J (1996). "Slow Motion Responses of Compliant
Offshore Structures." Proc. 6 a' Int. Symp. Offshore and Polar Engrg.,
ISOPE-96, Los Angeles, CA, Vol 1, pp 296-303.
Me!d,.a, BB, Johnson, CP and Roesset, ~A (1995). "Nonlinear
Response of a Spar in Deep Water: Different Hydrodynamic and
Structural Models." Proc. 5 th Int. Syrup. Offshore and Polar Engrg.,
ISOPE-95, The Hague, The Netherlands, Vol 3, pp 462-469.
Mekha, BB and Roesset, JM (1998). "Effect of Ramp Duration on the
Response of Spar Platforms to irregular Waves." Proc. 8 ~hint. Syrup.
Offshore and Polar Engrg., ISOPE-98, Montreal, Canada, Vol 1, pp
270-277.
Ran, Z, Kim, MH, Niedzwecki, JM and Johnson, RP (1995).
"Responses of a Spar Platform in Random Waves and Currents
(Experiment vs. Theory)." Proc. 5 th Int. Syrup. Offshore and Polar
Engrg., ISOPE-95, The Hague, The Netherlands, Vo! 3, pp 363-371.
429
01-JSC-308
I. Anarn
0.8
0.8
0.6
0.6
0.4
a= 0.4
0.2
0.2
g
-
0
+.a
0
, . i
0.1
,
,~
0,2
.,-,
0.3
1
t
0.4
0.5
,
0.4
0
0.5
O.l
0.2
0.3
freq (rad/s)
freq (rad/s)
Fig. 1 MN1 Surge: Ramp = 20 sec
Fig 2 MN1 Pitch: Ramp = 20 sec
Wait = 0
Wait = 0
0.8
0.8
0.6
0.6
,,._.,
0.4
0.4
O
0.2
0.2
,
0.1
0.2
0.3
0.4
.
I
0
0.5
0.1
0.2
0.3
0.4
freq (rad/s)
freq (rad/s)
Fig. 3 MN1 Surge: Ramp = 50 sec
Fig. 4 M N I Pitch: Ramp = 50 sec
Wait = 0
Wait = 0
0.8
0.8
0.6
0.5
-
0.6
,,,--,.
g
0.4
,=
0.4
. ,...~
0.2
0
0.2
~
0.1
,
,
~ ,
0.2
,
,
i
0.3
,
,I
,
0.4
]
,
0,5
0
freq (rad/s)
0.2
0.3
0.4
freq (rad/s)
Fig. 5 MN1 Surge: Ramp = 50 sec,
Wait = 950 sec
Fig 6 M N ] P i t c h Ramp = 50 sec,
Wait = 950 sec
430
01-JSC-308
0,1
I. Anam
0.5
0.8
0.8 -
0.6
0.6
~D
0.4
0.4
O
+.o
0.2
0.2
0
0.1
0.2
0.3
0.5
0.4
0.1
0.2
freq (rad/sec)
0.3
0.5
0.4
freq (rad/sec)
Fig. 7 MNI Surge: Ramp = 400 sec,
Fig. 8 MN1 P i t c h Ramp = 400 sec,
Wait = 600 sec
Wait = 600 sec
0.8
0.8
0.6
0.6
0.4
0.4
O
0.2
0.2
-.~
0
i
,i . . . .
0.1
0.2
i
,
0~
:
0.3
0
0.5
0.4
,I
0. t
,
i
0.2
. . . .
T
,~
,
0.3
0.4
0.5
freq (rad/s)
freq (rad/s)
Fig. 10 MN1 Pitch: Ramp = 900 sec,
Fig. 9 MN1 Surge: Ramp = 900 sec,
Wait = 100 sec
Wait = 100 sec
0.8
0.8
0.6
0.6
0.4
"-" 0.4
0.2
0.2
~D
. . .
0
,
0.1
..........
,
0.2
~
0.3
.
i
. . . . .
0.4
r
0
0.5
0.2
0.4
freq (rad/s)
freq (rad/s)
Fig. 12 MN1 Pitch: Frequency Domain
Fig. 11 MN1 Surge: Frequency Domain
431
01-JSC-308
I. A n a m
5
5