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Subharmonic motions and wave force-structure interaction
Marine Structures 5 (1992) 281-295
Subharmonic Motions and Wave Force-Structure Interaction
C. Y. L i a w , N. J o t h i S h a n k a r & K. S. C h u a Department of Civil Engineering, National University of Singapore, Singapore 0511 (Revised version accepted 19 August 1991)
ABSTRACT The problem of subharmonic oscillations attributable to the interaction between wave forces and the structural motion is investigated both analytically and experimentally. The sensitivity of this nonlinear phenomenon with respect to some system parameters and initial conditions is evaluated. The behaviour of an articulated tower model in subharmonic oscillations is studied, and qualitative comparisons between the experimental and numerical results are made. Key words: subharmonic oscillation, articulated tower, nonlinear
dynamics, wave force, wave force-structure interaction.
1 INTRODUCTION Offshore structures which are designed to experience significantly large motion are b e c o m i n g more c o m m o n as the search for new oil fields progresses into deeper ocean waters. In such conditions, these structures may undergo highly complex d y n a m i c motions in response to the environmental loads I-9. One important factor for such complex motion is the response-dependent loading of the structure. Obviously, in order to study the b e h a v i o u r of these excitation-response coupled systems, n o n l i n e a r d y n a m i c analysis models have to be used in the numerical simulations. 281 Marine Structures 0951-8339/92/$05.00 © 1992 Elsevier Science Publishers Ltd. England. Printed in Great Britain.
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It is a well known fact in n o n l i n e a r dynamics that m a n y n o n l i n e a r p h e n o m e n a , e.g. s u b h a r m o n i c oscillations and chaos, are very sensitive to small variations of the model used in the simulation. ~°-~2This means that, unless the precise system parameters and the complete mathematical representation of the physical problem are known, deterministic predictions of these nonlinear p h e n o m e n a with quantitative accuracy are almost impossible. However, recent advances in n o n l i n e a r dynamics ]]-]3 seem to indicate that some n o n l i n e a r behaviour is not only intrinsic but also universal. The exact quantitative response of a n o n l i n e a r system may not be predictable using the inevitably simplistic model a n d approximate parameters: its qualitative behaviour can nevertheless be studied and understood if the model can simulate the essence of the n o n l i n e a r behaviour. In this paper, it is intended to show, both experimentally and numerically, that one of the causes for the frequently encountered n o n l i n e a r behavior, namely s u b h a r m o n i c oscillations of m a n y offshore structures can be attributed to the wave force-structure interaction, and this b e h a v i o u r can be qualitatively modeled by the modified Morison's equation.
2 ANALYTICAL MODEL FOR LARGE MOTION OF OFFSHORE STRUCTURES In order to study the general three-dimensional motion, including large translation a n d rotation, of a structure, an analytical model based on the d y n a m i c s of large rigid-body motions has been developed using the Lagrangian formulation and Euler parameters ]4"15, i.e.
<,ro lT_ [ GT Lo4J
[o,-,]
=
+
Q
(1)
where T is the kinetic energy of the rigid body, A is the Lagrange multiplier, Q represents all the applied forces, q = < r, p > r and F~ is the constraint equation
prp
=
I
(2)
r is the translation displacement vector at the center of gravity of the rigid b o d y a n d p = < e0, el, e2, e3 > r; e~are the Euler parameters defined as 16 e0 = cos 0/2 el = l] sin 0/2
(3)
Subharmonic oscillations e2 =
12sin O/2
e3 =
13sin O/2
283
where 0 is the angle of rotation of the rigid body about the axis defined by the unit vector < l~, 12, 13 > r. The detailed derivation for the equation of motion is given by Liaw et al) 5, the final expression has the following form:
A
=
+
(4)
The matrix A depends on the mass, mass moment of inertia and the Euler parameters, p. The vector u is a function of the mass moment of inertia and p. The vector s is a function ofp and the positive vector of the constraint. The force vector Q includes the restoring force, damping force and wave loading. If the structure is assumed to be composed of small cylinders, the wave force per unit length of a cylinder can be written as D2 f L = CAMPrt T
D2 D (UL -- XL) q- CFKP J7 ~ IlL "1- CDP "~ (UL -- XL)[ UL -- XL [
(5) where UL and JIL are the water particle and the structural member velocities, respectively; the subscript L describing the vector terms in eqn (5) indicates that these are measured in the local member axes. The coefficients CAM and CFK are the added mass and the Froude-Krylov coefficients; p is the density of water; D is the diameter of the cylindrical member; and CD is the drag coefficient. The modified Morison's equation, eqn. (5), evaluated at the deformed position of the body, is used in the calculation of the hydrodynamic forces acting on the body. Since the wave force fL and consequently the force vector Q is a function of the response quantities, r and p, the governing equations of motion, eqn. (4), are nonlinear and can be solved numerically using the standard step-bystep time integration method. If the structural restoring force is assumed to be linear, then the only nonlinearity of the system described by eqn. (4) is due to the large motion of the system. Nonlinear systems can have many different types of responses which do not exist in linear systems and the one which is of interest in this paper is subharmonic oscillation.
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3 SUBHARMONIC OSCILLATIONS Subharmonic oscillations are the responses of a system having a period n times that of the period of the applied force, where n is an integer. Thus, when n is 2, the resulting oscillation is termed a subharmonic of order 2. Studies carried out on the subharmonic oscillations of offshore structures seem to indicate that there are two important causes of subharmonics: the nonlinear characteristics of the restoring force and the large motion of the structure (or the interaction effect). Chantrel and Marol 3 studied the harmonic pitch response of an articulated loading platform and found that the subharmonic responses obtained from the numerical simulations are due to the nonlinear stiffness characteristic of the hawser. Another work, by Thompson et al. 4, demonstrated the existence of subharmonic motions in articulated towers and attributed these motions to the high nonlinearity of the mooring line, which can become slack or taut during dynamic motions, thereby introducing a discontinuity in the restoring force. Essentially, the mooring line behaves as a bilinear spring. Their results also show the presence ofsubharmonic oscillations of order 2, 3 and higher, as well as chaotic motions. A previous study 5 by one of the authors looked at a simple single degree of freedom system with surge motion, acted upon by a linear harmonic wave. Using only a linear restoring stiffness, his results showed that the nonlinearity arising from the interaction effect alone can cause subharmonic surge oscillations. However, in that formulation the model was applicable only to the case of a single cylinder. With the threedimensional general formulation of the problem, described in Section 2 above, any structure with arbitrarily oriented cylindrical members subjected to the Morison type wave forces can now be modeled. In this paper, the numerical results obtained from the general formulation are used to reconfirm the earlier conclusion based on the previous formulation relating the subharmonics and the wave force - - structure interaction. Some of the observations are further verified in the laboratory.
4 N U M E R I C A L RESULTS An articulated tower model was chosen for both the numerical and experimental studies. Its motion is restricted to only one degree of freedom which is the rotation of the tower about its base. This singledegree-of-freedom model can, of course, also be modeled in a straightforward way involving only 0. However, as previously mentioned, one of
Subharmonic oscillations o.1 kg F-/'/t SWL ~
I
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~ K ~ _
Fig. 1. Articulated tower model for subharmonic analysis, Co = 1.0, CAM= CFK = 1.0, wave length is 0.12 m, wave period is 1,2 s the objectives of this study is to verify the general formulation described in Section 2. The Euler parameters a p p r o a c h is therefore used for the numerical study. A simple diagram of the model is shown in Fig. 1. The numerical model exhibits a clear s u b h a r m o n i c response of order 2 as shown in Fig. 2. For this case, the frequency ratio, p/to, is 2.0, where p = 2n/Tw; the wave period Tw -- 1.2 s; and to is the natural frequency of the partially submerged tower at its u n d e f o r m e d position in still water. The time history of the tower response (rotation) presented in Fig. 2 has a period of 2-4 s, twice the wave period. Similar s u b h a r m o n i c responses of order 2 can also be obtained, if the p/to ratio is varied within a range a r o u n d 2.0. This is accomplished by changing K, the stiffness of the linear spring, while keeping the wave period Tw fixed at 1.2 s. Since the system is nonlinear, it may behave differently in the downstream a n d upstream directions. Therefore, both positive and negative response amplitudes are plotted against the p/to LO 30 ~' 20 10 "~ 0 0
=:-10 -20 -30
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.L
L
2O
30
l.O
Time (sec)
Fig. 2. Subharmonic response of the analytical model with p/to = 2.0, ¢ = 0-0.
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1.8 2.2 ratio p/co
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3.0
Fig. 3. Frequency-response spectrum for the analytical model. The solid lines are for -- 0.0 and the dashed lines are for ¢ = 0-05. The solid dots are for ¢ = 0.0 without the large motion effect included.
ratio. Two pairs o f such curves are s h o w n in Fig. 3, one m a r k e d ~ = 0.0 a n d the o t h e r ~ = 0.05. T h e quantity ~ is the structural viscous d a m p i n g factor. It can be observed from Fig. 3 that nearp/to = 1.0 the curves show h a r m o n i c r e s o n a n c e responses similar to those o f any linear system; n e a r p/to = 2.0 (1.6-2.4), the system is in the s u b h a r m o n i c range a n d r e s p o n d s with a period which is twice that of the wave; the a m p l i t u d e s o f the s u b h a r m o n i c responses can be as large or larger t h a n those of the h a r m o n i c r e s o n a n c e responses. As in the case o f h a r m o n i c resonance, the structural viscous d a m p i n g has the effect o f r e d u c i n g s u b h a r m o n i c responses. T h e third pair o f curves is i n c l u d e d in Fig. 3 to e m p h a s i s e the effect o f large m o t i o n . Apparently, the s u b h a r m o n i c responses will not occur, if the structural m o t i o n is a s s u m e d to be small a n d the wave k i n e m a t i c s are evaluated at the u n d e f o r m e d , instead o f the deformed, position o f the structure. T h e m o d i f i e d Morison's e q u a t i o n which is used in the calculation o f the h y d r o d y n a m i c forces separates the loads into inertia a n d drag forces. Following the n o r m a l practice, the associated h y d r o d y n a m i c coefficients CAM, CrK a n d CD are a s s u m e d to be constants. However, for the p r o b l e m o f a structure oscillating in waves, there is no c o m m o n l y accepted way of d e t e r m i n i n g these coefficients, especially Co. C o m p a r i s o n s o f the results using different values Of CD are therefore m a d e to study the effects Of CD o n s u b h a r m o n i c responses. Figure 4 presents the f r e q u e n c y response curves in the range o f 1-4 < p/o9 < 3-0 for different values o f CD ranging from 0-5 to 1-2. It c a n be n o t e d that the s u b h a r m o n i c response is larger
Subharmonic oscillations 60
~
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l
l
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Fig. 4. Effect of C D o n subharmonic resonance. w h e n the values of C D a r e lower. T h e value o f Co appears to have relatively little effect on the u p p e r a n d lower limits of the frequency range for s u b h a r m o n i c responses. T h e s a m e observation also appears to h o l d for viscous structural d a m p i n g as s h o w n in Fig. 3. A n interesting b e h a v i o r discovered recently in n o n l i n e r d y n a m i c s is the initial-condition d e p e n d e n t characteristic of m a n y n o n l i n e a r systems. A typical case occurring in the m o d e l studied here is presented in Fig. 5. F o r values o f Co = 0.5 a n d p/to = 2-4, two different initial 6O 5O GO A ;I
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( deg )
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conditions, (00,00), were used to start the motion. For a starting condition of (0, 0), i.e. from rest, the motion of the tower converges to a h a r m o n i c response, while the initial condition of (20, 0) leads to a s u b h a r m o n i c motion. To have a clearer picture as to what initial condition leads to which response, a m a p depicting the domains of attraction for the analytical model has been generated for different values of(00, 00) and is presented in Fig. 6. In the map, the symbol '1' at a position, e.g. (0, 0), indicates that h a r m o n i c steady state motion will occur with the initial condition, (0, 0), while the symbol '2' is for the s u b h a r m o n i c response o f order 2. Although only sufficient points have been generated to m a k e the boundaries o f the two d o m a i n s distinct, it appears that the h a r m o n i c d o m a i n is small and is enclosed by the m u c h larger s u b h a r m o n i c domain. This leads to the observation that the probability of triggering a s u b h a r m o n i c oscillation is greater than that for a h a r m o n i c oscillation for this case.
5 EXPERIMENTAL INVESTIGATIONS The experimental investigation was conducted in a wave flume 36 m long, 2 m wide a n d 1.2 m deep. At one end of the flume is a wave generating system capable of generating regular waves. A schematic representation o f the experimental model is depicted in Fig. 7. The articulated tower model is m a d e from a hollow PVC cylinder 0-8 m long
289
Subharmonic oscillations
To pulleys T stainless steel wire
. ~ .
=
incoming wave
heavy b
acceleromeler
toad
hanger
vSWL
hollow PVC cylinder smooth bearings base of
wave flume Fig. 7. Schematic layout of experimental model.
with a diameter of 114 m m and a wall thickness of 7 mm. On the bottom end, it is waterproofed and connected to a heavy base plate through a stainless steel rod supported by two smooth bearings which allow only one degree of freedom, viz. rotation in the x-y plane. At the top of the cylinder, an accelerometer is mounted. The accelerometer (B&K 4370) measures the tangential acceleration at the top of the cylinder. The restoring force of the tower model is supplied, in addition to the buoyancy, by a stainless steel wire which is connected to the top of the cylinder; it then goes round a system of pulleys before ending with a load hanger. This combination of wire-pulley-hanger is used to simulate the effect of a linear stiffness. By varying the tension in the wire, the structural stiffness of the system can be adjusted, thus making it possible to alter the natural frequency of the system conveniently. The length of the wire measured from the top of the cylinder to the nearest pulley is made much longer than the height of the cylinder. This ensures that the stiffness remains linear even when the cylinder undergoes large motion, so that the only nonlinearity in the system arises from the wave forcestructure interaction. Each experimental simulation begins with the placing of a weight on the load hanger, thereby changing the natural frequency of the system. A regular wave train is then generated by the wave generator to interact with the model. When the response of the model has reached a steady state condition, the data collection is commenced and continued for a period of 20-48 s. To show a typical case of the subharmonic oscillations of the model, the time history and the frequency spectrum for the case whenp/to = 2.0
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0.30
8
0-25
0.10
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0-20 0-15 E
-6 -8
5
0
Time (sec)
0-5 1.0 1.5 frequency (Hzl
(o)
(b)
I0
15
20
20
Fig. 8. (a) Time history: and (b) frequency spectrum for Case 1, p/to -- 2.0. are depicted in Fig. 8. Figure 8a shows that the motion of the model w h e n it is undergoing steady s u b h a r m o n i c oscillations while Fig. 8b is obtained by performing the fast Fourier transform on the motion of the model. The frequency spectrum clearly shows two peaks, the larger one located at a frequency of 0.83 Hz and the smaller peak at 0-42 Hz. The larger peak corresponds to the m a i n h a r m o n i c c o m p o n e n t in the response of the structure while the smaller peak arises from the s u b h a r m o n i c c o m p o n e n t in the response. The peak at 0.42 Hz is not caused by a corresponding wave c o m p o n e n t , because the frequency spectrum of the m e a s u r e d wave profile shows only one distinct peak at 0-83 Hz a n d n o n e at 0.42 Hz. Small peaks at frequencies higher than 0.83 Hz nevertheless exist; the existence of high frequency components in the wave indicates that the regular waves generated are contaminated by some noise. The results obtained demonstrate that s u b h a r m o n i c responses of order 2, due to the interaction effect, can occur in an articulated tower with a linear restoring force, even if the excitation is not monochromatic. An interesting behavior was noticed while the model was having s u b h a r m o n i c responses. W h e n the model was held a n d m a i n t a i n e d in a stationary position for a while a n d then released at some arbitrary instant, it sometimes went back to the s u b h a r m o n i c motion mode, while at other times it responded in the h a r m o n i c mode. This confirmed the analytical prediction that the system can respond in either h a r m o n i c or subharrnonic mode, d e p e n d i n g on the initial conditions. The reported experimental results in the s u b h a r m o n i c frequency range included only
291
Subharmonic oscillations
those of s u b h a r m o n i c responses. D u r i n g the experiment, whenever a h a r m o n i c response was found to occur in the s u b h a r m o n i c range, the model was simply given an arbitrary disturbance which enabled it to revert to the s u b h a r m o n i c mode. This also agreed with the analytical observation for this model that subharmonics occur more readily than the h a r m o n i c responses. The frequency response curve for a particular wave condition can be obtained by adjusting the weight on the hanger. A total of four cases have been studied in the investigation. Cases 1 and 2 are identical in all respects with the wave period (Tw) 1.2 s, wave height (H) 10 cm and water depth 60 cm; the purpose of repeating the experiment for the same combination of wave parameters is to demonstrate the repeatability of the results. In the third case, the water depth was increased to 65 cm. This was to observe the sensitivity of the s u b h a r m o n i c characteristic to a small variation in the water depth. In the final case, the wave height was changed instead. The experimentally obtained frequency response curves for the two identical cases 1 and 2 are plotted in Fig. 9. These two curves, represented by the light and dark circles, are found to be very close to each other. This means that the experiment, w h e n repeated u n d e r the same wave conditions, will produce nearly identical results. In Fig. 9 a r o u n d the frequency ratio ofp/o9 ~- 2.0, the curves lying on the negative side of the Y-axis (i.e. responses in the upstream direction) show a distinctive h u m p occurring f r o m p / w -- 1.8 to 2-1. Within this frequency range, the experimental model has u n d e r g o n e s u b h a r m o n i c oscillations for the typical case presented in Fig. 8.
10.0
A ~ i ~
Tw = 1.2 s • CASF 1 o CASE 2
7.5 -~
5.0
2.5 ~0
0.0 -2.5
-5.0 -7.5 -10.0 0.5
i l 1.0 1.5 Frequency ratio
L 2.0 p/w
t 2.5
3.0
Fig. 9. Frequency responses of experimental model. The solid dots are for Case 1 while the open circles are for Case 2.
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7-5
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'c~
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Frequency ratio p/co Fig, 10. F r e q u e n c y response o f e x p e r i m e n t a l model, changing the water depth. The solid dots are for water depth 60 cm, while the open circles are for water depth 65 cm.
5.1 Effect of water depth T h e third case investigated the effect o f variation in the water d e p t h o n the s u b h a r m o n i c b e h a v i o u r of the e x p e r i m e n t a l model. T h e water d e p t h was increased from 60 to 65 c m a n d the e x p e r i m e n t was repeated. T h e results o b t a i n e d are plotted in Fig. 10 as f r e q u e n c y response curves. T h e two curves (60 c m a n d 65 c m water depth) a p p e a r to be close to each other. Usually, an increase in the water depth will lead to a c o r r e s p o n d i n g increase in the w a v e - i n d u c e d m o m e n t forces acting o n the model. However, a h i g h e r water d e p t h w o u l d also m e a n that the a d d e d mass o f the structure would increase which, in turn, w o u l d shift thep/t0 ratio to a h i g h e r value. T h e first effect increases the response o f the structure while the s e c o n d m a y decrease or increase it, d e p e n d i n g o n where the system is on the response curve. C o n s e q u e n t l y , w h e t h e r the response o f the structure increases or decreases d e p e n d s o n w h i c h effect is d o m i n a n t a n d o n the d y n a m i c characteristics o f the system. F o r this particular case, it w o u l d a p p e a r that these two trends m o r e or less cancel each other out a n d the m a g n i t u d e of the response r e m a i n s almost u n c h a n g e d .
5.2 Effect of wave height In the last case (Case 4), the wave height was r e d u c e d from 10 c m to 6 cm. T h e f r e q u e n c y response curves are given in Fig. 11. T h e acceleration values s h o w n have been n o r m a l i z e d with respect to HT~. As can be seen, the curves do not differ very m u c h from each o t h e r except below the
Subharmonic oscillations
:i o=
293
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• = height lOcm (expt) o = height 5 crn (expt}
50
25 0 -25 -50
'-
E
-75 I
-I00 0.5
1.0
I
1-5
I
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2.0
2-5
3.0
Frequency ratio p/co Fig. 11. Frequency response of experimental model, changing the wave h e i g h t The solid dots are for wave height 10 cm while the open circles are for wave height 6 cm.
frequency ratio of 1.0. Here the effect of reducing the wave height leads to an increase in the normalized response of the model. This result is probably due to the fact that the model has undergone large displacement in that region and hence the nonlinear effects are greater. These curves also exhibit the hump at p/to = 2-0 indicating the presence of subharmonic resonance. Interestingly, a reduction of the wave height does not significantly affect the subharmonic resonance behaviour of the model and the normalized subharmonic amplitudes remain approximately the same. This agrees with a previous conclusion based on a numerical s t u d y 9 that the subharmonic responses are not very sensitive to the wave height. An increasing wave height causes larger wave forces as well as larger damping effect, and these two effects tend to balance each other.
6 CONCLUSIONS A general formulation based on a previous work ~5 is used in the numerical simulations of an articulated structure subjected to the Morison-type wave forces. Subharmonic responses of order 2 due to the wave force-structure interaction have been demonstrated numerically and confirmed by the experimental study of a linearly constrained cylindrical tower. In a certain frequency range, the tower can respond either in a subharmonic or harmonic mode. The initial conditions of the motion apparently determine its final steady state response. Based on the prediction of the numerical results, subharmonic responses, rather than
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h a r m o n i c responses, are believed to be m o r e likely to occur in the model. T h e e x p e r i m e n t a l study also confirms this trend. T h e s u b h a r m o n i c m o t i o n s observed in this study are quite robust. Small variations in the wave height a n d the water depth as well as the high frequency noise c o n t a i n e d in the generated regular waves do not have a significant effect on the qualitative b e h a v i o u r of the s u b h a r m o n i c responses. F r o m the analytical a n d experimental investigations c o n d u c t e d in this study, it can be c o n c l u d e d that the s u b h a r m o n i c p h e n o m e n o n triggered by the interaction of wave force a n d structural m o t i o n is a realistic p r o b l e m w h i c h deserves attention from engineers engaged in the analysis of offshore structures.
ACKNOWLEDGEMENT This work was s u p p o r t e d by the National University o f Singapore u n d e r the Research Project RP93/85.
REFERENCES 1. Lean, G. H., Subharmonic motion of moored ships subjected to wave action, Trans. R. Inst.. Nav. Archit., 113, pp. 387-99. 2. Yoshida, K., Yoneya, T. & Oka, N., Response analysis of tension leg platforms, Comput. Methods Offshore Struct.. AMD, ASME, 37 (1980) pp. 77-85. 3. Chantrel, J. & Marol, P., Subharmonic response of articulated loading platform, J. Offshore Mech. Arct. Eng.. Trans. ASME, (1987) pp. 35-43. 4. Thompson, J. M. T., Bokaian, A. R. & Ghaffari, R., Subharmonic and chaotic motions of compliant offshore structures and articulated mooring towers, J. Energy Resour. TechnoL. Trans. ASME. 106 (1984) pp. 191-8. 5. Liaw, C. Y., Subharmonic response of offshore structures, J. Eng. Mech., ASCE, 113(3) (March 1987) pp. 366-77. 6. Liaw, C. Y., Bifurcations ofsubharmonic and chaotic motions of articulated towers, Eng. Struct., 10 (April 1988) pp. 117-24. 7. Jefferys, E. R., Nonlinear marine structures with random excitation, J. Offshore Mech. Arct. Eng., Trans. ASME, 110 (August 1988) pp. 246-53. 8. Liaw, C. Y., Chaotic and periodic responses of a coupled wave force and structure system, Comput. Struct., 30(4) (1988) pp. 985-93. 9. Liaw, C. Y., Numerical modeling and subharmonic bifurcations of a compliant cylinder exposed to waves, J. Offshore Mech. Arct. Eng., Trans. ASME, 111 (February 1989)pp. 29-36. 10. Nayfeh, A. H. & Mook, D. T.,NonlinearOscillations. Wiley, New York, (1979) pp. 161-257. 11. Thompson, J. M. T. & Stewart, H. B., Nonlinear Dynamics and Chaos. Wiley, Chichester, (1986) pp. 1-50.
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12. Moon, F. C., Chaotic Vibrations, Wiley, New York, 1987, pp. 1-23. 13. Hao, B.-L., Elementary Symbolic Dynamics and Chaos in Dissipative Systems, World Scientific, Singapore, (1989) pp. 253-279. 14. Chua. K. S.. Liaw. C. Y. & Nilakantan, J. S., Nonlinear dynamic motions of offshore structures, 7th Offshore South East Asia Conference, Singapore, 2-5 February 1988. Offshore South East Asia, Singapore, pp. 268-76. 15. Liaw. C. Y.. Shankar, N. J. & Chua. K. S.. Large motion analysis of compliant structures using Euler parameters. Ocean Eng, 16(5/6) (1989) pp. 545-57. 16. Beggs. J. S.. Kinematics, Springer-Verlag. New York. 1983. p. 30.
Subharmonic Motions and Wave Force-Structure Interaction
C. Y. L i a w , N. J o t h i S h a n k a r & K. S. C h u a Department of Civil Engineering, National University of Singapore, Singapore 0511 (Revised version accepted 19 August 1991)
ABSTRACT The problem of subharmonic oscillations attributable to the interaction between wave forces and the structural motion is investigated both analytically and experimentally. The sensitivity of this nonlinear phenomenon with respect to some system parameters and initial conditions is evaluated. The behaviour of an articulated tower model in subharmonic oscillations is studied, and qualitative comparisons between the experimental and numerical results are made. Key words: subharmonic oscillation, articulated tower, nonlinear
dynamics, wave force, wave force-structure interaction.
1 INTRODUCTION Offshore structures which are designed to experience significantly large motion are b e c o m i n g more c o m m o n as the search for new oil fields progresses into deeper ocean waters. In such conditions, these structures may undergo highly complex d y n a m i c motions in response to the environmental loads I-9. One important factor for such complex motion is the response-dependent loading of the structure. Obviously, in order to study the b e h a v i o u r of these excitation-response coupled systems, n o n l i n e a r d y n a m i c analysis models have to be used in the numerical simulations. 281 Marine Structures 0951-8339/92/$05.00 © 1992 Elsevier Science Publishers Ltd. England. Printed in Great Britain.
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It is a well known fact in n o n l i n e a r dynamics that m a n y n o n l i n e a r p h e n o m e n a , e.g. s u b h a r m o n i c oscillations and chaos, are very sensitive to small variations of the model used in the simulation. ~°-~2This means that, unless the precise system parameters and the complete mathematical representation of the physical problem are known, deterministic predictions of these nonlinear p h e n o m e n a with quantitative accuracy are almost impossible. However, recent advances in n o n l i n e a r dynamics ]]-]3 seem to indicate that some n o n l i n e a r behaviour is not only intrinsic but also universal. The exact quantitative response of a n o n l i n e a r system may not be predictable using the inevitably simplistic model a n d approximate parameters: its qualitative behaviour can nevertheless be studied and understood if the model can simulate the essence of the n o n l i n e a r behaviour. In this paper, it is intended to show, both experimentally and numerically, that one of the causes for the frequently encountered n o n l i n e a r behavior, namely s u b h a r m o n i c oscillations of m a n y offshore structures can be attributed to the wave force-structure interaction, and this b e h a v i o u r can be qualitatively modeled by the modified Morison's equation.
2 ANALYTICAL MODEL FOR LARGE MOTION OF OFFSHORE STRUCTURES In order to study the general three-dimensional motion, including large translation a n d rotation, of a structure, an analytical model based on the d y n a m i c s of large rigid-body motions has been developed using the Lagrangian formulation and Euler parameters ]4"15, i.e.
<,ro lT_ [ GT Lo4J
[o,-,]
=
+
Q
(1)
where T is the kinetic energy of the rigid body, A is the Lagrange multiplier, Q represents all the applied forces, q = < r, p > r and F~ is the constraint equation
prp
=
I
(2)
r is the translation displacement vector at the center of gravity of the rigid b o d y a n d p = < e0, el, e2, e3 > r; e~are the Euler parameters defined as 16 e0 = cos 0/2 el = l] sin 0/2
(3)
Subharmonic oscillations e2 =
12sin O/2
e3 =
13sin O/2
283
where 0 is the angle of rotation of the rigid body about the axis defined by the unit vector < l~, 12, 13 > r. The detailed derivation for the equation of motion is given by Liaw et al) 5, the final expression has the following form:
A
=
+
(4)
The matrix A depends on the mass, mass moment of inertia and the Euler parameters, p. The vector u is a function of the mass moment of inertia and p. The vector s is a function ofp and the positive vector of the constraint. The force vector Q includes the restoring force, damping force and wave loading. If the structure is assumed to be composed of small cylinders, the wave force per unit length of a cylinder can be written as D2 f L = CAMPrt T
D2 D (UL -- XL) q- CFKP J7 ~ IlL "1- CDP "~ (UL -- XL)[ UL -- XL [
(5) where UL and JIL are the water particle and the structural member velocities, respectively; the subscript L describing the vector terms in eqn (5) indicates that these are measured in the local member axes. The coefficients CAM and CFK are the added mass and the Froude-Krylov coefficients; p is the density of water; D is the diameter of the cylindrical member; and CD is the drag coefficient. The modified Morison's equation, eqn. (5), evaluated at the deformed position of the body, is used in the calculation of the hydrodynamic forces acting on the body. Since the wave force fL and consequently the force vector Q is a function of the response quantities, r and p, the governing equations of motion, eqn. (4), are nonlinear and can be solved numerically using the standard step-bystep time integration method. If the structural restoring force is assumed to be linear, then the only nonlinearity of the system described by eqn. (4) is due to the large motion of the system. Nonlinear systems can have many different types of responses which do not exist in linear systems and the one which is of interest in this paper is subharmonic oscillation.
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3 SUBHARMONIC OSCILLATIONS Subharmonic oscillations are the responses of a system having a period n times that of the period of the applied force, where n is an integer. Thus, when n is 2, the resulting oscillation is termed a subharmonic of order 2. Studies carried out on the subharmonic oscillations of offshore structures seem to indicate that there are two important causes of subharmonics: the nonlinear characteristics of the restoring force and the large motion of the structure (or the interaction effect). Chantrel and Marol 3 studied the harmonic pitch response of an articulated loading platform and found that the subharmonic responses obtained from the numerical simulations are due to the nonlinear stiffness characteristic of the hawser. Another work, by Thompson et al. 4, demonstrated the existence of subharmonic motions in articulated towers and attributed these motions to the high nonlinearity of the mooring line, which can become slack or taut during dynamic motions, thereby introducing a discontinuity in the restoring force. Essentially, the mooring line behaves as a bilinear spring. Their results also show the presence ofsubharmonic oscillations of order 2, 3 and higher, as well as chaotic motions. A previous study 5 by one of the authors looked at a simple single degree of freedom system with surge motion, acted upon by a linear harmonic wave. Using only a linear restoring stiffness, his results showed that the nonlinearity arising from the interaction effect alone can cause subharmonic surge oscillations. However, in that formulation the model was applicable only to the case of a single cylinder. With the threedimensional general formulation of the problem, described in Section 2 above, any structure with arbitrarily oriented cylindrical members subjected to the Morison type wave forces can now be modeled. In this paper, the numerical results obtained from the general formulation are used to reconfirm the earlier conclusion based on the previous formulation relating the subharmonics and the wave force - - structure interaction. Some of the observations are further verified in the laboratory.
4 N U M E R I C A L RESULTS An articulated tower model was chosen for both the numerical and experimental studies. Its motion is restricted to only one degree of freedom which is the rotation of the tower about its base. This singledegree-of-freedom model can, of course, also be modeled in a straightforward way involving only 0. However, as previously mentioned, one of
Subharmonic oscillations o.1 kg F-/'/t SWL ~
I
I
285
~ K ~ _
Fig. 1. Articulated tower model for subharmonic analysis, Co = 1.0, CAM= CFK = 1.0, wave length is 0.12 m, wave period is 1,2 s the objectives of this study is to verify the general formulation described in Section 2. The Euler parameters a p p r o a c h is therefore used for the numerical study. A simple diagram of the model is shown in Fig. 1. The numerical model exhibits a clear s u b h a r m o n i c response of order 2 as shown in Fig. 2. For this case, the frequency ratio, p/to, is 2.0, where p = 2n/Tw; the wave period Tw -- 1.2 s; and to is the natural frequency of the partially submerged tower at its u n d e f o r m e d position in still water. The time history of the tower response (rotation) presented in Fig. 2 has a period of 2-4 s, twice the wave period. Similar s u b h a r m o n i c responses of order 2 can also be obtained, if the p/to ratio is varied within a range a r o u n d 2.0. This is accomplished by changing K, the stiffness of the linear spring, while keeping the wave period Tw fixed at 1.2 s. Since the system is nonlinear, it may behave differently in the downstream a n d upstream directions. Therefore, both positive and negative response amplitudes are plotted against the p/to LO 30 ~' 20 10 "~ 0 0
=:-10 -20 -30
I0
.L
L
2O
30
l.O
Time (sec)
Fig. 2. Subharmonic response of the analytical model with p/to = 2.0, ¢ = 0-0.
C.Y. Liaw et al.
286 z,O ~
3O
~
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m
-~ -lo
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0.6
I
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I
I
1.8 2.2 ratio p/co
I
2.6
3.0
Fig. 3. Frequency-response spectrum for the analytical model. The solid lines are for -- 0.0 and the dashed lines are for ¢ = 0-05. The solid dots are for ¢ = 0.0 without the large motion effect included.
ratio. Two pairs o f such curves are s h o w n in Fig. 3, one m a r k e d ~ = 0.0 a n d the o t h e r ~ = 0.05. T h e quantity ~ is the structural viscous d a m p i n g factor. It can be observed from Fig. 3 that nearp/to = 1.0 the curves show h a r m o n i c r e s o n a n c e responses similar to those o f any linear system; n e a r p/to = 2.0 (1.6-2.4), the system is in the s u b h a r m o n i c range a n d r e s p o n d s with a period which is twice that of the wave; the a m p l i t u d e s o f the s u b h a r m o n i c responses can be as large or larger t h a n those of the h a r m o n i c r e s o n a n c e responses. As in the case o f h a r m o n i c resonance, the structural viscous d a m p i n g has the effect o f r e d u c i n g s u b h a r m o n i c responses. T h e third pair o f curves is i n c l u d e d in Fig. 3 to e m p h a s i s e the effect o f large m o t i o n . Apparently, the s u b h a r m o n i c responses will not occur, if the structural m o t i o n is a s s u m e d to be small a n d the wave k i n e m a t i c s are evaluated at the u n d e f o r m e d , instead o f the deformed, position o f the structure. T h e m o d i f i e d Morison's e q u a t i o n which is used in the calculation o f the h y d r o d y n a m i c forces separates the loads into inertia a n d drag forces. Following the n o r m a l practice, the associated h y d r o d y n a m i c coefficients CAM, CrK a n d CD are a s s u m e d to be constants. However, for the p r o b l e m o f a structure oscillating in waves, there is no c o m m o n l y accepted way of d e t e r m i n i n g these coefficients, especially Co. C o m p a r i s o n s o f the results using different values Of CD are therefore m a d e to study the effects Of CD o n s u b h a r m o n i c responses. Figure 4 presents the f r e q u e n c y response curves in the range o f 1-4 < p/o9 < 3-0 for different values o f CD ranging from 0-5 to 1-2. It c a n be n o t e d that the s u b h a r m o n i c response is larger
Subharmonic oscillations 60
~
50 Z,O 30 20 o=
.m
I0
287 CD 0.5 0.7 1.0 1.2
0
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I
I.£,
1.6
I-B
I
l
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l
l
2.6
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30
Fig. 4. Effect of C D o n subharmonic resonance. w h e n the values of C D a r e lower. T h e value o f Co appears to have relatively little effect on the u p p e r a n d lower limits of the frequency range for s u b h a r m o n i c responses. T h e s a m e observation also appears to h o l d for viscous structural d a m p i n g as s h o w n in Fig. 3. A n interesting b e h a v i o r discovered recently in n o n l i n e r d y n a m i c s is the initial-condition d e p e n d e n t characteristic of m a n y n o n l i n e a r systems. A typical case occurring in the m o d e l studied here is presented in Fig. 5. F o r values o f Co = 0.5 a n d p/to = 2-4, two different initial 6O 5O GO A ;I
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Fig. 5. Different initial conditions leading to different steady state responses.
C . Y . L i a w et al.
288
3.5 2 1 1
2
2.5
2
2 ~=
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1 1 2 1 1 1 2 2 2 1 1 2 221 1 12 2 1 12 2 I 1 I 2 2 1 112 2 2 1122
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I
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( deg )
Fig. 6. Domains of attraction for the analytical model w i t h CD = 0.5, p/to = 2.44, ' l ' = h a r m o n i c , '2" = subharmonic of order 2.
conditions, (00,00), were used to start the motion. For a starting condition of (0, 0), i.e. from rest, the motion of the tower converges to a h a r m o n i c response, while the initial condition of (20, 0) leads to a s u b h a r m o n i c motion. To have a clearer picture as to what initial condition leads to which response, a m a p depicting the domains of attraction for the analytical model has been generated for different values of(00, 00) and is presented in Fig. 6. In the map, the symbol '1' at a position, e.g. (0, 0), indicates that h a r m o n i c steady state motion will occur with the initial condition, (0, 0), while the symbol '2' is for the s u b h a r m o n i c response o f order 2. Although only sufficient points have been generated to m a k e the boundaries o f the two d o m a i n s distinct, it appears that the h a r m o n i c d o m a i n is small and is enclosed by the m u c h larger s u b h a r m o n i c domain. This leads to the observation that the probability of triggering a s u b h a r m o n i c oscillation is greater than that for a h a r m o n i c oscillation for this case.
5 EXPERIMENTAL INVESTIGATIONS The experimental investigation was conducted in a wave flume 36 m long, 2 m wide a n d 1.2 m deep. At one end of the flume is a wave generating system capable of generating regular waves. A schematic representation o f the experimental model is depicted in Fig. 7. The articulated tower model is m a d e from a hollow PVC cylinder 0-8 m long
289
Subharmonic oscillations
To pulleys T stainless steel wire
. ~ .
=
incoming wave
heavy b
acceleromeler
toad
hanger
vSWL
hollow PVC cylinder smooth bearings base of
wave flume Fig. 7. Schematic layout of experimental model.
with a diameter of 114 m m and a wall thickness of 7 mm. On the bottom end, it is waterproofed and connected to a heavy base plate through a stainless steel rod supported by two smooth bearings which allow only one degree of freedom, viz. rotation in the x-y plane. At the top of the cylinder, an accelerometer is mounted. The accelerometer (B&K 4370) measures the tangential acceleration at the top of the cylinder. The restoring force of the tower model is supplied, in addition to the buoyancy, by a stainless steel wire which is connected to the top of the cylinder; it then goes round a system of pulleys before ending with a load hanger. This combination of wire-pulley-hanger is used to simulate the effect of a linear stiffness. By varying the tension in the wire, the structural stiffness of the system can be adjusted, thus making it possible to alter the natural frequency of the system conveniently. The length of the wire measured from the top of the cylinder to the nearest pulley is made much longer than the height of the cylinder. This ensures that the stiffness remains linear even when the cylinder undergoes large motion, so that the only nonlinearity in the system arises from the wave forcestructure interaction. Each experimental simulation begins with the placing of a weight on the load hanger, thereby changing the natural frequency of the system. A regular wave train is then generated by the wave generator to interact with the model. When the response of the model has reached a steady state condition, the data collection is commenced and continued for a period of 20-48 s. To show a typical case of the subharmonic oscillations of the model, the time history and the frequency spectrum for the case whenp/to = 2.0
290
C.Y. Liaw et al. Io
0.30
8
0-25
0.10
II II I I II II II II
005
I I
0-20 0-15 E
-6 -8
5
0
Time (sec)
0-5 1.0 1.5 frequency (Hzl
(o)
(b)
I0
15
20
20
Fig. 8. (a) Time history: and (b) frequency spectrum for Case 1, p/to -- 2.0. are depicted in Fig. 8. Figure 8a shows that the motion of the model w h e n it is undergoing steady s u b h a r m o n i c oscillations while Fig. 8b is obtained by performing the fast Fourier transform on the motion of the model. The frequency spectrum clearly shows two peaks, the larger one located at a frequency of 0.83 Hz and the smaller peak at 0-42 Hz. The larger peak corresponds to the m a i n h a r m o n i c c o m p o n e n t in the response of the structure while the smaller peak arises from the s u b h a r m o n i c c o m p o n e n t in the response. The peak at 0.42 Hz is not caused by a corresponding wave c o m p o n e n t , because the frequency spectrum of the m e a s u r e d wave profile shows only one distinct peak at 0-83 Hz a n d n o n e at 0.42 Hz. Small peaks at frequencies higher than 0.83 Hz nevertheless exist; the existence of high frequency components in the wave indicates that the regular waves generated are contaminated by some noise. The results obtained demonstrate that s u b h a r m o n i c responses of order 2, due to the interaction effect, can occur in an articulated tower with a linear restoring force, even if the excitation is not monochromatic. An interesting behavior was noticed while the model was having s u b h a r m o n i c responses. W h e n the model was held a n d m a i n t a i n e d in a stationary position for a while a n d then released at some arbitrary instant, it sometimes went back to the s u b h a r m o n i c motion mode, while at other times it responded in the h a r m o n i c mode. This confirmed the analytical prediction that the system can respond in either h a r m o n i c or subharrnonic mode, d e p e n d i n g on the initial conditions. The reported experimental results in the s u b h a r m o n i c frequency range included only
291
Subharmonic oscillations
those of s u b h a r m o n i c responses. D u r i n g the experiment, whenever a h a r m o n i c response was found to occur in the s u b h a r m o n i c range, the model was simply given an arbitrary disturbance which enabled it to revert to the s u b h a r m o n i c mode. This also agreed with the analytical observation for this model that subharmonics occur more readily than the h a r m o n i c responses. The frequency response curve for a particular wave condition can be obtained by adjusting the weight on the hanger. A total of four cases have been studied in the investigation. Cases 1 and 2 are identical in all respects with the wave period (Tw) 1.2 s, wave height (H) 10 cm and water depth 60 cm; the purpose of repeating the experiment for the same combination of wave parameters is to demonstrate the repeatability of the results. In the third case, the water depth was increased to 65 cm. This was to observe the sensitivity of the s u b h a r m o n i c characteristic to a small variation in the water depth. In the final case, the wave height was changed instead. The experimentally obtained frequency response curves for the two identical cases 1 and 2 are plotted in Fig. 9. These two curves, represented by the light and dark circles, are found to be very close to each other. This means that the experiment, w h e n repeated u n d e r the same wave conditions, will produce nearly identical results. In Fig. 9 a r o u n d the frequency ratio ofp/o9 ~- 2.0, the curves lying on the negative side of the Y-axis (i.e. responses in the upstream direction) show a distinctive h u m p occurring f r o m p / w -- 1.8 to 2-1. Within this frequency range, the experimental model has u n d e r g o n e s u b h a r m o n i c oscillations for the typical case presented in Fig. 8.
10.0
A ~ i ~
Tw = 1.2 s • CASF 1 o CASE 2
7.5 -~
5.0
2.5 ~0
0.0 -2.5
-5.0 -7.5 -10.0 0.5
i l 1.0 1.5 Frequency ratio
L 2.0 p/w
t 2.5
3.0
Fig. 9. Frequency responses of experimental model. The solid dots are for Case 1 while the open circles are for Case 2.
292
C.Y. Liaw et al. 12-5 10.0 A
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Frequency ratio p/co Fig, 10. F r e q u e n c y response o f e x p e r i m e n t a l model, changing the water depth. The solid dots are for water depth 60 cm, while the open circles are for water depth 65 cm.
5.1 Effect of water depth T h e third case investigated the effect o f variation in the water d e p t h o n the s u b h a r m o n i c b e h a v i o u r of the e x p e r i m e n t a l model. T h e water d e p t h was increased from 60 to 65 c m a n d the e x p e r i m e n t was repeated. T h e results o b t a i n e d are plotted in Fig. 10 as f r e q u e n c y response curves. T h e two curves (60 c m a n d 65 c m water depth) a p p e a r to be close to each other. Usually, an increase in the water depth will lead to a c o r r e s p o n d i n g increase in the w a v e - i n d u c e d m o m e n t forces acting o n the model. However, a h i g h e r water d e p t h w o u l d also m e a n that the a d d e d mass o f the structure would increase which, in turn, w o u l d shift thep/t0 ratio to a h i g h e r value. T h e first effect increases the response o f the structure while the s e c o n d m a y decrease or increase it, d e p e n d i n g o n where the system is on the response curve. C o n s e q u e n t l y , w h e t h e r the response o f the structure increases or decreases d e p e n d s o n w h i c h effect is d o m i n a n t a n d o n the d y n a m i c characteristics o f the system. F o r this particular case, it w o u l d a p p e a r that these two trends m o r e or less cancel each other out a n d the m a g n i t u d e of the response r e m a i n s almost u n c h a n g e d .
5.2 Effect of wave height In the last case (Case 4), the wave height was r e d u c e d from 10 c m to 6 cm. T h e f r e q u e n c y response curves are given in Fig. 11. T h e acceleration values s h o w n have been n o r m a l i z e d with respect to HT~. As can be seen, the curves do not differ very m u c h from each o t h e r except below the
Subharmonic oscillations
:i o=
293
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• = height lOcm (expt) o = height 5 crn (expt}
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'-
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I
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3.0
Frequency ratio p/co Fig. 11. Frequency response of experimental model, changing the wave h e i g h t The solid dots are for wave height 10 cm while the open circles are for wave height 6 cm.
frequency ratio of 1.0. Here the effect of reducing the wave height leads to an increase in the normalized response of the model. This result is probably due to the fact that the model has undergone large displacement in that region and hence the nonlinear effects are greater. These curves also exhibit the hump at p/to = 2-0 indicating the presence of subharmonic resonance. Interestingly, a reduction of the wave height does not significantly affect the subharmonic resonance behaviour of the model and the normalized subharmonic amplitudes remain approximately the same. This agrees with a previous conclusion based on a numerical s t u d y 9 that the subharmonic responses are not very sensitive to the wave height. An increasing wave height causes larger wave forces as well as larger damping effect, and these two effects tend to balance each other.
6 CONCLUSIONS A general formulation based on a previous work ~5 is used in the numerical simulations of an articulated structure subjected to the Morison-type wave forces. Subharmonic responses of order 2 due to the wave force-structure interaction have been demonstrated numerically and confirmed by the experimental study of a linearly constrained cylindrical tower. In a certain frequency range, the tower can respond either in a subharmonic or harmonic mode. The initial conditions of the motion apparently determine its final steady state response. Based on the prediction of the numerical results, subharmonic responses, rather than
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h a r m o n i c responses, are believed to be m o r e likely to occur in the model. T h e e x p e r i m e n t a l study also confirms this trend. T h e s u b h a r m o n i c m o t i o n s observed in this study are quite robust. Small variations in the wave height a n d the water depth as well as the high frequency noise c o n t a i n e d in the generated regular waves do not have a significant effect on the qualitative b e h a v i o u r of the s u b h a r m o n i c responses. F r o m the analytical a n d experimental investigations c o n d u c t e d in this study, it can be c o n c l u d e d that the s u b h a r m o n i c p h e n o m e n o n triggered by the interaction of wave force a n d structural m o t i o n is a realistic p r o b l e m w h i c h deserves attention from engineers engaged in the analysis of offshore structures.
ACKNOWLEDGEMENT This work was s u p p o r t e d by the National University o f Singapore u n d e r the Research Project RP93/85.
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