Coupled Surge-Heave Motions of a Moored System. II: Stochastic Analysis and Simulations Solomon C. S. Yim, M.ASCE 1 ; and Huan Lin, A.M.ASCE Abstract: Analyses and simulations of the coupled surge-and-heave motions of a nonlinear, moored, experimental, submerged structure subjected to random waves are presented here. The random wave excitations examined include periodic waves with additive noise and narrow-band random waves. Characteristic experimental results include noisy subharmonic and superharmonic responses, and transition behaviors among multiple coexisting responses. This investigation applies a systematic, stochastic analysis procedure to further the deterministic study presented in Part I. Good agreement between the analytical predictions and experimental results is shown. Effects of random perturbations in waves on nonlinear response phenomena are examined, especially for the cases of multiple responses coexisting with chaos. It is found that chaotic responses are sensitive and of weak strength compared to other coexisting responses, and the system response trajectories mainly stay in the stronger, periodic attracting domains. Numerical results indicate perturbation-induced response transitions leading to very large-amplitude response beyond the experimental model limitations. DOI: XXXX CE Database subject headings: Stochastic models; Nonlinear systems; Coupling; Experimental data; Mooring; Damping; Simulation. Introduction Stochastic responses of moored structures have been frequently observed in the ocean fields because field environments including wind, waves, and current often contain significant random components e.g., Sarpkaya and Isaacson 1981; Naess and Johnsen 1993. The moored structures are usually highly nonlinear in restoring forces and fluid structure interactions, and their responses are often coupled. The presence of random wave components adds another degree of complexity in excitation and structural response, and necessitates a stochastic analysis of the randomly perturbed responses. However, due to the high degree of nonlinearity of the system involved, early studies had been conducted on simplified, single-degree-of-freedom SDOF models of only its predominant motion e.g., surge to keep analyses manageable. Wave excitations were often modeled by a simple sinusoidal function e.g., Bernitsas and Chung 1990; Falzarano et al. 199. Analysis results of the deterministic, simplified model could shed some light in the complicated response characteristics of the physical structural system. Studies of SDOF models subjected to deterministic, periodic excitations had shown the existence of underlying organized 1 Professor, Dept. of Civil, Construction, Environmental Engineering, Oregon State Univ., Corvallis, OR 97331 corresponding author. E-mail: sdomin.yim@oregonstate.edu Research Associate, Dept. of Civil, Construction, Environmental Engineering, Oregon State Univ., Corvallis, OR 97331. Note. Associate Editor: Joel P. Conte. Discussion open until November 1, 006. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on October 4, 004; approved on July 14, 005. This paper is part of the Journal of Engineering Mechanics, Vol. 13, No. 6, June 1, 006. ASCE, ISSN 0733-9399/006/6-1 XXXX/$5.00. transitions embedded in response bifurcation domains. These phenomena had been identified and facilitated a numerical search of possible routes to quasiperiodic and chaotic responses e.g., Gottlieb et al. 1997; Lin and Yim 005; Yim and Lin 005. Under the same system parameters with different initial conditions, coexisting and competing nonlinear ocean structural responses, had been found near resonances where local bifurcations occurred. Other studies on SDOF models incorporated additive random components in the wave models, which had been carried out to take into account the random nature of wave excitations e.g., Lin and Yim 1995. The presence of random perturbations led to transition and interaction behaviors among coexisting response attractors. Others examined nonlinear responses subjected to spectrum-specified random waves e.g., Hsieh et al. 1994. A preliminary deterministic study on surge-heave coupled model development and response-characteristic predictions was carried out by Gottlieb and Yim 1997. Highly nonlinear, complex responses were frequently observed in the simulations. Comparisons with field data and/or experimental results were to be conducted to validate analysis and numerical predictions. Incorporating an independent-flow-field hydrodynamic formulation, an improved model governing coupled surge and heave motions is employed and validated in Part I of this two-part study. The investigation focuses on examining various response characteristics of coupled motions as well as sensitivity studies on some seemingly critical parameters. Analytical predictions are in good agreement with existing experimental results. Complex nonlinear responses, including chaos, are numerically observed under strong coupling and light damping. This study represents a stochastic extension furthering the deterministic study in Part I. Here the coupled surge-and-heave experimental responses subjected to random wave excitations are examined. Excitations considered include periodic waves with perturbations and narrow-band random waves. A stochastic JOURNAL OF ENGINEERING MECHANICS ASCE / JUNE 006 / 1
extension of two-degree-of-freedom DOF, independent-flowfield IFF, Morison model governing coupled surge-and-heave motions is employed for numerical predictions and simulations. The DOF model consists of an IFF, Morison-type hydrodynamic drag, a three-term polynomial approximation of the nonlinear restoring force, and a nonlinear surge-heave coupling term. Randomly perturbed wave excitations are modeled as a dominant sinusoidal component with band-limited random perturbations e.g., Lin and Yim 005. The additive random perturbations are approximated by linearly filtered white noise. An optimal set of constant system and hydrodynamic parameters has been identified in Part I c.f., Table 1 in Part I based on a frequency domain identification technique e.g., Narayanan and Yim 004; Yim and Narayanan 004. This set of parameters is employed here for all examinations and comparisons. Results comparisons and interpretations are conducted for all tests in the time, frequency, and probability domains. Observed transition behaviors are verified and interpreted based on time domain simulations and response probability density functions PDFs solved from the associated Fokker Planck equation FPE. The DOF FPE is derived and extended from its SDOF counterpart Lin and Yim 004. Based on numerical results examined, causes of experimentally observed very large-amplitude random responses beyond model limitations are identified, and the possibility of observing chaotic responses in the field is also assessed. System Considered Multipoint moored, submerged structural systems subjected to random wave excitations are experimentally modeled as a submerged sphere in a wave channel. The sphere is moored by elastic mooring cables with geometric nonlinearity. Random waves considered in this study, including both periodic waves with random perturbations and narrow-band waves, are generated with specified wave spectra by a flip-board wave generator. Experimental model setup and configuration were reported by Yim et al. 1993. A brief description of the experiment is summarized in Part I. As discussed in Part I, because of the direction of incident waves and symmetry of experimental configuration, the sphere s movement was predominantly in a two-dimensional surge and heave fashion with negligible roll motions. Four measurements Channels 11 and 14 16 of the sphere s movements were decomposed into surge-and-heave components for analyses and comparisons. Measurements of the wave gage at 0.46 m upstream of the sphere are chosen to estimate incident wave conditions. Note that indices 1 and 3 represent the components in surge and heave, respectively. C S1 and C S3 denote the effective linear system damping coefficients = S1,3 C CR1,3 ; S1,3 =damping ratio and C CR1,3 =critical damping ; C D1 and C D3 =hydrodynamic damping coefficients; C 13 and C 31 =coupling coefficients of surge and heave; and F D1,3 and F I1,3 =drag and inertial components of the exciting force, respectively cf. Eqs. 3 and 4 in Part I. Taking into account the stochastic nature in the wave excitations observed and considered in the experiment, a wave model of combined periodic wave component with filtered white noise perturbations is employed. Eq. 1 can be then rewritten as ẋ s1 = x s ẋ s = R 1 x s1 C s1 x s C d1 x s x s + C d1 u 1 u 1 + C 13 x s1 x h1 + A 1 cos t + 1 + 1 t ẋ h1 = x h ẋ h = R 1 x h1 C s3 x h C d3 x h x h + C d3 u 3 u 3 + C 31 x h1 x s1 + A 3 sin t + 3 + 3 t where C d1,3 hydrodynamic viscous drag coefficients; C s1,3 structural damping coefficients; and u 1 and u 3 =water particle velocities in surge and heave, respectively. Wave amplitude, and phase are represented by A 1,3,, and 1,3, respectively. Polynomial approximations of restoring forces R 1 x s1 and R 3 x h1 are given as R 1 x s1 k 11 x s1 +k 1 x 3 s1 +k 13 x s1 and R 3 x h1 k 31 x h1 +k 3 x h1 +k 33 x 3 h1. Symbols 1,3 t, with variances 1,3 approximating the stochastic component in the waves are narrow-band processes, which can be obtained from a linearly filtered white noise. The linear filter is given by 1,3 t + n1,3 1,3 t + f 1,3 1,3 t = 1,3 t 3 where n1,3 denote spectral bandwidth parameters; f 1,3 =peak frequencies; and 1,3 =delta-correlated white noises of unity intensities 1 and 3, respectively. The wave excitation model describes the randomly perturbed periodic waves, and may also approximate narrow-band random waves with very narrow bandwidths. Response characteristics subjected to narrow-band random waves with slight randomnesscaused variations and/or transitions e.g., Falzarano et al. 199 may closely follow their deterministic counterparts, which are subjected to monochromatic waves of spectral peak frequency. Equations of Motion Employing independent-flow-field Morison damping, the equations of the surge-and-heave coupled motions of the cable-moored system subjected to stochastic wave excitations are derived as Mẍ s + C S1 ẋ s + C D1 ẋ s ẋ s + R 1 x s + C 13 x s x h = F D1 u 1 + F I1 u 1 1 Mẍ h + C S3 ẋ h + C D3 ẋ h ẋ h + R 3 x h + C 31 x h x s = F D3 u 3 + F I3 u 3 where M =mass of the structure; and x s, ẋ s, x h, and ẋ h denote the surge displacement and velocity, and heave displacement and velocity, respectively. R 1 and R 3 =nonlinear restoring forces. Fokker Planck Equation A set of stochastic differential equations as shown in Eq. can be converted to a statistically equivalent deterministic PDE, called the Fokker Planck FPE equation. A systematic FPE associated with a SDOF moored sphere model based on standard Morison formulation subjected to random wave excitations was first introduced by Lin and Yim 1997. The FPE of a SDOF IFF model was later derived Lin and Yim 004. The Fokker Planck formulation associated with the DOF moored sphere model Eq. can be straightforwardly expanded by incorporating additional state variables. The Fokker Planck equation and its path-integral solver are derived here to solve for response PDFs for later analysis and behavior predictions. / JOURNAL OF ENGINEERING MECHANICS ASCE / JUNE 006
Table 1. Model Tests Subjected to Periodic Waves with Random Noise Test case H p m T p s m Response type E09 0.183. 0.0010 Noisy subharmonic E11 0.146. 0.0007 Noisy subharmonic E1 0.305. 0.0014 Noisy subharmonic E14 0.35 6.67 0.004 Noisy superharmonic E15 0.333 6.67 0.0054 Noisy superharmonic Fokker Planck Equation and Path Integral Solution The Fokker Planck equation for the DOF system is given by P X,t = x P X,t R 1 x 1 C s1 x C t x 1 x d1 x x where + C d1 u 1 u 1 + C 13 x 1 x 5 + A 1 cos t + 1 + x 3 P X,t x 4P X,t + n1x 4 f 0 x 3 P X,t x 3 x 4 + 1 P X,t x 6 P X,t x 4 x 5 x 6 R 3 x 5 C s3 x 6 C d3 x 6 x 6 + C d3 u 3 u 3 + C 31 x 5 x 1 + A 3 sin t + 3 + x 7 P X,t x 8P X,t x 7 + n3x 8 f 0 x 7 P X,t + 3 P X,t x 8 x 8 4a x 1 = x s,x = ẋ s,x 3 = 1,x 4 = 1,x 5 = x h,x 6 = ẋ h,x 7 = 3,x 8 = 3 P X,t = P x 1,x,x 3,x 4,x 5,x 6,x 7,x 8,t and the path-integral solution is given by P X,dt N 1 dt 0 i=0 N Ndt t t 0 = lim N 1 4b... exp dt G X j+1,x j,t j ;dt P X 0,t 0 dx i j=0 4c where G X j+1,x j,t j,dt =short-time propagator. Response PDF P X,t can be obtained by applying Eq. 4c iteratively. Numerical Implementation Using a multidimensional histogram representation for the PDF, the path sum Eq. 4c can be evaluated numerically. The probability domain at time t is discretized into a finite number of elements. The short time propagator G X j+1,x j,t j,dt is also discretized into a short-time transition tensor T kl dt. The PDF of N state variables at time t+dt can be evaluated by summing all the probability mass propagated from time t and normalized afterward e.g., Lin and Yim 1995 Table. Model Tests Subjected to TWA Narrow-Band Random Waves Test case H max m T p s & a Response type E10 0.75. 0.05 & 5 Out of bound a and are governing bandwidth parameters of TWA narrow-band spectrum. Fig. 1. Experimental noisy subharmonic response with wave period of. s and noise perturbation of 0.0014 test E1 : a wave profile, b surge, and c heave and P k t + dt = T kl dt P l t T kl dt = x k1 i 1 + x k1 i... x kn j 1 + x kn j x k1 i + x k1 i / x k1 i x k1 i / x l1 i + x l1 i / x l1 i x l1 i / N x kn i + x kn i / dx kn dx k1... x kn i x kn i / x kn i + x ln i / 5a dx l1 dx ln P l,dt X k X l... x kn i x ln i / 5b Response PDFs of the desired state can be obtained by numerically iterating and updating Eqs. 5a and 5b. Note that the computational efforts increase exponentially with the number of state variables eight state variables, x 1...x 8 or N=8 in this study. Compared to the PDF simulations of a SDOF model with two state variables, the computational efforts for the DOF model are M 3 times, where M =number of grid elements for each JOURNAL OF ENGINEERING MECHANICS ASCE / JUNE 006 / 3
Fig. 3. Experimental response to narrow band random waves with peak frequency at 0.45 Hz test E10 : a wave profile and b surge response Fig.. Experimental noisy superharmonic response with wave period of 6.67 s and noise perturbation of 0.0056 test E15 : a wave profile, b surge, and c heave paired state-variable domain e.g., x 1 -x. When we performed the numerical work, all the DOF, PDF computations are conducted on an Ultra Sparc 80 workstation with dual processors of 450 MHz each. The typical run time for ten cycles of the dominant period is, on the average, over 400 h in real time under a single user mode. With faster machines available to date, the expected run time is still expected to be on the order of 100 h. Experimental Results and Comparisons Observations A total 6 DOF experimental tests subjected to various random wave excitations were performed, including five tests subjected to periodic waves with random perturbations E09, E11-1, and E14-15, and one test E10 subjected to narrow-band random waves Yim et al. 1993. For the tests of randomly perturbed periodic wave excitations, wave periods were chosen near the subharmonic. s and superharmonic 6.67 s resonances to examine in detail the effects of random perturbations on these two nonlinear responses. The variances of random perturbations varied from 10 to 40% of the energy of the dominant periodic component. For the only test of narrow-band random wave excitation, the dominant wave period is near the subharmonic domain i.e.,. s to examine transition behaviors induced by the variation in the excitation amplitude and frequency. A brief summary of these tests can be found in Tables 1 and. Wave profile, and typical noisy subharmonic surge and heave responses Test E1 are shown in Figs. 1 a, b, and c. Noiseinduced variations in the wave amplitude are noted Fig. 1 a, and notable subharmonic components in response are observed between 430 and 500 s as shown in Figs. 1 b and c. Figs. a, b, and c, respectively, show the wave profile, and typical noisy superharmonic surge and heave responses Test E15. Noiseinduced, high frequency disturbance is noted in the wave profile Fig. a, and both surge and heave responses show smoother superharmonic components Figs. b and c. Note that the surge motion is of large amplitude in comparison with the heave, therefore the most dominant DOF. Similar response characteristics in the surge-and-heave motions indicate a high degree of correlation. The wave profile of a narrow-band random excitation and the time history of response Test E10 are shown in Figs. 3 a and b, respectively. It is noted that the sphere response behaves in a narrow-band random fashion for about the first 900 s, and then transitions to a relatively large amplitude response. The test was terminated at around 1,000 s because the response amplitude exceeds the experimental model limitations. Possible causes of the transitions will be examined in a later section. 4 / JOURNAL OF ENGINEERING MECHANICS ASCE / JUNE 006
Fig. 4. Comparison of simulated and experimental noisy subharmonic response with wave period of. s and noise perturbation of 0.0014 test E1 : a surge and b heave; experimental results -, and simulations -- with C A1,3 =0.5, C D1,3 =0.0, k 11,31 =135.78 N/m, k 1,3 =191.60 N/m, k 13,33 =68.40 N/m 3, C D1,3 =0.0, S1,3 =1%, and C 13 =0.311 N/m 3, and C 31 =0.83 N/m 3 Fig. 5. Comparison of simulated and experimental noisy superharmonic response with wave period of 6.67 s and noise perturbation of 0.0056 test E15 : a surge and b heave; experimental results -, and simulations -- with C A1,3 =0.5, C D1,3 =0.0, k 11,31 =183 N/m, k 1,3 =191.60 N/m, k 13,33 =68.40 N/m 3, C D1,3 =0.0, S1,3 =1%, and C 13 =0.311 N/m 3, and C 31 =0.83 N/m 3 Comparisons in Frequency Domain Employing the optimal set of constant coefficients identified from the DOF, deterministic model cf. Table 1 in Part I, the simulations and experimental results are compared in the frequency domain. Comparisons have been performed on all test results, and overall good agreement is observed. Due to paper length limitation, only selected representative test results are presented here for demonstration purposes. Response characteristics of each test, and discrepancies between predictions and experimental results, are also discussed below. Comparisons of noisy subharmonic simulations and experimental results Test E1 in both surge and heave displacements are shown in Figs. 4 a and b. The wave height varies between around 0.47 and 0.55 m, and the wave frequency is 0.45 Hz. It is shown that the simulations of surge and heave are in good agreement with experimental results. However, it is noted that the harmonic response component smaller amplitude component is not perfectly in line with the experimental results. Comparisons of a noisy superharmonic simulation and experimental results Test E15 in both surge and heave displacements are shown in Figs. 5 a and b. The wave height varies between around 0.6 and 0.4 m with the wave frequency at 0.15 Hz. It is observed that the spectra of the simulations of surge and heave are in good agreement with experimental results. Discrepancies between predictions and experimental results in the details of the response characteristics are observed. Small adjustments in the system parameters for better agreement are also noted, and higher frequency dependency in hydrodynamic parameters in the lower frequency range is hence indicated. Numerical Predictions Frequency Response Diagram The frequency response diagram presented in Figs. 6 a and b in Part I, consisting of simulations subjected to deterministic wave conditions, is employed here to interpret stochastic, experimental, transitional response characteristics. The frequency response diagrams obtained by numerical simulations indicate an intricate nature of system response in both surge and heave displacements. Both experimental and numerical results indicate that the DOF model is lightly damped and strongly coupled in surge and heave motions. As shown in the diagrams in Part I, a secondary, superharmonic resonance tilting to the left is indicated by a hump in the low frequency range near 0.15 Hz. In the frequency range, superharmonic response is observed experimentally and verified numerically. Another hump is shown near 0.35 Hz, which JOURNAL OF ENGINEERING MECHANICS ASCE / JUNE 006 / 5
Fig. 6. Simulated coexisting nonlinear responses on Poincaré section, including harmonics,, and *, subharmonics o, and chaos : a surge and b heave; wave amplitude of 0.61 m and wave frequency at 0.71 Hz indicates multiple coexistence of steady-state responses. Coexisting harmonic and subharmonic responses are found near the wave frequency at 0.35 Hz. The primary resonance tilting to the right is found to be located near 0.5 Hz. In the frequency range of 0.5 0.8 Hz, multiple steady-state responses, including chaos are predicted to coexist. The coexisting surge and heave responses near a frequency of 0.71 Hz are shown in the Poincaré map in Figs. 6 a and b, respectively. Five various, distinct characteristic responses, including subharmonics, chaos, small, medium, and large amplitude harmonics are found. The highly nonlinear chaotic response shows random-like characteristics in its time history and a fractal nature in its attractor captured on the Poincaré map as shown in Figs. 6 a and b. Another secondary, subharmonic resonance tilting to the right is found to be located near 0.5 Hz. Multiple responses are also found to coexist near the subharmonic frequency range. Probability Domain The contour map and three-dimensional 3D presentation of the multiple coexisting surge response PDFs of wave frequency near 0.71 Hz, including subharmonics, chaos, small, medium, and large amplitude harmonics are shown in Figs. 7 a and b, respectively. The PDFs reflect the noise-bridged, coexisting subharmonic, small, medium, and large-amplitude harmonic responses. The presence of noise perturbations bridges the coexisting response attractors, and the concentrations in the PDF reflect the Fig. 7. Surge response PDF of coexisting response attractors on Poincaré section, including harmonics near 0,1, 0.7, 0., and 0.7, 3.0, subharmonics near 0.3, 0., and chaos not captured : a contour map and b 3D presentation; wave amplitude of 0.61 m and wave frequency at 0.71 Hz relative competing strength of each attractor Lin and Yim 1995. Note that the chaotic attractor is not captured in the PDF, which indicates that the chaotic response is of the weakest competing strength probability mass relatively negligible among the coexisting attractors. Based on the numerical results in Part I, the deterministic chaotic response exists when the damping is very weak 1% of the critical damping. It is also noted that the discretizations employed in computing response PDF using the path integral solution Eqs. 5 a and 5 b may impose additional damping in the numerical model. The additional damping may further decrease the strength of the attractor and/or hinder its occurrence. Therefore, in spite of the fact that chaotic responses are numerically predicted to exist in the deterministic DOF moored structural system, they may not be observed in the large-scale experiments or fields because of their sensitivity and weak competing strength. The transition behaviors of the multiple coexisting competing attractors are further examined in the time domain in the following subsection. Noise-Induced Transitions Transitions among coexisting response characteristics near a wave frequency of 0.71 Hz cf. Figs. 6 a and b caused by the 6 / JOURNAL OF ENGINEERING MECHANICS ASCE / JUNE 006
Fig. 8. Noise-induced transitions in coexisting responses demonstrated by simulations: a deterministic chaos 1,3 =0.0, b noisy chaos 1,3 =0.03, c noisy subharmonics 1,3 =0.04, d noisy subharmonics 1,3 =0.05, e noisy harmonics 1,3 =0.06, and f noisy harmonics =0.07 1,3 presence of random perturbations with various intensities are examined here. With initial conditions set near the chaotic domain, Figs. 8 a f show the structural response transitions from deterministic chaos 1,3 =0.0, to noisy chaos 1,3 =0.03, to noisy subharmonics 1,3 =0.04 and 0.05, respectively, and then to noisy harmonics 1,3 =0.06 and 0.07, respectively. With the presence of band-limited random perturbations, the response is transitioning from the chaotic domain, to the subharmonic and then remains in the small-amplitude harmonic. It is indicated that chaotic responses may be observed when the incident waves are strictly controlled deterministically near the identified chaotic domain. With small fluctuations in wave profiles, noisy harmonic responses are most likely observed under the wave frequencies considered in this study. JOURNAL OF ENGINEERING MECHANICS ASCE / JUNE 006 / 7
Concluding Remarks Analyses and simulations of the results of medium-scale, DOF, nonlinear, stochastic, moored ocean structural responses are presented here. Wave excitations considered here include periodic waves with random perturbations and narrow-band random waves. Some concluding remarks are accordingly drawn as follows: 1. Predictions of the proposed stochastic DOF, IFF model are in line with the experimental results in both frequency and probability domains. Validity of the model is then demonstrated.. Numerical results indicate the coexistence of high-order nonlinear responses, including chaos. The strength of the chaotic attractor is found the weakest compared with the other coexisting/competing attractors. 3. With the presence of random perturbations, transition behavior in a sampled case of coexisting response characteristics from noisy chaos, to noisy subharmonics, and then to noisy harmonic are numerically identified and demonstrated. Under the model configuration and wave conditions considered, the noisy harmonic response is most likely observed in the largescale experiments, and possibly in the field. 4. Numerical results indicate that variations in narrow-band wave amplitude may lead responses transitioning from a small amplitude domain to a coexisting large amplitude response. The numerical result may explain the observation of an experimental response reaching a large amplitude motion exceeding the experimental configuration limitations. Fig. 9. Nonlinear relationship between response amplitudes and wave amplitudes with wave frequency at 0.45 Hz: a surge and b heave motions Acknowledgment Financial support from the United States Office of Naval Research Grant Nos. N00014-9-11 and N00014-04-10008 is gratefully acknowledged. Large Amplitude Response in Narrow Band Test As shown in Fig. 3, in the test subjected to narrow-band random wave excitations, the response remains in a narrow-band steady state for about 900 s, and then transitions to a relatively large-amplitude response exceeding experimental configuration limitations. The observation can be interpreted based on the wave-and-response amplitude diagram shown in Fig. 9. The diagram is comprised of deterministic simulations of responses subjected to wave excitations with fixed frequency 0.45 Hz and varied amplitude from 0.01 to 0.75 m. Twenty five initial conditions are employed at each and every specified wave excitation condition. It is shown that the multiple response amplitudes exist when the wave amplitude is within 0.05 0.35 and 0.43 0.75 m. The test subjected to narrow-band random waves in Fig. 3 shows that the wave amplitude fluctuates between 0.075 and 0.5 m. When the wave amplitude reaches near 0.5 m, there is the possibility, depending on initial conditions, that the sphere reaches the large amplitude response existing when the wave amplitude is within 0.43 0.75 m. Therefore, when the narrowband wave amplitude reaches near 0.5 m, the experimental response as shown in Fig. 3 tends to transition into the largeamplitude attractor exceeding the model configuration limitations as observed. References Bernitsas, M. M., and Chung, J. S. 1990. Nonlinear stability and simulations of two-line ship: Towing and mooring. Appl. Ocean. Res., 11, 153 166. Falzarano, J., Shaw, S. W., and Troesch, A. W. 199. Application of global methods for analyzing dynamical systems to ship rolling motion and capsizing. Int. J. Bifurcation Chaos Appl. Sci. Eng.,, 101 115. Gottlieb, O., and Yim, S. C. S. 1997. Nonlinear dynamics of a coupled surge-heave small-body ocean mooring system. Ocean Eng., 4, 479 495. Gottlieb, O., Yim, S. C. S., and Lin, H. 1997. Analysis of bifurcated superstructure of nonlinear ocean system. J. Eng. Mech., 13 11, 1180 1187. Hsieh, S. R., Troesch, A. W., and Shaw, S. W. 1994. Nonlinear probabilistic method for predicting vessel capsizing in random beam seas. Proc. R. Soc. London, Ser. A, 445, 195 11. Lin, H., and Yim, S. C. S. 1995. Chaotic roll motion and capsizing of ships under periodic excitation with random noise. Appl. Ocean. Res., 17, 185 04. Lin, H., and Yim, S. C. S. 1997. Noisy nonlinear motions of moored system. I: Analysis and simulations. J. Waterw., Port, Coastal, Ocean Eng., 13 5, 87 95. Lin, H., and Yim, S. C. S. 004. Stochastic analysis of a single-degreeof-freedom nonlinear experimental moored system using an independent-flow-field model. J. Eng. Mech., 130, 161 169. 8 / JOURNAL OF ENGINEERING MECHANICS ASCE / JUNE 006
Lin, H., and Yim, S. C. S. 005. An IFF model for a SDOF nonlinear structural system. Part I: Modeling and comparisons. J. Offshore Mech. Arct. Eng., in press. Naess, A., and Johnsen, J. M. 1993. Response statistics of nonlinear, compliant offshore structures by the path integral solution method. Probab. Eng. Mech., 8, 91 106. Narayanan, S., and Yim, S. C. S. 004. Modeling and identification of a nonlinear SDOF moored structure. Part 1. Analytical models and identification algorithms. J. Offshore Mech. Arct. Eng., 16, 175 18. Sarpkaya, T., and Isaacson, M. 1981. Mechanics of wave forces on offshore structures, Van Nostrand Reinhold, New York. Yim, S. C. S., and Lin, H. 005. An IFF model for a SDOF nonlinear structural system. Part II: Analysis of complex responses. J. Offshore Mech. Arct. Eng., in press. Yim, S. C. S., Myrum, M. A., Gottlieb, O., Lin, H., and Shih, I.-M. 1993. Summary and preliminary analysis of nonlinear oscillations in a submerged mooring system experiment. Ocean Engineering Rep. No. OE-93-03, Oregon State Univ., Corvallis, Ore. Yim, S. C. S., and Narayanan, S. 004. Modeling and identification of a nonlinear SDOF moored structure. Part : Results and sensitivity study. J. Offshore Mech. Arct. Eng., 16, 183 190. JOURNAL OF ENGINEERING MECHANICS ASCE / JUNE 006 / 9