CHAOTIC AND RANDOM RESPONSES OF OCEAN STRUCTURES:
ANALYSIS OF MEDIUM-SCALE EXPERIMENTS
Solomon C.S. Yim
Ocean Engineering Program
Department of Civil, Construction and Environmental Engineering
Apperson Hall 202
Oregon State University
Corvallis, OR 97331-2302
phone: (541) 737-6894 fax: (541) 737-3052 email:
Award Nos.: N00014-92-J1221 & N00014-97-1-0581
www.orst.edu/ccee/~yims/onr
LONG-TERM GOAL
The long-term goal of the research is to develop a unified, systematic, reliability-based analysis and design methodology for dynamically sensitive nonlinear ocean structural systems incorporating the influence of a wide range of possible motion responses. The transition phenomena among various complex nonlinear motions including periodic (primary, sub- and super-harmonic resonance), quasi-periodic, chaotic, noisy periodic, noisy chaotic and purely random responses, and their effects on extreme excursions and fatigue behavior of structural systems will be emphasized.
OBJECTIVES
· Develop analytical models and corresponding (deterministic and stochastic) analysis techniques and prediction theories for selected nonlinear ocean structural systems.
· Conduct a medium-scale experiment to provide measured data to calibrate the predictive capability of selected analytical models.
· Perform comparisons of analytical model predictions with experimental results.
· Develop deterministic and stochastic extreme and fatigue response prediction techniques.
· Examine predictive results and explore possible improvements and/or alternatives on modeling, analysis and prediction techniques of both input waves and dynamic responses.
The first two long-term objectives have been accomplished in previous phases. The immediate objectives of the current phase are to: (1) analyze the measured data of the medium-scale nonlinear moored structural system experiment; (2) calibrate the predictive capabilities of the analytical models developed using the experimental results; and (3) study the transition phenomena among various attraction domains via parameter maps constructed from experimental and analytical results.
APPROACH
The approach is to first focus on modeling of the nonlinear system via the analysis of measured stochastic responses of a single-degree-of-freedom (SDOF) system. Using nonlinear system identification techniques being developed in this project, suitable dynamic models and ranges of system parameters will be identified. Taking advantage of the sensitive nature of the nonlinear response behavior, individual SDOF deterministic test results will be examined to pinpoint the system parameters. Experimental data from frequency scan runs will be analyzed to construct frequency-amplitude primary and secondary resonance response curves. The resulting nonlinear map will be calibrated against predictions from analytical models with system parameters identified from the experiment. Numerical results will be generated as necessary to complete the frequency-amplitude parametric map to fully delineate the nonlinear system response behavior. Corresponding analyses will be conducted on the associated multi-degree-of-freedom (MDOF) system to determine the coupling effects among various motion responses. Numerical predictions from stochastic techniques being developed for time and probability domain analyses will be calibrated with experimental and simulation results.
WORK COMPLETED
A nonlinear Reverse Multiple-Input Single-Output (R-MISO) system identification technique has been developed and applied to identify the parameters of several alternative SDOF and MDOF analytical models of the moored structural system using experimental data from the OSU Wave Lab medium-scale experiment [1]. With the identified system parameters, numerical simulations and parametric studies of the deterministic and stochastic responses of selected analytical models have been conducted and compared to the measured data [2]. Transition phenomena between various attraction domains of selected models of the nonlinear system have been examined. Semi-analytical techniques to predict the probability of intra-domain and inter-domain transitions have been developed [3]. SDOF and MDOF analytical models of nonlinear barge motions have been developed and analytical predictions are being compared to experimental data from the Naval Facilities Engineering Service Center (NFESC). A control technique that uses the sensitivity of the nonlinear response to an advantage has been developed and applied to the moored system and freestanding offshore equipment [4, 5].
RESULTS
Three alternative formulations of the hydrodynamic damping effects on the SDOF system (nonlinear-structure linear-damping (NSLD), nonlinear-structure coupled hydrodynamically-damped (NSCHD), and nonlinear-structure nonlinearly-damped (NSND)) were developed for the moored system [1]. The R-MISO technique was used to identify the system parameters. Numerical simulations of the dynamic responses to the measured wave excitations using the analytical models with corresponding identified parameters were compared to experimental data [6]. It is found that analytical predictions based on the NSND formulation best assimilate the experimental results (Figure 1). Using the NSND formulation, hydrodynamic properties including added mass, inertia and drag coefficients for the SDOF and MDOF systems have been identified.
Possible existence of noisy chaos has been observed in the experiment. Preliminary analysis indicates that the response may be a combination of multiple coexisting steady states. Special identification techniques are being developed to further classify and quantify the degree of nonlinearity of each attraction domain, including possible chaos. The mechanisms inducing transitions among co-existing attraction domains (Figure2) have been investigated from a stochastic perspective. Semi-analytical probability models have been derived to describe the inter- and intra-attraction domain transitions of response trajectories of the nonlinear system under narrow-band excitations. Focus of the analysis has been on response regions near the primary and secondary resonance. The theoretical basis of these semi-analytical procedures has been examined. Resulting predictions are found to agree well with the statistics of the corresponding time domain simulations. Comparisons of analytical predictions and experimental results are being conducted using the parameters identified for individual test cases of the nonlinear SDOF and MDOF moored systems.
The nonlinear system identification techniques and the stochastic attraction-domain transition analysis developed in this study are being applied to examine the behavior of moored barges. Experimental data from NFESC are employed to guide the development of analytical models. Preliminary analytical models identified so far appear to be able to predict the nonlinear response behaviors accurately.
A control procedure that uses the chaotic response of the nonlinear system to an advantage has been developed. By describing the nonlinear response with unstable periodic orbits, a locally linear map of the dynamics is obtained. The linear map is subsequently employed in a controller design and the controller is applied to the moored system. Robustness of the controller is investigated under noisy conditions and modifications to the controller are made in order to maintain control of the moored system [4]. The methods presented are equally applicable to most chaotic systems for which the response time history can be monitored. As an example, the technique has been successfully applied the control of rocking response of freestanding offshore equipment [5].
IMPACT/APPLICATIONS
The nonlinear response superstructure (being constructed in this study), together with the timed nature of the attraction-domain transitions, if found to be an invariant feature of the complex responses, would give hope that accurate extreme value and cumulative fatigue predictions of these responses may be possible. Completion of the analysis of the SDOF and MDOF experimental data may provide sufficient information for the development of a unified analytical procedure to predict the long-term extreme-value distributions and fatigue failure probabilities for the design of nonlinear, dynamically sensitive ocean structural systems.
TRANSITIONS
Several techniques developed in this research project have been found useful for other Naval structural system analyses. Specifically, the stochastic analysis techniques in the probability and time domains have been applied by contractors of NFESC to analyze the nonlinear stability and capsizing probability of transport barges.
RELATED PROJECTS
This research project complements those supported by other ONR programs on the study of physical systems including nonlinear ocean waves and structures. There are significant cross fertilization of ideas and development/implementation of numerical techniques on nonlinear stochastic and chaos analyses between this project and those under hydrodynamics, mathematical sciences, physics and other programs. This research may eventually benefit higher category programs when the resulting unified analysis methodology can be employed in the analysis of rogue waves and the design of Naval ships, barges, platforms and other special structures.
REFERENCES
1. S.C.S. Yim, M.A. Myrum, O. Gottlieb, H. Lin, and I-M. Shih, “Summary and Preliminary Analysis of Nonlinear Oscillations in a Submerged Mooring System Experiment,” Report No. OE-93-03, Oregon State University, Ocean Engineering Program, May 1993, 229pp.
2. H. Lin, S.C.S. Yim and O. Gottlieb, “Experimental Investigation of Response Stability and Transition Behavior of a Nonlinear Ocean Structural System,” International Journal of Ocean Engineering, Vol.25 4/5, 1998, pp.323-343.
2. O. Gottlieb and S.C.S. Yim, “Nonlinear Dynamics of a Coupled Surge-Heave Small-Body Ocean Mooring Systems,” International Journal of Ocean Engineering, Vol.24, No.5, 1997, pp.479-495.
3. I-M. Shih and S.C.S. Yim, “Stochastic Analysis of Complex Nonlinear System Response Under Narrow-Band Excitations,” Ocean Engineering Report, Oregon State University, June 1998.
4. P.E. King and S.C.S. Yim, "Primary Resonance Response of a Controlled Ocean System," Proceedings of the Nineth International Offshore and Polar Engineering Conference, Brest, France, 30 May – 4 Jun3, 1999, Vol.III, pp.596-600.
5. P.E. King and S.C.S. Yim, "Active Control of Noisy Nonlinear Oscillations in a Structural
System," Proceedings of the Second World Conference on Structural Control, Kyoto, Japan, 28
June ‑ 1 July 1998, Vol.3, pp.1931-1938.
6. S. Narayanan, S.C.S. Yim and P.A. Palo, "Nonlinear System Identification of a Moored
Structural Systems," Proceedings of the Eighth International Offshore and Polar Engineering
Conference, Montreal, Canada, 24‑29 May 1998, Vol.III, pp.478‑484.
Fig.1 Comparison of experimental data (o) and analytical predictions (+)
Fig.2a Different response behaviors under same excitation
Fig.2b Attraction domains (small amplitude harmonic, 1/3 subharmonic, 1/2 subharmonic and large harmonic response)
Fig.2c Response amplitude curves Fig.2d Inter-domain transitions
PUBLICATIONS
S.C.S. Yim and H. Lin, "A Methodology for Analysis and Design of Sensitive Nonlinear Ocean Systems," Chapter 4, Stochastically Excited Nonlinear Ocean Structures, edited by M. Shlesinger and T.F. Swean, World Scientific Publisher, 1998, pp.105‑128.
H. Lin, S.C.S. Yim and O. Gottlieb, "Experimental Investigation of Response Stability and
Transition Behavior of a Nonlinear Ocean Structural System," International Journal of Ocean
Engineering, Vol.25, No.4‑5, 1998, pp.323‑343.
H. Lin and S.C.S. Yim, "Unified Analysis of Complex Nonlinear Motions Via Densities," International Journal of Nonlinear Dynamics, in press.
S.C.S. Yim, and H. Lin, "Noisy Nonlinear Motions of a Moored System, II: an Experimental Study," Journal of Waterway, Port, Coastal and Ocean Engineering, ASCE, in press.
J.W. Leonard, K. Idris, and S.C.S. Yim, "Large Angular Motions of Tethered Surface
Buoys," International Journal of Ocean Engineering, in press.
P.E. King, and S.C.S. Yim, "Control of Nonlinear Rocking Responses of Free-Standing Rigid Blocks," Journal of Engineering Mechanics, ASCE, submitted.
P.E. King and S.C.S. Yim, "Active Control of Noisy Nonlinear Oscillations in a Structural
System," Proceedings of the Second World Conference on Structural Control, Kyoto, Japan, 28
June ‑ 1 July, 1998, Vol.3, pp.1931-1938 (appear in 1999).
H. Lin and S.C.S. Yim, "Stability of Nonlinear Rocking Responses to Noisy Excitations," Proceedings of Fourth International Conference on Stochastic Structural Dynamics, Notre Dame, IN, 6‑8 Aug. 1998, pp.323-328 (appear in 1999).
H. Lin and S.C.S. Yim, "Reliability Analysis of Rocking Responses to Random Excitations," Proceedings of Fourth International Conference on Stochastic Structural Dynamics, Notre Dame, IN, 6‑8 Aug. 1998, pp.355-360 (appear in 1999).
P.E. King and S.C.S. Yim, "Primary Resonance Response of a Controlled Ocean System," Proceedings of the Nineth International Offshore and Polar Engineering Conference, Brest, France, 30 May – 4 Jun3, 1999, Vol.III, pp.596-600.
P.E. King and S.C.S. Yim, "Periodic Dynamic Responses of a Controlled Ocean System," the Fifth International Experimental Chaos Conference, Orlando, Florida, (to appear).