Stochastically Excited Nonlinear Ocean Structures

The following sections are included:

• PREFACE

• CONTENTS

• LIST OF CONTRIBUTORS

The following sections are included:

• INTRODUCTION

• BIFURCATION IN MOMENTS
  • Dynamic Moment Equations
  • Moment Lyapunov Exponents
  • On-Off Intermittency

• NOISE-INDUCED TRANSITION
  • Bifurcation in Distribution
  • Noise-Induced Transition in One-Dimensional Systems
  • Noise-Induced Transition in One-Dimensional Mapping
  • Ocean Nonlinear Systems

• STABILIZATION OF UNSTABLE SYSTEMS BY PARAMETRIC NOISE
  • Mathematical Development
  • Stabilization of Nonlinear One-Dimensional Systems
  • Stabilization of the Inverted Pendulum

• CONCLUSIONS

• Acknowledgment

• References

In this paper we examine the question of chaos control and its application to engineering systems. In particular, we review a control method designed for systems that are embedded in time varying environments, where the environmental variations may be of relatively large amplitude, and may have a fairly irregular nature. We demonstrate the effectiveness of the method by applying it to (1) prevent chaos-induced capsizing of a ship, and to (2) eliminate uncertainties inherent in nonlinear oscillators, associated with sudden changes in attractor structures.

The following sections are included:

• Introduction

• The Volterra Theory of Nonlinear Systems

• Non-Gaussian Simulation Techniques
  • Spectral Correction
  • Multi-variate Non-Gaussian Simulation Using Spectral Correction
  • Conditional Non-Gaussian Simulation Using Spectral Correction
  • Estimation of the Probability Density Function
    • Maximum Entropy Method (MEM) of PDF Estimation
      • MEM Applied with System Equations
      • Example of MEM Using System Equations
      • Restriction of MEM in the Extreme Regions
      • Alternative MEM Constraints
    • Moment-based Hermite Transformation PDF Estimation
      • Limitations in the Standard Hermite Model
      • Example Application of Modified Hermite PDF Estimation Method

• Ringing

• Nonlinear Dynamical Systems & Chaos

• Nonlinear Random Wave Fields

• Conclusions

• Acknowledgement

• References

A methodology for systematic investigation and incorporation of highly nonlinear sensitive behaviors into reliability analysis and design of ocean structural systems using modern geometric and stochastic techniques is summarized in this paper. The representative systems examined are characterized by a nonlinear restoring force and a coupled fluid-structure interaction exciting force. Regular wave excitations are first examine to demonstrate the complex nature of nonlinear responses. A deterministic analysis procedure which includes the method of harmonic balance to solve for approximate solutions, a variational method for their stability analysis, and a Melnikov approach (global stability) to identify the existence of complex nonlinear (and possibly chaotic) motions is demonstrated. Locations of nonlinear primary and secondary resonances are identified in the parameter space. Highly nonlinear phenomena are identified and routes of response transition to chaos are demonstrated. The presence of random excitation components is next taken into account to assimilate nearly periodic wave excitations. A stochastic analysis procedure which includes a stochastic Melnikov approach and the Fokker-Planck formulation is then presented. A criterion based on the stochastic Melnikov process demonstrating the noise effects on chaotic response is developed. The Fokker-Planck equation is derived and solved for response probability density functions to interpret the response behavior from an ensemble perspective. The systematic stochastic analysis procedure is also applicable for response analysis to purely random excitations. An experimental study assimilating nonlinear behavior of an ocean system is briefly summarized and its results are compared against both deterministic and stochastic analytical predictions. Experimental results from deterministic models verify the existence of the primary and secondary resonances as analytically predicted. Nonlinear responses, e.g., harmonic, subharmonic and ultrasubharmonic, are also observed. Experimental results from stochastic model demonstrate noise-induced transitions between distinct response modes, e.g., harmonic and subharmonic. Engineering applications, e.g., system reliability, are demonstrated using probability density functions as a measure. The systematic analysis and design methodology provides a means to integrate existing seemingly independent deterministic and stochastic design procedures.

Three types of failures, the first-passage failure, the fatigue failure, and motion instability, are investigated theoretically for stochastically excited systems, of which the crucial response may be modeled as a one-dimensional Markov diffusive process. For engineering system of a higher dimension, a reduction in system dimensionality is necessary, which is possible using the method of stochastic averaging, or quasi-conservative averaging, or a modified version of quasiconservative averaging. The theories developed are then applied to several example for illustration.

Extreme values and high level excursions by stochastic processes have important bearing on structural reliability where failure can result from excessively high values of some crucial quantity such as stress, or from repeated smaller variations leading to fatigue-like failure. Such one dimensional (time variation) theory has been developed over many years from the early days of Extreme Value Theory for maxima and the early work of pioneers such as S.O. Rice in “level-crossing” problems.

More recently there has been an increasing interest (in engineering and environmental contexts) in modeling not only in time but also in spatial dimensions. This necessitates consideration of randomly varying functions of space or space/time, i.e. of random fields in place of the more traditional stochastic processes in one dimension.

One thrust of the work done in structural reliability in this research effort has been the development of extremal theory and related theory of excursions above high levels for random fields with emphasis on aspects having structural reliability implications. In this account we survey the relevant one-dimensional results, emphasizing aspects of concern for random field theory prior to a more detailed discussion of recently developed results for random fields and their extremal and excursion theory which bear on structural reliability.

For a class of deterministic systems the necessary condition for chaos is that the system's Melnikov function have simple zeros. The behavior of those systems' stochastic counterparts is similarly characterized by their Melnikov processes. We summarize: (1) recent research by the authors on the use of Melnikov processes to obtain weak upper bounds for rates of escape from a potential wall induced by a variety of types of noise, including dichotomous noise, and (2) applications of Melnikov processes in oceanography, the investigation of column snap-through induced by stochastic loads, nonlinear control, stochastic resonance, and the behavior of auditory nerve fibers.

Potential wells in Newtonian dynamical systems define homo/heteroclinic orbits composed of superposed stable and unstable invariant manifolds. The standard Melnikov criterion for these manifolds separated by weak external forcing requires both complete knowledge of the system potential and of the forcing. A new criterion—yielding a necessary condition for escape from a potential well—is given in terms of just the forcing amplitude and the ratio of two geometrical summary statistics of the potential well: the well width and the area of the safe region in state space contained by the orbit(s). The new criterion, though weaker in general than the standard Melnikov criterion, is nonetheless sharp. This establishes, in particular, that the standard Melnikov criterion is sharp. Using the new criterion, a broadly applicable upper bound is developed for the probability of escaping the potential well within a given time.

The following sections are included:

• Introduction

• Periodic Processes with Nonstationary Random Phase Modulation

• Stochastical Models for a Periodic Processes with Random Disturbances in Both Amplitude and Phase

• Random Fourier Series for Disordered General Periodic Processes

• Application 1: Reliability of Ocean Structures to a Spectrum-Consistent Excitation

• Application 2: Random Response of a Nonlinear Duffing Oscillator to Disordered Periodic Excitation

• Remarks

• Acknowledgment

• References

The following sections are included:

• Introduction

• Characteristic functions

• Wind-Wave spectra

• Extrema statistics

• References

This review paper describes a system identification technique called “Reverse MI/SO” that is capable of identifying an optimum system representation for a wide class of single-degree-of-freedom (SDOF) nonlinear systems using measured excitation and response time histories. Reverse MI/SO is straighforward to implement and outputs a series of one-dimensional frequency response functions that can be readily converted into a nonlinear integro-differential equation of motion. There are no restrictions regarding the system amplitude nonlinearities, or the distribution and spectral content of the excitation. Separate coherence functions are provided for each linear and nonlinear system path. Use of this technique is illustrated for analytical nonlinear systems with and without memory. Numerous studies have proven that the Reverse MI/SO technique is quite robust and practical, and that it produces correct results where other commonly used methods give erroneous results. A discussion is included on the importance of this new technique for nonlinear stability studies.

Tension Leg Platforms are compliant offshore structures particularly well suited for deepwater operation. Unlike fixed structures, its cost does not dramatically increase with water depth. The derivation for a coupled elastic tendon - rigid hull model is presented. The Lagrangian is fully developed and assumptions are analyzed. The method for determining the governing equations of motion and boundary conditions are detailed. Stochastic wave forces are applied to the Tension Leg Platform and are calculated based on Borgman's method. An example is presented for a model with a 470 m long tendon and a 50 m long hull. The analytical model developed in this chapter would be useful to the designers of Tension Leg Platforms in determining the overall dynamic response characteristics.

In previous work, a method for extracting information from the Fokker-Planck equation without having to solve the equation itself was explored. This method, called the van Kampen expansion, entails expanding the Fokker-Planck equation about the intensity of the forcing function. This results in a series of first order linear differential equations in time for the moments of the response of the system. These can be solved by any standard technique to give the moments of the response. This method was shown to give very good results for Duffing oscillators excited by white or colored (exponentially correlated) noise and for linked Duffing and linear oscillators excited by colored noise. The main drawback of this method has been the amount of work required to obtain the differential equations for the moments. For the simple case of two oscillators, one linear and one Duffing, excited by colored noise, the expansion ran to several pages. Application of the expansion to a larger system of 10 or more oscillators would not be practical. In this work, a rule set is developed whereby the differential equations can be written directly from the equations of motion. This reduces the amount of work required to perform the expansion to a manageable amount, even for larger systems. However, this rule base was developed with an eye toward implementation in either a symbolic processor such as Maple^™ or a mathematical spreadsheet such as MathCad^™.

The following sections are included:

• Index