Seon Han

Probability Models in Engineering and Science

INTRODUCTION Applications Units Organization of the Text Problems EVENTS AND PROBABILITY Sets Probability Concluding Summary Problems RANDOM VARIABLE MODELS Probability Distribution Function Probability Density Function Mathematical Expectation Variance USEFUL PROBABILITY DENSITIES Two Random Variables Concluding Summary Problems FUNCTIONS OF RANDOM VARIABLES Exact Functions of One Variable Functions of Two or More RVs General Case Approximate Analysis Monte Carlo Method Concluding Summary Problems The Standard Normal Table RANDOM PROCESSES Basic Random Process Descriptors Ensemble Averaging Stationarity Derivatives of Stationary Processes Fourier Series and Fourier Transforms Harmonic Processes Power Spectra Fourier Representation of a Random Process Borgmans Method of Frequency Discretization Concluding Summary Problems SINGLE DEGREE OF FREEDOM DYNAMICS Motivating Examples Deterministic SDoF Vibration SDoF: The Response to Random Loads Response to Two Random Loads Concluding Summary Problems MULTI DEGREE OF FREEDOM VIBRATION Deterministic Vibration Response to Random Loads Periodic Structures Inverse Vibration Random Eigenvalues Concluding Summary Problems CONTINUOUS SYSTEM VIBRATION Deterministic Continuous Systems Sturm-Liouville Eigenvalue Problem Deterministic Vibration Random Vibration of Continuous System Beams with Complex Loading Concluding Summary Problems RELIABILITY Introduction First Excursion Failure Fatigue Life Prediction Concluding Summary Problems NONLINEAR DYNAMIC MODELS Examples of Nonlinear Vibration Fundamental Nonlinear Equations Statistical Equivalent Linearization Perturbation Methods The van der Pol Equation Markov Process Based Models Concluding Summary Problems NONSTATIONARY MODELS Some Applications Envelope Function Model Nonstationary Generalizations Priestleys Model SDoF Oscillator Response Multi DoF Oscillator Response Nonstationary and Nonlinear Oscillator Concluding Summary Problems THE MONTE CARLO METHOD Introduction Random Number Generation Joint Random Numbers Error Estimates Applications Concluding Summary Problems FLUID INDUCED VIBRATION Ocean Currents and Waves Fluid Forces - In General Examples Available Numerical Codes Index

Three-Dimensional Vibration

The purpose of this chapter is to explore the three-dimensional vibration of an offshore platform. First, the equations of motion of the three-dimensional model are derived for two cases: a rigid structure and an elastic structure. Second, the free responses of the rigid and the elastic models are compared. The free response obtained using the rigid model allows us to gain confidence in the formulation and the numerical results obtained using the elastic model. Finally, the forced responses due to deterministic loads are investigated. Two cases are studied in particular. The first case is when a harmonic (in time) load is applied in one direction, and the second case is when steady and harmonic loads are applied in mutually perpendicular directions. The second case can be thought of as the three-dimensional loading arising from a current that also induces vortex shedding. The responses due to these simple fluid force models will prepare us for the study of responses due to more complicated fluid loading models. This subject has also been treated by Han and Benaroya [26].

Nonlinear Free Vibration of a Cable Against a Straight Obstacle

The purpose of this study is to investigate the nonlinear effect of gravity on the free vibration of a cable against a straight obstacle. The cable model is expressed in terms of nonlinearly coupled transverse and axial displacements. The penalty method is used to simulate the obstacle, which is equivalent to inserting a stiff elastic foundation. The first symmetric frequencies are obtained when the depth of the obstacle is 1/2 and 1/3 of the initial transverse displacement. The effects of varying amplitude and equilibrium curvature are investigated.Copyright

Bending Fatigue Analysis of Sea Floor Observatory Moorings

The purpose of this paper is to provide design criteria for seafloor observatory moorings where the principle mooring line element is an electro-mechanical (EM) cable, and the primary cause for failure is the bending fatigue at terminations. A single point mooring system that consists of a discus surface buoy, an S-tether, a subsurface buoy, and an anchor is considered for this analysis. The waves are assumed random and modeled using the Pierson-Moskowitz spectrum. Raoof’s contact stress-slip parameter at the connection between the surface buoy and the EM cable is used to evaluate a particular design and to provide general guidelines to arrive at optimal design.Copyright

Principle of Virtual Work, Lagrange’s Equation and Hamilton’s Principle

This chapter presents several of the most important concepts from analytical dynamics. The most important of these concepts to us is Lagrange’s equation and how it can be used for the derivation of governing equations of motion. The equation is especially useful for the derivation of the equations of motion for systems, discrete or continuous, with more than one degree of freedom, where the Newtonian free body diagrams become more difficult to apply. We will also derive Hamilton’s principle, an integral energy formulation, also applicable to both discrete and continuous systems, and see how it is connected to Lagrange’s equation.

Environmental Loading-Waves and Currents

This chapter is devoted to the formulation of the distributed transverse fluid load f (X, t) in an ocean environment. An offshore structure in an ocean environment is subjected to loadings due to wind, current, and waves. The Morison equation is used to model the in-plane fluid force. Random waves are modeled using the Airy linear wave theory and the Pierson-Moskowitz spectrum. A sample time history is generated using Borgman’s method. The out-of-plane fluid force due to vortex shedding is modeled as a simple sinusoid.

Overview of Transverse Beam Models

The purpose of this chapter is to provide an overview of the existing linear transverse beam models that can be used to model an offshore structure. The existing beam theories are the Euler-Bernoulli, Rayleigh, shear and Timoshenko. First, a review of the four beam theories are presented. Second, the underlying assumptions used in the beam models are delineated. Third, the equation of motion for each model and the expressions for boundary conditions are obtained using Hamilton’s variational principle. Fourth, the frequency equations are obtained for four sets of end conditions: free-free, clamped-clamped, hinged-hinged, and clamped-free. The roots of the frequency equations are presented in terms of normalized wave numbers. The normalized wave numbers for the other six sets of end conditions are obtained using the analysis of symmetric and antisymmetric modes from the wave numbers of the previous four sets of end conditions. For engineering purposes, the normalized wave numbers for each set of end conditions are tabulated or plotted as a function of geometric and physical parameters. Fifth, the orthogonality conditions of the eigenfunctions and the procedure to obtain the forced response using the method of eigenfunction expansion are presented. Finally, a numerical example is shown for a non-slender beam to illustrate the differences among the four beam models.

Coupled Axial and Transverse Vibration in Two Dimensions

For a vibrating beam, bending and rigid body motion are the primary components of the overall behavior. Therefore, it may be sufficient to use a single degree of freedom model or a linear transverse model such as the Euler-Bernoulli, Rayleigh, shear, or Timoshenko model. A discussion of linear transverse models can be found in Chapter 3. The linear transverse models assume that the coupling between the transverse and the axial motion is negligible. However, the coupling becomes more significant with increasing slenderness ratio (the ratio of length to the radius of gyration of the cross-sectional area). Therefore, the nonlinear coupling effect in long slender members of compliant towers may be important in the overall response. It has also been observed that there are significant high frequency nonlinear effects that appear to be caused by extensional or longitudinal vibration in some cases in an ocean environment. Therefore, it is necessary to investigate the couple transverse and longitudinal response of a beam under realistic environmental forces.