Haym Benaroya

Random eigenvalues, algebraic methods and structural dynamic models

A review of the literature of the study of structural vibration problems with random parameters is provided. There have been several approaches to this problem. These have been encapsulated in this paper for the benefit of those who need to assess such possibilities. In particular, the algebraic theory of random variables is delineated here with the application in mind being the determination or estimation of the statistics of the eigenvalues of linear dynamical systems. Various transformation techniques are summarized and discussed with simple examples.

Dynamics of Periodic and Near-Periodic Structures

A brief review of linear waves and dynamic behavior of both periodic and near-periodic structures is presented in this paper. Emphasis is placed on a summary of more recent studies of engineering structures subject to wave motion and the effects of near-periodicity on their dynamic behavior. Conclusions drawn regarding linear waves and dynamics serve as a foundation for further investigation into the area of nonlinear waves and dynamics of these structures.

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

Modelling vortex-induced fluid-structure interaction

The principal goal of this research is developing physics-based, reduced-order, analytical models of nonlinear fluid–structure interactions associated with offshore structures. Our primary focus is to generalize the Hamiltons variational framework so that systems of flow-oscillator equations can be derived from first principles. This is an extension of earlier work that led to a single energy equation describing the fluid–structure interaction. It is demonstrated here that flow-oscillator models are a subclass of the general, physical-based framework. A flow-oscillator model is a reduced-order mechanical model, generally comprising two mechanical oscillators, one modelling the structural oscillation and the other a nonlinear oscillator representing the fluid behaviour coupled to the structural motion. Reduced-order analytical model development continues to be carried out using a Hamiltons principle-based variational approach. This provides flexibility in the long run for generalizing the modelling paradigm to complex, three-dimensional problems with multiple degrees of freedom, although such extension is very difficult. As both experimental and analytical capabilities advance, the critical research path to developing and implementing fluid–structure interaction models entails formulating generalized equations of motion, as a superset of the flow-oscillator models; and developing experimentally derived, semi-analytical functions to describe key terms in the governing equations of motion. The developed variational approach yields a system of governing equations. This will allow modelling of multiple d.f. systems. The extensions derived generalize the Hamiltons variational formulation for such problems. The Navier–Stokes equations are derived and coupled to the structural oscillator. This general model has been shown to be a superset of the flow-oscillator model. Based on different assumptions, one can derive a variety of flow-oscillator models.

Response of a tension leg platform to stochastic wave forces

The equations of motion and the response of a Tension Leg Platform with a single tendon undergoing planar motion are presented. The hull is represented by a rigid cylindrical body and the tendon as a nonlinear elastic beam. Tendons have often been modeled as massless springs, which neglects contribution to the response by the wave forces on the tendons and its varying stiffness due to changes in its tension. The structure is subjected to random wave loading. Linear wave theory is applied and the random wave height power spectrum is transformed into a time history using Borgmans method. The surge and pitch responses for the hull, and the surge response along the tendon are presented for two cases. Case 1 represents a tendon with a hull mass two orders of magnitude smaller than the tendons mass. In Case 2, the hull mass is two orders of magnitude greater than the tendon. Inclusion of tendon forces were found to significantly increase the amplitude of the surge response for Case 1 but not for Case 2.

Reliability of Structures for the Moon

A brief review of lunar base structural concepts is presented. The subject of risk and reliability for lunar structures is introduced and critical issues deliberated. A tensile structure is considered as an example in order to become more specific on the general introductory comments.

Applied Mechanics of Lunar Exploration and Development

An overview of current concepts for lunar outposts is provided, with emphasis on identifying design issues that are also prospective research problems in applied mechanics. The authors believe that the conjugation of new applications and the unique features of the lunar environment will provide many interesting problems whose solution will provide fundamental insights into problems in applied mechanics.

Magnesium as an ISRU-Derived Resource for Lunar Structures

AbstractMagnesium is one of the most pervasive metals in lunar soil and has many characteristics that make it applicable to in situ refining and production. This somewhat overlooked alkaline earth metal is easily cast, used, and recycled, characteristics that are required in the Moons harsh environment. Moreover, alloys of this element have several properties fine-tuned for building shelters in a lunar environment, including several advantages over aluminum alloys. Magnesium alloys may prove to be the optimal choices for reinforcing lunar structures or manufacturing components as needed on the Moon. As such, further research on the in situ resource utilization (ISRU) of magnesium is imperative for the development of a self-sufficient lunar base. This paper brings together key properties of magnesium within the context of it being used as an in situ resource once the Moon, again, becomes a goal for permanent habitation and we require the ability to live there in perpetuity.

Comparison of linear and nonlinear responses of a compliant tower to random wave forces

Abstract A vertical member of a compliant offshore structure is modeled as a beam undergoing both bending and extension. The beam has a point mass and is subjected to a point axial load at the free end. The equations of motion for the axial and transverse displacements are nonlinear and coupled. A linear tension model is derived as a special case of the nonlinear coupled model with negligible axial displacement. The responses are obtained numerically for both models. A quarter of the International Ship and Offshore Structures Congress (ISSC) tension leg platform model is used as a numerical example. The free and forced responses obtained using the nonlinear coupled modeled are compared to those of the linear tension model.

Waves, normal modes and frequencies in periodic and near-periodic rods. Part II

Abstract A systematic approach to the study of normal modes and frequencies of disordered periodic rods is presented within a new transfer matrix framework proposed earlier by the authors. The normal frequency structure and mode localization of multiply-disorder periodic rods are investigated. The Monte Carlo and the perturbation method are applied to study the effects of material parameter uncertainties on normal modes and frequencies of randomly-disordered periodic rods. Some intricate aspects are investigated statistically, and it is shown that for this strongly-coupled elastic system, multiple and/or random disorders lead to more localized modes in or near stop-bands in a more complex way. In addition, high frequency wave localization is a typical feature of such a strongly-coupled but randomly-disordered periodic rod system.