R. A. Ibrahim

Liquid sloshing dynamics : theory and applications

The problem of liquid sloshing in moving or stationary containers remains of great concern to aerospace, civil, and nuclear engineers; physicists; designers of road tankers and ship tankers; and mathematicians. Beginning with the fundamentals of liquid sloshing theory, this book takes the reader systematically from basic theory to advanced analytical and experimental results in a self-contained and coherent format. It presents liquid sloshing effects on space vehicles, storage tanks, road vehicle tanks and ships, and elevated water towers under ground motion. The book is divided into four sections. Part I deals with the theory of linear liquid sloshing dynamics; Part II addresses the nonlinear theory of liquid sloshing dynamics, Faraday waves, and sloshing impacts; Part III presents the problem of linear and nonlinear interaction of liquid sloshing dynamics with elastic containers and supported structures; and Part IV considers the fluid dynamics in spinning containers and microgravity sloshing. This book will be invaluable to researchers and graduate students in mechanical and aeronautical engineering, designers of liquid containers, and applied mathematicians.

Friction-Induced Vibration, Chatter, Squeal, and Chaos—Part I: Mechanics of Contact and Friction

Friction force between sliding surfaces arises due to varied and complex mechanisms and can be responsible for undesirable dynamic characteristics in many mechanical systems. Controversies over the theory of friction have been reported in the literature. Friction laws are phenomenological in charcacter since they are based on observable and measurable quantities. The mechanics of contact and friction in metal-metal and elastomer-metal contact surfaces are reviewed. Unfortunately, there is no satisfactory method capable of determining or measuring the area of contact between sliding bodies. Both dry friction and lubricated friction are considered. The modeling of the friction force in mechanical systems depends on several factors. These include the material properties and geometry of the sliding surfaces, surface roughness, surface chemistry, sliding speed, temperature, and normal load. Other factors include the effect of normal and tangential vibrations on the static friction. Here the static friction is considered as a special case of kinetic friction. This background is essential for dynamicists studying friction-induced vibration, chatter, squeal and chaos topics which will be presented in the second part.

Parametric random vibration

PARAMETRIC random vibration is an applied scientific discipline that covers problems from the broad field of applied dynamics, e.g. structural dynamics, aerodynamics, naval architecture etc. The system equations are characterized by random perturbed parameters while, in many practical situations, non-linearities and random forcing terms create additional complications. Various textbooks have appeared covering the field of random vibration of time-invariant systems. This monograph, a state of the art presentation of parametric random vibration, based on an enormous number of published papers and reports, is a great credit to the author. In the first chapter the reader is introduced to the basic definitions of parametric and autoparametric instabilities, chaotic motion, pseudo-random excitation and crypto-deter-ministic systems and a brief review of parametric random vibration is given. In random vibrations, the emphasis is on the response and stability of systems under wide band random parametric excitations. Unfortunately system equations with physical wide band noise excitation are very difficult to handle, therefore physical Gaussian wide band noise is often replaced by idealized white noise, or the wide band noise is generated by a shaping filter driven by white noise. This is usually the point where many engineers get lost. They are referred to books on stochastic processes and stochastic differential equations and are encouraged to go into theories which are embedded in mathematical abstraction. For these engineers the root of all evil (which is at the same time a source of pleasure for many mathematicians) is the unbounded variation of the Brownian motion, which has white noise as its derivative, in a formal sense. For these processes a new stochastic calculus is needed. In Chapters 2-4 the author provides the necessary tools for deriving response statistical functions and techniques for examining stochastic parameter stability, as required in later chapters. The author has chosen an engineering approach without mathematical abstraction, which implies that some important theorems are verified in a heuristic way, while many others are only mentioned. The result is a nice reference frame for readers with a reasonable background in stochastic processes and stochastic differential equations. Readers without this background will certainly get into trouble, for example, when reading the definitions of random variables and random processes. Obviously, the author could not resist the temptation to give some flavour of mathematical abstraction by introducing a random variable as a function of a sample space f~, which is confusing in the context it is used. …

Liquid sloshing dynamics

The problem of liquid sloshing in moving or stationary containers remains of great concern to aerospace, civil, and nuclear engineers; physicists; designers of road tankers and ship tankers; and mathematicians. Beginning with the fundamentals of liquid sloshing theory, this book takes the reader systematically from basic theory to advanced analytical and experimental results in a self-contained and coherent format. It presents liquid sloshing effects on space vehicles, storage tanks, road vehicle tanks and ships, and elevated water towers under ground motion. The book is divided into four sections. Part I deals with the theory of linear liquid sloshing dynamics; Part II addresses the nonlinear theory of liquid sloshing dynamics, Faraday waves, and sloshing impacts; Part III presents the problem of linear and nonlinear interaction of liquid sloshing dynamics with elastic containers and supported structures; and Part IV considers the fluid dynamics in spinning containers and microgravity sloshing. This book will be invaluable to researchers and graduate students in mechanical and aeronautical engineering, designers of liquid containers, and applied mathematicians.

Progress in Structural Dynamics With Stochastic Parameter Variations: 1987-1998

This paper is an update of an earlier paper by Ibrahim (1987) and is aimed at reviewing the papers published during the last decade in the area of vibration of structures with parameter uncertainties. Analytical, computational, and experimental studies conducted on probabilistic modeling of structural uncertainties and free and forced vibration of stochastically defined systems are discussed. The review also covers developments in the areas of statistical modeling of high frequency vibrations and behavior of statistically disordered periodic systems.

Stochastic response of nonlinear dynamic systems based on a non-Gaussian closure

The random response of nonlinear dynamic systems involving stochastic coefficients is examined. A non-Gaussian closure scheme is outlined and employed to resolve an observed contradiction of the results obtained by other techniques. The method is applied to a system possessing nonlinear damping. The stationary response is obtained by numerical integration of the closed differential equations of the moments (up to fourth order). An interesting feature of the numerical results reveals the existence of a jump in the response statistic functions. This new feature may be attributed to the fact that the non-Gaussian closure more adequately models the nonlinearity, and thus results in characteristics that are similar to those of deterministic nonlinear systems. The results are compared with solutions derived by the Gaussian closure and stochastic averaging method.

AUTOPARAMETRIC RESONANCE IN A STRUCTURE CONTAINING A LIQUID, PART I: TWO MODE INTERACTION

An elastic structure carrying a rigid circular cylindrical tank containing a liquid with a free surface is considered. Autoparametric coupling between a single structural freedom and the first antisymmetric sloshing mode is investigated theoretically and experimentally. Under the condition of principal internal resonance (i.e., when the structure natural frequency equals twice the liquid sloshing frequency) the response of the system is obtained by an asymptotic approximation taken to the second order. Both theoretical and experimental results show that the coupling between liquid sloshing and vertical structure vibration is rather weak.

Excitation-induced stability and phase transition: A review

Dynamical systems may experience undesirable behavior or instability, which can be eliminated using feedback control means. However, in the absence of feedback control, the stability of some systems may be increased by imposing parametric excitation. In other cases, the exit time of the system response from the stable to the unstable domain may be prolonged by imposing external noise, a phenomenon termed noise-enhanced stability (NES). This article presents an assessment of the mechanisms of stabilization via multiplicative noise and noise-enhanced stability. The first part deals with stabilization via deterministic parametric excitation of gravity-defying systems such as the inverted simple and spherical pendulums, aeroelastic structures, human walking, and quantum nonlinear couplers. The second part introduces the concept of noise-induced transition (NIT) in one-dimensional nonlinear systems and ship roll motion. Stabilization of originally unstable systems via multiplicative noise is treated in the third part. The fourth part addresses the influence of additive noise in delaying the exit time of system response to an unstable domain. This topic is related to the phenomenon of stochastic resonance (SR) and NES of systems with one or more metastable states and fluctuating potential. Finally, this review article also introduces some applications in other fields such as the Ising model, ecosystems, and tumor-immune system models.

Experimental investigation of friction-induced noise in disc brake systems

This paper presents experimental and analytical investigations examining the influence of the interfacial forces between a rotating disc and a friction element on the generation of chatter and squeal. Due to inevitable misalignment between the element and disc surface, a kinematic constraint instability known as sprag-slip is created. The experimental measurements include time history records of normal and friction forces, time variation of the friction coefficient, and acceleration of the friction element. The time history records of interfacial forces revealed short periods of high frequency component. It is found that the friction force is non-Gaussian and that its power spectral density covers a wide frequency band. The dependence of the root mean square of the friction coefficient on the relative velocity is found to have a negative slope at lower disc speeds. Depending on the direction of disc rotation, it is found that the friction velocity curve for clockwise disc speed is completely different from counter-clockwise rotation. The associated noise is also different in pitch and frequency content. The analytical modelling emulates the dynamics of the friction element. The transverse motion of the friction element is described by a homogeneous partial differential equation with non-homogeneous boundary conditions. The analysis shows that the normal force appears as a coefficient of the stiffness term, while the friction force appears as a non-homogeneous term. Since the normal force varies randomly as observed experimentally, it acts as a parametric noise to the friction element, and results in parametric instability in the form of squeal or vibration.

Broad band random excitation of a two-degree-of-freedom system with autoparametric coupling

Abstract The response of a two-degree-of-freedom system with autoparametric coupling under the action of broad band random excitation is investigated. The system corresponds to the autoparametric vibration absorber and is also typical of many common structural configurations. A method based upon the Markov vector approach, together with an approximate treatment of third and higher statistical moments, is used to derive a set of fourteen coupled non-linear equations for the first and second moments of the system responses. A numerical integration procedure is used to obtain quantitative results for the system mean and mean square responses over a range of system parameters. The results show that large random motions of the coupled system may occur when the internal detuning parameter is close to the principal internal resonance, and that these motions may give rise to a suppression effect on the random motions of the main system. A feature of the results is that under conditions of internal resonance the random motions are found to be quasi-stationary, with steady oscillatory terms in the response moments. This suggests the possibility of entrainment of regular harmonic responses by the system random motions.