This review summarizes fundamental results and discoveries concerning vortex-induced vibration (VIV), that have been made over the last two decades, many of which are related to the push to explore very low mass and damping, and to new computational and experimental techniques that were hitherto not available. We bring together new concepts and phenomena generic to VIV systems, and pay special attention to the vortex dynamics and energy transfer that give rise to modes of vibration, the importance of mass and damping, the concept of a critical mass, the relationship between force and vorticity, and the concept of "effective elasticity," among other points. We present new vortex wake modes, generally in the framework of a map of vortex modes compiled from forced vibration studies, some of which cause free vibration. Some discussion focuses on topics of current debate, such as the decomposition of force, the relevance of the paradigm flow of an elastically mounted cylinder to more complex systems, and the relationship between forced and free vibration.

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Vortex-induced vibrations

Most of the signal processing that we will study in this course involves local operations on a signal, namely transforming the signal by applying linear combinations of values in the neighborhood of each sample point. You are familiar with such operations from Calculus, namely, taking derivatives and you are also familiar with this from optics namely blurring a signal. We will be looking at sampled signals only. Let's start with a few basic examples. Local difference Suppose we have a 1D image and we take the local difference of intensities, DI(x) = 1 2 (I(x + 1) − I(x − 1)) which give a discrete approximation to a partial derivative. (We compute this for each x in the image.) What is the effect of such a transformation? One key idea is that such a derivative would be useful for marking positions where the intensity changes. Such a change is called an edge. It is important to detect edges in images because they often mark locations at which object properties change. These can include changes in illumination along a surface due to a shadow boundary, or a material (pigment) change, or a change in depth as when one object ends and another begins. The computational problem of finding intensity edges in images is called edge detection. We could look for positions at which DI(x) has a large negative or positive value. Large positive values indicate an edge that goes from low to high intensity, and large negative values indicate an edge that goes from high to low intensity. Example Suppose the image consists of a single (slightly sloped) edge:

http://ieeexplore.ieee.org/iel5/5/31307/01456462.pdf

Linear systems

Numerical simulation of large-scale dynamical systems plays a fundamental role in studying a wide range of complex physical phenomena; however, the inherent large-scale nature of the models often leads to unmanageable demands on computational resources. Model reduction aims to reduce this computational burden by generating reduced models that are faster and cheaper to simulate, yet accurately represent the original large-scale system behavior. Model reduction of linear, nonparametric dynamical systems has reached a considerable level of maturity, as reflected by several survey papers and books. However, parametric model reduction has emerged only more recently as an important and vibrant research area, with several recent advances making a survey paper timely. Thus, this paper aims to provide a resource that draws together recent contributions in different communities to survey the state of the art in parametric model reduction methods. Parametric model reduction targets the broad class of problems for wh...

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A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems

A new method for performing a balanced reduction of a high-order linear system is presented. The technique combines the proper orthogonal decomposition and concepts from balanced realization theory. The method of snapshotsisused to obtainlow-rank,reduced-rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain.Theapproximationsarethenusedtoobtainabalancedreducedorder model. The method is particularly effective when a small number of outputs is of interest. It is demonstrated for a linearized high-order system that models unsteady motion of a two-dimensional airfoil. Computation of the exact grammians would be impractical for such a large system. For this problem, very accurate reducedorder models are obtained that capture the required dynamics with just three states. The new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition. Although further development is necessary, the concept also extends to nonlinear systems.

/pdf/balanced-model-reduction-via-the-proper-orthogonal-5c77iwc45j.pdf

Balanced Model Reduction via the Proper Orthogonal Decomposition

In this paper, we study the transverse vortex-induced vibrations of an elastically mounted rigid cylinder in a fluid flow. We use simultaneous force, displacement and vorticity measurements (using DPIV) for the first time in free vibrations. There exist two distinct types of response in such systems, depending on whether one has a high or low combined mass–damping parameter (m*ζ). In the classical high-(m*ζ) case, an ‘initial’ and ‘lower’ amplitude branch are separated by a discontinuous mode transition, whereas in the case of low (m*ζ), a further higher-amplitude ‘upper’ branch of response appears, and there exist two mode transitions.To understand the existence of more than one mode transition for low (m*ζ), we employ two distinct formulations of the equation of motion, one of which uses the ‘total force’, while the other uses the ‘vortex force’, which is related only to the dynamics of vorticity. The first mode transition involves a jump in ‘vortex phase’ (between vortex force and displacement), ϕvortex, at which point the frequency of oscillation (f) passes through the natural frequency of the system in the fluid, f ∼ fNwater. This transition is associated with a jump between 2S [harr ] 2P vortex wake modes, and a corresponding switch in vortex shedding timing. Across the second mode transition, there is a jump in ‘total phase’, phis;total , at which point f ∼ fNvacuum. In this case, there is no jump in ϕvortex, since both branches are associated with the 2P mode, and there is therefore no switch in timing of shedding, contrary to previous assumptions. Interestingly, for the high-(m*ζ) case, the vibration frequency jumps across both fNwater and fNvacuum, corresponding to the simultaneous jumps in ϕvortex and ϕtotal. This causes a switch in the timing of shedding, coincident with the ‘total phase’ jump, in agreement with previous assumptions.For large mass ratios, m* = O(100), the vibration frequency for synchronization lies close to the natural frequency (f* = f/fN ≈ 1.0), but as mass is reduced to m* = O(1), f* can reach remarkably large values. We deduce an expression for the frequency of the lower-branch vibration, as follows:formula herewhich agrees very well with a wide set of experimental data. This frequency equation uncovers the existence of a critical mass ratio, where the frequency f* becomes large: m*crit = 0.54. When m* < m*crit, the lower branch can never be reached and it ceases to exist. The upper-branch large-amplitude vibrations persist for all velocities, no matter how high, and the frequency increases indefinitely with flow velocity. Experiments at m* < m*crit show that the upper-branch vibrations continue to the limits (in flow speed) of our facility.

Modes of vortex formation and frequency response of a freely vibrating cylinder

Dynamics of transversely vibrating beams using four engineering theories

An overview of modeling and experiments of vortex-induced vibration of circular cylinders

Finite Element Methods in Probabilistic Structural Analysis: A Selective Review

Review of force reconstruction techniques

Introduction and Background Basic Concepts of Systems and Structures Basic Concepts of Vibration Basic Concepts of Random Vibration Types of System Models Basic Dynamics Units Concluding Summary Single Degree-of-Freedom Vibration: Discrete Models Motivating Examples Mathematical Modeling: Deterministic Undamped Free Vibration Harmonic Forcing with no Damping Concepts Summary Single Degree-of-Freedom Vibration: Discrete Models with Damping Damping Free Vibration with Viscous Damping Free Response with Coulomb Damping Forced Vibration with Viscous Damping Forced Harmonic Vibration Periodic but Not Harmonic Excitation Concepts Summary Single Degree-of-Freedom Vibration: General Loading and Advanced Topics Arbitrary Loading: Laplace Transform Step Loading Impulsive Excitation Arbitrary Loading Introduction to Lagrange.s Equation Notions of Randomness Notions of Control The Inverse Problem A Self-Excited System and its Stability Solution Analysis and Design Techniques A Model of a Bouncing Ball Concepts Summary Single Degree-of-Freedom Vibration: Probabilistic Forces Introduction Example Problems and Motivation Random Variables Mathematical Expectation Useful Probability Densities Two Random Variables Random Processes Random Vibration Concepts Summary Vibration Control Motivation Approaches to Controlling Vibration Feedback Control Performance of Feedback Control Systems Control of Response Sensitivity to Parameter Variations State Variable Models Concepts Summary Variational Principles and Analytical Dynamics Introduction Virtual Work Lagrange.s Equation of Motion Hamilton's Principle Lagrange's Equation with Damping Concepts Summary Multi Degree-of-Freedom Vibration: Introductory Topics Example Problems and Motivation The Concepts of Stiness and Flexibility Derivation of Equations of Motion Undamped Vibration Direct Method: Free Vibration with Damping Modal Analysis Real and Complex Modes Concepts Summary Multi Degree-of-Freedom Vibration: Advanced Topics Overview Unrestrained Systems The Geometry of the Eigenvalue Problem Periodic Structures Inverse Vibration Sloshing of Fluids in Containers Stability of Motion Multivariable Control MDOF Stochastic Response Stochastic Control Rayleigh.s Quotient Monte Carlo Simulation Concepts Summary Continuous Models for Vibration Continuous Limit of a Discrete Formulation Vibration of String Longitudinal (Axial) Vibration of Beams Torsional Vibration of Shafts 10.5 Transverse Vibration of Beams 10.6 Beam Vibration: Special Problems Concepts Summary Continuous Models for Vibration: Advanced Models Vibration of Membranes Vibration of Plates Random Vibration of Continuous Structures Approximate Methods Variables That Do Not Separate Concepts Summary Nonlinear Vibration Examples of Nonlinear Vibration The Phase Plane Perturbation Methods The Mathieu Equation The van der Pol Equation Motion in the Large Nonlinear Control Advanced Topics Concluding Summary Appendices A Mathematical Concepts for Vibration Complex Numbers Matrices Taylor Series and Linearization Ordinary Dierential Equations Laplace Transforms Fourier Series and Transforms Partial Dierential Equations Index

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Haym Benaroya

Papers

Dynamics of transversely vibrating beams using four engineering theories

An overview of modeling and experiments of vortex-induced vibration of circular cylinders

Finite Element Methods in Probabilistic Structural Analysis: A Selective Review

Review of force reconstruction techniques

Mechanical vibration : analysis, uncertainties, and control