Statistical flow-oscillator modeling of vortex-shedding

The very important engineering problem of modeling the fluid-structure interaction occurring during the shedding of vortices has defied, and will probably continue to defy, a closed form exact solution for the foreseeable future. Therefore, an attempt must be made to extract relevant information about the process in order to be able to have a basic understanding of it for the purpose of analysis. A useful method involves the flow-oscillator concepts of Hartlen and Currie [1] redefined here for stochastic processes. The fluid-structure system is assumed to be governed by the cross-coupled equations

$\text{x}\text{?}\text{(t)+2ξω}{}_{\mathrm{n}}\text{x}\text{?}\text{(t)+ω}{}^2{}_{\mathrm{n}}\text{=C}{}_{\mathrm{e}}\text{(t)pV}{}^2{}_0\text{(t)DL/2m (i)}$

$\text{C}\text{?}{}_{\mathrm{e}}\text{(t)+{α ? βC}{}^2{}_{\mathrm{e}}\text{(t)+γC}{}^4{}_{\mathrm{e}}\text{(t)}}\text{C}\text{?}{}_{\mathrm{e}}\text{(t)+ω}{}^2{}_0\text{C}{}_{\mathrm{e}}\text{(t)=b}\text{x}\text{?}\text{(t), (ii)}$

where these equations govern the structure and fluid oscillators, respectively. The fluid damping is non-linear. These equations are taken as stochastic differential equations because of the many unpredictable, random effects that determine the loading and response. The lift coefficient C_$\text{l}$(t) is assumed to be a zero mean, narrow band process and the velocity V₀, composed of a uniform, constant velocity current plus oscillating wave, a broad band process. The analysis is based on solving equation (i) for x(t) by using Duhamel's integral and substituting its derivative $\text{x}\text{?}\text{(t)}$ into equation (ii). This equation is then used to derive the Fokker-Planck equation for the process C_$\text{l}$(t). To obtain the Fokker-Planck equation, slowly varying variables are replaced by their long-time averages [2] and then the method of stochastic averaging is employed [3, 4]. The moment equation for the lift-oscillator process is derived from the Fokker-Planck equation and, as equation (ii) is non-linear, one finds the moment equation to be in terms of higher order moments. A truncation scheme [5] is used to derive the moment generating function. It is possible then to generate the first and second order statistics of the lift coefficient and the structure response in terms of the empirical parameters of fluid damping. This work was carried out in conjunction with an analysis of ocean wave-current forces with application to offshore fixed structures [6].     

The Kulinkovich hydroxycyclopropanation reaction in natural product synthesis

The Kulinkovich cyclopropanation reaction provides a flexible and convenient method for the synthesis of cyclopropanols. Together with the diverse chemistry of the cyclopropanol unit, it offers access to a wide range of functionalised unsaturated and saturated compounds. The successful use in the synthesis of natural compounds is outlined in this perspective.     

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

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.     

The van Kampen expansion for the Fokker-Planck equation of a duffing oscillator

In Rodríguez and van Kampen's 1976 paper a method of extracting information from the Fokker-Planck equation without having to solve the equation is outlined. The Fokker-Planck equation for a Duffing oscillator excited by white noise is expanded about the intensity of the forcing function. This expansion is carried to order . However, no studies are made of the effects of the order of the expansion or variation of the parameters, nor are comparisons made to experimental results. In the present paper, the expansion is carried to a higher order, , results are presented and compared to Monte Carlo experiments using both white and colored noise, and parametric studies are performed on the intensity of the forcing function and the damping coefficient. It is found that the expansion method works well for the case of white noise and for colored noise where the correlation time is less than 0.1 sec, but fails to give certain details. It is also found that the system behaves as expected when the parameters are varied.     

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

Longitudinal wave motions and localized normal modes in a rod system with periodically-alternating material properties are investigated in this paper. The energy injected into the rod system is shown either to be transported through the whole rod system in pass-bands or to be trapped near the excitation source in stop-bands. For this one-dimensional continuous model, the full power of linear system theory is utilized and a new transfer matrix method is proposed to get closed-form normal mode solutions. Localized normal modes in stop-bands in perfectly-periodic rods with asymmetric bays are identified. It is shown that for this strongly-coupled elastic system, a single small disorder may produce one or two additional modes in each stop-band, these modes are localized around the disordered bay. By understanding basic behavior of such a system, it is hoped ultimately that some insights can be achieved where closed-form results are not possible.     

Waves in periodic structures with imperfections

The study of wave propagation in structures and media has a significant history which follows from the examination of the dynamics of atomic and molecular lattices. Relatively recently, some of these ideas have been transferred and applied to the dynamic behavior of engineered structures. In particular, structures with periodic and almost periodic topologies and material properties have been extensively studied and important conclusions drawn regarding their energy-transmission properties. The attraction to wave propagation models is due to the efficient nature of the analytical tools available to study how energies of different frequency content are propagated or filtered by the structure. Such properties of a structure are profoundly affected by any imperfections or “near-periodicities”. It has been found that imperfections will have the effect of localizing energies about them, thus not allowing the development of normal modes of vibration as would be observed when assuming a perfect structure. It is envisioned that such understanding will permit the analyst to take advantage of localization effects to isolate locations experiencing loading. Additional applications possibly include the modeling of composite and layered structures and cracks. Also, one expects that structures with periodic boundary conditions will experience some sort of localization of energies in certain frequency ranges.     

Stationarity and stochastic differential equations

A linear differential equation of order N with stochastic process coefficients and excitation are studied. The objective of this paper is to demonstrate that, by using an expansion method, when the coefficients and excitation are strict sense stationary processes, the response is also a strict sense stationary process. Such problems occur frequently in the engineering sciences and are very important. Applications include parametric random vibrations, turbulent environment rotorcraft dynamics, and dynamics of axially loaded structural members, among others. An example application is provided.     

The van Kampen expansion for the Fokker-Planck equation of a duffing oscillator excited by colored noise

In Rodríguez and van Kampen's 1976 paper a method of extracting information from the Fokker-Planck equation without having to solve the equation is outlined. The Fokker-Planck equation for a Duffing oscillator excited by white noise is expanded about the intensity of the forcing function. In Weinstein and Benaroya, the effect of the order of expansion is investigated by carrying the expansion to a higher order. The effect of varying the system parameters is also investigated. All results are verified by comparison to Monte Carlo experiments. In this paper, the van Kampen expansion is modified and applied to the case of a Duffing oscillator excited by colored noise. The effect of the correlation time is investigated. Again the results are compared to those of Monte Carlo experiments. It is found that the expansion compares closely with those of the Monte Carlo experiments as the correlation time _c is varied from 0.001 to 10 sec. Examination of the results reveals that the colored noise can be categorized in one of four ways: (1) for the noise can be considered as white for all intents and purposes, (2) for the noise can be considered white for some purposes, (3) for the correlated nature of the noise must be considered in an analysis, and (4) for the noise can be considered as deterministic.     

Review of force reconstruction techniques

An important engineering problem is the recovery of the input of a system given its output. This is a difficult problem to solve in that it is often an ill-defined problem. Such ill-posedness is problematic since noise becomes very influential and results in inaccurate or non-unique solutions. To combat this ill-posedness, additional constraints are typically applied to redefine the problem, leading to a well-defined problem with a unique solution. Current input reconstruction methods span the spectrum of analysis and computation, and we have grouped them into three categories: Direct, Regularization, and Probabilistic/Statistical. Each of these groups is divided into several subsets that offer different perspectives in which to view the reconstruction problem. Our primary interests lie in the behavior of mechanical systems and, as such, we have focused on the literature in these fields. However, applicability includes other fields with the same and similar governing equations.