Dynamic stability of elastically supported pipes conveying pulsating fluid

The effect of support flexibility on the dynamic behaviour of pipes conveying fluid is investigated for both steady and pulsatile flows. The pipes are built-in at the upstream end and supported at the other by both a translational and a rotational spring. For the steady flow condition, the critical flow velocities, the frequencies and flow induced damping patterns that are associated with the different vibration modes of selected pipe systems are determined as functions of the flow velocity. The results from steady flow cases show that the pipes may first lose stability by either buckling or flutter, depending on the values of the rotational and translational spring constants and their relative magnitudes. In the case of pulsatile flow, the Floquet theory is utilized for the stability analysis of the selected pipe-fluid systems. Numerical results are presented to illustrate the effects of the amount of translational and rotational resiliences at the elastic support on the regions of parametric and combination resonances of the pipes. The results more of the interesting aspects of the behaviour of non-conservative systems.     

Dissipation effect on local and global stability of fluid-conveying pipes

In this article, the effect of dissipation on local and global stability of fluid conveying pipes is analyzed. The local approach refers to an infinite medium and uses wave propagation analyses without taking boundary conditions into account. The global approach refers to the same medium, but with finite length and associated with a given set of boundary conditions. The finite length system is generally studied by calculating its eigenmodes and eigenfrequencies. Criteria for local instability are derived in the first part of this paper, and dissipation is found to significantly affect local stability. Moreover, dissipation is found to have a stabilizing or destabilizing effect, depending on the other parameters. Next, numerical computations are presented for finite-length systems and results are analyzed and compared with local stability properties of the corresponding media. When the system is sufficiently long, it is found that critical velocity for global instability tends to a local criterion which can be that of local stability of the damped medium or a local transition criterion of the undamped medium, which is not necessarily the local instability criterion. Finally, a reasoning based on lengthscale ratios is developed. It allows to know which criterion is able to predict the global stability for long systems.     

On nonlinear model equations for the response of premixed flames to acoustic like accelerations

The response of a dynamical flame model to imposed acoustic accelerations is studied analytically and numerically. Through linear stability analyses, two analytical approximations for the primary and the parametric stability boundaries are found. The approximation for the primary instability boundary is accurate for any periodic accelerations, in the limit of large acoustic frequencies. The critical acoustic amplitude u _a for Landau–Darrieus instability suppression is identified and found to depend only on the density contrast and the shape of the periodic acoustic stimuli. The proposed model evolution equation is next integrated numerically with various imposed acoustic accelerations; the primary and parametric flame responses are identified. It is shown analytically and numerically that in the presence of a fully developed, yet weakened by acoustics, Landau–Darrieus (or primary) instability the wrinkle amplitude and the mean flame speed oscillate at the same frequency as the acoustic stimuli; the threshold for suppression of primary instability by acoustic forcing is determined exactly. Increasing the acoustic amplitude allows the flame to respond parametrically to the acoustics. This response is characterised by troughs and crests interchanging their roles while the mean flame speed again oscillates with the same frequency as the acoustic stimuli and at twice that of wrinkle amplitude oscillations.     

并联管内两相流密度波脉动线性均相模型

两相流不稳定性理论分析方法可以分为两大类:一类是数值解析法,另一类是近似分析法。数值分析法已经有很多学者进行了大量研究,而近似分析法研究相对少一些。本文提出了描述并联沸腾管内两相流密度波型脉动的线性均相模型。根据线性均相模型运用系统控制原理的方法导出了描述系统稳定性的无因次参数ε。运用参数ε判断系统的稳定性,计算值和实验值基本吻合。     

Dynamic stability of pipes conveying pulsating fluid

The dynamic stability of supported cylindrical pipes converying fluid, when the flow velocity is harmonically perturbed about a constant mean value, is considered in this paper. Explicit stability conditions for perturbations of small intensity are obtained by using the method of averaging. For large periodic excitation a numerical method based on the Floquet theory is used to extend the stability boundaries. The effects of the mean flow velocity, dissipative forces, boundary conditions, and virtual mass on the extent of the parametric instability regions are then discussed.     

Dimensional implications of dynamical data on manifolds to empirical KL analysis

We explore the approximation of attracting manifolds of complex systems using dimension reducing methods. Complex systems having high-dimensional dynamics typically are initially analyzed by exploring techniques to reduce the dimension. Linear techniques, such as Galerkin projection methods, and nonlinear techniques, such as center manifold reduction are just some of the examples used to approximate the manifolds on which the attractors lie. In general, if the manifold is not highly curved, then both linear and nonlinear methods approximate the surface well. However, if the manifold curvature changes significantly with respect to parametric variations, then linear techniques may fail to give an accurate model of the manifold. This may not be a surprise in itself, but it is a fact so often overlooked or misunderstood when utilizing the popular KL method, that we offer this explicit study of the effects and consequences. Here we show that certain dimensions defined by linear methods are highly sensitive when modeled in situations where the attracting manifolds have large parametric curvature. Specifically, we show how manifold curvature mediates the dimension when using a linear basis set as a model. We punctuate our results with the definition of what we call, a “curvature induced parameter,” $d_{CI}$. Both finite- and infinite-dimensional models are used to illustrate the theory.     

Nonlinear parametric instability of wind turbine wings

Nonlinear rotor dynamic is characterized by parametric excitation of both linear and nonlinear terms caused by centrifugal and Coriolis forces when formulated in a moving frame of reference. Assuming harmonically varying support point motions from the tower, the nonlinear parametric instability of a wind turbine wing has been analysed based on a two-degrees-of-freedom model with one modal coordinate representing the vibrations in the blade direction and the other vibrations in edgewise direction. The functional basis for the eigenmode expansion has been taken as the linear undamped fixed-base eigenmodes. It turns out that the system becomes unstable at certain excitation amplitudes and frequencies. If the ratio between the support point motion and the rotational frequency of the rotor is rational, the response becomes periodic, and Floquet theory may be used to determine instability. In reality the indicated frequency ratio may be irrational in which case the response is shown to be quasi-periodic, rendering the Floquet theory useless. Moreover, as the excitation frequency exceeds the eigenfrequency in the edgewise direction, the response may become chaotic. For this reason stability of the system has in all cases been evaluated based on a Lyapunov exponent approach. Stability boundaries are determined as a function of the amplitude and frequency of the support point motion, the rotational speed, damping ratios and eigenfrequencies in the blade and edgewise directions.     

Stability analysis with lumped mass and friction effects in elastically supported pipes conveying fluid

This paper presents an accurate finite element procedure for the stability analysis of elastically supported pipes conveying fluid. With consideration of effects of lumped masses, fluid pressure and friction, the equations of motion are derived based on Hamilton's principle for the mass transport system. The kinematics of the pipe is based on Timoshenko beam theory for which the transverse shear deformation and rotary inertia of the pipe are included. The material behaviour of the pipe is described by the Kelvin viscoelastic model. The dynamic stability behaviours obtained by the present work are more conservative as compared with those evaluated by conventional Euler-Bernoulli beam theory. Also, it is found that the lumped masses, fluid pressure and friction will destabilize the system while the elastic support may have either a stabilizing or destabilizing effect depending on its stiffness and location. To demonstrate the validity and accuracy of the technique developed, several numerical examples are illustrated.     

Parametric instability regions in multi-degree of freedom systems under quasi-periodic beating input excitation

The existence of three different types of unstable region in multi-degree of freedom linear systems undergoing beating two-frequency parametric loading is demonstrated.Two and five degree of freedom digital system models exhibiting simple and combination resonant response to quasi-periodic parametric loading are discussed. Two degree of freedom mathematical models subjected to beating input situations are analysed. Analytical, numerical, electronic analogue and experimental studies are described and the practicalities and relevance of the methods used are indicated.In the case of the beating input problem for two degree of freedom undamped linear systems it has been shown that numerical techniques will predict in size and position all the regions of instability which can reasonably be expected to occur. Approximate perturbational analyses applied to the beating input problem, while confirming the origins of the regions of instability and their general character, do not furnish adequate results as to the size and position of the regions for any useful range of values of the excitation parameters.     

DYNAMIC STABILITY OF CURVED PANELS WITH CUTOUTS

The parametric instability behaviour of curved panels with cutouts subjected to in-plane static and periodic compressive edge loadings are studied using finite element analysis. The first order shear deformation theory is used to model the curved panels, considering the effects of transverse shear deformation and rotary inertia. The theory used is the extension of dynamic, shear deformable theory according to Sanders' first approximation for doubly curved shells, which can be reduced to Love's and Donnell's theories by means of tracers. The effects of static and dynamic load factors, geometry, boundary conditions and the cutout parameters on the principal instability regions of curved panels with cutouts are studied in detail using Bolotin's method. Quantitative results are presented to show the effects of shell geometry and load parameters on the stability boundaries. Results for plates are also presented as special cases and are compared with those available in the literature.     

Solitary wave solutions as a signature of the instability in the discrete nonlinear Schrödinger equation

The effect of instability on the propagation of solitary waves along one-dimensional discrete nonlinear Schrödinger equation with cubic nonlinearity is revisited. A self-contained quasicontinuum approximation is developed to derive closed-form expressions for small-amplitude solitary waves. The notion that the existence of nonlinear solitary waves in discrete systems is a signature of the modulation instability is used. With the help of this notion we conjecture that instability effects on moving solitons can be qualitative estimated from the analytical solutions. Results from numerical simulations are presented to support this conjecture.     

Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings

The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed on-board rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/nonlinear. Thus the application of Lagrange's equations yields the linear/nonlinear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the nonlinear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.     

Quantitative measurement of variational approximations

Variational problems have long been used to mathematically model physical systems. Their advantage has been the simplicity of the model as well as the ability to deduce information concerning the functional dependence of the system on various parameters embedded in the variational trial functions. However, the only method in use for estimating the error in a variational approximation has been to compare the variational result to the exact solution. In this Letter, we demonstrate that one can computationally obtain estimates of the errors in a one-dimensional variational approximation, without any a priori knowledge of the exact solution. Additionally, this analysis can be done by using only linear techniques. The extension of this method to multidimensional problems is clearly possible, although one could expect that additional difficulties could arise.     

Dynamics of microscale pipes containing internal fluid flow: Damping, frequency shift, and stability

This paper initiates the theoretical analysis of microscale resonators containing internal flow, modelled here as microfabricated pipes conveying fluid, and investigates the effects of flow velocity on damping, stability, and frequency shift. The analysis is conducted within the context of classical continuum mechanics, and the effects of structural dissipation (including thermoelastic damping in hollow beams), boundary conditions, geometry, and flow velocity on vibrations are discussed. A scaling analysis suggests that slender elastomeric micropipes are susceptible to instability by divergence (buckling) and flutter at relatively low flow velocities of ∼10 m/s.     

An alternative approach for the analysis of nonlinear vibrations of pipes conveying fluid

Based on a novel extended version of the Lagrange equations for systems containing non-material volumes, the nonlinear equations of motion for cantilever pipe systems conveying fluid are deduced. An alternative to existing methods utilizing Newtonian balance equations or Hamilton's principle is thus provided. The application of the extended Lagrange equations in combination with a Ritz method directly results in a set of nonlinear ordinary differential equations of motion, as opposed to the methods of derivation previously published, which result in partial differential equations. The pipe is modeled as a Euler elastica, where large deflections are considered without order-of-magnitude assumptions. For the equations of motion, a dimensional reduction with arbitrary order of approximation is introduced afterwards and compared with existing lower-order formulations from the literature. The effects of nonlinearities in the equations of motion are studied numerically. The numerical solutions of the extended Lagrange equations of the cantilever pipe system are compared with a second approach based on discrete masses and modeled in the framework of the multibody software HOTINT/MBS. Instability phenomena for an increasing number of discrete masses are presented and convergence towards the solution for pipes conveying fluid is shown.     

Phase bunching of oscillators in parametric resonance

Phase bunching in an ensemble of oscillators is considered by solving the Cauchy problem for the Mathieu equation using the asymptotic method of averaging in the third approximation for the zeroth resonance zone and in the fourth approximation for the first, second, third, and fourth zones in instability regions, as well as in stability regions near the boundaries with instability regions. It is shown that the existence and regularities of bunching follow from analysis of the well-known physical phenomenon, viz., beats of two oscillations. By way of example, parametric oscillations of charges at a node of the electric field of a standing wave are considered.     

Slip boundary conditions and through flow effects on double-diffusive convection in internally heated heterogeneous Brinkman porous media

A model for double-diffusive convection in a heterogeneous porous layer with a constant throughflow is explored, with penetrative convection being simulated via an internal heat source using the Brinkman model. In particular, we analyse the effect of slip boundary conditions on the stability of the model. Because of the many applications in micro-electro-mechanical systems (MEMS) and other microfluidic devices, a study of this problem is necessary. Both linear instability analysis and nonlinear stability analysis are employed. We accurately analyse when stability and instability will commence and determine the critical Rayleigh number as a function of the slip coefficient.     

Analysis of Transient Optical Bistability and Stability in a Nonlinear Fiber Fabry-Perot Resonator Based on an Iterative Method

We perform a transient analysis, steady-state analysis, and linear stability analysis of a nonlinear Fabry-Perot resonator in order to examine the possibility of a fiber bistable device. We here develop two iterative methods for calculating the dynamics of the Fabry-Perot resonator containing a nonlinear medium with an instantaneous response time. The treatment of the counter-propagating field within the cavity is very important in estimating the nonlinear phase shift due to propagation. A trapezoid rule and midpoint rule are used here for the numerical integration. The iterative method using the trapezoid rule gives excellent agreement with the multiple-beam method developed by Bischofberger and Shen (Phys. Rev. A 19 (1979) 1169), which is more complicated than the proposed procedure. Unfortunately it is found that the midpoint approximation is numerically unstable. Assuming a conventional optical fiber, the switching power for optical bistability is less than 1 kW for a resonator length of 1-2 cm. On the basis of the iterative method, we perform a linear stability analysis to examine whether Ikeda instability affects bistable device application or not. The stability analysis shows that the instability threshold is two orders of magnitude larger than the switching power for optical bistability.     

Frequency domain analysis and identification of block-oriented nonlinear systems

Block-oriented nonlinear models including Wiener models, Hammerstein models and Wiener-Hammerstein models, etc. have been extensively applied in practice for system identification, signal processing and control. In this study, analytical frequency response functions including generalized frequency response functions (GFRFs) and nonlinear output spectrum of block-oriented nonlinear systems are developed, which can demonstrate clearly the relationship between frequency response functions and model parameters, and also the dependence of frequency response functions on the linear part of the model. The nonlinear part of these models can be a more general multivariate polynomial function. These fundamental results provide a significant insight into the analysis and design of block-oriented nonlinear systems. Effective algorithms are therefore proposed for the estimation of nonlinear output spectrum and for parametric or nonparametric identification of nonlinear systems. Compared with some existing frequency domain identification methods, the new estimation algorithms do not necessarily require model structure information, not need the invertibility of the nonlinearity and not restrict to harmonic inputs. Simulation examples are given to illustrate these new results.