TRANSIENT RESPONSE ANALYSIS OF LARGE-SCALE ROTOR-BEARING SYSTEM WITH STRONG NON-LINEAR ELEMENTS BY A TRANSFER MATRIX-NEWMARK FORMULATION ITEGRATION METHOD

This paper extends the transfer matrix technique (TMT) to the transient response analysis of a large complex non-linear rotor-bearing system by a transfer matrix-Newmark formulation itegration method. Firstly, the transfer matrix is obtained via the Newmark formulation. Secondly, the deflections and velocities at the stations, containing non-linear element, are determined by iteration. Finally, the deflections, velocities and accelerations of all stations are computed by TMT and the Newmark formulation consistent with the boundary conditions. In order to eliminate the numerical instability of TMT, the transfer vector {f^T ? ë^T}^T is used, instead of the traditional one {f^T ? ë^T}^T. Owing to the advantages of TMT and the Newmark formulation, this method can be applied to calculate the transient response of a large-scale rotor-bearing system with strong non-linear elements, and to analyze its stability. Two illustration examples are given, and the results agree well with those by Runge-Kutta method, and by modal synthetic method.     

SUPER AND COMBINATORIAL HARMONIC RESPONSE OF FLEXIBLE ELASTIC CABLES WITH SMALL SAG

The paper deals with the analysis of cables in stayed bridges and TV-towers, where the excitation is caused by harmonically varying in-plane motions of the upper support point with the amplitude U. Such cables are characterized by a sag-to-chord-length ratio below &0uml;02, which means that the lowest circular eigenfrequencies for in-plane and out-of-plane eigenvibrations, ω₁and ω₂, are closely separated. The dynamic analysis is performed by a two-degree-of-freedom modal decomposition in the lowest in-plane and out-of-plane eigenmodes. Modal parameters are evaluated based on the eigenmodes for the parabolic approximation to the equilibrium suspension. Superharmonic components of the ordern , supported by the parametric terms of the excitation and the non-linear coupling terms, are registered in the response for circular frequency ω?ω₁/n. At moderate U, the cable response takes place entirely in the static equilibrium plane. At larger amplitudes the in-plane response becomes unstable and a coupled whirling superharmonic component occurs. In the paper a first order perturbation solution to the superharmonic response is performed based on the averaging method. For ω?(m/n)ω₁, m<n, the geometrical non-linear restoring forces gives rise to a substantial combinatorial harmonic component with the circular frequency (n/m)ω. Both entirely in-plane and coupled in-plane and out-of-plane responses occur. Based on an initial frequency analysis of the response, an analytical model for these vibrations is formulated with emphasis on superharmonics of the order n=3 and combinatorial harmonics of the order (n, m)=(3,2). All analytical solutions have been verified by direct numerical integration of the modal equations of motion.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES. PART I: APPLICATION TO CLAMPED-CLAMPED AND SIMPLY SUPPORTED-CLAMPED BEAMS

In a previous series of papers (Benamar 1990 Ph.D. Thesis, University of Southampton; Benamaret al. 1991 Journal of Sound and Vibration149, 179-195;164, 399-424 [1-3]) a general model based on Hamilton's principle and spectral analysis has been developed for non-linear free vibrations occurring at large displacement amplitudes of fully clamped beams and rectangular homogeneous and composite plates. The results obtained with this model corresponding to the first non-linear mode shape of a clamped-clamped (CC) beam and to the first non-linear mode shape of a CC plate are in good agreement with those obtained in previous experimental studies (Benamaret al. 1991 Journal of Sound and Vibration 149, 179-195;164, 399-424 [2, 3]). More recently, this model has been re-derived (Azar et al. 1999 Journal of Sound and Vibration224, 377-395; submitted [4, 5]) using spectral analysis, Lagrange's equations and the harmonic balance method, and applied to obtain the non-linear steady state forced periodic response of simply supported (SS), CC, and simply supported-clamped (SSC) beams. The practical application of this approach to engineering problems necessitates the use of appropriate software in each case or use of published tables of data, obtained from numerical solution of the non-linear algebraic system, corresponding to each problem. The present work was an attempt to develop a more practical simple “multi-mode theory” based on the linearization of the non-linear algebraic equations, written on the modal basis, in the neighbourhood of each resonance. The purpose was to derive simple formulae, which are easy to use, for engineering purposes. In this paper, two models are proposed. The first is concerned with displacement amplitudes of vibrationW_max /H, obtained at the beam centre, up to about 0·7 times the beam thickness and the second may be used for higher amplitudes W_max/H up to about 1·5 times the beam thickness. This new approach has been successfully used in the free vibration case to the first, second and third non-linear modes shapes of CC beams and to the first non-linear mode shape of a CSS beam. It has also been applied to obtain the non-linear steady state periodic forced response of CC and CSS beams, excited harmonically with concentrated and distributed forces.     

Bifurcations to divergence and flutter in flow-induced oscillations: A finite dimensional analysis

The behaviours of a pipe conveying fluid and a fluid loaded panel are studied from the viewpoint of differentiable dynamics. Non-linear terms are included and it is shown how the partial differential equation of motion can be recast, by Galerkin's method and modal truncation, in the form of an ordinary differential equation in Euclidean n-space. This evolution equation is then analysed qualitatively, attention being paid to bifurcations which occur as the control parameters of axial force and flow velocity are varied. Bifurcations of fixed points occur when at least one of the eigenvalues of the linearized evolution equation crosses the imaginary axis in the complex plane. In this situation, centre manifold theory can be used to extract a low dimensional subsystem which completely captures the local bifurcational behaviour. Such essential models enable the onset of divergence and flutter to be analysed relatively simply and the inclusion of non-linear terms permits the global study of post-bifurcational behaviour. The general approach is illustrated by analysis of two mode models of a pipe and of a panel and some important omissions in previous treatments of linear and undamped systems are discussed.     

ROBUST VIBRATION CONTROL OF A BEAM USING THE H∞-BASED CONTROLLER WITH MODEL ERROR COMPENSATOR

The vibration control of a flexible beam subjected to arbitrary, unmeasurable disturbance forces is investigated in this paper. The beam is analyzed by using modal expansion theorem. The independent modal space control is adopted for the active vibration control. Discrete sensors and actuators are used here. The modal filters are used as the state estimator to obtain the modal co-ordinates and modal velocities for the state feedback control. Because of the existence of the disturbance forces, the vibration control only with the state feedback control law cannot suppress the vibration well. The method of disturbance forces cancellation is then added in the feedback loop. In order to implement the disturbance forces cancellation, the unknown disturbance forces must be observed. The model error compensator is employed to observe the unknown disturbance modal forces for the direct cancellation. After the implementation of the disturbance modal forces cancellation, there are still some residual disturbance modal forces which excite the beam. The disturbance attenuation problem is of concern in the design of the state feedback control law. For ensuring that influence of the residual disturbance modal forces is reduced to an acceptable level, the robust static H_∞ state feedback controller is designed. The vibration control performances of the feedback control with the H_∞ controller and the disturbance forces cancellation are discussed.     

Non-linear axisymmetric transient analysis of an orthotropic thin circular plate with an elastically restrained edge

Results are presented for the geometrically non-linear axisymmetric transient elastic stress and deflection responses of a cylindrically orthotropic thin circular plate with an elastically restrained edge, including both rotational and in-plane displacements. In the analysis the dynamic analogue of the von Kárman governing differential equations in terms of the normal displacement w and the stress function ψ are employed. The displacement w and stress function ψ are expanded in finite power series. The orthogonal point collocation method in the space domain and the Newmark-β scheme in the time domain are used. Four types of uniformly distributed transient loadings have been considered: step function, sinusoidal and N-shaped pulses, and exponentially decaying loads. The influence of the orthotropic parameter β and the elastic rotational and in-plane edge restraint parameters (K_b, K_i) on the large amplitude response has been investigated. The effect of a prescribed in-plane displacement on the non-linear transient response has also been studied.     

Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers

In order to evaluate a fiber amplifier’s output beam quality, we have explored the beam quality factor calculation. A theoretical beam quality factor calculation method is presented in this paper. The method bases on modal power decomposition, laser rate equations and Fourier diffraction theory. Through the rate equations, each modal power weight factor at the fiber output facet is obtained. By the modal power decomposition, the optical field at the fiber output facet is established. Finally, running Fourier transformation, the beam diverging state in free space and corresponding M ² parameter are obtained. The calculation process is verified with applying it to working out the beam M ² parameters of large-mode-area (LMA) fiber amplifier system under different conditions. The numerical results help us well know and predict the out beam property. Some obtained results are also accord with other reporting outcomes.     

The hysteresis loop of two-phase particulate magnetic composite with non-linear hard magnetic components

The method of calculating the B - H curve of two-phase ferromagnetic composite materials whose components have B - H relations of the form B = mH + Br is extended to include materials whose components have non-linear B - H relations. The functional form of the non-linear B - H curve of the components materials is simulated by hyperbolic tangent functions with adjustable parameters. The result of theoretical calculations are compared to experimental data on Mn-Al-C and Fe-Cr-Co particulate composites. Good agreement are obtained.     

Analysis of critical and post-critical behaviour of non-linear dynamical systems by the normal form method,Part I: Normalization formulae

A method, based on normal form theory, is presented to study the dynamical behaviour of a system in the neighbourhood of a nearly critical equilibrium state associated with a bifurcation condition. Explicit formulae for the normalization procedure are derived. These formulae can be numerically programmed, avoiding usual complicated algebraic calculations and making the method effectively applicable for n-dimensional systems. Rather general bifurcations can be included: e.g., non-linear flutter (Hopf bifurcation), divergence and internal resonance.     

Measurement of relevant elastic and damping material properties in sandwich thick plates

An easy-to-implement method to measure relevant elastic and damping properties of the constituents of a sandwich structure, possibly with a heterogeneous core, is proposed. The method makes use of a one-point dynamical measurement on a thick-plate. The hysteretic model for each (possibly orthotropic) constituent is written generically as “E(1+jη)” for all mechanical parameters. The estimation method of the parameters relies on a mixed experimental/numerical procedure. The frequencies and dampings of the natural modes of the plate are obtained from experimental impulse responses by means of a high-resolution modal analysis technique. This allows for considerably more experimental data to be used. Numerical modes (frequencies, dampings, and modal shapes) are computed by means of an extended Rayleigh-Ritz procedure under the “light damping” hypothesis, for given values of the mechanical parameters. Minimising the differences between the modal characteristics yields an estimation of the values of the mechanical parameters describing the hysteretic behaviour. A sensitivity analysis assesses the reliability of the method for each parameter. Validations of the method are proposed by (a) applying it to virtual plates on which a finite-element model replaces the experimental modal analysis, (b) some comparisons with results obtained by static mechanical measurements, and (c) by comparing the results on different plates made of the same sandwich material.     

Discrete-time norms of flexible structures

The paper presents and analyses the H₂, H_∞, and Hankel norms of flexible structures. The analysis is conducted for the discrete-time models of structures and compared with the continuous-time results. The structural state-space models are presented in modal co-ordinates. Closed-form expressions for norms of structural modes are obtained, and norms of a structure are determined from the modal norms. The relationships between the Hankel, H_∞, and H₂ modal norms are derived. In addition, the paper shows that the discrete-time Hankel and H_∞ norms converge to the continuous-time counterparts when the sampling time approaches zero; however, the H₂ norm does not.     

A method of implementation of frequency encoded all optical logic gates based on non-linear total reflectional switch at the interface

A novel method of implementation of frequency encoded logic gates NOT, OR, AND, NOR, NAND, X-OR, X-NOR is discussed. The frequency sources and physical requirements for the implementation are also discussed. The non-linear material (liquid) suitable for these operations to be performed should be of large non-linear coefficient, high reverse saturation absorption, large thermo-optic coefficient and low viscosity. The input controlling beams used to induce non-linearity in the switch are either of frequency υ₁ or υ₂ and the probe beam is a mixed signal of frequencies υ₁ and υ₂. Depending on the nature of the controlling inputs the output conditions of the probe can be adjusted to get different logic gates.     

Multiparametric magnetic immunoassays utilizing non-linear signatures of magnetic labels

A powerful route to utilizing magnetic nanoparticles as labels in magnetic immunoassays is to exploit their non-linear response when they are exposed to a multi-frequency alternating magnetic field. We have upgraded this non-linear method allowing for the detection, discrimination and quantification of particles of two kinds when mixed together, with no need for spatial resolution. Each kind of particle is characterized by a specific magnetic signature based on d²B(H)/dH². Appropriate data processing of the signature measured on a mixture of both particles allows for obtaining the amount of each particle. This will enable utilizing magnetic labels for multiparametric magnetic immunoassays.     

Experimental and Numerical Study of Structural Identification Using Non-Linear Resonant Decay Method

One of the practical approaches in identifying structures is the non-linear resonant decay method which identifies a non-linear dynamic system utilizing a model based on linear modal space containing the underlying linear system and a small number of extra terms that exhibit the non-linear effects. In this paper, the method is illustrated in a simulated system and an experimental structure. The main objective of the non-linear resonant decay method is to identify the non-linear dynamic systems based on the use of a multi-shaker excitation using appropriated excitation which is obtained from the force appropriation approach. The experimental application of the method is indicated to provide suitable estimates of modal parameters for the identification of non-linear models of structures.     

On predicting point mobility plots from measurements of other mobility parameters

Experimental modal analysis techniques are designed to exploit a property of linear vibration theory in order to construct a mathematical model of a structure from the minimum amount of measured mobility data. This property, derived from the orthogonality of normal modes, is that, in principle, all the elements of the full N × N mobility matrix can be derived from measurement of the elements in just one row or column of that matrix. The property is also used in procedural quality “checks”, in which one measures some of the derived mobilities as well as predicting them. The derivation process itself, however, gives rise to an inherent error when the frequency range covered does not include all the natural frequencies of the structure. The source of this error is discussed and some illustrations of its significance are quoted, demonstrating the need for caution when applying modal analysis methods to practical structures.     

Adaptive non-linear control of a clamped rectangular plate with PZT patches

The dynamic behavior and control of a clamped rectangular plate with bonded piezoelectric ceramic patches are investigated in this paper. The dynamic behavior is studied experimentally, showing that the plate exhibits dense modes, varying residual stress and non-linearity. An adaptive non-linear control scheme is then presented, which introduces a non-linear function into the normal adaptive feedforward control to non-linearize a reference signal. Vibration reductions using both the normal adaptive feedforward control and the adaptive non-linear control are compared in the cases of sinusoidal excitations at the first nine modal frequencies of the plate and a swept-frequency harmonic excitation below 100 Hz, indicating that the adaptive non-linear control can suppress not only the fundamental frequency vibration that the normal adaptive feedforward control can only attenuate, but also its higher harmonic components. Significant vibration reduction achieved by the adaptive non-linear control demonstrates its validity and reliability.     

Solutions of the full- and quasi-vector wave equations based on H-field for optical waveguides by using mapped Galerkin method

The mapped Galerkin method in solving the full-vector and quasi-vector wave equations in terms of transverse magnetic fields (H-formulation) for optical waveguides with step-index profiles is described. By transforming the whole x-y space onto a unit square and using two-dimensional Fourier series expansion, the modal distributions and propagation constants for optical waveguides are obtained in the absence of boundary truncation. Results for step-index circular fiber, buried rectangular waveguide, and optical rib waveguide are presented. Solutions are good agreed with exact solutions and numerical results by using vector nonlinear iterative method, Fourier operator transform method, and vector beam propagation method.     

Non-linear effect of wall linings on sound attenuation in ducts

The equivalent surface source method is extended to the analysis of high intensity sound propagation in a duct whose wall is partially treated with a sound absorbing material. The propagation of sound in the gas is assumed to be linear, but the acoustic resistance of the sound absorbing material is assumed to be a function of the normal acoustic velocity. The problem is reduced to a non-linear integro-differential equation for the fluid particle displacement at the lined wall surface, which can be solved by a successive approximation method. Numerical examples show that the non-linear effect decreases or increases the peak sound attenuation rate of the lowest mode depending upon the linear component of the resistance. The dependence of the attenuation spectrum on modal phase difference of multi-mode incident waves is heavily affected by the non-linear effect. In the case of incident waves of multi-circumferential modes, different circumferential modes are generated by the non-linear effect.