Identification of a continuous structure with a geometrical non-linearity. Part II: Proper orthogonal decomposition

Particular effort has been spent in the field of identification of multi-degree-of-freedom non-linear systems. The newly developed methods permit the structural analyst to consider increasingly complex systems. The aim of this paper and a companion paper is to study, by means of two methods, a continuous non-linear system consisting of an experimental cantilever beam with a geometrical non-linearity. In the companion paper (Part I) [1] the ability of the conditioned reverse path method, which is a frequency domain technique, to identify the behaviour of this structure is assessed. This paper (Part II) is devoted to the application of proper orthogonal decomposition, which is an updating technique, to the test example.     

Forced response of a cantilever beam with a dry friction damper attached,part I: Theory

A theoretical-experimental study of the forced vibration response of a cantilevered beam with Coulomb damping non-linearity is described. Particular emphasis is placed upon the types of non-linear behavior and system response spectral characteristics observed theoretically and experimentally, upon the design of the experimental apparatus and the utilization of associated instrumentation, and upon the methodology developed for system calibration. Several system configurations are explored, with similarities and deviations analyzed and discussed. This paper, Part I, is devoted to the theory and the companion paper, Part II [1] to the experiments including correlation with theory.     

Study of airfoil gust response alleviation using an electro-magnetic dry friction damper. Part 1: Theory

An electro-magnetic controllable dry friction damper has been designed and numerically simulated. The gust response of a three degree-of-freedom typical airfoil section with a control surface using this non-linear damper has been studied theoretically. The effects of the different gust excitations and parameter variations of the non-linear damper on the non-linear aeroelastic response are discussed. The numerical results show the present electro-magnetic dry friction damper can be used to alleviate the dynamic response to both a periodic and a linear frequency sweep gust excitation, especially for the plunge and pitch responses. The results are also verified by an experimental investigation in a wind tunnel presented in a companion paper, Part 2.     

Acoustic transmission in non-uniform ducts with mean flow,part I: The method of weighted residuals

3n this and a companion paper the problem of transmission of sound through non-uniform ducts carrying a high speed subsonic compressible flow is approached using the method of weighted residuals (MWR) in the form of a modified Galerkin method and the finite element method (FEM). The intent of the investigation is to carry out a careful evaluation of these methods in this computationally difficult problem. To this end both MWR and FEM have been limited to in-core computer implementations to generate useful results with relatively modest computational requirements. This paper (Part I) details the MWR formulation and presents numerical results establishing the degree of accuracy of MWR as compared to exact eigenvalue calculations and approximate one dimensional transmission calculations. The comparison of MWR and FEM results is carried out in the companion paper (Part II).     

The acoustic simulation and analysis of complicated reciprocating compressor piping systems,II: Program structure and applications

The main objectives of the investigation reported in this paper, Part II, and its companion paper, Part I, are (a) to provide a formulation, including the mean flow effects and suitable for digital computer automation, of the acoustics of complicated piping systems, and (b) to develop a comprehensive digital computer program for the simulation and analysis of complicated reciprocating compressor piping systems. In this paper, the digital computer program structure and applications of the program developed, written in Fortran IV, are described. It is concluded that the computer program is versatile and user-friendly. It is capable of providing a great deal of information from one set of input data, and is open-ended and modular for updating.     

On the use of exact modal analysis techniques in the design of damping devices for multi-conductor overhead power lines,Part I: The control of Aeolian vibration

A mathematical model of a multi-conductor overhead power line with spacer-dampers is presented. Advanced methods of exact modal and response analysis, requiring no more than single-precision computational accuracy, are used to develop a method for spacer-damper design which requires no more computational power than that of a standard microcomputer. In this paper an outline is presented of the application of this method to the control of Aeolian vibration by using self-damping spacers, and in the companion paper (Part II) subconductor oscillation is considered.     

Forced response of a cantilever beam with a dry friction damper attached,part II: Experiment

The experimental program is described along with comparison of the results to those obtained from the theory of the companion paper, Part I [1].     

Dislocation dynamics in cyclic plastic deformation

The paper presents a theoretical investigation of the slip avalanches (so-called strain bursts) which occur in single-glide-orientated face-centered cubic or hexagonal close-packed metals during stress-amplitude-controlled cyclic plastic deformation. The study is based on a model of the dynamics of dislocations that has been developed in a companion paper (Part I). It is shown that this model allows for a quantitative treatment of the strain-burst phenomenon. In particular, the scaling relations between different strain-burst-characteristic parameters which have been found by experiment are connected to the evolution of the dislocation microstructure and thus find a natural explanation.     

A global analysis of a non-linear system under parametric excitation

A strongly non-linear mechanical system consisting of a rigid damped bar subjected to a periodic parametric excitation is treated here in an exact manner. The emphasis is on the global behavior of this system which is carried out by using the point mapping and the cell-to-cell mapping methods. Even though the mechanical system is a simple one, yet it has a very complex global behavior. It is shown here that the newly developed theory of cell-to-cell mappings offers a tremendous advantage in obtaining the global domains of attraction of strongly non-linear dynamical systems.     

Index to volume 72

A numerical analysis of wave propagation in one dimensional inhomogeneous media is presented in this paper, the first (Part I) of two companion papers. The aim is to provide a natural and physical basis for the solution of the inverse scattering problem in one dimension. The paper includes (i) a discussion of some aspects of the numerical solution of the direct problem, (ii) typical results of calculation and (iii) a comparison between exact solutions and forward scattering approximation (FSA) solutions of the direct problem. It is shown that the FSA allows accurate calculations of the reflected wave and thus constitutes a sound basis for the solution of the inverse problem treated in Part II.     

The application of a semi-actuator disk model to sound transmission calculations in turbomachinery,part I: The single blade row

The work described in this paper generalizes the semi-actuator disk model of a blade row of Kaji and Okazaki [1, 2] to include a three-dimensional incident sound field. This field is equivalent, for example, to a radially fluctuating spinning mode in an annular compressor duct. The dependence of the earlier model upon the wavelength of the incident sound field is removed by adopting an average frequency approach. In this model, the acoustic energy entering the cascade of blades across one interface and leaving it across the other interface is summed over an infinite series of reflections within the cascade. This is equivalent, in effect, to the assumption of an incident delta function of pressure and thus gives an average frequency result. The validity of the model is demonstrated in a series of comparisons with data obtained from earlier models. The model is subsequently extended to include the effects of introducing cambered blades. In a companion paper (Part II), the model is applied to multiple blade rows.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped-clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.     

Comparison between the discrete and finite element methods for modelling an agricultural spray boom—Part 2: Automatic procedure for transforming the equations of motion from force to displacement input and validation

This paper validates the discrete element method for linear flexible multibody systems, elaborated in Part 1 of the paper, of which the flexible bodies are a composition of flexible beams. An automatic procedure is developed to convert the linear equations of motion of a multibody system from force to displacement input. By this procedure, support motions and displacements of actuators between the bodies can be employed as an input to the system. Furthermore, using this procedure, the methodology explained in Part 1, which was valid for tree structured systems can be extended to systems containing closed kinematic chains. The methodology of Part 1 is applied for the discrete and finite element approximations to model the horizontal behaviour of an agricultural spray boom. As the inputs to the spray boom are known under the form of positions, the equations of motions are converted from force to position inputs. The discrete and finite element approximations are compared based on accuracy and the complexity of the resulting models.     

Mean square response of a duffing oscillator to a modulated white noise excitation by the generalized method of equivalent linearization

The non-stationary random response of non-linear systems is considered. The technique of equivalent linearization is generalized for application to non-stationary non-linear random systems and several approximate methods of solution are presented. The example of a Duffing oscillator is studied in detail and its mean square response is evaluated and discussed.     

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.     

Non-linear vibration analysis of multilayer beams by incremental finite elements,Part I: Theory and numerical formulation

An incremental variational equation for non-linear motions of multilayer beams composed of n stiff layers and (n ? 1) soft cores is derived from the dynamic virtual work equation by an appropriate integration procedure. The kinematical hypotheses of Euler-Bernoulli and Timoshenko beam theories are used to describe the displacement fields of the stiff layers and cores respectively. An efficient solution procedure of incremental harmonic balance method type, with use of finite elements, is developed. To demonstrate its capability, some problems in free non-linear vibrations of multilayer beams are treated by using the procedure. Results are compared with those available in the literature. The effects of damping are also included in this investigation but are described in Part II [1] of this paper in which a number of undamped and damped forced non-linear vibration problems are studied. Results in the form of tables and plots are also presented and comparisons are made with those available in the literature.     

Interacting Fermi Liquid in Two Dimensions¶at Finite Temperature.¶Part I: Convergent Attributions

Using the method of a continuous renormalization group around the Fermi surface, we prove that a two-dimensional interacting system of Fermions at low temperature T is a Fermi liquid in the domain , where K is some numerical constant. According to [S1], this means that it is analytic in the coupling constant λ, and that the first and second derivatives of the self energy obey uniform bounds in that range. This is also a step in the program of rigorous (non-perturbative) study of the BCS phase transition for many Fermion systems; it proves in particular that in dimension two the transition temperature (if any) must be non-perturbative in the coupling constant. The proof is organized into two parts: the present paper deals with the convergent contributions, and a companion paper (Part II) deals with the renormalization of dangerous two point subgraphs and achieves the proof. Received: 27 July 1999 / Accepted: 31 May 2000     

Time-domain simulation of damped impacted plates. II. Numerical model and results

A time-domain model for the flexural vibrations of damped plates was presented in a companion paper [Part I, J. Acoust. Soc. Am. 109, 1422-1432 (2001)]. In this paper (Part II), the damped-plate model is extended to impact excitation, using Hertz's law of contact, and is solved numerically in order to synthesize sounds. The numerical method is based on the use of a finite-difference scheme of second order in time and fourth order in space. As a consequence of the damping terms, the stability and dispersion properties of this scheme are modified, compared to the undamped case. The numerical model is used for the time-domain simulation of vibrations and sounds produced by impact on isotropic and orthotropic plates made of various materials (aluminum, glass, carbon fiber and wood). The efficiency of the method is validated by comparisons with analytical and experimental data. The sounds produced show a high degree of similarity with real sounds and allow a clear recognition of each constitutive material of the plate without ambiguity.     

Application of Laplace transform technique to the solution of certain third-order non-linear systems

A number of papers have appeared on the application of operational methods and in particular the Laplace transform to problems concerning non-linear systems of one kind or other. This, however, has met with only partial success in solving a class of non-linear problems as each approach has some limitations and drawbacks. In this study the approach of Baycura has been extended to certain third-order non-linear systems subjected to non-periodic excitations, as this approximate method combines the advantages of engineering accuracy with ease of application to such problems. Under non-periodic excitations the method provides a procedure for estimating quickly the maximum response amplitude, which is important from the point of view of a designer. Limitations of such a procedure are brought out and the method is illustrated by an example taken from a physical situation.