Use of lagrange multipliers with polynomial series for dynamic analysis of constrained plates part I: Polynomial series

A theory for prediction of the dynamic response of a constrained plate is presented here. The boundaries of the plate may be partially fixed, its dynamic loading is due to elastically mounted vibrating machines and its constraints include beam-like stiffeners. The theory yields the eigenvalues and modal shapes of the plate and stiffeners which comprise the system. The solution, given in Part I, is based on Galerkin's method combined with use of special polynomial series presented by Kantorovich and Krylov. These eigenvalues are used in Part II [1] for response analysis of the complete system and the eigenvalues of the complete system will be obtained by the application of Lagrange equations and multipliers. The various coefficients used in the process are presented in the Appendices to the work. Comparisons with published results show good agreement.     

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].     

A unified quantum theory of mechanics and thermodynamics. Part IIa. Available energy

Part II of this three-part paper presents some of the most important theorems that can be deduced from the four postulates of the unified theory discussed in Part I. In Part IIa, it is shown that the maximum energy that can be extracted adiabatically from any system in any state is solely a function of the density operator associated with the state. Moreover, it is shown that for any state of a system, nonequilibrium, equilibrium or stable equilibrium, a unique propertyS exists which is proportional to the total energy of the system minus the maximum energy that can be extracted adiabatically from the system in combination with a reservoir. For statistically independent systems, propertyS is extensive, it is invariant during all reversible processes, and it increases during all irreversible processes.Part I of this paper appeared inFound. Phys. 6(1) (1976). The numbering of the sections, equations, and references in this part of the paper continues from those in Part I.     

Fokker action principles with self-action. II. Applications of the general theory to a single particle in uniform circular motion; modified Maxwell-Lorentz interaction; a model of the electron

Fokker action principles with self-action are studied. A program is outlined that offers long-odds hope for the construction of classical models of some elementary particles. Part I [31] gave an outline of the general theory, its possibilities, and its difficulties. Part II gives applications of the theory to circular motions of a single particle interacting with itself under modified Maxwell-Lorentz interaction. A specific, but ugly, model that arose in a thus-far unsuccessful attempt at a theory of the electron is discussed. Part III, which will be published shortly, will extend the results of Part II to a general interaction function Λ.     

The axisymmetrical response of a circular cylindrical double-shell system with internal damping

The axisymmetrical response of a circular cylindrical double-shell system with internal damping to a time-dependent surface load is determined by the matrix analysis method. For this purpose, the equations of vibration of the system based upon the Goldenveizer-Novozhilov theory are written as a coupled set of first order differential equations by the use of the state vector of the system. Once the vector has been determined by quadrature of the equations, the steady state response is calculated numerically together with the natural frequencies in terms of the elements of the transfer matrix of the system under any combination of boundary conditions. By the application of the method, the dynamic response and the resonant frequencies (the natural frequencies) are calculated numerically for a double-shell system simply supported at the edges.     

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.     

Identification of a continuous structure with a geometrical non-linearity. Part I: Conditioned reverse path method

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 this paper (Part I), the ability of the conditioned reverse path method, which is a frequency domain technique, to identify the behaviour of this structure is assessed. The companion paper (Part II) is devoted to the application of proper orthogonal decomposition, which is an updating technique, to the test example.     

Modified projective synchronization of a fractional-order hyperchaotic system with a single driving variable

In this paper, the modified projective synchronization (MPS) of a fractional-order hyperchaotic system is investigated. We design the response system corresponding to the drive system on the basis of projective synchronization theory, and determine the sufficient condition for the synchronization of the drive system and the response system based on fractional-order stability theory. The MPS of a fractional-order hyperchaotic system is achieved by transmitting a single variable. This scheme reduces the information transmission in order to achieve the synchronization, and extends the applicable scope of MPS. Numerical simulations further demonstrate the feasibility and the effectiveness of the proposed scheme.     

Dynamic response of composite plates with cut-outs,part II: Clamped-clamped plates

The forced and free dynamic response of plates with cut-outs formulated in Part I [1] is used to investigate the effect of cut-outs on the natural frequencies of clamped-clamped plates. The size, shape and location of the cut-out is expressed as a displacement dependent external loading. The plates considered are homogeneous and anisotropic. Lagrange's equations of motion lead to an infinite system of differential equations in time-dependent generalized co-ordinates with generalized forces which include the effects of the cut-outs. There is an infinite system of frequency equations for free vibrations. The infinite system is truncated to a finite system of equations depending upon the accuracy desired in frequency values. Results are given for square, clamped-clamped plates with centrally located square cut-outs for different modulus ratios. Good agreement is obtained when results for isotropic plates with cut-outs are compared with available theoretical and experimental results.     

Auditory physics. Physical principles in hearing theory. II

In the first part of this series of papers [Phys. Reports 62 (1980) 87–174] several physical problems are described that are relevant for the study of the auditory system. The present paper extends this treatment, it describes more facts upon which auditory theory is to be based and it delves considerably deeper into the mechanics of the cochlea (inner ear). The first two chapters treat nonlinear phenomena that are found in physiological and mechanical responses of the cochlea and in psychophysical experiments (listening tests carried out in human subjects). The main part of the paper is devoted to the mechanics of the cochlea. This part is preceded by an overview of the results of mechanical measurements on the cochlea. As it turns out, the newest experimental data present a specific challenge for cochlear mechanics.The central three chapters of the paper describe the development of a linear three-dimensional model of the cochlea. The main intention is to describe this model in a step-by-step fashion (hence the chapter headings borrowed from the field of architecture). Even in this simplified case, the solution for the response of the cochlear model is far from easy. Therefore, a few excursions are made into fields of physics and engineering in which related problems are worked out in analytical form. Just as in Part I of this series of papers, this extended treatment considerably deepens insight into the physical factors involved. Confrontation of the requirements for modelling described in the first few chapters with what has been achieved in the final part reveals how much study in the field of auditory physics remains to be done.     

Quantum mechanics of seeing

A human being viewing a defocused television tube with sweep voltages turned off will see point scintillations at sufficiently low intensities. We show that quantum mechanics predicts these scintillations. Furthermore, by assuming a response of the human nervous system of a type not inconsistent with experiment, measurement theory is used to show that these scintillations will be distributed in proportion to the magnitude squared of the electron wave function incident upon the television tube screen. This nervous system response is to break up the wave incident upon a spot on the retina into a number of similar waves transmitted by different nerves to the brain. The number of these waves is proportional to the incident energy density. Since the theory itself predicts the proper probability distribution, it is unnecessary to introduce a postulate for it.     

一类混沌系统的修正函数投影同步

针对一类混沌系统,通过构造合适的响应系统,提出一种修正函数投影同步方法.基于单向耦合同步原理,给出了两种响应系统的设计方案,由于只需向响应系统传送一个驱动变量即可实现混沌修正函数投影同步,因而实用性更强.利用Lyapunov稳定性理论给出了相应的证明. 最后以一个超混沌系统进行了数值仿真,仿真结果进一步表明该方法的有效性. 关键词： 混沌系统 修正函数投影同步 单向耦合     

NON-LINEAR AEROELASTIC ANALYSIS USING THE POINT TRANSFORMATION METHOD, PART 2: HYSTERESIS MODEL

This paper investigates the dynamic response of a two-dimensional aeroelastic system with structural non-linearity represented by hysteresis. The formulations of the point transformation method developed in Part 1 of this study for the aeroelastic system with a freeplay model is extended for a hysteresis model. These formulations can be applied not only to predict the amplitude and frequency of limit cycle oscillations, but also to detect complex aeroelastic responses such as periodic motion with harmonics, period doubling, chaotic motion and the coexistence of stable limit cycles. It is shown that the point transformation technique is the most suitable to analyze the aeroelastic response of systems containing piecewise continuous restoring forces.     

A WAVENUMBER APPROACH TO MODELLING THE RESPONSE OF A RANDOMLY EXCITED PANEL, PART I: GENERAL THEORY

Part I of this paper presents a self-contained analytical framework for determining the vibro-acoustic response of a plate to a large class of random excitations. The wavenumber approach is used, which provides an insight into the physical properties of the panel response and enables us to evaluate efficiently the validity of several simplifying assumptions. This formulation is used in Part II for predicting the statistical response of an aircraft panel excited by a turbulent boundary layer. In this paper, we first provide a general statement of the problem and describe how the spectral densities of the panel response can be obtained from an analysis of the system response to a harmonic deterministic excitation and a statistical model for the forcing field. The harmonic response of the system is then expanded as a series of the eigenmodes of the fluid-loaded panel and these fluid-loaded eigenmodes are approximated by a perturbation method. Then, we evaluate the conditions under which this series simplifies into a classical modal formulation in terms of the in vacuo eigenmodes.To illustrate the use of a wavenumber approach, we consider three examples, namely, the vibro-acoustic response of a panel excited by an incidence diffuse acoustic field, by a fully developed turbulent flow and by a pressure field which is spatially uncorrelated from one point to another. Convergence properties of the modal formulations are also examined.     

Study of airfoil gust response alleviation using an electro-magnetic dry friction damper. Part 2: Experiment

In Part 1 of this work, a theoretical simulation study of the non-linear gust response of a three degree-of-freedom typical airfoil section with a control surface using an electro-magnetic dry friction damper is presented. For validation of this theoretical model, an electro-magnetic dry friction damper has been designed and an experimental investigation of the gust response has been carried out in a wind tunnel. Results for both periodic and linear frequency sweep gust excitations have been computed and measured. The fair to good quantitative agreement between theory and experiment verifies that the present electro-magnetic dry friction damper can be used to alleviate the gust response, especially for the plunge and pitch responses. It also shows that the present theoretical method can be successfully applied to determine the non-linear gust response when an electro-magnetic dry friction damper is used in the linear aeroelastic system.     

Dynamic characterization of hysteresis elements in mechanical systems. I. Theoretical analysis

The pre-sliding-pre-rolling phase of friction behavior is dominated by rate-independent hysteresis. Many machine elements in common engineering use exhibit, therefore, the characteristic of "hysteresis springs," for small displacements at least. Plain and rolling element bearings that are widely used in motion guidance of machine tools are typical examples. While the presence of a hysteresis element may mark the character of the resulting dynamics, little is to be found about this topic in the literature. The study of the nonlinear dynamics caused by such elements becomes imperative if we wish to achieve accurate control of such machines. In this Part I of the investigation, we examine a single-degree-of-freedom mass-hysteresis-spring system and show that, while the free response case is amenable to an exact solution, the more important case of forced response has no closed form solution and requires other methods of treatment. We consider harmonic-balance analysis methods (which are common analysis tools in engineering) suitable for frequency-domain treatment, in particular the approximate describing function (DF) method, and compare those results with "exact" numerical simulations. The DF method yields basically a linear equation with amplitude-dependent modal parameters. We find that agreement in the frequency response function, between DF and exact solution, is good for small excitation amplitudes and for very large amplitudes. Intermediate values, however, show high sensitivity to amplitude variations and, consequently, no regular solution is obtainable by either approach. This appears to be an inherent property of the system pointing to the need for developing further analysis methods. Experimental verification of the analysis outlined in this Part I is given in Part II of the paper.     

Evidence of Dispersion Relations for the Nonlinear Response of the Lorenz 63 System

Along the lines of the nonlinear response theory developed by Ruelle, in a previous paper we have proved under rather general conditions that Kramers-Kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of Axiom A non equilibrium steady state systems to weak monochromatic forcings. We present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed Lorenz 63 system, by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions. Some new theoretical results, showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables, are also presented. Our findings confirm the conceptual validity of the nonlinear response theory, suggest that the theory can be extended for more general non equilibrium steady state systems, and shed new light on the applicability of very general tools, based only upon the principle of causality, for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals, in situations where the fluctuation-dissipation relation is not of great help.     

Onsager–Casimir reciprocity relations for open gaseous systems at arbitrary rarefaction IV. Rotating systems

The statistical approach to the nonequilibrium thermodynamics presented in Part I of this paper was generalized to rotating gaseous systems. Basing on the Boltzmann equation written for rotating reference frame the kinetic coefficients satisfying the Onsager–Casimir reciprocity relations are obtained in a general form. An example of the application of this theory is given. Some new phenomena arising only in rotating system are found.     

An experimental investigation of unsteady surface pressure on an airfoil in turbulence—Part 2: Sources and prediction of mean loading effects

An experimental investigation into the response of an airfoil in turbulence was undertaken and the results are presented in a two part series of papers. The effects of mean loading on the airfoil response are investigated in Part 1 with the likely origins discussed in this paper (Part 2). Unsteady pressure measurements were made on the surface of a NACA 0015 airfoil immersed in grid turbulence (λ/c=13%) for angles of attack α=0-20°. This paper (Part 2) presents the causes of the low-frequency reduction and high-frequency increase observed in measured lift and pressure spectral levels. Scaling lift spectra on the mean lift reveals the increase in lift spectral level for reduced frequencies greater than 10 is closely related to the airfoils mean pressure field. Based on analysis of the chordwise and spanwise pressure correlation length scale, the reduction in lift spectral level at low reduced frequency appears to result from distortion of the inflow by the mean velocity field. A possible model is developed that accurately predicts mean loading effects on lift spectra. This model uses a circular cylinder fit to the airfoil to compute effects of distortion on the inflow turbulence. The distorted inflow velocity spectrum is then used with Amiet's theory to predict the unsteady loading. This model successfully captures the reduction observed in measured lift spectra at low reduced frequencies. Furthermore, it is shown that the angle of attack effects arising from inflow distortion are significant only when the relative scale of the inflow turbulence to airfoil chord is sufficiently small (λ/c=13% for present experiment).     

Enclosed laser ablation of 20 nm aluminium films using a continuous wave diode laser: II—Theoretical heating considerations and proposed mechanisms

Laser ablation using a 120 W continuous wave diode laser was carried out upon a thin aluminium layer which had been hermetically sealed between two layers of polymer. The results (presented in Part I) have shown that the film could be partially transparentised but with an efficiency which was far less than that seen for a Q-switched laser. In this paper, the thermal effects within the films are analysed and a mechanism by which enclosed surface laser ablation occurs is proposed.