Simplified computation of the Volterra frequency response functions of non-linear systems

A new and more straightforward algorithm for computing the higher order frequency response functions of a broad class of non-linear systems is presented. The algorithm is not limited to specific harmonic inputs, and is applicable to a broad class of non-linear differential, difference, or time-delayed differential equation models. It is readily automated in software, and results are illustrated by means of an example.     

Interpretation of crack-induced rotor non-linear response using instantaneous frequency

Instantaneous frequency was introduced to describe the dependency of frequency components on time for non-stationary signals. A powerful alternative to the Fourier-based spectral analysis that provides insufficient resolution for the temporal progression of all frequency components, the concept of instantaneous frequency has rarely been applied to machine monitoring and diagnosis. The confusing fact that instantaneous frequencies determined could occasionally be negative and the associated amplitudes could be infinite for certain types of vibration signals inevitably limits the adaptability of the concept to fault detection as a result. Significant insight into the applicability of instantaneous frequency is gained through re-examining the fundamentals upon which the concept was first defined. It is found that the Hilbert-transform-based definition of instantaneous frequency is applicable only to signals of monocomponent, thus implying the need for separating a multicomponent signal into its monocomponent subsets. Misuse of the definition to multicomponent signals would result in the individual instantaneous frequency associated with each inherent monocomponent being averaged and thus obscure the underlying characteristics of the signal. A mathematically complete decomposition scheme effective in resolving a multicomponent signal into an orthogonal space spanned by its intrinsic monocomponents is explored. Several examples are considered to show that the scheme not just enables the removal of the difficulties commonly encountered in applying instantaneous frequency but also imparts valid renditions to the interpretation of fault-induced bifurcation and dynamic instability.     

An explicit frequency response function formulation for multi-degree-of-freedom non-linear systems

This paper presents a technique for the explicit formulation of frequency response functions (FRFs) for nonlinear MDOF systems, yielding closed-form analytical expressions. The technique produces FRFs at selected coordinates only, regardless of the system's size or the type of nonlinearity. The nonlinear formulation in this work is based on first-order describing functions, which represent the nonlinearities by amplitude-dependent coefficients. For the sake of clarity, the paper is focused on cubic stiffness and friction damping nonlinearities, although, given the generality of the formulation, the inclusion of other types should be straightforward. Conducted entirely in the physical domain, the approach is based on manipulating the coefficients stored in the explicitly formulated system matrices, thus the term “explicit”. An enhanced version, which can tackle large systems, is also presented and validated against real measurements taken from a test rig. It was found that the nonlinear behaviour was predicted with reasonable accuracy in all cases studied.     

A hybrid expansion method for frequency response functions of non-proportionally damped systems

This study is aimed at eliminating the influence of the higher-order modes on the frequency response functions (FRFs) of non-proportionally viscously damped systems. Based on the Neumann expansion theorem, two power-series expansions in terms of eigenpairs and system matrices are derived to obtain the FRF matrix. The relationships satisfied by eigensolutions and system matrices are established by combining the two power-series expansions. By using the relationships, an explicit expression on the contribution of the higher-order modes to FRF matrix can be obtained by expressing it as a sum of the lower-order modes and system matrices. A hybrid expansion method (HEM) is then presented by expressing FRFs as the explicit expression of the contribution of the higher-order modes and the modal superposition of the lower-order modes. The HEM maintains original-space without having to use the state-space equation of motion such that it is efficient in computational effort and storage capacity. Finally, a two-stage floating raft isolation system is used to illustrate the effectiveness of the derived results.     

Frequency response of non-linear single degree-of-freedom systems

Simple linear Kelvin or Maxwell models cannot predict adequately the response of many practical systems to vibration excitation. A more realistic model consisting of a mass between two preloaded non-linear (cubic elasticity springs) and restrained by a Coulomb and viscous damper is proposed. A harmonic motion solution, satisfying “on the average” the non-linear differential equation of motion of the model is developed whereby equations for frequency response curves are obtained. Expressions for relative and absolute transmissibility as well as their values at resonance are developed, these are shown to properly reduce to the respective exact expressions of a Kelvin model and friction damped linear spring. Although the model is intended mainly for quantitative design in packaging engineering problems, it is suitable for studying general vibration isolation problems as well.     

A new method for the accurate measurement of higher-order frequency response functions of nonlinear structural systems

Higher-order frequency response functions (FRFs) are important to the analysis and identification of structural nonlinearities. Though much research effort has been devoted recently to their potential applications, practical issues concerning the difficulty and accuracy of higher-order FRF measurement have not been rigorously assessed to date. This paper presents a new method for the accurate measurement of higher-order FRFs. The method is developed based on sinusoidal input, which is ideal for exciting a nonlinear structure into desired regimes with flexible control, and the correlation technique, which is a novel signal processing method capable of extracting accurate frequency components present in general nonlinear responses. The correlation technique adopted is a major improvement over Fourier transform based existing methods since it eliminates leakage and aliasing errors altogether and proves to be extremely robust in the presence of measurement noise. Extensive numerical case studies have been carried out to critically assess the capability and accuracy of the proposed method and the results achieved are indeed very promising. Interesting nonlinear behavior such as frequency shift and jump have been observed in first-, second- and third-order FRFs, as well as solitary islands which have been identified over which higher-order FRFs virtually do not change as input force amplitude varies. Higher-order FRFs over such solitary islands are essentially their theoretical counterparts of Volterra transfer functions which can be measured with very low input force and can be profitably employed for the identification of physical parameters of structural nonlinearities. Subsequently, a nonlinear parameter identification method has also been developed using measured higher-order FRFs and results are presented and discussed.     

Advanced vision systems for computer-integrated manufacture Part II

This paper describes the application of computer vision techniques to solve a variety of industrial problems. Systems for robot guidance, quality assurance and inspection, and strain measurement are described and details of hardware and software components of each application are provided.     

利用键合图的高频响插装阀非线性建模方法研究

高频响插装阀在系统频响要求较高的大流量凋速系统中应用比较广泛,通过分析二位三通高频响插装阀的工作原理,利用功率键合图建立二位三通插装阀的非线性动态模型,模型输出与实验数据的比较表明,该模型较好地反映了二位三通插装阀的动态特性.     

Decoupling analysis on nonlinear system based on the modified generalized frequency response functions

This paper presents a nonlinear decoupling approach based on the Modified Generalized Frequency Response Functions (MGFRFs) and the nonlinear feature of phase invariance, for the pure nonlinearity-input nonlinear system. The MGFRFs are defined by combining the ‘homotopy’ GFRFs and phase information of the system input. The nonlinear feature of phase invariance is extracted based on MGFRFs. The decoupling approach is proposed based on MGFRFs and extended from the pure tone excitation to the multi-tone excitations by considering phase invariance. Numerical simulation and experimental investigation were carried out, whose results have shown that nonlinear feature of phase invariance is correct and reasonable and the proposed decoupling approach is valid and feasible. The proposed decoupling approach can be employed to identify the excitation sources and to estimate nonlinear system parameters for the pure nonlinearity-input nonlinear vibration system.     

Modelling CIM systems part II: the generic functions

A generic function is an abstraction of the activities which may appear differently in various manufacturing industries but actually have the same types of inputs and outputs. The generic functions and their structure represent the fundamental manufacturing activities and their interrelationships and form the basic structure of CIM. This paper discusses the generic functions and their structure in the domain of the discrete, engineering manufacturing industry. It draws upon a current research programme with a major UK manufacturer together with other research programmes mainly the ICAM program in the USA and the ESPRIT project in Europe.     

The Huang Hilbert Transform for evaluating the instantaneous frequency evolution of transient signals in non-linear systems

Traditional approaches for the analysis of transient signals are generally based on the apriori knowledge of the system under test for choosing a preliminary set of waveforms; consequently, they use a mathematical algorithm to decompose the signal itself into a suitable combination of the chosen waveforms. Conversely, this work is aimed to investigate the possibility of extracting the features of transient signals through the evaluation of their instantaneous frequency evolution. For this aim, the Huang Hilbert Transform (HHT) has been exploited (i) to decompose the input signal into a set of Intrinsic Mode Functions (IMFs), (ii) to extract the IMFs analytical signals, (iii) to evaluate their amplitude and phase evolutions, (iv) to compute the instantaneous frequency of the input signal and (v) to extract the signal information searched for. In order to evaluate its performance, the proposed approach has been firstly applied to a synthesized signal with known instantaneous amplitude and frequency evolution. Successively, in order to assess the reliability of HHT results with signals acquired on experimental circuits, the current flowing in an actual RLC circuit during its free natural oscillation has been analyzed. With the aim of analyzing the performance gained also in the presence of evident non-linearities, a saturable inductor has been introduced in the test circuit. Also in this case, by comparing the achieved results with those shown by different traditional approaches, great advantages have been experienced in terms of accuracy. Furthermore, beyond the accurate frequency representation, the experimental results evidenced the intrinsic ability of the proposed approach to extract meaningful information related to the knowledge of the underlying process. Finally, it is worth noting that the results reported in this paper requested no apriori knowledge about the signal/process under test.     

Hydroforming of anisotropic aluminum tubes: Part II analysis

Part I presented an experimental investigation of hydroforming of Al-6260-T4 tubes and a simple two-dimensional model of the process. Relatively long, extruded circular tubes were formed against a square die with rounded corners, with simultaneous application of axial feeding. Localized wall thinning was reported to occur at mid-span which, accentuated by friction, led to burst. Part II presents fully 3D models of the process that include friction as well as more advanced constitutive models shown in previous studies to be essential for simulation of burst in free hydroforming of aluminum alloy tubes. The models are used to simulate several of the experiments of Part I, emphasizing the prediction of all aspects of the forming process, including wall thinning and its localization that lead to rupture. A shell element model is shown to capture the majority of the structural features of the process very successfully. However, even with the implementation of advanced constitutive models, it fails to reproduce correctly the localization of wall thinning. It is demonstrated that switching to solid elements coupled to non-quadratic yield functions results in accurate predictions of all aspects of the problem, including the onset of rupture. Apparently, slow growing depressions that develop at the interface between the flattened part of the cross section that is in contact with the die and the rounded part that is not, have a complex three dimensional stress state requiring accurate modeling offered by solid elements. Furthermore, the evolution of these depressions is only reproduced with accuracy when in addition non-quadratic yield functions are adopted.     

A Galerkin method for the steady state analysis of harmonically excited non-linear systems

A Galerkin method for the computation of the steady state of harmonically excited non-linear systems in the frequency domain is presented. The non-linear differential equations of motion are transformed, via the Galerkin technique, to a minimized system of non-linear algebraic equations in the frequency domain. These equations are then solved by a specially developed iteration procedure based on Powell's minimization method. The solution technique proves to be suitable for non-linear systems with arbitrary and numerous non-linearities (e.g. joints with clearance, non-linear springs and dampers).     

Stability of journal bearing systems using non-Newtonian lubricants: a non-linear transient analysis

The effect of the non-Newtonian behaviour of lubricants, resulting from the addition of polymers, on the performance of hydrodynamic journal bearings is investigated. The model of non-Newtonian lubricant developed by Dien and Elrod is taken into consideration. An attempt has been made to evaluate the mass parameter (a measure of stability) besides finding out the steady-state characteristics of finite journal bearings with non-Newtonian lubricants. A non-linear time transient analysis is carried out for the stability analysis.     

On the use of the transmissibility concept for the evaluation of frequency response functions

The dynamic characteristics of a structure are often derived from a set of measured frequency response functions (FRFs). However, it may happen that the measurement of certain FRFs is impossible, as they are related to some points of interest that may become physically inaccessible in operational conditions. In this circumstance, it is useful to have some tools that can provide the prediction of such dynamic information. The transmissibility concept can play an important role to circumvent these situations. In fact, there are important properties associated to the transmissibility—the relationship between two sets of responses, for a given set of applied forces, extended to a general multiple degree-of-freedom system. In this paper, some important properties of the transmissibility matrix will be presented. Additionally, it will be shown that if a modification is operated on the original system - using both theoretical and experimental models - it is possible to estimate the FRFs associated to the unknown co-ordinates, without the necessity of measuring the responses on those co-ordinates.     

Electromagnetic actuator for determining frequency response functions of dynamic modal testing on milling tool

Machine tools are the main driving forces of industrialization of a country. However, poor machinability because of chatter vibration results in poor surface quality, excessive noise, and reduced material removal rate. Modal testing is a useful method to investigate dynamic properties of a cutting tool system and improve material removal rate. However, at present, modal testing using impact hammer is limited by certain problems. This paper developed a non-contacting electromagnetic actuator (EMA) to determine frequency response functions (FRFs) under amplitude and speed dependencies of cutting milling tools. The geometry was designed using magnetic circuit analysis and generalized machined theory before finite element analysis was conducted using magnetostatic-ansys software. Next, EMA was used as a contacting and non-contacting exciter of a conventional milling machine to determine the FRFs and dynamic properties of milling tool with amplitude and speed dependencies including comparison with static FRFs. Subsequently, dynamic properties and FRFs are used to establish stability lobe diagram. Stability lobe diagram also shows an improvement of up to 5% of depth of cut at lower spindle speed. In conclusion, by generating force that applies to static and dynamic modal testing, an EMA can determine dynamic properties and stability lobe diagram for increasing material removal rate and production rate.     

Acoustic based modal filtering of orthogonal radiating functions for global error sensing: Part II. Implementation

This is the second of two companion papers, which summarise the theoretical and experimental work carried out to develop and compare a generalised acoustic based sensing strategy for radiated sound power. In this paper we focus on the implementation of an acoustic-centric sensing system; based on different orthogonal decompositions. An important consideration here is that the sensing strategy is capable of being applied to a large and complex noise source. A sensing array consisting of 128 microphones and a custom-built electronic filtering system used in the data fusion process is demonstrated. We examine different estimates of the global error criterion, radiated sound power, based on orthogonal multipole and radiation modes. The experiments are the extension of the simulations in the first part of this work, where each orthogonal constituent is filtered, in real time, from the 128 microphone signals. The estimates of radiated power are compared for a two-dimensional structure under two sets of boundary conditions. The sensing comparisons are extended to a comparison of the achieved global attenuation when each of the different orthogonal constituents are minimised by a feedforward control system.     

煤的二次离子质谱数据解析1

作为全面认识煤的二次离子质谱数据与结果的开始,本文展示了典型煤样的低质量端的二次离子质谱图。利用了含四位小数的准高分辨数据在1至20质量数据区域确定了20种正离子组成;在1至31质量数区域确定了26种负离子组成。其中,简单氮氢离子─—H3N （m／z17．0272）、H4N （m／z18．0355）、H2N-（m／z16.0187）、HN-（m／z15．0107）,以及破氮组成离子CN-（m／z26．0036）等许多离子的组成,在煤的二次离子质谱及激光电高质谱中均属于首次检出。本文还对同类化学组成形成正离子与负离子的差异,以及对合同位素组成离子的同位素比值利用的潜力,进行了详细讨论。     

Identification of damage on a substructure with measured frequency response functions

Recently the authors tried to find damage position only using measured frequency response functions. According to their work, it seems that the algorithm is very practical since it needs only measured frequency responses while other methods require exact analytic model. But when applying the method to a real structure, it requires lots of experiment. The authors, in this time, propose a method to reduce its experimental load by detecting damage within a substructure. This method searches damages not within an entire structure but within substructures. In addition, damage severity was treated in this paper since it is worthy to know damage severity. Optimization technique is used to estimate damage level using measured responses and damage model. Two test examples, a plate and a jointed structure, are chosen to verify the suggesting method.