PROPER ORTHOGONAL DECOMPOSITION FOR MODEL UPDATING OF NON-LINEAR MECHANICAL SYSTEMS

Proper orthogonal decomposition (POD), also known as Karhunen–Loeve (K–L) decomposition, is emerging as a useful experimental tool in dynamics and vibrations. The POD is a means of extracting spatial information from a set of time-series data available on a domain. The use of (K–L) transform is of great help in non-linear settings where traditional linear techniques such as modal-testing and power-spectrum analyses cannot be applied. These decomposition can be used as an orthogonal basis for efficient representation of the ensemble. The POM have been interpreted mainly as empirical system modes and the application of POD to measured displacements of a discrete structure with a known mass matrix leads to an estimation of the normal modes. We investigate the use of the proper orthogonal modes of displacements for the identification of parameters of non-linear dynamical structures with an optimisation procedure based on the difference between the experimental and simulated POM. A numerical example of a beam with a local non-linear component will illustrate the method.     

COUPLING OF NON-LINEAR SUBSTRUCTURES USING VARIABLE MODAL PARAMETERS

This paper presents a general methodology for the coupling analysis of systems with relatively weak non-linearities by assuming that the response remains harmonic under harmonic excitation. Standard coupling methods and their current shortcomings were discussed first. Two ways of obtaining non-linear modal parameter variations, namely profile constructing and parameter extracting, were presented next. The profile constructing method uses the system's spatial data directly, while the parameter extracting method is based on a non-linear modal analysis of measured response data. Through numerical test cases, it was shown that both methods yielded virtually identical results. An iterative algorithm for the coupling of non-linear subsystems was presented in a form compatible with profile building. A six-degree-of-freedom system with cubic stiffness non-linearity was chosen for a detailed numerical study. Two subsystems, one linear and the other non-linear, were coupled to obtain the modal parameter variations of the coupled system. Using the non-linear modal parameters, the response of the coupled system was predicted at various force levels and the findings were checked via direct simulations using the harmonic balance method. Finally, the methodology was validated by coupling experimentally derived non-linear modal models of two substructures. As for the numerical study, the response of the coupled non-linear structure was predicted at various force levels and the findings were checked against direct measurements. Very good agreement was obtained in all cases studied.     

SPECTRAL ANALYSES OF NON-LINEAR INTERACTIONS

Signals pertaining to motions of a non-linear oscillator and a set of quadratically coupled oscillators are studied using higher-order spectral analyses. The analyses are used to understand the role of phase coupling in non-linear interactions between two or more Fourier components. For certain motions of weakly non-linear systems, analytical approximations are obtained for the relevant higher-order spectra and coherence functions. The analytical approximations are useful in understanding the dependence of bispectra on the non-linearities. Numerical studies are conducted to verify analytical predictions and to illustrate the usefulness of spectral analyses for different cases.     

NON-LINEAR CONTROL OF BUCKLED BEAMS UNDER STEP LOADING

The present paper studies a strategy for the active non-linear control of the oscillations of simply supported buckled beams, in order to mitigate the effects of dynamic loading on the vibration amplitudes and prevent dangerous instability phenomena. First, an analysis of the symmetric non-linear behaviour of the structure without any control system is carried out. In order to control the non-linear vibrations of the beam, an active tendon control system is adopted. A control method based on non-linear optimal control using state feedback is developed and the solution of the non-linear optimal control problem is obtained by representing system non-linearities and performance indices by power series with the help of algebraic tensor theory. In this work, general polynomial representations of the non-linear control law are obtained up to the fifth order. This solution procedure is employed to analyse the influence of the resulting non-linear control laws on the dynamic behaviour of a buckled beam under a lateral step load. This arch-like structural system is highly non-linear and under compressive lateral loading may suffer snap-through buckling. This may cause undesirable stresses and/or displacements, leading as a rule to a failure of the structural system. So, special attention is given to the determination of the potential of the present control methodology for efficiently limiting extreme state responses and preventing the snap-through buckling. Numerical results indicate that the control algorithm can effectively increase the load-carrying capacity of the buckled beam without demanding large control forces. Also, this study can be used as a basis for the non-linear control of more complex structures and for the design of control systems.     

NON-LINEAR MECHANICAL SYSTEMS IDENTIFICATION USING LINEAR SYSTEMS WITH RANDOM PARAMETERS

A method to identify the parameters involved in the non-linear terms of randomly excited mechanical systems is presented. It is based on the minimisation of an index function which reflects the difference between an analytical approximation of the power-spectral density function response and the measured one. Using the concept of non-linear modes, an equivalent linear system with random parameters is used to approximate the power-spectral density. The method is applied to the ECL benchmark.     

FITTING MEASURED FREQUENCY RESPONSE USING NON-LINEAR MODES

The article is devoted to curve fitting of the measured frequency response functions (FRFs) of an actual beam with a non-linear component. A frequency response model, based on the non-linear normal modes (NNMs) approximated by the Ritz–Galerkin method and on a superposition assumption is built up. Identification of NNM is performed by minimising a cost function involving synthetised FRF and measured data and leads to natural frequencies, damping factors, mode shapes as functions of modal amplitudes.     

含有非线性摩擦阻尼的失谐叶盘系统受迫响应研究

基于谐波平衡法,提出一种可用于分析含有非线性摩擦阻尼叶盘振动响应的多谐波法,对含有非线性干摩擦阻尼的失谐叶盘系统的受迫响应进行仿真计算.研究失谐叶盘系统在不同的耦合强度、失谐程度、粘性阻尼系数、干摩擦强度等系统参数影响下的受迫响应特性.同时对叶片的非线性干摩擦阻尼的散乱失谐和刚度失谐的耦合影响进行研究,总结含有非线性干摩擦阻尼的失谐叶盘系统的受迫响应规律.     

NON-LINEAR SYSTEMS REPRESENTATION USING ARMAX MODELS WITH TIME-DEPENDENT COEFFICIENTS

An approach that models a non-linear process operating over a large dynamic range is developed and validated. This approach is based on stochastic Time-varying AutoRegressive Moving Average with eXogeneous inputs (TARMAX) models. The TARMAX model coefficients are explicit functions of time and vary in a deterministically organised fashion. A novel model parameter estimation method fully based on linear operations is presented. The estimation approach is characterised by a low computational complexity and requires no initial guess of the parameter values. The ability of the approach to model non-linear processes is validated by addressing problems dealing with improving the estimation of mass air flow going into an automobile engine. First, a TARMAX model is used to capture the dynamics of the engine process relating air flow provided by a laboratory grade sensor and three input signals available in the engine electronic controller. The TARMAX model is used to simulate the complex relationship between the output and the three input signals. Second, TARMAX models are used to anticipate the future response of a hot-wire-based mass air flow sensor (MAF) in order to obtain an accurate estimate of the cylinders air charge. The estimated TARMAX models prove to have good simulation and prediction capabilities. All models are estimated using actual production vehicle data.     

A FREQUENCY DOMAIN METHOD FOR ESTIMATING THE PARAMETERS OF A NON-LINEAR STRUCTURAL DYNAMIC MODEL THROUGH FEEDBACK

A unifying perspective of non-linear structural dynamic systems as linear in the open loop with non-linear feedback in the closed loop has recently been revisited by the authors. The authors have previously used feedback to derive a new formulation of frequency response function matrices of non-linear systems, which are described as modulations of nominal linear systems. The modulation creates a pseudo-separation of the linear and non-linear dynamics of the system. The present article derives a new method for estimating parameters of non-linear parametric models that uses internal feedback to account for non-linearities. The main advantage of the new formulation of non-linear system identification is its simplicity. Moreover, the method estimates the linear frequency response matrix and non-linear system parameters at forced and unforced degrees of freedom of general multiple-degree-of-freedom non-linear systems simultaneously. This article demonstrates the implementation of this method on simulated data from single- and multiple-degree-of-freedom lumped parameter models.     

NON-LINEAR DYNAMICS TOOLS FOR THE MOTION ANALYSIS AND CONDITION MONITORING OF ROBOT JOINTS

Time series from non-damaged and three types of damaged robot joints are considered and analysed from the viewpoint of non-linear dynamics. The embedding spaces for the four types of signals are recovered. The application of surrogate data tests is used to prove the presence of non-linearities in the joints. The results suggest a rise in unstable behaviour due to the introduction of backlash in robot joints. The chaotic behaviour gets stronger with the increase of the backlash extent. This is confirmed by the increase of the embedding dimension as well as by the increase of the Lyapunov exponents and the correlation dimension with the backlash increase. A straightforward method for condition monitoring using non-linear dynamics characteristics, based on a classification procedure, is suggested.     

完全参数激励下索梁耦合系统非线性动力学分析

研究完全参数激励下弹性拉索与悬臂梁耦合结构的非线性动力学问题。建立索梁耦合结构力学模型,利用Hamilton 原理建立索－梁耦合系统的非线性动力学方程,利用Galerkin方法将索－梁耦合系统的非线性运动偏微分方程离散为一组常微分方程。利用多尺度法分析研究索－梁耦合动力学系统的非线性振动,用Runge-Kutta法对数学模型进行数值计算,得到当梁与索的频率比为2:1时,系统发生严重的参数共振,同时探讨阻尼参数对索－梁耦合系统非线性动力学的影响,并提出对工程有实际意义的结论。     

Non-linear feature identifications based on self-sensing impedance measurements for structural health assessment

This paper presents the application of a non-linear feature identification technique for structural damage detection. This method is coupled with the impedance-based structural health monitoring (SHM) method, which utilises electromechanical coupling properties of piezoelectric materials. The non-linear feature examined in this study is in the form of autoregressive coefficients in the frequency domain autoregressive model with exogenous (ARX) inputs, which explicitly considers non-linear system input/output relationships. The applicability of this non-linear feature for damage identification is investigated in various frequency ranges using impedance signals measured from a laboratory-test structure. The performance of the non-linear feature is also compared with those of linear features typically used in impedance methods. This paper reinforces the utility of non-linear features in SHM and suggests that their sensitivity in different frequency ranges may be leveraged for certain applications.     

THE NON-LINEAR MECHANICAL PROPERTIES OF AN AIRSPRING

The paper deals with non-linear mechanical properties of an airspring. The geometry structures and characters in the changing shape of the airspring are examined. The changing volume of the airspring is obtained by using a numerical calculation method and the relationship curve between the force applied on the airspring and the displacement of the airspring are given. The method used for obtaining the non-linear mechanical properties of the airspring is practical and effective and may apply to other kinds of airsprings.     

Non-linear flap-lag-extensional vibrations of rotating, pretwisted, preconed beams including Coriolis effects

The effects of pretwist, precone, setting angle, Coriolis forces and second degree geometric non-linearities on the steady state deflections, coupled frequencies and mode shapes of rotating, torsionally rigid, cantilevered beams are studied in this investigation. The equations governing flap-lag-extensional motion are derived including the effects of large precone (a component of sweep) and retaining geometric non-linearities up to second degree. The Galerkin method, with non-rotating normal modes, is used for the solution of both steady state non-linear equations and linear perturbation equations. The results indicate that the second degree geometric non-linear terms, which vanish for zero precone, can produce frequency changes of engineering significance (of the order of 20% on the fundamental mode, and about ±4% on the second mode). The results further indicate that the linear and non-linear Coriolis effects must be included in analyzing thick blades while these effects can be neglected in analyzing thin blades, typical of advanced turboprop blade configurations. For those cases where the effect is significant, the linear and non-linear Coriolis effects oppose one another, the non-linear effects generally being stronger.     

NON-LINEAR DYNAMIC BEHAVIOR OF THERMOELASTIC CIRCULAR PLATE WITH VARYING THICKNESS SUBJECTED TO NON- CONSERVATIVE LOADING

The non-linear dynamic behaviors of thermoelastic circular plate with varying thickness subjected to radially uniformly distributed follower forces are considered. Two coupled non-linear differential equations of motion for this problem are derived in terms of the transverse deflection and radial displacement component of the mid-plane of the plate. Using the Kantorovich averaging method, the differential equation of mode shape of the plate is derived, and the eigenvalue problem is solved by using shooting method. The eigencurves for frequencies and critical loads of the circular plate with unmovable simply supported edge and clamped edge are obtained. The effects of the variation of thickness and temperature on the frequencies and critical loads of the thermoelastic circular plate subjected to radially uniformly distributed follower forces are then discussed.     

NON-LINEAR VIBROISOLATION PADS DESIGN,NUMERICAL FEM ANALYSIS AND INTRODUCTORY EXPERIMENTAL INVESTIGATIONS

Design and theoretical and experimental investigation of vibroisolation pads with non-linear static and dynamic responses is the objective of the paper. The analytical investigations are based on non-linear finite element analysis where the load–deflection response is traced against the shape and material properties of the analysed model of the vibroisolation pad. A new model of vibroisolation pad of antisymmetrical type was designed and analysed by the finite element method based on the second-order theory (large displacements and strains) with the assumption of material's non-linearities (Mooney–Rivlin model). Stability loss phenomenon was used in the design of the vibroisolators, and it was proved that it would be possible to design a model of vibroisolator in the form of a continuous pad with non-linear static and dynamic response, typical to vibroisolation purposes. The materials used for the vibroisolator are those of rubber, elastomers, and similar ones. The results of theoretical investigations were examined experimentally. A series of models made of soft rubber were designed for the test purposes. The experimental investigations of the vibroisolation models, under static and dynamic loads, confirmed the results of the FEM analysis.     

An improved tool–holder model for RCSA tool-point frequency response prediction

In addition to the precise kinematic motions of the machine tools and spindles, machining accurate parts necessitates controlling the dynamic behavior of the tool tip with respect to the workpiece. High-fidelity models of tool-tip dynamics can be used to select operating parameters that improve the accuracy by reducing the effect of vibrations. To effectively model the tool-tip dynamics for arbitrary tool-and-holder combinations using the receptance coupling substructure analysis (RCSA) technique, highly accurate and numerically efficient models of the tool–holder dynamics are needed. In this paper, we present a tool–holder model that incorporates a spectral-Tchebychev technique with the Timoshenko beam equation to obtain a completely parameterized solution. Comparison of the tool–holder model to a three-dimensional finite elements solution shows that the dynamic behavior is captured with sufficient accuracy. The tool–holder model is then coupled with the experimentally determined spindle–machine dynamics through RCSA to realize a model of the tool-tip dynamics. The coupled model is validated through experiments for three different tool overhang lengths. The presented technique can be used to predict the tool-tip dynamics for different tool-and-holder combinations and for optimization studies without the need for extensive experimentation.     

双层空竹超表面中基于BIC的局域太赫兹磁场增强的研究

设计了包含双层空竹的超表面,通过调节其晶格周期实现了不同阶晶格模式与超表面本征模式间的耦合,获得了3个强耦合区,并在两个晶格周期处实现了弗里德里希–温特根型连续域束缚态（Friedrich-Wintgen bound states in the continuum,FW-BIC）。基于耦合模理论对其进行分析,理论分析结果与模拟结果吻合,进一步证明了结构的有效性。讨论了在介质间隔层中基于强耦合和FW-BIC的局域磁场增强,发现最大磁场强度是入射太赫兹波磁场强度的41 209倍,且该值是单纯的由超表面电磁共振产生的磁场强度的4倍。这项研究将为强场太赫兹产生和太赫兹非线性研究提供参考。