一种改进的基于响应耦合子结构法的刀尖点频响函数预测方法

针对现有的基于响应耦合子结构法(RCSA)的刀尖点频响函数预测方法需要辨识主轴-刀柄、刀柄-刀具结合面参数以及需要自制刀柄模型等引起的预测误差和预测过程复杂等问题,提出一种改进的基于RCSA的铣刀刀尖点频响函数预测方法。该方法首先改进已有的子结构划分方法,将机床-主轴-刀柄-刀具系统划分为机床-主轴-刀柄-部分刀杆、剩余刀杆和刀齿三个子结构;然后改进主轴-刀柄处转动频响函数的计算方法,通过铣刀的模态锤击实验采用反向RCSA和有限差分法计算机床-主轴-刀柄-部分刀杆结构的转动频响函数,并基于Euler梁模型计算出剩余刀杆、刀齿子结构的频响函数;最后将三个子结构的频响函数耦合确定刀尖点的预测频响函数。以一立式加工中心为研究对象,应用所提出的方法对铣刀刀尖点的频响函数进行了预测,并与其实测频响函数进行对比。对比结果表明：刀尖点的预测频响函数与实测频响函数符合程度较高,其预测、实测前三阶固有频率之间的误差在6.9%以内,所提出的方法可行有效、简单方便,且可直接基于铣刀的模态实验计算主轴-刀柄的频响函数,避免了相关结合面参数的辨识和刀柄模型的制作。     

基于复合加工特征的航空结构件频响快速预测

航空结构件作为飞机中占比最大的零件，具有大尺寸、多槽腔、弱刚性和高材料去除率等特点，如何在保证加工过程稳定的前提下高效去除结构件材料是航空结构件加工的一大瓶颈问题，零件频响的快速预测是选择高效切削参数的前提。通过对典型航空结构件加工特征的分类和提取，构建了“槽腔-筋”复合加工特征，建立了其参数化的频响特性分析模型，并计算了复合加工特征若干刚度薄弱点的频响特性。将其与复合加工特征零件、整体零件相应位置的频响函数进行对比，结果表明模型预测精度满足要求，验证了使用复合加工特征模型频响特性代替整体零件模型频响特性的可行性。同时，复合加工特征有限元模型的自由度数量远小于整体零件有限元模型自由度数量，充分体现了所提出的方法预测结构件频响的快速性。     

基于试验频响函数刚体特性参数的计算及其应用

试验频响函数(FRF)正越来越成功、广泛地应用于动态和振动系统的不同领域中。基于模态试验的频响函数,尤其是频响函数中的质量线与刚体特性参数的内在关系,提出了一种简便、易行且精确的进行刚体惯量特性参数的计算方法。进行了相关理论的推导,通过实际试验验证分析了该方法。对大型复杂的结构,采用该方法具有一定的优势。     

Sensitivity analysis on ram speed of a direct extrusion process model using a porthole die through CEL method

Prediction of machine tool chatter requires the characterization of dynamic of the machine-tool-workpiece system by means of frequency response functions (FRFs). Uncertainties of the measured FRFs result in uncertainties of the calculated stability diagrams, therefore robustness of stability prediction against parameter perturbations is of high importance. Although there exist methods to determine robust stability in terms of stability radii, these methods either give a conservative estimate of the real uncertainties or are limited to perturbations of a few modal parameters, only. In this paper, a frequency-domain approach is presented to determine robust stability boundaries using the measured FRFs directly without any modal parameter identification. The method is based on an envelope fitting around the measured FRFs combined with some considerations of the single-frequency method. The application of the method is demonstrated in case of a turning operation, where the machine tool structure is characterized by a series of FRF measurements.     

Identification of modal parameters of unmeasured modes using multiple FRF modal analysis method

Current modal analysis methods seek to identify the modal parameters of some or all of the modes in the measured frequency range of interest. In many applications however, it will be very useful if modal parameters of some of the out-of-range modes can be identified during modal analysis. Such a goal is obviously theoretically possible since the raw measured frequency response functions (FRFs), upon which modal analysis is performed, do contain adequate information about the out-of-range modes in the form of residue contributions. In this paper, a new method for the estimation of modal parameters using multiple FRFs analysis is presented. In the process of modal identification, the proposed method not only presents accurate modal parameters of the modes which are present in the measurement frequency range, but also quite accurately identifies some of the modes which are not measured. The method calculates the required modal parameters by solving eigenvalue problem of an equivalent eigensystem derived from those measured FRF data. All measured FRFs are used simultaneously to construct the equivalent eigensystem matrices from which natural frequencies, damping loss factors and modeshape vectors of interest are solved. Since the identification problem is reduced to an eigenvalue problem of an equivalent system, natural frequencies and damping loss factors identified are consistent. Applications of the method to both numerically simulated and practically measured FRF data are given to demonstrate the practicality of the proposed method and the results have shown the method is capable of accurately identifying modal parameters of out-of-range modes.     

基于Morlet小波改善飞行颤振试验参数识别精度的方法

为了解决激励能量有限和现场测试数据量较少、噪声大，系统参数识别的准确度差的问题，采用Morlet小波时频滤波和频域参数识别相结合的方法进行参数识别来提高精度。基于Morlet小波函数建立特性滤波器组进行时频域滤波，讨论滤波参数的选取方法，采用有理正交多项式（RFOP）拟合算法进行频域参数识别，基于欧洲航空界广泛采用的GARTEUR飞机模型数据建立密频模态系统，进行飞行颤振的试验数据仿真。结果表明该方法在信号噪声较大时，可以有效地提高系统参数识别的精度。     

Frequency response function shape-based methods for structural damage localisation

The ultilisation of structural shape signals for damage localisation has shown some promise, especially in the applications where an accurate finite element model of the structure is not available. For this purpose, traditional shape signals, like mode shapes, flexibility matrices, uniform load surface (ULS) and operational deflection shapes (ODS) have been widely used. Using frequency response function (FRF) shapes for structural damage localisation is however, a relatively new but promising technique. Unlike mode shapes, ULS and ODS, FRF shapes are defined on broadband data and so have potential to reveal damage location more clearly. Another advantage of using FRF shapes is that the test data can be directly used without the necessity of conducting modal identification. Nevertheless, some problems associated with this approach still remain to be solved. No solid foundation or deduction about the use of FRF shapes for damage localisation has been given in any literature so far. In addition, it has been observed that this method only works for a low-frequency range. This limitation of FRF shapes has not been explained or well treated so far. In this study, a scheme of using FRF shapes for structural damage localisation is proposed. Methods within this scheme include some important modifications like using the imaginary parts of FRF shapes and normalising FRF shapes before comparison. The theoretical explanation of using FRF shapes for damage localisation is presented and the limitations of the previous FRF shape methods have been overcome. The proposed methods have shown great potential in structural damage localisation.     

An experimental investigation on the effects of exponential window and impact force level on harmonic reduction in impact-synchronous modal analysis

A novel method called Impact-synchronous modal analysis (ISMA) was proposed previously which allows modal testing to be performed during operation. This technique focuses on signal processing of the upstream data to provide cleaner Frequency response function (FRF) estimation prior to modal extraction. Two important parameters, i.e., windowing function and impact force level were identified and their effect on the effectiveness of this technique were experimentally investigated. When performing modal testing during running condition, the cyclic loads signals are dominant in the measured response for the entire time history. Exponential window is effectively in minimizing leakage and attenuating signals of non-synchronous running speed, its harmonics and noises to zero at the end of each time record window block. Besides, with the information of the calculated cyclic force, suitable amount of impact force to be applied on the system could be decided prior to performing ISMA. Maximum allowable impact force could be determined from nonlinearity test using coherence function. By applying higher impact forces than the cyclic loads along with an ideal decay rate in ISMA, harmonic reduction is significantly achieved in FRF estimation. Subsequently, the dynamic characteristics of the system are successfully extracted from a cleaner FRF and the results obtained are comparable with Experimental modal analysis (EMA).     

A DUAL FITTING ALGORITHM FOR MODAL PARAMETERS IDENTIFICATION

A dual fitting algorithm (DFA) for modal parameters identification is presented. The method is implemented in three steps: first, the coefficients of the Forsythe orthogonal polynomials for the rational fraction of the frequency response functions (FRFs) are obtained by fitting the experimental FRF data; then the coefficients of the orthogonal polynomials are converted into the coefficients of ordinary power polynomials by the fitting method again; and finally, the poles and residues of the rational fraction of FRFs in ordinary power polynomials are extracted to identify the modal parameters. Some notes to the definition and use of the recurrence formulation for the real half-function Forsythe orthogonal polynomials are introduced. An example is given to show the aspects of the present method.     

Locating damage in waveguides from the phase of point frequency response measurements

A method for detecting damage in uniform waveguide structures from two or more point frequency response functions (FRFs) is described. Attention is focussed on bending waves in beams although the method can in principle be applied to any waveguide structure. The input FRF is the superposition of directly injected waves and waves reflected from the damage and from other scattering regions in the structure. The phase of this FRF modulates with wavenumber, with the period of modulation in wavenumber space being related to the distances between the excitation point and scattering locations. The phase spectrum of the input FRF is found: the phase is determined, dispersive effects are removed by transforming from the frequency domain to the wavenumber domain, and the inverse Fourier transform from the wavenumber to the space domain found. Peaks in this phase spectrum indicate the distance to the scatterer. Two (or more) input FRFs can be used to determine the location of the scatterer unambiguously. Signal processing issues are discussed. Numerical results for a uniform beam with a breathing crack are given and experimental results for beams with a slot cut into them are presented. The approach lies in the middle ground between low frequency, modal methods and high frequency, ultrasonic methods. It allows one to interrogate a region of a structure rather than the complete structure and there is no requirement for a validated model of the structure in order to locate damage, apart from an estimate of the dispersion relation.     

Parameter estimation of Duffing oscillator using Volterra series and multi-tone excitation

Use of Volterra series in nonlinear system identification is well established now. The series represents response of a nonlinear system in a functional series form consisting of convolution integrals involving higher-order impulse response functions known as Volterra kernels. Multi-dimensional Fourier transforms of these Volterra kernels give the higher-order frequency response functions (FRFs). The measurement of these FRFs under harmonic excitation and their relationship with the first-order FRFs provide a basis for estimation of the nonlinear parameters. However, most of the methods employ single-tone excitation, which provide limited FRF measurement in a single experiment. In the present study, a novel procedure based on multi-tone excitation is presented for a typical Duffing oscillator and it is demonstrated that accurate estimation of both nonlinear and linear parameters is possible with fewer number of experiments.     

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.     

频响函数在结构损伤检测中应用的试验研究

结构动力测试中,频响函数可直接测得,不仅避免了在模态提取时的误差,而且在相同频段上可提供更多结构损伤信息,因此可方便地应用于结构损伤的检测中。对于结构基础的随机激励,试验结果表明频响函数法可有效地检测结构的损伤。同时说明传感器布置在结构的不同位置都可检测结构的损伤。并且随机振动的大小不影响结构损伤检测的效果。为频响函数法在现场实测结构的损伤提供了依据。     

On the estimate of the FRFs from operational data

In this paper, the effects of different mass loadings required for the estimation of the frequency response functions, FRFs, from data gained by the emerging technique of operational modal testing, is proposed. This technique allows the evaluation of the natural frequencies, mode shapes and damping ratios from operational data achieved from a first session of tests, then the scaling factors are derived from a further experimental investigation. The approach is based on the sensitivity of the eigenproperties to structural modifications, such as the mass and stiffness distribution. It is shown that the generalized modal parameters could be derived by the measurements of the natural frequency shifts due to a controlled mass variation in the structure, assuming negligible changes in the mode shapes. Such generalized modal parameters are finally used to estimate the FRFs. This mode shape scaling technique, together with the investigation of the effects of the mass positioning on the uncertainties in the estimates of the scaling factors will be experimentally investigated on simple aerospace structures.     

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.     

Adaptive mode superposition and acceleration technique with application to frequency response function and its sensitivity

An adaptive mode superposition and acceleration technique (AMSAT) is proposed and implemented into the computation of frequency response functions (FRFs) and their sensitivities. Based on the mode superposition and mode acceleration methods for the FRFs, m-version, s-version, and ms-version adaptive schemes are presented. In these schemes, the error resulted from the mode truncation and/or series truncation is, at first, estimated at every specific frequency, respectively. Then, one more mode (called m-version), or one more level of the series (called s-version), or the combination (called ms-version) is included in the computation of the FRF when its error is greater than the error tolerance. The new FRF is recalculated and its error is re-evaluated. This procedure is repeated until all the errors fall below the specified value. Although only the implementation of FRFs and their sensitivities is demonstrated in this paper, the proposed adaptive technique may be applied to the computation of dynamic responses in time domain and their sensitivities, sensitivity of eigenpairs, modal energy, etc. One numerical example is included to demonstrate the application of the proposed adaptive schemes. The results show that the present schemes work very well. The s- and ms-version adaptive schemes are much more efficient than m-version scheme. Since the intention of this paper is to propose these new procedures, the damping, particularly the non-classical damping, is not included due to the complexity.     

Identifying and quantifying structural nonlinearities in engineering applications from measured frequency response functions

Engineering structures seldom behave linearly and, as a result, linearity checks are common practice in the testing of critical structures exposed to dynamic loading to define the boundary of validity of the linear regime. However, in large scale industrial applications, there is no general methodology for dynamicists to extract nonlinear parameters from measured vibration data so that these can be then included in the associated numerical models. In this paper, a simple method based on the information contained in the frequency response function (FRF) properties of a structure is studied. This technique falls within the category of single-degree-of-freedom (SDOF) modal analysis methods. The principle upon which it is based is effectively a linearisation whereby it is assumed that at given amplitude of displacement response the system responds at the same frequency as the excitation and that stiffness and damping are constants. In so doing, by extracting this information at different amplitudes of vibration response, it is possible to estimate the amplitude-dependent ‘natural’ frequency and modal loss factor. Because of its mathematical simplicity and practical implementation during standard vibration testing, this method is particularly suitable for practical applications. In this paper, the method is illustrated and new analyses are carried out to validate its performance on numerical simulations before applying it to data measured on a complex aerospace test structure as well as a full-scale helicopter.     

Modal parameters estimation in the Z-domain

This paper aims to explain in a clear, plain and detailed way a modal parameter estimation method in the frequency domain, or similarly in the Z-domain, valid for multi degrees-of-freedom systems. The technique is based on the rational fraction polynomials (RFP) representation of the frequency-response function (FRF) of a single input single output (SISO) system but is simply extended to multi input multi output (MIMO) and output only problems. A least-squares approach is adopted to take into account the information of all the FRFs but, when large data sets are used, the solution of the resulting system of algebraic linear equations can be a long and difficult task. A procedure to drastically reduce the problem dimensions is then adopted and fully explained; some practical hints are also given in order to achieve well-conditioned matrices. The method is validated through numerical and experimental examples.     

Dynamic stiffness deterioration of a machining center based on relative excitation method

The tool point frequency response function(FRF) is commonly obtained by impacting test or semi-analytical techniques.Regardless of the approach,it is assumed that the workpiece system is rigid.The assumption is valid in common machining,but it doesn’t work well in the cutting processes of thin-wall products.In order to solve the problem,a multi-degree-of-freedom dynamic model is employed to obtain the relative dynamic stiffness between the cutting tool and the workpiece system.The relative direct and cross FRFs between the cutting tool and workpiece system are achieved by relative excitation experiment,and compared with the tool point FRFs at x and y axial direction.The comparison results indicate that the relative excitation method could be used to obtain the relative dynamic compliance of machine-tool-workpiece system more actually and precisely.Based on the more precise relative FRFs,four evaluation criterions of dynamic stiffness are proposed,and the variation trend curves of these criterions during the last six months are achieved and analyzed.The analysis results show that the lowest natural frequency,the maximum and the average dynamic compliances at x axial direction deteriorate more quickly than that at y axial direction.Therefore,the main cutting direction and the large-size direction of workpieces should be arranged at y axial direction to slow down the deterioration of the dynamic stiffness of machining centers.The compliance of workpiece system is considered,which can help master the deterioration rules of the dynamic stiffness of machining centers,and enhance the reliability of machine centers and the consistency of machining processes.     

TUNING OF NORMAL MODES BY MULTI-AXIAL BASE EXCITATION

Normal mode testing is commonly performed on large aerospace structures in order to identify their eigenfrequencies, mode shapes, modal damping values, and generalized masses. Normally, multiple-point sinusoidal excitation is applied. For each normal mode, the exciter configuration is adapted in location and magnitude in order to compensate for the internal damping forces of the test structure. An alternative to multiple-point excitation is a driven-base test using a multi-axial shaker table. Multi-axis shaker facilities have been developed for structural dynamic qualification. However, they can be used for normal mode testing as well, whereby the overall structure is accelerated in space in order to compensate for the internal damping forces. This requires a suitable combination of all six spatial dof of the base excitation. In the case of resonance, the structure responds in quadrature related to the base excitation. If this criterion is fulfilled, the structure vibrates in an eigenmode of vibration in the sense of a sdof system. This article gives the theoretical background of the test method with particular emphasis on the damping estimation, on the determination of the generalized mass, and on finding a suitable base axis combination. A suitable test rig for the normal mode testing by base excitation, DLR's Multi-Axis Vibration Simulator (MAVIS) in Göttingen (FRG), is presented. The test procedure and test data evaluation as applied to a test structure characterized by closely spaced eigenfrequencies is presented. The advantages and disadvantages of normal mode testing by means of base excitation are pointed out.