基于传递率函数的运行模态分析方法

传统的运行模态分析方法在推导过程中多假设激励为白噪声,造成在应用上有一定的局限性。提出了一种基于传递率函数的运行模态分析方法,无需采用白噪声假设,利用两种不同载荷情况下的传递率函数构造有理函数,通过有理分式Forsythe正交多项式法对其进行拟合,得到模态频率、阻尼和振型参数。最后,采用机翼模型仿真算例和悬臂梁实验,验证了在非白噪声激励情况下该方法的有效性与可靠性。     

Direct calculation of modal parameters from matrix orthogonal polynomials

The object of this paper is to introduce a new technique to derive the global modal parameter (i.e. system poles) directly from estimated matrix orthogonal polynomials. This contribution generalized the results given in Rolain et al. (1994) [5] and Rolain et al. (1995) [6] for scalar orthogonal polynomials to multivariable (matrix) orthogonal polynomials for multiple input multiple output (MIMO) system.Using orthogonal polynomials improves the numerical properties of the estimation process. However, the derivation of the modal parameters from the orthogonal polynomials is in general ill-conditioned if not handled properly. The transformation of the coefficients from orthogonal polynomials basis to power polynomials basis is known to be an ill-conditioned transformation. In this paper a new approach is proposed to compute the system poles directly from the multivariable orthogonal polynomials. High order models can be used without any numerical problems.The proposed method will be compared with existing methods (Van Der Auweraer and Leuridan (1987) [4] Chen and Xu (2003) [7]). For this comparative study, simulated as well as experimental data will be used.     

基于过渡矩阵的振动模态参数整体识别正交多项式算法

对正交多项式的振动模态参数整体识别方法进行了研究。针对Ｒｉｃｈａｒｄｓｏｎ整体识别方法中确定有理分式的分母多项式系数的错误,提出了基于过渡矩阵的正交多项式整体识别新算法。采用该法可显著提高模态参数的整体识别精度。     

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.     

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.     

On the frequency mapping of modal parameters identification

With borrowing the complex orthogonal polynomial (OP), the numerical condition of the normal matrix of the rational fractional polynomial method is insensitive to frequency scaling. The other two concerns are, the first, transforming coefficients based on OPs into coefficients based on monomials, and the second, computing polynomials. The leading term coefficients of OPs usually blast approximately exponentially as the order increases, so are the diagonal entries of the transitional matrix. This can be contributed to that the orthogonal relationship among the selected OPs is over the frequency band [0,1]. By examining the recursion relation of the Legendre polynomials, we found that mapping the actual frequency vector into [0,2] can efficiently avoid the aforementioned exponential trend. Moreover, in the case of sub-band fitting, an empirical formula for frequency mapping was proposed. Numerical simulation demonstrates that this formula, not only works well in the case of a uniform weight function, but also three typical cases of non-uniform weight cases. The second concern, overflow in computing polynomials with new mapping, can be overcome by the Horner's scheme for a general polynomial and three-term recurrent algorithm for an OP.     

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.     

An optimized rational fraction polynomial approach for modal parameters estimation from FRF measurements

This paper presents an Optimized Rational Fraction Polynomial (ORFP) approach for modal parameters estimation from the measurements of the Frequency Response Function (FRF). Although this approach is based on the Rational Fraction Polynomial (RFP) technique described in [1], it suggests the use of a constrained optimization scheme rather than the Forsythe method to overcome the shortcomings of the Forsythe method. The latter are the estimation of modal parameters that do not necessarily describe a stable system and the estimation of fictitious natural frequencies. The formulation of the constrained optimization problem is presented and discussed. The assessment of the performance of the ORFP approach showed that it is better than the RFP approach in terms of its ability to identify modal parameters that ensure a stable system and its flexibility in selecting and setting the natural frequencies of the system. Several illustrative examples are given to demonstrate the robustness of the ORFP approach.     

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。     

Orthogonal models of structure functions

Approximation of structure functions (SFs) using orthogonal polynomials in most cases, leads to an incorrect determination of the spectral power density for the obtained orthogonal model. A method for constructing orthogonal models of SFs is proposed based on preliminary shifting of the analyzed SF and using orthogonal functions as the bases. In this method, the expansion coefficients are determined using a numerical-analytical approach, which allows one to determine orthogonal models of SFs with a small approximation error and calculate spectral power density with greater accuracy. The implementation of the algorithms for constructing orthogonal models of SFs in the Laguerre and Legendre bases is considered.     

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.     

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.     

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.     

An identification method for joint structural parameters using an FRF-based substructuring method and an optimization technique

A new method is proposed to identify the joint structural parameters of complex systems using a frequency response function (FRF)-based substructuring method and an optimization technique. The FRF method is used to estimate the joint parameters indirectly by minimizing the difference between the reference and calculated responses using a gradient-based optimization technique with analytical gradient information. To assess the robustness of the identification method with respect to noisy input data, FRFs contaminated by uniformly distributed random noise were tested in a numerical example. The effects of the random noise and the magnitude of the connection stiffness values on the accuracy of the method were investigated while identifying the joint parameters. When the FRFs were contaminated with random noise, the proposed procedure performed well when used to identify the stiffness values, but the accuracy of identification is deteriorative when used to identify the damping coefficients. The joint parameters of a real bolted structure were also identified by the proposed method. The results show that it can be applied successfully to real structures, and that a hybrid approach using both calculated and measured FRFs in the substructure model can enhance the quality of the identification results.     

移动最小二乘法的时变结构模态参数辨识

研究时变结构模态参数辨识,基于泛函矢量时变自回归模型(Functional series vector time-dependent AR model,FS-VTAR)提出一种改进的移动最小二乘法的时变结构模态参数辨识方法。该方法源于无网格法中构造形函数进行局部近似的思想,引入带权正交基函数对移动最小二乘(Moving least square,MLS)的基函数进行改进,使得在辨识时间域内构造形函数矩阵过程中不再出现数值条件问题,从而提高了计算精度。把时变系数在形函数上线性展开,利用最小二乘法得到形函数的系数,从而得到时变系数。把时变模型特征方程转换为广义特征值问题提取出模态参数。利用时变刚度系统非平稳振动信号验证该方法,结果表明:改进的移动最小二乘法相比于传统的FS-VTAR模型能有效地避免基函数形式的选择和很高的基函数阶数且更加高效,相比于移动最小二乘法能有效地避免辨识过程中的数值问题,具有更高的模态参数辨识精度。     

Analysis of swept-sine runs during modal identification

Experimental modal analysis of large aerospace structures in Europe combine nowadays the benefits of the very reliable but time-consuming phase resonance method and the application of phase separation techniques evaluating frequency response functions (FRF). FRFs of a test structure can be determined by a variety of means. Applied excitation signal waveforms include harmonic signals like stepped-sine excitation, periodic signals like multi-sine excitation, transient signals like impulse and swept-sine excitation, and stochastic signals like random. The current article focuses on slow swept-sine excitation which is a good trade-off between magnitude of excitation level needed for large aircraft and testing time. However, recent ground vibration tests (GVTs) brought up that reliable modal data from swept-sine test runs depend on a proper data processing. The article elucidates the strategy of modal analysis based on swept-sine excitation. The standards for the application of slowly swept sinusoids defined by the international organisation for standardisation in ISO 7626 part 2 are critically reviewed. The theoretical background of swept-sine testing is expounded with particular emphasis to the transition through structural resonances. The effect of different standard procedures of data processing like tracking filter, fast Fourier transform (FFT), and data reduction via averaging are investigated with respect to their influence on the FRFs and modal parameters. Particular emphasis is given to FRF distortions evoked by unsuitable data processing. All data processing methods are investigated on a numerical example. Their practical usefulness is demonstrated on test data taken from a recent GVT on a large aircraft. The revision of ISO 7626 part 2 is suggested regarding the application of slow swept-sine excitation. Recommendations about the proper FRF estimation from slow swept-sine excitation are given in order to enable the optimisation on these applications for future modal survey tests of large aerospace structures.     

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。     

A frequency response based structural damage localization method using independent component analysis

Vibration-based structural damage detection has been the focus of attention by many researchers over the last few decades. However, most methods proposed for this purpose utilize extracted modal parameters or some indices constructed on the basis these parameters. Our literature review revealed that few papers have employed frequency response functions (FRFs) for detecting structural damage. In this paper, a technique is presented for damage detection which is based on measured FRFs. Independent component analysis (ICA) has been implemented on the spatiotemporal responses in each frequency in order to reduce the dimension of the data. This is based on the concept that the forced harmonic response of a linear vibrating system can be fully captured utilizing a single ICA mode. A different approach is also presented in which ICA is applied to the frequency domain data. Operational deflection shapes (ODSs) have been decomposed using ICA to localize the damage. The efficiency of both methods is demonstrated through some numerical and experimental case studies.     

OPTIMISATION OF DYNAMIC VIBRATION ABSORBERS OVER A FREQUENCY BAND

This paper is focused on the reduction of vibration levels of mechanical systems using dynamic vibration absorbers (DVAs). A general methodology is proposed for the optimum selection of DVA parameters so as to guarantee the efficiency of those devices over a previously selected frequency band. The presented methodology utilises a substructure coupling technique exploring frequency response functions (FRFs), which enables one to calculate the FRFs of the composite structure (primary system+DVAs), from the FRFs of the primary structure and the theoretical expressions of the FRFs of the DVAs. The FRFs of the composite structure, which are expressed as functions of the DVA parameters, are then used to define scalar performance indexes related to the vibration levels of the composite structure over the selected frequency band. These performance indexes are optimised with respect to the DVA parameters by solving a general non-linear constrained optimisation problem. The first part of the paper is devoted to the formulation of the substructure coupling method and the optimisation procedures. Numerical applications using experimentally acquired FRFs are then presented to illustrate the main features of the proposed methodology.     

Solution of ill-posed problems for identification of the dynamic parameters of bolted joints

The measured frequency response functions (FRFs) are directly used to identify the bolted joint properties. However, the noise effect and the matrix inverse operations create ill-posed problems, and a small noise level may cause the identified result to be faulty. A sensitivity method is developed to avoid the ill-posed problems for identification of the dynamic parameters of bolted joints in this paper. To calculate the sensitivity of stiffness and damping for bolted joints, the sensitive values are used to determine the frequency ranges before the identification. Then, the equivalent stiffness and damping of bolted joints are identified by FRFs method in this range. All results of simulation and experiment show that the proposed method can improve significantly the identification accuracy. Additionally, the sensitivity method can be used to avoid an ill-posed problem by eliminating ill-posed FRFs in some frequency range before identification.