Phase correction for frequency response function measurements

In modal testing, an impulse is often used to excite the structure and a linear transducer is used to measure the response. For these impact tests, two signals are measured: the impulsive force and the vibration response. Any lack of synchronization in the time domain acquisition of the two signals results in a frequency-dependent phase error in the frequency response function, or FRF. However, knowledge of the time delay may be used to correct the corresponding phase error. In this research, tests were conducted to measure the frequency-dependent phase error for a capacitive sensor and a frequency domain technique is proposed to correct the FRF. The method was validated using an FRF measurement of a cylindrical artifact mounted in a milling machine spindle.     

Error analysis of indirect force determination and a regularisation method to reduce force determination error

Accurate estimation of dynamic forces acting on a structure is very difficult, since, at certain frequency regions, the identified forces may contain an unacceptable amount of error. In this study, by investigating the characteristics of the inverse transfer function and frequency response function (FRF) error, the indirect force determination error is analysed statistically, and the special frequency regions where force determination error is very large are searched for each case of the number of response measurements and applied forces. It is shown that the large force determination error near the first kind of inverse pole frequencies is mainly due to the rank deficiency of the FRF submatrix, while the large error in the vicinity of resonance frequencies comes from the larger FRF error. The idea was tested both analytically and experimentally. Also a regularisation process to reduce the error especially near the first kind of inverse pole frequencies is proposed. Since the degree of singularity of the FRF submatrix near the first kind of inverse pole frequencies is mainly related to system damping, a regularisation procedure is suggested by adding additional damping. An optimal regularisation constant is also derived. It is shown that, the greater the order of magnitude of the FRF error is than that of modal damping ratio, the larger the force determination error reduction is obtained with the suggested regularisation procedure. The proposed regularisation method was tested experimentally and its effects examined.     

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.     

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.     

一种新的频响函数无偏估计方法

在对几种典型的频响函数估计进行分析的基础上，利用不相关信号的互功率谱多次平均趋于零的性质，提出了一种新的频响函数无偏估计方法。该估计方法不需要测试第3个信号即可达到无偏估计的目的，避免了测试系统过于复杂，减小了测试工作量和因过多环节的引入而降低估计精度的可能性。测试n组数据，可以进行n(n—1)／2次平均，平均次数是其他方法的(n—1)／2倍，提高了频响函数估计的精度。     

相关小波法在腕力传感器动态特性分析中的研究

在机器人腕力传感器的动态特性分析中，最关键的步骤之一是从传感器输入输出信号中提取精确的脉冲响应函数和频响函数。介绍了一种新的提取腕力传感器脉冲响应函数的方法——相关小波法。在建立腕力传感器的简化动力学模型的基础上，模拟了机器人与环境相互碰撞及机器人在强噪声环境中的典型工作条件，用相关小波法提取了模型系统的脉冲响应函数和频响函数，并与传统的FFT方法进行了比较。结果表明，相关小波法比FFT方法具有更明显的优势，获得了更高精度的脉冲响应函数和频响函数的提取。相关小波法为腕力传感器，同时也为其他传感器的动态特性分析提供了一种新的、有效的方法。     

Estimation of uncorrelated content in experimentally measured frequency response functions using three measurement channels

Experimental frequency response functions (FRF) measurements are normally made by measuring two signals: the force applied to the structure and the resulting response. If the coherence between these signals is less than unity, it is known only that there is uncorrelated content present in either or both signals.This paper presents, for the first time, a method for estimating the magnitude and source of any uncorrelated content present in a measurement system, provided that the original random signal is available for measurement. This is the case when a signal generator/amplifier/shaker set-up is used. However, this method cannot be applied to hammer impact testing.Measurements are made of the signals from the random noise generator, the force gauge, and the response transducer. From these measurements estimates can be made of uncorrelated content in the force and response measurements as well as in the amplification/shaker system. These estimates of uncorrelated content will be useful in improving the measurement system and in determining when the results from the and estimators are valid. They also can be used to estimate the s² term in .The uncorrelated content estimators are applied in several different FRF measurements. Estimates of uncorrelated content from an inefficient non-linear shaker-to-force-gauge connection, from a ground-loop problem, and from leakage are shown. Limits on dynamic range are discussed.     

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。     

A non-parametric approach for linear system identification using principal component analysis

This paper considers the applications of principal component analysis (PCA) for signal-based linear system identification. Linear time-invariant (LTI) single-input-single-output (SISO) and multi-input-multi-output (MIMO) system frequency response function (FRF) estimation problems are formulated on the basis of the eigen-value decomposition (EVD) of the input–output measurement spectral correlation matrix. It is demonstrated that resulting algorithms for the SISO and MIMO cases are equivalent to that of the maximum likelihood (ML) and the total least squares (TLS) approaches respectively. Originating from the proposed FRF estimation scheme, a moving-segment EVD procedure is developed for SISO time-varying transfer function estimation. Based on the sensitivity of the time-domain PCA to delays/shifts between signals, an extended lagged-covariance-matrix approach is introduced for delay detection from time series.     

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。     

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.     

基于频响函数 Neumann 级数展开的有限元模型修正

针对有限元模型修正中测试自由度不完备、求解不适定的问题,提出了一种基于频响函数Neumann级数展开的模型修正方法。首先,利用测试和有限元分析模态数据构造完整的实测频响函数。然后,根据Neumann级数展开式建立实测频响函数与有限元模型频响函数之间的关系式,以此构建模型修正目标函数。最后,采用改进鲸鱼优化算法求解目标函数获取修正结果。通过平面桁架数值算例表明,该方法具有较强的噪声鲁棒性,模型修正精度较好,15%噪声时修正后的平均相对误差不超过1%。利用4层剪力框架结构测试算例进一步验证所提方法,结果表明,修正后的模型能反映结构真实状况。     

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.     

DETERMINATION OF UNKNOWN IMPACT FORCE ACTING ON A SIMPLY SUPPORTED BEAM

This work presents a predictive model for determining the location and amplitude of an unknown impact force acting on a simply supported beam. Both time and frequency domain prediction methods are developed, respectively. The structural modal parameters can be first obtained by theoretical modal analysis (TMA) or by experimental modal analysis (EMA). The structural response at time and frequency domains due to an unknown impact force can then be measured and recorded. The predicted response can also be formulated and expressed as functions of amplitude and location of the impact force. The sum of square errors between the predicted and measured response is then defined as the objective function, while the amplitude and the location of the unknown impact force are defined as design variables. The optimisation problem is thereby constructed and can be solved for the amplitude of the impact force. The mode shape information associated to the location of the impact force can also be resolved and compared to the structural mode shapes to determine the location of the unknown impact force. Both numerical and experimental prediction results are presented. Results show that the predictive model is feasible and leads to the prediction of magnitude and location of the unknown impact force for arbitrary structures as well.     

Complex mode indication function and its applications to spatial domain parameter estimation

This paper introduces the concept of the Complex Mode Indication Function (CMIF) and its application in spatial domain parameter estimation. The concept of CMIF is developed by performing singular value decomposition (SVD) of the Frequency Response Function (FRF) matrix at each spectral line. The CMIF is defined as the eigenvalues, which are the square of the singular values, solved from the normal matrix formed from the FRF matrix, [H(jω)]^H[H(jω)], at each spectral line. The CMIF appears to be a simple and efficient method for identifying the modes of the complex system. The CMIF identifies modes by showing the physical magnitude of each mode and the damped natural frequency for each root. Since multiple reference data is applied in CMIF, repeated roots can be detected. The CMIF also gives global modal parameters, such as damped natural frequencies, mode shapes and modal participation vectors. Since CMIF works in the spatial domain, uneven frequency spacing data such as data from spatial sine testing can be used. A second-stage procedure for accurate damped natural frequency and damping estimation as well as mode shape scaling is also discussed in this paper.     

Optimization of the Tuned Mass Damper for Chatter Suppression in Turning

The tuned mass damper(TMD) has been successfully applied to the vibration control in machining,while the most widely adopted tuning is equal peaks,which splits the magnitude of the frequency response function(FRF) into equal peaks.However,chatter is a special self-excited problem and a chatter-free machining is determined by FRF at the cutting zone.A TMD tuning aiming at achieving the maximum chatter stability is studied,and it is formulated as an optimization problem of maximizing the minimum negative real part of FRF.By employing the steepest descend method,the optimum frequency and damping ratio of TMD are obtained,and they are compared against the equal peaks tuning.The advantage of the proposed tuning is demonstrated numerically by comparing the minimum point of the negative real part,and is further verified by damping a flexible mode from the fixture of a turning machine.A TMD is designed and placed on the fixture along the vibration of the target mode after performing modal analysis and mode shape visualization.Both of the above two tunings are applied to modify the tool point FRF by tuning TMD respectively.Chatter stability chart of the turning shows that the proposed tuning can increase the critical depth of cut 37% more than the equal peaks.Cutting tests with an increasing depth of cut are conducted on the turning machine in order to distinguish the stability limit.The tool vibrations during the machining are compared to validate the simulation results.The proposed damping design and optimization routine are able to further increase the chatter suppression effect.     

An efficient numerical solution for frequency response function of micromechanical resonator arrays

Array forms of MEMS resonator that uses a specific mid-frequency normal mode have been introduced for acquiring a wider bandwidth of frequency response function (FRF). A conventional frequency response solver based on a modal approach faces computational difficulties in obtaining the FRF of these array forms because of the increase in the order of a linear dynamic model and the number of retained normal modes. The computational difficulties can be resolved by using a substructuring-based model order reduction and a frequency sweep algorithm, which requires a smaller number of retained modes of a reduced dynamic system than the conventional solver. In computing the FRF of a single resonator and its array forms, the presented method shows much better efficiency than the conventional solution by ANSYS as the number of resonators increases. In addition, the effects of multiple resonators in the array forms on filter performance are discussed compared with experimental data.     

移动荷载作用下时变简支钢桥损伤识别

针对移动荷载作用下桥梁结构振动响应信号呈现非平稳性的特点,构建新的一阶本征函数自功率谱最大值变化比和一阶本征函数小波能量变化率两个指标来识别时变结构的损伤。首先,采用小波阀值去噪法对时变结构响应信号进行去噪处理;其次,运用解析模态分解定理提取响应信号的一阶本征函数并构建一阶本征函数自功率谱最大值变化比指标来识别结构的损伤位置,在识别结构损伤位置的基础上,将损伤位置处的加速度响应信号的一阶和二阶本征函数进行线性混叠后,采用快速独立成分分析进行分离,得到更有效的一阶本征函数;最后,基于连续小波变换和时间窗思想,提出一阶本征函数小波能量变化率指标来预测结构的时变损伤。通过移动荷载作用下的时变简支钢桥试验验证所提出的损伤指标,研究结果表明,提出的两个指标能够有效识别结构的损伤位置和时变损伤。     

基于实验模态分析的集中参数法建模

利用集中参数模型的建立,不仅拓展了实验模态分析(EMA)的使用范围,还能大量缩减原有模型的自由度并保持结构的动态特性。在对一立铣床进行模态测试以及振型可视化的基础上,采用集中质量、弹簧阻尼单元建立了该铣床的7自由度集中参数模型,并在该模型基础上合成刀尖频响函数及预测颤振稳定域,揭示出机床颤振与振型的联系。与原有测试频响函数的比较表明,该模型准确度较高。