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.     

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.     

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。     

相机结构系统的试验模态分析

介绍了锤击法试验模态分析技术在某卫星相机光机结构系统动态特性研究中的应用;阐述了多自由度系统模态分析的理论基础和模态参数识别原理;测量分析得到了相机整机结构和关键部件的固有频率,以及与之相对应的模态振型和模态阻尼,并对模态数据及其可能对相机工作的影响作了分析,在此基础上,给出了相机强度和振动试验的载荷谱,保证了试验的安全顺利进行;最后就相机结构设计问题作了讨论。     

Identification of bearing dynamics under operational conditions for chatter stability prediction in high speed machining operations

Chatter is a major problem causing poor surface finish, low material removal rate, machine tool failure, increased tool wear, excessive noise and thus increased cost for machining applications. Chatter vibrations can be avoided using stability diagrams for which tool point frequency response function (FRF) must be determined accurately. During cutting operations, due to gyroscopic moments, centrifugal forces and thermal expansions bearing dynamics change resulting in tool point FRF variations. In addition, gyroscopic moments on spindle–holder–tool assembly cause separation of modes in tool point FRF into backward and forward modes which will lead to variations in tool point FRF. Therefore, for accurate stability predictions of machining operations, effects of operational conditions on machine tool dynamics should be considered in calculations. In this study, spindle bearing dynamics are identified for various spindle rotational speeds and cutting forces. Then, for a real machining center, tool point FRFs under operating conditions are determined using the identified speed dependent bearing dynamics and the mathematical model proposed. Moreover, effects of gyroscopic moments and bearing dynamics variations on tool point FRF are examined separately. Finally, computationally determined tool point FRFs using revised bearing parameters are verified through chatter tests.     

Modal identification of spindle-tool unit in high-speed machining

The accurate knowledge of high-speed motorised spindle dynamic behaviour during machining is important in order to ensure the reliability of machine tools in service and the quality of machined parts. More specifically, the prediction of stable cutting regions, which is a critical requirement for high-speed milling operations, requires the accurate estimation of tool/holder/spindle set dynamic modal parameters. These estimations are generally obtained through Frequency Response Function (FRF) measurements of the non-rotating spindle. However, significant changes in modal parameters are expected to occur during operation, due to high-speed spindle rotation.The spindle's modal variations are highlighted through an integrated finite element model of the dynamic high-speed spindle-bearing system, taking into account rotor dynamics effects. The dependency of dynamic behaviour on speed range is then investigated and determined with accuracy. The objective of the proposed paper is to validate these numerical results through an experiment-based approach. Hence, an experimental setup is elaborated to measure rotating tool vibration during the machining operation in order to determine the spindle's modal frequency variation with respect to spindle speed in an industrial environment. The identification of natural frequencies of the spindle under rotating conditions is challenging, due to the low number of sensors and the presence of many harmonics in the measured signals. In order to overcome these issues and to extract the characteristics of the system, the spindle modes are determined through a 3-step procedure. First, spindle modes are highlighted using the Frequency Domain Decomposition (FDD) technique, with a new formulation at the considered rotating speed. These extracted modes are then analysed through the value of their respective damping ratios in order to separate the harmonics component from structural spindle natural frequencies. Finally, the stochastic properties of the modes are also investigated by considering the probability density of the retained modes. Results show a good correlation between numerical and experiment-based identified frequencies. The identified spindle-tool modal properties during machining allow the numerical model to be considered as representative of the real dynamic properties of the system.     

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.     

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.     

Operational modal analysis by updating autoregressive model

This paper presents improvements of a multivariable autoregressive (AR) model for applications in operational modal analysis considering simultaneously the temporal response data of multi-channel measurements. The parameters are estimated by using the least squares method via the implementation of the QR factorization. A new noise rate-based factor called the Noise rate Order Factor (NOF) is introduced for use in the effective selection of model order and noise rate estimation. For the selection of structural modes, an orderwise criterion called the Order Modal Assurance Criterion (OMAC) is used, based on the correlation of mode shapes computed from two successive orders. Specifically, the algorithm is updated with respect to model order from a small value to produce a cost-effective computation. Furthermore, the confidence intervals of each natural frequency, damping ratio and mode shapes are also computed and evaluated with respect to model order and noise rate. This method is thus very effective for identifying the modal parameters in case of ambient vibrations dealing with modern output-only modal analysis. Simulations and discussions on a steel plate structure are presented, and the experimental results show good agreement with the finite element analysis.     

滚筒洗衣机机箱有限元及试验模态分析

结合实验模态与有限元理论模态分析,对全自动滚筒洗衣机的机箱进行了动态特性研究.首先,在.Pro/Eingeer中建立机箱的三维实体模型;然后导入ANSYS中进行网格划分,得到其有限元模型;再采用兰索斯迭代法进行模态求解,得到前50阶计算模态参数.另外,用脉冲激励法在实验室对机箱进行模态实验,得到实验模态参数.通过比较理论模态与实验模态参数,表明结果吻合较好.基于对机箱固有特性的分析,对该滚筒洗衣机机箱振动特性的改善提供了一定的参考依据.     

Inverse identification of the mechanical parameters of a pipeline hoop and analysis of the effect of preload

To create a dynamic model of a pipeline system effectively and analyze its vibration characteristics, the mechanical characteristic parameters of the pipeline hoop, such as support stiffness and damping under dynamic load, must be obtained. In this study, an inverse method was developed by utilizing measured vibration data to identify the support stiffness and damping of a hoop. The procedure of identifying such parameters was described based on the measured natural frequencies and amplitudes of the frequency response functions (FRFs) of a pipeline system supported by two hoops. A dynamic model of the pipe-hoop system was built with the finite element method, and the formulas for solving the FRF of the pipeline system were provided. On the premise of selecting initial values reasonably, an inverse identification algorithm based on sensitivity analysis was proposed. A case study was performed, and the mechanical parameters of the hoop were identified using the proposed method. After introducing the identified values into the analysis model, the reliability of the identification results was validated by comparing the predicted and measured FRFs of the pipeline. Then, the developed method was used to identify the support stiffness and damping of the pipeline hoop under different preloads of the bolts. The influence of preload was also discussed. Results indicated that the support stiffness and damping of the hoop exhibited frequency-dependent characteristics. When the preloads of the bolts increased, the support stiffness increased, whereas the support damping decreased.     

基于随机减量技术的模态参数识别方法探讨

大型基础工程结构的特征参数识别通常是通过对环境载荷激励的结构响应进行分析来实现,随机减量(Random Decrement,RD)技术是环境激励下的模态参数识别方法中应用较广的方法。在实际应用中受环境、测量等条件的限制,信号常为含有某些优势频率的非平稳信号,常常导致随机减量技术在识别结构参数尤其是系统阻尼时带来较大误差。为提高随机减量技术在环境激励作用下识别结构参数的准确性,文中从分析随机减量信号频谱中的频率分布特性入手,结合随机减量函数产生的触发条件,给出了一种利用信号频谱的统计特征进行模态参数识别的方法。数值仿真结果表明该函数能准确识别在含有优势频率环境载荷作用下的结构参数。     

Improvement of damage detection methods based on experimental modal parameters

The objective of this paper is to introduce a new method for structural damage detection based on experimentally obtained modal parameters. The new method is suitable for detection of fatigue damage occurring in an aluminium cantilever beam. The damage has been practically realised as saw cuts of different sizes and at different locations. The first step of analysis included an attempt of damage identification with the most often used damage indicators based on measured modal parameters. For that purpose special signal processing technique has been proposed improving the effectiveness of indicators tested. However the results obtained have not been satisfactory. That was the motivation for defining new damage indicators (frequency change based damage indicator, Hybrid Damage Detection method), utilising the change of natural frequencies and any mode shape (measured or modelled) as the measurement of frequencies is much less time consuming in comparison to total mode shape measurement. It has been shown that the proposed technique is suitable for damage localisation in beam-like structures.     

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.     

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.     

结构应变模态辨识的特征系统实现方法

应变模态可以分析动载下结构的应力状态。文中将基于位移响应数据的特征系统实现算法应用于应力应变分析中，建立基于应变测量数据的结构动力学参数辨识方法。首先建立应变的状态空间描述方程。由于应变模态和位移模态是同一物理状态的两种不同表现方式，二者的数学表达式具有相似性，前者是后者的相似变换。在此基础上，给出应变格式的ERA识别方法。利用应变格式进行eigensystem realization algorithm(ERA)分析；不仅可以辨识出结构的频率、阻尼比等动力学参数，而且还可以辨识得到相应的结构应变模态，从而建立应力应变预测响应模型。数值仿真和实验验证表明，该方法对结构应变模态有快速准确的辨识能力。     

A framework for blind modal identification using joint approximate diagonalization

In this paper, a second-order statistical method employed in blind source separation (BSS) is adapted for use in modal parameter identification. Modal responses and mode shapes are estimated by the use of second-order blind identification (SOBI) on an expanded and pre-treated dataset. Frequency and damping can be obtained from the modal responses by simple single degree of freedom methods. Using this approach, a class of new non-parametric output-only modal identification algorithms is proposed and examples of its use are provided. It is demonstrated that the proposed methodology provides a novel and robust approach to modal identification. For the example shown, it is deduced that quality of the modal parameters produced by the method is competitive with the state of the art parametric methods.     

IDENTIFICATION OF TIME-VARYING MODAL PARAMETERS USING LINEAR TIME-FREQUENCY REPRESENTATION

A new method of parameter identification based on linear time-frequency representation andHilbert transform is proposed to identify modal parameters of linear time-varying systems frommeasured vibration responses. Using Gabor expansion and synthesis theory measured responses arerepresented in the time-frequency domain and modal components are reconstructed by time-frequencyfiltering. The Hilbert transform is applied to obtain time histories of the amplitude and phase angle ofeach modal component, from which time-varying frequencies and damping ratios are identified. The     

Improved independent component analysis based modal identification of higher damping structures

Structural modal parameter identification under ambient excitation has strong engineering value and theoretical significance. As the most popular tool for solving Blind Source Separation (BSS) problems, Independent Component Analysis (ICA) is able to directly extract the time-domain modal parameters, including frequencies, damping ratios and modal shapes. ICA, however, has a fatal flaw of failing to identify structures with higher damping. To overcome the flaw above, the paper proposes a new method named “ICA + IDT”. Firstly, free vibration response of a structure is obtained from structural outputs under ambient excitation. Inverse damping transfer (IDT) is employed to turn a highly damped signal into a low damping response signal without changing of frequencies and mode shapes. Then, structural modal parameters are extracted from the low damping response signal by ICA. Finally, the identified damping ratios are adjusted to eliminate the impact of IDT. To verify the effectiveness and applicability of IDT + ICA proposed herein, two numerical simulations—mass-spring model and simply supported concrete beam—and an experiment model of three-story steel frame are built, and the analysis results reveal that presented method can identify structures with higher damping effectively.     

基于奇异值分解的频响函数降噪方法

为了提高外场测试中频响函数的信噪比,提出了一种基于奇异值分解的频响函数降噪方法。该方法首先对脉冲响应函数在相空间内进行重构;然后对重构轨道矩阵进行奇异值分解达到降噪的目的。其中,降噪阶次通过奇异熵增量进行确定。采用GARTEUR飞机模型建立具有密集模态的仿真算例进行验证。结果表明,在噪声干扰较大时,该降噪方法能够显著改善模态参数的识别精度,尤其是阻尼的识别精度。