A new theoretical basis for the bandwidth method and optimal power ratios for the damping estimation

Using the well-known bandwidth formula and the half power bandwidth formula [R.E.D. Bishop, G.M.L. Gladwell, An investigation into the theory of resonance testing, Philosophical Transactions of the Royal Society of London A 255 (1963) 241–280], in particular, is the simplest way to estimate modal damping for engineers. By using the half power bandwidth formula, the damping factor is estimated to be approximately the half bandwidth at the half power points. One of the major limitations that restrict the use of this method is the coupling effect between closely spaced modes. In this paper, the dependence of the damping estimation accuracy on the selected power ratios is studied with both analytical and experimental data of frequency response functions. The results show that by selecting adequate power ratio values, the coupling effect can be minimized and the estimation accuracy can be significantly improved for closely spaced modes. A further improvement of accuracy can be obtained by applying the algorithm of mode isolation [H.P. Yin, D. Duhamel, Substraction technique and finite difference formulas for modal parameter estimation, Mechanical Systems and Signal Processing 18 (2004) 1497–1503; M.S. Allen, J.H. Ginsberg, A global, single-input-multi-output (SIMO) implementation of the algorithm of mode isolation and application to analytical and experimental data, Mechanical Systems and Signal Processing 20 (2006) 1090–1111]. Also an exact bandwidth formula in case of a single degree of freedom system is presented and the link between the exact formula and the classical approximated formula is indicated. The exact bandwidth formula provides a new theoretical basis of the bandwidth method for the damping estimation from frequency response functions.     

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 the effect of mechanical properties of liquid and geometrical parameters of cantilever on the frequency response function of AFM

The dynamic behavior of atomic force microscopy (AFM) cantilevers in liquid is completely different from its behavior in air due to the applied hydrodynamic force. Exciting cantilever with frequencies close to resonance frequency and primary position alignment are two critical issues that should be considered in deriving frequency response function (FRF). In this paper, the hydrodynamic force has been modeled with string of spheres and the effect of the damping and the added mass on the model has been analyzed. Afterward, this force is applied to the dynamic equation so that the dynamic behavior of AFM cantilevers is studied in liquids by analyzing the effect of some important parameters such as added mass, internal, and fluid damping. By simulations of the dynamic equations for a silicon cantilever, FRF is determined in both air and liquid. In addition, the effects of two significant parameters of liquid mechanical properties (liquid viscosity and density) and geometrical parameters of cantilever on FRF are studied. The results for string of spheres model are compared with the other hydrodynamic model and the experimental data. When length/width ratio decreases, it is found that string of spheres model has a better agreement than the other hydrodynamic model with experimental data.     

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.     

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.     

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。     

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.     

Model updating of damped structures using FRF data

Due to the important contribution of damping on structural vibration, model updating of damped structures becomes significant and remains an issue in most model updating methods developed to date. In this paper, the frequency response function(FRF) method, which is one of the most frequently referenced model updating methods, has been further developed to identify damping matrices of structural systems, as well as mass and stiffness matrices. In order to overcome the problem of complexity of measured FRF and modal data, complex updating formulations using FRF data to identify damping coefficients have been established for the cases of proportional damping and general non-proportional damping. To demonstrate the effectiveness of the proposed complex FRF updating method, numerical simulations based on the GARTEUR structure with structural damping have been presented. The updated results have shown that the complex FRF updating method can be used to derive accurate updated mass and stiffness modelling errors and system damping matrices.     

Response selection and dynamic damper application to improve the identification of multiple input forces of narrow frequency band

In identifying input forces by using FRF matrices and response measurements, improper selection of the response measurement position and the excitation frequency, if it coincides with the natural frequencies of the structure, may cause doubtful accuracy and effectiveness. This paper presents two methods to improve force identification. The first is by improving the conditioning of the system FRF matrix by a proper selection of the measurement positions. An expression for the covariance matrix representing the uncertainty on the force estimate is defined. Then the response selection algorithm, which is both systematic and effective, is derived. The basic strategy in selecting the response positions is to let the smallest singular value be as large as possible. The second method is to modify the structure by attaching a dynamic damper at a suitable location to minimise the ill-conditional nature of the FRF matrix especially near a resonance frequency. The methods suggested are tested numerically and experimentally in two examples: a beam structure and a rectangular plate structure. The test results show that the response selection and the addition of a dynamic damper are very effective in solving ill-conditioning problems, especially when the structure has a large degree of freedom and when the excitation frequencies are near one of the structure's natural frequencies.     

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。     

半功率点法估计阻尼比的误差分析

从快速傅里叶变换（FFT）离散谱线中难以恰好找到半功率点，通常是通过线性插值法近似确定半功率点来估计阻尼。文中采用理论分析与数值仿真相结合的方法研究基于线性插值的半功率点法的主要误差因素和各自特性，特别是其中窗阻尼变化规律。研究结果表明，①应尽量接近整周期采样。②无阻尼系统因进行FFT截断也会出现虚假的窗阻尼，且真实阻尼不等无表观阻尼减去窗阻尼。③阻尼的相对误差主要取决于采样周期数与阻尼比，其物理意义是截断窗口内的信号衰减程度。④要保证阻尼10％的精度，窗阻尼要小于表观阻尼的2／3。     

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.     

Chatter suppression in micro-milling using shank-mounted Two-DOF tuned mass damper

The tuned mass damper (TMD) has been used in machining processes for reducing forced vibration, suppressing chatter, and improving machined surface quality. In micro-milling process, the tiny size of the cutting tool-tip and the high rotating speed bring challenges in implementing the TMD. Besides, the TMD needs to have two degrees-of-freedom (DOFs) for reducing vibrations of micro-mill in two orthogonal directions. This paper presents the chatter suppression for micro-milling by attaching a two-DOF TMD to the tool shank and rotates with the cutting tool. The frequency response function (FRF) at the tip of the micro-mill clamped by an aerostatic spindle is predicted using receptance coupling analysis. A two-DOF TMD is designed via graphical approach based on the FRF result at the tool-tip. The natural frequencies and damping ratio of the TMD are optimized under different spindle speeds in order to enhance the cutting stability. The chatter stability of micro-milling is predicted considering the gyroscopic and centrifugal effects of the TMD structure. Modal tests and micro-milling experiments are conducted to validate the effect of the TMD on chatter stability. The results show that the TMD is able to improve the critical depth of cut by 13 folds, and satisfy the compact design requirement for micro-milling.     

基于频响函数相关性的灵敏度分析的有限元模型修正

有限元模型的修正对机械结构的动态特性进行准确而可靠的预测是很重要的。利用试验测试和预测的有限元模型计算得到的频响函数（FRF），引入两种频响函数相关性的判定标准，提出基于频响相关函数的灵敏度分析的修正方程。数值实例研究结果表明，该方法利用少量的测量数据，即使测试数据含附加噪声，也可在很宽的频率范围内得到接近真实结构的有限元模型修正解。本文的方法可适用于大型复杂结构的模型修正。     

Further experience with model updating incorporating damping matrices

Most of the model updating techniques do not employ damping matrices and hence cannot be used for accurate prediction of complex frequency response functions (FRFs) and complex mode shapes. In this paper, response function method (RFM) is extended to deal with the complexity of FRF by updating damping matrices along with mass and stiffness matrices. The effectiveness of the damped FE model updating procedure is demonstrated by actual laboratory experiments of an F-shaped test structure. The updated results have shown that the damped RFM model updating procedure can be used to derive accurate model of the system. This is illustrated by matching of the complex FRFs obtained from the updated model with the experimental data.     

振动台功率谱复现算法

传统的振动控制技术将初始辨识的系统频率响应函数贯穿使用于振动控制的过程中;针对液压振动台系统的时变特性,提出使用基于最小均方误差(least mean square,简称LMS)的自适应算法在线辨识系统的频响函数.平滑周期图功率谱估计法相对现代谱估计法分辨率较低,提出自回归(auto-regressive,简称AR)模型法对振动系统响应信号进行功率谱估计,利用尤利-沃克(Yule-Walker)方程求解AR模型参数,并给出AR模型阶次确定的方法.利用自行开发的基于DSP和ARM多处理器信号处理系统对功率谱复现进行软硬件仿真.结果表明,此方法对振动台功率谱进行复现,复现精度优于传统功率谱复现算法.     

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.     

Vibration of a three-layered viscoelastic sandwich circular plate

The equations of motion for a three-layer sandwich circular plate with viscoelastic core are derived. The Donnell–Mushtari–Vlasov assumptions follow to simplify. For the case of iso-symmetric annular plate, five equations are further reduced to three equations. The assumed-mode method then yields the free and the forced solutions being characterized by six generalized coordinates for each circumferential wave number n. Due to the splitting of transversal and in-plane terms the study of vibration response can be reduced to the consideration of the SDOF linear oscillator with hysteretic damping. Numerical results first illustrated the frequency response function (FRF) for two different visco models and slight difference is observed. The effects of frequencies changes and damping due to viscoelastic material (VEM) core for several core-to-face thickness ratios are studied. The results, via the FRF's, show that a relatively thin visco (VEM) core can reduce vibration effectively, but as the core thickness exceeds it the damping gradually reduces. The embedded VEM core decreases the frequencies as expected. The damping effects upon individual modes are looked into as well. The results show that the n=0 mode is affected the most by VEM core.     

一种模态弱响应且模态密集的参数识别方法

针对运行模态分析中响应信号数据样本较短、模态密集,以及弱响应信号淹没在大噪声中,系统模态难以全部辨识的问题,提出了联合相关函数与传递率识别系统模态参数的方法(联合方法),该方法先应用传递率近似频响函数获取系统弱响应频率特征函数,然后再通过小波变换进行模态识别。运用随机激励下的GARTEUR飞机模型仿真运行状态的输出进行了数值仿真实验。结果表明:相比于多参考最小二乘复频域法,联合方法不仅能提高模态频率的识别精度,而且还能极大地提高阻尼比的识别精度,尤其是传递率对模态密集的弱响应模态识别结果良好。