Identification of inertia properties from the results of output-only modal analysis

Accurate knowledge of the rigid body properties of a structure including the mass, the location of mass center and the moments of inertia is important in machine design, vibration analysis, optimization and modeling of mechanical systems. However, estimation of these properties through theoretical methods is difficult when the structure has a complicated shape. In practice, the inertia properties can be estimated using the conventional modal testing methods by extracting the rigid body modes when the structure is tested in free-free boundary condition. However, all the rigid body modes are not always detectable, due to this fact that the structure is not excited at all degrees of freedom. In order to obtain all of the rigid body modes, many activities have been conducted for selecting the type and location of excitation without much success. In operational modal analysis (OMA), the structure can be excited at any arbitrary point and in different directions. In this paper, a new approach is introduced for estimation of the inertia properties from OMA. The data from OMA are adequate to extract all the rigid body modes of structure. A modal method is used for estimating the inertia properties from the rigid body modes extracted from OMA. The suggested approach is applied to a numerical model of a two-dimensional steel beam as well as a numerical model of a 3D frame and the accuracy of results is evaluated. It is shown that OMA can provide enough data to extract the inertia properties. A real beam is also tested in order to evaluate the performance of the method in practice, needless of a complicated procedure as for conventional methods.     

基于有限测点信息的结构损伤识别柔度法

利用有限测点获得结构的模态参数,提出了基于有限测点的结构损伤识别的柔度法。该方法是通过仅考虑结构柔度的灵敏度分析,以结构各自由度的损伤信息为条件,选择对结构柔度变化敏感的自由度为测点,并利用有限测点的信息提出了结构完备模态振型的重建技术。在此基础上,对柔度矩阵做关于结构物理参数变化量的一阶泰勒展开,来确定结构单元的损伤因子对结构进行损伤识别。从而实现利用结构有限测点的模态信息来识别结构的损伤,解决了测试结构的模态振型的不完整给结构的损伤诊断带来的困难。通过数值算例说明了该方法的有效性。     

Free vibration analysis of a von Koch beam

In this paper, the free undamped motion of a cantilever von Koch beam is investigated. The reduction of the stiffness and mass matrices leads to simple analytical recursive relationships depending on the fractal dimension of the structure. Results are then extended to perform a detailed modal analysis, which suggests peculiar scaling laws for the natural frequencies and modal shapes of the structure. Energy considerations are also provided. Finally, the potentiality of the von Koch beam as a fractal antenna is examined in terms of resonant frequencies.     

Determination of mode shapes using wavelet transform of free vibration data

In this article, a new method is proposed to determine the mode shapes of linear dynamic systems with proportional viscous damping excited by an impact force. The time signals of responses and a priori knowledge of the natural frequencies are required in this method. The method is particularly suitable for the wavelet techniques which can precisely estimate the natural frequencies. A previously proposed method based on a modified Morlet wavelet function with an adjusting parameter is used to identify the natural frequencies and damping ratios of system, and the mode shapes are estimated using the proposed method in this work. It is shown that the extracted mode shapes are not scaled. Therefore, mass change method is used for scaling the mode shapes. Moreover, the effect of noise on the extracted modal parameters is investigated. The validity of method is demonstrated using numerical and experimental case studies.     

Operational modal filter and its applications

In the paper, the procedure for the estimation of modal filter coefficients from output-only data is presented. The basic concept of the procedure consists in frequency response functions synthesis based on the knowledge of an operational modal model. A method of operational mode shapes scaling is described. The method is then compared with the classical modal filter and with modal filtration of responses spectra, which is sometimes used as a solution for modal filtration based on the output-only data. Each solution is applied to load identification and damage detection. The study shows the method verification on data obtained from laboratory experiment.     

Dynamics of cable truss via polynomial shape functions

A model for the in-plane dynamic behaviour of a biconcave cable structure, subject to large static deformations and potentially slack harnesses is proposed, based on polynomial shape functions, in line with the classical Ritz method. The model provides a semi-analytical approach to the calculation of natural frequencies and modal shapes of the structure. The proposed formulation leads to an eigenvalue problem, based on a reduced number of degrees of freedom compared with equivalent FEM solutions, providing the basis for fast and accurate sensitivity analysis. The behaviour of the deformed structure is analysed in detail to understand the non-linear effects of non-symmetric mass and load distribution and slack harnesses on natural frequencies and corresponding modal shapes. Results confirm the relevance of the non-linear effects, due to the statically loaded configuration, on the linear vibrations of the structure, in particular evidencing the influence of the slackening of harnesses on modal shapes. Results are compared to analytical models, where available (single sagged cable), and to FEM solutions (for cable trusses with non-uniform mass and load distribution and potentially slack harnesses), providing good agreement.

     

Analytical and experimental modal analysis of nonuniformly ring-stiffened cylindrical shells

In this research, the free vibration analysis of cylindrical shells with circumferential stiffeners, i.e., rings with nonuniform stiffener eccentricity and unequal stiffener spacing, is investigated using analytical and experimental methods. The Ritz method is applied in analytical solution, while stiffeners are treated as discrete elements. The polynomial functions are used for Ritz functions. The effects of nonuniformity of stiffener distribution on natural frequencies are considered for free–free boundary conditions. Results show that, at constant stiffener mass, significant increments in natural frequencies can be achieved using nonuniform stiffener distribution. In experimental method, modal testing is performed to obtain modal parameters, including natural frequencies, mode shapes, and damping in each mode. Analytical results are compared with experimental ones, showing good agreement. Because of insufficient experimental modal data for nonuniform stiffener distribution, the results of modal testing obtained in this study could be a useful reference for validating the accuracy of other analytical and numerical methods for free vibration analysis.     

Identifying mode shapes and modal frequencies by operational modal analysis in the presence of harmonic excitation

Operational modal analysis techniques allow us to extract the modal properties of structures based on their response to non-measured stationary white noise, i.e., by considering only the system response to operational excitations. In this paper we outline a procedure to deduce modal parameters from operational response measurements. In particular, we discuss a novel approach to analyze operational responses due to unknown harmonic excitation in addition to noise. Structural eigenfrequencies and modal damping are computed using a modified least-squares complex exponential method. Once the poles of the system are identified, mode shapes are obtained by post-processing. The robustness and accuracy of the approach are illustrated by performing tests on a plate structure.     

基于改进遗传算法主动柔性结构压电元件位置优化

基于改进的遗传算法提出了一种解决多压电片布置在柔性结构上以实现振动主动控制的有效方法.本文采用D优化设计准则,即把Fisher信息矩阵行列式最大值作为目标函数的一种优化方法.通过对结构模态振型和动力特性的研究,使用一种简单的方法将所选择的低阶模态振型转化为归一化形式,最后通过结构模态振型可确定压电片最优位置.为了达到更好的振动控制效果,压电片布置在结构模态应变最大位置处.当在结构不同位置上布置压电片时,结构各阶模态振型也将随之发生不同程度的变化.本文建立了ANSYS软件和遗传算法的接口来实现对结构的模态重分析,从而提取各阶模态振型.     

Best achievable modal eigenvectors in structural damage detection

This paper reports a modal formulation of the original method presented by Lim and Kashangaky, based on the use of the best achievable eigenvectors in damage detection problems. The method requires the measurement of both frequencies and mode shapes. The structural damage is located by computing the Euclidean distances between the measured mode shapes and the best achievable modal eigenvectors. The method is able to detect loss of both stiffness and mass properties, even though in this paper only the loss of stiffness will be analyzed. A simple numerical example is reported to investigate the applicability of the modal formulation. Finally, an experimental validation is included using a 10-bay truss laboratory structure.     

A fluid–structure interaction model for stability analysis of shells conveying fluid

In this paper, a fluid–structure interaction model for stability analysis of shells conveying fluid is developed. This model is developed for shells of arbitrary geometry and structure and is based on incompressible potential flow. The boundary element method is applied to model the potential flow. The fluid dynamics model is derived by using an inflow/outflow model along with the impermeability condition at the fluid–shell interface. This model is applied to obtain the flow modes and eigenvalues, which are used for the modal representation of the flow field in the shell. Based on the mode shapes and natural frequencies of the shell obtained from an FEM model, the modal analysis technique is used for structural modeling of the shell. Using the linearized Bernoulli equation for unsteady pressure on the fluid–shell interface in combination with the virtual work principle, the generalized structural forces are obtained in terms of the modal coordinates of the fluid flow and the coupled field equations of the fluid–structure are derived. The obtained model is validated by comparison with results in the literature, and very good agreement is demonstrated. Then, some examples are provided to demonstrate the application of the present model to determining the stability conditions of shells with arbitrary geometries.     

基于改进残余力向量法的桁架结构损伤诊断

提出一种基于改进残余力向量法的桁架结构损伤诊断方法. 先由灵敏度分析, 求出结构刚度联系矩阵,再由刚度联系矩阵将损伤后的刚度摄动矩阵展开成对角矩阵,代入 残余力向量方程,得到由刚度联系矩阵表示的新的残余力向量方程,此方程可以直接求解, 即可诊断出桁架结构的损伤杆件及其损伤程度. 对于实测中难以获得完备振型的情况,采用 模态扩阶的方法来获得完备的测试振型. 最后以一桁架结构进行数值仿真分析,证实了该方 法的有效性.     

大范围运动细长柔性空间结构动力学特性分析

自由-自由边界无约束状态的细长柔性空间结构大范围运动时的动力学特性对整体结构运动分析和运动控制系统设计具有极其重要的作用。通过浮动坐标系建立结构的运动学关系;借助假设模态法对结构变形进行变量分离;利用Lagrange’s方程建立了结构的刚柔耦合振动方程;再通过Rayleigh-Ritz法,以无大范围运动时的振型函数作为基本解组,得到了大范围运动影响下的结构振动特征方程,求解该方程得到了结构频率和振型。通过几组数值算例的对比分析,指出了非耦合模型和耦合模型下结构频率及振型之间的差异。     

High-order layerwise mechanics and finite element for the damped dynamic characteristics of sandwich composite beams

A high-order discrete-layer theory and a finite element are presented for predicting the damping of laminated composite sandwich beams. The new layerwise laminate theory involves quadratic and cubic terms for approximation of the in-plane displacement in each discrete layer, while interlaminar shear stress continuity is imposed through the thickness. Integrated damping mechanics are formulated and both laminate and structural stiffness, mass and damping matrices are formed. A finite element method and a beam element are further developed for predicting the free vibration response, including modal frequencies, modal loss factors and through-thickness mode shapes. Numerical results and evaluations of the present model are shown. Modal frequencies and damping of sandwich composite beams are measured and correlated with predicted values. Finally, parametric studies illustrate the effect of core thickness and face lamination on modal damping and frequency values.     

Tikhonov's regularization approach in mode shape curvature analysis applied to damage detection

In this paper an on-going research effort aimed at detecting and localizing damage in plate-like structures by using mode shape curvature based damage detection algorithm is described. The proposed damage index uses exclusively mode shape curvature data from the damaged structure. This method was originally developed for beam-like structures. In this paper, the method is generalized to plate-like structures which are characterized by two-dimensional mode shape curvature. To calculate mode shape curvatures from the measured mode shapes, three approaches are proposed: the first one is the well-known central difference approximation, the other two are classical approaches based on Tikhonov's regularization technique with smoothing functional. The applicability and effectiveness of the proposed damage detection algorithms are demonstrated experimentally on an aluminium plate containing mill-cut damage. The validity of the method is assessed by comparing the identification results of the experimental test case to the results obtained from the simulated test case. The modal frequencies and the corresponding mode shapes of the aluminium plate are obtained via finite element models for numerical simulations and by using a scanning laser vibrometer (SLV) for the experimental study.     

一种有限元模型修正中的参数选择方法

有限元模型修正面临的首要问题是待修正参数的选择.目前主要是基于灵敏度分析的结果进行参数选择.本文从结构固有频率分析的能量法出发,论证了某阶固有频率主要受到同阶振型中发生弹性变形部位刚度的影响,据此提出一种待修正参数的选择方法,以振型中弹性变形为依据,选择弹性变形较大部位的参数作为待修正参数.用一个梁模型仿真验证了该方法的正确性,并将其应用于系杆拱桥的有限元模型修正中,修正后有限元计算结果与试验结果的最大误差缩小至3.3%.     

3-D inviscid self-excited vibrations of a blade row in the last stage turbine

Presented here is a three-dimensional (3-D) nonlinear time-marching method for the aeroelastic behaviour of an oscillating turbine blade row. The approach has been based in the solution of a coupled fluid–structure problem where the aerodynamic and structural dynamic equations are integrated simultaneously in time. This provides the correct formulation of a coupled problem, as the interblade phase angle (IBPA) at which stability (instability) would occur is also a part of solution. The ideal gas flow around multiple interblade passages (with periodicity in the entire annulus) is described by the unsteady Euler equations in conservative form, which are integrated by using the explicit monotonic second-order accurate Godunov–Kolgan finite-volume scheme and a moving hybrid H–O (or H–H) grid. The fluid and the structural equations are solved using the modal superposition method. An aeroelasticity prediction of a turbine blade of 0.765 m is presented. The natural frequencies and modal shapes of the blade were calculated by using 3-D finite element models. The instability regions for five mode shapes and the distribution of the aerodamping coefficient along the blade length were shown for harmonic oscillations with an assumed IBPA. The coupled fluid–structure oscillations in which the IBPA is part of the solution are shown.     

A numerical method for determining nonlinear normal modes

This paper examines a new approach for determining the nonlinear normal modes of undamped non-gyroscopic multiple degree-of-freedom systems. Unlike algebraic solutions that generally assume a solution in the form of a polynomial expansion, this method makes only the assumption of repetitive motion in numerically determining the mode shapes. The advantage of this approach is that the accuracy obtained in the mode shape identification is a function only of the accuracy of the numerical integration used and not of the number of terms in the power series expansion. The drawbacks are that invariance of the modal manifolds cannot be proven and mode bifurcations can be easily overlooked.     

带弹簧-质量系统的车-桥耦合演变随机振动

提取悬挂有弹簧-质量系统桥梁的模态函数,将车辆简化为一个两自由度的移动系统,以桥面不平度功率谱密度为输入,建立车辆与带弹簧-质量系统的桥梁的耦合系统力学模型. 结合状态空间理论和演变随机过程的一般理论,研究车-桥耦合演变随机振动的弹簧-质量系统控制. 通过数值算例可知:弹簧-质量系统对车辆的随机响应没有影响,而可以降低桥梁的随机响应. 最后,探讨了弹簧-质量系统与桥梁质量比、弹簧--质量系统个数对桥梁随机响应的影响.     

Free vibration characteristics of a rectangular plate carrying multiple three-degree-of-freedom spring–mass systems using equivalent mass method

This paper aims at presenting a technique to replace each three-degree-of-freedom (3-dof) spring–mass system (or substructure) by a set of equivalent masses so that the dynamic characteristics of a rectangular plate (or main structure) carrying any number of elastically mounted lumped masses may be obtained from the same plate carrying the same sets of rigidly attached equivalent masses. Because the three degrees-of-freedom (dof’s) of the substructure are embedded in its equivalent masses, the total dof of the entire vibrating system (i.e., the main structure together with all the substructures) is independent on the total number of the substructures attached to the main structure in the presented equivalent mass method (EMM). However, in the conventional finite element method (FEM), the total dof of the entire vibrating system increases by three when one more substructure is attached to the main structure. Compared with FEM, the merits of EMM are double: Firstly, since the total dof of the entire vibrating system in EMM is smaller than that in FEM, some computer storage memory may be saved. Secondly, since the dof’s for all the substructures are eliminated, the associated natural frequencies and mode shapes are excluded from those of the main structure and some effort required for the analysis of computer-output data may also be saved. It is evident that the last merits of EMM will be more predominant if the total number of substructures attached to the main structure is large.