USE OF AN UPDATED FINITE ELEMENT MODEL FOR DYNAMIC DESIGN

Model updating techniques are used to update a finite element model of a structure so that an updated model predicts more accurately the dynamics of a structure. The application of such an updated model in dynamic design demands that it also predict the effects of structural modifications with a reasonable accuracy. This paper deals with the updating of a finite element model of a structure using measured modal data and its subsequent use for predicting the effects of structural modifications. An updated model is obtained by employing a method of model updating based on the constrained optimisation. Structural modifications in terms of mass and beam modifications are then introduced to evaluate the updated model for its usefulness in dynamic design.     

Comparative study of damped FE model updating methods

Most of finite element (FE) 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, a detailed comparison of two approaches of obtaining damped FE model updating methods are evaluated with the objective that the FRFs obtained from damped updated FE models is able to predict the measured FRFs accurately. In the first method, damped updating FE model is obtained by complex parameter-based updating procedure, which is a single-step procedure. In the second method, damped updated model is obtained by the FE model updating with damping identification, which is a two-step procedure. In the first step, mass and stiffness matrices are updated and in the second step, damping matrix is identified using updated mass and stiffness matrices, which are obtained in the previous step. The effectiveness of both methods is evaluated by numerical examples as well as by actual experimental data. Firstly, a study is performed using a numerical simulation based on fixed–fixed beam structure with non-proportional viscous damping model. The numerical study is followed by a case involving actual measured data for the case of F-shaped test structure. The updated results have shown that the complex parameter-based FE model updating procedure gives better matching of complex FRFs with the experimental data.     

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.     

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.     

Free vibration analysis of sandwich beam carrying sprung masses

In this paper, free vibration of three-layered symmetric sandwich beam carrying sprung masses is investigated using the dynamic stiffness method and the finite element formulation. First the governing partial differential equations of motion for one element are derived using Hamilton’s principle. Closed form analytical solution of these equations is determined. Applying the effect of sprung masses by replacing each sprung mass with an effective spring on the boundary condition of the element, the element dynamic stiffness matrix is developed. These matrices are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the well known Wittrick–Williams algorithm. Free vibration analysis using the finite element method is carried out by increasing one degree of freedom for each sprung mass. Finally, some numerical examples are discussed using the dynamic stiffness method and the finite element formulation. After verification of the present model, the effect of various parameters such as mass and stiffness of the sprung mass is studied on the natural frequencies.     

步兵战车车体结构有限元模型修正

针对某型步兵战车整车刚柔耦合发射动力学中柔性车体有限元模型精度低的问题,基于模态试验数据,应用支持向量机响应面模型修正理论对车体结构有限元模型进行了修正。应用ANSYS有限元分析软件对车体结构进行模态分析,提取前6阶模态的固有频率和振型。为验证模型,设计了模态试验方案,实测了车体结构的模态信息。基于有限元模型数据与实测数据的相对误差,采用支持向量机响应面模型修正方法对车体结构弹性模量和密度进行修正。模型确认结果和动力学模型应用结果表明,修正后的车体有限元模型精度有了大幅度提高,能更加真实地反映车体的结构特征,为射击精度分析提供了准确的模型基础。     

Mechanical joint parameter estimation using frequency response functions and component mode synthesis

This paper treats the problem of finite element model updating of structures consisting of substructures connected through mechanical joints whose stiffness and damping properties are unknown. The model is updated by estimating the mechanical joint parameters via the curve fit of measured frequency response functions using a non-linear least-squares method. A damped component mode synthesis formulation is used to calculate the theoretical frequency response functions of the assembled structure given the joint stiffness and damping parameter values. It is shown that a good approximation of the sensitivity matrix may be obtained by finite differences at a smaller computational cost than the closed form expression. Identifiability criteria and estimation errors are addressed. A numerical example consisting of two free-free beams connected through axial springs and dampers illustrates the proposed method.     

Model updating of rotors supported on ball bearings and its application in response prediction and balancing

This work attempts experimental studies in finite element model updating of an actual rotor system mounted on ball bearings by using Inverse Eigen Sensitivity Method (IESM). The IESM is applied on state space representation of equations of motion and is used to identify bearing stiffness, damping and shaft material damping parameters. Non-proportional viscous damping model is used to model the bearing and shaft material damping. The experimental identification of viscous coefficient of shaft material damping was not found in the available literature and this work attempts the same as well. The updated model is validated for its accuracy by comparing the predicted frequency response with that obtained from the experiments. Finally, it is shown that the updated finite element model of the rotor system can be efficiently used to predict the unbalance in the rotor.     

AN ITERATIVE METHOD FOR DYNAMIC CONDENSATION OF STRUCTURAL MATRICES

Dynamic condensation techniques have been broadly applied to the domains of test-analysis-model correlation, vibration control, damage detection and so on to reduce the structural matrices (stiffness, mass and/or damping matrices) of finite element models. Based on the subspace iteration method in the eigenproblem, a dynamic condensation approach is derived in this paper. It is iterative. Comparing almost all the iterative schemes for dynamic condensation proposed in the past, the present approach has three advantages: (1) The convergence is much faster than all these methods, especially when the eigenpairs of the reduced model are close to those of the full model. (2) The convergence of the iterative scheme can be proved simply. (3) It is computationally efficient since it is unnecessary to calculate the stiffness and mass matrices as well as the eigensolutions of the condensed model in every iteration. Two iterative schemes, which are based on the convergence of the eigenvalues of the reduced model and the column vectors of the dynamic condensation matrix, respectively, are given in this paper. Not only the accuracy of eigenvalues, but also that of eigenvectors are considered in every iteration. Numerical examples are also presented to show the efficiency of the proposed method.     

用凝聚技术综合时序分析法和有限元法识别机床的结构参数

机床接触面刚度和阻尼的确定是对机床进行动态分析和优化设计的关键问题之一。本文提出了一种识别机床接触刚度和阻尼的新方法,它利用一种新的凝聚技术把时序分析法和有限元法结合起来,从而只要利用一、二个不完全的振型就可以识别机床接触面的结构参数。这方法由两大部分组成。首先利用时序分析法从实验数据序列建立随机的自回滑动平均向量（ARMAV）模型并进而确定机床的模态参数。然后把机床结构有限元模型在某一复频下进行精确凝聚。并根据从时序分析法和凝聚后的有限元模型得出的模态参数必须相等的条件,就可以识别未知的机床结构参数。利用计算机仿真技术对新提出的方法进行了验证,证明它具有很高的识别精度。最后进行了立柱模型实验,对立柱底部的接触刚度和阻尼成功地进行了识别。     

Reconstructing structural changes in a dynamic system from experimentally identified state-space models

This paper presents an experimental investigation of a recently developed Kronecker Product (KP) method to determine the type, location, and intensity of structural change from an identified state-space model of the system. Although this inverse problem appears to be highly nonlinear, the system mass, stiffness, and damping matrices are identified through a series of transformations, and with the aid of the Kronecker product, only linear operations are involved in the process. Since a state-space model can be identified directly from input-output data, an initial finite element model and/or model updating is not required. The test structure is a two-degree-of-freedom torsional system in which mass and stiffness are arbitrarily adjustable to simulate various conditions of structural change or damage. This simple apparatus illustrates the potential applicability of the system identification technique for damage detection problems by not only identifying the location and the extent of the damage, but also differentiating the nature of the damage. The results are successfully confirmed by laboratory tests.     

Distributed parameter model updating using the Karhunen–Loève expansion

Discrepancies between experimentally measured data and computational predictions are unavoidable for complex engineering dynamical systems. To reduce this gap, model updating methods have been developed over the past three decades. Current methods for model updating often use discrete parameters, such as thickness or joint stiffness, for model updating. However, there are many parameters in a numerical model which are spatially distributed in nature. Such parameters include, but are not limited to, thickness, Poisson's ratio, Young's modulus, density and damping. In this paper a novel approach is proposed which takes account of the distributed nature of the parameters to be updated, by expressing the parameters as spatially correlated random fields. Based on this assumption, the random fields corresponding to the parameters to be updated have been expanded in a spectral decomposition known as the Karhunen–Loève (KL) expansion. Using the KL expansion, the mass and stiffness matrices are expanded in series in terms of discrete parameters. These parameters in turn are obtained using a sensitivity based optimization approach. A numerical example involving a beam with distributed updating parameters is used to illustrate this new idea.     

表面阻尼处理薄板结构声辐射分析的有限元法

用整体划分单元法建立自由阻尼处理和约束阻尼处理薄板结构的有限元模型,给出结构的变形能表达式,并通过变形能原理推得单元的刚度和质量矩阵。在用有限元法分析结构动态响应特性的基础上,利用有限元分析得到结构表面振动速度分布,分析表面阻尼处理板结构的声辐射。算例表明,用整体划分单元法预测表面阻尼处理薄板结构的动态特性和声辐射是可行的,其精度满足工程要求,并有简单方便的特点。     

DAMAGE IDENTIFICATION USING CHANGES OF EIGENFREQUENCIES AND MODE SHAPES

In the present paper, we describe an approach to identify the location and the extent of the damage introduced into the steel frame, using a two-step procedure. In the first step, the measured dynamic response of the original undamaged structure was used to generate a reference finite element (FE) model of the structure. The selected parameters were identified by means of a mathematical optimisation algorithm (‘updating procedure'), minimising an objective function containing the test/analyses differences of eigenfrequencies and mode shapes. The uncertain model parameters had to be chosen with care in order to retain the physical significance of the updated model. In the next step, the experimental modal data of the damaged structure were used to identify the extent of the damages. This was based on comparing the changes of stiffness parameters identified from the undamaged and the damaged structure. With the identified parameters, the FE model was able to reproduce the experimental data as close as possible and allowed the identification of the extent of the damage.     

5R柔性并联机器人系统运动微分方程

为了描述平面5R柔性并联机器人的运动学和动力学特性,需要建立机器人的运动微分方程。针对刚性活动平台和柔性杆件的运动学耦合特点,改进了一套适用于刚体、柔性体耦合的有限元建模方法,推导出单元弹性广义坐标相对于系统弹性广义坐标的转换矩阵,综合考虑了科氏阻尼、离心刚度和几何非线性的影响,利用运动弹性动力学理论,建立了平面5R柔性并联机器人的运动微分方程,避免了采用运动学和动力学约束方程的弊端,提高了建模精度。计算实例表明,该方程反映了机器人的弹性振动特性,杆件的弹性变形对机器人的运动误差具有重要影响。     

橡胶减振器动刚度有限元数值预测方法研究

提出了一种基于有限元的橡胶减振器动刚度预测方法,对方法中的材料本构模型、有限元模型、动刚度预测流程进行了分析和研究。利用此方法对某惯导减振器进行预测,获得减振器动刚度和阻尼值,代入单自由度分析模型后预报出的动响应结果与试验结果基本一致。用该方法能捕捉到由于减振器刚度变化而引入的高频率共振峰,表明该方法具有更高的精度。     

使用有效模态质量和遗传算法的有限元模型修正

提出将模态频率和有效模态质量构造的残差作为遗传算法的目标函数进行结构动力学有限元模型修正的方法。有效模态质量不但可以为结构动力学响应分析提供一种判断模态贡献程度的方法,而且能够为有限元模型修正提供更多的信息量。介绍了有效模态质量的概念和基于遗传算法的结构动力学模型修正理论,在此基础上采用仿真算例验证了所提出方法的正确性和有效性。仿真结果显示,模型修正后参数最大误差为-0.062%,不管是在修正频段内还是修正频段外,频率和有效模态质量的均方误差都小于0.025%。研究表明,使用有效模态质量和遗传算法的结构动力学有限元模型修正是有效可行的。     

高频谐振疲劳机载荷测量误差建模分析及试验夹具优化设计

电磁谐振式疲劳裂纹扩展试验过程中,对动态载荷的高精度测量和控制是保证试验结果准确性和系统工作稳定性的重要前提,夹具作为夹持试件和传递载荷的重要机构,对其进行结构优化显得尤为重要。建立了紧凑拉伸(CT)试件连接刚度模型和三自由度有阻尼振动系统动力学模型,得到了动态载荷测量误差理论表达式;分析了夹具刚度对动态载荷测量误差的影响,设计了不同结构刚度的试件夹具,并通过有限元方法计算其刚度,得到了不同结构夹具对动态载荷测量误差的影响结果;实验验证了优化夹具设计的合理性。