On the use of damped updated FE model for dynamic design

Model updating techniques are used to update the finite element model of a structure, so that updated model predicts the dynamics of a structure more accurately. The application of such an updated model in dynamic design demands that it also predicts the effects of structural modifications with a reasonable accuracy. Most of the model updating techniques neglect damping and so these updated models cannot be used for predicting amplitudes of vibration at resonance and antiresonance frequencies. This paper deals with updating of the finite element model using the FRF data with damping identification using complex modal data and its subsequent use for predicting the effects of structure modifications. The updated model is obtained in two steps. In the first step, mass and stiffness matrices are updated using FRF-based model updating method. In the second step, damping is identified using updated mass and stiffness matrices, which are obtained in first step. 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.     

IDENTIFICATION OF NON-PROPORTIONAL MODAL DAMPING MATRIX AND REAL NORMAL MODES

Equations of motion for non-proportionally damped structures cannot be decoupled by using the real normal modes. For such structures, the complex normal modes are in common use for this purpose, but for the validation of finite element mass and stiffness matrices where physical damping matrices are not available, the related experimental real normal modes must be known. In previous publications, an identification theory using the real normal modes and the non-diagonal modal damping matrix for the non-proportionally damped system and some applications with the computer code ISSPA were presented. However, the theory cannot assure the symmetry of the identified modal damping matrix, which must be theoretically symmetric. In this paper, a method for identifying the symmetric non-proportional modal damping matrix using undamped modal parameters obtained from ISSPA is presented and the validity of the method is demon-strated through both numerical and experimental examples.     

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。     

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.     

Iterative finite element model updating in the time domain

An iterative time domain formulation for finite element model updating in structural dynamics is presented. The approach is supported on a derivation showing that the discrepancy between observations and model predictions can be expressed as a convolution between the state of the system and a sequence of pseudo-Markov parameters which are linear in the change of the free parameters. The approach is illustrated by updating all the stiffness and damping parameters of a twenty degree of freedom shear beam using four noise contaminated measurements.     

不确定声场分析的区间矩阵分解摄动有限元法

针对修正一阶区间摄动有限元法存在的一阶Taylor展开误差较大和求解摄动逆矩阵时计算效率不高的缺陷,提出区间矩阵分解摄动有限元法(Decomposed interval matrix perturbation finite element method, DIMPFEM)。该方法将系统动态刚度矩阵分解为若干系统子矩阵之和,每个系统子矩阵的摄动矩阵用摄动因子和常量矩阵的乘积表示,避免了摄动矩阵的Taylor展开误差;采用Epsilon算法求解摄动逆矩阵的修正Neumann级数,有效提高了计算效率。将DIMPFEM应用于具有区间参数的二维管道和二维商务车声腔模型的声压响应分析,分析结果表明,与修正一阶区间摄动有限元法比较,DIMPFEM获得了更高的计算精度和计算效率。     

Identification of prestress force from measured structural responses

A method for the identification of prestress force of a prestressed concrete bridge deck is presented using the measured structural dynamic responses. A Euler–Bernoulli beam finite element model is used to represent the bridge deck, and the prestress force is modelled as the axial prestress force in each beam element. The state-space approach is used to calculate the dynamic responses of the structure and the sensitivities of dynamic responses with respect to the structural parameters, such as the prestress force, flexural rigidity, etc. The prestress force in each beam element is taken to be a system parameter, and it is expressed explicitly in the system equation for forward analysis. The prestress force in each element is identified using a sensitivity-based finite element model updating method in the inverse analysis. Data obtained from a single or multiple accelerometers or strain gauges are used in the identification. Both sinusoidal and impulsive excitations are illustrated to give very good results. Two numerical simulations are presented to illustrate the effectiveness and robustness of the proposed method. Laboratory work on an axially prestressed concrete beam is also included as a practical application.     

考虑剪切变形的复杂刚架结构动力学特性分析

采用谱元法计算分析了复杂刚架结构的动力学响应特性,建立了直杆和Timoshenko梁的局部动力刚度阵,并组装得到了整体刚架结构的动力刚度阵。计算了刚架结构的频响曲线,并与有限元法得到的结果进行了比较,获得了刚架的固有频率和冲击载荷作用下的频域曲线。研究结果表明,谱元法可有效且准确地分析刚架结构的动力学响应特性,尤其适合中高频动态响应特性分析,与有限元方法相比,其精度高且用时短。       

Identification of the dynamic parameters of active magnetic bearings based on the transfer matrix model updating method

The stiffness and damping coefficients of Active magnetic bearings (AMBs) have a great impact on the dynamics of a high-speed rotor AMB system, from its bending critical speed to the modes of its vibration and stability. To accurately obtain the stiffness and damping coefficients of AMBs, this study proposes a new identification approach based on the transfer matrix model updating method. By minimizing the error between the unbalance response calculated through the transfer matrix approach and the experimental measurements, the stiffness and damping coefficients are obtained using the simplex optimization algorithm based on the updating method of the model. According to the experimental data, we identify the parameters from 20 Hz to 260 Hz (1200 rpm to 15600 rpm). To verify the identified results, a finite element rotor AMBs model is created, and the theoretical unbalance response is predicted using the identified parameters. The theoretical unbalance responses closely coincide with the experimental measurements, indicating the effectiveness of the proposed method.     

Optimum design of structures with stress and displacement constraints using the force method

A new structural analysis and optimization algorithm is developed to determine the minimum weight of structures with the truss and beam-type members under displacement and stress constraints. The algorithm combines the mathematical programming based on the sequential quadratic programming (SQP) technique and the finite element technique based on the integrated force method. The equilibrium matrix is generated automatically through the finite element analysis while the compatibility matrix is obtained directly using the displacement–deformation relations and the single value decomposition (SVD) technique. By combining the equilibrium and compatibility matrices with the force–displacement relations, the equations of equilibrium with the element forces as variables are obtained. The proposed method is extremely efficient to analyze and optimize the truss and beam structures under stress and displacement constraints. The computational effort required by the force method is found to be significantly lower than that of the displacement method. The effect of the geometric nonlinearity in the structural optimization problems under the stress and displacement constraints were also investigated and it is illustrated that the geometric nonlinearity is not an important issue in these types of problems and hence, it does not affect the final optimum solution significantly. Four examples illustrate the procedure and allow the results to be compared with those reported in the literatures.     

EVALUATION OF DAMPING NON-PROPORTIONALITY USING IDENTIFIED MODAL INFORMATION

This paper focuses on quantification of damping non-proportionality present in a discrete vibratory system. The study assumes that the information available is a set of identified system eigenvalues and eigenvectors and that the system parameters such as mass, stiffness, and damping matrices are unknown a priori. This set of modal parameters may be incomplete. The investigation is concentrated on how two existing analytical indices can be utilised when the modal damping matrix is not available. The quantification procedure starts with extraction of normal modes using a known algorithm. It is shown that two matrices, by-products of the normal model extraction, can be used to study damping non-proportionality. The first matrix is a scaled modal damping matrix. The paper shows that the indices developed from the scaled modal damping matrix preserves the properties of the indices based on the analytical modal damping matrix. The second matrix is a complex matrix which is obtained by expanding complex modes into the subspace of real modes. The off-diagonal elements of the complex matrix indicate coupling between modes due to damping non-proportionality. Based on this characteristic, three new indices are proposed. Numerical examples are presented to illustrate the use of the new indices and to compare them with the indices that are described in literature.     

不同自由度耦合斜齿轮转子系统的振动特性

以一个两对斜齿轮耦合的三平行轴转子系统为研究对象,考虑静态传递误差和齿轮几何偏心等因素的影响,建立了全自由度通用齿轮啮合动力学模型。将其与转子系统有限元模型进行耦合,建立了平行轴系齿轮转子系统有限元模型。转子系统采用梁单元模拟,齿轮之间的啮合通过啮合刚度矩阵和阻尼矩阵模拟,并分析了不同自由度耦合下系统的固有特性和振动响应特性。研究结果表明,考虑弯扭耦合和弯扭轴摆耦合会产生较多的弯扭耦合频率,响应计算结果出现的峰值点均对应系统的固有频率,而考虑弯扭轴摆耦合可以更好地表征系统的不同自由度的耦合振动情况。此研究结果可为齿轮耦合转子系统设计提供参考。     

五参量结构阻尼模型及其在弹性机构动力学中的应用

将阻尼合金视为粘弹性材料,利用五参量本构关系来描述阻尼合金材料的应力应变关系。在试验的基础上,利用优化算法拟合出本构关系式中的五个参量。导出了以五参量表示阻尼和刚度特性的单元运动微分方程。为便于计算,将包含卷积运算的微分方程转换成一个四阶微分方程,进而装配出含有阻尼合金构件的弹性连杆机构的系统动力学方程。利用状态空间法对导出的高阶时变微分方程组进行了数值求解。计算实例结果表明所提模型是正确、有效的。     

A finite thin circular beam element for out-of-plane vibration analysis of curved beams

A finite thin circular beam element for the out-of-plane vibration analysis of curved beams is presented in this paper. Its stiffness matrix and mass matrix are derived, respectively, from the strain energy and the kinetic energy by using the natural shape functions derived from an integration of the differential equations in static equilibrium. The matrices are formulated with respect to the local polar coordinate system or to the global Cartesian coordinate system in consideration of the effects of shear deformation and rotary inertias. Some numerical examples are analyzed to confirm the validity of the element. It is shown that this kind of finite element can describe quite efficiently and accurately the out-of-plane motion of thin curved beams. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Chang-Boo Kim received his B.S. degree in Mechanical Engineering from Seoul University, Korea in 1973. He then received his D.E.A., Dr.-Ing. and Dr.-es-Science degrees from Nantes University, France in 1979, 1981 and 1984, respectively. Dr. Kim is currently a Professor at the School of Mechanical Engineering at Inha University in Incheon, Korea. His research interests are in the area of vibrations, structural dynamics, and MEMS.     

Parametric instability of a spinning pretwisted beam under periodic axial force

This paper addresses the parametric instability of a cantilever pretwisted beam rotating around its longitudinal axis under a time-dependent conservative end axial force which contains a steady-state part and a small periodically fluctuating component. This structural element can be used to model fluted cutting tools such as the twist drill bit and the end milling cutter, etc. Using the Euler—Bernoulli beam theory and Hamilton's principle, the present study derives the equation of motion which governs the lateral vibration of a spinning pretwisted beam. Rotary inertia, structural viscous damping and conservative end axial force are included. The Galerkin method is then applied to obtain the associated finite element equations of motion. Due to the existence of the Coriolis force, the resulting finite element equations of motion are transformed into a set of first-order simultaneous differential equations by a special modal analysis procedure. This set of simultaneous differential equations is solved by the method of multiple scales, yielding the system response and expressions for the boundaries of the unstable regions. Numerical results are presented to demonstrate the effects of pretwist angle, spinning speed and steady-state part of the end axial force on the parametric instability regions of the present problem.     

A stochastic finite element dynamic analysis of structures with uncertain parameters

The real life structural systems are characterized by the inherent uncertainty in the definition of their parameters in the context of both space and time. In the present study a stochastic finite element method has been proposed in the frequency domain for analysis of structural dynamic problems involving uncertain parameters. The harmonic forces as well as earthquake-induced ground motion are treated as random process defined by respective power spectral density function. The uncertain structural parameters are modelled as homogeneous Gaussian stochastic field and discretized by the local averaging method. The discretized stochastic field is simulated by the Cholesky decomposition of respective covariance matrix. By expanding the uncertain dynamic stiffness matrix about its reference value the Neumann expansion method is introduced in the finite element procedure within the framework of Monte Carlo simulation. This approach involves only single decomposition of the dynamic stiffness matrix for entire simulated structure. Thus a considerable saving of computing time and the facility that several stochastic fields can be simultaneously handled are the basic advantages of the proposed formulation. Numerical examples are presented to elucidate the accuracy and efficiency of the proposed method with the direct Monte Carlo simulation.     

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.     

Passive vibration damping enhancement of piezoelectric shunt damping system using optimization approach

Piezoelectric materials can be used for structural damping because of their ability to efficiently transform mechanical energy to electrical energy and vice versa. The electrical energy may be dissipated through a connected load resistance. In this paper, a new optimization technique for the optimal piezoelectric shunt damping system is investigated in order to search for the optimal shunt electrical components of the shunt damping circuit connected to the piezoelectric patch on a vibrating structure for the structural vibration suppression of several modes. The vibration suppression optimization technique is based on the idea of using the piezoelectric shunt damping system, the integrated p-version finite element method (p-version FEM), and the particle swarm optimization algorithm (PSOA). The optimal shunt electrical components for the piezoelectric shunt damping system are then determined by wholly minimizing the objective function, which is defined as the sum of the average vibration velocity over a frequency range of interest. Moreover, the optimization technique is performed by also taking into account the inherent mechanical damping of the controlled structure with the piezoelectric patch. To numerically evaluate the multiple-mode damping capability by the optimal shunting damper, an integrated p-version FEM for the beam with the shunt damping system is modeled and developed by MATLAB. Finally, the structural damping performance of the optimal shunt damping system is demonstrated numerically and experimentally with respect to the beam. The simulated result shows a good agreement with that of the experimental result. This paper was recommended for publication in revised form by Associate Editor Eung-Soo Shin Jin-Young Jeon received his Ph.D. degree in Mechanical and Aerospace Engineering from Tokyo Institute of Technology in 2005. Dr. Jeon is currently a senior engineer at Digital Printing Division, Digital Media & Communications Business at Samsung Electronics Co., Ltd., Korea. His research interests are the areas of structural-acoustic optimization, sound quality, motion quality, and vibration control.