A new iterative model updating method using incomplete frequency response function data

A new iterative model updating method is proposed for reduced model using incomplete frequency response function (FRF) data. It uses a modified difference vector between the analytical and experimental FRF data to construct a linear sensitivity updating equation system. To improve the convergence performance of the proposed algorithm, a concept of pseudo master degree-of-freedom (DOF) is put forward and the finite element (FE) model is reduced to the measured and user selected pseudo DOFs. The FRFs at pseudo master DOFs are estimated using the impedance matrix of iteratively modified analytical model and the measured FRFs at master DOFs. They are only used to improve the sensitivity matrix and difference calculation between the analytical and experimental FRF data without introducing additional difference equation. At the end, a 25 truss structure is used to evaluate the performance of the proposed method.     

Assessment of uncertainty on structural dynamic responses with the short transformation method

The predictive capability of finite element models is limited by their deterministic nature: typically, not all model parameters are exactly known, while even small deviations may have significant effects on the predicted response. Parameter uncertainty should therefore be taken into account, e.g. with fuzzy arithmetic. The absence of fuzzy solvers led to interval arithmetic as a numerical alternative. The Transformation Method (TM), presented by M. Hanss, replaces interval arithmetic with a set of deterministic computations: for each interval, all parameter extrema are combined in every possible way. In a Design of Experiments terminology, the TM is a so-called Full Factorial design.The TM is applicable if the output is monotonic in the inputs. Unlike interval arithmetic, it does not overestimate the response uncertainty, as only parameter combinations are evaluated that actually occur. In this paper, the TM has been applied to visualise uncertain frequency response functions (FRFs), obtained with modal superposition. This yields accurate results when validated against Monte Carlo data, but the computation time is rather high. The Short Transformation Method (STM) is proposed as an attractive alternative to the original TM. A full set of deterministic computations, combining all interval extrema, is only performed at the lowest interval. For higher levels, a smaller set is evaluated. This allows reconstructing the fuzzy FRF from a much lower number of deterministic computations, with only a small reduction in the accuracy of FRFs. Both methods are demonstrated on a clamped plate and a car front cradle with uncertain design parameters.     

A new experimental modal analysis method based on double-exponential windowing and application to lightly damped bladed wheels

In this paper, a new method is presented for experimental modal analysis for lightly damped structures such as bladed wheels. Like most of existing methods, the one proposed here is also based on frequency response functions (FRFs). An FRF can be calculated to be the ratio of the Fourier transforms of a response and a force. For a pair of force–response time-histories generated by a force hammer impacting on a lightly damped structure, an exponential window is required to apply on both the force- and response-time histories when performing the Fourier transforms. This results in an approximate of the true FRF. Using a different decaying rate in the exponential window, a second approximated FRF is produced. These two approximated FRFs, instead of one as in existing experimental modal analysis methods, are then combined to abstract modal information and recover the true FRF.     

Modelling the spindle–holder taper joint in machine tools: A tapered zero-thickness finite element method

This study presents a tapered zero-thickness finite element model together with its parameter identification method for modelling the spindle–holder taper joint in machine tools. In the presented model, the spindle and the holder are modelled as solid elements and the taper joint is modelled as a tapered zero-thickness finite element with stiffness and damping but without mass or thickness. The proposed model considers not only the coupling of adjacent degrees of freedom but also the radial, tangential and axial effects of the spindle–holder taper joint. Based on the inverse relationship between the dynamic matrix and frequency response function matrix of a multi-degree-of-freedom system, this study proposes a combined analytical–experimental method to identify the stiffness matrix and damping coefficient of the proposed tapered zero-thickness finite element. The method extracts those parameters from FRFs of an entire specimen that contains only the spindle–holder taper joint. The simulated FRF obtained from the proposed model matches the experimental FRF quite well, which indicates that the presented method provides high accuracy and is easy to implement in modelling the spindle–holder taper joint.     

Analysis of Single-Walled Carbon Nanotubes Using a Chemical Bond Element Model

提出了一种纳米尺度的有限元方法,碳纳米管中的碳-碳化学键被模拟为键单元．按照平衡关系,根据有限元理论,作用于每个碳原子上的作用力可以写成键单元的刚度矩阵与每个碳原子位移的乘积．在分子力学的基本假设下,键单元刚度矩阵的每个元素可以写为分子力学中力场常数的函数,这样建立起了宏观力学方法(有限元)与纳米尺度力学方法(分子力学)之间的联系．应用该方法模拟了扶椅型与锯齿型单壁碳纳米管的力学行为从而验证了该方法的有效性．分析结果说明单壁碳纳米管的弹性模量与管厚度的选取直接相关．此外,弹性模量对所选取的分子力学中的力场常数非常敏感,管的弹性模量显示出对半径的尺度依赖性,但是管长度对弹性模量的影响小到可以被忽略．     

Measurement of the Elasticity of Biological Soft Tissue of Finite Thickness

Elasticity is of profound significance to evaluating the function of a biological soft tissue.When the elasticity of a tissue is macroscopically changed,it means that the biological function of the tissue is abnormal and some disease or injury may occur.In the present work,an elastometer is developed to measure the elasticity of biological soft tissues.The measurement is based on the indentation method and the force is measured by the bending of the cantilever.The force-indentation data of the soft tissue is experimentally measured by this elastometer and Young's modulus of the tissue is calculated using the Hertz-Sneddon model.For comparison,a numerical model for the indentation method is established using the finite element method.The difference between the actual modulus and the measured modulus is discussed.The effect of the thickness of the specimen on the measurement is investigated.Young's moduli of beef,porcine liver and porcine kidney are experimentally measured.The results indicate that our elastometer is effective in measuring Young's modulus of a soft tissue quantitatively.     

FINITE ELEMENT MODEL UPDATING USING ANTIRESONANT FREQUENCIES

This paper uses antiresonant frequencies in the finite element model updating of an experimental 6-m aluminum truss and analyzes the physical correctness of the updated model by using it to detect damage. Rigid elements are used to simplify the modelling of welded joints, and their dimensions are used as parameters in an iterative update based on eigenvalue and antiresonance sensitivities. An update using both natural frequencies and antiresonant frequencies is shown to produce a 48% better correlation to experimental frequency response functions (FRFs) than an update that uses only natural frequencies. The antiresonant updated model is used to predict FRFs for the truss in 112 damaged configurations. Pattern classification and curve-fit algorithms for damage detection are tested. The curve-fit method correctly identified damage 92·6% of the time compared to 76·1% for the pattern classifier. The high quality of the model is attributed to the use of rigid elements that are updated using antiresonant frequencies.     

Non-linear parameter estimation with Volterra series using the method of recursive iteration through harmonic probing

Volterra series provides a platform for non-linear response representation and definition of higher order frequency response functions (FRFs). It has been extensively used in non-parametric system identification through measurement of first and higher order FRFs. A parametric system identification approach has been adopted in the present study. The series response structure is explored for parameter estimation of polynomial form non-linearity. First and higher order frequency response functions are extracted from the measured response harmonic amplitudes through recursive iteration. Relationships between higher order FRFs and first order FRF are then employed to estimate the non-linear parameters. Excitation levels are selected for minimum series approximation error and the number of terms in the series is controlled according to convergence requirement. The problem of low signal strength of higher harmonics is investigated and a measurability criterion is proposed for selection of excitation level and range of excitation frequency. The procedure is illustrated through numerical simulation for a Duffing oscillator. Robustness of the estimation procedure in the presence of measurement noise is also investigated.     

驻波法测杨氏模量及误差分析

驻波法运用驻波原理,采用人为控制金属丝形变,测出金属丝驻波基频,得出其张力,从而求出金属丝杨氏模量.相对传统方法,驻波法避免了金属丝直径和形变的测量,其不确定度基本上仅由千分尺精度决定,故而具有操作简单,测量准确度高的特点.     

基于多项式混沌方法的场线耦合响应不确定度量化

在传输线场线耦合的计算中,由于辐射场可能从不同方向入射,入射方位角和入射仰角会在一定范围内变化,可以将其看作不确定变量,因此传输线的响应也呈现出不确定性。针对输入参数服从非典型分布的情况,应用多项式混沌方法对传输线场线耦合频域响应进行不确定度量化。结合入射方位角和仰角的物理意义,给出其服从的分布类型并构建相应的正交多项式基底,并对该模型的传输线方程进行多项式混沌展开。最后结合含两个不确定参数的传输线场线耦合算例,给出远端电流响应的统计信息,对比蒙特卡罗方法,验证了该方法的正确性和高效性。     

Uncertainty quantification of an inflatable/rigidizable torus

There is an increasing interest in lightweight inflatable structures for space missions. The dynamic testing and model updating of these types of structures present many challenges in terms of model uncertainty and structural nonlinearity. This paper presents an experimental study of uncertainty quantification of a 3m-diameter inflatable torus. Model uncertainty can be thought of as coming from two different sources, uncertainty due to changes in controlled conditions, such as temperature and input force level, and uncertainty associated with others random factors, such as measurement noise, etc. To precisely investigate and quantify model uncertainty from different sources, experiments, using sine-sweep excitation in the specified narrow frequency bands, are conducted to collect frequency response function (FRF) under various test conditions. To model the variation of the identified parameters, a singular value decomposition technique is applied to extract the principal components of the parameter change.     

Influence of static compression on mechanical parameters of acoustic foams

The modification of elastic properties of compressed acoustic foams is investigated. The porous sample is first submitted to a static compression and then to a dynamic excitation of smaller amplitude, corresponding to acoustical applications. The static compression induces the modification of the dynamic elastic parameters of the material. This work focuses on Young's modulus. The variation is measured with two different experimental methods: The classical rigidimeter and an absorption measurement. The effective Young's modulus is directly measured with the first method and is indirectly determined through the quarter-wave length resonance of the frame with the second one. The results of the two measurements are compared and give similar tendencies. The variation of the dynamic Young's modulus as a function of the degree of compression of the sample is shown to be separated in several zones. In the zones associated with weak compression (those usually zones encountered in practice), the variation of the effective Young's modulus can be approximated by a simple affine function. The results are compared for different foams. A simple model of the dependency of the Young's modulus with respect to the static degree of compression is finally proposed for weak compressions.     

Stress and magnetic field-dependent Young's modulus in single crystal iron-gallium alloys

The variability in Young's modulus of single crystal iron-gallium (Galfenol) alloys having 16, 17.5, 19, 24.7 and 29 at% gallium is investigated using experiments and simulations. Some of these alloys showed more than 60% change in Young's modulus along the 〈1 0 0〉 directions on varying their magnetization and stress states compared to their modulus at magnetic saturation. A function, ΔE(σ,H), is defined such that the variability of modulus is bound between 0% and 100%. The observations are related to the inherent magnetomechanical coupling in the material. An energy-based non-linear constitutive model is used to predict the variable modulus in Galfenol as a continuous function of stress and magnetic field. Model predictions showed good correlation with experimental results.     

杨氏模量实验中不确定度的评定方法

讨论了在杨氏模量实验的线性回归中,自变量有误差时的不确定度评定方法。     

OPTIMIZATION OF SIMPLIFIED MODELS MESHED WITH FINITE TRIANGULAR PLATE ELEMENTS

Designers often want to analyze more and more sophisticated structures, thus leading to very large finite element models (typically 10 00 000 degrees of freedom for a body car, for example). These models being too costly for the early stages of design and optimization can be reduced by a substructure analysis or a mesh simplification of the components. A methodology is proposed in this paper for simplifying finite triangular plate element models leading to a dramatic reduction in the number of degrees of freedom while preserving the dynamical properties of the initial system. In particular, the proposed method is developed for models composed of the plate element STIFF63 generated by the software ANSYS. The principle consists in determining the parameters (thickness, Young's modulus, density) of the triangular elements of a coarse model which replaces a large set of elements of the refined model. The simplified mesh must satisfy one of two criteria. The first requires that the mass and stiffness matrices of the simplified model be as close as possible to the Guyan condensed matrices of the refined model on the reduced node set, whilst the second requires that the dynamical properties of the global structure be preserved. The application of these approaches is illustrated on two test structures using the gradient method to solve the resulting optimization problem. The second approach is shown to give the best results. Typically, the size of the models can be reduced by a factor of 20 whilst preserving the dynamical properties of the structure at low frequencies.     

Debonding detection of honeycomb sandwich structures using frequency response functions

A vibration-based non-destructive evaluation (NDE) method is proposed to determine the location and size of debonding in honeycomb sandwich beams. Although most of the existing vibration-based NDE methods need many measurement points, the method proposed here only utilizes the frequency response function (FRF) measured at one point. A parameterized damaged Timoshenko beam model is developed with the method of reverberation-ray matrix (MRRM) for the first time, and combined with the genetic algorithm (GA) to inverse the damage parameters from the measured FRF. The detection of a honeycomb sandwich beam can be divided into two steps: (1) identifying the equivalent elastic moduli and other parameters of the intact sandwich beam. (2) Identifying the debonding location and size of the damaged sandwich beam with the predetermined parameters. It is demonstrated experimentally that the method can inverse damage parameters with acceptable precision.     

Measurement of adhesion energies and Young's modulus in thin polymer films using a novel axi-symmetric peel test geometry

We present a novel method of probing adhesion energies of solids, particularly polymers. This method uses the axi-symmetric deformation of a thin spincast polymer membrane brought into contact with a flat substrate to probe the work of adhesion. The use of a thin membrane minimizes uncertainty in the radius of contact, while the use of spincast films provides very smooth surfaces by means of a very simple method. The experimental profile of the deformed membrane shows good agreement with the expected logarithmic profile. The experimental setup enables the measurement of Young's modulus and the solid-solid work of adhesion for thin films. The value obtained for Young's modulus of polystyrene (PS) was found to be in agreement with other conventional measurement techniques. In addition, measurement of the work of adhesion at the PS/silicon oxide interface was possible. The apparatus is well suited to studying the dependence of Young's modulus, work of adhesion and fracture energy on membrane thickness, temperature, pulling rate, and ageing of the interface, and can readily be modified to study biologically relevant samples.     

高温高压下Zr2AlC的结构及热学性质的第一性原理

利用密度泛函理论研究了高温高压下Zr₂AlC的结构和热力学性质,计算得到Zr₂AlC的晶格参数与实验值符合较好．研究了Zr₂AlC的弹性常数、体模量、剪切模量和杨氏模量等力学性质随压力变化的趋势．同时研究了维氏硬度随压力的变化趋势．通过计算得到的杨氏模量预测了Zr₂AlC的弹性各向异性．最后,基于准简谐德拜模型,成功预测了Zr₂AlC的德拜温度、热容、热膨胀系数和Grüneisen参数随着压强和温度的变化关系.     

A fuzzy finite element procedure for the calculation of uncertain frequency-response functions of damped structures: Part 1—Procedure

This work introduces a numerical algorithm to calculate frequency-response functions (FRFs) of damped finite element (FE) models with fuzzy uncertain parameters. Part one of this paper describes the numerical algorithm for the solution of the underlying interval finite element (IFE) problem. First, the IFE procedure for the calculation of undamped envelope FRFs is discussed. Starting from the undamped procedure, a strategy is developed to analyse damped structures based on the principle of Rayleigh damping. This is achieved by analysing the effect of the proportional damping coefficients on the subsequent steps of the undamped procedure. This finally results in a procedure for the calculation of fuzzy damped FRFs based on an analytical extension of the undamped algorithm. Part one of this paper introduces the numerical procedure. Part two of this paper illustrates the application of the methodology on four numerical case studies.     

Estimation of dynamic elastic constants from the amplitude and velocity of Rayleigh waves

In this paper a method is proposed to characterize the elasticity of isotropic linear materials from the generation and detection of an acoustic surface wave. For the calculation of the elastic constants, it is sufficient that only one of the faces of the sample be accessible. The methodology is based on both the measurement of the Rayleigh wave velocity and on the determination of the normal to longitudinal amplitude ratio calculated from the normal and longitudinal components of the displacement of a point. The detection of two consecutive surface wave pulses using a single experimental setup permits the determination of the elastic constants. The method is applied to calculate Young's modulus and Poisson's ratio of an aluminum sample as well as their systematic uncertainties. The results obtained give a relative uncertainty for Young's modulus on the order of the sixth part of that calculated for Poisson's ratio.