含表面裂纹简支梁的非线性振动分析

对含表面裂纹简支梁在大幅振动下的几何非线性进行了理论分析,从建立了梁的非线性振动的半解析解.用Rayleigh方法将振型函数表示为线性模型振型函数的组合,建立了梁非线性振动的第一阶振型函数的显式表达式,数值模拟计算了不同的裂纹深度和给定不同第一函数系数a1对梁最大位移的影响.建立的显式方程简单,易于工程应用.     

Asymptotic non-linear normal modes for large-amplitude vibrations of continuous structures

Non-linear normal modes (NNMs) are used in order to derive accurate reduced-order models for large amplitude vibrations of structural systems displaying geometrical non-linearities. This is achieved through real normal form theory, recovering the definition of a NNM as an invariant manifold in phase space, and allowing definition of new co-ordinates non-linearly related to the initial, modal ones. Two examples are studied: a linear beam resting on a non-linear elastic foundation, and a non-linear clamped–clamped beam. Throughout these examples, the main features of the NNM formulation will be illustrated: prediction of the correct trend of non-linearity for the amplitude-frequency relationship, as well as amplitude-dependent mode shapes. Comparisons between different models—using linear and non-linear modes, different number of degrees of freedom, increasing accuracy in the asymptotic developments—are also provided, in order to quantify the gain in using NNMs instead of linear modes.     

Application of genetic algorithm in crack detection of beam-like structures using a new cracked Euler–Bernoulli beam element

In this paper, a crack identification approach is presented for detecting crack depth and location in beam-like structures. For this purpose, a new beam element with a single transverse edge crack, in arbitrary position of beam element with any depth, is developed. The crack is not physically modeled within the element, but its effect on the local flexibility of the element is considered by the modification of the element stiffness as a function of crack's depth and position. The development is based on a simplified model, where each crack is substituted by a corresponding linear rotational spring, connecting two adjacent elastic parts. The localized spring may be represented based on linear fracture mechanics theory. The components of the stiffness matrix for the cracked element are derived using the conjugate beam concept and Betti's theorem, and finally represented in closed-form expressions. The proposed beam element is efficiently employed for solving forward problem (i.e., to gain accurate natural frequencies of beam-like structures knowing the cracks’ characteristics). To validate the proposed element, results obtained by new element are compared with two-dimensional (2D) finite element results as well as available experimental measurements. Moreover, by knowing the natural frequencies, an inverse problem is established in which the cracks location and depth are identified. In the inverse approach, an optimization problem based on the new beam element and genetic algorithms (GAs) is solved to search the solution. The proposed approach is verified through various examples on cracked beams with different damage scenarios. It is shown that the present algorithm is able to identify various crack configurations in a cracked beam.     

Harmonic analysis of the vibrations of a cantilevered beam with a closing crack

In this article, the vibrational response of a cracked cantilevered beam to harmonic forcing is analysed. The study has been performed using a finite element model of the beam, in which a so-called closing crack model, fully open or fully closed, is used to represent the damaged element. Undamaged parts of the beam are modelled by Euler-type finite elements with two nodes and 2 d.f. (transverse displacement and rotation) at each node. Recently the harmonic balance method has been employed by other researchers to solve the resulting non-linear equations of motion. Instead, in this study, the analysis has been extended to employ the first and higher order harmonics of the response to a harmonic forcing in order to characterize the non-linear behaviour of the cracked beam. Correlating the higher order harmonics of the response with the forcing term the so-called higher order frequency response function (FRFs), defined from the Volterra series representation of the dynamics of non-linear systems, can be determined by using the finite element model to simulate the time domain response of the cracked beam. Ultimately the aim will be to employ such a series of FRFs, an estimate of which in practice could be measured in a stepped sine test on the beam to indicate both the location and depth of the crack, thus forming the basis of an experimental structural damage identification procedure.     

Detection of cracks in simply-supported beams by continuous wavelet transform of reconstructed modal data

This paper proposes a new approach for damage detection in beam-like structures with small cracks, whose crack ratio [r = H_c/H] is less than 5%, without baseline modal parameters. The approach is based on the difference of the continuous wavelet transforms (CWTs) of two sets of mode shape data which correspond to the left half and the right half of the modal data of a cracked simply-supported beam. The mode shape data of a cracked beam are apparently smooth curves, but actually exhibit local peaks or discontinuities in the region of damage because they include additional response due to the cracks. The modal responses of the damaged simply-supported beams used are computed using the finite element method. The results demonstrate the efficiency of the proposed method for crack detection, and they provide a better crack indicator than the result of the CWT of the original mode shape data. The effects of crack location and sampling interval are examined. The simulated and experimental results show that the proposed method has great potential in crack detection of beam-like structures as it does not require the modal parameter of an uncracked beam as a baseline for crack detection. It can be recommended for real applications.     

带边角裂纹悬臂Mindlin 板的振动特性研究

本文基于Mindlin板理论,应用Ritz法研究带边角裂纹Mindlin板的振动特性,分析了不同裂纹参数如裂纹位置,裂纹长度,裂纹角度对悬臂Mindlin板的固有频率和模态的影响.利用Ritz法求解固有频率和模态函数,本文构造了一个特殊的模态函数,其模态函数由两部分构成,一部分是用梁函数组合法得到的无裂纹理想完整矩形板的振型,另一部分是利用裂纹尖端奇异性理论,构造描述裂纹附近位移和转角不连续的角函数.通过高精度的数值计算软件Maple得出结果,并与有限元软件ANSYS分析的结果进行对比,验证本文计算结果的准确性.     

基于移动有限元法的裂纹梁振动分析

采用移动有限元法和局部柔度法对移动质量作用下含裂纹简支梁进行了振动计算分析.计算考虑了裂纹和移动质量的相对位置对梁固有频率的影响,以及移动质量在不同位置、速度情况下对裂纹梁的动力响应的影响.结果分析表明,裂纹与移动质量的存在会使得梁的动态位移有不同程度的增大,且随着移动质量位置和裂纹位置的改变会使得梁的固有频率变小.     

Vibrations of moderately thick rectangular plates in terms of a set of static Timoshenko beam functions

In this paper, a set of static Timoshenko beam functions is developed as the admissible functions to study the free vibrations of moderately thick rectangular plates using the Rayleigh–Ritz method. This set of beam functions is made up of the static solutions of a Timoshenko beam under a series of sinusoidal distributed loads. The beam is considered to be a unit width strip taken from the rectangular plate in a direction parallel to the edges of the plate. In addition, the geometric boundary conditions of the plate are exactly satisfied in this set of beam functions, and the effect of the shear correction factor on the admissible functions of the plate is also taken into account. It can be seen that the method is sound in theory and no complicated mathematical knowledge is needed. Each of the beam functions is only a third-order polynomial plus a sine function or a cosine function. Furthermore, a change of the boundary conditions of the plate only results in a change of the coefficients of the polynomial. The method is very simple and a unified computational program can be given for the plates with arbitrary boundary conditions and thickness. Comparison and convergency studies demonstrate the correctness and the accuracy of the method. It can be shown that using a small number of terms of the static Timoshenko beam functions can give rather accurate results for all cases. Finally, the effect of thickness–span ratio on the eigenfrequency parameters of Mindlin rectangular plates is studied in detail.     

Postural stability effects of random vibration at the feet of construction workers in simulated elevation

The risk of falls from height on a construction site increases under conditions which degrade workers’ postural control. At elevation, workers depend heavily on sensory information from their feet to maintain balance. The study tested two hypotheses: “sensory enhancement” – sub-sensory (undetectable) random mechanical vibrations at the plantar surface of the feet can improve worker’s balance at elevation; and “sensory suppression” – supra-sensory (detectable) random mechanical vibrations can have a degrading effect on balance in the same experimental settings.Six young (age 20–35) and six aging (age 45–60) construction workers were tested while standing in standard and semi-tandem postures on instrumented gel insoles. The insoles applied sub- or supra-sensory levels of random mechanical vibrations to the feet. The tests were conducted in a surround-screen virtual reality system, which simulated a narrow plank at elevation on a construction site. Upper body kinematics was assessed with a motion-measurement system. Postural stability effects were evaluated by conventional and statistical mechanics sway measures, as well as trunk angular displacement parameters.Analysis of variance did not confirm the “sensory enhancement” hypothesis, but provided evidence for the “sensory suppression” hypothesis. The supra-sensory vibration had a destabilizing effect, which was considerably stronger in the semi-tandem posture and affected most of the sway variables.Sensory suppression associated with elevated vibration levels on a construction site may increase the danger of losing balance. Construction workers at elevation, e.g., on a beam or narrow plank might be at increased risk of fall if they can detect vibrations under their feet. To reduce the possibility of losing balance, mechanical vibration to supporting structures used as walking/working surfaces should be minimized when performing construction tasks at elevation.     

多频简谐激励下裂纹梁的非线性振动响应

主要对含裂纹梁在振动与超声波联合激励下所出现的非线性动力响应的机理和特性进行研究.将疲劳裂纹在外加激励下的状态简化为周期性张开-闭合的非线性过程,基于圣维南原理,采用有限元方法建立了含非对称疲劳裂纹梁的非线性数值分析模型.利用非线性输出频率响应函数(NOFRFs)概念,对裂纹梁在高-低频简谐激励下所出现的非线性动力响应特性的机理进行了解释.具体以悬臂梁为例,仿真分析了裂纹深度和裂纹位置等参数的变化对系统非线性动力响应特性的影响规律.     

Time-dependent modelling of RC structures using the cracked membrane model and solidification theory

A non-linear finite element model is presented for the time-dependent analysis of reinforced concrete structures under service loads. For the analysis of members in plane stress, the model is based on the cracked membrane model using a rotating crack approach combined with solidification theory for modelling creep. The numerical results are compared with a variety of long-term laboratory measurements, including development of deflections and cracking with time in a reinforced concrete beam, time-dependent change in support reactions of a continuous beam subject to support settlement and creep buckling of columns. The numerical results are in good agreement with the test data.     

Buckling, flutter and vibration analyses of beams by integral equation formulations

This paper presents a model for the investigation of buckling, flutter and vibration analyses of beams using the integral equation formulation. A mathematical formulation based on Euler–Bernoulli beam theory is presented for beams with variable sections on elastic foundations and subjected to lateral excitation, conservative and non-conservative loads. Using the boundary element method and radial basis functions, the equation of motion is reduced to an algebraic system related to internal and boundary unknowns. Eigenvalue problems related to buckling and vibrations are formulated and numerically solved. Buckling loads, natural frequencies and associated eigenmodes are computed. The corresponding slope, bending and shear forces can be directly obtained. The load-frequency dependence is investigated for various elastic foundations and the divergence critical loads are evidenced. Under non-conservative loads, a dynamic stability analysis is illustrated numerically based on the coalescence of eigenfrequencies. The flutter load and instability regions with respect to the elastic concentrated and distributed foundations are identified. Using the eigenmodes, numerically computed, non-linear vibrations of beams are investigated based on one mode analysis. The presented model is quite general and the obtained numerical results are in agreement with available data.     

Solving a non-linear model: The importance of model specification for deriving a suitable solution

In this paper, we consider a macroeconomic model with alternative linear and non-linear specifications. One version of the model, expressed in levels, is highly non-linear and has at least two steady-state equilibria. One of these equilibria has an economically meaningful interpretation, while the other does not have a sensible economic interpretation. A second version of the model, expressed in logarithms, is linear and has a unique steady-state equilibrium, which corresponds to the economically meaningful equilibrium of the non-linear version of the model. The dynamic solution of each model version has a combination of stable and unstable eigenvalues so that any dynamic solution requires the calculation of appropriate “jumps” in endogenous variables. Attempts to solve these models, using forward-shooting and reverse-shooting algorithms, show that the forward-shooting algorithm chooses the “wrong” solution for the non-linear model, but the “right” solution for the linear model. The reverse-shooting algorithm chooses the “right” solution in both cases. We demonstrate how this result is driven by particular properties of the two versions of the model.     

Crack detection in beam-like structures using genetic algorithms

A fault diagnosis method based on genetic algorithms (GAs) and a model of damaged (cracked) structure is proposed. For modeling the cracked-beam structure an analytical model of a cracked cantilever beam is utilized and natural frequencies are obtained through numerical methods. Our method utilizes genetic algorithms to monitor the possible changes in the natural frequencies of the structure. The identification of the crack location and depth in the cantilever beam is formulated as an optimization problem, and binary and continuous genetic algorithms (BGA, CGA) are used to find the optimal location and depth by minimizing the cost function which is based on the difference of measured and calculated natural frequencies. Also we present a new cost function based on natural frequencies. The average values of location and depth prediction errors are 1.02% and 1.98%, respectively, using the BGA. These values become 0.73% and 1.11% for the CGA. To validate the proposed method and investigate the modeling and measurement errors some experimental results are also included. The average values of experimental location and depth prediction errors are 10.57% and 11.19%, respectively, for the BGA. These values become 10.21% and 10.39% for the CGA.     

A Review and Study on Ritz Method Admissible Functions with Emphasis on Buckling and Free Vibration of Isotropic and Anisotropic Beams and Plates

The first goal of this work is to present a literature review regarding the use of several sets of admissible functions in the Ritz method. The papers reviewed deal mainly with the analysis of buckling and free vibration of isotropic and anisotropic beams and plates. Theoretically, in order to obtain a correct solution, the set of admissible functions must not violate the essential or geometric boundary conditions and should also be linearly independent and complete. However, in practice, some of the sets of functions proposed in the literature present a bad numerical behavior, namely in terms of convergence, computational time and stability. Thus, a second goal of the present work is to compare the performance of several sets of functions in terms of these three features. To achieve this objective, the free vibration analysis of a fully clamped rectangular plate is carried out using six different sets of functions, along with the study of the convergence of natural frequencies and mode shapes, the computational time and the numerical stability.     

Vibration analysis of edge-cracked beam on elastic foundation with axial loading using the differential quadrature method

The eigenvalue problems of clamped-free and hinged-hinged Bernoulli-Euler beams on elastic foundation with a single edge crack, axial loading and excitation force were numerically formulated using the differential quadrature method (DQM). Appropriate boundary conditions accompanied the DQM to transform the partial differential equation of a Bernoulli-Euler beam with a single edge crack into a discrete eigenvalue problem. The DQM results for the natural frequencies of cracked beams agree well with other literature values. The sampling point number effect, the location of the crack effect and the depth of the crack effect on the accuracy variation of calculated natural frequencies are presented by using two elements in this work. The effects of axial loading, foundation stiffness, opening crack and closing crack are also studied.     

基于PVDF传感器和工作曲率变形实现裂纹梁裂纹检测

提出通过压电高分子薄膜PVDF(Polyvinylidene Fluoride)作为传感器直接测量结构的工作曲率变形(Curvature Operating Shape)实现结构裂纹检测.以悬臂裂纹梁为例,首先探讨基于PVDF传感器测量工作曲率变形,并用于结构损伤检测的可行性,然后介绍工作曲率变形用于损伤检测的理论基础,并以PVDF压电薄膜作为传感器,提出只通过损伤后结构的频率响应函数,直接得到工作曲率变形和损伤指标.最后通过对一条裂纹和两条裂纹悬臂梁的损伤检测实验,证明该方法在梁结构损伤识别和健康监测中的有效性,并且识别过程不需要健康结构的振动信息.     

Seismic stability of cracked concrete dams using rigid block models

Several gravity dams subjected to severe ground motions are likely to experience cracking and sliding in the upper section where dynamic amplification is important. A high acceleration spike realistically applies large inertia forces computed from mass times the acceleration. However, these impulsive inertia forces might not be applied in the same direction for a sufficient long period of time to induce significant rotational or sliding displacements detrimental to the seismic or post-seismic structural stability of the cracked components. When it is of interest to estimate residual sliding displacements, a convenient and simple tool is to perform transient rigid body “sliding block” analysis of the “cracked” component. However, this requires the definition of proper seismic input motions at the base of the block with due consideration of dynamic amplification. The possibility to compute in-structure response spectra (ISRS) at the base of the block to define suitable spectra compatible accelerograms is presented in this paper. An important conclusion is that it is not conservative to use accelerograms compatible with the linear (uncracked) dam ISRS to perform transient rigid body sliding response analyses. Dam base and upper joint cracking affects its dynamic properties such that there are modifications of the intensities and frequency content of seismic motions as they propagate over the dam’s height. An envelope of nonlinear ISRS computed from cracked beam models of the dam is recommended to obtain compatible accelerograms and to provide a conservative estimate of upper block residual sliding displacements.     

Solving macroeconomic models with “off-the-shelf” software: An example of potential pitfalls

When working with large-scale models or numerous small models, there can be a temptation to rely on default settings in proprietary software to derive solutions to the model. In this paper we show that, for the solution of non-linear dynamic models, this approach can be inappropriate. Alternative linear and non-linear specifications of a particular model are examined. One version of the model, expressed in levels, is highly non-linear. A second version of the model, expressed in logarithms, is linear. The dynamic solution of each model version has a combination of stable and unstable eigenvalues so that any dynamic solution requires the calculation of appropriate “jumps” in endogenous variables. We can derive a closed-form solution of the model, which we use as our “true” benchmark, for comparison with computational solutions of both linear and non-linear models. Our approach is to compare the “goodness of fit” of reverse-shooting solutions for both the linear and non-linear model, by comparing the computational solutions with the benchmark solution. Under the basic solution method with default settings, we show that there is significant difference between the computational solution for the non-linear model and the benchmark closed-form solution. We show that this result can be substantially improved using modifications to the solver and to parameter settings.     

Geometrically non-linear free vibrations of clamped simply supported rectangular plates. Part I: the effects of large vibration amplitudes on the fundamental mode shape

The geometrically non-linear free vibrations of thin isotropic and laminated rectangular composite plates with fully clamped edges have been successfully investigated in previous series of works using a theoretical model based on Hamilton’s principle and spectral analysis. The objective of this work is the extension of the above model to the case of clamped clamped simply supported simply supported rectangular plates, denoted by CCSSSSRP, in order to determine their fundamental non-linear mode shape, and associated amplitude-dependent resonant frequencies, and flexural stress distribution. Numerical data are given for both linear and non-linear analysis, for various plate aspect ratios and vibration amplitudes. Good agreement was found with previous published results.