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.     

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.     

基于多体系统动力学和有限元法的联合仿真在车桥耦合振动研究中的应用*

为了实现车桥耦合振动精细化仿真研究,利用多体系统动力学软件SIMPACK建立完整的车辆空间模型,采用空间杆系和板壳混合单元有限元方法建立桥梁的动力分析模型;然后将车辆和桥梁两个子系统在轮轨接触面离散的信息点上进行数据交换,实现车桥耦合振动联合仿真分析。以高速铁路上的简支梁桥为研究对象,采用基于多体系统动力学和有限元法结合的联合仿真技术,计算了弹性轮轨接触时动车组列车以不同车速通过桥梁的空间耦合振动响应,证明了该研究方法的可行性。     

移动质量-梁振动的时空有限元法数值分析

为了获得移动质量沿梁匀速运动的系统动态响应,建立了时空有限元数值求解模型.考虑移动质量惯性项,得到移动质量-梁时变系统的动力学方程.应用时空有限元法.得到了移动集中质量作用下Bernoulli-Euler梁离散单元的质量矩阵、刚度矩阵.与Newmark-β法、Wilson-θ法计算结果进行比较,时空有限无法计算梁的动态响应的精度更高.     

Identification of coexistent load and damage

Load reconstruction and damage identification are crucial problems in structural health monitoring. However, it seems there is not much investigation on identification of coexistent load and damage, although in practice they usually exist together. This paper presents a methodology to solve this problem based on the Virtual Distortion Method. A damaged structure is modeled by an equivalent intact structure subjected to the same loads and to virtual distortions which model the damages. The measured structural response is used to identify the loads, the distortions and to recover the stress-strain relationship of the damaged elements. This way both the damage type and extent are identified. The approach can be used off-line and online by repetitive applications in a moving time window. A numerical experiment of a truss with 5% measurement error validates that the two tested damage types (constant stiffness reduction and breathing crack) can be identified along with the loads.     

Wave propagation in delaminated beam

The wave propagation model investigated herein is based on the known fact that material discontinuities affect the propagation of elastic waves in solids. The change in certain material characteristics, such as a local change in stiffness or inertia caused by a crack or the presence of material damage, will affect the propagation of transmitted elastic waves and will modify the received signal. Wave frequencies associated with the highest detection sensitivity depend, among other things, on the type of structure, the type of material, and the type of damage. This paper presents a method of wave propagation, which can be further used to detect small delaminations in beam-like structures. The considered beam is modelled by spectral finite elements.     

基于ANSYS的车桥耦合振动分析

为准确分析车桥之间的相互作用,根据模态综合法推导车桥系统动力平衡方程,基于ANSYS前后处理器的桥梁建模和结果后处理功能,结合自主程序VBDIP(Vehicle Bridge Dynamic Interaction Program),形成1个通用工具用于分析车桥耦合振动问题.以单自由度质-弹系统通过简支梁桥模型为例,计算车桥的动力响应.所得结果与相关文献结果吻合良好,表明该方法正确有效,可用于分析各种车桥耦合振动问题.     

单侧主缆刚度损伤悬索桥的模态分析及1∶1内共振

悬索桥缆索在长期服役状态下损伤很难避免,单侧主缆刚度损伤会使系统出现主梁竖弯与扭转自由度间的耦合,这可能会进一步对悬索桥的模态和大振幅下的非线性响应造成影响.为此,建立了考虑单侧主缆刚度受损的七自由度悬索桥横截面模型,模态分析发现单侧主缆刚度损伤会使得系统前两阶固有频率曲线由交叉变为跃迁,导致两个本身相互独立的模态发生耦合.以此为基础,考虑悬索桥主缆刚度有损伤和未损伤两种工况,运用拓展的增量谐波平衡法(EIHB)计算系统在内共振条件下的非线性振动.研究结果表明：单侧主缆刚度有损伤的悬索桥在外部简谐激励下发生1:1内共振,系统能量表现出明显的转移现象;竖向和扭转简谐激励下,有损伤的悬索桥较未损伤工况响应幅值有所减小,但出现了两个共振响应峰值,对激励频率更为敏感.数值结果与利用Runge-Kutta法计算得到的结果吻合一致,验证了EIHB法的准确性.     

Dynamic response of a rectangular beam with a known non-propagating crack of certain or uncertain depth

In this paper the deterministic behaviour of a beam with a transverse on edge non-propagating crack is first studied. Moreover the stochastic setting pertaining the case in which the crack has an uncertain depth is investigated. The beam is discretized by finite elements in which a so-called closing crack model, with fully open or fully closed crack, is used to describe the damaged element. Once the mathematical model of the beam is defined, the dynamic response is evaluated by applying a numerical procedure based on the philosophy of structural systems with dynamic modification. In the stochastic case the improved perturbation method is modified in order to solve efficiently the stochastic non-linear differential equations.     

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.     

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.     

Finite element modal analysis of an HDD considering the flexibility of spinning disk–spindle, head–suspension–actuator and supporting structure

This paper presents a finite element method to analyze the free vibration of a flexible HDD (hard disk drive) composed of the spinning disk–spindle system with fluid dynamic bearings (FDBs), the head–suspension–actuator with pivot bearings, and the base plate with complicated geometry. Finite element equations of each component of an HDD are consistently derived with the satisfaction of the geometric compatibility in the internal boundary between each component. The spinning disk, hub and FDBs are modeled by annular sector elements, beam elements and stiffness and damping elements, respectively. It develops a 2-D quadrilateral 4-node shell element with rotational degrees of freedom to model the thin suspension efficiently as well as to satisfy the geometric compatibility between the 3-D tetrahedral element and the 2-D shell element. Base plate, arm, E-block and fantail are modeled by tetrahedral elements. Pivot bearing of an actuator and air bearing between spinning disk and head are modeled by stiffness elements. The restarted Arnoldi iteration method is applied to solve the large asymmetric eigenvalue problem to determine the natural frequencies and mode shapes of the finite element model. Experimental modal testing shows that the proposed method well predicts the vibration characteristics of an HDD. This research also shows that even the vibration motion of the spinning disk corresponding to half-speed whirl and the pure disk mode are transferred to a head–suspension–actuator and base plate through the air bearing and the pivot bearing consecutively. The proposed method can be effectively extended to investigate the forced vibration of an HDD and to design a robust HDD against shock.     

Dynamic response of double-FG porous beam system subjected to moving load

In this paper, the vibration response of the double-FG porous beam system (DFGPBS) acted by a moving load is investigated. The DFGPBS composed of two parallel FG porous beams with their material properties varying along both the axial and transverse directions, i.e., bi-directional FG material distribution, is taken into account. The porous imperfection is simulated by distributing the porosity along the beam thickness with even and uneven patterns. The governing equations of this bi-directional DFGPBS under a moving load are established with the aid of the Hamilton principle associated with the Timoshenko beam theory. The Ritz method is adopted to discrete the differential governing equations, which are solved by the Newmark-β approach. The validation of the present model is performed by comparing the numerical results with two previous works. Then, the parametric study is carried out to investigate the influences of bi-directional gradient indices, porosity volume fraction, boundary conditions, stiffness of elastic layer, and velocity of the moving load on the vibration response of bi-directional DFGPBSs excited by a moving load. It is demonstrated that the vibration response of the double-beam system subjected to moving loads can be governed by tailoring the distribution of the bi-directional FG materials. The present work can be used to guide the multi-functional design of a double-beam system under dynamic loadings.

     

Structural damage identification using system dynamic properties

A damage detection method is presented for the identification and quantification of damage that leads to a change in the structure’s mass and/or stiffness properties. The proposed method requires the use of finite element to model the structure in its undamaged state as well as information on the dynamic properties such as frequencies and mode shapes of the structure in its damaged state. The technique is applicable to any structure that can be accurately modeled using the finite element method and whose frequencies and mode shapes can be reliably measured. A structure pseudo force vector derived from the residual force method is described to locate the damaged regions in the structure. A matrix condensation approach in conjunction with a proportional damage model is then employed to quantify the damage by calculating the change in stiffness and mass properties of the damaged elements in the structure. The validity of the method is demonstrated by applying it to three structures: a beam, a frame and a plate. It is shown that if the amount of damage is not excessively large, the proposed method can be used to detect damage in these structures even when the measured system dynamic properties are slightly erroneous.     

Bead pattern optimization

Plane sheet panels exhibit poor stiffness and NVH (noise, vibration, and harshness) performance due to their flexibility. A common and cost-effective approach in the automotive industry to improve the stiffness and NVH peformance of sheet panels is the addition of beads. However, no systematic methodology is available for determining the optimal pattern of beads in sheet metal. This research explores the feasibility of applying topology optimization methods to the bead design of sheet panels. The approach starts with adding beam elements to the shell element model of the sheet panel to simulate the stiffness improvement of the structure and then uses the topology optimization method to obtain the optimal layout of the beam elements. A cantilever plate is used to perform a preliminary study for bead pattern design and a simplified vehicle structure is used to demonstrate the applicability of the proposed method.     

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.     

Finite element free vibration analysis of damped stiffened panels

Free vibration characleristics of a damped stiffened panel with applied viscoelastic damping on the flanges of the stiffeners are studied using finite element method. The complex nature of the rotational and transverse stiffnesses of the stringers is taken into consideration while deriving the stiffness and mass matrices of the damped stiffener element. The finite element method consists of representing the panel by rectangular plate elements of 12 d.o.f. and the stiffeners by beam elements of 8 d.o.f. which allow for bending, torsional and warping effects. Numerical results showing the effect of the geometric and material properties of the damping layer treatment on the resonant frequencies and loss factors of the composite panel are presented.     

Discrete element analysis of dynamic response of Timoshenko beams under moving mass

In this paper, dynamic response of Timoshenko beams under moving mass is analyzed using a numerical method called discrete element technique (DET). In DET, continuous flexible beam elements are replaced by a system of rigid bars and flexible joints. We present a DET model of Timoshenko beams under moving mass. The results of our DET model are compared with the solutions obtained by PAFEC (programs for automatic finite element calculations) for Euler–Bernoulli beams and finite difference method for Timoshenko beams. The effects of beam thickness and moving mass velocity on dynamic response of beams under moving mass are numerically studied.     

压电复合梁高阶有限元模型与主动振动控制研究

大型柔性空间结构的振动控制问题引起了广泛的关注.压电材料以其低质量、宽频带和适应性强等特点,非常适合于柔性空间结构的振动控制.本文针对上下表面粘贴有分布式压电传感器和作动器的智能层梁结构,提出了一种考虑压电材料对结构质量、刚度影响的高阶有限元模型.考虑到空间结构可能承受较大的热载荷,在模型中计及了压电材料的热电耦合效应.采用常增益负反馈控制方法、常增益速度负反馈控制方法、Lyapunov反馈控制方法和线性二次型调节器方法(LQR)设计主动控制器,实现了智能层梁结构脉冲激励下的振动主动控制.仿真结果表明,LQR方法更能有效的实现结构振动控制,并且具有更低的作动器峰值电压,但不能消除热载荷引起的结构静变形.     

Natural vibration analysis of rail track as a system of elastically coupled beam structures on Winkler foundation

An analytical model for analyzing the vertical free vibration of a rail track is presented. The track structure is represented as a system of elastically coupled beam structures resting on a Winkler foundation. The rail and the tie beams are described by any combination of the two existing beam theories, the Bernoulli-Euler type, and the Timoshenko type, while the rail is assumed to be periodically supported at discrete points on cross-track tie beams. A generalized track element, which consists of a rail span (beam segment), two adjacent ties, and the coupling spring stiffnesses, is established to discretize the track system into identical units. A concept of an equivalent frequency-dependent spring coefficient for the rail support system is introduced to formulate the dynamic stiffness matrix of the track element. Solutions are provided for the natural frequencies of the track and the associated mode shapes of the rail and the ties under transversely (cross-track) symmetric vibration. The free vibration results are used to obtain the dynamic receptance response of a typical field track and to compare them with an existing model and field experimental data.