Repair of a cracked Timoshenko beam subjected to a moving mass using piezoelectric patches

This paper presents an analytical method for the application of piezoelectric patches for the repair of cracked beams subjected to a moving mass. The beam equations of motion are obtained based on the Timoshenko beam theory by including the dynamic effect of a moving mass traveling along a vibrating path. The criterion used for the repair is altering the first natural frequency of the cracked beam towards that of the healthy beam using a piezoelectric patch. Conceptually, an external voltage is applied to actuate a piezoelectric patch bonded on the beam. This affects the closure of the crack so that the singularity induced by the crack tip will be decreased. The equations of motion are discretized by using the assumed modes method. The cracked beam is modeled as number of segments connected by two massless springs at the crack locations (one, extensional and the other, rotational). The relationships between any two spans can be obtained by considering the compatibility requirements on the crack section and on the ends of the piezoelectric patch. Using the analytical transfer matrix method, eigensolutions of the system can be calculated explicitly. Finally, numerical simulations are performed with respect to different conditions such as the moving load velocity. It is seen that when the piezoelectric patch is used, the maximum deflection of the cracked beam approaches maximum deflection of the healthy beam.     

Forced responses of cracked cantilever beams subjected to a concentrated moving load

An analytical method is developed to present the dynamic response of a cracked cantilever beam subject to a concentrated moving load. The cracked beam system is modeled as a two-span beam and each span of the continuous beam is assumed to obey Euler–Bernoulli beam theory. The crack is modeled as a rotational spring with sectional flexibility. Considering the compatibility requirements on the crack, the relationships between these two spans can be obtained. By using the analytical transfer matrix method, eigensolutions of this cracked system are obtained explicitly. The forced responses can be obtained by the modal expansion theory using the determined eigenfunctions. Some numerical results are shown to present the crack effects (crack extent, location of the crack) and are studied for different speeds of the moving load.     

气体动载荷下压电振子能量转化特性实验研究

针对当前利用化学电池为传感器供能具有的使用时间短及污染环境等问题，提出了一种可应用于气动系统中气体动载荷环境的能量转化与收集装置。利用压电材料的正压电效应对压电振子发电机理进行分析，分析结果表明，在气体动载荷激励下输出电能与压力变化量呈正比。仿真分析了压电振子所在容腔的内部流场变化，并分析了腔内压力及压力变化量随时间的变化规律。结合理论研究与仿真分析设计并制作了一种压电振子能量转化实验样机，搭建了实验测试系统。以动态气体载荷为激励源对不同流量、周期及负载条件的改变进行了实验测试。实验结果表明随着距离的增加，压电振子的输出电压逐渐下降；随着流量的增加峰值电压增加，当周期为1.2 s、流量为200 L/min、压力为0.3 MPa时最大的输出电压为79.60 V。     

Four-beam model for vibration analysis of a cantilever beam with an embedded horizontal crack

As one of the main failure modes, embedded cracks occur in beam structures due to periodic loads. Hence it is useful to investigate the dynamic characteristics of a beam structure with an embedded crack for early crack detection and diagnosis. A new four-beam model with local flexibilities at crack tips is developed to investigate the transverse vibration of a cantilever beam with an embedded horizontal crack; two separate beam segments are used to model the crack region to allow opening of crack surfaces. Each beam segment is considered as an Euler-Bernoulli beam. The governing equations and the matching and boundary conditions of the four-beam model are derived using Hamilton's principle. The natural frequencies and mode shapes of the four-beam model are calculated using the transfer matrix method. The effects of the crack length, depth, and location on the first three natural frequencies and mode shapes of the cracked cantilever beam are investigated. A continuous wavelet transform method is used to analyze the mode shapes of the cracked cantilever beam. It is shown that sudden changes in spatial variations of the wavelet coefficients of the mode shapes can be used to identify the length and location of an embedded horizontal crack. The first three natural frequencies and mode shapes of a cantilever beam with an embedded crack from the finite element method and an experimental investigation are used to validate the proposed model. Local deformations in the vicinity of the crack tips can be described by the proposed four-beam model, which cannot be captured by previous methods.     

Characterization of fracture behavior in repaired skin/stiffener structure with an inclined central crack

Finite element analysis for the stress intensity factor (SIF) at the skin/stiffener structure with inclined central crack repaired by composite stiffened panels is developed. A numerical investigation was conducted to characterize the fracture behavior and crack growth behavior at the inclined crack. In order to investigate the crack growth direction, maximum tangential stress (MTS) criterion are used. Also, this paper is to study the performance of the effective bonded composite patch repair of a plate containing an inclined central through-crack. The main objective of this research is the validation of the inclined crack patching design. In this paper, the reduction of stress intensity factors at the crack-tip and prediction of crack growth direction are determined to evaluate the effects of various non-dimensional design parameter, including; composite patch thickness and stiffener distance. We report, the results of finite element analysis on the stiffener locations and crack slant angles and discuss them in this paper. The research on cracked structure subjected to mixed mode loading is accomplished and concludes that more work using a different approaches is necessary. The authors hope the present study will aid those who are responsible for the repair of damaged aircraft structures and also provide general repair guidelines.     

周向裂纹管道在含扭转各种组合变形时的极限载荷分析

到目前为止，还没有文献给中向裂纹管道在非对称弯曲及扭转组合变形时的塑性极限载荷计算公式。文中根据净截面垮塌准则用沙堆比拟法分别求出埋藏裂纹、外表面裂纹、内表面裂纹、穿透裂纹管道发生扭转变形时的塑性极限扭矩；给出含周向裂纹薄壁管道横截面上的剪应力分布规律，其塑性极限扭矩等于一个闭口薄壁截面与一个开口薄壁截面圆环的塑性极限扭矩之和，闭口薄壁截面的壁厚为管道壁厚减裂纹深度；开口薄壁截面的壁厚为裂纹的深度。推导了各种周向裂纹管在内压、轴力、扭矩及非对称弯矩共同作用时的塑性极限载荷关系式，并由此给出其他一些组合变形时的极限载荷计算公式。本文结果可供管道安全评价时参考。     

Dynamic response of an anti-plane shear crack in a functionally graded piezoelectric strip

The dynamic response of a cracked functionally graded piezoelectric material (FGPM) under transient anti-plane shear mechanical and in-plane electrical loads is investigated in the present paper. It is assumed that the electroelastic material properties of the FGPM vary smoothly in the form of an exponential function along the thickness of the strip. The analysis is conducted on the basis of the unified (or natural) crack boundary condition which is related to the ellipsoidal crack parameters. By using the Laplace and Fourier transforms, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, the electric field, FGPM gradation, crack length, and electromechanical coupling coefficient.     

Nonlinear vibrational response of a single edge cracked beam

The nonlinear vibrational response of a breathing cracked beam was investigated. The study was done by using a new crack stiffness model to examine some of the nonlinear behaviors of a cantilever beam with a breathing crack. The quadratic polynomial stiffness equation of the cracked beam was derived based on the hypothesis that the breathing process of a crack depends on the vibration magnitude. The Galerkin method combined with the stiffness equation was used to simplify the cracked beam into a Single-degree-of-freedom (SDOF) lumped system with nonlinear terms. The multi scale method was adopted to analyze the nonlinear amplitude frequency response of the beam. The applicability of the stiffness model was discussed and parameter sensitivity studies on the dynamic response were carried out by the SDOF model for a cantilever beam. Results indicate that the new stiffness model provides an efficient tool to study the vibrational nonlinearities introuduced by the breathing crack. Therefore, it might be used to develop a nonlinear identification method of a crack in a beam.     

Crack detection in simply supported beams without baseline modal parameters by stationary wavelet transform

There are significant changes in the vibration responses of cracked structures when the crack depth is significant in comparison to the depth of the structure. This fact enables the identification of cracks in structures from their vibration response data. However when the crack is relatively small, it is difficult to identify the presence of the crack by a mere observation of the vibration response data. A new approach for crack detection in beam-like structures is presented and applied to cracked simply supported beams in this paper. The approach is based on finding the difference between two sets of detail coefficients obtained by the use of the stationary wavelet transform (SWT) of two sets of mode shape data of the beam-like structure. These two sets of mode shape data, which constitute two new signal series, are obtained and reconstructed from the modal displacement data of a cracked simply supported beam. They represent the left half and the modified right half of the modal data of the simply supported beam. SWT is a redundant transform that doubles the number of input samples at each iteration. It provides a more accurate estimate of the variances at each scale and facilitates the identification of salient features in a signal, especially for recognising noise or signal rupture. It is well known that the mode shape of a beam containing a small crack is apparently a single smooth curve like that of an uncracked beam. However, the mode shape of the cracked beam actually exhibits a local peak or discontinuity in the region of damage. Therefore, the mode shape ‘signal’ of a cracked beam can be approximately considered as that of the uncracked beam contaminated by ‘noise’, which consists of response noise and the additional response due to the crack. Thus, the modal data can be decomposed by SWT into a smooth curve, called the approximation coefficient, and a detail coefficient. The difference of the detail coefficients of the two new signal series includes crack information that is useful for damage detection. The modal responses of the damaged simply supported beams used are computed using the finite element method. For real cases, mode shape data are affected by experimental noise. Therefore, mode shape data with a normally distributed random noise are also studied. The results show that the proposed method has great potential in crack detection of beam-like structures as it does not require the modal parameters of an uncracked beam as a baseline for crack detection. The effects of crack size, depth and location, and the effects of sampling interval are examined.     

智能Timoshenko梁强迫振动响应的行波解

运用行波法求解智能Timoshenko梁的强迫振动响应.基于梁的振动行波法分析理论,介绍周期力和周期力矩激励下激励点处波的传播关系,推导在逆压电效应下,智能Timoshenko梁上某一点的响应函数,并以智能Timoshenko悬臂梁为例,得到梁不同位置的拾振点处的幅频和相频特性及不同受力状况下的智能梁的幅频特性,还得到了受迫振动的时域响应信号波形,最后将求解结果与有限元法求解结果做对比分析.分析结果表明,行波法分析可用于精确计算智能梁的强迫振动响应.     

基于有限元位移模式的含裂纹梁结构动力学模型

针对裂纹的存在将降低梁的刚度的实际情形,首先根据断裂力学理论,引入裂纹梁因裂纹扩展而释放的应变能表达式,然后根据金属材料的特点,运用有限元位移法建立裂纹梁单元的动力学模型,再在梁单元模型的基础上应用有限元位移法建立裂纹梁结构的动力学方程。研究表明：基于有限元位移模式所建立的动力学方程较好地体现了裂纹梁动态性能与其结构参数和裂纹参数之间的内在关系,反映了裂纹的位置及长度对含裂纹梁结构动态性能的影响,为建立含裂纹梁结构动力学模型提供了一种新的有效方法。最后通过实例对理论分析结果进行了验证。     

压电柔性梁裂缝损伤识别实验

针对压电柔性悬臂梁裂缝损伤检测与损伤程度识别问题,采用小波包分析和小波神经网络相结合的方法进行裂缝深度识别实验研究.利用小波包频带能量谱构造柔性悬臂梁裂缝损伤指标,即能量比相对变化量的H2范数,并建立压电柔性梁裂缝损伤实验装置.激励柔性梁的振动,记录两路压电传感器采集的振动信号,进行小波包分解并计算损伤指标.将这些损伤指标进行组合,作为小波神经网络的输入特征参数,进行裂缝深度即损伤程度的识别.实验结果表明:能量比相对变化量的H2范数对柔性梁的裂缝损伤敏感,对测试噪声不敏感;采用的小波神经网络可以精确识别柔性梁的裂缝深度.     

悬臂梁非线性振动的压电分阶最优控制

提出非线性的分阶最优控制策略,并将其应用于悬臂梁非线性振动的压电减振控制.建立悬臂梁非线性压电减振系统动力学模型,导出减振系统的非线性动力学运动微分方程.将梁振动挠度和压电驱动器的控制电压同时展开为小参数形式,利用摄动法实现非线性压电控制微分方程的线性化.通过空间解耦,得到状态空间方程.设计非线性分阶控制器,对该减振系统进行分阶最优控制.     

压电振动电阻尼与能量转换特性研究

根据压电构造方程和振动原理,建立压电振动能量收集的耦合场动力学模型。详细推导电阻尼与外接电阻和机电耦合系数之间的数学关系,并揭示外接电阻对系统固有弹性的作用效果。通过数值模拟研究电阻尼特性对谐振频率、振动幅值和功率的影响关系,并从能量转换效率的角度分析优化电阻与最大输出功率的关系。分别对多种外接电阻条件下压电梁的输出电压及功率进行实验测试,实验结果表明,电阻尼导致压电梁的谐振频率发生偏移,其大小与外接电阻值成正比,而且在外接优化电阻时输出功率最大。     

Dynamic analysis of structures with piezoelectric actuators based on thermal analogy method

The thermal analogy method is presented to predict the dynamic behavior of complex structures with piezoelectric actuators. Based on the analogy between the converse piezoelectric effect and thermoelastic effect, an applied electric field is modeled as a thermal load and piezoelectric strain coefficients characterizing an actuator are input as thermal expansion coefficients. Thus a voltage actuation can be exactly simulated using conventional tri-dimensional elastic elements with the thermal actuation rather than using piezoelectric elements. Finally, numerical results in terms of the transient response of a cantilever beam with surface bonded piezoelectric actuators demonstrates the validity of this method, which can be used to assist in the design of complex smart structures and implementation of piezoelectric control systems.     

Beam vibration control using cellulose-based Electro-Active Paper sensor

Cellulose based Electro-Active Paper (EAPap) has recently shown a great potential as an environment-friendly smart material due to its biodegradability, biocompatibility and flexibility. Lots of studies have been conducted to investigate the basic smart characteristics of EAPap, but its application has not yet developed well. In this paper, the possibility of cellulose-based Electro-Active Paper (EAPap) as a piezoelectric sensor was investigated by the vibration control of the cantilevered beam. The EAPap sample was attached at the root of the cantilevered beam and used as a vibration sensor. The piezoceramic patch was also attached at the root of the beam and played as an actuator. The voltage output of EAPap showed exact dynamic characteristics of the cantilevered beam. The frequency bandwidth and quality factor of EAPap were similar to those of piezoceramic patch, which results EAPap has similar sensing capability of piezoceramic patch. To find the application of EAPap sensor, beam vibration control was performed. EAPap sensor output was considered as a position error of the cantilevered beam and a simple PID controller was designed to suppress the vibration of the beam. The EAPap sensor output provided clear time response of the beam. The controlled system showed good vibration control performance of the beam. The results provided that the piezoelectric characteristic of EAPap has a great potential as a sensor and also as a new smart material.     

Crack detection in beams using experimental modal data and finite element model

In this paper, an analytical, as well as experimental approach to the crack detection in cantilever beams by vibration analysis is established. An experimental setup is designed in which a cracked cantilever beam is excited by a hammer and the response is obtained using an accelerometer attached to the beam. To avoid non-linearity, it is assumed that the crack is always open. To identify the crack, contours of the normalized frequency in terms of the normalized crack depth and location are plotted. The intersection of contours with the constant modal natural frequency planes is used to relate the crack location and depth. A minimization approach is employed for identifying the cracked element within the cantilever beam. The proposed method is based on measured frequencies and mode shapes of the beam.     

Structural damage identification of a cantilever beam using excitation force level control

Most studies in damage identification so far have concentrated on comparing modal parameters of a structure with an open crack with those of an intact structure. In this study, a new damage identification method for beam-like structures with a fatigue crack is proposed, which does not require comparative measurement on an intact structure but several measurements at different level of excitation forces on the cracked structure. The idea comes from the fact that dynamic behavior of a structure with a fatigue crack changes with the level of the excitation force. In other words, a beam with a real fatigue crack would behave as an intact beam at low excitation forces due to the crack closure. The 2nd spatial derivatives of frequency response functions along the longitudinal direction of a beam are used as the sensitive indicator of crack existence. Then, weighting function is employed in the averaging process in frequency domain to account for the modal participation of the differences between the dynamic behavior of beam with a fatigue crack at the low excitation and one at the high excitation. Subsequently, a damage index is defined such that the location and level of the crack may be identified. Finally, it is shown that damage identification method using the proposed damage index is very successful through experiment and finite element analysis.     

基于压电增益特性进行梁中缺陷的识别

驱动元件PZT片和传感元件PVDF膜粘贴于自由梁表面,通过测试压电增益,试验获取梁中不同缺陷尺寸下的固有频率,根据固有频率的变化,实现缺陷的识别,梁中的缺陷采用等效线性弹簧模拟,描绘出不同模态下刚度与缺陷可能位置曲线,根据曲线的交点,得出缺陷位置与尺寸,相比于实际的缺陷位置与尺寸,自由梁弯曲激振下识别的结果满足一定的精度。     

行波型超声电动机定子振动模型及其驱动设计

利用复合梁振动理论对典型结构的行波型超声波电动机(TWUSM)的振动特性进行了研究。根据定子结构的特点,将定子简化为一系列简支梁,由此建立了描述定子在逆压电效应作用下的强迫振动模型。此外,该模型还考虑了粘滑阻尼转矩和外部阻尼力的影响。最后借助于广义模态方法对该方程进行了求解,从而获得了在不同控制变量和不同结构参数下TWUSM的振动特性。为验证模型的准确性,设计制作了一台TWUSM样机和专用驱动器。样机试验结果与模型的仿真结果具有较好的一致性,从而验证了模型的有效性。