Repair of notched beam under dynamic load using piezoelectric patch

In this paper, the repair of a cracked beam under an external dynamic load employing the electro-mechanical characteristic of piezoelectric material to induce a local moment is presented. Conceptually, an external voltage is applied to actuate a piezoelectric patch bonded on the beam to effect closure of a crack so that the singularity at the crack tip under dynamic load may be decreased. Globally, this has the effect of altering the resonant frequency of the cracked beam towards that of the healthy beam, which is the criterion used for the repair. To demonstrate the repair methodology, a cantilever beam is used as an illustration, where the repair moment coefficient and the voltage required are mathematically derived. The relationship between repair moment coefficient, crack parameters and length of piezoelectric patch is investigated. The difference between the proposed repair criterion and an earlier published criterion for cracked beam under static load is also shown. A numerical example is used to study the effectiveness of the proposed repair methodology and its results are compared with those from 3-D finite element analyses using ABAQUS 6.4 as one means of verification.     

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.     

Dynamic responses and fuzzy control of a simply-supported beam subjected to a moving mass

This paper deals with the active vibration control of a simply-supported beam traversed by a moving mass using fuzzy control. Governing equations for dynamic responses of a beam under a moving mass are derived by Galerkin’s mode summation method, and the effect of forces (gravity force, Coliolis force, inertia force caused by the slope of the beam, transverse inertia force of the beam) due to the moving mass on the dynamic response of a beam is discussed. For the active control of dynamic deflection and vibration of a beam under the moving mass, the controller based on fuzzy logic is used and the experiments are conducted by VCM (voice coil motor) actuator to suppress the vibration of a beam. Through the numerical and experimental studies, the following conclusions were obtained. With increasing mass ratioy at a fixed velocity of the moving mass under the critical velocity, the position of moving mass at the maximum dynamic deflection moves to the right end of the beam. With increasing velocity of the moving mass at a fixed mass ratioy, the position of moving mass at the maximum dynamic deflection moves to the right end of the beam too. The numerical predictions of dynamic deflection of the beam have a good agreement with the experimental results. With the fuzzy control, more than 50% reductions of dynamic deflection and residual vibration of the tested beam under the moving mass are obtained.     

附有滑动质量的旋转柔性梁动力学特性研究

提出用弹簧-质量系统抑制旋转柔性梁的振动,建立Euler-Bernoulli梁的动力学模型.对方程进行无量纲化,并在质量为慢时变运动、旋转角速度和角加速度为梁变形同阶小量时,对非线性方程进行近似简化,分析弹簧-质量系统对梁振动特性的影响机理.利用多尺度方法对非线性方程近似求解,在主共振、内共振条件下,研究梁和质量运动幅值随质量位置和调协参数的变化趋势.     

Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions

An analytical approach for crack identification procedure in uniform beams with an open edge crack, based on bending vibration measurements, is developed in this research. The cracked beam is modeled as two segments connected by a rotational mass-less linear elastic spring with sectional flexibility, and each segment of the continuous beam is assumed to obey Timoshenko beam theory. The method is based on the assumption that the equivalent spring stiffness does not depend on the frequency of vibration, and may be obtained from fracture mechanics. Six various boundary conditions (i.e., simply supported, simple–clamped, clamped–clamped, simple–free shear, clamped–free shear, and cantilever beam) are considered in this research. Considering appropriate compatibility requirements at the cracked section and the corresponding boundary conditions, closed-form expressions for the characteristic equation of each of the six cracked beams are reached. The results provide simple expressions for the characteristic equations, which are functions of circular natural frequencies, crack location, and crack depth. Methods for solving forward solutions (i.e., determination of natural frequencies of beams knowing the crack parameters) are discussed and verified through a large number of finite-element analyses. By knowing the natural frequencies in bending vibrations, it is possible to study the inverse problem in which the crack location and the sectional flexibility may be determined using the characteristic equation. The crack depth is then computed using the relationship between the sectional flexibility and the crack depth. The proposed analytical method is also validated using numerical studies on cracked beam examples with different boundary conditions. There is quite encouraging agreement between the results of the present study and those numerically obtained by the finite-element method.     

Discussion on calculation of the local flexibility due to the crack in a pipe

The stress intensity factor plays an important role in the calculation of the local flexibility due to the crack in a cracked structure. Many researches on the stress intensity factors in a cracked beam or rotor have been made, but there are some difficulties in calculating the stress intensity factors in a cracked pipe. Maiti et al. obtained the local flexibility due to the crack in a pipe experimentally by deflection and natural frequency methods without calculating the stress intensity factor. In this paper, the stress intensity factor in a cracked pipe can be calculated by dividing a cracked pipe into a series of thin annuli, and a method to calculate the local flexibility due to the crack in a pipe is presented. The experiment results are given to verify the effectiveness of such a method.     

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.     

Semi-analytical approach for free vibration analysis of cracked beams resting on two-parameter elastic foundation with elastically restrained ends

In present study, free vibration of cracked beams resting on two-parameter elastic foundation with elastically restrained ends is considered. Euler-Bernoulli beam hypothesis has been applied and translational and rotational elastic springs in each end considered as support. The crack is modeled as a mass-less rotational spring which divides beam into two segments. After governing the equations of motion, the differential transform method (DTM) has been served to determine dimensionless frequencies and normalized mode shapes. DTM is a semi-analytical approach based on Taylor expansion series that converts differential equations to recursive algebraic equations. The DTM results for the natural frequencies in special cases are in very good agreement with results reported by well-known references. Also, the DTM procedure yields rapid convergence beside high accuracy without any frequency missing. Comprehensive studies to analyze the effects of crack location, crack severity, parameters of elastic foundation and boundary conditions on dimensionless frequencies as well as effects of elastic boundary conditions on cracked beams mode shapes are carried out and some problems handled for first time in this paper. Since this paper deals with general problem, the derived formulation has capability for analyzing free vibration of cracked beam with every boundary condition.     

Simulation on the motion of crankshaft with a slant crack in crankpin

A new model for vibration analysis of a crankshaft with a slant crack in crankpin is proposed, and the influence of crack depth on the transient response of a cracked crankshaft is investigated. A slant cracked shaft element is developed by deducing the local flexibility due to a slant crack. The frequently occurred slant crack in crankpin is studied, and a new finite element model of crankshaft including the slant crack in crankpin, which combines the slant cracked shaft element and Timoshenko beam elements, is derived. The support of engine block and the switching behaviour of the crack are considered, and the non-linear equation of motion for cracked crankshaft-bearing system is set up in a rotating coordinate system. The motion of a crankshaft of a four in-line cylinder engine with and without an initial crack is simulated. The influence of the crack depth on the transient response is investigated. The numerical simulation demonstrates that the current model is valid for simulating the motion of cracked crankshaft system. The results show that a useful foundation is laid for crack detection of crankshaft.     

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.     

An investigation on effects of traveling mass with variable velocity on nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions

In this paper, the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a traveling mass with variable velocity is investigated. The nonlinear coupled partial differential equations of motion for the bending rotation of cross-section, longitudinal and transverse displacements are derived using Hamilton’s principle. These nonlinear coupled PDEs are solved by applying Galerkin’s method to obtain dynamic response of the beam under the act of a moving mass. The appropriate parametric studies by taking into account the effects of the magnitude of the traveling mass, the velocity of the traveling mass with a constant acceleration/deceleration and effect of different beam’s boundary conditions are carried out. The beams’ large deflection has been captured by including the stretching effect of its mid-surface. It was seen that the existence of quadratic-cubic nonlinear terms in the governing coupled PDEs of motion renders hardening/stiffening behavior on the dynamic responses of the beam when traversed by a moving mass. In addition, the obtained nonlinear results are compared with those from the linear 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。     

Free vibration analysis of Euler-Bernoulli beam with double cracks

In this paper, the influence of two open cracks on the dynamic behavior of a double cracked simply supported beam is investigated both analytically and experimentally. The equation of motion is derived by using the Hamilton’s principle and analyzed by numerical method. The simply supported beam is modeled by the Euler-Bemoulli beam theory. The crack sections are represented by a local flexibility matrix connecting three undamaged beam segments. The influences of the crack depth and the position of each crack on the vibration mode and the natural frequencies of a simply supported beam are analytically clarified for the single and double cracked simply supported beam. The theoretical results are also validated by a comparison with experimental measurements.     

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 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.     

Dynamic propagation of a finite eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic propagation of an eccentric Griffith crack in a functionally graded piezoelectric ceramic strip under anti-plane shear is analyzed using the integral transform method. A constant velocity Yoffe-type moving crack is considered. Fourier transform is used to reduce the problem to a pair of dual integral equations, which is then expressed in a Fredholm integral equation of the second kind. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. The impermeable crack boundary condition is adopted. Numerical values on the dynamic stress intensity factors are presented for the functionally graded piezoelectric material to show the dependence of the gradient of material properties, crack moving velocity, and eccentricity. The dynamic stress intensity factors of a moving crack in functionally graded piezoelectric material increases when the crack moving velocity, eccentricity of crack location, material property gradient, and crack length increase. This paper was recommended for publication in revised form by Associate Editor Hyeon Gyu Beom Jeong Woo Shin received a B.S. and M.S. degree in Mechanical Engineering from Yonsei University in Seoul, Korea in 1998 and 2000, respectively. A major field of Mr. Shin is fracture mechanics. He is currently working on the KARI (Korea Aerospace Research Institute) as a senior researcher. He conducted load analysis of fixed wing aircraft and full scale airframe static test at the KARI. He is now developing landing gear in the KHP (Korea Helicopter Program) as a performance engineer.     

I-beam Crack Identification Based on Study of Local Flexibility due to Crack

Local flexibility of crack plays an important role in crack identification of structures.Analytical methods on local flexibility in a cracked beam with simple geometric crossing sections,such as rectangle,circle,have been made,but there are some difficulties in calculating local flexibility in a cracked beam with complex crossing section,such as pipe and I-beam.In this paper,an analytical method to calculate the local flexibility and rotational spring stiffness due to crack in I-beam is proposed.The local flexibility with respect to various crack depths can be calculated by dividing a cracked I-beam into a series of thin rectangles.The forward and inverse problems in crack detection of I-beam are studied.The forward problem comprises the construction of crack model exclusively for crack section and the construction of a numerically I-beam model to gain crack detection database.The inverse problem consists of the measurement of modal parameters and the detection of crack parameters.Two experiments including measurement of rotational spring stiffness and prediction of cracks in I-beam are conducted.Experimental results based on the current methods indicate that relative error of crack location is less than 3%,while the error of crack depth identification is less than 6%.Crack identification of I-beam is expected to contribute to the development of automated crack detection techniques for railway lines and building skeletons.     

Influence of tip mass on dynamic behavior of cracked cantilever pipe conveying fluid with moving mass

In this paper, we studied about the effect of the open crack and a tip mass on the dynamic behavior of a cantilever pipe conveying fluid with a moving mass. The equation of motion is derived by using Lagrange’s equation and analyzed by numerical method. The cantilever pipe is modelled by the Euler-Bernoulli beam theory. The crack section is represented by a local flexibility matrix connecting two undamaged pipe segments. The influences of the crack, the moving mass, the tip mass and its moment of inertia, the velocity of fluid, and the coupling of these factors on the vibration mode, the frequency, and the tip-displacement of the cantilever pipe are analytically clarified.     

半电极含金属芯压电纤维的驱动性能

建立了悬臂梁结构半电极含金属芯压电纤维新型压电弯曲驱动器的理论模型.根据第一类压电方程,推导出自由端位移、夹持力和弯曲共振频率的解析表达式,分析了金属芯性能和半径对这3个参数的影响,并把理论计算结果和有限元分析结果进行了比较.实验结果表明,悬臂梁结构半电极含金属芯压电纤维弯曲驱动器的自由端位移可达589 μm,夹持力可达427 μN,一阶弯曲共振频率为28 Hz,有限元分析结果和理论值基本吻合,说明这种驱动器有较大的端部位移、较小的夹持力和较低的弯曲共振频率.     

Dynamic behaviors of an elastically restrained beam carrying a moving mass

Dynamic responses of a simply supported beam with a translational spring carrying a moving mass are studied. Governing equations of motion including all the inertia effects of a moving mass are derived by employing the Galerkin’s mode summation method, and solved by using the Runge-Kutta integral method. Numerical solutions for dynamic responses of a beam are obtained for various cases by changing parameters of the spring stiffness, the spring position, the mass ratio and the velocity ratio of a moving mass. Some experiments are conducted to verify the numerical results obtained. Experimental results for the dynamic responses of the test beam have a good agreement with numerical ones.