含45度斜裂纹圆柱薄壳动态特性试验研究

测量了含45度斜裂纹圆柱薄壳的固有频率并拍摄了相应的激光全息振型图. 实验 表明斜裂纹比轴向和环向裂纹对壳体动态特性的影响更大，致使振型图发生了严重 畸变而显得相当复杂，利用传统思路难以找到裂纹长度对壳体动态特性的影响规律. 为此， 把裂纹周围的振动看作为一种独立的局部振动，从而把含斜裂纹壳体的各种复杂振型划 分为3类：纯局部振动振型、纯原振动振型、局部振动和原振动耦合振型. 其中前两种振型 的固有频率皆随裂纹的加长而降低，但对于耦合振型有时会出现``随裂纹加长频率反而升高 的现象'，这是由于把壳体原振动的频率和局部振动的频率相混淆而产生的错觉.     

含45°斜裂纹圆柱薄壳动态特性试验研究

测量了含45°斜裂纹圆柱薄壳的固有频率并拍摄了相应的激光全息振型图．实验表明斜裂纹比轴向和环向裂纹对壳体动态特性的影响更大,致使振型图发生了严重畸变而显得相当复杂,利用传统思路难以找到裂纹长度对壳体动态特性的影响规律．为此,把裂纹周围的振动看作为一种独立的局部振动,从而把含斜裂纹壳体的各种复杂振型划分为3类:纯局部振动振型、纯原振动振型、局部振动和原振动耦合振型．其中前两种振型的固有频率皆随裂纹的加长而降低,但对于耦合振型有时会出现“随裂纹加长频率反而升高的现象”, 这是由于把壳体原振动的频率和局部振动的频率相混淆而产生的错觉．     

Non-linear dynamics of a cracked cantilever beam under harmonic excitation

The presence of cracks in a structure is usually detected by adopting a linear approach through the monitoring of changes in its dynamic response features, such as natural frequencies and mode shapes. But these linear vibration procedures do not always come up to practical results because of their inherently low sensitivity to defects. Since a crack introduces non-linearities in the system, their use in damage detection merits to be investigated. With this aim the present paper is devoted to analysing the peculiar features of the non-linear response of a cracked beam.The problem of a cantilever beam with an asymmetric edge crack subjected to a harmonic forcing at the tip is considered as a plane problem and is solved by using two-dimensional finite elements; the behaviour of the breathing crack is simulated as a frictionless contact problem. The modification of the response with respect to the linear one is outlined: in particular, excitation of sub- and super-harmonics, period doubling, and quasi-impulsive behaviour at crack interfaces are the main achievements. These response characteristics, strictly due to the presence of a crack, can be used in non-linear techniques of crack identification.     

Erratum to: Vibration Analysis of Perforated Plates Using Time-Average Digital Speckle Pattern Interferometry

Digital Speckle Interferometry is a non invasive full-field coherent optical technique used in mechanical vibration measurement. In this research, it is used for tuning resonant frequencies of vibrating plates in order to investigate the dynamical behavior of perforated plates. The plate was excited to resonant vibration by a sinusoidal acoustic source. Fringe pattern produced during the time-average recording of the vibrating plate, for several resonant frequencies were registered. Results of plates fixed at one edge having internal holes and attached masses are presented. Experimental natural frequencies and modal shapes are compared to those obtained by an analytical approximate solution based on the Rayleigh–Ritz method with the use of orthogonal polynomials as coordinate function. A high degree of correlation between computational analysis and experimental results was observed, proving the potentiality of the optical technique as experimental validation of the numerical simulations.     

A NEW APPROACH TO FREE VIBRATION ANALYSIS OF A BEAM WITH A BREATHING CRACK BASED ON MECHANICAL ENERGY BALANCE METHOD

In this paper, a new approach to free vibration analysis of a cracked cantilever beam is proposed. By considering the effect of opening and closing the crack during the beam vibration, it is modeled as a fatigue crack. Also, local stiffness changes at the crack location are considered to be a nonlinear amplitude-dependent function and it is assumed that during one half a cycle, the frequencies and mode shapes of the beam vary continuously with time. In addition, by using the experimental tests, it is shown that the local stiffness at the crack location varies continuously between the two extreme values corresponding to the fully closed and the fully open cases of the crack. Then, by using the mechanical energy balance the dynamic response of the cracked beam is obtained at every time instant. The results show that for a specific crack depth, by approaching the crack location to the fixed end of the beam, more reduction in the fundamental frequency occurs. Furthermore, for a specific crack location, the fundamental frequency diminishes and the nonlinearity of the system increases by increasing the crack depth. In order to validate the results, the variations of the fundamental frequency ratio against the crack location are compared with experimental results.     

A comprehensive analytical solution for free vibration of rectangular plates with classical edge coditions: Experimental verification

The problem of obtaining free vibration frequencies and mode shapes of rectangular plates resting on combinations of classical (i.e., clamped, simply supported, or free) edge supports is one that has been investigated for more than one hundred years. More recently, the superposition method has been developed for obtaining accurate analytical-type solutions for this family of problems. The object of this paper is to report on the results of numerous experimental tests carefully performed in order to verify the superposition method and associated computer software. Experimental and computed results are compared for a wide range of plate configurations. Very good agreement between theory and experiment has been obtained with regard to both plate natural frequencies and mode shapes. It is concluded that this computational procedure constitutes a powerful new tool for analysis of rectangular plate vibration problems.     

Vibration of Shallow Shells with Two Adjacent Edges Clamped and the Others Free∗

ABSTRACT

Considerable information is available in the published literature on the free vibration frequencies and mode shapes of rectangular flat plates having two adjacent edges clamped and the other two free. However, no results appear to have been published previously for shallow shells having such edge conditions. The present work uses the Ritz method with displacement components in the form of algebraic polynomials to obtain accurate frequencies. Frequencies are determined for the first eight modes of shallow shells having spherical, cylindrical, and hyperbolic paraboloi-dal curvatures and square planforms. Beginning with the plate, the curvatures are incrementally increased in each case to the limits of shallow     

Vibration Analysis of Perforated Plates Using Time-Average Digital Speckle Pattern Interferometry

Digital Speckle Interferometry is a non invasive full-field coherent optical technique used in mechanical vibration measurement. In this research, it is used for tuning resonant frequencies of vibrating plates in order to investigate the dynamical behavior of perforated plates. The plate was excited to resonant vibration by a sinusoidal acoustic source. Fringe pattern produced during the time-average recording of the vibrating plate, for several resonant frequencies were registered. Results of plates fixed at one edge having internal holes and attached masses are presented. Experimental natural frequencies and modal shapes are compared to those obtained by an analytical approximate solution based on the Rayleigh–Ritz method with the use of orthogonal polynomials as coordinate function. A high degree of correlation between computational analysis and experimental results was observed, proving the potentiality of the optical technique as experimental validation of the numerical simulations.     

Analytical modeling for nonlinear vibration analysis of partially cracked thin magneto-electro-elastic plate coupled with fluid

A nonlinear analytical model for the transverse vibration of cracked magneto-electro-elastic (MEE) thin plate is presented using the classical plate theory (CPT). The MEE plate material selected is fiber-reinforced \(\hbox {BaTiO}_{3}\)–\(\hbox {CoFe}_{2}\hbox {O}_{4}\) composite, which contains a partial crack at the center. The CPT and the simplified line spring model for crack terms are modified to accommodate the effect of electric and magnetic field rigidities. The analysis considers in-plane forces for the MEE plate, which makes the model nonlinear. The derived governing equation is solved by expressing the transverse displacement in terms of modal coordinates. An approximate solution for forced vibration of cracked MEE plate is also obtained using a perturbation technique. The effect of part-through crack, volume fraction of the composite on the vibration frequencies and structure response is investigated. The frequency response curves presented shows the phenomenon of hard or soft spring. Furthermore, the devised model is extended to the case of cracked MEE plate submerged in fluid. Velocity potential function and Bernoulli’s equation are used to incorporate the inertia effect of surrounding fluid. Both partially and totally submerged plate configurations are considered. The validation of the present results is carried out for intact submerged plate as to the best of the author’s knowledge the literature lacks in results for submerged-cracked plates. New results for cracked MEE plate show that the vibration characteristics are affected by volume fraction, crack length, fluid level and depth of immersion.     

A hybrid energy approach for vibrational modelling of laminated trapezoidal plates with point supports

An investigation on the free flexural vibration of symmetric angle-ply thin trapezoidal plates continuous over arbitrarily distributed point supports is reported. A hybrid energy approach which combines the pb-2 Rayleigh-Ritz method with the Lagrangian multiplier method is proposed for the modelling of the aforementioned plate problem. The pb-2 Rayleigh-Ritz method uses a set of Ritz functions generated from the product of a two-dimensional polynomial and the equations of boundaries each raised to the power of 0, 1 or 2 corresponding to a free, simply-supported or clamped edge, respectively. The geometric boundary conditions associated with the point supports are satisfied through the use of Lagrangian multipliers. In this paper, some new solutions for the natural frequencies of several laminated trapezoidal plates with different stacking sequences and location of point supports are presented. The first known mode shapes by means of contour plots for such laminated plates are also included.     

Asymmetric vibrations and stability of a rotating annular plate loaded by a torque

The paper investigates transverse vibration of a thin annular plate clamped at its inner edge to a rigid shaft, while its outer edge is clamped to a rigid cylinder. The shaft and the outer edge of the plate are loaded by torques of the same intensity, but of opposite directions. The whole structure rotates at a constant angular speed. The solution has been determined using Galerkin’s method. The obtained results illustrate the impact of the torque, angular speed and inner and outer radia ratio to transverse asymmetric vibration frequency of the plate. Stability of the plate has been examined and critical values of angular speed and torque leading to the loss of stability of the plate have been determined. Some mode shapes have been drawn and the influence of torque and angular speed on nodal lines has been shown.     

含双裂纹圆截面悬臂梁的损伤识别研究

裂纹的萌生和扩展直接影响构件的振动响应,对构件的安全可靠性具有重要影响．本文以圆截面悬臂梁为对象,结合转角模态振型和模态频率等高线,研究了一种双裂纹识别技术．首先,基于应力强度因子和卡氏定理推导了无裂纹梁单元和含裂纹梁单元的刚度矩阵;在此基础上,建立了含裂纹圆截面悬臂梁的有限元动力学方程;然后,结合裂纹对梁转角模态振型和模态频率的影响,提出了双裂纹识别策略．最后,通过算例讨论了双裂纹识别策略的可行性．结果表明,圆截面悬臂梁的模态转角在裂纹位置出现突变,裂纹深度越大转角突变值越大;将识别出的裂纹位置作为已知参数,通过模态频率等高线法,可以准确地识别出双裂纹的深度．     

A new approach to on-line vibration monitoring and fault diagnosis

This paper presents a finite element method by deriving the stiffness matrix of cracked element from the stress intensity factor. The method can be used to evaluate the strain response of the cantilever beam with a single edge crack. By using the method of identification in time domain, the modal parameters are identified and at the same time the effect of crack closure is taken into consideration. Analysis has also been made of the sensitivity of the strain to the faults. Test results show that the strain response is more sensitive to faults than the displacement response. Then, a distinguishing function is introduced, which is more effective than any other existing ones in failure detection. Finally a new approach to on-line vibration monitoring and fault diagnosis is given.     

Experimental full field investigations of resonant vibrations for piezoceramic plates by an optical interferometry method

The experimental measurement of resonant frequencies for piezoelectric material is generally performed by impedance analysis. In this paper we employ an optical interferometry method, called amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI), to investigate the vibration characteristics of piezoceramic plates. This method demonstrates its advantages of combining noise reduction, like the subtraction method, and high fringe sensitivity, like the time-averaged method. As compared with the film recording and optical reconstruction procedures used for holographic interferometry, the interferometric fringes of AF-ESPI are produced instantly by a video recording system. Based on the fact that clear fringe patterns measured by the AF-ESPI method will be shown only at resonant frequencies, both the resonant frequencies and corresponding mode shapes are obtained experimentally at the same time. Excellent quality for the interferometric fringe patterns of the mode shapes is demonstrated. We find from experimental results that the out-of-plane vibration modes (type A) with lower resonant frequencies cannot be measured by impedance analysis and only the in-plane vibration modes (type B) will be shown. However, both the out-of-plane (bending) and in-plane (extension) vibration modes of piezoceramic plates are obtained by the AF-ESPI method. Finally, numberical finite element calculations are also performed, and the results are compared with the experimental measurements. Excellent agreement for the resonant frequencies and mode shapes are obtained from both results.     

Nonlinear dynamic response of a functionally graded plate with a through-width surface crack

A nonlinear vibration analysis of a simply supported functionally graded rectangular plate with a through-width surface crack is presented in this paper. The plate is subjected to a transverse excitation force. Material properties are graded in the thickness direction according to exponential distributions. The cracked plate is treated as an assembly of two sub-plates connected by a rotational spring at the cracked section whose stiffness is calculated through stress intensity factor. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the FGM plate are derived by using the Hamilton’s principle. The deflection of each sub-plate is assumed to be a combination of the first two mode shape functions with unknown constants to be determined from boundary and compatibility conditions. The Galerkin’s method is then utilized to convert the governing equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under the external excitation, which is numerically solved to obtain the nonlinear responses of cracked FGM rectangular plates. The influences of material property gradient, crack depth, crack location and plate thickness ratio on the vibration frequencies and transient response of the surface-racked FGM plate are discussed in detail through a parametric study.     

Effect of gravity on the vibration of vertical cantilevers

The free vibration of a vertically-oriented, thin, prismatic cantilever is influenced by weight. That is, the natural frequencies (and to a lesser extent, mode shapes) are affected by the application of a linearly varying axial load. A beam with an “upward” orientation, i.e., with the free end above the clamped end, will experience a de-stiffening effect, up to the point of self-weight buckling (at zero effective stiffness). A beam in a “downward” orientation will be stiffened by the weight of the beam. This technical note describes some simple experiments on very slender strips and their (vertical) orientation and shows a close correlation with theory.     

Experiments on crack identification in cantilever beams

In this paper we present an experimental investigation of the identification of crack location and size. By providing the first three natural frequencies through vibration measurements, curves of crack equivalent stiffness versus crack location are plotted, and the intersection of the three curves predicts the crack location and size. In the experiments, the cracked specimens were made using a wire-cut electrical discharge machine, and the cantilever beams were excited next to the free end by means of an impulse force hammer. In order to obtain the accurate natural frequencies from the transient signal measured, the method of zoom fast Fourier transform is adopted to improve frequency resolution. From experimental results, it is observed that the identification errors of crack location and size are less than 2% and 4%, respectively. The effectiveness of crack identification through vibration measurements is verified.     

厚圆板轴对称振动的弹性力学解

本文以轴对称三维弹性力学基本方程为基础,导出厚圆板强迫振动的状态方程式。利用Ｍａｃｌａｕｒｉｎ级数和Ｓｙｌｖｅｓｔｅｒ定理,厚圆板的位移和应力可以用中面位移和应力的微分算子表示。通过载荷分解和圆板表面条件,可以得到厚圆板在对称载荷与反对称载荷作用下的振动控制方程。求解了厚圆板在周边固支和简支条件下的对称与反对称的自由振动问题。通过数值计算得到了这两类自由振动的固有频率。本文的方法适用于求解厚圆板在     

Modal analysis of cracked continuous Timoshenko beam made of functionally graded material

Abstract

The article addresses development of the Transfer Matrix Method (TMM) for free vibration of cracked continuous Timoshenko beam made of Functionally Graded Material (FGM). The governing equations of free vibration are established for the beam based on the power law of material grading, actual position of neutral plane and double spring model of crack. There is conducted frequency equation of the beam with intermediate rigid supports using the TMM after the transverse displacements at rigid supports have been disregarded. Therefore, the frequency equation is simplified and becomes more useful to compute natural frequencies of continuous FGM Timoshenko beam with a number of cracks. The obtained numerical results show the essential effect of cracks, material properties and also number of spans on natural frequencies of the beam.