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.     

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.     

弹性连接旋转柔性梁动力学分析

采用Chebyshev 谱方法对考虑根部连接弹性的平面内旋转柔性梁动力学特性进行研究. 基于Gauss-Lobatto 节点与Chebyshev 多项式方法对柔性梁变形场进行离散,通过投影矩阵法施加固定及弹性连接边界条件. 利用Chebyshev 谱方法获得了系统固有频率和模态振型数值解,通过与有限元方法及加权残余法的比较,验证了方法的有效性. 分析了弹性连接刚度、角速度比率、系统径长比及梁的长细比等参数对系统固有频率及模态振型的影响. 研究发现:由于系统弯曲模态、拉伸模态的频率随各参数的变化规律不一致,将出现频率转向与振型转换现象;随着弹性连接刚度、角速度比率及系统径长比的增大,低阶弯曲模态频率增大并超过高阶拉伸模态频率,随着梁的长细比的增大,低阶拉伸模态频率增大并超过高阶弯曲模态频率.      

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.     

In-plane Operating Deflection Shape Measurement of an Aluminum Plate Using a Three-dimensional Continuously Scanning Laser Doppler Vibrometer System

Background

In-plane vibration is significant to a structure and has been accurately solved by many numerical methods; however, there are still not enough studies on its experimental measurement.

Objective

This work aims to propose a non-contact and fast way to measure dense full-field in-plane vibration of a plate structure, which has high frequencies and low response magnitudes.

Methods

A novel three-dimensional (3D) continuously scanning laser Doppler vibrometer (CSLDV) system that contains three CSLDVs is developed to conduct full-field scanning of a plate with free boundary conditions under sinusoidal excitation to measure its 3D vibrations. Calibration among the three CSLDVs in the 3D CSLDV system based on the geometrical model of its scan mirrors is conducted to adjust their rotational angles to ensure that three laser spots can continuously and synchronously move along the same two-dimensional scan trajectory on the plate. The demodulation method is used to process the measured response to obtain in-plane operating deflection shapes (ODSs) of the plate.

Results

Four in-plane ODSs are obtained in the frequency range of 0–5000 Hz. Modal assurance criterion (MAC) values between in-plane ODSs from 3D CSLDV and step-wise scanning laser Doppler vibrometer (SLDV) measurements are larger than 95%. MAC values between ODSs from 3D CSLDV measurements and corresponding mode shapes from the finite element model of the plate are larger than 91%.

Conclusions

Results from 3D CSLDV measurements have good accuracy compared to those from SLDV measurements and numerical calculation, and the 3D CSLDV system can scan much more measurement points in much less time than the SLDV system.

     

重频模态可控可观性的度量与主模态概念

本文研究重频模态的可控性和可观性的定量度虽以及不变子空间的可控性结构和可观性结构问题，提出了一种利用奇异值来度量重频模态可控性和可观性的方法，同时提出了如何获得一组新的重频模态向量，使得在这组模态坐标下，子空间的可控性结构或可观性结构有更加明了的表达．     

“Non-linear normal modes” and the generalized Ritz method in the problems of vibrations of non-linear elastic continuous systems

In the study of natural vibrations of non-linear elastic systems it is shown that the mode shape of the vibration can vary with the amplitude as well as the frequency, and that the amplitude frequency relation is strongly affected by constraints imposed on the mode shape in an approximate solution. A method is developed which assumes the approximate solution in the form of a truncated series in which, instead of the set of coefficients, the set of functions of spatial variables is unknown and then determined by a procedure that can be regarded as a generalization of the Ritz method. The problem of variations of the normal mode shapes and of the associated natural frequencies with the amplitude is illustrated by two examples of beams with non-linear boundary conditions, and the amplitude-frequency relation is compared to that corresponding to the a priori assumed linear normal mode solution. Further possible consequences of the mode shape amplitude variations in forced, resonant motion of nonconservative systems are also indicated.     

Static and dynamic responses of a microcantilever with a T-shaped tip mass to an electrostatic actuation

In this study, nonlinear static and dynamic responses of a microcantilever with a T-shaped tip mass excited by electrostatic actuations are investigated. The electrostatic force is generated by applying an electric voltage between the horizontal part of T-shaped tip mass and an opposite electrode plate. The cantilever microbeam is modeled as an Euler–Bernoulli beam. The T-shaped tip mass is assumed to be a rigid body and the nonlinear effect of electrostatic force is considered. An equation of motion and its associated boundary conditions are derived by the aid of combining the Hamilton principle and Newton's method.An exact solution is obtained for static deflection and mode shape of vibration around the static position. The differential equation of nonlinear vibration around the static position is discretized using the Galerkin method. The system mode shapes are used as its related comparison functions. The discretized equations are solved by the perturbation theory in the neighborhood of primary and subharmonic resonances.In addition, effects of mass inertia, mass moment of inertia as well as rotation of the T-shaped mass, which were ignored in previous works, are considered in the analysis. It is shown that by increasing the length of the horizontal part of the T-shaped mass, the amount of static deflection increases,natural frequency decreases and nonlinear shift of the resonance frequency increases. It is concluded that attaching an electrode plate with a T-shaped configuration to the end of the cantilever microbeam results in a configuration with larger pull-in voltage and smaller nonlinear shift of the reso-nance frequency compared to the configuration in which the electrode plate is directly attached to it.     

局部损伤复合材料层合板的振动分析

本文采用了一种改进方法对局部损伤复合材料层合板进行了振动分析,将复合材料板中的损伤模拟为局部刚度的削减,并取三个损伤因子来刻画损伤的特性．利用高阶摄动法对其自由振动方程进行求解,主要计算了损伤板的自然振动频率和振动模态．相较于一阶摄动展开法,该方法在计算局部较大损伤问题中具有更高的准确度和敏感度．最后对损伤问题进行了参数研究,分析了不同的损伤因子（包括局部损伤程度、方向、面积大小）对板自由振动频率的影响．该方法为二维板局部损伤检测提供了有效精确的理论依据,并为损伤的定量评价提供了一种思路．     

Full-Field High-Strain Gradient Evaluation from Wrapped ESPI Data Using Phasors

Electronic Speckle Pattern Interferometry (ESPI) is a sensitive optical method commonly used for making full-field measurements of surface displacements. It would be very desirable to be able to extend the technique also to determine surface strains. This would provide a full-field, non-contact strain measurement method that avoids the substantial installation burden of strain gauges. A mathematical approach is described where the ESPI data from an in-plane interferometer are numerically differentiated to determine surface strains. This is a challenging process because numerical differentiation is very sensitive to the presence of noise and ESPI data are inherently noisy. In addition, the phase information from ESPI data are wrapped modulo-2π. The resulting phase discontinuities make it difficult to use local averaging to smooth the data. A technique is described here where phasors are used to avoid the need for phase unwrapping. The effect of noise is reduced by a localized multiple smoothing technique that is effective in preserving spatial resolution, even near very high strain concentrations. Example measurements are shown and the effectiveness of the proposed method is illustrated.     

In-plane modal testing of a free isotropic rectangular plate

In-plane vibration modes of an aluminum panel were experimentally identified from frequency response tests. Responses were measured on the panel edges and at selected locations on the panel surface. The measurements on the surface were made by attaching accelerometers oriented parallel to the panel plane. Resonance frequencies, relative damping ratios and mode shapes were established for the lowest 12 in-plane modes found in the frequency range between 1600 and 7000 Hz. A damping ratio of less than 0.05 percent of critical damping is proved to be valid for the aluminum panel. A finite element software was used to calculate 12 corresponding theoretical in-plane eigenfrequencies and mode shapes. An outline for a nondestructive procedure is suggested to estimate the input data for the elastic constants of an isotropic plate model. Two of the modes were used in analogy with the flexural vibration of beams and plates. The modes illustrate the deformation pattern including shear deformations, through the thickness, for the bending modes of thick beams or plates. The Rayleigh-Timoshenko theory also was used for the calculation of these two eigenfrequencies.     

二元复合材料板的自由振动: 半解析法

二元复合材料板是超材料板结构中常见的单元之一. 针对由材料参数相差两个量级的基体和嵌入体组成的二元复合材料板, 提出结构自由振动的半解析模型, 并对其振动特性进行了研究. 基于区域分解法和二元材料的分布, 将二维平板分解成两个子区域. 通过在振型函数中附加区域试函数, 来描述复合材料板面内刚度突变引起局部位移和转角的非光滑性. 基于二元复合材料板的基本边界条件和两子区连接处的变形协调条件, 构造了新的振型函数. 基于经典薄板理论, 利用带特殊试函数的里兹法, 求得不同几何构型下二元复合材料板的固有频率和振型, 并研究了嵌入体的尺寸和位置对结构振动特性的影响规律. 通过收敛分析并与有限元仿真结果对比, 验证了本文方法的准确性. 研究结果表明: 传统的全局试函数在分析具有振动局部化的模态时会得到不准确的结果, 而附加区域试函数可以显著提高里兹法的收敛速度以及结果的准确性; 嵌入体位置对低阶固有频率的作用不明显, 却能显著改变低阶振型节线的分布和振动局部化发生的区域.      

热环境中旋转运动功能梯度圆板的强非线性固有振动

研究热环境中旋转运动功能梯度圆板的非线性固有振动问题．针对金属－陶瓷功能梯度圆板,考虑几何非线性、材料物理属性参数随温度变化以及材料组分沿厚度方向按幂律分布的情况,应用哈密顿原理推得热环境中旋转运动功能梯度圆板的非线性振动微分方程．考虑周边夹支边界条件,利用伽辽金法得到了横向非线性固有振动方程,并确定了静载荷引起的静挠度．用改进的多尺度法求解强非线性方程,得出非线性固有频率表达式．通过算例,分析了旋转运动功能梯度圆板固有频率随转速、温度等参量的变化情况．结果表明,非线性固有频率随金属含量的增加而降低;随转速和圆板厚度的增大而升高;随功能梯度圆板表面温度的升高而降低．     

A mathematical approach to study fluid-coupled vibration of eccentric annular plates

A mathematical method is proposed to study fluid-coupled vibration of axisymmetric plate structures with asymmetries due to either imperfection or practical reasons, e.g. the weight reduction of structure, natural frequency shifting, and accessibility. The suggested approach makes use of the separation of variables to determine general solutions of the partial differential equation of the plate transverse displacement, whilst defining multiple polar coordinate systems, each of which offers its own formulation of the plate deformation with respect to its coordinate system. Moreover, closed-form geometric equations and the chain rule for determining derivatives are implemented to move from one coordinate system to the other in order to satisfy boundary conditions. The mode shapes of the vibrating plate in the dry condition are determined and in turn used in the Rayleigh–Ritz method to characterize vibrational properties of the fluid-coupled plate structure. While implementing such an energy method, the fluid motion is formulated employing the velocity potential and solved using the separation of variables. Fluid–structure interaction is also taken into account satisfying the compatibility condition on the fluid–plate​ interface. The developed methodology to predict natural frequencies has been validated by comparison with results obtained by a commercial finite element program. It is also found that the eccentricity tends to reduce natural frequencies of the fluid-coupled system for the lower serial mode, but increases them for the higher serial modes regardless of the presence of liquid.     

基于应变梯度中厚板单元的石墨烯振动研究

基于应变梯度理论建立了单层石墨烯等效明德林(Mindlin) 板动力学方程,推导了四边简支明德林中厚板自由振动固有频率的解析解. 提出了一种考虑应变梯度的4 节点36 自由度明德林板单元,利用虚功原理建立了单层石墨烯的等效非局部板有限元模型. 通过对石墨烯振动问题的研究,验证了应变梯度有限元计算结果的收敛性. 运用该有限元法研究了尺寸、振动模态阶数以及非局部参数对石墨烯振动特性的影响. 研究表明,这种单元能够较好地适用于研究考虑复杂边界条件石墨烯的尺度效应问题. 基于应变梯度理论的明德林板所获得石墨烯的固有频率小于基于经典明德林板理论得到的结果. 尺寸较小、模态阶数较高的石墨烯振动尺度效应更加明显. 无论采用应变梯度理论还是经典弹性本构关系,考虑一阶剪切变形的明德林板模型预测的固有频率低于基尔霍夫(Kirchho) 板所预测的固有频率.      

Elasticity solutions for free vibrations of annular plates from three-dimensional analysis

This article, examines the vibrational characteristics of annular plates by using the three-dimensional elasticity theory. It aims to raise the quality of the investigation beyond that provided by the two-dimensional plate theories by resorting to a full three-dimensional analysis. A polynomials–Ritz model based on sets of orthogonally generated polynomial functions to approximate the spatial displacements of the plates in cylindrical polar coordinates is presented. The model is then used to extract the full vibration spectrum of natural frequencies and mode shapes. The vibration responses due to the variations of boundary conditions and thickness are investigated. Frequency parameters and three-dimensional deformed mode shapes are presented in vivid graphical forms. The accuracy of the method is validated through appropriate convergence and comparison studies.     

Experimental and numerical analysis of vibrating cracked plates at resonant frequencies

Owing to the advantages of noncontact and fullfield measurement, an optical system called the amplitude fluctuation electronic speckle pattern interferometry (AFESPI) method with an out-of-plane setup is employed to investigate the vibration of a cantilever square plate with a crack emanating from one edge. Based on the fact that clear fringe patterns will be shown by the AFESPI method only at resonant frequencies, both the resonant frequencies and the vibration mode shapes can be obtained experimentally at the same time. Three different crack locations will be discussed in detail in this study. One is parallel to the clamped edge, and the other two are perpendicular to the clamped edge. The numerical finite element calculations are compared with the experimental results, and good agreement is obtained for resonant frequencies and mode shapes. The influences of crack locations and lengths on the vibration behavior of the clamped cantilever plate are studied in terms of the dimensionless frequency parameter (λ ²) versus crack length ratio (a/L). The authors find that if the crack face displacements are out of phase, a large value of stress intensity factor may be induced, and the cracked plate will be dangerous from the fracture mechanics point of view. However, there are some resonant frequencies for which the crack face displacements are completely in phase, causing a zero stress intensity factor, and the cracked plate will be safe.     

Determination of mode shapes using wavelet transform of free vibration data

In this article, a new method is proposed to determine the mode shapes of linear dynamic systems with proportional viscous damping excited by an impact force. The time signals of responses and a priori knowledge of the natural frequencies are required in this method. The method is particularly suitable for the wavelet techniques which can precisely estimate the natural frequencies. A previously proposed method based on a modified Morlet wavelet function with an adjusting parameter is used to identify the natural frequencies and damping ratios of system, and the mode shapes are estimated using the proposed method in this work. It is shown that the extracted mode shapes are not scaled. Therefore, mass change method is used for scaling the mode shapes. Moreover, the effect of noise on the extracted modal parameters is investigated. The validity of method is demonstrated using numerical and experimental case studies.     

Initial value method for free vibration of axially loaded functionally graded Timoshenko beams with nonuniform cross section

Free vibration of nonuniform axially functionally graded Timoshenko beams subjected to combined axially tensile or compressive loading is studied. An emphasis is placed on the effect of tip and distributed axial loads on the natural frequencies and mode shapes for an inhomogeneous cantilever beam including material inhomogeneity and geometric non-uniform cross section. The initial value method is developed to determine the natural frequencies. The method’s effectiveness is verified by comparing our results with previous ones for special cases. Natural frequencies of standing/hanging Timoshenko beams are calculated for four different cross sections. The influences of shear rigidity, taper ratio, gradient index, tip force, and axially distributed loading on the natural frequencies of clamped-free beams are discussed. Material inhomogeneity and geometric non-uniform cross-section strongly affect higher-order vibration frequencies and mode shapes.     

The application of interferometric techniques to the nondestructive inspection of fiber-reinforced materials

A method for increasing the sensitivity of dynamic materials evaluation (DME) to localized damage in fiber-reinforced composites was examined. To obtain this improved sensitivity, different aspects of DME were examined. These included an increase in the frequency used to evaluate the dynamic properties, utilization of mode-shape information and different procedures for evaluating the experimental data.The extent of the internal damage was determined using measured changes in the dynamic properties of the system (loss factor, dynamic stiffness and mode shape). To obtain the response information at higher frequencies a modalanalysis system was built around the performance characteristics of a laser doppler vibrometer (LDV) and an electronic speckle pattern interferometer (ESPI). These two devices provided complementary information for the determination of the dynamic characteristics of each vibration mode. With this system, damage-induced changes in the dynamic characteristics of composite materials were measured at frequencies up to 10 kHz.The results of this study showed the following. (1) Torsion modes provide the most sensitivity to localized internal damage. (2) The evaluation of higher frequency NDI data requires the ability to correlate the measured loss factor and resonant frequencies with the actual mode shape. (3) The data obtained over the frequency range of the test could be reduced to a series of slopes that provide a sensitive indication of the material condition. (4) The sensitivity of the dynamic method to localized damage is limited by the measurement of the loss factor.