Design of shaped piezoelectric modal sensor for beam with arbitrary boundary conditions by using Adomian decomposition method

The theory for designing distributed piezoelectric modal sensors is well established for beam structures. However, the current modal sensor theory is limited in scope in that it can only be applied in the case of classical boundary conditions (i.e., either clamped, free, simply supported or sliding). In this paper a solution to the problem of finding the shape of piezoelectric modal sensors for a beam with arbitrary boundary conditions is proposed, using the Adomian decomposition method (ADM). A general expression for designing the shape of a piezoelectric modal sensor is presented, in which the output signal of the designed sensor is proportional to the response of the target mode. Other modes are filtered out. The modal sensor shape is expressed as a function of the second spatial derivative of the structural mode shape function. Based on the ADM and employing some simple mathematical operations, the closed-form series solution of the second spatial derivative of the mode shapes can be determined. Then the shapes of the designed modal sensors are obtained. Finally, some numerical examples are given to demonstrate the feasibility of the proposed modal sensors. It is shown that, for classical boundary conditions, the shapes of the modal sensors based on the ADM agree well with analytical and numerical results given in the literature. For general boundary conditions it is found that the shape of the modal sensors is influenced by the number of modes of interest because the second spatial derivatives of the mode shapes are not orthogonal to one another. The modal sensors for general boundary conditions can be considered as modal filters within a limited frequency band.     

Vibration response of stepped FGM beams with elastically end constraints using differential transformation method

In this paper, free vibration response of stepped beams made from functionally graded materials (FGMs) is investigated. The beams are supported by various types of elastically end constraints. The differential transformation method (DTM) is employed to solve the governing differential equations of such beams in order to obtain natural frequencies and mode shapes. The power law distribution is used and modified to describe material compositions across the thickness of the beams made of FGMs. Two main types of the stepped FGM beams in which their material compositions can be described by using the modified power law distribution are selected to investigate their vibration behaviour. The significant parametric studies such as step ratio, step location, boundary conditions, spring constants and material volume fraction are taken into investigation.     

Free vibration analysis of moderately thick asymmetric piezoelectric adaptive cantilever beams using the distributed transfer function approach

A semi-analytical distributed transfer function (DTF) approach is proposed for the free-vibration analysis of moderately thick cantilever beams with a single surface-bonded piezoelectric patch. The asymmetric piezoelectric adaptive structure is decomposed into three segments; the first and third segments are bare beam parts before and after the patch, while the second segment contains the beam part with attached piezoelectric patch bonded to its upper surface. The theoretical formulation assumes first-order shear deformation kinematics and linear electric potential through the patch thickness with an electrode equipotential physical condition, and uses the extended Hamilton?s principle to derive the equations of motion and electromechanical boundary conditions. The latter, together with the continuity and equilibrium conditions at the segments interfaces, are then transformed into a first-order state space equation that is solved using the DTF approach. The electrodes of the piezoelectric patch are considered either in short-circuit (SC) or open-circuit (OC); this leads to two free-vibration problems to be solved for the corresponding SC and OC frequencies, from which the Electro-Mechanical Coupling Coefficient (EMCC) is post-treated. Four benchmarks from the open literature are simulated in order to validate the proposed approach. Very satisfactory correlations are obtained for all examples with maximum errors less thank 5 percent in all results. For future reference, an additional benchmark is proposed to assess the influence of the patch-to-composite host width ratio on the effective modal EMCC. It was found that the latter is mode-dependent (as expected) and decreases with increasing the former.     

Design of passive vibration controls for internally damped beams by modal control techniques

Modern modal control techniques are applied to the design of dynamic absorbers for the control of transient vibrations of internally damped thin uniform beams. Parameters and locations of single and dual absorbers are chosen so that the overall system approximates a desired characteristic polynomial.     

Coupled bending-bending vibrations of pretwisted tapered cantilever beams treated by the Reissner method

Theoretical natural frequencies and mode shapes of the first four coupled modes of a uniform pretwisted cantilever blade and the first five coupled flexural frequencies of pretwisted tapered blading are determined by using the Reissner method. The shape functions for the bending moments and deflections are developed in series form and with these used in the dynamic Reissner functional, the frequency equation is obtained by minimizing it through the Ritz process. A convergence study made in the case of the pretwisted uniform blade indicates that there appears to be a quicker convergence of the natural frequencies and that a five-term solution yields a set of results that are in good agreement with the theoretical and experimental values of other authors, available in the literature. The mode shapes obtained from the present analysis are compared with those from an earlier investigation and the effect of ignoring the shear deflection and rotary inertia in the analysis is discussed. The effects of breadth taper and depth taper on the vibration characteristics of pretwisted cantilever blading are discussed from the results obtained in the present limited study and it is observed that an extensive investigation appears to be necessary to draw positive conclusions covering wide ranges of pretwisted blade parameters.     

Transverse vibrations of beams with variable moment of inertia and an intermediate support

The above problem is tackled using the Ritz method and approximating the deflection function by means of polynomial co-ordinate functions which satisfy the essential boundary conditions. The presence of an axial force and concentrated masses is considered. Fundamental frequency coefficients are determined in the case of a rather complex structure taking into account different types of boundary conditions.     

Solution of singular two-point boundary value problems using differential transformation method

In this Letter, we implemented relatively new, exact series method of solution known as the differential transform method for solving singular two-point boundary value problems. Several illustrative examples are given to demonstrate the effectiveness of the present method.     

A new method for accurately determining the modal equivalent stiffness ratio of bonded piezoelectric structures

This paper describes new methods for measuring the modal equivalent stiffness ratios and modal electromechanical coupling coefficients of piezoelectric elements attached to a host structure such as a beam. Modal equivalent stiffness ratios and modal electromechanical coupling coefficients are essential for estimating the performance and determining an optimum design of active vibration control and passive vibration suppression systems that use piezoelectric elements. Accurate determination of these modal parameters is also useful for other systems including piezoelectric sensors and energy generators. This paper not only describes the measurement methods but also presents the theoretical formulations derived by taking into account the effect of adhesive bonds. The formulations in this paper demonstrate the necessity of experimental measurements and the accuracy enhancements that the theoretical estimations can provide. Conventional methods for obtaining the modal equivalent stiffness ratios are sensitive to measurement errors, which result in the loss of accuracy, rendering these methods unreliable for many practical applications. The proposed methods use an inductor instead of an open circuit to address the abovementioned issue and, thereby, provide significant improvement in the accuracy. Because the loss factors of the experimental apparatus tend to compromise the accuracy of the proposed methods, a method using a negative resistor is proposed, theoretically analyzed, and confirmed to eliminate some of the errors introduced by loss factors. The advantages of the proposed methods and the effectiveness of theoretical analysis, considering the effect of adhesive bonds, are verified experimentally.     

基于傅里叶模方法的KDP晶体镀膜减反技术

采用傅里叶模方法,分析了单点金刚石铣削后KDP晶体表面小尺度波纹的周期和幅值对单层增透膜折射率、厚度以及透射率的影响。研究表明：膜层最佳折射率在1.22左右,在此折射率条件下,保证透射率大于99%的单层增透膜的理想厚度范围应为180~220 nm,并且折射率和膜厚值的选取基本不受晶体表面小尺度波纹周期和幅值的影响。若只考虑SPDT法加工后KDP晶体表面小尺度波纹周期和幅值的实际范围,透射率基本不受波纹周期的影响,但却会随波纹幅值的增大而加速下降。理想镀膜条件下透射率最大值大于99%,并且通常在99.67%~99.94%之间。     

基于傅里叶模方法的KDP晶体镀膜减反技术

 采用傅里叶模方法,分析了单点金刚石铣削后KDP晶体表面小尺度波纹的周期和幅值对单层增透膜折射率、厚度以及透射率的影响。研究表明：膜层最佳折射率在1.22左右,在此折射率条件下,保证透射率大于99%的单层增透膜的理想厚度范围应为180~220 nm,并且折射率和膜厚值的选取基本不受晶体表面小尺度波纹周期和幅值的影响。若只考虑SPDT法加工后KDP晶体表面小尺度波纹周期和幅值的实际范围,透射率基本不受波纹周期的影响,但却会随波纹幅值的增大而加速下降。理想镀膜条件下透射率最大值大于99%,并且通常在99.67%~99.94%之间。     

Design of piezoelectric actuators using a multiphase level set method of piecewise constants

This paper presents a multiphase level set method of piecewise constants for shape and topology optimization of multi-material piezoelectric actuators with in-plane motion. First, an indicator function which takes level sets of piecewise constants is used to implicitly represent structural boundaries of the multiple phases in the design domain. Compared with standard level set methods using n scalar functions to represent 2ⁿ phases, each constant value in the present method denotes one material phase and 2ⁿ phases can be represented by 2ⁿ pre-defined constants. Thus, only one indicator function including different constant values is required to identify all structural boundaries between different material phases by making use of its discontinuities. In the context of designing smart actuators with in-plane motions, the optimization problem is defined mathematically as the minimization of a smooth energy functional under some specified constraints. Thus, the design optimization of the smart actuator is transferred into a numerical process by which the constant values of the indicator function are updated via a semi-implicit scheme with additive operator splitting (AOS) algorithm. In such a way, the different material phases are distributed simultaneously in the design domain until both the passive compliant host structure and embedded piezoelectric actuators are optimized. The compliant structure serves as a mechanical amplifier to enlarge the small strain stroke generated by piezoelectric actuators. The major advantage of the present method is to remove numerical difficulties associated with the solution of the Hamilton–Jacobi equations in most conventional level set methods, such as the CFL condition, the regularization procedure to retain a signed distance level set function and the non-differentiability related to the Heaviside and the Delta functions. Two widely studied examples are chosen to demonstrate the effectiveness of the present method.     

Condition assessment of reinforced concrete beams using dynamic data measured with distributed long-gage macro-strain sensors

A new condition assessment strategy of reinforced concrete (RC) beams is proposed in this paper. This strategy is based on frequency analysis of the dynamic data measured with distributed long-gage macro-stain sensors. After extracting modal macro-strain, the reference-based damage index is theoretically deducted in which the variations of modal flexural rigidity and modal neutral axis height are considered. The reference-free damage index is also presented for comparison. Both finite element simulation and experiment investigations were carried out to verify the proposed method. The manufacturing procedure of long-gage fiber Bragg grating (FBG) sensor chosen in the experiment is firstly presented, followed by an experimental study on the essential sensing properties of the long-gage macro-strain sensors and the results verify the excellent sensing properties, in particular the measurement accuracy and dynamic measuring capacity. Modal analysis results of a concrete beam show that the damage appearing in the beam can be well identified by the damage index while the vibration testing results of a RC beam show that the proposed method can not only capture small crack initiation but its propagation. It can be concluded that distributed long-gage dynamic macro-strain sensing technique has great potential for the condition assessment of RC structures subjected to dynamic loading.     

外加磁场压电悬臂梁能量采集系统的磁化电流法磁力研究

以外加磁场压电悬臂梁能量采集系统结构为研究对象, 根据磁化电流方法探讨了具有悬臂梁特征的系统结构的磁场作用力及其计算方法, 给出了相应的磁力计算模型, 并将计算结果与实验数据进行了对比. 研究表明, 磁化电流方法导出的磁力计算模型存在偏差, 其磁力计算误差随着磁铁间距缩小而增大. 通过引入悬臂梁末端磁铁的偏转角度, 对磁化电流法计算模型进行改进, 得到合理的外加磁场压电悬臂梁能量采集系统的磁力计算模型, 为该能量采集系统的进一步研究提供了可靠的磁力计算理论依据.     

Flapwise bending vibration analysis of a rotating tapered cantilever Bernoulli–Euler beam by differential transform method

This paper studies the vibration characteristics of a rotating tapered cantilever Bernoulli–Euler beam with linearly varying rectangular cross-section of area proportional to xⁿ, where n equals to 1 or 2 covers the most practical cases. In this work, the differential transform method (DTM) is used to find the nondimensional natural frequencies of the tapered beam. Numerical results are tabulated for different taper ratios, nondimensional angular velocities and nondimensional hub radius. The effects of the taper ratio, nondimensional angular velocity and nondimensional hub radius are discussed. The accuracy is assured from the convergence of the natural frequencies and from the comparisons made with the studies in the open literature. It is shown that the natural frequencies of a rotating tapered cantilever Bernoulli–Euler beam can be obtained with high accuracy by using DTM.     

Vibration analysis of a circular arch with variable cross-section using differential transformation and generalized differential quadrature

Vibration analysis of circular arches is an important subject in mechanics due to its various applications. In particular, circular arches with variable cross-section have been widely used to satisfy modern architectural and structural requirements. Recently, the generalized differential quadrature method (GDQM) and differential transformation method (DTM) were proposed by Shu and Zhou, respectively. In this study, GDQM and DTM are applied to vibration analysis of circular arches with variable cross-section. The governing equation of motion is derived and the non-dimensional natural frequencies are obtained for various boundary conditions. The concepts of differential transformation and generalized differential quadrature are briefly introduced. The results obtained by these methods are compared with previously published works. GDQM and DTM showed fast convergence, accuracy and validity in solving the vibration problem for circular arches with variable cross-sections.     

Active suppression of panel flutter with piezoelectric actuators using eigenvector orientation method

This paper examines the use of eigenvector orientation method to detect the onset of subsonic and supersonic flutter of panels modeled by finite elements. The accuracy of the eigenvector orientation method for prediction of the flutter boundary (indicated by a gradual loss of orthogonality between two eigenvectors) is demonstrated by using the examples of a swept-back cantilever plate model at subsonic speed and a simply supported plate model at supersonic speed. Piezoelectric layers are assumed to be bonded to the top and bottom surfaces of the simply supported plate in order to provide bending moments to control motions of each finite element. An approach of optimal control design is presented to actively suppress the possible flutter based on linear quadratic regulator theory and the nonlinear modal equations of motions. To illustrate the applicability and effectiveness of using the piezoelectric layers as controllers, several cases are studied and presented. The effects of varying locations of control moments are studied so as to fulfill the objective of adjusting the flutter speed to be within a desirable range. The results illustrate that the control moment manipulation can offset the flutter occurrence and additionally generate a lead time for possibly executing flutter control.     

Design of square-shaped heat flux cloaks and concentrators using method of coordinate transformation

A square-shaped heat flux cloak and a square-shaped heat flux concentrator have been designed theoretically according to the invariance symmetry of steady state thermal conductive equation. The direction of heat flux in these devices can be modulated as desired. Using the method of coordinate transformation, the inhomogeneous and anisotropic thermal conductivity in the transformation region have been acquired. Two-dimensional finite element simulations were performed to confirm the theoretical results.     

使用离焦望远镜系统合成轴上平顶光束的一种新方法

提出了用平顶多高斯光束作为理想光束和使用离焦望远镜系统合成轴上平顶光束的一种新方法.在近轴近似下,基于柯林斯衍射积分公式推导出轴上光场和入射光场的解析表达式,并做了物理分析.光束阶数N,平顶长度参数L,平顶中心位置z_c和纵向中心空间频率S_c对轴上光强分布的影响用数值计算例做了说明.比较表明,该方法优于文献中使用矩形函数模拟理想光束合成轴上平顶光束的方法. 关键词： 轴上平顶光束 平顶多高斯光束 离焦望远镜系统 傅里叶变换