Transfer matrix method for free and forced vibrations of multi-level functionally graded material stepped beams with different boundary conditions

Functionally graded materials(FGMs)are a novel class of composite materials that have attracted significant attention in the field of engineering due to their unique mechanical properties.This study aims to explore the dynamic behaviors of an FGM stepped beam with different boundary conditions based on an efficient solving method.Under the assumptions of the Euler-Bernoulli beam theory,the governing differential equations of an individual FGM beam are derived with Hamilton’s principle and decoupled via the separation-of-variable approach.Then,the free and forced vibrations of the FGM stepped beam are solved with the transfer matrix method(TMM).Two models,i.e.,a three-level FGM stepped beam and a five-level FGM stepped beam,are considered,and their natural frequencies and mode shapes are presented.To demonstrate the validity of the method in this paper,the simulation results by ABAQUS are also given.On this basis,the detailed parametric analyses on the frequencies and dynamic responses of the three-level FGM stepped beam are carried out.The results show the accuracy and efficiency of the TMM.

Transverse free vibration analysis of a tapered Timoshenko beam on visco-Pasternak foundations using the interpolating matrix method

The characteristics of transverse free vibration of a tapered Timoshenko beam under an axially conservative compression resting on visco-Pasternak foundations are investigated by the interpolating matrix method. The research is executed in view of a three-parameter foundation which includes the eff ects of the Winkler coeffi cient, Pasternak coeffi cient and damping coeffi cient of the elastic medium. The governing equations of free vibration of a non-prismatic Timoshenko beam under an axially conservative force resting on visco-Pasternak foundations are transformed into ordinary diff erential equations with variable coeffi cients in light of the bending rotation angle and transverse displacement. All the natural frequencies orders together with the corresponding mode shapes of the beam are calculated at the same time, and a good convergence and accuracy of the proposed method is verifi ed through two numerical examples. The infl uences of foundation mechanical characteristics together with rotary inertia and shear deformation on natural frequencies of the beam with diff erent taper ratios are analyzed. A comprehensive parametric numerical study is carried out emphasizing the primary parameters that describe the dynamic property of the beam.

Method of reverberation ray matrix for static analysis of planar framed structures composed of anisotropic Timoshenko beam members

Based on the method of reverberation ray matrix（MRRM）, a reverberation matrix for planar framed structures composed of anisotropic Timoshenko（T） beam members containing completely hinged joints is developed for static analysis of such structures.In the MRRM for dynamic analysis, amplitudes of arriving and departing waves for joints are chosen as unknown quantities. However, for the present case of static analysis, displacements and rotational angles at the ends of each beam member are directly considered as unknown quantities. The expressions for stiffness matrices for anisotropic beam members are developed. A corresponding reverberation matrix is derived analytically for exact and unified determination on the displacements and internal forces at both ends of each member and arbitrary cross sectional locations in the structure. Numerical examples are given and compared with the finite element method（FEM） results to validate the present model. The characteristic parameter analysis is performed to demonstrate accuracy of the present model with the T beam theory in contrast with errors in the usual model based on the Euler-Bernoulli（EB） beam theory. The resulting reverberation matrix can be used for exact calculation of anisotropic framed structures as well as for parameter analysis of geometrical and material properties of the framed structures.

CRACK IDENTIFICATION IN STEPPED CANTILEVER BEAM COMBINING WAVELET ANALYSIS WITH TRANSFORM MATRIX

This paper illustrates the crack identification method combining wavelet analysis with transform matrix. Firstly, the fundamental vibration mode was applied to wavelet analysis. The crack location was found by the peaks of the wavelet coefficients. Secondly, based on the identified crack locations, a simple transform matrix method requiring only the first two tested natural frequencies was used to further identify the crack depth. The present method can be used for crack identification in a complex structure. Numerical results of crack identification of a stepped cantilever beam show that the suggested method is feasible.

Compact Multibeam Array with Miniaturized Butler Matrix for 5G Applications

This paper presents the design and implementation of a miniaturized beam steering network that produces broadside beams when it is fed with a compact antenna array.Butler Matrix(BM)was used as the beam steering network.It was completely built from a miniaturized 3 dB hybrid-couplers in planar microstrip technology.It was configured by feeding the BM with a Planar Inverted-E Antenna(PIEA)array separated at 0.3λas against the 0.5λseparation.This makes the BM produce a major radiation pattern at the broadside.Apart from the miniaturization,no modification was made from the BM side.However,employing effective mutual coupling reduction techniques helped to design the compact PIEA array.The validity of this BM based multibeam PIEA array was demonstrated by comparing the simulation results of the reflection coefficients,transmissions coefficients and the radiation pattern with measurements.The measurement results showed good agreement with simulations.

BEAM BROADENING SYNTHESIS FOR SAR ANTENNA ARRAY BASED ON PROJECTION MATRIX ALGORITHM

A new beam broadening synthesis technique for Synthetic Aperture Radar(SAR) antenna array, namely Projection Matrix Algorithm(PMA) is presented. The theory of PMA is introduced firstly, and then the iterative renewed manner is improved to resolve the unbalance problem under amplitude and phase control. In order to validate the algorithm correct and effective, an actual engineering application example is investigated. The beam synthesis results of 1.0~4.5 times broadening under the phase only control and the amplitude and phase control using improved PMA are given. The results show that the beam directivity, the beam broadening, and the side-lobe level requirements were met. It is demonstrated that the improved PMA was effective and feasible for SAR application.

Matrix beam LED智能光通量输出系统研究

分析一种前沿的Matrix beam发光二极管(LED)智能光通量输出系统,侧重于对现阶段汽车车灯的Matrix beam发光二极管的研究,可自由设置各发光二极管的电流参数,因此达到差异化光通量输出的目的,进而匹配于发光二极管车灯造型及配光的要求。

用于夹层梁静动力及屈曲分析的新型组合结构单元

推导出新型组合结构单元,用于考虑界面滑移的3层部分作用夹层组合梁的静动力及屈曲特性分析.基于铁木辛柯梁理论,建立考虑夹层梁部分作用效应的能量原理.针对其受力特性,在节点位移、横截面转角和界面滑移插值时均采用含内部自由度的高阶插值函数,以解决含界面有限元数值分析中常遇到的“滑移锁定”现象.通过变分原理得到夹层梁的刚度矩阵、质量矩阵以及几何刚度矩阵的显示表达式.基于MATLAB编译相应夹层结构的有限元程序并验证其准确性.对不同截面3层夹层组合梁进行不同荷载条件和边界条件下的静动力及屈曲特性分析,并探讨不同夹层组合梁跨高比和不同界面抗剪连接刚度下的计算结果及其误差的变化规律.所推导的显示表达式新型组合结构单元计算效率高,并便于推广应用于其他有限元程序或商业软件子程序中.

振动梁离散模型的一类模态反问题

讨论了一类对系统的刚度主子阵存在一定约束的条件下,部分或缺损的模态信息构造振动梁模型刚度矩阵的模态反问题.由特征信息得到矩阵方程组,根据给定的频率数据是否属于系统,分三种情形求解.分别利用牛顿-康托洛维奇定理和线性方程组的基本理论,得到了模态反问题存在唯一解的充要条件,并给出了解的显式表达式和数值算法;证明了对应算法的收敛性和数值稳定性分析,结合工程实例,通过数值实验表明了理论的正确性和算法的有效性,并分析说明了计算过程是数值稳定的.

考虑剪切变形的欧拉梁单元一致质量矩阵

为明确考虑剪切变形的欧拉梁单元一致质量矩阵推导方法,以形函数为基础,基于虚功原理,区分伸缩、扭转以及是否考虑剪切变形的弯曲受力状态,给出了欧拉梁单元一致质量矩阵表达式。研究表明:不考虑剪切变形时,欧拉梁受弯状态的一致质量分析中一般不计单元上微元体水平方向的惯性力,此时仅需弯曲变形引发的竖向位移形函数即可;考虑剪切变形时,则需计入水平方向的惯性力,且同时需要完整的竖向位移形函数和纯弯曲转角位移形函数;对于变截面欧拉梁单元,其一致质量矩阵表达式过于复杂,可在等截面梁的基础上,根据元素位置对矩阵元素匹配左右端或平均截面面积及截面极惯性矩近似给出实用的质量矩阵表达式;考虑剪切变形时,欧拉梁的刚度矩阵亦可经竖向位移和纯弯曲转角位移形函数计算得到,这一过程与使用纯弯曲竖向位移和纯剪切竖向位移形函数的过程在本质上是一致的。

埃尔米特梁单元的分块对角与高阶质量矩阵

埃尔米特梁单元常用的集中质量矩阵,是由挠度自由度对应的一致质量矩阵元素通过行求和或节点积分构造.然而,数值结果表明该集中质量矩阵在求解包含自由端的梁振动问题时,会出现频率精度掉阶现象.本文首先从保障质量矩阵最优收敛性的数值积分精度出发,分别针对三次和五次梁单元,发展了质量矩阵的梯度增强节点积分方案.利用梯度增强节点积分方案,可以得到具有分块对角形式的单元质量矩阵,而其组装的整体质量矩阵除边界节点外仍然呈现对角形式.对于两种单元,其分块对角质量矩阵分别具有4阶最优精度和6阶次优精度.再者,将标准一致质量矩阵和具有同阶精度的梯度增强节点积分质量矩阵进行优化组合,建立了具有超收敛特性的高阶质量矩阵.最后,通过数值算例系统验证了三次和五次单元的分块对角与高阶质量矩阵的频率计算精度.

FREE VIBRATION ANALYSIS AND PHYSICAL PARAMETER IDENTIFICATION OF NON-UNIFORM BEAM CARRYING SPRING-MASS SYSTEMS

To analyze a multibody system composed of non-uniform beam and spring-mass subsystems, the model discretization is carried on by utilizing the finite element method（FEM）, the dynamic model of non-uniform beam is developed by using the transfer matrix method of multibody system（MS-TMM）, the transfer matrix of non-u- niform beam is derived, and the natural frequencies are computed. Compared with the numerical assembly method （NAM）, the results by MS-TMM have good agreement with the results by FEM, and are better than the results by NAM. When using the high precision method, the global dynamic equations of the complex multibody system are not needed and the orders of involved system matrices are decreased greatly. For the investigation on the re- verse problem of the physical parameter identification of multibody system, MS-TMM and the optimization tech- nology based on genetic algorithms（GAs） are combined and extended. The identification problem is exchanged for an optimization problem, and it is formulated as a global minimum solution of the objective function with respect to natural frequencies of multibody system. At last, the numerical example of non-uniform beam with attach- ments is discussed, and the identification results indicate the feasibility and the effectivity of the proposed aop- proach.

New tangent stiffness matrix for geometrically nonlinear analysis of space frames

A three-dimensional beam element is derived based on the principle of stationary total potential energy for geometrically nonlinear analysis of space frames. A new tangent stiffness matrix, which allows for high order effects of element deformations, replaces the conventional incremental secant stiffness matrix. Two deformation stiffness matrices due to the variation of axial force and bending moments are included in the tangent stiffness. They are functions of element deformations and incorporate the coupling among axial, lateral and torsional deformations. A correction matrix is added to the tangent stiffness matrix to make displacement derivatives equivalent to the commutative rotational degrees of freedom. Numerical examples show that the proposed dement is accurate and efficient in predicting the nonlinear behavior, such as axial-torsional and flexural-torsional buckling, of space frames even when fewer elements are used to model a member.

分数阶粘弹性地基上裂纹梁的自由振动

本文研究了分数阶粘弹性Pasternak地基上含多裂纹Euler梁的自由振动问题.首先,引入分数阶导数概念,建立粘弹性Pasternak地基模型,推导出地基应力-应变本构的复模量以及地基反力.其次,将裂纹等效成无质量扭转弹簧,借助于裂纹能量释放率和应力强度因子的关系,推导出梁的局部柔度.然后,基于传递矩阵方法,建立含多裂纹梁的分段传递矩阵进而得到总体传递矩阵.最后,基于梁的边界条件,建立求解裂纹梁的复固有频率及振型的线性代数方程组,数值求解得到含多裂纹梁的固有频率及振型.以含双裂纹的两端简支Euler梁为例,数值计算了复固有频率和振型.并基于曲率模态分析裂纹位置对复固有频率和振型的影响.数值结果揭示了粘弹性地基的分数阶系数、粘性系数以及裂纹位置和裂纹深度对复固频率和振型的影响规律.

基于BPMS的砂浆和混凝土细观模型研究

混凝土是由砂浆和骨料构成的非均质材料。在细观层面上评估混凝土的断裂过程有助于阐明混凝土的材料特性。采用C++编制的梁-粒子模型求解器(Beam Particle Model Solver,简称BPMS)对砂浆和混凝土进行二维数值分析。为了模拟混凝土的断裂,采用点阵法高效快速地建立细观尺度的颗粒模型。数值研究较好地呈现了砂浆和混凝土在单轴压缩条件下的破坏行为以及混凝土三点弯在冲击条件下的断裂过程,从而证实了BPMS的强健性及稳定性。

基于传递矩阵法的Timoshenko裂纹梁自振特性分析

首先基于传递矩阵法,将裂纹截面假定一无质量扭转弹簧,通过扭转弹簧建立起各段子梁在裂纹截面处的矩阵传递关系,推导出含任意裂纹Timoshenko梁及无裂纹Timoshenko梁的传递矩阵,引入边界条件简化矩阵方程,并利用Matlab对方程进行求解。其次讨论了裂纹位置和相对深度对单裂纹简支梁自振频率的影响,与文献结果进行对比,误差最多不超过1%;然后通过ABAQUS建立悬臂梁及两端固支梁有限元模型,分析相对裂纹深度对自振频率的影响,计算结果与有限元结果进行对比,误差不超过3.86%;最后研究了不同跨高比下相对裂纹深度对Timoshenko简支梁自振频率的影响,计算结果与文献最大误差为4.56%,验证了本文方法的有效性及适用性。

基于幅相解耦调制的透射型声超表面的声波调控

文章构造了一种复合型无源结构,利用该结构设计的声学超表面能够实现透射相位和幅度的解耦调控。研究表明,当声波入射至该复合结构后,多腔体结构具有接近于1的透射率和接近于2π的全相位变化,而开孔结构单元可以提供恒定不变的相位和0~1范围内变化的透射率。此外,根据Airy函数的波曲线,选择合适的结构参数,构造出的超表面可以在空气中生成Airy波束,同时利用两个对称的超表面可以实现具有自愈性声聚焦特性的透镜,通过与纯相位调控方法生成的波束现象进行对比,进一步说明幅相解耦方法对于声波精确调控的重要性。文章对于幅相解耦超表面的研究在声波定向传输、全息图像、医学超声成像等方面具有潜在的应用前景。

基于光Blass矩阵的光控二维多波束产生

多波束是电子信息装备实现多功能、多任务、大容量信息传输与处理的关键。围绕新一代相控阵系统对宽带多波束形成网络的应用需求,提出一种基于光Blass矩阵的两级级联结构产生二维多波束的实现方法。通过光域相干波束形成技术完成第一级水平方向的波束合成,再利用光域非相干波束形成技术实现第二级垂直方向的波束合成,进而获得空域二维波束。采用该架构,一方面可突破光波长数量不足对光控波束形成网络阵列规模的限制,可将光控相控阵系统的单元规模从数十个提升至数百个甚至更多;另一方面可大幅减少光控多波束芯片的数量,对于阵元数N×N、波束数量M×M的多波束系统,波束芯片数量可从M^(2)个减少到(N+M)个。同时该方案可采用光子集成技术手段实现,在简化多波束系统结构与提升系统可靠性方面具有无可比拟的优势。

多裂缝形式下沥青混合料梁裂缝扩展研究

采用接近实际路面受力形式的平面支撑方式,对从参考标准高速公路搅拌、摊铺、碾压方式形成的沥青混凝土路面切割下的沥青混合料梁试件,开展表面、底部裂缝共存形式下路面面层裂缝扩展模拟,以期研究沥青路面面层顶部、底部裂缝共存形式下的裂缝扩展行为;针对沥青混凝土路面为包含沥青基质、骨料、空隙等的多相复合材料特点,基于ABAQUS平台建立沥青混合料梁细观异质模型,探讨了该模型下裂缝偏离跨中不同距离下裂缝扩展行为。在此基础上,进一步对复合小梁均质、细观异质模型偏裂缝形式下裂缝扩展过程中应力分布进行了对比分析,得出两者应力状态存在明显的不同,均质模型应力分布总体较均匀、突变不明显;细观异质模型则刚好相反,应力分布极不均匀,骨料周边应力突变较明显。两种模型对比而言,细观异质模型能够更好地解释试验过程中梁裂缝扩展沿着骨料周边进行,并且试验过程中骨料周边出现多裂纹的现象。梁细观异质模型数值模拟为建立更准确的路面裂缝扩展模型提供了参考。

局域共振夹芯超结构梁带隙特性及实验研究

本文提出了一种新型局域共振夹芯超结构梁,该结构由夹芯梁基体和由软材料包裹质量块组成的局域共振单元构成。分别采用传递矩阵法和有限元法计算了该结构的带隙分布和振动传输特性,搭建了局域共振夹芯超结构梁的振动传输测试实验平台,并与数值计算结果进行对比验证,最后分析了结构参数对结构带隙分布特性的调控规律。研究结果表明,该结构在低频范围内共出现两条弯曲波带隙,带隙频率分别为196.1~339.0 Hz和377.2~655.6 Hz。而完全带隙位于弯曲波带隙内,频率范围为380.0~423.6 Hz。且在这两条弯曲波带隙范围内,弯曲振动传输会发生明显的减弱现象。另外,局域共振夹芯超结构梁的结构参数会对带隙分布特性产生较大影响,通过合理选择结构参数,可以实现局域共振夹芯超结构梁低频率和高带宽的减振要求。