压电功能梯度层合梁的力-电-热耦合梁单元及最优振动控制

给出了一个压电功能梯度层合梁振动分析的两节点力-电-热耦合梁单元,并将其用于功能梯度层合梁的振动最优控制。在这个多场耦合梁单元中,功能梯度材料的等效力学性能用Voigt或Mori-Tanaka模型表征;梁的位移场用Shi改进的三阶剪切变形板理论描述;压电层的电势场用Layer-wise理论分层表征,且呈高阶非线性电势场的压电层可离散成数个子层。用Hamilton原理推导了压电功能梯度梁的力-电-热耦合单元列式,用拟协调元法给出了多场耦合梁单元的高计算效率的显式单元刚度矩阵,以及采用线性二次型(LQR)最优控制算法进行压电功能梯度层合梁的最优振动控制。使用所得力-电-热耦合梁单元进行了压电功能梯度层合梁的静力和动力分析。数值算例表明,所得力-电-热耦合梁单元可靠、准确和高效,LQR最优控制算法得到最优控制电压可有效抑制功能梯度梁的振动且实现控制系统能量的优化。     

Evaluation of nonlocal higher order shear deformation models for the vibrational analysis of functionally graded nanostructures

Small scale effects in the functionally graded beam are investigated by using various nonlocal higher-order shear deformation beam theories. The material properties of a beam are supposed to vary according to power law distribution of the volume fraction of the constituents. The nonlocal equilibrium equations are obtained and an exact solution is presented for vibration analysis of functionally graded (FG) nanobeams. The accuracy of the present model is discussed by comparing the results with previous studies and a parametric investigation is presented to study the effects of power law index, small-scale parameter, and aspect ratio on the vibrational behavior of FG nanostructures.     

Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load

In this paper, free vibration characteristics and the dynamic behavior of a functionally graded simply-supported beam under a concentrated moving harmonic load are investigated. The system of equations of motion is derived by using Lagrange’s equations under the assumptions of the Euler–Bernoulli beam theory. Trial functions denoting the transverse and the axial deflections of the beam are expressed in polynomial forms. The constraint conditions of supports are taken into account by using Lagrange multipliers. It is assumed that material properties of the beam vary continuously in the thickness direction according to the exponential law and the power-law form. In this study, the effects of the different material distribution, velocity of the moving harmonic load, the excitation frequency on the dynamic responses of the beam are discussed. Numerical results show that the above-mentioned effects play very important role on the dynamic deflections of the beam.     

Non-linear free vibration of a functionally graded doubly-curved shallow shell of elliptical plan-form

The non-linear free vibration of a functionally graded doubly-curved shallow shell of elliptical plan-form is investigated using the p-version of the finite element method in conjunction with the blending function method. The effects of transverse shear deformations, rotary inertia, and geometrical non-linearity are taken into account. It is assumed that the material properties vary through the thickness according to a power law distribution. The harmonic balance method is used to derive the equations of free motion. The resultant non-linear equations are solved iteratively using the linearized updated mode method. The efficiency of the method is demonstrated through convergence study and comparison with published results. Three types of functionally graded doubly-curved shallow shells of elliptical plan-form are considered. The effects of the volume fraction exponent and thickness ratio on the linear and non-linear frequencies are discussed. It is shown that these parameters influence the hardening behaviour.     

湿-热-机-弹耦合FGM梁的稳定性及振动特性

研究了初始轴向机械载荷作用下Winkler-Pasternak弹性地基上功能梯度材料（FGM）梁在湿-热环境中的稳定性及振动特性。假设温度和湿度沿梁厚度方向稳态分布,材料的物性依赖于温度且按Voigt混合幂律模型连续分布。首先,基于一种扩展的n阶广义梁理论,应用Hamilton原理,统一建立了以轴向位移、弯曲变形项挠度及剪切变形项挠度为基本未知函数FGM梁的屈曲及自由振动方程,采用Navier解法获得了FGM简支梁静动态响应的精确解。其次,通过算例验证并给出了该广义梁理论阶次n的理想取值,丰富梁理论的同时,可供验证或改进其他各种剪切变形梁理论。最后,着重探讨了3种湿-热分布下湿度与温度增加、初始轴向机械载荷、跨厚比、地基刚度、梯度指标等诸多参数对FGM梁稳定性和振动特性的影响。     

非保守力和热载荷作用下FGM梁的稳定性

研究了在热载荷和切向均布随从力作用下FGM梁的稳定性问题。假设材料常数(即弹性模量和密度)随温度及沿截面高度连续变化,且材料常数按各材料的体积分数以幂率变化,温度分布满足一维热传导方程,计算了不同梯度指标和不同温度下FGM梁的弹性模量随截面高度变化情况。基于Euler-Bernoulli梁理论,建立梁的控制微分方程,用小波微分求积法(WDQ法)求解,分析了梯度指标、温度、随从力等参数对简支FGM梁振动特性与稳定性的影响。     

轴向运动功能梯度悬臂梁动力学分析

针对轴向运动悬臂梁振动会影响系统的安全性、稳定性问题,对功能梯度悬臂梁振动特性进行分析,利用广义哈密尔顿原理及假设模态法导出系统动力学方程。结果表明,功能梯度悬臂梁的横向位移与轴向位移耦合,功能梯度材料在厚度方向按体积分数函数呈指数变化,且梁自由端有集中质量块。并讨论材料指数及末端集中质量大小对振动影响,分析梁在伸展、收缩时的运动特性。所得结论可为类似结构的动力学分析、设计提供依据。     

Hierarchical theories for the free vibration analysis of functionally graded beams

This work addresses a free vibration analysis of functionally graded beams via several axiomatic refined theories. The material properties of the beam are assumed to vary continuously on the cross-section according to a power law distribution in terms of the volume fraction of the material constituents. Young’s modulus, Poisson’s ratio and density can vary along one or two dimensions all together or independently. The three-dimensional kinematic field is derived in a compact form as a generic N-order polynomial approximation. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the Principle of Virtual Displacements. They are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted. Higher-order displacements-based theories that account for non-classical effects are formulated. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. Bending, torsion and axial modes are investigated. Slender as well as short beams are considered. Numerical results highlight the effect of different material distributions on natural frequencies and mode shapes and the accuracy of the proposed models.     

Nonlinear forced vibration analysis of clamped functionally graded beams

Multiple time scale solutions are presented to study the nonlinear forced vibration of a beam made of symmetric functionally graded (FG) materials based on Euler?CBernoulli beam theory and von Kármán geometric nonlinearity. It is assumed that material properties follow either exponential or power law distributions through the thickness direction. A Galerkin procedure is used to obtain a second-order nonlinear ordinary equation with cubic nonlinear term. The natural frequencies are obtained for the nonlinear problem. The effects of material property distribution and end supports on the nonlinear dynamic behavior of FG beams are discussed. Also, forced vibrations of the system in primary and secondary resonances have been studied, and the effects of different parameters on the frequency-response have been investigated.     

Thermal buckling analysis of functionally graded material beams

Buckling of beams made of functionally graded material under various types of thermal loading is considered. The derivation of equations is based on the Euler–Bernoulli beam theory. It is assumed that the mechanical and thermal nonhomogeneous properties of beam vary smoothly by distribution of power law across the thickness of beam. Using the nonlinear strain–displacement relations, equilibrium equations and stability equations of beam are derived. The beam is assumed under three types of thermal loading, namely; uniform temperature rise, nonlinear, and linear temperature distribution through the thickness. Various types of boundary conditions are assumed for the beam with combination of roller, clamped and simply-supported edges. In each case of boundary conditions and loading, a closed form solution for the critical buckling temperature for the beam is presented. The formulations are compared using reduction of results for the functionally graded beams to those of isotropic homogeneous beams given in the literature.     

受初应力作用的轴向运动功能梯度梁的动力学分析

基于Euler梁模型研究了初始应力作用下轴向运动功能梯度材料梁的横向振动问题。假设材料性质沿梁的厚度方向按幂指数形式连续变化,利用Hamilton原理建立了系统的控制方程,应用复模态法求得了其固有频率和模态函数,接着分析了轴向运动速度、梯度指数、初应力大小等因素对梁的动力响应的影响。结果表明：梯度指数和轴向速度的增大都会导致固有频率降低,轴向初应力的增大则使得固有频率升高。     

Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length

The nonlinear free vibration of an axially translating viscoelastic beam with an arbitrarily varying length and axial velocity is investigated. Based on the linear viscoelastic differential constitutive law, the extended Hamilton’s principle is utilized to derive the generalized third-order equations of motion for the axially translating viscoelastic Bernoulli–Euler beam. The coupling effects between the axial motion and transverse vibration are assessed under various prescribed time-varying velocity fields. The inertia force arising from the longitudinal acceleration emerges, rendering the coupling terms between the axial beam acceleration and the beam flexure. Semi-analytical solutions for the governing PDE are obtained through the separation of variables and the assumed modes method. The modified Galerkin’s method and the fourth-order Runge–Kutta method are employed to numerically analyze the resulting equations. Further, dynamic stabilization is examined from the system energy standpoint for beam extension and retraction. Extensive numerical simulations are presented to illustrate the influences of varying translating velocities and viscoelastic parameters on the underlying dynamic responses. The material viscosity always dissipates energy and helps stabilize the transverse vibration.     

A study of the vibration of delaminated beams using a nonlinear anti-interpenetration constraint model

In this paper, the vibration of a beam with an embedded delamination is studied within the formalism of Timoshenko beam theory. As the free-body model results in dramatic interpenetration of the delaminated sublaminates that is physically impossible, a novel nonlinear constraint model is introduced to prevent the interpenetration. Unlike the previous model in the literature, the present constraint model automatically produces the zero or a proper contact traction following any given contact law without specifying whether the sublaminates are in contact or not before solving the problem. The resulted nonlinear partial differential equations are solved numerically. It is found that the predicted vibration of the beam with the constraint model is remarkably different from that without it. Moreover, the vibration mode of the beam depends upon the type of the contact function.     

A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities

This article proposes a four-variable shear deformation refined beam theory for thermo-mechanical vibration characteristics of porous, functionally graded (FG) beams exposed to various kinds of thermal loadings by using an analytical method. Thermo-mechanical properties of functionally graded material (FGM) beams are supposed to vary through the thickness direction, and are estimated through the modified power-law rule in which the porosities with even and uneven types are approximated. The material properties of FGM beams are supposed to be temperature dependent. Porosities possibly occur inside FGMs during fabrication because of technical problems that lead to the creation of microvoids in these materials. The variation of pores along the thickness direction influences the mechanical properties. Thus, it is incumbent to predict the effect of porosities on the thermo-mechanical vibration behavior of FG beam in the present study. Four types of thermal loading, namely, uniform, linear, nonlinear, and sinusoidal temperature rises through the z-axis direction are discussed. The governing differential equations and boundary conditions of FG porous beams subjected to thermal loadings are formulated through Hamilton's principle, based on a four-variable refined theory that considers a constant transverse displacement and higher order variation of axial displacement through the depth of the beam without the need of any shear correction factors. An analytical solution procedure is used to achieve the natural frequencies of porous FG beams subjected to various temperature fields. The impact of several specific parameters such as power-law exponent, porosity volume fraction, different porosity distribution, and thermal effect on the vibration of the porous FG beams is perused and discussed in detail. It is deduced that these parameters play a notable role on the thermo-dynamic behavior of porous FG beams. Presented numerical results can serve as benchmarks for the future analyses of FG beams with porosity phases.     

MRVE夹层梁随机振动的最优跳变参数控制

研究MRVE夹层梁随机振动的最优参数控制。建立夹层梁的运动微分方程,运用伽辽金法转化为含非线性参数控制项的振动方程;考虑控制参数的有界性,建立系统最优参数控制问题,应用随机动态规划原理与Bang-Bang策略确定HJB方程并得到最优有界非线性跳变参数控制律,最后通过数值结果说明该最优控制对于MRVE夹层梁随机振动能够达到显著控制效果。     

A Quasi-Exact Dynamic Finite Element for Free Vibration Analysis of Sandwich Beams

A Dynamic Finite Element (DFE) model for the vibration analysis of three-layered sandwich beams is presented. The governing differential equations of motion of the sandwich beam for the general case, when the properties of each layer are dissimilar, are exploited. Displacement fields are imposed such that the face layers follow the Rayleigh beam assumptions, while the core is governed by Timoshenko beam theory. The DFE model is then used to examine the free-vibration characteristics of an asymmetric soft-core sandwich beam with steel face layers and a rubber core. The natural frequency results for the first four modes, in this case, show the exact match between the DFE and ‘exact’ Dynamic Stiffness Matrix (DSM) formulations, using only a one-element mesh, justifying the use of Quasi-Exact (QE-DFE) title. Convergence-wise, the QE-DFE formulation also outperforms the conventional FEM, which makes it useful in benchmarking other studies or the examination of high frequency response where FEM requires the use of large number of elements in order to achieve better accuracy. The application of the DFE to a lead-core sandwich beam is also discussed.     

Analysis of a rotating elastic beam with piezoelectric films as an angular rate sensor

The flexural vibration of an elastic beam with surface-bonded piezoelectric films rotating about its axis is studied. One-dimensional equations governing the motion of the beam are developed, including the effects of Coriolis and centrifugal forces. The equations are used in the analysis of the flexural vibration of the beam under the excitation of an alternating electric voltage. The forced vibration solution is obtained. The beam can be used as a gyroscope for detecting the angular rate of the rotation. Voltage sensitivity and its dependence on various geometric and physical parameters are examined     

Nonlinear free vibration of non-prismatic single-walled carbon nanotubes by a non-local shear deformable beam p-element

The nonlinear free vibration of non-prismatic single-walled carbon nanotubes (SWNTs) is studied using a new non-local shear deformable beam p-element. The effects of the internal length scale parameter, transverse shear deformation, rotary inertia, and geometrical nonlinearity are taken into account. The principle of virtual displacements and the harmonic balance method are used to derive the nonlinear equations of motion, which are solved iteratively by the linearized updated mode method to obtain the fundamental nonlinear frequencies and mode shapes of H–H, C–H, and C–C SWNTs with uniform, linear, and quadratic radius variation. The convergence and accuracy of the non-local shear deformable beam p-element are demonstrated through comparison with other methods. It is shown that the non-uniformity parameters influence significantly the backbone curves and mode shapes of non-prismatic SWNTs.     

索-梁组合结构非线性时滞减振研究

采用时滞减振技术对索-梁组合结构进行了振动控制分析。通过Hamilton原理建立了索-梁组合结构的运动控制方程,引入时滞减振技术,应用多尺度摄动方法得到了主共振和1/3亚谐波共振的解的近似表达式。结果表明,时滞减振技术的两个主要参数时滞和控制增益能有效调节阻尼和频率。通过调节控制增益和时滞值,可增大阻尼比,避免共振域,从而对索-梁组合结构实现减振。     

热载荷作用下嵌入SMA丝复合材料梁的横向自由振动

基于形状记忆合金Brinson一维热力学本构方程,采用复合材料细观力学分析方法,建立了热载荷作用下嵌入SMA丝复合材料梁的一维热弹性本构关系。其次利用Euler-Bernoulli梁的轴线可伸长几何非线性理论和自由振动理论,建立了嵌入SMA丝复合材料梁在均匀升温场内自由振动的动力学控制方程,导出了热过屈曲构形附近嵌入SMA丝复合材料梁微幅横向自由振动的模型。最后通过打靶法求解了两端固定约束条件下嵌入形状记忆合金丝复合材料梁在加热过程中的振动响应,获得了梁的前四阶固有频率在不同SMA相对体积含量时随温度变化的特征关系曲线。数值结果表明,SMA丝相变过程中的回复应力和弹性模量变化对梁在过屈曲前后的各阶固有频率均有影响,是实现梁自振频率主动控制的一种有效方法。