Vibration suppression of FGM shells using embedded magnetostrictive layers

Analytical solutions of functionally graded material (FGM) shells with embedded magnetostrictive layers are presented in this study. These magnetostrictive layers are used for the vibration suppression of the functionally graded shells. The first-order shear deformation shell theory (FSDT) is employed to study the vibration suppression characteristics. The exact solution for the FGM shell with simply supported boundary conditions is based on the Navier solution procedure. Negative velocity feedback control is used. The parametric effect of the location of the magnetostrictive layers, material properties, and control parameters on the suppression effect are investigated in detail. It is found that (i) the shortest vibration suppression time is achieved by placing the actuating layers farthest from the neutral axis, (ii) the use of thinner smart material layers leads to better vibration attenuation characteristics, and, (iii) the vibration suppression time is longer for a smaller value of the feedback control coefficient.     

Stabilization of beam parametric vibrations with shear deformations and rotary inertia effects

The purpose of this theoretical work is to present a stabilization problem of beam with shear deformations and rotary inertia effects. A velocity feedback and particular polarization profiles of piezoelectric sensors and actuators are introduced. The structure is described by partial differential equations with time-dependent coefficient including transverse and rotary inertia terms, general deformation state with interlaminar shear strains. The first order deformation theory is utilized to investigate beam vibrations. The beam motion is described by the transverse displacement and the slope. The almost sure stochastic stability criteria of the beam equilibrium are derived using the Liapunov direct method. If the axial force is described by the stationary and continuous with probability one process the classic differentiation rule can be applied to calculate the time-derivative of functional. The particular problem of beam stabilization due to the Gaussian and harmonic forces is analyzed in details. The influence of the shear deformations, rotary inertia effects and the gain factors on dynamic stability regions is shown.     

Nonlinear vibration of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations in thermal environments

Modeling and nonlinear vibration analysis of graphene-reinforced composite (GRC) laminated beams resting on elastic foundations in thermal environments are presented. The graphene reinforcements are assumed to be aligned and are distributed either uniformly or functionally graded of piece-wise type along the thickness of the beam. The motion equations of the beams are based on a higher-order shear deformation beam theory and von Kármán strain displacement relationships. The beam–foundation interaction and thermal effects are also included. The temperature-dependent material properties of GRCs are estimated through a micromechanical model. A two-step perturbation approach is employed to determine the nonlinear-to-linear frequency ratios of GRC laminated beams. Detailed parametric studies are carried out to investigate the effects of material property gradient, temperature variation, stacking sequence as well as the foundation stiffness on the linear and nonlinear vibration characteristics of the GRC laminated beams.     

Vibration analysis of a stepped laminated composite Timoshenko beam

The goal of this study is to investigate the vibration characteristics of a stepped laminated composite Timoshenko beam. Based on the first order shear deformation theory, flexural rigidity and transverse shearing rigidity of a laminated beam are determined. In order to account for the effect of shear deformation and rotary inertia of the stepped beam, Timoshenko beam theory is then used to deduce the frequency function. Graphs of the natural frequencies and mode shapes of a T300/970 laminated stepped beam are given, in order to illustrate the influence of step location parameter exerts on the dynamic behavior of the beam.     

Nonlinear low-velocity impact on damped and matrix-cracked hybrid laminated beams containing carbon nanotube reinforced composite layers

This paper investigates the low-velocity impact response of a shear deformable laminated beam which contains both carbon nanotube reinforced composite (CNTRC) layers and carbon fiber reinforced composite (CFRC) layers. The effect of matrix cracks is considered, and a refined self-consistent model is selected to describe the degraded stiffness caused by the damage. The beam including damping effects rests on a two-parameter elastic foundation in thermal environments. Based on a higher-order shear deformation theory and von Kármán nonlinear strain–displacement relationships, the motion equations of the beam and impactor are established and solved by means of a two-step perturbation approach. The material properties of both CFRC layers and CNTRC layers are assumed to be temperature-dependent. To assess engineering application of this hybrid structure, two conditions for outer CNTRC layers and outer CFRC layers are compared. Besides, the effects of the crack density, volume fraction of carbon nanotube, temperature variation, the foundation stiffness and damping on the nonlinear low-velocity impact behavior of hybrid laminated beams are also discussed in detail.

     

Non-linear response of laminated composite plates under thermomechanical loading

The non-linear response of laminated composite plates under thermomechanical loading is studied using the third-order shear deformation theory (TSDT) that includes classical and first-order shear deformation theories (CLPT and FSDT) as special cases. Geometric non-linearity in the von Kármán sense is considered. The temperature field is assumed to be uniform in the plate. Layers of magnetostrictive material, Terfenol-D, are used to actively control the center deflection. The negative velocity feedback control is used with the constant gain value. The effects of lamination scheme, magnitude of loading, layer material properties, and boundary conditions are studied under thermomechanical loading.     

Analytical modeling of sandwich beam for piezoelectric bender elements

Piezoelectric bender elements are widely used as electromechanical sensors and actuators.An analytical sandwich beam model for piezoelectric bender elements was developed based on the first-order shear deformation theory(FSDT),which assumes a single rotation angle for the whole cross-section and a quadratic distribution function for coupled electric potential in piezoelectric layers,and corrects the effect of transverse shear strain on the electric displacement integration.Free vibration analysis of simply- supported bender elements was carried out and the numerical results showed that,solu- tions of the present model for various thickness-to-length ratios are compared well with the exact two-dimensional solutions,which presents an efficient and accurate model for analyzing dynamic electromechanical responses of bender elements.     

复合材料叠层梁和金属梁的固有振动特性

对根据三种梁理论得到的金属梁和复合材料叠层梁的固有振动特性进行了对比性的研究对常用的三种梁理论在弹性碰撞分析中的应用进行了分析和比较     

Transient responses of magnetostrictive plates by using the GDQ method

We used the generalized differential quadrature (GDQ) method to compute the transient response of thermal stresses and center displacement in laminated magnetostrictive plates under thermal vibration. We obtained the GDQ solutions in a three-layer (0°^m/90°/0) and a 10-layer (0°^m/90°/0°/90°/0)_s laminated magnetostrictive plate with four simply supported edges. We presented the transient responses of thermal stress and center displacement with and without velocity feedback control, respectively. The advantage of the GDQ method used provide us with an efficient method to compute the results including shear deformation effect with a few grid points. These GDQ results had its potential that could be used and considered as basic data in the future magnetostrictive laminate studies.     

深梁理论的研究现状与工程应用

综述了深梁理论、截面剪切修正系数计算理论、深梁线性与几何非线性有限元、深梁材料非线性分析、深梁振动理论、深梁稳定理论、箱梁结构分析中弯曲、剪力滞、畸变分析时考虑剪切变形影响的计算理论、钢腹板桥梁考虑剪切变形的研究成果、弹性地基深梁、深梁理论在工程结构中的应用等. 提出了杆系结构的静力、振动和稳定分析方法都可用Timoshenko 深梁理论进行重建和重写.     

Nonlinear active control of damaged piezoelectric smart laminated plates and damage detection

Considering mass and stiffness of piezoelectric layers and damage effects of composite layers, nonlinear dynamic equations of damaged piezoelectric smart laminated plates are derived. The derivation is based on the Hamilton＇s principle, the higher- order shear deformation plate theory, von Karman type geometrically nonlinear straindisplacement relations, and the strain energy equivalence theory. A negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to realize the active control and damage detection with a closed control loop. Simply supported rectangular laminated plates with immovable edges are used in numerical computation. Influence of the piezoelectric layers＇ location on the vibration control is in- vestigated. In addition, effects of the degree and location of damage on the sensor output voltage are discussed. A method for damage detection is introduced.     

基于一种广义梁理论的弹性地基FGM简支梁自由振动解析解

基于一种扩展的n阶广义剪切变形梁理论(n-GBT),应用Hamilton原理,建立了以轴向位移、横向位移及转角为未知函数的Winkler-Pasternak弹性地基功能梯度材料(FGM)梁的自由振动方程,采用Navier法获得了弹性地基FGM简支梁自由振动的精确解。与多种梁理论预测结果进行比较,讨论并给出了GBT阶次n的理想取值;分析了梯度指标、跨厚比及地基刚度对FGM梁频率的影响。结果表明:本文方法有效且适用范围广,若采用高阶剪切梁理论模型,宜取n≥3的奇数;FGM梁的自振频率随材料梯度指标的增大而减小;随跨厚比的增加而增大,但当跨厚比大于20,跨厚比增加对频率的影响很小;随地基刚度的增加而增大,地基刚度足够大时,频率趋于收敛。     

基于修正偶应力和高阶剪切变形理论的变截面微梁的自由振动

基于修正偶应力和高阶剪切理论建立了仅含有一个尺度参数的Reddy变截面微梁的自由振动模型,研究了变截面微梁自由振动问题的尺度效应和横向剪切变形对自振频率计算的影响。基于哈密顿原理推导了动力学方程与边界条件,并采用微分求积法求解了各种边界条件下的自振频率。算例结果表明,基于偶应力理论预测的变截面微梁的自振频率均大于经典梁理论的预测结果,即捕捉到了尺度效应。另外,梁的几何尺寸与尺度参数越接近,尺度效应就越明显,而梁的长细比越小,横向剪切变形对自振频率的影响就越明显。     

Refined theories may be needed for vibration analysis of structures with overhang

The effect of shear deformation and rotary inertia terms on the free vibration of a beam with overhang was investigated. A recently proposed modified Timoshenko-type equations of motion were used to analyze the vibration of the structure. Two different sets of boundary conditions, with either a fixed or hinged end support, were studied. The results were compared with those obtained for the classical Bernoulli–Euler beam theory. The comparison shows that for a hinged end beam with very long overhang, where the span between the supports is less than one tenth of the overall beam length, the classical theory significantly overestimates the values of the fundamental natural frequencies, even for isotropic shear rigidity. On the other hand, the span effect is reversed for the clamped end beam, for which a relatively significant difference between the classical theory and shear theory results may occur only for a long span. For transversely isotropic beams, the refined theory predictions of the fundamental natural frequencies may be much smaller than those obtained through the rigid shear theory, especially for short span hinged end beams and long span clamped end beams.     

Mechanics and fracture of crack tip deformable bi-material interface

Novel interface deformable bi-layer beam theory is developed to account for local effects at crack tip of bi-material interface by modeling a bi-layer composite beam as two separate shear deformable sub-layers with consideration of crack tip deformation. Unlike the sub-layer model in the literature in which the crack tip deformations under the interface peel and shear stresses are ignored and thus a “rigid” joint is used, the present study introduces two interface compliances to account for the effect of interface stresses on the crack tip deformation which is referred to as the elastic foundation effect; thus a flexible condition along the interface is considered. Closed-form solutions of resultant forces, deformations, and interface stresses are obtained for each sub-layer in the bi-layer beam, of which the local effects at the crack tip are demonstrated. In this study, an elastic deformable crack tip model is presented for the first time which can improve the split beam solution. The present model is in excellent agreements with analytical 2-D continuum solutions and finite element analyses. The resulting crack tip rotation is then used to calculate the energy release rate (ERR) and stress intensity factor (SIF) of interface fracture in bi-layer materials. Explicit closed-form solutions for ERR and SIF are obtained for which both the transverse shear and crack tip deformation effects are accounted. Compared to the full continuum elasticity analysis, such as finite element analysis, the present solutions are much explicit, more applicable, while comparable in accuracy. Further, the concept of deformable crack tip model can be applied to other bi-layer beam analyses (e.g., delamination buckling and vibration, etc.).     

Effects of shear stresses and rotary inertia on the stability and vibration of sandwich cylindrical shells with FGM core surrounded by elastic medium

The vibration and stability of axially loaded sandwich cylindrical shells with the functionally graded (FG) core with and without shear stresses and rotary inertia resting Pasternak foundation are investigated. The dynamic stability is derived based on the first order shear deformation theory (FSDT) including shear stresses. The axial load and dimensionless fundamental frequency for FG sandwich shell with shear stresses and rotary inertia and resting on the Pasternak foundation. Finally, the influences of variations of FG core, elastic foundations, shear stresses and rotary inertia on the fundamental frequencies and critical axial loads are investigated.     

Surface and thermal effects on vibration of embedded alumina nanobeams based on novel Timoshenko beam model

This paper deals with the free vibration analysis of circular alumina （Al2O3） nanobeams in the presence of surface and thermal effects resting on a Pasternak foun- dation. The system of motion equations is derived using Hamilton＇s principle under the assumptions of the classical Timoshenko beam theory. The effects of the transverse shear deformation and rotary inertia are also considered within the framework of the mentioned theory. The separation of variables approach is employed to discretize the governing equa- tions which are then solved by an analytical method to obtain the natural frequencies of the alumina nanobeams. The results show that the surface effects lead to an increase in the natural frequency of nanobeams as compared with the classical Timoshenko beam model. In addition, for nanobeams with large diameters, the surface effects may increase the natural frequencies by increasing the thermal effects. Moreover, with regard to the Pasternak elastic foundation, the natural frequencies are increased slightly. The results of the present model are compared with the literature, showing that the present model can capture correctly the surface effects in thermal vibration of nanobeams.     

Nonlinear resonance responses of geometrically imperfect shear deformable nanobeams including surface stress effects

In this work, a thorough investigation is presented into the nonlinear resonant dynamics of geometrically imperfect shear deformable nanobeams subjected to harmonic external excitation force in the transverse direction. To this end, the Gurtin–Murdoch surface elasticity theory together with Reddy’s third-order shear deformation beam theory is utilized to take into account the size-dependent behavior of nanobeams and the effects of transverse shear deformation and rotary inertia, respectively. The kinematic nonlinearity is considered using the von Kármán kinematic hypothesis. The geometric imperfection as a slight curvature is assumed as the mode shape associated with the first vibration mode. The weak form of geometrically nonlinear governing equations of motion is derived using the variational differential quadrature (VDQ) technique and Lagrange equations. Then, a multistep numerical scheme is employed to solve the obtained governing equations in order to study the nonlinear frequency–response and force–response curves of nanobeams. Comprehensive studies into the effects of initial imperfection and boundary condition as well as geometric parameters on the nonlinear dynamic characteristics of imperfect shear deformable nanobeams are carried out through numerical results. Finally, the importance of incorporating the surface stress effects via the Gurtin–Murdoch elasticity theory, is emphasized by comparing the nonlinear dynamic responses of the nanobeams with different thicknesses.     

剪切流作用下层合梁非线性振动特性研究

针对剪切流中层合梁的大变形非线性振动问题, 采用高阶剪切变形锯齿理论和冯·卡门应变描述层合梁的变形模式和几何非线性效应, 构建了大变形层合梁非线性振动有限元数值模型; 采用基于任意拉格朗日?欧拉方法的有限体积法求解不可压缩黏性流体纳维-斯托克斯方程, 结合层合梁和流体的耦合界面条件建立了剪切流作用下层合梁流固耦合非线性动力学数值模型, 采用分区并行强耦合方法对层合梁的流致非线性振动响应进行了迭代计算. 研究了不同速度分布的剪切流作用下单层梁和多层复合材料梁的振动响应特性, 并验证了本文数值建模方法的有效性. 结果表明: 剪切流作用下单层梁的振动特性与均匀流作用下的情况不同, 梁的运动轨迹受剪切流影响向下偏斜, 随着速度分布系数增加, 尾部流场中的涡结构发生改变; 刚度比对剪切流作用下层合梁的振动特性有显著影响, 随着刚度比的增加, 层合梁振动的振幅增大, 主导频率下降, 运动轨迹由‘8’字形逐渐变得不对称; 发现了不同厚度比和铺层角度情况下, 层合梁存在定点稳定模式、周期极限环振动模式和非周期振动模式三种不同的振动模式, 改变层合梁铺层角度可实现层合梁周期极限环振动模式向非周期振动模式转变.      

Dynamic response to a moving load of a Timoshenko beam resting on a nonlinear viscoelastic foundation

The present paper investigates the dynamic response of finite Timoshenko beams resting on a sixparameter foundation subjected to a moving load. It is for the first time that the Galerkin method and its convergence are studied for the response of a Timoshenko beam supported by a nonlinear foundation. The nonlinear Pasternak foundation is assumed to be cubic. Therefore, the efects of the shear deformable beams and the shear deformation of foundations are considered at the same time. The Galerkin method is utilized for discretizing the nonlinear partial differential governing equations of the forced vibration. The dynamic responses of Timoshenko beams are determined via the fourth-order Runge–Kutta method. Moreover, the efects of diferent truncation terms on the dynamic responses of a Timoshenko beam resting on a complex foundation are discussed. The numerical investigations shows that the dynamic response of Timoshenko beams supported by elastic foundations needs super high-order modes. Furthermore, the system parameters are compared to determine the dependence of the convergences of the Galerkin method.