基于旋转场曲率的二维剪切梁单元建模

对二维剪切梁单元进行研究,利用平面旋转场理论推导了精确曲率模型.采用几何精确梁理论构建了剪切梁单元弹性力矩阵.通过绝对节点坐标方法建立了系统的非线性动力学方程,提出基于旋转场曲率的二维剪切梁单元,并分别引入经典二维剪切梁单元和基于位移场曲率的二维剪切梁单元进行比较研究.首先,静力学分析证明了所提模型的正确性;其次,特征频率分析验证了模型可与理论解符合,收敛精度高,并且能准确地预测单元固有频率对应的振型;最后,在非线性动力学问题上,通过与ANSYS结果对比分析,证明了该模型可有效处理柔性大变形问题,并且与经典二维剪切梁单元相比具有缓解剪切闭锁的优势.因此,本文提出的基于旋转场曲率的二维剪切梁单元在处理几何非线性问题中具有较大的应用潜力.     

基于绝对节点坐标法的大变形柔性梁几种动力学模型研究

对在平面内大范围转动的大变形柔性梁动力学进行了研究, 基于绝对节点坐标法建立了一种新的大变形柔性梁的非线性动力学模型. 该动力学模型中考虑了柔性梁的轴向拉伸变形和横向弯曲变形, 利用Green-Lagrangian应变张量计算柔性梁的轴向应变及应变能, 利用曲率的精确表达式计算柔性梁的横向弯曲变形能. 运用拉格朗日恒等式给出了柔性梁横向弯曲变形能新的表达式, 该变形能表达式更加简洁, 通过新的变形能表达式得到了新的弹性力模型, 由此得到的动力学方程可以精确地描述柔性梁的几何大变形问题. 通过与高次耦合模型以及ANSYS中BEAM188非线性梁单元模型的比较, 验证了本模型在计算大变形时的正确性以及高次耦合模型在处理大变形问题时的不足. 进一步研究发现, 新的广义弹性力模型可以适当地简化, 给出了两种简化模型, 根据不同模型的计算效率以及计算精度的比较确定了不同模型的适用范围.     

Dynamic modeling of beams with non-material,deformation-dependent boundary conditions

In conventional problems of structural mechanics, both kinematic boundary conditions and external forces are prescribed at fixed material points that are known in advance. If, however, a structure may move relative to its supports, the position of the imposed constraint relations generally changes in the course of motion. A class of problems which inherently exhibits this particular type of non-material boundary conditions is that of axially moving continua. Despite varying in time, the positions of the supports relative to the material points of the body have usually assumed to be known a priori throughout the deformation process in previous investigations. This requirement is abandoned in the present paper, where the dynamic behavior of a structure is studied, which may move freely relative to one of its supports. As a consequence, the position of such a non-material boundary relative to the structure does not only change in time but also depends on the current state of deformation of the body. The variational formulation of the equilibrium relations of a slender beam that may undergo large deformations is presented. To this end, a theory based on Reissner's geometrically exact relations for the plane deformation of beams is adopted, in which shear deformation is neglected for the sake of brevity. Before a finite element scheme is developed, a deformation-dependent transformation of the beam's material coordinate is introduced, by which the varying positions of the constraint relations are mapped onto fixed points with respect to the new non-material coordinate. By means of this transformation, additional convective terms emerge from the virtual work of the inertia forces, whose symmetry properties turn out to be different from what has previously been presented in the literature. In order to obtain approximate solutions, a finite element discretization utilizing absolute nodal displacements as coordinates is subsequently used in characteristic numerical examples, which give an insight into the complex dynamic behavior of problems of this type. On the one hand, the free vibrations of a statically pre-deformed beam are investigated, on the other hand, an extended version of the sliding beam problem is studied.     

A tapered beam finite element for rotor dynamics analysis

The stiffness, mass and gyroscopic matrices of a rotating beam element are developed, a cubic function being used for the transverse displacement. Shear deflection is included by use of end nodal variables of shear strain, along with transverse displacement and cross-section rotation; rotatory inertia effects are included in the energy functional to provide a Timoshenko beam formulation. The gyroscopic effects for small perturbations are linearized as a skew symmetric damping matrix. The formulation is implemented by numerical integration for a linearly tapered circular beam. A technique of reduction of the shear nodal variable prior to global assembly is shown to provide little loss in accuracy with reduced system bandwidth. Numerical comparisons for three previously published beam models are included, with results presented for the case of forward and reverse precession to verify the gyroscopic effects. The utility of the element in a general program for rotor dynamics analysis is identified.     

Dynamics of a belt-drive system using a linear complementarity problem for the belt–pulley contact description

The aim of this study was to develop an efficient and realistic numerical model in order to predict the dynamic response of belt drives. The belt was modeled as a planar beam element based on an absolute nodal coordinate formulation. A viscoelastic material was adopted for the belt and the corresponding damping and stiffness matrices were determined. The belt–pulley contact was formulated as a linear complementarity problem together with a penalty method. This made it possible for us to accurately predict the contact forces, including the stick and slip zones between the belt and the pulley. The belt-drive model was verified by comparing it with the available analytical solutions. A good agreement was found. Finally, the applicability of the method was demonstrated by considering non-steady belt-drive operating conditions.     

Higher order representation of the beam cross section deformation in large displacement finite element analysis

Most existing beam formulations assume that the cross section of the beam remains rigid regardless of the amplitude of the displacement. The absolute nodal coordinate formulation (ANCF); however, allows for the deformation of the cross section and leads to a more general beam models that capture the coupling between different modes of displacement. This paper examines the effect of the order of interpolation on the modes of deformation of the beam cross section using ANCF finite elements. To this end, a new two-dimensional shear deformable ANCF beam element is developed. The new finite element employs a higher order of interpolation, and allows for new cross section deformation modes that cannot be captured using previously developed shear deformable ANCF beam elements. The element developed in this study relaxes the assumption of planar cross section; thereby allowing for including the effect of warping as well as for different stretch values at different points on the element cross section. The displacement field of the new element is assumed to be cubic in the axial direction and quadratic in the transverse direction. Using this displacement field, more expressions for the element extension, shear and the cross section stretch can be systematically defined. The change in the cross section area is measured using Nanson’s formula. Measures of the shear angle, extension, and cross section stretch can also be systematically defined using coordinate systems defined at the element material points. Using these local coordinate systems, expressions for a nominal shear angle are obtained. The differences between the cross section deformation modes obtained using the new higher order element and those obtained using the previously developed lower order elements are highlighted. Numerical examples are presented in order to compare the results obtained using the new finite element and the results obtained using previously developed ANCF finite elements.     

基于IB-LBFS和绝对节点坐标法的降落伞柔性结构流固耦合数值模拟

在空气动力学、水动力学和生物流体力学领域中,大变形柔性结构的流固耦合现象是一个重要的非线性力学问题.对该系统的数值模拟是分析这一问题的有效手段.将近年提出的一种Descartes流场求解器,即浸润边界-格子Boltzmann通量求解器(immersed boundary-lattice Boltzmann flux solver,IB-LBFS)作为流场求解方法,并引入绝对节点坐标法(absolute nodal coordinate formulation,ANCF)作为大变形结构分析手段,构建了流固耦合求解器以模拟三维流场中的大变形柔性体运动.使用三维来流中的旗帜摆动算例对该流固耦合求解器进行了验证计算.基于该流固耦合求解器对三维不可压流场中的矩形降落伞和十字形降落伞的展开过程进行了非定常流固耦合数值模拟.     

弹性压扭直杆的Greenhill公式对精确模型的推广

将圆截面Kirchhoff弹性压扭直杆的Greenhill公式推广到精确模型.基于平面截面假定,在弯扭的基础上增加了拉压和剪切变形,将弹性杆的位形表达为截面的弧坐标历程.由弹性杆精确模型的平衡微分方程,得到了两端受力螺旋作用时对应于直线平衡状态的特解,导出了线性化扰动方程及其通解,再根据两端为铰支时的边界条件以及积分常数存在非零解的条件导出弹性直杆精确模型的Greenhill公式.结果表明,由力螺旋表示的稳定域为一对称的封闭区域,拉压和剪切对稳定性的影响取决于拉压柔度与剪切柔度之差、抗弯刚度和杆长这三个因素.     

Mechanics and morphology of single-walled carbon nanotubes: from graphene to the elastica

Abstract

The elastica is referred to the shape of the curve into which the centreline of a flexible lamina is bent. Hence, single-walled carbon nanotubes (SWCNTs) are treated as the elastica obtained from bending of graphene. The corresponding large deformation accompanies both the material and geometrical non-linearities. The morphology of the free-standing SWCNTs such as the natural angle of twist, bond lengths, tube radius and wall thickness are determined. Moreover, it is shown that the induced self-equilibriated strain field has a remarkable impact on the mechanical behaviour of the nanotube. Utilization of an appropriate non-linear continuum constitutive relation for graphene leads to exact formulation of the governing equations of SWCNTs. Subsequently, through perturbation analysis, the asymptotic solutions of the initial elastic fields for the SWCNTs are presented. By performing ab initio calculations, the components of the fourth and sixth-order elastic moduli tensors in the constitutive model of graphene needed in this study are computed.     

Single-beam analysis of damaged beams: Comparison using Euler–Bernoulli and Timoshenko beam theory

A new general formulation that is applicable to the damaged, linear elastic structures ‘unified framework’ is used to obtain analytical expressions for natural frequencies and mode shapes. The term mode shapes is used to mean the displacement modes, the section rotation modes, the sectional bending strain modes and sectional shear strain modes. The formulation is applicable to damaged elastic self-adjoint systems. The formulation has two unique aspects: First, the theory is mathematically rigorous since no assumptions are made regarding the physical behavior at a damage location, therefore there is no need to substitute the damage with a hypothetical elastic element such as a spring. Since the beam is not divided at the damage location, rather than an 8 by 8, only a 4 by 4 matrix is solved to obtain the natural frequencies and mode shapes. Second, the inertia effects due to damage which have till now been neglected by researchers are accounted for. The formulation uses a geometric damage model, perturbation of mode shapes and natural frequencies, and a modal superposition technique to obtain and solve the governing differential equation. Timoshenko beam theory is then taken as an example, and its results are compared with results using Euler–Bernoulli beam theory and finite element models. The range of applicability of the two theories is ascertained for damage characteristics such as depth and extent of damage and beam characteristics such as slenderness ratio and Poisson?s ratio. The paper considers rectangular notch like non-propagating damage as an example of the damage.     

Assessment of rod behaviour theories used in spectral finite element modelling

In this work different theories of rods have been discussed and compared. The investigated theories are widely used in spectral finite element modelling of rod behaviour associated with propagation of symmetric longitudinal waves. These are various single, two-mode and three-mode theories including the elementary, classical Love and Mindlin-Herrmann approaches as well as new two, three and four-mode theories proposed by the authors. Dispersion curves associated with each theory, obtained by the use of Hamilton's principle, have been presented and discussed in the paper. The investigation programme carried out by the authors aimed to show major differences and similarities between the rod theories and to discuss certain numerical aspects of their application. Great attention has been paid on properties, limitations as well as difficulties associated with the use of the theories. The results obtained from a wide program on numerical tests allowed the authors to draw certain general conclusions that are valid not only in the field of the spectral finite element method but also in the field of dynamics of engineering rod structures.     

On the flexural vibration of cylinders under axial loads: Numerical and experimental study

The flexural vibration of a homogeneous isotropic linearly elastic cylinder of any aspect ratio is analysed in this paper. Natural frequencies of a cylinder under uniformly distributed axial loads acting on its bases are calculated numerically by the Ritz method with terms of power series in the coordinate directions as approximating functions. The effect of axial loads on the flexural vibration cannot be described by applying infinitesimal strain theory, therefore, geometrically nonlinear strain–displacement relations with second-order terms are considered here. The natural frequencies of free–free, clamped–clamped, and sliding–sliding cylinders subjected to axial loads are calculated using the proposed three-dimensional Ritz approach and are compared with those obtained with the finite element method and the Bernoulli–Euler theory. Different experiments with cylinders axially compressed by a hydraulic press are carried out and the experimental results for the lowest flexural frequency are compared with the numerical results. An approach based on the Ritz formulation is proposed for the flexural vibration of a cylinder between the platens of the press with constraints varying with the intensity of the compression. The results show that for low compressions the cylinder behaves similarly to a sliding–sliding cylinder, whereas for high compressions the cylinder vibrates as a clamped–clamped one.     

非惯性系下考虑剪切变形的柔性梁的动力学建模

研究非惯性坐标系下考虑剪切变形的柔性梁的动力学建模. 首先借鉴Euler-Bernoulli梁的几何非线性变形模式,考虑了Timoshenko梁弯曲以及剪切变形产生的几何非线性效应对纵向、横向变形位移的影响,在考虑两个方向的变形耦合项后,利用有限元法对柔性梁进行了离散,采用Lagrange方程建立了柔性梁的动力学模型,首次建立了包含变形二次耦合量的Timoshenko梁的动力学方程. 关键词： 非惯性坐标系 剪切变形 柔性梁 动力学建模     

A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach

Large-amplitude (geometrically non-linear) vibrations of circular cylindrical shells subjected to radial harmonic excitation in the spectral neighbourhood of the lowest resonances are investigated. The Lagrange equations of motion are obtained by an energy approach, retaining damping through Rayleigh's dissipation function. Four different non-linear thin shell theories, namely Donnell's, Sanders-Koiter, Flügge-Lur’e-Byrne and Novozhilov's theories, which neglect rotary inertia and shear deformation, are used to calculate the elastic strain energy. The formulation is also valid for orthotropic and symmetric cross-ply laminated composite shells. The large-amplitude response of perfect and imperfect, simply supported circular cylindrical shells to harmonic excitation in the spectral neighbourhood of the lowest natural frequency is computed for all these shell theories. Numerical responses obtained by using these four non-linear shell theories are also compared to results obtained by using the Donnell's non-linear shallow-shell equation of motion. A validation of calculations by comparison with experimental results is also performed. Both empty and fluid-filled shells are investigated by using a potential fluid model. The effects of radial pressure and axial load are also studied. Boundary conditions for simply supported shells are exactly satisfied. Different expansions involving from 14 to 48 generalized co-ordinates, associated with natural modes of simply supported shells, are used. The non-linear equations of motion are studied by using a code based on an arclength continuation method allowing bifurcation analysis.     

Boundary conditions for the bending of a piezoelectric beam

For beam bending in transversely isotropic piezoelectric media, the reciprocal theorem and the general solution of piezoelasticity are applied in a novel way to obtain the appropriate stress and mixed boundary conditions accurate to all orders for the beam of general edge geometry and loadings. By generalizing the method developed by Gregory and Wan, a set of necessary conditions on the edge-data for the existence of a rapidly decaying solution is established. The prescribed edge-data of the beam must satisfy these conditions in order that they could generate a decaying state within the beam. When stress and mixed conditions are imposed on the beam edge, these decaying state conditions for the case of bending deformation of piezoelectric beam are derived explicitly. They are then used for the correct formulation of boundary conditions for the beam theory solution (or the interior solution). Besides, an analytical solution of elastic beam is formulated to verify validity of our boundary conditions. For the stress data, our boundary conditions coincide with those obtained in conventional forms of beam theories. More importantly, the appropriate boundary conditions with two sets of mixed edge-data are obtained for the first time.     

Dynamic characterization of beam type structures: Analytical, numerical and experimental applications

A general algorithm for the free vibration analysis of stepped and tapered beam type structures with multiple elastic supports is developed in this work. The analytical formulation is based on the Ritz method and on the use of orthogonal polynomials within the framework of the first order shear deformation beam theory. To verify the validity and convergence of the general algorithm several numerical examples are analyzed. A further example concerned with the determination of the dynamical properties of a bell tower is also presented and compared with the finite element method and experimental results.     

Improvement of efficiency of piezoelectric element attached to beam based on mechanical impedance matching

This paper describes new methods that improve the efficiency of a piezoelectric element attached to a beam based on mechanical impedance matching. Piezoelectric elements are often used to suppress bending vibration. They are also used as sensors or energy-harvesting sources. In such cases, the piezoelectric element is usually bonded onto the host structure by an adhesive bond. The efficiency of the piezoelectric element depends on the bonding location. When the efficiency is insufficient despite a good location, the size or number of piezoelectric elements is increased. However, the efficiency of the piezoelectric element is usually insufficient even if these methods are applied. In order to enhance the efficiency of the piezoelectric elements without using active methods, this paper proposes a mechanical impedance matching method that uses spacers or tuning for the size of the piezoelectric element. Because the attached piezoelectric element and host structure in this region behave as springs in parallel to the bending deformation, the stored strain energy in the piezoelectric element is maximized under the condition that their spring constants match. The proposed methods were theoretically investigated with consideration for the effects of the bonding layer, spacers, and host structure. The optimum conditions for the proposed methods were theoretically formulated, and the effectiveness of the proposed methods and theoretical analysis was verified through simulations and experiments.     

Effect of Comatic Aberration on the Propagation Characteristics of Laguerre--Gaussian Beams

Seidel comatic aberration is an important cause of deformation for a Laguerre--Gaussian (LG) beam. In addition, mono-axial comatic aberration, whose phase modulation depends only on one transverse coordinate, is also an important cause of beam deformation. Deformation of an LG beam by such aberrations is analyzed through numerical simulation based on the angular spectrum method. It is also shown that for holographically generated LG beams quadratic spatial variation of grating pitch can produce seidel and mono-axial comatic aberrations. An example of an experimentally generated LG beam with mono-axial comatic aberration is reported.     

Predicting the nonlinear tensile behavior of carbon nanotubes using finite element simulation

Carbon nanotubes (CNTs) possess extremely high mechanical properties and could be the ultimate reinforcing materials for the development of nanocomposites. In this work, a Finite Element (FE) model based on the molecular mechanics theory was developed to evaluate tensile properties of single-walled carbon nanotubes (SWCNTs). The deformation and fracture of carbon nanotubes under tensile strain conditions were studied by common FE software, Ansys. In this model, individual carbon nanotube was simulated as a frame-like structure, and the primary bonds between two nearest-neighboring atoms were treated as beam elements. The beam element properties were determined via the concept of energy equivalence between molecular dynamics and structural mechanics. So far, several researches have studied the elastic behavior of CNTs, and its nonlinearity is not well understood. The novelty of the model lies on the use of nonlinear beam elements to evaluate SWNTs tensile failure. The obtained calculated mechanical properties show good agreement with existing numerical and experimental results.     

An inertia-capacitance beam substructure formulation based on the bond graph method with application to rotating beams

In this paper, a novel inertia-capacitance (IC) beam substructure formulation based on the IC-field presentation from the bond graph method is developed. The IC beam provides a modular, systematic and graphical approach to beam modeling. These features allow the modeler to focus more on the modeling and less on the mathematics. As such, the IC beam is proposed as an alternative to the many existing types of beam models available in the literature. The IC beam is formulated in the center of mass body fixed coordinate system allowing for easy interfacing in a multibody system setting. This floating frame approach is also computationally cheap. Elastic deformations in the IC beam are assumed to be small and described by modal superposition. The formulation couples rigid body and elastic deformations in a nonlinear fashion. The formulation is also compact and efficient. Detailed derivations for a two-dimensional planar IC beam with bending modes are presented. A modal acceleration method based on the decoupling of bending modes is proposed for use in the IC beam. The rotating beam spin-up maneuver problem is solved. The Karnopp-Margolis method is applied to ensure complete integral causality for an efficient numerical system. Geometric substructuring technique is applied to model large deflections. The IC beam is shown to be capable of solving the rotating beam problem accurately and efficiently.