Exact dynamic stiffness matrix of non-symmetric thin-walled beams on elastic foundation using power series method

Exact dynamic element stiffness matrix for the flexural–torsional free vibration analysis of the shear deformable thin-walled beam with non-symmetric cross-section on two-types of elastic foundation is newly presented using power series method based on the technical computing program Mathematica. For this, the shear deformable beam on elastic foundation theory is developed by introducing Vlasov's assumption and applying Hellinger–Reissner principle. This beam includes the shear deformation effects due to the shear forces and the restrained warping torsion and due to the coupled effects between them, and rotary inertia effects and the flexural–torsional coupling effects due to the non-symmetric cross-sections. And then equations of motion and force–deformation relations are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components and the exact dynamic element stiffness matrix is determined using force–deformation relationships. In order to verify the accuracy of this study, the numerical solutions are presented and compared with the analytical solutions and the finite element solutions using the isoparametric beam elements. Particularly the influences of the coupled shear deformation on the vibrational behavior of non-symmetric beam on elastic foundation are investigated.     

A general three-dimensional L-section beam finite element for elastoplastic large deformation analysis

In this paper, the development of a general three-dimensional L-section beam finite element for elastoplastic large deformation analysis is presented. We propose the generalized interpolation scheme for the isoparametric formulation of three-dimensional beam finite elements and the numerical procedure is developed for elastoplastic large deformation analysis. The formulation is general and effective for other thin-walled section beam finite elements. To show the validity of the formulation proposed, a 2-node three-dimensional L-section beam finite element is implemented in an analysis code. As numerical examples, we first perform elastic small and large deformation analyses of a cantilever beam structure subjected to various tip loadings, and elastoplastic large deformation analysis of the same structure under reversed cyclic tip loading. We then analyze the failures of simply supported beam structures of different lengths and slenderness ratios under elastoplastic large deformation. The same problems are solved using refined shell finite element models of the structures. The numerical results of the L-section beam finite element developed here are compared with the solutions obtained using shell finite element analyses. We also discuss the numerical solutions in detail.     

空间薄壁梁单元面向对象程序的实现

针对传统有限元分析软件主要面向过程设计,其可维护性和可扩展性等较差的问题,基于面向对象程序设计方法,建立具有内部节点的空间薄壁截面梁单元模型,给出线弹性空间薄壁梁单元的UML类图,介绍矩阵类、截面类、材料类、节点类、单元类和结构类等6种类成员的主要属性和方法.用C#编制相应的有限元程序,通过T形框架算例比较和验证其位移和弯曲转角计算值、理论解和ANSYS的BEAM 189梁单元的数值解,结果表明该程序精度良好,可用于空间薄壁结构的有限元分析.     

Free vibration analysis of beams on elastic foundation

A simple finite element method is developed and applied to treat the free vibration analysis of beams supported on elastic foundations. The entire analysis is programmed to run on a microcomputer and with few elements modelling the beam, gives quick and reliable results. Numerical examples pertaining to the free vibration of beams in some special situations are considered, such as a stepped beam on an elastic foundation, beam on a stepped elastic foundation and a continuous beam on an elastic foundation. Present results compare very well with those obtained from existing solutions, wherever possible.     

Dynamic stiffness matrix of non-symmetric thin-walled curved beam on Winkler and Pasternak type foundations

Using the technical computing program Mathematica, the dynamic stiffness matrix for the spatially coupled free vibration analysis of thin-walled curved beam with non-symmetric cross-section on two-types of elastic foundation is newly presented based on the power series method. For this, the elastic strain energy considering the axial/flexural/torsional coupled terms, the kinetic energy including the rotary inertia effect, and the energy due to the elastic foundation are introduced. Then, equations of motion are derived from the energy principle and explicit expressions for displacement parameters are derived based on power series expansions of displacement components. Finally, the exact dynamic stiffness matrix is determined using force–displacement relations. In order to demonstrate the validity and the accuracy of this study, the natural frequencies of thin-walled curved beams with mono-symmetric and non-symmetric cross-sections are evaluated and compared with the analytical solutions and finite element solutions using Hermitian curved beam elements and ABAQUS’s shell elements. In addition, some results by a parametric study are reported.     

A nonlinear two-node superelement for use in flexible multibody systems

A nonlinear two-node superelement is proposed for the modeling of flexible complex-shaped links for use in multibody simulations. Assuming that the elastic deformations with respect to a corotational reference frame remain small, substructuring methods may be used to obtain reduced mass and stiffness matrices from a linear finite element model. These matrices are used in the derivation of potential and kinetic energy expressions of the nonlinear two-node superelement. By evaluating Lagrange’s equations, expressions for the internal and external forces acting on the superelement can be obtained. The inertia forces of the superelement are derived in terms of absolute nodal velocities and accelerations, which greatly simplifies the dynamic formulation. Three examples are included. The first two examples are used to validate the method by comparing the results with those obtained from nonlinear beam element solutions. We consider a benchmark simulation of the spin-up motion of a flexible beam with uniform cross-section and a similar simulation in which the beam is simultaneously excited in the out-of-plane direction. Results from both examples show good agreement with simulation results obtained using nonlinear finite beam elements. In a third example, the method is applied to an unbalanced rotating shaft, illustrating the potential of the proposed methodology for a more complex geometry.     

Static, dynamic and stability analysis of structures composed of tapered beams

A new numerical method is proposed for the static, dynamic and stability analysis of linear elastic plane structures consisting of beams with constant width and variable depth. It is a finite element method based on an exact flexural and axial stiffness matrix and approximate consistent mass and geometric stiffness matrices for a linearly tapered beam element with constant width. Use of this method provides the exact solution of the static problem with just one element per member of a structure with linearly tapered beams and excellent approximate solutions of the dynamic and stability problems with very few elements per member of the structure in a computationally very efficient way. Very detailed comparison studies of the proposed method against a number of other known finite element methods with respect to accuracy and computational efficiency for cantilever tapered beams of rectangular and I cross section clearly favor the proposed method. A continuous beam, a gable frame and a portal frame consisting of tapered members are analyzed by the proposed method as well as by other known methods to illustrate the use of the method to structures composed of tapered beams.     

A beam element for coupled torsional-flexural vibration of doubly unsymmetrical thin walled beams axially loaded

A coupled torsional-bending finite element with shear deformations and rotatory inertia for vibration of nonsymmetric thin walled beams axially loaded is developed. The equations of motion are based on Vlasov’s theory of thin-walled beams, which are modified to include an axial load. The formulation is also applicable to solid beams. The Hermite cubic polynomials are adopted as shape functions. Mass, elastic stiffness and geometrical stiffness matrices of unsymmetrical cross-section beams are presented. In order to verify the accuracy of this theory and the corresponding beam element developed, a numerical study is presented and compared with the literature and experimental tests.     

Flexible Multibody Systems Models Using Composite Materials Components

The use of a multibody methodology to describe the large motion of complex systems that experience structural deformations enables to represent the complete system motion, the relative kinematics between the components involved, the deformation of the structural members and the inertia coupling between the large rigid body motion and the system elastodynamics. In this work, the flexible multibody dynamics formulations of complex models are extended to include elastic components made of composite materials, which may be laminated and anisotropic. The deformation of any structural member must be elastic and linear, when described in a coordinate frame fixed to one or more material points of its domain, regardless of the complexity of its geometry. To achieve the proposed flexible multibody formulation, a finite element model for each flexible body is used. For the beam composite material elements, the sections properties are found using an asymptotic procedure that involves a two-dimensional finite element analysis of their cross-section. The equations of motion of the flexible multibody system are solved using an augmented Lagrangian formulation and the accelerations and velocities are integrated in time using a multi-step multi-order integration algorithm based on the Gear method.     

基于欧拉 伯努利梁和铁木辛柯梁理论的功能梯度材料模量测定

为测定功能梯度材料的弹性模量和剪切模量,引入梁理论并将梁沿长度方向离散,建立单元平衡方程后可得到弹性模量和剪切模量分布;假设弹性模量为沿长度方向的线性函数或指数函数,用有限元软件仿真计算功能梯度材料梁单元节点处的挠度和转角,然后用插值法构造变形特征函数,并计算得出弹性模量和剪切模量,且计算值与理论值的误差较小.计算结果还表明,采用铁木辛柯梁理论不仅可以得到弹性模量,还可以计算剪切模量,且弹性模量计算结果比用欧拉-伯努利梁计算结果更接近真实值,但铁木辛柯梁理论中需测定转角,对测定过程的要求会更加严格。     

中心刚体-柔性梁刚柔耦合动力学模型降阶研究

有限单元法被广泛的采用来描述柔性体的弹性变形,然而有限元节点坐标数目庞大,将会给动力学方程求解带来巨大的计算负担.如何降低柔性体的自由度,是当前柔性多体系统动力学研究的一个重要命题.本文以中心刚体-柔性梁系统为例,采用Krylov方法和模态方法进行降价.然后分别采用有限元全模型、Krylov降阶模型和模态降阶模型,对中心刚体-柔性梁进行刚-柔耦合动力学仿真.仿真结果表明,与采用模态降阶方法相比,采用Krylov模型降阶方法只需要较低的自由度,就可以得到与采用有限元方法完全一致的结果.说明Krylov模型降阶方法能够有效的用于柔性多体系统的模型降价研究.     

High-precision finite element analysis of cylindrical shells

This paper presents the finite element formulation to study the free vibration of cylindrical shells. The displacement function for the high-precision shell element with 16 degrees of freedom is approximated by a Hermitian polynomial of beam function type. The explicit formulation for the high-precision element is extremely efficient. For the purpose of comparison, the subject element is used to study the sample case of free vibration of a shell structure. The results are in good agreement with those published. The study shows that solution accuracy with fewer elements is assured and that accurate solutions are obtainable in the high-frequency range.     

An efficient computational method for dynamic stress analysis of flexible multibody systems

This paper presents an efficient computational method of dynamic stress history calculation for a general three-dimensional flexible body by combining flexible multibody dynamic simulation and quasi-static finite element analysis (FEA). In the dynamic simulation of flexible multibody systems, flexible components can undergo nonsteady gross motion and small elastic deformation that is described with respect to the body reference frame by using the assumed mode method. D'Alembert inertia loads from the gross body motion and the elastic deformation are expressed as a combination of space-dependent and time-dependent terms that are obtained from the dynamic simulation. D'Alembert inertia loads that are associated with each unit value of the time-dependent terms are then distributed to all finite element nodes in order to compute a corresponding stress influence coefficient through quasi-static structural analyses. Total dynamic stresses due to D'Alembert inertia loads are obtained by multiplying actual magnitude of time-dependent terms with the associated stress influence coefficients. By the proposed method, it is shown that, for a general three-dimensional component, the required number of FEAs can be significantly reduced.     

Lead-lag vibrational frequencies of a rotating beam with end mass

Transverse vibrations of the lead-lag motion of a uniform rotating beam with tip mass have been investigated. A finite element consistent mass formulation is developed. This formulation accounts for the centrifugal force field and the centripetal acceleration effects. The generalized eigenvalue problem is defined and numerical solutions are generated. Both fixed and hinged end conditions are considered while solutions are obtained for lead-lag as well as flapping frequencies. Numerical results for the first 12 eigenfrequencies are presented, in a useful tabular form, for a wide range of parameter changes.     

Riccati discrete time transfer matrix method for elastic beam undergoing large overall motion

An efficient method for dynamics simulation for elastic beam with large overall spatial motion and nonlinear deformation, namely, the Riccati discrete time transfer matrix method (Riccati-DT-TMM), is proposed in this investigation. With finite segments, continuous deformation field of a beam can be decomposed into many rigid bodies connected by rotational springs. Discrete time transfer matrices of rigid bodies and rotational springs are used to analyze the dynamic characteristic of the beam, and the Riccati transform is used to improve the numerical stability of discrete time transfer matrix method of multibody system dynamics. A predictor-corrector method is used to improve the numerical accuracy of the Riccati-DT-TMM. Using the Riccati-DT-TMM in dynamics analysis, the global dynamics equations of the system are not needed and the computation time required increases linearly with the system’s number of degrees of freedom. Three numerical examples are given to validate the method for the dynamic simulation of a geometric nonlinear beam undergoing large overall motion.     

Finite element analysis of a compressible dynamic viscoelastic sphere using FFT

An extension to a compressible dynamic viscoelastic hollow sphere problem with both finite and infinite outer radius is performed. The governing viscoelastic equations of motion are transformed into the Laplace domain via the elastic–viscoelastic correspondence principle. Real and imaginary parts of the nodal displacements are obtained by solving a non-symmetric matrix equation in the complex Laplace domain. Inversion into the time domain is performed using the discrete inverse Fourier transform. Use is made of an infinite element in the infinite sphere problem. Numerical solutions are compared to both the exact Laplace and time domain solutions wherever possible.     

Finite element dynamic analysis of geometrically exact planar beams

We present a new strain-based finite element formulation for the dynamic analysis of highly flexible elastic planar beams. The formulation employs the geometrically exact Reissner planar beam theory which accounts for finite displacements and rotations, and finite membrane, shear and bending strains. The system of semi-discrete dynamic equations of motion is derived from the modified Hamilton principle in which only the strain variables are interpolated. Such a choice of the interpolated variables is an advantage over approaches, in which the displacements and rotations are interpolated, since the field consistency problem and related locking phenomena do not arise. The performance and accuracy of the formulation are illustrated by several numerical examples.     

Dynamic response of frameworks by fast fourier transform

A general numerical method for determining the dynamic response of linear elastic plane frameworks to dynamic shocks, wind forces or earthquake excitations is presented. The method consists of formulating and solving the dynamic problem in the frequency domain by the finite element method and of obtaining the response by a numerical inversion of the transformed solution with the aid of the fast Fourier transform algorithm. The formulation is based on the exact solution of the transformed governing equation of motion of a beam element and it consequently leads to the exact solution of the problem. Flexural, and axial motion of the framework members are considered. The effects of damping (external viscous or internal viscoelastic), axial forces on bending, rotatory inertia and shear deformation on the dynamic response are also taken into account. Numerical examples to illustrate the method and demonstrate its advantages over other methods are presented.     

Moving finite element analysis for the elastic beams in contact problems

The contact problem of a linear elastic beam with a rigid barrier is considered, in which the contact surface is assumed to be frictionless. The problem contains the characteristic of moving boundaries as the contact regions expand or reduce in size when the external loads alter. Starting from the variational principles, we derived the interface equations as well as the governing equation. In addition to matching the deflections and the slopes with the rigid barrier at the marginal nodes of the contact regions, the interface equations also have to be satisfied there. In essence, the interface equations are designed to locate the unknown and moving marginal nodes. Although it is confined within the framework of small deformation and linear elastic material behavior, the problem exhibits high nonlinearity due to the moving nature of the marginal nodes. In this paper, we present a moving finite element analysis employing incremental procedures and an iterative numerical scheme to tackle the problem. A fixed number of two-node beam elements are used in the moving finite element analysis and the size of the elements varies as the loads alter. A couple of examples, whose exact solutions are obtainable, are chosen to demonstrate the accuracy and efficiency of the proposed numerical algorithm.