移动质量-梁振动的时空有限元法数值分析

为了获得移动质量沿梁匀速运动的系统动态响应,建立了时空有限元数值求解模型.考虑移动质量惯性项,得到移动质量-梁时变系统的动力学方程.应用时空有限元法.得到了移动集中质量作用下Bernoulli-Euler梁离散单元的质量矩阵、刚度矩阵.与Newmark-β法、Wilson-θ法计算结果进行比较,时空有限无法计算梁的动态响应的精度更高.     

基于移动有限元法的裂纹梁振动分析

采用移动有限元法和局部柔度法对移动质量作用下含裂纹简支梁进行了振动计算分析.计算考虑了裂纹和移动质量的相对位置对梁固有频率的影响,以及移动质量在不同位置、速度情况下对裂纹梁的动力响应的影响.结果分析表明,裂纹与移动质量的存在会使得梁的动态位移有不同程度的增大,且随着移动质量位置和裂纹位置的改变会使得梁的固有频率变小.     

Discrete element response of beams with traveling mass

This article extends a procedure that has been used to discretize the static physical system following the assumption that a continuous flexible beam can be replaced by a system of rigid bars and joints which resists relative rotation of the attached bars. We call this procedure the “discrete element response of beams”. The object of this article is to present and formulate a new simple, practical and inexpensive approximate technique for determining the time response of beams, with different boundary conditions, carrying a moving mass. To verify the results, other solutions are obtained by continuous simulation systems, CSMP, and dynamic finite element, PAFEC. This algorithm is shown to be much more efficient computationally.     

Dynamic analysis of eccentrically prestressed viscoelastic Timoshenko beams under a moving harmonic load

The dynamic response of eccentrically prestressed viscoelastic Timoshenko beams under a moving harmonic load is studied by using Lagrange equations. In the study, for using the Lagrange equations, trial functions denoting the deflection of the beam and the rotation of the cross-sections are expressed in polynomial forms. The constraint conditions of supports are taken into account by using Lagrange multipliers. The effects of the value of the eccentricity of the compressive load, the excitation frequency, the constant velocity of the transverse moving harmonic load and viscous damping of the material of beams are studied in detail. Convergence studies are made. The validity of the obtained results is demonstrated by comparing them with exact solutions based on the Euler–Bernoulli beam theory obtained for the special cases of the investigated problem.     

Reality of virtual damage identification based on neural networks and vibration analysis of a damaged bridge under a moving vehicle

Presented here is a reality of virtual damage detection and vibration behaviour study of a discrete beam-like bridge with one or several non-propagating edge cracks subjected to a moving vehicle. In this model, the simply supported beam elements are replaced by a range of rigid bars, which are connected by transverse and rotational springs, while the mass and rotational moment of inertia may be lumped at various points along the beam. The adopted vehicle model here is a four degrees-of-freedom, two axes half-vehicle model with tires flexibility and linear suspensions. Damage can be modelled by altering the spring stiffness equation at the crack position according to predictions, which allows the inclusion of simple or complex damage. To simplify, damage is represented here by an open crack, and stiffness of a given element with damage is calculated by fracture mechanics. Both the discrete element and finite element methods are used to investigate vibration analysis of a discrete beam model subjected to a moving vehicle to confirm model feasibility in vibration analysis under a moving vehicle. Besides, some dynamic response laws are obtained. Considering an irregular road profile, the effects of the moving vehicle velocity, the moving vehicle mass, the crack location and the crack depth on dynamic response of a beam-like bridge are analysed by a numerical example, combining a vehicle–bridge coupled vibration MATLAB program with ANSYS. In addition, the neural network is used to identify the damage of the structure. Numerical results of the numerical model predictions, compared with those obtained from the continuous elements beam, support the accuracy of the discrete elements beam model in both cases of undamaged beam and damaged one. The evidence for condition assessment and damage identification of bridge is obtained from this simulation as obtaining the vibrational characteristics of the damaged beam structure subjected to a moving vehicle. And the inversion results show that the neural network method can identify the injury location and injury size of the structure accurately.

     

Motion control and shape optimization of a suitlike flexible arm

A method of motion control as well as shape optimization is proposed for the preliminary design of a suitlike flexible arm, which is composed of some variable-length and fixed-length beams. The large deformation, variable geometry and motion of the flexible structure are calculated by dynamic finite element analysis (FEA) using the step by step time integration method. A neural-networks-inverse-model, which learns nonlinear behaviour of the flexible structure, has been applied for the motion control as an inverse model of the flexible arm. For this geometrically nonlinear structure and time response problem, the optimum shape of the cross-section has been calculated under constraints of stress, stiffness and minimum weight with FEA and sensitivity analysis combined with fuzzy rules. This method has been applied for the design of a flexible arm, which simulates a process of lifting a human body and moving it. The calculated optimum shape has a much higher stiffness with decrease in weight in comparison with the initial shape. Moreover, the calculated motion agrees well with the one aimed for and the flexible arm reduces the impact force.     

Dynamic response of double-FG porous beam system subjected to moving load

In this paper, the vibration response of the double-FG porous beam system (DFGPBS) acted by a moving load is investigated. The DFGPBS composed of two parallel FG porous beams with their material properties varying along both the axial and transverse directions, i.e., bi-directional FG material distribution, is taken into account. The porous imperfection is simulated by distributing the porosity along the beam thickness with even and uneven patterns. The governing equations of this bi-directional DFGPBS under a moving load are established with the aid of the Hamilton principle associated with the Timoshenko beam theory. The Ritz method is adopted to discrete the differential governing equations, which are solved by the Newmark-β approach. The validation of the present model is performed by comparing the numerical results with two previous works. Then, the parametric study is carried out to investigate the influences of bi-directional gradient indices, porosity volume fraction, boundary conditions, stiffness of elastic layer, and velocity of the moving load on the vibration response of bi-directional DFGPBSs excited by a moving load. It is demonstrated that the vibration response of the double-beam system subjected to moving loads can be governed by tailoring the distribution of the bi-directional FG materials. The present work can be used to guide the multi-functional design of a double-beam system under dynamic loadings.

     

Natural vibration analysis of rail track as a system of elastically coupled beam structures on Winkler foundation

An analytical model for analyzing the vertical free vibration of a rail track is presented. The track structure is represented as a system of elastically coupled beam structures resting on a Winkler foundation. The rail and the tie beams are described by any combination of the two existing beam theories, the Bernoulli-Euler type, and the Timoshenko type, while the rail is assumed to be periodically supported at discrete points on cross-track tie beams. A generalized track element, which consists of a rail span (beam segment), two adjacent ties, and the coupling spring stiffnesses, is established to discretize the track system into identical units. A concept of an equivalent frequency-dependent spring coefficient for the rail support system is introduced to formulate the dynamic stiffness matrix of the track element. Solutions are provided for the natural frequencies of the track and the associated mode shapes of the rail and the ties under transversely (cross-track) symmetric vibration. The free vibration results are used to obtain the dynamic receptance response of a typical field track and to compare them with an existing model and field experimental data.     

Dynamics modeling for a rigid-flexible coupling system with nonlinear deformation field

In this paper, a moving flexible beam, which incorporates the effect of the geometrically nonlinear kinematics of deformation, is investigated. Considering the second-order coupling terms of deformation in the longitudinal and transverse deflections, the exact nonlinear strain-displacement relations for a beam element are described. The shear strains formulated by the present modeling method in this paper are zero, so it is reasonable to use geometrically nonlinear deformation fields to demonstrate and simplify a flexible beam undergoing large overall motions. Then, considering the coupling terms of deformation in two dimensions, finite element shape functions of a beam element and Lagrange’s equations are employed for deriving the coupling dynamical formulations. The complete expression of the stiffness matrix and all coupling terms are included in the formulations. A model consisting of a rotating planar flexible beam is presented. Then the frequency and dynamical response are studied, and the differences among the zero-order model, first-order coupling model and the new present model are discussed. Numerical examples demonstrate that a ‘stiffening beam’ can be obtained, when more coupling terms of deformation are added to the longitudinal and transverse deformation field. It is shown that the traditional zero-order and first-order coupling models may not provide an exact dynamic model in some cases.     

Nonlinear feedback control of slewing beams

This paper is concerned with feedback control of slewing beams. It allows for the hub to have finite rotation which leads to a fully nonlinear system. It uses the method of state-dependent Riccati equation (SDRE) to obtain feedback control laws. The model assumes finite rigid body rotations and infinitesimal (small) elastic displacements. It considers both the classical Euler–Bermoulli model and the Timoshenko model for the beam. It uses finite element method to obtain approximate equations that can be used to construct feedback control laws.     

基于UM的磁浮列车-轨道梁耦合振动仿真程序开发

基于大型通用多体动力学仿真分析平台Universal Mechanism(UM),开发用于磁浮列车 轨道梁耦合振动仿真的专用程序UM Maglev,其中：磁浮列车设置为多刚体模型,弹簧和阻尼器的刚度和阻尼视为线性或非线性力元;轨道梁设置为三维铁木辛柯梁模型,或从外部有限元软件导入模态分析结果;轨道线路包含平面和纵断面曲线、超高和轨面随机不平顺;悬浮和导向系统控制采用PID模型;多体动力学系统微分 代数方程求解采用Park刚性稳定法。该程序可用于考察磁浮列车的曲线通过性能、运行平稳性和乘坐舒适度,研究悬浮/导向气隙与磁浮控制系统参数优化,分析轨道梁在动态电磁力作用下的振动响应。     

几何非线性的损伤粘弹性Timoshenko梁的动力学行为

从考虑损伤的粘弹性材料的一种卷积型本构关系出发,建立了在有限变形下损伤粘弹性Timoshenko梁的控制方程．利用Galerkin方法对该组方程进行简化。得到一组非线性积分一常微分方程．然后应用非线性动力学数值分析方法,如相平面图,Poincare截面分析了载荷参数对非线性损伤粘弹性Timoshenko梁动力学性能的影响．特别考察了损伤对粘弹性梁的动力学行为的影响．     

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.     

柔性多体系统动力学在直升机旋翼桨叶上的应用

提出了一种全新的计算旋翼桨叶动力响应的方法。以柔性多体系统动力学为基础，结合有限单元法和拉格朗日方程，推导出绕动轴转动的多柔体动力学质量矩阵，建立了直升机旋翼桨叶的动力学控制方程及其系数矩阵的解析表达式，采用Rung-Kutta法进行数值积分．分析了挥舞运动在强耦合状态下的运动规律。仿真结果表明采用此方法建立旋翼桨叶的动力学控制方程。可以更真实的反映桨叶的实际运动状态。     

Finite element formulation for dynamics of planar flexible multi-beam system

In some previous geometric nonlinear finite element formulations, due to the use of axial displacement, the contribution of all the elements lying between the reference node of zero axial displacement and the element to the foreshortening effect should be taken into account. In this paper, a finite element formulation is proposed based on geometric nonlinear elastic theory and finite element technique. The coupling deformation terms of an arbitrary point only relate to the nodal coordinates of the element at which the point is located. Based on Hamilton principle, dynamic equations of elastic beams undergoing large overall motions are derived. To investigate the effect of coupling deformation terms on system dynamic characters and reduce the dynamic equations, a complete dynamic model and three reduced models of hub-beam are prospected. When the Cartesian deformation coordinates are adopted, the results indicate that the terms related to the coupling deformation in the inertia forces of dynamic equations have small effect on system dynamic behavior and may be neglected, whereas the terms related to coupling deformation in the elastic forces are important for system dynamic behavior and should be considered in dynamic equation. Numerical examples of the rotating beam and flexible beam system are carried out to demonstrate the accuracy and validity of this dynamic model. Furthermore, it is shown that a small number of finite elements are needed to obtain a stable solution using the present coupling finite element formulation.     

Absolute nodal coordinate plane beam formulation for multibody systems dynamics

A new plane beam dynamic formulation for constrained multibody system dynamics is developed. Flexible multibody system dynamics includes rigid body dynamics and superimposed vibratory motions. The complexity of mechanical system dynamics originates from rotational kinematics, but the natural coordinate formulation does not use rotational coordinates, so that simple dynamic formulation is possible. These methods use only translational coordinates and simple algebraic constraints. A new formulation for plane flexible multibody systems are developed utilizing the curvature of a beam and point masses. Using absolute nodal coordinates, a constant mass matrix is obtained and the elastic force becomes a nonlinear function of the nodal coordinates. In this formulation, no infinitesimal or finite rotation assumptions are used and no assumption on the magnitude of the element rotations is made. The distributed body mass and applied forces are lumped to the point masses. Closed loop mechanical systems consisting of elastic beams can be modeled without constraints since the loop closure constraints can be substituted as beam longitudinal elasticity. A curved beam is modeled automatically. Several numerical examples are presented to show the effectiveness of this method.     

Transverse impact problems by higher order beam finite element

A higher order beam finite element is derived and shown to be very efficient in solving the transient dynamic problem. The finite element thus developed is then applied to impact problems concerning the response of beams to striking elastic masses. Due to the local indentation the motion of the beam is nonlinearly coupled with the motion of the mass. An efficient iterative procedure is proposed in this paper to integrate the time variable. Both elastic impact and impact with permanent indentations are considered. The finite element solutions are found in good agreement with some existing solutions.     

Finite element analysis of unbalance response of high-speed polygon mirror scanner motor considering the flexibility of the structure

This paper presents the finite element method and the mode superposition method to analyze the unbalance response of a high speed polygon mirror scanner motor supported by sintered bearing and flexible supporting structures. Finite element equations of each component of the polygon mirror scanner motor and the flexible supporting structures are consistently derived by satisfying the geometric compatibility in the internal boundary between each component. The rotating polygon mirror is modeled by annular sector element, and its rigid body motion is also considered. The rotating components except for the polygon mirror are modeled by Timoshenko beam element including the gyroscopic effect. The flexible supporting structures are modeled by using a 4-node tetrahedron element and 4-node shell element with rotational degrees of freedom. The rigid link constraints are imposed at the interface between sleeve and sintered bearing to describe the physical motion at this interface. A global matrix equation obtained by assembling the finite element equations of each substructure is transformed to a state-space matrix-vector equation, and both damped natural frequencies and modal damping ratios are calculated by solving the associated eigenvalue problem by using the restarted Arnoldi iteration method. Unbalance responses are calculated by superposing the eigenvalues and eigenvectors of the free vibration analysis. The validity of the proposed method is verified by comparing the simulated unbalance response with the experimental results. This research also shows that the flexibility of supporting structures plays an important role in determining the unbalance response of the polygon mirror scanner motor.     

A penalty model for the analysis of curved composite beams

A simple one-dimensional mechanical model for curved laminated beams is presented. The laminae composing the beam are modelled as Timoshenko beams, perfectly bonded at the interfaces. Because the laminae can rotate differently from one to the other, the cross-sections of the composite beam can warp. The elasto-static problem of the beam is formulated through the principle of stationary potential energy, imposing constraint conditions between the displacements of adjacent laminae by a penalty technique. This approach produces an approximation of radial and tangential interactions between adjacent laminae. By using four-node isoparametric finite elements, numerical values of interlaminar stresses in straight and curved laminated beams are given. They are compared with the results obtained by other authors under different conditions.     

Nonlinear dynamics of viscoelastic flexible structural systems by finite element method

This article presents a nonlinear displacement based finite elements model to study and analyze the nonlinear dynamic response of flexible double wishbone structural vehicle suspension system considering damping effect which was not previously discussed elsewhere. Due to large deflection and moderate rotation encountered during passing over road bumps, the kinematic nonlinearity is included through von Kármán strain component. Elastic undamped as well as viscous and viscoelastic damping mechanism are considered and compared. Considering the viscoelastic damping mechanism, the viscoelastic damping mechanism is modeled based on the integral constitutive form, which is recast into an incremental form suitable for finite element implementation. Additionally, the revolute joint element is adopted to incorporate the joint flexibility in the double wishbone system. The plane frame element is adopted to model the suspension links by using Timoshenko beam theory. The developed nonlinear finite element equations of motion are solved through the incremental iterative Newmark technique. The developed procedure is verified by comparing the obtained results with analytical solution and excellent agreement is observed. The applicability of the developed procedure is demonstrated by conducting parametric studies to show the effects of the road irregularities profiles, the vehicle speed, and the material damping coefficients on the nonlinear vibrations response of the double wishbone suspension systems. The obtained results are supportive in the design and manufacturing processes of these structural systems.