Parallel finite element simulation of mooring forces on floating objects

The coupling between the equations governing the free‐surface flows, the six degrees of freedom non‐linear rigid body dynamics, the linear elasticity equations for mesh‐moving and the cables has resulted in a fluid‐structure interaction technology capable of simulating mooring forces on floating objects. The finite element solution strategy is based on a combination approach derived from fixed‐mesh and moving‐mesh techniques. Here, the free‐surface flow simulations are based on the Navier–Stokes equations written for two incompressible fluids where the impact of one fluid on the other one is extremely small. An interface function with two distinct values is used to locate the position of the free‐surface. The stabilized finite element formulations are written and integrated in an arbitrary Lagrangian–Eulerian domain. This allows us to handle the motion of the time dependent geometries. Forces and momentums exerted on the floating object by both water and hawsers are calculated and used to update the position of the floating object in time. In the mesh moving scheme, we assume that the computational domain is made of elastic materials. The linear elasticity equations are solved to obtain the displacements for each computational node. The non‐linear rigid body dynamics equations are coupled with the governing equations of fluid flow and are solved simultaneously to update the position of the floating object. The numerical examples includes a 3D simulation of water waves impacting on a moored floating box and a model boat and simulation of floating object under water constrained with a cable. Copyright © 2003 John Wiley & Sons, Ltd.     

带集中质量的旋转柔性曲梁动力学特性分析

本文对带集中质量的平面内旋转柔性曲梁动力学特性进行了研究.基于绝对节点坐标法推导出曲梁单元,其中该曲梁单元采用Green-Lagrangian应变,并根据曲梁变形前后的曲率变化和曲率的精确表达式计算了曲梁单元弹性力所作的虚功.通过虚功原理,利用$\delta$函数和中心刚体与悬臂曲梁之间的固支边界条件,建立了带集中质量的旋转柔性曲梁非线性动力学模型.基于该模型,本文仿真计算了悬臂曲梁的纯弯曲问题和带有刚柔耦合效应的旋转柔性曲梁动力学响应问题,以此分别讨论了所提出曲梁单元的收敛性和动力学模型的正确性.进一步应用D'Alembert原理,将旋转曲梁等效为带离心力的无旋转曲梁,通过线性摄动处理得到系统的特征方程,以此分别研究了旋转角速度、初始曲率和集中质量对曲梁动力学特性的影响.最后重点分析了旋转曲梁的频率转向和振型切换问题,并阐述了两者之间的相互关系.研究结果表明:随着旋转角速度的增大,曲梁的频率特性与直梁的频率特性相近,以及曲梁拉伸变形占主导的模态振型会提前.      

Nonlinear dynamics of oriented elastic solids

Based on a continuum model for oriented elastic solids the set of nonlinear dispersive equations derived in Part I of this work allows one to investigate the nonlinear wave propagation of the soliton type. The equations govern the coupled rotation-displacement motions in connection with the linear elastic behavior and large-amplitude rotations of the director field. In the one-dimensional version of the equations and for two simple configurations an exhaustive study of solitons is presented. We show that the transverse and/or longitudinal elastic displacements are coupled to the rotational motion so that solitons, jointly in the rotation of the director and the elastic deformations, are exhibited. These solitons are solutions of a system of linear wave equations for the elastic displacements which are nonlinearly coupled to a sine-Gordon equation for the rotational motion. For each configuration, the solutions are numerically illustrated and the energy of the solitions is calculated. Finally, some applications of the continuum model and the related nonlinear dynamics to several physical situations are given and additional more complex problems are also evoked by way of conclusion.     

考虑刚体运动与弹性运动耦合影响的旋转叶片振动有限元分析

本文研究了旋转叶片的纵向振动和双向横振动,考虑了刚体运动和弹性振动的耦合关系,利用有限元法推导出离散系统动力学方程,从而引出陀螺特征值问题。本文就某一特例了计算了在不同转速时叶片振动的自然频率,讨论了转速对振动频率的影响。     

一类刚-柔耦合系统的建模与稳定性研究

对于由中心刚体带有柔性梁附件组成的这一类简单刚 柔耦合系统，目前文献广泛采用的Ｅｕｌｅｒ Ｂｅｒｎｏｕｌｉ梁模型中考虑的刚 柔运动耦合项有严重的缺陷．本文对于物理本构关系线性的有限变形梁，分别采用微元法和变分法建立了该系统大挠度非线性动力学方程组．本文使用严格的方法来研究此非线性耦合动力学模型，采用能量 动量矩组合方法构成Ｌｉａｐｕｎｏｖ函数，严格证明了此非线性系统平凡解的积分范数稳定性以及具有鲜明物理意义的最大模范数稳定性．本文对文献中引用的三类线性化模型，采用假设模态法，对中心刚体匀速转动时梁的振动作了数值仿真，进一步验证了本文的结论．上述结果，对选择刚 柔耦合系统正确的动力学模型是有益的．     

Rigid body coupled rotation around no intersecting axes

In this paper rigid body dynamic with coupled rotation around axes that are not intersecting is described by vectors connected to the pole and the axis. These mass moment vectors are defined by K. Hedrih. Dynamic equilibrium of rigid body dynamics with coupled rotations is described by vector equations. Also, they are used for obtaining differential equations to the rotor dynamics. In the case where one component of rotation is programmed by constant angular velocity, the non-linear differential equation of the system dynamics in the gravitational field is obtained and so is the corresponding equation of the phase trajectory. Series of phase trajectory transformations in relation with changes of some parameters of rigid body are presented.     

Effect of imperfect bonding on the dynamic response of a pre-stressed sandwich plate-strip with elastic layers and a piezoelectric core

In this paper,we consider the dynamic response of a pre-stressed sandwich plate-strip with a piezoelectric core and elastic layers under the action of a time-harmonic force resting on a rigid foundation.The investigation is carried out within the framework of the piecewise homogeneous body model by utilizing the exact equations of motion and relations of the linear theory of electro-elasticity.It is assumed that there is a shear-spring-type imperfect contact between the layers,but a complete contact between the plate-strip and the rigid foundation.A mathematical model of the problem is constructed,and the governing equations of motion are solved by employing the finite element method(FEM).Numerical results illustrating the influence of a change in the value of the shear-spring parameter on the dynamic response of the plate-strip are then presented.     

DYNAMIC ANALYSIS OF A SPATIAL COUPLED TIMOSHENKO ROTATING SHAFT WITH LARGE DISPLACEMENTS

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Control structure interaction in the nonlinear analysis of flexible mechanical systems

The effect of the control structure interaction on the feedforward control law as well as the dynamics of flexible mechanical systems is examined in this investigation. An inverse dynamics procedure is developed for the analysis of the dynamic motion of interconnected rigid and flexible bodies. This method is used to examine the effect of the elastic deformation on the driving forces in flexible mechanical systems. The driving forces are expressed in terms of the specified motion trajectories and the deformations of the elastic members. The system equations of motion are formulated using Lagrange's equation. A finite element discretization of the flexible bodies is used to define the deformation degrees of freedom. The algebraic constraint equations that describe the motion trajectories and joint constraints between adjacent bodies are adjoined to the system differential equations of motion using the vector of Lagrange multipliers. A unique displacement field is then identified by imposing an appropriate set of reference conditions. The effect of the nonlinear centrifugal and Coriolis forces that depend on the body displacements and velocities are taken into consideration. A direct numerical integration method coupled with a Newton-Raphson algorithm is used to solve the resulting nonlinear differential and algebraic equations of motion. The formulation obtained for the flexible mechanical system is compared with the rigid body dynamic formulation. The effect of the sampling time, number of vibration modes, the viscous damping, and the selection of the constrained modes are examined. The results presented in this numerical study demonstrate that the use of the driving forees obtained using the rigid body analysis can lead to a significant error when these forces are used as the feedforward control law for the flexible mechanical system. The analysis presented in this investigation differs significantly from previously published work in many ways. It includes the effect of the structural flexibility on the centrifugal and Coriolis forces, it accounts for all inertia nonlinearities resulting from the coupling between the rigid body and elastic displacements, it uses a precise definition of the equipollent systems of forces in flexible body dynamics, it demonstrates the use of general purpose multibody computer codes in the feedforward control of flexible mechanical systems, and it demonstrates numerically the effect of the selected set of constrained modes on the feedforward control law.     

A node-based smoothed point interpolation method for dynamic analysis of rotating flexible beams

We proposed a mesh-free method, the called node-based smoothed point interpolation method (NS-PIM), for dynamic analysis of rotating beams. A gradient smoothing technique is used, and the requirements on the consistence of the displacement functions are further weakened. In static problems, the beams with three types of boundary conditions are analyzed, and the results are compared with the exact solution, which shows the effectiveness of this method and can provide an upper bound solution for the deflection. This means that the NS-PIM makes the system soften. The NS-PIM is then further extended for solving a rigid-flexible coupled system dynamics problem, considering a rotating flexible cantilever beam. In this case, the rotating flexible cantilever beam considers not only the transverse deformations, but also the longitudinal deformations. The rigid-flexible coupled dynamic equations of the system are derived via employing Lagrange’s equations of the second type. Simulation results of the NS-PIM are compared with those obtained using finite element method (FEM) and assumed mode method. It is found that compared with FEM, the NS-PIM has anti-ill solving ability under the same calculation conditions.     

Parametric identification of crystals having a cubic lattice with negative Poisson’s ratios

A two-dimensional model of an anisotropic crystalline material with cubic symmetry is considered. This model consists of a square lattice of round rigid particles, each possessing two translational and one rotational degree of freedom. Differential equations that describe propagation of elastic and rotational waves in such a medium are derived. A relationship between three groups of parameters is found: second-order elastic constants, acoustic wave velocities, and microstructure parameters. Values of the microstructure parameters of the considered anisotropic material at which its Poisson’s ratios become negative are found.     

Computational dynamics of a 3D elastic string pendulum attached to a rigid body and an inertially fixed reel mechanism

A high fidelity model is developed for an elastic string pendulum, one end of which is attached to a rigid body while the other end is attached to an inertially fixed reel mechanism which allows the unstretched length of the string to be dynamically varied. The string is assumed to have distributed mass and elasticity that permits axial deformations. The rigid body is attached to the string at an arbitrary point, and the resulting string pendulum system exhibits nontrivial coupling between the elastic wave propagation in the string and the rigid body dynamics. Variational methods are used to develop coupled ordinary and partial differential equations of motion. Computational methods, referred to as Lie group variational integrators, are then developed, based on a finite element approximation and the use of variational methods in a discrete-time setting to obtain discrete-time equations of motion. This approach preserves the geometry of the configurations, and leads to accurate and efficient algorithms that have guaranteed accuracy properties that make them suitable for many dynamic simulations, especially over long simulation times. Numerical results are presented for typical examples involving a constant length string, string deployment, and string retrieval. These demonstrate the complicated dynamics that arise in a string pendulum from the interaction of the rigid body motion, elastic wave dynamics in the string, and the disturbances introduced by the reeling mechanism. Such interactions are dynamically important in many engineering problems, but tend be obscured in lower fidelity models.     

作大范围运动矩形板的动力学建模理论研究

研究了初应力法的作大范围运动矩形板的建模理论。根据连续介质理论,考虑应变-位移中的非线性项,用一致质量有限元法对柔性板进行离散,基于Jourdain速度变分原理导出定轴转动下大范围运动为自由的柔性板刚-柔耦合动力学方程。从其刚柔耦合动力学方程出发,考虑在大范围运动已知情况下的结构动力学方程。通过引入准静态概念,把其结构动力学方程转化为准静态方程。对纵向和横向变形节点坐标进行坐标分离,解出与纵向变形相关的准静态方程,得到准静态时的纵向应力表达式,从而获得附加刚度项;并对此非惯性系下作大范围运动柔性板的结构动力学方程进行数值仿真,验证了采用初应力法柔性板的动力学建模方法来计算经历大范围运动的不规则柔性板的动力学响应是可行的,体现了初应力法对柔性板建模的优越性。     

压电结构振动控制及压电片位置优化的遗传算法

采用比例反馈控制,由线弹性压电层合结构有限元动力方程,推导了压电智能结构的振动控制方程.采用三维八节点实体耦合单元(Solid5)模拟压电致动器/感应器,三维实体单元(Solid45)模拟主结构,利用ANSYS中参数化语言(APDL)编写了压电层合智能结构的有限元程序.以零一最优问题来描述致动器/感应器的位置优化问题,...     

基于初应力法的作大范围运动柔性梁动力学建模理论研究

研究了初应力法的作大范围运动柔性梁的建模理论.根据连续介质理论,考虑应变-位移中的非线性项,用一致质量有限元法对柔性梁进行离散,基于Jourdain速度变分原理导出定轴转动下大范围运动为自由的柔性梁刚-柔耦合动力学方程.从其刚柔耦合动力学方程出发,考虑在大范围运动已知情况下的结构动力学方程.通过引入准静态概念,把其结构动力学方程转化为准静态方程.对纵向和横向变形节点坐标进行坐标分离,解出与纵向变形相关的准静态方程,得到准静态时的纵向应力表达式,从而获得附加刚度项.并对此非惯性系下作大范围运动柔性梁的结构动力学方程进行数值仿真,对零次近似模型、一次近似模型、初应力法动力学模型的仿真结果进行分析,揭示三种模型的动力学性质的差异.     

Geometrically exact beam theory without Euler angles

In modeling highly flexible beams undergoing arbitrary rigid–elastic deformations, difficulties exist in describing large rotations using rotational variables, including three Euler angles, two Euler angles, one principal rotation angle plus three direction cosines of the principal rotation axis, four Euler parameters, three Rodrigues parameters, and three modified Rodrigues parameters. The main problem is that such rotational variables are either sequence-dependent and/or spatially discontinuous because they are not mechanics-based variables. Hence, they are not appropriate for use as nodal degrees of freedom in total-Lagrangian finite-element modeling. Moreover, it is difficult to apply boundary conditions on such discontinuous and/or sequence-dependent rotational variables. This paper presents a new geometrically exact beam theory that uses no rotation variables and has no singular points in the spatial domain. The theory fully accounts for geometric nonlinearities and initial curvatures by using Jaumann strains, exact coordinate transformations, and orthogonal virtual rotations. The derivations are presented in detail, fully nonlinear governing equations and boundary conditions are presented, a finite element formulation is included, and the corresponding governing equations for numerically exact analysis using a multiple shooting method is also derived. Numerical examples are used to illustrate the problems of using rotational variables and to demonstrate the accuracy of the proposed geometrically exact displacement-based beam theory.     

应变能中耦合项对旋转悬臂梁振动频率的影响

大范围运动悬臂梁的动力学建模问题对动力学特性分析及控制系统设计具有极其重要的作用. 当前研究多采用一次近似模型,其忽略了由轴向和横向变形所产生的应变能中的耦合项,然而这些项对动力学特性会产生影响. 通过讨论应变能的选取方式,计入了应变能中的耦合项;利用哈密尔顿原理建立结构的耦合振动模型;再借助瑞利—里兹法,以无大范围运动时的振型函数作为基本解组,得到了结构振动广义特征方程并求解. 通过数值算例对比分析,指出考虑应变能耦合项得到的频率与不考虑应变能耦合项得到的频率存在明显差别.      

Limit angular velocities of variable thickness rotating disks

Elastic and plastic limit angular velocities are calculated for rotating disks of variable thickness in power function form. Analytical solution is obtained and used to calculate elastic limit angular velocities of variable thickness rotating annular disks and annular disks with rigid inclusion. An efficient numerical solution procedure is designed and used to obtain the elastic limit angular velocities of variable thickness rotating solid disks. Von Mises yield criterion and its flow rule is used to estimate plastic limit angular velocities. Both linear and nonlinear hardening material behaviors are treated numerically. The results are verified by comparing with those of uniform thickness rotating solid disks available in the literature. Elastic and plastic limit angular velocities are found to increase with the reduction of the disk thickness at the edge as well as the reduction in the disk mass due to the shape of the profile.     

Finite amplitude, free vibrations of an axisymmetric load supported by a highly elastic tubular shear spring

The finite amplitude, free vibrational characteristics of a simple mechanical system consisting of an axisymmetric rigid body supported by a highly elastic tubular shear spring subjected to axial, rotational, and coupled shearing motions are studied. Two classes of elastic tube materials are considered: a compressible material whose shear response is constant, and an incompressible material whose shear response is a quadratic function of the total amount of shear. The class of materials with constant shear response includes the incompressible Mooney-Rivlin material and certain compressible Blatz-Ko, Hadamard, and other general kinds of models. For each material class, the quasi-static elasticity problem is solved to determine the telescopic and gyratory shearing deformation functions needed to evaluate the elastic tube restoring force and torque exerted on the body. For all materials with constant shear response, the differential equations of motion are uncoupled equations typical of simple harmonic oscillators. Hence, exact solutions for the forced vibration of the system can be readily obtained; and for this class, engineering design formulae for the load-deflection relations are discussed and compared with experimental results of others'. For the quadratic material, however, the general motion of the body is characterized by a formidable, coupled system of nonlinear equations. The free, coupled shearing motion for which either the axial or the azimuthal shear deformation may be small is governed by a pair of equations of the Duffing and Hill types. On the other hand, the finite amplitude, pure axial and pure rotational motions of the load are described by the classical, nonlinear Duffing equation alone. A variety of problems are solved exactly for these separate free vibrational modes, and a number of physical results are presented throughout.     

Experimental evidence of non-standard bifurcations in non-smooth oscillator dynamics

Analytical and experimental investigations are performed in order to characterize the dynamic behaviour of a non-smooth rotational oscillator, which exhibits multiple discontinuity boundaries in the phase space. The physical system consists of a rotating body subjected to an elastic restoring force and in contact with one or two rough discs rotating with constant driving velocities. The presence of multiple discontinuity boundaries caused by frictional contacts leads to non-standard bifurcations that are studied by means of a simple mechanical model.A test set-up has then been built to investigate the correctness of modelling of the friction force and the validity of the proposed model for technical applications: the experimental measurements qualitatively and quantitatively capture the basic scenarios anticipated by the model while a strong robustness of the phenomena pointed out by the theoretical analyses has been revealed in the experiments.