Experimental validation of rigid-flexible coupling dynamic formulation for hub–beam system

The aim of this paper is to compare the accuracy of the absolute nodal coordinate formulation and the floating frame of reference formulation for the rigid-flexible coupling dynamics of a three-dimensional Euler–Bernoulli beam by numerical and experimental validation. In the absolute nodal coordinate formulation, based on geometrically exact beam theory and considering the torsion effect, the material curvature of the beam is derived, and then variational equations of motion of a three-dimensional beam are obtained, which consist of three position coordinates, two slope coordinates, and one rotational coordinate. In the floating frame of reference formulation, the displacement of an arbitrary point on the beam is described by the rigid-body motion and a small superimposed deformation displacement. Based on linear elastic theory, the quadratic terms of the axial strain are neglected, and the curvatures are simplified to the first order. Considering both the linear damping and the quadratic air resistance damping, the equations of motion of the multibody system composed of air-bearing test bed and a cantilevered three-dimensional beam are derived based on the principle of virtual work. In order to verify the results of the computer simulation, two experiments are carried out: an experiment of hub–beam system with large deformation and a dynamic stiffening experiment. The comparison of the simulation and experiment results shows that in case of large deformation, the frequency result obtained by the floating frame of reference formulation is lower than that obtained by the experiment. On the contrary, the result obtained by the absolute nodal coordinate formulation agrees well with that obtained by the experiment. It is also shown that the floating frame of reference formulation based on linear elastic theory cannot reveal the dynamic stiffening effect. Finally, the applicability of the floating frame of reference formulation is clarified.     

基于一阶剪切板理论的FGM板刚柔耦合动力学建模与仿真

对作大范围运动功能梯度材料(functionally graded materials,FGM)厚板的刚柔耦合动力学问题进行了研究,基于一阶剪切变形理论,从连续介质理论出发,计及了变形位移场中的二次耦合变形量,利用Lagrange方法推导了FGM厚板的刚柔耦合动力学方程,该方程适用于普通均质板和FGM板的动力学分析.采用20自由度矩形单元对变形场进行离散,对不同转速下的悬臂板进行动力学仿真,比较了本文建立的基于一阶剪切理论的模型和基于经典薄板理论的模型,验证了本文模型的正确性以及经典薄板理论的一些不足.研究了不同功能梯度指数下,FGM厚板的横向变形、速度响应频率和固有频率.结果表明,随着转速增大,剪切项对结构动力学行为影响变大;考虑横向剪切项的情况下,计算结果更偏柔性.     

考虑刚-柔-热耦合的板结构多体系统的动力学建模

对热载荷作用下中心刚体与大变形薄板多体系统的动力学建模问题进行研究.基于Kirchhoff假设,从格林应变和曲率与绝对位移的非线性关系式出发,推导了非线性广义弹性力阵,用绝对节点坐标法建立了大变形矩形薄板的有限元离散的动力学变分方程.为了考虑刚体姿态运动、弹性变形和温度变化的相互耦合作用,推导了热流密度与绝对节点坐标之间的关系式.引入系统的运动学约束方程,建立了中心刚体-矩形板多体系统的考虑刚-柔-热耦合的热传导方程和带拉格朗日乘子的第一类拉格朗日动力学方程.为了有效地提高计算效率,将改进的中心差分法和广义-α法相结合,求解热传导方程和动力学方程,差分后的方程通过牛顿迭代法耦合求解.对刚-柔耦合和刚-柔-热三者耦合两种模型的仿真结果进行比较表明,刚体运动对温度梯度和热变形的影响显著.此外,本文建模方法考虑了几何非线性项,因此也考虑了热膨胀引起的轴向变形对横向变形的影响.     

Analysis of Large Flexible Body Deformation in Multibody Systems Using Absolute Coordinates

To consider large deformation problems in multibody system simulations afinite element approach, called absolute nodal coordinate.formulation,has been proposed. In this formulation absolute nodal coordinates andtheir material derivatives are applied to represent both deformation andrigid body motion. The choice of nodal variables allows a fullynonlinear representation of rigid body motion and can provide the exactrigid body inertia in the case of large rotations. The methodology isespecially suited for but not limited to modeling of beams, cables andshells in multibody dynamics.This paper summarizes the absolute nodal coordinate formulation for a 3D Euler–Bernoulli beam model, in particular the definition of nodal variables, corresponding generalized elastic and inertia forces and equations of motion. The element stiffness matrix is a nonlinear function of the nodal variables even in the case of linearized strain/displacement relations. Nonlinear strain/displacement relations can be calculated from the global displacements using quadrature formulae.Computational examples are given which demonstrate the capabilities of the applied methodology. Consequences of the choice of shape.functions on the representation of internal forces are discussed. Linearized strain/displacement modeling is compared to the nonlinear approach and significant advantages of the latter, when using the absolute nodal coordinate formulation, are outlined.     

柔性梁线接触碰撞的动力学建模和实验研究

研究作旋转运动的柔性梁的线接触正碰撞问题.基于Goldsmith的线接触撞击力模型,分别用基于小变形的混合坐标法和基于大变形的绝对坐标法建立了柔性梁的动力学方程,考虑了几何非线性效应.在此基础上,进一步考虑非线性阻尼项的影响,将Hunt,Crossley的阻尼模型推广到线接触问题.介绍了柔性梁线接触碰撞的实验方法.计算结果显示,在考虑阻尼的情况下,计算结果与实验结果吻合很好.比较了混合坐标法和绝对坐标法的撞击力计算结果,与实验结果对比表明,绝对坐标法更适用于大变形的撞击问题.     

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.     

考虑几何非线性和热效应的刚柔耦合系统的频域特性研究

研究温度场中旋转刚体-梁系统的刚-柔耦合动力学特性.考虑几何非线性和热效应,从精确的应变-位移关系式出发,用虚功原理和有限单元法建立了旋转刚体-梁系统的刚-柔耦合动力学方程.由于非线性刚度阵与变形的高次项有关,将非线性刚度阵的各元素表示为广义坐标阵和常值阵的乘积.数值计算表明,该方法可避免重复积分,提高计算效率.在此基础上研究了在温度递增的情况下几何非线性对系统的刚-柔耦合动力学特性的影响,用频谱分析方法研究了系统的固有频率随中心刚体转动惯量和温度的变化.     

一种在轨可展开天线的多柔体动力学建模与计算

研究了一种考虑重力梯度的大型空间可展开天线桁架的动力学建模与计算问题.使用绝对节点坐标法和绝对节点坐标参考节点法建立了可展开结构的刚柔耦合动力学模型.利用离散方向导数构建了动力学方程的保能量动量时间积分算法,通过计算获得了含有多个模块的大型空间可展开天线在轨展开以及展开后轨道机动过程轨道-姿态-变形耦合动响应.通过数值计算结果与经典算例结果、商业软件ADAMS计算结果的对比分析验证了提出方法的正确性.     

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.     

Coupled thermo-structural analysis of a bimetallic strip using the absolute nodal coordinate formulation

A bimetallic strip consists of two different metal pieces that are bonded together. Due to the different coefficients of thermal expansion, exposing the strip to temperature induces thermal stresses that cause the structure to bend. Most often, incremental finite-element methods that introduce element nodal coordinates have been successfully applied to analyze the thermally induced vibrations in such systems. The exposure of these bimetallic strips to high temperatures results in large deflections and deformations, where the effects of the rigid-body motion and large rotations must be taken into account. For classic, non-isoparametric elements such as beams and plates the incremental methods do not result in zero strains under arbitrary, rigid-body motion. Therefore, in this paper a new model of a bimetallic strip is proposed based on a coupled thermo-structural analysis using the absolute nodal coordinate formulation. The applied, non-incremental, absolute nodal coordinate formulation uses a set of global displacements and slopes so that the beam and the plate elements can be treated as isoparametric elements. In order to simulate the bimetallic strip’s dynamic response, the formulation of the shear-deformable beam element had to be extended with thermally induced stresses. This made it possible to model the coupled thermo-structural problem and to represent the connectivity constraints at the interface between the two strips of metal. The proposed formulation was verified by comparing the responses using a general-purpose finite-element software.     

大范围运动柔性机械臂斜碰撞动力学分析

对作大范围运动柔性机械臂系统,进行斜碰撞动力学分析.基于柔性多体系统刚柔耦合动力学理论,计入耦合变形项,全面考虑大范围刚体运动与弹性小变形运动的耦合,建立系统连续动力学方程.引入斜碰撞力学模型,将法向和切向碰撞力以广义力的形式加入动力学方程中,对系统进行斜碰撞动力学建模分析.法向碰撞模型选取基于连续接触力法的非线性弹簧阻尼模型,切向碰撞模型选取一种修正Coulomb摩擦模型,对切向摩擦力进行统一描述.给出接触、分离判据,实现不同状态的动力学模型转换与求解.对斜碰撞全局动力学进行了仿真验证,分析了柔性机械臂全局过程的动力学特性变化以及碰撞对大范围运动和小变形运动的作用,并对比了不同碰撞方向对大范围运动、变形、机械能、碰撞力等动力学参数的影响.     

A geometrically exact beam element based on the absolute nodal coordinate formulation

In this study, Reissner’s classical nonlinear rod formulation, as implemented by Simo and Vu-Quoc by means of the large rotation vector approach, is implemented into the framework of the absolute nodal coordinate formulation. The implementation is accomplished in the planar case accounting for coupled axial, bending, and shear deformation. By employing the virtual work of elastic forces similarly to Simo and Vu-Quoc in the absolute nodal coordinate formulation, the numerical results of the formulation are identical to those of the large rotation vector formulation. It is noteworthy, however, that the material definition in the absolute nodal coordinate formulation can differ from the material definition used in Reissner’s beam formulation. Based on an analytical eigenvalue analysis, it turns out that the high frequencies of cross section deformation modes in the absolute nodal coordinate formulation are only slightly higher than frequencies of common shear modes, which are present in the classical large rotation vector formulation of Simo and Vu-Quoc, as well. Thus, previous claims that the absolute nodal coordinate formulation is inefficient or would lead to ill-conditioned finite element matrices, as compared to classical approaches, could be refuted. In the introduced beam element, locking is prevented by means of reduced integration of certain parts of the elastic forces. Several classical large deformation static and dynamic examples as well as an eigenvalue analysis document the equivalence of classical nonlinear rod theories and the absolute nodal coordinate formulation for the case of appropriate material definitions. The results also agree highly with those computed in commercial finite element codes.     

A Lagrange–Eulerian formulation of an axially moving beam based on the absolute nodal coordinate formulation

In the scope of this paper, a finite-element formulation for an axially moving beam is presented. The beam element is based on the absolute nodal coordinate formulation, where position and slope vectors are used as degrees of freedom instead of rotational parameters. The equations of motion for an axially moving beam are derived from generalized Lagrange equations in a Lagrange–Eulerian sense. This procedure yields equations which can be implemented as a straightforward augmentation to the standard equations of motion for a Bernoulli–Euler beam. Moreover, a contact model for frictional contact between an axially moving strip and rotating rolls is presented. To show the efficiency of the method, simulations of a belt drive are presented.     

Topology optimization based on level set for a flexible multibody system modeled via ANCF

A topology optimization methodology is proposed for the flexible multibody system undergoing both large overall motion and large deformation. The system of concern is modeled via the absolute nodal coordinate formulation. The equivalent static load method is employed to transform the topology optimization of the nonlinear dynamic response of the system into a static one, and evaluated to adapt to the absolute nodal coordinate formulation by splitting the elastic deformations of the flexible components from the overall motions of those components. During the static topology optimization, the material interface is implicitly described as the zero level set of a higher-dimensional scalar function. Then, the semi-implicit level set method with the additive operator splitting algorithm is employed to solve the corresponding Hamilton-Jacobi partial differential equation. In addition, the expert evaluation method of weights based on the grey theory is utilized to define the objective function, and a modified augmented Lagrange multiplier method is proposed to treat the inequality volume constraint so as to avoid the oscillation and drift of the volume. Finally, two numerical examples are provided to validate the proposed methodology.     

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.     

A corotational procedure that handles large rotations of spatial beam structures

A practical motion process of the three dimensional beam element is presented to remove the restriction of small rotations between two successive increments for large displacement and large rotation analysis of space frames using incremental-iterative methods. In order to improve convergence properties of the equilibrium iteration, an n-cycle iteration scheme is introduced.

The nonlinear formulation is based on the corotational formulation. The transformation of the element coordinate system is assumed to be accomplished by a translation and two successive rigid body rotations: a transverse rotation followed by an axial rotation. The element formulation is derived based on the small deflection beam theory with the inclusion of the effect of axial force in the element coordinate system. The membrane strain along the deformed beam axis obtained from the elongation of the arc length of the beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations. Two methods, referred to as direct method and incremental method, are proposed in this paper to calculate the total deformational rotations.

An incremental-iterative method based on the Newton-Raphson method combined with arc length control is adopted. Numerical studies are presented to demonstrate the accuracy and efficiency of the present method.     

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.     

Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams

Most existing formulations for structural elements such as beams, plates and shells do not allow for the use of general nonlinear constitutive models in a straightforward manner. Furthermore, such structural element models, due to the nature of the generalized coordinates used, do not capture some Poisson modes such as the ones that couple the deformation of the cross section of the structural element and stretch and bending. In this paper, beam models that employ general nonlinear constitutive equations are presented using finite elements based on the nonlinear absolute nodal coordinate formulation. This formulation relaxes the assumptions of the Euler–Bernoulli and Timoshenko beam theories, and allows for the use of general nonlinear constitutive models. The finite elements based on the absolute nodal coordinate formulation also allow for the rotation as well as the deformation of the cross section, thereby capturing Poisson modes which can not be captured using other beam models. In this investigation, three different nonlinear constitutive models based on the hyper-elasticity theory are considered. These three models are based on the Neo–Hookean constitutive law for compressible materials, the Neo–Hookean constitutive law for incompressible materials, and the Mooney–Rivlin constitutive law in which the material is assumed to be incompressible. These models, which allow capturing Poisson modes, are suitable for many materials and applications, including rubber-like materials and biological tissues which are governed by nonlinear elastic behavior. Numerical examples that demonstrate the implementation of these nonlinear constitutive models in the absolute nodal coordinate formulation are presented. The results obtained using the nonlinear and linear constitutive models are compared in this study. These results show that the use of nonlinear constitutive models can significantly enhance the performance and improve the computational efficiency of the finite element models based on the absolute nodal coordinate formulation. The results also show that when linear constitutive models are used in the large deformation analysis, singular configurations are encountered and basic formulas such as Nanson’s formula are no longer valid. These singular deformation configurations are not encountered when the nonlinear constitutive models are used.     

Effect of the centrifugal forces on the finite element eigenvalue solution of a rotating blade: a comparative study

In this study, the effect of the centrifugal forces on the eigenvalue solution obtained using two different nonlinear finite element formulations is examined. Both formulations can correctly describe arbitrary rigid body displacements and can be used in the large deformation analysis. The first formulation is based on the geometrically exact beam theory, which assumes that the cross section does not deform in its own plane and remains plane after deformation. The second formulation, the absolute nodal coordinate formulation (ANCF), relaxes this assumption and introduces modes that couple the deformation of the cross section and the axial and bending deformations. In the absolute nodal coordinate formulation, four different models are developed; a beam model based on a general continuum mechanics approach, a beam model based on an elastic line approach, a beam model based on an elastic line approach combined with the Hellinger–Reissner principle, and a plate model based on a general continuum mechanics approach. The use of the general continuum mechanics approach leads to a model that includes the ANCF coupled deformation modes. Because of these modes, the continuum mechanics model differs from the models based on the elastic line approach. In both the geometrically exact beam and the absolute nodal coordinate formulations, the centrifugal forces are formulated in terms of the element nodal coordinates. The effect of the centrifugal forces on the flap and lag modes of the rotating beam is examined, and the results obtained using the two formulations are compared for different values of the beam angular velocity. The numerical comparative study presented in this investigation shows that when the effect of some ANCF coupled deformation modes is neglected, the eigenvalue solutions obtained using the geometrically exact beam and the absolute nodal coordinate formulations are in a good agreement. The results also show that as the effect of the centrifugal forces, which tend to increase the beam stiffness, increases, the effect of the ANCF coupled deformation modes on the computed eigenvalues becomes less significant. It is shown in this paper that when the effect of the Poisson ration is neglected, the eigenvalue solution obtained using the absolute nodal coordinate formulation based on a general continuum mechanics approach is in a good agreement with the solution obtained using the geometrically exact beam model.