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.     

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 nonlinear visco-elastic constitutive model for large rotation finite element formulations

Although all known materials have internal damping that leads to energy dissipation, most existing large deformation visco-elastic finite element formulations are based on linear constitutive models or on nonlinear constitutive models that can be used in the framework of an incremental co-rotational finite element solution procedure. In this investigation, a new nonlinear objective visco-elastic constitutive model that can be implemented in non-incremental large rotation and large deformation finite element formulations is developed. This new model is based on developing a simple linear relationship between the damping forces and the rates of deformation vector gradients. The deformation vector gradients can be defined using the decomposition of the matrix of position vector gradients. In this paper, the decomposition associated with the use of the tangent frame that is equivalent to the QR decomposition is employed to define the matrix of deformation gradients that enter into the formulation of the viso-elastic constitutive model developed in this investigation. Using the relationship between the deformation gradients and the components of the Green–Lagrange strain tensor, it is shown that the damping forces depend nonlinearly on the strains and linearly on the classical strain rates. The relationship between the damping forces and strains and their rates is used to develop a new visco-elastic model that satisfies the objectivity requirements and leads to zero strain rates under an arbitrary rigid body displacement. The linear visco-elastic Kelvin–Voigt model frequently used in the literature can be obtained as a special case of the proposed nonlinear model when only two visco-elastic coefficients are used. As demonstrated in this paper, the use of two visco-elastic coefficients only leads to viscous coupling between the deformation gradients. The model developed in this investigation can be used in the framework of large deformation and large rotation non-incremental solution procedure without the need for using existing co-rotational finite element formulations. The finite element absolute nodal coordinate formulation (ANCF) that allows for straightforward implementation of general constitutive material models is used in the validation of the proposed visco-elastic model. A comparison with the linear visco-elastic model is also made in this study. The results obtained in this investigation show that there is a good agreement between the solutions obtained using the proposed nonlinear model and the linear model in the case of small deformations.     

Study of the ligament tension and cross-section deformation using nonlinear finite element/multibody system algorithms

In this investigation, the effects of the knee-joint movements on the ligament tension and cross-section deformation are examined using large displacement nonlinear finite element/multibody system formulations. Two knee-joint models that employ different constitutive equations and significantly different deformation kinematics are developed and implemented to analyze the ligament dynamics in a computational solution procedure that integrates large displacement finite element and multibody system algorithms. The first model employs a lower fidelity large displacement cable element that does not capture the cross-section deformations and allows for using only nonlinear classical beam theory with a linear Hookean material law instead of a general continuum mechanics approach. In the second model, a higher fidelity large displacement beam model that captures more coupled deformation modes including Poisson modes as well the cross-section deformation is used. This higher fidelity model also allows for a straight forward implementation of general nonlinear constitutive models, such as Neo Hookean material laws, based on a general continuum mechanics approach. Cauchy stress tensor and Nanson’s formula are used to obtain an accurate expression for the ligament tension forces, which as shown in this investigation depend on the ligament cross section deformation. The two models are implemented in a general multibody system algorithm that allows introducing general constraint and force functions. The finite element/multibody system computational algorithm used in this investigation is based on an optimum sparse matrix structure and ensures that the kinematic constraint equations are satisfied at the position, velocity, and acceleration levels. The results obtained in this investigation show that models that ignore coupled deformation modes including some Poisson modes and the cross-section deformations can lead to inaccurate prediction of the ligament forces. These simpler models, as demonstrated in this investigation, can be used to obtain only simplified expressions for the ligament tensions. A three-dimensional knee-joint model that consists of five bodies including two flexible bodies that represent the medial collateral ligament (MCL) and lateral collateral ligament (LCL) is used in the numerical comparative study presented in this paper. The large displacement procedure presented in this investigation can be applied to other types of Ligaments, Muscles, and Soft Tissues (LMST) in biomechanics applications.     

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.     

A new locking-free formulation for planar,shear deformable,linear and quadratic beam finite elements based on the absolute nodal coordinate formulation

Many widely used beam finite element formulations are based either on Reissner’s classical nonlinear rod theory or the absolute nodal coordinate formulation (ANCF). Advantages of the second method have been pointed out by several authors; among the benefits are the constant mass matrix of ANCF elements, the isoparametric approach and the existence of a consistent displacement field along the whole cross section. Consistency of the displacement field allows simpler, alternative formulations for contact problems or inelastic materials. Despite conceptional differences of the two formulations, the two models are unified in the present paper.     

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.     

Definition of the Elastic Forces in the Finite-Element Absolute Nodal Coordinate Formulation and the Floating Frame of Reference Formulation

The equivalence of the finite-element formulations used inflexible multibody dynamics is the focus of this investigation. Thisequivalence will be used to address several fundamental issues related tothe deformations, flexible body coordinate systems, and the geometriccentrifugal stiffening effect. Two conceptually different finite-elementformulations that lead to exact modeling of the rigid body dynamics will beused. The first one is the absolute nodal coordinateformulation in which beams and plates can be treated as isoparametricelements. This formulation leads to a constant and symmetric mass matrix andhighly nonlinear elastic forces. In this study, it is demonstrated thatdifferent element coordinate systems which are used for the convenience ofdescribing the element deformations lead to similar results as the elementsize is reduced. In particular, two element frames are used;the pinned and the tangent frames. The pinned frame has one ofits axes passing through two nodes of the element, while the tangent frame isrigidly attached to one of the ends of the element. Numerical resultsobtained using these two different frames are found tobe in good agreement as the element size decreases. The relationshipbetween the coordinates used in the absolute nodal coordinate formulationand the floating frame of reference formulation is presented. Thisrelationship can be used to obtain the highly nonlinear expression of thestrain energy used in the absolute nodal coordinate formulation from thesimple energy expression used in the floating frame of referenceformulation. It is also shown that the source of the nonlinearityis due to the finite rotation of the element. The result of the analysispresented clearly demonstrates that the instability observedin high-speed rotor analytical models due to the neglect of the geometriccentrifugal stiffening is not a problem inherent to a particular finite-element formulation. Such a problem can only be avoided by considering the known linear effect of the geometric centrifugal stiffening or by using a nonlinear elastic model as recently demonstrated. Fourier analysis of the solutions obtained in this investigation also sheds new light on the fundamental problem of the choice of the deformable body coordinate system in the floating frame of reference formulation. Another method forformulating the elastic forces in the absolute nodal coordinate formulationbased on a continuum mechanics approach is also presented.     

A study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation

Finite element analysis using plate elements based on the absolute nodal coordinate formulation (ANCF) can predict the behaviors of moderately thick plates subject to large deformation. However, the formulation is subject to numerical locking, which compromises results. This study was designed to investigate and develop techniques to prevent or mitigate numerical locking phenomena. Three different ANCF plate element types were examined. The first is the original fully parameterized quadrilateral ANCF plate element. The second is an update to this element that linearly interpolates transverse shear strains to overcome slow convergence due to transverse shear locking. Finally, the third is based on a new higher order ANCF plate element that is being introduced here. The higher order plate element makes it possible to describe a higher than first-order transverse displacement field to prevent Poisson thickness locking. The term “higher order” is used, because some nodal coordinates of the new plate element are defined by higher order derivatives. The performance of each plate element type was tested by (1) solving a comprehensive set of small deformation static problems, (2) carrying out eigenfrequency analyses, and (3) analyzing a typical dynamic scenario. The numerical calculations were made using MATLAB. The results of the static and eigenfrequency analyses were benchmarked to reference solutions provided by the commercially available finite element software ANSYS. The results show that shear locking is strongly dependent on material thickness. Poisson thickness locking is independent of thickness, but strongly depends on the Poisson effect. Poisson thickness locking becomes a problem for both of the fully parameterized element types implemented with full 3-D elasticity. Their converged results differ by about 18 % from the ANSYS results. Corresponding results for the new higher order ANCF plate element agree with the benchmark. ANCF plate elements can describe the trapezoidal mode; therefore, they do not suffer from Poisson locking, a reported problem for fully parameterized ANCF beam elements. For cases with shear deformation loading, shear locking slows solution convergence for models based on either the original fully parameterized plate element or the newly introduced higher order plate element.     

Simulation of a viscoelastic flexible multibody system using absolute nodal coordinate and fractional derivative methods

This paper employs a new finite element formulation for dynamics analysis of a viscoelastic flexible multibody system. The viscoelastic constitutive equation used to describe the behavior of the system is a three-parameter fractional derivative model. Based on continuum mechanics, the three-parameter fractional derivative model is modified and the proposed new fractional derivative model can reduce to the widely used elastic constitutive model, which meets the continuum mechanics law strictly for pure elastic materials. The system equations of motion are derived based on the absolute nodal coordinate formulation (ANCF) and the principle of virtual work, which can relax the small deformation assumption in the traditional finite element implementation. In order to implement the viscoelastic model into the absolute nodal coordinate, the Grünwald definition of the fractional derivative is employed. Based on a comparison of the HHT-I3 method and the Newmark method, the HHT-I3 method is used to solve the equations of motion. Another particularity of the proposed method based on the ANCF method lies in the storage of displacement history only during the integration process, reducing the numerical computation considerably. Numerical examples are presented in order to analyze the effects of the truncation number of the Grünwald series (fading memory phenomena) and the value of several fractional model parameters and solution convergence aspects. An erratum to this article can be found at     

Reduction of system matrices of planar beam in ANCF by component mode synthesis method

A method of reducing the system matrices of a planar flexible beam described by an absolute nodal coordinate formulation (ANCF) is presented. In this method, we focus that the bending stiffness matrix expressed by adopting a continuum mechanics approach to the ANCF beam element is constant when the axial strain is not very large. This feature allows to apply the Craig–Bampton method to the equation of motion that is composed of the independent coordinates when the constraint forces are eliminated. Four numerical examples that compare the proposed method and the conventional ANCF are demonstrated to verify the performance and accuracy of the proposed method. From these examples, it is verified that the proposed method can describe the large deformation effects such as dynamic stiffening due to the centrifugal force, as well as the conventional ANCF does. The use of this method also reduces the computing time, while maintaining an acceptable degree of accuracy for the expression characteristics of the conventional ANCF when the modal truncation number is adequately employed. This reduction in CPU time particularly pronounced in the case of a large element number and small modal truncation number; the reduction can be verified not only in the case of small deformation but also in the case of a fair bit large deformation.     

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.     

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.     

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.     

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.     

Rigid-flexible coupling dynamics of three-dimensional hub-beams system

In the previous research of the coupling dynamics of a hub-beam system, coupling between the rotational motion of hub and the torsion deformation of beam is not taken into account since the system undergoes planar motion. Due to the small longitudinal deformation, coupling between the rotational motion of hub and the longitudinal deformation of beam is also neglected. In this paper, rigid-flexible coupling dynamics is extended to a hub-beams system with three-dimensional large overall motion. Not only coupling between the large overall motion and the bending deformation, but also coupling between the large overall motion and the torsional deformation are taken into account. In case of temperature increase, the longitudinal deformation caused by the thermal expansion is significant, such that coupling between the large overall motion and the longitudinal deformation is also investigated. Combining the characteristics of the hybrid coordinate formulation and the absolute nodal coordinate formulation, the system generalized coordinates include the relative nodal displacement and the slope of each beam element with respect to the body-fixed frame of the hub, and the variables related to the spatial large overall motion of the hub and beams. Based on precise strain-displacement relation, the geometric stiffening effect is taken into account, and the rigid-flexible coupling dynamic equations are derived using velocity variational principle. Finite element method is employed for discretization. Simulation of a hub-beams system is used to show the coupling effect between the large overall motion and the torsional deformation as well as the longitudinal deformation. Furthermore, conservation of energy in case of free motion is shown to verify the formulation.