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.     

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.     

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.     

Study of the Centrifugal Stiffening Effect Using the Finite Element Absolute Nodal Coordinate Formulation

The finite element absolute nodal coordinate formulation is usedin this investigation to study the centrifugal stiffening effect onrotating two-dimensional beams. It is demonstrated that the geometricstiffening effect can be automatically accounted for in the above mentionedfinite element formulation by using an expression for the elastic forcesobtained with a general continuum mechanics approach. The Hill equation thatgoverns the vibration of the rotating beam is obtained in terms of a set ofgeneralized coordinates that describe the beam displacements and slopes.Under the assumption of small deformation, the Hill equation is linearized,and the complete solution is obtained and used to demonstrate analyticallythat such a solution does not exhibit instabilities as the angular velocityof the beam increases. The results obtained using this finite elementprocedure are compared with the results reported in the literature.     

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.     

Definition of the Slopes and the Finite Element Absolute Nodal Coordinate Formulation

In the finite element literature, beams and platesare not treated as isoparametric elements becausethe displacement gradients at the nodes are used todescribe infinitesimal rotations. Such alinearization of the slopes leads to incorrect rigidbody equations of motion when the beams and platesrotate as rigid bodies. A simple and efficient absolute nodal coordinate formulation, in which noinfinitesimal or finite rotations are used as nodalcoordinates, can be systematically developed andefficiently used in many large deformationapplications as well as in the analysis of curvedstructures. This formulation leads to a constantmass matrix, and as a consequence, an efficientprocedure can be used for solving for the nodalaccelerations. In the absolute nodal coordinateformulation, the displacement gradients at the nodesare determined in the undeformed referenceconfiguration using simple rigid body kinematics.The problems that arise from the linearization ofthe slopes and the use of the finite rotations asnodal coordinates are discussed in this paper.     

A 3D Finite Element Method for Flexible Multibody Systems

An efficient finite element (FE) formulation for the simulation of multibody systems is derived from Hamilton's principle. According to the classical assumptions of multibody systems, a large rotation formulation has been chosen, where large rotations and large displacements, but only small deformations of the single bodies are taken into account. The strain tensor is linearized with respect to a co-rotated frame. The present approach uses absolute coordinates for the degrees of freedom and forms an alternative to the floating frame of reference formulation that is based on relative coordinates and describes deformation with respect to a co-rotated frame. Due to the modified strain tensor, the present formulation distinguishes significantly from standard nodal based nonlinear FE methods. Constraints are defined in integral form for every pair of surfaces of two bodies. This leads to a small number of constraint equations and avoids artificial stress singularities. The resulting mass and stiffness matrices are constant apart from a transformation based on a single rotation matrix for each body. The particular structure of this transformation allows to prevent from the usually expensive factorization of the system Jacobian within implicit time--integration methods. The present method has been implemented and tested with the FE-package NGSolve and specific 3D examples are verified with a standard beam formulation.     

Method for a Fast and Simple Dynamic Analysis of 2D and 3D Mechanisms

Here, an approach to describe the dynamics of 2D and 3D mechanisms that leads to a simpleand fast analysis is presented. Any position of an elastic mechanism may be defined usingonly rotations. Because in space the finite angles are not vectors, the Euler–Rodriguesparameters were adopted to describe the 3D rotations. The assembling process of the links intothe whole mechanism is natural and follows the standard finite element scheme. Lagrange multipliersare used only to impose the closed loop conditions. The approach developed in this work can be applied to 3D mechanisms composed by beams and having internal hinges as kinematics pairs, but itcan be generalized to other link shapes. The method is used either for mechanisms with rigid links,or with elastic ones, even for very deformable links that completely change the initial configuration.Both the floating and the absolute reference frame approach may be used, depending on the problem;passing from one formulation to the other is quite natural.If the links may be considered beams, the method starts from the exact equations written forthe deformed shape of each link and this provides a good accuracy. In this paper, a special finiteelement is presented: the unknowns of the problems being the nodal rotations or nodal Euler–Rodriguesparameters. Few nodes are requested for good accuracy. In general, as the number degrees of freedomper node is smaller than in the classical finite element approach, an important reducing of the totalnumber of nodal unknowns is obtained leading to an important reducing of the computer time. TheEuler–Bernoulli beam model was adopted, but the implementation of the Timoshenko beam model thattakes the shear efforts into account, is not difficult.     

The Absolute Coordinate Formulation with Elasto-Plastic Deformations

The present work contributes to the field of multibody systems with respect to the absolute coordinate formulation with a reduced expression of the strain energy and a non-linear constitutive model. Standard methods for multibody systems lead to highly non-linear terms either in the mass matrix or in the stiffness matrix and the most expensive part in the solution of the equations of motion is the assembling of these matrices, the computation of the Jacobian of the non-linear system and the solution of a linear system with the system matrices. In the present work, a consistent simplification of the equations of motion with respect to small deformations but large rigid-body motions is performed. The absolute coordinate formulation is used, therefore the total displacements are the unknowns. This formulation leads to a constant mass matrix while the non-linear stiffness matrix is composed of the constant small strain stiffness matrix rotated by the underlying rigid body rotation. Plastic strains are introduced by an additive split of the strain into an elastic and a plastic part, a yield condition and an associative flow rule. The decomposition of strain has to be performed carefully in order to obey the principle of objectivity for plasticity under large rigid body rotations. As an example, a two-dimensional plate which is hinged at one side and driven by a harmonic force at the opposite side is considered. Plastic deformation is assumed to occur due to extreme environmental influences or due to failure of some attached parts like a defect bearing.     

Large Deflection Analysis of a Thin Plate: Computer Simulations and Experiments

Many previous studies have conducted computer-aided simulations ofelastic bodies undergoing large deflections and deformations, but therehave not been many attempts to validate their numerical results. Thesubject of this paper is a thin clamped plate undergone large vibrationdue to attached end-point weight. The main aim of this paper is to showthe validity of the absolute nodal coordinate formulation (ANCF) bycomparing to the real experiments. Large oscillations of thin plates arestudied in the paper with taking into account effects of an attachedend-point weight and aerodynamic damping forces. The physicalexperiments are carried out using a high-speed camera and dataacquisition system. For numerical modeling of the plate, the absolutenodal coordinate formulation is used.     

On the use of absolute interface coordinates in the floating frame of reference formulation for flexible multibody dynamics

In this work a new formulation for flexible multibody systems is presented based on the floating frame formulation. In this method, the absolute interface coordinates are used as degrees of freedom. To this end, a coordinate transformation is established from the absolute floating frame coordinates and the local interface coordinates to the absolute interface coordinates. This is done by assuming linear theory of elasticity for a body’s local elastic deformation and by using the Craig–Bampton interface modes as local shape functions. In order to put this new method into perspective, relevant relations between inertial frame, corotational frame and floating frame formulations are explained. As such, this work provides a clear overview of how these three well-known and apparently different flexible multibody methods are related. An advantage of the method presented in this work is that the resulting equations of motion are of the differential rather than the differential-algebraic type. At the same time, it is possible to use well-developed model order reduction techniques on the flexible bodies locally. Hence, the method can be employed to construct superelements from arbitrarily shaped three dimensional elastic bodies, which can be used in a flexible multibody dynamics simulation. The method is validated by simulating the static and dynamic behavior of a number of flexible systems.     

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.     

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.     

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.     

Slope discontinuities in the finite element absolute nodal coordinate formulation: gradient deficient elements

In this paper, the treatment of the slope discontinuities in the finite element absolute nodal coordinate formulation (ANCF) is discussed. The paper explains the fundamental problems associated with developing a constant transformation that accounts for the slope discontinuities in the case of gradient deficient ANCF finite elements. A procedure that allows for the treatment of slope discontinuities in the case of gradient deficient finite elements which do not employ full parameterization is proposed for the special case of commutative rotations. The use of the proposed procedure leads to a constant orthogonal element transformation that describes the element initial configuration. As a consequence, one obtains in the case of large deformation and commutative rotations, a constant mass matrix for the structures. In order to achieve this goal, the concept of the intermediate finite element coordinate system is invoked. The intermediate finite element coordinate system used in this investigation serves to define the element reference configuration, follows the rotation of the structure, and maintains a fixed orientation relative to the structure coordinate system. Since planar rotations are always commutative, the procedure proposed in this investigation is applicable to all planar gradient deficient ANCF finite elements.