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.     

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.     

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.     

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 Non-Incremental Finite Element Procedure for the Analysis of Large Deformation of Plates and Shells in Mechanical System Applications

In this investigation, a non-incremental solution procedure for the finite rotationand large deformation analysis of plates is presented. The method, whichis based on the absolute nodal coordinate formulation, leads to plateelements capable of representing exact rigid body motion. In thismethod, continuity conditions on all the displacement gradients areimposed. Therefore, non-smoothness of the plate mid-surface at the nodalpoints is avoided. Unlike other existing finite element formulationsthat lead to a highly nonlinear inertial forces for three-dimensionalelements, the proposed formulation leads to a constant mass matrix, andas a result, the centrifugal and Coriolis inertia forces are identicallyequal to zero. Furthermore, the method relaxes some of the assumptionsused in the classical and Mindlin plate models and automatically satisfiesthe objectivity requirements. By using a generalcontinuum mechanics approach, a relatively simple expression for theelastic forces is obtained. Generalization of the formulation to thecase of shell elements is discussed. As examples of the implementationof the proposed method, two different plate elements are presented; oneplate element does not guarantee the continuity of the displacementgradients between the nodal points, while the other plate elementguarantees this continuity. Numerical results are presented in order todemonstrate the use of the proposed method in the large rotation anddeformation analysis of plates and shells. The numerical results, whichare compared with the results obtained using existing incrementalprocedures, show that the solution obtained using the proposed methodsatisfies the principle of work and energy. These results are obtainedusing explicit numerical integration method. Potential applications ofthe proposed method include high-speed metal forming, vehiclecrashworthiness, rotor blades, and tires.     

A generalized geometrically nonlinear formulation with large rotations for finite elements with rotational degrees of freedoms

A generalized geometrically nonlinear formulation using total Lagrangian approach is presented for the finite elements with translational as well as rotational degrees of freedoms. An important aspect of the formulation presented here is that the restriction on the magnitude of the nodal rotations is eliminated by retaining true nonlinear nodal rotation terms in the definition of the element displacement field and the consistent derivation of the element properties based on this displacement field. The general derivation and the formulation steps are applicable to any element with translational and rotational nodal degrees of freedoms. The specific forms of the formulation for axisymmetric shells, two-dimensional isoparametric beams, curved shells, two-dimensional transition elements and solid-shell transition elements can be easily derived by considering the explicit forms of the nonlinear nodal rotations for the element at hand. The specific forms of this formulation have already been well tested and applied to various two- and three-dimensional elements, the results for some of which are presented here. Currently it is being applied to the three-dimensional isoparametric beam elements.     

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.     

A simple method to impose rotations and concentrated moments on ANC beams

Recently introduced ANC beam elements furnish a simple formulation that allows to solve nonlinear problems of beams, including those with large displacements and strains, as well as complex nonlinear (inelastic) materials. The success and simplicity of these finite elements is mainly due to the fact that the only nodal degrees of freedom that they employ are displacements, and rotations are thus completely avoided. This in turn makes it very difficult to apply concentrated moments or to impose rotations at specific nodes of a finite element mesh. In this article, we present a simple enhancement to this beam formulation that allows to apply these two types of boundary conditions in a simple manner, making ANC beam elements more versatile for both multibody and structural applications.     

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.     

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.     

Three- and four-noded planar elements using absolute nodal coordinate formulation

This paper investigates two new types of planar finite elements containing three and four nodes. These elements are the reduced forms of the spatial plate elements employing the absolute nodal coordinate approach. Elements of the first type use translations of nodes and global slopes as nodal coordinates and have 18 and 24 degrees of freedom. The slopes facilitate the prevention of the shear locking effect in bending problems. Furthermore, the slopes accurately describe the deformed shape of the elements. Triangular and quadrilateral elements of the second type use translational degrees of freedom only and, therefore, can be utilized successfully in problems without bending. These simple elements with 6 and 8 degrees of freedom are identical to the elements used in conventional formulation of the finite element method from the kinematical point of view. Similarly to the famous problem called “flying spaghetti” which is used often as a benchmark for beam elements, a kind of “flying lasagna” is simulated for the planar elements. Numerical results of simulations are presented.     

Substructure synthesis methods for dynamic analysis of multi-body systems

The formulation for the dynamic analysis of flexible multi-body systems that undergo large rigid body motion, leads to geometrically non-linear inertia properties due to large rotations. These inertia non-linearities that represent the coupling between gross rigid body motion and small elastic deformation, are dependent on the assumed displacement field. As alternatives to the finite element methods, deformable body shape functions and shape vectors are commonly employed to describe elastic deformation of linear structures. In this paper, substructure shape functions and shape vectors are used to describe elastic deformation of non-linear inertia-variant multi-body systems. This leads to two different representations of inertia nonlinearities; one is based on a consistent mass formulation, while the other is a lumped mass technique. The multi-body systems considered are collections of interconnected rigid and flexible bodies. Open and closed loop systems are permitted.     

Nonlinear finite element analysis of elastic frames

A very simple and effective formulation and numerical procedure to remove the restriction of small rotations between two successive increments for the geometrically nonlinear finite element analysis of in-plane frames is presented. A co-rotational formulation combined with small deflection beam theory with the inclusion of the effect of axial force is adopted. A body attached coordinate is used to distinguish between rigid body and deformational rotations. The deformational nodal rotational angles are assumed to be small, and the membrane strain along the deformed beam axis obtained from the elongation of the arc length of the deformed beam element is assumed to be constant. The element internal nodal forces are calculated using the total deformational nodal rotations in the body attached coordinate. The element stiffness matrix is obtained by superimposing the bending and the geometric stiffness matrices of the elementary beam element and the stiffness matrix of the linear bar element. An incremental iterative method based on the Newton-Raphson method combined with a constant arc length control method is employed for the solution of the nonlinear equilibrium equations. In order to improve convergence properties of the equilibrium iteration, a two-cycle iteration scheme is introduced. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.     

Arbitrary active constrained layer damping treatments on beams: Finite element modelling and experimental validation

This paper concerns the analytical formulation and finite element modelling of arbitrary active constrained layer damping (ACLD) treatments applied to beams. A partial layerwise theory is utilized to define the displacement field of beams with an arbitrary number of elastic, viscoelastic and piezoelectric layers attached to both surfaces, and a fully coupled electro-mechanical theory is considered for modelling the behavior of the piezoelectric layers. The damping of the viscoelastic layers is modelled by the complex modulus approach. The weak forms of the analytical formulation, governing the motion and electric charge equilibrium, are presented. Based on the weak forms, a one-dimensional finite element (FE) model is developed, with the nodal mechanical degrees of freedom being the axial displacement, transverse displacement and the rotation of the mid-plane of the host beam and the rotations of the individual layers, and the electrical elemental degrees of freedom being the electrical potential difference of each piezoelectric layer. Frequency response functions were measured experimentally and evaluated numerically for a freely suspended aluminium beam with an ACLD patch. In order to validate the FE model the results are presented and discussed.     

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.     

On large displacement-small strain analysis of structures with rotational degrees of freedom

The matrix displacement analysis of geometrically nonlinear structures becomes an intricate task as soon as finite elements in space with rotational degrees of freedom are considered. The fundamental reason for these difficulties lies in the non-commutativity of successive finite rotations about fixed axes with different directions. In order to circumvent this difficulty, a new definition of rotations — the so-called semitangential rotations — is introduced in this paper. Our new definition leads to a reformulation of the theory of [1,2]which in itself is clearly consistent and correct.In contrast to rotations about fixed axes these semitangential rotations which correspond to the semitangential torques of Ziegler [3]possess the most important property of being commutative. In this manner, all complexities involved in the standard definition of rotations are avoided ab initio.A specific aspect of this paper is a careful exposition of semitangential torques and rotations, as well as the consequences of the semitangential definitions for the geometrical stiffness of finite elements. In fact, these new definitions permit a very simple and consistent derivation of the geometrical stiffness matrices. Moreover, the semitangential definition automatically leads to a symmetric geometrical stiffness which clearly expresses that the nonlinear strain-displacement relations must satisfy the condition of conservativity of the structure itself — independently of any loading.The general theory of geometrical stiffness matrices as evolved in this paper is applied to beams in space. The consistency of the theory is demonstrated by a large number of numerical examples not only of straight beams but also of the lateral and torsional buckling and post-buckling behaviour of stiff-joined frames. Most of the former developments appear to be inadequate.     

Spatial joint constraints for the absolute nodal coordinate formulation using the non-generalized intermediate coordinates

In this investigation, a systematic procedure that can be used for modeling joint constraints for the absolute nodal coordinate formulation is developed. To this end, the non-generalized intermediate coordinates are introduced to derive a mapping between the generalized gradient coordinates and the non-generalized rotation parameters. With this mapping, a wide variety of joint constraints can be defined for the absolute nodal coordinate formulation in terms of the non-generalized reference coordinates and, therefore, existing well-developed constraint libraries formulated for the rigid body reference coordinates can be directly employed without significant modifications in existing codes. Furthermore, in order to define a rigid surface at the joint definition point, a set of orthonormality conditions is imposed on the gradient coordinates. This leads to not only accurate modeling of interface to mechanical joint, but also a simpler definition of the joint coordinate system obtained by the orthonormal gradient vectors. For this reason, a simpler form of constraint Jacobian and quadratic velocity vectors can be obtained as compared to those of the existing approach which requires the use of highly nonlinear joint coordinate system. A systematic procedure for eliminating the non-generalized coordinates and the dependent Lagrange multipliers associated with the coordinate mapping equations from the equations of motion is presented. As a result, a standard augmented form of the equations of motion can be obtained in terms of the generalized coordinates only. Several numerical examples are presented in order to demonstrate the use of the joint constraint formulation developed in this investigation.     

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.     

A geometrical nonlinear eccentric 3D-beam element with arbitrary cross-sections

In this paper a finite element formulation of eccentric space curved beams with arbitrary cross-sections is derived. Based on a Timoshenko beam kinematic, the strain measures are derived by exploitation of the Green-Lagrangean strain tensor. Thus, the formulation is conformed with existing nonlinear shell theories. Finite rotations are described by orthogonal transformations of the basis systems from the initial to the current configuration. Since for arbitrary cross-sections the centroid and shear center do not coincide, torsion bending coupling occurs in the linear as well as in the finite deformation case. The linearization of the boundary value formulation leads to a symmetric bilinear form for conservative loads. The resulting finite element model is characterized by 6 degrees of freedom at the nodes and therefore is fully compatible with existing shell elements. Since the reference curve lies arbitrarily to the line of centroids, the element can be used to model eccentric stiffener of shells with arbitrary cross-sections.