Bassam A. Hussein

Coupled Deformation Modes in the Large Deformation Finite-Element Analysis: Problem Definition

In the classical formulations of beam problems, the beam cross section is assumed to remain rigid when the beam deforms. In Euler-Bernoulli beam theory, the rigid cross section remains perpendicular to the beam centerline; while in the more general Timoshenko beam theory the rigid cross section is permitted to rotate due to the shear deformation, and as a result, the cross section can have an arbitrary rotation with respect to the beam centerline. In more general beam models as the ones based on the absolute nodal coordinate formulation (ANCF), the cross section is allowed to deform and it is no longer treated as a rigid surface. These more general models lead to new geometric terms that do not appear in the classical formulations of beams. Some of these geometric terms are the result of the coupling between the deformation of the cross section and other modes of deformations such as bending and they lead to a new set of modes referred to in this paper as the ANCF-coupled deformation modes. The effect of the ANCF-coupled deformation modes can be significant in the case of very flexible structures. In this investigation, three different large deformation dynamic beam models are discussed and compared in order to investigate the effect of the ANCF-coupled deformation modes. The three methods differ in the way the beam elastic forces are calculated. The first method is based on a general continuum mechanics approach that leads to a model that includes the ANCF-coupled deformation modes; while the second method is based on the elastic line approach that systematically eliminates these modes. The ANCF-coupled deformation modes eliminated in the elastic line approach are identified and the effect of such deformation modes on the efficiency and accuracy of the numerical solution is discussed. The third large deformation beam model discussed in this investigation is based on the Hellinger-Reissner principle that can be used to eliminate the shear locking encountered in some beam models. Numerical examples are presented in order to demonstrate the use and compare the results of the three different beam formulations. It is shown that while the effect of the ANCF-coupled deformation modes is not significant in very stiff and moderately stiff structures, the effect of these modes can not be neglected in the case of very flexible structures.

Use of the floating frame of reference formulation in large deformation analysis: Experimental and numerical validation

Abstract The finite-element floating frame of reference (FFR) formulation is used, for the most part, in the small deformation analysis of flexible bodies that undergo large reference displacements. This formulation allows for filtering out systematically complex shapes associated with high frequencies that have no significant effect on the solution in the case of small deformations. The resulting low-order FFR models have been widely used to obtain efficient and accurate solutions for many engineering and physics applications. In this investigation, the use of the FFR formulation in the large deformation analysis is examined, and it is demonstrated that large deformation FFR models can be accurate in applications, where the deformation can be described using simple shapes as it is the case in robot system manipulators. In these cases, the standard finite-element FFR formulation must be used with non-linear strain—displacement relationships that account for the geometric non-linearities. The results obtained using the large deformation FFR models are compared with the results obtained using the large deformation absolute nodal coordinate formulation (ANCF), which does not allow for the use of linear modes. The ANCF models are developed using two different methods for formulating the elastic forces: the basic continuum mechanics approach (ANCF-BC) and the elastic line method (ANCF-EL). While the explicit Adams method can be used to obtain the numerical solution of the FFR model, two implicit integration methods are implemented in order to be able to obtain an efficient solution of the FFR and ANCF models. These implicit integration methods are the RADAU5 method and the Hilber—Hughes—Taylor (HHT) method. In the case of simple large deformation shapes, the simulation results obtained in this study show a good agreement between the FFR and the ANCF solutions. The results also show that, in the case of thin and stiff beams, the coupled deformation modes that result from the use of the ANCF-BC can be a source of numerical and locking problems, as reported in the literature. These ANCF-BC numerical problems can be circumvented using the implicit HHT integration method. Nonetheless, the HHT integrator does not capture high-frequency FFR axial modes which are necessary in order to obtain accurate solutions for high-speed rotating beams. In addition to the comparison with the ANCF solutions, experimental results of a forward dynamics model are used in this study to validate the large deformation FFR numerical solutions. The experimental set-up used in the validation of the numerical solutions is also described in this investigation.

Coupled Deformation Modes in the Large Deformation Finite Element Analysis: Generalization

The effect of the absolute nodal coordinate formulation (ANCF)-coupled deformation modes on the accuracy and efficiency when higher order three-dimensional beam and plate finite elements are used is investigated in this study. It is shown that while computational efficiency can be achieved in some applications by neglecting the effect of some of the ANCF-coupled deformation modes, such modes introduce geometric stiffening/ softening effects that can be significant in the case of very flexible structures. As shown in previous publications, for stiff structures, the effect of the ANCF-coupled deformation modes can be neglected. For such stiff structures, the solution does not strongly depend on some of the ANCF-coupled deformation modes, and formulations that include these modes lead to numerical results that are in good agreement with formulations that exclude them. In the case of a very flexible structure, on the other hand, the inclusion of the ANCF-coupled deformation modes becomes necessary in order to obtain an accurate solution. In this case of very flexible structures, the use of the general continuum mechanics approach leads to an efficient solution algorithm and to more accurate numerical results. In order to examine the effect of the elastic force formulation on the efficiency and the coupling between different modes of deformation, three different models are used again to formulate the elastic forces in the absolute nodal coordinate formulation. These three methods are the general continuum mechanics approach, the elastic line (midsurface) approach, and the elastic line (midsurface) approach with the Hellinger-Reissner principle. Three-dimensional absolute nodal coordinate formulation beam and plate elements are used in this study. In the general continuum mechanics approach, the coupling between the cross section deformation and the beam centerline or plate midsurface displacement is considered, while in the approaches based on the elastic line and the Hellinger-Reissner principle, this coupling is neglected. In addition to the fully parametrized beam element used in this study, three different plate elements, two fully parametrized and one reduced order thin plate elements, are used. The numerical results obtained using different finite elements and elastic force formulations are compared in this study.

FLOATING FRAME OF REFERENCE AND ABSOLUTE NODAL COORDINATE FORMULATIONS IN THE LARGE DEFORMATION ANALYSIS OF ROBOTIC MANIPULATORS: A COMPARATIVE EXPERIMENTAL AND NUMERICAL STUDY

This paper describes the use of flexible multibody system approaches in the dynamic modeling of interconnected rigid-elastic robotic manipulators. Two approaches are used to establish the flexible robot dynamic model; the floating frame of reference formulation (FFR) and the absolute nodal coordinate formulation (ANCF). The ANCF is used with two different methods for formulating the elastic forces; basic continuum mechanics approach (ANCF-BC) and elastic line method (ANCF-EL). The simulation results show that the use of the nonlinear FFR and the ANCF-EL improves the performance of the beam element in the modeling of flexible robotic manipulators. In the case of simple large deformation shape, the simulation results obtained show a good agreement between the FFR and the ANCF solutions. In the case of thin and stiff beams, the coupled deformation modes that result from the use of the ANCF-BC can be a source of numerical problems. These problems can be avoided using the implicit Hilber-Hughes-Taylor (HHT) integration method. On the other hand, HHT integrator does not capture high frequency axial modes when the FFR is used; RADAU5 method is used instead. The experimental results of the direct dynamics model are effectively used in this study to validate the numerical solutions.Copyright

Absolute Nodal Coordinate Formulation Coupled Deformation Modes

The finite element absolute nodal coordinate formulation (ANCF) leads to beam and plate models that relax the assumption of the classical Euler-bernoulli and Timoshenko beam and Mindlin plate theories. In these more general models, the cross section is allowed to deform and it is no longer treated as a rigid surface. The coupling between the bending and cross section deformations leads to the new ANCF-coupled deformation modes that are examined in this study. While these coupled deformation can be source of numerical and convergence problems when thin and stiff beam models are considered, the inclusion of the effect of these modes in the dynamic model is necessary in the case of very flexible structures. In order to examine the effect of these coupled deformation modes in this investigation, three different large deformation dynamic beam models are discussed. Two of these models, which differ in the way the beam elastic forces are calculated in the absolute nodal coordinate formulation, allow for systematically eliminating the coupled deformation modes, while the third allows for including these modes. The first of these models is based on a general continuum mechanics approach that leads to a model that includes the ANCF-coupled deformation modes; while the second and third methods that can be used to eliminate the coupled deformation modes are based on the elastic line approach and the Hellinger-Reissner principle. It is shown in this study that the inclusion of the ANCF-coupled deformation modes introduces geometric stiffening effects that can not be captured using other finite element models.Copyright