Finite element analysis of the geometric stiffening effect. Part 1: A correction in the floating frame of reference formulation

2005 ◽  
Vol 219 (2) ◽  
pp. 187-202 ◽  
Author(s):  
D García-Vallejo ◽  
H Sugiyama ◽  
A A Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Frame Of Reference ◽  
Reference Formulation ◽  
Absolute Nodal Coordinate ◽  
Floating Frame Of Reference ◽  
Elastic Forces ◽  
Floating Frame ◽  
The Absolute ◽  
Finite Element Formulations

The fact that incorrect unstable solutions are obtained for linearly elastic models motivates the analytical study presented in this paper. The increase in the number of finite elements only leads to an increase in the critical speed. Crucial in the analysis presented in this paper is the fact that the mass matrix and the form of the elastic forces obtained using the absolute nodal coordinate formulation remain the same under orthogonal coordinate transformation. The absolute nodal coordinate formulation, in contrast to conventional finite element formulations, does account for the effect of the coupling between bending and extension. Based on the analytical results obtained using the absolute nodal coordinate formulation, a new correction is proposed for the finite element floating frame of reference formulation in order to introduce coupling between the axial and bending displacements. In this two-part paper, two- and three-dimensional finite element models are used to study the problem of rotating beams. The models are developed using the absolute nodal coordinate formulation that allows for accurate representation of the axial strain, thereby avoiding the ill-conditioning problem that arises when classical displacement-based finite element formulations are used. In the first part of the paper, the case of linear elasticity is considered and assumptions used in the finite element floating frame of reference formulation are investigated. In the second part of the paper, non-linear elasticity is considered. A rotating helicopter blade is simulated, and the complexity of the motion suggests the inclusion of rotary inertia, shear deformation, and non-linear elastic forces in order to obtain an accurate solution that does not suffer from the instability problem regardless of the number of finite elements used.

Elastic Forces and Coordinate Systems in the Finite Element Formulations

1999 ◽  
Author(s):  
Marcello Berzeri ◽  
Marcello Campanelli ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Frame Of Reference ◽  
Coordinate Systems ◽  
Absolute Nodal Coordinate ◽  
Floating Frame Of Reference ◽  
Elastic Forces ◽  
Floating Frame ◽  
The Absolute ◽  
Finite Element Formulations

Abstract The equivalence of the elastic forces of finite element formulations used in flexible multibody dynamics is the focus of this investigation. Two conceptually different finite element formulations that lead to exact modeling of the rigid body dynamics will be used. These are the floating frame of reference formulation and the absolute nodal coordinate formulation. It is demonstrated in this study that different element coordinate systems, which are used for the convenience of describing the element deformations in the absolute nodal coordinate formulation, lead to similar results as the element size is reduced. The equivalence of the elastic forces in the absolute nodal coordinate and the floating frame of reference formulations is shown. The result of this analysis clearly demonstrates that the instability observed in high speed rotor analytical models due to the neglect of the geometric centrifugal stiffening is not a problem inherent to a particular finite element formulation but only depends on the beam model that is used. 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. A new method is presented and used to obtain a simple expression for the elastic forces in the absolute nodal coordinate formulation. This method, which employs a nonlinear elastic strain-displacement relationship, does not result in an unstable solution when the angular velocity is increased.

On the Accuracy and Computational Costs of the Absolute Nodal Coordinate and the Floating Frame of Reference Formulation in Deformable Multibody Systems

2007 ◽  
Author(s):  
Markus Dibold ◽  
Johannes Gerstmayr ◽  
Hans Irschik
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Frame Of Reference ◽  
Reference Formulation ◽  
Absolute Nodal Coordinate ◽  
Computational Costs ◽  
Floating Frame Of Reference ◽  
Computational Performance ◽  
Floating Frame ◽  
The Absolute

In the present paper, a comparison of the absolute nodal coordinate formulation (ANCF) and the floating frame of reference formulation (FFRF) is performed for standard static and dynamic problems, both in the small and large deformation regime. Special emphasis is laid on the converged solutions and a comparison to analytical and numerical solutions from the literature. In addition to the work of previous authors, the computational performance of both formulations is studied for the dynamic case, where detailed information is provided concerning the different effects influencing the single parts of the computation time. In case of the ANCF finite element, a planar formulation based on the Bernoulli-Euler theory is utilized, consisting of two position and two slope coordinates in each node only. In the FFRF beam finite element, the displacements are described by the rigid body motion and a small superimposed transverse deflection. The latter is described by means of two static modes for the rotation at the boundary and a user-defined number of eigenmodes of the clamped-clamped beam. In numerical studies, the accuracy and computational costs of the two formulations are compared for a cantilever beam, a pendulum and a slider-crank mechanism. It turns out that both formulations have comparable performance and that the choice of the optimal formulation depends on the problem configuration. Recent claims in the literature that the ANCF would have deficiencies compared to the FFRF thus can be refuted.

A Detailed Comparison of the Absolute Nodal Coordinate and the Floating Frame of Reference Formulation in Deformable Multibody Systems

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Markus Dibold ◽  
Johannes Gerstmayr ◽  
Hans Irschik
Keyword(s):  
Finite Element ◽  
Numerical Solutions ◽  
Detailed Comparison ◽  
Frame Of Reference ◽  
Reference Formulation ◽  
Absolute Nodal Coordinate ◽  
Floating Frame Of Reference ◽  
Computational Performance ◽  
Floating Frame ◽  
The Absolute

In extension to a former work, a detailed comparison of the absolute nodal coordinate formulation (ANCF) and the floating frame of reference formulation (FFRF) is performed for standard static and dynamic problems, both in the small and large deformation regimes. Special emphasis is laid on converged solutions and on a comparison to analytical and numerical solutions from the literature. In addition to the work of previous authors, the computational performance of both formulations is studied for the dynamic case, where detailed information is provided, concerning the different effects influencing the single parts of the computation time. In case of the ANCF finite element, a planar formulation based on the Bernoulli–Euler theory is utilized, consisting of two position and two slope coordinates in each node only. In the FFRF beam finite element, the displacements are described by the rigid body motion and a small superimposed transverse deflection. The latter is described by means of two static modes for the rotation at the boundary and a user-defined number of eigenmodes of the clamped-clamped beam. In numerical studies, the accuracy and computational costs of the two formulations are compared for a cantilever beam, a pendulum, and a slider-crank mechanism. It turns out that both formulations have comparable performance and that the choice of the optimal formulation depends on the problem configuration. Recent claims in literature that the ANCF would have deficiencies compared with the FFRF thus can be refuted.

Multibody System Modeling of Leaf Springs

2004 ◽  
Vol 10 (11) ◽  
pp. 1601-1638 ◽  
Author(s):  
Mohamed A. Omar ◽  
Ahmed A. Shabana ◽  
Aki Mikkola ◽  
Wei-Yi Loh ◽  
Rena Basch
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Frame Of Reference ◽  
Leaf Spring ◽  
Reference Formulation ◽  
Order Model ◽  
Reduced Order ◽  
Absolute Nodal Coordinate ◽  
Leaf Springs ◽  
Floating Frame

Leaf springs are essential elements in the suspension systems of vehicles including sport utility vehicles, trucks, and railroad vehicles. Accurate modeling of the leaf springs is necessary in evaluating ride comfort, braking performance, vibration characteristics, and stability. In order to accurately model the deformations and vibrations of the leaf springs, nonlinear finite-element procedures, which account for the dynamic coupling between different modes of displacement, are employed. Two finite-element methods that take into account the effect of the distributed inertia and elasticity are discussed in this investigation to model the dynamics of leaf springs. The first is based on a floating frame of reference formulation, while the second is an absolute nodal coordinate formulation. The floating frame of reference formulation allows for using a reduced-order model by employing component mode synthesis techniques, while the absolute nodal coordinate formulation enables more detailed finite-element models for the large deformation of very flexible leaf springs. Methods for modeling the contact and friction between the leaves of the spring are discussed. A comparison is also presented between the results obtained using the proposed method and simplified approaches presented in the literature. While there are many issues that can be important in leaf spring modeling, the analysis presented in this paper is focused on a few key issues that include the computer implementation, the effect of the dynamic load on the spring stiffness, the selection of the vibration modes in the reduced-order model, and the effect of the structural damping on the response of the leaf spring.

Performance of the Incremental and Non-Incremental Finite Element Formulations in Flexible Multibody Problems

1999 ◽  
Vol 122 (4) ◽  
pp. 498-507 ◽  
Author(s):  
Marcello Campanelli ◽  
Marcello Berzeri ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Multibody Systems ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Multibody Systems ◽  
Large Displacement ◽  
Body Motion ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
The Absolute ◽  
Finite Element Formulations

Many flexible multibody applications are characterized by high inertia forces and motion discontinuities. Because of these characteristics, problems can be encountered when large displacement finite element formulations are used in the simulation of flexible multibody systems. In this investigation, the performance of two different large displacement finite element formulations in the analysis of flexible multibody systems is investigated. These are the incremental corotational procedure proposed in an earlier article (Rankin, C. C., and Brogan, F. A., 1986, ASME J. Pressure Vessel Technol., 108, pp. 165–174) and the non-incremental absolute nodal coordinate formulation recently proposed (Shabana, A. A., 1998, Dynamics of Multibody Systems, 2nd ed., Cambridge University Press, Cambridge). It is demonstrated in this investigation that the limitation resulting from the use of the infinitesmal nodal rotations in the incremental corotational procedure can lead to simulation problems even when simple flexible multibody applications are considered. The absolute nodal coordinate formulation, on the other hand, does not employ infinitesimal or finite rotation coordinates and leads to a constant mass matrix. Despite the fact that the absolute nodal coordinate formulation leads to a non-linear expression for the elastic forces, the results presented in this study, surprisingly, demonstrate that such a formulation is efficient in static problems as compared to the incremental corotational procedure. The excellent performance of the absolute nodal coordinate formulation in static and dynamic problems can be attributed to the fact that such a formulation does not employ rotations and leads to exact representation of the rigid body motion of the finite element. [S1050-0472(00)00604-8]

Application of Plasticity Theory and Absolute Nodal Coordinate Formulation to Flexible Multibody System Dynamics

2003 ◽  
Vol 126 (3) ◽  
pp. 478-487 ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Plasticity Theory ◽  
Position Vector ◽  
Element Formulation ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
The Absolute ◽  
Return Algorithm ◽  
Finite Element Formulations

The objective of this investigation is to develop a nonlinear finite element formulation for the elastic-plastic analysis of flexible multibody systems. The Lagrangian plasticity theory based on J2 flow theory is used to account for the effect of plasticity in flexible multibody dynamics. It is demonstrated that the principle of objectivity that is an issue when existing finite element formulations using rate-type constitutive equations are used is automatically satisfied when the stress and strain rate are directly calculated in the Lagrangian descriptions using the absolute nodal coordinate formulation employed in this investigation. This is attributed to the fact that, in the finite element absolute nodal coordinate formulation, the position vector gradients can completely define the state of rotation and deformation within the element. As a consequence, the numerical algorithm used to determine the plastic deformations such as the radial return algorithm becomes much simpler when the absolute nodal coordinate formulation is used as compared to existing finite element formulations that employ incrementally objective algorithms. Several numerical examples are presented in order to demonstrate the use of the formulations presented in the paper.

scholarly journals A concise nodal-based derivation of the floating frame of reference formulation for displacement-based solid finite elements

2019 ◽  
Vol 49 (3) ◽  
pp. 291-313 ◽  
Author(s):  
Andreas Zwölfer ◽  
Johannes Gerstmayr
Keyword(s):  
Finite Element ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Frame Of Reference ◽  
Body Motion ◽  
Reference Formulation ◽  
Lumped Mass ◽  
Floating Frame Of Reference ◽  
Mass Approximation ◽  
Floating Frame

AbstractThe Floating Frame of Reference Formulation (FFRF) is one of the most widely used methods to analyze flexible multibody systems subjected to large rigid-body motion but small strains and deformations. The FFRF is conventionally derived via a continuum mechanics approach. This tedious and circuitous approach, which still attracts attention among researchers, yields so-called inertia shape integrals. These unhandy volume integrals, arising in the FFRF mass matrix and quadratic velocity vector, depend not only on the degrees of freedom, but also on the finite element shape functions. That is why conventional computer implementations of the FFRF are laborious and error prone; they require access to the algorithmic level of the underlying finite element code or are restricted to a lumped mass approximation. This contribution presents a nodal-based treatment of the FFRF to bypass these integrals. Each flexible body is considered in its spatially discretized state ab initio, wherefore the integrals are replaced by multiplications by a constant finite element mass matrix. Besides that, this approach leads to a simpler and concise but rigorous derivation of the equations of motion. The steps to obtain the inertia-integral-free equations of motion (in 2D and 3D spaces) are presented in a clear and comprehensive way; the final result provides ready-to-implement equations of motion without a lumped mass approximation, in contrast to the conventional formulation.

Improved Description of Elastic Forces for the Absolute Nodal Coordinate Based Plate Element

2005 ◽  
Author(s):  
Marko K. Matikainen ◽  
Aki M. Mikkola
Keyword(s):  
Finite Element ◽  
Continuum Mechanics ◽  
Absolute Nodal Coordinate Formulation ◽  
Three Dimensional ◽  
Plate Element ◽  
Absolute Nodal Coordinate ◽  
Finite Element Software ◽  
Elastic Forces ◽  
The Absolute ◽  
The Continuum

In this study, the improved description of elastic forces for the absolute nodal coordinate based plate element is introduced. The absolute nodal coordinate formulation, which utilizes global displacements and slope coordinates as nodal variables, can be used in large rotation and deformation dynamic analysis of beam and plate structures. The formulation avoids difficulties that arise when a rotation is interpolated in three-dimensional applications. In the absolute nodal coordinate formulation, a continuum mechanics approach has become the dominating procedure when elastic forces are defined. It has recently been perceived, however, that the continuum mechanics based absolute nodal coordinate elements suffer from serious shortcomings, including Poisson’s locking and poor convergence rate. These problems can be circumvented by modifying the displacement field of a finite element in the definition of elastic forces. This allows the use of the mixed type interpolation technique, leading to accurate and efficient finite element formulations. This approach has been previously applied to two- and three-dimensional absolute nodal coordinate based finite elements. In this study, the improved approach for elastic forces is extended to the absolute nodal coordinate plate element. The introduced plate element is compared in static examples to the continuum mechanics based absolute nodal coordinate plate element, as well as to commercial finite element software.

Use of Plasticity Theory in Flexible Multibody System Dynamics

2003 ◽  
Author(s):  
Hiroyuki Sugiyama ◽  
Ahmed A. Shabana
Keyword(s):  
Finite Element ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Multibody Dynamics ◽  
Plasticity Theory ◽  
Position Vector ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
The Absolute ◽  
Return Algorithm ◽  
Finite Element Formulations

The objective of this investigation is to develop a general nonlinear finite deformation formulation for the elastic-plastic analysis of flexible multibody systems. The Lagrangian plasticity theory based on J2 flow theory is used to account for the effect of plasticity in flexible multibody dynamics. In addition, it is demonstrated that the principle of objectivity that is an issue when existing finite element formulations using ratetype constitutive equations are used can be fully satisfied when the stress and strain rate are directly calculated in the Lagrangian descriptions using the absolute nodal coordinate formulation employed in this investigation. This is attributed to the fact that, in the finite element absolute nodal coordinate formulation, the position vector gradients can completely define the state of rotation and deformation within the element. As a consequence, the numerical algorithm used to determine the plastic deformations such as the Radial Return Algorithm becomes much simpler when the absolute nodal coordinate formulation is used as compared to existing finite element formulations that employ incrementally objective algorithms. Several numerical examples are presented in order to demonstrate the use of the formulations presented in the paper.

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