Karin Nachbagauer

A Spatial Thin Beam Finite Element Based on the Absolute Nodal Coordinate Formulation Without Singularities

A three-dimensional nonlinear finite element for thin beams is proposed within the absolute nodal coordinate formulation (ANCF). The deformation of the element is described by means of displacement vector, axial slope and axial rotation parameter per node. The element is based on the Bernoulli-Euler theory and can undergo coupled axial extension, bending and torsion in the large deformation case. Singularities — which are typically caused by such parameterizations — are overcome by a director per element node. Once the directors are properly defined, a cross sectional frame is defined at any point of the beam axis. Since the director is updated during computation, no singularities occur. The proposed element is a three-dimensional ANCF Bernoulli-Euler beam element free of singularities and without transverse slope vectors. Detailed convergence analysis by means of various numerical examples and comparison to analytical solutions shows the performance and accuracy of the element.Copyright

A 3D Shear Deformable Finite Element Based on the Absolute Nodal Coordinate Formulation

The absolute nodal coordinate formulation (ANCF) has been developed for the modeling of large deformation beams in two or three dimensions. The absence of rotational degrees of freedom is the main conceptual difference between the ANCF and classical nonlinear beam finite elements that can be found in literature. In the present approach, an ANCF beam finite element is presented, in which the orientation of the cross section is parameterized by means of slope vectors. Based on these slope vectors, a thickness as well as a shear deformation of the cross section is included. The proposed finite beam element is investigated by an eigenfrequency analysis of a simply supported beam. The high frequencies of thickness modes are of the same magnitude as the shear mode frequencies. Therefore, the thickness modes do not significantly influence the performance of the finite element in dynamical simulations. The lateral buckling of a cantilevered right-angle frame under an end load is investigated in order to show a large deformation example in statics, as well as a dynamic application. A comparison to results provided in the literature reveals that the present element shows accuracy and high order convergence.

Optimal input design for multibody systems by using an extended adjoint approach

We present a method for optimizing inputs of multibody systems for a subsequently performed parameter identification. Herein, optimality with respect to identifiability is attained by maximizing the information content in measurements described by the Fisher information matrix. For solving the resulting optimization problem, the adjoint system of the sensitivity differential equation system is employed. The proposed approach combines these two well-established methods and can be applied to multibody systems in a systematic, automated manner. Furthermore, additional optimization goals can be added and used to find inputs satisfying, for example, end conditions or state constraints.

HOTINT: A Script Language Based Framework for the Simulation of Multibody Dynamics Systems

The multibody dynamics and finite element simulation code has been developed since 1997. In the past years, more than 10 researchers have contributed to certain parts of HOTINT, such as solver, graphical user interface, element library, joint library, finite element functionality and port blocks. Currently, a script-language based version of HOTINT is freely available for download, intended for research, education and industrial applications. The main features of the current available version include objects like point mass, rigid bodies, complex point-based joints, classical mechanical joints, flexible (nonlinear) beams, port-blocks for mechatronics applications and many other features such as loads, sensors and graphical objects. HOTINT includes a 3D graphical visualization showing the results immediately during simulation, which helps to reduce modelling errors. In the present paper, we show the current state and the structure of the code. Examples should demonstrate the easiness of use of HOTINT.Copyright

On the rotational equations of motion in rigid body dynamics when using Euler parameters

Many models of three-dimensional rigid body dynamics employ Euler parameters as rotational coordinates. Since the four Euler parameters are not independent, one has to consider the quaternion constraint in the equations of motion. This is usually done by the Lagrange multiplier technique. In the present paper, various forms of the rotational equations of motion will be derived, and it will be shown that they can be transformed into each other. Special attention is hereby given to the value of the Lagrange multiplier and the complexity of terms representing the inertia forces. Particular attention is also paid to the rotational generalized external force vector, which is not unique when using Euler parameters as rotational coordinates.

The Absolute Nodal Coordinate Formulation

The key idea of the absolute nodal coordinate formulation (ANCF) is to use slope vectors in order to describe the orientation of the cross-section of structural mechanics components, such as beams, plates or shells. This formulation relaxes the kinematical assumptions of Bernoulli–Euler and Timoshenko beam theories and enables a deformation of the cross-sections. The present contribution shows how to create 2D and 3D structural finite elements based on the ANCF by employing different sets of slope vectors for approximating the cross-sections’ orientation. A specific aim of this chapter is to present a unified notation for structural mechanics and continuum mechanics ANC formulations. Particular focus is laid on enhanced formulations for such finite elements that circumvent severe issues like Poisson or shear locking. The performance of these elements is evaluated and a detailed assessment comprising the convergence order, the number of iterations, and Jacobian updates for large deformation benchmark problems is provided.

Nonlinear Finite Element Modelling of Moving Beam Vibrations Controlled by Distributed Actuators

In many applications, nonlinear beams undergoing bending, axial and shear deformation are important structural elements. In the present paper, a shear deformable beam finite element is presented for such applications. Since displacements and displacement gradients are chosen as the nodal degrees of freedom, an equivalent displacement and rotation interpolation is retrieved. The definition of strain energy is based on Reissner’s nonlinear rod theory with special strain measures for axial strain, shear strain and bending strain. Furthermore, a thickness deformation is introduced by adding an according term to the virtual work of internal forces. This underlying formulation is extended for piezo-electric actuation. The obtained beam finite elements are applied to a two-link robot with two flexible arms with tip masses. Distributed and concentrated masses cause flexural vibrations, which are compensated by means of piezo-electric actuators attached to the arms. A numerical example of a highly flexible robot with piezo-electric actuation and feedforward control is presented to show the applicability of the finite element.

A frequency domain approach for parameter identification in multibody dynamics

The adjoint method shows an efficient way to incorporate inverse dynamics to engineering multibody applications, as, e.g., parameter identification. In case of the identification of parameters in oscillating multibody systems, a combination of Fourier analysis and the adjoint method is an obvious and promising approach. The present paper shows the adjoint method including adjoint Fourier coefficients for the parameter identification of the amplitude response of oscillations. Two examples show the potential and efficiency of the proposed method in multibody dynamics.

The discrete adjoint method for parameter identification in multibody system dynamics

The adjoint method is an elegant approach for the computation of the gradient of a cost function to identify a set of parameters. An additional set of differential equations has to be solved to compute the adjoint variables, which are further used for the gradient computation. However, the accuracy of the numerical solution of the adjoint differential equation has a great impact on the gradient. Hence, an alternative approach is the discrete adjoint method, where the adjoint differential equations are replaced by algebraic equations. Therefore, a finite difference scheme is constructed for the adjoint system directly from the numerical time integration method. The method provides the exact gradient of the discretized cost function subjected to the discretized equations of motion.

An Efficient Treatment of Parameter Identification in the Context of Multibody System Dynamics Using the Adjoint Method

In multibody system dynamics, a wide range of parameters can occur where some of them may not be known a priori. Therefore, this work presents an efficient adjoint method for parameter identification that can be utilized in multibody simulation software. Compared to standard system sensitivity based approaches the adjoint method has the major advantage of being independent on the number of parameters to identify. Especially when dealing with large and probably flexible multibody systems this characteristic is crucial. Formulating parameter identification as an automatable procedure, of course, leads to a complicated structure of the involved matrices and equations. However, during a forward simulation of the system many of the matrices needed for solving the so called “adjoint system equations” are already evaluated. Adopting the functionality of the forward solver for the adjoint system solver therefore results in little additional effort. In order to illustrate the performance of the adjoint method two examples are presented. A planar example shows the possibility of identifying non-linear control parameters and a three-dimensional example is presented for identifying time-invariant inertia parameters.