Daniel García-Vallejo

Efficient Evaluation of the Elastic Forces and the Jacobian in the Absolute Nodal Coordinate Formulation

This paper develops a new procedure for evaluating the elastic forces, the elastic energy and the jacobian of the elastic forces in the absolute nodal coordinate formulation. For this procedure, it is fundamental to use some invariant sparse matrices that are integrated in advance and have the property of transforming the evaluation of the elastic forces in a matrix multiplication process. The use of the invariant matrices avoids the integration over the volume of the element for every evaluation of the elastic forces. Great advantages can be achieved from these invariant matrices when evaluating the elastic energy and calculating the jacobian of the elastic forces as well. The exact expression of the jacobian of the differential system of equations of motion is obtained, and some advantages of using the absolute nodal coordinate formulation are pointed out. Numerical results show that there is important time saving as a result of the use of the invariant matrices.

Finite element analysis of the geometric stiffening effect. Part 2: Non-linear elasticity

Abstract In the first part of this paper, the relationship between the number of finite elements used to model the dynamics of rotating beams and the critical speed at which an incorrect solution is obtained when using linear elasticity theory is discussed. The increase in the number of finite elements leads to an increase in the critical speed when linear elasticity is used and no measures are taken, as recommended in the literature, to account for the effect of the coupling between the bending and axial displacements. In this part of the paper, a non-linear finite element model based on the absolute nodal coordinate formulation is used to study the dynamics of rotating beams. It is shown that, when the non-linear elasticity theory is used, a stable solution is always obtained regardless of the number of finite elements used. Numerical results of various simulations are presented in order to compare the solution of a three-dimensional rotating beam that is obtained using the absolute nodal coordinate formulation with the results previously reported in the literature. A finite element numerical study of the dynamics of a helicopter rotor blade is also presented in this investigation. It is shown that, when the finite element absolute nodal coordinate formulation is used in the analysis of helicopter blades, the problem of ill-conditioning that characterizes many of the existing formulations is not encountered.

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

Abstract 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.

A fast feedback controlled magnetic drive for the ASDEX Upgrade fast-ion loss detectors

A magnetically driven fast-ion loss detector system for the ASDEX Upgrade tokamak has been designed and will be presented here. The device is feedback controlled to adapt the detector head position to the heat load and physics requirements. Dynamic simulations have been performed taking into account effects such as friction, coil self-induction, and eddy currents. A real time positioning control algorithm to maximize the detector operational window has been developed. This algorithm considers dynamical behavior and mechanical resistance as well as measured and predicted thermal loads. The mechanical design and real time predictive algorithm presented here may be used for other reciprocating systems.

Finite element analysis of the geometric stiffening effect using the absolute nodal coordinate formulation

In this paper, the limitations of the linear elasticity finite element solutions in describing the coupling between the extensional and bending displacements are discussed. The fact that incorrect unstable solutions are obtained for models with more than one finite element using the linear elasticity theory motivates the analytical study of the rotating beam presented in this paper. It is shown, as documented in the literature, that the instability of the incorrect solution is directly related to the singularity of the stiffness matrix, and such an instability occurs when the angular velocity reaches the first bending fundamental frequency of the beam. The increase in the number of finite elements only leads to an increase of 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. A similar concept can be incorporated into the finite element floating frame of reference formulation in order to introduce coupling between the axial and bending displacements. Nonetheless, when the linear theory of elasticity is used and no special measures are taken to account for the coupling effect as proposed in the literature, there always exists a critical velocity regardless of the number of finite elements used.Copyright

Describing Rigid-Flexible Multibody Systems Using Natural and Absolute Nodal Coordinates

This paper deals with the dynamic description of interconnected rigid and flexible bodies. The absolute nodal coordinate formulation is used to describe the motion of flexible bodies and natural coordinates are used to describe the motion of the rigid bodies. The absolute nodal coordinate formulation is a non-incremental finite element procedure specially suitable for the dynamic analysis of flexible bodies exhibiting rigid body motion and large deformations. Nodal coordinates, that include global position vectors and global slopes, are all defined in a global inertial coordinate system. The advantages of using the absolute nodal coordinate formulation include constancy in the mass matrix, and the need for only a minimal set of non-linear constraint equations when connecting different flexible bodies with kinematic joints. When bodies within the system can be considered rigid, the above-mentioned advantages of the equations of motion can be preserved provided natural coordinates are used. In the natural coordinates method, the coordinates used to describe rigid bodies include global position vectors of basic points and global unit vectors. As in the absolute nodal coordinate formulation, rotational coordinates are avoided and the mass matrix is also constant. This paper provides computer implementation of this formulation that only uses absolute coordinates for general two-dimensional multibody systems. The constraint equations needed to define kinematic joints between different bodies can be linear or non-linear. The linear constraint equations, that include those needed to define rigid connections and revolute joints, are used to define constant connectivity matrices that reduce the size of the system coordinates. These constant connectivity matrices are also used to obtain the system mass matrix and the system generalized forces. However, the non-linear constraint equations that account for sliding joints, require the use of the Lagrange multipliers technique. Numerical examples are provided and compared to the results of other existing formulations.Copyright

A rotary and reciprocating scintillator based fast-ion loss detector for the MAST-U tokamak

The design and unique feature of the first fast-ion loss detector (FILD) for the Mega Amp Spherical Tokamak - Upgrade (MAST-U) is presented here. The MAST-U FILD head is mounted on an axially and angularly actuated mechanism that makes it possible to independently adapt the orientation [0°, 90°] and radial position [1.40 m, 1.60 m] of the FILD head, i.e., its collimator, thus maximizing the detector velocity-space coverage in a broad range of plasma scenarios with different q95. The 3D geometry of the detector has been optimized to detect fast-ion losses from the neutral beam injectors. Orbit simulations are used to calculate the strike map and predict the expected signals. The results show a velocity-space range of [4 cm, 13 cm] in gyroradius and [30°, 85°] in pitch angle, covering the entire neutral beam ion energy range. The optical system will provide direct sight of the scintillator and simultaneous detection with two cameras, giving high spatial and temporal resolution. The MAST-U FILD will shed light on the dominant fast-ion transport mechanisms in one of the worlds two largest spherical tokamaks through absolute measurements of fast-ion losses.

Role of Link Flexibility and Variable Stiffness Actuator on Collision Safety for Service Robots

The use of robots that share their workspace with humans in cooperative tasks, involves new risks for human safety. To ensure safety of the user, flexible robots and variable stiffness actuators are growing in interest. In this paper, a dynamic model of the collision between a 1 d.o.f. robot arm and a human head is presented. This model incorporates the many times neglected link and gear transmission flexibility. The contribution of the link flexibility and the variable stiffness actuator to human safety and to robot joint protection is evaluated. The head injury criterion and fracture force of cranial bones have been used as safety criteria for the human head.

Formulation of Three-Dimensional Rigid-Flexible Multibody Systems

Multibody systems generally contain solids the deformations of which are appreciable and which decisively influence the dynamics of the system. These solids have to be modeled by means of special formulations for flexible solids. At the same time, other solids are of such a high stiffness that they may be considered rigid, which simplifies their modeling. For these reasons, for a rigid-flexible multibody system, two types of formulations co-exist in the equations of the system. Among the different possibilities provided in bibliography on the material, the formulation in natural coordinates and the formulation in absolute nodal coordinates are utilized in this article to model the rigid and flexible solids, respectively. This article contains a mixed formulation based on the possibility of sharing coordinates between a rigid solid and a flexible solid. In addition, the fact that the matrix of the global mass of the system is shown to be constant and that many of the constraint equations obtained upon utilizing these formulations are linear and can be eliminated. In this work, the formulation presented is utilized to simulate a mechanism with both rigid and flexible components.Copyright

Problemas Numéricos en la Determinación de la Tensión de Cierre en Fatiga Mediante Elementos Finitos

Se analizan y discuten los problemas numericos que aparecen en la determinacion de la tension de cierre en fatiga mediante el metodo de los elementos finitos. El cierre de grieta y su determinacion tiene una especial importancia para la estimacion correcta de la vida a fatiga de la grieta. Sin embargo, la complejidad de modelar este proceso implica que la validez de los resultados sea dependiente de la metodologia y aproximaciones realizadas, implicando la aparicion de diversos problemas numericos. Los resultados muestran que para la determinacion exacta de las tensiones de apertura o cierre se requieren tamanos de malla muy finos, en un numero muy superior a las recomendaciones aceptadas hasta la fecha.