Alberto Cardona

Time integration of the equations of motion in mechanism analysis

The formulation of time integration algorithms for mechanism analysis problems is discussed. Particular aspects to be considered are the treatment of constraints and of the finite rotation associated terms. It is shown that in order to time integrate constrained systems, the algorithmic damping at infinite frequency is of utmost importance. It is demonstrated that the Newmark trapezoidal rule is unconditionally unstable in the presence of constraints. The use of second order accurate dissipative algorithms like the Hilber-Hughes-Taylor scheme is recommended. Finally, the integration of finite rotation associated terms is performed by projecting the equations onto the tangent space to the rotation group. In this way, well-known results of accuracy and convergence can be applied. Several examples illustrating the covered topics are presented.

A reduction method for nonlinear structural dynamic analysis

Abstract A computational algorithm for predicting the nonlinear dynamic response of a structure is presented. The nonlinear system of ordinary differential equations resulting from the finite element discretization is highly reduced by means of a Rayleigh-Ritz analysis. The basis vectors are chosen to be the current tangent eigenmodes together with some modal derivatives that indicate the way in which the spectrum is changing. Only a few basis updatings are required during the whole time integration. The truncation error introduced at every change of basis is pointed out as the cause for a divergence-type behaviour, and some means for eliminating it are discussed. Results for examples involving large displacements are shown and compared to the results obtained by integrating the complete system of equations.

Kinematics and dynamics of rigid and flexible mechanisms using finite elements and quaternion algebra

A technique for representing large finite rotations in terms of only three independent parameters, the conformal rotation vector, is described and applied to the finite element formulation of 3-D mechanisms problems. A beam finite element that takes into account large finite rotations and various types of rigid joints have been developed. Some test examples which demonstrate the applicability of the proposed technique are presented.

A LOAD-DEPENDENT BASIS FOR REDUCED NONLINEAR STRUCTURAL DYNAMICS

Abstract A computational algorithm for predicting the dynamical response of a nonlinear structure by means of a reduction scheme is described. In it, the nonlinear system of ordinary differential equations obtained from the finite element discretization is reduced, by employing a Rayleigh-Ritz technique. A new criterion for the computation of the basis vectors is proposed. It is based on a sequence of basis vectors proposed by Wilson et al. for linear dynamics problems, which was augmented by adding derivatives of these vectors with respect to the generalized displacement amplitudes. The later vectors allow the treatment of nonlinear problems.

On the Use of Lie Group Time Integrators in Multibody Dynamics

This paper proposes a family of Lie group time integrators for the simulation of flexible multibody systems. The method provides an elegant solution to the rotation parametrization problem. As an extension of the classical generalized-a method for dynamic systems, it can deal with constrained equations of motion. Second-order accuracy is demonstrated in the unconstrained case. The performance is illustrated on several critical benchmarks of rigid body systems with high rotation speeds, and second-order accuracy is evidenced in all of them, even for constrained cases. The remarkable simplicity of the new algorithms opens some interesting perspectives for real-time applications, model-based control, and optimization of multibody systems.

Thermomechanical model of a continuous casting process

A model for the analysis of thermal stresses arising at the early stage of a continuous casting process is proposed. The model is used to simulate the casting of round billets assuming axial symmetry. Thermal analysis takes into account phase-change in the material and heat transfer through the mould. The heat balance equation is solved using an Eulerian formulation for the billet domain, while the mould is analyzed using a Lagrangian one. Heat transfer from billet to mould is simulated assuming non-linear heat conduction through the air gap formed between them. Air conductivity is taken as a function of temperature. Two mechanical models are built assuming elastoplastic and viscoplastic hardening materials. Both Lagrangian and Arbitrary Lagrangian Eulerian (ALE) techniques have been implemented and compared. In the latter, advection of the history-dependent material parameters is taken into account using an appropriate finite volume integration scheme. Stresses and strains obtained with both models are compared, showing a good agreement.

Design of a new finite element programming environment

This paper presents the architecture for a new finite element program written in the C++ programming language. A powerful command interpreter allows the user not only to introduce data, but also to define the algorithms that will treat this data to obtain the desired results. In this way, the program can be very easily configured to new computational strategies. By following an object‐oriented programming technique, we expect the program would not fall into the “stagnation” state that affects large finite element codes currently in use.

Time-Step-Size-Independent Conditioning and Sensitivity to Perturbations in the Numerical Solution of Index Three Differential Algebraic Equations

We propose a simple preconditioning for the equations of motion of constrained mechanical systems in index three form. The scaling transformation is applied to the displacement-velocity-multiplier and to the reduced displacement-multiplier forms. The analysis of the transformed system shows that conditioning and sensitivity to perturbations become independent of the time step size, as in the case of well-behaved ordinary differential equations. The new scaling transformation is simple to implement and does not require the rewriting of the system equations as other approaches do. The theoretical analysis is confirmed by numerical examples.

Kinematic and dynamic analysis of mechanisms with cams

Abstract We present a formulation for describing cam/follower devices in the three-dimensional analysis of flexible mechanisms. The set of holonomic constraint equations that defines the behaviour of cams is developed. Cam and follower are treated indistinctly, leading to a general formulation that allows to model almost any shape of components. The cams shape can be expressed in either local cartesian or cylindrical coordinates. Contact between cam and follower may be intermittent, allowing to predict jump phenomena. Coulomb friction forces due to relative sliding motion is also considered.

Superelements Modelling in Flexible Multibody Dynamics

We present a new implementation of substructuring methods forflexible multibody analysis. In previous developed formulations, wefixed the local axes of the superelement to one node. In thisformulation, the reference frame is floating and close, in some sense,to the body center. The local frame is selected based on the positionsof the interface nodes of the superelement, and completely independentof the order in which the nodes of the superelement are given.Therefore, the superelement itself depends only on the nodes positions,and on the mass and stiffness properties, thus allowing a very easyinterfacing between the finite element program which computed thesuperelement and the mechanism analysis program.