We present a new methodology for the integration of general non-linear multibody systems within a finite-element framework, with special attention to numerical robustness. The outcome is a non-linearly unconditionally stable algorithm with dissipation properties. This algorithm exactly preserves the total linear and angular momenta of holonomically constrained multibody systems, which implies the satisfaction of Newton's Third law of Action and Reaction. Furthermore, the scheme strictly dissipates the total mechanical energy of the system. This is accomplished by selective damping of the unresolved high-frequency components of the response. We derive the governing equations relying on the 6-D compact representation of motion and we employ a parameterization based on the Cayley transform which ensures geometric invariance of the resulting numerical schemes. We present some numerical tests in order to illustrate the main features of the methodology, and to demonstrate the properties predicted in the analysis.

J. Chróscielewski, J. Makowski and W. Pietraszkiewicz. Non-linear dynamics of flexible shell structures. CAMES 2002 (9)

The initial-boundary value problem in the weak form is formulated for the general six-field non-linear theory of branched shell structures. The extended time-stepping algorithm of the Newmark type is worked out for the non-linear dynamic analysis on the configuration space containing the rotation group *SO*(3). Within the finite element approximation, an accurate indirect *C0* interpolation procedure on *SO*(3) with a transport of approximation domain is developed. Numerical simulations by the finite element method of 2D and 3D large overall motions of several flexible elastic shell structures are presented. It is shown that values of potential and kinetic energies may oscillate in time, but the total energy remains conserved during the free motion of the structures in space.

M. Gofron. Dynamics of a sprayer with large flexible boom. CAMES 2002 (9)

The quality of spraying chemicals in a field depends on the distance between the boom of the sprayer and the canopy. Keeping that distance relatively constant enables even distribution of chemicals over the field. However, because the boom has a significant moment of inertia due to its length, (commonly 30 [m] and above) the vehicle has a tendency to roll. The excessive rolling significantly decreases the quality of spraying and can even cause damage to the boom if the tip of the boom hits the ground. The boom itself deflects significantly due to its flexibility and can increase the total amplitude of the boom tip point movement during spraying operation. In this study the effect of the boom flexibility on vehicle rolling, boom rolling and boom reaction forces is evaluated. Also the effect of number of modes selected to represent flexible model, on the boom tip point deflection is analyzed. The simulation model of the sprayer is developed in DADS multibody code and mode shapes of the boom are obtained from I-DEAS code. The simulation model of the sprayer is driven over a ramp and a numerical representation of the NATC (Nevada Automotive Test Center) track.

J. Gonçalves and J. Ambrósio. Advanced modelling of flexible multibody systems using virtual bodies. CAMES 2002 (9)

When new formulations for the description of flexible multibody systems are proposed, often they imply the use of new sets of generalized coordinates, even if the finite element method is used to describe the system flexibility. The adoption of such formulations implies that an additional effort must be made to describe the kinematic constraints that involve flexible bodies. The commercial multibody codes generally have good kinematic joint libraries for rigid bodies, but they are limited in the type of joints available in what flexible bodies are concerned. This work proposes and demonstrates that such limitations can be overcome by using virtual rigid bodies. The idea is to develop a single kinematic joint that restricts all relative degrees of freedom between one or more nodes of the flexible body and a rigid body. The designation of virtual body derives from assuming that it is a massless rigid body. In this form any of the kinematic joints between rigid bodies available in the multibody code libraries, can be used. In the process it is shown that the interaction of the user with the multibody code is much simpler. The numerical problems resulting from ill-conditioned mass matrix, due to the null inertias of the virtual bodies, are avoided by using a sparse matrix solver for the solution of the equations of motion. The proposed formulation is applied to a complex flexible multibody system, represented by the model of a road vehicle with flexible chassis, the results are presented and the discussion on the relative virtues and drawbacks of the current methodologies is made with emphasis on the models and algorithms used.

S.-S. Kim. Nordsieck form of multirate integration method for flexible multibody dynamic analysis. CAMES 2002 (9)

A Nordsieck form of multirate integration scheme has been proposed for flexible multibody dynamic systems of which motions are represented by large gross motion coupled with small vibration. Based on the conventional flexible multibody dynamics formulation, vibrational modal coordinates with floating reference frame and relative joint coordinates are employed to describe the motion in this research. In the multirate integration, the fast variables of the flexible multibody system are integrated with smaller stepsize, whereas the slow variables are integrated with larger stepsize. It is assumed that vibrational modal coordinates are treated as fast variables, whereas the relative joint coordinates are treated as slow variables to apply multirate integration method. A method that decomposes the equations of motion for flexible multibody systems into a fast system with flexible coordinates and a slow system with joint relative coordinates has been also proposed. The proposed multirate integration method is based on the Adams-Bashforth-Moulton predictor-corrector method and implemented in the Nordsieck vector form. The Nordsieck form of multirate integration method provides effective step-size control and at the same time, inherits the efficiency from the Adams integration method. Simulations of a flexible gun and turret system of a military tank have been carried out to show the effectiveness and efficiency of the proposed method.

W. Wagner and F. Gruttmann. Modeling of shell-beam transitions in the presence of finite rotations. CAMES 2002 (9)

A finite element formulation for a transition element between shells and beam structures is described in this paper. The elements should allow changes between models in an `optimal' way without or with little disturbances which decrease rapidly due to the principle of Saint-Venant. Thus, the constraints are formulated in such a way that a transverse contraction within the coupling range is possible. The implementation of the coupling conditions is done with the Penalty Method or the Augmented Lagrange Method. The element formulation is derived for finite rotations. Same rotational formulations are used in beam and shell elements. Rotational increments up to an angle of 2*pi* are possible without singularities based on a multiplicative update procedure. It can be shown that the transition to rigid bodies can be derived with some modifications. Examples prove the reliability of the transition formulation. Here simple element tests and practical applications are shown.