1. Introduction

In this work, we address the issues pertaining to dynamics of constrained finite rotations as a follow-up from the previous considerations in statics (Ibrahimbegovi? et al., 2001). The present considerations are of direct interest for nonlinear dynamics of smooth shells where, according to the classical shell theory (Naghdi, 1972), one can eliminate the rotation component around the shell-director and retain only two rotation parameters. Rotations in classical shell models are therefore unrestricted in size, but constrained in the space in the direction of the shell-director. There exists a number of possibilities as a choice for the rotation parameters of this kind (B chter and Ramm, 1992; Betsch et al., 1998; Brank and Ibrahimbegovi?, 2001) and related works. Among them, we believe, the prominent role is played by the incremental rotation vector (Ibrahimbegovi?, 1997a) as the most suitable parameter for the standard incremental solution strategy as well as for the construction of the time-stepping schemes. The latter is examined in more detail in this work in the context of Newmark implicit time-stepping schemes. A number of illustrative examples show a very satisfying performance of the proposed schemes. For the discretization in space a four-noded isoparametric shell finite element with continuum-consistent interpolations is used. This kind of interpolations allow one to treat shell finite rotations exclusively at the element nodes.

The outline of the paper is as follows. In Section 2 we recall governing equations for the dynamics of stress resultant geometrically exact shells. Equations are...