On the geometrically exact low-order modelling of a flexible beam: formulation and numerical tests

Abstract

This paper proposes a low-order geometrically exact flexible beam formulation based on the utilization of generic beam shape functions to approximate distributed kinematic properties of the deformed structure. The proposed nonlinear beam shapes approach is in contrast to the majority of geometrically nonlinear treatments in the literature in which element-based—and hence high-order—discretizations are adopted. The kinematic quantities approximated specifically pertain to shear and extensional gradients as well as local orientation parameters based on an arbitrary set of globally referenced attitude parameters. In developing the dynamic equations of motion, an Euler angle parametrization is selected as it is found to yield fast computational performance. The resulting dynamic formulation is closed using an example shape function set satisfying the single generic kinematic constraint. The formulation is demonstrated via its application to the modelling of a series of static and dynamic test cases of both simple and non-prismatic structures; the simulated results are verified using MSC Nastran and an element-based intrinsic beam formulation. Through these examples, it is shown that the nonlinear beam shapes approach is able to accurately capture the beam behaviour with a very minimal number of system states.

Efficient aeroelastic beam modelling and the selection of a structural shape basis

“…The work of this study is based around the nonlinear beam shapes formulation detailed in [9], allowing for the treatment of a geometric (and materially) nonlinear beam structure using a small set of describing states. The formulation is constructed using a global referenced attitude parameterisation of a body fixed local coordinate system; this coordinate system follows the deforming structure at each spanwise location along the beam.…”

“…The formulation is constructed using a global referenced attitude parameterisation of a body fixed local coordinate system; this coordinate system follows the deforming structure at each spanwise location along the beam. This concept is demonstrated by the deflected beam depicted in figure 1, also taken from reference [9]. The vector Γ(s) is used to denote the beam reference line in the global (X, Y, Z) system, where s is the curvilinear spanwise coordinate.…”

“…Given this parameterisation the beam reference line is recovered via spanwise integration with respect to this vector triad. In the absence of shear and extensional strain (see [9] for details of these terms) Γ [A] may be written as…”

“…: Positional vectors and coordinate systems used in expressing the deflected beam geometry; taken from [9].…”

“…Therefore, although a number of element-based structural codes have address the requirement for nonlinear modelling (e.g. absolute nodal coordinate [16], geometrically exact [17,18], co-rotational [5,4], intrinsic beam [8,13], strain-based element [6,19] to name but a few), this paper instead considers the nonlinear beam shapes formulation derived in [9] which uses global shape functions to significantly reduce the number of states over which the geometrically nonlinear problem is expressed. The choice of a suitable admissible shape set is central to realising this efficient representation.…”