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.
Highlights
Within the field of flexible structural modelling, one-dimensional beam models display a surprisingly wide applicability to the representation of numerous structural applications such as the modelling of aircraft wings, flexible satellites, turbine blades, communication towers, energy harvesters, atomic force microscopes and DNA molecules, to name but a few
The kinematic description of the beam follows from knowledge of this local material deformation, as well as the span-varying orientation of the local body system, which may comprise a large rotation from the initial undeformed configuration; a global formulation of the flexible beam is attained in which geometric nonlinearity is inherently captured
The resulting dynamic equations are suitable for the representation of beam problems displaying generic span-varying geometric and material properties that typically exist in real structures; the underlying shape functions are kept general with candidate sets requiring only the satisfaction of a single kinematic constraint
Summary
Within the field of flexible structural modelling, one-dimensional beam models display a surprisingly wide applicability to the representation of numerous structural applications such as the modelling of aircraft wings, flexible satellites, turbine blades, communication towers, energy harvesters, atomic force microscopes and DNA molecules, to name but a few. Rather than reducing a high-order (large number of states) element-based model, the focus is cast instead on the low-order (minimal state) approximation of the full geometrically nonlinear dynamic problem To this end, a nonlinear beam shapes approach is employed which approximates the kinematic quantities distributed along the full length of the flexible structure without subdivision into smaller elements. Throughout these test cases, additional results are generated as a baseline for comparison using both MSC Nastran (a well-established finite-element package capable of treating geometric nonlinearity) and the intrinsic beam formulation of [13] implemented as a finite-element code [23,24] The latter modelling approach forms a suitable base of comparison as, in addition to constituting a well-documented and validated nonlinear treatment, the resulting equations of motion take the form of a high-order system with fairly sparse system matrices and maximum order 2 nonlinearity.
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