Repetitive structures : Eigenanalysis continuum beam modelling and pre-twisted form

Abstract

Eigenanalysis is applied to three major types of repetitive structures: rigid-jointed, asymmetric, and pre-twisted.

For a 2-D rigid-jointed repetitive structure, the general idea of the State Variable Transfer Matrix Method is presented. This provides not only an exposition of the approach, but also justification for the treatment of rigid-jointed structures as pin-jointed.

For an asymmetric repetitive structure, a generalised eigenproblem is developed due to the non-existence of the conventional transfer matrix. Analysis of a 3-D NASA truss reveals couplings of tension-torsion, and bending-shear, and the equivalent continuum beam properties are determined. To fully understand these coupling effects, a 2-D planar asymmetric framework representing a single face of the 3-D truss is also analysed. Further, the continuum beam dynamic theories for the tension-torsion and bending-shear couplings are derived through the application of Hamilton's principle, and natural frequency predictions of the 3-D NASA truss are compared with those from FEM.

For a pre-twisted repetitive structure with uniform pre-twist rate, the Floquet system is transformed into an autonomous system by introducing a local coordinate system to define the transfer matrix, prior to eigenanalysis. For the multiple complex unity eigenvalues, near diagonal Jordan decompositions are employed to determine the simplest eigen- and principal vectors. Equivalent continuum properties including coupling coefficients are determined. The tension-torsion coupling agrees with established pre-twisted beam theory, but the bending and shear vectors cannot be fully explained according to existing approximate bending theory for pre-twisted structures. An in-depth study of tension-torsion coupling, both static and dynamic, is presented for structures with pre-twist angles per cell over the range of 0° to 180. Variations of the equivalent continuum properties are also evaluated over this range.

Moreover, an alternative analytical approach is developed for the continuum modelling of a symmetric repetitive structure, based on minimisation of potential energy of a single cell.

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