A computationally efficient non-linear beam model

B.A. Freno, P.G.A. Cizmas
International Journal of Non-Linear Mechanics Volume 46, Issue 6, July 2011, Pages 854-869

Abstract

Aerospace structures with large aspect ratio, such as airplane wings, rotorcraft blades, wind turbine blades, and jet engine fan and compressor blades, are particularly susceptible to aeroelastic phenomena. Finite element analysis provides an effective and generalized method to model these structures; however, it is computationally expensive. Fortunately, the large aspect ratio of these structures is exploitable as these potential aeroelastically unstable structures can be modeled as cantilevered beams, drastically reducing computational time.

In this paper, the non-linear equations of motion are derived for an inextensional, non-uniform cantilevered beam with a straight elastic axis. Along the elastic axis, the cross-sectional center of mass can be offset in both dimensions, and the principal bending and centroidal axes can each be rotated uniquely. The Galerkin method is used, permitting arbitrary and abrupt variations along the length that require no knowledge of the spatial derivatives of the beam properties. Additionally, these equations consistently retain all third-order non-linearities that account for flexural–flexural–torsional coupling and extend the validity of the equations for large deformations.

Furthermore, linearly independent shape functions are substituted into these equations, providing an efficient method to determine the natural frequencies and mode shapes of the beam and to solve for time-varying deformation.

This method is validated using finite element analysis and is extended to swept wings. Finally, the importance of retaining cubic terms, in addition to quadratic terms, for non-linear analysis is demonstrated for several examples.

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