2.2 Uniform Beam Torsional Dynamics

Now that the fundamental aspects of structural dynamics analysis have been considered for the uniform string problem, these concepts will be applied to the dynamics of beam torsional deformation. The beam has many more of the characteristics of typical aeronautical structures. Indeed, high-aspect-ratio wings and helicopter rotor blades are frequently idealized as beams, especially in preliminary design. Even for low-aspect-ratio wings, although a plate model is more realistic, the bending and torsional deformation can be approximated by use of beam theory with adjusted stiffness coefficients.

In an effort to retain a level of simplicity that promotes tractability, the torsional rigidity of St. Venant theory, denoted GJ, will be taken as given. For homogeneous and isotropic beams, G denotes the shear modulus and J is a constant that depends only on the geometry of the cross section. For such beams J can be determined by solving a boundary-value problem over the cross-sectional area, which requires finding the cross-sectional warping caused by torsion. Although analytical solutions for this problem are available for some simple cross-sectional geometries, solving for the cross-sectional warping and torsional stiffness is not a trivial exercise in general. For nonhomogeneous, anisotropic beams one may also use the symbol GJ to denote an effective torsional rigidity, which can be determined by solving a far more involved boundary-value problem over the cross-sectional area.