Three-dimensional free vibration of pretwisted beams

A method for analyzing the free vibration of pretwisted structural members of arbitrary but uniform cross section is presented. This method is based on three-dimensional elasticity theory and three-dimensional finite element formulation. The equations of elasticity and the governing equations of motion are obtained using a rotating coordinate system that rotates along with the cross section. The relationship between the fixed Cartesian coordinate system and the rotating coordinate system is based on the pretwist angle rate. This method can capture warping, Poisson effects, and shear effects. Hamilton's principle is used to derive the equations of motion that, in turn, are used to obtain the natural frequencies and mode shapes. Results showing the relationship between the rate of pretwist and the natural frequencies of prismatic members with rectangular cross sections and different lengths are presented. The results also show coupling between the two bending modes and coupling between the axial and torsional modes. If the axis of rotation and the centroid of the cross section are not coincident, coupling among all four basic modes and higher modes may occur. This behavior is difficult or sometimes impossible to capture with simpler technical theories. Because the method is based on three-dimensional elasticity theory it can accurately capture the behavior of long or short beams, thin or thick sections, arbitrary geometry, and inhomogeneous, laminated (anisotropic) materials.