Doubly Symmetric Tube Structures. II: Dynamic Analysis

This paper presents simplified analytical methods and closed‐form approximate solutions for free and forced vibrations of a doubly symmetric tube structure to use in preliminary stages of the design. The governing equations proposed in the companion paper are modified to a simplified differential equation expressed in terms of the lateral deflections. It includes the effect of the bending, transverse shear deformation, and shear lag. First, the natural frequencies for a uniform tube structure are presented in closed form. Natural frequencies for variable tube structures are formulated by means of the Galerkin method and in the closed‐form approximate solutions. Second, a dynamic analysis for a variable tube structure is proposed in closed‐form approximate solution. The validity of these solutions is shown from numerical results.