Beam-type modeling for the free vibration of a long thick laminated conical tube

The free vibration of a long thick laminated conical tube is studied in this paper with a beam-type model. Taken as a beam, the tube is divided into elements by cross-sections. Each cross-section is a node. In the beam-type model, the nodal forces are assumed to be uniformly distributed along the middle circle, which is the intersection of the cross-section with the middle surface of the tube. The nodal torsional moment is assumed to be produced by a uniformly distributed tangential force along the middle circle. The nodal bending moments are assumed to be produced by sinusoidally distributed axial forces along the middle circle. Three nodal displacements and three nodal rotations corresponding to these nodal forces and moments, respectively, are determined by the energy principle and the assumed deformation pattern. Thus, in the beam-type model, there are only six degrees of freedom at each node. The stiffness matrix in terms of these nodal degrees of freedom is derived from a thick-shell theory together with a semi-analytical method. Based on the stiffness matrix, the frequencies of free vibration are calculated. The numerical results are checked by experimental measurements with good agreement.