Non-linear coupled transverse and axial vibration of a compliant structure, part 1: Formulation and free vibration

A compliant tower in the ocean environment is modelled as a beam undergoing coupled transverse and axial motion. The equations of motion are non-linear and coupled and are derived here. The beam is assumed to be supported by a linear-elastic torsional spring at one end and with a point mass at the other end. Such a model is representative of numerous applications. The equations of motion and boundary conditions are obtained using Hamilton's variational principle. It is assumed that strains are small but the rotation is moderate compared to the strain so that the equations of motion for the axial and transverse motion are non-linearly coupled. The free response in vacua and the free response in water are considered in particular. The fluid forces, the added mass and drag forces, are modelled using a semi-empirical Morison equation. The resulting non-linear coupled partial differential equations are solved numerically using the finite difference approach. In Part 2 of this work, various forced responses are studied. (C) 2000 Academic Press.