Finite element analysis of the geometric stiffening effect.: Part 2:: non-linear elasticity

In the first part of this paper, the relationship between the number of finite elements used to model the dynamics of rotating beams and the critical speed at which an incorrect solution is obtained when using linear elasticity theory is discussed. The increase in the number of finite elements leads to an increase in the critical speed when linear elasticity is used and no measures are taken, as recommended in the literature, to account for the effect of the coupling between the bending and axial displacements. In this part of the paper, a non-linear finite element model based on the absolute nodal coordinate formulation is used to study the dynamics of rotating beams. It is shown that, when the non-linear elasticity theory is used, a stable solution is always obtained regardless of the number of finite elements used. Numerical results of various simulations are presented in order to compare the solution of a three-dimensional rotating beam that is obtained using the absolute nodal coordinate formulation with the results previously reported in the literature. A finite element numerical study of the dynamics of a helicopter rotor blade is also presented in this investigation. It is shown that, when the finite element absolute nodal coordinate formulation is used in the analysis of helicopter blades, the problem of ill-conditioning that characterizes many of the existing formulations is not encountered.