Simple and effective equilibrium models for vibration analysis of curved rods

New finite-element models for curved beam vibration analysis are derived from classical complementary variational principles of elastodynamics. The use of a spline approximation of the axis line (as previously introduced by the writers in the static case) allows for the a priori satisfaction of the dynamic differential equilibrium equations in a simple and effective way. More precisely, starting from the Hellinger-Reissner principle and making use of a linear interpolation of displacements and momentum fields, a very simple hybrid-mixed model is obtained that can be easily linked with general-purpose finite element packages. Alternatively, fully equilibrated models are derived from the complementary energy principle assuming as unknowns either the momentum or the stress resultant fields; in both cases highly accurate finite element models are obtained for which upper and lower bounds on eigenvalue estimates are readily available. Several examples are worked out that are capable of showing the efficiency and the wide spectrum of applicability of the proposed method. The comparison with two general-purpose finite element packages of large diffusion let us assess the high level of performance of the complementary energy models for curved elements.