Vibration analysis of curved frames

In many structural applications like bridges, arches, turbo-machineries blades, etc. curved frame structures are used and it is important to study its dynamic behavior. The use of a curvilinear coordinate system to solve such problems generates higher-order, complicated differential equations. Finite Element Method can be used to determine the dynamics of such structures. However, the high frequency simulation using FEM is inefficient and thus restricted in its applicability. The wave-based methods are advantageous in this regard. The wave methods have been welldeveloped for dealing with frame structures comprising of straight Euler-Bernoulli members with joints at an arbitrary angle. In such structures, the presence of the joint induces a coupling between the longitudinal and transverse dynamics. In the present work, we extend the wave-based technique to analyze frame structures comprising of filleted or circularly curved joints. The propagation matrix in the straight portion of the structure is well documented in the literature. In the present work, the curved portion of the structures is discretized into small linear segments, wherein each segment subtends a small angle with the neighboring segment. Using the continuity relations and equilibrium conditions at the joints, the reflection and transmission matrices at the joints can be obtained. The assembly of reflection, transmission and propagation matrices and the incorporation of the boundary conditions is in line with the standard wave method. Modal Analysis and Harmonic Analysis are conducted using the present approach and the results were found to correlate with those reported in literature as also with FEM simulations. The characterization of circular fillets in terms of its transmission and reflection effects is presented. The present method is computationally efficient for high frequency calculation in comparison to FEM simulation.