Computing Dispersion Diagrams and Forced Responses of Arbitrarily Varying Waveguides

Periodic structures can be used for vibration control thanks to their vibration attenuation stop bands. The dispersion diagram obtained for the structural unit cell can be used to characterize such wave attenuation bands. Several designs using spatially varying unit cells have recently been reported in the literature. Spatially varying properties can also be used to investigate the effect of variability on finite structures. Therefore, various methodologies to model spatially varying structures have been recently developed. This work presents new numerical methodologies to compute (i) dispersion diagrams and (ii) forced responses of one-dimensional spatially varying periodic waveguides. The first method extends previous work using a state-space formulation to compute dispersion relations using a linear time-varying system dynamics approach. The methodology is applied to elementary rods, Saint-Venant shafts, Euler–Bernoulli beams, and Timoshenko beams, and the results are validated with discretized spectral element models and the plane wave expansion method. The proposed approach is shown to be more efficient in cases where the plane wave expansion method presents a slow convergence. The second method is a recursive dynamic condensation to compute forced responses using the dynamic stiffness matrix of the unit cell. The results and computational times of the proposed recursive condensation are compared with the direct assembly of a global dynamic stiffness and its standard dynamic condensation. The two proposed methods are shown to be more efficient and accurate in the computation of dispersion diagrams and forced responses of one-dimensional structures for which the elastodynamic equations are known.