Analysis of Damped Bloch Waves by the Rayleigh Perturbation Method

Abstract

Bloch waves in viscously damped periodic material and structural systems are analyzed using a perturbation method originally developed by Rayleigh for vibration analysis of finite structures. The extended method, called the Bloch–Rayleigh perturbation method here, utilizes the Bloch waves of an undamped unit cell as basis functions to provide approximate closedform expressions for the complex eigenvalues and eigenvectors of the damped unit cell. In doing so, we circumvent the solution of a quadratic Bloch eigenvalue problem and subsequent computationally intensive transformation to first order/statespace form. Dispersion curves of a onedimensional damped springmass chain and a twodimensional phononic crystal with square inclusions are calculated using the statespace method and the proposed method. They are compared and found to be in excellent quantitative agreement for both proportional and nonproportional viscous damping models. The perturbation method is able to capture anomalous dispersion phenomena—branch overtaking, branch cuton/cutoff, and frequency contour transformation—in parametric ranges where statespace formulations encounter numerical issues. Generalization to other linear nonviscous damping models is permissible.