On the efficient evaluation of modal density and damping of one-dimensional periodic structures using a dynamic homogenization method

Modal density and damping are key parameters for structural vibration analysis. However, the current methods for calculating these properties of periodic structures in high-frequency environment are still insufficient in accuracy and efficiency. In this article, a semi-analytical form for the modal density and damping of longitudinal vibration of one-dimensional periodic structures is proposed based on a dynamic homogenization method. By virtue of asymptotic perturbation expansion, explicit expression is obtained for the dispersion relation. And then, the modal density is obtained in terms of the semi-analytical form by differentiating frequency with respect to wave number. By noting that the high-frequency homogenization method is valid only in the neighborhood of the standing wave frequencies, a weighted technique is introduced to compensate this deficiency. Based on the mode strain energy method, the damping loss factor is also obtained using the high-frequency homogenization results. Because explicit expression can be obtained analytically, the shortage of low computational efficiency and accuracy faced by traditional analysis is significantly made up by the proposed method, which is confirmed by numerical examples.