Free vibration of cellular structures using continuum modes

This paper is concerned with free vibration of elastic structures made up of cellular material. A network of cells in a plane is considered. Cell walls are modelled as beams. The problem of calculating the first few natural frequencies and the corresponding mode shapes is addressed. An approximate method based on the assumption that cellular solids behave as a continuum for low-frequency dynamics is presented. A direct application of the assumed modes method incorporating continuum modes as basis leads to difficulties. The source of this difficulty is identified. A method that uses inverse power iterations is proposed to pre-condition the assumed modes appropriately. The proposed method of using continuum modes as basis for model order reduction leads to substantial computational saving while maintaining good accuracy. Two examples are given to illustrate the proposed procedure of using the continuum modes in conjunction with pre-conditioning.