A variational method for band-gap analysis of metamaterial plates with local resonators

To establish an efficient solution method is the foremost issue when studying the band-gap properties of acoustic metamaterials.For this purpose,this paper presents a variational method for band-gap analysis of metamaterial plates with local resonators.This method is based on the Lagrangian functional of a unit cell and orthogonal polynomial expansions.Specifically,the Lagrangian functional is built by combining energy terms of the plate and the local resonator,to which the variational operation is performed to derive the governing equations of a unit cell.These equations are discretized to a set of generalized coordinates by introducing the Chebyshev orthogonal polynomials of the first kind as admissible functions.To overcome the difficulty that the Periodic Boundary Condition (PBC) cannot be directly applied to these generalized coordinates,the proposed method weakens the PBC to a number of pre-selected collocation points,thus allowing the continuous boundary conditions to be imposed in a more straightforward,though discrete,way.The linear constraints given by this procedure is enforced by Lagrange multipliers method.Numerical examples show that the proposed method is of good accuracy and efficiency and accommodates to composite structures like double-leaf metamaterial plates with local resonators.It can also be extended to band-gap analysis of metamaterial plates with other configurations,providing technical support for the design and optimization of their vibration and noise reduction characteristics.