A semi-analytical model for the modal density of periodic mediums based on the symplectic method

In this paper, a semi-analytical approach is provided for the modal density of periodic mediums based on the symplectic method. For two-dimensional periodic mediums with a plate component and one-dimensional periodic mediums with a beam component and truss component, the symplectic method is introduced to describe the conditions of continuity and periodicity of the unit cell. And then by virtue of the adjoint symplectic orthogonal relations, an eigenproblem is first established for the dispersion relation of the periodic mediums. The group velocity is then obtained semi-analytically by differentiating the eigenproblem with respect to frequency. Since the expressions of the kinematic and the kinetic variables of the unit cell involved in derivation processes are expressed in terms of symplectic analytical waves, the modal density of periodic mediums can be obtained with high efficiency and with high accuracy. Numerical examples including two-dimensional periodic mediums with a plate component and one-dimensional periodic mediums with a beam component and truss component are provided. The comparison of the present results with the results obtained from the finite element model confirms the effectiveness of the proposed method.