Ma Yongbin, Deng Zichen
School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi'an, 710072, People's Republic of China.
J Acoust Soc Am. 2021 Mar;149(3):1955. doi: 10.1121/10.0003800.
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.
本文基于辛方法为周期介质的模态密度提供了一种半解析方法。对于具有板部件的二维周期介质以及具有梁部件和桁架部件的一维周期介质，引入辛方法来描述单胞的连续性和周期性条件。然后借助伴随辛正交关系，首先为周期介质的色散关系建立一个特征值问题。接着通过对特征值问题关于频率求导半解析地得到群速度。由于推导过程中涉及的单胞运动学和动力学变量的表达式是以辛解析波的形式给出的，所以可以高效且高精度地获得周期介质的模态密度。给出了包括具有板部件的二维周期介质以及具有梁部件和桁架部件的一维周期介质的数值算例。将本文结果与有限元模型得到的结果进行比较，证实了所提方法的有效性。
