含裂纹纳米板振动问题的哈密顿体系方法

基于裂纹处范德华力效应，采用非局部弹性理论构造纳米板模型，并通过导入哈密顿体系建立含裂纹纳米板振动问题的对偶正则控制方程组。在全状态向量表示的哈密顿体系下，将含裂纹纳米板的固有频率和振型问题归结为广义辛本征值和本征解问题。利用哈密顿体系具有的辛共轭正交关系，得到问题解的级数解析表达式。结合边界条件，得到固有频率与辛本征值的代数方程关系式，进而直接给出固有频率的表达式。数值结果表明，非局部尺寸参数和裂纹长度对纳米板振动的各阶固有频率有直接的影响。对比表明，辛方法是准确且可靠的，可为工程应用提供依据。

The method of Hamiltonian system for vibration problem of cracked nanoplates

Based on theory of nonlocal elasticity and the van der Waals force effect at the crack location, Hamiltonian system is introduced into the vibration problem of cracked nanoplates and the Hamiltonian dual equations are represented.In the Hamiltonian system, which is represented by the full state vector, the natural frequencies and modes of the cracked nanoplates are reduced to the problem of the symplectic eigenvalues and symplectic eigensolutions.The expression of analytical solutions for the problem can be obtained by the series of symplectic eigenfunctions using the adjoint symplectic relationships of orthogonality in the Hamiltonian system.Considering the boundary conditions, the relationship between the natural frequencies and the symplectic eigenvalues are obtained, and then the frequency equations can be given directly.The numerical results indicate that the nonlocal parameter and the crack length have a direct effect on all the natural frequencies of the nanoplates.It is shown that the symplectic method has high accuracy and reliability by comparison of the results.Meanwhile, the method provides a basis for engineering applications.