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.
Hamiltonian systemcracked nanoplatenonlocal theoryvibrationnatural frequency
万方数据
维普
