Eigenfunciton Expansions And Variational Principle Of Rectangular Nanoplates Under Different Dynamic Behavior

In this thesis,the free vibration and buckling of rectangular nanoplates are studied.The biorthogonal expansion theorem,the variational principle and the completeness theorem of dynamic problems are obtained.The first chapter mainly introduces the research significance of rectangular nanoplates,the research background and research status of biorthogonal expansions,variational principle,and symplectic expansion.In the second chapter,the two dynamic problems are respectively derived to the Hamiltonian system,and the corresponding orthogonal symmetric biorthogonal relations and biorthogonal expansions theorem are obtained.And numerical examples are used to illustrate the validity of the theorem.Furthermore,under the general boundary conditions,the Hamiltonian mixed energy variational principles of the vibration and buckling value problem are established respectively,and the complete functional formulae are given.The third chapter is based on the Chapter 2,with the help of Mathematica software,the eigenvalues and eigenfunction vectors of the off-diagonal infinite-dimensional Hamiltonian operators are solved,and the symplectic orthogonality of the eigenfunction system is verified.The most important is to prove the completeness theorem of the eigenfunction system and the general solutions to the problem are given.Finally,a numerical example is given to illustrate the validity of the results.