Taking the Konopelchenko-Dubrovsky system as a simple example, some familles of rational formal hyperbolic function solutions, rational formal triangular periodic solutions, and rational solutions are constructed by u...Taking the Konopelchenko-Dubrovsky system as a simple example, some familles of rational formal hyperbolic function solutions, rational formal triangular periodic solutions, and rational solutions are constructed by using the extended Riccati equation rational expansion method presented by us. The method can also be applied to solve more nonlinear partial differential equation or equations.展开更多
A rotating axisymmetric circular nanoplate is modeled by the Mindlin plate theory.The Mindlin plate theory incorporates the nonlocal scale and strain gradient effects.The shear deformation of the circular nanoplate is...A rotating axisymmetric circular nanoplate is modeled by the Mindlin plate theory.The Mindlin plate theory incorporates the nonlocal scale and strain gradient effects.The shear deformation of the circular nanoplate is considered and the nonlocal strain gradient theory is utilized to derive the governing differential equation of motion that describes the out-of-plane free vibration behaviors of the nanoplate.The differential quadrature method is used to solve the governing equation numerically,and the natural frequencies of the out-of-plane vibration of rotating nanoplates are obtained accordingly.Two kinds of boundary conditions are commonly used in practical engineering,namely the fixed and simply supported constraints,and are considered in numerical examples.The variations of natural frequencies with respect to the thickness to radius ratio,the angular velocity,the nonlocal characteristic scale and the material characteristic scale are analyzed in detail.In particular,the critical angular velocity that measures whether the rotating circular nanoplate is stable or not is obtained numerically.The presented study has reference significance for the dynamic design and control of rotating circular nanostructures in current nano-technologies and nano-devices.展开更多
基金The project partially supported by the State Key Basic Research Program of China under Grant No. 2004CB318000
文摘Taking the Konopelchenko-Dubrovsky system as a simple example, some familles of rational formal hyperbolic function solutions, rational formal triangular periodic solutions, and rational solutions are constructed by using the extended Riccati equation rational expansion method presented by us. The method can also be applied to solve more nonlinear partial differential equation or equations.
基金supported by the Natural Science Foundation of China(No.11972240)the China Postdoctoral Science Foundation(No.2020M671574)the University Natural Science Research Project of Anhui Province(No.KJ2018A0481).
文摘A rotating axisymmetric circular nanoplate is modeled by the Mindlin plate theory.The Mindlin plate theory incorporates the nonlocal scale and strain gradient effects.The shear deformation of the circular nanoplate is considered and the nonlocal strain gradient theory is utilized to derive the governing differential equation of motion that describes the out-of-plane free vibration behaviors of the nanoplate.The differential quadrature method is used to solve the governing equation numerically,and the natural frequencies of the out-of-plane vibration of rotating nanoplates are obtained accordingly.Two kinds of boundary conditions are commonly used in practical engineering,namely the fixed and simply supported constraints,and are considered in numerical examples.The variations of natural frequencies with respect to the thickness to radius ratio,the angular velocity,the nonlocal characteristic scale and the material characteristic scale are analyzed in detail.In particular,the critical angular velocity that measures whether the rotating circular nanoplate is stable or not is obtained numerically.The presented study has reference significance for the dynamic design and control of rotating circular nanostructures in current nano-technologies and nano-devices.
