Vibration Of Fractional Viscoelastic Nanobeams And Rods Based On Non-local Strain Gradient Theory

With the rapid development of nanotechnology,various new types of nanomaterials are increasingly valued for their unique properties.And the mathematical modeling and engineering applications of fractional calculus have been widely used.In this paper,we use the concept of fractional calculus and non-local strain gradient theory to establish fractional viscoelastic nanobeam and rod models,and solve the different vibration problems of nanobeam and rod under the influence of different parameters.The main work of this topic is as follows:(1)Firstly,We study a model of nonlinear fractional nonlocal strain gradient based on fractional derivative and nonlocal strain gradient theory,and study the geometric nonlinear free vibration problem of fractional viscoelastic nanobeams.The ordinary differential equation converts to the resulting partial differential equation,and Matlab,the influence of non-local parameters,strain gradient parameter,viscoelastic coefficient,damping coefficient and fractional order of fractional on the vibration.(2)Based on the previous theory of nonlocal strain gradient,we then studied the axial vibration characteristics of fractional nanorod,still use the galerkin residual function method of transform equation,because the equation is special,we can use the Laplace transform and Laplace inverse transformation to solve the equation,the axial displacement of nano rod.The effects of damping coefficient and fractional order on axial vibration of nanorod are investigated.This paper aims to explore the effects of strain gradient parameters,non-local parameters,fractional order,viscoelastic coefficient,length-to-thickness ratio and damping coefficient on the vibration of nanobeam and rod,and shows that they have significant effects on vibration.