Fractional Modeling And Numerical Simulation Of Three Kinds Of Viscoelastic Beams Based On Bernstein Polynomials

In recent years,researchers found that the fractional differential theory has been wide?ly used in various fields,especially in material mechanics.The fractional-order model has good memory.It's much better than classical integer-order or non-linear viscoelastic models.Fractional viscoelastic constitutive relation and viscoelastic beam can better describe the me?chanical properties of viscoelastic materials.By introducing time parameters,the fractional order models of three kinds of viscoelastic beams are established.We used the Bernstein polynomial to simulate the material properties in time domain.Firstly,the background and the basic knowledge of viscoelastic beams are introduced.The fractional order model of viscoelastic Euler-Bernoulli beam is established by using the dynamic equation of Euler-Bernoulli beam,the constitutive relation for viscoelastic mate?rial and geometric relation.We derived various differential operator matrices of Bernstein polynomial.The equation is transformed into a system of equations by the discrete variable method.In this paper,numerical simulation is carried out in time domain.We obtained the displacement solutions of Polybutadiene beam and Butyl B252 beam under no-load,uniform load and harmonic load.The mechanical properties of viscoelastic materials are analyzed theoretically.Secondly,the fractional order model of viscoelastic cantilever beam with variable cross-section is established according to the motion equation of the beam,the constitutive relation-ship of viscoelastic material and the strain-displacement relationship.The Bernstein poly-nomial is used for numerical simulation.The numerical solutions of three kinds of variable cross-section beams with circular cross-section,rectangular cross-section and blade cross-section under different loads are given by numerical examples.The effects of load,taper ratio of cross section and material on displacement are studied by numerical examples.Finally,the fractional model of viscoelastic rotating beams is derived according to the constitutive relation,strain-displacement relation and Hamilton principle of viscoelastic ma-terials.The numerical simulation is carried out by using Bernstein polynomial as the basis function.The numerical results give the displacement solutions under uniform and linear loads.The results show the influence of load,rotational speed and material on displacement.In this paper,the fractional model of three kinds of viscoelastic beams is established in time domain for the first time.The Bernstein polynomial is used for numerical simu-lation.Indirect analysis of damping properties of viscoelastic materials from the aspects of viscoelastic beams.