Fractional Order Modeling And Numerical Analysis Of Two Types Of Viscoelastic Beams Based On Shifted Chebyshev Polynomial Algorithm

With the development of fractional calculus theory,it has been widely concerned and successful in many fields of science and engineering,especially in material mechanics.It is of great significance to study the dynamic response of viscoelastic beams for understanding the actual working conditions,which has been studied by many scholars.At present,most of the numerical algorithms for viscoelastic beams transform the problem into frequency domain to give solvability,and then back to time domain analysis.However,there is little research on the algorithm of solving this problem directly in time domain.Based on this,the fractional governing equations of two kinds of viscoelastic beams are established by introducing time parameters,and the shifted Chebyshev polynomials algorithm is used for numerical analysis for the first time.In particular,different from the existing results,we can directly obtain the displacement and stress numerical solutions of the governing equations in time domain.Firstly,based on the dynamic equation,the fractional Element constitutive model and the strain displacement relationship,the fractional control equation of the viscoelastic beam with two ends fixed is derived,and the numerical solution is carried out by the shifted Chebyshev polynomial algorithm.According to Caputo’s idea of fractional differentiation and function approximation,the displacement function is approximated by shifted Chebyshev polynomials,and various differential operator matrices are derived.Then,the variables are discretized and the displacement numerical solution of the control equation is obtained in time domain.In order to obtain the displacement solution with high accuracy,we propose the theory of error correction.The displacement solutions of fixed beams at both ends of HDPE and PEEK under different loads are compared by numerical examples,and the mechanical properties of viscoelastic materials are analyzed.Secondly,the paper establishes the fractional order control equation of the fixed beam at both ends of the fractional order Kelvin-Voigt model,and applies the shifted Chebyshev polynomial algorithm to approximate the displacement function.Based on the concept of fractional differential and operator matrix,the control equations are simulated in time domain.The numerical results of displacement and stress of HDPE fixed beam under different uniform load,harmonic load and linear load are given.In addition,the displacement of HDPE two end fixed beam under different models is compared and analyzed.Finally,based on the fractional Element constitutive model,the fractional Kelvin-Voigt model,Hamilton principle and the relationship between strain and displacement,two fractional governing equations of viscoelastic rotating beams are established.Moreover,the shifted Chebyshev polynomials are used as the basis functions,which are combined with the idea of function approximation and the method of collocation.In addition,the convergence is analyzed.The numerical results show the displacement and stress of HDPE and PEEK rotating beams under different speed and load,and analyze the influence of speed,load and material on displacement.