Solving and Numerical Simulations of Fractional-Order Governing Equation for Micro-Beams

Fractal and Fractional

article

Jingguo Qu¹  Yuhuan Cui¹  Yiming Chen⁶  Aimin Yang¹  Qunwei Zhang¹ 

[1] College of Science, North China University of Science and Technology;Hebei Key Laboratory of Data Science and Application, North China University of Science and Technology;The Key Laboratory of Engineering Computing in Tangshan City, North China University of Science and Technology;Tangshan Intelligent Industry and Image Processing Technology Innovation Center, North China University of Science and Technology;Hebei Engineering Research Center for the Intelligentization of Iron Ore Optimization and Ironmaking Raw Materials Preparation Processes, North China University of Science and Technology;School of Science, Yanshan University;LE STUDIUM, Loire Valley Institute for Advanced Studies Orléans & Tours

关键词: nonlinear-fractional order differential equations;    shifted Chebyshev polynomial algorithm;    viscoelastic micro-beams;    operator matrix;    numerical simulation;   

DOI  :  10.3390/fractalfract7020204

学科分类：社会科学、人文和艺术（综合）

来源: mdpi

【 摘 要 】

This paper applies a recently proposed numerical algorithm to discuss the deflection of viscoelastic micro-beams in the time domain with direct access. A nonlinear-fractional order model for viscoelastic micro-beams is constructed. Before solving the governing equations, the operator matrices of shifted Chebyshev polynomials are derived first. Shifted Chebyshev polynomials are used to approximate the deflection function, and the nonlinear fractional order governing equation is expressed in the form of operator matrices. Next, the collocation method is used to discretize the equations into the form of algebraic equations for solution. It simplifies the calculation. MATLAB software was used to program this algorithm to simulate the numerical solution of the deflection. The effectiveness and accuracy of the algorithm are verified by the numerical example. Finally, numerical simulations are carried out on the viscoelastic micro-beams. It is found that the viscous damping coefficient is inversely proportional to the deflection, and the length scale parameter of the micro-beam is also inversely proportional to the deflection. In addition, the stress and strain of micro-beam, the change of deflection under different simple harmonic loads, and potential energy of micro-beam are discussed. The results of the study fully demonstrated that the shifted Chebyshev polynomial algorithm is effective for the numerical simulations of viscoelastic micro-beams.

【 授权许可】

CC BY