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

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 shi...

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Published in:	Fractal and fractional Vol. 7; no. 2; p. 204
Main Authors:	Yang, Aimin, Zhang, Qunwei, Qu, Jingguo, Cui, Yuhuan, Chen, Yiming
Format:	Journal Article
Language:	English
Published:	Basel MDPI AG 01.02.2023
Subjects:	Algorithms Analysis Approximation Calculus Chebyshev approximation Collocation methods Computer simulation Deflection Deformation Finite element analysis Force and energy Iterative methods Mathematical functions Mathematical models Mechanical properties Microbeams nonlinear-fractional order differential equations Numerical analysis numerical simulation operator matrix Operators (mathematics) Polynomials Potential energy Rheology shifted Chebyshev polynomial algorithm Simulation methods viscoelastic micro-beams Viscoelasticity Viscous damping
ISSN:	2504-3110, 2504-3110
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Abstract	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.
AbstractList	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.
Audience	Academic
Author	Cui, Yuhuan Chen, Yiming Zhang, Qunwei Yang, Aimin Qu, Jingguo
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SubjectTerms	Algorithms Analysis Approximation Calculus Chebyshev approximation Collocation methods Computer simulation Deflection Deformation Finite element analysis Force and energy Iterative methods Mathematical functions Mathematical models Mechanical properties Microbeams nonlinear-fractional order differential equations Numerical analysis numerical simulation operator matrix Operators (mathematics) Polynomials Potential energy Rheology shifted Chebyshev polynomial algorithm Simulation methods viscoelastic micro-beams Viscoelasticity Viscous damping
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Title	Solving and Numerical Simulations of Fractional-Order Governing Equation for Micro-Beams
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