Time-Domain Dynamics of Fractional Viscoelastic Spinning Disks via Shifted Legendre Polynomials

This paper presents a novel algorithm for the dynamic analysis of fractional-order viscoelastic spinning disks in the time domain. The novelty mainly lies in the use of the shifted Legendre polynomial algorithm for the direct time-domain numerical analysis of displacement in two directions for a three-dimensional viscoelastic rotating disk, tackling a more complex and strongly coupled problem than those addressed in previous studies. By using the fractional-order Kelvin–Voigt model to describe the viscoelastic properties of the disk, a system of governing equations with three independent variables is established. For the two ternary unknown functions in the equations, a fractional-order differential operator matrix based on Shifted Legendre polynomials is derived, transforming the original equations into two sets of algebraic equations that are easier to solve. This paper presents an in-depth analysis of the convergence of the Legendre polynomial algorithm, complemented by an investigation of its error characteristics using numerical examples, thereby verifying the method’s accuracy and feasibility. This study can be applied to the dynamic analysis of viscoelastic rotating structures under body force density. The findings provide theoretical support for the optimization and safety assessment of load-bearing rotating components in engineering. And the algorithm demonstrates high accuracy and applicability in handling fractional-order equations in science and engineering.