Analytical Analysis on Nonlinear Parametric Vibration of an Axially Moving String with Fractional Viscoelastic Damping

Abstract

The nonlinear parametric vibration of an axially moving string made by rubber-like materials is studied in the paper. The fractional viscoelastic model is used to describe the damping of the string. Then, a new nonlinear fractional mathematical model governing transverse motion of the string is derived based on Newton's second law, the Euler beam theory, and the Lagrangian strain. Taking into consideration the fractional calculus law of Riemann-Liouville form, the principal parametric resonance is analytically investigated via applying the direct multiscale method. Numerical results are presented to show the influences of the fractional order, the stiffness constant, the viscosity coefficient, and the axial-speed fluctuation amplitude on steady-state responses. It is noticeable that the amplitudes and existing intervals of steady-state responses predicted by Kirchhoff 's fractional material model are much larger than those predicted by Mote's fractional material model.

Dynamic Analysis of a Plate on the Generalized Foundation with Fractional Damping Subjected to Random Excitation

“…donates the fractional derivative operator and according to Riemann-Liouville's definition is represented by the following [40,41]:…”

Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise

“…According to the boundary conditions (12), the solution of system (11) can be expressed as the Fourier series:…”

“…The fractional derivative can accurately describe the constitutive relation of viscoelastic materials with fewer parameters, so the studies of fractional differential equations on the typical mechanical properties and the influences of fractional order parameters on the system are very necessary and have important significance. In recent years, many scholars have done a lot of work and achieved fruitful results in this field: Li and Tang studied the nonlinear parametric vibration of an axially moving string made by rubber-like materials, a new nonlinear fractional mathematical model governing transverse motion of the string is derived based on Newton's second law, the Euler beam theory, and the Lagrangian strain, and the principal parametric resonance is analytically investigated via applying the direct multiscale method [12]. Liu et al introduced a transfer entropy and surrogate data algorithm to identify the nonlinearity level of the system by using a numerical solution of nonlinear response of beams, the Galerkin method was applied to discretize the dimensionless differential governing equation of the forced vibration, and then the fourth-order Runge-Kutta method was used to obtain the time history response of the lateral displacement [13].…”

Nonlinear Vibrations of an Axially Moving Beam with Fractional Viscoelastic Damping

“…Consequently, the fractional calculus [21] has been introduced in constitutive relations to obtain a satisfactory solution for the real viscoelastic responses of the materials over a large range of frequency [22,23]. Although there remain some mathematical issues unsolved, the fractional calculus-based modern viscoelasticity problems are becoming the focus of attention [24][25][26][27][28][29][30][31][32].…”