Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model

Abstract

Nonlinear parametric vibration of axially accelerating viscoelastic strings is investigated via an approximate analytical approach. The standard linear solid model using the material time derivative is employed to describe the string viscoelastic behaviors. A coordinate transformation is introduced to derive Mote's model of transverse motion from the governing equation of the stationary string. Mote's model leads to Kirchhoff's model by replacing the tension with the averaged tension over the string. An asymptotic perturbation approach is proposed to study principal parametric resonance based on the two models. The amplitude and the existence conditions of the steady-state responses are determined by locating the nonzero fixed points in the modulation equations resulting from the solvability condition. Numerical results are presented to highlight the effects of the material parameters, the axial-speed fluctuation amplitude, and the initial stress on steady-state responses.

Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beam constituted by standard linear solid model

“…The standard linear solid model is adopted to describe the viscoelastic property of the beam material. The stress-strain relationship of the model is expressed in a differential form as [20][21][22][23] …”

A general solution procedure for vibrations of systems with cubic nonlinearities and nonlinear/time-dependent internal boundary conditions

“…In the method of multiple timescales, we assume asymptotic expansions in the form [23][24][25][26][27][28][29][30][31][32][33]:…”

Free Vibration of Axially Functionally Graded Beam

“…The asymptotic development method, which is a kind of perturbation analysis method, is always used to solve nonlinear vibration equations. For example, Chen et al [71,72] studied the nonlinear dynamic behavior of axially accelerated viscoelastic beams and strings based on the asymptotic perturbation method. Ding et al [73,74] studied the influence of natural frequency of transverse vibration of axially moving viscoelastic beams and the steady-state periodic response of forced vibration of dynamic viscoelastic beams based on the multi-scale method.…”