Complex-mode Galerkin approach in transverse vibration of an axially accelerating viscoelastic string

Abstract

Under the consideration of harmonic fluctuations of initial tension and axially velocity, a nonlinear governing equation for transverse vibration of an axially accelerating string is set up by using the equation of motion for a 3-dimensional deformable body with initial stresses. The Kelvin model is used to describe viscoelastic behaviors of the material. The basis function of the complex-mode Galerkin method for axially accelerating nonlinear strings is constructed by using the modal function of linear moving strings with constant axially transport velocity. By the constructed basis functions, the application of the complex-mode Galerkin method in nonlinear vibration analysis of an axially accelerating viscoelastic string is investigated. Numerical results show that the convergence velocity of the complex-mode Galerkin method is higher than that of the real-mode Galerkin method for a variable coefficient gyroscopic system.

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

“…Because increasingly used materials such metallic or ceramic reinforced materials exert inherently viscoelastic behaviors, researchers investigate free vibration [2,3], forced vibration [4], and parametric vibrations [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] of axially moving viscoelastic strings. The viscoelastic strings they studied are differential-type materials described by the Kelvin model [2,4,7,8,11,12,14,15,18,20,21,23] or the standard linear solid model [3,6,17,22], as well as integral-type materials defined by the stress relaxation as an exponential function [5,9,10,16,19] or a power function [13]. Actually, the standard linear solid model, which can describe the behavior of linear viscoelastic materials of solid type with limited creep deformation, covers the Kelvin model and the integral-type constitution relation with an exponential relaxation function as special cases.…”

“…Actually, the standard linear solid model, which can describe the behavior of linear viscoelastic materials of solid type with limited creep deformation, covers the Kelvin model and the integral-type constitution relation with an exponential relaxation function as special cases. When differential-type constitutive laws are incorporated, some investigators used the partial time derivative in the viscoelastic constitutive relations [2][3][4][6][7][8]11,12,14,15,20,21,23,24]. However, Mochensturm and Guo [18] demonstrated that the material time derivative should be used to account for the additional "steady state" dissipation of an axially moving viscoelastic string.…”

Dynamic Stability of Axially Accelerating Viscoelastic Plate

“…There are comprehensive studies on such systems [1][2][3][4][5]. In recent years, much attention has been paid to parametric vibration of traveling systems and transverse parametric vibration of axially accelerating systems has been extensively analyzed [6,7]. Pakdemirli et al [8] conducted a stability analysis using the Floquet theory for sinusoidal transporting velocity function.…”

Nonlinear dynamic behaviors of viscoelastic shallow arches

“…Therefore, many studies [1][2][3][4][5][6][7][8][9][10][11][12][13] have been performed on the nonlinear dynamics of viscoelastic structures. Chen et al [2][3][4][5] investigated the dynamic stability and chaotic motion of a nonlinear viscoelastic beam and column by applying the Leaderman constitutive relation with Galerkin numerical integration.…”

“…Also, other constitutive relations [6][7][8] are adopted to investigate the dynamics of the viscoelastic structures. Moreover, there has been considerable interest in the study of the nonlinear dynamics of the beam [2][3][6][7][8] , column [4][5] , pile [9][10] , plate [11] , shell [12] , and string [13] etc. However, few studies are focused on the nonlinear dynamic behavior of a viscoelastic shallow arch.…”