Nonlinear dynamic analysis of the viscoelastic string with a harmonically varying transport speed

doi:10.1016/S0045-7949(98)00001-7

Computers & Structures

Volume 66, Issue 6, 1 March 1998, Pages 777-784

https://doi.org/10.1016/S0045-7949(98)00001-7 Get rights and content

Abstract

Qualitative aspects of parametric excitation due to the non-constant traveling velocity of a viscoelastic string are investigated. The problem considered is an initially stressed viscoelastic string subjected to steady-state and harmonic variation of axially traveling motion. The string material is considered as a Violet element in series with a spring (three-parameter model). The partial differential equation of motion is derived first, and then is reduced to be a set of third-order nonlinear ordinary differential equations by applying Galerkin's method. Finally, the effects of elastic and viscoelastic parameters, constant and non-uniform transport speed, wave propagation speed ratio, and nonlinear terms on the transient amplitudes are investigated numerically.

Section snippets

Notation

A	area of cross section of the string
E₁, E₂	spring coefficients for viscoelastic models
L	length of the string
k₂	wave speed ratio
k₁, k₃	material parameter
t	time
T	initial tension of the string
x	axial coordinate
x_t	traveling speed of the string
V	transverse displacement in y direction
σ	perturbed axial stress
ρ	mass density per unit volume
ε	axial strain of the string
η₂	damping coefficients for viscoelastic model
V	dimensionless transverse displacement in y direction
ξ	dimensionless axial coordinate
τ	dimensionless time

Equation of motion

The problem of transverse vibrations of a viscoelastic string in a state of uniform initial stress is shown in Fig. 1. The transverse deformations in y direction are considered and body force is neglected. The equation of motion in the y direction is $T A +σ ν_{xx} +ν_{x} σ_{x} =ρ d^{2} ν d t^{2},$ where σ is perturbed axial stress, ν is the displacement in the transverse direction and A is area of cross section of the string. The uniform initial tension force T provides the required initial stress for the string. The

Differential constitutive law

The one-dimensional constitutive equation[6] of a differential type material (called Revlon material) obeys the relation $∑ j=0 R a_{j} d^{j} σ d t^{j} = ∑ j=0 P b_{j} d^{j} ε d t^{j}, (a_{0} ≠0, b_{0} =1).$ Eq. (3)may also be written as $A σ= B ε,$ where A and B are differential operators defined as $A = ∑ j=0 R a_{j} d^{j} d t^{j}, B = ∑ j=0 P b_{j} d^{j} d t^{j} .$

Choosing the three-parameter viscoelastic model as shown in Fig. 2, the differential constitutive law of a linear viscoelastic material can be written as $σ ̇ + E_{1} +E_{2} η_{2} σ=E_{1} ε ̇ + E_{1} E_{2} η_{2} ε.$ The form of viscoelastic property is

Dimensionless form and Galerkin's approximation

Define the nondimensional parameters $V= ν L, ξ= x L, τ= c_{2} t L, k_{1} = E_{1} E_{2} L ρη_{2} c^{3}_{2}, k_{2} = c_{1} c_{2}, k_{3} = E_{1} +E_{2} η_{2} c_{2} L,$ where L is the length of the string, c₁= $E_{1} /ρ$ , c₂= $T/(Aρ)$ . By the definitions of ξ, τ and ξ_τ=x_t/c₂, the general equation of motion (Eq. (6)) can be transformed into the following nondimensional form $k_{3} V_{ξξ} + V_{ξξτ} +ξ_{τ} V_{ξξξ} + 32 k_{1} V^{2}_{ξ} V_{ξξ} +k^{2}_{2} (2V_{ξ} V^{2}_{ξξ} ξ_{τ} + 2V_{ξ} V_{ξξ} V_{ξτ} +V^{2}_{ξ} V_{ξξξ} ξ_{τ} +V^{2}_{ξ} V_{ξξτ})=k_{3} (V_{ττ} + 2ξ_{τ} V_{τξ} +ξ^{2}_{τ} V_{ξξ} +ξ_{ττ} V_{ξ})+V_{τττ} +3ξ_{τ} V_{ττξ} + ξ^{3}_{τ} V_{ξξξ} +3ξ_{ττ} V_{τξ} +3ξ_{τ} ξ_{ττξξ} +ξ_{τττ} V_{ξ} + 3ξ^{2}_{τ} V_{τξξ} .$

From Eq. (8), some observations are noticed as follows.

Numerical results and discussions

The numerical results of the traveling string shown here are with constantly and non-constant traveling velocities.

Constant traveling velocity

The parameter values are chosen as follows: T₀=100 N, ρ=7860 kg/m³, L=1 m, E₁=E₂=3000 Mpa and η₂=300 Mpa days. In the present analysis, up to two-term approximations based upon the eigenfunctions of the stationary string are considered (i.e., n=1, 2). Taking two terms, the resulting two ordinary differential equations are coupled. While a qualitative agreement is evident among the two-term approximations, increasing the number of terms in the approximations improves the quantitative result[10]. It

Non-constant traveling velocity

The dynamic behavior of axially moving systems such as conduits, band saws, and magnetic tapes has received considerable attention[12]. In many works, the axial velocity was taken to be constant. However, when a system is subject to acceleration, it may alter the stability of the system. The equation of motion for an accelerating elastic string was derived by Pakdemirli et al.[10].

In this section, we investigate the transverse vibrations of an axially accelerating viscoelastic string. The

Conclusion

This paper investigates the vibration responses of a viscoelastic string with constantly and non-constant axial velocities. The general form of differential equations is derived with a differential constitutive law for the linear viscoelastic string. A three-parameter model is adopted in the linear and nonlinear vibration systems. The partial differential equation governing the motion is discretized using Galerkin's method. Taking two terms leads to a set of gyroscopic and non-linear coupled

Acknowledgements

The authors are greatly indebted to the National Science Council of R.O.C. for the support of the research through contract No. NSC 84-2212-E-033-012.

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