Kelvin–Voigt versus Bürgers internal damping in modeling of axially moving viscoelastic web

doi:10.1016/S0020-7462(01)00142-1

International Journal of Non-Linear Mechanics

Volume 37, Issue 7, October 2002, Pages 1147-1161

https://doi.org/10.1016/S0020-7462(01)00142-1 Get rights and content

Abstract

Stability and oscillation characteristics of two-dimensional axially moving web have been investigated. The application of one-dimensional beam-like models of the web allows the identification of instability regions and the estimation of the critical speed. For the beam material two different models, i.e., Kelvin–Voigt and Bürgers have been considered. The numerical solutions of full non-linear and linearized equations have been compared. The effects of axially travelling speed and the internal damping on dynamical stability of axially moving web have been studied in details. Our numerical studies of Kelvin–Voigt and Burger's models show that both models give similar results for small values of internal damping and can be used to describe the dynamics of axially moving webs made from materials with internal damping coefficient smaller than 3×10⁻⁵. For the materials with larger damping coefficient the Bürgers model gives more reliable results.

Introduction

The continuous development of mechanics and its engineering applications in our days have increased remarkably the interest in non-linear stability analysis of structures. Particularly a great number of studies consider two general models, the systems under followed loading [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] and elastic continua translating at high speed [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. In the following dynamical behavior of the second model is studied.

Elastic continua translating at high speed such as band saw blades, magnetic types, paper webs, plastic sheets, films, transmission cables are present in various industrial applications. Generally, axially moving continuum in the form of thin, flat rectangular shape material with small flexural stiffness is called a web. Webs are moving at high speed, for example, in paper production the paper webs are transported with longitudinal speeds of up to $3000 m / min$ . Above the critical speed one can expect various dynamical instabilities mainly of divergent and flutter type. These instabilities can decrease the quality of products and their performance. The excessive oscillations of a computer tape degrade the signal and can affect the data storage. The instability of a band saw results in low surface quality, unsatisfactory cutting performance and leads to the loss of raw materials. In paper production, the machines instabilities, resonance oscillations and the flutter of the web can cause the wrinkling or even a breaking of the web. To ensure that the operating system is under stable working conditions, full analysis of its dynamics has to be performed. Complete knowledge of the dynamical behavior allows the prediction and control of instabilities.

The results of the dynamical analysis of the axially moving web shows that it is impossible to obtain the mathematical model, which is valid for the whole range of transport speeds. It has been necessary to introduce a number of simplifications in the considered models. The validation of such models is restricted to specific intervals of transport speed and usually they describe only the most significant features of the dynamical behavior.

In modeling the axially moving webs one can use one-dimensional beam theory (e.g. [24]) or two-dimensional plate theory (e.g. [20]). Historically, also the string theory has been used but nowadays it is considered to give too rough approximation of the real phenomena as it has not considering plate stiffness of the web. Although the plate theory gives the most accurate description of the physical phenomena that occur in the web, it is very complicated mathematically and requires time-consuming calculations. Our previous studies show that for the large class of the practically important webs with small flexural stiffness the beam theory gives equally accurate results as the plate theory [20]. In particular, the plot of critical speed s_cr of the web (defined as the ratio of the transport speed to the wave speed) versus dimensionless plate stiffness ψ (defined as Eh³/[12(1−ν²)Pl²]) is shown in Fig. 1.

One can observe that for the large range of stiffness ψ and the slenderness ratio l/b (length over width of the web) the results of the beam and the plate theories are very close.

The other important problem one can meet in considering axially moving web is how to model web material. Particularly, how to model the internal damping of the material and this is the main problem, which we stress in this paper. Generally, two different models, namely Kelvin–Voigt [22] and Bürgers [23] are commonly used and we try to answer the question, which of them better describes the dynamics of the web. Additionally, we compare the results obtained from the analysis of the linearized equations with the results of the integration of the full non-linear equations.

The paper is organized as follows. In Section 2 basing on the beam theory, we derive the equations of motion of the axially moving web. Section 3 describes differential constitutive equations obtained from two-parameter Kelvin–Voigt and four parameter Bürgers models. In Section 4 we give full mathematical models of the web with Kelvin–Voigt (Section 4.1) and Bürgers damping (Section 4.2). In Section 5, we discuss the results of our numerical investigations of two-parameter Kelvin–Voigt (Section 5.1) and four-parameter Bürgers (Section 5.2) models. The conclusions are presented in Section 6.

Section snippets

Equations of motion

A viscoelastic axially moving web of the length l is considered. The web moves at axial velocity c. The geometry of the system and the introduced co-ordinates are shown in Fig. 2.

The problem of transverse oscillations of the axially moving continua in a state of uniform initial stress was investigated [20]. In the case of thin web, the results of earlier studies show that the beam models can approximate the dynamical behavior of the web (Fig. 3). The application of this model gives the

Differential constitutive equations

The models of internal damping introduced by Kelvin–Voigt and Bürgers are shown in Fig. 4.

For the two-parameter viscoelastic model of material—Kelvin–Voigt element (Fig. 4a), the differential constitutive equation can be written as $a_{0} σ=b_{1} ε_{,t} +b_{o} ε,$ where $a_{0} =1, b_{0} =E, b_{1} =γ.$

The four-parameter viscoelastic model of material in the form of the Bürgers element (Fig. 4b) was taken into account. The differential constitutive equation of the model material can be written as $a_{2} σ_{,tt} +a_{1} σ_{,t} +a_{0} σ=b_{2} ε_{,tt} +b_{1} ε_{,t},$ where $a_{2}$

Two-parameter model of material

To obtain mathematical description of the viscoelastic beam model one should multiply Eq. (1) with operator Γ. The bending moment M is given $M=−E J_{z} w_{,xx} −J_{z} γ w_{,xxt} .$ Using , , one receives $w_{,tt} +2cw_{,xt} +c^{2} w_{,xx} + EJ_{z} ρA_{z} w_{,xxxx} + J_{z} γ ρA_{z} w_{xxxxt}$ $− P_{0} ρA_{z} w_{,xx} − 32 E ρ w_{,x}^{2} w_{,xx}$ $− 2 γ ρ (w_{,x} w_{,xt} w_{,xx} +cw_{,x} w_{,xx}^{2})$ $− γ ρ (w_{,x}^{2} w_{,xxt} +cw_{,x}^{2} w_{,xxx})=0.$ The boundary conditions $w(0,t)=w(l,t)=0, w_{,xx} (0,t)=w_{,xx} (l,t)=0.$ Let the dimensionless parameters be $z= w h_{z}, ξ= x l, s= c c_{f} =c A_{z} ρ P_{0},$ $τ=t c_{f} l = t l P A_{z} ρ, c_{f} = P A_{z} ρ .$ The substitution of Eq. (12) into Eq. (10) gives the

Numerical results

Numerical investigations have been carried out for the beam model of the steal web. Parameters data: length $l=1 m$ , width $b=0.2 m$ , thickness $h=0.0015 m$ , mass density $ρ=7800 kg / m^{3}$ , Young's modulus along $x : E_{x} =0.2 10^{12} N / m^{2}$ , initial stress $N_{0} =2500 N / m$ , n=3. Initial conditions: q₁=1, q_1,t=0,…,q_3,ttt=0. The Runge–Kutta method was used to integrate ordinary differential equations and analyze the dynamic behavior of the system.

Conclusions

Dynamic investigations of beam-like models of the axially moving web with constant axial stress are carried out in this paper. The beam model material as the Kelvin–Voigt element (two-parameter model) and the Bürgers element (four-parameter model) are considered. The general forms of differential equations of transverse oscillations of the systems are derived together with the differential constitutive law for their rheologic models.

The numerical investigations have been carried out for the

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Kelvin–Voigt versus Bürgers internal damping in modeling of axially moving viscoelastic web

Abstract

Introduction

Section snippets

Equations of motion

Differential constitutive equations

Two-parameter model of material

Numerical results

Conclusions

Int. J. Eng. Sci.

Int. J. Eng. Sci.

J. Sound Vibration

J. Sound Vibrations

J. Sound Vibrations

J. Sound Vibration

J. Sound Vibrations

J. Sound Vibration

J. Sound Vibration

Int. J. Non-Linear Mech.

J. Sound Vibration

J. Sound Vibration