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

Fung, Rong‐Fong; Huang, Jianguo; Chen, Ying-Chang; Yao, Chih-Min

doi:10.1016/s0045-7949(98)00001-7

Cited by 38 publications

(24 citation statements)

References 10 publications

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“…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] …”

Section: Governing Equations Of Motionmentioning

confidence: 99%

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

王波

Wang

2012

Appl. Math. Mech.-Engl. Ed.

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The weakly forced vibration of an axially moving viscoelastic beam is investigated. The viscoelastic material of the beam is constituted by the standard linear solid model with the material time derivative involved. The nonlinear equations governing the transverse vibration are derived from the dynamical, constitutive, and geometrical relations. The method of multiple scales is used to determine the steady-state response. The modulation equation is derived from the solvability condition of eliminating secular terms. Closed-form expressions of the amplitude and existence condition of nontrivial steady-state response are derived from the modulation equation. The stability of nontrivial steady-state response is examined via the Routh-Hurwitz criterion.

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Section: Governing Equations Of Motionmentioning

confidence: 99%

“…In this section, we shall use the method of multiple scales to solve the governing equation (18) with (19) and (20) subjected to the boundary condition (21), defining a slow time scale T = εt and looking for the asymptotic solutions of the form, respectively, i.e.,…”

Section: Approximate Solution For Primary Resonance Of Nth Modementioning

confidence: 99%

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

王波

Wang

2012

Appl. Math. Mech.-Engl. Ed.

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“…Furthermore, if one adopts the Kelvin viscoelastic model (cf., [22][23][24]28,29]), there will be no higher-order time derivative for the stress; therefore, the third-order time derivative for the equations of motion will also disappear. Though Fung et al [21] and Hou and Zu [31] employed the SLS model, Fung et al [21] assumed the viscosity constant to be zero for simplicity, and Hou and Zu [31] applied the multiple scale method in a different fashion. Here, we define I in Eq.…”

Section: Q (T) + M(t)q(t) + C(t)q(t) + K(t) Q(t)mentioning

confidence: 99%

“…Therefore, the application of viscoelasticity attracts substantial attention within the axially moving continuum field. For example, Fung et al [21] explored the transverse vibrations of an axially moving viscoelastic string subjected to an initial stress on the uniform cross section; they applied the Galerkin's method to solve the equations of motion. The multiple scales method was presented by Yang and Chen [22] for obtaining the near-and exact-resonant steady-state response of the forced vibration of a simply supported axially moving viscoelastic beam.…”

Section: Introductionmentioning

confidence: 99%

Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length

Wang

Zhong

et al. 2010

Acta Mech

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The nonlinear free vibration of an axially translating viscoelastic beam with an arbitrarily varying length and axial velocity is investigated. Based on the linear viscoelastic differential constitutive law, the extended Hamilton's principle is utilized to derive the generalized third-order equations of motion for the axially translating viscoelastic Bernoulli-Euler beam. The coupling effects between the axial motion and transverse vibration are assessed under various prescribed time-varying velocity fields. The inertia force arising from the longitudinal acceleration emerges, rendering the coupling terms between the axial beam acceleration and the beam flexure. Semi-analytical solutions for the governing PDE are obtained through the separation of variables and the assumed modes method. The modified Galerkin's method and the fourth-order RungeKutta method are employed to numerically analyze the resulting equations. Further, dynamic stabilization is examined from the system energy standpoint for beam extension and retraction. Extensive numerical simulations are presented to illustrate the influences of varying translating velocities and viscoelastic parameters on the underlying dynamic responses. The material viscosity always dissipates energy and helps stabilize the transverse vibration.

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“…using a string model [20]. Extending their work, they studied the viscous damping effect in their later research [21]. Viscoelastic strings and beams have recently been studied extensively, see e.g.…”

Section: Introductionmentioning

confidence: 99%

The origin of in-plane stresses in axially moving orthotropic continua

Kurki

Jeronen

Saksa

et al. 2016

International Journal of Solids and Structures

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Highlights• In-plane motion of an axially travelling orthotropic sheet was modelled• Velocity difference between supports generates strain due to mass conservation• In a travelling viscoelastic material the axial strain varies along the length• Small deformations of free edges due to the Poisson effect affect the velocity• Inertial effects lead to additional cross-directional contraction AbstractIn this paper, we address the problem of the origin of in-plane stresses in continuous, two-dimensional high-speed webs. In the case of thin, slender webs, a typical modeling approach is the application of a stationary in-plane model, without considering the effects of the in-plane velocity field. However, for high-speed webs this approach is insufficient, because it neglects the coupling between the total material velocity and the deformation experienced by the material. By using a mixed Lagrange-Euler approach in model derivation, the solid continuum problem can be transformed into a solid continuum flow problem. Mass conservation in the flow problem, and the behaviour of free edges in the two-dimensional case, are both seen to influence the velocity field. We concentrate on solutions of a steady-state type, and study briefly the coupled nature of material viscoelasticity and transport velocity in one dimension. Analytical solutions of the one-dimensional equation are presented with both elastic and viscoelastic material models. The two-dimensional elastic problem is solved numerically using a nonlinear finite element procedure. An important new fundamental feature of the model is the coupling of the driving velocity field to the deformation of the material, while accounting for small deformations of the free edges. The results indicate that inertial effects produce an additional contribution to elastic contraction in unsupported, free webs.

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

Cited by 38 publications

References 10 publications

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

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

Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length

The origin of in-plane stresses in axially moving orthotropic continua

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