Non-linear forced vibration of axially moving viscoelastic beams

Abstract

The non-linear forced vibration of axially moving viscoelastic beams excited by the vibration of the supporting foundation is investigated. A non-linear partial-differential equation governing the transverse motion is derived from the dynamical, constitutive equations and geometrical relations. By referring to the quasi-static stretch assumption, the partial-differential non-linearity is reduced to an integro-partial-differential one. The method of multiple scales is directly applied to the governing equations with the two types of non-linearity, respectively. The amplitude of near-and exact-resonant steady state is analyzed by use of the solvability condition of eliminating secular terms. Numerical results are presented to show the contributions of foundation vibration amplitude, viscoelastic damping, and nonlinearity to the response amplitude for the first and the second mode.KEY WORDS axially moving beam, viscoelasticity, non-linear forced vibration, method of multiple scales

Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams

“…Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams 427 The present investigation also differentiates from the previous analytical study on transverse forced vibration of axially moving viscoelastic beams by Yang and Chen [15] in the constitutive relation and boundary conditions. To describe the viscoelastic behavior of the beam material, the partial time derivative in the Kelvin model was used in Ref.…”

“…The beam treated here is constrained by a rotating sleeve with a rotational spring at each end [17]. The boundary conditions are more general than the fixed ends used by Yang and Chen [15] and the simple supports used by Wang and Chen [14]. The generalization is necessary for understanding some practical engineering problems.…”

“…To describe the viscoelastic behavior of the beam material, the partial time derivative in the Kelvin model was used in Ref. [15]. Mockensturm and Guo [16] convincingly argued that the Kelvin model generalized to axially moving materials should contain the material time derivative to account for the energy dissipation in steady motion.…”

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

“…For an elastic beam, there will be no such third-order time derivative in the equations of motion. 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.…”

“…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. Zhang and Zu [23,24] attempted to describe the mechanical energy dissipation using a viscoelastic model for the belt, and utilized the perturbation techniques to predict the non-linear response.…”