Nonlinear dynamic response of a simply-supported Kelvin-Voigt viscoelastic beam, additionally supported by a nonlinear spring

The free and forced vibrations of a Kelvin-Voigt viscoelastic beam, supported by a nonlinear spring are analytically investigated in this paper. The governing equations of motion along with the compatibility conditions are obtained employing Newton’s second law of motion and constitutive relations. The viscoelastic beam material is constituted by the Kelvin-Voigt rheological model, which is a two-parameter energy dissipation model. The method of multiple timescales, a perturbation technique, is employed which ultimately leads to approximate analytical expressions for vibration response, and provides better insight into how the system parameters influence the vibration response. Finally, the effect of system parameters on the linear and nonlinear natural frequencies, vibration responses and frequency-response curves of the system is characterized.     

Long‐time behavior of a nonlinear viscoelastic beam equation with past history

This paper is concerned with the existence of a global attractor for the nonlinear viscoelastic beam equation with past history memory where g(u_t) is a damping like | u_t | ^ru_t and f(u) is a source term like | u | ^αu ? | u | ^βu, by considering 0 ≤ β < α and r > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

Nonlinear vibrations of a viscoelastic beam

Averaging [2, 3] is used to examine the free transverse oscillations of a viscoelastic beam supported at the ends, the rheological properties being described by a certain cubic integral relationship. The forced principal and fractional resonant oscillations are considered for n=1, these being excited by a transverse periodic force; the limiting oscillations are determined that correspond to the principal resonance.M. T. Urazbaev Institute of Building Mechanics and Seismic Stability, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 5, pp. 939–943, September–October, 1973.     

Stabilization of a viscoelastic Timoshenko beam

A viscoelastic Timoshenko beam is investigated. We prove an exponential decay of solutions for a large class of kernels with weaker conditions than the existing ones in the literature. This will allow the use of other types of viscoelastic material for Timoshenko type beams than the usually used ones.     

Boundary controllability of a viscoelastic beam

It is shown thai, a viscoelastic beam with long memory is exactly controllable by L₂ (0 ,T)-boundary controls in any time T > 0. Furthermore, a weak bang-hang principle is derived In the elastic case     

Free vibrations of a viscoelastic beam

The method of averaging is used to investigate the free transverse vibrations of a viscoelastic beam supported at the ends. The material properties of the beam are described by a certain nonlinear integral equation of Volterra type.Institute of Mechanics and Seismic Stability of Structures, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, No. 4, pp. 752–756, July–August, 1971.     

Two-dimensional nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed

In the present study, the coupled nonlinear dynamics of an axially moving viscoelastic beam with time-dependent axial speed is investigated employing a numerical technique. The equations of motion for both the transverse and longitudinal motions are obtained using Newton’s second law of motion and the constitutive relations. A two-parameter rheological model of the Kelvin–Voigt energy dissipation mechanism is employed in the modelling of the viscoelastic beam material, in which the material time derivative is used in the viscoelastic constitutive relation. The Galerkin method is then applied to the coupled nonlinear equations, which are in the form of partial differential equations, resulting in a set of nonlinear ordinary differential equations (ODEs) with time-dependent coefficients due to the axial acceleration. A change of variables is then introduced to this set of ODEs to transform them into a set of first-order ordinary differential equations. A variable step-size modified Rosenbrock method is used to conduct direct time integration upon this new set of first-order nonlinear ODEs. The mean axial speed and the amplitude of the speed variations, which are taken as bifurcation parameters, are varied, resulting in the bifurcation diagrams of Poincaré maps of the system. The dynamical characteristics of the system are examined more precisely via plotting time histories, phase-plane portraits, Poincaré sections, and fast Fourier transforms (FFTs).     

Forced nonlinear vibrations of a viscoelastic body

An approximate solution of the problem of the forced, geometrically nonlinear vibrations of an arbitrary viscoelastic body is found in the form of an expansion in eigenfunctions of the corresponding linear elastic problem. With the aid of the virtual displacement principle the problem is reduced to a system of nonlinear integro-differential equations whose periodic solution is constructed by the small-parameter method.     

Global attractor for a nonlinear viscoelastic model

In this paper we establish the existence of global attractors of a semiflow and a semigroup for a nonlinear 1-d viscoelasticity in two frameworks. We exploit the properties of the analytic semigroup to show the compactness for the semiflow and semigroup generated by the weak global solutions.     

Calculation of nonlinear vibrations of a viscoelastic bar

The problem of forced nonlinear vibrations of a viscoelastic bar is reduced by the Bubnov—Galerkin method to a nonlinear integro-differential equation. Transverse vibrations with small amplitude are considered, and therefore the approximate solution is constructed by the small-parameter method.     

Mathematical justification of viscoelastic beam models by asymptotic methods

The authors derive and justify two models for the bending-stretching of a viscoelastic rod by using the asymptotic expansion method.     

Transient contraction flow of a nonlinear viscoelastic fluid

The unsteady behavior of a Phan-Thien-Tanner fluid in a pressure driven contraction flow is investigated. Calculations are performed using the log-conformation reformulation and block coupled pressure correction. Studies are carried out for the Deborah number De of the process and the model's extension parameter α. It is shown that for very small and very large De the fluid responds newtonian whereas for moderate De damped oscillations of the volume flux can be observed. The studies for α show that the flow is shear dominant despite the extensional flow character of contractions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stabilization of a nonlinear Timoshenko beam

We study the asymptotic behavior of the system governing the nonlinear vibrations of a Timoshenko beam.     

Vibrations of a shock-absorbing device on a nonlinear viscoelastic base

A method of calculating a viscoelastic shock-absorbing device or structure by the method of averaging systems of integro-differential equations is considered. The principal quadratic theory of hereditary viscoelasticity is used as the physical relationship between forces and displacements of the shock-absorbing base or structure.Moscow Institute of Electronic Machine Construction. Translated from Mekhanika Polimerov, No. 4, pp. 730–737, July–August, 1972.     

Flow of a nonlinear viscoelastic liquid in cylindrical channels

The problem of nonrectilinear steady-state flow of a nonlinear viscoelastic liquid in an arbitrary cylindrical channel is examined. On the assumption that the cross flows are insignificant as compared with the longitudinal flows an equation of state is derived for the flow regime in question. A variational principle established for steady-state flows of the investigated media is proposed as the basis of a method of solving problems of the flow of polymer materials in arbitrary cylindrical channels. The flow of a polymer solution in rectangular channels is investigated.Institute of Mechanics, AS UkrSSR, Kiev. Translated from Mekhanika Polimerov, Vol. 4, No. 6, pp. 1103–1111, November–December, 1968.