A harmonic balance method for the non-linear vibration of viscoelastic shells

Abstract

In this Note, we deal with the non-linear vibration of viscoelastic shell structures. Coupling a harmonic balance method with a one mode Galerkin's procedure, one obtains an amplitude equation depending on two complex coefficients. These are determined by solving a classical eigenvalue problem and two linear problems. This permits us to characterize the evolution of the loss factor with the vibration amplitude. To validate our approach, the amplitude-frequency and the amplitude-loss factor relationships are illustrated in the case of a circular ring.

Nonlinear forced vibration of damped plates by an asymptotic numerical method

“…An amplitude equation, based on an approximated harmonic balance method and Galerkin's procedure, was proposed by Daya et al [5] to study sandwich beams and plates with central viscoelastic layers. The same approach was extended to the analysis of circular sandwich rings by Boutyour et al [6]. Touzé and Amabili built reduced-order models for damped geometrically nonlinear systems [7].…”

“…Injecting expressions (1)-(4) into (6), taking into account the damping coefficients and using Hamilton's principle, the governing equation is obtained in the following form:…”

Non-linear vibrations of sandwich viscoelastic shells

“…For instance, these studies concern sandwich viscoelastic structures with simple geometry as beams or plates [23][24][25][26]. As it is well known, the non-linear geometrical effects induce some dependence between the frequencies and the loss factors with respect to the amplitude [25,27]. Recently, Boumediene et al [28] developed a reduction method based on a high-order Newton algorithm and reductions techniques to determine the modal characteristics of viscoelastic sandwich structures.…”

“…The approach is based on a coupling of an approximated harmonic balance method with a Galerkin's procedure with one mode. The non-linear modal relationship giving the frequency (free and forced) and the loss factor, with respect to the displacement, are obtained by solving a classical eigenvalue problem and two linear ones [24,27]. To validate our approach, one gives an application to a sandwich viscoelastic ring.…”

Geometry Effects on the Nonlinear Oscillations of Viscoelastic Cylindrical Shells

“…After, a general methodology looking to describe the non-linear vibrations of viscoelastic shell structures, considering periodic or damped responses through the coupling of the harmonic balance method with one mode Galerkin discretization was considered [6]. The radial motions of compressible non-linearly viscoelastic cylindrical and spherical shells under lateral time-dependent pressures [7].…”