An amplitude equation for the non-linear vibration of viscoelastically damped sandwich beams

• Published in: Journal of sound and vibration Vol. 271; no. 3; pp. 789 - 813
• Main Authors: Daya, E.M., Azrar, L., Potier-Ferry, M.
• Format: Journal Article
• Language: English
• Published: London Elsevier Ltd 06.04.2004; Elsevier
• Subjects: Exact sciences and technology; Fundamental areas of phenomenology (including applications); Physics; Solid mechanics; Structural and continuum mechanics; Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...); Constitutive equation; Viscoelasticity; Frequency dependence; Forced vibration; Sandwich structure; Numerical method; Harmonic balance; Beam(mechanics); Viscous damping; Complex variable method; Complex number; Coupled modes; Response frequency; Galerkin method; Frequency response; Structural analysis; Non linear vibration
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/S0022-460X(03)00754-5

An elementary theory for non-linear vibrations of viscoelastic sandwich beams is presented. The harmonic balance method is coupled with a one mode Galerkin analysis. This results in a scalar complex frequency–response relationship. So the non-linear free vibration response is governed by only two complex numbers. This permits one to recover first the concept of linear loss factor, second a parabolic approximation of the backbone curve that accounts for the amplitude dependence of the frequency. A new amplitude–loss factor relationship is also established in this way. The forced vibration analysis leads to resonance curves that are classical within non-linear vibration theory. They are extended here to any viscoelastic constitutive behaviour. This elementary approach could be extended to a large class of structures and in a finite element framework. The amplitude equation is obtained in closed form for a class of sandwich beams. The effects of the boundary conditions and of the temperature on the response are discussed.