Irregular instability boundaries of axially accelerating viscoelastic beams with 1:3 internal resonance. (November 2017)

• Record Type: Journal Article
• Authors: Zhang, Deng-Bo; Tang, You-Qi; Chen, Li-Qun
• Is Part Of: International journal of mechanical sciences. Volume 133(2017)
• Page Start: 535
• Page End: 543
• Publication Date: 2017-11
• Subjects: Axially moving beam -- Irregular instability boundary -- Nonhomogeneous boundary condition -- Parametric resonance -- Differential quadrature scheme; Mechanical engineering -- Periodicals; Génie mécanique -- Périodiques; Mechanical engineering; Maschinenbau; Mechanik; Zeitschrift; Periodicals; 621.05
• DOI: 10.1016/j.ijmecsci.2017.08.052
• Languages: English
• ISSNs: 0020-7403

Abstract

Highlights: Irregular instability boundaries of axially accelerating beams with 1:3 internal resonance are analytically and numerically investigated. Both combination and principal parametric resonances are treated. The effects of the nonhomogeneous boundaries are highlighted. By the method of multiple scales, the solvability conditions in summation and principal parametric resonances are established by some different manipulations in the process of the classical multiple scales method. Numerical integrations via the differential quadrature support results via the method of multiple scales. Abstracts: Irregular instability boundaries of axially accelerating beams with 1:3 internal resonance are analytically and numerically investigated in this paper. The distributed parameter is due to the small simple harmonic axial speed. The viscoelastic characteristic of the beam is described by the Kelvin–Voigt model in which the material time derivative is used. A linear partial-differential equation with the variable coefficient and the relevant boundary conditions governing the transverse motion is presented. The effects of the nonhomogeneous boundaries are highlighted. By the method of multiple scales, the solvability conditions in summation and principal parametric resonances are established by some different manipulations in the process of the classical multiple scales method. The Routh–Hurwitz criterion is used to determine the instability boundaries. The effects of viscoelastic coefficient and the viscous damping coefficient are examined on the instability boundaries. Irregular instability boundaries appeared when the 1:3 internal resonance is introduced. It is shown that the numerical calculations by the differential quadrature scheme can verify the approximate analytical results. Graphical abstract: Irregular instability boundaries of axially accelerating beams with 1:3 internal resonance are analytically and numerically investigated in this paper. Irregular instability boundaries appeared when the 1:3 internal resonance is introduced. The effects of the viscoelastic coefficients on the stability boundaries have been studied.