Abstract
Transverse non-linear vibration is investigated in principal parametric resonance of an axially accelerating viscoelastic beam. The axial speed is characterized as a simple harmonic variation about a constant mean speed. The material time derivative is used in the viscoelastic constitutive relation. The transverse motion can be governed by a non-linear partial-differential equation or a non-linear integro-partial-differential equation. The method of multiple scales is applied to the governing equations to determine steady-state responses. It is confirmed that the mode uninvolved in the resonance has no effect on the steady-state response. The differential quadrature schemes are developed to verify results via the method of multiple scales. It is demonstrated that the straight equilibrium configuration becomes unstable and a stable steady-state emerges when the axial speed variation frequency is close to twice any linear natural frequency. The results derived for two governing equations are qualitatively the same, but quantitatively different. Numerical simulations are presented to examine the effects of the mean speed and the variation of the amplitude of the axial speed, the dynamic viscosity, the non-linear coefficients, and the boundary constraint stiffness on the instability interval and the steady-state response amplitude.
Highlights
Many engineering devices can be modeled as an axially moving beam, which is a typical gyroscopic continuum
This paper is devoted to steady-state response in principal parametric resonance of axially accelerating viscoelastic beams
The method of multiple scales is applied to a non-linear partial-differential equation and a non-linear integropartial-differential equation to determine steady-state responses
Summary
Many engineering devices can be modeled as an axially moving beam, which is a typical gyroscopic continuum. As exact solutions are usually unavailable for non-linear differential equations, approximate analytical methods are widely applied to investigate non-linear vibration of axially moving beams. In spite of the fact that there have been many approximate analytical investigations on non-linear vibration of axially moving beams, there are very limited researches on the topic via the direct numerical approaches. To address the lack of research in the above-mentioned two aspects, the present investigation studies the principal parametric resonance of an axially accelerating viscoelastic beam with the emphasis on understanding the effects of the mode uninvolved in the resonance and verifying the analytical results by the differential quadrature method. The present investigation differentiates from the previous analytical study on axially accelerating viscoelastic beams [14] in the constitutive relation and the boundary conditions.
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