TY - JOUR
T1 - Principal parametric resonance of axially accelerating viscoelastic beams
T2 - Multi-scale analysis and differential quadrature verification
AU - Chen, Li-Qun
AU - Ding, Hu
AU - Lim, C. W.
PY - 2012
Y1 - 2012
N2 - 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. © 2012-IOS Press and the authors. All rights reserved.
AB - 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. © 2012-IOS Press and the authors. All rights reserved.
KW - axially accelerating beam
KW - differential quadrature method
KW - method of multiple scales
KW - nonlinearity
KW - Vibration
KW - viscoelasticity
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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84866493844&origin=recordpage
U2 - 10.3233/SAV-2011-0648
DO - 10.3233/SAV-2011-0648
M3 - 21_Publication in refereed journal
VL - 19
SP - 527
EP - 543
JO - Shock and Vibration
JF - Shock and Vibration
SN - 1070-9622
IS - 4
ER -