Affiliations: Department of Mechanics, Shanghai Key Laboratory of Mechanics in Energy Engineering,
Shanghai University, Shanghai, China | Shanghai Institute of Applied Mathematics and
Mechanics, Shanghai, China | Department of Building and Construction, City
University of Hong Kong, Kowloon, Hong Kong, China
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.
