Abstract: The motion of an axially moving beam with rotating prismatic joint with a tip mass on the end is analyzed to investigate the nonlinear vibration and dynamic stability of the beam. The beam is moving with a harmonic axially and rotating velocity about a constant mean velocity. A time-dependent partial differential equation and boundary conditions with the aid of the Hamilton principle are derived to describe the beam lateral deflection. After the partial differential equation is discretized by the Galerkin method, the method of multiple scales is applied to obtain analytical solutions. Frequency response curves are plotted for the super harmonic resonances of the first and the second modes. The effects of non-linear term and mean velocity are investigated on the steady state response of the axially moving beam. The results are validated with numerical simulations.
Abstract: In this paper, using the method of multiple scales, the second sub-harmonic resonance in vortex-induced vibrations (VIV) of a marine pipeline close to the seabed is investigated based on a developed wake oscillator model. The amplitude-frequency equations are also derived. It is found that the oscillation will increase all the time when both discriminants of the amplitude-frequency equations are positive while the oscillation will decay when the discriminants are negative.
Abstract: This paper investigates the parametric stability of an
axially moving web subjected to non-uniform in-plane edge
excitations on two opposite, simply-supported edges. The web is
modeled as a viscoelastic plate whose constitutive relation obeys the
Kelvin-Voigt model, and the in-plane edge excitations are expressed
as the sum of a static tension and a periodical perturbation. Due to the
in-plane edge excitations, the moving plate may bring about
parametric instability under certain situations. First, the in-plane
stresses of the plate due to the non-uniform edge excitations are
determined by solving the in-plane forced vibration problem. Then,
the dependence on the spatial coordinates in the equation of transverse
motion is eliminated by the generalized Galerkin method, which
results in a set of discretized system equations in time. Finally, the
method of multiple scales is utilized to solve the set of system
equations analytically if the periodical perturbation of the in-plane
edge excitations is much smaller as compared with the static tension of
the plate, from which the stability boundaries of the moving plate are
obtained. Numerical results reveal that only combination resonances
of the summed-type appear under the in-plane edge excitations
considered in this work.
Abstract: The effect of small non-parallelism of the base flow
on the stability of slightly curved mixing layers is analyzed in the
present paper. Assuming that the instability wavelength is much
smaller than the length scale of the variation of the base flow we
derive an amplitude evolution equation using the method of multiple
scales. The proposed asymptotic model provides connection between
parallel flow approximations and takes into account slow
longitudinal variation of the base flow.
Abstract: Method of multiple scales is used in the paper in order
to derive an amplitude evolution equation for the most unstable mode
from two-dimensional shallow water equations under the rigid-lid
assumption. It is assumed that shallow mixing layer is slightly curved
in the longitudinal direction and contains small particles. Dynamic
interaction between carrier fluid and particles is neglected. It is
shown that the evolution equation is the complex Ginzburg-Landau
equation. Explicit formulas for the computation of the coefficients of
the equation are obtained.