Abstract: A method for determining the stress distribution of a rectangular plate subjected to two pairs of arbitrarily distributed in-plane edge shear loads is proposed, and the free vibration and buckling of such a rectangular plate are investigated in this work. The method utilizes two stress functions to synthesize the stress-resultant field of the plate with each of the stress functions satisfying the biharmonic compatibility equation. The sum of stress-resultant fields due to these two stress functions satisfies the boundary conditions at the edges of the plate, from which these two stress functions are determined. Then, the free vibration and buckling of the rectangular plate are investigated by the Galerkin method. Numerical results obtained by this work are compared with those appeared in the literature, and good agreements are observed.
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 free and forced in-plane vibrations of axially
moving plates are investigated in this work. The plate possesses an
internal damping of which the constitutive relation obeys the
Kelvin-Voigt model, and the excitations are arbitrarily distributed on
two opposite edges. First, the equations of motion and the boundary
conditions of the axially moving plate are derived. Then, the extended
Ritz method is used to obtain discretized system equations. Finally,
numerical results for the natural frequencies and the mode shapes of
the in-plane vibration and the in-plane response of the moving plate
subjected to arbitrary edge excitations are presented. It is observed that
the symmetry class of the mode shapes of the in-plane vibration
disperses gradually as the moving speed gets higher, and the u- and
v-components of the mode shapes belong to different symmetry class.
In addition, large response amplitudes having shapes similar to the
mode shapes of the plate can be excited by the edge excitations at the
resonant frequencies and with the same symmetry class of distribution
as the u-components.