Approximate and numerical analysis of nonlinear forced vibration of axially moving viscoelastic beams

Steady-state periodical response is investigated for an axially moving viscoelastic beam with hybrid supports via approximate analysis with numerical confirmation. It is assumed that the excitation is spatially uniform and temporally harmonic. The transverse motion of axially moving beams is governed by a nonlinear partial-differential equation and a nonlinear integro-partial-differential equation. The material time derivative is used in the viscoelastic constitutive relation. The method of multiple scales is applied to the governing equations to investigate primary resonances under general boundary conditions. It is demonstrated that the mode uninvolved in the resonance has no effect on the steady-state response. Numerical examples are presented to demonstrate the effects of the boundary constraint stiffness on the amplitude and the stability of the steady-state response. The results derived for two governing equations are qualitatively the same,but quantitatively different. The differential quadrature schemes are developed to verify those results via the method of multiple scales.

Multiplicative Derivations on Rank-s Matrices for Relatively Small s

In this paper,we describe multiplicative derivations on the set of all rank-s matrices of Mn(K)over a field K with a relatively small integer s.Concretely,for fixed integers n,s satisfying 1≤s≤n/2 and n≥2,we prove that if a map d:Mn(K)→Mn(K)satisfiesδ(xy)=δ(x)y+xδ(y)for any two rank-s matrices,y∈Mn(K),then there exists a derivation D of Mn(K)such thatδ(x)=D(x)for each rank-k matrix a E Mn(K)with 0≤k≤s.As an application,we prove that a multiplicative derivation on a special subset of Mn(K)must be a derivation.