Asymptotic analysis of the stability of a cylindrical viscoelastic shell under the action of a longitudinal periodic load
type: Item
instance of: scholarly article
title: Asymptotic analysis of the stability of a cylindrical viscoelastic shell under the action of a longitudinal periodic load
zbMATH Open document ID: 0724.73098
DOI: 10.1016/0021-8928(88)90099-8
author: L. Kh. Belen'kaya
published in: Journal of Applied Mathematics and Mechanics
publication date: 1988-01-01T00:00:00Z
review text: Stability of the rectilinear form of a viscoelastic, orthotropic cylindrical shell acted upon by a longitudinal periodic load is considered in the case of high-frequency modulation and near the resonance frequencies. The boundary conditions at the ends of the shell allow a periodic continuation over the spatial variables, so that the problem can be reduced to a system of ordinary integrodifferential equations with periodic coefficients. We obtain an asymptotic formula for the boundary of stability in the case when \(\omega\to \infty\) and the wave numbers are fixed. In the case of fractional exponential relaxation kernels the critical load and neutral oscillations are sought in the form of series in fractional powers of the parameter \(\epsilon\) \((\epsilon =1/\omega)\). For the differential equations in the case when the relaxation kernel has no singularities at the zero, and for the integrodifferential equations, the asymptotic expansions are constructed in integral powers of \(\epsilon\). It is shown that at high modulation frequencies the critical value of the load is close to its stationary value. Further, the Lyapunov-Schmidt method is used to study the behaviour of the system in the case when the frequency \(\omega\) is close to the resonance frequency \(\omega_ k\) \((\omega_ k=2\omega_*/k\), \(k=1,2,3,...;\omega_*\) is the natural frequency of oscillation) and the coefficients of viscosity are small. It is shown that if the mean value of the load \(<\phi >\neq 0\), then at the higher-order resonances \((k=2,3,...)\) it strongly shifts the frequency of natural oscillations and a stable state exists near the k-th resonance \((k=2,3,...)\) when the viscosity is low, i.e. the instability near the higher-order resonances \((k=2,3,...)\) is quenched by the viscosity. If on the other hand \(<\phi >=0\), then a state of instability exists near the k-th resonance \((k=2,3,...)\).
Mathematics Subject Classification ID: 74G60
Mathematics Subject Classification ID: 74K15
Mathematics Subject Classification ID: 45M05
Mathematics Subject Classification ID: 45J05
zbMATH DE Number: 4195361
zbMATH Keywords: high-frequency modulation
zbMATH Keywords: near the resonance frequencies
zbMATH Keywords: system of ordinary integrodifferential equations with periodic coefficients
zbMATH Keywords: fractional exponential relaxation kernels
zbMATH Keywords: critical load
zbMATH Keywords: neutral oscillations
zbMATH Keywords: Lyapunov-Schmidt method
MaRDI profile type: Publication
full work available at URL: 0021-8928(88)90099-8
OpenAlex ID: W2105178093
cites work: Q5682682
description: scientific article; zbMATH DE number 4195361