Asymptotic analysis of the stability of a cylindrical viscoelastic shell under the action of a longitudinal periodic load
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 \((k=2,3,...)\).
system of ordinary integrodifferential equations with periodic coefficients
fractional exponential relaxation kernels
scientific article; zbMATH DE number 4195361