Stability of a linear integro-differential equation with periodic coefficients

Stability of a linear integro-differential equation with periodic coefficients is studied. Such an equation arises in the dynamics of thin-walled viscoelastic elements of structures under periodic compressive loading. The equation under consideration has a specific peculiarity which makes its analysis difficult: in the absence of the integral term it is only stable, but not asymptotically stable. Therefore, in order to derive stability conditions we have to introduce some specific restrictions on the behavior of the kernel of the integral operator. These restrictions are taken from the analysis of the relaxation measures for a linear viscoelastic material. We suggest an approach to the study of the stability based on the direct Lyapunov method and construct new stability functionals. Employing these techniques we derive some new sufficient stability conditions which are close enough to the necessary ones. In particular, when the integral term vanishes, our stability conditions pass into the well-known stability criterion for a linear differential equation with periodic coefficients. In the general case, the proposed stability conditions have the following mechanical meaning: a viscoelastic structure under periodic excitations is asymptotically stable if the corresponding elastic structure is stable and the material viscosity is sufficiently large. As an example, the stability problem is considered for a linear viscoelastic beam compressed by periodic-in-time forces. Explicit limitations on the material parameters are obtained which guarantee the beam stability, and the dependence of the critical relaxation rate on the material viscosity is analysed numerically for different frequencies of the periodic compressive load.