Explicit stability conditions for integro-differential equations with periodic coefficients

New explicit stability conditions are derived for a linear integro-differential equation with period operator coefficients. The equation under consideration describes oscillations of thin-walled viscoelastic structural members driven by periodic loads. To develop stability conditions two approaches are combined. The first is based on the direct Lyapunov method of constructing stability functionals. It allows stability conditions to be derived for unbounded operator coefficients, but fails to correctly predict the critical loads for high-frequency excitations The other approach is based on transforming the equations under consideration stability. Stability conditions for the transformed equation are obtained by using a technique of integral estimates. This method provides acceptable estimates of the critical forces for periodic loads, but can be applied to equations with bounded coefficients only. Combining these two approaches, we derive explicit stability conditions which are close to the Floquet criterion when the integral term vanishes. These conditions are applied to the the stability problem for viscoelastic bar compares by periodic forces. The effect of material and structural parameters on the critical load is studied numerically.