The torsion of a viscoelastic plate in an unsteady flow

The stability of the equilibrium position of a viscoelastic plate, subjected to torsional strain and effect of the free airflow is investigated. The unsteadiness of the flow is taken into account by introducing integral terms into the moments of the aerodynamic forces acting on the plate. In a neighbourhood of the equilibrium position, a general solution of a Volterra-type integro-differential equation with partial derivatives is constructed in the form of a Fourier series, as a function of the longitudinal coordinate of the plate with coefficients that are the power series in the small parameters introduced. The stability of the plate equilibrium in the unstrained state is analysed in the case when there are small perturbations (possibly, discontinuous) of the flow velocity. The stability under persistent perturbations of the equilibrium of the strained plate with respect to non-linear perturbing forces and perturbations of its shape at the instant of time preceding the specified initial instant is also investigated.