Bistability and stochastic jumps in an airfoil system with viscoelastic material property and random fluctuations

The purpose of this paper is to explore analytically the influences of random fluctuations on a two-degrees-of-freedom (TDOF) airfoil model with viscoelastic terms. To begin with, a convolution integral over an exponentially decaying kernel function is employed to establish a constitutive relation of the viscoelastic material. Then the corresponding TDOF airfoil model with viscoelastic terms and random excitations is introduced. Subsequently, a theoretical analysis for the proposed airfoil model is achieved through a multiple-scale method together with a perturbation technique. All of the obtained approximate analytical solutions are verified by numerical simulation results, and a good agreement is observed. Meanwhile, we also find that both high-amplitude and low-amplitude oscillations coexist within a certain range of the excitation frequency or amplitude, which is regarded as a bi-stable behavior. In addition, effects of the viscoelastic terms and the random excitations on the system responses are investigated in detail. We uncover that the viscoelastic terms have a considerable influence on the system dynamics, which can simultaneously affect the structural damping and stiffness of the airfoil system. More interestingly, stochastic jumps between high-amplitude and low-amplitude oscillations can be induced due to random fluctuations, which are further illustrated through time history and steady-state probability density function. The jumps are considered as a transition from one probable state to another or vice versa. These results indicate that the external random fluctuations have a remarkable influence on dynamics of the TDOF airfoil model with viscoelastic material property.