Nonlinear dynamic analysis of a viscoelastic rotating cantilever beam under stochastic perturbation

The dynamic characteristics of rotating beams significantly affect the performance of equipment with rotating components, such as flexible wires of spacecrafts, steam turbines and helicopter rotors. The coupling of nonlinear deformation and viscoelastic materials will lead to integral-containing nonlinear viscoelastic terms, which have been overlooked in previous studies. Departing from conventional studies, this paper investigated the nonlinear dynamics of a rotating viscoelastic cantilever beam under stochastic excitation. A new nonlinear dynamic equation is established by integrating nonlinear deformation, rotation effect, and viscoelastic constitution in the modeling process, which is transformed into a set of nonlinear stochastic differential equations with integral viscoelastic terms using the assumed mode method. Numerical simulations and stochastic linearization method are used to analyze the multimodal response of the rotating cantilever beam, in which the results showed that the primary mode vibration dominates the dynamic response. Thus, a theoretical method based on stochastic averaging method is proposed to derive the approximate responses of the primary mode. The nonlinear viscoelastic terms resulting from the coupling of nonlinear deformation and viscoelasticity are converted into a combination of the amplitude-dependent modified damping and conservative forces. Analytical responses are obtained by solving the Fokker-Planck-Kolmogorov (FPK) equation. Finally, the impacts of excitation intensity, damping ratio, rotational angular velocity and viscoelastic parameters on the system response are comprehensively analyzed. The high consistency between the theoretical predictions and numerical simulations validates the effectiveness of the proposed analytical method, which facilitates a deeper understanding the dynamic behavior of viscoelastic rotating beams.