Dynamical systems analysis of electrostatic and aerodynamic forced vibrations of a thin flexible electrode

Transverse vibrations of an electrostatically actuated thin flexible cantilever perturbed by low-speed air flow is studied using both experiments and numerical modeling. In the experiments the dynamic characteristics of the cantilever are studied by supplying a DC voltage with an AC component for electrostatic forcing and a constant uniform air flow around the cantilever system for aerodynamic forcing. The maximum voltage applied varies from 1 - 9 kV and air flow speeds range from 0.224 - 3.58 m/s (0.5 - 8 mile/hr). The Reynolds numbers for these speeds lie in the range of 1000 - 20000. A range of control parameters leading to stable vibrations are established using the Strouhal number as the operating parameter whose inverse values change from 100 - 2500. The Numerical results are validated with experimental results. Assuming the amplitude of vibrations are small, then a non-linear dynamic Euler-Bernoulli beam equation with viscous damping and gravitational effects is used to model the vibrations of the dynamical system. Aerodynamic forcing is modeled as a temporally sinusoidal and uniform force acting perpendicular to the beam length. The forcing amplitude is found to be proportional to square of air flow velocity by obtaining relationship between the experimental amplitude of vibrations and air flow velocity. Numerical results strongly agree with those of experiments predicting accurate vibration amplitudes, displacement frequency and quasi-periodic displacements of the cantilever tip.