Suppression of a nonplanar motion in an externally excited string by the addition of viscosity based on velocity feedback control

This study concerns the out-of-plane motion of a vertically disposed string with one end fixed and the other subjected to a harmonic excitation. Excited in its primary resonance, the string can vibrate in the direction orthogonal to the excitation direction through nonlinear coupling effects and has out-of-plane motion. To suppress this out-of-plane motion, we propose a boundary control method that modifies the boundary condition of the fixed upper end, i.e., we move the fixed upper end in the direction orthogonal to the external excitation. The unstable range for trivial steady state motion in this direction, which exists in the neighborhood of the primary resonance condition, changes to a stable range under suitable feedback gain. The out-of-plane motion is then suppressed for any excitation frequency. Experiments conducted using a simple apparatus confirmed the validity of the proposed method adopting boundary control.