Variable mass modeling and velocity stability analysis of axially moving tether

Airborne wind energy (AWE) technology uses tethers to connect high-altitude parafoils or another type of aircraft to harness wind energy for offshore wind power generation or ship propulsion. Therefore, an AWE system's stability and productivity are directly influenced by the multivariate axial tensions and velocities of tethers. In view of this, this study conducts the varying-mass tether modeling and the Lagrangian stability analysis of an axially moving tether. First, the Hamilton principle is modified by adding a semi-open boundary of the control surface, which can accurately describe a varying-mass tether's behavior when the tether is released or recovered. Considering the control surface behavior, the expressions of Lagrange stability analysis and virtual work are defined to achieve accurate dynamic tether modeling. Second, for an axially moving tether system with non-conservative total energy, the vibration response of conservative energy at the axial critical velocity value is obtained by determining the conservative Euler and Lagrangian functionals. Further, the critical value and stability of a tether system's axial velocity are analyzed using the quasi-static eigenvalues, considering several possible scenarios with different frictional conditions. Finally, numerical simulations are conducted to evaluate the eigenvalue analysis results under different frictional conditions, and the values of the friction coefficients and tether axial velocity are provided for the stable condition of the tether vibration displacement.