Modeling Viscoelastic Planetary Aerobot Tethers Using Fractional Derivatives and Anelastic Displacement Fields

Aerobots are expected to play a key role in future missions to explore the atmospheres of Venus and other planetary bodies. Proposed missions to Venus involve lowering instruments on tethers toward the inhospitable surface from balloons at altitudes on the order of 50 km. Accurately capturing the viscoelastic behavior of the tethers in system design models is essential for managing response and loads during dynamic events (deployment, gusts, flight control) and following long-term aging in extreme environments. The behavior of materials likely to be used in tethers, such as Kevlar, is well-described by a fractional derivative viscoelastic model. One of the shortcomings of such models is the challenge of numerical time integration of the system equations of motion. In this work, fractional derivative behavior is approximated using a multi-anelastic displacement fields (multi-ADF) model. The multi-ADF model augments the instantaneous displacement field with multiple discrete internal displacement fields that individually evolve following first-order relaxation processes. Notably, while the fractional derivative model is nonlocal in time, requiring the entire history for evaluation, the first-order version is local, essentially capturing the history in the present values of internal state variables. A multi-ADF finite element model of a tether with an instrument gondola is developed, and a Newmark time integration scheme is used to obtain the dynamic response of the system during a drop test. The multi-ADF model is shown to be very accurate for creep, dynamics, and gondola-drop simulations, and it runs in a small fraction of the time required by a direct fractional-derivative model.