Deformation and Flexibility Equations for Curved, End-Loaded, Planar Elastica

The International Space Station relies on the active rack isolation system as the central component of an integrated, station-wide strategy to isolate microgravity space-science experiments. The isolation system uses electromechanical actuators to isolate an international standard payload rack from disturbances due to the motion of the station. Disturbances to microgravity experiments on isolated racks are transmitted primarily via the isolation-system power and vacuum umbilicals. Experimental tests indicate that these umbilicals resonate at frequencies outside the controller's bandwidth, at levels of potential concern for certain microgravity experiments. Reduction in the umbilical resonant frequencies could help to address this issue. Toward that end, this work documents the development and verification of static equations for the in-plane deflections and flexibilities of an idealized umbilical (thin, flexible, elastic, inextensible, prismatic cantilever beam) under endpoint, in-plane loading (inclined force and moment). Gravity is neglected due to the on-orbit application. The analysis assumes that the umbilical experiences large static deflections from a reference curve describing its relaxed configuration, for which the slope follows a quadratic function of arc length. The treatment is applicable to the power and vacuum umbilicals, under the indicated assumptions.