Inflation mechanics of a membrane reflector supported by an inflated toroidal rim

A stack of two identical flat circular membranes, bonded along the periphery can be inflated to form a reflector. To maintain the desired shape of the reflector surface, a supporting outer rim is required. An inflatable membrane reflector supported by an outer inflated toroidal rim structure is considered. Both the reflector and the toroidal rim are considered to be geometrically flat in the uninflated state. Initially, the circular membranes are pre-stretched and joined with the inner equator of the inflated torus, causing an inward radial force on the toroidal rim. The inflated shapes of the circular membranes (reflectors) under uniform pressure are obtained by an iterative solution scheme. The inflation problem is solved to determine the inflated shapes and the possible occurrence of wrinkling instability of the reflector or pull-in instability of the toroidal rim. The shape of the reflector is found to be close to a paraboloid whose focal length depends on the level of inflation and pre-stretch. Lower inflation pressure of the inflatable reflector is found to result in a better parabolic approximation of the reflecting surface.