Analytical Investigation of Dynamics of Inflatable Parabolic Membrane Reflector

An analytical method for computing the dynamic characteristics of a lenticular inflatable parabolic reflector supported by an inflatable torus is presented. The system is modeled by a combined structure consisting of two pressurized paraboloidal membranes with an elastic circular ring support. Equations of motion for the transverse free vibration of the pressurized paraboloidal membrane are derived first by using the nonlinear theory of shallow shells of revolution. Then, the analytic solutions for those equations of motion are formulated. Afterward, according to the force balance and the displacement continuity between the membranes and the circular ring, the displacements of the ring are calculated as boundary conditions of the paraboloidal membranes. As a result, the frequency equation and analytic mode shape functions of the inflatable reflector are obtained, and the influences of support stiffness on the dynamic characteristics of the reflector are analyzed. The validity of these analytical modal solutions is verified by finite-element simulations.