Nonlinear Deployment Dynamics and Wrinkling of a Membrane Attached to Two Axially Moving Beams

The out-of-plane and in-plane deployment dynamics of a flexible space structure, namely, a solar sail quadrant consisting of a membrane attached to two support booms, are considered. The equations of motion of the system are obtained using a time-varying generalization of the extended Hamilton’s principle. They are then discretized via quasi-modal expansion of the deflections as truncated series involving both time- and space-dependent basis functions. Because all directions are accounted for and plate strains are used to capture the potentially significant effect of even a small stiffness on the dynamics, nonlinear terms appear in the discretized equations. To increase computational efficiency, coordinate transformations and linear algebraic manipulations are performed to make all spatial integrals time invariant. In addition, attempts are made to predict wrinkling using the Miller–Hedgepeth model: a coarse mesh is defined, the instantaneous state of each region is determined using a wrinkling criterion and averaged principal stresses, and constitutive relation of each region is adjusted based on its wrinkling state. Numerical simulations provide basic validation, sample deployment results, and a comparison against the results of an earlier linear model with only out-of-plane deflections. The stress predictions are also partially validated using previous results based on constant-size loaded membrane experiments.