Analytical Model for Transverse Vibrations of Spinning Membranes

Abstract

Spinning membrane structures are of benefit to space applications because they can be deployed without such mass resources as booms. On the other hand, the lack of supporting structures can lead to undesirable deformations that are dominated by a complex partial differential equation. In this paper, an analytical description for transverse vibrations of spinning membranes is presented. It is shown that the eigenfunctions can be written using the Zernike polynomials; in the presented framework, however, a more practical representation that focuses on the physical characteristics of the vibrations is developed. Furthermore, response to external forces, or controls, is formulated by introducing an extended form of an impulse response function (IRF) that includes spatial information in addition to time information. The deformation under arbitrary forces can be calculated through a convolution integral with the extended IRF. Moreover, the Fourier transform of the IRF gives the frequency response function, which can be useful in designing a vibration control system. To verify the validity of the developed model, the analytical solution is compared to the numerical solution in the finite element analysis. Results demonstrate that both solutions agree in the modal analysis, frequency response analysis, and transient response analysis.