Vibration Of A Long, Tip Pulled Deflected Beam

Long, flexible deployable structures are used in space exploration vehicles in low-gravity environment applications (Puig et al. Tibert and Pellegrino). The use of the same types of probes in the exploration of planets with a large gravitational field would be of interest to reach regions of scientific interest of difficult access, such as cliff sides and terrain slopes. Vibration control in this case is a sensitive issue that can be addressed by the use of a reliable model of dynamics of the long exploration probe. The present work addresses the problem of building a simple dynamic model of a long beam deformed by a tip pulling cable. The simple model is sought for future use in a vibrations control strategy. The exact static deformation shape of a long beam is obtained through the solution of the nonlinear beam governing differential equations. The exact deformed configuration for beams with a uniform cross section and transversal end load is found through the recursive solution of elliptic integrals (Frish–Fay and Timoshenko and Gere). The deformed configuration of beams with an inclined pulling force can be found via numerical approximations, using adequate methods such as the Runge–Kutta (Ohtsuki and Shvartsman), the shooting method (Holland et al.), the finite element method (Howell), and the quasi-linearization finite difference method (Al-Sadder and Al-Rawi). A strategy to calculate the static deformation of a long beam pulled by a tip cable is summarized in the work of Yau. The dynamic behavior of the long beam can be obtained through the analysis of small vibration about the static equilibrium of a finite elements model of the structure (Ferris and Afonta, Sallstrom et al. and Santillan et al.). Vibration analysis and experimental validation of a finely meshed finite element model of a tip pulled beam was carried out by Holland et al. The present work proposes the combination of a nonlinear static deformation model of the same structure and a courser finite element model of the deformed beam in order to assess its dynamic characteristics. Results of the different modeling strategies are compared and discussed.