Results on Transversal and Axial Motions of a System of Two Beams Coupled to a Joint Through Two Legs

J. A. BURNS, E. M. CLIFF, Z. Y. LIU AND R D. SPIES

Budapest University of Technology and Economics, Budapest, Hungría.

Conferencia; International Conference on Nonlinear Problems in Aviation and Aerospace, ICNPAA,; 2006

Budapest University of Technology and Economics

Truss structures are common in Earth-borne constructions, such as bridges and cranes, as well as in space applications. In recent years there has been renewed interest in in°atable-rigidizable space structures because of the e±ciency they o®er in packaging during boost-to-orbit (C. H. M. Jenkins (ed.), Gossamer Spacecraft: Membrane and In°atable Structures Technology for Space Applications", Progress in Aeronautics and Astronautics, 191, AIAA Pubs., 2001). However, much research is still needed to better understand dynamic response characteristics, including inherent damping, of truss structures fabricated with these advanced material systems. We report some results of an ongoing research related to a model consisting of an assembly of two beams with Kelvin-Voight damping, coupled to a simple joint through two legs. An explicit form for the energy of the system is found and its dissipativeness is proved. The system can be written as a second order di®erential equation in an appropriate Hilbert space in which well-posedness, exponential stability as well as other regularity properties of the solutions can be obtained. While the beams are clamped at one end, they satisfy a much di®erent type of boundary conditions at the other end. In fact, these are dynamic and non-homogeneous and take into account all compatibility conditions at the beam-leg interfaces. This conditions result in an ODE coupling boundary terms of both beams. Two di®erent approaches are followed to develop ¯nite dimensional approximations for the solutions of the system. The ¯rst approach uses a projection method to enforce the dynamic boundary conditions while the second approach consist in enforcing these boundary conditions into the basis functions. Numerical results are presented and comparisons are made.Gossamer Spacecraft: Membrane and In°atable Structures Technology for Space Applications", Progress in Aeronautics and Astronautics, 191, AIAA Pubs., 2001). However, much research is still needed to better understand dynamic response characteristics, including inherent damping, of truss structures fabricated with these advanced material systems. We report some results of an ongoing research related to a model consisting of an assembly of two beams with Kelvin-Voight damping, coupled to a simple joint through two legs. An explicit form for the energy of the system is found and its dissipativeness is proved. The system can be written as a second order di®erential equation in an appropriate Hilbert space in which well-posedness, exponential stability as well as other regularity properties of the solutions can be obtained. While the beams are clamped at one end, they satisfy a much di®erent type of boundary conditions at the other end. In fact, these are dynamic and non-homogeneous and take into account all compatibility conditions at the beam-leg interfaces. This conditions result in an ODE coupling boundary terms of both beams. Two di®erent approaches are followed to develop ¯nite dimensional approximations for the solutions of the system. The ¯rst approach uses a projection method to enforce the dynamic boundary conditions while the second approach consist in enforcing these boundary conditions into the basis functions. Numerical results are presented and comparisons are made.