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Well‐posedness, stability and numerical results for the thermoelastic behavior of a coupled joint‐beam PDE‐ODE system modeling the transverse motions of the antennas of a space structure
Author(s) -
Cliff Eugene M.,
Herdman Terry L.,
Liu Zhuangyi,
Spies Ruben D.
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700938
Subject(s) - thermoelastic damping , ordinary differential equation , partial differential equation , mathematical analysis , boundary value problem , beam (structure) , mathematics , transverse plane , differential equation , physics , thermal , structural engineering , engineering , meteorology , optics
A mathematical model for both axial and transverse motions of two beams with cylindrical cross‐sections coupled through a joint is presented and analyzed. The motivation for this problem comes from the need to accurately model damping and joint dynamics for the next generation of inflatable/rigidizable space structures. Thermo‐elastic damping is included in the two beams and the motions are coupled through a joint which includes an internal moment. Thermal response in each beam is modeled by two temperature fields. The first field describes the circumferentially averaged temperature along the beam, and is linked to the axial deformation of the beam. The second describes the circumferential variation and is coupled to transverse bending. The resulting equations of motion consist of four, second‐order in time, partial differential equations, four, first‐order in time, partial differential equations, four second order ordinary differential equations, and certain compatibility boundary conditions. The system is written as an abstract differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections and temperature fields, and a finite‐dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove that the system is well‐posed, and that with positive damping parameters the resulting semigroup is exponentially stable. Steady states are characterized and several numerical approximation results are presented. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)