In this paper we develop and analyze a mathematical model for combined axial and transverse motions of two Euler-Bernoulli beams coupled through a joint composed of two rigid bodies. The motivation for this problem comes from the need to accurately model damping and joints for the next generation of inflatable/rigidizable space structures. We assume Kelvin-Voigt damping in the two beams whose motions are coupled through a joint which includes an internal moment. The resulting equations of motion consist of four, second-order in time, partial differential equations, four second-order ordinary differential equations, and certain compatibility boundary conditions. The system is re-cast as an abstract second-order differential equation in an appropriate Hilbert space, consisting of function spaces describing the distributed beam deflections, and a finite-dimensional space that projects important features at the joint boundary. Semigroup theory is used to prove the system is well posed, and that with positive damping parameters the resulting semigroup is analytic and exponentially stable. The spectrum of the infinitesimal generator is characterized.
|Original language||English (US)|
|Number of pages||15|
|Journal||Journal of Mathematical Analysis and Applications|
|State||Published - Mar 1 2008|
Bibliographical noteFunding Information:
✩ Research supported in part by DARPA/STO, NASA Langley Research Center, and the National Institute for Aerospace under Grant NIA 2535, and in part by AFOSR Grants F49620-03-1-0243 and FA9550-07-1-0273. * Corresponding author. E-mail addresses: email@example.com (J.A. Burns), firstname.lastname@example.org (E.M. Cliff), email@example.com (Z. Liu), firstname.lastname@example.org (R.D. Spies).
- Abstract differential equations
- Mechanics of deformable solids
- Semigroups and linear evolution equations