ANALYTICITY OF SEMIGROUP ASSOCIATED WITH A LAMINATED COMPOSITE BEAM Scott Hansen* Department of Mathematics Iowa State University, Ames, lA 50011, USA shansenCimath.iastate.edu Zhuangyi Liu Department of Mathematics & Statistics University of Minnesota, Duluth, MN 55812, USA. zliu@d.umn.edu Abstract: We consider a system of coupled partial differential equations that describe the vibrations of laminated beam in which the layers are bonded to- gether by a medium that dissipates energy at a rate proportional to the shear. We show that for the simplest model, in which only transverse inertial energy is accounted for, the associated semigroup is analytic. 1 INTRODUCTION In this paper we study the damping characteristics of a laminated beam model introduced (for plates) in Hansen [3]. The model is derived under the assump- tion that the laminated beam consists of 2n layers of Euler-Bernoulli beams bonded to one another by 2n - 1 "adhesive layers" which resist shear, but otherwise have negligible physical characteristics. We suppose that damping is also included in the adhesive layers so that a force opposing shear, proportional to the rate of shear exists within the adhesive layers. No damping is included elsewhere in the model. It was noticed in Hansen and Spies [5), that in the case of one adhesive layer, with constant coefficients and simply supported boundary conditions, the spectrum associated with the generator of the semigroup exhibits frequency- proportional damping characteristics {see Russell [16]). Many systems with this "Research partially supported the National Science Foundation under grant DMS-9504822 for arbitrary values of the coupling parameter a. © The original version of this chapter was revised: The copyright line was incorrect. This has been corrected. The Erratum to this chapter is available at DOI: 10.1007/978-0-387-35359-3_40 IFIP International Federation for Information Processing 1999 S. Chen et al. (eds.), Control of Distributed Parameter and Stochastic Systems