QUARTERLY OF APPLIED MATHEMATICS VOLUME LXIII, NUMBER 4 DECEMBER 2005, PAGES 681–690 S 0033-569X(05)00979-4 Article electronically published on September 27, 2005 ON AN IMPROVED ELASTIC DISSIPATION MODEL FOR A CANTILEVERED BEAM By M. A. ZARUBINSKAYA (Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands ) and W. T. VAN HORSSEN (Delft Institute of Applied Mathematics, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands ) Abstract. In this paper we will study an improved elastic dissipation model for a cantilevered beam, where the damping is assumed to be proportional to the bending rate of the beam. For an earlier formulated dissipation model for the cantilevered beam it has been recently shown that damping will not always be generated. However, for the improved dissipation model it will be shown in this paper that damping will always be generated. 1. Introduction. For many years different approaches were used to describe energy dissipation in oscillating elastic bodies such as beams (see [1]-[4]). However, many ap- proaches (such as molecular theories) are too complicated to use in practice. As a result, different phenomenological models are used in mechanics. At the end of the nineteenth century, Kelvin and Voigt noted that damping rates tend to increase with frequency. At the end of the last century, Chen and Russell proposed the following dissipation model (see [4]): ¨ x + B ˙ x + Ax =0, (1.1) where A is an elastic operator, and where B is related in various ways to the positive square root, A 1/2 , of A . For beam equations this approach was generalized and developed further by Russell in [1], [2]. Russell studied a new phenomenological dissipation model for a beam, where the damping is assumed to be proportional to the bending rate of the beam u tt − δu txx + u xxxx =0, Received February 2, 2005 and, in revised form, on March 16, 2005. 2000 Mathematics Subject Classification. Primary 35B05, 35Q72, 74H45. E-mail address : maria@dv.twi.tudelft.nl E-mail address : W.T.vanHorssen@ewi.tudelft.nl c 2005 Brown University Reverts to public domain 28 years from publication 681 License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf