1 Copyright © 2005 by ASME Proceedings of 2005 ASME International Mechanical Engineering Congress and Exposition November 11-15, 2005 Orlando, Florida DRAFT IMECE 2005 - 80377 TRANSVERSE VIBRATION OF TWO AXIALLY MOVING BEAMS CONNECTED BY AN ELASTIC FOUNDATION Mohamed Gaith and Sinan Müftü Northeastern University Department of Mechanical and Industrial Engineering Boston, MA 02115 ABSTRACT Transverse vibration of two axially moving beams connected by a Winkler elastic foundation is analyzed analytically. The system is a model of paper and paper-cloth (wire-screen) used in paper making. The two beams are tensioned, translating axially with a common constant velocity, simply supported at their ends, and of different materials and geometry. Due to the effect of translation, the dynamics of the system displays gyroscopic motion. The Euler-Bernoulli beam theory is used to model the deflections, and the governing equations are expressed in the canonical state form. The natural frequencies and associated mode shapes are obtained. It is found that the natural frequencies of the system are composed of two infinite sets describing in-phase and out-of-phase vibrations. In case the beams are identical, these modes become synchronous and asynchronous, respectively. Divergence instability occurs at the critical velocity; and, the frequency-velocity relationship is similar to that of a single traveling beam. The effects of the mass, flexural rigidity, and axial tension ratios of the two beams, as well as the effects of the elastic foundation stiffness are investigated. 1. INTRODUCTION Axially moving materials have many engineering applications like magnetic tape systems, fiber winders, power transmission belts, textile and paper web handling machinery [1]. Axially moving materials typically are modeled as a string or as an Euler-Bernoulli beam [2, 3]. Web is a generic name used for thin, flexible continuous materials such as magnetic tapes and papers. Paper making is one of the oldest of the industries involved with web handling, with more than a century of history. In the papermaking process, paper fibers are mixed with water, and this pulp slurry is sprayed onto a large, flat, fast-moving wire-screen, sometimes called the paper-cloth. As the wire-screen translates along the paper machine, the water drains out, and the fibers bond together. The paper web is pressed between rolls in order to squeeze out more water and it is further dried by heated rollers. The stiffness of paper increases as it is dried along the path of the machine. The paper is eventually rolled and removed from the machine. Vibration problems can arise during transport of the paper-wire system, where excessive vibration could cause the paper to separate from the wire-screen prematurely. In this work the translating wire/paper system is modeled as two translating beams, connected by an elastic foundation. The elastic foundation is used, without much justification, to represent the bonding between the wire and the paper. To the best of the authors’ knowledge the vibration of such a system has not been considered in the literature. On the other hand, vibration of translating single beams/strings and vibration of non-translating, double beam/string systems have been studied extensively. The vibration of a translating string supported by an elastic foundation was studied by Perkins [5], Wickert [4] and Parker [6]. Perkins studied the axially moving, string supported by a distributed elastic foundation and obtained the natural frequencies and corresponding mode shapes and examined the subcritical frequencies [5]. Parker found that the elastic foundation does not alter the lowest critical speed, but the supercritical stability is changed by the elastic foundation [6]. Oz et al. investigated the transition from string to beam for an axially moving material and obtained the natural frequencies expressions [7]. Tabarrok et al. derived the governing equation of motion of an axially moving beam including the effect of flexural rigidity of the beam [8]. Barakat investigated the transverse vibrations of moving thin rod, and obtained the natural frequencies and corresponding mode shapes for fixed and simply supports at ends [9]. Wickert and Mote presented a closed form solution for axially moving continua, subjected to