Friction Induced Transverse Vibrations of an Axially Accelerating String

The purpose of this study is to investigate the dynamic response of axially translating continua undergoing both the effect of friction and axial acceleration. The axially moving continuum is initially modeled as a string, neglecting its flexural stiffness; the response, with particular interest given to transverse vibrations and dynamic stability, is studied through numerical methods. A finite element method is employed to discretize the space domain and an implicit α−method is employed to integrate the resulting matrix equation in the time domain. Results are given through time history diagrams and stability considerations. INTRODUCTION Translating continua are encountered in numerous machinery such as power transmission belts, band saw blades, and processing systems such as magnetic tapes, thread lines, paper and photographic film. Many researchers have addressed the axially moving continuum problems, but most of them have considered conservative systems without friction [1,2]. However, in many applications stationary frictional guides, which are potential sources of undesired vibrations and dynamic instability, exist. The response of a frictional, non-conservative system has been studied by Chen [3], and Cheng and Perkins [4], among others, who considered a string moving with constant axial velocity under the effect of a stationary dry friction load. Accelerations and decelerations, as well as the undesirable fluctuations of the translational speed could have significant effects on the vibrational behavior of the translating systems. Miranker derived the governing equations of an accelerating string using energy methods [5]. Solutions of this and similar systems have been studied by Pakdemirli et al. [6], among others. To the best of our knowledge the combined effect of frictioned guides and acceleration has not been addressed. In systems requiring frequent start-stop operations, such as high capacity data tape-recorders, simultaneous consideration of friction and acceleration is necessary. Here a generic model and a numerical method are presented to address such a system. The primary goal of this paper is to show the validity of numerical model by comparing the numerical results with that of reference [6]. EQUATION OF MOTION A tensioned flexible string is translating with time dependent velocity V(t) between two fixed supports at x = 0 and x = L. At a distance D from the left support there is a frictional guide resting on two springs. The surfaces of the guides are assumed to remain in contact with the string because of a static force N and the stiffnesses K1 and K2 of the guide supports. F1 and F2 are the top and the bottom friction forces due to the guide. The longitudinal and the transverse displacements are u and w, respectively. In this paper the equation of motion is derived using the extended Hamilton’s principle, assuming a time dependent F2 F1