ENOC 2014, July 6-11, 2014, Vienna, Austria On Transverse Vibrations of a Damped Traveling String with Boundary Damping Nick V. Gaiko * , Wim T. van Horssen * * Department of Mathematical Physics, Delft University of Technology, Delft, The Netherlands Summary . In this paper, a damped axially moving string with two different types of boundary conditions, such as homogeneous Dirichlet (or fixed) boundary conditions and non-classical (spring-mass-dashpot type of) boundary conditions, is studied. The solution of a string-like (or wave) problem with fixed boundary conditions shows that the axial velocity and the material damping reduce the amplitude of the oscillations. For the non-classical boundary conditions, the damping rates and the frequencies of all oscillation modes of the initial-boundary value problem are determined by the approximate solution. Introduction Axially moving systems are found in different engineering mechanisms, such as power transmission belts [1], conveyor belts [2, 3, 4, 5], elevator cables [6, 7], and many others. Oscillations affecting these systems are of great interest for researchers due to theoretical and industrial importance. Transversal and longitudinal vibrations arising in the moving material are caused by the eccentricity of a pulley, the irregular speed of the driving motor, non-uniform material prop- erties, environmental disturbances, or several of these combined. In this work, we focus on transversal vibrations since longitudinal ones are negligible for a considered axially moving material. Here, an axially moving uniform string of linear mass density ρ (unit kg / m) is studied. The string travels with constant speed v (unit m / s) between two pulleys which are separated by distance l (unit m). Transverse vibrations are denoted by the vertical displacement u (unit m); it depends on the position x (unit m) along the string and the time t (unit s). Additionally, an external force F (unit N / m) and a viscous damping force act on the system. Naturally, the tension T (unit N) arises in the string. Furthermore, for the sake of simplicity of the problem it is necessary to assume that bending stiffness and the effects due to gravity are neglected. Apart from these, the transverse displacements are sufficiently small, such that the nonlinear terms in the governing equation of motion can be neglected as well. Mathematical Models The equation of the transverse motion for an axially moving, damped string is given by ρ(u tt +2vu xt + v 2 u xx ) − Tu xx + η(vu x + u t )= F, 0 <x<l,t> 0, where η (unit kg / s) is the viscous damping factor of the material. The initial displacement and the initial transverse velocity are given by u(x, 0) = φ(x), and u t (x, 0) = ψ(x), 0 <x<l. Additionally, two types of boundary conditions are considered. In the first case, the string is fixed at its ends (see Figure 1). Consequently, homogeneous Dirichlet boundary conditions are given by u(0,t)=0, and u(l,t)=0,t ≥ 0. u =0 v v x u 0 l Figure 1. A schematic model of an axially moving string with clamped ends. In the second case, we apply boundary damping at one end of the string to suppress its transverse vibrations. More precisely, we consider the string which is clamped at x =0, and a linear spring and a dashpot are attached at x = l (see Figure 2). In addition, the right pulley has a mass m (unit kg). As a result, the boundary conditions are derived as follows u(0,t)=0, and mu tt (l,t)+ ku(l,t)+ cu t (l,t)+ Tu x (l,t) − ρvu t (l,t) − ρv 2 u x (l,t)=0,t ≥ 0, where k (unit kg / s 2 ) is an elasticity modulus of the linear spring and c (unit kg / s) is the damping coefficient of the dashpot. Note that in the second initial-boundary value problem we consider c = O(ǫ) and v = O(ǫ), where ǫ is a small positive parameter (i.e., 0 <ǫ ≪ 1), in order to have not too fast decaying oscillations and a reflection of wave from the boundaries in the string.