In this paper an initial-boundary value problem for a linear, nonhomogeneous axially moving string equation will be considered. The velocity of the string is assumed to be constant, and the nonhomogeneous terms in the string equation are due to external force acting on the string. The Laplace transform method will be used to construct the solution of the problem. It will turn out that the method has considerable, computational advantages compared to the usually applied method of modal analysis based on eigenfunction expansions.

In this paper an initial-boundary value problem for a linear equation describing an axially moving string will be considered for which the bending stiffness will be neglected. The velocity of the string is assumed to be time-varying and to be of the same order of magnitude as the wave speed. A two time-scales perturbation method and the Laplace transform method will be used to construct formal asymptotic approximations of the solutions. It will be shown that the linear axially moving string model already has complicated dynamical behavior and that the truncation method can not be applied to this problem in order to obtain approximations which are valid on long time-scales.Keywords Axially moving string . Asymptotics . Internal resonances . Oscillations . Two-timescales perturbation method Formulation of the problemIn this paper the dynamic behavior of an axially moving string without bending stiffness will be studied (see Fig. 1 The following linear equation of motion for a moving string will be consideredwhere, u(x, t): the displacement of the string in vertical direction, V (t): the time-varying string speed, c: the wave speed, x: the coordinate in horizontal direction, t: the time, and, l: the distance between the pulleys, and, in which T 0 and ρ are assumed to be the constant tension and the constant mass density of the string, respectively. In this paper the case V 0 < c is considered and it is assumed that V (t) = V 0 + εα sin(ωt), where V 0 , ω and α are some positive constants, and where ε is a small parameter with 0 < ε 1. The term εα sin(ωt) can be seen as a small perturbation of the main belt speed V 0 due to different kinds of imperfections of the belt system. At the pulleys it is assumed that there is no displacement of the string in vertical direction. Equation (1) can also be found in [1], but now it is assumed that V 0 is not necessarily small compared to the wave speed c. Consequently (1) becomes:

In this paper an initial-boundary value problem for a linear, nonhomogeneous axially moving string equation will be considered. The velocity of the string is assumed to be constant, and the nonhomogeneous terms in the string equation are due to external forces acting on the string. The Laplace transform method will be used to construct the solution of the problem. It will turn out that the method has considerable, computational advantages compared to the usually applied method of modal analysis based on eigenfunction expansions.

In this paper an initial-boundary value problem for a linear equation describing an axially moving stretched beam will be considered. The velocity of the beam is assumed to be time-varying. since the order of magnitude of the bending stiffness terms depends on the vibrations modes and the frequencies involved a that combination of two simplified models (a string equation and a beam with string effect equation) will be used to describe the transversal vibrations of the system accurately. Based on the calculations of the natural frequencies the regions of applicability of these models will be determined. A two time-scales perturbation method will be used to construct formal asymptotic approximations of the solutions. It will be shown that the linear axially moving “string to beam” model already has complicated dynamical behavior.

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