Abstract
This study investigates a linear homogeneous initial-boundary value problem for a traveling string under linear viscous damping. The string is assumed to be traveling with constant speed, while it is fixed at both ends. Physically, this problem represents the vertical (lateral) vibrations of damped axially moving materials. The axial belt speed is taken to be positive, constant and small in comparison with a wave speed, and the damping is also considered relatively small. A two timescale perturbation method together with the characteristic coordinate’s method will be employed to establish the approximate analytic solutions. The damped amplitude-response of the system will be computed under specific initial conditions. The obtained results are compared with the finite difference numerical technique for justification. It turned out that the introduced damping has a significant effect on the amplitude-response. Additionally, it is proven that the mode-truncation is applicable for the damped axially traveling string system on a timescale of order ε -1
Highlights
This study investigates a linear homogeneous initial-boundary value problem for a traveling string under linear viscous damping
Gaiko and van Horssen [5] used the method of Laplace transform and two timescale perturbation method to investigate the transversal vibrations of traveling string with the boundary damping
T exhibits the amplitude-response of the system with the initial displacement: u (x) = sin( x) and the initial velocity: u (x) = 0.5sin( x)
Summary
This study investigates a linear homogeneous initial-boundary value problem for a traveling string under linear viscous damping. The string is assumed to be traveling with constant speed, while it is fixed at both ends This problem represents the vertical (lateral) vibrations of damped axially moving materials. Oving structures have vast applications in Mvarious engineering disciplines such as civil, mechanical and aerospace engineering These systems represent many engineering devices, for instances; conveyor belts, elevator cables, chair lifts, pipes transporting liquids and gases, power transformation lines, and serpentine belts are few of them.Axially moving systems are mainly divided into two categories, that is, traveling string and EulerBernoulli beam. Gaiko and van Horssen [5] used the method of Laplace transform and two timescale perturbation method to investigate the transversal vibrations of traveling string with the boundary damping. Only few modes were taken in [13] to examine the behavior of the amplitude-response of the damped traveling string system
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