ABSTRACT

Axially moving continua are classified as axially moving string system and axially moving beam system. Many engineering devices such as conveyor belt, elevator cables, Power transmission lines and band-saw blades are few examples of axially moving continua. It is widely observed that the vibrations, specifically transverse vibrations have always limited their applications. In this thesis, the (in)stability of an axially moving string subject to the time-dependent axial velocity under the effect of viscous damping is examined. Mathematically, the transverse vibrations of an axially moving string under the effect of damping are described by second order homogenous linear PDF with variable coefficients. The asymptotic approximations for the solutions of the equations of motion are obtained via the application of a two timescales perturbation method together with Fourier-expansion method and Laplace transforms method. In this work, two different cases of the axial velocity are considered, that is the Harmonically low mean time-varying velocity and the Harmonically high mean time-varying velocity. In low mean axial velocity case, it turned out that the system is stable for two cases of damping when 5 — 2m and 5 > 2m and mode-truncation for these cases in not problematic, however, for < 2m (where m is positive odd integer) the energy of system and mode-truncation have different behaviour, so mode-truncation is not applicable for this case. In addition to this, damped string-like equation is also investigated for near resonance case and it is observed that for ICl > 2m (where a is detuning parameter) the energy of system is bounded, however for< 2m with ö > the energy is tending to zero for large time, however for ö < the energy grows exponentially and for 101 = 2m with 5 > 0 the energy of system clearly damped out but for = 0 the energy grows polynomially. In high mean axial speed case, it is found that the truncation is only applicable when = and (5 > however for case < 02 + Y2 truncation is not allowed due to different behaviour of the energy and mode- truncation. Furthermore, the (in) stability of an axially moving system with constant axial speed under the effect of damping at the boundary is examined. The closed form solutions are obtained via the application of a two timescales perturbation method together with the method of characteristics coordinates. The effects of boundary damping are clearly seen in the transversal displacement of the system. It is shown that the motion of the traveling string in terms of transversal displacement is damped out by increasing the damping in the system.
Keywords: Viscous and boundary damping, Asymptotic approximations, Internal