On the asymptotic approximation of the solution of an equation for a non-constant axially moving string

The transverse vibrations of axially moving strings with time dependent velocity and its instability are examined in this paper. The velocity function is assumed to be a harmonically varying function about a constant low mean speed. Approximate solutions are sought using the two timescales perturbation method in conjunction with the method of characteristic coordinates. Moreover, it will turn out that infinitely many resonances in the system can arise, when the velocity fluctuation frequency is equal (or close) to an odd multiple of the lowest natural frequency of the constant velocity system. The dynamic displacement-response and the energy of the belt system obtained using the method of characteristic coordinates are compared numerically. It is shown that the energy of the belt system can become large in a resonance case. In addition, the amplitude of vibration is also found to be growing on long timescales, that is, on a timescale of order ε−1.