Transverse Vibrations of an Axially Travelling String

In this paper, the transverse vibration of axially travelling string is analysed. The axial velocity of the string is periodically varying about an average value. Applying direct perturbation method (MMS), an analytical solution is found. An analysis of principal parametric resonances is carried out when changing frequency of the axial velocity is zero, close to zero and twice the natural frequency. Mathematical analysis is carried out to determine the stability and instability zones. The results show that instability occurs when changing frequency of the axial velocity is close to two times the natural frequency, whereas no instability occurs when changing frequency is close to zero. A case study of bandsaw is discussed. The stability and instability zones are plotted for the first five natural frequencies.