Stability characteristics of an axially accelerating string supported by an elastic foundation

Stability characteristics in principal and combination parametric resonance of an axially accelerating string supported by a partial elastic foundation are analytically investigated. The model is considered as a three part system; one is subjected to an elastic foundation, and two are free to vibrate. The equations of orders one and epsilon are derived using the method of multiple scales (MMS). The axial velocity is assumed to be a mean velocity, along with small harmonic fluctuations. The mode shapes and natural frequencies of the system are derived using the equation of order one, and satisfying the compatibility conditions between three segments of the string. For the principal and combination resonance cases, including summation and difference types, the stability conditions are investigated analytically. Eventually, the effects of system parameters on natural frequencies and stability boundaries of the system are presented through numerical simulations.