Abstract
The dynamic response of an axially accelerating string is investigated. The time dependent velocity is assumed to vary harmonically about a constant mean velocity. Approximate analytical solutions are sought using two different approaches. In the first approach, the equations are discretized first and then the method of multiple scales is applied to the resulting equations. In the second approach, the method of multiple scales is applied directly to the partial differential system. Principal parametric resonances and combination resonances are investigated in detail. Stability boundaries are determined analytically. It is found that instabilities occur when the frequency of velocity fluctuations is close to two times the natural frequency of the constant velocity system or when the frequency is close to the sum of any two natural frequencies. When the velocity variation frequency is close to zero or to the difference of two natural frequencies, however, no instabilities are detected up to the first order of perturbation. Numerical results are presented for a band-saw and a threadline problem.
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