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 us
ing two different approaches. In the first approach, the equations are
discretized first and then the method of multiple scales is applied t
o the resulting equations. In the second approach, the method of multi
ple scales is applied directly to the partial differential system. Pri
ncipal parametric resonances and combination resonances are investigat
ed in detail. Stability boundaries are determined analytically. It is
found that instabilities occur when the frequency of velocity fluctuat
ions is close to two times the natural frequency of the constant veloc
ity system or when the frequency is close to the sum of any two natura
l frequencies. When the velocity variation frequency is close to zero
or to the difference of two natural frequencies, however, no instabili
ties are detected up to the first order of perturbation. Numerical res
ults are presented for a band-saw and a threadline problem. (C) 1997 A
cademic Press Limited.