Free and Forced Vibration of an Axially Moving String With an Arbitrary Velocity Profile

The exact response of an axially moving string with an arbitrary velocity profile is determined under general initial conditions and external excitation. In terms of the curvilinear characteristic coordinates, the time-dependent governing equation reduces to one with constant coefficients. The domain of interest in the characteristic coordinate plane is bounded by two monotonic curves with the same shape corresponding to the spatial boundaries of the string. Solutions of the transformed equation are derived for the free and forced responses. The amplitude of the free response is shown to be bounded under arbitrary variation of the transport speed in the subcritical regime. The unbounded displacement of the string predicted by Floquet theory for the spatially discretized models does not occur. The method is demonstrated on two illustrative examples with piecewise constant and sinusoidal accelerations.