Parametric Instability of Axially Moving Media Subjected to Multifrequency Tension and Speed Fluctuations

This work investigates the stability of axially moving media subjected to parametric excitation resulting from tension and translation speed oscillations. Each of these excitation sources has spectral content with multiple frequencies and arbitrary phases. Stability boundaries for primary parametric instabilities, secondary instabilities, and combination instabilities are determined analytically through second-order perturbation. The classical result that primary instability occurs when one of the excitation frequencies is close to twice a natural frequency changes as a result of multiple excitation frequencies. Unusual interactions occur for the practically important case of simultaneous primary and secondary instabilities. While sum type combination instabilities occur, no difference type instabilities are detected. The nonlinear limit cycle amplitude that occurs under primary instability is derived using the method of multiple scales.