Coupling between dissimilar modes in an asymmetrically forced string

It is well known that a stretched string subjected to symmetric harmonic excitation in a single plane may exhibit large amplitude, coupled motion between the nth in-plane mode and the nth out-of-plane mode (n:n coupling). Significant coupling between modes of different indices (n:m coupling) is not expected in the steady state; if the nth in-plane mode is directly excited, it can be shown that the amplitude of the mth mode (m ≠ n), both in-plane and out-of-plane, will decay monotonically, for any m. The present study illustrates that steady-state n: m coupled motion of the string may exist, provided that a constant in-plane bias force is superimposed on the harmonic excitation, thereby breaking the symmetry of the problem. In particular, the case of 2:1 coupling between in-plane and out-of-plane modes is examined using a perturbation approach. Results are confirmed by numerical simulations and physical experiments.