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Models of non-li near systems frequently introduce forces with bounded continuity resulting in nonsmooth (even discontinuous) flow. Examples include systems with clearances, backlash, friction, and impulses. Asymptotic methods require smooth (differentiable) flow and are therefore ill-suited for analyzing non-smooth systems. In these cases, the traditional harmonic balance method may be used to obtain approximate periodic solutions, but the method suffers from extremely slow convergence in general. Generalizations of the traditional harmonic balance method are introduced in this paper that result in superior convergence rates and superior modes of convergence. These improvements derive from the introduction of one or more expansion functions that possesses the same degree of continuity as the exact solution. In particular, forming an infinite series of such functions results in an expansion in the same function space of the exact solution. This expansion converges pointwise to the exact solution and to all derivatives thereof. These improvements are illustrated by example upon re-evaluating a classical single degree-of-freedom model for friction-induced vibration.

Analytical studies of vortex-induced vibration (VIV) of cables during lock-in have considered small amplitude and relatively fast dynamic responses about an equilibrium configuration. However, this equilibrium may change as a result of the significantly increased mean drag created during lock-in. In response to increased drag, the cable may slowly drift downstream causing appreciable changes in cable geometry and tension. The resonance conditions for lock-in may be preserved during this slow drift or they may be disrupted. A nonlinear cable/fluid model is discussed that captures both fast (small amplitude) motions due to VIV and slow (large amplitude) motions due to drift.