Towards design of a nonlinear vibration stabilizer for suppressing single-mode instability

Many mechanical systems exhibit destabilization of a single mode as some parameters are varied. In such situations, the destabilizing effect is often modeled using a negative damping term, e.g., in flow induced galloping instability of a pipe in a cross-flow. A small, nonlinear, not precisely tuned, lightly damped, secondary system mounted on the primary system can stabilize the primary system over some useful range of parameter values. We study such a system, modeling the primary system as a linear, negatively damped, single-degree-of-freedom oscillator and modeling the small secondary system using small linear damping and significantly nonlinear stiffness. Numerical simulations show useful behavior over a range of parameters even without precise tuning of the secondary system. Analytical treatment using harmonic balance followed by an informal averaging calculation yields amplitude and phase equations that capture the dynamics accurately over a range of parameters. Although intermediate expressions involved are long and unwieldy, numerical and analytical treatment of the equations lead to simplifying insights which in turn allow us to construct useful approximate formulas that can be used by practitioners. In particular, simple approximate expressions are obtained for oscillation amplitudes of the primary and secondary systems and for the negative damping range over which effective vibration stabilization occurs, in terms of system parameters.