Scaling law for the slow flow of an unstable mechanical system coupled to a nonlinear energy sink

In this paper one first shows that the slow flow of a mechanical system with unstable mode coupled to Nonlinear Energy Sink (NES) can be reduced, in neighborhood fold point its critical manifold, normal form dynamic saddle-node bifurcation. This allows us then obtain scaling law for dynamics and improve accuracy theoretical prediction mitigation limit NES previously obtained as part zeroth-order approximation. For purpose, governing equations are simplified using reduced-order model primary structure by keeping only modal coordinates. The is derived means complexification-averaging method and, presence small perturbation parameter related mass ratio between structure, it appears fast-slow system. center manifold theorem finally used reduced which solved analytically leading law. latter reveals nontrivial dependence respect near point, involving fractional exponents 1/3 2/3. Finally, new deduced from end, proposed methodology exemplified validated numerically an aeroelastic aircraft wing NES.